UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations   Help 
Material Information
Notes
Record Information

Full Text 
A MULTIPHASE SUPERSONIC JET IMPINGEMENT FACILITY FOR THERMAL MANAGEMENT By RICHARD RAPHAEL PARKER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 @ 2012 Richard Raphael Parker ACKNOWLEDGMENTS Throughout my time in graduate school I have been helped by many people, although I will try my best to include everyone, inevitably I will leave some people out. For that I am sorry. First and foremost I would like to thank my parents as without them this would not be possible. Dad, you taught me what it meant to be a man, and I can't express my gratitude enough. There have been a multitude of people in my lab that have helped me immensely, Dr. Patrick Garrity, Dr. Ayyoub Mehdizadeh, Dr. Jameel Khan, and Dr. Fadi Alnaimat (my current roommate) who have all graduated and are going places in the world, you have provided help and motivation. Bradley Bon, Fotouh AIRagom (the lab mother), and ChengKang (Ken) Guan who are currently preparing for their dissertation defense, we have all helped motive each other. Ben Greek, Prasanna Venvanalingam, Kyle Allen, Like Li, and Nima Rahmatian, who are currently completing their degree requirements, have helped me develop some of my presentation skills. I hope you have learned as much from me as I have from you. My friends in the Interdisciplinary Microsystems Group (IMG) who have provided many opportunities to escape the pressures of graduate studies. I would specifically like to thank my old roommate, Dr. Drew Wetzel, who let me bounce ideas off of him, Dr. Matt Williams, who always ran the group football competitions and helped me reminisce about my roots in South Carolina, and Brandon Bertolucci our official social chair. Thank you all and good luck. I would also like to thank my professors at the University of South Carolina. Dr. Jamil Khan, whose heat transfer classes helped shape the foundation of my knowledge in the thermal and fluid sciences, Dr. David Rocheleau who provided me with contacts at the University of Florida, Dr. Phillip Voglewede who encouraged me to attend the University of Florida, and finally Dr. Abdel Bayoumi, my undergraduate research advisor who saw potential in me and provided many opportunities for me to grow. Last, but not least, I would like to thank my committee. Dr. Orazem has helped me learn things outside of my field and my comfort zone and for that I am grateful. Dr. Hahn, your rigorous heat conduction class helped build the foundation for much of my studies. Dr. Mei, you have helped me realize my potential for numerical studies and how to know when a solution is good enough or when perfection is essential. Lastly I would like to thank my chair, Dr. James F Klausner. Dr. Klausner, you have kept me around, even as I struggled, because you saw the potential in me. You've help keep some levity in the lab with football talk and going to some excellent concerts. Most importantly, you've helped me grow both professionally and as a person. For this you have my most sincere gratitude. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ............... 3 LIST O FTABLES ..................... ................. 8 LIST OF FIGURES .................... ................. 9 NOMENCLATURE .................... ................. 13 ABSTRACT .................... ................... .. 17 CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW ................... 19 1.1 Literature Review ................... ............ 20 1.1.1 SinglePhase Jet Impingement .................. 20 1.1.2 Mist and Spray Cooling ....................... 24 1.1.3 Supersonic Jet Impingement ................ ... 25 1.2 Summary .................... ................ 34 2 JET IMPINGEMENT FACILITY ........................... 36 2.1 Impinging Jet Facility Systems ................ ........ 36 2.1.1 Air Storage System ........................... 36 2.1.2 Water Storage and Flow Control System ..... 36 2.1.3 Air Pressure Control System .. .. 39 2.1.4 Air Mass Flow Measuring System ..... 39 2.1.5 Temperature and Pressure Measurements .... 42 2.1.6 ConvergingDiverging Nozzle . ... 42 2.1.7 Data Acquisition System ..... .... 43 2.2 Analysis of Impingement Facility ... 44 2.2.1 Temperature and Pressure Upstream of the Nozzle ... 44 2.2.2 Nozzle Exit Pressure Considerations ..... 46 2.2.3 Oblique Shock Waves at Nozzle Exit ... 48 2.2.4 Complete Shock Structure of an Overexpanded Jet ... 50 2.3 Sum m ary ... . .. .. 51 3 STEADY STATE EXPERIMENTS ........................ 53 3.1 Heater Construction .. .. .. .. .. ... 56 3.1.1 Physical Description .. .. .. .. .. .. 56 3.1.2 Theoretical Concerns ......................... 57 3.2 Experimental Procedure .. ............... ....... ..59 3.2.1 TwoPhase Experiments .... .. .. .. .. ..59 3.2.2 SinglePhase Experiments ..... .... 62 3.3 Experimental Results .. ............. 3.3.1 Uncertainty Analysis .. .......... 3.3.2 SinglePhase Results .. ......... 3.3.3 TwoPhase Results .. ........... 3.3.4 Evaporation Effects .. ........... 3.4 Comparison between Single and TwoPhase Jets 3.5 Discussion . . 3.6 Sum m ary . . 4 DETERMINATION OF HEAT TRANSFER COEFFICIENT USING AN INVERSE HEAT TRANSFER ANALYSIS ............................ 4.1 Inverse Problems ........ 4.2 Introduction to Inverse Problem Method with Adjoint Problem . Solution Using the Conjugate Gradient 4.2.1 The Direct Problem . 4.2.2 The Measurement Equation . 4.2.3 The Indirect Problem . 4.2.4 The Adjoint Problem . 4.2.5 Gradient Equation . 4.2.6 Sensitivity Equation . 4.2.7 The Conjugate Gradient Method . 4.3 Factors Influencing Inverse Heat Transfer Problems . 4.3.1 Boundary Condition Formulation Effects . 4.3.2 Sensor Location Effects . 4.3.3 Thermocouple Insertion Effects . 4.4 Inverse Heat Transfer Problem Formulation . 4.4.1 Direct Problem ................. 4.4.2 Measurement Equation . 4.4.3 Indirect Problem ................ 4.4.4 Adjoint Problem .............. 4.4.5 Gradient Equation ............... 4.4.6 Sensitivity Problem . 4.4.7 Conjugate Gradient Method . 4.4.8 Stopping Criteria . 4.4.9 Algorithm . . 4.5 Numerical Method and Limitations . 4.5.1 Alternating Direction Implicit Method . 4.5.2 Grid Stretching in the ZDirection . 4.5.3 Time step size complications . 4.6 Deconvolution for Thermocouple Impulse Response 4.6.1 Direct Problem ................. 4.6.2 Indirect Problem .. .. .. .. . 4.6.3 Adjoint Problem . . . . . . . . . . ...... . . ...... .. .. ...... . . . . . . ...... . . . . . . . . Functions ...... .. .. 4.6.4 Gradient Equation .. .................. 82 82 83 83 85 85 86 88 88 91 92 95 96 97 98 98 101 101 102 103 104 105 105 106 108 108 109 110 110 4.6.5 Sensitivity Problem .. .. .. .. .. .. 110 4.6.6 Conjugate Gradient Method .... 110 4.6.7 Stopping C riteria . 111 4.6.8 A lgorithm . . 111 4.6.9 TestCase ................... .......... .112 4.7 Sum m ary . . 114 5 SENSOR DYNAMICS AND THE EFFECTIVENESS OF THE INVERSE HEAT TRANSFER ALGORITHM ................ ............... 116 5.1 Thermocouple Measurement Dynamics . ... 116 5.1.1 Low Biot Number Thermocouple Models ... 116 5.1.2 High Biot Number Thermocouple Models ... 121 5.1.3 Design of Experiment ......................... 122 5.1.4 Experimental Results ..........................126 5.1.5 Comparison to Established Models ... 127 5.2 Inverse Heat Transfer Algorithm Verification ... 131 5.2.1 Inverse Quenching Parametric Study Setup ... 132 5.2.2 Error Assessment Methods . 136 5.2.3 Parametric Study Results ... 137 5.2.4 Heat Loss/Gain Effects ......................... 141 5.2.5 Effectiveness of the Inverse Heat Transfer Algorithm ... 142 5.3 Sum m ary . . 144 6 CO NCLUSIO NS .. .. .. .. .. .. .. .. 147 APPENDIX A COMPLETE STEADY STATE TWOPHASE HEAT TRANSFER JET RESULTS 151 B COMPLETE INVERSE HEAT TRANSFER ALGORITHM ERROR ASSESSMENT CONTOUR PLOTS .................................. 166 C IMAGES OF ORIFICES USED DURING EXPERIMENTS ... 172 REFERENC ES . . .. 175 BIOGRAPHICAL SKETCH ................................ 184 LIST OF TABLES Table page 21 Water Mass Flowrate and Average Water Velocity for Different Regulator Pressures and O rifice Sizes. . .. 40 22 Area, temperature, and pressure ratios at various points in the jet impingement facility.... .. .. .. . ... 46 23 Nozzle exit pressure for various regulator pressures. ... 47 31 Reynolds number and corresponding heat fluxes. ... 62 51 Curve Fitting Constants for Rabin and Rittel's thermocouple impulse response m odel, from [114] . . 121 52 RMS errors for L = 10 mm, and TDAQ = 0.2 C. ... 138 53 RMS errors for L = 10 mm, oDAQ = 1 OC, and M = 8. ... 139 54 RMS errors for L = 5 mm and 0DAQ = 0.2 C. . ... 140 55 RMS error for L = 10 mm,DAQ = 0.2 oC, M = 8, and various values of a for the Biot number distribution. .. ......... .. 140 56 RMS error for L = 10 mm and M = 8 with different actual and simulated time co nstants . . 14 1 LIST OF FIGURES *e Figur 11 12 13 14 15 16 21 22 23 24 25 26 27 28 29 210 211 31 32 33 34 35 36 37 38 Solution of stagnation point flow .................... An illustration of the shock structure in the wall jet region. . Grease streak photograph ...................... Numerical results of a flow field with a plate shock. . Jet centerline pressure fluctuations with and without moisture. . Adiabatic and heated temperature variation with z/D. . Illustration of the jet impingement facility. . Cross section of the mixing chamber . Thread details of the mixing chamber cross section . Orifice cross section ................... ........ Theoretical vs measured mir. ... .... ................ Cross section of nozzle ......................... Simplified view of the air flow path in the facility. . Illustration of the limiting cases for shock waves in the nozzle. . Illustration of an oblique shock wave at the nozzle exit. . Variation of flow properties downstream of an oblique shock wave.. Structure of oblique shock waves .................. Stainless steel heater assembly. . . Copper heater assembly. ........................ Illustration of heater assembly used for steady state experiments. .. Ice formation at adiabatic conditions. . . Measured singlephase NuD spatial variation at different heat fluxes. Spatial heater temperature variation at different applied heat fluxes. Spatial variation of NuD at different ReD, a) unsealed and b) scaled. Singlephase NuD at various nozzle height to diameter ratios. . page . 2 1 . 26 . 28 . 3 1 . 32 . 34 . 37 . 38 . 38 . 39 . 4 1 . 43 . 45 . 48 . 49 . 50 . 52 . 54 . 55 . 57 . 60 . 64 . 65 . 66 . 67 39 Measured twophase NuD spatial variation at different heat fluxes without ice formation ........... ... .. ....................... 68 310 Measured twophase NuD spatial variation at different heat fluxes with and w without ice form ation. . .. 69 311 Spatial heater temperature variation, twophase jet results. ... 70 312 Twophase NuD at various liquid mass fractions. ... 71 313 Twophase NuD number at various nozzle height to diameter ratios. 72 314 Heat transfer enhancement ratio at various liquid mass fractions. ... 74 315 Heat transfer enhancement ratio at various nozzle height to diameter ratios. 76 41 1dimensional solid for the sensitivity problem. ... 89 42 Relative step sensivity coefficients at x* = 0.1 as a function of time. ...... 92 43 Relative step sensivity coefficient for a heat flux input. ... 93 44 Relative step sensivity coefficient for a temperature input. ... 93 45 Illustration of the heat transfer physics of the inverse problem formulation. 96 46 Comparison of true temperature vs inverse results. ... 104 47 Effects of grid stretching. a) real domain and b) computational domain. 107 48 Time step results. .................................. 109 49 True and estimated impulse response function. ... 113 410 Convergence history.. .. .. .. .. .. .. .. .. 113 411 O utput com prison. . . 114 51 Illustration of the first order slab. ... .. 118 52 Illustration of the second order slab. ... .... .. 119 53 Example of a first and second order impulse response function. ... 120 54 Impulse response functions using the model of Rabin and Rittel, adapted from [114]. .. .. .. .. ....................... .... 122 55 Diagram of the copper disc assembly. ... 123 56 Illustration of the experimental setup. .. .... ... 124 57 Back wall temperatures with nonideal insulations. ... 125 58 59 510 511 512 513 514 515 516 517 518 519 Results of three separate impulse response experiments. . Inverse method deconvolution results . . First order response function comparison. ... ... Best fit results using a first order impulse response function . Comparison of model to deconvolution results . . Best fit results using the model of [114] response function. .. Comparison of the 2 exponential model to the deconvolution algorithm results. Best fit results using the 2 exponential model. . . Biot number distribution showing the effects of Bimax.. . Biot number distribution showing the effects of . . Truncation of Biot number ................... .......... Error Contours for L = 10 mm, 7DAQ = 0.2 o C, and 8 measurement points. . . 126 . 127 . 128 . 129 . 129 . 130 131 . 132 . 134 . 135 S137 S139 520 Comparison of the effects of heat gain. .... 143 521 Centerline NuD results from the inverse heat transfer algorithm. ... 144 522 Spatial NuD results from the inverse heat transfer algorithm. ... 145 A1 Twophase NuD results for various Z/D, nominal ReD = 4.46 x 10s ...... .152 A2 Twophase NuD results for various Z/D, nominal ReD 7.27 x 10 ... 153 A3 Twophase NuD results for various Z/D, nominal ReD 1.01 x 106 ... 154 A4 Twophase NuD results for various liquid mass fractions and Z/D = 2.0. ...... 155 A5 Twophase NuD results for various liquid mass fractions and Z/D = 4.0. ...... 156 A6 Twophase NuD results for various liquid mass fractions and Z/D = 6.0. ...... 157 A7 Twophase NuD results for various liquid mass fractions and Z/D = 8.0. ...... 158 A8 Twophase 0 results for various Z/D, nominal ReD = 4.46 x 10 ... 159 A9 Twophase 0 results for various Z/D, nominal ReD 7.27 x 10 ... 160 A10 Twophase 0 results for various Z/D, nominal ReD 1.01 x 106 ... 161 A11 Twophase 0 A12 Twophase d results results for various liquid for various liquid mass fractions and Z/D = 2.0. mass fractions and Z/D = 4.0. . 162 . 163 A13 Twophase 0 results for various liquid mass fractions and Z/D = 6.0. 164 A14 Twophase 0 results for various liquid mass fractions and Z/D = 8.0. 165 B1 Error Contours for L = 5 mm, ODAQ = 0.2 o C, and 8 measurement points 167 B2 Error Contours for L = 5 mm, ODAQ = 0.2 o C, and 16 measurement points 168 B3 Error Contours for L = 10 mm, ODAQ = 0.2 o C, and 8 measurement points 169 B4 Error Contours for L = 10 mm, (DAQ = 0.2 o C, and 16 measurement points 170 B5 Error Contours for L = 10 mm, (DAQ = 1 o C, and 8 measurement points 171 C1 The 0.33 m m orifice . ... .. 172 C2 The 0.37 m m orifice .. .. .. .. .. .. .. .. 173 C3 The 0.41 m m orifice . . 173 C4 The 0.51 m m orifice . . 174 NOMENCLATURE Variables A Bi C D L M M Nu P R R Re S T V X Cp f h h k Area [m2] Biot number = hL/k Measurement equation operator Diameter [m] Lagrangian Mach number, Chapter 2 Total number of measurements Nusselt number = hL/k Pressure [Pa] Gas constant for dry air [J/kg K, Chapter 2 Residual of modeling equation Reynolds number = 4m/7r D Least squares value Temperature [oC or K] Volume [m3] Dimensionless relative sensitivity coefficient Specific heat capacity at constant pressure [J/kg K] General function Heat transfer coefficient [W/m2 K], Chapter 3 Impulse response function [s] Thermal conductivity [W/m K] in q"'1 r t u u v w x y y z Greek letters a /3 /3 6 7 7 6 6 C rI Mass flowrate [kg/s] Internal heat generation rate [W/m3] Radial coordinate [m] time [s] Velocity [m/s] Dummy variable for partial differential equation, Chapter 4 Dummy variable for partial differential equation, Chapter 4 Liquid mass fraction Length coordinate [m] Width coordinate [m] Generalize output variable, Chapter 4 Height coordinate [m] Thermal diffusivity [m2/s] Step size for Conjugate Gradient Method Grid stretching parameter Ratio of specific heats, Chapter 2 Conjugation coefficient for Conjugate Gradient Method, Chapter 4 Deflection angle in radians, Chapter 2 Thickness [m], Chapter 3 Dirac delta function, Chapter 4 Stopping criteria value Transformed z coordinate 0 0 A A p p 7 7 w Subscripts D T a air exp f I m mix mod Oblique Shock wave angle in radians, Chapter 2 Dimensionless temperature Lagrange multiplier, Chapter 4 Effective time constant, Chapter 5 viscosity [Pas] General parameter to be estimated, Chapter 4 Dummy integration variable, Chapter 5 Density [kg/m3] Dummy variable of integration Time constant [s] Heat transfer enhancement ratio, Chapter 3 Dimensionless step response, Chapter 4 Humidity ratio Solid domain Thermocouple measured quantity Adiabatic quantity Air quantity Experimental measurement Fluid quantity Liquid quantity Measurement quantity Mixture quantity Modified quantity 0 0 rms s sat sim snd v w Superscripts Stagnation quantity, Chapter 2 Initial quantity, Chapter 4 Rootmeansquare value Surface quantity Value at saturation conditions Simulated measurement Quantity at the speed of sound Vapor quantity Wall quantity Critical quantity, Chapter 2 Dimensionless quantity Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A MULTIPHASE SUPERSONIC JET IMPINGEMENT FACILITY FOR THERMAL MANAGEMENT By Richard Raphael Parker May 2012 Chair: James F Klausner Cochair: Renwei Mei Major: Mechanical Engineering This study investigates the heat transfer characteristics of a multiphase supersonic jet impingement heat transfer facility. In this facility water droplets are injected upstream of a convergingdiverging nozzle designed for Mach 3.26 air flow. The nozzle is operated in an overexpanded mode. Upon exiting the nozzle, the high speed air/water mixture impinges onto a heated surface and provides cooling. Steady state heat transfer measurements have been performed with peak heat transfer coefficients exceeding 200,000 W/m2. These heat transfer coefficients are on the same order as some of the highest heat transfer coefficients ever recorded in the literature. Remarkably these heat transfer coefficients are obtained using liquid flowrates ranging from 0.2 to 0.7 g/s, in contrast to the several kg/s flowrates seen in studies that produce similarly high heat transfer coefficients. During steady state operation it is noted that no evidence of phase change was experimentally observed. Preliminary investigations indicate that it may not be possible to obtain evaporative heat transfer in the current facility. In order to investigate this possibility higher surface temperatures are needed. However, designing a steady state experiment to achieve high temperature operation is rife with difficulties and is likely to be prohibitively expensive. In order to overcome these challenges a transient inverse heat transfer (IHT) method has been developed. One of the important issues revealed during this investigation is that sensor dynamics will impact the measurements, thus diminishing the measurement reliability. To alleviate this issue, a method of incorporating sensor dynamics into the IHT method was developed. This type of method is not explicitly found in the literature to the author's knowledge. A method for accurately determining the impulse response function of the thermocouples used in the transient IHT experiments yields good experimental results. Heat loss is discovered to be a critical factor in the IHT method, and a difference in temperature of 3 C between that measured and the ideal case renders the IHT results unusable. A parametric study was performed to determine the effects of: disc height, impulse response function, magnitude and shape of the heat transfer coefficient distribution, the number of temperature sensors used, and the magnitude of the error in the data acquisition system. It was discovered that the method was insensitive to noise levels found in laboratory conditions and the accuracy increases for a decreasing disc height. The relative slowness of the impulse response functions did affect the accuracy of the IHT method as long as the time constant of the functions is accurately known. CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW Jet impingement produces high heat transfer coefficients, up to approximately 105 W/m2 K. Liquid impinging jets have supported some of the highest recorded surface heat fluxes, ranging from 100 to 400 MW/m2, [1]. Oh et al. [2] and Lienhard and Hadaeler [3] have studied liquid jets and arrays that can produce heat transfer coefficients of 200 kW/m2K. These high heat transfer rates are accompanied by high liquid flowrates of up to several kg/s of water, and such high water consumption may be undesirable in some industrial settings. The current study proposes to use a supersonic multiphase jet impingement facility designed after an experiment by Klausner et al. [4], which uses the addition of liquid droplets to the impinging airstream to enhance the heat removal rate of the supersonic jet. The liquid flowrate will be orders of magnitude lower than that used by the studies mentioned above, with less than one g/s, which may be very desirable in applications where minimal water consumption is a concern. Supersonic twophase jet heat transfer is a field that has not been previously studied. The contribution of the current study will largely consist of characterizing the heat transfer capabilities of such a system including the effects of air and liquid mass flowrates and nozzle spacing. Additionally, evaporative heat transfer capabilities of the jet will be studied; in this scenario the latent heat of vaporization could potentially greatly enhance the heat removal capabilities of the facility. However, it is not known whether or not liquid evaporation can be achieved due to the high stagnation pressure. Due to high impact pressures near the jet centerline, phase change is not likely in this region; however, the conditions far removed from the impingement point may allow phase change to occur. 1.1 Literature Review Jet impingement heat transfer is a very diverse field and consists of singlephase heat transfer and evaporative heat transfer, spray/mist cooling, and supersonic jet impingement heat transfer. A brief review of jet impingement heat transfer is provided. 1.1.1 SinglePhase Jet Impingement The analytical study of stagnation point flows largely begins with Hiemenz [5] who studied the flow field of a laminar impinging jet by modifying the Blausis boundary layer solution. Homann [6] extended this analysis to axisymmetric flows. These flows are part of the FalknerSkan boundary layer equations, which take the general form f"' + of"f 3 (1 f'2) = 0 where (11) f(O) = f'(O) =0 and f(oo) = 1. The velocities u and v, and the similarity variable, Tl, are defined as u = axf' (r) v = poVf avf ) where a is a proportionality constant and v is the kinematic velocity of the fluid. Note that in the radial case the variable x is the radial distance from the origin. The particular values of 3 and 3o are 1 and 1 for Hiemenz flow and 1 and 2 for Homann flow. The variable f" is proportional to the shear stress, f' is the nondimensional velocity u/U,, and f is the stream function. Equation (11) represents a nonlinear ordinary differential equation (ODE) which must be solved numerically. The shooting type method is generally used as the value of f" is unknown at the origin. The values of f" at the origin Figure 11. Solution of a) Hiemenz stagnation point flow and b) Homann stagnation point flow. obtained numerically are 1.2325 for Hiemenz flow and 1.3120 for Homann flow, as found in [7]. The solution for Hiemenz and Homann flow are shown in Figure 11. It is evident that the the flow fields behave very similarly; however, the free stream velocity and shear stress are reached for smaller values of the similarity variable for axisymmetric stagnation point flow. A full derivation of these and other stagnation point flows using a similarity type approach as that above can be found in the book by Schlichting and Gersten [8]. While the above analysis is sufficient for completely laminar impinging jets, jets which find industrial application must deal with the boundary layer approaching the free surface of the jet far removed form the centerline as well as the transition to turbulence. Due to the different flow regimes the jet analysis is typically broken up into several different regions and analyzed through the use of a von Karmam momentum integral analysis; for an analysis of stagnation region see [9]. The analysis of the temperature field within the boundary layer for these types of flows is complicated due to the behavior of the thermal boundary layer that develops on the surface. It is further complicated by the nature of the fluid itself as flows with larger Prandlt (Pr) number behave very differently than flows with small ones. As the boundary layer moves away from the centerline, the hydrodynamic boundary layer reaches the free surface before the thermal boundary layer for Pr < 1 while the converse is true for Pr > 1. Liu et al. have analyzed the flow in each of these regions for single phase jets with constant surface temperature and heat fluxes, mostly through the use of the von KarmanPohlhausen integral solution [10, 11]. They were able to model the transition to turbulence and the subsequent turbulent flow as well. Their solutions agree exceptionally well with experimental results. In general the solutions have the form of The analysis of the above flow field is not limited to integral solutions or to constant boundary conditions. Wang et al. [12, 13] studied the effects of a spatially varying surface temperature and heat flux on the solution using a perturbation method. They found that the direction of increasing temperature affects the Nusselt number of the flows, notably that increasing the wall temperature or heat flux with radial distance from the origin will decrease the Nusselt number in the stagnation zone. Conversely it increases the Nusselt number in the boundary layer region. Wang et al. additionally studied the conjugate heat transfer problem where the temperature field is determined in the liquid and solid simultaneously, [14]. They found that thickness of the heater can be a contributing factor for the heat removal capability of the jet. There are several additional phenomena that impact the cooling rates of impinging jets. These include the effects of the jet nozzle diameter [15], hydraulic jump [16], and the splattering of liquid from the resulting free surface [17]. In cases with sufficiently high surface temperature, phase change can be observed under impinging jets including the regions of nucleate boiling, departure from nucleate boiling, and transition boiling [18]. While most studies of liquid jet impingement find no appreciable effect on the nozzle height above the heat surface, Jambunathan et al. [19] noted that some studies do show an effect most notably at higher Reynolds numbers. An empirical correlation based on heat transfer data available in the literature was proposed however, it provides no physical insight of the flow field and heat transfer taking place. Liquid jet impingement can support exceptionally high surface heat fluxes. Liu and Lienhard [1] used a liquid jet with velocities exceeding 100 m/s, liquid supply pressures of up to approximately 9 MPa, and flowrates of approximately 300 g/s to remove heat fluxes of at least 100 MW/m2. These experiments were novel in the fact that they used a plasma torch as a heat source. Surface temperatures were determined by coating the top surface with a material of known melting temperature and completing several experimental runs until the surface temperature could be isolated to lie within a range of temperatures. Melting of the heated surface occurred due to the use of the torch and the back wall temperature was assumed to be essentially the melting temperature of the solid. The heat flux was determined by using the minimum thickness of the solid where it had melted and then assuming a linear temperature profile. Because of the coarse nature of the measurements, uncertainty is hard to quantify and heat transfer coefficients were not reported. However, the heat fluxes measured are the highest steady state valuesrecorded in the literature. To further enhance the study of high heat flux removal Michels, Hadeler, and Lienhard [20] and Lienhard and Napolitano [21] designed thin film heaters using vacuum plasma spraying and high velocity oxygen fuel spraying. These heaters are supplied with dc electrical power of up to 3,000 A and 24 V, producing heat fluxes of up to 17 MW/m2. Lienhard and Hadeler [3] were able to construct an array of liquid jets with liquid supply pressures of approximately 2 MPa and flowrates of approximately 4 kg/s. These impinging jet arrays were able to support heat fluxes of 17 MW/m2 with an average heat transfer coefficient of 200 kW/m2 with uncertainties of 20%. Similar results were found in a study by Oh et al. [2]. These studies help illustrate the high heat removal capabilities of jet impingement technology. For an extensive review of the subject of liquid jet impingement the author recommends the review articles of Liendhard [22], Webb and Ma [23], and Martin [24]. These articles offer a complete review of the subject and include many effects not discussed in this brief literature survey. 1.1.2 Mist and Spray Cooling Mist/spray cooling of a heated surface is largely different from jet impingement due to the fact that the liquid impinging on the surface is in the form of disperse droplets. These droplets are usually generated by forcing liquid through very small orifices within a nozzle which atomizes the liquid. The primary benefit of using this technique is that the disperse droplets generally allow for evaporative heat transfer to dominate. Mist/spray cooling produces heat transfer coefficients on the order of those found during pool boiling. However, the critical heat flux can be several times higher [25]. The droplet size is an important parameter in mist/spray cooling, Estes and Mudawar [26] correlated the critical heat flux (CHF) with the Sauter mean diameter of the sprays; they also found that the apparent density of the spray can be an important factor in mist/spray cooling as denser sprays are less effective. Mist/spray cooling can also be applied to surfaces lower than the boiling point of the liquid. The reduced evaporation can lead to a buildup of a liquid film on the heated surface which can have a thickness less than the size of the droplets. Graham and Ramadhyani [27] performed an experimental study which shows that increasing the amount of droplets on the surface can lead to thicker films which may increase the thermal resistance at the surface; however, this thick film may be able to convect the heat away better due to an increased velocity. They were able to develop a simple model of the thin film dynamics and the resulting heat transfer which had approximately 4% error for heat flux predictions with an air/methanol mixture but, only provided qualitative agreement when used with air/water data. It is noted here that because of the evaporation taking place in mist/spray cooling, the heat transfer coefficient does not vary appreciably with the radial distance, a feature quite different than that found in jet impingement heat transfer. Readers desiring a comprehensive review of mist spray cooling are encouraged to consult the review article of Bolle and Moureau [28]. 1.1.3 Supersonic Jet Impingement The flow field of a supersonic jet exhibits very complex phenomena. The nature of the flow can change dramatically as the nozzle exit to ambient pressure ratio changes, sometimes quantified by the stagnation to ambient pressure ratio. When the nozzle exit pressure is lower than the ambient pressure oblique shock waves form at the edge of the nozzle in order to compress the flow. These shock waves become PrandtlMeyer expansion fans when they meet at the jet centerline. This process leads to a series of reflected shock waves and expansion fans forming in the flow field at the exit of the nozzle, including the formation of normal shocks in the flow known as Mach disks. When the jet exit pressure is larger than the ambient pressure, PrandtlMeyer expansion fans form at the exit of the nozzle and a similar series of events take place. The flow field of the jet changes dramatically in the axial direction. Zapryagaev et al. [29] noted that for overexpanded jets the radial pressure distribution upstream of the first shock cell contains several local maxima with very sharp discontinuities present. These discontinuities disappear downstream of the first shock cell and generally a non centerline maximum appears in the pressure distribution. These features are present in underexpanded jets as well, [30]. While these features are complex they can be Figure 12. An illustration of the shock structure in the wall jet region. [Reprinted with permission from Carling, J. C. and Hunt, B. L., The Near Wall Jet of a Normally Impinging, Uniform, Axisymmetric, Supersonic Jet, Journal of Fluid Mechanics 66 (1974) 159176 (Page 174 Figure 9(b)), Cambridge University Press] modeled somewhat accurately by a method of characteristics approach as noted by Pack, [31, 32] and Chu [33], among others. Underexpanded impinging jets have been extensively studied in the literature as they pertain to the launching of rockets and spacecraft, whereas overexpanded impinging jets are relatively uncommon in industrial settings. When a jet impinges upon a flat plate, a complex shock structure is formed. This shock structure forms several complex features including a triple shock structure, where three shock waves intersect near the impingement surface, a bow shock, also known as a plate shock in the literature, as the flow must come to rest in the stagnation point on the surface, and the shock waves which radiate from the triple shock point and slow the flow along the plate to subsonic speeds. These features appear in all types of supersonic impinging jets including underexpanded, ideally expanded, and overexpanded. The bow shock, if curved, can form a recirculating stagnation region in the area of the center of the jet to the edge of the nozzle. An illustration of the complex shock structure of the flow at larger radial distances along the impingement plate is shown in Figure 12. The stagnation region is very complex due to the formation of the above mentioned bow shock and stagnation bubble. Donaldson and Snedeker [30] studied underexpanded jets from a converging nozzle and performed many different measurements to help characterize some of the important features of the flow including impingement angle and nozzle pressure ratio. They were able to observe stagnation bubbles forming, but noted that this phenomenon did not occur in every experiment. Schlieren photographs were taken as well as total pressure measurements along the jet centerline, and it was observed that the velocity and pressure vary greatly in the axial direction. The velocity in the radial wall jet region was measured via the use of pitotstatic pressure tube measurements along the impingement surface, the effects of surface curvature were also characterized. Gummer and Hunt [34] also studied the flow of uniform axisymmetric ideally expanded supersonic jets with low nozzle to height spacing and noted the presence of the bow shock and complex shock structure in the wall jet region. They attempted to use a polynomial and integral relation method to model the bow shock height and the pressure distribution under the nozzle. Some success was seen for high Mach numbers but not in the region of the triple shock. Low Mach number calculations contained as much as 60% error. Carling and Hunt [35] performed a theoretical and experimental investigation using the nozzles of Gummer and Hunt. Their study mostly comprised of the region just outside of the nozzle along the impingement plate. They were able to note the presence of the stagnation bubble for some of their experiments, but not all. The presence of the stagnation bubble can severely influence the pressure distribution on the plate and an annular maximum is possible for some jet spacings. Attempts were made to model the shock structure in the wall jet region using the method of characteristics. Qualitative features of the flow were able to be reproduced. However, there appears to be some error in the region near the triple shock region. The pressure variation along the plate was measured which showed several regions of unfavorable pressure gradient. Carling and Hunt were able to investigate these regions by coating the impingement plate with a type of grease. When the jet is impinged upon the plate these unfavorable pressure gradients cause the grease to be removed due to local separation of the boundary layer. A photograph of one of these experiments is shown Figure 13. Grease streak type photograph from [35], the dark areas contain no grease and are areas of high wall shear stress. [Reprinted with permission from Carling, J. C. and Hunt, B. L., The Near Wall Jet of a Normally Impinging, Uniform, Axisymmetric, Supersonic Jet, Journal of Fluid Mechanics 66 (1974) 159176 (Plate 3 Figure 6(d)), Cambridge University Press] in Figure 13. The dark regions represent where no grease is present; note the very dark region near a radial distance of 2 nozzle diameters from the center where evidence of separation is clearly evident. The separation phenomenon was noted by several investigators including Donaldson and Snedeker, [30]. Kalghatgi and Hunt provide a qualitative analysis experimental study of overexpanded jets which concentrated on the triple shock problem near the edge of the bow shock. Their analysis suggests that flat bow shocks are a possibility and schlieren photographs of overexpanded impinging jets with Mach numbers ranging from approximately 1.5 to 2.8 largely confirmed their qualitative analysis. They also note that the formation of a flat bow shock is a phenomenon that is hard to predict. Lamont and Hunt performed a comprehensive experimental study on underexpanded jets oriented normally and obliquely to a flat plate which includes pressure measurements and schlieren photographs. The stagnation bubble phenomenon was noted as well as some unsteadiness in the jet. Velocity and pressure profiles were seen to vary greatly with the nozzle to plate distance, and it was noted that the local shock structure has a strong influence on the flow field. Unsteadiness of the impinging jet is caused by a feedback phenomenon which has been extensively studied due to its importance in air vehicle takeoff, including the launching of rockets and short/vertical take off and landing vehicles, such as the Joint Strike Fighter. This mechanism was successfully modeled by Powell, [36, 37]. The mechanism is caused by acoustic phenomena occurring at the edge of the nozzle. These acoustic waves cause vortical structures to be generated in the shear layer of the jet and are convected towards the impingement point. Upon encountering the region near the plate, these structures interact with the shock waves near the plate generating strong acoustic waves, which travel upstream towards the nozzle where were they interact with the nozzle edge generating more acoustic waves which then repeat the process [38]. Krothapalli [39] was able to predict the frequencies generated by a supersonic impinging rectangular jet using Powell's model, thus validating the theory. The effects of the unsteadiness on the flow field will be detailed below. Due to the complex shock structure and unsteady phenomena in impinging jets, numerical simulations are often used to help enhance the knowledge in this area. Alvi et al. [40] modeled the impingement of moderately underexpanded jets and used Particle Image Velocimetry (PIV) to help verify their results. Their method had reduced temporal accuracy, but was able to reproduce major flow features including the stagnation bubble and wall jet region, although the region of the triple shock point had some disparity between the numerical and experimental results. Klinkov et al. [41] compared numerical results of the velocity, pressure, and density fields to experimental results in the form of schlieren photographs and surface pressure measurements. Their study focused on overexpanded jets with Mach numbers in the range of 2.6 to 2.8 at approximately ambient stagnation temperatures. They found that the location of the bow shock can change significantly with nozzle to plate spacing, with several regions of a near constant shock height followed by an almost discontinuous change to another height. Regions of high shock height represent a convex bow shock and regions of low shock height represent a flat bow shock with unsteadiness noted as the shock transforms to from a convex shock to a flat shock. They also noted that a stagnation bubble region is typical of a convex bow shock and that regular stagnation flow accompanies a flat bow shock. An illustration is shown in Figure 14. The behavior of the bow shock is significantly affected by the unsteady feedback mechanism as it is seen to oscillate back and forth along the axis of the jet. This causes correspondingly large fluctuations in the surface pressure on the impingement plate. Kawai et al. performed a computational aeroacoustic study which was 2nd order accurate in time and 7th order accurate in space. This study was done to determine the effects of the presence or absence of a hole in a launch pad configuration and primarily focused on large nozzle to plate spacings and the effect of Reynolds number on the unsteady phenomena. It was seen that high Reynolds numbers can significantly increase the sound power levels of the jet and the magnitude of its oscillations. Their numerical code produced results which agreed well with historical sound power level data maintained by NASA. This study is useful in illustrating the complexity of the problem under study and how very complex numerical simulations are needed to accurately reproduce the features of the flow. The addition of moisture in the form of water vapor to the air supply of an impinging jet can have a noticeable impact on the flow field. This was observed experimentally by Baek and Kwon [42] who performed studies of air with varying degrees of supersaturation of water vapor for a supersonic jet issuing into quiescent air. They found that the location of the Mach disk was located further upstream in the flow for moist air jets and its size was reduced. Empirical correlations for the location of quantities such as the size and location of the Mach disk and the location of the jet boundary were proposed, although little mechanistic insight to the flow was gained. Numerical studies by Alam et al. [43, 44] and Otobe et al. [45] were performed for air with various values of supersaturation of water vapor for a supersonic jet impinging on a flat plate. They attempted to model the nonequilibrium condensation taking place in the region after the Figure 14. Numerical results of a flow field with a flat plate shock (left) and curved flat plate shock (right). [Reprinted with permission from Klinkov K. et al, Behavior of Supersonic Overexpanded Jets on Plats, in: H.J. Rath, C. Holze, H.J. Heinemann, R. Henke, H. Hnlinger (Eds.), New Results in Numerical Fluid Mechanics V, volume 92 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer, 2006, pp. 168175 (Page 173 Figure 3] first Mach disk in the flow. Their model assumes no slip between the liquid droplets that condense and that these droplets do not influence the pressure of the flow downstream. The flow field displays some noticeable differences than that of dry air. The authors propose that this is due to the addition of the latent heat of condensation to the air by the condensing water vapor. Unsteady behavior due to the acoustic feedback mechanism by Powell was seen in the simulations. This unsteadiness was not present upstream of the first Mach disk, but was seen down stream of it. The presence of condensate particles combined with the addition of the latent heat reduces the magnitude of the pressure fluctuations seen in the downstream portion of the flow which is illustrated in Figure 15. The authors attempted to verify their simulations with experimental data, i Phic I Lmtc S: rTA4 Pc Static pressure I :: ppT S 3T4 T14 0.9 .. ,: 0.9  :t= 2 S. : t=/ 3T,4 0.6 A Jw, ,,0.6 1 " 0 II 0 1 2 3 4 o D 1 2 3 x 1 2 3 4 L/ID=4.0 xiA)e a b Figure 15. Jet centerline pressure fluctuations with a) no moisture and b) 40% supersaturation of water vapor. [Reprinted with permission from Ashraful Alam, M. M. et al., Effect of NonEquilibrium Homogeneous Condensation on the SelfInduced Flow Oscillation of Supersonic Impinging Jets, International Journal of Thermal Sciences 49 (2010) 20782092 (Page 2086 Figure 10(b) and Page 2088 Figure 13(c)), Elsevier] mostly consisting of comparing the shock structure as seen in schlieren photographs like those in the study by Baek and Kwon, along with noise tones for dry air generated by the acoustic feedback mechanism. This proposed validation is weak because there is a lack of experimental data of which to compare to in the literature. The study of supersonic impinging jet heat transfer jets has been studied extensively in the literature. Unfortunately most of these studies have focused on the heat transfer from a rocket exhaust to a launch pad facility. Donaldson et al. [46] performed an experimental study of impinging sonic jets and their turbulent structure. The authors were able to develop a correlation for Nusselt number based on applying a turbulent correction factor to laminar impinging jet theory near the stagnation point and further away in the wall jet region. While good agreement was found for their correlation it is for sonic or just slightly supersonic impinging jets and does not apply to the highly supersonic jets previously mentioned. The unsteady acoustic feedback phenomenon previously discussed causes interactions between acoustic waves and the shock structure of the impingement region. This results in local cooling to occur in the region of the jet edge and is very noticeable in the measurement of the adiabatic wall temperature. This phenomenon is termed cooling by shockvortex interaction by Fox and Kurosaka [47] who investigated this phenomenon. Kim et al. [48] studied the surface pressure and adiabatic wall temperature of an underexpanded supersonic impinging jet. They noted that the acoustic vortical structure interaction significantly affects the adiabatic wall temperature and surface pressure which also varies greatly with nozzle height. The presence of a stagnation bubble, which enhances the cooling directly below the nozzle, was noted as well. Rahimi et al. studied the heat transfer of underexpanded impinging jets onto a heated surface. The temperature of the impingement surface with uniform applied heat flux is noted to change dramatically with radial distance as well as with nozzle spacing as shown in Figure 16. They note that Nusselt number scales approximately with the square root of Reynolds number and that high heat transfer rates are encountered in the stagnation zone when a stagnation bubble is present. Due to the complexity of the problem, they note that a general correlation of Nusselt number should be a function of not only Reynolds number and Prandtl number, as is common, but also a function of Mach number and nozzle spacing. Yu et al. performed a similar study and noticed similar trends; their measured Nusselt numbers exceed 1,500. Studies of the heat transfer characteristics of supersonic moist impinging jets are not found in the literature. They are likely to show very complex phenomena as evidenced by the differences in the shock structure and general behavior of the relevant flow quantities in the jet and along the impingement plate. The current study uses discrete liquid droplets that are injected into the air upstream of the nozzle. This will likely result in an air stream supersaturated with water vapor which is further complicated by the behavior of the liquid droplets and their effects on the flow. As Figure 16. 80 90 "o P,,jj 91 1 I 1I 1 I 1  70 0 P,, P =5.08 ~~~ P, P. , 5 0 60 70  50 60 0 40  J50 30 40 .. o 20 30    0 10 0 0 1 2 3 4 5 6 7 8 9 1011 0 1 2 3 4 5 6 7 8 9 1011 a b Adiabatic (circles) and heated (diamonds) wall temperature for a nozzle spacing of a) z/D = 3.0 and b) z/D = 6.0. [Reprinted with permission from Rahimi M. et al, Impingement Heat Transfer in an UnderExpanded Axisymmetric Air Jet, International Journal of Heat and Mass Transfer 46 (2003) 263272 (Page 267 Figures 6(a) and 6(b)), Elsevier] elucidated by the literature survey the flow structure associated with this technology is very complex, and essentially no analytical solutions are available for the flow field and heat transfer. The available empirical correlations do not cover twophase supersonic impinging jets. Numerical studies may provide some qualitative insight, but in most instances they do not adequately capture all of the physics taking place in the flow field. 1.2 Summary In this Chapter an introduction to the study was made and the relevance of multiphase supersonic impinging jets was introduced. The contributions of this study were also described, mainly that this is a technology that has not been studied until now. A brief literature review of the different types of impingement heat transfer was presented. Liquid and singlephase heat transfer was introduced starting with the classic work of Hiemenz and Homann. The development of accurate Nusselt number correlations based on von KarmanPohlhausen integral method were detailed. The agreement between theory and experiments is exceptional for these correlations. Other aspects such as the flow hydrodynamics, transition to turbulence, nozzle height, and nonuniform boundary conditions effects were discussed. High heat flux removal technologies that are capable of heat transfer coefficients as high as 200 kW/m2 were detailed as well. The study of supersonic underexpanded and overexpanded impinging jets was described as well. This field is complicated by the complex flow structure generated by shock waves which form when a nozzle is operated away from its designed pressure ratio. The details of these shock waves including the effects of the curvature of the bow shock just above the impingement plate were discussed. Stagnation bubbles formed just below the bow shock were discussed and their impact on the flow field was detailed as well. Shock waves near the impingement region cause an unsteady feedback phenomenon caused by the interaction of acoustic waves with the edge of the nozzle. The effects of this feedback phenomenon and the unsteadiness it causes and relevant changes in the local flow field were detailed. Moisture in the air stream and how it changes the relevant flow field was briefly explored as it is a relatively new area of study in the literature. The temperature profile on the impingement plate and how it changes with the presence of the stagnation bubble and acoustic feedback mechanism were discussed. Numerous experimental studies in the literature which include pressure and temperature measurements, particle image velocimetry, and schlieren photographs along with relevant numerical studies in the literature that discovered and confirmed these phenomena were discussed where relevant. Lastly the complexity of the current study was discussed. It is noted that an analytical solution to the problem will not be attainable and a predictive numerical study is not feasible as well. The contributions of this study will be in the form of developing an understanding of the mechanisms taking place as the multiphase supersonic jet removes heat from a surface. CHAPTER 2 JET IMPINGEMENT FACILITY The supersonic multiphase jet facility should possess several traits in order to be useful for an experimental apparatus. It should have sufficient air storage capacity so that experiments can be run at steady state. The stored air should be pressurized to such an extent that the desired Mach number can be achieved. Lastly it should contain sufficient water, and a means to control the flow, so that the impinging jet will remain in multiphase operation during experiments. The design for the current setup is based on a similar experiment by Klausner et al. [4]. The impinging jet consists of the following systems to be described below: air storage system, water storage and flow control system, air pressure control system, air mass flowrate measuring system, temperature and pressure measurement system, convergingdiverging nozzle, and data acquisition system (DAQ). A schematic diagram illustrating the configuration of the jet impingement facility is shown in Figure 21. 2.1 Impinging Jet Facility Systems 2.1.1 Air Storage System The air storage system consists of 9 'K' sized bottles which give a total volume of 0.45 m3 and are kept at a pressure of 14 MPa. The air storage system is filled with air from a model UE3 compressor from Bauer Corporation. The compressor is powered by a 3phase 240 V power supply and is capable of supplying 0.1 m3/min of approximately moisture free air to the air cylinders, thus the air storage facility can be charged to capacity in approximately 4.5 hrs. 2.1.2 Water Storage and Flow Control System Water for the facility is contained in a stainless steel vessel with a capacity of 2 L and a pressure rating of 12.4 MPa. Water is forced into the mixing chamber by the difference in pressure between the top of the water vessel (which is acted on the the full force of the air supply pressure) and the pressure inside the mixing chamber, which is From Air 14/2.8 MPa Pressure Compressor Air Reducer Transducer T lowmeter Flowmeter Water Chamber Air Storage Thermocouple (T) i 1Mixing Chamber Drain Orifice Chamber Connection T Nozzle TwoPhase Jet Figure 21. Illustration of the jet impingement facility. lower due to a change in area and because of the friction acting in the system tubing. A drawing the of the mixing chamber is found in Figures 22 and 23. The flowrate of the water is controlled by means of an orifice between the water vessel and mixing chamber and the air pressure in the system. Orifice diameters of 0.33, 0.37, 0.41, and 0.51 mm are used during experiments, and a drawing of the orifice design is found in Figure 24. While there is some variance in flowrate between each experiment for a given orifice size this effect is eliminated in the analysis due to the fact the the flowrate of water into the mixing chamber is measured during each experiment. This is accomplished by recording the elapsed time of each experiment and measuring the difference in the mass of liquid in the water vessel. Table 21 shows the nominal flowrate of liquid and the 50.80 25.40 Figure 22. Cross section of the mixing chamber. 9/16 18 THREAD 10.16 mm DEEP  3/8 NPT THREAD 12.7 mm DEEP 1/2 20 THREAD 7.62 mm DEEP ' / 0 6.35 Figure 23. Thread details of the mixing chamber cross section. average liquid velocity for the pressures and orifice sizes used during the experiments. It is noted that the flowrate of the 0.37 mm orifice is less than that of the 0.33 mm orifice, this is due to the fact that the 0.37 mm orifice is not perfectly circular. This condition is also seen in the 0.51 mm orifice, but it is not as severe as that in the 0.37 mm orifice. Pictures of each orifice taken with an optical microscope are shown in Appendix C. Ax^^ 012.7 I Orifice Diameter oo 0 f 1 1 6cc) c o 70 2.1.3 Air Pressure Control System The air pressure control system consists of an air regulator located between the air storage tanks and the inlet to the air mass flowmeter and the top of the water storage tank. The regulator is capable of reducing air pressure from 14 MPa down to a maximum of 2.8 MPa. Air pressures of 1.0, 1.7, and 2.4 MPa are used during experiments. As discussed later, these air pressures result in the convergingdiverging nozzle operating in an overexpanded manner. 2.1.4 Air Mass Flow Measuring System Measurement of the air mass flowrate is accomplished using an Annubar Diamond II model DNT10 mass flowmeter located downstream of the regulator and before the mixing chamber. The diamond cross section of the flowmeter is such that it has a fixed separation point and also reduces pressure loss. The flowmeter senses differential pressure which is measured with a differential pressure (DP) transducer, which is Table 21. Water Mass Flowrate and Average Water Velocity for Different Regulator Pressures and Orifice Sizes. Orifice Regulator Water Mass Average Water Size Pressure Flowrate Velocity (mm) (MPa) (kg/s) (m/s) 1.0 3.08 x 104 3.60 0.33 1.7 3.99 x 104 4.67 2.4 4.73 x 104 5.53 1.0 2.67 x 104 2.50 0.37 1.7 3.31 x 104 3.11 2.4 3.97 x 104 3.73 1.0 4.61 x 104 3.55 0.41 1.7 5.72 x 104 4.41 2.4 6.79 x 104 5.24 1.0 4.85 x 104 2.39 0.51 1.7 6.90 x 104 3.00 2.4 7.20 x 104 3.55 calibrated to measure pressure differences of equation for the flow meter is up to 2.21 x 103 MPa. The calibration m = 58.283KD2 APf (21) where A P is the measured differential pressure in kPa, D is the diameter of the flowmeter in mm, in this case 15.80 mm, K is a gage factor of 0.6, and pf is the density of the flowing air, in kg/m3 calculated via pf = 539.5 (22) Tf where Pf is the pressure of the flowing air in kPa, as measured by the pressure transducer upstream of the mass flowmeter and Tf is the temperature of the flowing air in Kelvin as measured by the the thermocouple upstream of the mass flowmeter. 0.035 Experiment  theoretical 0.03  +20% 20% I 0.025 0.02  0.015  0.01  0.005 0.01 0.015 0.02 0.025 7mtheo Figure 25. A comparison of the theoretical and experimentally measured air mass flowrate. The theoretical mass flowrate through the nozzle for 1D isentropic flow is calculated using mass flow, m = pAV, the Mach number, M = a/V (the speed of sound for a perfect gas is a = vYRT), and the ideal gas law, P = pRT, m= D2MP (23) 4 RT Knowing that at the throat of the nozzle the Mach number is one and using Temperature and Pressure stagnation ratios of T/To = 0.8333 and P/Po = 0.5282 Equation (23) reduces to m = 0.4545D2P /o (24) PRT, Equation (24) neglects the effects of friction and heat transfer, which affect the flowrate of air through the nozzle. A comparison of the air mass flowrate measured during the course of experiments with the theoretical air mass flowrate is shown in Figure 25. The agreement between theory and experiment is within 20%. 2.1.5 Temperature and Pressure Measurements The temperature and pressure of the jet impingement system is monitored during system operation for calculating various quantities of interest. The temperature measurements are accomplished by the use of E type thermocouple probes which are inserted into 'T' junction compression fittings at a depth such that the tip of the probe is at the centerline of the fitting. The thermocouple probes used are grounded and sheathed in stainless steel and have a nominal diameter of 1.59 mm. Temperature measurements are taken at the following points: the outlet of the pressure regulator, the outlet of the water reservoir, and at the outlet of the mixing chamber. Pressure is measured at the outlet of the pressure regulator just before the location of the air mass flowmeter. The pressure measurement is made using a strain gage type pressure transducer, which has a range of 0 2.8 MPa. The output signal of the pressure transducer is a current which varies between 4 20 mA; because the DAQ system used in the experiments only senses voltages a resistor of 520 Q is used to convert this current into a voltage in the 0 10 V range needed. 2.1.6 ConvergingDiverging Nozzle The convergingdiverging nozzle is where the mixture of liquid and air are expanded to supersonic speeds. The nozzle is constructed of stainless steel with a throat diameter of 2.38 mm and an exit diameter of 5.56 mm, giving an exit Mach number of 3.26. The nozzle is attached to a size 10 DN (1/2" NPS) stainless steel pipe with an internal diameter of 13.51 mm, which is connected to a braided stainless steel hose approximately 9 m long with an inner diameter of 9.53 mm and is connected to the outlet of the mixing chamber. Although the hose adds some small amount of pressure 6 31.75 Figure 26. Cross sectional view of the convergingdiverging nozzle used in experiments. loss, it allows the nozzle to be located away from the air storage cylinders and near the impingement heat transfer targets. A cross sectional view of the nozzle is shown in Figure 26. 2.1.7 Data Acquisition System The DAQ used during the course of steady state heat transfer experiments is a DAS 1601 data acquisition PCI card and a CIOEXP32 analog to digital converter board, both made by Measurement and Computing Inc. This DAQ consists of 32 16bit double ended channels and channel gains of 1, 10, 100, 200, and 500 are selectable. The system has a maximum reliable sampling rate of 50 Hz and the software Softwire, produced by Measurement and Computing Inc is used for programming data collection. For transient measurements on heated targets during inverse heat transfer experiments, the DAQ system is supplemented with a National Instruments (NI), NI USB6210 system which has 8 double ended 16bit channels and has a maximum aggregate sampling rate of 250 kHz. This system uses Labview software produced by NI which and is also able to interface with the Measurement and Computing DAQ via the use of an NI supplied .dll library. 2.2 Analysis of Impingement Facility Some analysis of the jet impingement facilities are warranted. The behavior of the system upstream of the nozzle is examined to determine if there are any corrections that need to be applied to the thermocouple or pressure transducer readings. Additionally, the following is examined: the pressure required to operate the nozzle in a perfectly expanded manner, the minimum and maximum pressure that cause a shock wave to form inside the nozzle, and the nozzle exit pressure when operating at various regulator pressures. Lastly the shock wave angles forming at the nozzle exit for various operating pressure are calculated as well. Onedimensional gas dynamic relations are used to investigate the quantities of interest. Here it is noted that the analysis used has limitations, the one dimension gas dynamic relations are isentropic in nature, with the exception of shock wave calculations. The jet impingement facility experiences friction and heat transfer during operation, thus the isentropic assumption is not met. Additionally after the mixing chamber the flow will contain water droplets which are not compressible. The quantities calculated below will have some inherent error however, they do provide a reasonable approximation of the physics taking place in the facility. 2.2.1 Temperature and Pressure Upstream of the Nozzle Calculating the temperature and pressure at various points upstream of the nozzle is a simple matter; the crosssectional area of the points in the system are required for this analysis; Figure 27 provides an illustration of the jet impingement facility and the diameters of the points of interest. Using the commonly known onedimensional gas dynamics relationships found in various compressible flow textbooks, such as Liepmann and Roshko [49] or John and Keith [50], the pressure and temperature ratios as well as the Mach number of the flow in these areas can be determined. In the following SSection 4 D = 6.22 mm S Section 3 D = 15.80mm Section 2 SD = 6.22 mm  Section 1 D = 13.51 mm Throat  hD = 2.38 mm SNozzle Exit D = 5.56 mm Figure 27. Simplified view of the air flow path in the facility. equations 7 is the ratio of specific heats and is a constant equal to 1.4. To determine the flow Mach number the following relationship is used A 1 2 9 11 2(1) A* M 1 2 ) M2)\ (25) where the superscript denotes the critical area where Mach = 1. Note that Equation (25) is a quadratic equation in M and has two solutions thus careful attention must be paid in selecting the proper Mach number given an area ratio, in the present case all Mach numbers upstream of the throat of the convergingdiverging nozzle are subsonic. Once the Mach number of the given section is determined the pressure and temperature ratios can be determined from the following S=1+ 1M2 (26) T 2 P = l1Y M2) (27) where the subscript o is the stagnation property, which is simply the particular property with zero velocity. The analysis results using Equations (25) to (27) are shown in Table 22. The results show that the temperature and pressure upstream of the nozzle throat differ from their stagnation point properties by less than 1%; no correction due the the velocity of the flow is needed. Table 22. Area, temperature, and pressure ratios at various points in the iet Point Exit Throat 1 2 3 4 impingement facility. A/A, T/To P/Po M 5.44 0.3193 0.01840 3.26 1.00 0.5283 0.8333 1.00 32.20 0.9999 0.9998 0.018 6.83 0.9986 0.9950 0.085 44.02 0.9999 1.0000 0.013 6.83 0.9986 0.9950 0.085 2.2.2 Nozzle Exit Pressure Considerations There are a few theoretical considerations that need to be explored at the nozzle exit. First, the necessary regulator pressure in order to obtain a perfect expansion at the nozzle exit is needed; then the actual exit pressures based on the regulator pressure during experiments are determined. The results from the previous calculations listed in Table 22 show the stagnation pressure ratio at the exit of the nozzle, simply carrying the requisite algebra and assuming a back pressure of 101.4 kPa yields the nozzle exit pressure, see the results of these calculations in Table 23. From these results it is observed that the pressure necessary for ideal expansion is approximately twice the pressure the regulator of the system can provide, and thus during normal operation of the jet impingement facility the nozzle operates in an overexpanded manner. Table 23. Nozzle exit pressure for various regulator pressures. Regulator Nozzle Exit Nozzle Pressure Pressure Operation (MPa) (MPa) 5.5 0.1014 perfectly expanded 2.8 0.0508 overexpanded 2.4 0.0443 overexpanded 1.7 0.0317 overexpanded 1.0 0.0190 overexpanded Due to the fact that the nozzle exit pressure is below the ambient pressure some concern about a shock wave forming in the nozzle will be addressed. There are two limiting pressures for this case, one is the pressure at which at shock wave forms at the throat of the nozzle, the other is the pressure that a shock wave forms at the nozzle exit; Figure 28 shows an illustration for both of these two cases. In the limiting case, a shock wave occurring at the throat (where the Mach number is equal to unity), the stagnation pressure ratio is 0.992. Assuming that the back pressure is atmospheric pressure, the stagnation pressure that will cause a shock to be located at the throat is 0.102 MPa. To calculate the limiting case of a shock wave occurring at the nozzle exit is just slightly more complicated. When it is assumed that a shock wave is located at the exit plane of the nozzle, the stagnation pressure ratio and Mach number just before the exit plane can be found from Table 22. The normal shock wave relations for Mach number and static pressure ratio across a shock (Equations (28) and (29)) can then be applied, /M2(y7 1) + 2 M2 = M 1) (28) P2 2 M 7 1 P2 2 1 (29) P1 7<+1 7+1' Nozzle Exit Throat Po I Patm I I Figure 28. Illustration of the limiting cases for shock waves in the nozzle. Using Equations (28) and (29) the Mach number just past the shock wave is found to be 0.461 and the static pressure ratio is 12.268. Performing the requisite algebra yields a back pressure of 0.449 MPa. Thus the nozzle will have a shock wave located inside for a stagnation (regulator) pressure in the range of 0.102 to 0.449 MPa. Since the minimum regulator pressure used during experiments is 1.0 MPa there is little concern that a shock wave will form inside the nozzle. 2.2.3 Oblique Shock Waves at Nozzle Exit It is known that the converging diverging nozzle operates in an overexpanded manner; the exit conditions of the nozzle should be considered. When a nozzle is overexpanded oblique shock waves form at the outlet of the nozzle, see [49] for instance. These shock waves compress the air such that it is then equal to the nozzle back pressure; the first oblique shock wave coming out of the nozzle can be modeled using the standard one dimensional gas dynamic relations parameters such as the shock angle, deflection angle, temperature ratio, stagnation pressure ratio, and down stream Mach number. Figure 29 shows an illustration of the shock wave at the nozzle exit. To determine these exit quantities, first the nozzle exit pressure should be determined using the regulator pressure and Equation (27). The shock wave angle can Nozzle SOblique Shock Wave ST,,PM,/ Figure 29. Illustration of an oblique shock wave at the nozzle exit. then be calculated using the following equation since the Mach number at the nozzle exit is known from the design conditions, and the static pressure ratio can be calculated, P2 2y M2 sin2 1 U1 (210) The deflection angle, 6 can be determined from the following M2sin20 1 tan6 = 2cotO (211) M2 (y + cos20) + 2 The downstream Mach number, stagnation pressure ratio, and static temperature ratio are easily calculated via the following equations: 1 1+ M2 sin20 M2 = 1 (212) sin (0 6) M sin2 (212) Po1 2 M sin2 1 Y (21 3) Po2 [1+ 2M2 sin2O L 27 M2 sin2O (213) TPo2 1 M2 sin20 ) ( M2 sin20 )_ (214) T1 (7+1)'] M2 sin20 h  ) 40 22 20 35 18 16 30 14 12 25 10 1 1.5 2 2.5 1 1.5 2 2.5 Pol (MPa) Po (MPa) a b 1 2 0.95 1.9 0.9 1.8 0.85 1.7 0.8 1.6 0.75 / 1.5 0.7 1.4 0.65 k 1.3 1 1.5 2 2.5 1 1.5 2 2.5 Pol (MPa) Pol (MPa) c d Figure 210. The variation of a) shock angle, b) deflection angle, c) stagnation pressure ratio and d) static temperature ratio as a function of upstream stagnation pressure, downstream of an oblique shock wave. It is noted that the stagnation temperature across the shock is constant. Results of these calculations are shown in Figure 210. It is briefly mentioned that the stagnation pressure ratio reflects a loss of momentum going across a shock wave and that this loss of energy is lessened at higher upstream pressure ratios. 2.2.4 Complete Shock Structure of an Overexpanded Jet The first shock wave at the nozzle exit is easily modeled as shown above; however, the subsequent behavior of those shock waves is quite complex. The oblique shock waves can intersect at a point downstream or can merge and form a normal shock area known as a Mach disk. Mach disks typically form after relatively strong shocks which are typical for nozzles operating far removed from the ideal pressure ratio. These shock waves compress the flow causing the formation of PrandtlMeyer expansion fans which turn the flow and lower the pressure. When expansion fans intersect the shear layer which is formed at the boundary of the jet, they are reflected back as oblique shock waves. This series of events causes the formation of "shock diamonds" in the flow and is repeated until the combination of viscous effects and the influx of low momentum fluid cause the jet to become subsonic, or when the jet interacts with an obstacle. Figure 211 provides and illustration of the shock structure typically seen at the exit of an overexpanded jet. The pressure, temperature, and velocity of the flow in the down stream of the nozzle changes very rapidly and is difficult to model analytically. Zapryagaev et al. [29] performed experiments with an overexpanded nozzle and performed schlieren photography as well. Their results show that the pressure in the flow upstream of the first Mach disk varies greatly in the radial and axial direction with several sharp discontinuities present. Downstream of the first Mach disk the variation in pressure is still present. However, the discontinuities are no longer present. Many authors have extensively studied underexanded jets [30, 34, 35, 46, 51, 52] and similar trends as the above are observed. 2.3 Summary In this Chapter the construction of the jet impingement facility has been explored. The facility systems include the air storage, water storage and flow control, air pressure control, air mass flow measuring, convergingdiverging nozzle, and data acquisition. A comparison between the mass flow rate measured during experiments and the theoretical mass flowrate based on onedimensional isentropic gas dynamic relations was performed and the results differed by less than 20%. The pressure, temperature, and Mach number various sections of the jet impingement facility were determined and Oblique Shock Wave Viscous Shear Layer i I      PrandtlMeyer Mach Disk Expansion Fan Nozzle Exit Figure 211. Illustration of the structure of oblique shock waves at the exit of an overexpanded nozzle. are shown in Table 22. The nozzle exit pressure was calculated for each regulator pressure used during the experiments, and it was found that the nozzle operates in an overexpanded manner for the entire pressure range. No shock wave is expected inside the nozzle. Finally, the shock structure at the exit of the nozzle has been described. CHAPTER 3 STEADY STATE EXPERIMENTS Jet impingement facilities are capable of very high heat flux removal, and an experimental procedure to determine the heat transfer coefficient for the facility in Chapter 2 is developed. Several different experiments were initially tested without success before a successful experimental configuration was developed to measure the steadystate heat transfer coefficient over a range of operating conditions. Initially it was believed that phase change heat transfer would occur during operation of the jet impingement facility to such an extent that no radial variation of heat transfer coefficient would be observed. As such an experiment was designed in which a thin sheet of stainless steel was machined into a metal blank and heated via Joule heating and insulated at the bottom where the temperature was measured via a thermocouple. The area of the strip was quite small (approximately 15 mm2) and as such the heat fluxes produced during the experiment were high. The problem experienced with this setup is that the back wall temperature measured by the thermocouple is not very sensitive to changes in the heat transfer coefficient with the applied heat fluxes and the metal thickness used. An illustration of the heater assembly used is shown in Figure 31. In order to alleviate this problem an experiment was conducted where thermocouples were embedded inside of a copper cylinder which was heated from the bottom and insulated along the side. The jet was allowed to impinge on the top of the cylinder and the temperatures inside the copper piece were measured. Onedimensional heat conduction in the axial direction was assumed and due to the absence of internal heat generation, a linear fit of the measured temperature was then used to extrapolate the temperature to the surface and allowed the determination of heat flux. Upon analyzing the data gained from these experiments it was observed that the heat transfer coefficient Thickness 0.3mm . x *  Area 3 x 5 mm Figure 31. Stainless steel heater assembly. for the jet impingement facility varies significantly in the radial direction. An illustration of the copper heater assembly used is shown in Figure 32. In order to gain some insight into the radial variation of the heat transfer coefficient the heater in Figure 32 was modified to include temperature sensitive paint on the top surface of the copper test piece. Steady state temperature distributions at the top surface where then used as input to an inverse heat transfer algorithm to determine the radially varying heat transfer coefficient. There were several drawbacks to this study. First the temperature sensitive paint is very brittle and had to be protected from the impinging jet via the use of thick clear coat applications to the surface of the paint or via the use of transparent tape. These protective layers could not be neglected in the inverse heat transfer analysis and complicated the algorithm. Lastly, and most importantly, the impinging jet partially obscures visual observation of the temperature Insulation Copper Test Piece Thermocouple Cartridge Heater Insulation Figure 32. Copper heater assembly. sensitive paint. These complications render reliable experimental results difficult to obtain. The third experimental configuration tested consists of a thin sheet of nichrome which is heated by Joule heating. It is insulated at the bottom and 9 thermocouples are used to measured the spatial variation of the back wall temperature. The area of the nichrome strip is large and the heat fluxes produced are considerably smaller than those applied to the stainless steel setup described earlier. This experimental configuration proved to give reliable measurements of heat transfer coefficient, and the abundance of thermocouples allows the spatial variation of heat transfer coefficient to be determined. This experiment is now be described in detail. Later Sections will describe the experimental procedure and explore the heat transfer results. 3.1 Heater Construction 3.1.1 Physical Description The heater design used during the course of the following experiments is inspired by the work of Rahimi et al. [51]. It is constructed using a thin nichrome strip which is 0.127 mm thick with an exposed area of 50.8 x 25.4 mm. The heater has 9 total E type thermocouples attached to the backside of the strip. One thermocouple is attached at the center of the strip and 7 additional thermocouples are attached every 3.79 mm towards one side of the strip. Additionally one thermocouple is attached at 12.7 mm from the center on the opposite side. This thermocouple is used to ensure the jet is centered over the heater by verifying symmetry. The nichrome strip is then epoxied on top of a Garolite slab that is 140 x 140 x 6.35 mm. Small holes are drilled in the slab so that the thermocouples penetrate it and avoid deforming the flat surface of the nichrome strip. The slab of Garolite has a thermal conductivity of 0.27 W/mK compared with 13 W/mK for the nichrome and thus acts to insulate the backside of the nichrome strip. Electrical power is supplied to the nichrome via two copper bus bars with dimensions of 43 x 25 x 2 mm attached to the top of the strip. During operation of the twophase jet, liquid flows towards the bus bars and accumulates at the edge. This liquid film of accumulation could affect the heat transfer physics; to lessen this effect the edges of the bus bars are filed to an angle of approximately 300. An illustration of the heater assembly is shown in Figure 33. Power is supplied to the nichrome strip via a high current, low voltage dc power supply. The power supply is capable of supplying 4.5 kW of power through a voltage range of 030V and a current limit of 125 A. The maximum voltage seen during experiments is approximately 5 V at 125 A. Copper Bus Bar I Garolite Base N nnhr, : rr, > Healcr z 6 1 \300 Power Supply Thermocouples Electrode Figure 33. Illustration of heater assembly used for steady state experiments. 3.1.2 Theoretical Concerns The thermocouples used for the experiment measure the temperature on the back wall of the Nichrome strip. To determine the heat transfer coefficient for the impinging jet the surface temperature of the nichrome is needed. Typical heat transfer coefficients for impinging jets will yield Biot numbers (Bi = h6/k) much greater than unity thus a lumped system assumption is not valid for the current experimental configuration. The following analysis provides a method for evaluating the spatially resolved surface temperature. First the steady state heat equation in Cartesian coordinates with internal heat generation is examined. 2 T T2 T2 q'" X2 2 + Z2 + 0 (31) ax2 ay2 az2 k When Equation (31) is nondimensionalized the following results: ()6 22 () 6 \2 2 9 z*22 0 y2 + l = 0, (32) where STk x y z 0= x = y= z = q'562 D D' 6 Here D is the nozzle diameter, and 6 is the thickness of the strip. The coefficients in the first 2 terms of Equation (32) are quite small, and three dimensional effects can be neglected. As such the governing equation and boundary conditions are d02 dz*2 dO d = 0 (33) dz* z*_ dO kD  kDNUD (r*) (0 0, (r*)) dz* kuDr k,6 where r* = r/D is the nondimensional radius with the origin at the centerline of the jet, NuD (r*) = h(r*) D/ k, is the Nusselt number, k, is the thermal conductivity of water, and 0o(r*) is the reference temperature. Due to the extreme difficulty in measuring the expanding jet fluid temperature, the adiabatic wall temperature is commonly used as a reference for purposes of computing a heat transfer coefficient [53] associated with impinging jets. Note that the effects of the radial variation of heat transfer coefficient come from the boundary conditions only. The solution of this ordinary differential equation is: S(r*, z*) = 1 z*2) +D w(r*) (34) 2 kD NUD(r*) or in dimensional form T (r, z) = q (62 2) h + T. (r), (35) 2k h (r) where Ta,w is the adiabatic wall temperature. It is observed that the difference in temperature between the top and bottom surfaces is the quantity q"'62/2k, in dimensionless form it is equal to 1/2. The maximum volumetric heat generation in the experiments is 3.66 x 109 W/m3. Using a value of 13 W/mK for the thermal conductivity of nichrome the maximum temperature difference between the upper and lower surface observed in experiments is on the order of 2 C and cannot be neglected. However, the solution for the heat transfer coefficient and thus the Nusselt number can be calculated knowing the internal heat generation rate, back wall temperature, and the adiabatic wall temperature: NUD(r*) NNUD,w(r*) 1 k D NuD,(r*) kw6 2 where (36) 1 NUDw(r*) = D[O(r*, 0) e,(r*)] 3.2 Experimental Procedure During the course of the experiments, the Measurement and Computing data acquisition system is used Also note that the heat flux removed from the top of the heater assembly is the quantity q"'6, since the heat generated internally can only be removed from the top surface. 3.2.1 TwoPhase Experiments During the course of experiments it was observed that running the impinging jet would cause ice to form on the surface of the heater at low heat fluxes. At adiabatic conditions the ice would begin to form into a cone shape, eventually this cone would be broken off and a new cone would form in its place in a periodic manner, as shown in Figure 34. At low but, nonzero heat fluxes, a thin sheet of ice would form that would exhibit similar periodic behavior as the ice cones. This ice formation affects heat transfer since this layer of ice is stationary and acts as an insulator. To avoid this condition a minimum heat flux of 300 kW/m2 was chosen such that ice formation is not visually observed during operation of the impinging jet. Figure 34. Ice formation at adiabatic conditions. Note the conical shape of the ice structure. To run a complete experiment, the nozzle is aligned near the center of the heater surface at a given height and the impinging jet is initiated with the pressure regulator set at a desired pressure. The power supply to the heater is turned on and a heat flux of approximately 470 kW/m2 is supplied to the heater. To ensure the impinging jet is centered over the heater, the position of the heater is moved such that the temperatures 60 measured by the single thermocouple on one side matches the temperature of the corresponding thermocouple on the opposite side to within approximately 0.5 OC. Once this condition is achieved the heater is allowed to reach steady state, which is determined by observing a graph of the heater temperatures vs. time. Upon reaching steady state operation, the heater thermocouples are logged at a sampling rate of 50 Hz for approximately 2 minutes. Afterwards this same procedure is performed for heat fluxes of approximately 430, 390, 350, and 315 kW/m2. These heat fluxes are chosen to be as high as reasonably achievable with the given equipment for two reasons. First to help prevent the formation of ice on the surface of the heater and such that the heater wall temperature difference is as high as possible to minimize the uncertainty in the heat transfer coefficient measurement. The data reported for this study are averaged values from 100 samples taken over the course of 2 minutes. While it is possible for high frequency oscillations in the measured temperatures to exist due to the multitude of droplets impinging onto the surface, this is not likely to be observed for several reasons. First, the data are averaged which will lessen any transient effects. Second, the heater, although thin, does have a finite thickness. So it will tend to act like a low pass filter and dampen fluctuations. Lastly it is believed that a thin liquid layer exists on the surface of the heater. Any liquid drops that impinge onto the heater will tend to coalesce with this liquid film, thus minimizing transients of the local heat transfer coefficient. The measurement of the adiabatic wall temperature is accomplished by a similar procedure as above except after ensuring the jet is centered, the power supply to the heater is turned off. The heater temperatures are allowed to reach steady state and then measurement of the adiabatic wall temperature is commenced. It could take several minutes (on the order of 5 minutes) for the heater to reach steady state. Because of the absence of heat transfer, the measured temperature at the back wall is equal to the surface adiabatic wall temperature of the jet. It is noted that the formation of ice was observed on the surface of the heater during the adiabatic wall temperature measurements, but because of the reasons listed above for neglecting fluctuations due to multiple droplets impinging onto the surface, it is believed that this will have little to no effect on the measurement of adiabatic wall temperature. In computing the jet Reynolds number. the viscosity is based on the nozzle exit air temperature and pressure as calculated by onedimensional gas dynamic relations. In computing the Nusselt number for the singlephase jet, thermal conductivity is evaluated based on air and the adiabatic wall temperature. During twophase jet impingement, a thin liquid film exists on the heater surface, and thin liquid film dynamics dominate the heat transfer physics. Thus the water thermal conductivity based on the adiabatic wall temperature is used for Nusselt number calculations of the twophase jet. Typical values for air viscosity, water thermal conductivity, and air thermal conductivity respectively are 6.45 x 106 Pas with less than 1% variation, 0.588 W/mK with a 3% variation, and 0.0243 W/mK with a 5% variation. 3.2.2 SinglePhase Experiments To provide a comparison for reference to the twophase jet results singlephase experiments are carried out with the jet impingement facility using only air expanding through the nozzle, water flow is cut off. The experimental procedure is essentially identical to that for the twophase jet expect the applied heat flux is reduced. The heat fluxes used during experiments are lower than those in the twophase experiment to ensure that the heater does not overheat and delaminate from the Garolite base. Table 31 displays the corresponding heat fluxes for a given Reynolds number. 3.3 Experimental Results 3.3.1 Uncertainty Analysis Uncertainty analysis for the experiments is done using the method of Kline and McClintock [54]. The uncertainty of the calculated Nusselt number ranges from 2.0 to 4.0% for the singlephase jet and 2.5 to 18% for the twophase jet near the centerline Table 31. Reynolds number and corresponding heat fluxes. Nominal lowest mid highest ReD Heat Flux Heat Flux Heat Flux (kW/m2) (kW/m2) (kW/m2) 4.5 x105 35 50 65 7.3 x105 60 80 100 1.0 x106 80 100 120 and 0.3 to 1.0% at the outer extents of the domain. The Reynolds number uncertainty ranges from 2 to 3% and does not vary appreciable between the single and twophase experiments. The uncertainty in the temperature measurements are 0.2 OC. To help ascertain the error in the experiments by assuming that the three dimensional effects were neglected, a numerical analysis was conducted. This analysis uses a second order accurate finite difference scheme to solve Equation (32) on a three dimensional grid of 2" x 2" x 2" where n = 4, 5, 6, and 7. The top boundary is modeled as having a Nusselt number distribution found in the experiments and the remaining sides are modeled as being insulated. After completion of each simulation the Nusselt number is calculated using Equation (36). To characterize the error in using a one dimensional assumption the rootmeansquare error is calculated via the following equation f f (Nuexp Nsim)2 dxdy errorrms 0 0 L W (37) ff (Nuexp)2 dxdy o o the use of the different grid sizes allows the extrapolation of the error using Richardson's extrapolation method [55, 56]. Using the highest Nusselt number distribution found during the experiments, results in an rms error of 1.21% with a peak error of 3% located near the origin, while using the lowest Nusselt number distribution results in an rms error of 0.83 % with a peak error of 3.25% located near the edge of the domain. These errors 2500 q' = 65.3 kW/m2 0 q' = 49.3 kW/m2 2000 0 q' = 33.4 kW/m2 z/D = 2.0 1500 I ReD = 4.45 x 105 1000  e 500 0 I I I I I I I I 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D Figure 35. Measured singlephase NuD spatial variation at different heat fluxes. are less than the uncertainty contained in the experimental measurements and thus the onedimensional treatment for evaluating the Nusselt number is deemed satisfactory. 3.3.2 SinglePhase Results During the course of the experiments it was observed that the heat transfer coefficient is independent of heat flux, as expected. Results for a typical experiment are shown in Figure 35. To illustrate the amount of uncertainty in the data error bars have been included in this figure. However, to facilitate ease of viewing they are not shown in the rest of this Section. Figure 36 shows the measured thermocouple temperature profiles at various heat fluxes for a nozzle spacing of 4 nozzle diameters for a singlephase experiment. Note that the adiabatic wall temperature corresponds to that measured with zero applied heat flux. Figure 37 shows that Nusselt number scales with ReD as reported by 1 1.5 2 r/D 2.5 3 3.5 Figure 36. Spatial heater temperature variation at different applied heat fluxes. Donaldson et al. [46] and Rahim et al. [51], among others. As a matter of reference, a singlephase Nusselt number on the order of 2,500 corresponds to a heat transfer coefficient on the order of 11,000 W/m2K in the present study. A single nozzle was used in the experiments and thus the overexpansion pressure ratio and Reynolds number are not independent of each other and the effects of overexpansion ratio could not be isolated. Figure 38 compares the local Nusselt number for nozzle heights and Reynolds numbers used during the experiments. For r/D > 0.5 and H/D > 2 the Nusselt number distribution is not strongly dependent on the nozzle height. However, for r/D < 0.5 there is a small variation in Nusselt number. This behavior results from the complex shock structure at the nozzle exit and its interaction with the heater surface. For H/D < 2 Nusselt number is slightly elevated, but this effect appears to lessen at higher Reynolds I I I I I * q"= 102 kW/m2 0 q"= 79 kW/m2 0 q"= 62 kW/m2 S q"= 0 kW/m2 Z/D = 4.0 ReD = 7.50 x 105 O i i i ii i* 0 0.5 ReD =4 57x105 4000 A Re = 7.355X 10s  O ReD 1.05x10' 3500 z/D 6.0 3000 O 0 2500 z A 2000 o A 1500 A 2 * 500 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D a 4.5 ReD =4.57x10 4 A ReD =755x10n O ReD 1 05106 3.5 z/D = 6 0 3a Q 0 S2.5 1.5 0.5 0 0 0.5 1 15 2 25 3 35 4 r/D b Figure 37. Spatial variation of NuD at different ReD, a) unsealed and b) scaled. numbers. A possible explanation for this behavior is that due to the low temperature of the flowing air, the nozzle becomes cooled. This will cause moisture in the surrounding air to condense on the nozzle which can become entrained in the jet. This entrained moisture will increase the amount of heat removed from the surface of the heater and thus elevate the measured Nusselt number. To combat this issue the nozzle is insulated to the best extent possible. With the added insulation, it is believed that the moisture condensation effects are minimal but, nonzero. 0 0.5 1 1.5 2 r/D a 2.5 3 3.5 4 0 05 1 15 25 3 35 4 0 0.5 1 1.5 2 r/D 2.5 3 3.5 4 Figure 38. Singlephase NUD at various nozzle height to diameter ratios, a) ReD = 4.57 x 105, b) ReD = 7.55 x 105, and c) ReD = 1.05 x 106. 3.3.3 TwoPhase Results The twophase jet experiments are performed in the same manner as the singlephase jet with the exception of water being added to the airstream. In order to quantify the effect of the water on the heat transfer properties, the mass fraction of water in the jet is calculated as ml w m + mar (38) H/D = 2 0 A H/D 4 SH/D = 6.0 + H/D = b. ReD = 4.37 x103 ? * * * 9 H/D 2.0 A H/D =4.0 0 H/D = 6.0 + H/D 8.0 * HeD = 7.55 x10' 6 * A 1* * * ai H/D 2.0 A H/D = 4. 0 H/D = .0 + H/D = 3.0 RePD 1.05 x10" Y A 9 P * ? * q" = 478 kW/m2 0 q" = 440 kW/m2 0 q" = 397kW/m2 A q" 355 kW/m2 0 q" = 317 kW/m" 800 Z/D 2.0 w 0.0236 eD 4.50 x 105 600 400 200 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D Figure 39. Measured twophase NuD spatial variation at different heat fluxes without ice formation. In general, the twophase heat transfer coefficient is found to be independent of heat flux. However, as previously mentioned, when the heat flux at the heater surface is too low, ice formation affects the heat transfer measurements. In order to combat ice formation a minimum heat flux of 315 kW/m2 is used. Nevertheless, there are a few cases where icing is observed in heat fluxes up 350 kW/m2. To identify and help mitigate these effects, the mean and standard deviation of the heat transfer coefficient as a function of space is taken. When the standard deviation of the experimental values exceeded 20%, then heat fluxes of 470, 430, and 390 kW/m2 are used in the averaging calculations. These heat fluxes are selected because the higher heat fluxes will result in higher surface temperatures and inhibit ice formation. Also the higher temperatures will result in a larger AT and less uncertainty in the computed heat transfer coefficient. Approximately 20% of the measurements taken require these corrective measures, and in all cases the resulting standard deviation is less than 20% of the mean. See Figure 39 for an example of an experiment where the heat transfer coefficient is clearly independent of heat flux and Figure 310 where a reduction in the heat flux used was necessary. 2500 q' = 474 kW/m2 Sq" = 439 kW/m A 0 q 395 kW/m2 2000 A q 360 kW/m2 o q = 313 kW/m2 Z/D 8.0 1500 w 0.02 1500 ReD 1.02 x 10" 1000 A 500 0, * 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D a 2000 Sq = 474 kW/m' 1800 0 q" =439 kW/m 0 q = 395 kW/m 1600 Z/D = 8.0 w = 0.0255 1400 ReD =1.02 x 10" 1200 1000 800 600 400 200 * 0 0.5 1 15 2 25 3 35 4 r/D b Figure 310. Measured twophase NuD spatial variation at different heat fluxes, a) ice effects present and b) after removal of lowest heat fluxes. Figure 311 shows the radial variation of measured thermocouple temperature for various heat fluxes for a twophase experiment. Note that the zero heat flux condition represents the adiabatic wall temperature. Figure 312 shows the radial variation of Nusselt number for the twophase jet at different water mass fractions and constant Reynolds number and nozzle height, Figure 313 shows the variation with nozzle height with a constant Reynolds number with a nominally constant mass fraction of liquid. Note that it is not possible in the current study to vary Reynolds number and the liquid mass 45 q"= 477 kW/m2 0 q"= 433 kW/m2 40 O q"= 391 kW/m2 O + q"= 354 kW/m2 O 35 q"= 315 kW/m2 q"= 0 kW/m2 O 30 w = 0.0255 0 + ReD = 1.01 x 106 O 0 S25 O o 0 0 20 0 0 15 + 10 5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D Figure 311. Spatial heater temperature variation at different applied heat fluxes, twophase jet results. fraction independently of each other; thus it is not possible to show how the Nusselt number scales with Reynolds number. Nusselt number generally increases with increasing Reynolds number and increasing water mass fraction near the interior of the jet. For r/D > 1.5 there does not appear to be a noticeable dependence of Nusselt number on the nozzle height. There is some variation of Nusselt number with nozzle height in the jet interior, but a definite trend is not apparent. For reference purposes, a twophase Nusselt number on the order of 2,000 corresponds to a heat transfer coefficient on the order of 200,000 W/m2 K. More experimental results than those presented in this Chapter are presented in Appendix A. I UU I I UU ,',',',: + ',1,1.i S0 .. .::::: 1400 .. 1000 Z/D = 2 0 Z/D =60 ReD =1. 42 x 105 1200 R,,eD = 4.45 x 105 800 . 1000 600 o 800 + A 600 0 400 O A + 400 A 200* 200 JC 20 0 0.5 1 1.5 2 2.5 3 3.5 4 0 05 1 15 2 25 3 3.5 4 r/D r/D a b 2500 w= .01fi A = 00189 Sw = 0.0255 + + w 0.0301 2000 Z/D = 6.0 ReD = 7.24 x 105 0 1500 + 1000 A 500 A 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D C Figure 312. Twophase NUD at various liquid mass fractions. a) Z/D = 2.0, ReD = 4.42 x 105, b) Z/D = 6.0, ReD = 4.45 x 105, and c) Z/D = 6.0, ReD = 7.24 x 105. Heat transfer coefficients exceeding 400,000 W/m2K are observed in Figure 313c, which are on the same order as the highest liquid jet heat transfer coefficients, see [2], to the author's knowledge. While there is more experimental uncertainty at these high heat transfer rates (8 to 18%), the efficacy of the twophase jet for high heat transfer applications is clearly demonstrated. It is briefly noted that the orifice for the 0.37 mm orifice had a defect and hence is not perfectly circular; the liquid flowrate delivered was less than that for the 0.33 mm orifice. The Nusselt number results for the 0.37 mm orifice are noticeably smaller that 16006 Z/D=20 A Z/D = 4,0 1400 0 Z/D = 6.0 + Z/D = 8.0 w = 0.0375 1200 ReD 442x105 A 1000 800 0 600 4OO 400 + 200 0 05 1 1 2 2 3 0 0.5 1 1.5 2 2.5 3 r/D a 4000 3500 3000 2500 2000 1500 1000 500 3.5 4 1800 1600 1400 1200 S1000 800 600 400 200 0 05 1 1.5 SZ/D = 2.0 o A Z/D = 4.0 SZ/D = 6.0 + Z/D = 8.0 w 0.0248 RoD = 1.01x10'6 0 + t 4  0 0.5 1 1.5 2 2.5 3 3.5 4 r/D C Figure 313. Twophase NUD number at 0.0375, ReD = 4.42 x 105, 0.0248, ReD = 1.01 x 106. Z/D =2.0 A Z/D = 4.0 0 Z/D = 6.0 0 + Z/D = 8.0 w= 0.0273 ReD 7.23x105 0 * 25b 2 25 3 r/D b 3.5 4 various nozzle height to diameter ratios. a) w = b) w = 0.0273, ReD = 7.23 x 105, and c) w = that of the 0.33 mm orifice and do not follow the expected trend. This is believed to be due to the eccentricity of the orifice causing different behavior in the mixing chamber and effecting the resulting droplet size/distribution at the nozzle exit. While the 0.51 mm orifice does have some eccentricity, it is not as severe as that found in the 0.37 mm orifice, and it does not seem to have a noticeable effect on the Nusselt number measurements. Pictures of each orifice are shown in Appendix C. 0 3.3.4 Evaporation Effects To help quantify the effect of evaporation on the heat transfer coefficient the saturated humidity ratio at the impingement site (r = 0) and at the edge of the measurement location (r = 50.8 mm) is carried out. The saturated humidity ratio is calculated from Wsat = 0.622 PV,sat (39) P Pv,sat The vapor saturation pressure, Pv,sat is calculated from [57] Pv,a = exp 64796 [7.85951783v + 1.84408259v15 11.78664977v3 22.6807411v35 15.9618719v4 1.80122502v7. (310) / (310) where T v= 1 647.096 T has units of Kelvin, P has units of Pascals, and v is nondimensional. At the jet impingement zone the temperature is on the order of 10 C and the pressure is approximately the stagnation pressure. Results for the saturated humidity ratio for the three separate stagnation pressures used are all on the order of 104; thus any effects due to evaporation near the centerline are considered negligible. The pressure at the edge of the heater as well as the temperature at the surface of the liquid film are unknown and a similar analysis cannot be performed. However, no visual observation of phase change at the highest temperatures seen during experiments is seen. It is observed in Figures 312 and 313, that Nusselt number remains essentially constant near the edge of the heater. Evaporation would further enhance heat transfer resulting in an increase in Nusselt number in this region thus it is believed that evaporation is likely negligible in this area as well. 3.4 Comparison between Single and TwoPhase Jets To gain an appreciation of the twophase jet heat transfer enhancement, the measured heat transfer coefficient is compared to the that for the singlephase case. The heat transfer enhancement factor is defined here as hmix S= mx (311) hair 11 A F w 0 0211 10 9 Z/D = 2.0 ReD = 4 42 x 105 A + 0 * * 0 0.5 1 1.5 2 rD a 25 3 3.5 4 0 05 1 15 25 3 35 4 0 0 0.5 1 1.5 2 rin Z/D = 2.0 ReD = 1.01 x 106 2.5 3 3.5 4 C Figure 314. Heat transfer enhancement ratio at various liquid mass fractions. a) Z/D = 2.0, ReD = 4.42 x 105, b) Z/D = 4.0, ReD = 7.35 x 105, and c) Z/D = 2.0, ReD= 1.01 x 106. 18 Sw 0152 A 0 w 0.0189 16 a w0.0277 + w 0.0277 SZ/D 4.0 14 IteD 7.35 x 10  12 + 100 0 O * 10 O 6 2 Figure 314 show the heat transfer enhancement with the variation in water mass fraction and Reynolds number. It is observed that the enhancement increases with increasing mass fraction; the increase is diminished with increasing Reynolds number. It is observed that in Figure 314 that for r/D < 0.5 there is a marked increase in the measured heat transfer enhancement. At higher Reynolds numbers the maximum enhancement occurs near the edge of the jet (r/D = 0.5). It is observed in Figure 315 that the variation of the heat transfer enhancement with nozzle height is similar to that for water mass fraction, it shows an increasing trend at lower Reynolds numbers but the effect is damped at higher Reynolds numbers. 3.5 Discussion One of the features apparent in all experiments is that at radial distances of 1.5 to 2 nozzle diameters the Nusselt number and heat transfer enhancement ratios approach a near constant value which is indicative of film flow heat transfer. Various studies [34, 40] have noted that there is a large adverse pressure gradient in this region which likely causes boundary layer separation [35]. During the course of the present experiments, two "rings" indicative of a separation region are observed on the heater surface. One of which corresponds to the edge of the nozzle where there is a shock wave present, and the other is located approximately 1.5 nozzle diameters from the jet centerline. Inside of these region the heat transfer coefficient is affected by nozzle height and the water mass fraction indicating that jet impingement is the dominating heat transfer mechanism for r/D < 1.5. All of the experiments reported are carried out at relatively low surface temperatures; the highest temperature is on the order of 70 OC. Phase change due to mass transfer of the liquid into the impinging airstream is considered negligible for reasons discussed in Section 3.3.4. Additionally the work of Benardin and Mudawar [58] explore the Leidenfrost model for impinging drops and sprays. Their model predicts the pressure in droplets using onedimensional elastic impact theory [59], and a correction factor due to 1' '1 SZ/D =2 0 20A Z/D = 2.0 O Z/D =4.0 0 Z/D = 4.0 14 A A Z/D =6.0 18 A Z/D =6.0 + Z/ 8.0 A + Z/D = 8.0 12 w = 0.0205 16+ w 0.0290 ReD 4.54 x 10 UeD 7 24 x 10" 104 120 O 8O 10 6 8 0 A + 6 4 A I A A A6 0 0 0 4 22 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 05 1 15 2 25 3 3.5 4 r/D r/D a b 30 F Z/D = 2.0 O ZiD = 4.0 + A ZiD = 0.0 25 + /D = 8.0 w 0.0233 A RcD = 1.02 x 10" 20 15 0 15 A 10 o 0 0.5 1 1.5 2 2.5 3 3.5 4 r/D C Figure 315. Heat transfer enhancement ratio at various nozzle height to diameter ratios. a) w = 0.0205, ReD = 4.54 x 105, b) w = 0.0290, ReD = 7.24 x 105, and c) w = 0.0233, ReD = 1.02 x 106. Engel [60, 61] gives good results. The pressure rise at the impingement surface can be modeled as AP = 0.20piuoUsnd (312) where Uo is the droplet velocity and Usnd is the speed of sound in the liquid. Using this equation it is seen that for any droplet velocity above approximately 7% of the speed of the air in the impinging jet (approximately Mach 3) will yield surface pressures I I I I I I above that of the critical pressure for water. Thus even if phase change occurs at the impingement point, the latent heat of vaporization is zero and no enhancement of heat transfer will occur. Because of the complex shock structures occurring when the jet impinges onto the surface, making similar arguments for regions far removed from the impingement zone are not reliable and thus are not attempted. However, it is noted that most of the liquid droplets impinging onto the surface will occur near the centerline; thus the pressure far removed from the jet centerline will be lower and evaporation may still be possible at elevated surface temperatures and heat fluxes. One of the current limitations of the current results is they lack information on the liquid drop size distribution. Such measurements are not available at the current time and future work is planned to address this deficiency. The current heat transfer measurements are compared to the singlephase liquid jet heat transfer measurements of Oh et al. [2], and the measured heat transfer coefficients are on the same order of magnitude. The liquid flow rate in the current experiments is very small when compared to those experiments, up to 0.7 g/s (542.5 g/m2 s, referenced to heated area) compared with 4.3 kg/s (2.55 x 106 g/m2 s, referenced to heated area), a feature which has significant industrial advantages. In the study performed by Oh et al., liquid to vapor phase change is not observed, and those experiments were performed at much higher heat fluxes (up to 30 times the heat fluxes reported in the current study). Future investigations will explore higher heat flux regimes. 3.6 Summary In this Chapter heat transfer enhancement measurements using twophase overexpanded supersonic impinging jets were presented for a wide range of Reynolds numbers. These jets twophase jets are generated by the addition of water droplets upstream of a convergingdiverging nozzle. Heat transfer measurements using a singlephase jet is used for comparison. It is observed that the addition of water droplets into the air flow significantly enhances the heat transfer rate. Enhancement is significant near the jet centerline the enhancement factor exceeds 10 in most cases. The mass fraction of water added to the jet is observed to by an important parameter for heat transfer, generally increasing Nusselt follows increasing water mass fraction. However, its influence diminishes at higher Reynolds numbers. Nozzle height appears to have a small impact on the observed heat transfer rates. CHAPTER 4 DETERMINATION OF HEAT TRANSFER COEFFICIENT USING AN INVERSE HEAT TRANSFER ANALYSIS As was seen in Chapter 3 the use of steady state measurement techniques yields low surface temperatures which are not suitable for evaporating the liquid film and the low AT between the wall and liquid film which creates uncertainty in evaluating the heat transfer coefficient. In order to alleviate some of these effects a transient approach involving an inverse heat transfer quenching problem is developed. 4.1 Inverse Problems There are two basic paradigms in heat transfer. The most well known paradigm is the solution of the temperature field within a medium subject to constraints such as a given thermal conductivity, thermal diffusivity, and known boundary conditions. If all of the constraints are known, then the resulting temperature field within the medium of interest can be solved; in many cases an analytical solution can be determined. However, if these conditions are not known then the problem is not unique, is underspecified, and no solution can be determined. The second paradigm in heat transfer, named inverse heat transfer, is when the temperature at specific points inside of a medium are known and a constraint needs to be determined e.g. contact resistance between two surfaces or a boundary condition. Because of unavoidable measurement errors in the temperature field this problem is illposed and can be difficult to solve. The difficulties of this problem can be circumvented in very special circumstances, for example if one desires to determine the heat flux applied at a boundary of a onedimensional bar at steady state one can ensemble average temperature measurements at a few locations along the length of the bar and determine the temperature gradient via linear regression. With a known thermal conductivity, Fourier's law can be used to determine the applied heat flux and the results can be quite accurate. Unfortunately these simple problems do not come about in practice often. For example if the heat flux varies in time, then the above method would not be applicable and a different method would be needed. Inverse Problems, in general, fall into one of two categories: parameter estimation, in which one or more desired parameters are determined using experimental data (e.g. thermal conductivity of a solid and a applied heat flux) and function estimation, in which a desired function is to be estimated using experimental data (e.g. a boundary condition which varies in space and time). It should be noted that many function estimation problems can be formulated in terms of a parameter estimation problem if the functional form of the of desired function is known, for instance if thermal conductivity is a quadratic function of temperature the problem can be reduced to determining the coefficients of the governing equation. This approach can yield good results, see Flach and Oziik [62] for example, however, if the form of the equation is not known a priori then this approach may not be useful. Inverse Problems and Inverse Heat Transfer (IHT) problems have been studied extensively in the literature and have been in use since at least the 1950's. Tikhonov, [6365] among others, was one of the first to tackle the challenge of inverse problems and take into account measurement errors. His technique titled Tikhonov's regularization minimized the least square error by adding a regularization term that penalizes unwanted oscillations in the estimated function. Tikhonov's method can be related to damped least squares methods, most notably the method due to Levenberg [66] and Marquardt [67], known as the LevenbergMarquardt method. These methods are only suitable for parameter estimation. Stoltz [68] used a function estimation technique based on Duhamel's principle and two simultaneous thermocouple measurements to determine the surface heat flux in a onedimensional problem. This process is known as exact matching and does not take into account any measurement errors. Beck [6971] used a method similar to that by Stotlz; however, temperatures at future times are used to provide regularization and reduce instabilities in the method. This method can be used for parameter or function estimation but, can become unstable for small time steps and thus highly transient phenomena cannot be accurately reproduced. The Monte Carlo method can be used to estimate a parameter or function as was demonstrated by HajiSheikh and Buckingham [72]; a good review of the technique can be found in [73]. A method that is suitable for small time steps and performs parameter or function estimation is Alifanvo's Iterative Regularization Method [74]. This method is also known as parameter/function estimation with the adjoint problem and conjugate gradient method, and is the method used for the present study. This method will be able to determine a time and space varying heat transfer coefficient produced by a multiphase supersonic impinging jet, as well as any temperature dependence due to any evaporation of the liquid film. 4.2 Introduction to Inverse Problem Solution Using the Conjugate Gradient Method with Adjoint Problem Dealing with inverse problems, which by their nature are illposed, usually involves some type of regularization technique or an optimization technique which inherently regularizes the solution. The technique used in the current study is an optimization technique known as function estimation using the conjugate gradient method with adjoint problem. As the name implies this method uses the conjugate gradient method to minimize the error in the least squares sense between the estimated output of an equation/system of equations and the measured output which has been corrupted with noise. The method will be described below in its general form to familiarize the reader. Many references exist for functional estimation with the adjoint problem and conjugate gradient method including Oziik [75], Oziik and Orlande [76], Alifanov [74], and the Chapter by Jarny [77]. Much of the following analysis follows that of Jarny as the author found that particular reference to be mathematically rigorous, thorough, general in nature, and easy to follow. Specific implementations of this method will be discussed where needed. 4.2.1 The Direct Problem The direct problem is the model equation(s) for the system of interest. It can be an algebraic, integral, ordinary differential, or partial differential equation or system of equations or some combination therein. y(x, t) = f(x, t, ) (41) where y is the output of the system, is taken to be a parameters) or function to be estimated and x and t are the independent variables. Note that although both space and time are independent variables in this example it is not necessary for the output to depend on both of them. 4.2.2 The Measurement Equation The measurement equation exists due to the discrete nature of a sampling process and due to changes brought about in data due to sensor dynamics. Although in modern data acquisition systems it is possible to measure quantities at a near continuous rate taking measurements still is an inherently discrete process. Bendat and Piersol [78] have written a good reference on data measurement and analysis which includes sensor/system dynamics. Sensor dynamics can greatly effect the measurements of a system and their effects can be quite significant. This process can be simplified if the sensor dynamics can be approximated by a linear time invariant (LTI) system, which most sensors fall under. In an LTI system the output of a sensor is the result of a convolution of its input with the sensor's impulse response function. The impulse response function is the response of the sensor, initially at rest (or zero), to an impulse input. The measurement equation can be mathematically expressed as t Ym= h(t T)y(T)dT (42) (  J where Y, is the measured output, h is the impulse response function, and y is the true output of the system. If the sensor is perfect and the goal is to simply denote its discrete nature the impulse response function would simply be a delta function. For ease of viewing the measurement equation can also be discussed in an operator form such that Equation (42) is equal to Ym = Cy (43) 4.2.3 The Indirect Problem The indirect problem is actually the statement of the least squares criteria. When solving an inverse problem with the current method the parameter or function sought is the one which minimizes the least squares criteria. Simply stated the indirect problem is M t 5(o) = m.,i Ciyi(t,)]2 dt (44) i=1 0 where S is the integrated squares (note that in the case of discrete data this would be the sum of squares), i is the sensor number, and M is the total number of sensors. The spatial dependence of y is left out of Equation (44) because it is assumed that the sensors are placed at varying distances in space, thus the measurement operator, C, would only operate on measurements at a location, xi. It can be useful to think of the least squares criteria in the form of a norm operator, Ilull or sometimes(u, v), for instance Equation (44) is equal to S(O = II Ym Cy(t, 0)11 (45) 4.2.4 The Adjoint Problem Formulating the adjoint problem correctly is a crucial step in the solution process. Essentially this is where the optimization portion of the problem comes into play. To do this the indirect problem is considered the modeling equation and the direct problem is considered as a constraint such that the following equation holds, R(y,) = y f (x, t, ) These are then joined together through the use of a Lagrange multiplier. L(y, A) = II Y Cy(t, ) I (A, R(y, )) (46) where L is the Lagrangian variable and A is the adjoint variable (also known as the Lagrange multiplier) which, in general, can be a function of space and time. When the correct parameter/function is inserted into Equation (46) the resulting Lagrangian is zero for perfect measurements. Real world measurements will be corrupted with noise and the resulting Lagrangian will be the minimum least squares criteria. To determine the proper the Lagrangian must be minimized. If the adjoint variable is treated as fixed the the differential of the Lagrangian is dL = (VyS, Ay) (A, VyR(y, )Ay) (A, VR(y, )A) (47) or expressed in a more convenient form dL = (VyS A [VyR(y, )], Ay) (A [VR(y, A)], A ) (48) Because the choice of the adjoint variable is not constrained it is chosen to be the solution of VyS A [VyR(y,)]= 0 (49) Equation (49) is known as the adjoint equation. Note that this is implicit in A, through mathematical operation (usually involving integration by parts) it can be expressed as an explicit function of the adjoint variable. 4.2.5 Gradient Equation Note that the adjoint equation renders the first term in Equation (48) to be zero. At the solution point where is equal to the true value, the Lagrangian is equal to the minimum of the least squares criteria dL = dS = (VS((), A), (410) and comparing the remainder of Equation (48) to Equation (410) the gradient equation results VS= A [V R(y, )]. (411) The gradient equation is used in the conjugate gradient minimization algorithm to determine a step size and descent direction in order to minimize the least squares criteria. 4.2.6 Sensitivity Equation As mentioned one of the parameters needed to find the minimum of the indirect problem is the step size. This parameter can take a few different forms depending on whether the inverse problem is linear or nonlinear. In the interest of presenting a general method, the form for nonlinear problems are presented. The step size to be determined is a perturbation in the parameter/function _, which is to be estimated. To derive this quantity we simply perturb the direct problem y +Ay= f(x, t, + A); (412) generally the right hand side of Equation (412) is linearized such that (413) When Equation (41) is subtracted from the above equation the sensitivity equation is the result Ay = f(x, t, t, A) (414) Note that the second term in Equation (413) and the right hand side of Equation (414) contain both and Ai. Inverse problems in which the sensitivity equation contains both parameters/functions and Ai are nonlinear. Not all inverse problems are nonlinear in nature, and this form is used here for the sake of generality. The sensitivity equation simply states that a perturbation in the parameter to be estimated will result in a perturbation of the computed output. 4.2.7 The Conjugate Gradient Method The conjugate gradient method is an optimization problem for solving linear or nonlinear equations. There are several references which detail the mathematics behind this tool, for instance the books by Rao [79] and Fletcher [80], among others. As such readers interested in a rigorous derivation of the method are encouraged to consult these references. The essential steps of the method are that a guess for is chosen, the above equations are solved and a search direction, d< which is Cconjugate to the previous direction is calculated using a conjugation coefficient, 7. The search direction is then multiplied by the step size, 3 and is added to the previous guess for _. This iterative process continues until the error between the measured output and calculated output reaches a predetermined tolerance. There are several different forms of the conjugation coefficient, 7 in the literature such as the HestenesStiefel [81], PolakRibiere [82], and FletcherReeves [83], among others. All of the mentioned forms are equivalent for linear equations; however it is y + ay = f (x, t, +) f (x,t, t, a)O. discussed in the literature [84, 85] that the PolakRibiere form of the equation has better convergence properties for nonlinear equations and as such will be used in the present study unless otherwise noted. The PolakRibiere form of the conjugation coefficient is M Z(VSk, VSk VSk1) 7k = i= for k = 1,2,3... (415) and Yk = O for k = 0 The search direction is then calculated by the following A(k = VSk + kAk1. (416) The step size is defined by the following M tf / = arg min [S( /3A)] = arg min [Yi Cyi(t, /A)]2 dt. (417) i 10 The output of the direct problem is expanded in a Taylor series as y(t, pAZ) y(t, ) POY(t) A y(t, ) 3Ay(t, A), (418) and substituting the above into Equation (417) yields M tf = arg min [Y,i C,yi(t, ) +/CAy(t, A)]2 dt. (419) i=1 0 Performing the differentiation with respect to /, setting the result equal to zero and solving for 6 yields the final form of the equation for the step size M tf E [Cy,(x, t, k) m,i(t)] CAy(t, Ak)dt Mk = i10 (420) M tr E J [Cayi(t, ak)]2 dt i=1 0 'i 1 It is noted that the above equation can be simplified for linear problems when the least squares criteria (the indirect problem) is cast in quadratic form. However, for the sake of generality, the above equation will be used throughout the current Chapter. 4.3 Factors Influencing Inverse Heat Transfer Problems There are several factors which can influence the solution of an inverse problem. Some of these factors are discussed below. 4.3.1 Boundary Condition Formulation Effects To perform a function estimation inverse problem to determine a spatially and temporally varying heat transfer coefficient, the choice of the boundary condition formulation is very important. The boundary condition can be formulated as a Dirichlet (specified temperature) boundary condition where the surface temperature is determined and the resulting heat flux is calculated in order to determine the heat transfer coefficient, as a Neumann (specified heat flux) boundary condition where the heat flux is determined and the resulting surface temperature is calculated, or as a Robin (convection) type boundary condition where the heat transfer coefficient is directly determined. At first glance, the Robin type boundary condition seems to be the best choice as the underlying physics taking place are convective in nature. Upon further analysis this is actually the worst choice. In order to demonstrate this an example using a onedimensional heat transfer problem with a time varying heat transfer coefficient will be used because of the ease of calculation. The same concepts apply to a twodimensional problem with a spatially and temporally varying heat transfer coefficient. The following analysis can be found in [86] but is reproduced here for clarity and to correct some errors contained therein. Suppose a time varying boundary condition is 30 X L Boundary Conditions Specified: : ::: T,q",orhTr.: : : Figure 41. 1dimensional solid for the sensitivity problem. applied to the x = 0 surface of a one dimensional solid with an insulated boundary at x = L, see Figure 41. The solution of this problem can be determined by using Duhamel's principle for each type of boundary condition previously discussed. Essentially the time varying boundary is convolved with the impulse response function of the slab. The impulse response function is determined by solving the heat equation for the solid with a boundary condition of unity (a step response function) and then taking the derivative of that function with respect to time. For instance the solution for a time varying heat flux in dimensionless form is .t* 8 (x*, t* ) 0 (x*, t*) = 0 q* (7) (t* d where Oq in dimensionless form is: (421) 1 x X* 2 1 2 ,q (x*, t*) = t* + x* 1 2 exp (n72t*) cos (nrx*) 0 \ ^ / / n I In order to analyze which type of boundary condition should be used in the inverse problem formulation, a sensitivity analysis should be performed. The sensitivity analysis is accomplished through the use of relative step sensitivity coefficients. First the governing equations and boundary conditions are nondimensionalized and their solution obtained. The derivative of this solution is taken with respect to the nondimensional input parameter for the boundary condition (nondimensional temperature, heat flux, or Biot number). The result of these operations is the step sensitivity coefficient, although the magnitude of the coefficient for the convection case varies depending on the magnitude of the input Biot number. To allow direct comparison of these sensitivity coefficients they are multiplied by their boundary condition inputs transforming them to relative step sensitivity coefficients, denoted as Xipt. With nondimensionalization of the problem, the net effect is only seen in the convection case. Xq (x*, t*) = q (*, t*) (422a) ) 21 12] (422b)1 1 2 X (x*, sin n 7x* exp n t* (422b) n= 1 22 n1 n 2 2 XB (x*, t*)= BIB OBi =Bi exp (At*) cos (An(1 x*)) n=l1 Cn [2Ant* cos (An(1 x*)) + (1 x*) sin (A(1 x*)] where tAn 1 (422c) 8Bi tanAn + AnSec2An and aCn 9An f 4cos(An) 8sin(An) [1+ cos(2An)] OBi OBi 2An + sin(2An) [2An + sin(2An)]2 Antan(An) = Bi Note that Equation (422c) depends on the input parameter (Bi) and thus the inverse problem is nonlinear in nature and can be difficult to solve. Also note that this equation is different than that found in [86]; the equation in that reference contains x* as opposed to 1 x*. A plot of the sensitivity coefficients at x* = 0.1 is presented in Figure 42; it should be mentioned that the magnitudes of the sensitivity coefficients are plotted and the coefficients for Bi are actually negative and this is not clarified for the plot in Reference [86]. A few points of interests should be pointed out. First, note that the coefficients for Bi are lower than all of the others and that as Bi number increases its sensitivity coefficient decreases. The coefficients for a temperature and heat flux input are much larger than those for Bi number with the coefficients for temperature being larger than those for heat flux until a value of t* z 0.75. Clearly an inverse problem formulated in terms of an unknown convection coefficient is not a good choice. 4.3.2 Sensor Location Effects The sensitivity coefficients also depend on position. The sensitivity coefficients for a heat flux input are plotted in Figures 43 and 44. It is easily seen that the closer 1.8 Xe  Xq 1.6 X XBi (Bi = 1) 1.4  XBi (Bi = 10) 1.2 XBi (Bi = 100) `   0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t* Figure 42. Relative step sensivity coefficients at x* = 0.1 as a function of time. a temperature sensor is placed to the boundary of interest the more sensitive it is to changes of that boundary condition, as one would expect from simple physical reasoning. Comparing Figures 43 and 44 one can see that the magnitude of the relative step sensitivity coefficient is larger at the back wall for an unknown surface temperature formulation than for an unknown heat flux formulation. This characteristic will be exploited in this study in order to minimize the effects of thermocouple insertion on the inverse problem. 4.3.3 Thermocouple Insertion Effects In order to perform temperature measurements, solid thermocouples are commonly used as they are a robust, inexpensive method to the measurement. As was demonstrated in Section 4.3.2, the closer to the surface of interest a sensor is placed, better sensitivities to changes in the boundary condition are achieved. This can be accomplished 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X* Figure 43. Relative step sensivity coefficient for a heat flux input. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X* Figure 44. Relative step sensivity coefficient for a temperature input. by drilling holes in the solid and inserting thermocouples inside the solid. Placing holes in the solid can have adverse effects on the heat transfer dynamics. t* 0 t* 0.1 t* = 0.5 t = 0.8 5 .5 Several researchers have studied the problem of thermocouples inserted into a solid and how they distort the thermal field as well as how this affects inverse problems. Chen and Li [87] studied the problem numerically and found the the error produced by the thermocouple insertion is proportional to the hole size and that the magnitude of the error decreases in time. Chen and Danh [88] expounded upon the research in [87] by performing experiments which confirmed some of the predicted results from numerical simulations, these studied focused on thermocouples inserted parallel to the direction of heat flow. Beck [89] used Duhamel's theorem to determine a correction kernel for thermocouples inserted normal to a low thermal conductivity surface to compensate for the insertion effects. Woodbury and Gupta [90] used numerical methods to study thermocouple insertion and the effects on inverse heat transfer problems. Woodbury and Gupta [91] also developed a simple onedimensional sensor model to numerically correct the effects for the thermocouple holes; this study also included the fin effect from the thermocouple wires and is applicable to a thermocouple of any orientation to the surface. Attia et al. [92] performed a very comprehensive numerical and experimental study which helped quantify the error that the thermocouple insertion produces on measurements including wire effects, filler material effects, and nonideal contact situations in which the thermocouple is inserted at an angle in the hole. Li and Wells [93] performed numerical and experimental work studying the different factors affecting the error due to thermocouple insertion. Interestingly they found that for a thermocouple inserted perpendicular to the direction of heat flow (i.e. parallel to the surface of interest) there would be no effect on the temperature measurements but, thermocouples oriented parallel to the direction of heat flow would have noticeable effects on measurements or on an inverse heat transfer analysis. Caron, Wells, and Li [94] continued this study and found a correction model called the equivalent depth. This model implies that the temperatures measured from an inserted thermocouple can be put into an inverse analysis as measurements taken from a different position; this new position is the one which would would experience the temperature transients recorded if there were no thermocouples inserted. The correction model was only able to accurately reproduce surface heat flux histories. Franco, Caron, and Wells [95] continued the work and developed correction models which accurately reproduce surface temperatures. At first inspection the work of Li and Wells [93] appears to give an ideal result, that orienting the thermocouple in a specified direction will cause it to have no noticeable impact. The author attempted to use this information and designed an inverse experiment with sheathed thermocouples inserted 2 mm below the top surface of a copper cylinder at a angular spacing every 45. After many trials to determine the impulse response functions of these embedded thermocouples it was concluded that thermocouples significantly impacted the heat flow and temperature field. This experimental finding is contrary to the study by Li and Wells, but it could be due to differing factors such as different types of thermocouples used, different solid material (copper for the author's experiment, aluminum for Li and Wells), and the fact there were many thermocouples inserted versus one for Li and Wells. One commonality for the above work cited is that the correction models can be quite complicated and they only assess the effects of a single thermocouple being inserted into the solid. Because of these difficulties it was decided to use simple weldedbead type thermocouples and silver solder them to the back of the solid (no insertion). This configuration eliminated all insertion effects because there are no holes drilled. This will affect the nature of the inverse problem because measurements performed at the back surface will cause the sensitivity of the inverse method to decrease. This limitation can be overcome by formulating the problem as an unknown surface temperature instead of an unknown surface heat flux or convection coefficient. 4.4 Inverse Heat Transfer Problem Formulation As was demonstrated in Section 4.3 the best choice for formulating an inverse problem for determining the heat transfer coefficient is a specified temperature Ts (rt0 . . ...............\.............. . . ..:* :*: *: :*: *:* *:* :*: :*: *:* . . . . ..^:^ : . ..:: : :: ::: ..* '* *: : '* *: : * :. ^ :^ *:*:.. . Figure 45. Illustration of the heat transfer physics of the inverse problem formulation. formulation with thermocouples measuring temperature at the back wall. Once the temperature at the impingement surface is known the heat flux at the surface and heat transfer coefficient can be determined. A schematic diagram illustrating the problem formulation is shown in Figure 45. The inverse problem equations from Section 4.2 will now be cast into the proper from for an IHT problem for a cylinder at an initial temperature that is exposed to a time and space varying surface temperature. 4.4.1 Direct Problem The Direct problem in nondimensional form is formulated as, ai 1 ai0 0 20\ ^ r*+( (423a) at* r* ar* r*) az*2 (423a) 0(r*,z* = 0, t*) = s(r*, t*) (423b) 8e S = 0 (423c) 9Z* *=1 = 0 (423d) Or* r*=0 =0 0 (423e) Or* r*=1 0 (r*,z*, t* = 0)= 0 (423f) where the following nondimensionalization is used r z at r* z*= t* R' L' R2 To T r To Ts(r*, t*) TO To 4.4.2 Measurement Equation The measurement equation is, in its rigorous form t* 1i m,i= JI I hi(t T)O(r*,z*, )6(r* r)6(z* z) r* dr* dz* d. (424) T=O r*=0z =0 Note that the delta functions in Equation (424) merely take into account the discrete nature of the measurements. Noting this point, the measurement equation can now be cast as t* m,=i = hi(t T)0j(r)dT (425) The subscript i in the equation denotes the measurement location, of which there are 7 total measurement points. Also note that each thermocouple can have its own impulse response function and hence the subscript. This equation can take on an operator form similar to Equation (43). 4.4.3 Indirect Problem The corresponding indirect problem is t S (Os) = [Zm (t*) C, (r*, z*, t*, s)]2 dt*. (426) 0 Note that the operator from of the measurement equation is used. 4.4.4 Adjoint Problem The development of the adjoint problem is quite involved mathematically. Because of the sensor dynamics involved in the measurement equation the form of the adjoint equation will look different than many of those found in the literature such as [96100] for example. To the author's knowledge there are no references in the literature that explicitly take into account the sensor dynamics, except [86], which merely discusses the convolution of the delta function to account for the discrete nature of the measurements. Also Marquardt's analysis[101] which accounts for sensor dynamics, but uses a state and disturbance observers model, which is different than using the adjoint problem such as used for the current analysis. To begin the derivation of the adjoint problem, the necessary substitutions are carried out for Equation (46) t* t M L(O, O0, A) = [Ym,i CiO(r*, z*, t*, 0)]2 0 il (427) A r 1 a ( *o\ 2 dt. at* r* r* r (r*) z*2 dt Here the norm in the second term of Equation (427) is equal to 1 R (u, v) = I Iu v r* dr* dz* (428) r*0 z*0 Next the second term of Equation (427) is integrated by parts. This allows for an explicit function of the adjoint variable to appear. After using the boundary and initial conditions of the direct problem, Equation (423), the result is the following t] L(, 0s, A) = /{ [Ym CiO(r*, z*,t*Os)]2 X. O Id \ O 2X \ S0 t* r* Or* Or*) )z*2 (429) 0A +0OA 0OA ,+0sA z*o+ t* t Next the derivative of Equation (429) is taken with respect to 0 and 0s. dL(O, s,A) = (2(Ym,i CO(r*, z*, t*, s)), CiAO) Mi {=1 AO r* a0 t* r* Or* r*j 9z*2 / (430) _AO ,9 + AO ,_AO OA + 9 Or r*=1 r r*0 Oz z*= Az z =0 A + AAO I dt. az *=o t=t The goal is to specify the adjoint equation as the solution to the terms in Equation (430) involving AO. However, the first term involves the measurement operator and AO. To rectify this the adjoint of the measurement equation is sought such that (ei(t*), CiA) = (Cjei(t*),AO) where ei(t*) = 2 [Ym(t*) m(t*)]. (431) The operator Cq is known as the adjoint operator of C,. To solve for this operator examination of the left hand side of Equation (431) gives (ei(t*), CiA0) tT t* Sei(t*) Jhi(t* T)AO(T) dTdt t*=0 T=0 (432) = A0(r) hi(t* T)e,(t*) dt dr. T=0 t*=0 Comparing Equations (431) and (432) it is observed that t* C*e,(t*)= h,(t* T)ei(t*)dt*. 0 Taking into account causality, it is known that (433) (434) hi(t* T) = 0. Therefore the equation for the operator C* is Cje,(t*) = hi(t* )ei(t*)dt* (435) Now knowing the adjoint operator of the measurement equation, the adjoint problem is selected to be the solution of A OA r* / ,a\ at* r* ar* \ r* 02A Oz*2 (436a) 100 for T > t*, C [Ym (t*) m (t*)] 
Full Text 
PAGE 1 AMULTIPHASESUPERSONICJETIMPINGEMENTFACILITYFORTHERMAL MANAGEMENT By RICHARDRAPHAELPARKER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012 PAGE 2 c 2012RichardRaphaelParker 2 PAGE 3 ACKNOWLEDGMENTS ThroughoutmytimeingraduateschoolIhavebeenhelpedbymanypeople, althoughIwilltrymybesttoincludeeveryone,inevitablyIwillleavesomepeopleout. ForthatIamsorry. FirstandforemostIwouldliketothankmyparentsaswithoutthemthiswouldnot bepossible.Dad,youtaughtmewhatitmeanttobeaman,andIcan'texpressmy gratitudeenough. Therehavebeenamultitudeofpeopleinmylabthathavehelpedmeimmensely, Dr.PatrickGarrity,Dr.AyyoubMehdizadeh,Dr.JameelKhan,andDr.FadiAlnaimat mycurrentroommatewhohaveallgraduatedandaregoingplacesintheworld,you haveprovidedhelpandmotivation.BradleyBon,FotouhAlRagomthelabmother,and ChengKangKenGuanwhoarecurrentlypreparingfortheirdissertationdefense,we haveallhelpedmotiveeachother.BenGreek,PrasannaVenvanalingam,KyleAllen, LikeLi,andNimaRahmatian,whoarecurrentlycompletingtheirdegreerequirements, havehelpedmedevelopsomeofmypresentationskills.Ihopeyouhavelearnedas muchfrommeasIhavefromyou. MyfriendsintheInterdisciplinaryMicrosystemsGroupIMGwhohaveprovided manyopportunitiestoescapethepressuresofgraduatestudies.Iwouldspecicallylike tothankmyoldroommate,Dr.DrewWetzel,wholetmebounceideasoffofhim,Dr. MattWilliams,whoalwaysranthegroupfootballcompetitionsandhelpedmereminisce aboutmyrootsinSouthCarolina,andBrandonBertolucciourofcialsocialchair.Thank youallandgoodluck. IwouldalsoliketothankmyprofessorsattheUniversityofSouthCarolina.Dr. JamilKhan,whoseheattransferclasseshelpedshapethefoundationofmyknowledge inthethermalanduidsciences,Dr.DavidRocheleauwhoprovidedmewithcontacts attheUniversityofFlorida,Dr.PhillipVoglewedewhoencouragedmetoattendthe 3 PAGE 4 UniversityofFlorida,andnallyDr.AbdelBayoumi,myundergraduateresearchadvisor whosawpotentialinmeandprovidedmanyopportunitiesformetogrow. Last,butnotleast,Iwouldliketothankmycommittee.Dr.Orazemhashelped melearnthingsoutsideofmyeldandmycomfortzoneandforthatIamgrateful.Dr. Hahn,yourrigorousheatconductionclasshelpedbuildthefoundationformuchofmy studies.Dr.Mei,youhavehelpedmerealizemypotentialfornumericalstudiesandhow toknowwhenasolutionisgoodenoughorwhenperfectionisessential.LastlyIwould liketothankmychair,Dr.JamesF.Klausner.Dr.Klausner,youhavekeptmearound, evenasIstruggled,becauseyousawthepotentialinme.You'vehelpkeepsomelevity inthelabwithfootballtalkandgoingtosomeexcellentconcerts.Mostimportantly, you'vehelpedmegrowbothprofessionallyandasaperson.Forthisyouhavemymost sinceregratitude. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................3 LISTOFTABLES......................................8 LISTOFFIGURES.....................................9 NOMENCLATURE.....................................13 ABSTRACT.........................................17 CHAPTER 1INTRODUCTIONANDLITERATUREREVIEW..................19 1.1LiteratureReview................................20 1.1.1SinglePhaseJetImpingement....................20 1.1.2MistandSprayCooling........................24 1.1.3SupersonicJetImpingement.....................25 1.2Summary....................................34 2JETIMPINGEMENTFACILITY...........................36 2.1ImpingingJetFacilitySystems........................36 2.1.1AirStorageSystem...........................36 2.1.2WaterStorageandFlowControlSystem...............36 2.1.3AirPressureControlSystem......................39 2.1.4AirMassFlowMeasuringSystem...................39 2.1.5TemperatureandPressureMeasurements..............42 2.1.6ConvergingDivergingNozzle.....................42 2.1.7DataAcquisitionSystem........................43 2.2AnalysisofImpingementFacility.......................44 2.2.1TemperatureandPressureUpstreamoftheNozzle.........44 2.2.2NozzleExitPressureConsiderations.................46 2.2.3ObliqueShockWavesatNozzleExit.................48 2.2.4CompleteShockStructureofanOverexpandedJet.........50 2.3Summary....................................51 3STEADYSTATEEXPERIMENTS..........................53 3.1HeaterConstruction..............................56 3.1.1PhysicalDescription..........................56 3.1.2TheoreticalConcerns.........................57 3.2ExperimentalProcedure............................59 3.2.1TwoPhaseExperiments........................59 3.2.2SinglePhaseExperiments.......................62 5 PAGE 6 3.3ExperimentalResults.............................63 3.3.1UncertaintyAnalysis..........................63 3.3.2SinglePhaseResults.........................64 3.3.3TwoPhaseResults...........................67 3.3.4EvaporationEffects...........................73 3.4ComparisonbetweenSingleandTwoPhaseJets..............74 3.5Discussion...................................75 3.6Summary....................................77 4DETERMINATIONOFHEATTRANSFERCOEFFICIENTUSINGANINVERSE HEATTRANSFERANALYSIS............................79 4.1InverseProblems................................79 4.2IntroductiontoInverseProblemSolutionUsingtheConjugateGradient MethodwithAdjointProblem.........................81 4.2.1TheDirectProblem...........................82 4.2.2TheMeasurementEquation......................82 4.2.3TheIndirectProblem..........................83 4.2.4TheAdjointProblem..........................83 4.2.5GradientEquation...........................85 4.2.6SensitivityEquation...........................85 4.2.7TheConjugateGradientMethod...................86 4.3FactorsInuencingInverseHeatTransferProblems.............88 4.3.1BoundaryConditionFormulationEffects...............88 4.3.2SensorLocationEffects........................91 4.3.3ThermocoupleInsertionEffects....................92 4.4InverseHeatTransferProblemFormulation.................95 4.4.1DirectProblem.............................96 4.4.2MeasurementEquation........................97 4.4.3IndirectProblem............................98 4.4.4AdjointProblem.............................98 4.4.5GradientEquation...........................101 4.4.6SensitivityProblem...........................101 4.4.7ConjugateGradientMethod......................102 4.4.8StoppingCriteria............................103 4.4.9Algorithm................................104 4.5NumericalMethodandLimitations......................105 4.5.1AlternatingDirectionImplicitMethod.................105 4.5.2GridStretchingintheZDirection...................106 4.5.3Timestepsizecomplications.....................108 4.6DeconvolutionforThermocoupleImpulseResponseFunctions......108 4.6.1DirectProblem.............................109 4.6.2IndirectProblem............................110 4.6.3AdjointProblem.............................110 4.6.4GradientEquation...........................110 6 PAGE 7 4.6.5SensitivityProblem...........................110 4.6.6ConjugateGradientMethod......................110 4.6.7StoppingCriteria............................111 4.6.8Algorithm................................111 4.6.9TestCase................................112 4.7Summary....................................114 5SENSORDYNAMICSANDTHEEFFECTIVENESSOFTHEINVERSEHEAT TRANSFERALGORITHM..............................116 5.1ThermocoupleMeasurementDynamics...................116 5.1.1LowBiotNumberThermocoupleModels...............116 5.1.2HighBiotNumberThermocoupleModels..............121 5.1.3DesignofExperiment.........................122 5.1.4ExperimentalResults..........................126 5.1.5ComparisontoEstablishedModels..................127 5.2InverseHeatTransferAlgorithmVerication.................131 5.2.1InverseQuenchingParametricStudySetup.............132 5.2.2ErrorAssessmentMethods......................136 5.2.3ParametricStudyResults.......................137 5.2.4HeatLoss/GainEffects.........................141 5.2.5EffectivenessoftheInverseHeatTransferAlgorithm........142 5.3Summary....................................144 6CONCLUSIONS...................................147 APPENDIX ACOMPLETESTEADYSTATETWOPHASEHEATTRANSFERJETRESULTS151 BCOMPLETEINVERSEHEATTRANSFERALGORITHMERRORASSESSMENT CONTOURPLOTS..................................166 CIMAGESOFORIFICESUSEDDURINGEXPERIMENTS............172 REFERENCES.......................................175 BIOGRAPHICALSKETCH................................184 7 PAGE 8 LISTOFTABLES Table page 21WaterMassFlowrateandAverageWaterVelocityforDifferentRegulatorPressures andOriceSizes...................................40 22Area,temperature,andpressureratiosatvariouspointsinthejetimpingement facility..........................................46 23Nozzleexitpressureforvariousregulatorpressures................47 31Reynoldsnumberandcorrespondingheatuxes..................62 51CurveFittingConstantsforRabinandRittel'sthermocoupleimpulseresponse model,from[114]...................................121 52RMSerrorsforL=10mm,and DAQ =0.2 C...................138 53RMSerrorsforL=10mm, DAQ =1 C,andM=8................139 54RMSerrorsforL=5mmand DAQ =0.2 C....................140 55RMSerrorforL=10mm, DAQ =0.2 C,M=8,andvariousvaluesof for theBiotnumberdistribution..............................140 56RMSerrorforL=10mmandM=8withdifferentactualandsimulatedtime constants........................................141 8 PAGE 9 LISTOFFIGURES Figure page 11Solutionofstagnationpointow..........................21 12Anillustrationoftheshockstructureinthewalljetregion.............26 13Greasestreakphotograph..............................28 14Numericalresultsofaoweldwithaplateshock.................31 15Jetcenterlinepressureuctuationswithandwithoutmoisture..........32 16Adiabaticandheatedtemperaturevariationwithz/D................34 21Illustrationofthejetimpingementfacility.......................37 22Crosssectionofthemixingchamber........................38 23Threaddetailsofthemixingchambercrosssection................38 24Oricecrosssection..................................39 25Theoreticalvsmeasured m air .............................41 26Crosssectionofnozzle................................43 27Simpliedviewoftheairowpathinthefacility...................45 28Illustrationofthelimitingcasesforshockwavesinthenozzle...........48 29Illustrationofanobliqueshockwaveatthenozzleexit...............49 210Variationofowpropertiesdownstreamofanobliqueshockwave........50 211Structureofobliqueshockwaves..........................52 31Stainlesssteelheaterassembly...........................54 32Copperheaterassembly...............................55 33Illustrationofheaterassemblyusedforsteadystateexperiments.........57 34Iceformationatadiabaticconditions.........................60 35MeasuredsinglephaseNu D spatialvariationatdifferentheatuxes.......64 36Spatialheatertemperaturevariationatdifferentappliedheatuxes.......65 37SpatialvariationofNu D atdifferentRe D ,aunscaledandbscaled.......66 38SinglephaseNu D atvariousnozzleheighttodiameterratios...........67 9 PAGE 10 39MeasuredtwophaseNu D spatialvariationatdifferentheatuxeswithoutice formation........................................68 310MeasuredtwophaseNu D spatialvariationatdifferentheatuxeswithand withouticeformation.................................69 311Spatialheatertemperaturevariation,twophasejetresults............70 312TwophaseNu D atvariousliquidmassfractions..................71 313TwophaseNu D numberatvariousnozzleheighttodiameterratios.......72 314Heattransferenhancementratioatvariousliquidmassfractions.........74 315Heattransferenhancementratioatvariousnozzleheighttodiameterratios...76 411dimensionalsolidforthesensitivityproblem...................89 42Relativestepsensivitycoefcientsatx =0.1asafunctionoftime........92 43Relativestepsensivitycoefcientforaheatuxinput...............93 44Relativestepsensivitycoefcientforatemperatureinput.............93 45Illustrationoftheheattransferphysicsoftheinverseproblemformulation....96 46Comparisonoftruetemperaturevsinverseresults.................104 47Effectsofgridstretching.arealdomainandbcomputationaldomain.....107 48Timestepresults...................................109 49Trueandestimatedimpulseresponsefunction...................113 410Convergencehistory..................................113 411Outputcomparison..................................114 51Illustrationoftherstorderslab...........................118 52Illustrationofthesecondorderslab.........................119 53Exampleofarstandsecondorderimpulseresponsefunction..........120 54ImpulseresponsefunctionsusingthemodelofRabinandRittel,adaptedfrom [114]..........................................122 55Diagramofthecopperdiscassembly........................123 56Illustrationoftheexperimentalsetup.........................124 57Backwalltemperatureswithnonidealinsulations.................125 10 PAGE 11 58Resultsofthreeseparateimpulseresponseexperiments.............126 59Inversemethoddeconvolutionresults........................127 510Firstorderresponsefunctioncomparison......................128 511Besttresultsusingarstorderimpulseresponsefunction............129 512Comparisonofmodeltodeconvolutionresults...................129 513Besttresultsusingthemodelof[114]responsefunction.............130 514Comparisonofthe2exponentialmodeltothedeconvolutionalgorithmresults.131 515Besttresultsusingthe2exponentialmodel....................132 516BiotnumberdistributionshowingtheeffectsofBi max ................134 517Biotnumberdistributionshowingtheeffectsof ..................135 518TruncationofBiotnumber...............................137 519ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints....139 520Comparisonoftheeffectsofheatgain.......................143 521CenterlineNu D resultsfromtheinverseheattransferalgorithm..........144 522SpatialNu D resultsfromtheinverseheattransferalgorithm............145 A1TwophaseNu D resultsforvariousZ/D,nominalRe D =4.46 10 5 .......152 A2TwophaseNu D resultsforvariousZ/D,nominalRe D 7.27 10 5 ........153 A3TwophaseNu D resultsforvariousZ/D,nominalRe D 1.01 10 6 ........154 A4TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=2.0......155 A5TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=4.0......156 A6TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=6.0......157 A7TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=8.0......158 A8Twophase resultsforvariousZ/D,nominalRe D =4.46 10 5 .........159 A9Twophase resultsforvariousZ/D,nominalRe D 7.27 10 5 ..........160 A10Twophase resultsforvariousZ/D,nominalRe D 1.01 10 6 ..........161 A11Twophase resultsforvariousliquidmassfractionsandZ/D=2.0.......162 A12Twophase resultsforvariousliquidmassfractionsandZ/D=4.0.......163 11 PAGE 12 A13Twophase resultsforvariousliquidmassfractionsandZ/D=6.0.......164 A14Twophase resultsforvariousliquidmassfractionsandZ/D=8.0.......165 B1ErrorContoursforL=5mm, DAQ =0.2 C,and8measurementpoints....167 B2ErrorContoursforL=5mm, DAQ =0.2 C,and16measurementpoints...168 B3ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints...169 B4ErrorContoursforL=10mm, DAQ =0.2 C,and16measurementpoints..170 B5ErrorContoursforL=10mm, DAQ =1 C,and8measurementpoints....171 C1The0.33mmorice..................................172 C2The0.37mmorice..................................173 C3The0.41mmorice..................................173 C4The0.51mmorice..................................174 12 PAGE 13 NOMENCLATURE Variables AArea[m 2 ] BiBiotnumber=hL/k CMeasurementequationoperator DDiameter[m] LLagrangian MMachnumber,Chapter2 MTotalnumberofmeasurements NuNusseltnumber=hL/k PPressure[Pa] RGasconstantfordryair[J/kgK,Chapter2 RResidualofmodelingequation ReReynoldsnumber=4 m / D SLeastsquaresvalue TTemperature[ CorK] VVolume[m 3 ] XDimensionlessrelativesensitivitycoefcient c p Specicheatcapacityatconstantpressure[J/kgK] fGeneralfunction hHeattransfercoefcient[W/m 2 K],Chapter3 hImpulseresponsefunction[s] kThermalconductivity[W/mK] 13 PAGE 14 _ m Massowrate[kg/s] q 000 Internalheatgenerationrate[W/m 3 ] rRadialcoordinate[m] ttime[s] uVelocity[m/s] uDummyvariableforpartialdifferentialequation,Chapter4 vDummyvariableforpartialdifferentialequation,Chapter4 wLiquidmassfraction xLengthcoordinate[m] yWidthcoordinate[m] yGeneralizeoutputvariable,Chapter4 zHeightcoordinate[m] Greekletters Thermaldiffusivity[m 2 /s] StepsizeforConjugateGradientMethod Gridstretchingparameter Ratioofspecicheats,Chapter2 ConjugationcoefcientforConjugateGradientMethod,Chapter4 Deectionangleinradians,Chapter2 Thickness[m],Chapter3 Diracdeltafunction,Chapter4 Stoppingcriteriavalue Transformedzcorrdinate 14 PAGE 15 ObliqueShockwaveangleinradians,Chapter2 Dimensionlesstemperature Lagrangemultiplier,Chapter4 Effectivetimeconstant,Chapter5 viscosity[Pas] Generalparametertobeestimated,Chapter4 Dummyintegrationvariable,Chapter5 Density[kg/m 3 ] Dummyvariableofintegration Timeconstant[s] Heattransferenhancementratio,Chapter3 Dimensionlessstepresponse,Chapter4 Humidityratio Subscripts DSoliddomain TThermocouplemeasuredquantity aAdiabaticquantity airAirquantity expExperimentalmeasurement fFluidquantity lLiquidquantity mMeasurementquantity mixMixturequantity modModiedquantity 15 PAGE 16 oStagnationquantity,Chapter2 oInitialquantity,Chapter4 rmsRootmeansquarevalue sSurfacequantity satValueatsaturationconditions simSimulatedmeasurement sndQuantityatthespeedofsound vVaporquantity wWallquantity Superscripts *Criticalquantity,Chapter2 *Dimensionlessquantity 16 PAGE 17 AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy AMULTIPHASESUPERSONICJETIMPINGEMENTFACILITYFORTHERMAL MANAGEMENT By RichardRaphaelParker May2012 Chair:JamesF.Klausner Cochair:RenweiMei Major:MechanicalEngineering Thisstudyinvestigatestheheattransfercharacteristicsofamultiphasesupersonic jetimpingementheattransferfacility.Inthisfacilitywaterdropletsareinjectedupstream ofaconvergingdivergingnozzledesignedforMach3.26airow.Thenozzleisoperated inanoverexpandedmode.Uponexitingthenozzle,thehighspeedair/watermixture impingesontoaheatedsurfaceandprovidescooling.Steadystateheattransfer measurementshavebeenperformedwithpeakheattransfercoefcientsexceeding 200,000W/m 2 .Theseheattransfercoefcientsareonthesameorderassomeofthe highestheattransfercoefcientseverrecordedintheliterature.Remarkablytheseheat transfercoefcientsareobtainedusingliquidowratesrangingfrom0.2to0.7g/s,in contrasttotheseveralkg/sowratesseeninstudiesthatproducesimilarlyhighheat transfercoefcients. Duringsteadystateoperationitisnotedthatnoevidenceofphasechangewas experimentallyobserved.Preliminaryinvestigationsindicatethatitmaynotbepossible toobtainevaporativeheattransferinthecurrentfacility.Inordertoinvestigatethis possibilityhighersurfacetemperaturesareneeded.However,designingasteadystate experimenttoachievehightemperatureoperationisrifewithdifcultiesandislikelyto beprohibitivelyexpensive. 17 PAGE 18 InordertoovercomethesechallengesatransientinverseheattransferIHT methodhasbeendeveloped.Oneoftheimportantissuesrevealedduringthis investigationisthatsensordynamicswillimpactthemeasurements,thusdiminishing themeasurementreliability.Toalleviatethisissue,amethodofincorporatingsensor dynamicsintotheIHTmethodwasdeveloped.Thistypeofmethodisnotexplicitlyfound intheliteraturetotheauthor'sknowledge.Amethodforaccuratelydeterminingthe impulseresponsefunctionofthethermocouplesusedinthetransientIHTexperiments yieldsgoodexperimentalresults.HeatlossisdiscoveredtobeacriticalfactorintheIHT method,andadifferenceintemperatureof3 Cbetweenthatmeasuredandtheideal caserenderstheIHTresultsunusable. Aparametricstudywasperformedtodeterminetheeffectsof:discheight,impulse responsefunction,magnitudeandshapeoftheheattransfercoefcientdistribution, thenumberoftemperaturesensorsused,andthemagnitudeoftheerrorinthedata acquisitionsystem.Itwasdiscoveredthatthemethodwasinsensitivetonoiselevels foundinlaboratoryconditionsandtheaccuracyincreasesforadecreasingdischeight. Therelativeslownessoftheimpulseresponsefunctionsdidaffecttheaccuracyofthe IHTmethodaslongasthetimeconstantofthefunctionsisaccuratelyknown. 18 PAGE 19 CHAPTER1 INTRODUCTIONANDLITERATUREREVIEW Jetimpingementproduceshighheattransfercoefcients,uptoapproximately 10 5 W/m 2 K.Liquidimpingingjetshavesupportedsomeofthehighestrecorded surfaceheatuxes,rangingfrom100to400MW/m 2 ,[1].Ohetal.[2]andLienhard andHadaeler[3]havestudiedliquidjetsandarraysthatcanproduceheattransfer coefcientsof200kW/m 2 K.Thesehighheattransferratesareaccompaniedbyhigh liquidowratesofuptoseveralkg/sofwater,andsuchhighwaterconsumptionmaybe undesirableinsomeindustrialsettings.Thecurrentstudyproposestouseasupersonic multiphasejetimpingementfacilitydesignedafteranexperimentbyKlausneretal.[4], whichusestheadditionofliquiddropletstotheimpingingairstreamtoenhancethe heatremovalrateofthesupersonicjet.Theliquidowratewillbeordersofmagnitude lowerthanthatusedbythestudiesmentionedabove,withlessthanoneg/s,whichmay beverydesirableinapplicationswhereminimalwaterconsumptionisaconcern. Supersonictwophasejetheattransferisaeldthathasnotbeenpreviously studied.Thecontributionofthecurrentstudywilllargelyconsistofcharacterizingthe heattransfercapabilitiesofsuchasystemincludingtheeffectsofairandliquidmass owratesandnozzlespacing.Additionally,evaporativeheattransfercapabilitiesofthe jetwillbestudied;inthisscenariothelatentheatofvaporizationcouldpotentiallygreatly enhancetheheatremovalcapabilitiesofthefacility.However,itisnotknownwhether ornotliquidevaporationcanbeachievedduetothehighstagnationpressure.Dueto highimpactpressuresnearthejetcenterline,phasechangeisnotlikelyinthisregion; however,theconditionsfarremovedfromtheimpingementpointmayallowphase changetooccur. 19 PAGE 20 1.1LiteratureReview Jetimpingementheattransferisaverydiverseeldandconsistsofsinglephase heattransferandevaporativeheattransfer,spray/mistcooling,andsupersonicjet impingementheattransfer.Abriefreviewofjetimpingementheattransferisprovided. 1.1.1SinglePhaseJetImpingement TheanalyticalstudyofstagnationpointowslargelybeginswithHiemenz[5]who studiedtheoweldofalaminarimpingingjetbymodifyingtheBlausisboundarylayer solution.Homann[6]extendedthisanalysistoaxisymmetricows.Theseowsarepart oftheFalknerSkanboundarylayerequations,whichtakethegeneralform f 000 + o f 00 f )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F22 11.9552 Tf 5.48 9.684 Td [(1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(f 0 2 =0 where f = f 0 =0 and f 1 =1. Thevelocitiesuandv,andthesimilarityvariable, ,aredenedas u = axf 0 v = )]TJ/F25 11.9552 Tf 9.298 0 Td [( o p a f = y r a whereaisaproportionalityconstantand isthekinematicvelocityoftheuid.Note thatintheradialcasethevariablexistheradialdistancefromtheorigin.Theparticular valuesof and o are1and1forHiemenzowand1and2forHomannow.The variablef 00 isproportionaltotheshearstress,f 0 isthenondimensionalvelocityu/U 1 andfisthestreamfunction.Equation1representsanonlinearordinarydifferential equationODEwhichmustbesolvednumerically.Theshootingtypemethodis generallyusedasthevalueoff 00 isunknownattheorigin.Thevaluesoff 00 attheorigin 20 PAGE 21 a b Figure11.SolutionofaHiemenzstagnationpointowandbHomannstagnation pointow. obtainednumericallyare1.2325forHiemenzowand1.3120forHomannow,asfound in[7].ThesolutionforHiemenzandHomannowareshowninFigure11.Itisevident thatthetheoweldsbehaveverysimilarly;however,thefreestreamvelocityand shearstressarereachedforsmallervaluesofthesimilarityvariableforaxisymmetric stagnationpointow.Afullderivationoftheseandotherstagnationpointowsusing asimilaritytypeapproachasthatabovecanbefoundinthebookbySchlichtingand Gersten[8]. 21 PAGE 22 Whiletheaboveanalysisissufcientforcompletelylaminarimpingingjets,jets whichndindustrialapplicationmustdealwiththeboundarylayerapproachingthefree surfaceofthejetfarremovedformthecenterlineaswellasthetransitiontoturbulence. Duetothedifferentowregimesthejetanalysisistypicallybrokenupintoseveral differentregionsandanalyzedthroughtheuseofavonK arm ammomentumintegral analysis;forananalysisofstagnationregionsee[9]. Theanalysisofthetemperatureeldwithintheboundarylayerforthesetypesof owsiscomplicatedduetothebehaviorofthethermalboundarylayerthatdevelops onthesurface.Itisfurthercomplicatedbythenatureoftheuiditselfasowswith largerPrandltPrnumberbehaveverydifferentlythanowswithsmallones.Asthe boundarylayermovesawayfromthecenterline,thehydrodynamicboundarylayer reachesthefreesurfacebeforethethermalboundarylayerforPr < 1whiletheconverse istrueforPr > 1 .Liuetal.haveanalyzedtheowineachoftheseregionsforsingle phasejetswithconstantsurfacetemperatureandheatuxes,mostlythroughtheuse ofthevonK arm anPohlhausenintegralsolution[10,11].Theywereabletomodelthe transitiontoturbulenceandthesubsequentturbulentowaswell.Theirsolutionsagree exceptionallywellwithexperimentalresults.Ingeneralthesolutionshavetheformof Theanalysisoftheaboveoweldisnotlimitedtointegralsolutionsortoconstant boundaryconditions.Wangetal.[12,13]studiedtheeffectsofaspatiallyvarying surfacetemperatureandheatuxonthesolutionusingaperturbationmethod.They foundthatthedirectionofincreasingtemperatureaffectstheNusseltnumberofthe ows,notablythatincreasingthewalltemperatureorheatuxwithradialdistance fromtheoriginwilldecreasetheNusseltnumberinthestagnationzone.Conversely itincreasestheNusseltnumberintheboundarylayerregion.Wangetal.additionally studiedtheconjugateheattransferproblemwherethetemperatureeldisdeterminedin theliquidandsolidsimultaneously,[14].Theyfoundthatthicknessoftheheatercanbe acontributingfactorfortheheatremovalcapabilityofthejet. 22 PAGE 23 Thereareseveraladditionalphenomenathatimpactthecoolingratesofimpinging jets.Theseincludetheeffectsofthejetnozzlediameter[15],hydraulicjump[16],and thesplatteringofliquidfromtheresultingfreesurface[17].Incaseswithsufcientlyhigh surfacetemperature,phasechangecanbeobservedunderimpingingjetsincludingthe regionsofnucleateboiling,departurefromnucleateboiling,andtransitionboiling[18]. Whilemoststudiesofliquidjetimpingementndnoappreciableeffectonthenozzle heightabovetheheatsurface,Jambunathanetal.[19]notedthatsomestudiesdoshow aneffectmostnotablyathigherReynoldsnumbers.Anempiricalcorrelationbased onheattransferdataavailableintheliteraturewasproposedhowever,itprovidesno physicalinsightoftheoweldandheattransfertakingplace. Liquidjetimpingementcansupportexceptionallyhighsurfaceheatuxes.Liuand Lienhard[1]usedaliquidjetwithvelocitiesexceeding100m/s,liquidsupplypressures ofuptoapproximately9MPa,andowratesofapproximately300g/storemoveheat uxesofatleast100MW/m 2 .Theseexperimentswerenovelinthefactthattheyused aplasmatorchasaheatsource.Surfacetemperaturesweredeterminedbycoating thetopsurfacewithamaterialofknownmeltingtemperatureandcompletingseveral experimentalrunsuntilthesurfacetemperaturecouldbeisolatedtoliewithinarange oftemperatures.Meltingoftheheatedsurfaceoccurredduetotheuseofthetorchand thebackwalltemperaturewasassumedtobeessentiallythemeltingtemperatureof thesolid.Theheatuxwasdeterminedbyusingtheminimumthicknessofthesolid whereithadmeltedandthenassumingalineartemperatureprole.Becauseofthe coarsenatureofthemeasurements,uncertaintyishardtoquantifyandheattransfer coefcientswerenotreported.However,theheatuxesmeasuredarethehighest steadystatevaluesrecordedintheliterature.Tofurtherenhancethestudyofhighheat uxremovalMichels,Hadeler,andLienhard[20]andLienhardandNapolitano[21] designedthinlmheatersusingvacuumplasmasprayingandhighvelocityoxygen fuelspraying.Theseheatersaresuppliedwithdcelectricalpowerofupto3,000Aand 23 PAGE 24 24V,producingheatuxesofupto17MW/m 2 .LienhardandHadeler[3]wereable toconstructanarrayofliquidjetswithliquidsupplypressuresofapproximately2MPa andowratesofapproximately4kg/s.Theseimpingingjetarrayswereabletosupport heatuxesof17MW/m 2 withanaverageheattransfercoefcientof200kW/m 2 with uncertaintiesof 20%.SimilarresultswerefoundinastudybyOhetal.[2].These studieshelpillustratethehighheatremovalcapabilitiesofjetimpingementtechnology. Foranextensivereviewofthesubjectofliquidjetimpingementtheauthor recommendsthereviewarticlesofLiendhard[22],WebbandMa[23],andMartin [24].Thesearticlesofferacompletereviewofthesubjectandincludemanyeffectsnot discussedinthisbriefliteraturesurvey. 1.1.2MistandSprayCooling Mist/spraycoolingofaheatedsurfaceislargelydifferentfromjetimpingementdue tothefactthattheliquidimpingingonthesurfaceisintheformofdispersedroplets. Thesedropletsareusuallygeneratedbyforcingliquidthroughverysmallorices withinanozzlewhichatomizestheliquid.Theprimarybenetofusingthistechnique isthatthedispersedropletsgenerallyallowforevaporativeheattransfertodominate. Mist/spraycoolingproducesheattransfercoefcientsontheorderofthosefoundduring poolboiling.However,thecriticalheatuxcanbeseveraltimeshigher[25]. Thedropletsizeisanimportantparameterinmist/spraycooling,Estesand Mudawar[26]correlatedthecriticalheatuxCHFwiththeSautermeandiameter ofthesprays;theyalsofoundthattheapparentdensityofthespraycanbeanimportant factorinmist/spraycoolingasdenserspraysarelesseffective.Mist/spraycooling canalsobeappliedtosurfaceslowerthantheboilingpointoftheliquid.Thereduced evaporationcanleadtoabuildupofaliquidlmontheheatedsurfacewhichcanhavea thicknesslessthanthesizeofthedroplets.GrahamandRamadhyani[27]performedan experimentalstudywhichshowsthatincreasingtheamountofdropletsonthesurface canleadtothickerlmswhichmayincreasethethermalresistanceatthesurface; 24 PAGE 25 however,thisthicklmmaybeabletoconvecttheheatawaybetterduetoanincreased velocity.Theywereabletodevelopasimplemodelofthethinlmdynamicsandthe resultingheattransferwhichhadapproximately4%errorforheatuxpredictionswithan air/methanolmixturebut,onlyprovidedqualitativeagreementwhenusedwithair/water data.Itisnotedherethatbecauseoftheevaporationtakingplaceinmist/spraycooling, theheattransfercoefcientdoesnotvaryappreciablywiththeradialdistance,afeature quitedifferentthanthatfoundinjetimpingementheattransfer.Readersdesiringa comprehensivereviewofmistspraycoolingareencouragedtoconsultthereviewarticle ofBolleandMoureau[28]. 1.1.3SupersonicJetImpingement Theoweldofasupersonicjetexhibitsverycomplexphenomena.Thenature oftheowcanchangedramaticallyasthenozzleexittoambientpressureratio changes,sometimesquantiedbythestagnationtoambientpressureratio.When thenozzleexitpressureislowerthantheambientpressureobliqueshockwavesform attheedgeofthenozzleinordertocompresstheow.Theseshockwavesbecome PrandtlMeyerexpansionfanswhentheymeetatthejetcenterline.Thisprocessleads toaseriesofreectedshockwavesandexpansionfansformingintheoweldatthe exitofthenozzle,includingtheformationofnormalshocksintheowknownasMach disks.Whenthejetexitpressureislargerthantheambientpressure,PrandtlMeyer expansionfansformattheexitofthenozzleandasimilarseriesofeventstakeplace. Theoweldofthejetchangesdramaticallyintheaxialdirection.Zapryagaevetal. [29]notedthatforoverexpandedjetstheradialpressuredistributionupstreamofthe rstshockcellcontainsseverallocalmaximawithverysharpdiscontinuitiespresent. Thesediscontinuitiesdisappeardownstreamoftherstshockcellandgenerallyanon centerlinemaximumappearsinthepressuredistribution.Thesefeaturesarepresent inunderexpandedjetsaswell,[30].Whilethesefeaturesarecomplextheycanbe 25 PAGE 26 Figure12.Anillustrationoftheshockstructureinthewalljetregion.[Reprintedwith permissionfromCarling,J.C.andHunt,B.L.,TheNearWallJetofa NormallyImpinging,Uniform,Axisymmetric,SupersonicJet,JournalofFluid Mechanics66159Page174Figure9b,CambridgeUniversity Press] modeledsomewhataccuratelybyamethodofcharacteristicsapproachasnotedby Pack,[31,32]andChu[33],amongothers. Underexpandedimpingingjetshavebeenextensivelystudiedintheliterature astheypertaintothelaunchingofrocketsandspacecraft,whereasoverexpanded impingingjetsarerelativelyuncommoninindustrialsettings.Whenajetimpinges uponaatplate,acomplexshockstructureisformed.Thisshockstructureforms severalcomplexfeaturesincludingatripleshockstructure,wherethreeshockwaves intersectneartheimpingementsurface,abowshock,alsoknownasaplateshockin theliterature,astheowmustcometorestinthestagnationpointonthesurface,and theshockwaveswhichradiatefromthetripleshockpointandslowtheowalongthe platetosubsonicspeeds.Thesefeaturesappearinalltypesofsupersonicimpinging jetsincludingunderexpanded,ideallyexpanded,andoverexpanded.Thebowshock,if curved,canformarecirculatingstagnationregionintheareaofthecenterofthejetto theedgeofthenozzle.Anillustrationofthecomplexshockstructureoftheowatlarger radialdistancesalongtheimpingementplateisshowninFigure12. Thestagnationregionisverycomplexduetotheformationoftheabovementioned bowshockandstagnationbubble.DonaldsonandSnedeker[30]studiedunderexpanded jetsfromaconvergingnozzleandperformedmanydifferentmeasurementstohelp 26 PAGE 27 characterizesomeoftheimportantfeaturesoftheowincludingimpingementangle andnozzlepressureratio.Theywereabletoobservestagnationbubblesforming,but notedthatthisphenomenondidnotoccurineveryexperiment.Schlierenphotographs weretakenaswellastotalpressuremeasurementsalongthejetcenterline,anditwas observedthatthevelocityandpressurevarygreatlyintheaxialdirection.Thevelocity intheradialwalljetregionwasmeasuredviatheuseofpitotstaticpressuretube measurementsalongtheimpingementsurface,theeffectsofsurfacecurvaturewere alsocharacterized.GummerandHunt[34]alsostudiedtheowofuniformaxisymmetric ideallyexpandedsupersonicjetswithlownozzletoheightspacingandnotedthe presenceofthebowshockandcomplexshockstructureinthewalljetregion.They attemptedtouseapolynomialandintegralrelationmethodtomodelthebowshock heightandthepressuredistributionunderthenozzle.Somesuccesswasseenforhigh Machnumbersbutnotintheregionofthetripleshock.LowMachnumbercalculations containedasmuchas60%error.CarlingandHunt[35]performedatheoreticaland experimentalinvestigationusingthenozzlesofGummerandHunt.Theirstudymostly comprisedoftheregionjustoutsideofthenozzlealongtheimpingementplate.They wereabletonotethepresenceofthestagnationbubbleforsomeoftheirexperiments, butnotall.Thepresenceofthestagnationbubblecanseverelyinuencethepressure distributionontheplateandanannularmaximumispossibleforsomejetspacings. Attemptsweremadetomodeltheshockstructureinthewalljetregionusingthemethod ofcharacteristics.Qualitativefeaturesoftheowwereabletobereproduced.However, thereappearstobesomeerrorintheregionnearthetripleshockregion.Thepressure variationalongtheplatewasmeasuredwhichshowedseveralregionsofunfavorable pressuregradient.CarlingandHuntwereabletoinvestigatetheseregionsbycoating theimpingementplatewithatypeofgrease.Whenthejetisimpingedupontheplate theseunfavorablepressuregradientscausethegreasetoberemovedduetolocal separationoftheboundarylayer.Aphotographofoneoftheseexperimentsisshown 27 PAGE 28 Figure13.Greasestreaktypephotographfrom[35],thedarkareascontainnogrease andareareasofhighwallshearstress.[Reprintedwithpermissionfrom Carling,J.C.andHunt,B.L.,TheNearWallJetofaNormallyImpinging, Uniform,Axisymmetric,SupersonicJet,JournalofFluidMechanics66 159Plate3Figure6d,CambridgeUniversityPress] inFigure13.Thedarkregionsrepresentwherenogreaseispresent;notethevery darkregionneararadialdistanceof2nozzlediametersfromthecenterwhereevidence ofseparationisclearlyevident.Theseparationphenomenonwasnotedbyseveral investigatorsincludingDonaldsonandSnedeker,[30].KalghatgiandHuntprovidea qualitativeanalysisexperimentalstudyofoverexpandedjetswhichconcentratedonthe tripleshockproblemneartheedgeofthebowshock.Theiranalysissuggeststhatat bowshocksareapossibilityandschlierenphotographsofoverexpandedimpinging jetswithMachnumbersrangingfromapproximately1.5to2.8largelyconrmed theirqualitativeanalysis.Theyalsonotethattheformationofaatbowshockisa phenomenonthatishardtopredict.LamontandHuntperformedacomprehensive experimentalstudyonunderexpandedjetsorientednormallyandobliquelytoaatplate whichincludespressuremeasurementsandschlierenphotographs.Thestagnation bubblephenomenonwasnotedaswellassomeunsteadinessinthejet.Velocityand pressureproleswereseentovarygreatlywiththenozzletoplatedistance,anditwas notedthatthelocalshockstructurehasastronginuenceontheoweld. 28 PAGE 29 Unsteadinessoftheimpingingjetiscausedbyafeedbackphenomenonwhich hasbeenextensivelystudiedduetoitsimportanceinairvehicletakeoff,includingthe launchingofrocketsandshort/verticaltakeoffandlandingvehicles,suchastheJoint StrikeFighter.ThismechanismwassuccessfullymodeledbyPowell,[36,37].The mechanismiscausedbyacousticphenomenaoccurringattheedgeofthenozzle. Theseacousticwavescausevorticalstructurestobegeneratedintheshearlayer ofthejetandareconvectedtowardstheimpingementpoint.Uponencounteringthe regionneartheplate,thesestructuresinteractwiththeshockwavesneartheplate generatingstrongacousticwaves,whichtravelupstreamtowardsthenozzlewherewere theyinteractwiththenozzleedgegeneratingmoreacousticwaveswhichthenrepeat theprocess[38].Krothapalli[39]wasabletopredictthefrequenciesgeneratedbya supersonicimpingingrectangularjetusingPowell'smodel,thusvalidatingthetheory. Theeffectsoftheunsteadinessontheoweldwillbedetailedbelow. Duetothecomplexshockstructureandunsteadyphenomenainimpingingjets, numericalsimulationsareoftenusedtohelpenhancetheknowledgeinthisarea.Alviet al.[40]modeledtheimpingementofmoderatelyunderexpandedjetsandusedParticle ImageVelocimetryPIVtohelpverifytheirresults.Theirmethodhadreducedtemporal accuracy,butwasabletoreproducemajorowfeaturesincludingthestagnationbubble andwalljetregion,althoughtheregionofthetripleshockpointhadsomedisparity betweenthenumericalandexperimentalresults.Klinkovetal.[41]comparednumerical resultsofthevelocity,pressure,anddensityeldstoexperimentalresultsintheform ofschlierenphotographsandsurfacepressuremeasurements.Theirstudyfocused onoverexpandedjetswithMachnumbersintherangeof2.6to2.8atapproximately ambientstagnationtemperatures.Theyfoundthatthelocationofthebowshockcan changesignicantlywithnozzletoplatespacing,withseveralregionsofanearconstant shockheightfollowedbyanalmostdiscontinuouschangetoanotherheight.Regions ofhighshockheightrepresentaconvexbowshockandregionsoflowshockheight 29 PAGE 30 representaatbowshockwithunsteadinessnotedastheshocktransformstofrom aconvexshocktoaatshock.Theyalsonotedthatastagnationbubbleregionis typicalofaconvexbowshockandthatregularstagnationowaccompaniesaat bowshock.AnillustrationisshowninFigure14.Thebehaviorofthebowshockis signicantlyaffectedbytheunsteadyfeedbackmechanismasitisseentooscillateback andforthalongtheaxisofthejet.Thiscausescorrespondinglylargeuctuationsinthe surfacepressureontheimpingementplate.Kawaietal.performedacomputational aeroacousticstudywhichwas2 nd orderaccurateintimeand7 th orderaccuratein space.Thisstudywasdonetodeterminetheeffectsofthepresenceorabsenceof aholeinalaunchpadcongurationandprimarilyfocusedonlargenozzletoplate spacingsandtheeffectofReynoldsnumberontheunsteadyphenomena.Itwasseen thathighReynoldsnumberscansignicantlyincreasethesoundpowerlevelsofthe jetandthemagnitudeofitsoscillations.Theirnumericalcodeproducedresultswhich agreedwellwithhistoricalsoundpowerleveldatamaintainedbyNASA.Thisstudyis usefulinillustratingthecomplexityoftheproblemunderstudyandhowverycomplex numericalsimulationsareneededtoaccuratelyreproducethefeaturesoftheow. Theadditionofmoistureintheformofwatervaportotheairsupplyofanimpinging jetcanhaveanoticeableimpactontheoweld.Thiswasobservedexperimentallyby BaekandKwon[42]whoperformedstudiesofairwithvaryingdegreesofsupersaturation ofwatervaporforasupersonicjetissuingintoquiescentair.Theyfoundthatthe locationoftheMachdiskwaslocatedfurtherupstreamintheowformoistairjets anditssizewasreduced.Empiricalcorrelationsforthelocationofquantitiessuch asthesizeandlocationoftheMachdiskandthelocationofthejetboundarywere proposed,althoughlittlemechanisticinsighttotheowwasgained.Numericalstudies byAlametal.[43,44]andOtobeetal.[45]wereperformedforairwithvariousvalues ofsupersaturationofwatervaporforasupersonicjetimpingingonaatplate.They attemptedtomodelthenonequilibriumcondensationtakingplaceintheregionafterthe 30 PAGE 31 Figure14.Numericalresultsofaoweldwithaatplateshockleftandcurvedat plateshockright.[ReprintedwithpermissionfromKlinkovK.etal, BehaviorofSupersonicOverexpandedJetsonPlats,in:H.J.Rath,C. Holze,H.J.Heinemann,R.Henke,H.HnlingerEds.,NewResultsin NumericalFluidMechanicsV,volume92of NotesonNumericalFluid MechanicsandMultidisciplinaryDesign ,Springer,2006,pp.168Page 173Figure3] rstMachdiskintheow.Theirmodelassumesnoslipbetweentheliquiddropletsthat condenseandthatthesedropletsdonotinuencethepressureoftheowdownstream. Theowelddisplayssomenoticeabledifferencesthanthatofdryair.Theauthors proposethatthisisduetotheadditionofthelatentheatofcondensationtotheairbythe condensingwatervapor.Unsteadybehaviorduetotheacousticfeedbackmechanism byPowellwasseeninthesimulations.Thisunsteadinesswasnotpresentupstream oftherstMachdisk,butwasseendownstreamofit.Thepresenceofcondensate particlescombinedwiththeadditionofthelatentheatreducesthemagnitudeofthe pressureuctuationsseeninthedownstreamportionoftheowwhichisillustratedin Figure15.Theauthorsattemptedtoverifytheirsimulationswithexperimentaldata, 31 PAGE 32 a b Figure15.Jetcenterlinepressureuctuationswithanomoistureandb40% supersaturationofwatervapor.[ReprintedwithpermissionfromAshraful Alam,M.M.etal.,EffectofNonEquilibriumHomogeneousCondensation ontheSelfInducedFlowOscillationofSupersonicImpingingJets, InternationalJournalofThermalSciences492078Page2086 Figure10bandPage2088Figure13c,Elsevier] mostlyconsistingofcomparingtheshockstructureasseeninschlierenphotographslike thoseinthestudybyBaekandKwon,alongwithnoisetonesfordryairgeneratedbythe acousticfeedbackmechanism.Thisproposedvalidationisweakbecausethereisalack ofexperimentaldataofwhichtocomparetointheliterature. Thestudyofsupersonicimpingingjetheattransferjetshasbeenstudiedextensively intheliterature.Unfortunatelymostofthesestudieshavefocusedontheheattransfer fromarocketexhausttoalaunchpadfacility.Donaldsonetal.[46]performedan experimentalstudyofimpingingsonicjetsandtheirturbulentstructure.Theauthors wereabletodevelopacorrelationforNusseltnumberbasedonapplyingaturbulent correctionfactortolaminarimpingingjettheorynearthestagnationpointandfurther awayinthewalljetregion.Whilegoodagreementwasfoundfortheircorrelationit isforsonicorjustslightlysupersonicimpingingjetsanddoesnotapplytothehighly 32 PAGE 33 supersonicjetspreviouslymentioned.Theunsteadyacousticfeedbackphenomenon previouslydiscussedcausesinteractionsbetweenacousticwavesandtheshock structureoftheimpingementregion.Thisresultsinlocalcoolingtooccurinthe regionofthejetedgeandisverynoticeableinthemeasurementoftheadiabaticwall temperature.ThisphenomenonistermedcoolingbyshockvortexinteractionbyFoxand Kurosaka[47]whoinvestigatedthisphenomenon.Kimetal.[48]studiedthesurface pressureandadiabaticwalltemperatureofanunderexpandedsupersonicimpinging jet.Theynotedthattheacousticvorticalstructureinteractionsignicantlyaffectsthe adiabaticwalltemperatureandsurfacepressurewhichalsovariesgreatlywithnozzle height.Thepresenceofastagnationbubble,whichenhancesthecoolingdirectlybelow thenozzle,wasnotedaswell.Rahimietal.studiedtheheattransferofunderexpanded impingingjetsontoaheatedsurface.Thetemperatureoftheimpingementsurfacewith uniformappliedheatuxisnotedtochangedramaticallywithradialdistanceaswell aswithnozzlespacingasshowninFigure16.TheynotethatNusseltnumberscales approximatelywiththesquarerootofReynoldsnumberandthathighheattransferrates areencounteredinthestagnationzonewhenastagnationbubbleispresent.Duetothe complexityoftheproblem,theynotethatageneralcorrelationofNusseltnumbershould beafunctionofnotonlyReynoldsnumberandPrandtlnumber,asiscommon,butalso afunctionofMachnumberandnozzlespacing.Yuetal.performedasimilarstudyand noticedsimilartrends;theirmeasuredNusseltnumbersexceed1,500. Studiesoftheheattransfercharacteristicsofsupersonicmoistimpingingjets arenotfoundintheliterature.Theyarelikelytoshowverycomplexphenomena asevidencedbythedifferencesintheshockstructureandgeneralbehaviorofthe relevantowquantitiesinthejetandalongtheimpingementplate.Thecurrentstudy usesdiscreteliquiddropletsthatareinjectedintotheairupstreamofthenozzle. Thiswilllikelyresultinanairstreamsupersaturatedwithwatervaporwhichisfurther complicatedbythebehavioroftheliquiddropletsandtheireffectsontheow.As 33 PAGE 34 a b Figure16.Adiabaticcirclesandheateddiamondswalltemperatureforanozzle spacingofaz/D=3.0andbz/D=6.0.[Reprintedwithpermissionfrom RahimiM.etal,ImpingementHeatTransferinanUnderExpanded AxisymmetricAirJet,InternationalJournalofHeatandMassTransfer46 263Page267Figures6aand6b,Elsevier] elucidatedbytheliteraturesurveytheowstructureassociatedwiththistechnologyis verycomplex,andessentiallynoanalyticalsolutionsareavailablefortheoweldand heattransfer.Theavailableempiricalcorrelationsdonotcovertwophasesupersonic impingingjets.Numericalstudiesmayprovidesomequalitativeinsight,butinmost instancestheydonotadequatelycaptureallofthephysicstakingplaceintheoweld. 1.2Summary InthisChapteranintroductiontothestudywasmadeandtherelevanceof multiphasesupersonicimpingingjetswasintroduced.Thecontributionsofthisstudy werealsodescribed,mainlythatthisisatechnologythathasnotbeenstudieduntilnow. Abriefliteraturereviewofthedifferenttypesofimpingementheattransferwas presented.Liquidandsinglephaseheattransferwasintroducedstartingwiththe classicworkofHiemenzandHomann.ThedevelopmentofaccurateNusseltnumber correlationsbasedonvonK arm anPohlhausenintegralmethodweredetailed.The agreementbetweentheoryandexperimentsisexceptionalforthesecorrelations. Otheraspectssuchastheowhydrodynamics,transitiontoturbulence,nozzleheight, 34 PAGE 35 andnonuniformboundaryconditionseffectswerediscussed.Highheatuxremoval technologiesthatarecapableofheattransfercoefcientsashighas200kW/m 2 were detailedaswell. Thestudyofsupersonicunderexpandedandoverexpandedimpingingjetswas describedaswell.Thiseldiscomplicatedbythecomplexowstructuregeneratedby shockwaveswhichformwhenanozzleisoperatedawayfromitsdesignedpressure ratio.Thedetailsoftheseshockwavesincludingtheeffectsofthecurvatureofthe bowshockjustabovetheimpingementplatewerediscussed.Stagnationbubbles formedjustbelowthebowshockwerediscussedandtheirimpactontheoweld wasdetailedaswell.Shockwavesneartheimpingementregioncauseanunsteady feedbackphenomenoncausedbytheinteractionofacousticwaveswiththeedgeof thenozzle.Theeffectsofthisfeedbackphenomenonandtheunsteadinessitcauses andrelevantchangesinthelocaloweldweredetailed.Moistureintheairstreamand howitchangestherelevantoweldwasbrieyexploredasitisarelativelynewarea ofstudyintheliterature.Thetemperatureproleontheimpingementplateandhowit changeswiththepresenceofthestagnationbubbleandacousticfeedbackmechanism werediscussed.Numerousexperimentalstudiesintheliteraturewhichincludepressure andtemperaturemeasurements,particleimagevelocimetry,andschlierenphotographs alongwithrelevantnumericalstudiesintheliteraturethatdiscoveredandconrmed thesephenomenawerediscussedwhererelevant. Lastlythecomplexityofthecurrentstudywasdiscussed.Itisnotedthatan analyticalsolutiontotheproblemwillnotbeattainableandapredictivenumericalstudy isnotfeasibleaswell.Thecontributionsofthisstudywillbeintheformofdeveloping anunderstandingofthemechanismstakingplaceasthemultiphasesupersonicjet removesheatfromasurface. 35 PAGE 36 CHAPTER2 JETIMPINGEMENTFACILITY Thesupersonicmultiphasejetfacilityshouldpossessseveraltraitsinordertobe usefulforanexperimentalapparatus.Itshouldhavesufcientairstoragecapacityso thatexperimentscanberunatsteadystate.Thestoredairshouldbepressurizedto suchanextentthatthedesiredMachnumbercanbeachieved.Lastlyitshouldcontain sufcientwater,andameanstocontroltheow,sothattheimpingingjetwillremainin multiphaseoperationduringexperiments. ThedesignforthecurrentsetupisbasedonasimilarexperimentbyKlausneret al.[4].Theimpingingjetconsistsofthefollowingsystemstobedescribedbelow:air storagesystem,waterstorageandowcontrolsystem,airpressurecontrolsystem, airmassowratemeasuringsystem,temperatureandpressuremeasurementsystem, convergingdivergingnozzle,anddataacquisitionsystemDAQ.Aschematicdiagram illustratingthecongurationofthejetimpingementfacilityisshowninFigure21. 2.1ImpingingJetFacilitySystems 2.1.1AirStorageSystem Theairstoragesystemconsistsof9`K'sizedbottleswhichgiveatotalvolumeof 0.45m 3 andarekeptatapressureof14MPa.Theairstoragesystemislledwithair fromamodelUE3compressorfromBauerCorporation.Thecompressorispoweredby a3phase240Vpowersupplyandiscapableofsupplying0.1m 3 /minofapproximately moisturefreeairtotheaircylinders,thustheairstoragefacilitycanbechargedto capacityinapproximately4.5hrs. 2.1.2WaterStorageandFlowControlSystem Waterforthefacilityiscontainedinastainlesssteelvesselwithacapacityof2 Landapressureratingof12.4MPa.Waterisforcedintothemixingchamberbythe differenceinpressurebetweenthetopofthewatervesselwhichisactedonthethefull forceoftheairsupplypressureandthepressureinsidethemixingchamber,whichis 36 PAGE 37 Figure21.Illustrationofthejetimpingementfacility. lowerduetoachangeinareaandbecauseofthefrictionactinginthesystemtubing.A drawingtheofthemixingchamberisfoundinFigures22and23.Theowrateofthe wateriscontrolledbymeansofanoricebetweenthewatervesselandmixingchamber andtheairpressureinthesystem.Oricediametersof0.33,0.37,0.41,and0.51mm areusedduringexperiments,andadrawingoftheoricedesignisfoundinFigure24. Whilethereissomevarianceinowratebetweeneachexperimentforagivenorice sizethiseffectiseliminatedintheanalysisduetothefactthetheowrateofwater intothemixingchamberismeasuredduringeachexperiment.Thisisaccomplished byrecordingtheelapsedtimeofeachexperimentandmeasuringthedifferenceinthe massofliquidinthewatervessel.Table21showsthenominalowrateofliquidandthe 37 PAGE 38 Figure22.Crosssectionofthemixingchamber. Figure23.Threaddetailsofthemixingchambercrosssection. averageliquidvelocityforthepressuresandoricesizesusedduringtheexperiments.It isnotedthattheowrateofthe0.37mmoriceislessthanthatofthe0.33mmorice, thisisduetothefactthatthe0.37mmoriceisnotperfectlycircular.Thisconditionis alsoseeninthe0.51mmorice,butitisnotassevereasthatinthe0.37mmorice. PicturesofeachoricetakenwithanopticalmicroscopeareshowninAppendixC. 38 PAGE 39 Figure24.Oricecrosssection. 2.1.3AirPressureControlSystem Theairpressurecontrolsystemconsistsofanairregulatorlocatedbetween theairstoragetanksandtheinlettotheairmassowmeterandthetopofthewater storagetank.Theregulatoriscapableofreducingairpressurefrom14MPadown toamaximumof2.8MPa.Airpressuresof1.0,1.7,and2.4MPaareusedduring experiments.Asdiscussedlater,theseairpressuresresultintheconvergingdiverging nozzleoperatinginanoverexpandedmanner. 2.1.4AirMassFlowMeasuringSystem MeasurementoftheairmassowrateisaccomplishedusinganAnnubarDiamond IImodelDNT10massowmeterlocateddownstreamoftheregulatorandbeforethe mixingchamber.Thediamondcrosssectionoftheowmeterissuchthatithasaxed separationpointandalsoreducespressureloss.Theowmetersensesdifferential pressurewhichismeasuredwithadifferentialpressureDPtransducer,whichis 39 PAGE 40 Table21.WaterMassFlowrateandAverageWaterVelocityforDifferentRegulator PressuresandOriceSizes. OriceRegulatorWaterMassAverageWater SizePressureFlowrateVelocity mmMPakg/sm/s 1.03.08x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.60 0.331.73.99x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 4.67 2.44.73x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 5.53 1.02.67x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 2.50 0.371.73.31x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.11 2.43.97x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.73 1.04.61x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.55 0.411.75.72x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 4.41 2.46.79x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 5.24 1.04.85x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 2.39 0.511.76.90x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.00 2.47.20x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.55 calibratedtomeasurepressuredifferencesofupto2.21 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 MPa.Thecalibration equationfortheowmeteris m =58.283 KD 2 p P f where PisthemeasureddifferentialpressureinkPa,Disthediameterofthe owmeterinmm,inthiscase15.80mm,Kisagagefactorof0.6,and f isthedensity oftheowingair,inkg/m 3 calculatedvia f =539.5 P f T f whereP f isthepressureoftheowingairinkPa,asmeasuredbythepressure transducerupstreamofthemassowmeterandT f isthetemperatureoftheowing airinKelvinasmeasuredbythethethermocoupleupstreamofthemassowmeter. 40 PAGE 41 Figure25.Acomparisonofthetheoreticalandexperimentallymeasuredairmass owrate. Thetheoreticalmassowratethroughthenozzlefor1Disentropicowis calculatedusingmassow, m = AV,theMachnumber,M=a/Vthespeedofsoundfor aperfectgasisa= p RT ,andtheidealgaslaw,P= RT, m = 4 D 2 MP r RT KnowingthatatthethroatofthenozzletheMachnumberisoneandusing TemperatureandPressurestagnationratiosof T = T o =0.8333 and P = P o =0.5282 Equation2reducesto m =0.4545 D 2 P o r RT o 41 PAGE 42 Equation2neglectstheeffectsoffrictionandheattransfer,whichaffecttheowrate ofairthroughthenozzle.Acomparisonoftheairmassowratemeasuredduringthe courseofexperimentswiththetheoreticalairmassowrateisshowninFigure25.The agreementbetweentheoryandexperimentiswithin20%. 2.1.5TemperatureandPressureMeasurements Thetemperatureandpressureofthejetimpingementsystemismonitoredduring systemoperationforcalculatingvariousquantitiesofinterest.Thetemperature measurementsareaccomplishedbytheuseofEtypethermocoupleprobeswhich areinsertedinto`T'junctioncompressionttingsatadepthsuchthatthetipofthe probeisatthecenterlineofthetting.Thethermocoupleprobesusedaregrounded andsheathedinstainlesssteelandhaveanominaldiameterof1.59mm.Temperature measurementsaretakenatthefollowingpoints:theoutletofthepressureregulator,the outletofthewaterreservoir,andattheoutletofthemixingchamber. Pressureismeasuredattheoutletofthepressureregulatorjustbeforethelocation oftheairmassowmeter.Thepressuremeasurementismadeusingastraingage typepressuretransducer,whichhasarangeof02.8MPa.Theoutputsignalofthe pressuretransducerisacurrentwhichvariesbetween420mA;becausetheDAQ systemusedintheexperimentsonlysensesvoltagesaresistorof520 isusedto convertthiscurrentintoavoltageinthe010Vrangeneeded. 2.1.6ConvergingDivergingNozzle Theconvergingdivergingnozzleiswherethemixtureofliquidandairareexpanded tosupersonicspeeds.Thenozzleisconstructedofstainlesssteelwithathroat diameterof2.38mmandanexitdiameterof5.56mm,givinganexitMachnumber of3.26.Thenozzleisattachedtoasize10DN 1 / 2 NPSstainlesssteelpipewithan internaldiameterof13.51mm,whichisconnectedtoabraidedstainlesssteelhose approximately9mlongwithaninnerdiameterof9.53mmandisconnectedtothe outletofthemixingchamber.Althoughthehoseaddssomesmallamountofpressure 42 PAGE 43 Figure26.Crosssectionalviewoftheconvergingdivergingnozzleusedin experiments. loss,itallowsthenozzletobelocatedawayfromtheairstoragecylindersandnear theimpingementheattransfertargets.Acrosssectionalviewofthenozzleisshownin Figure26. 2.1.7DataAcquisitionSystem TheDAQusedduringthecourseofsteadystateheattransferexperimentsisa DAS1601dataacquisitionPCIcardandaCIOEXP32analogtodigitalconverter board,bothmadebyMeasurementandComputingInc.ThisDAQconsistsof3216bit doubleendedchannelsandchannelgainsof1,10,100,200,and500areselectable. Thesystemhasamaximumreliablesamplingrateof50HzandthesoftwareSoftwire, producedbyMeasurementandComputingIncisusedforprogrammingdatacollection. Fortransientmeasurementsonheatedtargetsduringinverseheattransferexperiments, theDAQsystemissupplementedwithaNationalInstrumentsNI,NIUSB6210system whichhas8doubleended16bitchannelsandhasamaximumaggregatesamplingrate of250kHz.ThissystemusesLabviewsoftwareproducedbyNIwhichandisalsoable 43 PAGE 44 tointerfacewiththeMeasurementandComputingDAQviatheuseofanNIsupplied.dll library. 2.2AnalysisofImpingementFacility Someanalysisofthejetimpingementfacilitiesarewarranted.Thebehaviorofthe systemupstreamofthenozzleisexaminedtodetermineifthereareanycorrectionsthat needtobeappliedtothethermocoupleorpressuretransducerreadings.Additionally, thefollowingisexamined:thepressurerequiredtooperatethenozzleinaperfectly expandedmanner,theminimumandmaximumpressurethatcauseashockwave toforminsidethenozzle,andthenozzleexitpressurewhenoperatingatvarious regulatorpressures.Lastlytheshockwaveanglesformingatthenozzleexitforvarious operatingpressurearecalculatedaswell.Onedimensionalgasdynamicrelationsare usedtoinvestigatethequantitiesofinterest.Hereitisnotedthattheanalysisused haslimitations,theonedimensiongasdynamicrelationsareisentropicinnature,with theexceptionofshockwavecalculations.Thejetimpingementfacilityexperiences frictionandheattransferduringoperation,thustheisentropicassumptionisnotmet. Additionallyafterthemixingchambertheowwillcontainwaterdropletswhicharenot compressible.Thequantitiescalculatedbelowwillhavesomeinherenterrorhowever, theydoprovideareasonableapproximationofthephysicstakingplaceinthefacility. 2.2.1TemperatureandPressureUpstreamoftheNozzle Calculatingthetemperatureandpressureatvariouspointsupstreamofthenozzle isasimplematter;thecrosssectionalareaofthepointsinthesystemarerequiredfor thisanalysis;Figure27providesanillustrationofthejetimpingementfacilityandthe diametersofthepointsofinterest.Usingthecommonlyknownonedimensionalgas dynamicsrelationshipsfoundinvariouscompressibleowtextbooks,suchasLiepmann andRoshko[49]orJohnandKeith[50],thepressureandtemperatureratiosaswell astheMachnumberoftheowintheseareascanbedetermined.Inthefollowing 44 PAGE 45 Figure27.Simpliedviewoftheairowpathinthefacility. equations istheratioofspecicheatsandisaconstantequalto1.4.Todeterminethe owMachnumberthefollowingrelationshipisused A A = 1 M 2 +1 1+ )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 M 2 +1 2 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 wherethe*superscriptdenotesthecriticalareawhereMach=1.NotethatEquation2 isaquadraticequationinMandhastwosolutionsthuscarefulattentionmustbepaid inselectingtheproperMachnumbergivenanarearatio,inthepresentcaseallMach numbersupstreamofthethroatoftheconvergingdivergingnozzlearesubsonic.Once 45 PAGE 46 theMachnumberofthegivensectionisdeterminedthepressureandtemperatureratios canbedeterminedfromthefollowing T o T =1+ )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 M 2 P o P = 1+ )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 M 2 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 wherethesubscriptoisthestagnationproperty,whichissimplytheparticularproperty withzerovelocity.TheanalysisresultsusingEquations2to2areshownin Table22.Theresultsshowthatthetemperatureandpressureupstreamofthenozzle throatdifferfromtheirstagnationpointpropertiesbylessthan1%;nocorrectionduethe thevelocityoftheowisneeded. Table22.Area,temperature,andpressureratiosatvariouspointsinthejet impingementfacility. PointA/A T/T o P/P o M Exit5.440.31930.018403.26 Throat1.000.52830.83331.00 132.200.99990.99980.018 26.830.99860.99500.085 344.020.99991.00000.013 46.830.99860.99500.085 2.2.2NozzleExitPressureConsiderations Thereareafewtheoreticalconsiderationsthatneedtobeexploredatthenozzle exit.First,thenecessaryregulatorpressureinordertoobtainaperfectexpansionatthe nozzleexitisneeded;thentheactualexitpressuresbasedontheregulatorpressure duringexperimentsaredetermined.Theresultsfromthepreviouscalculationslistedin Table22showthestagnationpressureratioattheexitofthenozzle,simplycarrying therequisitealgebraandassumingabackpressureof101.4kPayieldsthenozzle exitpressure,seetheresultsofthesecalculationsinTable23.Fromtheseresultsit isobservedthatthepressurenecessaryforidealexpansionisapproximatelytwicethe 46 PAGE 47 pressuretheregulatorofthesystemcanprovide,andthusduringnormaloperationof thejetimpingementfacilitythenozzleoperatesinanoverexpandedmanner. Table23.Nozzleexitpressureforvariousregulatorpressures. RegulatorNozzleExitNozzle PressurePressureOperation MPaMPa 5.50.1014perfectlyexpanded 2.80.0508overexpanded 2.40.0443overexpanded 1.70.0317overexpanded 1.00.0190overexpanded Duetothefactthatthenozzleexitpressureisbelowtheambientpressuresome concernaboutashockwaveforminginthenozzlewillbeaddressed.Therearetwo limitingpressuresforthiscase,oneisthepressureatwhichatshockwaveformsatthe throatofthenozzle,theotheristhepressurethatashockwaveformsatthenozzleexit; Figure28showsanillustrationforbothofthesetwocases.Inthelimitingcase,ashock waveoccurringatthethroatwheretheMachnumberisequaltounity,thestagnation pressureratiois0.992.Assumingthatthebackpressureisatmosphericpressure,the stagnationpressurethatwillcauseashocktobelocatedatthethroatis0.102MPa.To calculatethelimitingcaseofashockwaveoccurringatthenozzleexitisjustslightly morecomplicated.Whenitisassumedthatashockwaveislocatedattheexitplaneof thenozzle,thestagnationpressureratioandMachnumberjustbeforetheexitplanecan befoundfromTable22.ThenormalshockwaverelationsforMachnumberandstatic pressureratioacrossashockEquations2and2canthenbeapplied, M 2 = s M 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1+2 2 M 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 P 2 P 1 = 2 M 2 1 +1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 +1 47 PAGE 48 Figure28.Illustrationofthelimitingcasesforshockwavesinthenozzle. UsingEquations2and2theMachnumberjustpasttheshockwaveis foundtobe0.461andthestaticpressureratiois12.268.Performingtherequisite algebrayieldsabackpressureof0.449MPa.Thusthenozzlewillhaveashockwave locatedinsideforastagnationregulatorpressureintherangeof0.102to0.449MPa. Sincetheminimumregulatorpressureusedduringexperimentsis1.0MPathereislittle concernthatashockwavewillforminsidethenozzle. 2.2.3ObliqueShockWavesatNozzleExit Itisknownthattheconvergingdivergingnozzleoperatesinanoverexpanded manner;theexitconditionsofthenozzleshouldbeconsidered.Whenanozzle isoverexpandedobliqueshockwavesformattheoutletofthenozzle,see[49]for instance.Theseshockwavescompresstheairsuchthatitisthenequaltothe nozzlebackpressure;therstobliqueshockwavecomingoutofthenozzlecanbe modeledusingthestandardonedimensionalgasdynamicrelationsparameterssuch astheshockangle,deectionangle,temperatureratio,stagnationpressureratio,and downstreamMachnumber.Figure29showsanillustrationoftheshockwaveatthe nozzleexit.Todeterminetheseexitquantities,rstthenozzleexitpressureshouldbe determinedusingtheregulatorpressureandEquation2.Theshockwaveanglecan 48 PAGE 49 Figure29.Illustrationofanobliqueshockwaveatthenozzleexit. thenbecalculatedusingthefollowingequationsincetheMachnumberatthenozzleexit isknownfromthedesignconditions,andthestaticpressureratiocanbecalculated, P 2 P 1 = 2 M 2 1 sin 2 +1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 +1 Thedeectionangle, canbedeterminedfromthefollowing tan =2 cot M 2 1 sin 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 M 2 1 + cos 2 +2 ThedownstreamMachnumber,stagnationpressureratio,andstatictemperatureratio areeasilycalculatedviathefollowingequations: M 2 = 1 sin )]TJ/F25 11.9552 Tf 11.955 0 Td [( s 1+ )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 M 2 1 sin 2 M 2 1 sin 2 )]TJ/F26 7.9701 Tf 13.15 5.256 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 P o 1 P o 2 = +1 2 M 2 1 sin 2 1+ )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 M 2 1 sin 2 # )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 1 2 +1 M 2 1 sin 2 )]TJ/F26 7.9701 Tf 13.151 5.256 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 +1 # 1 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 T 2 T 1 = )]TJ/F22 11.9552 Tf 5.479 9.684 Td [(1+ )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 M 2 1 sin 2 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 M 2 1 sin 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 h +1 2 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i M 2 1 sin 2 49 PAGE 50 a b c d Figure210.Thevariationofashockangle,bdeectionangle,cstagnationpressure ratioanddstatictemperatureratioasafunctionofupstreamstagnation pressure,downstreamofanobliqueshockwave. Itisnotedthatthestagnationtemperatureacrosstheshockisconstant.Resultsof thesecalculationsareshowninFigure210.Itisbrieymentionedthatthestagnation pressureratioreectsalossofmomentumgoingacrossashockwaveandthatthisloss ofenergyislessenedathigherupstreampressureratios. 2.2.4CompleteShockStructureofanOverexpandedJet Therstshockwaveatthenozzleexitiseasilymodeledasshownabove;however, thesubsequentbehaviorofthoseshockwavesisquitecomplex.Theobliqueshock wavescanintersectatapointdownstreamorcanmergeandformanormalshockarea 50 PAGE 51 knownasaMachdisk.Machdiskstypicallyformafterrelativelystrongshockswhich aretypicalfornozzlesoperatingfarremovedfromtheidealpressureratio.Theseshock wavescompresstheowcausingtheformationofPrandtlMeyerexpansionfanswhich turntheowandlowerthepressure.Whenexpansionfansintersecttheshearlayer whichisformedattheboundaryofthejet,theyarereectedbackasobliqueshock waves.Thisseriesofeventscausestheformationofshockdiamondsintheowand isrepeateduntilthecombinationofviscouseffectsandtheinuxoflowmomentum uidcausethejettobecomesubsonic,orwhenthejetinteractswithanobstacle. Figure211providesandillustrationoftheshockstructuretypicallyseenattheexitofan overexpandedjet. Thepressure,temperature,andvelocityoftheowinthedownstreamofthe nozzlechangesveryrapidlyandisdifculttomodelanalytically.Zapryagaevetal. [29]performedexperimentswithanoverexpandednozzleandperformedschlieren photographyaswell.Theirresultsshowthatthepressureintheowupstreamof therstMachdiskvariesgreatlyintheradialandaxialdirectionwithseveralsharp discontinuitiespresent.DownstreamoftherstMachdiskthevariationinpressureis stillpresent.However,thediscontinuitiesarenolongerpresent.Manyauthorshave extensivelystudiedunderexandedjets[30,34,35,46,51,52]andsimilartrendsasthe aboveareobserved. 2.3Summary InthisChaptertheconstructionofthejetimpingementfacilityhasbeenexplored. Thefacilitysystemsincludetheairstorage,waterstorageandowcontrol,airpressure control,airmassowmeasuring,convergingdivergingnozzle,anddataacquisition. Acomparisonbetweenthemassowratemeasuredduringexperimentsandthe theoreticalmassowratebasedononedimensionalisentropicgasdynamicrelations wasperformedandtheresultsdifferedbylessthan20%.Thepressure,temperature, andMachnumbervarioussectionsofthejetimpingementfacilityweredeterminedand 51 PAGE 52 Figure211.Illustrationofthestructureofobliqueshockwavesattheexitofan overexpandednozzle. areshowninTable22.Thenozzleexitpressurewascalculatedforeachregulator pressureusedduringtheexperiments,anditwasfoundthatthenozzleoperatesinan overexpandedmannerfortheentirepressurerange.Noshockwaveisexpectedinside thenozzle.Finally,theshockstructureattheexitofthenozzlehasbeendescribed. 52 PAGE 53 CHAPTER3 STEADYSTATEEXPERIMENTS Jetimpingementfacilitiesarecapableofveryhighheatuxremoval,andan experimentalproceduretodeterminetheheattransfercoefcientforthefacilityin Chapter2isdeveloped.Severaldifferentexperimentswereinitiallytestedwithout successbeforeasuccessfulexperimentalcongurationwasdevelopedtomeasurethe steadystateheattransfercoefcientoverarangeofoperatingconditions. Initiallyitwasbelievedthatphasechangeheattransferwouldoccurduring operationofthejetimpingementfacilitytosuchanextentthatnoradialvariationof heattransfercoefcientwouldbeobserved.Assuchanexperimentwasdesignedin whichathinsheetofstainlesssteelwasmachinedintoametalblankandheatedvia Jouleheatingandinsulatedatthebottomwherethetemperaturewasmeasuredvia athermocouple.Theareaofthestripwasquitesmallapproximately15mm 2 and assuchtheheatuxesproducedduringtheexperimentwerehigh.Theproblem experiencedwiththissetupisthatthebackwalltemperaturemeasuredbythe thermocoupleisnotverysensitivetochangesintheheattransfercoefcientwiththe appliedheatuxesandthemetalthicknessused.Anillustrationoftheheaterassembly usedisshowninFigure31. Inordertoalleviatethisproblemanexperimentwasconductedwherethermocouples wereembeddedinsideofacoppercylinderwhichwasheatedfromthebottomand insulatedalongtheside.Thejetwasallowedtoimpingeonthetopofthecylinder andthetemperaturesinsidethecopperpieceweremeasured.Onedimensionalheat conductionintheaxialdirectionwasassumedandduetotheabsenceofinternalheat generation,alineartofthemeasuredtemperaturewasthenusedtoextrapolatethe temperaturetothesurfaceandallowedthedeterminationofheatux.Uponanalyzing thedatagainedfromtheseexperimentsitwasobservedthattheheattransfercoefcient 53 PAGE 54 Figure31.Stainlesssteelheaterassembly. forthejetimpingementfacilityvariessignicantlyintheradialdirection.Anillustrationof thecopperheaterassemblyusedisshowninFigure32. Inordertogainsomeinsightintotheradialvariationoftheheattransfercoefcient theheaterinFigure32wasmodiedtoincludetemperaturesensitivepaintonthe topsurfaceofthecoppertestpiece.Steadystatetemperaturedistributionsatthetop surfacewherethenusedasinputtoaninverseheattransferalgorithmtodetermine theradiallyvaryingheattransfercoefcient.Therewereseveraldrawbackstothis study.Firstthetemperaturesensitivepaintisverybrittleandhadtobeprotectedfrom theimpingingjetviatheuseofthickclearcoatapplicationstothesurfaceofthepaint orviatheuseoftransparenttape.Theseprotectivelayerscouldnotbeneglectedin theinverseheattransferanalysisandcomplicatedthealgorithm.Lastly,andmost importantly,theimpingingjetpartiallyobscuresvisualobservationofthetemperature 54 PAGE 55 Figure32.Copperheaterassembly. sensitivepaint.Thesecomplicationsrenderreliableexperimentalresultsdifcultto obtain. Thethirdexperimentalcongurationtestedconsistsofathinsheetofnichrome whichisheatedbyJouleheating.Itisinsulatedatthebottomand9thermocouples areusedtomeasuredthespatialvariationofthebackwalltemperature.Thearea ofthenichromestripislargeandtheheatuxesproducedareconsiderablysmaller thanthoseappliedtothestainlesssteelsetupdescribedearlier.Thisexperimental congurationprovedtogivereliablemeasurementsofheattransfercoefcient,andthe abundanceofthermocouplesallowsthespatialvariationofheattransfercoefcienttobe determined.Thisexperimentisnowbedescribedindetail.LaterSectionswilldescribe theexperimentalprocedureandexploretheheattransferresults. 55 PAGE 56 3.1HeaterConstruction 3.1.1PhysicalDescription Theheaterdesignusedduringthecourseofthefollowingexperimentsisinspired bytheworkofRahimietal.[51].Itisconstructedusingathinnichromestripwhichis 0.127mmthickwithanexposedareaof50.8x25.4mm.Theheaterhas9totalEtype thermocouplesattachedtothebacksideofthestrip.Onethermocoupleisattached atthecenterofthestripand7additionalthermocouplesareattachedevery3.79mm towardsonesideofthestrip.Additionallyonethermocoupleisattachedat12.7mm fromthecenterontheoppositeside.Thisthermocoupleisusedtoensurethejetis centeredovertheheaterbyverifyingsymmetry. ThenichromestripisthenepoxiedontopofaGaroliteslabthatis140x140x 6.35mm.Smallholesaredrilledintheslabsothatthethermocouplespenetrateitand avoiddeformingtheatsurfaceofthenichromestrip.TheslabofGarolitehasathermal conductivityof0.27W/mKcomparedwith13W/mKforthenichromeandthusactsto insulatethebacksideofthenichromestrip. Electricalpowerissuppliedtothenichromeviatwocopperbusbarswithdimensions of43x25x2mmattachedtothetopofthestrip.Duringoperationofthetwophasejet, liquidowstowardsthebusbarsandaccumulatesattheedge.Thisliquidlmof accumulationcouldaffecttheheattransferphysics;tolessenthiseffecttheedgesof thebusbarsareledtoanangleofapproximately30 .Anillustrationoftheheater assemblyisshowninFigure33. Powerissuppliedtothenichromestripviaahighcurrent,lowvoltagedcpower supply.Thepowersupplyiscapableofsupplying4.5kWofpowerthroughavoltage rangeof030Vandacurrentlimitof125A.Themaximumvoltageseenduring experimentsisapproximately5Vat125A. 56 PAGE 57 Figure33.Illustrationofheaterassemblyusedforsteadystateexperiments. 3.1.2TheoreticalConcerns Thethermocouplesusedfortheexperimentmeasurethetemperatureontheback walloftheNichromestrip.Todeterminetheheattransfercoefcientfortheimpingingjet thesurfacetemperatureofthenichromeisneeded.Typicalheattransfercoefcientsfor impingingjetswillyieldBiotnumbersBi=h /kmuchgreaterthanunitythusalumped systemassumptionisnotvalidforthecurrentexperimentalconguration. Thefollowinganalysisprovidesamethodforevaluatingthespatiallyresolved surfacetemperature.FirstthesteadystateheatequationinCartesiancoordinateswith internalheatgenerationisexamined. @ 2 T @ x 2 + @ T 2 @ y 2 + @ T 2 @ z 2 + q 000 k =0 WhenEquation3isnondimensionalizedthefollowingresults: D 2 @ 2 @ x 2 + D 2 @ 2 @ y 2 + @ 2 @ z 2 +1=0, where 57 PAGE 58 = Tk q 000 2 x = x D y = y D z = z HereDisthenozzlediameter,and isthethicknessofthestrip.Thecoefcientsinthe rst2termsofEquation3arequitesmall,andthreedimensionaleffectscanbe neglected.Assuchthegoverningequationandboundaryconditionsare d 2 dz 2 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 d dz z =0 =0 )]TJ/F39 11.9552 Tf 12.68 8.088 Td [(d dz z =1 = kD k w Nu D r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 r wherer =r/Disthenondimensionalradiuswiththeoriginatthecenterlineofthejet, Nu D r =hr D/k w istheNusseltnumber,k w isthethermalconductivityofwater, and 1 r isthereferencetemperature.Duetotheextremedifcultyinmeasuringthe expandingjetuidtemperature,theadiabaticwalltemperatureiscommonlyusedas areferenceforpurposesofcomputingaheattransfercoefcient[53]associatedwith impingingjets.Notethattheeffectsoftheradialvariationofheattransfercoefcient comefromtheboundaryconditionsonly.Thesolutionofthisordinarydifferential equationis: r z = 1 2 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(z 2 + 1 kD k w Nu D r + a w r orindimensionalform T r z = q 000 2 k )]TJ/F25 11.9552 Tf 5.48 9.684 Td [( 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(z 2 + q 000 h r + T a w r whereT a,w istheadiabaticwalltemperature.Itisobservedthatthedifferencein temperaturebetweenthetopandbottomsurfacesisthequantity q 000 2 = 2 k ,indimensionless formitisequalto 1 / 2 .Themaximumvolumetricheatgenerationintheexperimentsis 58 PAGE 59 3.66 10 9 W/m 3 .Usingavalueof13W/mKforthethermalconductivityofnichrome themaximumtemperaturedifferencebetweentheupperandlowersurfaceobservedin experimentsisontheorderof2 Candcannotbeneglected.However,thesolutionfor theheattransfercoefcientandthustheNusseltnumbercanbecalculatedknowingthe internalheatgenerationrate,backwalltemperature,andtheadiabaticwalltemperature: Nu D r = Nu D w r 1 )]TJ/F40 7.9701 Tf 13.459 4.707 Td [(kD k w Nu D w r 2 where Nu D w r = 1 kD k w [ r ,0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( a w r ] 3.2ExperimentalProcedure Duringthecourseoftheexperiments,theMeasurementandComputingdata acquisitionsystemisused.Alsonotethattheheatuxremovedfromthetopofthe heaterassemblyisthequantityq 000 ,sincetheheatgeneratedinternallycanonlybe removedfromthetopsurface. 3.2.1TwoPhaseExperiments Duringthecourseofexperimentsitwasobservedthatrunningtheimpingingjet wouldcauseicetoformonthesurfaceoftheheateratlowheatuxes.Atadiabatic conditionstheicewouldbegintoformintoaconeshape,eventuallythisconewould bebrokenoffandanewconewouldforminitsplaceinaperiodicmanner,asshownin Figure34.Atlowbut,nonzeroheatuxes,athinsheetoficewouldformthatwould exhibitsimilarperiodicbehaviorastheicecones.Thisiceformationaffectsheattransfer sincethislayeroficeisstationaryandactsasaninsulator.Toavoidthisconditiona minimumheatuxof300kW/m 2 waschosensuchthaticeformationisnotvisually observedduringoperationoftheimpingingjet. 59 PAGE 60 Figure34.Iceformationatadiabaticconditions.Notetheconicalshapeoftheice structure. Torunacompleteexperiment,thenozzleisalignednearthecenteroftheheater surfaceatagivenheightandtheimpingingjetisinitiatedwiththepressureregulator setatadesiredpressure.Thepowersupplytotheheateristurnedonandaheatux ofapproximately470kW/m 2 issuppliedtotheheater.Toensuretheimpingingjetis centeredovertheheater,thepositionoftheheaterismovedsuchthatthetemperatures 60 PAGE 61 measuredbythesinglethermocoupleononesidematchesthetemperatureofthe correspondingthermocoupleontheoppositesidetowithinapproximately 0.5 C. Oncethisconditionisachievedtheheaterisallowedtoreachsteadystate,whichis determinedbyobservingagraphoftheheatertemperaturesvs.time.Uponreaching steadystateoperation,theheaterthermocouplesareloggedatasamplingrateof50 Hzforapproximately2minutes.Afterwardsthissameprocedureisperformedforheat uxesofapproximately430,390,350,and315kW/m 2 .Theseheatuxesarechosento beashighasreasonablyachievablewiththegivenequipmentfortworeasons.Firstto helppreventtheformationoficeonthesurfaceoftheheaterandsuchthattheheater walltemperaturedifferenceisashighaspossibletominimizetheuncertaintyintheheat transfercoefcientmeasurement. Thedatareportedforthisstudyareaveragedvaluesfrom100samplestaken overthecourseof2minutes.Whileitispossibleforhighfrequencyoscillationsinthe measuredtemperaturestoexistduetothemultitudeofdropletsimpingingontothe surface,thisisnotlikelytobeobservedforseveralreasons.First,thedataareaveraged whichwilllessenanytransienteffects.Second,theheater,althoughthin,doeshavea nitethickness.Soitwilltendtoactlikealowpasslteranddampenuctuations.Lastly itisbelievedthatathinliquidlayerexistsonthesurfaceoftheheater.Anyliquiddrops thatimpingeontotheheaterwilltendtocoalescewiththisliquidlm,thusminimizing transientsofthelocalheattransfercoefcient. Themeasurementoftheadiabaticwalltemperatureisaccomplishedbyasimilar procedureasaboveexceptafterensuringthejetiscentered,thepowersupplytothe heateristurnedoff.Theheatertemperaturesareallowedtoreachsteadystateandthen measurementoftheadiabaticwalltemperatureiscommenced.Itcouldtakeseveral minutesontheorderof5minutesfortheheatertoreachsteadystate.Because oftheabsenceofheattransfer,themeasuredtemperatureatthebackwallisequal tothesurfaceadiabaticwalltemperatureofthejet.Itisnotedthattheformationof 61 PAGE 62 icewasobservedonthesurfaceoftheheaterduringtheadiabaticwalltemperature measurements,butbecauseofthereasonslistedaboveforneglectinguctuationsdue tomultipledropletsimpingingontothesurface,itisbelievedthatthiswillhavelittletono effectonthemeasurementofadiabaticwalltemperature. IncomputingthejetReynoldsnumber.theviscosityisbasedonthenozzleexitair temperatureandpressureascalculatedbyonedimensionalgasdynamicrelations.In computingtheNusseltnumberforthesinglephasejet,thermalconductivityisevaluated basedonairandtheadiabaticwalltemperature.Duringtwophasejetimpingement,a thinliquidlmexistsontheheatersurface,andthinliquidlmdynamicsdominatethe heattransferphysics.Thusthewaterthermalconductivitybasedontheadiabaticwall temperatureisusedforNusseltnumbercalculationsofthetwophasejet.Typicalvalues forairviscosity,waterthermalconductivity,andairthermalconductivityrespectivelyare 6.45 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(6 Paswithlessthan1%variation,0.588W/mKwitha3%variation,and 0.0243W/mKwitha5%variation. 3.2.2SinglePhaseExperiments Toprovideacomparisonforreferencetothetwophasejetresultssinglephase experimentsarecarriedoutwiththejetimpingementfacilityusingonlyairexpanding throughthenozzle,waterowiscutoff.Theexperimentalprocedureisessentially identicaltothatforthetwophasejetexpecttheappliedheatuxisreduced.Theheat uxesusedduringexperimentsarelowerthanthoseinthetwophaseexperimentto ensurethattheheaterdoesnotoverheatanddelaminatefromtheGarolitebase. Table31displaysthecorrespondingheatuxesforagivenReynoldsnumber. 3.3ExperimentalResults 3.3.1UncertaintyAnalysis UncertaintyanalysisfortheexperimentsisdoneusingthemethodofKlineand McClintock[54].TheuncertaintyofthecalculatedNusseltnumberrangesfrom2.0to 4.0%forthesinglephasejetand2.5to18%forthetwophasejetnearthecenterline 62 PAGE 63 Table31.Reynoldsnumberandcorrespondingheatuxes. Nominallowestmidhighest Re D HeatFluxHeatFluxHeatFlux kW/m 2 kW/m 2 kW/m 2 4.5 10 5 355065 7.3 10 5 6080100 1.0 10 6 80100120 and0.3to1.0%attheouterextentsofthedomain.TheReynoldsnumberuncertainty rangesfrom2to3%anddoesnotvaryappreciablebetweenthesingleandtwophase experiments.Theuncertaintyinthetemperaturemeasurementsare0.2 C. Tohelpascertaintheerrorintheexperimentsbyassumingthatthethreedimensional effectswereneglected,anumericalanalysiswasconducted.Thisanalysisusesa secondorderaccuratenitedifferenceschemetosolveEquation3onathree dimensionalgridof2 n 2 n 2 n wheren=4,5,6,and7.Thetopboundaryismodeled ashavingaNusseltnumberdistributionfoundintheexperimentsandtheremaining sidesaremodeledasbeinginsulated.AftercompletionofeachsimulationtheNusselt numberiscalculatedusingEquation3.Tocharacterizetheerrorinusingaone dimensionalassumptiontherootmeansquareerroriscalculatedviathefollowing equation error rms = v u u u u u u t L R 0 W R 0 Nu exp )]TJ/F39 11.9552 Tf 11.956 0 Td [(Nu sim 2 dxdy L R 0 W R 0 Nu exp 2 dxdy theuseofthedifferentgridsizesallowstheextrapolationoftheerrorusingRichardson's extrapolationmethod[55,56].UsingthehighestNusseltnumberdistributionfound duringtheexperiments,resultsinanrmserrorof1.21%withapeakerrorof3%located neartheorigin,whileusingthelowestNusseltnumberdistributionresultsinanrmserror of0.83%withapeakerrorof3.25%locatedneartheedgeofthedomain.Theseerrors 63 PAGE 64 Figure35.MeasuredsinglephaseNu D spatialvariationatdifferentheatuxes. arelessthantheuncertaintycontainedintheexperimentalmeasurementsandthusthe onedimensionaltreatmentforevaluatingtheNusseltnumberisdeemedsatisfactory. 3.3.2SinglePhaseResults Duringthecourseoftheexperimentsitwasobservedthattheheattransfer coefcientisindependentofheatux,asexpected.Resultsforatypicalexperiment areshowninFigure35.Toillustratetheamountofuncertaintyinthedataerrorbars havebeenincludedinthisgure.However,tofacilitateeaseofviewingtheyarenot shownintherestofthisSection. Figure36showsthemeasuredthermocoupletemperatureprolesatvariousheat uxesforanozzlespacingof4nozzlediametersforasinglephaseexperiment.Note thattheadiabaticwalltemperaturecorrespondstothatmeasuredwithzeroapplied heatux.Figure37showsthatNusseltnumberscaleswith p Re D asreportedby 64 PAGE 65 Figure36.Spatialheatertemperaturevariationatdifferentappliedheatuxes. Donaldsonetal.[46]andRahimetal.[51],amongothers.Asamatterofreference, asinglephaseNusseltnumberontheorderof2,500correspondstoaheattransfer coefcientontheorderof11,000W/m 2 Kinthepresentstudy.Asinglenozzlewasused intheexperimentsandthustheoverexpansionpressureratioandReynoldsnumber arenotindependentofeachotherandtheeffectsofoverexpansionratiocouldnotbe isolated. Figure38comparesthelocalNusseltnumberfornozzleheightsandReynolds numbersusedduringtheexperiments.Forr/D > 0.5andH/D > 2theNusseltnumber distributionisnotstronglydependentonthenozzleheight.However,forr/D < 0.5there isasmallvariationinNusseltnumber.Thisbehaviorresultsfromthecomplexshock structureatthenozzleexitanditsinteractionwiththeheatersurface.ForH/D 2 Nusseltnumberisslightlyelevated,butthiseffectappearstolessenathigherReynolds 65 PAGE 66 a b Figure37.SpatialvariationofNu D atdifferentRe D ,aunscaledandbscaled. numbers.Apossibleexplanationforthisbehavioristhatduetothelowtemperatureof theowingair,thenozzlebecomescooled.Thiswillcausemoistureinthesurrounding airtocondenseonthenozzlewhichcanbecomeentrainedinthejet.Thisentrained moisturewillincreasetheamountofheatremovedfromthesurfaceoftheheaterand thuselevatethemeasuredNusseltnumber.Tocombatthisissuethenozzleisinsulated tothebestextentpossible.Withtheaddedinsulation,itisbelievedthatthemoisture condensationeffectsareminimalbut,nonzero. 66 PAGE 67 a b c Figure38.SinglephaseNu D atvariousnozzleheighttodiameterratios,aRe D =4.57 10 5 ,bRe D =7.55 10 5 ,andcRe D =1.05 10 6 3.3.3TwoPhaseResults Thetwophasejetexperimentsareperformedinthesamemannerasthe singlephasejetwiththeexceptionofwaterbeingaddedtotheairstream.Inorder toquantifytheeffectofthewaterontheheattransferproperties,themassfractionof waterinthejetiscalculatedas w = m l m l +_ m air 67 PAGE 68 Figure39.MeasuredtwophaseNu D spatialvariationatdifferentheatuxeswithoutice formation. Ingeneral,thetwophaseheattransfercoefcientisfoundtobeindependentof heatux.However,aspreviouslymentioned,whentheheatuxattheheatersurface istoolow,iceformationaffectstheheattransfermeasurements.Inordertocombat iceformationaminimumheatuxof315kW/m 2 isused.Nevertheless,therearea fewcaseswhereicingisobservedinheatuxesup350kW/m 2 .Toidentifyandhelp mitigatetheseeffects,themeanandstandarddeviationoftheheattransfercoefcient asafunctionofspaceistaken.Whenthestandarddeviationoftheexperimentalvalues exceeded20%,thenheatuxesof470,430,and390kW/m 2 areusedintheaveraging calculations.Theseheatuxesareselectedbecausethehigherheatuxeswillresult inhighersurfacetemperaturesandinhibiticeformation.Alsothehighertemperatures willresultinalarger Tandlessuncertaintyinthecomputedheattransfercoefcient. Approximately20%ofthemeasurementstakenrequirethesecorrectivemeasures, andinallcasestheresultingstandarddeviationislessthan20%ofthemean.See Figure39foranexampleofanexperimentwheretheheattransfercoefcientisclearly independentofheatuxandFigure310whereareductionintheheatuxusedwas necessary. 68 PAGE 69 a b Figure310.MeasuredtwophaseNu D spatialvariationatdifferentheatuxes,aice effectspresentandbafterremovaloflowestheatuxes. Figure311showstheradialvariationofmeasuredthermocoupletemperaturefor variousheatuxesforatwophaseexperiment.Notethatthezeroheatuxcondition representstheadiabaticwalltemperature.Figure312showstheradialvariationof Nusseltnumberforthetwophasejetatdifferentwatermassfractionsandconstant Reynoldsnumberandnozzleheight,Figure313showsthevariationwithnozzleheight withaconstantReynoldsnumberwithanominallyconstantmassfractionofliquid.Note thatitisnotpossibleinthecurrentstudytovaryReynoldsnumberandtheliquidmass 69 PAGE 70 Figure311.Spatialheatertemperaturevariationatdifferentappliedheatuxes, twophasejetresults. fractionindependentlyofeachother;thusitisnotpossibletoshowhowtheNusselt numberscaleswithReynoldsnumber. NusseltnumbergenerallyincreaseswithincreasingReynoldsnumberand increasingwatermassfractionneartheinteriorofthejet.Forr/D 1.5theredoes notappeartobeanoticeabledependenceofNusseltnumberonthenozzleheight. ThereissomevariationofNusseltnumberwithnozzleheightinthejetinterior,buta denitetrendisnotapparent.Forreferencepurposes,atwophaseNusseltnumber ontheorderof2,000correspondstoaheattransfercoefcientontheorderof200,000 W/m 2 K.MoreexperimentalresultsthanthosepresentedinthisChapterarepresented inAppendixA. 70 PAGE 71 a b c Figure312.TwophaseNu D atvariousliquidmassfractions.aZ/D=2.0,Re D =4.42 10 5 ,bZ/D=6.0,Re D =4.45 10 5 ,andcZ/D=6.0,Re D =7.24 10 5 Heattransfercoefcientsexceeding400,000W/m 2 KareobservedinFigure313c, whichareonthesameorderasthehighestliquidjetheattransfercoefcients,see[2], totheauthor'sknowledge.Whilethereismoreexperimentaluncertaintyatthesehigh heattransferratesto18%,theefcacyofthetwophasejetforhighheattransfer applicationsisclearlydemonstrated. Itisbrieynotedthattheoriceforthe0.37mmoricehadadefectandhenceis notperfectlycircular;theliquidowratedeliveredwaslessthanthatforthe0.33mm orice.TheNusseltnumberresultsforthe0.37mmoricearenoticeablysmallerthat 71 PAGE 72 a b c Figure313.TwophaseNu D numberatvariousnozzleheighttodiameterratios.aw= 0.0375,Re D =4.42 10 5 ,bw=0.0273,Re D =7.23 10 5 ,andcw= 0.0248,Re D =1.01 10 6 thatofthe0.33mmoriceanddonotfollowtheexpectedtrend.Thisisbelievedtobe duetotheeccentricityoftheoricecausingdifferentbehaviorinthemixingchamber andeffectingtheresultingdropletsize/distributionatthenozzleexit.Whilethe0.51 mmoricedoeshavesomeeccentricity,itisnotassevereasthatfoundinthe0.37 mmorice,anditdoesnotseemtohaveanoticeableeffectontheNusseltnumber measurements.PicturesofeachoriceareshowninAppendixC. 72 PAGE 73 3.3.4EvaporationEffects Tohelpquantifytheeffectofevaporationontheheattransfercoefcientthe saturatedhumidityratioattheimpingementsiter=0andattheedgeofthemeasurement locationr=50.8mmiscarriedout.Thesaturatedhumidityratioiscalculatedfrom sat =0.622 P v sat P )]TJ/F39 11.9552 Tf 11.955 0 Td [(P v sat Thevaporsaturationpressure,P v,sat iscalculatedfrom[57] P v sat =exp 647.096 T )]TJ/F22 11.9552 Tf 9.298 0 Td [(7.85951783 v +1.84408259 v 1.5 )]TJ/F22 11.9552 Tf 11.955 0 Td [(11.78664977 v 3 +22.6807411 v 3.5 )]TJ/F22 11.9552 Tf 11.955 0 Td [(15.9618719 v 4 +1.80122502 v 7.5 where v =1 )]TJ/F39 11.9552 Tf 28.595 8.088 Td [(T 647.096 ThasunitsofKelvin,PhasunitsofPascals,andvisnondimensional.Atthejet impingementzonethetemperatureisontheorderof10 Candthepressureis approximatelythestagnationpressure.Resultsforthesaturatedhumidityratiofor thethreeseparatestagnationpressuresusedareallontheorderof10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 ;thusany effectsduetoevaporationnearthecenterlineareconsiderednegligible. Thepressureattheedgeoftheheateraswellasthetemperatureatthesurface oftheliquidlmareunknownandasimilaranalysiscannotbeperformed.However, novisualobservationofphasechangeatthehighesttemperaturesseenduring experimentsisseen.ItisobservedinFigures312and313,thatNusseltnumber remainsessentiallyconstantneartheedgeoftheheater.Evaporationwouldfurther enhanceheattransferresultinginanincreaseinNusseltnumberinthisregionthusitis believedthatevaporationislikelynegligibleinthisareaaswell. 73 PAGE 74 3.4ComparisonbetweenSingleandTwoPhaseJets Togainanappreciationofthetwophasejetheattransferenhancement,the measuredheattransfercoefcientiscomparedtothethatforthesinglephasecase. Theheattransferenhancementfactorisdenedhereas = h mix h air a b c Figure314.Heattransferenhancementratioatvariousliquidmassfractions.aZ/D= 2.0,Re D =4.42 10 5 ,bZ/D=4.0,Re D =7.35 10 5 ,andcZ/D=2.0, Re D =1.01 10 6 74 PAGE 75 Figure314showtheheattransferenhancementwiththevariationinwatermass fractionandReynoldsnumber.Itisobservedthattheenhancementincreaseswith increasingmassfraction;theincreaseisdiminishedwithincreasingReynoldsnumber. ItisobservedthatinFigure314thatforr/D < 0.5thereisamarkedincreaseinthe measuredheattransferenhancement.AthigherReynoldsnumbersthemaximum enhancementoccursneartheedgeofthejetr/D=0.5. ItisobservedinFigure315thatthevariationoftheheattransferenhancementwith nozzleheightissimilartothatforwatermassfraction,itshowsanincreasingtrendat lowerReynoldsnumbersbuttheeffectisdampedathigherReynoldsnumbers. 3.5Discussion Oneofthefeaturesapparentinallexperimentsisthatatradialdistancesof1.5to2 nozzlediameterstheNusseltnumberandheattransferenhancementratiosapproacha nearconstantvaluewhichisindicativeoflmowheattransfer.Variousstudies[34,40] havenotedthatthereisalargeadversepressuregradientinthisregionwhichlikely causesboundarylayerseparation[35].Duringthecourseofthepresentexperiments, tworingsindicativeofaseparationregionareobservedontheheatersurface.Oneof whichcorrespondstotheedgeofthenozzlewherethereisashockwavepresent,and theotherislocatedapproximately1.5nozzlediametersfromthejetcenterline.Insideof theseregiontheheattransfercoefcientisaffectedbynozzleheightandthewatermass fractionindicatingthatjetimpingementisthedominatingheattransfermechanismfor r/D < 1.5. Alloftheexperimentsreportedarecarriedoutatrelativelylowsurfacetemperatures; thehighesttemperatureisontheorderof70 C.Phasechangeduetomasstransferof theliquidintotheimpingingairstreamisconsiderednegligibleforreasonsdiscussed inSection3.3.4.AdditionallytheworkofBenardinandMudawar[58]explorethe Leidenfrostmodelforimpingingdropsandsprays.Theirmodelpredictsthepressurein dropletsusingonedimensionalelasticimpacttheory[59],andacorrectionfactordueto 75 PAGE 76 a b c Figure315.Heattransferenhancementratioatvariousnozzleheighttodiameterratios. aw=0.0205,Re D =4.54 10 5 ,bw=0.0290,Re D =7.24 10 5 ,andc w=0.0233,Re D =1.02 10 6 Engel[60,61]givesgoodresults.Thepressureriseattheimpingementsurfacecanbe modeledas P =0.20 l u o u snd whereu o isthedropletvelocityandu snd isthespeedofsoundintheliquid.Using thisequationitisseenthatforanydropletvelocityaboveapproximately7%ofthe speedoftheairintheimpingingjetapproximatelyMach3willyieldsurfacepressures 76 PAGE 77 abovethatofthecriticalpressureforwater.Thusevenifphasechangeoccursatthe impingementpoint,thelatentheatofvaporizationiszeroandnoenhancementofheat transferwilloccur.Becauseofthecomplexshockstructuresoccurringwhenthejet impingesontothesurface,makingsimilarargumentsforregionsfarremovedfromthe impingementzonearenotreliableandthusarenotattempted.However,itisnotedthat mostoftheliquiddropletsimpingingontothesurfacewilloccurnearthecenterline;thus thepressurefarremovedfromthejetcenterlinewillbelowerandevaporationmaystill bepossibleatelevatedsurfacetemperaturesandheatuxes. Oneofthecurrentlimitationsofthecurrentresultsistheylackinformationonthe liquiddropsizedistribution.Suchmeasurementsarenotavailableatthecurrenttime andfutureworkisplannedtoaddressthisdeciency. Thecurrentheattransfermeasurementsarecomparedtothesinglephaseliquidjet heattransfermeasurementsofOhetal.[2],andthemeasuredheattransfercoefcients areonthesameorderofmagnitude.Theliquidowrateinthecurrentexperimentsis verysmallwhencomparedtothoseexperiments,upto0.7g/s.5g/m 2 s,referenced toheatedareacomparedwith4.3kg/s.55 10 6 g/m 2 s,referencedtoheated area,afeaturewhichhassignicantindustrialadvantages.Inthestudyperformedby Ohetal.,liquidtovaporphasechangeisnotobserved,andthoseexperimentswere performedatmuchhigherheatuxesupto30timestheheatuxesreportedinthe currentstudy.Futureinvestigationswillexplorehigherheatuxregimes. 3.6Summary InthisChapterheattransferenhancementmeasurementsusingtwophase overexpandedsupersonicimpingingjetswerepresentedforawiderangeofReynolds numbers.Thesejetstwophasejetsaregeneratedbytheadditionofwaterdroplets upstreamofaconvergingdivergingnozzle.Heattransfermeasurementsusinga singlephasejetisusedforcomparison.Itisobservedthattheadditionofwaterdroplets intotheairowsignicantlyenhancestheheattransferrate.Enhancementissignicant 77 PAGE 78 nearthejetcenterlinetheenhancementfactorexceeds10inmostcases.Themass fractionofwateraddedtothejetisobservedtobyanimportantparameterforheat transfer,generallyincreasingNusseltfollowsincreasingwatermassfraction.However, itsinuencediminishesathigherReynoldsnumbers.Nozzleheightappearstohavea smallimpactontheobservedheattransferrates. 78 PAGE 79 CHAPTER4 DETERMINATIONOFHEATTRANSFERCOEFFICIENTUSINGANINVERSEHEAT TRANSFERANALYSIS AswasseeninChapter3theuseofsteadystatemeasurementtechniquesyields lowsurfacetemperatureswhicharenotsuitableforevaporatingtheliquidlmandthe low Tbetweenthewallandliquidlmwhichcreatesuncertaintyinevaluatingtheheat transfercoefcient.Inordertoalleviatesomeoftheseeffectsatransientapproach involvinganinverseheattransferquenchingproblemisdeveloped. 4.1InverseProblems Therearetwobasicparadigmsinheattransfer.Themostwellknownparadigm isthesolutionofthetemperatureeldwithinamediumsubjecttoconstraintssuch asagiventhermalconductivity,thermaldiffusivity,andknownboundaryconditions. Ifalloftheconstraintsareknown,thentheresultingtemperatureeldwithinthe mediumofinterestcanbesolved;inmanycasesananalyticalsolutioncanbe determined.However,iftheseconditionsarenotknownthentheproblemisnotunique, isunderspecied,andnosolutioncanbedetermined. Thesecondparadigminheattransfer,namedinverseheattransfer,iswhen thetemperatureatspecicpointsinsideofamediumareknownandaconstraint needstobedeterminede.g.contactresistancebetweentwosurfacesoraboundary condition.Becauseofunavoidablemeasurementerrorsinthetemperatureeldthis problemisillposedandcanbedifculttosolve.Thedifcultiesofthisproblemcanbe circumventedinveryspecialcircumstances,forexampleifonedesirestodetermine theheatuxappliedataboundaryofaonedimensionalbaratsteadystateonecan ensembleaveragetemperaturemeasurementsatafewlocationsalongthelengthofthe baranddeterminethetemperaturegradientvialinearregression.Withaknownthermal conductivity,Fourier'slawcanbeusedtodeterminetheappliedheatuxandtheresults canbequiteaccurate.Unfortunatelythesesimpleproblemsdonotcomeaboutin 79 PAGE 80 practiceoften.Forexampleiftheheatuxvariesintime,thentheabovemethodwould notbeapplicableandadifferentmethodwouldbeneeded. InverseProblems,ingeneral,fallintooneoftwocategories:parameterestimation, inwhichoneormoredesiredparametersaredeterminedusingexperimentaldata e.g.thermalconductivityofasolidandaappliedheatuxandfunctionestimation,in whichadesiredfunctionistobeestimatedusingexperimentaldatae.g.aboundary conditionwhichvariesinspaceandtime.Itshouldbenotedthatmanyfunction estimationproblemscanbeformulatedintermsofaparameterestimationproblemifthe functionalformoftheofdesiredfunctionisknown,forinstanceifthermalconductivity isaquadraticfunctionoftemperaturetheproblemcanbereducedtodeterminingthe coefcientsofthegoverningequation.Thisapproachcanyieldgoodresults,seeFlach and Ozisik[62]forexample,however,iftheformoftheequationisnotknown apriori thenthisapproachmaynotbeuseful. InverseProblemsandInverseHeatTransferIHTproblemshavebeenstudied extensivelyintheliteratureandhavebeeninusesinceatleastthe1950's.Tikhonov, [6365]amongothers,wasoneofthersttotacklethechallengeofinverseproblems andtakeintoaccountmeasurementerrors.HistechniquetitledTikhonov'sregularization minimizedtheleastsquareerrorbyaddingaregularizationtermthatpenalizes unwantedoscillationsintheestimatedfunction.Tikhonov'smethodcanberelated todampedleastsquaresmethods,mostnotablythemethodduetoLevenberg[66] andMarquardt[67],knownastheLevenbergMarquardtmethod.Thesemethodsare onlysuitableforparameterestimation.Stoltz[68]usedafunctionestimationtechnique basedonDuhamel'sprincipleandtwosimultaneousthermocouplemeasurementsto determinethesurfaceheatuxinaonedimensionalproblem.Thisprocessisknown asexactmatchinganddoesnottakeintoaccountanymeasurementerrors.Beck [6971]usedamethodsimilartothatbyStotlz;however,temperaturesatfuturetimes areusedtoprovideregularizationandreduceinstabilitiesinthemethod.Thismethod 80 PAGE 81 canbeusedforparameterorfunctionestimationbut,canbecomeunstableforsmall timestepsandthushighlytransientphenomenacannotbeaccuratelyreproduced. TheMonteCarlomethodcanbeusedtoestimateaparameterorfunctionaswas demonstratedbyHajiSheikhandBuckingham[72];agoodreviewofthetechniquecan befoundin[73].Amethodthatissuitableforsmalltimestepsandperformsparameter orfunctionestimationisAlifanvo'sIterativeRegularizationMethod[74].Thismethodis alsoknownasparameter/functionestimationwiththeadjointproblemandconjugate gradientmethod,andisthemethodusedforthepresentstudy.Thismethodwillbe abletodetermineatimeandspacevaryingheattransfercoefcientproducedbya multiphasesupersonicimpingingjet,aswellasanytemperaturedependenceduetoany evaporationoftheliquidlm. 4.2IntroductiontoInverseProblemSolutionUsingtheConjugateGradient MethodwithAdjointProblem Dealingwithinverseproblems,whichbytheirnatureareillposed,usuallyinvolves sometypeofregularizationtechniqueoranoptimizationtechniquewhichinherently regularizesthesolution.Thetechniqueusedinthecurrentstudyisanoptimization techniqueknownasfunctionestimationusingtheconjugategradientmethodwith adjointproblem.Asthenameimpliesthismethodusestheconjugategradientmethod tominimizetheerrorintheleastsquaressensebetweentheestimatedoutputofan equation/systemofequationsandthemeasuredoutputwhichhasbeencorruptedwith noise.Themethodwillbedescribedbelowinitsgeneralformtofamiliarizethereader. Manyreferencesexistforfunctionalestimationwiththeadjointproblemandconjugate gradientmethodincluding Ozisik[75], OzisikandOrlande[76],Alifanov[74],andthe ChapterbyJarny[77].MuchofthefollowinganalysisfollowsthatofJarnyastheauthor foundthatparticularreferencetobemathematicallyrigorous,thorough,generalin nature,andeasytofollow.Specicimplementationsofthismethodwillbediscussed whereneeded. 81 PAGE 82 4.2.1TheDirectProblem Thedirectproblemisthemodelequationsforthesystemofinterest.Itcanbe analgebraic,integral,ordinarydifferential,orpartialdifferentialequationorsystemof equationsorsomecombinationtherein. y x t = f x t whereyistheoutputofthesystem, istakentobeaparametersorfunctiontobe estimatedandxandtaretheindependentvariables.Notethatalthoughbothspace andtimeareindependentvariablesinthisexampleitisnotnecessaryfortheoutputto dependonbothofthem. 4.2.2TheMeasurementEquation Themeasurementequationexistsduetothediscretenatureofasamplingprocess andduetochangesbroughtaboutindataduetosensordynamics.Althoughinmodern dataacquisitionsystemsitispossibletomeasurequantitiesatanearcontinuousrate takingmeasurementsstillisaninherentlydiscreteprocess.BendatandPiersol[78] havewrittenagoodreferenceondatameasurementandanalysiswhichincludes sensor/systemdynamics. Sensordynamicscangreatlyeffectthemeasurementsofasystemandtheireffects canbequitesignicant.Thisprocesscanbesimpliedifthesensordynamicscanbe approximatedbyalineartimeinvariantLTIsystem,whichmostsensorsfallunder.In anLTIsystemtheoutputofasensoristheresultofaconvolutionofitsinputwiththe sensor'simpulseresponsefunction.Theimpulseresponsefunctionistheresponseof thesensor,initiallyatrestorzero,toanimpulseinput.Themeasurementequationcan bemathematicallyexpressedas Y m = t Z 0 h t )]TJ/F25 11.9552 Tf 11.955 0 Td [( y d 82 PAGE 83 whereY m isthemeasuredoutput,histheimpulseresponsefunction,andyisthetrue outputofthesystem.Ifthesensorisperfectandthegoalistosimplydenoteitsdiscrete naturetheimpulseresponsefunctionwouldsimplybeadeltafunction.Foreaseof viewingthemeasurementequationcanalsobediscussedinanoperatorformsuchthat Equation4isequalto Y m = Cy 4.2.3TheIndirectProblem Theindirectproblemisactuallythestatementoftheleastsquarescriteria.When solvinganinverseproblemwiththecurrentmethodtheparameterorfunctionsoughtis theonewhichminimizestheleastsquarescriteria.Simplystatedtheindirectproblemis S = M X i =1 t f Z 0 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i y i t ] 2 dt whereSistheintegratedsquaresnotethatinthecaseofdiscretedatathiswould bethesumofsquares,iisthesensornumber,andMisthetotalnumberofsensors. ThespatialdependenceofyisleftoutofEquation4becauseitisassumedthat thesensorsareplacedatvaryingdistancesinspace,thusthemeasurementoperator, C i wouldonlyoperateonmeasurementsatalocation, x i .Itcanbeusefultothinkof theleastsquarescriteriaintheformofanormoperator, k u k orsometimes h u v i ,for instanceEquation4isequalto S = k Y m )]TJ/F39 11.9552 Tf 11.956 0 Td [(Cy t k 4.2.4TheAdjointProblem Formulatingtheadjointproblemcorrectlyisacrucialstepinthesolutionprocess. Essentiallythisiswheretheoptimizationportionoftheproblemcomesintoplay.Todo 83 PAGE 84 thistheindirectproblemisconsideredthemodelingequationandthedirectproblemis consideredasaconstraintsuchthatthefollowingequationholds, R y = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(f x t ThesearethenjoinedtogetherthroughtheuseofaLagrangemultiplier. L y = k Y m )]TJ/F39 11.9552 Tf 11.955 0 Td [(Cy t k)222(h R y i whereListheLagrangianvariableand istheadjointvariablealsoknownasthe Lagrangemultiplierwhich,ingeneral,canbeafunctionofspaceandtime. Whenthecorrectparameter/function isinsertedintoEquation4theresulting Lagrangianiszeroforperfectmeasurements.Realworldmeasurementswillbe corruptedwithnoiseandtheresultingLagrangianwillbetheminimumleastsquares criteria. Todeterminetheproper theLagrangianmustbeminimized.Iftheadjointvariable istreatedasxedthethedifferentialoftheLagrangianis dL = hr y S y i)222(h r y R y y i)222(h r R y i orexpressedinamoreconvenientform dL = hr y S )]TJ/F25 11.9552 Tf 11.955 0 Td [( [ r y R y ] y i)222(h [ r R y ] i Becausethechoiceoftheadjointvariableisnotconstraineditischosentobethe solutionof r y S )]TJ/F25 11.9552 Tf 11.955 0 Td [( [ r y R y ] =0 84 PAGE 85 Equation4isknownastheadjointequation.Notethatthisisimplicitin throughmathematicaloperationusuallyinvolvingintegrationbypartsitcanbe expressedasanexplicitfunctionoftheadjointvariable. 4.2.5GradientEquation NotethattheadjointequationrenderstherstterminEquation4tobezero. Atthesolutionpointwhere isequaltothetruevalue,theLagrangianisequaltothe minimumoftheleastsquarescriteria dL = dS = hr S i andcomparingtheremainderofEquation4toEquation4thegradient equationresults r S = )]TJ/F25 11.9552 Tf 9.299 0 Td [( [ r R y ] Thegradientequationisusedintheconjugategradientminimizationalgorithmto determineastepsizeanddescentdirectioninordertominimizetheleastsquares criteria. 4.2.6SensitivityEquation Asmentionedoneoftheparametersneededtondtheminimumoftheindirect problemisthestepsize.Thisparametercantakeafewdifferentformsdepending onwhethertheinverseproblemislinearornonlinear.Intheinterestofpresentinga generalmethod,theformfornonlinearproblemsarepresented. Thestepsizetobedeterminedisaperturbationintheparameter/function ,which istobeestimated.Toderivethisquantitywesimplyperturbthedirectproblem y + y = f x t + ; generallytherighthandsideofEquation4islinearizedsuchthat 85 PAGE 86 y + y = f x t + f x t WhenEquation4issubtractedfromtheaboveequationthesensitivityequationis theresult y = f x t NotethatthesecondterminEquation4andtherighthandsideofEquation4 containboth and .Inverseproblemsinwhichthesensitivityequationcontainsboth parameters/functions and arenonlinear.Notallinverseproblemsarenonlinearin nature,andthisformisusedhereforthesakeofgenerality.Thesensitivityequation simplystatesthataperturbationintheparametertobeestimatedwillresultina perturbationofthecomputedoutput. 4.2.7TheConjugateGradientMethod Theconjugategradientmethodisanoptimizationproblemforsolvinglinearor nonlinearequations.Thereareseveralreferenceswhichdetailthemathematicsbehind thistool,forinstancethebooksbyRao[79]andFletcher[80],amongothers.Assuch readersinterestedinarigorousderivationofthemethodareencouragedtoconsult thesereferences. Theessentialstepsofthemethodarethataguessfor ischosen,theabove equationsaresolvedandasearchdirection, d whichisCconjugatetotheprevious directioniscalculatedusingaconjugationcoefcient, .Thesearchdirectionisthen multipliedbythestepsize, andisaddedtothepreviousguessfor .Thisiterative processcontinuesuntiltheerrorbetweenthemeasuredoutputandcalculatedoutput reachesapredeterminedtolerance. Thereareseveraldifferentformsoftheconjugationcoefcient, intheliterature suchastheHestenesStiefel[81],PolakRibi ere[82],andFletcherReeves[83],among others.Allofthementionedformsareequivalentforlinearequations;howeveritis 86 PAGE 87 discussedintheliterature[84,85]thatthePolakRibi ereformoftheequationhasbetter convergencepropertiesfornonlinearequationsandassuchwillbeusedinthepresent studyunlessotherwisenoted.ThePolakRibi ereformoftheconjugationcoefcientis k = M P i =1 hr S k r S k )222(r S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i kr S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 k for k =1,2,3... and k =0 for k =0 Thesearchdirectionisthencalculatedbythefollowing k = r S k + k k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Thestepsizeisdenedbythefollowing =argmin [ S )]TJ/F25 11.9552 Tf 11.955 0 Td [( ] =argmin M X i =1 t f Z 0 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i y i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( ] 2 dt TheoutputofthedirectproblemisexpandedinaTaylorseriesas y t )]TJ/F25 11.9552 Tf 11.956 0 Td [( y t )]TJ/F25 11.9552 Tf 11.955 0 Td [( @ y t @ y t )]TJ/F25 11.9552 Tf 11.955 0 Td [( y t andsubstitutingtheaboveintoEquation4yields =argmin M X i =1 t f Z 0 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i y i t + C i y t ] 2 dt Performingthedifferentiationwithrespectto ,settingtheresultequaltozeroand solvingfor yieldsthenalformoftheequationforthestepsize 87 PAGE 88 k = M P i =1 t f R 0 C i y i x t k )]TJ/F39 11.9552 Tf 11.955 0 Td [(Y m i t C i y i t k dt M P i =1 t f R 0 [ C i y i t k ] 2 dt Itisnotedthattheaboveequationcanbesimpliedforlinearproblemswhenthe leastsquarescriteriatheindirectproblemiscastinquadraticform.However,forthe sakeofgenerality,theaboveequationwillbeusedthroughoutthecurrentChapter. 4.3FactorsInuencingInverseHeatTransferProblems Thereareseveralfactorswhichcaninuencethesolutionofaninverseproblem. Someofthesefactorsarediscussedbelow. 4.3.1BoundaryConditionFormulationEffects Toperformafunctionestimationinverseproblemtodetermineaspatiallyand temporallyvaryingheattransfercoefcient,thechoiceoftheboundarycondition formulationisveryimportant.TheboundaryconditioncanbeformulatedasaDirichlet speciedtemperatureboundaryconditionwherethesurfacetemperatureisdetermined andtheresultingheatuxiscalculatedinordertodeterminetheheattransfer coefcient,asaNeumannspeciedheatuxboundaryconditionwheretheheat uxisdeterminedandtheresultingsurfacetemperatureiscalculated,orasaRobin convectiontypeboundaryconditionwheretheheattransfercoefcientisdirectly determined.Atrstglance,theRobintypeboundaryconditionseemstobethebest choiceastheunderlyingphysicstakingplaceareconvectiveinnature.Uponfurther analysisthisisactuallytheworstchoice.Inordertodemonstratethisanexample usingaonedimensionalheattransferproblemwithatimevaryingheattransfer coefcientwillbeusedbecauseoftheeaseofcalculation.Thesameconceptsapply toatwodimensionalproblemwithaspatiallyandtemporallyvaryingheattransfer coefcient. Thefollowinganalysiscanbefoundin[86]butisreproducedhereforclarityandto correctsomeerrorscontainedtherein.Supposeatimevaryingboundaryconditionis 88 PAGE 89 Figure41.1dimensionalsolidforthesensitivityproblem. appliedtothex=0surfaceofaonedimensionalsolidwithaninsulatedboundaryatx= L,seeFigure41.ThesolutionofthisproblemcanbedeterminedbyusingDuhamel's principleforeachtypeofboundaryconditionpreviouslydiscussed.Essentiallythetime varyingboundaryisconvolvedwiththeimpulseresponsefunctionoftheslab.The impulseresponsefunctionisdeterminedbysolvingtheheatequationforthesolidwitha boundaryconditionofunityastepresponsefunctionandthentakingthederivativeof thatfunctionwithrespecttotime.Forinstancethesolutionforatimevaryingheatuxin dimensionlessformis x t = o + Z t 0 q @ q x t )]TJ/F25 11.9552 Tf 11.956 0 Td [( @ t d where q indimensionlessformis: q x t = t + 1 3 + x x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 )]TJ/F22 11.9552 Tf 13.547 8.088 Td [(2 1 X n =1 1 n 2 exp )]TJ/F21 11.9552 Tf 5.479 9.684 Td [()]TJ/F39 11.9552 Tf 9.299 0 Td [(n 2 2 t cos n x 89 PAGE 90 Inordertoanalyzewhichtypeofboundaryconditionshouldbeusedintheinverse problemformulation,asensitivityanalysisshouldbeperformed.Thesensitivity analysisisaccomplishedthroughtheuseofrelativestepsensitivitycoefcients. Firstthegoverningequationsandboundaryconditionsarenondimensionalized andtheirsolutionobtained.Thederivativeofthissolutionistakenwithrespectto thenondimensionalinputparameterfortheboundaryconditionnondimensional temperature,heatux,orBiotnumber.Theresultoftheseoperationsisthestep sensitivitycoefcient,althoughthemagnitudeofthecoefcientfortheconvectioncase variesdependingonthemagnitudeoftheinputBiotnumber.Toallowdirectcomparison ofthesesensitivitycoefcientstheyaremultipliedbytheirboundaryconditioninputs transformingthemtorelativestepsensitivitycoefcients,denotedasX input .With nondimensionalizationoftheproblem,theneteffectisonlyseenintheconvectioncase. X q x t = q x t a X x t =1 )]TJ/F22 11.9552 Tf 13.547 8.088 Td [(2 1 X n =1 1 n )]TJ/F23 7.9701 Tf 13.151 4.707 Td [(1 2 sin n )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(1 2 x exp )]TJ/F30 11.9552 Tf 11.291 16.856 Td [( n )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(1 2 2 2 t # b 90 PAGE 91 X Bi x t = Bi @ @ Bi = Bi 1 X n =1 exp )]TJ/F21 11.9552 Tf 5.479 9.684 Td [()]TJ/F25 11.9552 Tf 9.299 0 Td [( 2 n t n @ C n @ Bi cos n )]TJ/F39 11.9552 Tf 11.955 0 Td [(x )]TJ/F39 11.9552 Tf 229.465 35.714 Td [(C n @ n @ Bi [2 n t cos n )]TJ/F39 11.9552 Tf 11.955 0 Td [(x + )]TJ/F39 11.9552 Tf 11.955 0 Td [(x sin n )]TJ/F39 11.9552 Tf 11.955 0 Td [(x ] o where @ n @ Bi = 1 tan n + n sec 2 n and @ C n @ Bi = @ n @ Bi 4 cos n 2 n + sin n )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(8 sin n [ 1+ cos n ] [ 2 n + sin n ] 2 n tan n = Bi c NotethatEquation4cdependsontheinputparameterBiandthustheinverse problemisnonlinearinnatureandcanbedifculttosolve.Alsonotethatthisequation isdifferentthanthatfoundin[86];theequationinthatreferencecontains x asopposed to 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x Aplotofthesensitivitycoefcientsat x =0.1 ispresentedinFigure42;itshould bementionedthatthemagnitudesofthesensitivitycoefcientsareplottedandthe coefcientsforBiareactuallynegativeandthisisnotclariedfortheplotinReference [86].Afewpointsofinterestsshouldbepointedout.First,notethatthecoefcients forBiarelowerthanalloftheothersandthatasBinumberincreasesitssensitivity coefcientdecreases.Thecoefcientsforatemperatureandheatuxinputaremuch largerthanthoseforBinumberwiththecoefcientsfortemperaturebeinglargerthan thoseforheatuxuntilavalueof t 0.75 .Clearlyaninverseproblemformulatedin termsofanunknownconvectioncoefcientisnotagoodchoice. 4.3.2SensorLocationEffects Thesensitivitycoefcientsalsodependonposition.Thesensitivitycoefcients foraheatuxinputareplottedinFigures43and44.Itiseasilyseenthatthecloser 91 PAGE 92 Figure42.Relativestepsensivitycoefcientsatx =0.1asafunctionoftime. atemperaturesensorisplacedtotheboundaryofinterestthemoresensitiveitis tochangesofthatboundarycondition,asonewouldexpectfromsimplephysical reasoning. ComparingFigures43and44onecanseethatthemagnitudeoftherelative stepsensitivitycoefcientislargeratthebackwallforanunknownsurfacetemperature formulationthanforanunknownheatuxformulation.Thischaracteristicwillbe exploitedinthisstudyinordertominimizetheeffectsofthermocoupleinsertiononthe inverseproblem. 4.3.3ThermocoupleInsertionEffects Inordertoperformtemperaturemeasurements,solidthermocouplesarecommonly usedastheyarearobust,inexpensivemethodtothemeasurement.Aswasdemonstrated inSection4.3.2,theclosertothesurfaceofinterestasensorisplaced,better sensitivitiestochangesintheboundaryconditionareachieved.Thiscanbeaccomplished 92 PAGE 93 Figure43.Relativestepsensivitycoefcientforaheatuxinput. Figure44.Relativestepsensivitycoefcientforatemperatureinput. bydrillingholesinthesolidandinsertingthermocouplesinsidethesolid.Placingholes inthesolidcanhaveadverseeffectsontheheattransferdynamics. 93 PAGE 94 Severalresearchershavestudiedtheproblemofthermocouplesinsertedintoa solidandhowtheydistortthethermaleldaswellashowthisaffectsinverseproblems. ChenandLi[87]studiedtheproblemnumericallyandfoundthetheerrorproducedby thethermocoupleinsertionisproportionaltotheholesizeandthatthemagnitudeofthe errordecreasesintime.ChenandDanh[88]expoundedupontheresearchin[87]by performingexperimentswhichconrmedsomeofthepredictedresultsfromnumerical simulations,thesestudiedfocusedonthermocouplesinsertedparalleltothedirection ofheatow.Beck[89]usedDuhamel'stheoremtodetermineacorrectionkernelfor thermocouplesinsertednormaltoalowthermalconductivitysurfacetocompensate fortheinsertioneffects.WoodburyandGupta[90]usednumericalmethodstostudy thermocoupleinsertionandtheeffectsoninverseheattransferproblems.Woodbury andGupta[91]alsodevelopedasimpleonedimensionalsensormodeltonumerically correcttheeffectsforthethermocoupleholes;thisstudyalsoincludedtheneffectfrom thethermocouplewiresandisapplicabletoathermocoupleofanyorientationtothe surface.Attiaetal.[92]performedaverycomprehensivenumericalandexperimental studywhichhelpedquantifytheerrorthatthethermocoupleinsertionproduceson measurementsincludingwireeffects,llermaterialeffects,andnonidealcontact situationsinwhichthethermocoupleisinsertedatanangleinthehole.LiandWells [93]performednumericalandexperimentalworkstudyingthedifferentfactorsaffecting theerrorduetothermocoupleinsertion.Interestinglytheyfoundthatforathermocouple insertedperpendiculartothedirectionofheatowi.e.paralleltothesurfaceofinterest therewouldbenoeffectonthetemperaturemeasurementsbut,thermocouplesoriented paralleltothedirectionofheatowwouldhavenoticeableeffectsonmeasurements oronaninverseheattransferanalysis.Caron,Wells,andLi[94]continuedthisstudy andfoundacorrectionmodelcalledtheequivalentdepth.Thismodelimpliesthatthe temperaturesmeasuredfromaninsertedthermocouplecanbeputintoaninverse analysisasmeasurementstakenfromadifferentposition;thisnewpositionistheone 94 PAGE 95 whichwouldwouldexperiencethetemperaturetransientsrecordediftherewereno thermocouplesinserted.Thecorrectionmodelwasonlyabletoaccuratelyreproduce surfaceheatuxhistories.Franco,Caron,andWells[95]continuedtheworkand developedcorrectionmodelswhichaccuratelyreproducesurfacetemperatures. AtrstinspectiontheworkofLiandWells[93]appearstogiveanidealresult, thatorientingthethermocoupleinaspecieddirectionwillcauseittohaveno noticeableimpact.Theauthorattemptedtousethisinformationanddesignedan inverseexperimentwithsheathedthermocouplesinserted2mmbelowthetopsurface ofacoppercylinderataangularspacingevery45 .Aftermanytrialstodetermine theimpulseresponsefunctionsoftheseembeddedthermocouplesitwasconcluded thatthermocouplessignicantlyimpactedtheheatowandtemperatureeld.This experimentalndingiscontrarytothestudybyLiandWells,butitcouldbedueto differingfactorssuchasdifferenttypesofthermocouplesused,differentsolidmaterial copperfortheauthor'sexperiment,aluminumforLiandWells,andthefacttherewere manythermocouplesinsertedversusoneforLiandWells. Onecommonalityfortheaboveworkcitedisthatthecorrectionmodelscanbequite complicatedandtheyonlyassesstheeffectsofasinglethermocouplebeinginserted intothesolid.Becauseofthesedifcultiesitwasdecidedtousesimpleweldedbead typethermocouplesandsilversolderthemtothebackofthesolidnoinsertion.This congurationeliminatedallinsertioneffectsbecausetherearenoholesdrilled.Thiswill affectthenatureoftheinverseproblembecausemeasurementsperformedattheback surfacewillcausethesensitivityoftheinversemethodtodecrease.Thislimitationcan beovercomebyformulatingtheproblemasanunknownsurfacetemperatureinsteadof anunknownsurfaceheatuxorconvectioncoefcient. 4.4InverseHeatTransferProblemFormulation AswasdemonstratedinSection4.3thebestchoiceforformulatinganinverse problemfordeterminingtheheattransfercoefcientisaspeciedtemperature 95 PAGE 96 Figure45.Illustrationoftheheattransferphysicsoftheinverseproblemformulation. formulationwiththermocouplesmeasuringtemperatureatthebackwall.Oncethe temperatureattheimpingementsurfaceisknowntheheatuxatthesurfaceandheat transfercoefcientcanbedetermined.Aschematicdiagramillustratingtheproblem formulationisshowninFigure45.TheinverseproblemequationsfromSection4.2 willnowbecastintotheproperfromforanIHTproblemforacylinderataninitial temperaturethatisexposedtoatimeandspacevaryingsurfacetemperature. 4.4.1DirectProblem TheDirectprobleminnondimensionalformisformulatedas, 96 PAGE 97 @ @ t = 1 r @ @ r r @ @ r + @ 2 @ z 2 a r z =0, t = s r t b @ @ z z = L R =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e r z t =0 =0 f wherethefollowingnondimensionalizationisused r = r R z = z L t = t R 2 = T o )]TJ/F39 11.9552 Tf 11.955 0 Td [(T T o s r t = T o )]TJ/F39 11.9552 Tf 11.956 0 Td [(T s r t T o 4.4.2MeasurementEquation Themeasurementequationis,initsrigorousform m i = t Z =0 1 Z r =0 L R Z z =0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( r z r )]TJ/F39 11.9552 Tf 11.956 0 Td [(r i z )]TJ/F39 11.9552 Tf 11.955 0 Td [(z i r dr dz d NotethatthedeltafunctionsinEquation4merelytakeintoaccountthediscrete natureofthemeasurements.Notingthispoint,themeasurementequationcannowbe castas m i = t Z 0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( i d 97 PAGE 98 Thesubscriptiintheequationdenotesthemeasurementlocation,ofwhichthere are7totalmeasurementpoints.Alsonotethateachthermocouplecanhaveitsown impulseresponsefunctionandhencethesubscript.Thisequationcantakeonan operatorformsimilartoEquation4. 4.4.3IndirectProblem Thecorrespondingindirectproblemis S s = t f Z 0 [ Z m t )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s ] 2 dt Notethattheoperatorfromofthemeasurementequationisused. 4.4.4AdjointProblem Thedevelopmentoftheadjointproblemisquiteinvolvedmathematically.Because ofthesensordynamicsinvolvedinthemeasurementequationtheformoftheadjoint equationwilllookdifferentthanmanyofthosefoundintheliteraturesuchas[96100] forexample.Totheauthor'sknowledgetherearenoreferencesintheliteraturethat explicitlytakeintoaccountthesensordynamics,except[86],whichmerelydiscussesthe convolutionofthedeltafunctiontoaccountforthediscretenatureofthemeasurements. AlsoMarquardt'sanalysis[101]whichaccountsforsensordynamics,butusesastate anddisturbanceobserversmodel,whichisdifferentthanusingtheadjointproblemsuch asusedforthecurrentanalysis. Tobeginthederivationoftheadjointproblem,thenecessarysubstitutionsare carriedoutforEquation4 L s = t f Z 0 M X i =1 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s ] 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( @ @ t )]TJ/F22 11.9552 Tf 15.133 8.088 Td [(1 r @ @ r r @ @ r )]TJ/F25 11.9552 Tf 15.358 8.088 Td [(@ 2 @ z 2 v dt HerethenorminthesecondtermofEquation4isequalto 98 PAGE 99 h u v i v = 1 Z r =0 L R Z z =0 uvr dr dz NextthesecondtermofEquation4isintegratedbyparts.Thisallowsforan explicitfunctionoftheadjointvariabletoappear.Afterusingtheboundaryandinitial conditionsofthedirectproblem,Equation4,theresultisthefollowing L s = t f Z 0 M X i =1 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s ] 2 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( @ @ t )]TJ/F22 11.9552 Tf 15.133 8.087 Td [(1 r @ @ r r @ @ r )]TJ/F25 11.9552 Tf 14.833 8.087 Td [(@ 2 @ z 2 v )]TJ/F25 11.9552 Tf 11.956 0 Td [( @ @ r r =1 + @ @ r r =0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( @ @ z z = L R + s @ @ z z =0 )]TJ/F25 11.9552 Tf 13.151 0 Td [( @ @ z z =0 + t = t f dt NextthederivativeofEquation4istakenwithrespectto and s dL s = t f Z 0 M X i =1 h)]TJ/F22 11.9552 Tf 13.948 0 Td [(2 Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s C i i )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( @ @ t )]TJ/F22 11.9552 Tf 15.133 8.088 Td [(1 r @ @ r r @ @ r )]TJ/F25 11.9552 Tf 14.833 8.088 Td [(@ 2 @ z 2 v )]TJ/F22 11.9552 Tf 11.955 0 Td [( @ @ r r =1 + @ @ r r =0 )]TJ/F22 11.9552 Tf 11.955 0 Td [( @ @ z z = L R + s @ @ z z =0 )]TJ/F25 11.9552 Tf 13.15 0 Td [( @ @ z z =0 + t = t f dt ThegoalistospecifytheadjointequationasthesolutiontothetermsinEquation4 involving .However,therstterminvolvesthemeasurementoperatorand .To rectifythistheadjointofthemeasurementequationissoughtsuchthat 99 PAGE 100 h e i t C i i = h C i e i t i where e i t = )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 [ Y m i t )]TJ/F25 11.9552 Tf 11.956 0 Td [( m t ] Theoperator C i isknownastheadjointoperatorof C i .Tosolveforthisoperator examinationofthelefthandsideofEquation4gives h e i t C i i = t f Z t =0 e i t t Z =0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( d dt = t f Z =0 t Z t =0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( e i t dtd ComparingEquations4and4itisobservedthat C i e i t = t Z 0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( e i t dt Takingintoaccountcausality,itisknownthat for > t h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( =0. Thereforetheequationfortheoperator C i is C i e i t = t Z h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( e i t dt Nowknowingtheadjointoperatorofthemeasurementequation,theadjointproblem isselectedtobethesolutionof )]TJ/F25 11.9552 Tf 14.829 8.088 Td [(@ @ t = 1 r @ @ r r @ @ r + @ 2 @ z 2 + C i [ Y m t )]TJ/F25 11.9552 Tf 11.955 0 Td [( m t ] a 100 PAGE 101 z =0 =0 b @ @ z z = L R =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e r z t = t f =0, f whereEquation4isanalboundaryvalueproblem.Totransformittoaninitial boundaryvalueproblemthefollowingsubstitutioncanbeperformed, = t )]TJ/F39 11.9552 Tf 11.955 0 Td [(t f 4.4.5GradientEquation AftertheadjointproblemhasbeenspeciedthedifferentialoftheLagrangian becomes dL = t f Z 0 s @ @ z z =0 dt UsingEquation4theequationforthegradientis r S = @ @ z z =0 4.4.6SensitivityProblem UsingtheoperationssetoutinSection4.2.6thesensitivityproblemis @ @ t = 1 r @ @ r r @ @ r + @ 2 @ z 2 a z =0 = s r t b @ @ z z = L R =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e 101 PAGE 102 r z t =0 =0 f Notethattheformofsensitivityproblemisthesameasthatoftheadjointproblemand thedirectproblem,thusthesamenumericalsolvercanbeusedforallthreeproblems. 4.4.7ConjugateGradientMethod ThetheorybehindtheconjugategradientfromSection4.2.7remainsunchanged. Thefollowingaretheequationsspecictotheproblemathand.Therstequation iteratesforthesurfacetemperature k +1 s r t = k s r t )]TJ/F25 11.9552 Tf 11.955 0 Td [( k s r t andthenextequationprovidesforthesearchdirection k s r t = r S k r t + k k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 s r t Thenextequationgivestheconjugationcoefcient, k = t f R t =0 1 R r =0 r S k r t r S k r t )222(r S k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r t r dr dt t f R t =0 1 R r =0 [ r S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r t ] 2 r dr dt and k =0 for k =0. Thenalequationgivesthestepsize, k = M P i =1 t f R 0 C i r z t k s )]TJ/F39 11.9552 Tf 11.955 0 Td [(Y m i t C i i t k s dt M P i =1 t f R 0 [ C i i t k s ] 2 dt 102 PAGE 103 4.4.8StoppingCriteria Theconjugategradientmethodisaniterativeprocedureandtheterminationpoint mustbepredened,thediscrepancyprincipleisusedforthispurpose.Thediscrepancy principlestatesthatthestoppingpointforthecalculationiswhenthevalueoftheleast squaresfunctionseeEquation4isequaltothenormoftheuncertaintyofthe inputtemperatures.Usingthisprinciplethefollowingcriterionisdened: 2 = M X i =0 t f Z 0 2 i t dt where i isthestandarddeviationofthedataatmeasurementpointi.Forconstant uncertaintythefollowingholds 2 = M 2 t f andthusthealgorithmisterminatedwhen S k s 2 Onethingofnoteisthattheadjointproblemisanalvalueproblemwithanal conditionequaltozero;thustheLagrangemultiplierfortheoptimizationproblemiszero andchangestotheinitialguessatthenaltimearenotpossible.Thiscancausesome errorinthedeterminationoftheinverseproblemasshowninFigure46,noticethat nearthenaltimetheerrorbetweentheactualtemperatureversusthatreturnedbythe inversealgorithmislarge.Thislargeerrorcancausethealgorithmtohaveconvergence problems.Toovercomethisissue,theerrorcalculationofEquation4usesa truncatedsampleofthedata.Forinstanceiftheinputdatais60,000timestepslong only50,000timestepswouldbeusedintheerrorcalculation,andthusonlytheresults inthetruncatedsampleareconsideredreliable. 103 PAGE 104 Figure46.Comparisonofthetruetemperatureversusthetemperaturereturnedbythe inversealgorithm.Notethedisagreementnearthenaltime. 4.4.9Algorithm Alloftheequationsneededtosolvetheinverseheattransferproblemhavebeen developed.Thefollowingcomputationalalgorithmhasbeendevelopedtoobtaina solution: 1.Set s r t totheinitialguessusually1andset k =0 2.Solvethedirectproblem,Equation4usingthecurrentvalueof s r t and recordthetemperaturesatthemeasurementpoints. 3.SolvethemeasurementEquation4 m t 4.DetermineifthestoppingcriterionismetusingEquations4and4; terminatethealgorithmifthecriterionismet,otherwisecontinue. 5.Usingthemeasuredandpredictedtemperaturessolvetheadjointproblem, Equation4. 6.DeterminethevalueofthegradientusingEquation438. 7.Determine k fromEquation4and s r t fromEquation4. 104 PAGE 105 8.Solvethesensitivityproblem,Equation4andobtain atthemeasurement points. 9.Determine k fromEquation4 10.Determine k +1 s r t viaEquation4,set k = k +1 andreturntostep2. 4.5NumericalMethodandLimitations ThesolutionoftheDirect,Sensitivity,andAdjointproblemsneedtobefound, whileanalyticalsolutionsfortheseproblemsmayexistinsomespecialcircumstances generallysuchsolutionsarenotavailableandhencetheyaresolvednumerically. 4.5.1AlternatingDirectionImplicitMethod TheAlternatingDirectionImplicitADImethodofPeacemanandRachford[102] isacommonmethodusedforsolvingtheheatequation.TheADImethodissecond orderaccurateinspaceandtime,unconditionallystable,andiswellsuitedforsolving theseproblems.However,becauseofitsimplicitnatureitcanconsumeagreatdealof computingpoweriftheselectedtimestepisverysmall.Thebasicalgorithmforsolving thedirectproblembeginswiththerststep. n +1 = 2 i j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n i j t 2 = 1 i )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 r n +1 = 2 i +1, j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r + n +1 = 2 i +1, j )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 n +1 = 2 i j + n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r 2 + n i j +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n i j + n i j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 z 2 andthesecondstepis n +1 i j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n +1 = 2 i j t 2 = 1 i )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 r n +1 = 2 i +1, j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r + n +1 = 2 i +1, j )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 n +1 = 2 i j + n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r 2 + n +1 i j +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n +1 i j + n +1 i j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 z 2 whereiandjaretheradialandaxialgridpointsindexes,respectivelyandnmarksthe timestep.Notethattheseindexesstartat1fortheaboveequations. 105 PAGE 106 4.5.2GridStretchingintheZDirection Toaccuratelygetanestimateoftheheattransfercoefcienthighmeasurement samplingratesandcorrespondinglysmallnumericaltimestepsareused.Duringthe initialtransientthesolidbehaveslikeasemiinnitemediumastheinitialeffectsofthe quenchingarenotgreatlyfeltbeyondthethermaldiffusionlengthwhichisproportional to p t .Toaccuratelycapturethethermalgradientsnearthesurfaceanemeshnear thesurfaceisdesired.Toaccomplishthistask,thefollowinggridtransformationisused, z = L R 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( tan )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 [ )]TJ/F25 11.9552 Tf 11.955 0 Td [( tan = ] where isthetransformedcoordinateand isastretchingparameter.Whileitis theoreticallypossibletostretchthegridinthezdirectionagreatdeal,itisgenerally unwisetodosomorethanisnecessary.Thiscausesthegridspacing r and z todifferfromeachothersignicantlywhichcancreateerror.InthecaseoftheADE methodthiscancauseunusablesolutionstobegenerated[103].Forthecurrent problemastretchingparameterof =1.1 ischosen.Figure47illustratestheeffectsof gridstretching. UsingthegridstretchingtransformationgivenbyEquation4,thedirect problemwillbetransformedto, @ @ t = 1 r @ @ r r @ @ r + h 2 2 @ 2 @ 2 + g 2 @ @ a r =0, t = s r t b @ @ =1 =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e r t =0 =0 f 106 PAGE 107 a b Figure47.Effectsofgridstretching.arealdomainandbcomputationaldomain. h 2 = @ @ z g 2 = @ 2 @ z 2 Similartransformationsoftheadjointandsensitivityproblemsresultaswell.Alsonote thatanyuxquantitywillbetransformedas @ u @ z z = z p = h 2 z = z p @ u @ z = z p where z p isthezcoordinatewheretheuxcalculationisbeingcarriedout. 107 PAGE 108 4.5.3Timestepsizecomplications Inordertochoosethesizeofthetimesteptouseintheinverseheattransfer algorithm,somesamplecalculationsarecarriedout.Asatestproblemaonedimensional slabataninitialtemperatureof0issubjectedtoanondimensionaltemperatureofunity. TheADImethodisusedona32x32gridwithtimestepsof5 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 and5 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 Courantnumberof0.512and0.0512respectivelyforatotalof800timestepsineach case.TheresultsofthecalculationsareshowninFigure48.Itisclearlyseenthat theADImethodreproducestheexactsolutionfortemperatureforbothtimesteps remarkablywell.Itisalsoevidentthatattimestepsof5 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 oscillationsappearin theresultingheatux.TheoscillationswerenotedbyKropf[104]fortheADImethod duetolargetimestepsandthediscontinuityofthesurfacetemperatureattheinitialtime step.Theseoscillationsarenotpresentinthesmallertimestepcaseandamaximum timestepof5 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(5 isselectedforimplementingtheinverseheattransferalgorithm. 4.6DeconvolutionforThermocoupleImpulseResponseFunctions Aninherentproblemforsolvingtheinverseheattransferproblemjustsetforthis knowingtheimpulseresponsefunctionsfortheinstalledthermocouples.Classically thermocouplesarethoughtofasrstordersystemshowever,theycansometimesbe thoughtofassecondorhigherordersystems[105]. However,giventhetooloftheinverseproblemtheimpulseresponsefunctionof eachthermocouplecanbedeterminedwithoutan apriori knowledgeofthefunctional formofthesefunctions.ThefollowingSectionswillformulatethenecessaryinverse problem.Theproblemisformulatedsuchthattheinputintothethermocoupleis knownviaanexactsolutionandthethermocoupleoutputisrecorded.Morethan oneexperimentalrecordcanbeusedinthefollowinganalysissimultaneouslyifdesired, eachexperimentalrecordisdenotedbythesubscripti.Thefollowinganalysisisaslight modicationtothatfoundinreference[77] 108 PAGE 109 a b c d Figure48.ComputedtemperatureandheatuxfordifferenttimestepsusingtheADI method. 4.6.1DirectProblem Thedirectproblemissimplytheconvolutionoftheinput,x,withtheimpulse responsefunctionh,todeterminetheoutputsignal,y, y i t = t Z 0 x i h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( d NotethesubscriptiinEquation4denotesameasurementchannel. 109 PAGE 110 4.6.2IndirectProblem TheindirectproblemisessentiallyunchangedfromtheonederivedinSection4.2.3. However,forcompletenessisitincludedbelow. S = M X i =0 1 2 t f Z 0 [ Y m i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y i t ] 2 dt 4.6.3AdjointProblem OncetheLagrangianhasbeenformed,whichintheinterestofspaceisnot demonstrated,theadjointequationischosenas t i = Y m i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y i t NotethatEquation4isasimpleexpressionandtheaboveformwhaschoseto facilitateeaseofsolution. 4.6.4GradientEquation Similartothetheadjointofthemeasurementoperatorthegradientequationis r S i t = t f Z i t h t )]TJ/F25 11.9552 Tf 11.955 0 Td [( dt NotethatEquation4isnotaconvolutionoperation. 4.6.5SensitivityProblem Usingthemethodoutlinedearlierthesensitivityproblemis y i t = t f Z 0 x i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( h d Note,onceagain,thatthesubscript,i,denotesthemeasurementchannel. 4.6.6ConjugateGradientMethod Theconjugategradientmethodforthedeconvolutionproblemissimilartothe previousderivation.However,thisparticularimplementationusestheFletcherReeves 110 PAGE 111 equationforthestepsize[83].Recallthatthedifferentstepsizesareequivalentfor linearproblems,ofwhichthecurrentdeconvolutionproblemisone.Theequationforthe iterationsis h k +1 t = h k t + k h k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ; theequationforthesearchdirectionis h k +1 t = )]TJ/F40 7.9701 Tf 15.895 14.944 Td [(M X i =1 r S k i t + k h k t ; theequationfortheconjugationcoefcientis k = M P i =1 t f R 0 r S k t 2 dt M P i =1 t f R 0 [ r S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ] 2 dt ; and k =0 for k =0 andtheequationforthesearchdirectionis k = M P i =1 t f R 0 [ x i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(Y m i t ] h k t dt t f R 0 [ h k t ] 2 dt 4.6.7StoppingCriteria Thestoppingcriteriaremainsunchanged.Equations4and4stillapply andarenotrepeatedhere. 4.6.8Algorithm Alloftheequationsneededtoperformthedeconvolutionproblemhavebeen developed.Thefollowingisthecomputationalalgorithm: 111 PAGE 112 1.Set h t toaninitialguessandset k =0 2.Solvethedirectproblem,Equation4usingthecurrentvalueof h t 3.DetermineifthestoppingcriteriaismetusingEquations4and4, terminatethealgorithmifthecriteriaismet,otherwisecontinue. 4.Usingthemeasuredandpredictedvaluesofysolvetheadjointproblem, Equation4. 5.DeterminethevalueofthegradientusingEquation4. 6.Determine k fromEquation4and h t fromEquation4. 7.Solvethesensitivityproblem,Equation4andobtain y i t 8.Determine k fromEquation4 9.Determine h k +1 t viaEquation4,set k = k +1 andreturntostep2. 4.6.9TestCase Inordertodemonstratethecapabilitiesoftheabovedeconvolutionmethodasimple exampleispresented.Inthisexamplearstorderimpulseresponsefunctionischosen torepresentthesensor h = 1 exp )]TJ/F39 11.9552 Tf 10.977 8.088 Td [(t whereatimeconstantof =0.2sisused.Threedifferentinputsarechosen x =sin t x =1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(t x = p t theseinputsareconvolvedwiththeimpulseresponsefunction;Equation4and noisehavingastandarddeviationof =0.1isadded.Theresultsofthedeconvolution aredisplayedinFigures49,410,and411. 112 PAGE 113 Figure49.Trueandestimatedimpulseresponsefunction. Figure410.Convergencehistory. ClearlyasevidencedbyFigure49thetrueimpulseresponsefunctionisreproduced quiteaccuratelyforthegivennoiselevel.Figure410showstheconvergencehistory,S vsiterationnumberk.LastlyFigure411showstheresultswhentherealinputfunction, x,isconvolvedwiththeestimatedimpulseresponsefunction. 113 PAGE 114 Figure411.Simulatedoutputlinesandoutputofxconvolvedwithestimatedimpulse responsefunctionsymbols. Onenoteworthypointisthatwhileattemptingtodeterminetheimpulseresponse functionofthermocouples,multipleexperimentalrecordsaregenerallynotusedinthis analysis.Thereasonisthatitisnotfeasibletousemanydistinctinputsignalsintothe systemgenerallyastepchangeintemperatureatasurfaceisused.Thusmultiple experimentalsamplerecordswillbenearlyidenticalandtheadvantagesofusing multipleinputswillnotberealized. 4.7Summary InthisChapterinverseproblemsandinverseheattransferareintroduced.General factorsaffectinganinverseanalysissuchassensordynamics,sensorlocation,and problemtypeformulationwereexploredaswellaswaystobestusetheanalysis. Problemsspecictothepresentcasewerealsoexplored,suchashowathermocouple insertedintoadrilledholeinthesolidcandistortthethermaleldarounditandimpede heattransfer.Asacorrectiveactionaninverseheattransferproblemwasdesignedthat usesnoinsertionholes,andtheproblemhasbeenformulatedtobeofanunknowntime 114 PAGE 115 andspacevaryingsurfacetemperatureinordertoovercomelimitationsassociatedwith suchaformulation. Numericalmethodstosolvethedirect,adjoint,andsensitivityproblemshavebeen explored.Oscillationscausedmainlybyusinglargetimestepsforadiscontinuityin thesurfacetemperatureattheinitialtimewerenotedandasuitabletimesteplimit wasfound.TheAlternatingDirectionImplicitmethodwaschosentobethesolution techniqueduetothehighspatialandtemporalaccuracy. Adeconvolutiontechniqueusingthefunctionestimationwithadjointproblemand conjugategradienttechniquewasdeveloped.Atestcaseillustratingthepowerofthe techniquewasperformedusingseveraldifferentsimulatedinputstoarstordersensor withnoisydata.Thistechniquewillbeusefulindeterminingtheimpulseresponse functionsofthermocouplesusedinexperiments. 115 PAGE 116 CHAPTER5 SENSORDYNAMICSANDTHEEFFECTIVENESSOFTHEINVERSEHEAT TRANSFERALGORITHM ThepurposeofthisChapteristoinvestigateamodelforthesensordynamicsof thermocouplesandverifyitviaanexperiment.Theeffectivenessoftheinverseheat transferalgorithmwillalsobepresented.Theeffectsofdischeight,numberofsensors, magnitudeofthenoisepresentinthemeasurementsystem,magnitudeoftheBiot numberdistribution,andthemagnitudeandaccuracyoftheimpulseresponsefunction modelwillbeexplored.Lastlytheeffectsofnonidealinsulationwhichcausesthediscto gainheatduringanexperimentwillalsobedetailed. 5.1ThermocoupleMeasurementDynamics AswasstatedinChapter4,measurementdynamicsareanimportantfactorin inverseheattransferproblems.Assuchmucheffortwastakentoinvestigatedifferent modelsforthermocoupledynamics. 5.1.1LowBiotNumberThermocoupleModels Theresponsemodelofweldedbeadtypethermocouplesusedinthepresentstudy isaeldthathasbeenextensivelystudied[106110].Forthecaseofthermocouples inauidenvironment,amodelfortheresponsetimecanbedevelopedusingrst principles.Firstitisassumedthatthethermocouplecanbeapproximatedasasphere withconstantthermalproperties.WhentheBiotnumberBi=hD/kismuchless thanunity,thetemperatureproleinsideoftheweldedbeadcanbeapproximatedas constantthroughoutitsradius[111];duetothelowBiotnumberandwithoutlossof generality,thespherecanbemodeledasaslab.Figure51showsanillustrationofthe rstordersystemforaslab.Performinganenergybalanceontheslabresultsinthe following Vc p dT dt = )]TJ/F39 11.9552 Tf 9.298 0 Td [(hA T )]TJ/F39 11.9552 Tf 11.955 0 Td [(T 1 116 PAGE 117 Thetemperatureisnondimensionalized,andtheinitialdimensionlesstemperature istakentobezeroattheinitialtime.Thesolutiontotheordinarydifferentialequation ODEaboveis =1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F39 11.9552 Tf 9.299 0 Td [(t where =TT o /T 1 T o isthedimensionlesstemperature,T o istheinitialtemperature, and = Vc p /hAisthetimeconstantofthethermocouple.Theaboveequationisknown asthestepresponseequation;todeterminetheimpulseresponseofarstorder system,thederivativewithrespecttotimeofthestepresponseequationistaken,and thustheimpulseresponseforarstorderthermocoupleis: h t = 1 exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(t Thetimeconstantcanbethoughtofasthetimeneededforthethermocoupletoreach 63.2%ofthevalueofT 1 .After5timeconstantsthetemperatureofthethermocoupleis essentiallyT 1 Foratimevaryingfreestreamtemperaturethethermocouplewillindicatea differenttemperatureaspertheconvolutionintegraldetailedinChapter4.Theequation governingthisbehaviorisrepeatedhereforcompleteness. t = t Z 0 1 exp )]TJ/F39 11.9552 Tf 10.494 8.088 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [( d ForasolidembeddedthermocouplewithaBiotnumbermuchlessthanunity,thesame logicisapplied,excepthisnolongertheconvectionheattransfercoefcient,butnow theproducthArepresentsthecontactconductance. Similartotheaboverstordersystem,asecondordersystemfortheresponseofa thermocoupleisderived.Manyofthedetailsforasecondorderthermalsystemcanbe foundin[112];onlythesalientdetailsareincludedhere.Whenthethermocouplebead 117 PAGE 118 Figure51.Illustrationoftherstorderslab. iscoatedwithsomesortofintermediatematerialepoxyforexamplethiscanformtwo rstordersystemsinseries,anillustrationofthissystemisshowninFigure52.The followingsecondorderODEwillresult: d 2 T 1 dt 2 +2 n dT 1 dt + 2 n T 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(T 1 =0 where 2 n = 1 1 + 1 2 + h 1 h 2 2 2 n = 1 1 2 i = Vc p i hA i Usuallythesystemdenedabovehastwotimeconstants,whichisotherwiseknownas overdamped[105].SolvingtheaboveODEusingtheprocedureoutlinedforarstorder systemwillresultinthefollowingimpulseresponsefunction: 118 PAGE 119 Figure52.Illustrationofthesecondorderslab. h t = 1 2 n p 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 [ exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( 1 t )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( 2 t ] 1,2 = n p 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. Thedifferenceindynamicsbetweenarstandsecondordersystemisquitesubstantial asillustratedinFigure53.Noticethatforordershigherthanonetheimpulseresponse functionisequaltozeroattheinitialtime.Thisobservationisusefulindetermining whetherthesystemisofhigherorderwhenameasurementoftheimpulseresponse functionofthesystemisavailable. Thereisathirdmodelofnoteandthatisadiffusiveelementwhichisfollowedbya 2 nd ordersystem,hereafterreferredtoasthe2exponentialmodel.Thedetailsofthis modelcanbefoundinreference[113].Thediffusiveelementwillcausetheresulting impulseresponsefunctiontoconsistofthefollowingform h t = aexp )]TJ/F39 11.9552 Tf 9.298 0 Td [(b + cexp )]TJ/F39 11.9552 Tf 9.299 0 Td [(d 119 PAGE 120 Figure53.Exampleofarstandsecondorderimpulseresponsefunction. wherea,b,c,anddareconstants.Thisimpulseresponsefunctionbehavesverysimilar tothatofarstordersystem. WhiletheabovemodelsarequitegoodwhenthethermocoupleBiotnumberis lessthanunity,itcanproduceresultsthatdeviatefromidealwhentheBiotnumber approachesorexceedsunity.ForthecurrentexperimentsusingEtypethermocouples, whichhaveathermalconductivityof19.5W/mKwithacharacteristiclengthof approximately1mm,theBiotnumberwillbe0.1orlesswhenheattransfercoefcient islessthanapproximately20,000W/m 2 K.Theuseofsmallerthermocouplesenables theuseofthelowBiotnumberassumptionwithhigherlimitsofheattransfercoefcients. Inthepresentstudysilversolderisusedtoattachweldedbeadthermocouplestoa heatedsample.Theuseofsilversolderwillresultincontactconductancesoneormore ordersofmagnitudehigherthanthelimitingcasepreviouslymentioned;thustheuse ofamodelwhichreliesonasmallBiotnumberassumptionforthepresentstudy,may producesomeerror. 120 PAGE 121 5.1.2HighBiotNumberThermocoupleModels Developingameasurementmodelforthermocouplesthatdoesnotrelyonalow Biotnumberassumptionisdifcult.Thereareseveralparameterswhichcanaffectthe impulseresponsefunction,suchasthethermaldiffusivityofthethermocoupleaswell asthethatofthesolidthethermocoupleisembeddedin,amongothers.Rabinand Rittel[114]havecompletedanumericalstudywherethelowBiotnumberassumptionis notnecessary.Toaccomplishthistheymodeledathermocoupleasasphereforbead typethermocouplesorcylinderforthermocoupleprobesembeddedinasolidthat undergoesastepchangeintemperature.Thenumericalexperimentsarecarriedout overalargenumberofthermaldiffusivityratiosandacurvetisusedtoexpressthe resultingimpulseresponsefunction.Theimpulseresponsefunctionhasthefollowing form h t =exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(B D t R 2 n whereBandnarecurvettingparameters,Ristheradiusofthesphere/cylinder,and D isthethermaldiffusivityofthesolid.Atableofthevariousconstantsisfoundin Table51. Table51.CurveFittingConstantsforRabinandRittel'sthermocoupleimpulseresponse model,from[114] CylindricalCaseSphericalCase TC = D BnBn 11.5820.562.7990.61 101.7240.453.1930.52 1001.8210.453.2090.50 3001.8300.453.2290.50 10001.8330.453.2360.50 Afewnoteworthypointsarethattheindicatedthermocoupletemperatureused inthenumericalexperimentsisthevolumeaveragedtemperatureinsideofthe thermocouple.RabinandRittelnotedthatthetemperatureinsideofthethermocouple 121 PAGE 122 Figure54.ImpulseresponsefunctionsusingthemodelofRabinandRittel,adapted from[114]. wouldbeverynonuniformforthermaldiffusivityratioslowerthanapproximately300. Alsoitisbrieynotedthattheheattransferphysicsbeingmodeleddonotincludeany effectsofthermalcontactresistancebetweenthesolidandthermocouple.Agraphof theimpulseresponsefunctionsforsphericalthermocouplesisshowninFigure54. 5.1.3DesignofExperiment Theexperimentusedtodeterminetheimpulseresponsefunctionsisperformed insitu .Thethermocouplesarelocatedonthebottomofadiscusedinaquenching experiment.Thethermocouplesconsistof7weldedbeadEtypethermocouplesmade from26AWGwire.405mmdiameter.Thesethermocouplesarespacedevery45 withtheinitialthermocouplelocated1.59mmfromthecenterofthecopperdiscwiththe remainderofthethermocouplesplacedevery3.18mmradiallyfromitsneighbor.There areatotalof8thermocouplessilversolderedtothebackofthecopperdisc.Oneofthe thermocouplesisusedasagroundforthesystem.Adiagramofthecopperdiscusedin theexperimentsisshowninFigure55.Thediscwasinsulatedonallsidesexceptthe topbyaceramicinsulationmaterialfromCotronicsInc.knownasRescor750,aSiO 2 122 PAGE 123 Figure55.Diagramofthecopperdiscassembly. basedceramicwithathermaldiffusivityof8 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(7 m 2 /sandathermalconductivityof 0.58W/m 2 K. Oneofthechallengesfacedindeterminingtheimpulseresponsefunctionof thethermocouplesisaccuratelydeterminingtheinputtothethermocouples.Several differenttechniqueswereused.Oneoftheearlytechniquesistohaveasinglephase liquid,turbulentjetimpingeonthesurfaceofthediscafterithadbeenheatedto approximately90 C.Theheattransfercoefcientforsuchajetisdeterminedusingthe correlationofLiuetal.[11].However,oneoftheconditionsassumedforthecorrelation isthatthesurfaceheatuxremainsconstantthroughouttheexperiment;thisconditionis notmet,eveninaquasisteadysense.Theseexperimentsdidnotproducereliabledata. Anothermethodutilizedistousedryicetosimulateaconstanttemperatureboundary conditionatthesurfaceafterheatingthedisctoapproximately100 C.Whilethisis goodintheory,inpracticethesublimationofCO 2 wouldcauseabuildupofgasnear thesurfacewhichseemedtocausesomethermalresistance.Thenalexperiment 123 PAGE 124 Figure56.Illustrationoftheexperimentalsetup. attemptedistouseiceH 2 Otoproduceaconstantsurfacetemperature.Initiallya cylinderoficeformedwithinaStyrofoamcupmoldthatislargerindiameterthanthe copperdiscwasused;however,itwasrealizedafterexperimentsthattheicewould meltandleaveanimpressionofthecopperdiscintheice.Thismeltingoftheicewould causenonuniformcontactonthecopperdiscwhichwouldrendertheassumptionof aconstantboundarytemperatureinvalid.Toovercomethischallenge,acylinderof iceofthesamediameterasthecopperdisc,madeusingaPVCpipe,isused.The moldofthecylinderhousedtheiceandaplungerisusedtohelpensureevenand constantpressureontheicecylinder.Adiagramofthisexperimentalsetupisshownin Figure56. Theheattransferphysicsgoverningthisexperimentareonedimensional,transient conduction,andassuchthetemperatureexperiencedinsideofthecopperismodeled 124 PAGE 125 Figure57.Comparisonofbackwalltemperaturesofthecopperdisc,accountingforthe effectsofnonidealinsulation. asinEquation4b.Toassesstheeffectsoftheinsulationonthesystemdynamics, anumericalstudyiscarriedoutmodelingthediscasatwodimensionalslabandis comparedtotheresultsofanideal,onedimensionalslabwithperfectinsulation.The resultsofthisstudyareshowninFigure57.Theeffectsofnonidealinsulationare essentiallynegligibleforshorttimeshowever,thereissomeerrorpresentattimes approaching0.5secondsanditvarieswiththeradialpositionofthethermocouples.This deviationfromidealcouldcausesomeerrorinthedeterminationofimpulseresponse functions. Thenalissueindeterminingtheimpulseresponsefunctionsistodetermine whentheiceisactuallyrstpressedagainstthesurface,thereissometimedelayin theresponseofthethermocoupleswhichprecludesusingtherstsignoftemperature changingastheinitialpointintheexperiment.Toovercomethisissueamicrophoneis placedontheceramicinsulationusedanditsoutputsignalisrecorded.Whentheice makestheinitialcontactwiththecopperdiscadistinctsoundisheardasmicrofractures 125 PAGE 126 Figure58.Resultsoftwoseparateimpulseresponseexperiments,notethe repeatabilityoftheresultsforshorttimes. appearintheice.Usingtheoutputofthemicrophonethetimedelayforthethermal wavetopropagatetothethermocouplesisdeterminedtobeapproximately80ms. 5.1.4ExperimentalResults Oncethedataarerecorded,theinversedeconvolutiontechniquediscussedin Section4.6isusedtodeterminethethermocoupleimpulseresponsefunctions.The samplingrateusedintheexperimentsissetat8kHz.AswasmentionedinSection4.6 theuseofmultipleexperimentalrunsisnotnecessaryastheyallcloselyresembled eachother;thisisexpectedsincethesameinputsignalisessentiallyusedineach experiment.TherepeatabilityoftheexperimentsisclearlydemonstratedinFigure58 forshorttimeswheretheconstanttemperatureboundaryconditionismostlikelymet. Figure59showstheimpulseresponsefunctionasobtainedusingtheinverse deconvolutiontechniquefortherstchannelintheDAQ.Similarresultsareseenfor eachchannel,andintheinterestofavoidingredundancytheyarenotshown.Oneof theimmediatefeaturesnoticeableinthisgraphisthattheimpulseresponsefunctionis 126 PAGE 127 Figure59.Inversemethoddeconvolutionresults. nonzeroattheinitialtime;thussecondorderandhigherimpulseresponsefunctions canberuledout. 5.1.5ComparisontoEstablishedModels Todeterminethebestimpulseresponsemodelforthethermocouples,acomparison ismadetoarstorderresponse,a2exponentialresponse,aswellasthemodelof RabinandRittel.Thisisaccomplishedviatheuseofanonlinearleastsquarescurve tusingthefunction lsqcurvet availableinMatLab,whichusesatrustregionreective algorithm.Aftertheparametersofthemodelaredetermined,acomparisonofthe outputoftheimpulseresponsemodelwiththeactualtemperaturemeasuredduring experimentsismade;acomparisonofthemodelwiththeimpulseresponsefunction determinedviatheinversedeconvolutionalgorithmisalsomade. Toremaingeneral,therstordersystemmodelismodiedsuchthattwoparameters areusedinsteadofthesingletimeconstantoftherigorousmodelofEquation5,this newequationis h t = a exp )]TJ/F39 11.9552 Tf 9.299 0 Td [(bt 127 PAGE 128 Figure510.Comparisonoftherstorderresponsefunctiontothedeconvolved impulseresponsefunction. TheresultingparametersdeterminedfortherstchanneloftheDAQarea=0.532 andb=0.623;thisisequivalenttimeconstantofapproximately1.75seconds.A comparisonoftheresultingfunctionwiththatproducedviatheinversealgorithm isshowninFigure510.Itisevidentthatthebehaviorofthefunctioniscaptured marginallybyarstorderresponse.Thedifferencesbetweentheresponsemodel outputandtheexperimentaldataareshowninFigure511.Whiletheoveralltrend iscapturedquitewell,thereisnoticeableerrorbetweenthepredictedandmeasured temperatureresults. Toremaingeneral,themodelofRabinandRittelismodiedaswellsuchthatthree parametersareusedinsteadofthetwoinEquation5.Thisnewequationis h t = c exp )]TJ/F39 11.9552 Tf 9.299 0 Td [(dt n Theresultingparametersfortherstthermocouplechannelafterdeconvolutionarec= 1.775,d=0.216,andn=0.369assumingthattheradiusofthethermocoupleis0.5mm andusingthethermaldiffusivityofcopper.Acomparisonoftheresultingfunctionwith 128 PAGE 129 Figure511.Besttresultsusingarstorderimpulseresponsefunction. Figure512.Comparisonofthemodelof[114]tothedeconvolvedimpulseresponse function. thatproducedviatheinversealgorithmfortherstchannelisshowninFigure512.This modelcompareswellwiththeinversemethoddeconvolution.However,thereissome disagreementneart=0.Thedifferencesbetweentheresponsemodeloutputandthe experimentaldataareshowninFigure513.Thetrendiscapturedquitewell. 129 PAGE 130 Figure513.Besttresultsusingthemodelof[114]responsefunction. ComparisonoftheparameterdtoBinTable51showsthatthereissome departurefromthemodelofRabinandRittelastheyareofdifferentordersofmagnitude. Thevalueoftheexponent,nisofthesameorderofmagnitude.Itisnotedthatthe thermalconductivityratioforthethermocoupletothecopperdiscisontheorderof10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 welloutsidetherecommendedrange,andsothecomparisonisgeneralinnature. AsobservedinFigure513themodelofRabinandRittelproducesgoodresults despitethedifferencesinthecoefcientsofthecurvet.Therearesomeconditions intheexperimentsthatarenotaccountedforinthemodel.Namely,therewillbesome nitethermalresistancebetweenthethermocoupleandthecopperdisc,althoughit isbelievedtobesmallduetotheuseofsilversolderinattachingthethermocouples. Additionally,thethermocouplesintheexperimentsarenotfullyembeddedinthesolid, buttheyaresolderedtothebackofthecopperdisc.Thisassumptionmaynotbebad duetothefactthatthesilversolderwilltendtoencasethethermocouples. Thenalcomparisonisthatofthe2exponentialmodel.Theresultingmodel parametersfortherstchanneloftheDAQarea=0.658,b=0.402,c=0.228,and 130 PAGE 131 Figure514.Comparisonofthe2exponentialmodeltothedeconvolutionalgorithm results. d=2.961.Acomparisonoftheresultingfunctionwiththeresultsofthedeconvolution algorithmisshowninFigure514;theagreementoftheresultsisexcellent.These resultstranslateintogoodagreementbetweentheoutputmodelandtheexperimental dataasshowninFigure515. TheresultsofthecomparisonofthethreemodelsshowthatthemodelofRabinand Rittelandthe2exponentialmodelproducethebestresults.However,the2exponential modelisselectedasthebestduetothefactthatexcellentagreementisobserved betweenthecurvettedanddeconvolutionresults. 5.2InverseHeatTransferAlgorithmVerication Thepurposeofthisportionofthecurrentstudyistodevelopaninverseheat transferalgorithmtodeterminetheheattransfercoefcientofanimpingingjet.However, duetosensitivityissuesassociatedwithhavingsensorsplacedonthebacksurfaceof thetestsample,directestimationofthesurfaceheattransfercoefcientisdifcult,if notimpossible,forhighBiotnumbers.Assuchtheinverseheattransferalgorithmwas formulatedsuchthatthesurfacetemperatureisdeterminedandthesurfaceheatuxis 131 PAGE 132 Figure515.Besttresultsusingthe2exponentialmodel. thenestimated,afterwhichthesurfaceheattransfercoefcientisestimated.Inorderto determinetheaccuracyoftheproposedalgorithmaparametricstudyisconsidered. 5.2.1InverseQuenchingParametricStudySetup Thereareseveraldifferentfactorswhichcanaffecttheinverseheattransferstudy whichare:thematerialpropertiesofthedisci.e.thethermaldiffusivity,theimpulse responsefunctions,theheight,L,ofthedisc,theaspectratioofthediscL/R,theBiot numberdistribution,thenumberofmeasurementpointsonthebacksideofthedisc,and themagnitudeofthenoiseoftheDAQ.Differentvaluesforthesevariablesareselected fortheparametricstudyasdetailedbelow. Thesetupfortheinverseheattransferparametricstudywillsimulatethecopper discassemblyoutlinedinSection5.1.3.Thecopperdiscisataninitialconditionofzero indimensionlesstemperatureandthenthejetisinitiatedwithaspeciedBiotnumber distributionandanadiabaticwalltemperatureofunity.Theheattransfercoefcientofan impingingjetcanbemodeledasaGaussianfunctionasseenbelow 132 PAGE 133 Bi = Bi max 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(ratio exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(r 2 2 2 + ratio whereBi max isthemaximumBiotnumberseen,ristheradialcoordinate, isthe standarddeviationwhichaffectsthewidthoftheGaussianpeak,andratioistheratioof themaximumtominimumBiotnumber.Figure516showstheBiotnumberdistribution fordifferentvaluesofBi max ,whileFigure517showstheBiotnumberdistributionfor differentvaluesof ,inbothguresthevalueofratiois0.1.ThreedifferentBi max values of15,5,and0.5wereusedintheparametricstudy,theserepresentpeakheattransfer coefcientsof234,78,and7.8kW/m 2 K,respectively.Onepointofnoteisthatfor highervaluesof theBiotnumberdistributionapproachesthatofaconstantBiot number,makingtheproblembehaveonedimensional,whichisasimplerestimation problem.Thiseffectisalsoseenforhighervaluesoftheparameter,ratio.Toadequately determinetheeffectivenessoftheinverseheattransferalgorithmtheratioquantity wassetto0.1andformostnumericalexperiments wassetto0.1aswell,although studiesathighervaluesof werecarriedoutinordertoverifythatgoodresultscouldbe obtained. Theeffectsoftheimpulseresponsefunction,andthemagnitudeofthenoise arealsoexplored.WhileitwasfoundinSection5.1.5thatthe2exponentialmodel producedthebestresultsexperimentally,itisdifculttoassesstheeffectofvarying eachparameterofthemodel.Assuchtherstorderimpulseresponsefunctionin Equation5waschosentosimulatethethermocouplemeasurementdynamicsas itdoesafairjobofapproximatingthedynamics,anditonlycontainsoneconstantto vary.Thedifferenttimeconstantschosenfortheparametricstudyare0.1,1,and5 secondswhichallowforcomparisontobemadeforshortandlongtimedelaysofthe thermocouples. Todeterminetheeffectofthedischeightandaspectratioontheinverseheat transferproblem,themaximumradiusofthedisc,R,wassetto25.4mmandthree 133 PAGE 134 Figure516.BiotnumberdistributionshowingtheeffectsofBi max differentvaluesoftheheightwereselected:15,10,and5mmgivingaspectratiosof 0.591,0.394,and0.197,respectively.Whileperformingtheparametricstudyitwas foundthatforheightsof15mm,theinverseheattransferalgorithmwouldnotconverge; thusnoresultsfromthisheightarereportedinthestudy.Theeffectofthediscthermal diffusivity,theimpulseresponsefunction,andtheheightofthedisccanbecollapsed intoadimensionlesstimeconstantdenedbelow = L 2 Toassesstheeffectsofmeasurementnoise,whiteGaussiannoisewasadded tothesimulatedmeasurements.Themagnitudeofthenoiseindimensionlessspace dependsontwoseparatevariables,theinitialtemperatureandthemagnitudeofthe noiseintheDAQ.Thisrelationshipisshownbelow = DAQ T o 134 PAGE 135 Figure517.Biotnumberdistributionshowingtheeffectsof where isthedimensionlessnoise, DAQ isthestandarddeviationofthemeasurement noiseoftheDAQ,and T o istheinitialtemperature.Theinitialtemperaturewasset to150 CandtwodifferentstandarddeviationsoftheDAQwerechosen:0.2 C,the measurementnoiseseenintheDAQusedfordeterminingtheimpulseresponse function,and1 C. Inadditiontotheaboveconditions,otherconditionsweresetfortheparametric study.Thesamplingfrequencywassetto8kHzandthegridresolutionwassetto32 by32forL=10mm.OscillationsofheatuxsimilartotheonesseeninSection4.5.3 werepresentintheL=5mmcaseandwereeliminatedbyusingagridresolutionof 64by64.Thetotaltimedurationofthesimulatedexperimentswassetto30,000time steps,25,000ofwhichwereusedinthestoppingcriteria,exceptfortheBi max =0.5,L =5mmsimulationswhere60,000timestepswereused,50,000ofwhichwereusedin thestoppingcriteria.ThereasonforthisdisparityisthatthelowerBiotnumbercaused aslowerresponseinthemeasuredtemperature.Additionally,thenumberofsimulated 135 PAGE 136 measurementstakenwasselectedtobe8and16,themeasurementlocationswere denedbythefollowingformula r i meas = R i pts )]TJ/F22 11.9552 Tf 21.407 8.088 Td [(1 2 pts i =1,2,3,... M where r i meas isthelocationofmeasurementi,Ristheradiusofthedisc,ptsisthe numberofmeasurementpoints,andMisthetotalnumberofmeasurements. 5.2.2ErrorAssessmentMethods Inordertoquantifytheerrorbetweentheinverseheattransferalgorithmresults andthetrueBiotnumberanerrordenitionmustbemade.Thealgorithmistechnically designedtoestimatethesurfacetemperatureandthusasurfacetemperatureerror shouldbeusedtoestimatetheeffectiveness.Additionally,becausetheBiotnumberis thequantityofinterestitmustbeestimatedaswell.Theerrorassessmentchosenis thatofarootmeansquareerror,thermserrorequationforthesurfacetemperatureand Biotnumberdistributionisdenedbelow e rms = r R t R r act )]TJ/F25 11.9552 Tf 11.956 0 Td [( inv 2 r dr dt r R t R r act 2 r dr dt where iseitherthedimensionlesssurfacetemperatureorBiotnumber,thesubscript actdenotestheinputvariable,andthesubscriptinvdenotestheoutputvariablereturned bytheinversemethod. WhileperformingtheparametricstudyitwasfoundthattheestimationoftheBiot numbernear t =0 wasgrosslyinerror,beingseveralordersofmagnitudegreaterthan theactualBiotnumber.ThelargeBiotnumbersneartheinitialtimearenotbelievable. However,afterthisinitialspike,theBiotnumberestimatedwillgenerallymaintainanear constantvalueforaportionoftheresultsanditwasdecidedthattheBiotnumberrms errorwouldbeassessedsuchthatonlythisportionoftheBiotnumberdistributionwould beusedintheerrorcalculations,thisisshowninFigure518. 136 PAGE 137 Figure518.ComparisonoftheinverseresultsofBiotnumberatthecenterlineofthe discforBi max =15, =0.1.Theasterisksdenotewheretheresultsare truncated. Additionally,becauseofthenearconstantBiotnumberaftertheinitialtimeitwas decidedthattheBiotnumberwouldbeaveragedtemporallybetweenthetruncation points.ThisaverageBiotnumber,denotedasBi ave ,wasalsousedtodetermineanrms errorasdenedbelow e Bi ave rms = r R r Bi ave real )]TJ/F39 11.9552 Tf 11.955 0 Td [(Bi ave inv 2 r dr r R r Bi ave real 2 r dr 5.2.3ParametricStudyResults TheresultsfortheparametricstudyforL=10mmand DAQ =0.2 Carelistedin Table52.Itisclearlyevidentthattheinversemethodaccuratelydeterminesthesurface temperaturedistribution,whichitisdesignedtodo.Theaccuracyrangesfrom1to3 %anddoesnotseemtobeaffectedbyincreasingthenumberofmeasurementpoints. ThetruncatedBiotnumbererrorisnoticeablyhigher,rangingfrom15to48%forthe8 137 PAGE 138 Table52.RMSerrorsforL=10mm,and DAQ =0.2 C. e s rms e Bi rms truncatede Bi ave rms 0.1121.1235.6170.1121.1235.6170.1121.1235.617 M=8 150.0120.0180.0180.1530.1670.1760.1390.1530.162 Bi max 50.0130.0170.0190.2610.3040.3800.2510.2930.341 0.50.0150.0070.0140.4290.3960.4750.3300.3650.424 M=16 150.0120.0160.0200.1370.1510.3030.1290.0880.258 Bi max 50.0160.0170.0250.3120.2950.3740.3090.2770.344 0.50.0180.0100.0170.4970.6080.7160.4420.5940.654 measurementpointscase;theerrortendstoincreaseforalowerBi max .Contourplots oftheerrorsfortheL=10mm, DAQ =0.2 C,and8measurementpointsareshownin Figure519;intheinterestofneatnesscontourplotsoftheothercasesarenotshownin thisChapterandcanbefoundinAppendixB.For16measurementpoints,theerroris inthesamerangeforthetwohighestBi max cases,butisnoticeablyhigherforthelowest case.TheerrorforthetemporallyaveragedBiotnumbererrorisslightlylowerthanthat ofthetruncatedBiotnumbererrorforboth8and16measurementpoints.Fromthese dataitappearsthatforL=10mm,increasingthemeasurementpointsbeyond8has onlyanmarginalincreaseintheaccuracyoftheinversemethodfortheseconditions. DatafortheL=10mm, DAQ =1 C,and8measurementpointsisfoundin Table53.Themagnitudeforallthreeerrorsisnearthesamerangeasthatfoundforthe DAQ =0.2 Ccase;thustheinversemethodisinsensitivetonoiseofapproximately0.7 %.Simulationsfor16measurementpointswerenotperformedforthiscaseduetothe resultsseenfor DAQ =0.2 C. TheerrorresultsforL=5mmand DAQ =0.2 CisfoundinTable54.Itisseen thatthesurfacetemperatureerrorforboth8and16measurementpointsisverylow,1 %orless.However,theerrorforBi max =0.5isanorderofmagnitudegreaterthanthe errorfoundfortheothertwocases.TheerrorforthetruncatedBiotnumberdistribution 138 PAGE 139 a b c Figure519.ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints. a e Bi ave rms ,b e Bi rms truncated,andc e s rms Table53.RMSerrorsforL=10mm, DAQ =1 C,andM=8. e s rms e Bi rms truncatede Bi ave rms 0.1121.1235.6170.1121.1235.6170.1121.1235.617 150.0170.0220.0150.2040.2530.2230.1660.1650.173 Bi max 50.0170.0170.0190.3410.3020.3710.3280.2870.327 0.50.0120.0120.0150.5020.6970.5290.4080.3280.475 with8measurementpointsrangesfrom10to30%formaximumBiotnumbersof 5and15,buttheerrorisverymuchincreasedfortheBi max =0.5case,essentially demonstratingthattheestimatedBiotnumberdistributionisuseless.Thetruncated 139 PAGE 140 Table54.RMSerrorsforL=5mmand DAQ =0.2 C. e s rms e Bi rms truncatede Bi ave rms 0.4494.49322.470.4494.49322.470.4494.49322.47 M=8 150.0030.0070.0110.1430.0940.3160.0220.0520.108 Bi max 50.0030.0050.0060.1660.1560.1210.0300.0600.051 0.50.0120.0110.0141.3811.0061.0711.1140.9600.984 M=16 150.0020.0070.0100.1700.1330.2340.0300.0360.136 Bi max 50.0020.0040.0060.1030.1000.1580.0210.0390.094 0.50.0050.0060.0090.3340.2080.2910.1740.1440.190 Table55.RMSerrorforL=10mm, DAQ =0.2 C,M=8,andvariousvaluesof forthe Biotnumberdistribution. Bi max Bi e s rms e Bi rms e Bi ave rms truncated 150.255.620.0180.1300.039 50.255.620.0150.0700.023 50.55.620.0190.0560.021 50.755.620.0210.0900.033 Biotnumberresultsfor16measurementpointsshowanerrorrangeof10to33% fortheentirerangeofmaximumBiotnumbers,indicatingthatforlowBiotnumber distributionsanincreaseinthenumberofmeasurementpointsiswarranted.The temporallyaveragedBiotnumbererrorshowssimilartrends,asexpected. Table55showserrorresultsfortheBiotnumberdistributionwith 6 = 0.1.The magnitudeoftheerrorforsurfacetemperatureissimilartothatseeninthe =0.1case. However,thetruncatedBiotnumberandtemporallyaveragedBiotnumbershowalarge increaseinaccuracy.Thereasonforthischangewasmentionedpreviously.Witha largervalueof theproblembehavesmorelikeaonedimensionalestimationproblem, whichlesscomplex. Whendeterminingtheimpulseresponsefunctionsforthermocouplesthereis apotentialforerrorbetweentherealimpulseresponsefunctionofthesensorand 140 PAGE 141 Table56.RMSerrorforL=10mmandM=8withdifferentactualandsimulatedtime constants. Bi max Bi act sim e s rms e Bi rms e Bi ave rms truncated 150.10.1120.1070.0120.1480.133 150.10.1120.1180.0120.1470.129 150.11.1231.0110.0470.2900.262 150.11.1231.0670.0370.2410.207 150.15.6175.0550.0790.5990.568 150.15.6175.5050.0260.3930.346 150.255.6175.5050.0520.5040.426 150.255.6175.7290.0450.2020.135 50.11.1231.1800.0410.5040.473 50.11.1231.0670.0400.4750.459 theresultsofexperiments.Theseerrorscanbeproducedduetonotbeingableto accuratelydeterminewhentheexperimentwasinitiated,orerrorsduetotheboundary conditiononthetopsurfacenotbeingpreciselyknown.Todeterminetheaffectthese errorshaveondeterminingtheBiotnumberdistribution,simulationswerecarried outsuchthatthetimeconstantfortheinversealgorithm sim wouldbedifferentthan theactualtimeconstantforthesensors act .ThisstudywasrestrictedtotheL=10 mm, DAQ =0.2 Ccase,andtheerrorssimulatedrangedfrom1to10%.Theresults fromthisstudycanbefoundinTable56.Itisnoticedthatforsmalltimeconstants, thedifferenceisnotverynoticeable.However,asthemagnitudeofthetimeconstant increases,sodoesthethemagnitudeoftheerror.Foratimeconstantof5secondsthe resultsareessentiallyunusable.Clearlycaremustbetakentoensurethattheimpulse responsefunctionsareestimatedaccurately,especiallywhenthetimeconstantsare large. 5.2.4HeatLoss/GainEffects Whenperformingatransientjetimpingementheattransferexperimenttheeffectsof nonidealinsulationwillbepresent,asbrieydiscussedinSection5.1.3.Theseeffects willcausethedisctogainheatfromtheinsulationafterthejetisinitiated,withmore 141 PAGE 142 heatenteringonthebottomnearthemaximumradialdistanceduetotherebeingmore insulationtouchingthediscinthatarea.Whilethisheatgainisverylowinmagnitude andcausesonaslighterrorapproximately3%att=1seconditwilltendtohave alargerimpactonthedeterminationofthesurfacetemperatureandhenceheatux andBiotnumber.Theseeffectshavebeendemonstratedbysimulatingthediscbeing ataninitialdimensionlesstemperatureofzeroandhavingasurfacetemperatureof unitybeingappliedtoitandrecordingthetemperatureat8locationsonthebackside ofthedisc;thissimulationincludestheceramicinsulationandallowsheatexchange betweenitandthecopperdisc.Thesimulatedmeasurementsaretheninputintorst orderimpulseresponsemodel,andwhiteGaussiannoiseof0.2 Cthedimensional initialtemperatureis150 Cisadded.Thesedataaretheninputintoaninverseheat transferalgorithmwhichassumesidealinsulation. Theerrorresultsforthesurfacetemperatureare0.050,0.044,and0.047fortime constantsof0.1,1,and5seconds,respectively.Whilethiserrorissmalltherealeffect liesintheestimationoftheheatux,whichhasanonuniformdistribution.Figure520 showsthedifferencebetweentheinverseandexactsolutions.Clearlytheerrorwill producenoticeableerrorinthecalculationofBiotnumber. 5.2.5EffectivenessoftheInverseHeatTransferAlgorithm Allofthemajorfactorsaffectingtheinverseheattransferhavebeenexploredand quantied.Fromtheresultsitisseenthatfortimesneart=0thereissignicanterrorin thereturnedsurfacetemperaturewhichiscarriedoverinthedeterminationofsurface heatuxandBiotnumber.Thisshowsthatthedevelopedheattransferalgorithmwillnot beeffectiveatdetectingphasechangeheattransferneartheinitialinstantintime,where itismostlikelytooccur,withoutmodication.Apossiblemethodtoovercomesomeof theseproblemswouldbetoinsertthethermocouplesintothethediscandhavethem locatedclosertothetopsurface.However,caremustbetakentoensurethatanyeffects ofthethermocoupleinsertionareeitherminimizedorareexplicitlytakenintoaccountin 142 PAGE 143 Figure520.Comparisonoftheresultsbetweenexactsolutionandinverseresultsfor thesurfaceheatuxduetoastepchangeinsurfacetemperature.Time= 0.375seconds, =1second. theinverseheattransferalgorithm.Whenthethermocouplesarelocatedclosertothe topsurfacethentheproblemcanalsobeformulatedintermsofanunknownsurface heatux,whichwillprovideamoreaccurateestimateofthesurfaceheatux,andboth inverseproblemscanbesolved.TheBiotnumbercanthenbedeterminedusingthis moreaccurateinformation. Theeffectsduetoerrorsintheestimatedimpulseresponsefunctionwereexplored anditisimperativethattheybedeterminedaccurately,especiallyifthetimeconstant ofthesystemisabove1secondorelsegrosserrorwillengulftheresults.Theeffects duetothemagnitudeofthenoisepresentwerefoundtobenegligibleformagnitudes approximatingthosefoundinalaboratorysetting.Thenumberofsensorsdoesnot seemtobeacontributingfactorunlessthediscisverythinhasasmallL/Rratio.The timeconstantofthemeasurementsystemdoesnothaveamajoreffectontheaccuracy ofthemethodunlessithassomeerrorinitsestimate. 143 PAGE 144 Figure521.CenterlineNu D forZ/D=6.0andRe D =7.25 10 5 asreturnedfromthe inverseheattransferalgorithm. Measurementsforthesupersonicimpingingjetfacilityweretakenandanattempt wasmadetodeterminetheBiotnumberdistribution.Figure521showsresultsforthe centerlineNu D withZ/D=6.0andRe D =7.25 10 5 ,Figure522showsthespatial distributionofNu D forthesameexperiment,noticethatduetotheheatgainedfrom insulationalongthesideofthedisccausesalargeovershootnearr =1.Similarresults werefoundinallexperimentsperformedandtheestimatesofNusseltnumberare grosslyinerror. 5.3Summary InthisChapterthethermocoupledynamicswereinvestigated.Fourseparate modelsforthemeasurementdynamicswereexplored:rstorder,secondorder,Rabin andRittel,andthetwoexponentialmodelwereintroduced.Anexperimentalsetup todeterminetheproperimpulseresponsemodelwasconductedandthemodelof RabinandRittelandthetwoexponentialordermodelproducedexcellentresultsby comparingwellbetweenthetheoreticaloutputandtheexperimentalmeasurement.The 144 PAGE 145 Figure522.SpatialdistributionofNu D forZ/D=6.0,Re D =7.25 10 5 ,andt =2.11as returnedfromtheinverseheattransferalgorithm. twoexponentialordermodelwaschosenasthebestmodeldoitsagreementwiththe impulseresponsefunctionasdeterminedbytheinversedeconvolutionmethod. Aparametricstudyoftheinversemethodwasconductedtoexploretheeffects of:discheight,theimpulseresponsefunction,themagnitudeandshapeoftheBiot numberdistribution,thenumberoftemperaturesensorsused,andthemagnitude ofthenoisefromthedataacquisitionsystem.Theerrorwasassessedusingthe rootmeansquareerrorofthesurfacetemperaturedistributionaswellasatruncated BiotnumberdistributionandatemporallyaveragedBiotnumber.Itwasfoundthat increasingthenumberofthermocouplesforadischeightof10mmhadlittleinuence ontheaccuracyofthealgorithm.Foradischeightof5mmitwasseenthatincreasing thenumberofthermocoupleswoulddecreasetheerroroftheinversemethod.However, estimatingBiotnumberdistributionwithasmallmagnitudeprovedtohavesignicant error. Therelativeslownessofthesensordynamics,characterizedbyatimeconstant, didnotseemtohaveaneffectontheaccuracyoftheinversealgorithm.However,itwas 145 PAGE 146 seenthatforslowresponsetimes,asmallamountoferrorinthetimeconstantestimate willproducelargeerrorsintheBiotnumberdistribution.Thiseffectwasnotpresentin thecaseofsmalltimeconstants,itwasseentohaveanegligibleeffect.Theeffectofthe dataacquisitionsystemnoisehadnoeffectatmagnitudesexperiencedinalaboratory setting. Theeffectsofheatgainontheinverseproblemwereexplored.Itwasfoundthat althoughtheerrorinthesurfacetemperatureestimatewaslowapproxiamtely5%the errorinthesubsequentsurfaceheatuxcalculationwasverynoticeable.Measurements performedonthesupersonicjetimpingementfacilitywereunabletoproducereliable estimatesoftheBiotnumberdistribution. 146 PAGE 147 CHAPTER6 CONCLUSIONS Amultiphasesupersonicimpingingjetfacilityforthermalmanagementhasbeen constructed.Thefacilityoperatesbytakinghighpressureairandreducingittoa pressurebetween1.0and2.4MPawhereitactsontopofawaterstoragetankand passesthroughamassowmeterandamixingchamber.Differentialpressurebetween thetopofthewaterstoragetankandthemixingchamberforcesthewaterthrough aregulatingoricewhereitbecomesmixedwiththeairinthemixingchamber;the resultingmixtureisthenexpandedthroughaconvergingdivergingnozzle,withadesign Machnumberof3.24.Itisthendirectedtoasurfacewhereitimpingesandremoves heat.Thefacilitysubsystemsinclude:airstorage,waterstorageandowcontrol, airpressurecontrol,airmassowmeasuring,convergingdivergingnozzle,anddata acquisition.Temperature,pressure,andMachnumberwerecalculatedthroughout thefacilityusingonedimensionalgasdynamicrelations.Itwasfoundthatatnormal operatingconditionsnoshockwavewouldbepresentintheconvergingdivergingnozzle andthustheexitingair/liquidmixtureisatsupersonicspeeds.However,becausethe operationalsupplypressureislowerthanideal,thenozzleoperatesinanoverexpanded manner. Steadystateheattransfermeasurementswereperformedfortheimpinging jetfacility.Theseexperimentswereperformedbyallowingthejettoimpingeon anichromestripwithcurrentpassingthroughitandthermocouplesrecordingthe backwalltemperatures.Thedatawerethentimeaveragedtoeliminatenoiseand turbulentuctuations.Bothsinglephaseandmultiphaseexperimentswereperformed inordertoassesstheheatremovalcapabilitiesofthemultiphasejetandtoquantify heattransferenhancementwiththeadditionofthesmallamountofdispersedliquid droplets.Oncethedropletsimpingeonthesurface,athinliquidlmformsontopofthe surface.Theheattransfercharacteristicsofthejetaredifferentneartheimpingement 147 PAGE 148 zone,wheredropletimpactdominates.Awayfromtheimpingementzone,thinlm dynamicsdominatetheheattransferprocess.Thepeakheattransfercoefcientsfor themultiphasejetexceed200,000W/m 2 Knearthecenterline,matchingsomeof thehighestheattransferratesreportedintheliterature,whilesimultaneouslyhaving asignicantlyreducedliquidowrate.Thisreducedowratecanbeadvantageousin someindustrialsettings. ItwasfoundthatincreasingtheReynoldsnumberofthejetandtheliquidmass fractionofwaterincreasestheNusseltnumberofthejetnearthecenterline.These effectsarediminishedfarremovedfromtheimpingementzone.Nozzleheightisseen tohaveaslighteffectoftheNusseltnumberneartheinteriorofthejet,butnodenite trendisapparent,andisseentohavenosignicantimpactawayformthecenterline. ThesinglephaseimpingingjetNusseltnumberscomparewellwithdataintheliterature. Comparisonbetweenthesingleandmultiphasejetsshowsthattheadditionoftheliquid dropletstothejetenhancesheattransferbyanorderofmagnitudeintheinteriorofthe jetandbyafactoroftwotoveawayformthecenterline. Thereisnoevidenceofsignicantphasechangeheattransferoccurringduringthe jetimpingementstudies.Analysisofthesaturatedhumidityrationearthecenterlineof thejetshowsthatlittleevaporationispossibleanditseffectsareessentiallynegligible. Determinationofthehumidityratiofarremovedformthecenterlineofthejetisnot possibleduetothelackofinformationregardingthesurfacetemperatureofthelm. However,theNusseltnumberinthisregionremainsessentiallyconstant;signicant evaporationwouldresultinincreasedNusseltnumberinthisregion.Thelackofthistype oftrendsupportsthecontentionthatthereisnosignicantphasechangeheattransfer takingplace.Theuseofdropletimpacttheorysuggeststhatheattransferenhancement inthejetinteriorduetophasechangemaynotbepossible.Thesaturationpressure issignicantlyelevatedduetothehighimpactvelocityofthedropletsandwillexceed thecriticalpressureforthenominalexitvelocityofthenozzle.Thusanyphasechange 148 PAGE 149 occurringwouldnotresultinadditionalheatremovalasthelatentheatofvaporization abovethecriticalpressureiszero. Aninverseheattransferalgorithmusingtheconjugategradientmethodwithadjoint problemwasdevelopedwhichcandeterminetheNusseltnumberforanimpinging jet.Thismethodexplicitlytakesintoaccountsensordynamics,afeaturethatisnot presentininversemethodsfoundintheliterature.Aproceduretodeterminetheimpulse responsefunctionofthermocouplesattachedtothesurfaceofasolidwasdeveloped usingicetocauseastepchangeinsurfacetemperature.Theresultingdataisthen inputintoadeconvolutionalgorithmwhichreturnstheimpulseresponsefunctionviaan inversemethod.Thisallowsinsightintotheunderlyingfunctionalformandallowsthe properimpulseresponsefunctionmodeltobefound.Fourdifferentimpulseresponse functionmodelswereexplored:arstordermodel,asecondordermodel,themodel ofRabinandRittel,andthetwoexponentialmodel.Itwasfoundthatthemodelof RabinandRittelandthetwoexponentialmodelproducedthebestagreementwhen comparedtoexperimentalresults.However,duetothefactthatthetwoexponential modelproducedbetteragreementbetweentheimpulseresponsefunctionsdetermined viacurvetandinversealgorithm,itwaschosenasthebestmodelforthesurface mountedthermocouplesusedinthoseexperiments. Aparametricstudywasperformedusingsimulateddatatoexploretheeffectsofthe discheightandaspectratio,themagnitudeandshapeoftheBiotnumberdistribution, thenumberofsensorsused,theimpulseresponsefunction,andthemagnitudeof thenoiseofthedataacquisitionsystemontheerrorassociatedwiththeIHTmethod. Toassesstheerrorintheinverseheattransferalgorithmthreedifferenterrorswere considered,thermserrorbetweeninputsurfacetemperatureandthatreturnedbythe inversemethod,thermserrorbetweentheinputBiotnumberandthatreturnedbythe inversemethodduringatruncatedtimedomainwhenthebehaviorofthereturnedBiot numberwasnearlyconstant,andthermserrorbetweentheinputBiotnumberandthe 149 PAGE 150 temporallyaverageBiotnumberoverthesametruncatedtimedomain.Theinverseheat transferalgorithmisspecicallydesignedtoestimatethesurfacetemperatureandas suchtheerrorinitsestimatewasverysmall,ontheorderofafewpercent.Theerror forestimatingBiotnumberwasnoticeablyhigher,rangingfrom10%toover80%in somecases.Mostofthefactorsinvestigatedappeartohavelittleeffectontheinverse heattransfermethod.However,themagnitudeoftheBiotnumberdistributionhada largeeffect.AlowerBiotnumberproducessmallchangesinmeasuredtemperatureand thusisadifcultestimationproblemandanyerrorinthesurfacetemperatureestimation iscarriedoverandampliedinthesurfaceheatuxcalculation.Errorsintheimpulse responsefunctionwereseentohavealargeeffectincertaincases.Iftheoveralldelay intheimpulseresponsefunctionisshort,thenthemethodcantoleratesomeerror. However,iftheoveralldelayisashighas5seconds,thenevenanerrorassmallasone percentcanrendertheresultsoftheinverseheattransferalgorithmuseless. Finallytheeffectsofheatloss/gainfrominsulationontheimpingementtarget wereinvestigated.Thedifferenceintemperatureatthesimulatedmeasurementpoints betweenanidealizeddiscandonewithheattransferto/fromitsinsulationwassmall, approximately3%.However,thissmallerrorismagniedbytheinversealgorithm, causingittooverestimatetheheatuxneartheedgeofthedisc.Thesendingwere veriedbyperformingexperimentsusingthemultiphasejetimpingementfacility. Inorderfortheinversemethodtobeeffective,theheatexchangewiththe insulationmustbetakenintoaccount.Theeffectsoftheheatexchangecouldalso belessenedbylocatingthetemperaturesensorsclosertothetopsurface.However, insertingthermocouplesinsideofthediscislikelytodistorttheheatowandsosome compromisemustbemade. 150 PAGE 151 APPENDIXA COMPLETESTEADYSTATETWOPHASEHEATTRANSFERJETRESULTS Therearemanyheattransferdatasetsforthesteadystateexperimentsdiscussed inChapter3whicharenotpresentedduetospaceconsiderationsandbecauseallof theresultsfollowthetrendsdiscussedtherein.Forthesakeofcompleteness,theheat transferresultsforthesteadystatetwophaseimpingingjetresultsarepresentedinthis Appendix. ForFiguresA1throughA3thesameoriceisusedforeachexperiment,thus keepingthewatermassfractionnominallyconstant.JetReynoldsnumberisheld constantaswell,andthenozzlespacingisvaried. ForFiguresA4throughA7thenozzlespacingandReynoldsnumberareconstant. Theoriceischangedtovarywatermassfraction. ForFiguresA8throughA10thesameoriceisusedforeachexperiment,thus keepingthewatermassfractionandjetReynoldsnumbernominallyconstant,andthe nozzlespacingisvaried. ForFiguresA11throughA14thenozzlespacingandReynoldsnumberare constant,andtheoriceischangedtovarywatermassfraction. 151 PAGE 152 a b c d FigureA1.TwophaseNu D resultsforvariousnozzleheighttodiameterratios.aw= 0.0205,Re D =4.54 10 5 ,bw=0.0240,Re D =4.47 10 5 ,cw=0.0357, Re D =4.42 10 5 ,anddw=0.0375,Re D =4.42 10 5 152 PAGE 153 a b c d FigureA2.TwophaseNu D resultsforvariousnozzleheighttodiameterratios.aw= 0.0156,Re D =7.34 10 5 ,bw=0.0189,Re D =7.29 10 5 ,cw=0.0273, Re D =7.23 10 5 ,anddw=0.0290,Re D =7.24 10 5 153 PAGE 154 a b c d FigureA3.TwophaseNu D resultsforvariousnozzleheighttodiameterratios.aw= 0.0131,Re D =1.02 10 6 ,bw=0.0163,Re D =1.01 10 6 ,cw=0.0234, Re D =1.02 10 6 ,anddw=0.0248,Re D =1.01 10 6 154 PAGE 155 a b c FigureA4.TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=2.0.a Re D =4.42 10 5 ,bRe D =7.20 10 5 ,andcRe D =1.01 10 6 155 PAGE 156 a b c FigureA5.TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=4.0.a Re D =4.43 10 5 ,bRe D =7.35 10 5 ,andcRe D =1.03 10 6 156 PAGE 157 a b c FigureA6.TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=4.0.a Re D =4.45 10 5 ,bRe D =7.24 10 5 ,andcRe D =1.02 10 6 157 PAGE 158 a b c FigureA7.TwophaseNu D resultsforvariousliquidmassfractionsandZ/D=6.0.a Re D =4.56 10 5 ,bRe D =7.31 10 5 ,andcRe D =1.01 10 6 158 PAGE 159 a b c d FigureA8.Twophaseenhancementratioresultsforvariousnozzleheighttodiameter ratios.aw=0.0205,Re D =4.54 10 5 ,bw=0.0240,Re D =4.47 10 5 cw=0.0357,Re D =4.42 10 5 ,anddw=0.0375,Re D =4.42 10 5 159 PAGE 160 a b c d FigureA9.Twophaseenhancementratioresultsforvariousnozzleheighttodiameter ratios.aw=0.0156,Re D =7.34 10 5 ,bw=0.0189,Re D =7.29 10 5 cw=0.0273,Re D =7.23 10 5 ,anddw=0.0290,Re D =7.24 10 5 160 PAGE 161 a b c d FigureA10.Twophaseenhancementratioresultsforvariousnozzleheighttodiameter ratios.aw=0.0131,Re D =1.02 10 6 ,bw=0.0163,Re D =1.01 10 6 cw=0.0234,Re D =1.02 10 6 ,anddw=0.0248,Re D =1.01 10 6 161 PAGE 162 a b c FigureA11.TwophaseenhancementresultsforvariousliquidmassfractionsandZ/D =2.0.aRe D =4.42 10 5 ,bRe D =7.20 10 5 ,andcRe D =1.01 10 6 162 PAGE 163 a b c FigureA12.Twophaseenhancementratioresultsforvariousliquidmassfractionsand Z/D=4.0.aRe D =4.43 10 5 ,bRe D =7.35 10 5 ,andcRe D =1.03 10 6 163 PAGE 164 a b c FigureA13.Twophaseenhancementratioresultsforvariousliquidmassfractionsand Z/D=4.0.aRe D =4.45 10 5 ,bRe D =7.24 10 5 ,andcRe D =1.02 10 6 164 PAGE 165 a b c FigureA14.Twophaseenhancementratioresultsforvariousliquidmassfractionsand Z/D=6.0.aRe D =4.56 10 5 ,bRe D =7.31 10 5 ,andcRe D =1.01 10 6 165 PAGE 166 APPENDIXB COMPLETEINVERSEHEATTRANSFERALGORITHMERRORASSESSMENT CONTOURPLOTS Therearemanydatasetsforthethreedifferenterrorsusedtoassessthe effectivenessoftheinverseheattransferalgorithmandassuchallofthemwere notincludedinChapter5.Intheinterestofcompletenesstheyareincludedinthis Appendix.FigureB1showstheerrorcontoursfortheL=5mm, DAQ =0.2 C,and8 measurementpointscase.FigureB2showstheerrorcontoursfortheL=5mm, DAQ = 0.2 C,and16measurementpointscase.FigureB3showstheerrorcontoursfortheL =10mm, DAQ =0.2 C,and8measurementpointscase.Lastly,FigureB5showsthe errorcontoursfortheL=10mm, DAQ =1 C,and8measurementpointscase. 166 PAGE 167 a b c FigureB1.ErrorContoursforL=5mm, DAQ =0.2 C,and8measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 167 PAGE 168 a b c FigureB2.ErrorContoursforL=5mm, DAQ =0.2 C,and16measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 168 PAGE 169 a b c FigureB3.ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 169 PAGE 170 a b c FigureB4.ErrorContoursforL=10mm, DAQ =0.2 C,and16measurementpoints. a e Bi ave rms ,b e Bi rms truncated,andc e s rms 170 PAGE 171 a b c FigureB5.ErrorContoursforL=10mm, DAQ =1 C,and8measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 171 PAGE 172 APPENDIXC IMAGESOFORIFICESUSEDDURINGEXPERIMENTS Duringoperationofthejetfacilityitwasnoticedthatthe0.38mmoriceprovided alowerliquidowratethanthe0.33mmorice.Afterexaminingthedifferentorices underanopticalmicroscopeitwasdiscoveredthatthe0.38and0.51mmoriceswere notperfectlycircular.Thiseccentricityisverynoticeableinthe0.38mmoriceand mayresultindifferentspraycharacteristics,whichcouldhaveaffectedtheheattransfer results.Theeccentricityisnotassevereinthe0.51mmoriceanddidnotseemto affecttheheattransferresults.FiguresC1throughC4showimagesoftheorices under4Xmagnication. FigureC1.The0.33mmorice. 172 PAGE 173 FigureC2.The0.37mmorice,notethehighdegreeofeccentricityintheorice FigureC3.The0.41mmorice. 173 PAGE 174 FigureC4.The0.51mmorice,notetheeccentricityintheorice 174 PAGE 175 REFERENCES [1]X.Liu,J.H.LienhardV,ExtremelyHighHeatFluxesbeneathImpingingLiquid Jets,JournalofHeatTransfer115472. [2]C.H.Oh,J.H.LienhardV,H.F.Younis,R.S.Dahbura,D.Michels,Liquid JetArrayCoolingModulesforHighHeatFluxes,AIChEJournal44 769. [3]J.H.LienhardV,J.Hadeler,HighHeatFluxCoolingbyLiquidJetArrayModules, ChemicalEngineeringandTechnology22867. [4]J.F.Klausner,R.Mei,S.Near,R.Stith,TwoPhaseJetImpingementfor NonVolatileResidueRemoval,ProceedingsoftheIMECHEPartEJournal ofProcessMechanicalEngineering212271. [5]K.Hiemenz,DieGrenzschichtanEinemindenGleichf ormigenFl ussigkeitsstrom EingetauchtenGeradenKreiszylinder,DinglerspolytechnischesJournal326 321. [6]F.Homann,Dereinussgrosserz ahigkeitbeiderstr omungumdenzylinderund umdiekugel,Zeitschriftf urAngewandteMathematikundMechanik16 153. [7]C.Y.Wang,StagnationowswithSlip:ExactSolutionsoftheNavierStokes Equations,ZeitschriftfrAngewandteMathematikundPhysikZAMP54 184.10.1007/PL00012632. [8]H.Schlichting,K.Gersten,BoundaryLayerTheory,Springer,8 th edition,2000. [9]A.A.Kendoush,TheoryofStagnationRegionHeatandMassTransfertoFluid JetsImpingingNormallyonSolidSurfaces,ChemicalEngineeringandProcessing 37223228. [10]X.Liu,J.H.LienhardV,LiquidJetImpingementHeatTransferonaUniformFlux Surface,HeatTransferPhenomenainRadiation,CombustionandFiresASME HTD106523. [11]X.Liu,J.H.LienhardV,J.S.Lombara,ConvectiveHeatTransferbyImpingement ofCircularLiquidJets,JournalofHeatTransfer113571. [12]X.Wang,Z.Dagan,L.Jiji,HeatTransferbetweenaCircularFreeImpingingJet andaSolidSurfacewithNonUniformWallTemperatureorWallHeatFlux. SolutionfortheStagnationRegion,InternationalJournalofHeatandMass Transfer3213511360. [13]X.Wang,Z.Dagan,L.Jiji,HeatTransferbetweenaCircularFreeImpingingJet andaSolidSurfacewithNonUniformWallTemperatureorWallHeatFlux. 175 PAGE 176 SolutionfortheBoundaryLayerRegion,InternationalJournalofHeatandMass Transfer3213611371. [14]X.S.Wang,Z.Dagan,L.M.Jiji,ConjugateHeatTransferBetweenaLaminar ImpingingLiquidJetandaSolidDisk,InternationalJournalofHeatandMass Transfer3221892197. [15]D.H.Lee,J.Song,M.C.Jo,TheEffectsofNozzleDiameteronImpingingJet HeatTransferandFluidFlow,JournalofHeatTransfer126554. [16]J.Baonga,H.LouahliaGualous,M.Imbert,ExperimentalStudyofthe HydrodynamicandHeatTransferofFreeLiquidJetImpingingaFlatCircular HeatedDisk,AppliedThermalEngineering2611251138. [17]J.H.LienhardV,X.Liu,L.A.Gabour,SplatteringandheatTransferduring ImpingementofaTurbulentLiquidJet,JournalofHeatTransfer114 362. [18]D.E.Hall,F.P.Incropera,R.Viskanta,JetImpingementBoilingfromaCircular FreeSurfaceJetDuringQuenching:Part1SinglePhaseJet,JournalofHeat Transfer123901. [19]K.Jambunathan,E.Lai,M.Moss,B.Button,AReviewofHeatTransferDatafor SingleCircularJetImpingement,InternationalJournalofHeatandFluidFlow13 106115. [20]D.Michels,J.Hadeler,J.H.Lienhard,HighHeatFluxResistanceHeaters fromVPSandHVOFThermalSpraying,ExperimentalHeatTransfer11 341. [21]J.H.LienhardV,D.Napolitano,ThermalStressLimitsofPlatesSubjectedto ExtremelyHighHeatFlux,ProceedingsoftheASMEHeatTransferDivision Volume2ASMEHTDVolume33323. [22]J.H.LienhardV,AnnualReviewofHeatTransfer,BegellHouse,pp.199. [23]B.Webb,C.F.Ma,AdvancesinHeatTransfer,volume26,Elsevier,pp.105217. [24]H.Martin,AdvancesinHeatTransfer,volume13,Elsevier,pp.1. [25]I.A.Kopchikov,G.I.Voronin,T.A.Kolach,D.A.Labuntsov,P.D.Lebedev,Liquid BoilinginaThinFilm,InternationalJournalofHeatandMassTransfer12 791. [26]K.A.Estes,I.Mudawar,CorrelationofSauterMeanDiameterandCriticalHeat FluxforSprayCoolingofSmallSurfaces,InternationalJournalofHeatandMass Transfer3829852996. 176 PAGE 177 [27]K.M.Graham,S.Ramadhyani,ExperimentalandTheoreticalStudiesofMistJet ImpingementCooling,JournalofHeatTransfer118343. [28]L.Bolle,J.Moureau,SprayCoolingofHotSurfaces,MulitphaseScienceand Technology11. [29]V.I.Zapryagaev,A.N.Kudryavtsev,A.V.Lokotoko,A.V.Solotchin,A.A.Pavlov, H.A.,AnExperimentalandNumericalStudyofaSupersonicJetShockWave Structure,in:ProceedingsofWestEastHighSpeedFlowFields:Aerospace ApplicationsfromHighSubsonictoHypersonicRegime,Marseilles,France,pp. 346. [30]C.d.Donaldson,R.S.Snedeker,AStudyofFreeJetImpingement.Part1.Mean PropertiesofFreeandImpingingJets,JournalofFluidMechanics45 281. [31]D.C.Pack,OntheFormationofShockWavesinSupersonicGasJets,The QuarterlyJournalofMechanicsandAppliedMathematics11. [32]D.C.Pack,TheReexionandDiffractionofShockWaves,JournalofFluid Mechanics18549. [33]C.W.Chu,CompatibilityRelationsandaGeneralizedFiniteDifference ApproximationforThreeDimensionalSteadySupersonicFlow,AIAAJournal 5493. [34]J.H.Gummer,B.L.Hunt,TheImpingementofaUniform,Axisymmetric, SupersonicJetonaPerpendicularFlatPlate,AeronauticalQuarterly22 403. [35]J.C.Carling,B.L.Hunt,TheNearWallJetofaNormallyImpinging,Uniform, Axisymmetric,SupersonicJet,JournalofFluidMechanics66159. [36]A.Powell,OntheMechanismofChokedJetNoise,ProceedingsofthePhysical Society.SectionB661039. [37]A.Powell,OnEdgeTonesandAssociatedPhenomena,Acoustica3 233. [38]C.K.W.Tam,K.K.Ahuja,TheoreticalModelofDiscreteToneGenerationby ImpingingJets,JournalofFluidMechanics21467. [39]A.Krothapalli,DiscreteTonesGeneratedbyanImpingingUnderexpanded RectangularJet,AIAAJournal231910. [40]F.S.Alvi,W.W.Bower,J.A.Ladd,ExperimentalandComputationalInvestigation ofSupersonicImpingingJets,AIAAJournal40599. 177 PAGE 178 [41]K.Klinkov,A.ErdiBetchi,M.Rein,BehaviorofSupersonicOverexpanded JetsImpingingonPlates,in:H.J.Rath,C.Holze,H.J.Heinemann,R.Henke, H.HnlingerEds.,NewResultsinNumericalandExperimentalFluidMechanics V,volume92of NotesonNumericalFluidMechanicsandMultidisciplinaryDesign SpringerBerlin/Heidelberg,2006,pp.168. [42]S.c.Baek,S.B.Kwon,StudyofModeratelyUnderexpandedSupersonicMoist AirJets,AIAAJournal441624. [43]M.M.AshrafulAlam,S.Matsuo,T.Setoguchi,EffectofNonEquilibrium HomogeneousCondensationontheSelfInducedFlowOscillationofSupersonic ImpingingJets,InternationalJournalofThermalSciences4920782092. [44]M.M.AshrafulAlam,S.Matsuo,T.Setoguchi,SupersonicMoistAirJet ImpingementsonFlatSurface,JournalofThermalScience1951. 10.1007/s1163001000513. [45]Y.Otobe,S.Matsuo,M.Tanaka,H.Kashimura,H.D.Kim,T.Setoguchi,Astudy ofunderExpandedMoistAirJetImpingingonaFlatPlate,JournalofThermal Science14334.10.1007/s1163000500547. [46]C.D.Donaldson,R.S.Snedeker,D.P.Margolis,AStudyofFreeJet Impingement.Part2.FreeJetTurbulentStructureandImpingementHeatTransfer, JournalofFluidMechanics45477. [47]M.D.Fox,M.Kurosaka,SupersonicCoolingbyShockVortexInteraction,Journal ofFluidMechanics308363. [48]B.G.Kim,M.S.Yu,Y.I.Cho,H.H.Cho,DistributionsofRecoveryTemperature onFlatPlatebyUnderexpandedSupersonicImpingingJet,Journalof ThermophysicsandHeatTransfer16425. [49]H.W.Liepmann,A.Roshko,ElementsofGasdynamics,JohnWiley&SonsInc, 1957. [50]J.E.John,T.G.Keith,GasDynamics,PearsonPrenticeHall,3 rd edition,2006. [51]M.Rahimi,I.Owen,J.Mistry,ImpingementHeatTransferinanUnderExpanded AxisymmetricAirJet,InternationalJournalofHeatandMassTransfer46 263. [52]M.S.Yu,B.G.Kim,H.H.Cho,HeatTransferonFlatSurfaceImpingedbyan UnderexpandedSonicJet,Journalofthermophysicsandheattransfer19 448. [53]R.Gardon,J.Cobonpue,HeatTransferBetweenaFlatPlateandJetsofAir Impingingonit,in:InternationalDevelopmentsinHeatTransfer,ASME,NewYork, pp.454. 178 PAGE 179 [54]S.J.Kline,F.A.McClintock,DescribingUncertaintiesinSingleSample Experiments,MechanicalEngineering13. [55]L.F.Richardson,TheApproximateArithmeticalSolutionbyFiniteDifferences ofPhysicalProblemsIncludingDifferentialEquations,withanApplicationtothe StressesinaMasonryAam,PhilosophicalTransactionsoftheRoyalSocietyof London.SeriesA210307. [56]L.F.Richardson,J.A.Gaunt,TheDeferredApproachtotheLimit,Philosophical TransactionsoftheRoyalSocietyofLondon.SeriesA22629961. [57]W.Wagner,A.Pru,TheIAPWSFormulation1995fortheThermodynamic PropertiesofOrdinaryWaterSubstanceforGeneralandScienticUse,Journalof PhysicalandChemicalReferenceData31387. [58]J.D.Bernardin,I.Mudawar,ALeidenfrostPointModelforImpingingDropletsand Sprays,JournalofHeatTransfer126272. [59]F.J.Heymann,HighSpeedImpactbetweenaLiquidDropandaSolidSurface, JournalofAppliedPhysics405113. [60]O.G.Engel,WaterdropCollisionswithSolidSurfaces,JournalofResearchofthe NationalBureauofStandards54281. [61]O.G.Engel,NoteonParticleVelocityinCollisionsBetweenLiquidDropsand Solids,JournalofResearchoftheNationalBureauofStandardsA.Physicsand Chemistry64A497. [62]G.P.Flach,M.N. Ozisik,PeriodicBSplineBasisforQuasiSteadyPeriodic InverseHeatConduction,InternationalJournalofHeatandMassTransfer30 869. [63]A.N.Tikhonov,OnStabilityofInverseProblems,DokladyAkademiiNaukSSSR 39195. [64]A.N.Tikhonov,OnSolvingIncorrectlyPosedProblemsandMethodof Regularization,DokladyAkademiiNaukSSSR151501. [65]A.N.Tikhonov,InverseProblemsinHeatConduction,JournalofEngineering Physics29816. [66]K.Levenberg,AMethodfortheSolutionofCertainNonLinearProblemsinLeast Squares,TheQuarterlyofAppliedMathematics2164. [67]D.W.Marquardt,AnAlgorithmforLeastSquaresEstimationofNonlinear Parameters,JournaloftheSocietyforIndustrialandAppliedMathematics11 431. 179 PAGE 180 [68]G.J.Stoltz,NumericalSolutionstoanInverseProblemofHeatConductionfor SimpleShapes,JournalofHeatTransfer8220. [69]J.V.Beck,CalculationofSurfaceHeatFluxfromanInternalTemperatureHistory, in:ASMEPaper62HT46. [70]J.V.Beck,ParameterEstimationinEngineeringandScience,JohnWiley&Sons Inc,1977. [71]J.V.Beck,B.F.Blackwell,C.St.Clair,InverseHeatConduction:IllPosed Problems,WileyInterscience,1985. [72]A.HajiSheikh,F.P.Buckingham,MultidimensinalInverseHeatConductionUsing theMonteCarloMethods,JournalofHeatTransfer1526. [73]A.HajiSheikh,InverseEngineeringHandbook,CRCPress,pp.327. [74]O.M.Alifanov,InverseHeatTransferProblems,SpringerVerlag,1995. [75]M.N. Ozisik,HeatConduction,JohnWiley&SonsInc,1993. [76]M.N. Ozisik,H.R.B.Orlande,InverseHeatTransfer:Fundamentalsand Applications,Taylor&Francis,2000. [77]Y.Jarny,InverseEngineeringHandbook,CRCPress,NewYork,pp.103. [78]J.S.Bendat,A.G.Piersol,RandomData,JohnWiley&SonsInc,NewYork,3 rd edition,2000. [79]S.S.Rao,EngineeringOptimization:TheoryandPractice,JohnWiley&SonsInc, 3 rd edition,1996. [80]R.Fletcher,PracticalMethodsofOptimization,JohnWiley&SonsInc,2 nd edition, 2000. [81]M.R.Hestenes,E.Stiefel,MethodsofConjugateGradientsforSolvingLinear Systems,JournalofResearchoftheNationalBureauofStandards49 409. [82]E.Polak,G.Ribi ere,NotesurlaConvergencedeM ethodsdeDirections Conjugr ees,RevueFrancaiseInformationRechercheOperationnelle16 35. [83]R.Fletcher,C.M.Reeves,FunctionMinimizationbyConjugateGradients,The ComputerJournal7149. [84]W.H.Press,B.F.Flannery,S.A.Teukolsky,W.T.Wtterling,NumericalRecipes, CambridgeUniversityPress,1989. [85]J.W.Daniel,TheApproximateMinimizationofFunctionals,PrenticeHall,1971. 180 PAGE 181 [86]K.A.Woodbury,InverseEngineeringHandbook,CRCPress,NewYork,pp. 41. [87]C.J.Chen,P.Li,TheoreticalErrorAnalysisofTemperatureMeasurementbyan EmbeddedThermocouple,LettersinHeatandMassTransfer11710. [88]C.J.Chen,T.M.Danh,TransientTemperatureDistortioninaSlabDueto ThermocoupleCavity,AIAAJournal14979. [89]J.V.Beck,DeterminationofUndisturbedTemperaturesfromThermocouple MeasurementsUsingCorrectionKernals,NuclearEngineeringandDesign7 9. [90]K.A.Woodbury,A.Gupta,EffectofDeterministicThermocoupleErrorsBias ontheSolutionoftheInverseHeatConductionProblem,in:Proceedingsofthe 5thInternationalConferenceonInverseProblemsinEngineering:Theoryand Practice. [91]K.A.Woodbury,A.Gupta,ASimple1DSensorModeltoAccountfor DeterministicThermocoupleErrorsBiasintheSolutionoftheInverseHeat ConductionProblems,InverseProblemsinScienceandEngineering16 21. [92]M.H.Attia,A.Cameron,L.Kops,DistortioninThermalFieldaroundInserted ThermocouplesinExperimentalInterfacialStudies,Part4:EndEffect,Journalof ManufacturingScienceandEngineering124135. [93]D.Li,M.Wells,EffectofSubsurfaceThermocoupleInstallationonthe DiscrepancyoftheMeasuredThermalHistoryandPredictedSurfaceHeat FluxduringaQuenchOperation,MetallurgicalandMaterialsTransactionsB36B 343. [94]E.Caron,M.Wells,D.Li,ACompensationMethodfortheDisturbanceinthe TemperatureFieldCausedbySubsurfaceThermocouples,Metallurgicaland MaterialsTransactionsB37B475. [95]G.Franco,E.Caron,M.Wells,QuanticationoftheSurfaceTemperature DiscrepancyCausedBySubsurfaceThermocouplesandMethodsfor Compensation,MetallurgicalandMaterialsTransactionsB38B94956. [96]W.L.Chen,Y.C.Yang,H.L.Lee,InverseProbleminDeterminingConvection HeatTransferCoefcientofanAnnularFin,EnergyConservationand Management481081. [97]M.J.Colaco,H.R.B.Orlande,ComparisonofDifferentVersionsoftheConjugate GradientMethodofFunctionEstimation,NumericalHeatTransfer,PartA36 229. 181 PAGE 182 [98]Y.Jarny,M. Ozisik,J.Bardon,AGeneralOptimizationMethodUsingAdjoint EquationforSolvingMultidimensionalInverseHeatConduction,International JournalofHeatandMassTransfer342911. [99]H.L.Lee,H.M.Chou,Y.C.Yang,TheFunctionEstimationinPredictingHeat FluxofPinFinswithVariableHeatTransferCoefcients,EnergyConservation andManagement451794. [100]T.Loulou,E.Artioukhine,NumericalSolutionof3DUnsteadyNonlinearInverse ProblemofEstimatingSurfaceHeatFluxforCylindricalGeometry,Inverse ProblemsinScienceandEngineering1439. [101]W.Marquardt,AnObserverBasedSolutionofInverseHeatConductionProblems, InternationalJournalofHeatandMassTransfer331545. [102]D.W.Peaceman,J.Rachford,H.H.,TheNumericalSolutionofParabolicand EllipticDifferentialEquations,JournaloftheSocietyforIndustrialandApplied Mathematics328. [103]K.Fukuyo,ConditionalStabilityofLarkinMethodswithNonUniformGrids, JournalofTheoreticalandAppliedMechanics37139. [104]R.F.Kropf,OscillatorySolutionsofthePeacemanRachfordAlternatingDirection ImplicitMethodandaComparisonofMethodsfortheSolutionoftheTwo DimensionalHeatDiffusionEquation,Master'sthesis,AirForceInstituteof Technology,1985. [105]J.A.Dantzig,ImprovedTransientResponseofThermocoupleSensors,Reviewof ScienticInstruments56723. [106]A.E.Segall,SolutionsfortheCorrectionofTemperatureMeasurementsBased onBeadedThermocouples,InternationalJournalofHeatandMassTransfer55 2801. [107]Y.Tashiro,T.Biwa,T.Yazaki,DeterminationofaResponseFunctionofa ThermocoupleUsingaShortAcousticPulse,JournaloftheAcousticalSocietyof America1211956. [108]L.Krebs,K.Bremhorst,ComparisonofComputedFrequencyResponses ofIntrinsicandEncapsulatedBeadTypeThermocouplesinLiquidSodium, InternationalJournalofHeatandMassTransfer291417. [109]M.Tagawa,T.Shimoji,Y.Ohta,ATwoThermocoupleProbeTechniquefor EstimatingThermocoupleTimeConstantsinFlowswithCombustion: InSitu ParameterIndenticationofaFirstOrderLagSystem,ReviewofScientic Instruments693370. 182 PAGE 183 [110]P.C.Hung,R.J.Kee,G.W.Irwin,S.F.McLoone,BlindDeconvolutionfor TwoThermocoupleSensorCharacterization,JournalofDynamicsSystems, Measurement,andControl129194. [111]F.P.Incropera,D.P.Dewitt,FundamentalsofHeatandMassTransfer,JohnWiley &SonsInc,2002. [112]J.H.LienhardIV,J.H.LienhardV,AHeatTransferTextbook,PhlogistonPress, Cambridge,MA,4 th edition,2011. [113]L.Michalski,K.Eckersdorf,J.Kucharski,J.McGhee,TemperatureMeasurement, JohnWiley&SonsInc,2 nd edition,2001. [114]Y.Rabin,D.Rittel,AModelfortheTimeResponseofSolidEmbedded Thermocouples,ExperimentalMechanics39132. 183 PAGE 184 BIOGRAPHICALSKETCH RichardParkerwasborninOrlando,FLandpromptlymovedtotheLouisville,KY area.AfterhisfamilyreceivedatransfertoColumbia,SCwhenhewas9yearsoldhe spenttheremainderofhischildhoodthere.Duringhissenioryearinhighschoolhe decidedtojointheUnitedStatesNavyasitwouldhelphimattainsomemuchneeded discipline. WhileintheUSNavyRichardwaspartoftheNavalNuclearPowerProgramand graduatedfromthenuclearmechanicprogramandwasthenselectedfornuclear chemistryschool.Aftercompletinghisnearlytwoyearlongschoolprogramhewas stationedinPearlHarbor,HIonboardtheUSSKeyWestSSN722.Whileservingin ReactorLaboratoriesDivisionheexcelledathisdutiesandwasawardedthreeNavy andMarineCorpsAchievementMedalsalongwithotheraccolades,includingbeing promotedtoPettyOfcerFirstClassE6inunder6yearsofservice.Whileonboard theKeyWestRichardtookpartinOperationEnduringFreedomaftertheSeptember11, 2001terrorattacksonUSsoil. RealizingthathelikedtoexplorescienceandmathledRichardtoleavetheNavy afterhisinitialcommitmentwasconcluded.Havingfoundheenjoyedtheareaof uidmechanicsandheattransferhedecidedtostudymechanicalengineeringatthe UniversityofSouthCarolinawherehegraduated summacumlaude in2006.Realizing therewasmuchmoretolearnRicharddecidedtopursuegraduatestudiesatthe UniversityofFloridainmechanicalengineeringwhereheearnedhisPh.D.inthermal anduidsciences. 184 xml version 1.0 encoding UTF8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ETURZ3R2R_RU1U30 INGEST_TIME 20130122T13:25:21Z PACKAGE AA00013289_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 