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E-raamat: Fundamentals of Turbulent and Multiphase Combustion

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  • Ilmumisaeg: 03-Jul-2012
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  • ISBN-13: 9781118099292
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Fundamentals of Turbulent and Multiphase CombustionSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 03-Jul-2012
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118099292
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"This book is a follow-on to the author's bestseller, Principles of Combustion, Second Edition published in 2005. The text covers advanced topics of combustion and flame that are not covered anywhere else. Kuo provides a multiphase systems approach beginning with more common topics and moving to higher level applications such as reacting boundary layer flows, ignition of homogeneous mixtures, flame extinction phenomena, and detonation processes in condensed phase materials. As with Kuo's earlier book, large numbers of examples and problems and a solutions manual are provided"--



Detailed coverage of advanced combustion topics from the author of Principles of Combustion, Second Edition

Turbulence, turbulent combustion, and multiphase reacting flows have become major research topics in recent decades due to their application across diverse fields, including energy, environment, propulsion, transportation, industrial safety, and nanotechnology. Most of the knowledge accumulated from this research has never been published in book form until now. Fundamentals of Turbulent and Multiphase Combustion presents up-to-date, integrated coverage of the fundamentals of turbulence, combustion, and multiphase phenomena along with useful experimental techniques, including non-intrusive, laser-based measurement techniques, providing a firm background in both contemporary and classical approaches. Beginning with two full chapters on laminar premixed and non-premixed flames, this book takes a multiphase approach, beginning with more common topics and moving on to higher-level applications.

In addition, Fundamentals of Turbulent and Multiphase Combustion:

  • Addresses seven basic topical areas in combustion and multiphase flows, including laminar premixed and non-premixed flames, theory of turbulence, turbulent premixed and non-premixed flames, and multiphase flows

  • Covers spray atomization and combustion, solid-propellant combustion, homogeneous propellants, nitramines, reacting boundary-layer flows, single energetic particle combustion, and granular bed combustion

  • Provides experimental setups and results whenever appropriate

Supported with a large number of examples and problems as well as a solutions manual, Fundamentals of Turbulent and Multiphase Combustion is an important resource for professional engineers and researchers as well as graduate students in mechanical, chemical, and aerospace engineering.

Preface xix
1 Introduction and Conservation Equations
1(42)
1.1 Why Is Turbulent and Multiphase Combustion Important?
3(1)
1.2 Different Applications for Turbulent and Multiphase Combustion
3(5)
1.2.1 Applications in High Rates of Combustion of Materials for Propulsion Systems
5(2)
1.2.2 Applications in Power Generation
7(1)
1.2.3 Applications in Process Industry
7(1)
1.2.4 Applications in Household and Industrial Heating
7(1)
1.2.5 Applications in Safety Protections for Unwanted Combustion
7(1)
1.2.6 Applications in Ignition of Various Combustible Materials
8(1)
1.2.7 Applications in Emission Control of Combustion Products
8(1)
1.2.8 Applications in Active Control of Combustion Processes
8(1)
1.3 Objectives of Combustion Modeling
8(1)
1.4 Combustion-Related Constituent Disciplines
9(1)
1.5 General Approach for Solving Combustion Problems
9(2)
1.6 Governing Equations for Combustion Models
11(3)
1.6.1 Conservation Equations
11(1)
1.6.2 Transport Equations
11(1)
1.6.3 Common Assumptions Made in Combustion Models
11(1)
1.6.4 Equation of State
12(1)
1.6.4.1 High-Pressure Correction
13(1)
1.7 Definitions of Concentrations
14(2)
1.8 Definitions of Energy and Enthalpy Forms
16(3)
1.9 Velocities of Chemical Species
19(4)
1.9.1 Definitions of Absolute and Relative Mass and Molar Fluxes
20(3)
1.10 Dimensionless Numbers
23(1)
1.11 Derivation of Species Mass Conservation Equation and Continuity Equation for Multicomponent Mixtures
23(6)
1.12 Momentum Conservation Equation for Mixture
29(4)
1.13 Energy Conservation Equation for Multicomponent Mixture
33(7)
1.14 Total Unknowns versus Governing Equations
40(3)
Homework Problems
41(2)
2 Laminar Premixed Flames
43(82)
2.1 Basic Structure of One-Dimensional Premixed Laminar Flames
46(1)
2.2 Conservation Equations for One-Dimensional Premixed Laminar Flames
47(21)
2.2.1 Various Models for Diffusion Velocities
49(1)
2.2.1.1 Multicomponent Diffusion Velocities (First-Order Approximation)
49(5)
2.2.1.2 Various Models for Describing Source Terms due to Chemical Reactions
54(12)
2.2.2 Sensitivity Analysis
66(2)
2.3 Analytical Relationships for Premixed Laminar Flames with a Global Reaction
68(18)
2.3.1 Three Analysis Procedures for Premixed Laminar Flames
77(3)
2.3.2 Generalized Expression for Laminar Flame Speeds
80(1)
2.3.2.1 Reduced Reaction Mechanism for HC-Air Flame
81(1)
2.3.3 Dependency of Laminar Flame Speed on Temperature and Pressure
82(2)
2.3.4 Premixed Laminar Flame Thickness
84(2)
2.4 Effect of Flame Stretch on Laminar Flame Speed
86(17)
2.4.1 Definitions of Stretch Factor and Karlovitz Number
86(8)
2.4.2 Governing Equation for Premixed Laminar Flame Surface Area
94(1)
2.4.3 Determination of Unstretched Premixed Laminar Flame Speeds and Markstein Lengths
95(8)
2.5 Modeling of Soot Formation in Laminar Premixed Flames
103(22)
2.5.1 Reaction Mechanisms for Soot Formation and Oxidation
104(2)
2.5.1.1 Empirical Models for Soot Formation
106(2)
2.5.1.2 Detailed Models for Soot Formation and Oxidation
108(1)
2.5.1.3 Formation of Aromatics
109(1)
2.5.1.4 Growth of Aromatics
110(2)
2.5.1.5 Migration Reactions
112(1)
2.5.1.6 Oxidation of Aromatics
113(1)
2.5.2 Mathematical Formulation of Soot Formation Model
114(10)
Homework Problems
124(1)
3 Laminar Non-Premixed Flames
125(81)
3.1 Basic Structure of Non-Premixed Laminar Flames
128(1)
3.2 Flame Sheet Model
129(1)
3.3 Mixture Fraction Definition and Examples
130(12)
3.3.1 Balance Equations for Element Mass Fractions
134(4)
3.3.2 Temperature-Mixture Fraction Relationship
138(4)
3.4 Flamelet Structure of a Diffusion Flame
142(9)
3.4.1 Physical Significance of the Instantaneous Scalar Dissipation Rate
145(2)
3.4.2 Steady-State Combustion and Critical Scalar Dissipation Rate
147(4)
3.5 Time and Length Scales in Diffusion Flames
151(2)
3.6 Examples of Laminar Diffusion Flames
153(19)
3.6.1 Unsteady Mixing Layer
153(2)
3.6.2 Counterflow Diffusion Flames
155(10)
3.6.3 Coflow Diffusion Flame or Jet Flames
165(7)
3.7 Soot Formation in Laminar Diffusion Flames
172(34)
3.7.1 Soot Formation Model
173(1)
3.7.1.1 Particle Inception
174(1)
3.7.1.2 Surface Growth and Oxidation
174(1)
3.7.2 Appearance of Soot
175(1)
3.7.3 Experimental Studies by Using Coflow Burners
176(2)
3.7.3.1 Sooting Zone
178(4)
3.7.3.2 Effect of Fuel Structure
182(1)
3.7.3.3 Influence of Additives
183(3)
3.7.3.4 Coflow Ethylene/Air Laminar Diffusion Flames
186(5)
3.7.3.5 Modeling of Soot Formation
191(13)
Homework Problems
204(2)
4 Background in Turbulent Flows
206(77)
4.1 Characteristics of Turbulent Flows
210(3)
4.1.1 Some Pictures
212(1)
4.2 Statistical Understanding of Turbulence
213(4)
4.2.1 Ensemble Averaging
214(1)
4.2.2 Time Averaging
215(1)
4.2.3 Spatial Averaging
215(1)
4.2.4 Statistical Moments
215(1)
4.2.5 Homogeneous Turbulence
216(1)
4.2.6 Isotropic Turbulence
217(1)
4.3 Conventional Averaging Methods
217(30)
4.3.1 Reynolds Averaging
218(4)
4.3.1.1 Correlation Functions
222(3)
4.3.2 Favre Averaging
225(2)
4.3.3 Relation between Time Averaged-Quantities and Mass-Weighted Averaged Quantities
227(1)
4.3.4 Mass-Weighted Conservation and Transport Equations
228(1)
4.3.4.1 Continuity and Momentum Equations
228(2)
4.3.4.2 Energy Equation
230(1)
4.3.4.3 Mean Kinetic Energy Equation
231(1)
4.3.4.4 Reynolds-Stress Transport Equations
232(2)
4.3.4.5 Turbulence-Kinetic-Energy Equation
234(2)
4.3.4.6 Turbulent Dissipation Rate Equation
236(6)
4.3.4.7 Species Mass Conservation Equation
242(1)
4.3.5 Vorticity Equation
243(3)
4.3.6 Relationship between Enstrophy and the Turbulent Dissipation Rate
246(1)
4.4 Turbulence Models
247(2)
4.5 Probability Density Function
249(7)
4.5.1 Distribution Function
250(2)
4.5.2 Joint Probability Density Function
252(2)
4.5.3 Bayes' Theorem
254(2)
4.6 Turbulent Scales
256(10)
4.6.1 Comment on Kolmogorov Hypotheses
260(6)
4.7 Large Eddy Simulation
266(13)
4.7.1 Filtering
268(2)
4.7.2 Filtered Momentum Equations and Subgrid Scale Stresses
270(4)
4.7.3 Modeling of Subgrid-Scale Stress Tensors
274(5)
4.8 Direct Numerical Simulation
279(4)
Homework Problems
280(3)
5 Turbulent Premixed Flames
283(119)
5.1 Physical Interpretation
289(2)
5.2 Some Early Studies in Correlation Development
291(13)
5.2.1 Damkohler's Analysis (1940)
292(3)
5.2.2 Schelkin's Analysis (1943)
295(1)
5.2.3 Karlovitz, Denniston, and Wells's Analysis (1951)
296(1)
5.2.4 Summerfield's Analysis (1955)
297(1)
5.2.5 Kovasznay's Characteristic Time Approach (1956)
298(1)
5.2.6 Limitations of the Preceding Approaches
299(5)
5.3 Characteristic Scale of Wrinkles in Turbulent Premixed Flames
304(6)
5.3.1 Schlieren Photographs
305(1)
5.3.2 Observations on the Structure of Wrinkled Laminar Flames
305(2)
5.3.3 Measurements of Scales of Unburned and Burned Gas Lumps
307(3)
5.3.4 Length Scale of Wrinkles
310(1)
5.4 Development of Borghi Diagram for Premixed Turbulent Flames
310(14)
5.4.1 Physical Interpretation of Various Regimes in Borghi's Diagram
311(1)
5.4.1.1 Wrinkled Flame Regime
311(1)
5.4.1.2 Wrinkled Flame with Pockets Regime (also Called Corrugated Flame Regime)
311(2)
5.4.1.3 Thickened Wrinkled Flames
313(1)
5.4.1.4 Thickened Flames with Possible Extinctions/Thick Flames
314(1)
5.4.2 Klimov-Williams Criterion
314(2)
5.4.3 Construction of Borghi Diagram
316(2)
5.4.3.1 Thick Flames (or Distributed Reaction Zone or Well-Stirred Reaction Zone)
318(1)
5.4.4 Wrinkled Flames
318(2)
5.4.4.1 Wrinkled Flamelets (Weak Turbulence)
320(2)
5.4.4.2 Corrugated Flamelets (Strong Turbulence)
322(2)
5.5 Measurements in Premixed Turbulent Flames
324(1)
5.6 Eddy-Break-up Model
324(13)
5.6.1 Spalding's EBU Model
335(1)
5.6.2 Magnussen and Hjertager's EBU Model
336(1)
5.7 Intermittency
337(2)
5.8 Flame-Turbulence Interaction
339(3)
5.8.1 Effects of Flame on Turbulence
341(1)
5.9 Bray-Moss-Libby Model
342(26)
5.9.1 Governing Equations
349(4)
5.9.2 Gradient Transport
353(1)
5.9.3 Countergradient Transport
354(3)
5.9.4 Closure of Transport Terms
357(1)
5.9.4.1 Gradient Closure
357(1)
5.9.4.2 BML Closure
358(3)
5.9.5 Effect of Pressure Fluctuations Gradients
361(3)
5.9.6 Summary of DNS Results
364(4)
5.10 Turbulent Combustion Modeling Approaches
368(1)
5.11 Geometrical Description of Turbulent Premixed Flames and G-Equation
368(8)
5.11.1 Level Set Approach for the Corrugated Flamelets Regime
371(3)
5.11.2 Level Set Approach for the Thin Reaction Zone Regime
374(2)
5.12 Scales in Turbulent Combustion
376(4)
5.13 Closure of Chemical Reaction Source Term
380(1)
5.14 Probability Density Function Approach to Turbulent Combustion
381(21)
5.14.1 Derivation of the Transport Equation for Probability Density Function
386(5)
5.14.2 Moment Equations and PDF Equations
391(1)
5.14.3 Lagrangian Equations for Fluid Particles
392(3)
5.14.4 Gradient Transport Model in Composition PDF Method
395(2)
5.14.5 Determination of Overall Reaction Rate
397(1)
5.14.6 Lagrangian Monte Carlo Particle Methods
398(1)
5.14.7 Filtered Density Function Approach
398(1)
5.14.8 Prospect of PDF Methods
399(1)
Homework Problems
400(1)
Project No. 1
400(1)
Project No. 2
401(1)
6 Non-premixed Turbulent Flames
402(107)
6.1 Major Issues in Non-premixed Turbulent Flames
404(2)
6.2 Turbulent Damkohler number
406(1)
6.3 Turbulent Reynolds Number
407(1)
6.4 Scales in Non-premixed Turbulent Flames
407(7)
6.4.1 Direct Numerical Simulation and Scales
411(3)
6.5 Turbulent Non-premixed Combustion Regime Diagram
414(4)
6.6 Turbulent Non-premixed Target Flames
418(38)
6.6.1 Simple Jet Flames
419(1)
6.6.1.1 CH4/H2/N2 Jet Flame
420(10)
6.6.1.2 Effect of Jet Velocity
430(2)
6.6.2 Piloted Jet Flames
432(16)
6.6.2.1 Comparison of Simple Jet Flame and Sandia Flames D and F
448(4)
6.6.3 Bluff Body Flames
452(3)
6.6.4 Swirl Stabilized Flames
455(1)
6.7 Turbulence-Chemistry Interaction
456(6)
6.7.1 Infinite Chemistry Assumption
456(1)
6.7.1.1 Unity Lewis Number
457(1)
6.7.1.2 Nonunity Lewis Number
458(1)
6.7.2 Finite-Rate Chemistry
458(4)
6.8 Probability Density Approach for Turbulent Non-premixed Combustion
462(14)
6.8.1 Physical Models
465(1)
6.8.2 Turbulent Transport in Velocity-Composition Pdf Methods
466(1)
6.8.2.1 Stochastic Mixing Model
467(1)
6.8.2.2 Stochastic Reorientation Model
468(1)
6.8.3 Molecular Transport and Scalar Mixing Models
469(2)
6.8.3.1 Interaction by Exchange with the Mean Model
471(1)
6.8.3.2 Modified Curl Mixing Model
471(1)
6.8.3.3 Euclidean Minimum Spanning Tree Model
472(4)
6.9 Flamelet Models
476(4)
6.9.1 Laminar Flamelet Assumption
477(1)
6.9.2 Unsteady Flamelet Modeling
478(1)
6.9.3 Flamelet Models and PDF
479(1)
6.10 Interactions of Flame and Vortices
480(12)
6.10.1 Flame Rolled Up in a Single Vortex
482(1)
6.10.2 Flame in a Shear Layer
483(1)
6.10.3 Jet Flames
483(1)
6.10.4 Karman Vortex Street/V-Shaped Flame Interaction
484(1)
6.10.5 Burning Vortex Ring
484(1)
6.10.6 Head-on Flame/Vortex Interaction
485(1)
6.10.7 Experimental Setups for Flame/Vortex Interaction Studies
486(1)
6.10.7.1 Reaction Front/Vortex Interaction in Liquids
486(1)
6.10.7.2 Jet Flames
487(1)
6.10.7.3 Counterflow Diffusion Flames
488(4)
6.11 Generation and Dissipation of Vorticity Effects
492(1)
6.12 Non-premixed Flame-Vortex Interaction Combustion Diagram
493(3)
6.13 Flame Instability in Non-premixed Turbulent Flames
496(4)
6.14 Partially Premixed Flames or Edge Flames
500(9)
6.14.1 Formation of Edge Flames
501(1)
6.14.2 Triple Flame Stabilization of Lifted Diffusion Flame
502(1)
6.14.3 Analysis of Edge Flames
503(3)
Homework Problems
506(1)
Project No. 6.1
506(1)
Project No. 6.2
507(1)
Project No. 6.3
507(2)
7 Background in Multiphase flows with Reactions
509(67)
7.1 Classification of Multiphase Flow Systems
512(2)
7.2 Practical Problems Involving Multiphase Systems
514(1)
7.3 Homogeneous versus Multi-component/Multiphase Mixtures
515(1)
7.4 CFD and Multiphase Simulation
516(4)
7.5 Averaging Methods
520(13)
7.5.1 Eulerian Average---Eulerian Mean Values
522(1)
7.5.2 Lagrangian Average---Lagrangian Mean Values
523(1)
7.5.3 Boltzmann Statistical Average
524(1)
7.5.4 Anderson and Jackson's Averaging for Dense Fluidized Beds
525(8)
7.6 Local Instant Formulation
533(3)
7.7 Eulerian-Eulerian Modeling
536(14)
7.7.1 Fluid-Fluid Modeling
536(2)
7.7.1.1 Closure Models
538(2)
7.7.2 Fluid-Solid Modeling
540(1)
7.7.2.1 Closure Models
541(6)
7.7.2.2 Dense Particle Flows
547(2)
7.7.2.3 Dilute Particle Flows
549(1)
7.8 Eulerian-Lagrangian Modeling
550(5)
7.8.1 Fluid-Solid Modeling
551(1)
7.8.1.1 Fluid Phase
551(1)
7.8.1.2 Solid Phase
552(3)
7.9 Interfacial Transport (Jump Conditions)
555(6)
7.10 Interface-Tracking/Capturing
561(12)
7.10.1 Interface Tracking
563(1)
7.10.1.1 Markers on Interface (Surface Marker Techniques)
564(3)
7.10.1.2 Surface-Fitted Method
567(1)
7.10.2 Interface Capturing
568(1)
7.10.2.1 Markers in Fluid (MAC Formulation)
568(1)
7.10.2.2 Volume of Fluid Method
569(4)
7.11 Discrete Particle Methods
573(3)
Homework Problems
575(1)
8 Spray Atomization and Combustion
576(147)
8.1 Introduction to Spray Combustion
578(2)
8.2 Spray-Combustion Systems
580(2)
8.3 Fuel Atomization
582(2)
8.3.1 Injector Types
582(2)
8.3.2 Atomization Characteristics
584(1)
8.4 Spray Statistics
584(10)
8.4.1 Particle Characterization
584(1)
8.4.2 Distribution Function
585(3)
8.4.2.1 Logarithmic Probability Distribution Function
588(1)
8.4.2.2 Rosin-Rammler Distribution Function
588(1)
8.4.2.3 Nukiyama-Tanasawa Distribution Function
589(1)
8.4.2.4 Upper-Limit Distribution Function of Mugele and Evans
589(1)
8.4.3 Transport Equation of the Distribution Function
590(1)
8.4.4 Simplified Spray Combustion Model for Liquid-Fuel Rocket Engines
591(3)
8.5 Spray Combustion Characteristics
594(8)
8.6 Classification of Models Developed for Spray Combustion Processes
602(3)
8.6.1 Simple Correlations
602(1)
8.6.2 Droplet Ballistic Models
603(1)
8.6.3 One-Dimensional Models
603(1)
8.6.4 Stirred-Reactor Models
604(1)
8.6.5 Locally Homogeneous-Flow Models
605(1)
8.6.6 Two-Phase-Flow (Dispersed-Flow) Models
605(1)
8.7 Locally Homogeneous Flow Models
605(29)
8.7.1 Classification of LHF Models
606(3)
8.7.2 Mathematical Formulation of LHF Models
609(1)
8.7.2.1 Basic Assumptions
609(1)
8.7.2.2 Equation of State
609(6)
8.7.2.3 Conservation Equations
615(4)
8.7.2.4 Turbulent Transport Equations
619(1)
8.7.2.5 Boundary Conditions
620(1)
8.7.2.6 Solution Procedures
620(6)
8.7.2.7 Comparison of LHF-Model Predictions with Experimental Data
626(8)
8.8 Two-Phase-Flow (Dispersed-Flow) Models
634(66)
8.8.1 Particle-Source-in-Cell Model (Discrete-Droplet Model)
637(2)
8.8.1.1 Models for Single Drop Behavior
639(15)
8.8.2 Drop Breakup Process and Mechanism
654(1)
8.8.2.1 Drop Breakup Process
654(5)
8.8.2.2 Multi-component Droplet Breakup by Microexplosion
659(3)
8.8.3 Deterministic Discrete Droplet Models
662(2)
8.8.3.1 Gas-Phase Treatment in DDDMs
664(2)
8.8.3.2 Liquid-Phase Treatment in DDDMs
666(1)
8.8.3.3 Results of DDDMs
667(2)
8.8.4 Stochastic Discrete Droplet Models
669(2)
8.8.5 Comparison of Results between DDDMs and SDDMs
671(11)
8.8.6 Dense Sprays
682(1)
8.8.6.1 Introduction
682(2)
8.8.6.2 Background
684(6)
8.8.6.3 Jet Breakup Models
690(9)
8.8.6.4 Impinging Jet Atomization
699(1)
8.9 Group-Combustion Models of Chiu
700(6)
8.9.1 Group-Combustion Number
701(2)
8.9.2 Modes of Group Burning in Spray Flames
703(3)
8.10 Droplet Collison
706(4)
8.10.1 Droplet-Droplet Collisions
707(1)
8.10.2 Droplet-Wall Collision
708(2)
8.10.3 Interacting Droplet in a Many-Droplet System
710(1)
8.11 Optical Techniques for Particle Size Measurements
710(7)
8.11.1 Types of Optical Particle Sizing Methods
711(1)
8.11.2 Single Particle Counting Methods
711(1)
8.11.2.1 Scattering Ratio Technique
712(1)
8.11.2.2 Intensity Deconvolution Method
713(1)
8.11.2.3 Interferometric Method (Phase-Shift Method)
713(1)
8.11.2.4 Visibility Method Using a Laser Doppler Velocimeter LDV
713(1)
8.11.2.5 Phase Doppler Sizing Anemometer
713(1)
8.11.3 Ensemble Particle Sizing Techniques
714(1)
8.11.3.1 Extinction Measurement Techniques
714(1)
8.11.3.2 Multiple Angle Scattering Technique
714(1)
8.11.3.3 Fraunhofer Diffraction Particle Analyzer
715(1)
8.11.3.4 Integral Transform Solutions for Near-Forward Scattering
716(1)
8.12 Effect of Droplet Spacing on Spray Combustion
717(6)
8.12.1 Evaporation and Combustion of Droplet Arrays
717(3)
Homework Problems
720(3)
Appendix A Useful Vector and Tensor Operations 723(28)
Appendix B Constants and Conversion Factors Often Used in Combustion 751(4)
Appendix C Naming of Hydrocarbons 755(4)
Appendix D Detailed Gas-Phase Reaction Mechanism for Aromatics Formation 759(36)
Appendix E Particle Size-U.S. Sieve Size and Tyler Screen Mesh Equivalents 795(4)
Bibliography 799(70)
Index 869
KENNETH K. KUO is Distinguished Professor of Mechanical Engineering and Director of the High Pressure Combustion Laboratory (HPCL) in the Department of Mechanical and Nuclear Engineering of the College of Engineering at Pennsylvania State University. Professor Kuo established the HPCL and is recognized as one of the leading researchers and experts in propulsion-related combustion.

RAGINI ACHARYA is Senior Research Scientist at United Technologies Research Center. She received her PhD from Pennsylvania State University in December, 2008. Dr. Acharyas research expertise includes development of multi-physics, multi-scale, multiphase models, fire dynamics, numerical methods, and scientific computing. She has authored or coauthored multiple technical articles in these areas.
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