# «A Dissertation Presented to The Academic Faculty by Kybeom Kwon In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the ...»

## A NOVEL NUMERICAL ANALYSIS OF HALL EFFECT

## THRUSTER AND ITS APPLICATION IN SIMULTANEOUS DESIGN

## OF THRUSTER AND OPTIMAL LOW-THRUST TRAJECTORY

A Dissertation

Presented to

The Academic Faculty

by

Kybeom Kwon

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy in the

School of Aerospace Engineering

Georgia Institute of Technology

August 2010

COPYRIGHT © 2010 BY KYBEOM KWON

## A NOVEL NUMERICAL ANALYSIS OF HALL EFFECT

## THRUSTER AND ITS APPLICATION IN SIMULTANEOUS DESIGN

## OF THRUSTER AND OPTIMAL LOW-THRUST TRAJECTORY

**Approved by:**

Dr. Dimitri N. Mavris Dr. Justin Koo Advisor (Committee Chair), Professor Propulsion Directorate School of Aerospace Engineering Air Force Research Laboratory Director of Aerospace Systems Design Edwards AFB Laboratory Georgia Institute of Technology Dr. Mitchell L. R. Walker Dr. Taewoo Nam Co-Advisor, Assistant Professor Research Engineer II School of Aerospace Engineering School of Aerospace Engineering Director of High Power Electric Aerospace Systems Design Laboratory Propulsion Laboratory Georgia Institute of Technology Georgia Institute of Technology Dr. Ryan P. Russell Co-Advisor, Assistant Professor School of Aerospace Engineering Space Systems Design Laboratory Georgia Institute of Technology Date Approved July 2, 2010 U

## DEDICATION

To the space pioneers and my wife iii## ACKNOWLEDGEMENTS

I would like to thank my committee for their advice and support. This work couldn’t have been done without their efforts. Most of all, I truly appreciate that Dr.Mavris, my advisor, accepted me as his student, supported me and allowed me to do this research. It is really an honor for me to be one of your students. I am also grateful that my co-advisors, Dr. Walker and Dr. Russell, have generously shared their priceless accumulated knowledge with me. Thanks to Dr. Koo for your thorough review on my proposal and thesis documents. I remember fondly the discussions with Dr. Nam and his kind guidance.

I am also grateful to my friend, Gregory. Our friendship extended beyond that of a collaborator of this thesis. I will always be grateful for your help and I will always feel that I have done little for you in return. I cannot find the words to express how important your friendship has been and I will never forget what you have done for me depite the cultural difference. I will never forget what you have done for me.

I would like to sincerely thank the Republic of Korea Air Force for the support.

There are also many colleagues to which I would like to extend my appreciation for;

Korean colleagues, the ASDL project teams I have worked on, people in other Aerospace Engineering labs. I thank you all.

I also want to give warm hugs to my beloved wife Insoon, who gave me full support during hard times over the past 5 years while bringing up two sons, and my loving parents who still think me as their child to be taken care of.

DEDICATION

ACKNOWLEDGEMENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF SYMBOLS

LIST OF ABBREVIATIONS

SUMMARY

CHAPTER 1 INTRODUCTION

1.1 Space Propulsion

1.2 Electric Propulsion (EP)

1.3 Hall Effect Thruster

1.4 Interim Summary

1.5 General Remarks on a New HET Design

1.6 Previous Design Activities for HET

1.6.1 Case Study I – University of Michigan / AFRL P5 5 kW class HET Design

1.6.2 Case Study II – More Recently Suggested Scaling Laws and Design Process

1.6.3 Concluding Remarks on Case Studies

1.7 Low-Thrust Trajectory Optimization

1.8 Motivation

1.9 Research Objectives

1.10 Research Questions

CHAPTER 2 THEORETICAL FOUNDATION

2.1 Understanding Basic Mathematical Modeling for HET

2.2 Understanding Basic Low-Thrust Trajectory Optimization

## CHAPTER 3 LITERATURE REVIEW AND PHYSICS-BASED ANALYSIS

TOOL IDENTIFICATION FOR HET3.1 Criteria for Conceptual Physics-Based Analysis Tool for HET

3.2 Previous Work on HET Numerical Modeling

3.2.1 Full Kinetic Modeling

3.2.2 Hybrid Modeling

3.2.3 Full Fluid Modeling

3.2.4 Other Methods

3.3 Tool Identification

## CHAPTER 4 PHYSICS-BASED ANALYSIS TOOL DEVELOPMENT FOR

HET4.1 Hypotheses for an Intended Tool

4.2 Ideas to Meet the Criteria

4.2.1 Assurance of Numerical Efficiency

4.2.2 Assurance of Numerical Robustness

4.2.3 Assurance of Self-Consistency

4.2.4 Assurance of Physics Representativeness

4.3 Development of Physics-based Analysis Tool

4.3.1 Analysis Domain

4.3.2 Assumptions

4.3.3 Expected Solution Structure for Electric Potential Distribution................73 4.3.4 Anode Sheath Region

4.3.6 Ionization/Acceleration Region

4.3.7 Matching Two Solutions

4.3.8 Non-Dimensionalization

CHAPTER 5 VALIDATION AND TOOL CAPABILITY STUDY

5.1 Point Validation with the SPT-100

5.1.1 SPT-100 Thruster

5.1.2 Comparisons of the Performance Metrics

5.1.3 Convergence Characteristics

5.1.4 Plasma Structures

5.2 Limitations of the Developed Tool

5.2.1 Accuracy of the Plasma Structures

5.2.2 Variation of Magnetic Field Distribution

5.3 Validation at Other Operating Points of the SPT-100

5.3.1 Remarks on the Proposed Modeling of the Anomalous Coefficients......110 5.3.2 Redefinition of Performance Metrics

5.3.3 Validation with Fixed Anomalous Coefficients

5.3.4 Classification of Solutions Obtained from the Developed Tool..............117 5.3.5 Construction of Design of Experiment (DOE) Environment

5.3.6 Numerical Exploration for the Ranges of the Anomalous Coefficients...122 5.3.7 Validation with Optimum Anomalous Coefficients

5.4 Pseudo-Validation with the High Power Class HETs

5.4.1 Validation with the T-220 Hall Effect Thruster

5.4.2 Validation with the NASA-457M Hall Effect Thruster

5.5 Sensitivity Studies for the SPT-100

vii

5.6 Approximation of the Radial Magnetic Field Distribution with the Given Performance Goals

CHAPTER 6 DESIGN SPACE EXPLORATION FOR HET

6.1 Need of Design Space Exploration

6.2 Design Space Exploration for the HET

6.2.1 Selection of the Design Space

6.2.2 Design Space Exploration Strategy

6.2.3 Constraints on Feasible Thruster Operation

6.2.4 Analysis of the DSE Results

CHAPTER 7 CONSTRUCTION OF SURROGATE MODELS FOR HET.........179

7.1 Surrogate Models

7.2 Surrogate Models for Performance Metrics using Response Surface Methodology

7.3 Neural Network Implementation for Performance Metric Surrogate Models....183

7.4 Surrogate Models for Constraints

7.4.1 General Considerations on Constraints

7.4.2 Use of Response Surface Methodology

7.4.3 Constraints as a Classification

7.4.4 Support Vector Machine Classifier as a Constraint Function..................197

## CHAPTER 8 SIMULTANEOUS DESIGN OPTIMIZATION FOR AN

ELECTRIC ORBIT RAISING MISSION BY COLLABORATION WORK.........2068.0 Acknowledgement

8.1 Mission Selection

8.2 Electric Orbit Raising Mission Description

8.3 Simultaneous Design Optimization Environment

8.4 Results and Comparisons

8.4.2 Case of d lower limit 3 for the SVM Classifier Constraint Limit................222 8.4.3 Optimal Low-Thrust Trajectory Calculation with the SPT-100 Thruster

8.4.4 Comparison with Pure Chemical Transfer

CHAPTER 9 CONCLUSIONS AND FUTURE WORK

9.1 Conclusions

9.2 Contributions

9.3 Future Work

## APPENDIX A. APPROXIMATION OF AZIMUTHAL ELECTRON MEAN

VELOCITY USING LANGEVIN’S APPROACHAPPENDIX B. CURVE-FIT EQUATIONS FOR REACTION RATES................251

## APPENDIX C. REFERENCE VALUES FOR NON-DIMENSIONALIZATION

OF VARIABLES## APPENDIX D. DESCRIPTION OF THE DEVELOPED TOOL:

HOW-TO-USE

REFERENCES

VITA

Table 1.1: Available Space Propulsion Technology Options [1] - [3]

** Table 1.2: Available Electric Propulsion Options and Their Characteristics [1] - [3].**

.......7 Table 1.3: Metrics and Parameters of Interest in HET Design

Table 2.1: Continuous Nonlinear Optimal Controller [58]

Table 3.1: Tool Identification based on the Criteria

Table 5.1: Geometry and Input Parameters of SPT-100

Table 5.2: Comparisons of Calculated Metrics

Table 5.3: Experimental Performance Data of the SPT-100 [16]

Table 5.4: Summary of Fit for Thrust

Table 5.5: Parameter Estimates and Associated Pareto Plot for Thrust

** Table 5.6: Parameter Estimates and Associated Pareto Plot for Discharge Current.**

......126 Table 5.7: Summary of Fit for Total Efficiency

** Table 5.8: Parameter Estimates and Associated Pareto Plot for Total Efficiency.**

..........127

Table 5.10: Operation Conditions and Geometry of the T-220

Table 5.11: Ranges of Magnetic Field Parameters

Table 5.12: Validation Results for the T-220 Design Operating Point

Table 5.13: Performance Metric Distributions

Table 5.14: Performance Metric Distributions at Optimum Magnetic Field Distribution

Table 5.15: Operation Conditions and Geometry of the NASA-457M (est: estimation)

** Table 5.16: Experimental Data of the NASA-457M [128] (Thrust Error = ± 1%).**

........149

Table 5.18: Validation Results for the NASA-457M

Table 5.19: Validation Results for the NASA-457M with All 4 Points

Table 5.20: Effect on Multiply-Charged Ions

Table 5.21: Variables and Ranges of Sensitivity Studies for the SPT-100

Table 5.22: Operation Conditions and Geometry of the P5

Table 5.23: The P5 Performance Metrics at Design Operation Point

Table 5.24: Ranges of Magnetic Field Parameters for the P5

** Table 5.25: Results of Finding Candidate Radial Magnetic Field Distribution.**

.............162 Table 6.1: Ranges of Variables

Table 6.2: DOE Cases and Results

Table 6.3: Performance Metric Distributions of the Design Space

Table 7.1: Goodness of Fit Results

Table 7.2: Validation and Test Data Sets

Table 7.3: Goodness of Fit for the Constraints

Table 7.4: Test Results of Constraint Surrogate Models

Table 7.5: Goodness of Fit with All Cases

Table 7.6: Test Results of Constraint Surrogate Models with All Cases

Table 8.1: Initial and Final Orbits of EOR

Table 8.2: Constraints for Each Trajectory Segment

Table 8.3: SNOPT options

Table 8.4: Design Results for d lower limit 1

Table 8.5: Convergence Property for d lower limit 3

Table 8.6: Design Results for d lower limit 3

Table 8.7: Results of Monte Carlo Simulation

Table 8.9: Comparison of Design Results with SPT-100

** Table 8.10: Comparison with Bipropellant Liquid Rocket Transfer for EOR.**

...............236 Table 8.11: Comparison of Pure Chemical Transfer and C-EOR

Table 8.12: GEO Delivery Cost Estimation [151]

** Figure 1.1: Final Mass Fraction Comparison (Isp of EP = 2000 sec, Isp of CP = 400 sec).**

.5 Figure 1.2: Baseline Dawn Mission [4]

Figure 1.3: The Schematic of Hall Effect Thruster (SPT)

Figure 1.4: The Sectional View of a Hall Effect Thruster (SPT)

Figure 1.5: Relation between Thruster Power and Specific Impulse [22]

** Figure 1.6: Relation between Expected Efficiency and Specific Impulse [22].**

................18 Figure 1.7: Relation between Expected Discharge Chamber Diameter Squared and Propellant Mass Flow Rate [22]

Figure 1.8: Lengths to be Determined [22]

** Figure 1.9: Relation between Expected Thrust and Nominal Discharge Power [24].**

.......21 Figure 1.10: Relation between Expected Propellant Mass Flow Rate and Nominal Discharge Power [24]

Figure 1.11: Relation between Expected Discharge Current and Propellant Mass Flow Rate [24]

Figure 1.12: Relation between Expected Specific Impulse and Nominal Discharge Power [24]

** Figure 1.13: Relation between Expected Efficiency and Specific Impulse [24].**

..............23 Figure 1.14: Relation between Expected Channel Diameter and Nominal Discharge Power [24]

Figure 1.15: Relation between Expected Thruster Mass and Nominal Discharge Power [24]

** Figure 1.16: Notional Technology S Curves on Improvement of HET Design Process.**

..29 Figure 1.17: Collaboration Framework

Figure 2.1: Mathematical Modeling based on Kn [45]

Figure 2.2: Cyclotron Motion of Charged Particles [46]

Figure 2.4: Schematic of the Sheath [54]

Figure 2.5: Plasma and Sheath Approximation [55]

Figure 4.1: Solution Curves for Isothermal Euler Equations [88]

Figure 4.2: Two Types of Anode Fall [94]

Figure 4.3: Ion Current from Experimental Measurement [96]

** Figure 4.4: Electron Trajectories in Magnetic Field and Uniform Electric Field [84].**

.....66 Figure 4.5: Schematic of Analysis Domain