Ablation Study of Tungsten-Based Nuclear Thermal Rocket Fuel by Tabitha Elizabeth Rose Smith B.A. in Physics, May 2010, West

Ablation Study of Tungsten-Based Nuclear Thermal Rocket Fuel

by Tabitha Elizabeth Rose Smith

B.A. in Physics, May 2010, West Virginia University B.A. in Sociology & Anthropology, May 2010, West Virginia University M.A. in International Science and Technology Policy, May 2012, The George Washington University

A Dissertation submitted to

The Faculty of School and Engineering and Applied Sciences Of The George Washington University In partial fulfillment of the requirements For the degree of Doctor Philosophy

May 18, 2014

Dissertation directed by

Michael Keidar Professor of Mechanical and Aerospace Engineering

The School of Engineering and Applied Sciences of The George Washington University certifies that Tabitha Elizabeth Rose Smith has passed the Final Examination for the degree of Doctor of Philosophy or Doctor of Science as of May 5, 2014. This is the final and approved form of the dissertation.

Ablation Study of Tungsten-Based Nuclear Thermal Rocket Fuel

Tabitha Elizabeth Rose Smith

Dissertation Research Committee:

Michael Keidar, Professor of Mechanical and Aerospace Engineering, Dissertation Director

Alexey Shashurin, Research Scientist, The George Washington University, Committee Member

Chunlei Liang, Assistant Professor of Engineering and Applied Sciences, Committee Member

William Emrich, Senior Engineer, National Aeronautics and Space Administration Marshall Spaceflight Center, Committee Member

Ronald Litchford, Principal Investigator, National Aeronautics and Space Administration Langley Research Center, Committee Member

Dedication

The author wishes to dedicate the work conducted for this Doctoral dissertation to her husband John and future children, including our baby boy who will be born this summer.

iii

Acknowledgements

The author wishes to firstly acknowledge Dr Michael Keidar, her advisor for the three and a half years she spent conducting the Doctoral research. Dr Keidar was supportive, kind, and always there when needed. He offered guidance and insight into matters of all subjects, maintained interest in the various research ventures of the author, and was consistently focused, providing a guiding light which was needed in order to finish the research. The author has changed as a person after working with Dr Keidar, and will forever be grateful that he was her advisor.

Secondly, the author wishes to acknowledge her husband John and his mother,

Leona. Since they were 15 years old, John has been a pillar of strength and best friend, regardless of the circumstances for the author. He, his mother, and his family have been constant variables in a continuously changing environment that was necessary in order for the author to maintain a stable reference point to focus on the future. The author will always remember the many situations in which her husband John demonstrated his dedication, friendship, and love through trying times.

Thirdly, the author wishes to acknowledge NASA, NASA Marshall Spaceflight

Center, and the NTR community. There are dozens of unique, passionate, and special people that the author has met within this community, and she was greatly honored to be included as one of them after receiving funding from NASA to do research on the NTR.

Specifically, the author would like to thank Ronald Litchford, William Emrich, Harold

Gerish, Steve Howe, Jeramie Broadway, Omar Mireles, Joe Sholtis, Steven Bowen, Mike

Houts, Gwyn Rosaire, Brad Appel, and last but not least, Lou Qualls. It has been a

iv tremendous honor to simply know all of these special people and many more, and the author hopes to remain in the NTR community in some way, for many years.

Fourthly, the author would like to express thanks to the GWU machine shop and the students at the mpnl laboratory. Those at the machine shop, William Rutkowski,

Nicholas Batista, and Tom Punte, were instrumental to assisting in fabricating vital pieces of metalwork that were needed in order for the experiments to be conducted with the arc jet. Their friendship and company was also invaluable, as was their instruction in the skills of machining. Within the mpnl lab, George Teel and Alexey Shashurin provided vital help with their expert knowledge on conducting experiments with the arc jet. It is with sincere gratitude that the author expresses her thanks for their guidance and patience as she learned what was needed in order to complete the dissertation research.

Abstract of Dissertation

Ablation Study of Tungsten-Based Nuclear Thermal Rocket Fuel

The research described in this thesis has been performed in order to support the materials research and development efforts of NASA Marshall Space Flight Center

(MSFC), of Tungsten-based Nuclear Thermal Rocket (NTR) fuel. The NTR was developed to a point of flight readiness nearly six decades ago and has been undergoing gradual modification and upgrading since then. Due to the simplicity in design of the

NTR, and also in the modernization of the materials fabrication processes of nuclear fuel since the 1960’s, the fuel of the NTR has been upgraded continuously. Tungsten-based fuel is of great interest to the NTR community, seeking to determine its advantages over the Carbide-based fuel of the previous NTR programs.

The materials development and fabrication process contains failure testing, which is currently being conducted at MSFC in the form of heating the material externally and internally to replicate operation within the nuclear reactor of the NTR, such as with hot gas and RF coils. In order to expand on these efforts, experiments and computational studies of Tungsten and a Tungsten Zirconium Oxide sample provided by NASA have been conducted for this dissertation within a plasma arc-jet, meant to induce ablation on the material.

Mathematical analysis was also conducted, for purposes of verifying experiments and making predictions. The computational method utilizes Anisimov’s kinetic method of plasma ablation, including a thermal conduction parameter from the Chapman Enskog

vi expansion of the Maxwell Boltzmann equations, and has been modified to include a tangential velocity component. Experimental data matches that of the computational data, in which plasma ablation at an angle shows nearly half the ablation of plasma ablation at no angle. Fuel failure analysis of two NASA samples post-testing was conducted, and suggestions have been made for future materials fabrication processes. These studies, including the computational kinetic model at an angle and the ablation of the NASA sample, could be applied to an atmospheric reentry body, reentering at a ballistic trajectory at hypersonic velocities.

vii

Table of Contents

Dedication ...... iii

Acknowledgments ...... iv

Abstract of Dissertation ...... vi

List of Figures ...... x

List of Tables ...... xiii

List of Symbols/Nomenclature ...... xiv

Chapter 1: Introduction ...... 1

1.1 Historical Development of Nuclear Propulsion and Fuel ...... 1

1.2 Plasma Ablation as a Method of Materials Failure Testing ...... 5

1.3 Objectives and Summary of Dissertation ...... 8

Chapter 2: Literature Review and Development of the Kinetic Model ...... 10

2.1 Materials Development and Failure Testing ...... 10

2.1.1 Literature Review ...... 11

2.1.1.1 Thermal Cycling and Oxides ...... 15

2.1.1.2 Stoichiometry ...... 15

2.1.2 Understanding Failure Mechanisms ...... 17

2.1.2.1 Mechanical Deformation, Defects, and Fractures ...... 17

2.1.2.2 Imperfections and Fatigue ...... 19

vii

2.1.1.3 Developing upon the Current Fuel Failure Studies ...... 23

2.2 Plasma Ablation as a Method of Failure Testing ...... 26

2.2.1 The Kinetic Model ...... 26

2.2.2 Chapman Enskog Expansion and the Thermal Conduction Parameter ...... 29

Chapter 3: Computational and Experimental Set Up ...... 34

3.1 Including Tangential Velocity ...... 34

3.1.1 Modification of the Kinetic Model Equations ...... 34

3.1.1.1 Calculations for Mass Conservation ...... 35

3.1.1.2 Calculations for Momentum Conservation ...... 36

3.1.1.3 Calculations for Energy Conservation ...... 37

3.1.2 Implementing the New Derivation into the Code ...... 40

3.1.3 Tangential Velocity Orientation ...... 42

3.2 Set Up for the Experiments within the Arc Jet ...... 45

Chapter 4: Experimental Results ...... 55

4.1 Trials for 99.9% Tungsten Foil Samples ...... 56

4.1.1 Observations and Issues ...... 57

4.2 Trials for NASA Samples ...... 59

4.3 Results of Computational Simulation ...... 60

4.3.1 Modifying the Thermal Conduction Parameter ...... 65

Chapter 5: Concluding Remarks ...... 69

viii

5.1 Plasma Ablation of the 99.9% Tungsten Samples ...... 69

5.2 Failure Testing of NASA Fabricated Samples ...... 70

5.2.1 Failure Analysis of NASA’s Tungsten-Based Fuel ...... 71

5.2.1.1 Tungsten Hafnium Nitride Sample ...... 71

5.2.1.2 Tungsten Zirconium Oxide Sample ...... 73

5.3 Potential Areas of Further Study ...... 77

References ...... 79

Appendices ...... 83

Appendix A Atmospheric Reentry ...... 83

Appendix B Materials Testing Procedure ...... 88

Appendix C Experimental and Computational Data in Table Format ...... 92

Appendix D Computer Code ...... 98

Appendix E Mathematical Derivations ...... 102

List of Figures

Figure 1 NERVA Engine Being Tested ...... 1

Figure 2 NTR Engine Schematic ...... 3

Figure 3 Current NASA MSFC Fuel Sample ...... 4

Figure 4 Simple Arc Jet Schematic ...... 7

Figure 5 Microcracks Forming in WUO2 Fuel ...... 11

Figure 6 NASA MSFC HIP Fabrication Device ...... 14

Figure 7 Pourbaix Diagram for Tungsten ...... 22

Figure 8 NASA MSFC’s NTREES ...... 25

Figure 9 Layers 0, 1, and 2 of the Kinetic Model ...... 27

Figure 10 Particles Leaving the Surface of a Material into a Vapor ...... 32

Figure 11 Updated Schematic to Include ‘y’ Component of Velocity ...... 42

Figure 12 Orientation of Holder and Sample to the Plasma ...... 43

Figure 13 The Use of Smart Protractor to Set the Angle of the Sample ...... 43

Figure 14 Sample at a Resting Position of θ = 0o ...... 44

Figure 15 NASA MSFC Arc Jet ...... 46

Figure 16 The GWU Arc Jet ...... 47

Figure 17 Solving the Cathode Spot Problem ...... 48

Figure 18 Preparation of the Tungsten Samples ...... 49

Figure 19 Tungsten Zirconium Oxide Sample from NASA MSFC ...... 50

Figure 20 Fragments of the WZrO2 Sample ...... 51

Figure 21 Tungsten Sample Tested over 50 Seconds ...... 52

Figure 22 Holder Configuration for Samples at an Angle ...... 53

Figure 23 WZrO2 Sample Set-Up ...... 54

Figure 24 Tungsten Sample undergoing Testing ...... 55

Figure 25 Experimental Results for Tungsten Samples at 0 and 30 Degrees ...... 56

Figure 26 Tungsten Sample Melting the Thermocouples ...... 57

Figure 27 Example of Shading on the Sample from the Titanium Cathode ...... 58

Figure 28 WZrO2 Sample undergoing Testing ...... 59

Figure 29 Experimental Results for WZrO2 Samples ...... 60

Figure 30 Computational Ablation Contours for 0 Degrees ...... 61

Figure 31 Computational Ablation Contours for 30 Degrees ...... 62

Figure 32 Angles Versus Maximum Ablation Rate ...... 62

Figure 33 Comparison of Computational Results for 0 and 30 Degrees ...... 63

Figure 34 Simulation Adjusting Molecular Mass to Reflect WZrO2 ...... 64

Figure 35 Simulation using Molecular Mass of WZrO2 with Saturation Pressure

Profiles for Zirconium and Tungsten ...... 65

Figure 36 a1 and a2 Versus α for τ Much Less than 1 ...... 66

Figure 37 Previous Research Showing Relationship between a1, a2, and τ ...... 67

Figure 38 a1 and a2 Versus α for τ Equal to 0.1 ...... 67

Figure 39 Computational Ablation Rate for τ Equal to 0.25 ...... 69

Figure 40 WHfN Control Rod and Rod that was Tested ...... 71

Figure 41 Cracks in WHfN Rod ...... 72

Figure 42 Crack Selected for Failure Analysis in WHfN Rod ...... 73

Figure 43 SEM Image of WZrO2 Rod before Testing ...... 74

Figure 44 SEM Image Showing Grain Divisions in WZrO2 Fuel ...... 75

Figure 45 Area of a Sample Chosen for SEM and EDM Analysis ...... 76

Figure 46 Comparison of Colored and Discolored Regions in WZrO2 Sample ...... 76

xii

List of Tables

Table 1 GE 710 and ANL program WUO2 fuel failure modes and solutions ...... 12

Table 2 Failure Scenarios for the Ulysses Spacecraft ...... 84

Table 3 Experimental Data at 0 Degrees ...... 92

Table 4 Experimental Data at 0 Degrees, Subtracting Holder Area ...... 93

Table 5 Experimental Data at 30 Degrees ...... 94

Table 6 Experimental Data for the WZrO2 Fragments ...... 95

Table 7 Computational Data used for Graph Comparison ...... 96

Table 8 Computational Data 60%W40%ZrO2 Saturation Pressure Comparison ...97

xiii

List of Symbols / Nomenclature

1. a1 Density ratio, N1/N0

2. a2 Temperature ratio T1/T0

3. α Mach number

4. αtan ‘y’ velocity component of α

5. cp Specific heat capacity, constant pressure

6. cv Specific heat capacity, constant volume

7. Gs Flux of particles entering from the surface

8. Gb Flux of particles entering from the outer boundary of the layer

9. Gr Flux of particles returned to the surface during a time step

10. Gf Flux of particles crossing the outer boundary of the layer

11. Hlv Enthalpy of Vaporization

12. k Thermal conductivity

13. kB Boltzmann's constant

14. λ Mean free path

15. mW Mass of Tungsten

16. N0 Equilibrium density, at the ablating surface

xiv

17. N1 Density in the hydrodynamic layer

18. N2 Density of plasma bulk

19. P0 Surface Pressure

20. Pc Characteristic pressure

21. PS Saturation Pressure

22. qe Electron charge

23. R Specific gas constant

24. ρW Density of Tungsten

25. σ Stress

26. T0 Temperature at the surface of an ablating material

27. T1 Temperature of backflux

28. T2 Temperature of plasma bulk

29. Tamb Ambient temperature

30. Tc Characteristic temperature

29. Te Electron temperature

30. Ts Surface temperature

31. Tw Wall temperature

32. τ Thermal conduction parameter

33. vm,s Most probable thermal speed at the ablated surface

34. V0 Velocity at the ablating surface

35. V1 Velocity of backflux

36. V2 Velocity of plasma bulk

37. Vtan Tangential velocity component

38. VT Thermal velocity

xvi

Chapter 1: Introduction

1.1 Historical Development of Nuclear Propulsion and Fuel

The peak of human spaceflight was reached when human beings walked on the

Moon during the Apollo program. A more powerful spacecraft was needed for development after the Saturn V rocket to carry humans beyond the Earth-Moon system, and thus from approximately 1958 to 1972, a program known as Nuclear Engine for

Rocket Vehicle Applications (NERVA) was underway in order to transport humans to

Mars and beyond 1.

Figure 1 - A NERVA engine being tested in Jackass Flats, NV. The engine had reached a stage in its development during the program where it was ready for flight testing.

Whereas a chemical rocket will take nearly two years to reach Mars, a rocket with a

NERVA engine would only take less than a year. The salient features of a NERVA engine are a proven Isp of more than 900 sec, backed by an extensive design, development, and test program in which thrusts of 210,000 lbs were demonstrated, as well as feasibility for 10 hour operation and 60 restarts, meaning that NERVA was flight qualified 1,2,14. A shift in the political climate with the inauguration of President Nixon reorganized the NASA human spaceflight mission directorate from a Mars mission towards funding the Space Shuttle program.

Since that time, small research and development projects for nuclear propulsion have continued under different agencies, in lieu of a mission in which to use it. The main operative components of an NTR consist of a simple design of a hot nuclear core, through which liquid Hydrogen propellant is fed via turbopumps, as seen in Figure 2.

Improvements to the fuel elements of the NTR have been sought and have been one of the main focuses of continuing research. Some improvements to the design of the fuel rods were explored, such as the twisted ribbon nuclear rods used in the Russian nuclear rocket programs and the pebble bed nuclear engine concept of the US Air Force’s

Timberwind project. The primary composition of the nuclear fuel used in the NERVA project was Uranium Dioxide enclosed within a matrix of Carbon particles, but there are other compositions of nuclear fuel that have been explored for use in an NTR. Tungsten- based NTR fuel has been a recurrent candidate for use in a modern-day nuclear spacecraft, and it underwent major periods of research in the past during the GE 710

2 nuclear aircraft research program and the Argonne National Laboratory (ANL) nuclear rocket program.

Figure 2 - NTR engine schematic. Liquid Hydrogen is passed directly into holes in fuel channels; the heat from nuclear fission causes the liquid hydrogen to turn into gas, providing thrust for the spacecraft.

The research presented in this dissertation is focused on failure testing of

Tungsten based fuel, as it applies to Tungsten Uranium Oxide (WUO2) fuel that will be designed to work within a NTR engine core for purposes such as a manned Mars mission and other deep space missions. The use of Tungsten is of interest, because it is a high- temperature material that keeps its strength, has good thermal conductivity, very low coefficient of thermal expansion, retains fission products, and it is also relatively non- reactive with most coolants 3. Tungsten burns hotter than Carbon, which helps to raise the temperature of the Hydrogen (H2) propellant which passes through the channels,

3 increasing the specific impulse of the spacecraft.

Previous efforts to construct WUO2 using ceramic metal composite (cermet) fuel fabrication technologies were met with a variety of problems such as excess Oxygen causing WO2 stringers and dimensional instability - due to an increase in the ductile to brittle temperature transition (DBTT) at temperatures higher than 1589 K 4,5. Problems associated with fuel fabrication were due to less sophisticated methods utilized during these time periods (before and up until the 1960's), and these problems may be mitigated with today's developed UO2 ceramic microsphere fabrication and processing techniques that are able to create materials which have higher-quality, uniform ceramic oxide particles between 50-6000μm 3. A variety of other advantages exist today for fuel fabrication such as Auto Ignition Combustion Synthesis and Spark Plasma Sintering 3,4,6.

Research is currently being conducted at the Marshall Space Flight Center

(MSFC) on the fabrication and failure testing of WUO2 fuel. An example of their fabrication capability is seen in Figure 3.

Figure 3 - An example of a hexagonal 99.9% Tungsten-based fuel sample from MSFC that will undergo failure testing. Eventually throughout the material development process,

4 the fuel will evolve to become comprised of WUO2.

The combination of studying fuel fabrication, testing, and failure encompasses the entirety of the materials and processing cycle. There are multiple ways through which a material can fail, such as mechanical, chemical, thermal, or electric. Sometimes, failure can begin internally due to due issues such as molecular deformities and atomic irregularities, and sometimes failure can be brought on by external means, such as by an externally applied stress on the material. Since the vehicle will be traveling through the air and above the Earth, individuals have concerns for the retention of fission products – thus, a concern is how well will a coating of UO2 with Tungsten help to retain fission products? The research conducted within this dissertation was supported by NASA

MSFC specifically to support their fuels development effort and has been underway for approximately three years.

1.2 Plasma Ablation as a Method of Materials Failure Testing

Plasma ablation is useful for understanding the behaviors of materials under high temperature duress. One practical application for the results of this research would be accidental atmospheric re-entry scenarios, which would require a study of how WUO2 fuel reacts when engulfed within plasma, ascertaining the structural fidelity of the WUO2 fuel. The kinetic model describing how plasma ablates the surface of the material has been successful in determining important properties of that material when it is introduced to situations of high temperature wear, using Maxwell-Boltzmann’s theories, which describe the distribution of particles as a function of velocity, according to the

5 distribution function seen in Eq. (1.1). The velocity of the gas passing over the surface of an incoming object is one of the most important variables to consider for atmospheric reentry situations at hypersonic conditions.

3 2 2 2 푚 2 −푚(𝑣푥 + 𝑣푦 + 𝑣푧 ) 𝑓푣(𝑣푥, 𝑣푦, 𝑣푧) = ( ) exp ( ) (1.1) 2𝜋푘퐵푇 2푘퐵푇

Atmospheric reentry at hypersonic velocities has been studied in the past, mostly with regards to the chemical processes which occur to convert air into hot plasma, and the thermodynamics involved in heat transfer between the hot plasma and the surface of the reentering body. Past tests have explored other elements of atmospheric reentry, such as a material’s susceptibility to heat penetration and inability to withstand thermal shock 6.

The convective aspects of atmospheric re-entry have been extensively studied, and reentry temperature profiles have been calculated in the range of up to 3E+5 Kelvin 8.

The research conducted within this dissertation focuses on the conductive processes of plasma ablation, whereby air molecules are kinetically interacting with the surface material of the re-entering spacecraft. Applying the Chapman-Enskog expansion to the

Maxwell-Boltzman distribution introduces a thermal conduction parameter in terms of a self-diffusion coefficient 9.

Various accident scenarios have been studied in the past for nuclear spacecraft, but most of the failure testing was conducted for shrapnel impacting and penetrating the case surrounding the reactor and furthermore focused on failure scenarios during lift-off and ascent. These tests concluded that the reactor shell was able to withstand shrapnel impact, and thus tests were not explored for an exposed reactor core. However, without a heat shield, the reactor’s case could be vulnerable to ablation from re-entry, thus leaving the nuclear fuel core exposed. The accidental atmospheric reentry of NTR Tungsten-

6 based fuel and the resultant ablation behavior is of interest to the NTR community, but extensive experimental fuel failure testing and analysis has not been conducted. The research conducted in this thesis supports the fuel development efforts at MSFC, by conducting experimental plasma ablation studies with both 99.9% pure Tungsten samples and surrogate Tungsten-based samples provided by NASA. Plasma was generated within the vacuum chamber of an arc jet, which operates by focusing electrical discharge between an anode and cathode with a magnetic coil, as seen in Figure 4.

Figure 4 - Arc jet schematic. This is the method of plasma ablation that was used for failure testing, in this dissertation.

Physical properties on various elements of the experiment, such as of plasma generated by a Copper cathode and of the pressure-temperature relationships for Tungsten as it is being vaporized are well-known, and aided in the analysis and verification of the

7 experimental results. The experiments were verified with computational simulations that are based on the kinetic ablation theory for plasmas.

1.3 Objectives and Summary of Dissertation

This dissertation begins with a literature review on past Tungsten-based NTR fuels development, failure testing, and improvements. Some observations are made as to what additional failure testing could be conducted, as well as to suggest an expansion on plasma ablation testing, which is the main focus of the dissertation. After conducting failure testing, analysis is conducted on the material to understand weaknesses which may have arisen due to fabrication methods or from the testing environment. Various, applicable failure modes are explained in detail, with applications to the material in question. Thus, the first objective is developed:

(1) What are the ablation rates and failure modes for Tungsten and for Tungsten-

based, NASA-fabricated fuel samples, specifically so these results may be

applied to the scenario of atmospheric reentry of an NTR?

The set-up of the experiments and computational analysis are explained in detail, elaborating on the kinetic method of plasma ablation that will be used in the computation.

Firstly, the model is described in historical terms, as first developed by Anisimov and then most recently developed upon to include the Chapman Enskog expansion.

Modifications made to the distribution function to include a tangential velocity component, and the set-up for the arc jet into which the samples are tested, are explained.

The addition of a ‘y’ component to the kinetic model thus introduces the next objective:

(2) What happens when the sample is ablated at an angle relative to the incoming

plasma plume, and can the computational model be further updated and

validated to reflect the experimental data?

The results of the experiments and the numerical analysis are presented in graphical form in the dissertation and tabular form in the Appendix. An observation is made, in which the sample’s ablation rate drops to about half when it faces the plasma plume at an angle compared to the sample which faces the plasma head-on. Ablation was recorded for the NASA-fabricated sample, and analysis was conducted in the SEM. The results of these analyses are described in the conclusion and compared to the analyses of another Tungsten-based, NASA-fabricated sample which was tested in a different failure testing chamber. The Appendix contains additional information on atmospheric reentry temperature profiles and some discussion on past atmospheric reentry and failure studies performed on other nuclear powered spacecraft.

Chapter 2: Literature Review and Development of the Kinetic Model

2.1 Materials Development and Failure Testing

The present-day materials and processing development procedure for NTR fuels involves fuel fabrication, fuel testing in the presence of a hot hydrogen environment to simulate the presence of propellant, and the identification and the analysis of failure mechanisms via tests such as radio-frequency (RF) coil heating. A literature review of previous fuel failure experiments and the review of modern materials theory and practices for WUO2 fuel testing procedures can assist in developing new testing procedures. Some preliminary suggestions are outlined for modernizing and improving upon the process for testing and some methods for analysis, with a focus on the expansion of plasma ablation studies. After conducting these failure tests and analyzing the results, new samples can be created based upon lessons-learned in order to avoid future failure.

2.1.1 Literature Review

Different equipment has been used in the past for failure testing, such as thermogravimetric apparati capable of static testing up to 2900⁰C, hot hydrogen test loop with seven-hole ‘simulated’ fuel samples at 2450⁰C with a Hydrogen flow rate of 33.3 x

103 liters/hr and pressure of 52 kg/cm2, high temperature and high vacuum furnace made entirely of Tungsten, and a hydrostatic press for the consolidation of powders 10.

Different material-properties were studied, as the fuel underwent environmental effects characterized by their thermal expansivity, Young’s modulus, bend ductility, and creep susceptibility.

There are also various methods for fabricating fuel, such as to roll material into plates 11. These plates were created with 80% Tungsten and 20% UO2 via sintering and hot rolling. After failure testing, four effects were observed that can account for the fuel loss: damage to the interconnection of fuel particles; diffusion of fuel through micropores and microcracks high local concentration of impurities; partial decomposition of the fuel, as seen in Figure 5.

Figure 5 – 80% Tungsten 20% Uranium Oxide material that has undergone failure testing, exhibiting microcracks 11. This image clearly shows cracks that have developed

11 in the Tungsten, through which Uranium Dioxide travels.

These researchers also analyzed samples that had different types of surface cladding, via either vapor deposition or plasma spraying, for Tungsten-coated fuel particles 11.

Cladding is a popular failure mitigation technique for attempting to prevent the creation of cracks and the migration of material throughout the fuel.

Past research for the development of WUO2 fuel was mostly underway during the time period in which the GE 710 nuclear aircraft and ANL nuclear rocket program occurred. Table 1 demonstrates a summary of these two programs’ WUO2 fuel failure modes and their physical mechanisms, including solutions adopted to address these issues

12. One of the main issues to overcome with Tungsten, is that the brittleness of the metal can sometimes lead to cracking and rupturing of the fuel element, thus unit cohesion is a major issue 3. Fuel development efforts of the past improved upon their materials by creating finer, more spherical particles with a more uniform mixture of powder and uniform sintering capabilities. This helped to eliminate some internal cracking and failure within the ceramic matrix.

Fuel Failure Mode Physical Mechanism Solution Adopted

Loss of Oxygen Tantalum clad transparency to medium/high temp Clad material changed to Tungsten

Volumetric expansion and cracking Coefficient of thermal expansion mismatch Tungsten alloyed with Re, Mo-Re

Void formation Insufficient autoclaving W alloys to improve ductility

Fission product release Building up pressure, dislocation weaknesses Sinter to 84% of theoretical density

Vaporization of Mo Vapor pressure of Mo significant above 2470K Problem not solved

Table 1 - Different GE 710 fuel failure modes, mechanisms, and adopted solutions.

The practice of cladding (coating) the particles in a different material can also help to insulate them from impurities. ANL conducted burnup analysis for various different specimens of WUO2 and Molybdenum Uranium Oxide (MoUO2), with different cladding such as Tungsten Rhenium (W-25% Re) and W-30%Re-30%Mo, both of which were generally leak-tight at the end of the tests at 2400o to 3000⁰C, but they exhibited some deformation and diameter changes up to 2% due to internal pressure build 12. Specimens with a vapor-deposited W cladding resulted in leaks, with no dimensional change. The

ANL nuclear rocket program’s 13 metallurgy research showed W or W-25 w/o Re alloy cladding on stabilized UO2 may help with the release of the UO2 during many thermal cycles in pure hydrogen.

The Russian research group (LUTCH) also experimented with cladding, creating a cermet candidate fuel formed of uranium zirconium carbo-nitride (U,Zr)CN within a tungsten matrix and clad in tungsten 12. This particular fuel was used in the Russian nuclear rocket program, being developed into a twisted ribbon carbide fuel. The success of this type of fuel could be explained by the inhibition of carbon migration, as indicated by the sharp clear boundary between the cladding and the matrix. This prevents problems that could arise between the carbon and tungsten, as tungsten and carbon tend to form a eutectic at moderate temperatures.

Building upon a heritage of Tungsten-based NTR fuels research, NASA MSFC is currently undergoing the Hot Isostatic Pressing (HIP) method of materials fabrication, in which powders are pressed into a rod at 2000oC and a pressure of 30 ksi 14, using the chamber seen in Figure 6. These fuel rods include the various strengthening elements

13 needed to mitigate failure, such as cladding around the fuel channels and placing a spherical coating of Tungsten around individual UO2 particles. The addition of oxides is underway as well, for stabilizing effects.

Figure 6 – The NASA MSFC method of fabricating NTR fuel samples, using the HIP method. The canister on the right is filled with powders and then placed inside the device, seen on the left, whereby high pressure and temperature press the powders together, forming a sample. This picture was taken in 2012, and the facilities have been upgraded since then.

2.1.1.1 Thermal Cycling and Oxides

The heating and cooling thermal cycle involved in the operation of an NTR engine could prove problematic, since Tungsten tends to crack once cooled, due to its brittle nature and aforementioned issues of the DBTT. Pre-heat treatment can be an important procedure to use during failure testing for materials undergoing failure testing in a thermal environment. To do so, specimens are prepared for thermal cycling tests with

15 pre-heat treatment in order to remove surface oxides and exposed UO2 particles .

Another way to mitigate the failure of WUO2 fuel, is the addition of oxides.

Previously, the losses of surface-cladding on WUO2 composites that occurred during thermal cycling could not be prevented, and so Thorium dioxide (ThO2) was added to the

Tungsten matrix, as well as solid solution additions of various Oxides (such as CaO,

14 13 ZrO2, or ThO2) to the UO2 . The ANL nuclear rocket program also found that an addition of certain oxides (such as GdO1.5, DyO1.5, and YO1.5) suppresses WUO2-

Hydrogen reactions and helps to increase the structural stability of the fuel. The Yttrium oxide additive prevents the most fuel loss, with 10 molar percent of Y2O3 in the UO2 giving the longest life. The investigators determined that thermal-cycling induced fuel migration occurs when the difference in the thermal expansiveness of each of the elements drives the Uranium-rich phase from their restraints and into the grain boundaries of the Tungsten; this was mitigated with the addition of the additives 11.

2.1.1.2 Stoichiometry

A stoichiometric molecule of UO2 is a fluorite lattice which contains Oxygen atoms, and a molecule is hypostoichiometric if it contains 2-x Oxygen molecules 12. The

15 decomposition of UO2 contributes to the loss of Uranium during thermal cycling, as the

Oxygen separates itself from the molecule in a substoichiometric event,

푈𝑂2 ↔ 푈𝑂2−푥 + 𝑥[𝑂], (2.1) in which the Oxygen molecules will leave their molecular positions to be replaced by reduced U4+ ions. Then a second reaction during the cooling phase occurs, in which

𝑥 𝑥 푈𝑂 ↔ (1 − ) 푈𝑂 + ( ) 푈, (2.2) 2−푥 2 2 2 the free Uranium will migrate through the Tungsten grain boundaries, leading to cracking and mechanical degeneration.

One proposed reason 15 for fuel-loss stabilization via the addition of oxides, is that oxygen vacancies are filled by the ions of the respective additives, and they are introduced into the fluorite lattice without corresponding reduction of U4+. This is because there are lower cation valences of the additive’s respective ions, such as for Ca2+

3+ or Y . However, this theory cannot be supported for ThO2, since vacancies do not occur with the presence of this additive; furthermore, due to other factors that affect ThO2’s potential as a candidate, the additive was not considered 15. Another possibility that could account for the stabilizing effect other than the oxygen vacancy defect has to do with each of the solid solutions oxidizing to different degrees. Due to a lack of a suitable analytical method, the oxygen-metal ratio was not controlled. An assumption can be made however that the oxygen-metal ratio was in excess of the stoichiometric value. The

ANL nuclear rocket program 13 also addressed the importance for a new method of coating and blending UO2, for optimum sintering conditions that keep the stoichiometry of UO2. They also discuss how the presence of Carbon and/or free Uranium within vapor- deposited cladding contributed to the early failure of the WUO2 fuel.

2.1.2 Understanding Failure Mechanisms

It is useful to review some engineering terms, as applied to determining fuel failure mechanisms for WUO2, and to understand how to mitigate them, combined with an understanding of the performed literature review.

2.1.2.1 Mechanical Deformation, Defects, and Fractures

Firstly, mechanical deformation, defects, and fractures are caused by external, applied, and exponentially dependent forces, dependent on temperature and activation energy. The failure from the forces’ flux will provide a flux divergence, which is proportional to degradation rate. Various models assume a power law or exponential dependence on the applied stress, such as the “Arrhenius” Mechanism, in which there is an exponential dependence on temperature and activation energy 16.

The WUO2 fuel can be developed by understanding the response of the material through different external stimuli, with the goal of learning how the material responds for these stimuli, describing the structure at every level (electron, macro structure, and so forth), and noting the difference between intensive and extensive responses. Various stimulus types that the fuel could encounter are mechanical (load, force), electrical

(electric, electromagnetic), magnetic, thermal, and other (acoustic, radiological). Each of these different stimulus types are responsible for different types of stress (torsion, compression, hydrostatic, etc.), engineering strain, and tensile strain.

Due to the nature of a cermet, there is a low tolerance for the Tungsten material to undergo plastic deformation, leading to brittle fracture through elastic energy. This tolerance is quantified through exploring the yield strength, as described by Hooke’s Law

𝜎 = 퐸휀, (2.3) where E is the inverse modulus of elasticity and ε is constant, the Young’s Modulus, as the ANL research describes 10,13. The material can be strengthened by raising the toughness level of the material, the energy to break a unit volume of the material, and the

17 resilience - the ability of the material to store energy - of the WUO2 material . As seen in the literature review, microvoids (void nucleation) due to low values in each of these variables could develop at times, which lead to some UO2 meandering through the

Tungsten matrix, developing cracks that led to fuel failure.

Different methods may be explored to raise the toughness factor for WUO2, through methods such as heat treating and pre-stressing, as described in the literature review. Post-experiment pre-stressing of materials for residual stress, in the form of compression and tension in order to achieve balanced internal forces, and the post- experiment dissection, such as, x-ray techniques to view residual stress, are important methods for locating different stresses and failure mechanisms. Since metals expand and contract when heated and cooled the WUO2 fuel will undergo residual stress since it is restrained within the pressure vessel of the reactor’s container. Additionally, although they are meant to be eventually removed for operable use, the fuel rods fabricated using the HIP-method of fabrication at NASA MSFC still have the can surrounding the material. This will lead to ductile and brittle fractures during failure testing, as seen in the conclusion section of this dissertation. Brittle fracture is highly undesirable, because it happens suddenly, due to the successive and repeated breakage of atomic bonds along the crystal planes.

The mechanisms for brittle fracture can be controlled by measuring and

18 controlling the cracks within the material, and there are various methods for doing so. For instance, the Griffith theory of brittle fracture 18 locates a point of stress concentration by studying a crack tip, and thereafter can calculate the critical stress which is needed for crack propagation in a brittle material:

2퐸훾 𝜎 = √ 𝑠 , (2.4) 𝑐 𝜋𝑎

where ‘a’ is ½ length of the internal crack, and γs is specific surface energy. The stress concentration that leads to cracks can be lowered by lowering the inverse modulus of elasticity ‘E’, in order to counter the application of any specific surface energy.

2.1.2.2 Imperfections and Fatigue

Imperfections and defects in the material that are introduced during the fabrication stage of WUO2 development could lead to reliability problems, since localized imperfections could lead to fatigue, creep, and failure. Defects in the material can be thought of in terms of dimensionality. One-dimensional imperfections are point defects, which have to do with atomic vacancies or extra atoms (impurities) within the matrix of the fuel element. The creation of vacancies within the matrix can lead to microvoids, which can become stress concentrators, as demonstrated in the literature review. For a two-dimensional defect, an entire row of atoms is taken out of the matrix.

These are also dislocations, or missing planes of atoms. Here, the deformation is permanent. Some key principles of thermodynamics that drive all processes are considered in order to understand these processes, and in general thermodynamics states that the system seeks to minimize the free energy of a system, and increase disorder,

19 entropy, and failure.

A ceramic is a compound between metal and non-metal, joining a cation and anion, and throughout the material defection process, charge neutrality has to be maintained amongst the molecules. Ceramics contain two different kinds of ions, so defects for each type of ion can occur, such as the Frenkel defect which is a vacancy- interstitial pair in the molecular lattice and the Schottky defect which is a vacancy- vacancy pair 19. The slip plane within the molecular lattice usually contains the highest density of atoms, and it determines the direction of the material’s movement. Inherently, metals are easy to slip because they are a sea of electrons. It is important to apply strengthening mechanisms to prevent failure through the control of dislocation motion.

This can be done by controlling physical aspects of the material, such as grain size specifications. In the case of WUO2 fuel, small grains are good, as the literature review describes for fuel performance.

Failure can also be prevented by removing obstacles to dislocation motion, through alloying and purposeful imperfections, such as with the addition of oxides to the

WUO2 fuel. Fatigue failure from cyclical loading occurs for 90% of mechanical engineering failures, and thermal cycling was a major cause for failure within the literature for WUO2 fuel. In the literature, UO2 mass transport between W grain boundaries occurred, in particular during thermal cycling, and this contributed to failure of the material.

To classify erosion, one may start with analyzing the visual appearance of the

WUO2 fuel after testing, for instance: is there pitting, cracking, or uniform corrosion?

Electrochemical corrosion includes processes of oxidation and then reduction, and can

20 also be explained in terms of aqueous destruction 17. Oxidation forms metal ions that go into the solution during an anodic reaction, whereby an element loses at least one electron. Reduction takes place during the consumption of electrons in a cathodic reaction, whereby an element gains at least one electron, and a half-reaction, when oxidation and reduction occur simultaneously 16. These two reactions take place at the same time as corrosion. This is then paired to the corrosive media, which may range from marine, atmosphere, and acids, or in the case of the WUO2 fuel failure tests, Hydrogen,

Helium, or cathodic arc jet plasma. These observations can then lead one to correlate a corrosion process and protection mechanism.

There are different cells within the material, such as: the galvanic cell where dissimilar metals are physically joined and the more anodic one corrodes; concentration cell; and stress cell 19. A higher temperature means more oxidation because the cell becomes less negative in terms of free energy. This can be observed in thermal cycling, such as, when one raises the temperature, WUO2 absorbs shocks better, and as the temperature goes down, the fuel becomes more brittle like glass. In order to avoid this, one could attempt to ensure that the material’s ductile-to-brittle transition temperature is lower than its operating temperature. It is also useful to take into consideration that irradiation has an effect on the ductile to brittle transition, as well, since the fuel and the surrounding reactor will be undergoing nuclear radiation. The influence of the corroding agent’s concentration and temperature on the cell’s potential can also be quantified. For the WUO2 fuel, UO2 stringers 3 traveling through the tungsten matrix are a mechanism driving intergranular corrosion, which can be driven by the potential difference between different grains. To look at susceptibility, chemical engineers may look at the ditch

21 structure within the matrix with oxalic acid etch. This involves a lengthy process of boiling the sample in for several hours.

Since the Hydrogen is flowing with some mass flow rate, the specimen can undergo a form of erosion corrosion, caused by turbulence, impingement, and cavitation.

This type of corrosion begins with a break in the passive film. The electrical resistance of the material can help to determine the corrosion rate, with the ideal state of the molecules being a passive one, since passivity is a corrosion resistant state. A material’s passive state can be determined with the help of a Pourbaix diagram, as seen in Figure 7.

Figure 7 - Tungsten’s Pourbaix diagram. This demonstrates an ideal balance between the pH voltage potential of Tungsten that would keep the material at a passive state, which would prevent it from corroding.

The absorption of anions at the oxide surface enhances catalytic breakdown of the

22 passive layer 16. Once a brittle oxidation layer forms, the galvanic potential between dissimilar metals and irradiation increases the conductivity of oxides, accelerating the breakdown process. Another corrosion process is stress corrosion cracking (SCC), which leads to pitting, which can then lead to erosion corrosion 18. Proposed mechanisms for

SCC are hydrogen embrittlement, absorption induced cleavage, atomic surface mobility and surface diffusion, film rupture and film induced cleavage, and localized surface

18,20 plasticity (metal weakened by anodic dissolution) . The WUO2 fuel could undergo each of these mechanisms that lead to stress corrosion cracking, such as Hydrogen embrittlement, in which molecules from the hydrogen propellant may lodge within the material, wedging between lattice atoms and prying them apart 20.

2.1.1.3 Developing upon the Current Fuel Failure Studies

One of the first suggestions that can be made after doing the literature review and reviewing available analytical methods, is that the oxygen-metal ratio should be

15 controlled for further WUO2 fuel testing in order to compare this ratio to the stoichiometric values for the fluorite lattice. This is in order to prevent a ternary eutectic, which has a low melting point. The prevention of eutectic region development is important for improving the robustness of the fuel, as was achieved by the Russian

LUTCH fuel development program.

Another suggestion that can be made is a more sophisticated quantification and analysis of crack propagation throughout the microstructure, with available (and supposedly more sophisticated) imaging technology. This can help to indicate a derived time to failure and propagation rate for the crack for a number of given thermal cycles.

This could tie into the next suggestion, which is an analysis of the molecular structure of the WUO2 and the fluorite lattice as it is undergoes different stoichiometric phases, in combination with various ways that this molecular structure is affected by atomic processes, such as embrittlement from the Hydrogen propellant 20, the regulation of free energy between the elements (in order to improve the ductile to brittle transition for the cermet during thermal cycling), and measurement of the potential difference between grains with the use of tools such as Pourbaix diagrams.

NASA MSFC has two main failure testing chambers for use on WUO2 fuel - the

Nuclear Thermal Rocket Element Environmental Simulator (NTREES) and CFEET.

These test chambers operate by heating the fuel from the inside out – NTREES passes hot

Hydrogen through the holes of the sample, while CFEET heats the sample using a surrounding RF coil 14. NTREES is currently being upgraded 21 to be capable of providing 5 MW of power as seen in Figure 8, in order to test fuel elements to near- prototypical operational conditions. Post-testing failure analysis of a sample tested in

NTREES was conducted for the purposes of this dissertation as an introductory understanding of how NASA MSFC fuel undergoes failure, and the results of these analyses are presented in the conclusion. Additionally, time was spent participating in the development of CFEET, and for the refinement of the failure testing process. Suggestions for future fuel failure testing procedures based on this experience with CFEET are discussed in Appendix B.

Figure 8 – NASA MSFC’s NTREES testing facility, mounted above its power units 21.

A 1 MWe arc heater for plasma ablation testing was also constructed at NASA

MSFC which operated by passing hot Hydrogen plasma at 3160oK over the sample 14,22.

The original purpose of the arc heater was for preliminary vetting of material samples; however, it was not able to conduct further studies of ablation on fuel elements. Since extensive experimental plasma ablation studies are not present in the literature as a method of failure testing for WUO2 fuel and follow-up tests have not been conducted at

NASA MSFC for arc heater tests on its Tungsten-based fuel samples, a suggestion is made to conduct more thorough and rigorous failure testing on the material using plasma ablation.

2.2 Plasma Ablation as a Method of Failure Testing

The kinetic theory of gas transport, describes the distribution of particles according to a Maxwell-Boltzman distribution function. The distribution function of particles is a function of velocity, but the distribution of the particles within a space depends only on the speed of the gas 23.

2.2.1 The Kinetic Model

The phenomena involved with the surface ablation of a material have a detailed history of research and understanding. Exposure to an ablative agent, such as a laser or a plasma beam, will exert high levels of pressure and temperature to the surface of a material. This will then cause an inequality of pressures at the molecular level, as particles are no longer held to the surface and seek to travel outwards and into the surrounding vapor. Plasma ablation can be comparative to a metal being ablated with a laser, which uses the same assumptions about metal boiling - pressure equilibrium in the material with the outer pressure - and metal vaporization. This assumption was made by 24, who developed the kinetic model of ablation for his experiments on a material undergoing laser ablation. The model was developed by utilizing the Maxwell-Boltzmann distribution equations with an assumption that the distribution existed in two layers, as seen in Eq (2) and Eq (3). The original ablation model 25 utilizes the following two functions: firstly, the function of returned particles near wall

3 2 푚 2 −푚풗 𝑓0(풗) = 푛0 ( ) exp ( ) , 𝑣푥 > 0 (2.5) 2휋푘퐵푇0 2푘퐵푇0 and secondly, the function of returned particles at outer boundary of kinetic layer

3 2 2 2 푚 2 −푚((푣푥−푢) +푣푦+푣푧 ) 𝑓1(풗) = 푛1 ( ) exp ( ). (2.6) 2휋푘퐵푇1 2푘퐵푇1

The kinetic model can also be used to calculate ablation rate of the material, in analyzing the net ablated flux and properties of the plasma bulk outside of the region understudy

25,26,27. Therefore, the model was developed to include a third layer, as seen in Figure 9.

Firstly is the layer near the surface known as the Knudsen layer, with the parameters (n0,

T0, V0); this is where the ablation occurs. Secondly is the layer that is roughly a few mean free paths λ away from the surface known as the hydrodynamic non-equilibrium layer

(n1, T1, V1), which contains a mixture of particles being ablated from the surface and a backflux of ablated particles being pushed in by the main plasma plume 25,26,27. The main plasma plume is indicated by the subscripts 2.

Figure 9 - Layers of the kinetic model, representing the surface (subscript 0), the hydrodynamic layer including the backflux (subscript 1), and the main plasma plume

(subscript 2).

The objective of solving these problems was to uncover the unknown variables within the hydrodynamic layer, using known parameters from the surface of the material and the plasma bulk. Before conducting simulations, a simplifying assumption can be made at the boundary between the plasma bulk and the hydrodynamic layer 25 in which

푛1 1 = 2푛2 (2. )

2 2 푛1푘푇1 + 푚푛1 1 = 푛2푘푇2 + 푚푛2 1. (2. )

This helps to the equation for the Mach number α, which is a ratio of velocities at that boundary layer,

푇 푛 푛 (( 2 2)− 1) 2 2푇 2 푣1 1 𝛼 = √ = √ 2 (2.9) (2푘퐵푇1)/푚 푛1 (푛1− ) 푛2

The goal is to calculate the distribution of particles within the various distribution regions, as a surface is subjected to an oncoming plasma flow. The equilibrium of particle fluxes are represented as follows

𝑠 + = + (2.10)

1 flux of particles entering from the surface = , subscript 0 , flux of particles 𝑠 2√휋 from the outer boundary layer = incoming from the plume, flux of particles returned to the surface during the time step, subscript 0 for vx < 0 , the backflux is

2 𝑣1 푛1 [ 𝑓 ( ) − 𝑥 ( 2)], (2.11) 2 𝑣1 2√𝜋 𝑣1 and flux of particles crossing the outer boundary of the layer. A question may be raised about the existence of an additional shock layer. Previous research using the GWU arc jet has concluded that during hypersonic experiments there was no observation of a

28 significant shock layer; however, the study suggested that the visual evidence of a shock layer may be seen by raising the background pressure above 0.1 Torr 45.

2.2.2 Chapman Enskog Expansion and the Thermal Conduction Parameter

The Chapman Enskog expansion 28 of the Maxwell Boltzman equation uses the following equation to include a degree of departure from local translational equilibrium, ξ

휕 휕 휕 휕 ̂ ̂ ̂ ̂ ξ [ (푛̂𝑓) + 푗̂ (푛̂𝑓) + 퐹̂푗 (푛̂𝑓)] = [ (푛̂𝑓)] (2.12) 휕𝑡̂ 휕𝑥̂푗 휕 푗̂ 휕𝑡̂ 𝑐표푙푙

푛 to derivate the distribution function f as a power series with 훷1 = ξ 휙푛

𝑓 = 𝑓0(1 + 훷1 + 훷2 + ∙∙∙) (2.13)

Which for the first order approximation, for small ξ and 훷1 ≪ 1, is represented as

𝑓 = 𝑓0(1 + 훷1) (2.14)

Where the following conditions must be met, where C is a velocity function

1 2푘푇 휕 훷1 = − [√ 𝐴푗 (푙푛푇)], (2.15) 푛 푚 휕𝑥푗

𝐴푖 = 𝐴(𝐶, 푇)𝐶푖, (2.16)

5 𝐼[𝐴𝐶 ] = 𝑓 𝐶 (𝐶2 − ). (2.1 ) 푖 0 푖 2

This expansion can be utilized to create an updated distribution function to include a thermal conduction parameter , which occurs in the boundary layer at ablative surface 29,

휆푚 푝 = , (2.1 ) 훿𝑥푇 where 휆푚 푝is the gas mean-free path and 훿𝑥푇 is the characteristic gradient length. The

29 updated distribution functions are represented within two primary regions – one region annotated as 𝑓 (→) is near the surface within the Knudsen layer, which has a thickness 푣 of a few mean free paths, and the other region is the plasma bulk, annotated as 𝑓푢 (→). 푣

The 𝑓 (→) is similar to the original model and describes the distribution of 푣 particles inside of the Knudsen layer, for particles ablating from the surface in the positive x direction and for particles returning to the surface traveling in the negative x direction.

1 3 2 ( ) exp(−(𝑣) ) , 𝑣푥 > 0 √𝜋 𝑓 (→) = 3 (2.1 ) 푣 2 1 (𝑣푥 − ) 푛1 ( ) exp (− 2 ) , 𝑣푥 0 { 𝑣1√𝜋 𝑣1

𝑓푢 (→) describes the distribution of particles beyond the Knudsen layer within the plasma 푣

28 bulk, and it includes the Chapman Enskog expansion for thermal velocity 𝑣푇

2 2 2 𝑣푇𝑣1 (𝑣푥 − ) (𝑣푥 − ) + 𝑣푦 + 𝑣푧 5 𝑓푢 (→) = 푛1𝑓 (→) {1 − [ ( − ) (푙푛푇)]} (2.20) 푣 푣 2 𝑣𝑐 𝑣1 𝑣1 2 𝑥

3 2 1 [(𝑣푥 − ) ] 𝑓 (→) = 푛1 ( ) exp (− ), (2.21) 푣 2 𝑣1√𝜋 𝑣1 for a thermal conduction parameter with a collision frequency vcf

(푙푛푇) 𝑣 = ( 푇 1). (2.22) 𝑥 𝑣𝑐

Ref. 29 demonstrated how a modification to the kinetic model would indicate a temperature gradient going towards the surface, compared to an outward temperature gradient without conduction, and that the calculations worked best with a small temperature conduction parameter, much less than 1. In order to obtain the derivations for

30 these fluxes, the following relationship is employed, for the distribution of particles divided into two main regions – near the Knudsen layer and in the plasma bulk

𝑓 (𝑥,→) = 훿(𝑥)𝑓 (→) + [1 − 훿(𝑥)]𝑓푢 (→). (2.23) 푣 푣 푣

These calculations can represent particle distribution coming off the surface when

훿(𝑥) = 훿(0) = 1, in which case 𝑓 (→) is represented for our analyses. The operator 푣

훿(𝑥) is set to 훿( ) = 0 in order to derive the distribution of particles beyond the

Knudsen layer integrating the function 𝑓푢 (→), in order to find the constants c1, c2, and c3 푣 of the equations (for mass, momentum, and energy respectively).

The surface distribution function is normalized for mass, momentum, and energy and uses the following relations 29

푥 = 푚푛0 𝑣푇 1, (2.24)

2 푥 = 푚푛0𝑣푇 2, (2.25)

𝑣3 퐸 = 푚푛 푇 . (2.26) 푥 0 2 3

The equations for mass, momentum, and energy distribution can be used for integrating the velocity distribution functions 25,26,27,29, according to:

+ + 𝑎 1 = ∫ 𝑓 (𝑥, →) 𝑣푥 𝑣푥 ∫ 𝑣푦 , (2.2 ) 푣 − −

+ + 2 푚 푛𝑡 푚 2 = ∫ 𝑓 (𝑥, →) 𝑣𝑥 𝑣𝑥 ∫ 𝑣 , (2.2 ) 𝑣 − −

+ + 2 퐸푛 3 = ∫ 𝑓 (𝑥, →) 𝑣 𝑣푥 𝑣푥 ∫ 𝑣푦 . (2.2 ) 푣 − −

The variable ‘v’ without a subscript is the magnitude of the velocity vector, or the root- mean-square. The thermal parameter only manifests itself in the constant for the energy

31 equation for c3, and it does not appear in the mass and momentum equations, in calculating c1 and c2. All three constants will be modified though, according to the new tangential velocity component.

The equilibrium density N0 is a known range of values, due to the known properties of a material that is being converted to a vapor, as seen in Figure 10. This is directly applicable to the kinetic model of plasma ablation, for which particles are leaving the surface of a material due to the pressure and temperature changes it is being subjected to.

Figure 10 - Particles leaving the surface of a material into a vapor. Once the pressure of the vapor is equal to the pressure of the particles at the surface, that pressure is said to be saturated, and a correlation can be drawn between the saturation pressure and the saturation temperature.

Once the pressure is quantified, then the equilibrium density can be found using the gas equation

0 = , (2.30) 푘퐵푇 for a given surface temperature range. One must firstly use the Clausius-Clayperion

30 equation in order to find the pressure P0,

푙푣(푇 − 푇 ) 𝑠 = 0 exp { }, (2.31) 푇푇

31 Where Hlv is the enthalpy of vaporization (which is 4.35E+06 J/kg for Tungsten) , R is the specific gas constant, which can be calculated by R = cp – cv = 45.15 (for Tungsten at

32 3400 K) , and Tb is the boiling point of Tungsten, at 5828 K.

Previous evaporation research 33 has been conducted by Tungsten in order to determine the relationship between its saturation pressure and saturation temperature. The following relationship between the saturated pressure and temperature was derived using the Clausius-Clayperion relationship and verified against experimental data and was used in the computational simulations:

4 ,440 푙 10 𝑠 = 15.502 − − 0. 푙 10푇𝑠 . (2.32) 푇𝑠

Chapter 3 Computational and Experimental Set-Up

3.1 Including Tangential Velocity

The previous kinetic ablation model for the modified Anisimov plasma ablation code, to include Chapman Enskog thermal conduction, did not consider the tangential component of plasma ablation. This is applicable to the scenario of atmospheric reentry, where the reentering object is most oftentimes reentering at an angle.

3.1.1 Modification of the Kinetic Model Equations

Drawing from the current kinetic model for plasma ablation and including the

Chapman Enskog distribution function, a tangential component may be added to the distribution functions, such that there is now an ‘x’ and a ‘y’ velocity component. This modification can be applied with the underlying assumption that the reentering body is not blunt and has angular sides, – which may result from the fracturing of the core as it breaks into fragments. It could also help to develop plasma ablation models for 2- dimensional plasmas around a surface which is not facing the incoming plasma directly.

The integration of the distribution functions within the Knudsen layer remain the same, while the distribution function for the particles beyond the Knudsen layer incorporate a tangential velocity component. The reason that the region within the

Knudsen layer does not incorporate the tangential velocity is because it is too small and too close to the surface to be affected by anomalous effects from the plasma bulk flow.

The distribution function with the additional tangential velocity component becomes:

2 2 𝑣푇𝑣1 (𝑣푥 − ) (𝑣푥 − ) + (𝑣푦 − 푛) 5 𝑓푢 (→) = 푛1𝑓 (→) {1 − [ ( − ) (푙푛푇)]} 푣 푣 2 𝑣푚 𝑣1 𝑣1 2 𝑥

(3.1)

3 2 2 1 [(𝑣푥 − ) + (𝑣푦 − 푛) ] 𝑓 (→) = 푛1 ( ) exp (− ). (3.2) 푣 2 𝑣1√𝜋 𝑣1

The distribution functions still maintain the assumption of a backflux of particles ablating from the surface, while omitting plasma flow in the z direction.

3.1.1.1 Calculations for Mass Conservation

The initial derivations 25,26,27,29 for the particle distribution assume a backflux in the x direction and the velocity distribution function is integrated for ‘x’ for the function representing the particle distribution beyond the Knudsen layer, giving a constant value of

3 1 푛1 1 = ( ) . (3.3) 𝑣1 𝜋

Now with a tangential velocity applied in the ‘y’ direction, an extra integral is added

∬ → 𝑣푥𝑓 (𝑥,→) = 1. (3.4) 푣 푣 −

The mass, momentum, and energy equations were calculated by using an integral substitution of

(𝑣 − ) = 푦 푛 ⁄ , (3.5) 𝑣1 in order to compute the exponential within the integral. The new constant becomes

푢 3 푛 1 푛 1 푣1 1 = ( ) 푛1 [−√𝜋 𝑓 ( ) ] , (3.6) 푢 𝑣1 𝑣1 2 푛 − 푣1

And after inserting the limits of ∞ and -∞ into erfc(∞) = 0 and erfc(-∞) = 2

3 1 푛1 1 = ( ) . (3. ) 𝑣1 √𝜋

Thus, the constant remains the same as before after the normalization of √𝜋 with the momentum and energy constants, which will also have this term.

3.1.1.2 Calculations for Momentum Conservation

The original calculations 25,26,27,29 for the integration of the distribution function, employing only the ‘x’ direction, gave a constant of

3 2 1 푛1 2 𝑣1 2 = ( ) ( + ), (3. ) 𝑣1 𝜋 2

Now employing the double integral

2 ∬ → 𝑣푥 𝑓 (𝑥,→) = 2, (3. ) 푣 푣 −

The modified momentum constant becomes

3 2 푧 1 푛1 2 𝑣1 1 2 = ( ) ( + ) [√𝜋 𝑓( ) ] , (3.10) 𝑣1 𝜋 2 2 푧 − and with the limits of erf(∞) = 1 and erf(-∞) = -1

3 2 1 푛1 2 𝑣1 2 = ( ) ( + ), (3.11) 𝑣1 √𝜋 2 which is also equal to the original derivation for the momentum constant, as was also demonstrated for the mass conservation constant, after normalization with √𝜋.

3.1.1.3 Calculations for Energy Conservation

The energy equation is the only one that differs from the previous calculations, due to the v2 term, which is the square of the magnitude of the velocity vector, and the

2 2 full extent of the derivations are in Appendix E. Before, when v = vx , the original derivation for the energy conservation equation gave the constant 29

3 2 ( ) ( ) 3 1 푛1 2 5𝑣1 𝑣푇𝑣1 푙푛푇 5𝑣1 3 = ( ) [ ( + ) − ]. (3.12) 𝑣1 𝜋 2 𝑣푚 𝑥 4

2 2 2 Now, v = vx + vy , which means that there is an additional grouping of integrals to calculate for the energy term. The integral now becomes:

+ + 3 2 ∬ 𝑓 (𝑥,→) 𝑣푥 풗 + ∬ 𝑓 (𝑥,→) 𝑣푦 𝑣푥 풗 = 3. (3.13) 푣 푣 − −

Recalling the structure of the function 𝑓푢 (→), which can be broken into two main parts, 푣 the new constant becomes:

3 = (퐹 𝑡 𝑡 푚 − 𝐶 𝑎 푚𝑎푛 푇 푚 ) = 𝐴 − , (3.14)

𝐴 = 𝑓 𝑡 3 + 𝑓 𝑡 2 = 𝑎 + , (3.15) 푣푥 푚𝑠 푣푦푣푥 푚𝑠

3 2 = (𝑣푥 𝑡 푚 + 𝑣푦 𝑣푥𝑡 푚 ) = ( + ), (3.16) where τ is the thermal conduction parameter

(푙푛푇) 𝑣 = ( 푇 1), (3.1 ) 𝑥 𝑣푚 which is assumed 29 to be much less than 1.

The calculations performed for the integrals in order to solve for the new energy constant are as follows:

Beginning with the integration of the First terms = a + b,

Calculations for a will result in:

2 3 2 [(𝑣푥 − ) + (𝑣푦 − 푛) ] 1 3 𝑎 = ∬ ( ) 푛1 exp (− 2 ) 𝑣푥 풗, (3.1 ) − 𝑣1√𝜋 𝑣1

1 3 푛 5𝑣2 𝑎 = ( ) 1 ( 2 + 1 ), (3.1 ) 𝑣1 √𝜋 2

Calculations for b will result in:

2 3 2 [(𝑣푥 − ) + (𝑣푦 − 푛) ] 1 2 = ( ) 푛1 ∬ exp (− 2 ) 𝑣푦 𝑣푥 𝑣푦 𝑣푥, (3.20) 𝑣1√𝜋 − 𝑣1

3 1 푛1 1 2 = ( ) [ (𝑣1 + 2 푛)] . (3.21) 𝑣1 √𝜋 2

Combining the results of a and b, will give

3 2 1 2 5𝑣1 1 2 𝐴 = 푛1 ( ) [√𝜋푛1 ( + ) √𝜋 + [ √𝜋(𝑣1 + 2 푛)] √𝜋 ] . (3.22) 𝑣1√𝜋 2 2 Now the results of the integration of the Chapman Enskog term incorporating the thermal conduction parameter can be acquired, performing the following integration:

1 3 = ( ) 푛1 √𝜋𝑣1

2 ( )2 ( ) 2 2 + [ 푣푥−푢 + 푣푦− 푢 푛 ] (푣푥−푢) (푣푥−푢) +(푣푦−푢 푛) ∬ exp (− 2 ) [ ( 2 − )] 𝑣⃗𝑣푥 풗, (3.23) − 푣1 푣1 푣1 2

Utilizing the following grouping for terms in the integration of the Chapman Enskog distribution function:

3 1 3 2 = ( ) 푛1 ∬ (𝑣푥 𝑡 푚 + 𝑣푦 𝑣푥𝑡 푚 ) 풗 = + , (3.24) √𝜋𝑣1 −

The integration of the Chapman terms = τ(c + d), results in:

Results of the calculations for c:

2 3 2 1 [(𝑣푥 − ) + (𝑣푦 − 푛) ] = ( ) 푛1 ∬ exp (− 2 ) 𝑣1√𝜋 − 𝑣1

2 ( ) (𝑣 − )2 + (𝑣 − ) 𝑣푥 − 푥 푦 푛 5 3 [ ( 2 − )] 𝑣푥 𝑣푥 𝑣푦 𝑣1 𝑣1 2

(3.25)

3 1 3 2 2 1 3 2 2 = ( ) 푛1 [ 𝑣1(5𝑣1 + 6 ) 𝜋 − 𝑣1(𝑣1 + 2 )𝜋] , (3.26) 𝑣1√𝜋 2 4

Results of the calculations for d:

2 3 2 1 [(𝑣푥 − ) + (𝑣푦 − 푛) ] = ( ) 푛1 ∬ exp (− 2 ) 𝑣1√𝜋 − 𝑣1

2 ( ) (𝑣 − )2 + (𝑣 − ) 𝑣푥 − 푥 푦 푛 5 2 [ ( 2 − )] 𝑣푥𝑣푦 𝑣푥 𝑣푦 , (3.2 ) 𝑣1 𝑣1 2

3 1 3 1 2 2 5 1 2 2 = ( ) 푛1 [ √𝜋𝑣1 ( √𝜋(3𝑣1 + 2 푛)) − √𝜋𝑣1 √𝜋(𝑣1 + 2 푛)] . (3.2 ) 𝑣1√𝜋 4 4 4 2

Combining c and d into the equation for B, will generate the following equation:

3 1 3 2 2 1 3 2 2 3 1 2 = ( ) 푛1 [ 𝑣1(5𝑣1 + 6 ) 𝜋 − 𝑣1(𝑣1 + 2 )𝜋 + [ √𝜋𝑣1 ( √𝜋(3𝑣1 + 푣1 2

1 2 2 )) − √𝜋𝑣 √𝜋(𝑣2 + 2 2 )]] (3.2 ) 푛 1 2 1 푛

The new constant from the energy integration equation is therefore:

3 2 푛1 1 2 5𝑣1 1 2 1 2 5 2 3 = ( ) [[푛1 ( + ) + [ (𝑣1 + 2 푛)] ] − [𝑣1 ( 𝑣1 − )]] √𝜋 𝑣1 2 2 4

(3.30)

Since is assumed to be much less than 1, the other terms grouped together with this parameter are called B and are also assumed to not influence the calculations for the particle distribution in a major way 29. The new constant from the energy integration equation can therefore be considered as:

3 2 푛1 1 2 5𝑣1 1 2 3 = ( ) [[푛1 ( + ) + [ (𝑣1 + 2 푛)] ]], √𝜋 𝑣1 2 2

(3.31) where B is much less than 1.

3.1.2 Implementing the New Derivation into the Code

The original version of the ablation code re-arranges the three main flux equilibrium equations, such that u1 is divided out in order to have equations represented as

𝛼 = , (3.32) 1

푛1 𝑎1 = , (3.33) 푛0

푇1 𝑎2 = . (3.34) 푇0

These ratios are used in order to interpolate through values of self-evolving α to solve for the unknowns u, n1, and T1. Taking the mass, momentum, and energy equations, a2 is first solved for using the quadratic equation after grouping the various equations into bunches

2 containing a2 and a2. Then, a1 is solved for. In order to calculate α, using a1 and a2, the following groupings were previously utilized in the kinetic ablation code, before introducing the Chapman-Enskog terms and the tangential ablation terms

𝑓 = 1 − e (𝛼), (3.35)

= − exp(−𝛼2) + 𝛼√𝜋 𝑓 , (3.36) 𝛼 = 𝑓 + 2 , (3.3 ) √𝜋

훷 = 2 + 4𝛼2 − 2𝛼√𝜋 , (3.4 )

𝛼2 √𝜋 = (1 + ) + 𝛼 𝑓 , (3.3 ) 2 4

5 = √𝜋 (𝛼𝛼2 + 𝛼) − (2 + 4𝛼2). (3.40) 2

The artifacts of c3 are in the term, which was created during the grouping process, utilizing the mass, momentum, and energy equations, for the purpose of solving for the unknowns. The term is updated to include the new energy terms, which are divided by

3 𝑣1 in order to put the unknown velocities u and utan in terms of α and αtan

5 1 1 5 = [(𝛼3 + 𝛼) √𝜋 + √𝜋 (1 + 2𝛼 )𝛼 − [ − 𝛼2] ] − (2 + 4𝛼2) (3.41) 2 2 푛 4

The computer code is provided at the end of this dissertation, in Appendix D.

3.1.3 Tangential Velocity Orientation

The previous model for using the kinetic theory of particle distribution for application for plasma ablation, assumed the following arrangement of layers above an ablating surface. The incoming plume has only an ‘x’ component of velocity. Once the sample has been set at an angle relative to the plasma plume, the ‘x’ component of the plume velocity is modified to become αnew and a ‘y’ component of velocity is included

αtan , as seen in Figure 11. These two components are now what make up α. Note – that in order to depict this configuration, the x and y axis on this schematic is rotated, compared to the original schematic.

Figure 11 – Updated schematic to include ‘y’ component of velocity, which separates the plasma plume into a normal and tangential component. The introduction of the ‘y’ component of the plasma velocity is only considered in the plasma bulk.

Since the cathode – and hence the incoming plasma plume - could not be rotated due to the arc jet hardware holding it in place, the tungsten foil was rotated relative to the incoming plasma plume, as seen in Figure 12. The Tungsten foil is held to a straight steel screw, going along in the ‘z’ axis, into the page.

Figure 12 - Orientation of the holder and sample relative to the plasma.

To set the foil at an angle, the ‘Smart Protractor’ phone application was utilized. Gravity travels down the negative ‘z’ direction, and a tilt of the phone relative to gravity will give precise angle measurements. A schematic of how the Smart Protractor was used is shown in Figure 13.

Figure 13 - The use of Smart Protractor to set the angle of the sample

Once inside the chamber, the sample and holder assembly is placed on a strut within the chamber, which maintains the axis orientations and thus, the fidelity of the angle measurement for the Tungsten foil.

Figure 14 - At θ = 90o, the sample is resting on the y axis and is perpendicular to the incoming plume, with a velocity component only in the ‘x’ direction, α.

The angle of the sample relative to the x axis can be used to determine the angle θ needed to obtain the new values for α, according to the geometrical relationships seen above.

Then the αtan component of the plasma velocity can be determined using geometrical angle relations and the two known quantities θ and α, which is now the hypotenuse and is known for a range between 0 < α < 4.

The sample at a resting position of 90 degrees is seen in Figure 14. The incoming plasma plume is the fixed reference point for the (x, y) axis since the cathode is fixed, relative to the sample. This is based on a geometrical understanding that a triangle’s

44 hypotenuse is a vector containing the information of both an ‘x’ and ‘y’ vector. The value of the hypotenuse does not change, even as the angle of the sample (and hence the lengths of αtan and αnew ) changes relative to the plume.

𝛼푥 = ( ) 𝛼. (3.42)

Once the sample has been tilted relative to the y axis, the ‘y’ component of the plasma becomes

𝛼 푛 = ( )𝛼. (3.43)

In studying these angle relations, as θ approaches 90o, the y component of the velocity, approaches 0 m/s, and this is the angle by which the sample is facing the plume directly.

As the sample rotates, the angle of the plume relative to the surface of the tungsten foil approaches 0o, which means that the ‘x’ component of the incoming plasma plume approaches 0 m/s.

3.2 Set Up for the Experiments within the Arc Jet

Previously, NASA MSFC had constructed a plasma arc jet and for purposes of doing ablation studies in order to support the nuclear fuels development program for

NTR. The inspiration for the research in this dissertation is based off of previous research conducted by the NASA MSFC plasma arcjet 22, which is shown in Figure 15. This multi-gas arc heater facility performed at 1 Mwe and tested rods composed mostly of

Tungsten using surrogates such as W-5%Re40%HfN fabricated using the HIP/sintering procedure at NASA MSFC 22. It revealed the ability for NASA's arc jet to run at durations of 30 minutes and demonstrated long duration tests at which the temperature of the specimen was raised to 2789 K 14,22.

Figure 15 - The NASA MSFC arc jet. The parts seen on the outside are the exhaust and the testing chamber. Inside the building are the power units.

While originally, the research conducted in this thesis was meant to utilize the capabilities of the multi-gas arc heater facility, an improvisation for using the GWU arc jet turned out to be an equally suitable candidate.

The plasma of the GWU arc jet is sustained by a welding unit power supply at

135-138 Amperes and in generated from a copper cathode that is 2 inches diameter with a

0.7 inch diameter cone, as seen in Figure 16. The anode which sustains an ongoing plume between the cathode and itself consists of the inner walls of the plasma chamber and a simple screw, which by air pneumatics triggers the arc jet by striking the cathode directly.

The magnetic coil is powered by an additional power supply at 20 Amperes at roughly

10~15 Volts with a magnetic field at the coil center of approximately 370 Gauss.

Figure 16 – The GWU arc jet in operation seen on the right, and the Copper cathode used to generate plasma, seen on the left.

Keeping in mind that one of the objectives of this research is the simulation of accidental atmospheric reentry of an NTR rocket with an exposed Tungsten-based nuclear core to air in plasma form, one may question the use of a metal-based cathodic plasma plume over a gas-based plasma plume. The arc jet being utilized by the GWU in the present work has been used in the past for investigating the effects of plasma on vehicles traveling at hypersonic speeds 34. An important phenomenon which is required to study the effects of plasma ablation via atmospheric reentry is the speed of the molecules which are interacting with the object, and the cathodic vacuum arc plasmas can thus be utilized as a source for a hypersonic jet. Experiments have shown that the plasma

47 plumes from Copper cathodes reach temperatures of 2 – 3.5 eV and velocities on the scale of 1E+4 m/s to 1E+5 m/s 35,36.

As testing is underway, some modifications need to be considered in order to maintain appropriate plasma conditions for ablation. After the protrusion of the cathode begins to wear away, there are two major phenomena that occur that need to be addressed during the experiments. Firstly, the distribution of the density of the plasma is more dispersed, due to the beam being more scattered and less focused. Secondly, the formation of cathode spots 37 and the erosion of the central protrusion were also problematic, leading to migration of plasma filaments from the central plume, which caused a range of values for the plasma density and the ablation rates. Throughout the duration of these tests, four cathodes were used. Once a cathode would begin to be populated with cathode spots, it would need to be faced in the lathe, which would restore the surface of the cathode to a smooth polish, as seen in Figure 17. This would also help with the problem of the plasma filaments arcing backwards onto the Teflon insulation material surrounding the cathode.

Figure 17 - The protrusion of the cathode would eventually be eaten away over time, due to the migration of plasma filaments over cathode spots. Therefore, the cathode would

48 need to be re-faced in the lathe periodically in order to produce the desirable density of plasma.

The samples tested in the ablation studies were either in the form of Tungsten foil or Tungsten Zirconium Oxide (WZrO2) fragments. As the sample is ablated, it will be taking on a layer of Copper. However, the Copper, with a melting point of 1358o K, will be re-evaporated when the Tungsten foil's temperature, with a melting point of 3695o K, is high, similar to the hypersonic experiments on Molybdenum conducted by the GWU lab. Furthermore, by increasing pressure in the chamber there should be less effect of

Copper and more of ionized gas.

Figure 18 – An example of the Tungsten samples that were tested in the experiments are seen on the left, and the method of taking the mass measurements is seen on the right.

The Tungsten foil was purchased from Alfa Aesar in sheets that were approximately 100mm by 100mm and a mass distribution of approximately 2.4 g per

50mm by 50 mm. The sheets were cut into small rectangles approximately 15 mm by 20 mm, as seen in Figure 18. The samples were handled with gloves, cleaned, and weighed

49 before and after testing with a Sartorius CPA225D scale, which has a capability to weigh samples with masses of 1E-08 kg. The weights were taken multiple times for one sample and averaged for accuracy.

The NASA samples came from a short rod comprised of Tungsten-Zirconium-

Oxide rod approximately 10 mm in diameter, with 7 holes approximately 1.76 mm in diameter, as seen in Figure 19.

Figure 19 – The Tungsten Zirconium Oxide fuel that was fabricated at NASA MSFC using the HIP fabrication method which was obtained for the purpose of failure testing for this dissertation. The holes, which were filled with Molybdenum rods, were etched out, and the parts of the rod that have the surrounding HIP can were not used.

The rods were fabricated with Molybdenum mandrels in order to form the holes and with a HIP can enclosing the material and holding the bottom of it. Thus, the rod needed to be submerged in an acid bath of nitric-sulfuric acid to remove the Molybdenum mandrels and to be broken apart in order to remove the parts of the sample that still contained the hip can. The acid bath took approximately two days. Then, the sample was

50 broken into fragments with a chisel. The pieces that were free of the mandrels and the hip can were selected for testing, as seen in Figure 20.

Figure 20 – Fragments of the WZrO2 rod that were selected for testing in the arc jet. As pictured, none of these samples have fragments of the HIP can on them, nor do they have fragments of the Molybdenum rods on them.

The samples were mounted on a stainless steel holder that protruded approximately 17.7 cm into the center of the vacuum chamber, directly into the middle of the plasma plume coming from the cathode. Furthermore, since the rod itself was connected directly to the chamber wall, which is considered to be the anode, it did not have a floating potential. The chamber would be pumped down to a vacuum of approximately 2E-04 Torr, which would take 30 – 60 minutes. After the testing was complete, the chamber and the sample were allowed to cool for approximately 5-10 minutes before extraction and processing.

Problems encountered in initial testing determined how the set-up of the experiments were to be conducted. For instance, the time duration of the test was

51 maintained at less than 30 seconds. At times approaching 50 seconds, the sample would be completely destroyed, as seen in Figure 21.

Figure 21 - Sample was destroyed at 51 seconds. Tests with subsequent foil samples were tested up to a maximum of 40.7 seconds to avoid destruction. The results from this test were not included in the data set

Furthermore, the sample was held in place by a steel holder, which would begin to melt when the holder was adjusted at an angle in order to conduct the tangential velocity experiments. This is because the foil sits inside a thin slot of steel, and once the holder was turned at an angle, the heat coming off of the Tungsten foil would enter the slots at an angle, causing the steel to melt onto it.

Therefore, the sample was perched at the very top of the holder as seen in Figure

22, with the slots adjusted to be very short. This fixed the problem of the steel melting and fusing onto the foil. It is interesting to note, that the steel became super-heated due to

52 the temperature of the Tungsten foil. Once the Tungsten foil was removed and perched to the very top of the holder, the steel no longer reached the melting temperature, because the influence of the heat of the Tungsten foil was removed.

Figure 22 – The set-up for ablation tests at an angle required that the sample be perched at the very top of the holder, as seen on the right. Otherwise, the sample would become very hot on one side of the holder, melting the steel onto itself, as seen on the left.

Since the NASA samples were irregularly shapes, the ends of the holder were pinched down and the sides of the slots pulled away in order to hold them. This

53 configuration did not seem to incur any issues, and thus this method of holding the sample during the testing of the NASA samples was used. The important parameter to monitor during the testing of the NASA samples was the density of the plasma plume, which needed to be kept at just the right conditions. If the plasma plume was too severe, it would cause the sample to become too hot, melting the holder onto itself. If the plasma plume was too weak, it would simply coat the sample in Copper, as seen in Figure 23.

Figure 23 – NASA sample on the left (7) coated in Copper, due to a plume which was too weak, and the NASA sample on the right (8) has a bit of melted steel on it, since the sample became too hot and melted the parts of the holder (seen on right).

Chapter 4 Experiment Results

Figure 24: A sample undergoing plasma ablation . The hydrodynamic layer is visible as a green glow.

Fifty-eight Tungsten foil samples were tested. The results of six samples were not used, because they were not able to achieve ablation. This is because these samples were tested with a Titanium cathode that had previously been used for plasma coating, and that set-up was not suitable for ablation experiments. The results of eight samples were not included, because they were used to determine the best experimental setup for the tangential holder, and thus these samples were mostly melted and destroyed along with some holders. The results of seven samples were not included in the analysis of this research, because they were at an angle other than 30 degrees.

The results for the experiments of the 37 remaining, suitable sample candidates are discussed here. These samples were tested at angles of 0 and 30 degrees. Eight fragments of the WZrO2 rod were also tested. The data from six of these fragments were

55 used, while the data from two of these fragments were omitted, since the plasma conditions were not good enough to include the data in the analysis.

4.1 Trials for 99.9% Tungsten Foil Samples

The results for the ablation of the 99.9% Tungsten foil samples are presented in in

Figure 25, for samples facing the plume at 0o and samples facing the plume at 30o. The samples tested at 0o faced the plasma plume for an average of 20.6 seconds, while the samples facing the plume at 30o were tested for an average of 20.9 seconds.

180 Ablation Rate 160 (kg/m^2s) 140 μ = 89.46 120 σ = 36.77 100 80 60 40 μ = 43.91 20 σ = 32.23 0 Samples

Figure 25: Comparison of experimental data points for samples tested in a direct plume

(sample at θ = 0o) and a plume with a tangential component (sample at θ = 30o) with data points showing error bars around two average lines – with the green representing the samples tested at 0 degrees and the blue representing the samples tested 30 degrees. The average ablation rate for the direct plume is 89.46 kg/m2s with a standard deviation of

36.77, while the average for the samples tested at an angle is 43.91 kg/m2s with a standard deviation of 32.23.

4.1.1 Observations and Issues

Some general observations were made about the material properties of Tungsten, before and after testing. Firstly, the DBTT of Tungsten took place during testing, and this was apparent after ablation, which caused the samples to become very brittle and to shatter easily when slight physical pressure was applied. The samples were more ductile before testing and required strong shears to cut them. The brittle nature of the Tungsten after testing was the most striking feature of the material, and the ability of Tungsten to become very hot and to destroy other materials that were nearby was the second general observation. This was observed when the Tungsten destroyed various steel holders when it became too hot, and the Tungsten also was able to quickly destroy type-K (chromel- alumel) thermocouples as seen in Figure 26, which were designed to take temperature measurements of up to 1623 Kelvin.

Figure 26 - Thermocouple set-up to measure the temperature of the surface of Tungsten.

The wires were thermocouple type K and were coated with small ceramic bits. However, the surface of the Tungsten became too hot and caused the wires to melt at the point where they joined with the Tungsten, and thus the thermocouples were destroyed.

The tests conducted for the samples at 0 degrees involved the method of holding the entire sample within the holder slot. A question was raised for this holder method, when the holder method was being investigated for 30 degrees, using the Titanium cathode. This is because shading occurred under the holder slot, for tests involving the

Titanium cathode, as seen in Figure 27.

Figure 27 - Example of holder shading: This image shows a sample tested under conditions where the plasma density was too low to initiate ablation, with an untested sample on the left for comparison. When attempting to ablate the sample with a titanium cathode (on the right) – you can see the titanium ions are able to reach everywhere on the sample but the areas under the holder.

This may simply be because the conditions of Titanium cathode were unable to conduct ablation, and the plasma coating occurred on all areas of the surface except for those under the holder area. If this is the case, it may indicate that ablation does occur for the area under the holder slot area, for a sample facing a copper cathode at 0 degrees.

Alternatively, the average ablation rate may have to be calculated by subtracting the area of the holder. If this alternative method of calculating the ablation rate for the theta=0 experiments is adopted, the ablation rates become: 121.24 kg/(m2s).

4.2 Trials for NASA Samples

Figure 28 - NASA sample being tested

An example of a NASA fuel fragment being tested in the plasma plume is seen in

Figure 28. The number of tests that could be performed on the WZrO2 material was limited, due to the limited amount of usable fragments that were candidates for arc jet testing. Since these fragments were denser than the Tungsten foil samples, they were held in front of the plasma plume for longer times, for an average of 64.4 seconds. The results of the experiments conducted on the WZrO2 samples are presented in Figure 29.

210 Ablation Rate 190 kg/m2s

170

150

130 μ = 157.91 110 σ = 22.58

70 NASA Samples

Figure 29 - Ablation rate data for 6 NASA samples, showing error bars around an average line. The average ablation rate for these samples is 157.91 kg/m2s with a standard deviation of 22.58.

4.3 Results of Computational Simulation

The kinetic ablation code with the Chapman Enskog conduction parameter was utilized in order to calculate and compare the results of the two experiment sets, as well as to numerically quantify the results from the WZrO2 tests. The thermal conduction parameter has a slight influence, but it was kept much less than 1, as in prior research 29, which ascertained that the molecular mean-free path was much smaller than the characteristic scale of the temperature change. Keeping τ much less than 1 is also needed for the Chapman-Enskog expansion method and to keep the energy constant valid.

The computation was conducted firstly for theta at 0 degrees. Since α evolves self-consistently for a set range of equilibrium densities No, a range of N2 was selected in order for the computational results to match that of the experimental results. The ablation

60 rate profile for theta at 0 degrees is seen in Figure 30. It is interesting to note, that the shape of the contours indicate an increase of the backflux of ablating particles to the surface of the material, and thus a decrease in the ablation rate, as the density increases.

Figure 30 - Ablation rate for θ at 0 degrees.

Computational simulation was then conducted for theta at 30 degrees, using the same density ranges as for theta at 0 degrees. This is because both of the samples face the sample plume with the same density and velocity conditions, and they are both made of the same material. The ablation rate profile for ablation at 30 degrees is seen in Figure 31, using the modified energy constant with an added tangential velocity in the ‘y’ direction, which is equal to αtan = α sinθ. A range of angles was also explored in the computational analysis, and the resultant maximum ablation rate for a range of angles is presented in

Figure 32.

Figure 31 - Ablation rate results for θ at 30 degrees, using the same plasma parameters as used in the computational simulation for θ at 0 degrees.

140 Maximum 120 Ablation Rate kg/m2s 100 80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Angle in Degrees

Figure 32 – A range of angles and their resultant maximum ablation rates, calculated in the computational simulation.

The simulation data can be compared to experimental data by selecting an array of data points, chosen at a fixed temperature, near the boiling point of Tungsten at

3695oK, for a range of plasma plume densities starting at 1E+21m-3 with a step size of

30E+22m-3as seen in Figure 33. This figure compares two ablation rate arrays – one for a sample at 0o and another array of ablation rate data at 30o.

140 Ablation Rate (kg/(m2s) 120 μ = 93.582 σ = 19.245 100

40 μ = 48.406 σ = 15.813 20

0 3.95 23.2 42.45 61.7 80.95 100.2 Plasma Plume Density x1023 m-3

Figure 33 - Comparison of data points taken from two computational simulations, the first which simulated ablation at 0o and the second which simulated ablation at 30o. The green line for ablation at 0o shows a mean ablation rate of 93.582 kg/m2s with a standard deviation of 19.245, while the blue line for ablation at 30o shows a mean ablation rate of

48.406 kg/m2s and standard deviation of 15.813.

Even though there is data on the saturation pressure-temperature relationship for

Tungsten, Zirconium, and Oxygen separately, there is not enough data on the WZrO2

63 material in order to conduct computational ablation rate simulations. The equation would demonstrate the vapor pressure behavior of 60%-W 40%-ZrO2 at high temperatures. A very rough estimate of the ablation rate was conducted, using the same pressure profile for Tungsten and adjusting the molecular mass of the vapor to reflect the molecular composition of the sample. The resultant ablation rate is seen in Figure 34.

Figure 34 – Adjusting the molecular mass of the computational simulation to reflect a sample composed of 60%-W 40%ZrO2. This is a very preliminary computational simulation, as it does not include the pressure profile for the molecular mixture of

60%W40%ZrO2.

This data can be compared to a simulation for the ablation of the material using the

38 molecular mass of WZrO2 and the saturation pressure profile of pure Zirconium ,

0.0 𝑠 = 2610.2 푇𝑠 (4.1)

Using the values for Zirconium’s enthalpy of vaporization 54,157.5 J/kg and boiling point Tb = 4650 to calculate the equilibrium pressure profile 39, as demonstrated in

Equation 2.31. The results of this simulation, compared to the simulation from Figure 33 are seen in Figure 35. A higher rate of ablation is seen when changing the saturation pressure profile from Tungsten’s to Zirconium’s.

160 For Zirconium saturation pressure profile, Average Ablation Rate is 138.75 kg/m2s

140

s) 120 2 100

60 For Tungsten saturation pressure profile, Average Ablation Rate is 79.56 kg/m2s AblationRate kg/(m 40

0 3.09 3.86 4.63 5.4 6.17 6.94 7.71 8.48 9.25 10.02 10.79 Number Density x1023 m -3

Figure 35 – A comparison of ablation rates for computational simulations with two different pressure profiles – Tungsten and Zirconium. Both simulations were run using the molecular mass mixture of WZrO2.

4.3.1 Modifying the Thermal Conduction Parameter

The usual relationship between the density ratio a1 and the temperature ratio a2 can be seen in Figure 36, where a downward slope from 1, for all values of α is shown.

Introducing additional variables within the energy constant affects the ranges for both the density ratio a1 and the temperature ratio a2.

Figure 36 - Relationship between a2 (T0/T1) and σtan (on the left) and a1 (N0/N1) and σtan

(on the right). Both of these plots were created with a thermal conduction parameter much less than 1.

Once the energy constant is modified within the equation set to include the ‘y’ component of velocity the slopes and ratio values change. This is because a2 is solved using the quadratic equation method (as seen in Chapter 2), for which the ‘b’ value changes when the energy constant is integrated to include the tangential components.

Previous research conducted for τ much less than 1 shows that as τ approaches 1, the thermal temperature ratio a2 and velocity ratio α will go up, while the density ratio a1 and backflux go down, as seen in Figure 37.

Figure 37 - Previous research 29, showing the relationship between the various parameters, for changing τ. As τ increases, the ratio for T1/T0 increases, while the ratio for n1/n0 decreases.

Figure 38 - Figure 34 - Relationship between a2 (T0/T1) and σtan (on the left) and a1

(N0/N1) and σtan (on the right). Both of these plots were created with a thermal conduction parameter equal to 0.1.

In our case, raising the thermal conduction parameter shows a decrease in the density ratio, and the temperature ratio and velocity ratio increases when raising the thermal

67 conduction parameter. This is consistent with the previous research, which calculated a drop in a1 and an increase in a2 and α for higher values of τ. Raising the thermal conduction parameter will also raise the ablation rate, as seen in Figure 39.

Figure 39 - Ablation rate contours, for ablation at an angle, setting the thermal conduction parameter to 0.25. Raising the thermal conduction parameter shows an increase in the ablation rate.

Chapter 5 Concluding Remarks

5.1 Plasma Ablation of the 99.9% Tungsten Samples

The experiments that were conducted for the ablation of Tungsten at 0o and for

Tungsten at 30o produced ablation rates with an average ablation rate of 89.46 kg/(m2s) and 43.91 kg/(m2s), respectively. A standard deviation of 36.77 was calculated for the samples at 0o, and a standard deviation of 32.23 was recorded for the samples at 30o. It is postulated that this range in ablation rates has a correlation with a range in plasma plume densities. A conclusion can be made, that the ablation rate has a dependence on the tangential component of the plasma plume velocity when the ablating material is at an angle relative to the plasma plume.

This idea is supported with data from the computational simulation, producing a range of ablation rates for a range of plasma plume density at a constant temperature.

Keeping the conditions of the plasma plume density range and temperature constant, an average ablation rate of 93.582 kg/(m2s) and standard deviation of 19.245 was calculated for 0o, and an average ablation rate of 48.406 kg/(m2s) and standard deviation of 15.813 was calculated for 30o. Therefore, the drop in the ablation rate seen in the data from experiments of approximately half was matched in the computational kinetic model code, when integrating the particle distribution function over a second dimension of velocity in the ‘y’ direction. The kinetic model also demonstrated that the ablation rate of a material at an angle relative to the plasma plume has a dependence on the tangential velocity component of the plasma.

Computational results also explored the effects of the thermal conduction parameter on the newly integrated distribution functions, mathematically demonstrating

69 the rationale for keeping τ much less than 1. A question was raised during the set-up of the tests conducted at an angle, as to whether the holder caused a shadowing to occur on the samples at 0o, as was seen for the samples that were coated with Titanium. The computational results which show a drop of approximately 50% in ablation rate for samples at an angle seem to support the idea that shadowing does not occur on samples which are able to undergo ablation.

5.2 Failure Testing of NASA Fabricated Samples

The ablation tests for the fragments of the WZrO2 rod showed an average ablation rate of 157.91 kg/(m2s) and a standard deviation of 22.58. This ablation rate is much higher than the ablation for pure Tungsten, and it is not reflected in the computational results, which show an estimated ablation rate of less than 100 kg/(m2s). This is mostly due to the fact that ZrO2 has a lower mass than Tungsten, which brings the average molecular mass down. This may demonstrate that the samples are being destroyed at a faster rate than predicted, due to imperfections in the material which cause more mass ablate per (m2s)-1.

The saturation pressure profile for the molecular mixture of 60%W40%ZrO2 was not available in the literature, and therefore, a precise computational simulation for the ablation rate of this material was not achievable. Two simulations were conducted, utilizing the molecular mass mixture of 60%W40%ZrO2, for a saturation pressure profile of Tungsten compared to Zirconium. Paired with a higher molecular mass, the ablation rate using the Zirconium pressure profile was higher than that of the ablation rate for

Tungsten – 138.75 kg/m2s and 79.56 kg/m2s, respectively. The experimental data show

2 an average ablation rate of 157.9 kg/m s for the WZrO2 rod, which is closer numerically to the simulation run for a saturation pressure profile of Zirconium, using the molecular mass mixture of 60%W40%ZrO2.

5.2.1 Failure Analysis of NASA’s Tungsten-Based Fuel

5.2.1.1 Tungsten Hafnium Nitride Sample

In the spring of 2012, a Tungsten Hafnium Nitride (WHfN) rod was tested in

14,21 NTREES by NASA MSFC . NTREES tests flowed H2 through the rod’s fuel channels at a nominal temperature of 1873o K 21. A vigorous understanding of post-failure testing analysis was needed for the purposes of this dissertation research, and thus the WHfN rod was acquired by myself for processing. The tested rod was cut and polished in order to study a surface of the material that was preserved within the rod and to provide a flat surface for the SEM and EDM to analyze. Then, SEM and EDM analysis was conducted as well as visual analysis of the rod. A control rod was also obtained, which was a part of the rod which had been kept aside and had not been tested, as see in Figure 4.

Figure 40 - Untested WHfN sample fabricated by NASA MSFC on left, and the part of the rod tested in NTREES on the right (shiny outer perimeter is the HIP can).

The first step in the failure analysis was to study the rod with the aid of the human eye.

An observation of Figure 41 reveals what appear to be clean granular cuts coating the surface of the rod and crisscrossed on rest of surrounding area. The crack would appear to have occurred at room temperature, which would point to issues in thermal cycling, which Tungsten based fuel is known to have.

Figure 41 - The entire rod broke in half – the left side of the image shows the jagged edges of the sample. This was thought to be due to the cooling down of the sample.

After some visual inspection of the sample, the sample was cut and polished for processing in the SEM. First, we focused on inspecting the cracks that had been observed with the human eye. Specifically, the crack in Figure 42 was chosen.

The SEM analysis revealed voids and gaps between the Tungsten particles, intergranular cracks at triple points (which would indicate that the cracks were chemical, not mechanical) and cracks in Hafnium Nitride are transgranular. The presence of the

72 flowing H2 could react with Flourine (from hexa-WFl), and cause discoloration and crack propagation from the flow channel. Secondly, we chose areas of discolorations for study in the EDM. In areas of discoloration, the Hf/W ratio was greater than 1 with a lower percentage of F. In areas of no discoloration, the Hf/W ratio was less than 1, with a higher percentage of higher F. The SEM also revealed areas of less and more W coating throughout sample, indicated by unsmooth chemical lumps. The F and W migrate along imperfections in the material, leading to HF migration which may generate cracks, demonstrating the porosity of HfN. Hydrofluoric acid created holes and etching in HfN, which could indicate that cracks were present while being tested under temperature.

Figure 42 – This particular part of the rod (the discolored crack) was chosen for SEM and

EDM analysis.

5.2.1.2 Tungsten Zirconium Oxide Sample

An SEM image of the WZrO2 rod before testing, shows the distribution of

Tungsten and ZrO2 particles within the rod, as seen in Figure 43. After testing, the

73 surfaces of the tested samples exposed to plasma were studied under the SEM and EDM.

Figure 44 shows a zoomed in region of the sample, which demonstrates the differences in grain structure between the W and ZrO2 molecules. A discolored region of the sample was chosen for EDM analysis, and the particular area of the sample that was selected is shown in Figure 45. The EDM results for the fragments of the WZrO2 were not as satisfactory as the failure analysis conducted for the WHfN sample, since the WZrO2 fragments were too small to polish. However, the SEM images could be comparable, in recording differences between grain structure of the colored versus the uncolored areas, as seen in Figure 46.

Figure 43 - The WZrO2 rod, before testing, magnified 25 times. The black particles show

Tungsten, while the white spaces indicate ZrO2. This image demonstrates a roughly

60%W-40%ZrO2 mixture throughout the rod.

Figure 44 – These images shows the transgranular divisions between the Tungsten particles and the Zirconium Oxide particles. Figure 46 shows the center of the bottom image zoomed in by 10 times.

Figure 45 – This particular spot enclosed within the red circle was of interest to study in the SEM and EDM, since it was an area of discoloration, which would indicate it was affected by the plasma more severely than surrounding areas. It was compared to the area of normal coloration, in the region of the orange circle.

Figure 46 - A comparison of zoomed-in regions of the sample, on the scale of 101 μm.

These images are meant to compare the discrepancy of grain boundaries between a grain of ZrO2 and W for a region, for an area of normal coloration (image on the left, which is the bottom image of Figure 44 magnified by 10 times) and discoloration (image on the right). The sharper boundaries depict the Tungsten sample, which seems to have been burned to a rounder shape in the discolored image. The ‘bubbles’ within the ZrO2 seem to

76 have expanded and gotten larger when exposed to the plasma, which may lead to an indication of rupturing the sample from within, as the molecules travel throughout the grain boundaries.

5.3 Potential Areas of Further Study

There are a number of areas of further study which could develop upon the research discussed in this thesis.

(1) A project could be conducted which studies the reach of plasma around small

corners, to reach under the parts of the sample that are being held by a holder

– to demonstrate whether or not the same ablation occurs regardless of its

presence.

(2) The ablation rate describes the amount of mass lost over time, for a given

area. Thus, as the mass increases, the ablation rate should also increase,

푘 according to the relationship: . Since the ablation rate is a function of 푚2𝑠

temperature, further study could investigate how the added mass (from a 3D,

volumetric perspective) would affect the ablation rate. This is because it may

not be accurate to assume that samples of different sizes are of a uniform

temperature (although, their temperatures may average for their respective

areas).

(3) Further plasma ablation tests could be conducted with more NASA samples,

this time cutting the rod fragments with the appropriate diamond saw. This

would be important for accurate EDM spectral analysis.

(4) Lastly, it would be useful to experimentally demonstrate and verify the

saturation pressure behavior for 60%W40%ZrO2. This pressure profile would

be used in computational analysis for comparison to experimental ablation

rate studies, to provide a more accurate prediction of what the ablation rate for

this molecular mass mixture should be. More specifically, this profile should

be obtained for the particular mixture of Tungsten Uranium Oxide that NASA

plans on testing in an engine.

References

1 J. R. Dewar. To the End of the Solar System: The Story of the Nuclear Rocket (The University Press of Kentucky, 2004).

2 G. L. Bennett et al. "Development and Use of Nuclear Power Sources for Space Applications," The Journal of the Astronautical Sciences, (1981): Vol. XXIX, No. 4 (October-December 1981).

3 D. E. Burkes, D.M. Wachs, J.E.Werner, S.D.Howe, "An Overview of Current and Past W-UO2 CERMET Fuel Fabrication Technology," Nuclear Space Conference 2007 (2007)

4 P.E. Blackburn, M. Hock and H.L. Johnston, "The Vaporization of Molybdenum and Tungsten Oxides," J. Phys. Chem. 62, 769 (1958)

5 R. A. Smith, D. E. Kizer, E. O. Spiedel and D. L. Keller, "Phase Equilibria in the U-W- O System," Battelle Memorial Institute, BMI-1755, 11 (1966).

6 A. Jain, K. Ananthasivan, S. Anthonysamy, P.R. Vasudeva Rao, "Synthesis and sintering of (U0.72Ce0.28)O2 solid solution," J. Nucl. Mater., 345, 245 (2005)

7 L. Steg and H. Lew, “Hypersonic Ablation,” Presented at AGARD Hypersonic Conference TCEA, Rhode-St. Geneva, Belgium. (April 3-6, 1962).

8 J.N. Moss and Scott, C. D. Thermophysical Aspects of Re-Entry Flows. Progress in Astronautics and Aeronautics, Vo. 103.

9 B. C. Eu, “Transport Coefficients of Fluids,” Springer Series in Chemical Physics (Book 82), Springer (2006).

10 Argonne National Laboratory. Nuclear Rocket Program Terminal Report. ANL-7236 Nuclear Reactors for Rocket Propulsion (M-3679, 54th Ed.) AEC Research and Development Report. June 30, 1966 (Published February 1968).

11 M. Gedwill, and P. Sikora: Fuel Retention Properties of Tungsten-Uranium Dioxide Composites NASA Technical Memorandum NASA TM X-1059. (February 1965).

12 S. Anghaie and Paul Harris. Evaluation of Cermet Fuels Test Data. Proceedings of ICAPP. Paper 4373. (June 13-17, 2004).

13 Fuels Materials Research Program, Chapter Five, the Argonne National Laboratory: Nuclear Rocket Program Terminal Report (1966)

14 Houts, M. G., et al. "Nuclear Thermal Propulsion for Advanced Space Exploration." (2012).

15 R. Gluyas and M. Gedwill. Stabilization of Tungsten – Uranium Dioxide Composites Under Thermal Cycling (1966)

16 E. Suhir, Y.C. Lee, and C.P. Wong Micro- and Opto-Electronic Materials and Structures: Physics, Mechanics, Springer, (2007).

17 T. Tinga,. “Principles of Loads and Failure Mechanisms.” Springer Series in Reliability Engineering, (2013)

18 A.F. Liu “Mechanics and Mechanisms of Fracture: An Introduction”, ASM International, (2005).

19 R.J.D. Tilley, “Understanding Solids: The Science of Materials,” Wiley 2nd Edition, (2013).

20 R. Gibala and R.F. Hehemann, “Hydrogen Embrittlement and Stress Corrosion Cracking,” Library of Congress in Publication Data, 6th Edition, (2002)

21 W. J. Emrich,, R. P. Moran, and J. B. Pearson. "Nuclear Thermal Rocket Element Environmental Simulator (NTREES) Upgrade Activities." 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. (2012).

22 R.J. Litchford, et al. Proceedings of Nuclear and Emerging Technologies for Space 2011,Albuquerque, NM, February 7-10, 2011, Paper 3331 (2011).

23 P.L. Houston, “Chemical Kinetics and Reaction Dynamics”, Courier Dover Publications, (2012).

24 S.I. Anisimov, Vaporization of Metal Absorbing Laser Radiation, Soviet Physics JETP, Vol. 27, No. 1, (July 1968).

25 M. Keidar, I.D. Boyd and I.I. Beilis, “On the model of Teflon ablation in an ablation- controlled discharge,” J. Phys. D: Appl. Phys., 34 pp. 1675-1677, (2001).

26 M. Keidar, I. D. Boyd, I.I. Beilis, “Vaporization of Heated Materials into Discharge Plasmas,” J. Appl. Phys., 89, p. 3095-3098 (2001).

27 M. Keidar, I.D. Boyd and I.I. Beilis, "Model of an Elecrothermal Pulsed Plasma Thruster,” J. Prop. Power, 19 pp. 424-430, 2003.

28 W. G. Vincenti, C.H. Kruger, Introduction to Physical Gas Dynamics, Krieger Publishing Co, (June 1975).

29 L. Pekker, M. Keidar, and J.L. Cambier, Effect of Thermal Conductivity on the Knudsen Laer at Ablative Surfaces, Journal of Applied Physics, 103, (2008).

30 Xu, X., Willis, D. A. Non-Equilibrium Phase Change in Metal Induced by Nanosecond Pulsed Laser Irradiation. Journal of Heat Transfer. April 2002, Vol. 124/293-298

31 Midwest Tungsten Service, http://www.tungsten.com/moreinfo.html

32 White, G. K, Collocott, S.J. Heat Capacity of Reference Materials: Cu and W. J. Phys. Chem. Ref. Data, Vol. 13, No. 4, 1984, pp. 1251 – 1257.

33Langmuir, I. “The Vapor Pressure of Metallic Tungsten”, The Physical Review, second series, November 1913, Vol II, No. 5

34 Keidar, M., Kim, M., Boyd, I. D. “Electromagnetic Reduction of Plasma Density During Atmospheric Reentry and Hypersonic Flights,” J. of Spacecraft and Rockets, 45, 3, pp. 445-453 (2008).

35 A. Shashurin, et al. Laboratory Modeling of the Plasma Layer at Hypersonic Flight, Journal of Spacecraft and Rockets, In Press, (2013).

36 A. Anders, Ion Flux from Vacuum Arc Cathode Spots in the Absence and Presence of a Magnetic Field, Journal of Applied Physics, Vol. 91, No. 8, (2002).

37 Juttner, B. Puchkarev, V. Hantzsche, E. Beilis, I. Handbook of Vacuum Arc Science and Technology, Ch. 3 Cathode Spots, Noyes Publication, (1995).

38 I. Schnell and Albers RC, "Zirconium under pressure: phase transitions and thermodynamics". Journal of Physics: Condensed Matter (Institute of Physics) 18 (5): 16, (January 2006).

39 G. B. Skinner, J.W. Edwards, and H. L. Johnston, The Vapor Pressure of Inorganic Substances. V Zirconium between 1949 and 2054. Journal of the American Chemical Society 73.1 174-176. (1951).

40 The White House, “Scientific or Technological Experiments with Possible Large Scale Adverse Environmental Effects and Launch of Nuclear Systems into Space,” Presidential Directive/National Security Council Memorandum, PD/NSC-25, (14 December 1977).

41 INSRP, “Interagency Nuclear Safety Review Panel, Safety Evaluation Report for Ulysses,” INSRP 90-01, Volumes I, II, and III, (July 1990).

42 Sholtis, J. A. et al, “Conduct and Results of the Interagency Nuclear Safety Review Panel's Evaluation of the Ulysses Space Mission,” AIP Conf. Proc. 217, pp. 132-139 (6−10 Jan 1991).

43 Regan, F. J. “Re-entry Vehicle Dynamics”, American Institute of Aeronautics and Astronautics, Inc., New York New York 1984

44 Finke, R. G. “Calculation of Reentry-Vehicle Temperature History”, Institute for Defense Analysis, IDA Paper P-2395, 1990

45 Shashurin, A., Zhuang, T., Teel, G., Keidar, M., “Modeling the Plasma Layer in the Vicinity of a Hypersonic Vehicle with a Laboratory Experiment,” Journal of Spacecraft and Rockets, accepted for publication 21 Nov. 2013, in press.

Appendix A Atmospheric Reentry

A 1977 Presidential Directive 40 mandated that flight safety reviews be conducted for spacecraft carrying nuclear power sources that have the potential to reenter the atmosphere and adversely affect the surface and atmosphere of Earth. The Ulysses spacecraft required a Jupiter flyby in order to reach the Sun, and since the spacecraft was so far from Earth, it required the use of a nuclear source which utilized 11 kg of

Plutonium Oxide.

The Interagency Nuclear Safety Review Panel (INSRP) determined that there would be no credible scenario in which radioactive materials would be released, unless the surrounding aeroshell module was struck by a very hard surface while in flight, which was very unlikely 41. Table 2 contains the various accident scenarios for reentry studied

42 for the Ulysses spacecraft, which contained Plutonium. The studies consider the release of the nuclear material due to various kinetic explosion-type events. The research contained in this dissertation would focus on those scenarios after the spacecraft has taken off.

A failure scenario during post-liftoff and ascent would have the spacecraft returning to Earth at velocities less than 10 km/s. Once the spacecraft exits the atmosphere, there may be failure due to the NTR’s failure to start in orbit, and it simply drops back to the Earth traveling at speeds of approximately 10 km/s. The most extreme and unlikely form of re-entry, would be if the spacecraft had started its engines in orbit, and the navigation system delivered it to Earth in a powered-re-entry at speeds greater than 10 km/s.

Phase Accident Type Source Term Releas Max Population Collective Collective Total Frequency of

Air/Ground e Prob Individual Potentially Dose in Organ Dose Health Health

in Curies Dose in Exposed Person-rem in Person- Effects Effects in

rem over 50 rem Events/Missi

Yr on

1 0-10s Near- ~/1.2 2.9x10- 9.4x10-4 6.3x105 20 32 0.015 65 0.008 1.2x10-8

Pad External 6 370 29

Tank

Explosion

1 0-10s Near- 24/50 5.2x10- 2.0x10-2 7.4x105 460 610 320 1400 0.2 2.7x10-6

Pad SRB 6 8100 650

Random

Failure

(Air/Ground

Release)

1 0-10s Near- ~/23 4.5x10- 7.0x10-3 6.1x105 610 188 160 530 0.07 5.3x10-6

Pad SRB 5 3100 250

Random

Failure

(Ground

Release Only)

1 10-20s Early 16/30 1.4x10- 9.7x10-3 1.3x106 240 320 350 510 0.09 7.0x10-7

Ascent SRB 6 3400 270

Random

Failure

(Air/Ground

Release)

1 10-20s Early ~/100 1.7x10- 2.2x10-2 6.2x105 500 590 490 1600 0.2 7.0x10-7

Ascent SRB 6 9600 770

Random

Failure

(Ground

Release Only)

1 20-57s Early, 14/0.6 1.6x10- 2.3x10-4 6.2x105 6.7 130 18 17 93 75 0.05 8.8x10-7

Mid-Ascent 6

SRB Random

Failure

1 57-105s Late, 72/- 1.5x10- - Worldwide -/ 2900 - - - - 0.08 1.1x10-6

Mid-Ascent 6

SRB Random

Failure

1 105-120s Late 280/- 3.6x10- - Worldwide -/ 1.1x104 - - - - 3 9.1x10-7

Ascent SRB 6

Random

Failure

2,3, or 4 Inadvertent -/0.4 6.2x10- 8.6x10-1 1.96x103 5.9 7.2 - - - - 0.002 3.5x10-5

Reentry and 4

Land Impact

Table 2 - Various failure scenarios 42, involving the spread of nuclear materials based on different failure scenarios. The scenarios that would be applicable to this dissertation are the ones that involve late ascent or re-entry, which show a worldwide distribution of nuclear materials for late ascent and a population distribution of 1.96x103 people being affected for re-entry. A potentially lower number of people would be affected for a reentry of a nuclear material that has a higher ablation rate, than the fuel used in this failure study (which is Plutonium based).

The velocity of a reentering vehicle can vary over the range of flight path angles from 0 to 90, depending on the range angle in degrees. The following analysis assumes that the NTR fuel is considering to be on a ballistic re-entry trajectory at a flight path angle of approximately 30o, which is a typical parameter used in re-entry studies. The reentry angle affects the time of flight for the spacecraft as it re-enters the atmosphere, and an estimation can be made for how long the material has to burn-up in the atmosphere

43 before returning to Earth. The time of flight is a function of the range angle θi , the

85 radius vector r0 = RE + z (radius of the Earth plus altitude) , the flight path angle γ , and the velocity magnitude at boost termination V,

𝑡𝑎푛훾(1 − ) + (1 − 휆) 푛 𝑡 𝑓 = 0 푖 푖 훾 1 − (훾 + ) (2 − 휆) {[ 푖] + [ 푖 ]} 훾 훾2 훾 {

1 2 2 2 훾 [( ) − 1] + 𝑡𝑎푛−1 휆 (𝐴. 1) 3 2 2 훾 ( ) − 푛훾 휆 [( ) − 1] 2 휆 [ ]}

2 o Where λ is a nondimensional value = V /(μ/r0). For a body re-entering at γ =30 and θi =

75o, the flight time can be estimated as approximately 1750 seconds. An ablation rate of

100 kg/(m2s) would indicate that a mass under 175,000 kg would burn-up before impacting the Earth’s surface.

The surface of a re-entering body will have an equilibrium temperature which is a function of the changing properties of atmospheric density 44.

2 (1 + ) 𝑣 √ 3.1 푇 = 1000 [1 3.4 2 ( ) /√ 푛] (𝐴. 2) 𝑐 10

This equilibrium temperature can then be applied to the heat equation, in order to find the wall temperature of the reentering body.

푇 푘 = − (𝐴. 3) 𝑥 ( 𝑡)𝐴

As the object is re-entering the atmosphere, the temperature radiating off of the surface is dependent upon the velocity, Stefan Boltzmann constant σ, and temperature-dependent emissivity, ‘e’

2 푇 𝑣 − 6. 5 10 ( ) 푇 2 푘 = −𝜎푇 ( 2 ) + 𝜎푇 (𝐴. 4) 𝑥 𝑣 − 6. 5 10

An estimation of the wall temperature can now be made by solving this equation, using the implicit differencing technique, which adds together wall temperature measurements at various thicknesses. For these calculations, the thermal conductivity ‘k’ is needed, which is 174 W/(mK)

휕푇 푘휕2푇 = (𝐴. 5) 휕𝑥 휕𝑥2

푛+1 푛 푇푗 − 푇푗 푘 𝑡 = [(푇푛 − 2푇푛 + 푇푛 ) + (푇푛+1 − 2푇푛+1 + 푇푛+1)] (𝐴. 6) 𝑡 2( 𝑥)2 푗+1 푗 푗−1 푗+1 푗 푗−1

Using this equation, the wall temperature for the Tungsten material as a function of the time step Δt, is seen to reach over 5000 K when re-entering the atmosphere. Compared to other materials, like glass fibre phenolic, the temperature of Tungsten takes longer to reach temperatures of 4500 K and above. This means that Tungsten remains below its melting point for a greater period of time than other materials, as it is reentering the atmosphere. This may help to support the claim that Tungsten could retain the Uranium product better as it is re-entering the atmosphere, than other materials.

Appendix B Materials Testing

Fuel Testing and Analysis Procedures

(includes NASA MSFC’s CFEET testing chamber).

1. Obtain a sample that is ready to test. Data about sample is recorded. (This will be the briefest section of information – because I’ll focus mostly on testing/analysis procedures).

a. Element composition

b. Shape, length/width, weight, holes (or none)

c. How it was fabricated:

i. HIPing procedure

ii. Temps used in fabricating, chemicals used to etch

2. Testing in CFEET

a. Put sample inside RF coil-heated CFEET

b. Ideal STATIC test procedure 1

First, the pre-heat treatment

i. Purge: Helium for 5+ minutes, then Hydrogen for 5+ minutes

ii. Heat to 1650 o C in 2 minutes, hold for 30 minutes

iii. Cool to room temp in Helium

iv. Weigh and measure samples

Then, the testing to simulate NTR environment

v. Purge again (per step one) vi. Heat to 2500o C in 1 minute, hold for at least 2 hours

1 Information on static tests in papers: `CVD W cladding of W-UO2 Cermets`, ` NASA TM X 1445 Process for WUO2 Honeycomb Cermets2`, ` NASA TM X1028 Grisaffe Plasma Sprayed Claddings for WUO2 Cermets2`,

vii. Cool to 1095o C in Hydrogen

viii. Purge with Helium. ix. Cool to room temperature in Helium.

x. After final steps, carefully bag for Analysis c. Ideal CYCLICAL test procedure 2,3 Procedures for RAPID CYCLING and LONG-

DURATION CYCLING

First, the pre-heat treatment

i. Purge: Helium for 5+ minutes, then Hydrogen for 5+ minutes

ii. Heat to 1650o C in 2 minutes, hold for 30 minutes

iii. Cool to room temp in Helium

iv. Weigh and measure samples

Then, the testing to simulate the NTR environment

For RAPID CYCLING (25 cycles)

v. Purge again (per step one) vi. Heat to 2500o C in 1 minute, hold for 10 minutes

vii. Cool to below 500o C in 1-2 min, hold room temp 10-15 min

viii. Repeat Steps vi and vii

2 Mars mission burn profile numbers (Four cycles: 39.4 minutes full power Cool down 17.8 minutes full power Cool down 15 minutes full power Cool down 23.5 minutes full power Cool down 3 Information on Cyclical test in papers: `Lietzke TWMR Fuel Elements and Materials`, ` NASA TM X-1296`, ` NASA CR 54835 WANL CD W cladding of WUO2 cermets`, ` NASA-CR 72711`,

ix. Weight and measure after cycles 10, 15, 20 and 25

x. After final steps, carefully bag for Analysis

For LONG-DURATION CYCLING (10 cycles)4

v. Purge again (per step one)

vi. Heat to 2500o C, hold for 40 minutes

vii. Cool to below 500o C in 1-2 min, hold room temp 10-15 min

viii. Repeat Steps vi and vii

ix. Weight and measure after cycles 5 and 10

x. After final steps, carefully bag for Analysis

3. Testing in arc-jet (at GWU) a. Put sample inside of arc-jet

b. Ideal arc-jet testing procedures 5

i. (< 10 km/s re-entry velocity)

ii. (~ 10 km/s re-entry velocity)

iii. (> 10 km/s re-entry velocity)

*. After final steps, carefully bag for Analysis

4. Post-test analysis

a. Observe variables obtained from 1. + (2. or 3.), and capture sample from

completed test.

i. Remove sample from bag, inspect under optical microscope, note color

changes, cracks, surface composition, etcetera

4 Suggested by Hickman, Robert, NASA MSFC June 26 2012 5 1.) Re-entry due to failure of chemical rocket prior to reaching orbit 2.) Re-entry due to NTR’s failure to fire in orbit 3.) Powered re-entry, due to navigational failure

90 ii. Prepare sample for SEM

- Cut sample in saw

- Polish/grind sample iii. Place sample in SEM and perform analysis iv. Additional analytical devices/steps may be necessary. v. Conclude and report potential failure mechanisms for fabrication group.

Appendix C: Experimental Data

Trial Mass Initial Mass Time (s) Mass Loss Area (m2) Ablation (g) Final (g) (g/s) Rate (kg/(m2 s) ) 1 0.261110 0.260940 7 0.000024 0.000250 94.444444 2 0.254530 0.254360 10 0.000017 0.000250 68.686869 3 0.261240 0.260710 18 0.000029 0.000250 115.217391 4 0.268400 0.267990 15 0.000027 0.000250 107.894737 5 0.283540 0.283210 23 0.000014 0.000250 56.410256 6 0.263730 0.263210 31 0.000017 0.000250 68.196721 7 0.262790 0.262290 26 0.000019 0.000250 77.821012 8 0.260330 0.259000 35 0.000038 0.000250 150.708215 9 0.274500 0.273200 41 0.000032 0.000250 127.764128 10 0.256820 0.256580 20 0.000012 0.000250 47.290640 11 0.266460 0.266200 12 0.000021 0.000250 84.552846 12 0.252120 0.251900 14 0.000015 0.000250 61.111111 13 0.269690 0.269320 16 0.000023 0.000240 95.164609 14 0.281860 0.281420 18 0.000024 0.000240 100.732601 15 0.251790 0.251310 24 0.000020 0.000240 82.644628 16 0.241010 0.240580 11 0.000040 0.000240 167.445483 18 0.271450 0.270990 11 0.000040 0.000240 168.128655 19 0.252290 0.252010 17 0.000016 0.000240 67.437380 20 0.395625 0.394885 41 0.000018 0.000380 47.496791 Average 0.269962 0.269479 20.647368 0.000024 0.000254 94.165711

Table 3 – Experimental data for 99.9% pure Tungsten foil samples, tested at 0 degrees.

These results are plotted and discussed in Chapter 4.

new area Ablation Rate (kg/(m2 s) ) 0.000190 124.269006 0.000190 90.377459 0.000190 151.601831 0.000190 141.966759 0.000190 74.224022 0.000190 89.732528 0.000190 102.396068 0.000190 198.300283 0.000190 168.110694 0.000190 62.224527 0.000190 111.253744 0.000190 80.409357 0.000180 126.886145 0.000180 134.310134 0.000180 110.192837 0.000180 223.260644 0.000180 224.171540 0.000180 89.916506 0.000320 56.402439 Avg 124.210870

Table 4 – Experimental data for 99.9% pure Tungsten foil samples tested at 0 degrees with the area of the holder subtracted, such the ablation rate is affected to reflect the new mass flow per area, accordingly.

2 Trial mi (grams) mf (grams) time dm/dt Area m AbRate (kg/(sm2))

1 0.334290 0.334220 19 0.000004 0.000265 14.292938 2 0.377980 0.377715 9 0.000031 0.000309 99.778687 3 0.351340 0.351300 11 0.000004 0.000272 13.135806 4 0.362073 0.361680 13 0.000031 0.000348 88.916756 5 0.350485 0.350285 21 0.000010 0.000330 29.416674 6 0.399475 0.398465 22 0.000046 0.000378 121.419938 7 0.348840 0.348660 23 0.000008 0.000339 23.464883 8 0.380175 0.379580 27 0.000022 0.000371 60.095004 9 0.330727 0.330600 15 0.000009 0.000315 27.765639 10 0.378465 0.377997 24 0.000020 0.000371 53.740107 11 0.377170 0.376893 28 0.000010 0.000363 26.962485 12 0.450805 0.450577 33 0.000007 0.000443 15.751103 13 0.362650 0.362470 19 0.000009 0.000352 26.364942 14 0.423107 0.422814 31 0.000009 0.000419 22.529343 15 0.373715 0.373320 19 0.000021 0.000360 58.924414 16 0.367963 0.367743 22 0.000010 0.000362 27.611295 17 0.446875 0.446520 23 0.000016 0.000426 36.365201 Average 0.37741971 0.37710813 21 1.5636E-05 0.00036487 43.913836

Table 5 – Experimental data for 99.9% pure Tungsten foil samples tested at 30 degrees.

These results are displayed graphically and discussed in Chapter 4.

Trial mass initial (g) mass final (g) time (s) dm/dt Area Ablation Rate (m2) (kg/(m2 s) ) 1 1.13241 1.13203 57.6 6.59722E-06 57 115.7407407

2 0.28158 0.2809175 62 1.06855E-05 64 166.9606855 3 0.748756 0.748203 65.2 8.4816E-06 60 141.3599182 4 0.411095 0.41035 69.9 1.06581E-05 64 166.5325465 5 0.575346667 0.574306667 58.9 1.7657E-05 100 176.5704584 6 0.20589 0.205245 73 8.83562E-06 49 180.3187028 Average 0.559179611 0.558508694 64.43 1.04858E-05 65.6 157.913842

Table 6 – The experimental results for the Tungsten Zirconium Oxide fragments, that were broken off of the rod fabricated by NASA MSFC. The results of these tests are displayed graphically in Chapter 4, and images of the samples are displayed and discussed in Chapter 5.

AbRate 0 AbRate 30 Density 5.164494 4.252109 3.95E+22 30.338151 24.978144 7.80E+22 42.056707 34.626738 1.17E+23 50.131659 41.275677 1.55E+23 56.277492 46.336551 1.94E+23 61.123086 50.327128 2.32E+23 64.987345 53.509964 2.71E+23 68.058948 56.040402 3.09E+23 70.459690 58.018723 3.48E+23 72.272525 59.513229 3.86E+23 73.555801 60.571969 4.25E+23 74.351172 61.229254 4.63E+23 74.688279 61.509506 5.02E+23 74.587602 61.429614 5.40E+23 74.062212 61.000373 5.79E+23 73.118776 60.227318 6.17E+23 71.758001 59.111088 6.56E+23 69.974574 57.647391 6.94E+23 67.756594 55.826535 7.33E+23 65.084327 53.632423 7.71E+23 61.928022 51.040774 8.10E+23 58.244171 48.016085 8.48E+23 53.969007 44.506351 8.87E+23 49.006480 40.433323 9.25E+23 43.203873 35.672804 9.64E+23 36.294332 30.008420 1.00E+24 27.722749 22.992636 1.04E+24 15.726230 13.228422 1.08E+24 Average 58.737122 48.406035

Table 7 – Computational data, for a temperature of 3700o Kelvin and a range of densities.

This data is plotted in Chapter 4.

Zirconium Tungsten Density Pressure Pressure (x10^23) Profile Profile 147.641719 96.5082813 3.09E+00 146.466173 102.119408 3.86E+00 144.015846 103.934219 4.63E+00 141.609495 102.689598 5.40E+00 139.490259 98.7958244 6.17E+00 137.676535 92.4756981 6.94E+00 136.135243 83.8138489 7.71E+00 134.825357 72.7593496 8.48E+00 133.709325 59.0816557 9.25E+00 132.755470 42.2275946 1.00E+01 131.937799 20.7770999 1.08E+01 Average 138.751202 79.562052

Table 8 – Computational data for a molecular mass mixture of 60%W40% ZrO2, comparing the results for saturation pressure profile of Tungsten and Zirconium.

Appendix D: Computer Code

% The original ablation code was written for the ablation of Teflon, in python % It has been edited by Tabitha Smith and rewritten for matlab % to study the affects of ablation on Tungsten % The most recent version shows a tangential velocity component %

% Initialize values %%%%%%%%%%%%%%%%%%% kb=1.38E-23; m=9.1E-31; mw =3.1E-25; % mw = 2.1E-25; for the mixture 60%W40%ZrO2 rho=19300;

%Temperature of the Plasma, converted to Kelvin Te= 2; %read from arc jet qe=1.6e-19; %charge q of electron T2=2*qe/kb;

%Density of the plasma N2= 1E+21:.385E+23:6E+25; for k=1:100; dTs=7; Ts0 =3000; % melting point of Tungsten, ish 3695. Plasma (copper) T2= 40580

Ts(k) = Ts0 + dTs*k;

P4(k) = 10^(15.502 - 47440/Ts(k) - 0.9*log10(Ts(k)))*133.32; %P4(k) = 2610.2*Ts(k)^(0.0465); for Zirconium

Tb1(k) = 5828; % Tb1(k) = 4650; for Zirconium

dH=4.35E+6; %[J/kg], = 800 kJ/mol, for Tungsten -> 183.85 g/mol %dH = 54157.5; %[J/kg] --> 142146 cal/mol --> 595137 J/mol, for Zirconium

R = 45.15; % = cp - cv = 262.7 - 217.57 (at 3400K)

N0(k) = P4(k)*exp(-dH*(Ts(k)-Tb1(k))/(R*Ts(k)*Tb1(k)))/(kb*Ts(k)); end tempextra = .001;

% IMPORTANT - For a Sample being tested at 0 degrees, the value “alpha” needs to be set for alpha = alpharange, and the value for alpha = sin*alpharange needs to be commented out. for i=1:100; theta=30; alpharange(1) = 1E-9; alpha(i) = sind(theta)*alpharange(i); alphatan(i) = alpharange(i)*cosd(theta); %alpha(i) = alpharange(i); %alphatan(i) = sqrt((alpharange(i))^2 - (alpha(i))^2); erfca(i)= erfc(alpha(i)); f1(i)=-exp(-(alpha(i)).^2)+alpha(i)*sqrt(pi)*erfca(i); f2(i)=erfca(i)+2*alpha(i)/sqrt(pi)*f1(i); % {fi}^M f3(i)=2+4*(alpha(i).^2)-2*alpha(i)*sqrt(pi)*f2(i)/f1(i); % {FI}^M f4(i)=(1+(alpha(i).^2)/2)*f1(i)+alpha(i)*sqrt(pi)/4*erfca(i); % {theta}^M

% IMPORTANT This f5 is for samples at 0 degrees. % To use it, uncomment it out, and comment out the other f5

%f5(i)=sqrt(pi)*(alpha(i)*(alpha(i).^2)+5/2*alpha(i))-f4(i)/f2(i)*(2+4*(alpha(i).^2)); % {sigma}^M

%New block of code for f5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

first(i) = (sqrt(pi)*((alpha(i)*alpha(i).^2)+(5/2)*alpha(i))) + (sqrt(pi)*(alpha(i)*(alphatan(i)+1/2)));

second(i) = 1/8 – 5/4*(alpha(i)^2);

Chapman(i) = (tempextra*(second(i)));

c3(i) = first(i) + (Chapman(i)); f5(i) = c3(i) - f4(i)/f2(i)*(2+4*(alpha(i)).^2); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Using the quadratic equation in order to find the ratio, a2 %a1 will then be calculated from a2

a10(i)=f5(i)*f2(i)/f1(i); b10(i)=-(f4(i)*f3(i)/f2(i)+f5(i)); c10(i)=f3(i); x10(i)=(-b10(i)-sqrt((b10(i))^2-4*a10(i)*c10(i)))/(2*a10(i));

a2(i)=(x10(i))^2; a1(i)=(1/a2(i)-f2(i)/(f1(i)*sqrt(a2(i))))/f3(i); Beta(i)=(2*alpha(i)*sqrt(pi)-1/(a1(i)*sqrt(a2(i))))/f1(i); a1n(i) = a1(i)*N0(i);

%Now to calculate alpha alpha2(i)=(N2(i)*T2/(2*a2(i)*Ts(i))-a1(i)*N0(i)/2)/(N0(i)*a1(i)- (a1(i)*N0(i))^2/N2(i));

if alpha2(i) > 0 && alpha2(i) < 36 alpharange(i+1)=(sqrt(alpha2(i)));

else alpharange(i+1) = 1e-9; end end for l=1:100 for j=1:100

AbRate(l,j)=(2*a1(l)*mw*N2(j)*alpha(l)*sqrt(2*kb/mw*Ts(l)*a2(l)));

end end n2small = N2(1:100); Tsmall = Ts(1:100); figure(1); contourf(Tsmall, n2small,AbRate); axis([3008 3700 1E+21 1.5E+24]); xlabel('T0'); ylabel('N2'); title('normalized AbRate'); hold on; colorbar; plotalpha=alphatan(1:100); figure(2) plot(plotalpha, a1); xlabel(‘alphatan'); ylabel('a1 (density ratio)'); figure (3) plot(plotalpha, a2); xlabel('alphatan'); ylabel('a2 (tempratio)');

100 figure(4) plot(alphaplot, a1); xlabel('alpha'); ylabel('a1 (density ratio)'); %axis([1 6 0 0.05]); figure (5) plot(alphaplot, a2); xlabel('alpha'); ylabel('a2 (tempratio)'); %axis([1 6 0 200]);

101

Appendix E: Mathematical Derivations

Starting with the main particle distribution functions

1 3 2 ( ) exp(−(𝑣) ) 𝑣푥 > 0 √𝜋 𝑓 (→) = 3 (퐸. 1) 푣 2 1 (𝑣푥 − ) 푛1 ( ) exp (− 2 ) 𝑣푥 0 { 𝑣1√𝜋 𝑣1

2 2 𝑣푇𝑣1 (𝑣푥 − ) (𝑣푥 − ) + (𝑣푦 − 푛) 5 𝑓푢 (→) = 푛1𝑓 (→) {1 − [ ( − ) (푙푛푇)]} 푣 푣 2 𝑣푚 𝑣1 𝑣1 2 𝑥 (퐸. 2)

3 2 2 1 [(𝑣푥 − ) + (𝑣푦 − 푛) ] 𝑓 (→) = 푛1 ( ) exp (− ) (퐸. 3) 푣 2 𝑣1√𝜋 𝑣1

Upon integration, the following integral substitution will be utilized

(𝑣 − ) = 푦 푛 ⁄ (퐸. 4) 𝑣1

The distribution of the particles at all layers are combined and represented as

𝑓 (𝑥,→) = 훿(𝑥)𝑓 (→) + [1 − 훿(𝑥)]𝑓푢 (→) (퐸. 5) 푣 푣 푣

The operator 훿(𝑥) is set to 훿( ) = 0 in order to derive the distribution of particles beyond the Knudsen layer integrating the function 𝑓푢 (→), in order to find the constants 푣 c1, c2, and c3 of the equations (for mass, momentum, and energy respectively).

102

𝑓 (𝑥,→) = 𝑓푢 (→) (퐸. 5) 푣 푣 The mass and momentum derivations reveal unchanged constants, whereas the energy constant changes. This is due to an extra term in the integral for v2.

E.1 Energy Derivations

3 2 ( ) ( ) 3 1 푛1 2 5𝑣1 𝑣푇𝑣1 푙푛푇 5𝑣1 3 = ( ) [ ( + ) − ] (퐸. 6) 𝑣1 𝜋 2 𝑣푚 𝑥 4

+ + 3 2 ∬ 𝑓(𝑥,풗)𝑣푥 풗 + ∬ 𝑓(𝑥,풗)𝑣푦 𝑣푥 풗 = 3 (퐸. ) − −

The integrations can be broken into two main parts, A and B

3 = (퐹 𝑡 𝑡 푚 − 𝐶 𝑎 푚𝑎푛 푇 푚 ) = 𝐴 − (퐸. )

𝐴 = 𝑓 𝑡 3 + 𝑓 𝑡 2 = 𝑎 + (퐸. ) 푣푥 푚𝑠 푣푦푣푥 푚𝑠

3 2 = (𝑣푥 𝑡 푚 + 𝑣푦 𝑣푥𝑡 푚 ) = ( + ) (퐸. )

The calculations for a are as follows: 2 3 2 [(𝑣푥 − ) + (𝑣푦 − 푛) ] 1 3 𝑎 = ∬ ( ) 푛1 exp (− 2 ) 𝑣푥 풗 (퐸. 10) − 𝑣1√𝜋 𝑣1

2 3 [(𝑣푦 − 푛) ] 1 2 3 𝑎 = ( ) 푛1 ∬ exp(− ) ( 𝑣1 + ) exp (− 2 ) 𝑣푦 (퐸. 11) 𝑣1√𝜋 − 𝑣1

103

3 1 1 2 2 2 2 𝑎 = ( ) 푛1 [ exp(− ) (√𝜋 exp( ) e ( ) (3𝑣1 + 2 ) 𝑣1√𝜋 4 2 2 − 2𝑣1(( + 1)𝑣1 + 3 𝑣1 2 푧 (𝑣 − ) 2 푦 푛 + 3 ))] ∫ exp (− 2 ) 𝑣푦 푧 − − 𝑣1 (퐸. 12)

2 3 2 (𝑣 − ) 1 2 5𝑣1 푦 푛 𝑎 = ( ) √𝜋푛1 ( + ) ∫ exp (− 2 ) 𝑣푦 (퐸. 13) 𝑣1√𝜋 2 − 𝑣1

3 2 1 2 5𝑣1 2 𝑎 = ( ) √𝜋푛1 ( + ) ∫ exp(− ) (퐸. 14) 𝑣1√𝜋 2 −

3 2 푧 1 2 푣1 1 𝑎 = ( ) √𝜋푛1 ( + ) [√𝜋 𝑓( ) ] (퐸. 15) 푣1√휋 2 2 푧 −

3 2 1 2 5𝑣1 𝑎 = ( ) √𝜋푛1 ( + ) √𝜋 (퐸. 16) 𝑣1√𝜋 2

The calculations for b are as follows:

2 3 2 [(𝑣푥 − ) + (𝑣푦 − 푛) ] 1 2 = ( ) 푛1 ∬ exp (− 2 ) 𝑣푦 𝑣푥 𝑣푦 𝑣푥 (퐸. 1 ) 𝑣1√𝜋 − 𝑣1

3 ( )2 1 2 2 − 𝑣푥 − = ( ) 푛1 ∬ exp(− ) ( 𝑣1 + 푛) exp ( 2 ) 𝑣푥 𝑣푥 (퐸. 1 ) 𝑣1√𝜋 − 𝑣1

104

3 1 1 2 2 2 2 = ( ) 푛1 [ √𝜋exp(− ) (exp( ) e ( ) (𝑣1 + 2 푛 ) 𝑣1√𝜋 4 푧 2 −(𝑣푥 − ) − 2𝑣1( 𝑣1 + 2 푛))] ∫ exp ( 2 ) 𝑣푥 𝑣푥 (퐸. 1 ) 푧 − − 𝑣1

3 1 1 2 = ( ) 푛1 [ √𝜋 e ( ) (𝑣1 𝑣1√𝜋 4

푧 [( )2] 2 𝑣푥 − + 2 푛)] [ ∫ exp (− 2 ) 𝑣푥 𝑣푥] (퐸. 20) 푧 − 𝑣1 −

3 1 1 2 2 = ( ) 푛1 [ √𝜋(𝑣1 + 2 푛)] [ ∫ exp(− ) ( 𝑣1 + ) ] (퐸. 21) 𝑣1√𝜋 2 −

3 푧 1 1 2 1 1 2 = ( ) 푛1 [ √𝜋(𝑣1 + 2 푛)] [ √𝜋 e ( ) − exp (− )𝑣1] (퐸. 22) 𝑣1√𝜋 2 2 2 푧 −

3 1 1 2 = ( ) 푛1 [ √𝜋(𝑣1 + 2 푛)] √𝜋 (퐸. 23) 𝑣1√𝜋 2

Combining the results of a and b, will give

3 2 1 2 5𝑣1 1 2 𝐴 = 푛1 ( ) [√𝜋푛1 ( + ) √𝜋 + [ √𝜋(𝑣1 + 2 푛)] √𝜋 ] (퐸. 24) 𝑣1√𝜋 2 2

105

The constant B can be obtained by utilizing the following grouping for terms in the integration of the Chapman Enskog distribution function: 1 3 = ( ) 푛1 √𝜋𝑣1

2 ( )2 ( ) 2 2 + [ 푣푥−푢 + 푣푦− 푢 푛 ] (푣푥−푢) (푣푥−푢) +(푣푦−푢 푛) ∬ exp (− 2 ) [ ( 2 − )] 𝑣⃗𝑣푥 풗 − 푣1 푣1 푣1 2 (퐸. 25)

3 2 = ∬ (𝑣푥 𝑡 푚 + 𝑣푦 𝑣푥𝑡 푚 ) 풗 = + , (퐸. 26) −

The calculations for c are as follows:

2 3 2 1 [(𝑣푥 − ) + (𝑣푦 − 푛) ] = ( ) 푛1 ∬ exp (− 2 ) 𝑣1√𝜋 − 𝑣1

2 ( ) (𝑣 − )2 + (𝑣 − ) 𝑣푥 − 푥 푦 푛 5 3 [ ( 2 − )] 𝑣푥 𝑣푥 𝑣푦 (퐸. 2 ) 𝑣1 𝑣1 2

2 3 [(𝑣푦 − 푛) ] 1 2 = ( ) 푛1 ∬ exp (− ) exp (− 2 ) 𝑣1√𝜋 − 𝑣1

2 (𝑣 − ) 2 푦 푛 5 3 [ ( 2 − )] ( 𝑣1 + ) 𝑣푦 (퐸. 2 ) 𝑣1 2

106

2 3 [(𝑣푦 − 푛) ] 1 2 = ( ) 푛1 [∬ exp(− ) exp (− 2 ) 𝑣1√𝜋 − 𝑣1

2 (𝑣 − ) 3 푦 푛 3 ( 2 ) ( 𝑣1 + ) 𝑣푦 𝑣1

[(𝑣푦 − 푛) ] 2 5 3 − ∬ exp(− ) exp (− 2 ) ( 𝑣1 + ) 𝑣푦] − 𝑣1 2

(퐸. 2 )

3 1 1 2 2 2 2 = ( ) 푛1 [[ exp(− ) (3√𝜋 exp( ) 𝑣1 e ( ) (5𝑣1 + 6 ) 𝑣1√𝜋 16

3 2 3 3 2 2 2 3 − 2(4 𝑣1 + 12 𝑣1 + 2 (5𝑣1 + 6𝑣1 ) + 4 (6𝑣1 + ) 3 2 + 3 (5𝑣1 + 6𝑣1 ) 2 + 4(6𝑣1 2 2 푧 [(𝑣푦 − 푛) ] (𝑣 − ) 3 푦 푛 + )))] ∫ exp (− 2 ) ( 2 ) 𝑣푦 푧 − − 𝑣1 𝑣1

1 − [ exp(− 2) (3√𝜋 exp( 2) 𝑣 𝑓( ) (𝑣2 + 2 2) 1 1 3 3 2 2 3 2 2 − 2(2 𝑣1 + 6 𝑣1 + 3 𝑣1 + 6 𝑣1 + 6𝑣1 2

푧 [(𝑣푦 − 푛) ] 3 + 2 ))] ∫ exp (− 2 ) 𝑣푦] 푧 − − 𝑣1

(퐸. 30)

3 1 3 2 2 2 2 = ( ) 푛1 [ 𝑣1(5𝑣1 + 6 )√𝜋 ∫ exp(− ) ( ) 𝑣1√𝜋 −

3 2 2 2 − 𝑣1(𝑣1 + 2 )√𝜋 ∫ exp(− ) ] (퐸. 31) 4 −

107

3 푧 1 3 2 2 1 1 2 = ( ) 푛1 [ 𝑣1(5𝑣1 + 6 )√𝜋 [ √𝜋 e ( ) − exp(− ) ] 𝑣1√𝜋 4 2 푧 − 푧 3 2 2 1 − 𝑣1(𝑣1 + 2 )√𝜋 [ √𝜋 e ( )] ] (퐸. 32) 4 2 푧 −

3 1 3 2 2 1 3 2 2 = ( ) 푛1 [ 𝑣1(5𝑣1 + 6 ) 𝜋 − 𝑣1(𝑣1 + 2 )𝜋] (퐸. 33) 𝑣1√𝜋 2 4

The calculations for d are as follows:

2 3 2 1 [(𝑣푥 − ) + (𝑣푦 − 푛) ] = ( ) 푛1 ∬ exp (− 2 ) 𝑣1√𝜋 − 𝑣1

2 ( ) (𝑣 − )2 + (𝑣 − ) 𝑣푥 − 푥 푦 푛 5 2 [ ( 2 − )] 𝑣푥𝑣푦 𝑣푥 𝑣푦 (퐸. 34) 𝑣1 𝑣1 2

2 3 2 [(𝑣푦 − 푛) ] (𝑣 − ) 1 2 2 푦 푛 = ( ) 푛1 ∬ exp (− ) exp (− 2 ) [ ( 2 𝑣1√𝜋 − 𝑣1 𝑣1

5 − )] ( 𝑣 + )𝑣2 𝑣 (퐸. 35) 2 1 푦 푦

108

2 3 2 [(𝑣푦 − 푛) ] (𝑣 − ) 1 2 3 푦 푛 = ( ) 푛1 [∬ exp(− ) exp (− 2 ) ( 2 ) ( 𝑣1 𝑣1√𝜋 − 𝑣1 𝑣1

2 + )𝑣푦 𝑣푦 2

[(𝑣푦 − 푛) ] 2 5 − ∬ exp(− ) exp (− 2 ) ( 𝑣1 − 𝑣1 2

2 + )𝑣푦 𝑣푦] (퐸. 36)

1 3 3 = ( ) 푛1 [[ √𝜋𝑣1 e ( ) 𝑣1√𝜋

3𝑣 2 3 𝑣 + exp(− 2) (− 1 − − 1 2 2 4

2 푧 2 [(𝑣푦 − 푛) ] (𝑣푦 − 푛) − )] ∫ exp (− ) ( ) 𝑣2 𝑣 2 𝑣 2 𝑣2 푦 푦 푧 − − 1 1

5 1 − [ ( √𝜋𝑣 e ( ) 2 4 1

2 푧 𝑣 [(𝑣푦 − 푛) ] + exp(− 2) (− 1 − ))] ∫ exp (− ) 𝑣2 𝑣 ] 2 2 𝑣 2 푦 푦 푧 − − 1

(퐸. 3 )

3 1 3 2 2 2 = ( ) 푛1 [ √𝜋𝑣1 ∫ exp(− ) ( 𝑣1 + 푛) 𝑣1√𝜋 4 −

5 2 2 − √𝜋𝑣1 ∫ exp(− ) ( 𝑣1 + 푛) ] (퐸. 3 ) 4 −

109

3 1 3 1 2 2 2 2 = ( ) 푛1 [[ √𝜋𝑣1 [ exp(− ) (√𝜋 exp( ) e ( ) (3𝑣1 + 2 푛) 𝑣1√𝜋 4 푧 3 2 2 2 2 − 2(2 𝑣1 + 4 𝑣1 푛 + 3 𝑣1 + 2 푛 + 4𝑣1 푛))] ] 푧 − 5 1 − √𝜋𝑣 [ exp(− 2) (√𝜋 exp( 2) e ( ) (𝑣2 + 2 2 ) 4 1 4 1 푛 푧 − 2𝑣1( 𝑣1 + 2 푛))] ] 푧 − (퐸. 3 )

3 1 3 1 2 2 5 1 2 2 = ( ) 푛1 [ √𝜋𝑣1 ( √𝜋(3𝑣1 + 2 푛)) − √𝜋𝑣1 √𝜋(𝑣1 + 2 푛)] (퐸. 40) 𝑣1√𝜋 4 4 4 2

3 1 3 1 2 2 5 1 2 2 = ( ) 푛1 [ √𝜋𝑣1 ( √𝜋(3𝑣1 + 2 푛)) − √𝜋𝑣1 √𝜋(𝑣1 + 2 푛)] (퐸. 41) 𝑣1√𝜋 4 4 4 2

Combining the results for A and B results in the equation for the energy equation

3 2 푛1 1 2 5𝑣1 1 2 1 2 5 2 3 = ( ) [[푛1 ( + ) + [ (𝑣1 + 2 푛)] ] − [𝑣1 ( 𝑣1 − )]] √𝜋 𝑣1 2 2 4 (퐸. 42)

E.2 Mass Derivations

3 1 2 1 = ( ) 푛1 ∬ 𝑣푥 exp(−(풗) ) 𝑣푥 𝑣푦 (퐸. 43) 𝑣1√𝜋 −

110

3 2 1 (𝑣 − ) = ( ) 푛 𝜋 ∫ exp (− [ 푦 푛 ⁄ ] ) 𝑣 (퐸. 44) 1 1√ 𝑣1 푦 𝑣1√𝜋 −

3 1 2 1 = ( ) 푛1√𝜋 ∫ exp(− ) (퐸. 45) 𝑣1√𝜋 −

1 3 1 푧 1 = ( ) 푛1√𝜋 [√𝜋 𝑓( ) ] (퐸. 46) 𝑣1√𝜋 2 푧 −

3 1 푛1 1 = ( ) (퐸. 4 ) 𝑣1 √𝜋

E.3 Momentum Derivations

2 2 = ∬ 𝑣푥 𝑓(𝑥,풗) 풗 (퐸. 4 ) −

3 1 2 2 2 = ( ) 푛1 ∬ 𝑣푥 exp(−(𝑣) ) 𝑣푥 𝑣푦 (퐸. 4 ) 𝑣1√𝜋 −

111

3 2 1 𝑣2 (𝑣 − ) = ( ) 푛 𝜋 ( 2 + 1 ) ∫ exp (− [ 푦 푛 ⁄ ] ) 𝑣 (퐸. 50) 2 1√ 𝑣1 푦 𝑣1√𝜋 2 −

3 2 1 2 𝑣1 2 2 = ( ) 푛1√𝜋 ( + ) ∫ exp(− ) (퐸. 51) 𝑣1√𝜋 2 −

3 2 푧 1 2 𝑣1 1 2 = ( ) 푛1√𝜋 ( + ) [√𝜋 𝑓( ) ] (퐸. 52) 𝑣1√𝜋 2 2 푧 −

2 푛1 2 𝑣1 2 = ( + ) (퐸. 53) √𝜋 2

112