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A theoretical and experimental investigation of cathode processes in electric thrusters

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A theoretical and experimental investigation of cathode processes in electric thrusters

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF CATHODE PROCESSES IN ELECTRIC THRUSTERS by K EITH DAVID GOODFELLOVV A Dissertation Presented to the FACULTY O F T H E GRADUATE SCHOOL UNIVERSITY O F SOUTHERN CALIFORNIA In P artial Fulfillment of the Requirem ents for the Degree D O C TO R O F PH ILO SO PH Y (Aerospace Engineering) May 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9636338 UMI Microform 9636338 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 90007 This dissertation, written by K e ith D avid G o o d fello w under the direction of h.i.5....... Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School in partial fulfillment of re quirements for the degree of DOCTOR OF PHILOSOPHY Dean of Graduate Studies Date DISSERTATION COMMITTEE Chairperson Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To Jennifer, Shelby an d A nnette Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A cknow ledgm ents T he author would like to thank the num erous people th a t m ade this work pos sible. T he author is grateful for the suggestions and guidance from his graduate com m ittee members nam ely, Professors Dan Erw in, Joe K une, Alan M cCurdy, and Phillip M untz. M any inform ative conversations were held w ith my associates a t JP L namely, John Blandino, John Biophy, R obert Frisbee, Charles G am er, Stephanie Leifer, Nick M oore, Laura Newlin, and Thom as P ivirotto. In particular th e author wishes to thank Jay Polk for his m any hours of discussions, collaborations, and contributions to this work. T he au th o r also wishes to thank A1 Owens, Lew Pless, Bill Thogm artin, and Bob Toom ath for th eir abilities to build anything, fix anything, and walk on w ater when necessary. A ssistance w ith the optical diagnostics was provided by John Schilling and Jeff P obst of the University of Southern California (U SC), and Ron Spores of the A ir Force Phillips Laboratory. Assistance w ith th e im aging pyrom eter was provided by Miguel Cerezo of the JP L O ptical Calibration G roup. T he author is grateful to everyone for their help. T he work described in this report was carried out a t the Je t Propulsion Labo ratory, California In stitu te of Technology, under contract with the N ational Aero nautics and Space A dm inistration. T he au th o r wishes to thank G ary B ennett and Earl Van Landingham for their support. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C on ten ts List o f Figures vii List o f Tables xvii Abstract xix 1 Introduction 1 1.1 Role of Electric Propulsion ............................................................................. 2 1.2 Electric T hrusters .............................................................................................. 4 1.2.1 Ion E n g i n e ............................................................................................. 6 1.2.2 Arcjet T h r u s te r ...................................................................................... 8 1.2.3 M agnetoplasm adynam ic (M PD ) T h ru s te rs ................................... 11 1.3 Establishing Lifetim e and R e lia b ility ............................................................ 15 1.4 C athode O p e ra tio n .............................................................................................. 20 1.4.1 Ion Engine C a t h o d e ............................................................................ 22 1.4.2 Arcjet C a th o d e ...................................................................................... 25 1.4.3 M PD T h ru ster C a th o d e ..................................................................... 31 1.4.4 C athode M o d e l s ................................................................................... 35 1.5 Role of this T h e s i s ......................................................................... 38 2 Near-Cathode Plasm a M odel 42 2.1 C athode Surface/Recom bination R e g io n ..................................................... 45 2.2 Sheath Region ..................................................................................................... 49 2.3 Presheath and Ionization R eg io n s.................................................................. 64 2.4 Ionization Region Species D eterm in atio n ..................................................... 72 2.5 Overall N ear-C athode Plasm a M o d e l........................................................... 77 2.6 M agnetic Pressure E f f e c ts ................................................................................ 85 2.7 Boundary Layer R e g io n s................................................................................... 87 2.8 Arc Column Region .......................................................................................... 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Cathode Thermal M odel 92 3.1 Quasi-Two-Dimensional Therm al Models .................................................. 93 3.2 Two-Dimensional Axisymmetric Therm al Model ........................................ 102 3.3 Two-Dimensional T ip A pproxim ation M o d el.................................................107 4 Combined M odel Solutions 110 4.1 Quasi-Two-Dimensional M o d e l .........................................................................I l l 4.2 Two-Dimensional M o d e l...................................................................................... 128 5 Experim ental Facility and Diagnostics 133 5.1 C athode Test Facility ..........................................................................................134 5.2 D ia g n o s tic s .............................................................................................................. 137 6 Pure Tungsten Cathode Experiments 142 6.1 Axial Tem perature D is tr ib u tio n s .....................................................................143 6.2 T he 488 nm Argon II Line E m is s io n .............................................................. 150 6.3 Electron Tem perature M easu rem en ts.............................................................. 157 6.4 C alculated R e s u l t s ................................................................................................ 161 7 Thoriated Tungsten Cathode Experim ents 170 7.1 C athode Surface E m itta n c e ............................................................................... 171 7.2 Axial Tem perature D is tr ib u tio n s .....................................................................172 7.3 M ass Flow R ate E f f e c t s ...................................................................................... 180 7.4 T h e 488 nm Argon II Line E m is s io n ..............................................................182 7.5 Electron Tem perature M easu rem en ts.............................................................. 194 7.6 C alculated R e s u l t s ................................................................................................ 205 7.7 C athode T ip Pressure M easu rem en ts..............................................................214 7.8 Surface M icrostructure and Chemical S t a t e ................................................221 8 Comparison o f M odel w ith Experiments 238 8.1 A rcjet T hruster Com parisons and Model P re d ic tio n s ...............................239 8.2 P u re Tungsten E x p e rim e n ts ...............................................................................250 8.2.1 Quasi-Two-Dimensional C o m p a r is o n s ..............................................250 8.2.2 Two-Dimensional Model R e s u l t s ........................................................258 8.3 T horiated Tungsten E x p e rim e n ts .................................................................... 261 8.4 C athode E v ap o ratio n .............................................................................................272 9 Conclusions and Recom mendations 278 A List o f Variables with Typical U nits 284 B Grid Transformation 291 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C The Tem perature Measurement System 204 C .l T he Im aging P y r o m e te r ......................................................................................294 C.2 C alibration of the P y r o m e te r ........................................................................... 296 C.3 A pplication in the E x p e rim e n ts........................................................................301 C.4 U ncertainty in the Tem perature M e a su re m e n ts........................................... 302 References 307 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f F igu res 1.1 Schematic diagram of the Hughes 13 cm ion engine................................. 7 1.2 Schem atic diagram of the JP L 30-kW class am m onia arcjet................. 9 1.3 D lustration of an arc colum n............................................................................ 9 1.4 Schem atic of a self-field M PD th ru ste r......................................................... 12 1.5 Schematic of a lithium -propellant Russian applied-field M PD th ru ster. 12 1.6 Probability of failure as a function of operating tim e....................... 16 1.7 T he probabilistic failure assessm ent m ethodology............................ 18 1.8 Diagram o f th e cathode erosion m odel................................................... 21 1.9 D lustration of a typical ion engine hollow cathode............................ 23 1.10 Photom icrograph of the eroded tip from a 10 kW am m onia arcjet 1470 hour te s t.................................................................................................. 26 1.11 Photograph of an arcjet cathode tip crater after 573 hours of opera tion w ith am m onia propellant a t 26 kW ................................................ 27 1.12 Photograph of an arcjet cathode tip before and after 1462 hours of continuous operation with am m onia propellant a t 10 kW ............... 28 1.13 Photograph of an arcjet cathode tip after 707 operation cycles (702 hours) w ith am m onia propeUant a t 10 kW ........................................... 28 1.14 D lustration o f a high-pressure and a low-pressure cathode arc attach m en t........................................................................................................ 32 1.15 Photograph of an M PD th ru ster cathode after testing in the Univer sity of S tu ttg a rt ZT1 th ru ster.................................................................... 33 1.16 Photograph of a sectioned M PD th ru ster cathode after testing in the University of S tu ttg a rt ZT1 th ru ste r....................................................... 34 2.1 N ear-cathode plasm a regions..................................................................... 44 2.2 Effect of m aterial work function on therm ionic emission current. . . 48 2.3 Effect of surface electric field on therm ionic emission cu rren t....... 49 2.4 Normalized charge density as a function of normalized distance with Jfc as a param eter, t]c = 10.0, = 0.3, c,/, = 0.0............................. 56 2.5 Normalized voltage as a function of normalized distance w ith Jb as a param eter, = 10.0, Ebo — 0.3, c,/, = 0.0............................................ 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.6 Normalized electric field as a function of normalized distance with Jb as a param eter, rje = 10.0, £j* , = 0.3, = 0.0............................... 58 2.7 Normalized charge density as a function of normalized distance with Eho as a param eter, rfe = 10.0, Jb = 0.1, c,* = 0.0................................... 59 2.8 Normalized electric field as a function of normalized distance with Ebo as a param eter, rjc = 10.0, Jb = 0.1, t , b = 0.0................................... 59 2.9 Normalized particle num ber densities as a function of norm alized dis tance, Jb = 0.1, t)c = 10.0, Ebo = 0.3, c,h = 0.0............................... 60 2.10 Normalized to tal current density as a function of norm alized sheath voltage...................................................................................................................... 61 2.11 Normalized to tal current density as a function of norm alized sheath voltage for large Jb values.................................................................................. 61 2.12 Normalized surface electric field as afunction of normalized therm ionic current w ith norm alized sheath voltage as a param eter.......................... 63 2.13 Normalized surface electric field as a function of normalized therm ionic current for sm all normalized sheath voltages.................................... 63 2.14 Illustration of ionization region.............................................................. 65 2.15 Normalized therm ionic current density as a function of norm alized sheath voltage w ith Ebo as a param eter for a singly-charged argon gas. 70 2.16 Normalized therm ionic current density as a function of normalized sheath voltage for different gas types................................................... 70 2.17 Species mole fractions as functions of electron tem p eratu re........ 76 2.18 M agnitude of current density com ponents as functions of cathode surface tem p eratu re................................................................................... 78 2.19 M agnitude of heat flux components as functions of cathode surface tem perature.................................................................................................. 78 2.20 C urrent density as a function of cathode surface tem p eratu re with sheath voltage as a param eter................................................................. 80 2.21 Heat flux as a function of cathode surface tem perature w ith sheath voltage as a p aram eter.............................................................................. 80 2.22 Electron T em perature as a function of cathode surface tem perature with sheath voltage as a param eter....................................................... 81 2.23 C urrent density as a function of cathode surface tem perature with work function as a param eter.................................................................. 82 2.24 Heat flux as a function of cathode surface tem perature w ith work function as a param eter............................................................................ 82 2.25 Electron tem p eratu re as a function of cathode surface tem perature with work function as a param eter........................................................ 83 2.26 Current density as a function of cathode surface tem perature with pressure as a p aram eter............................................................................. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.27 H eat flux as a function of cathode surface tem perature w ith pressure as a p aram eter....................................................................................................... 84 2.28 Electron T em perature as a function of cathode surface tem perature w ith pressure as a param eter............................................................................ 85 2.29 Radial tem perature profiles for the arc colum n.......................................... 89 2.30 Axial electric field as a function of arc column radius.............................. 91 3.1 Heat flux as a function of tip surface tem perature with current as a param eter................................................................................................................ 96 3.2 Cylindrical cathode tem perature distribution with constant electrical resistivity................................................................................................................. 97 3.3 H eat flux as a function of tip surface tem perature with current as a param eter for a cylindrical cathode w ith a conical tip ............................. 97 3.4 Cylindrical cathode tem perature distribution with conical tip and w ith constant electrical resistivity................................................................... 98 3.5 Illustration of a cylindrical cathode w ith a flattened conical tip. . . 98 3.6 Therm al conductivity and electrical resistivity for tungsten as a func tion of tem p eratu re.............................................................................................. 100 3.7 Cylindrical cathode tem perature distribution w ith tem perature de pendent electrical resistivity. ........................................................................ 101 3.8 H eat flux as a function of tip surface tem perature for a cylindrical cathode w ith and w ithout tem perature dependent m aterial properties..................................................................................................................... 101 3.9 Energy inputs for the two-dimensional therm al m odel.................................104 3.10 Two-dimensional tip approxim ation geom etry................................................ 108 3.11 Two-dimensional tip approxim ation surface fit...............................................109 4.1 H eat flux as a function of surface tem perature with therm al model solutions................................................................................................................... 112 4.2 Peak and zero point tem peratures as functions of normalized sheath voltage w ith pressure as a param eter for first model version................. 113 4.3 H eat flux as a function of cathode surface tem perature w ith therm al model solutions for second model version.........................................................115 4.4 A ttachm ent area as a function of cathode tip tem perature with pres sure as a p aram eter.............................................................................................. 117 4.5 Electron tem p eratu re as a function of cathode tip tem perature with pressure as a param eter...................................................................................... 117 4.6 Sheath voltage as a function of cathode tip tem perature w ith pressure as a param eter....................................................................................................... 118 4.7 Effective work function as a function of cathode tip tem perature with pressure as a param eter...................................................................................... 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X 4.8 A ttachm ent area as a function of sheath voltage with pressure as a param eter................................................................................................................ 119 4.9 A ttachm ent area as a function of cathode tip tem perature w ith work function as a param eter...................................................................................... 121 4.10 Electron tem p eratu re as a function of cathode tip tem perature with work function as a param eter........................................................................... 122 4.11 Sheath voltage as a function of cathode tip tem perature w ith work function sis a param eter...................................................................................... 122 4.12 Work function lowering as a function o f cathode tip tem perature with work function as a param eter........................................................................... 123 4.13 A ttachm ent area as a function of sheath voltage with work function as a p aram eter....................................................................................................... 123 4.14 A ttachm ent area as a function of cathode tip tem perature w ith cur rent as a p aram eter.............................................................................................. 125 4.15 Electron tem perature as a function of cathode tip tem p eratu re with current as a param eter............................................................................................ 125 4.16 Sheath voltage as a function of cathode tip tem perature w ith current as a p aram eter....................................................................................................... 126 4.17 Effective work function as a function of cathode tip tem perature with current as a p aram eter............................................................................................ 126 4.18 A ttachm ent area as a function of sheath voltage with current as a param eter................................................................................................................ 127 4.19 A ttachm ent area as a function of cathode tip tem perature for different gases.......................................................................................................................... 128 4.20 Electron tem perature as a function of cathode tip tem perature for different gases......................................................................................................... 129 4.21 Sheath voltage as a function of cathode tip tem perature for different gases.......................................................................................................................... 129 4.22 Effective work function as a function of cathode tip tem perature for different gases......................................................................................................... 130 4.23 A ttachm ent area as a function of sheath voltage for different gases. . 130 4.24 Radial C athode tip tem perature distributions..................................................132 5.1 Diagram of the cathode test facility..................................................................... 134 5.2 Schematic of th e electrode configuration............................................................135 6.1 Pure tungsten cathode profiles...............................................................................143 6.2 Pure tungsten cathode axial tem perature profiles for a current of 1400 A w ith surface emissivity as a p aram eter........................................... 144 6.3 Pure tungsten cathode axial tem perature profiles for a tank pressure of 1.5 kP a with current as a param eter............................................................. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 P ure tungsten cathode axial tem perature profiles for a tank pressure of 3.0 kP a w ith current as a param eter......................................................... 147 6.5 P ure tungsten cathode axial tem perature profiles for a current of 600 A with tank pressure as a p aram eter..................................................... 148 6.6 Pure tungsten cathode axial tem perature profiles for a current of 1000 A with tank pressure as a param eter.................................................. 149 6.7 P ure tungsten cathode axial tem perature profiles for a current of 1400 A w ith tank pressure as a p aram eter.................................................. 149 6.8 Photograph of th e 488 nm A r II line intensity distribution a t 1000 A and 1.5 k P a for a pure tungsten cathode...................................................... 151 6.9 Contours of th e 488 nm A r II line intensity distribution a t 1000 A and 1.5 kP a for a pure tungsten cathode......................................................151 6.10 D istribution of th e 488 nm A r II line intensity distribution at 600 A and 1.5 kP a for a pure tungsten cathode...................................................... 153 6.11 D istribution of th e 488 nm A r II line intensity distribution a t 1000 A and 1.5 k P a for a pure tungsten cathode...................................................... 153 6.12 D istribution of th e 488 nm Ar II line intensity distribution a t 1400 A and 1.5 k P a for a pure tungsten cathode..................................................... 154 6.13 D istribution of th e 488 nm A r II line intensity distribution a t 600 A and 3.0 kP a for a pure tungsten cathode...................................................... 154 6.14 D istribution of th e 488 nm Ar II line intensity distribution a t 1000 A and 3.0 kP a for a pure tungsten cathode...................................................... 155 6.15 D istribution of th e 488 nm A r II line intensity distribution a t 1400 A and 3.0 kP a for a pure tungsten cathode..................................................... 155 6.16 Boltzm ann fit to electron tem perature d a t a . .............................................158 6.17 Electron tem perature as a function of radius for a pressure of 1.5 kPa and a current level of 600 A with axial position as a param eter (pure tungsten)................................................................................................................. 160 6.18 Electron tem perature as a function of radius for a pressure of 1.5 kP a and a current of 1000 A with axial position as a param eter (pure tungsten)................................................................................................................. 161 6.19 Electron tem perature as a function of radius for a pressure of 1.5 kP a and a current of 1400 A with axial position as a param eter (pure tungsten)................................................................................................................. 162 6.20 Electron tem perature as a function of radius for a pressure of 3.0 kP a and a current level of 600 A with axial position as a param eter (pure tungsten)................................................................................................................. 163 6.21 Electron tem perature as a function of radius for a pressure of 3.0 kPa and a current level of 1000 A with axial position as a param eter (pure tungsten)................................................................................................................. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.22 Electron tem perature as a function of radius for a pressure of 3.0 kPa and a current level of 1400 A with axial position as a param eter (pure tungsten)................................................................................................................. 165 6.23 Arc attachm ent areas for pure tungsten cathode operation.......................168 6.24 Brightness of the 488 nm Argon II line em issivity as a function of current density for th e pure tungsten tests.................................................. 169 7.1 C athode surface em ittance measured a t three axial locations...................172 7.2 C athode surface em ittance as a function of surface tem perature. . . 173 7.3 Axial cathode tem perature distribution for a tan k pressure of 1.5 kP a w ith current as a param eter.............................................................................. 174 7.4 Axial cathode tem perature distribution for a tan k pressure of 3.0 kP a w ith current as a param eter.............................................................................. 174 7.5 Axial cathode tem perature distribution for a tank pressure of 4.5 kP a w ith current as a p aram eter.................................................................................. 175 7.6 Axial cathode tem perature distribution for a tan k pressure of 6.0 kP a with current as a param eter.............................................................................. 175 7.7 Axial cathode tem perature distribution for a current of 600 A with tank pressure as a param eter........................................................................... 176 7.8 Axial cathode tem perature distribution for a current of 1000 A with tank pressure as a p aram eter............................................................................ 176 7.9 Axial cathode tem perature distribution for a current of 1400 A with tank pressure as a param eter............................................................................ 177 7.10 Variation in cathode tem perature with operating tim e.............................. 179 7.11 Axial cathode tem perature distribution for a tan k pressure of 1.5 kPa, a current of 1000 A, and all mass flow rates................................................ 183 7.12 Axial cathode tem perature distribution for a tan k pressure of 1.5 kPa, a current of 1000 A, and selected mass flow rates..................................... 183 7.13 Axial cathode tem perature distribution for a tan k pressure of 1.5 kPa, a current of 1000 A, and a mass flow rate of 0.290 g /s ........................... 184 7.14 Axial cathode tem perature distribution for a tan k pressure of 3.0 kPa, a current of 1000 A, and all mass flow rates............................................... 184 7.15 Axial cathode tem perature distribution for a tan k pressure of 4.5 kPa, a current of 1000 A, and all mass flow rates............................................... 185 7.16 Axial cathode tem perature distribution for a tan k pressure of 6.0 kPa, a current of 1000 A, and all mass flow rates............................................... 185 7.17 Photograph of the 488 nm A r II line intensity distribution a t 1000 A and 1.5 kP a for a thoriated tungsten cathode............................................. 186 7.18 C ontours of the 488 nm A r II line intensity distribution a t 1000 A and 1.5 kP a for a thoriated tungsten cathode............................................. 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.19 D istribution of the 488 nm A r II line emissivity distribution at 600 A and 1.5 k P a ..................................................................................................................188 7.20 D istribution of the 488 nm A r II line emissivity distribution a t 1000 A and 1.5 k P a ..................................................................................................................188 7.21 D istribution of the 488 nm A r II line emissivity distribution a t 1400 A and 1.5 k P a ............................................................................................................. 189 7.22 D istribution of the 488 nm A r II line emissivity distribution a t 600 A and 3.0 k P a ..................................................................................................................189 7.23 D istribution of the 488 nm A r II line emissivity distribution a t 1000 A and 3.0 k P a ..................................................................................................................190 7.24 D istribution of the 488 nm A r II line emissivity distribution a t 1400 A and 3.0 k P a..................................................................................................................190 7.25 D istribution of the 488 nm A r II line emissivity distribution a t 600 A and 4.5 k P a ..................................................................................................................191 7.26 D istribution of the 488 nm A r II line emissivity distribution a t 1000 A and 4.5 k P a ..................................................................................................................191 7.27 D istribution of the 488 nm A r II line emissivity distribution a t 1400 A and 4.5 k P a ............................................................................................................. 192 7.28 D istribution of the 488 nm A r II line emissivity distribution a t 600 A and 6.0 k P a ..................................................................................................................192 7.29 D istribution of th e 488 nm A r II line emissivity distribution a t 1000 A and 6.0 k P a ..................................................................................................................193 7.30 D istribution of the 488 nm A r II line emissivity distribution a t 1400 A and 6.0 k P a ..................................................................................................................193 7.31 Electron tem perature as a function of radius for a pressure of 1.5 kP a and a current level of 600 A with axial position as a param eter. . . . 195 7.32 Electron tem perature as a function of radius for a pressure of 3.0 kP a and a current level of 600 A with axial position as a param eter. . . . 196 7.33 Electron tem perature as a function of radius for a pressure of 4.5 kP a and a current level of 600 A w ith axial position as a param eter. . . . 196 7.34 Electron tem perature as a function of radius for a pressure of 6.0 kP a and a current level of 600 A w ith axial position as a param eter. . . . 197 7.35 Electron tem perature as a function of radius for a pressure of 1.5 kP a and a current level of 1000 A w ith axial position as a param eter. . . 197 7.36 Electron tem perature as a function of radius for a pressure of 3.0 kPa and a current level of 1000 A w ith axial position as a param eter. . . 198 7.37 Electron tem perature as a function of radius for a pressure of 4.5 kPa and a current level of 1000 A w ith axial position as a param eter. . . 198 7.38 Electron tem perature as a function of radius for a pressure of 6.0 kP a and a current level of 1000 A w ith axial position as a param eter. . . 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.39 Electron tem perature as a function of radius for a pressure of 1.5 kPa and a current level of 1400 A w ith axial position as a param eter. . . 199 7.40 Electron tem perature as a function of radius for a pressure of 3.0 kP a and a current level of 1400 A w ith axial position as a param eter. . . 200 7.41 Electron tem perature as a function of radius for a pressure of 4.5 kPa and a current level of 1400 A w ith axial position as a param eter. . . 200 7.42 Electron tem perature as a function of radius for a pressure of 6.0 kPa and a current level of 1400 A w ith axial position as a param eter. . . 201 7.43 Electron tem perature as a function of axial position from the cathode tip for a current level of 600 A w ith pressure as a param eter....................202 7.44 Electron tem perature as a function of axial position from th e cathode tip for a current level of 1000 A w ith pressure as a p aram eter 202 7.45 Electron tem perature as a function of axial position from the cathode tip for a current level of 1400 A w ith pressure as a param eter 203 7.46 Electron tem perature as a function of radius for a tan k pressure of 1.5 k P a and a current level of 1400 A upstream of th e cathode tip w ith axial position as a param eter......................................................................204 7.47 Brightness of the 488 nm Argon II line emissivity as a function of current density for th e thoriated tungsten tests............................................. 211 7.48 Brightness of the 488 nm Argon II line em issivity as a function of current density for the thoriated tungsten tests a t 6.0 k P a........................ 212 7.49 C alculated work function for th e 1.5 kP a tests...............................................212 7.50 C alculated work function for th e 6.0 kP a tests............................................... 213 7.51 Schem atic diagram of the cathode containing th e pressure taps. . . . 215 7.52 Pressure m easurem ents during long duration test a t a current level of 1000 A w ith the pressure ta p cathode.......................................................... 217 7.53 Pressure m easurem ents for operation at 1000 A and 1.5 kP a for short and long durations tests..........................................................................................218 7.54 Pressure m easurem ents for operation at 1.5 kP a w ith current as a p aram eter.................................................................................................................... 219 7.55 Pressure m easurem ents for operation at 3.0 kP a w ith current as a param eter.................................................................................................................... 220 7.56 Pressure m easurem ents for operation at 4.5 kP a w ith current as a param eter................................................................................ 221 7.57 Pressure m easurem ents for operation a t 600 A with tan k pressure as a param eter.................................................................................................................222 7.58 Pressure m easurem ents for operation at 1000 A w ith tank pressure as a p aram eter........................................................................................................... 223 7.59 Pressure m easurem ents for operation at 1400 A w ith tank pressure as a p aram eter........................................................................................................... 224 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X V 7.60 Variation of work function w ith thorium coverage on tungsten. . . . 224 7.61 Photom icrograph of thorium deposits on floor of crater found after 12 s of operation................................................................................................... 226 7.62 Photom icrograph of sm ooth tungsten crystals with deposits of tho rium found after 1 hour of operation ............................................................. 227 7.63 Photom icrograph of craters clogged w ith tungsten crystals after 1 hour of operation...................................................................................................... 228 7.64 Photom icrograph of tungsten and thorium structures found on the cathode tip after 5 hours of operation............................................................... 229 7.65 Photom icrograph of the fern-like stru ctu res.................................................... 230 7.66 Photom icrograph of tiny tungsten depressions filled w ith thorium m etal............................................................................................................................. 230 7.67 Photom icrograph of cathode tip after two hours of operation at 1000 A and 6 k P a ............................................................................................................... 231 8.1 H eat flux as a function of cathode tem perature for a Ve value of 11.8 V ........................................................................................................................... 240 8.2 Sheath voltage as a function of cathode tem perature for 10 kW and 25 kW am m onia arcjets.......................................................................................... 241 8.3 Electron tem perature as a function of cathode tem p eratu re for 10 kW and 25 kW am m onia arcjets..................................................................................241 8.4 A ttachm ent area as a function of cathode tem perature for a 10 kW am m onia arcjet.......................................................................................................... 243 8.5 A ttachm ent area as a function of cathode tem perature for a 25 kW am m onia arcjet..........................................................................................................244 8.6 Sheath voltage as a function of cathode tem perature for 100 A arcjets........................................................................................................................... 246 8.7 Electron tem perature as a function of cathode tem p eratu re for 100 A arcjets........................................................................................................................... 246 8.8 A ttachm ent area as a function of cathode tem perature for 100 A arcjets........................................................................................................................... 247 8.9 Sheath voltage as a function of cathode tem perature for 100 A am m onia arcjets..............................................................................................................248 8.10 Electron tem perature as a function of cathode tem p eratu re for 100 A am m onia arcjets........................................................................................................ 249 8.11 A ttachm ent area as a function of cathode tem perature for 100 A am m onia arcjets........................................................................................................ 249 8.12 Effective work function as a function of cathode tem p eratu re for 100 A am m onia arcjets........................................................................................................ 250 8.13 Sheath Voltage as a function of cathode tem perature for a pure tung sten cathode and a pressure of 1.5 k P a with current as a param eter. 252 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8.14 Electron tem perature as a function of cathode tem perature for a pure tungsten cathode and a pressure of 1.5 kPa with current as a p aram eter.....................................................................................................................253 8.15 A ttachm ent area as a function of cathode tem perature for a pure tungsten cathode and a pressure of 1.5 kP a with current as a p aram eter.....................................................................................................................253 8.16 Effective work function as a function of cathode tem perature for a pure tungsten cathode and a pressure of 1.5 kP a with current as a p aram eter.....................................................................................................................254 8.17 Sheath Voltage as a function of cathode tem perature for a pure tung sten cathode and a pressure of 3.0 k P a with current as a param eter. 256 8.18 Electron tem perature as a function o f cathode tem perature for a pure tungsten cathode and a pressure of 3.0 kP a with current as a p aram eter.....................................................................................................................256 8.19 A ttachm ent area as a function of cathode tem perature for a pure tungsten cathode and a pressure of 3.0 kP a with current as a p aram eter.....................................................................................................................257 8.20 Effective work function as a function of cathode tem perature for a pure tungsten cathode and a pressure of 3.0 kP a w ith current as a p aram eter.....................................................................................................................257 8.21 C athode surface and centerline tem peratures as a function of axial position for a pure tungsten cathode and a pressure of 1.5 kPa. . . . 259 8.22 C athode surface tem perature as a function of axial position for a pure tungsten cathode and a pressure of 1.5 k P a .................................................. 260 8.23 C athode surface tem perature as a function of axial position for a pure tungsten cathode and a pressure of 3.0 k P a .................................................. 261 8.24 M easured and predicted axial tem p eratu re profiles for low currents. . 263 8.25 M easured and predicted axial tem perature profiles for high currents. 263 8.26 C urrent density as a function of cathode tem perature..................... 270 8.27 Effective work function as a function of cathode tem p eratu re .271 8.28 M aximum evaporation rate for tu n g sten .......................................................... 274 8.29 C athode evaporation rates for pure tungsten cathode te sts.......................275 8.30 C athode evaporation rates as a function of current for thoriated tung sten cathode tests with pressure as a param eter............................................ 276 8.31 C athode evaporation rates as a function of pressure for thoriated tungsten cathode tests with current as a param eter..................................... 277 C .l Diagram of im aging pyrom etry system .............................................................295 C.2 C ID TEC cam era calibration.................................................................................300 C.3 C athode axial intensity distribution.................................................................. 302 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Tables 6.1 Experim ental effective work function values for pure tungsten cathode operation.................................................................................................................. 162 6.2 Experim ental values for pure tungsten cathode operation a t 1.5 kP a and 600 A .....................................................................................................................166 6.3 Experim ental values for tungsten pure cathode operation a t 1.5 kP a and 1000 A .............................................................................................................. 166 6.4 Experim ental values for pure tungsten cathode operation a t 1.5 kP a and 1400 A .............................................................................................................. 166 6.5 Experim ental values for pure tungsten cathode operation a t 3.0 kP a and 600 A ................................................................................................................ 167 6.6 Experim ental values for pure tungsten cathode operation a t 3.0 kPa and 1000 A.............................................................................................................. 167 6.7 Experim ental values for pure tungsten cathode operation a t 3.0 kPa and 1400 A .............................................................................................................. 167 7.1 Experim ental effective work function values for T horiated tungsten cathode operation..................................................................................................... 205 7.2 Experim ental values for thoriated tungsten cathode operation at 1.5 kP a and 600 A .................................................................................................... 206 7.3 Experimental values for thoriated tungsten cathode operation at 1.5 kP a and 1000 A .................................................................................................. 206 7.4 Experim ental values for thoriated tungsten cathode operation at 1.5 kP a and 1400 A .................................................................................................. 207 7.5 Experim ental values for thoriated tungsten cathode operation at 3.0 kP a and 600 A .................................................................................................... 207 7.6 Experim ental values for thoriated tungsten cathode operation at 3.0 k P a and 1000 A .................................................................................................. 207 7.7 Experim ental values for thoriated tungsten cathode operation at 3.0 kP a and 1400 A .................................................................................................. 208 7.8 Experim ental values for thoriated tungsten cathode operation at 4.5 kP a and 600 A .................................................................................................... 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.9 Experim ental values for thoriated tungsten cathode operation a t 4.5 kP a and 1000 A ............................................................................................. 7.10 Experim ental values for thoriated tungsten cathode operation at 4.5 kP a and 1400 A............................................................................................. 7.11 Experim ental values for thoriated tungsten cathode operation a t 6.0 kP a and 600 A ................................................................................................ 7.12 Experim ental values for thoriated tungsten cathode operation a t 6.0 k P a and 1000 A ............................................................................................. 7.13 Experim ental values for thoriated tungsten cathode operation a t 6.0 kP a and 1400 A ............................................................................................. 8.1 Model solutions for arcjet thruster configuration with different propellants.............................................................................................................. 8.2 Model solutions for arcjet thruster configuration at different pressures.................................................................................................................. 8.3 Model solutions and experim ental d a ta for the pure tungsten cathode configuration a t 1.5 k P a ................................................................................... 8.4 Model solutions and experimental d a ta for the pure tungsten cathode configuration a t 3.0 k P a .................................................................................... 8.5 Experim ental d a ta for tem perature profile com parisons.......................... 8.6 Model predictions for tem perature profile comparisons using 98 per cent enclosed current area m atch m ethod.................................................... 8.7 Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 1.5 k P a ................................................................................... 8.8 Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 3.0 k P a .................................................................................... 8.9 Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 4.5 kP a..................................................................................... 8.10 Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 6.0 k P a .................................................................................... XVIII 208 209 209 209 210 245 248 254 258 265 266 268 268 269 269 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract C athode erosion is one of the life-limiting m echanism s in several classes of elec tric thrusters. Since cathode erosion depends strongly on the cathode tem perature, a quantitative understanding of the effects of cathode operation on the cathode tem p eratu re is required. A mode! describing the near-cathode plasm a was developed for determ ining the heat loads to th e cathode as functions of th e various free-stream plasm a param eters. T his model is combined w ith a cathode therm al m odel in order to provide a com plete and integrated picture of electric th ru ster cathode operation. Several com putational exam ples are used to illu strate th e combined m odel. Therm al models w ith different levels of com plexity were developed to determ ine th e tem per atu re distributions w ithin the cathode. A two-dimensional tip approxim ation model was added to a quasi-two-dimensional therm al m odel, allowing the use of arc a t tachm ent areas sm aller th an the total tip area. T his allows the attachm ent area to be changed so th a t the effect of operating conditions (pressure, gas type, geom etry, etc.) a t a constant total current could be com puted. It also provides a new stable low -tem perature solution th a t agrees well with th e experim ental d ata. A database of axial tem perature distributions on a cylindrical, two percent thoriated tungsten cathode has been collected for current levels of 600-1400 A, argon mass flow rates of 0.074 to 0.878 g /s and am bient gas pressures ranging from 1.5-6.0 kP a. At the higher pressures the cathode tem perature increases monotonically from th e base to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the tip while a t the lower pressures the maximum tem perature was located further back on the cathode. T he changes in th e flow rate at constant pressure were found to have no significant affect on the axial cathode tem peratures. A dditional tests were perform ed over a pressure range of 1.5-3.0 kP a and a current range of 600- 1400 A with a pure tungsten cathode. For both sets of experim ents, increases in the operating current a n d /o r decreases in the pressure resulted prim arily in increases in the arc attachm ent area with small increases in the cathode tip tem perature. T he pure tungsten cathode tem peratures reached equilibrium w ithin a m inute while the thoriated tungsten tem peratures were observed to drift over several hours. T his drift is probably a result of thorium m igration on the cathode affecting the local work function. Electron tem perature m easurem ents were m ade utilizing the m ethod of relative line intensity ratios. T he radial electron tem perature profiles are flat for the low-pressure case and increase radially for the higher-pressure cases. T he variation of the attachm ent area w ith current and pressure was characterized by m easuring the intensity distribution of an argon ion line near th e cathode surface. For all of the pressures considered the arc is attached in an annular ring on the cathode tip and not on the centerline. M aterials analysis of the cathode following a test a t 1000 A and 6.0 kPa for two hours revealed th a t the thorium tends to accum ulate at th e tip, is depleted on th e shaft, and a transition occurs in between. A m inimum in th e work function therefore occurs in an annulus around the cathode tip. Tests perform ed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with a cathode containing two pressure taps revealed th at th e pressure w ithin the arc attachm ent area can be significantly lower than the am bient pressure. Excellent comparisons were seen between the model predictions and th e experim ents for both high-pressure (100 kPA ), and low-pressure (1.5-6.0 kPa) operation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 C hapter 1 In trod u ction Historically, the exploration of new worlds has been lim ited by th e transportation system s available, w hether it was exploring th e Aegean sea or th e “race to the m oon.” New technologies are required to explore new frontiers. Sim ilarly today, space exploration is lim ited by the use of chemical rockets which are in tu rn lim ited by th e nature of their chemical reaction energies. This lim itation requires the usage of huge quantities of propellant. For exam ple, am bitious missions to the outer plan ets, other th an fly-bys and small orbiters, are im practical w ith chemical propulsion based on the-required launch masses alone. One m ethod around this lim it is the use of electric propulsion. T hat is, electrical energy rather th an chemical energy is used to accelerate the propellant. Typically, th e electrical power would be supplied by solar or nuclear sources. The exit velocities, or specific im pulses, available from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 electric thrusters are up to an order of m agnitude larger th an those of chemical rockets, therefore, the am ount of propellant required for a specific mission is sig nificantly less. T h a t is, electric thrusters allow more energy to be added per unit m ass of propellant. The developm ent of electric propulsion, or som e o th er advanced propulsion system , is therefore crucial for expanded exploration and colonization of th e solar system . One of th e m ajor life-lim iting mechanisms in electric propulsion th ru sters is cathode erosion. T he objective of this work is to characterize the nature of th e plasm a-cathode interaction which can then be used for estim ating th e erosion characteristics. 1.1 Role of Electric Propulsion M any studies from the 1950’s to present have shown the benefits of using electric th ru sters instead of chemical thrusters. Electric propulsion offers tw o types of ben efits: first, some missions are enabled by the significant perform ance im provem ents, and second, costs are reduced by the sm aller required launch masses a n d /o r extended spacecraft lifetim e. Low-power missions would use solar electric propulsion (SEP) system s, while high-power missions would use nuclear electric propulsion (N EP) system s. For exam ple, a mission using an SE P system with xenon ion engines to th e m ain belt asteroids can deliver a 920 kg payload (including a 10 kW solar array) w ith a 2-year flight tim e for S230M (including the S130M Atlas HAS launch vehicle), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. com pared to a 556-kg payload delivered in 1.3 years for S465M (upgraded T itan IV launch vehicle) using a traditional chemical system [1]. In other words, 50 percent more payload mass can be delivered with SEP for half the cost of a chemical system. In the current era of lim ited funding, the m onetary savings for electric propulsion may actually enable some missions. M ost space missions could be improved through the use of available electric propulsion systems. Usually solar power is best for small (< 100 kW ) system s while nuclear power is best for the large system s. Low-power system s provide increased capabilities for near-earth space missions. Hydrazine 1.8 kW arcjets are currently replacing hydrazine chemical thrusters for north-south station keeping system s for th e General Electric 7000 com m unications satellite series. T he hydrazine arcjet system will extend the life of the satellite by about 3 years w ithout m ajor propulsion system m odifications [2]. Arcjets provide a significant im provem ent in o th er near- earth applications such as orbit circularization and orbit raising [2,3]. T he former Soviet Union has used stationary-plasm a thrusters (S P T ) since the early 1970’s [4,5]. Recently, laboratories in the U nited States have become interested in these thrusters and have begun testing [6,7,8]. Large-scale missions such as m anned M ars, lunar cargo, or M ars cargo missions require large power system s (> 1 M W ). Efficient and light-weight th ru sters (effi ciency > 0.4 and total system specific m ass < 10 kg/kW ) with specific impulses Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between 2000 and 7000 s are required to support these missions [9], B oth N EP and SEP system s out-perform chemical system s for M ars cargo missions because of both reduced launch m ass and shortened trip tim es [10]. Although the perform ance of SEP system s is lower than for N EP ones, they are still com petitive because they do not involve the political com plications of space nuclear power. However, SEP system s a t these power levels may not be practical because of the large solar arrays required. Nuclear electric propulsion system s a t m oderate-power levels also ou tp er form nuclear therm al rockets (N T R ) for missions such as the M ars cargo, where trip tim e is not as im portant as it is for m anned missions; and m ultim egaw att (10 to 200 M W ) N EP system s are com petitive w ith N TR system s for th e piloted m is sions [11]. A review of N EP spacecraft and N EP system s is given in Ref. [12]. An overview of both th ru ster and power system s for current N EP designs is given in Ref. [13]. T h e only developed th ru ster capable of processing power a t th e level of 100’s of kilow atts is th e m agnetoplasm adynam ic (M PD ) thruster, described below. 1.2 Electric Thrusters Electric propulsion thrusters are usually categorized by their accelerating m echa nism, electrostatic, electrotherm al or electrom agnetic; and their power level. E xam ples of all three types are discussed below. T he th ru st, Ft , is related to th e th ru ster perform ance (specific im pulse, ftp , and efficiency, ijr) and the th ru ster power, Pt , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where g0 is th e acceleration of gravity on E arth. The th ru ster efficiency, tfj- is the ratio of th e je t kinetic power to th e input electrical power. A trade-off exists be tween operation a t high specific im pulse or “high” th ru st for a given power level and efficiency. Usually the th ru ster specific im pulse is “tuned” to the optim um specific im pulse for th e mission. T he specific impulses of electric thrusters are sig nificantly higher th an for chemical system s bu t they also have much lower th ru st levels. Therefore, electric thrusters m ust operate for significantly longer periods to achieve the sam e velocity change. T he lifetim e of the th ru ster is therefore as im por tan t as the perform ance. One of the prim ary life-limiting mechanisms for all three types of th ru sters is cathode erosion. For exam ple, typical deep-space missions of small robotic spacecraft require th ru ster lifetim es from 8000 to 15,000 hours because of the low -thrust values associated with electric thrusters [14,15]. The m ost applica ble th ru ster for these missions is th e ion engine because of high efficiency and high specific im pulse. O ften missions can be done with trip tim es less than or equal to those for chemical system s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2.1 Ion Engine An ion engine is an exam ple of an electrostatic thruster, where the propellant (ions) is accelerated by a static electric held. A schem atic of a typical ion engine is shown in Fig. (1.1). T h e engine consists of three main subsystem s, namely, th e discharge cham ber th a t produces the ions, the accelerating grids th a t accelerate th e ions to produce th ru st, and the neutralizer th a t therm ionically em its electrons into the plum e to provide electrical neutrality. Ion engines are capable of both high efficiency (70 to 80 percent) and large specific im pulse (3000 to 10,000 s). W hile cesium and m ercury propellants have been used in the past, noble gases are prim arily used today. Although the efficiency and specific im pulse are large, th e th ru st density is quite small. A typical 30-cm xenon ion engine produces 92 mN of th ru st w ith a specific im pulse of 3350 s and an efficiency of 6 6 percent [16]. A dditional perform ance inform ation for noble-gas propellants are given in Refs. [17] and [18]. T his low th ru st density lim its the am ount of power th a t can processed in a given area. Ion enjpne lifetim es are estim ated to be about 8,000-15,000 hours. The principal failures are due to accelerator grid erosion, discharge cathode failure, or erosion of other com ponents w ithin the engine [19]. W hile long-duration tests have been perform ed in the p ast on m ercury propellant, test d a ta for noble gas propellants are lim ited. T he m ost successful test was a 4200-hour full-power test of a J-series engine ( J l ) [20] and another test was performed on the J5 engine for 5000 hours bu t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PROPELLANT ELECTRICAL ISOLATOR MAGNETIC RETURN PATH PROPELLANT. PLENUM PERMANENT MAGNETS ELECTRICAL' INSULATOR NEUTRALIZER SUBASSEMBLY GROUND SCREEN LECTRODE APERTURES (3145) MASK ION-EXTRACTION ELECTRODES (3) CATHODE/ KEEPER SUBASSEMBLY PERMANENT MAGNETS Figure 1.1: Schematic diagram of the Hughes 13 cm ion engine. a t 1/4 power [21]. A 10,000-hour test of the 700-series 30-cm engine was performed b u t it could only m aintain a fraction of its operating power tow ards the end of the test and required significant operator intervention to m aintain its operation [22]. All of these tests were perform ed with m ercury propellant. Recent testing using noble gases has revealed new erosion problem s (for exam ple, accelerator grid erosion) and decreased lifetim e of already critical com ponents such as th e cathode. T he lifetim e and reliability im provem ents for noble-gas ion engines are currently being investigated. A ccelerator grids have been tested on xenon propellant for 890 hours a t the NASA Lewis Research C enter (LeRC) [23] and for 900 hours at the NASA J e t Propulsion L aboratory (JP L ) [24]. Both tests were voluntarily term inated and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. showed significant accelerator grid erosion. Later tests of a sim ilar th ru ster design, bu t operated a t a derated power level of 2.3 kW from 5.0 kW , showed a significant reduction in the accelerator grid erosion [25,26]. C urrently, an ion engine test with a planned duration of 8000 hours is being conducted a t JP L . A discharge cathode has been operated for 5000 hours w ithout failure on xenon propellant [27]. All of the long-duration tests to d ate w ith noble-gas propellants, however, are still well short of th e requirem ents. A lthough ion engines have been studied continuously since the early 1960s, sufficient engine lifetim e and reliability have not been dem onstrated w ith noble-gas propellants. 1.2.2 A rcjet T h ru ster An arcjet th ru ster is an exam ple of an electrotherm al th ru ster w here the propel lan t acceleration is achieved by expanding an electrically heated gas through a converging-diverging nozzle. A schem atic of the JP L 30-kW class am m onia arc je t is shown in Fig. (1.2). The propellant gas is heated by passing it through an arc struck from th e tip of th e cathode through the constrictor to the diverging por tion of the nozzle/anode. An illustration of the arc column is shown in Fig. (1.3). Hydrogen, am m onia and hydrazine are the principal arcjet propellants although re search work has been done on noble gases and nitrogen as well. A good review of hydrogen arcjet work is given in Ref. [28]. The th ru st, specific im pulse and efficiency for a typical 30-kW class hydrogen arcjet are 1.545 N, 1282 s, and 32.2 percent at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 PROPELLANT INLET 2%T1J0fUATED TUNGSTEN CATHODE BORON NITRIDE HtOPEULANT INJECTOR GRAPHITE GASKETS PURE TUNGSTEN PLENUM CHAMBER. CONSTRICTOR AND NOZZLE (ANODE) MOLYBDENUM BODY ANNULAR PROPELLANT FLOW PATH y tr SWAGELOCK FTTTTNO INCONEL6GO FEEDTHROUGH SPIRAL PROPELLANT GROOVE BORON NITRIDB ELBCTROOB INSULATOR STAINLESS STEEL EXPANDER MOLYBDENUM ASSEMBLY NUT , NOCHLE 7* TAPER JOINT Figure 1.2: Schematic diagram of the JP L 30-kW class am m onia arcjet. Lanvnc' Q 'C C O fufT > n .node ✓ Figure 1.3: Illustration of an arc column. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 30 kW , and 0.613 N, 1027 s, and 33.3 percent at 9 kW [29]. By comparison 30 kW class am m onia arcjet perform ance values are 2.4 N, 800 s and 31 percent a t 30 kW , and about 1 .1 N, 650 to 675 s and 36 to 39 percent at 10 kW [30,31]. Low-power hydrogen arcjet perform ance values are 181 mN, 1140 s and 32.4 percent a t 3 kW , and 95 m N , 864 s and 34.1 percent a t 1 kW [32]. A rcjet th ru ste r operating lim its have been exclusively established by endurance testing. Recently, the Rocket Research Company (RRC) 1 .8 -kW hydrazine arcjet was certified for 1258 hours (183 sta rts w ith 50-hour nominal on-tim e blocks) and 870 hours (1-hour on-tim e and 1/2-hour off-time blocks) operation as p a rt of the flight qualification procedure for th e General Electric 7000-series satellite and both thrusters showed lim ited wear [33]. Long-duration testing of low-power arcjets has also been perform ed a t LeRC. A test of a 1.2 kW arcjet on sim ulated hydrazine propellant (H 2 and Nj in a 2:1 ratio) for 500 cycles (2 hours of on-tim e and 2 hours of off-time) produced an average cathode mass loss of 6 . 6 /ig /h r [34]. T he operating tim e of high-power arcjets is more lim ited. Recent endurance testing of high-power am m onia arcjets (above 10 kW ) has shown poor results at about 30 kW and reasonable results a t 10 kW . Tests of the 30-kYV am m onia arcjet a t JP L have been performed for 573 hours [35] and 413 hours [36]. T he average cathode wear rates for these tests were 3.4 and 5.2 m g/hr, respectively. These tests fall far short of the required 1500 hours or more of operation th a t are required Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for typical missions. Testing of the 30-kW class am m onia arcjet a t 10 kW has, however, produced reasonable results of 1462 hours for continuous operation [30] and 707 cycles for cyclic operation (one cycle represents one hour of “on-tim e” and one-half hour of “ofT-time” ) [37]. T he longest tests of a hydrogen arcjet at 30 kW are a 723 hour test [38] and a 500 hour test performed in 1964 [39]. Both tests were voluntarily term inated. A lthough some past tests have shown reasonable arcjet operating tim es, these tests are still short of the requirem ents and tests perform ed in the 1960s have not been recently repeated. 1.2.3 M agnetoplasm adynam ic (M PD ) T hrusters T he M PD th ru ste r is an exam ple of an electrom agnetic thru ster, th a t is, the pro pellant is accelerated by electrom agnetic forces (Lorentz force). T he sim plest M PD th ru ster is th e self-field configuration shown in Fig. (1.4). T he term “self-field” indicates th a t th e m agnetic field w ithin the th ru ster is generated by th e imposed current. T he curvature of the current lines produces two force com ponents. The “blowing” or axial force acts to accelerate the particles axially out of the th ru ster while the “pum ping” or radial force acts to squeeze the particles towards th e cen terline. T he pressure w ithin the th ru ster is therefore higher near the cathode tip, which affects th e arc discharge. An exam ple of an applied-field M PD thruster is shown in Fig. (1.5) [40]. The electrom agnet surrounding the arc-head provides a solenoid a! m agnetic field. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 VVB 0 (blowing) Figure 1.4: Schematic of a self-field M PD th ru ster. con Figure 1.5: Schem atic of a lithium -propcllant Russian applied-field M PD th ru ster. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. applied-field acts w ith the electric field associated with the current lines to accelerate the charged particles created within the arc-head. Note th a t the self-field component is still present but is usually much sm aller than the applied-field. T he effect of the applied-field on th e current attachm ent a t the electrodes is not fully known. Also, the pressure near th e cathode can be significantly reduced when the applied m agnet field is stronger th a n about 1 0 0 0 G [41]. M agnetoplasm adynam ic thrusters currently represent the only technology capa ble of processing large am ounts of power (100’s kW to 10’s M W ). T he M PD th ru st effect was discovered in 1964 when the spedfic impulse of an arcjet th ru ster be ing tested began to anomalously increase while the propellant m ass flow rate was drastically decreased [42]. For electrotherm al thrusters, the specific im pulse alm ost always decreases w ith decreasing flow rate. A review of th e work on several select designs can be found in Ref. [43]. A very extensive body of current work can be found in references [44], [45] and [46]. In Ref. [44] current inform ation on applied-field MPD thrusters is used to outline a potential design for a 2.5 M W thruster. In Ref. [45] a review of M PD thruster research, both em pirical and theoretical, is presented and in Ref. [46] the effects of geometry, test-facility back pressure and propellant type are given. In general this work has been im peded by a lack of understanding of the complex fluid-dynamic, therm odynam ic and electrom agnetic processes occurring, both in the plasm a and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the engine com ponents. These processes m ust be controlled if this device is to becom e efficient and practical. These thrusters are operated in either a steady-state m ode or in a quasi-steady m ode. In the steady-state m ode, both th e electrom agnetic and the therm al proper ties w ithin the th ru ster are in a steady-state condition. For quasi-steady operation, the electrom agnetic properties are in a steady-state condition, but th e therm al prop erties are not. Q uasi-steady operation may consist of 10’s of kA currents for pulse w idths of the order of 1 m s [47]. Propellants th a t have been used in past tests include hydrogen, am m onia, noble gases, and alkali m etals while m ost of the experi m ental work has been perform ed using argon. T he efficiency for typical steady-state th ru sters is 0.43 and greater than 0.70 a t a specific impulse of ab o u t 5000 s for hy drogen and lithium propellants, repectively [48,49,50]. O ther propellants (A t, N2 , NH3 ) have produced efficiencies of 0.10 to 0.35 w ith specific im pulses from 1000 to 4500 s [48]. Lifetimes for M PD thrusters are the least known of the th ru ster types. Most M PD th ruster testing has been for quasi-steady, or pulsed, operation with cold electrodes where the current tends to concentrate in small areas called m icro-spots. T his micro-spot operation produces severe cathode erosion due to the ejection of m olten droplets of cathode m aterial resulting from the large localized currents and heat fluxes [51], Long-duration testing of steady-state thrusters has been lim ited. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The longest test to d ate is a 508-hour test at a nominal power of only 32 kW with an am m onia propellant flow rate of 30 m g/s performed in 1969 [52]. T he specific im pulse was about 1900 s, th e applied m agnetic field strength was 0.14 T and the cathode erosion was 2.43 g (2.9 percent of th e total cathode mass). This erosion corresponds to an average of 4.79 m g /h r over th e entire test. 1.3 Establishing Lifetime and Reliability T he lifetim e and reliability of a thruster are as im portant as its perform ance. These are of particular im portance to electric propulsion because low -thrust sys tem s require long operating tim es to achieve th e sam e total im pulse as chemical system s. For exam ple, planetary missions typically require engine lifetim es of 8000 to 10,000 hours [9]. Improved lifetime and reliability can also improve th e spacecraft perform ance by reducing th e spacecraft mass. T h a t is, fewer (or lighter) thrusters can be used and the need for redundant systems may also be reduced. T he lifetime of a th ru ster can be limited by two different types of failures. First, an event-consequent failure occurs when the device is operated beyond its lim its, for exam ple severely m elted or broken electrodes. This failure mode is independent of the operating history. Secondly, a failure may result from dam age accum ulation associated with repeated operation, for exam ple, electrode erosion o r crack propa gation resulting from therm al cycling. These failures may appear suddenly, in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 O Suspended Tests • Failures Requited Life Failure Probability Distribution Operating Time Figure 1.6: Probability of failure as a function of operating tim e. case of fractures, or may be seen gradually, for exam ple, gradual electrode erosion resulting in starting or operating problems. A lthough the thruster may still be operational, it may have worn such th at its perform ance is no longer sufficient or to a condition where it fails to operate (or sta rt) properly. One of the principal life-lim iting mechanisms in electric thrusters is cathode erosion. Lifetime is a prob abilistic quantity as shown in Fig. (1.6). The distribution represents the inherent uncertainties in the operation, m aterial properties, com ponent tolerances, and en vironm ent. T he tests in the left corner represent failures th at occur early in the testing program (infant m ortality). These failures are usually the result of im proper design or operation and are corrected, and therefore, these failures do not contribute Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 to the reliability assessment. Tests may be either voluntarily term inated or may be operated to failure, depending on the consequences of “failure.” Expensive or dan gerous system tests (for exam ple, large rocket m otors) are usually term inated before failure. T his m ethod, however, severely lim its the inform ation from the test, since the exact failure mode and tim e are not determ ined [53]. Therefore, additional te st ing is required to achieve th e sam e confidence level. It is desirable to have th ru ster design requirem ents fall far enough out in the left-hand tail of the distribution to have sufficient reliability, as well as having sufficient lifetim e. It is also im portant not to severely “over design” th e th ru ster since this results in increased m ass and complexity. T h e probability-of-failure distribution provides a quantitative m ethod for determ ining the design’s reliability. Generally there are insufficient d a ta to determ ine th e failure distribution due to th e com plexities and costs of testing. For these cases w ith sparse d ata, a rigorous qu antitative m ethod such as Probabilistic Failure Analysis (PFA ), is required. The PFA m ethodology has been successfully applied to reusable liquid rocket m otors [54], launch vehicle structures [55,56], and crack propagation [56]. This m ethodology has also been used for assessing ion engine accelerator grid life [57,58]. T he details of th e m ethodology are given in Refs. [59] and [60]. T he PFA methodology provides a rigorous means of using all of the existing d ata, both theoretical and experim ental, to provide a system failure risk estim ate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 QUANTITATIVE FAILURE MODEL 1 PROBABILISTIC FAIUJRE MODELING ! UNCERTAINTY OF ENGINEERING ANALYSIS PARAMETERS AND MODELS ENGINEERING ANALYSIS PRIOR FAILURE RISK DISTRIBUTION BAYESIAN STATISTICAL- ANALYSIS PROBABILITY DISTRIBUTIONS ' FOR FAILURE MODE(S) PARAMETER INFORMATION SUCCESS/FAILURE DATA TEST/FLIGHT EXPERIENCE MISSION PROFILES I MI SSI ON A N A L Y S I S I AGGREGATE FAILURE RISK ESTIMATE FOR SELECTED FAILURE MODES FAILURE MODE AGGREGATION Figure 1.7: T he probabilistic failure assessment methodology. An illustration from Ref. [53] of th e probabilistic failure assessm ent is shown in Fig. (1.7). T he m ethodology has three m ajor elements: probabilistic failure m od eling, a Bayesian statistical analysis to include success/failure d ata, and a mission analysis in which th e probability distributions for a num ber of relevant failure modes can be com bined. Models are developed for each of the failure m odes, and the un certainties iir th e model param eters and the model validity are determ ined. Ex perim ental d a ta or past experience is used a t this stage to help characterize the uncertainties in th e model input param eters. T he prior failure risk distribution is then modified in a Bayesian analysis incorporating the success/failure d a ta to pro Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vide the final distribution for each failure mode. This final distribution contains all of the d a ta (both theoretical and experim ental). The critical failure probability distributions are then aggregated to yield a failure risk assessm ent for the system . Since this m ethod can be used at any tim e during the design process, it can provide insight into which param eters are the m ost significant. Testing can then be directed to provide inform ation about these param eters. Past th ru ster work has been prim arily perform ance-oriented, but usually per form ance and lifetim e are coupled. T h a t is, generally a trade-off exists between increased perform ance and increased lifetim e. Lifetime experience has been exclu sively based on th e trial and error approach. T h at is, select a th ru ster configuration and operating condition, usually selected to maximize perform ance, and test the th ru ster to failure. W hile this approach reveals design flaws th a t lead to early fail ures (during assembly or in the first few hundred hours of testing) it does not allow for predicting the failure mechanism or tim e for a different configuration or o perat ing condition. These tests typically last from 1000 to 5000 or m ore hours and are expensive, requiring vacuum -fadlity operation for 3 to 7 m onths plus personnel tim e for m onitoring the experim ent and the facility. Also, the num ber of configurations and operating conditions th a t can be explored is limited by the tim e required for each test. Therefore m ethods, such as PFA , are needed for predicting the lifetimes of key th ru ster com ponents and eventually th e lifetime of th e entire thruster. An Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 illustration of th e com ponents of a model for cathode erosion is shown in Fig. ( 1 .8 ). T his work has focused on the near-cathode plasm a model and th e cathode therm al model (shaded boxes) using both theoretical and experim ental techniques. The ero sion m odels, the m ass tran sp o rt models, and the prelim inary work function model have been developed by Polk [51]. 1.4 Cathode Operation C athodes operate in two distinct arc discharge modes depending upon w hether they are sufficiently hot to be capable of therm ionic emission. C old-cathode operation is characterized by m any localized hot-spots while hot-cathode operation is character ized by large global tem peratures. Cold or nontherm ionic, cathodes operate with m any cathode “m icro-spots” th a t move rapidly over the surface. T hese m icro-spots have large current densities (about 107 to 10® A /cm 2) and produce small pits or craters on the surface. T he large localized current density, along w ith the associ ated heat flux, produces localized m elting or pitting of the surface. T he m easured erosion rates for th is type of operation are very large (0.3 p g /C to 15 p g /C ) com pared to hot, or therm ionic, cathode operation (0.038 n g /C to 180 n g /C ) in M PD thrusters [51]. T he difference in cathode lifetim es associated w ith erosion rates of 1.0 /ig /C and 1.0 n g /C is several hours versus about 3000 hours. T his assumes a 1 0 percent total m ass loss of a 2 -cm -diam eter 1 0-cm-long cathode operating at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1 CURRENT, PROPELLANT FLOW RATE, GAS PROPERTIES, ENGINE GEOMETRY PLASMA PROPERTIES CURRENT DENSITY SHEATH VOLTAGE CURRENT EMISSION PARAMETERS HEAT FLUX CURRENT DENSITY TEMPERATURE TEMPERATURE GROSS EROSION RATE INPUT NET EROSION RATE MATERIAL PROPERTIES. ' m ateriaiT ' PROPERTIES, WORK FUNCTION MODEL THERMAL MODEL MASSTRANSPORT MODEL EROSION MODEL PLASMA FLOW MODEL NEAR-CATHODE PLASMA MODEL v ANALYTICAL OR NUMERICAL MODELS TOTAL MASS LOSS OR GEOMETRY CHANGE Figure 1 .8 : Diagram of the cathode erosion model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 kA [51]. Cold-cathode thrusters are therefore not practical for system s requiring long life. H ot-cathode erosion is prim arily from the evaporation of m aterial [51]. The m icro-spot mode is prim arily associated w ith start-up transients for steady- state operation or for pulsed operation. Steady-state th ruster cathodes operate in the hot cathode m ode w ith tem peratures ranging from about 2000 K to 3600 K for thoriated tungsten. T he hot-cathode arc has a diffuse attachm ent th a t may only attach a t the tip or may envelop the first few centim eters of the tip depending on the operating conditions. The specific physics associated with th e cathode and the near-cathode plasm a are discussed in detail in the next chapter. Tw o physical types of cathodes are used in electric thrusters; solid rod designs, usually m ade of thoriated tungsten, and hollow cylindrical designs. 1.4.1 Ion Engine C athode Ion engines use hollow cathodes such as the one shown in Fig. (1.9) [27]. The cathode insert is im pregnated w ith a low work function m aterial, such as barium - calcium -alum inate (m olar ratio 4:1:1), which provides the necessary therm ionic elec tron emission a t reasonable tem peratures (<1800 K). An externa] h eater is used to preheat th e cathode to near its operating tem perature (> 1100 K) to prevent the dam age th a t occurs from arc spots during cold starts. Propellant flows through the cathode to m aintain the discharge a t a suitable pressure. T he mass flow rate and the orifice diam eter are selected for the desired current level. Some of the Russian Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 GAS FLOW INSERT ™ A TUBE BODY HEATER ORIFICE PLATE BASE FLANGE (NOMINAL DIMENSIONS IN MM) Figure 1.9: Illustration of a typical ion engine hollow cathode. Hall effect thrusters use sim ilar hollow cathodes. T his cathode design is also used for some plasm a-contactor designs for controlling spacecraft charging [61]. Ion engine cathode developm ent has been prim arily through testing w ith some theoretical work. T he model developed by Siegfried and W ilbur [62] com bines a sheath model [63] w ith an ionization zone, surface current and energy balances, and empirical relations for the flow field w ithin the hollow cathode. A similar model is presented by Salhi and Turchi [64], except th a t the empirical model is replaced with an isotherm al flow m odel. C athodes using m ercury and xenon propellants have been tested in excess of 10,000 hours w ithout failure [65,61]. The erosion rates for noble Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 gas cathodes are higher than for m ercury and d a ta are lim ited. A 5021 hour test of an ion engine cathode on xenon propellant at an emission current of 25 A was performed a t JP L [27]. T he cathode operation was not significantly different a t the end of the test than a t the beginning. Currently, a long duration test of a hollow cathode is being performed a t NASA LeRC [6 6 ]. To d ate, th is test has com pleted more than 20,000 hours. W hile these tests indicate th a t long operation of hollow cathodes is possible, their reliability has not been established. Ultim ately, hollow cathode lifetim e (aside from heater failure, orifice p late ero sion, etc.) is lim ited by the exhaustion of the low work function im pregnate from th e insert. Typically a loss of 25 to 50 percent of the im pregnate is assum ed to be a conservative “end-of-life” [67,68]. An empirical relation for a rough estim ate of this lifetim e is given by where M l is the fraction of the im pregnate m aterial lost, 7 / is the insert tem per ature, and C and a are empirical constants th a t depend on th e m aterials. Typical 1.61 x 104 K , respectively [67]. N ote the strong dependence of the lifetime on the insert tem perature. It is critical th a t this tem perature be kept low for sufficient cathode lifetim e. A lthough, if th e cathode tem perature is to o low for sufficient therm ionic emission, the cathode will operate in a “spot” m ode th a t also has a high ( 1.2) values of C and a for the im pregnate mix mentioned previously are 400 hr 1 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 erosion rate. T his model assumes th a t the insert tem perature is uniform, th a t all of the im pregnate evaporated is lost (no longer useful), and th a t the cathode will fail when a specific fraction of the im pregnate is lost. None of these simplifying as sum ptions are accurate. T he discharge will probably not be uniform and therefore the insert tem perature will not be uniform . M aterial evaporated from one p a rt of the insert will probably condense on colder regions of the insert or cathode tube. A lthough tem peratures of the cathode tu b e and the orifice p late have been readily m easured, insert tem peratures are less well known. In application, this model pro vides a very conservative estim ate. For example, the m easured insert tem perature and the above model produced an estim ated life of only 321 hours (M l = 0.67 and Tj = 1770 K) for th e cathode tested for over 5000 hours (27). T h e work in this study will provide a m eans of estim ating the insert tem perature distribution based on the discharge characteristics and the cathode therm al characteristics. This tem perature distribution would provide a more accurate estim ate of the im pregnate loss. 1.4.2 A rcjet C athode A rcjets use solid-rod thoriated tungsten cathodes, generally w ith conical tip shapes. O ther tip shapes have been investigated and indicate only small effects on perfor m ance and erosion [69]. These cathodes are typically started cold and quickly reach the required tem peratures for stable therm ionic emission. T his starting technique causes increased erosion to the cathode which may be im p o rtan t for missions such Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Figure 1.10: Photom icrograph of the eroded tip from a 10 kW am m onia arcjet 1470 hour test. as north-south station keeping, th a t require m any starts relative to th e steady-state operating tim e. For “long-bum ” missions, such as orbit raising, th e erosion from the occasional starts m ay be insignificant relative to the erosion from th e long periods of steady-state operation. T he work here will focus on the steady-state erosion. These cathodes operate with the arc attached to a small “spot” a t th e cathode tip which is typical of “high-pressure” arcs (pressure > 1 atm ) as shown in Fig. (1.3). Post-test analysis of arcjet cathodes usually reveals a small spot th a t was m olten prior to arc extinction as seen in Fig. (1.10) [70]. Also, for longer tests craters have been formed a t the cathode tip as seen in Fig. (1.11) [71,72]. Sometimes dendrites are formed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Figure 1.11: Photograph of an arcjet cathode tip crater after 573 hours of operation w ith am m onia propellant a t 26 kW . on the crater rim in conjunction with the crater form ation. It is suspected th a t the dendrites m ay cause a short between the electrodes producing a th ru ste r failure. Testing has also revealed th a t cathode erosion is not necessarily linear w ith time. For exam ple, tests with the same th ru ster configuration and operating conditions had an average erosion ra te of 1.4 m g /h r for 28 hours of operation, 2.0 m g /h r for 100 hours of operation, and 5.2 m g /h r for 413 hours of operation [36]. By comparison, testin g of th e sam e arcjet design with m inor nozzle m odifications a t 10 kW has produced little cathode tip erosion as seen in Figs. (1.12) and (1.13). T he erosion rates for the two long-duration tests revealed th a t the cycled th ru ster had 0.31 g Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 Figure 1.12: Photograph of an arcjet cathode tip before and after 1462 hours of continuous operation w ith am m onia propellant a t 10 kW . Figure 1.13: Photograph of an arcjet cathode tip after 707 operation cycles (702 hours) w ith am m onia propellant at 10 kW . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 (0.44 m g /h r average) of erosion m ass loss after 707 cycles while the steady-state test lost only 0.24 g (0.164 m g /h r) after 1462 hours. For this test repeated starts increased th e average cathode erosion by a factor of 2.7. It is suspected th a t the reduction of the thruster operating current from 215 to 100 A produced a lower cathode tip tem perature and therefore a lower erosion rate between th e 25 and 10 kW steady-state tests. Tests of low-power arcjets have also revealed some tests w ith significant cathode erosion and others with only m inor dam age. Testing of a 2 kW arcjet on sim ulated hydrazine produced 3 m g of cathode erosion in 300 hours (three 100 hour blocks) of operation for an average erosion rate of 10 /ig /h r (73). In co ntrast, two tests of a 1.8 kW hydrazine arcjet for 1258 hours (50 hour blocks) and 870 hours (one hour on-tim e cycles) revealed negligible cathode wear [33]. Similarly, a 1000 hour test (500 cycles) of a 1.2 kW arcjet using sim ulated hydrazine produced lim ited erosion loss of 6 . 6 m g for an average of 6 . 6 /ig /h r [34]. For this te st, th e cathode mass loss ra te decreased as the test progressed in contrast to the results previously m entioned for 30 k\V am m onia arcjets. Two possible explanations are th a t the sharp point of the cathode wears rapidly back to a more blunt geom etry, and second th a t under th e flow conditions for this arcjet, the m aterial eroded from th e tip was redeposited further back on the cathode and th a t this effect increased as th e tip crater enlarged. This type of redeposition could yield a lower total m ass loss, but Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. still have significant tip erosion. T he flow characteristics have some effect on th e tip erosion rates, b u t the n ature of this effect is not presently understood. One possible explanation is th a t a com bination of axial pressure gradients resulting from th e arc attachm ent and from self-field electrom agnetic forces produces a je t flowing tow ards the cathode tip . T he stagnation of this flow then tran sp o rts tungsten vapor from the center radially outw ard resulting in th e form ation of a crater [71]. Since th e flow characteristics of the plasm a near th e cathode tip will also affect this recirculation, changes in th e th ru ster geom etry (by design or through w ear) may significantly affect the erosion rate. For exam ple, the presence of crater dendrites for some conditions and not for others may be a result of th e evaporated m aterial being carried away from the cathode in some cases and deposited to form dendrites in others. M aterial evaporated from the cathode may be deposited on the anode/nozzle. Tests w ith thoriated tungsten cathodes in a water-cooled arcjet sim ulator have shown th a t the anode mass gains are com parable w ith cathode losses in some cases [74]. Different cathode m aterials have also been investigated. Since th e m ajor cathode loss m echanism is thought to be evaporation [51,75], a lower cathode tem p eratu re should reduce the mass loss. Two m ethods of achieving this are to increase the m aterial m elting tem perature or reduce the m aterial surface work function. T he m ajority of tests have been perform ed using two percent thoriated tungsten which has a work function th a t can range from 2.6-4.5 eV depending on the thorium Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. surface coverage [76] and a m elting tem perature of 3680 K. T he work function for pure tungsten is 4.5 eV. A test using a hafnium carbide cathode with a work function of 3.5 eV and m elting tem perature of 4163 K, was unsuccessful due to th e poor cathode m echanical properties, and chemical reactions w ith the sim ulated hydrazine propellant [75]. O ne-hundred-hour-tests of four other m aterials W -Y 2 O3 , \V -B aO 2 , W -LaBg, and W -L a 2C > 3 , all with two percent im pregnate, revealed th a t all of these m aterials produced factors of 6 to 35 greater erosion than th e thoriated tungsten [74]. W hile tests of 100 hours or less have been shown to produce inconclusive results for thoriated tungsten cathode erosion [36], the large erosion rates of the new m aterials in these experim ents are clearly evident. However, this erosion d a ta for th e thoriated tungsten cathodes may be questionable. 1.4.3 M PD T h ru ster C athode A wide variety of cathode shapes including both solid and hollow designs have been used in M PD thrusters. T he results of tests with several different designs are presented in Ref. [77]. W hile solid designs are the most utilized, hollow designs have received renewed interest recently [50,64]. T he startin g techniques for th e M PD th ru ster are sim ilar to those used for arcjets. These cathodes can operate in either the tip-attachm ent mode or a m ore diffuse attachm ent, illustrated in Fig. (1.14), depending on th e discharge pressure. These cathodes lack the tip spot or crater seen in arcjet cathodes. T he cathode surface usually has m any small pits associated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 High-Pressure Arc Attachment Low-Pressure Arc Attachment Figure 1.14: Illustration of a high-pressure and a low-pressure cathode arc a tta c h m ent. w ith the m icro-spots during start-up as seen by the rough surface of the cathode in Fig. (1.15). These cathodes are subjected to large therm al loads th a t can cause severe dam age. Two cathodes with conical tips were severely m elted after testing in th e JP L applied-field radiation-coded th ru ster [78]. Both cathodes had a m ore hem ispherical-shaped tip after the tests. A subsequent therm al analysis by this au th o r revealed th a t th e m elting was due to the much larger therm al gradients for the conical shape than for the hemispherical shape. A different type of cathode failure is shown in Fig. (1.16). This cathode was tested in the Z T l th ru ster at University Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Figure 1.15: Photograph of an M PD th ru ster cathode after testin g in the University o f S tu ttg a rt Z T l thru ster. o f S tu ttg a rt a t up to 8 kA on argon propellant. This cathode apparently m elted internally, and ruptured due to excessive Joule heating leaving th e bulge shown in Fig. (1.15) and the hollow area shown in Fig. (1.16) [79]. The m ajority of th e work on M PD th ru ster cathodes has focused on quasi steady or pulsed operation w ith cold cathodes. An excellent survey of both the experim ental and the theoretical studies is presented in Ref. [51]. W ork done on preheating cathodes for pulsed thrusters has shown th a t the preheating decreases th e term inal voltage while increasing both the specific im pulse and the efficiency (2300 s and 27 percent versus 800 s and 14 percent) [80]. Therefore, the cathode Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.16: Photograph of a sectioned M PD th ru ster cathode after testing in the University of S tu ttg a rt Z T l thruster. tem perature has a significant effect on th e overall th ru ster characteristics, so th a t cold and hot-cathode th ru sters cannot be considered sim ilar, as has been done in the past. T he work here will focus on hot-cathode operation. Excellent results have been reported using m ultichannel hollow cathodes. This cathode geom etry was initially suggested by Delcroix [81] and has been widely exploited in alkali m etal th ru sters in Russia [82,83]. In this type of cathode th e gas is injected through a bundle of rods or tu b es packed into a larger tube. T he interiors of the tubes and th e interstices between tubes or rods function as small hollow cathodes. W ith this geom etry th e effective em itting area can be increased m any tim es over th a t of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 com parably sized rod or single-channel hollow cathodes, depending on the size and num ber of tubes or rods in th e interior. These cathodes offer additional advantages, including a more stable attachm ent and lower operating voltages, th an large single channel cathodes [81] and a higher probability of recapturing m aterial evaporated from th e em itting surfaces. 1.4.4 C athode M odels Several different approaches have been taken in th e past to characterize th e nature of the hot-cathode arc physics. Typically two m odels are considered, one describing the plasm a near the cathode and th e other modeling o f the therm al characteristics of the cathode. P ast works have focused prim arily on one or th e o th er of these models, but both are needed since they are closely coupled. T h a t is, the plasm a model provides the heat loads (boundary conditions) for the therm al model, and the therm al model provides the surface tem perature which strongly affects the plasm a near the cathode through therm ionic emission. Additionally, an elem entary approach to an overall model is to consider an energy balance for the M PD th ru ster [84], T he difficulty w ith this approach is in determ ining the heat loads to the electrodes. Typically a fraction of the electrical power is used. For exam ple, a th ru ster operating a t 1 0 0 kW of electrical power, 40 percent (or 40 kW ) will be lost as anode heating and 20 percent (or 20 kW ) will be lost as cathode heating. T his approach can, however, be useful for exam ining experim ental d ata. Shih, et al. [85] used calorim etry on the anode, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 cathode and plume to determ ine the energy distribution w ithin the M PD thruster. T hen, using the input current, they determ ined “equivalent voltage drops” for the anode and cathode. T h at is, therm al characteristics were used to help describe the discharge characteristics. C om pared to th e plasm a, a therm al model of th e cathode is straight-forw ard but may be num erically challenging. Simple quasi-two-dimensional m odels considering axial tem perature distribution w ith radial heat inputs have been presented by Bade and Yos [8 6 ], M ehta [87] and Dorodnov, et al. [8 8 ]. M ehta’s and D orodnov’s models include radial convection and radiation, Ohmic heating, and tem p eratu re indepen dent constant m aterial properties for specified tip and base tem peratures. Note th at neither of these models are capable of including radial arc attachm ent. T he model by W eng and Seldin [89] considers the axial and radial tem perature distributions w ithin th e cathode of an electric steel furnace. W hile the therm al model may seem sim ple, numerical difficulties arise from the nonlinearities, specifically, tem perature dependent m aterial properties, radiation, and arc heating. Simple models have been developed for the combined plasm a-cathode heating solution. B ade and Yos [8 6 ] developed a model which combines a simplified sheath model w ith a simple therm al model for arcjet cathodes considering arc attachm ent a t the tip only. T he model presented by King [90] for an M PD th ru ste r includes a simplified sheath m odel, neglecting the plasm a electron contribution, w ith a sur Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. face energy balance. W hile this model is appropriate for high cathode voltages, neglecting the plasm a electron contribution can significantly affect the results for low voltages (< 10 V) which are m ore appropriate for steady-state M PD thrusters. Moizhes and Nemchinskii considered a cylindrical arc attachm ent to a semi-infinite solid cathode enabling an analytic expression for the radial cathode tem p eratu re to be used [91,92,93]. T he plasm a model for this work includes a sim ple collisionless sheath and a more detailed energy balance in the ionization region than th e Bade and Yos m odel. A combined m odel is also presented by Zhu, et al. [94] and Lowke, e t al. [95] based on the plasm a model of Morrow and Lowke [96]. T hese models combine th e cathode, sheath and arc models b u t assume a collisional sheath th a t is only applicable to high-pressure discharges. It would not be appropriate for MPD th ru sters bu t may be useful for arcjet thrusters. Experim ental work has focused on m easuring the cathode tem perature. Two problem s typically arose during testing. F irst, the cathode tem perature changed over tim e scales (from m inutes to hours) th a t were much longer than were expected for the cathode to reach its steady-state tem perature (few seconds to m inutes) [51, 77,79,86]. T his is probably a result of thorium depletion raising the m aterial work function and hence the surface tem perature for the same current density [76]. The work function for the thoriated tungsten cathodes from all of the experim ents is not accurately known. T he second problem is asym m etric arc attachm ent which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 disallows using axisym m etric m odels [77,86]. During som e tests, the arc has been observed m oving from the cathode tip region towards th e insulator backplate. This phenom ena usually results in severe insulator erosion [97]. Also from the sam e set of tests, severe cathode m elting was observed using helium propellant, and hollow cathode designs produced higher specific im pulse values than th e solid-rod designs. W hile this discussion has focused on ion engine, arcjet, and M PD th ru ster cath odes, o th er devices can have sim ilar arc characteristics. P lasm a torches o perate similarly to an arcjet thruster. Arc-lam p cathode operation can also be sim ilar to arcjet or M PD th ru ster operation [98,99]. 1.5 Role o f this Thesis The objective of this thesis is to characterize the operating characteristics of cath odes in gaseous discharges with specific applications to electric th rusters. T he m ajor life-limiting mechanism for steady-state cathode operation is erosion from evapora tion which is strongly dependent on the cathode tem perature [51]. Therefore, de term ining the effects of th ru ster operational param eters such as discharge pressure, current level and th ru ster geometry, on the cathode tem perature distribution is the key to estim ating the cathode lifetime. T he m ajority of this work was therefore focused on evaluating the cathode tem perature. This work consists of two p arts; a theoretical model of the cathode discharge including a therm al model of the cathode Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 and a model of th e near-cathode plasm a, and a series of experim ents to verify the model predictions and provide additional insights. T he cathode tem perature is determ ined by heat conduction and Ohmic heating w ithin th e cathode as well as the surface heat fluxes. A therm al model determ ines th e tem p eratu re distribution w ithin th e cathode while a model of the near-cathode plasm a determ ines th e surface heat flux and current density. For this work, both of these m odels were developed (Recall Fig. (1.8)). A first-principles near-cathode plasm a m odel, described in C hapter Two, was developed th a t contains term s ne glected in previous work which become significant under certain operating condi tions, particularly for lithium propellant and low-pressure operation. T he near cathode plasm a can be described by a series of plasm a regions such as the sheath and the boundary layers. T he models for th e individual regions can then be linked to provide th e near-cathode plasm a m odel. The numerical difficulties involved in coupling th e near-cathode plasm a model to the cathode therm al model required a series of therm al models w ith increasing sophistication, presented in C hapter T hree, to be developed. T h e solutions of the sim pler models were used as startin g points for the m ore complex models. T he overall cathode model, discussed in C hapter Four, combines th e model for the near-cathode plasm a w ith the therm al model for the cathode. T h e overall cathode model can be used to estim ate the m aterial losses through evaporation, provide the cathode boundary conditions for a th ru ster Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 flow model, and provide a theoretical basis for establishing cathode reliability and lifetim e using m ethods such as PFA. A series of experim ents described in C hapters Six and Seven were performed to complement the m odeling effort. A cathode test facility (C T F ), discussed in C hapter Five, was developed to allow optical diagnostic access to the discharge. Tests in this facility were designed to sim ulate the discharge characteristics within M PD thrusters. W ithin lim its, different operating param eters were individually varied in the system and their effects noted. T he trends and m agnitudes of these effects were then com pared with the m odel. Significant variations in the cathode tem perature distributions were observed in tests with thoriated tungsten cathodes due to thorium m igration on the cathode surface affecting the surface work function. A series of tests using a pure tungsten cathode, discussed in C hapter Six, were perform ed to elim inate this problem and th e results of these tests agreed well with th e experim ents. Comparisons between the experim ents and the model predictions are presented in C hapter Eight. C athode erosion m easurem ents and model predictions are also presented. Excellent agreem ent was observed between the model predictions and th e experim ental d ata. T his verifies th a t th e modeling approach developed here is applicable to low-pressure discharges. Excellent comparisons of th e model w ith data from long-duration high-power arcjet th ru ste r tests were also seen, confirming the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m odel’s applicability to high-pressure discharges. Conclusions from this study and recom m endations for fu tu re work are discussed in C hapter Nine. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 C hapter 2 N ear-C ath od e P lasm a M odel C athode erosion, one of th e prim ary life-limiting m echanism s, has been shown to depend strongly on the cathode tem perature [100]. Therefore, the m ajority of this study was intended to provide a simple m eans of predicting the cathode tem perature for various th ru ster operating conditions. In addition, the therm al characteristics of the electrodes must be known to com pute the therm al loads to the rest of the th ru ster and to the spacecraft. T his model also provides the appropriate boundary conditions a t the cathode surface for models of the operating characteristics of a com plete th ru ster. For exam ple, the current contours w ithin the M PD th ru ster cannot be specified independently of the cathode tem perature distribution because the m ajority of the current is from therm ionic emission. Since the cathode model boundary conditions also depend on the characteristics of the main plasm a, th e two Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 models m ust ultim ately be coupled to obtain an overall model of th e cathode region of the th ru ster. T he cathode model consists of two parts, namely a near-cathode plasm a model and a therm al model of the cathode. T he near-cathode plasm a model connects the properties of th e m ain plasm a w ith th e cathode. Specifically, given the plasm a prop erties w ithin an ionization m ean-free-path of the surface, th e near-cathode model predicts the heat flux and current density to the cathode surface. For this study a one-dim ensional plasm a model was developed for variations normal to th e cath ode surface. W ith these boundary conditions and the traditional therm al transport m echanism s, th e therm al model can predict the tem perature distribution w ithin the cathode. Because of the interdependency of the two m odels, they m ust be solved sim ultaneously. An illustration of the near-cathode plasm a is shown in Fig. (2 . 1 ). T h e Debye length, mean free path , and therm al, concentration and m om entum boundary layer thicknesses are represented by L d , L tl and respectively. For this study, only the surface, sh eath , presheath and ionization regions are m odeled. An illustration of the relative m agnitudes of the ion and plasm a electron currents is also shown for the near-cathode regions. In the m ain body of the plasma, th e current is predom inantly carried by th e electrons, while in th e sheath region the ion current may dom inate. To m atch these regions an ionization region, which produces the required num ber Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Solid \ i Sheath Presheath Boundary Layers Main Plasma Recombination region — l d Ionization region ■ ~ L I Lei • "^1 L T.C.M Position Figure 2.1: N ear-cathode plasm a regions. of ions for the sheath region, is required between the sheath and the main plasm a body. Similarly, a recom bination region exists at the cathode surface to produce a transition to pure electron conduction in the solid. At th e surface, ions are also converted to neutrals, which then return to the plasma. Each of th e regions will be discussed in the following sections. Since this model of the cathode and th e near-cathode plasm a is to ultim ately be integrated into a model of the overall th ru ster, it is desirable th a t the model be as simple and rapidly soluble as possible b u t still retain all of th e im portant physics. It will provide the cathode boundary conditions to the model describing the main plasm a flow and discharge. It is also im p o rtan t to have a sim ple model so th a t the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 effects of different characteristics can be clearly identified, particularly in com paring with experim ental d ata. The model m ust also be versatile since it will be used for both rod and hollow cathode analysis, and both high- and low-pressure discharges. Many of the m odels investigated are either to o complex, and case specific or over simplified. T he models developed here are based on the ideas presented in many of th e over-simplified m odels, but the im portant term s neglected in those models are included here. T he model is given below startin g from first principles so th a t the characteristics can be more easily identified. 2.1 Cathode Surface/Recom bination Region In general th e cathode surface is characterized by the m aterial, the surface finish and the tem perature. For this model, the recombination region is assum ed to be infinitesim ally thin and is considered a surface effect. Incident particles from the sheath heat th e surface while em itted particles cool the surface. T h e energy balance a t the surface balances the energy deposited and removed by th e particles with heat conduction into the solid, and radiative, convective and m ass (surface erosion) tran sp o rt to th e surroundings. T he net heat flux to the surface due to th e plasm a composed of two species, 1 and 2 , is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 9tot = 9f.I + 9.1,1 - 9n,l + 9i,2 + 9.1,2 - 9n,2 “ 96 + 9e 2 ♦ = E [— (e^ + cV rB + fM - &ff) + ^ (2eVc + eVB + <«,. - 2 f o ) .=i 1 c - E 2kT'} - T + 2*TC) + ^ (4 ff + 2kTe ) (2.1) *=i e where B oltzm ann's constant is expressed in units of eV /K . T he net current is given by jtot = J i.l + i .i .l + i.,2 + if,-.2 + J6 ~ ie - (2 .2 ) T he first and fourth m ajor term in Eq. (2.1) represents the energy from the singly- charged ions, the second and fifth term s from th e doubly-charged ions, the third and sixth term s is the therm al energy removed by the neutrals, th e seventh term is from the therm ionic electrons, and the eighth term is the energy from the plasm a electrons. In the ion term s, the subterm s represent the energy gained from the voltage drop through th e sheath and presheath regions plus the energy gained from th e recom bination of th e ions a t the surface. T h e plasm a electron term contains the energy gained from condensation of an electron on the m aterial plus the therm al energy of the electrons. T he therm ionic electron term consists of the energy required by an em itted electron to escape from the surface work function barrier plus its therm al energy. T he surface is assumed sufficiently rough to be fully accom m odating so th a t the em itted neutral flux for each gas species is equal to th e sum of the ion currents, j i t, +j,-f,I/2 = eF„,c,,. The em itted particles are assum ed to be Maxwellian w ith energies proportional to the surface tem perature. W hile the term s representing Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the therm al energy removed by the therm ionic electrons and the neutrals are small com pared to the other term s, they are not negligible and can be on th e order of tens of percent of the other term s under certain conditions, such as low sheath voltages or for gases with low ionization potentials such as lithium . In addition, radiative and convective tran sp o rt from the surface as well as conduction into th e m aterial are also considered for the overall energy balance, depending on the specific problem . These effects are included in th e therm al m odel. N ote th a t there is a typographical error in this equation in Refs. [101] and [102]. For high cathode tem peratures, therm ionic emission is the dom inant current conduction mechanism in the near-cathode region [90]. Therm ionic emission is de scribed by th e Richardson-D ushm an relation shown in Eq. (2.3). T h e therm ionic emission is extrem ely sensitive to the values of th e Richardson coefficient, A n , and the m aterial work function 4 > . A 0.5 eV change in work function can produce more th an an order of m agnitude change in current density as seen in Fig. (2.2). The Richardson coefficient and the work function may also be tem p eratu re depen dent [76]. However, only constant values are used here. Further, an em pirical A n value of 60 A /cm 2 /K 2 was used for all of the tungsten cathode calculations [103]. Only th e work function w ’ as varied for simplicity. In addition, the surface electric field acts to enhance the emission, a phenom enon known as the Schottky effect [98]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 As = 60 A/cm /K Woik Function (cV) 3.0 • — 3.5 4.0 - - - 4.5 2000 | I I I I I I I | I 2400 2800 3200 Cathode Temperature (K) 3600 Figure 2.2: Effect of m aterial work function on therm ionic emission current. T he effect is shown as a lowering of the m aterial work function in Eq. (2.4). * eE c = 4 >o - 4 jt c0 (2.3) (2.4) T he m agnitude of the electric field a t the cathode surface is prim arily determ ined by the characteristics of the sheath region. T he Schottky effect can significantly change the therm ionic emission current density as seen in Fig. (2.3). For the conditions of interest in this study, the Schottky effect may change the work function by several tenths of an electron volt, resulting in significant changes in th e therm ionic current. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 I £ e £ 3 o • 5 .2 Electric Field (V/cm) 0 - - 10 o £ 2000 2400 2800 3200 3600 Cathode Temperature (K) Figure 2.3: Effect of surface electric field on therm ionic emission current. 2.2 Sheath Region M any of the earlier sheath models were found to be inadequate for the conditions present in electric thrusters. T he m ajority of the models were developed for high- pressure ( 1 atm or higher) discharges in noble gases or nitrogen [86,90]. Although th is work is applicable to arcjets, it is not appropriate for the low-pressure discharges in the ion engine hollow cathode and in the M PD thruster. O ften th e plasm a electron current, j e, is neglected to obtain an analytical solution. T his assum ption is good for high sheath voltages (> 10 V) bu t fails for low voltages. O th er models included th e j t term bu t neglected other term s [63,98]. It is also common to neglect therm al Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 energy of the therm ionic electrons, 2kTet usually because the sheath voltage is much greater th an the therm al energy of the therm ionic electrons [63,86,98]. However, in an M PD th ru ster the sheath voltages may be low (less than 5 volts), and therefore th is effect is no longer negligible. Also, these term s are m ore im p o rtan t for gases w ith a lower ionization energy. For exam ple, the ionization energy term is less dom inant for lithium (5.392 eV) com pared to th e noble gases ( > 1 2 eV). T he sheath region is assumed to contain collisionless particles with constant to tal energy (potential plus kinetic). Six species are considered: monoenergetic therm ionic or beam electrons, singly- and doubly-charged m onoenergetic ions for two m onatom ic gases, and Maxwellian electrons originating in th e plasm a [63,104]. Doubly-charged ions were added to a previously developed model [101,102] because it has been suggested th a t cathode heating from doubly-charged ions m ay be significant a t low pressures for high-current discharges [98]. Furtherm ore, th e sheath thickness is assum ed to be much less than the Larm or radii of the particles, and therefore, m agnetic field effects on the particle trajectories are negligible. T h e model developed here is sim ilar to the one presented in Ref. [63] bu t also includes the.therm al energy of the therm ionic electrons, 2kTc, doubly-charged ions and two m onatom ic gas types. This model also normalizes the variables such th at th e norm alized therm ionic current density, •/(,, is independent of the normalized sheath voltage, t jc. T he energy equation for the therm ionic electrons is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Eq. (2.5), which relates the total energy, equal to the em itted therm al energy, as the sum of the kinetic and potential energies. i m'Vb2 - e ( V e - V ) = 2kTc (2.5) Note th a t th e m agnitude (not signed) of the sheath voltage drop is used throughout the calculations for simplicity. The flux or current density of the therm ionic electrons is given by jk = = const. (2 .6 ) Similarly, th e energy and current equations for th e singly-charged ions of gas type “s” are given respectively by, - e V = | m tf4r li0>J2 = eVB (2.7) and ji. * = en{',vi,, = eni,0fSVi,0,s = const. (2 .8 ) and for th e doubly-charged ions, - 2eV = r „ >0>>2 = eVB (2.9) and JiM = 2 en,;iJt> l;i, = 2 en;I,0,t r l,,0,J = const. ( 2 . 1 0 ) For a stable sheath to exist, the ions m ust enter the sheath w ith energies equal to or g reater th an the Bohm minimum energy [105], All of the ions here are assum ed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 to enter the sheath with energies equal to the Bohm minimum energy which is represented as the Bohm potential, Vb . The Bohm minimum ion energy is the sam e for both singly- and doubly-charged ions, and is independent of ion m ass. A simple proof of this is to consider a sheath with only doubly-charged ions and calculate the m inim um ion energy required for the correct electrostatic solution [106]. A lthough the doubly-charged ions have twice the charge and hence twice th e acceleration-, their num ber density is half w hat singly-charged ions would be to m aintain charge neutrality a t the sheath edge. T he plasm a electrons are assumed to be Maxwellian and have a num ber density given by where n e ,0 is th e total plasm a electron num ber density a t the sheath edge given by, These electrons fall into two classes, those with sufficient kinetic energy to overcome ficient energy th a t are repelled back to the main plasm a. T he corresponding flux of the high-energy electrons constitutes the plasm a electron current which is given by (2.11) (2.12) the sheath retarding potential and reach the cathode surface and those w ith insuf- (2.13) T he one-dim ensional Poisson charge equation is used to describe the electric field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 and the electric potential and is given by (P V d E ( f e ~ ~ ~ ~ ~ ~ ^n,>1 ^ n,'J ^ 2 nii,2 “ nb ~ »>e) • (2-14) Solving Eqs. (2.5) and (2.6) for the therm ionic electron num ber density, Eqs. (2.7) through (2.10) for the ion num ber densities and using Eq. (2.11) for th e plasm a electron density completes the right hand side of Eq. (2.14). Equation (2.14) is then normalized using the norm alization variables shown in Eqs. (2.15) through (2.20), resulting in Eq. (2.21). In addition it has been assum ed th a t th e m asses of the singly- and doubly-charged ions are equal (m,-,, = m,•;,,). « l-M III B - (2.15) I H H £ (2.16) H 1 Q H H III (2.17) L d = a/ - 2— \ - Debye Length Y (2.18) j _ 3b 1 m e 6 “ e ne ,0 V 2kTc (2.19) p - 2kTc bo~ kTe (2.20) d ? d t P -Mric-V+Eior'^-e-' (2.21) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Note th a t the addition of th e E)*, term to this model also prevents a singularity from occurring in the therm ionic electron term in this equation. Expressions for the norm alized ion num ber densities are given in Eq. (2.22) through Eq. (2.24). g ! w s * + * ( * + * . ) - ' » (2 22) «e.P ( 1 + 21%.,)+ f S f/J (l+ 2 .*ifJ) via = ~ ~ = * * (2*23) **e,o i | (i ^ (2.24) ni,< M T he mole fraction of singly-charged ions for gas type $ is given by Yi.,, and is discussed in more detail later. N ote th a t will never be exactly zero although it may be small. T h a t is, there will always be some fraction of singly-charged ions present. The ratio of th e partial pressures of the two gas types is given by 0 = 2 - (2.25) T he normalized Bohm m inim um ion energy can be determ ined by solving for the value of tjb th a t satisfies f r ° < 2-2 < i) a t the sheath edge where t j = 0 [107]. The solution of Eq. (2.26) using Eq. (2.21) is _ eVB Vi.i (1 + 4//,,.!) + Via (1 + ^ .i.2 ) VB = ----- (2 .2 *) e 2 - Jbfa + Efco) where j/,,i and v,a are related through Eq. (2.23). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 E quation (2.21) can then be integrated using the integration factor dtj/d£ to give Eq. (2.28). T he voltage a t the sheath edge was set to zero as a reference point and the electric field is assum ed to be (,h a t the sheath edge. Solving Eq. (2.28) for the electric field a t the cathode surface yields Eq. (2.29) which is used to com pute the Solving Eq. (2.28) num erically provides the spatial distributions of the electric potential w ithin the sheath. Using Eqs. (2.5) through (2.20) the rem aining sheath properties can be determ ined [104]. Sample distributions of the normalized param eters are shown in Figs. (2.4) through (2.9) for a single gas species and for only singly-charged ions. A dditional distributions of unnormalized param eters can be found in Refs. [104] and [108]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S chottky effect and com pletes the description of the therm ionic emission current given in Eqs. (2.3) and (2.4). - 4 Jk [(ijc + E io ) 1 /2 - (r,c - r, + E *,)1' 2] + 2 e " ” - 2 + cth 2 (2.28) 56 0.4 0.7 0.6 0.5 0.4 0.3 0.1 0 5 10 15 20 Normalized Position Figure 2.4: Normalized charge density as a function of normalized distance with Jh as a param eter, rjc = 10.0, E t0 = 0.3, c,/, = 0.0. If th e therm ionic current, Jt,, is sufficiently large, a double sheath is form ed. T h at is, two charge separation layers are form ed, an electron layer next to th e cathode followed by an ion layer. T his can be seen in Fig. (2.4) for values of Jj, greater than abo u t 0.1. N ote th a t the m axim um electric field within a double sheath is not a t the cathode surface, as seen in Fig. (2.6). As increases the value of th e normalized electric field a t the surface decreases and therefore decreases the significance of the Schottky effect. T h at is, as the therm ionic current density begins to dom inate, the effect o f the sheath on th e electron emission is decreased, assum ing th e cathode tem p eratu re is unaffected. These counteracting effects require a coupled solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 »*v\ — 0.7 • • 0.6 * * * 0.5 - - 0.4 — 0.3 o eo w 0 > 1 g o Z 0 5 10 15 20 Normalized Position Figure 2.5: Normalized voltage as a function of normalized distance w ith as a param eter, t j c = 10.0, = 0.3, (,h = 0.0. T he effect of on the distributions of normalized charge density and normalized electric field is seen in Figs. (2.7) and (2.8). T h e significance of including can be seen by com paring th e values of the normalized surface electric field for £{,0 values of 0 and 0.3 assum ing all of the o th er param eters are unchanged. An Eko value of 0.3 changes the value of the norm alized surface electric field 9 percent for a sheath -voltage of 10 volts, and a 17 percent change for a sheath voltage of 3 volts com pared to the case where is neglected. T he voltage distribution is not significantly affected. Changes in the electric field a t the surface are the most significant because they can dram atically affect the emission current. Also, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 2.5 0.7 0.6 0.5 0.4 0 3 0. 1 •3 2 .0 2 0.5- 0.0 0 5 10 15 20 Normalized Position Figure 2.6: Normalized electric field as a function of normalized distance with J t> as a param eter, t j c = 10.0, Ebo = 0.3, e,* = 0.0. effect of on the distributions is small for large sheath voltages (10 V or more) b u t becom es increasingly significant as the sheath voltage decreases. Since a current is present, the electric field a t the sheath edge cannot be zero. It may be estim ated as where j tot is the current density w ithin the main discharge plasm a (prim arily electron current) and at is the scalar electrical conductivity. The effect of th e normalized electric field, t,h , will depend on the nature of the discharge and will be different for each th ru ster type. Calculations using a range of values for (,h showed it to have a small effect on the overall near cathode plasm a. Therefore, c,h was set equal to zero for this study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 0 .4 - I : a 0.2. o e? c d g 0 .0 . i -0.2. 0.6 0 3 -0 .4 . 0.0 0.5 1.0 2.5 3.0 1.5 2.0 Normalized Position Figure 2.7: Normalized charge density as a function of normalized distance w ith E ^ as a param eter, » fc = 10.0, Jb = 0.1, (t h = 0.0. 2.4 ? 2.2- E ! 2 .0 - E ■ o M £ 1.8 * i o 25 1. 6 - 0.6 0 3 0.0 0.5 1.0 2.5 1.5 2.0 3.0 Normalized Position Figure 2.8: Norm alized electric field as a function of normalized distance w ith Ebo as a param eter, rjt = 10.0, Jb — 0.1, c,/, = 0.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 0.7 0.4 Z 0-3 V N ^ 0.2 2 0.1 0.0 0 5 10 15 20 Normalized Position Figure 2.9: Normalized particle num ber densities as a function of norm alized dis tance, Jb = 0.1, t ] c = 10.0, Ebo = 0.3, e,h = 0.0. Typical spatial distributions of th e various particle num ber densities are shown in Fig. (2.9). B oth the ion and the plasm a electron num ber densities decrease as they move towards th e cathode surface. T he ions are accelerated tow ards the cathode increasing th eir velocity and therefore decreasing their num ber density for a fixed total current shown in Eq. (2.8). T he plasm a electron num ber density decreases because of the increasing sheath potential barrier. T he density of the therm ionic electrons is highest at the surface and quickly decreases as the electrons are acceler ated away from th e surface. One advantage of including Ebo is th a t the therm ionic electron num ber density rem ains finite a t the surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 0.01 - - - 10 •o o 0.001 - 0 5 10 15 20 Normalized Sheath Voltage Figure 2.10: Norm alized total current density as a function of normalized sheath voltage. 0 . 1- 0. 1 0.2 0 3 0.4 0 3 - - - 0.6 0.7 ■ g 0 .0 1 , 0.001 8 0 2 4 6 10 Normalized Sheath Voltage Figure 2.11: Norm alized total current density as a function of normalized sheath voltage for large Jf, values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 T he sheath region presents two types of lim iting conditions for th e overall near cathode plasm a m odel. T he normalized to tal current given by is shown as a function of the normalized sheath voltage in Figs. (2.10) and (2.11). T he points where th e to tal current goes to zero represent the “floating potential.” Therefore, a m inim um normalized voltage value exists for specific values of norm al ized therm ionic current and normalized therm ionic electron therm al energy. The floating potential is reduced for larger therm ionic currents. For th e special case of no therm ionic current, a single gas type, and singly-charged ions, the normalized floating voltage can be calculated from For argon this value is about 4.681. T he curves are also fairly flat for larger values of sheath norm alized voltage, indicating th a t a large change in voltage is required to increase the to tal current for a fixed value of the normalized therm ionic current. This is due to th e weak dependence of the normalized ion current density (first term on the right-hand-side of Eq. (2.30)) on th e normalized sheath voltage. For small values of Jb, the ion current dom inates as can be seen in Fig. (2.10) by com paring the curves for Jb values of 10-4 and 10- 5 . As Jb increases it becomes dom inant, i.e., th e therm ionic electrons carry m ost of the current. (2.30) (2.31) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 Normalized sheath voltage 15 E 0.1 •o 0.01 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Thermionic Current Figure 2.12: Normalized surface electric field as a function of norm alized therm ionic current with normalized sheath voltage as a param eter. C J Ebo = 0.3 Normalized Sheath Voltage 5 0. 1- — 0.8 - - 0.7 — 0.6 -0 .5 .... 0 49 - 0.48 - 0.47 - -0.46 0.01 0.05 0.10 0.15 0.20 0.25 0 .3 0 Normalized Thermionic Current Figure 2.13: Normalized surface electric field as a function of normalized therm ionic current for small normalized sheath voltages. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he second lim iting condition can be seen in Figs. (2.12) and (2.13) which shows the normalized electric field a t the cathode surface as a function of therm ionic current. T he points w here th e electric field goes to zero represent conditions where the electric field reverses sign and a minimum voltage exists in the sheath. T h at is, a space charge lim it occurs. Because of this m odel’s lim itations (only m onotonic change in voltage is allowed), these points represent an upper lim it on the value for th e therm ionic current for specific values of T ) c and Ebo- For the conditions shown in Figs. (2.4) through (2.6) ( Ebo = 0.3, f,* = 0 and = 10) the m axim um value of Jb is 0.70034. T he norm alized electric field in Fig. (2.6) for a Jb value of 0.7 can be seen to go to approxim ately zero at the cathode surface. T he points where the curves are truncated a t th e left correspond to the floating potential. Low values of th e norm alized sheath voltage can significantly affect th e possible range of Jb values; th a t is, a more lim ited range of solutions exist. As the normalized sheath voltage is decreased the m axim um value for Jb decreases and the value of Jb corresponding to the floating potential increases. At a Jb value of 0.1752 and a jfc value of 0.456 these tw o lim its cross indicating an absolute lim it exists. 2.3 Presheath and Ionization Regions T he ionization and presheath regions connect the sheath region w ith th e main plasm a body [86,98]. T he purpose of the presheath region is to accelerate the ions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 Solid Sheath Main Plasma Combined Ionization and Presheath Regions Figure 2.14: Illustration of ionization region. so th a t they enter the sheath region w ith the m inim um energy required for a sta ble sheath (Bohm energy) [105]. For this model, the presheath region is combined w ith the ionization region by requiring th a t ions leave the ionization region with the Bohm energy. The ionization region generates the required num ber of ion and electron pairs to m atch the sheath and main plasm a body values. An illustration of the particles entering and leaving the ionization region is shown in Fig. (2.14). Electrical neutrality is given by = n.,i + 2n„,i + nIi2 + 2n„,2 = n e,0 + n& (2.32) and th e ionization rate for each gas type is given by ne> J — n,,, + 2n;lt# (2.33) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 6 Note th a t these two equations ate not independent. C onservation of heavy species generation for th e ionization region is given by «n,j + = 0 (2.34) and for the recom bination region a t the cathode surface by e F n W = J i,, + ^ (2.35) assum ing th a t th e cathode surface is fully accom m odating for each gas type. T he balance of species fluxes in and ou t of this region and th e generation w ithin the ionization region are given by Eqs. (2.36) through (2.38) w here d is the thickness of the ionization region. e ” i,.d — ji.t (2.36) 2e Tiu,,d — jn tt ~ ja,p,i (2.37) - n n,*d = Fn > ct, + F„,PfJ (2.38) T he conservation of current density in both the sheath and free-plasm a regions is given by ./lot = ii.l + iii.l + ji,3 + Jii.2 + jb - je = ji,p, 1 f Jli.p.l f ji,p,2 f jiij>,2 f Je.p* (2.39) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he energy balance for this region is given by (F„.c., + Fn,e,2) 2kTe + ^ (eVc + 2kTe) - ( j,,, + ^ + j,.2 + ^ + + ~ 2 ~ + J i.p .2 + “ 2~ J — “ - ^ 2 * T e + (F „iPt, + F „ iPi2 ) 2 * 7 \ - ^ (2 M ; + eVc) ~ (” i,l<i.l + "ti.tfii.l + ” i',2fi,2 + ” ii,2fu,2)d = 0. (2.40) The predom inant term s are th e energy added to the region by th e therm ionic elec trons, the energy consumed by ionization, the energy removed by the ions to the sheath, and the energy removed by th e plasm a electrons to b o th the sheath (fourth term ) and th e m ain plasm a (sixth term ). Note th a t the plasm a electron therm al energy contains tw o term s because only th e tail electrons of the Boltzm ann dis tribution are considered. T he retarding sheath potential serves to reject plasm a electrons w ith insufficient kinetic energy. T h e energy flux to th e surface is therefore obtained by integrating over th e population of electrons th a t have sufficient energy to overcome th e sheath potential, resulting in the two term s shown in Eq. (2.1) for th e plasm a electron therm al energy. T he relative sizes of th e current densities determ ine which energy-removal term dom inates. The other term s are significant under special circum stances, for exam ple, a t low sheath voltages. Using Eqs. (2.34) through (2.38) gives the expression - e F 0,P ,t = ji,p.t + 3 ^ f - (2.41) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is used to w rite Eq. (2.40) as a function of current densities. T he flux of the region from th e m ain plasma. T h a t is, the conservation of heavy species at the cathode surface determ ines Eq. (2.41) for th e conservation of heavy species for the ionization region. It has been assum ed th a t the neutral and ion therm al energies are equal and therefore their term s cancel. For th e m ain plasm a, the ion currents can be approxim ated using the relations for a fully ionized plasm a [109]. T he ra tio of the plasm a electron current, j e< p , to the plasm a ion current, j i tP, can be estim ated from their respective electrical conductivities and is of the order of m ,/m e [109]. As expected, th e ion currents are ion pairs th a t can be produced by ionization is determ ined by th e energy balance. T he energy equation (2.40) can be w ritten as Eq. (2.44) using these assum ptions, neutrals from the ionization region to the m ain plasm a is equal to th e ion current to TTli,. Jtol (2.42) rrtg Bji,w . (2.43) small com pared to th e electron current. T h e num ber density of th e electron and Eqs. (2.35) through (2.38), and Eqs. (2.41) through (2.43). ^ [2Jtre + eVe] - ^ (2*7; - C ) = 0 (2.44) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 where Using Eq. (2.39) w ith the norm alization variables given previously, Eq. (2.44) can be w ritten as This equation contains three unknowns, »)t, J* and Ebo for a given gas, ijt,,, 7 7,,,, and m ti,. Recall th a t and tjb are functions of r ]c, Jb and E\,0 only. Solutions to Eq. (2.46) are given in Fig. (2.15) for singly-charged argon gas with Ebo as a param eter and in Fig. (2.16) for different gas types w ith a fixed value of Ebo. N ote in Fig. (2.15) th a t Ebo can have a significant effect, particularly a t low norm alized sheath voltages. T he values of Jb for different gases scale approxim ately by and therefore argon and lithium have sim ilar results. T he solution of Eq. (2.46) along with th e norm alization variables is used to determ ine th e electron tem perature in the ionization region. Recall from the sheath solution th a t lim its exist for com binations of Jb and %. (ifc + B ) = 0 (2.46) where (2.47) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 4 ^ c E 4: 5 2 . u “ 0. 1 - Argon 0.6 0.3 0 5 10 15 20 Normalized Sheath Voltage Figure 2.15: Normalized therm ionic current density as a function of normalized sheath voltage w ith as a param eter for a singly-charged argon gas. * a » c £ u Hydrogen Helium Lithium Argon o 0 1 < L > £ ■o s 1 0 .0 1 - fc s: 5 V 20 0 5 10 15 Normalized Sheath Voltage Figure 2.16: Normalized therm ionic current density as a function of normalized sheath voltage for different gas types. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Previous authors have used a simpler form of Eq. (2.44), jb Vc = ji (i (2.48) T h at is, all of th e energy gained by the therm ionic electrons in the sheath is used to create ions [86,98]. This equation provides an algebraic relation between th e sheath voltage and the therm ionic current as com pared to Eq. (2.44) which is transcen dental. N ote th a t although Eq. (2.44) is more com plicated than Eq. (2.48) it does not contain any new variables (including the sheath region). Neum ann considered argon and helium a t 1 atm [98]. A semi-infinite thoriated tungsten cathode ( < f> = 2.6 eV, A n = 1 A /cm 2/K 2), w ith current densities of 3536 to 3837 A /cm 2 and with sheath voltages of 31.09 and 4.66 volts, respectively was analyzed. For th is range, th e plasm a electron current ranges from negligible at 31.09 volts to 30 percent of the total current a t 4.66 volts, and the products in Eq. (2.48) range from 37.6 to 17.8 k\V /cm 2, respectively. Clearly, the plasm a electron current cannot be neglected for low-voltage situations. A com parison of the relative sizes of the term s in Eq. (2.44) reveals th eir signif icance. Consider a norm alized version of Eq. (2.44) where all term s are divided by JtotkJ’ e which yields I* + E bo] - - £ -[ 2 + ifc]---?!-(ite + ij f- £ io ) J tot Jtot J tot - ^ - ( V B + V i i - E k o ) - 2 + B = 0. (2.49) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he last term , B, is of the order of which is much less than 2. T he value of 2 represents the therm al energy carried out of the region by the electrons to the m ain discharge. This term is of th e same order as the largest parts of the various current term s and therefore not negligible as considered in previous works. T he relative m agnitudes of the current densities determ ine th e relative m agnitudes of the term s, since the current densities can vary by several orders of m agnitude. The normalized Bohm energy, t )b , is of order of 1/2 and Ebo is of order 1/3 in m ost cases while the norm alized sheath voltage, tjt, can range from ab o u t 1 to 50 depending on th e discharge characteristics. Also note th a t for large sheath voltages the plasm a electron contribution (second term ) can also be neglected. There are m any other effects th a t have been neglected in this form ulation, for exam ple, radiation to/from both the main discharge and the surface, and nonequi librium effects. Also, th e ions w ithin an M PD th ru ster can achieve axial velocities greater th an 50 k m /s which may significantly change th e ionization characteristics along the cathode from the base to the tip. 2.4 Ionization Region Species D eterm ination Two m ethods were used for calculating the plasm a electron /io n num ber densities in the ionization region. T he first m ethod calculated the electron/ion density using th e ionization region energy balance, Eq. (2.44), for a specified electron tem perature Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. until the plasm a reached full ionization. Only singly-charged single species ions were considered [86,101,102]. Once full ionization was reached the num ber density was determ ined from Although this sim ple approach predicted reasonable results a t low pressures, it over- predicted the cathode tem perature and therefore the therm ionic emission current density a t high pressures. For the special case of tip only attachm ent only the fully correct this problem , a tw o-tem perature Saha equation was added to determ ine the species fractions in the ionization region. Using this m ethod, the energy balance in the ionization region is used to calculate the electron tem perature, and then th e Saha equation is used to calculate the species densities. M olecular dissociation is not included since the dissociation tem perature is much lower than the expected tem peratures (0.5 to 2 eV). The general reaction rate for ionization/recom bination is given for a tw o-tem perature plasm a by where Yt is the species mole fraction, Z , is the partition function, 8 is the ratio of th e electron and heavy tem peratures and s ranges from 0 for the neutral to 3 for P n t,m a x — ttj,nuui — _ (2.50) ionized solution is stable. The details of this are discussed in C hapter Four. To (2.51) th e third ion [110]. T he mole fractions are defined for each gas type. For exam ple Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 th e mole fraction of the electrons for gas type “g” is given by ^ = n n • ^ n < j + 2-i= o n t j T he partition function for the electron including the degeneracy for the two spin states is given by 2 V ^ r m ^ 13 (2 5 3 ) Since only m onatom ic heavy species are considered the partition functions for the n eutrals and the ions reduce to only th e electronic com ponent given by * vA Zs = 9o + 9i e17* (2-54) (=1 w here < 7; is the degeneracy of the ith level. T he num ber of levels included is set such th a t E x < a - A E (2.55) where A E is the ionization energy lowering due to the presence of th e plasm a [1 1 1 ]. T his energy lowering is given by Z e2 A E = --------- (2.56) 4 v (0rD w here Z is the charge of the species being ionized and, rp is th e Debye shielding distance for m ultiple charges which is given by 2c 0jfcrey / 2 -(¥&) V e2ndr / r0 = (2-57) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 where ne(r = nc + £ Z ,2n , (2.58) or, w ritten in term s of mole fractions, neff = (Ye + Yi + 4 Yu + 9 ^ , , ) ^ . (2.59) To decrease the num erical complexity, th e energy levels were consolidated into bands [110]. T hree ionization levels are considered, although only first and second levels are used in the m odel, to verify th a t the fraction of third ions is sm all. All species are considered as ideal gases w ith the total pressure given by P = n t kTt + £ n ,'tkT h = n totkTe . (2.60) Q uasi-neutrality can be expressed as Ye,g — Y{% g + ‘ i-Ya^g + 3V)„i5 (2.61) and th e sum of the mole fractions for each gas type must add up to unity Yc,g + K 0^ + Yi< g + Yiitg + Yuij, = 1. (2.62) Using Eqs. (2.61) and (2.62) along with the equations for the reaction rates for each ionization level, Eq. (2.51), a polynomial for the electron mole fraction can be w ritten as YefYej29 + 2A'i YeaYt J + 3 A', K 2Yc,g - A', Yt J - 2 h \ K 2 = 0. (2.63) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 a r g o n P = lOkPa • 3 u 0.01, 0.001 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Electron Temperature (eV) Figure 2.17: Species mole fractions as functions of electron tem perature. E quation (2.63) can be solved using N ew ton’s m ethod to determ ine Ye and then the o th er mole fractions can be determ ined using the reaction ra te equations. An exam ple solution is shown in Fig. (2.17). Note th a t Bose [110] recom m ends th a t 6 be less th an 2 or incorrect values may be calculated. Also, Richley and T um a [112] recom m end th a t the difference between Te and 7* not exceed 2000 K for sim ilar reasons. However, some authors violate both of these recom m endations and set Th to th e cathode tem perature of 2500 to 3700 K while Tt is typically around 11,000 K [113]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 2.5 Overall Near-Cathode Plasm a M odel T he surface, sheath and presheath/ionization region models are combined to form th e overall near-cathode plasm a model. T he heat flux and the current density to the surface are determ ined for values of the cathode tem perature, Te, the sheath voltage, Ve, the pressure, P, the surface m aterial and the gas type. W hile th e norm alized param eters are useful for exam ining the characteristics of each region, they are not useful for the combined m odel. T his is because the term in the relation for th e therm ionic emission current (Rjchardson-D ushm an equation) for the Schottky effect does not norm alize in the sam e m anner as the other equations. T herefore the com putations here were performed using the unnorm alized variables. T he current density com ponents and the heat flux com ponents are shown in Figs. (2.18) and (2.19). The conditions selected for this exam ple, argon gas with Vc = 6.0, < f > = 3.5 eV and P = 10 kPa, are w hat m ight be found in an M PD th ru ster w ith a thoriated tungsten cathode. T he m ajority of the current density is from therm ionic emission. In general, as the cathode tem perature increases (and therefore th e therm ionic current increases) more energy is added to th e ionization region and th e electron tem perature increases. T he species num ber densities will increase along w ith the increase in electron tem perature. Up to ab o u t 3150 K th e total current is prim arily composed of the therm ionic current and th e singly- charged ion current. At this point, the plasm a electron current begins to exceed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Components > < * I a * ■ * e g 5 2600 2800 3000 3200 34 0 0 3600 3800 Cathode Temperature (K) Figure 2.18: M agnitude of current density com ponents as functions of cathode sur face tem perature. Components argon q u > t Ve = 6.0 qi < t > = 3.5 eV qb P = lOkPa e o £ & X 3 E cs o s 2600 2800 3000 3200 3 4 0 0 3600 3800 Cathode Temperature (K) Figure 2.19: M agnitude of heat flux com ponents as functions of cathode surface tem perature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th e singly-charged ion current which results in total current being less th an the therm ionic current. At about 3500 K the contribution of th e doubly-charged ions exceeds th a t of th e singly-charged ions as the singly-charged ion num ber density begins to decrease. T he total heat flux is initially positive or heating the surface, b u t becomes negative when the cooling effect from the therm ionic electrons begins to dom inate th e heating effects for th e ions and the plasm a electrons. Generally th e solution of th e one-dimensional therm al m odel, which is presented in the next chapter, will occur near the values w here the heat flux is close to zero in these plots. This is due to the lim ited am ount of heat th a t the cathode can conduct away from th e surface. W ith this in m ind, the first order effects of these param eters can be seen by observing w hat effect th e param eter has on the zero intercept of the heat flux curve. The effect of the sheath voltage on th e plasm a properties is shown in Figs. (2.20) through (2.22). For a given cathode tem perature, increasing the sheath voltage results in increases in both the current density and the heat flux. Also, increasing th e voltage shifts th e peak heat flux value to lower cathode tem peratures. For th e larger voltages, it can be observed th a t the heat flux initially increases w ith cathode tem perature then decreases, and finally begins to increase again as heating from th e doubly-charged ions becomes significant. T he heat flux will continue to increase beyond this point. Solutions are presented for only a small portion of this region Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 aigon P = lOkPa $ = 3.5 cV n E s w Sheath Voltage I lc?i O 6 e / E < 3 . 26 0 0 2800 3000 3400 3200 Cathode Temperature (K) Figure 2.20: C urrent density as a function of cathode surface tem perature with sheath voltage as a param eter. Sheath Voltage argon P = lOkPa 6 = 3.5 cV / ' X E I X s E C Q O X -4- 2 6 0 0 2800 3000 3400 3200 Cathode Temperature (K) Figure 2.21: H eat flux as a function of cathode surface tem perature with sheath voltage as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 2.5 Sheath Voltage argon — 4 r :: 2.0 8- 1 .5- - 11 § 1.0- 0.5 2600 2800 3000 3200 3400 Cathode Temperature (K) Figure 2.22: Electron T em perature as a function of cathode surface tem perature with sheath voltage as a param eter. due to numerical difficulties with the solution of the Saha equation with rapidly increasing electron tem peratures. If the sheath voltage is sufficiently small, the cathode is cooled for all cathode tem peratures of interest. T he effect of the surface work function on the plasm a properties can be seen in Figs. (2.23) through (2.25). As expected, decreasing the work function significantly decreases the cathode tem perature needed to achieve a specific current density. T h e electron tem perature is similarly affected since Te is prim arily affected by the product of jb and V^. Decreasing the work function also shifts th e heat flux curves to the left and increases the peak value. In an earlier model, which did not include Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 E $ Work Function (eV) 3.0 3.5 4.0 4.5 c g 5 2500 3500 3000 4000 4500 Cathode Temperature (K) Figure 2.23: C urrent density as a function of cathode surface tem p eratu re w ith work function as a param eter. E 2 •Si £ * 1 x a E n < 3 0 u X - 1. -2 Aigon A P = lOkPa \ Vc = 6 Work Function (cV) ----- 3.0 ---- .......4.0 - - - 4 3 V ^ \ 1 / I 1 \ \ \ \ \ \ \ 1 t \ * * * \ S \ * • \ 1 t I \ 1 \ ...... * 2600 2800 3000 3200 3400 3600 3800 4000 Cathode Temperature (K) Figure 2.24: H eat flux as a function of cathode surface tem perature w ith work function as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 Work Function (eV) 26 0 0 2800 3000 3200 3400 3600 3800 4000 Cathode Temperature (K) Figure 2.25: Electron tem perature as a function of cathode surface tem perature w ith work function as a param eter. equilibrium ionization/recom bination or S aha equation, the curves were only shifted to the left [101]. T he effect of the pressure on the plasm a properties can be seen in Figs. (2.26) through (2.28). A decrease in pressure significantly shifts the zero intercept of the heat flux curves toward lower tem peratures indicating th a t a t lower pressures the cathode will operate at lower tem peratures. Correspondingly the arc attachm ent area will increase due to a decrease in current density. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8-1 E $ N — ✓ Pressure (kPa) -*50 — 100 2 6 0 0 2800 3000 3 2 0 0 3400 3600 3800 4000 Cathode Temperature (K) Figure 2.26: C urrent density as a function of cathode surface tem perature with pressure as a param eter. Pressure (kPa) -*50 — 100 9 E CO o S 26 0 0 2800 3000 32 0 0 3400 3600 3800 4000 Cathode Temperature (K) Figure 2.27: H eat flux as a function of cathode surface tem perature w ith pressure as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 Pressure (kPa) > / Argon Vc = 6 4 > = 3.5eV 0.5 I ■ • ' I 2600 28 0 0 3000 3200 3400 3600 3800 4000 Cathode Temperature (K) Figure 2.28: Electron T em perature as a function of cathode surface tem perature w ith pressure as a param eter. 2.6 M agnetic Pressure Effects A radial pressure gradient is present across th e tip of the cathode due to m agnetic “pum ping.” T he interaction of the axial current with the induced azim uthal m ag netic field produces an inw ard radial body force on the plasm a. This force m ust be counteracted by an increase in pressure to m aintain equilibrium. Maxwell’s V x f l provides the relation for th e purely azim uthal m agnetic field given by (2.64) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 6 where / enc is the enclosed current and r is the radial position. T he balance between the m agnetic “pum ping” and the pressure is given by j x f y = (2.65) where j z is the axial current density. For the special case of constant current density where 7tot = j t * r2, Eq. (2.65) can be integrated to provide <«•> where Pa is the pressure a t the outer edge of the cathode and rc is th e cathode radius [42]. To determ ine th e average pressure across the cathode tip segm ent, Eq. (2.66) is evaluated a t (2 .« T ) where r, and r0 are the inner and outer radii, respectively. At this radial location th e inner and outer areas are equal. For the case where r,- = 0, Eq. (2.66) becomes p = n + = A + ^ (2 68) The tip pressure is therefore dependent on both the current, or current density, and the tip area. For a given geom etry the quadratic increase in pressure with current can significantly change the arc attachm ent characteristics. Com parisons of solutions w ith and w ithout the m agnetic pressure effect are presented later. For the case where the current density is not constant across the tip , a different approach is required. T he tip of the cathode can be discretized into a series of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 concentric rings w ith a constant current density for each ring. Equation (2.65) can be integrated for a ring segm ent to provide r , - R + [ % * ( r . ’ - r,1) + ( /, - i t i T T ?) In ( | ) ] (2.69) where /, is the enclosed current up to the inner radius. T he second term is the correction for the nonuniform current density. 2.7 Boundary Layer Regions All of the th ru ste r configurations contain flowing plasm as and therefore th e mom en tum , therm al and concentration boundary layers should also be considered. The large gradients across the boundary layers could provide significantly different val ues a t th e sheath edge, near th e surface com pared to the main flow. Since the boundary layer characteristics are dependent on the flow characteristics w ithin the thru ster, they may change for different thruster configurations and operating con ditions. O nly crude estim ates can be made w ithout knowing the detailed flow and discharge characteristics. T he surface work function for thoriated tungsten cathodes is strongly dependent on the characteristics of th e thorium concentration boundary layer [51]. T he net evaporation of thorium from the surface is governed by th e surface tem perature, the diffusion o f thorium through th e tungsten lattice, and the diffusion rate of thorium from the surface. The la tte r is strongly affected by the boundary layer character Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 8 istics. Also, the evaporation of tungsten from the surface, which is the dom inant erosion m echanism , is strongly dependent on the characteristics of the tungsten con centration boundary layer [51]. None of these effects are included in this work but should be considered for fu tu re study. 2.8 Arc Column Region A description of the arc colum n is required to com plete the cathode plasm a model. T he com bination of the arc column and the contraction region near the cathode if one exists, would add th e additional inform ation needed to relate th e sheath voltage w ith th e attachm ent area. A ttem pts to include this region in this study were, however, not successful. Previous works have been able to solve this problem correctly for the special case of a constricted arc [114], where th e arc diam eter is physically constrained by m aterial walls. To a first approxim ation only radial variations are required for the arc column model. T his assumes th at th e arc column diam eter is constant for each axial po sition. T h e solution of the energy equation will determ ine the radial tem perature profile. Given th e radial tem perature profile, the electrical resistivity and current density can be calculated for each radial position. The arc column diam eter can then be set as th e radial position th a t encloses a specified fraction, perhaps 95 percent, of th e current. H eat is supplied through Ohmic heating while plasm a continuum radia- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 10000 Arc Radius (mm) argon P = 100 kPa I = 10 A T.in = 6000 K 9000 H 3 G & 8000 E s c G | 7000 Su 6000 0 5 10 20 15 Arc Radius (mm) Figure 2.29: Radial tem perature profiles for th e arc column. tion removes heat. T he radial geom etry sets a zero tem perature gradient boundary a t the centerline. T he second boundary condition is provided by specifying the tem perature a t a specified radius. N ote, th at specifying a gradient for the second boundary condition does not work well because it does not set the tem perature range. For th e special case of the constricted arc, the second boundary condition is well defined by the physical wall in the device. The wall tem perature can be reasonably estim ated and th e wall position is known. However, for th e unconstricted arc case, it is not clear w hat the position and value for the second boundary condition should be. A series of cases were run to determ ine the sensitivity of these values for argon gas Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. operating a t 10 A, and 100 kP a [91]. As long as the edge tem perature was sufficiently small (< 5000 K ), it had little effect on the radial tem perature distribution. This is a result of th e strong dependence of the electrical resistivity on the electron tem perature. T h a t is, electrical conductivity and the corresponding current density decrease rapidly w ith a decrease in tem perature. T he radial position, however, had a significant effect on th e arc column tem perature characteristics as seen in Fig. (2.29). As the radial edge position was increased, the arc filled the entire region, even when th e radius was much larger th an those m easured experim entally. T he addition of the m agnetic pressure had little effect on th e results for two reasons. F irst, the plasm a tran sp o rt properties are not as sensitive to pressure as they are to tem perature [110, 115]. Second, th e small current densities do not create a strong enough induced m agnetic field for the Lorentz force to provide significant contraction. The axial electrical field was also found to decrease monotonically with increasing arc radius as seen in Fig. (2.30). This indicates th a t the lowest energy/voltag e solution, which has been used for the constricted arc cases, would exist for an arc of infinite radius. Clearly, th is arc model is not com plete for the unconstricted arc cases and further work is required. Tim e and financial constraints prevented fu rth er investigation in this study. T he arc attachm ent areas in this study were estim ated from the experim ental d ata. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 E 0.1 - o • 0.01 20 Arc Radius (mm) Figure 2.30: Axial electric field as a function of arc column radius. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 C hapter 3 C ath od e T h erm al M odel For a given set of boundary conditions, the therm al model describes th e tem pera tu re distribution w ithin th e cathode. T here are several orders of approxim ation by which the therm al model can be done ranging from simple one-dimensional analyt ical models to complex two-dimensional axisym m etric numerical ones. T he nonlin earities of the near-cathode plasm a model can cause numerical difficulties for the therm al model. T he heat flux to the surface from the plasm a model is sensitive to the surface tem perature. T h e therm al model m ust be resilient enough to handle large boundary condition changes during the numerical iterations. W hile the simple models do not have th e capabilities for solving the problems of interest, they provide useful insights into the solution trends and provide good first approxim ations for th e startin g conditions for th e m ore advanced models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 3.1 Quasi-Two-Dimensional Thermal M odels The sim plest therm al model is one-dimensional heat conduction, given by , = ^ t (r tip _ r hMe). (3 .i) A slightly more advanced therm al model includes radial external heat removal where the cathode is assumed to be uniformly cooled by convection or linearized radiation. These models are often referred to as quasi-two-dimensional models. A cathode with a large length to diam eter ratio, L /d , can be effectively modeled as an “infinite fin” w ith a heat flux given by Eq. (3.2) and th e axial tem perature distribution by Eq. (3.3) (116). q = M (3.2) T = ( r b w - T 0O ) e x p ( - m c ) + Tb w (3.3) where M = (Ttip - Too) (3.4) and 2 _ 4A«mv r , m . (3.5) For cathodes with sm aller L /d ratios it is more appropriate to specify either th e base tem perature or the heat flux from the base. These models are more appropriate for cathodes with w ater cooling a t their base, such as th e cathode in the S tu ttg a rt ZT1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 th ru ste r or the JP L cathode test facility. If the base tem p eratu re is prescribed then th e tip heat flux and the cathode tem perature distribution are given by cosh ( m i ) - - « = M r - r y - (3.6) s in h (m l) and T = (Th w - T 00) » I T 1 sinh(m z) + sinh m ( L — z) 4 -T b ^ . (3.7) sinh(m L) Solutions for other electrode base conditions can also be found in Ref. [116]. W hile th is model adds radial heat removal, it is restricted to one value for the heat transfer coefficient, hcom. For an actual th ru ster, the heat transfer coefficient may vary significantly along th e length o f the cathode. These m odels provide a good first approxim ation for arcs with only tip attachm ent, such as arcjets. For an accurate therm al model of the M PD thruster cathode, the Ohmic heating m ust be included. Calculations for the large currents required in M PD thrusters show th a t Ohmic heating is the dom inant electrode heating m echanism . For exam ple, a 12 mm diam eter tungsten cathode 65 mm long produces 1.4 kW by Ohmic heating at 2000 A (pe = 60 x 10-8 fi-m ). A quasi-two-dimensional numerical solution for a cathode th a t includes radial radiation and convection, and Ohmic heating, for a cylindrical rod w ith a conical tip is presented in Ref. [87] for the steady-state case and in Ref. [117] for the transient case. T his model assumes constant properties: convection coefficient, emissivity, therm al conductivity, and electrical resistivity. The pair of second-order ordinary Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 differential equations arc solved numerically with constant tem p eratu re boundary conditions a t both ends. The heat removal from the cathode base could be used instead of the base tem perature depending on which p aram eter is known. The equation for the cylindrical portion is given by d2# P p 'L 3 _ 2crtrr Z271ip3 dZ1 kth Ttip[ff(rC yi)2]2 kih r* y ' I*- Isr=-Il =® (3-8> U fey] [ ' " ( t ^ ) ] = 0 for Zi < Z < 1 where 0 = T /7 u p , Z = z /L , and Zi is the axial location of the interface of the conical and the cylindrical portions. The equation for the conical portion is given by <P6 t 2 tan q d0 t L 2pt 4" . r . + dZ* cos o (Z tan a + r ^ / L ) d Z kth r lip[jr (Z tan a + rcy, / I ) ] 2 cos a {Z taxi a + r„ i/X ) 2Aconv i ' - © I H k ) \ - 0 (3.9) cos a kth {Z tan a + tcy\/L ) for 0 < Z < Zi. These second-order equations can be split into pairs of first-order equations using the transform ation Y\ = 6 and = (d 8 /d Z ). T hese sets of first- order equations for this boundary value problem are solved num erically using a shooting technique employing the fourth-order R unge-K utta m ethod coupled with th e secant m ethod. Note th a t the equations in both Refs. [87] and [117] contain errors; the correct relationships are given by Eqs. (3.8) and (3.9). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 2 Current (A) Cylindrical Tip 0 e =0.5 Tor = 500 K Tb«e= 1500 K P -2 - m i ■ | 1 * ■ 2000 2500 3000 3500 4 0 0 0 Tip Temperature (K) Figure 3.1: Heat flux as a function of tip surface tem perature w ith current as a param eter. Solutions for a cathode w ith a 9 mm diam eter and 65 mm length are shown in Fig. (3.1) and Fig. (3.2) for a cylindrical geom etry and in Fig. (3.3) and Fig. (3.4) for th e cylindrical cathode with a flattened conical tip shown in Fig. (3.5) (r,;p = 2 m m , and a = 30°). For these cases, only radiation is considered; th a t is, convective cooling is neglected and the base tem perature is 1500 K. Cases w ith only convective cooling are presented in Ref. [102]. For the cylindrical geometry, th e zero current case is the sam e as the pin-fin case previously considered. T he allowable heat load to th e cathode tip decreases w ith increasing current due to the heat generation from O hm ic heating. For a given current value and tip tem perature, th e allowable tip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 5000 ~ 4 0 0 0 3000 Current (A) 0 1000 u V | 2000 3000 4000 68 1000 -100 -80 -60 -40 0 -20 Distance from Tip (mm) Figure 3.2: Cylindrical cathode tem perature distribution w ith constant electrical resistivity. 0.8 0.6 0.4 § o i : - I Q . P -0.2: Current (A) 0 1000 2000 -0 .4 : 2000 2500 3500 3000 4 0 0 0 Tip Temperature (K) Figure 3.3: Heat flux as a function of tip surface tem perature w ith current as a param eter for a cylindrical cathode w ith a conical tip. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 5000 4 0 0 0 3000 Cunent (A) — 1000 / — 2000 - - 3000 i/ -•- 4000 I ' ■o 5 2000 1000 *100 -80 -40 -60 -20 0 Distance from Up (mm) Figure 3.4: Cylindrical cathode tem perature distribution with conical tip and with constant electrical resistivity. Figure 3.5: Illustration of a cylindrical cathode w ith a flattened conical tip. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 heat flux is fu rth er decreased w ith the addition of a conical tip to the cathode. This is due to the decreased cross-sectional area and therefore the larger tem perature gradient necessary for a given heat flux. T he tem perature distribution along the cathode is strongly dependent on th e cur rent value as seen in Figs. (3.2) and (3.4). For large currents, the m axim um tem per ature within th e cathode is not a t the tip bu t is towards the middle, which m ay ex plain the M PD th ru ster cathode m elting observed a t the University of S tu ttg a rt [79]. The larger tem p eratu re gradients associated w ith the conical tip can also be seen. One significant im provem ent to this model is the inclusion of the tem p eratu re dependent electrical resistivity. W hile the therm al conductivity rem ains relatively constant, th e electrical resistivity changes significantly over the range of tem pera tures expected for these cathodes, as seen in Fig. (3.6) [118]. A linear curve fit of the electrical resistivity provides the relation pt = (-1 2 .4 5 7 + 0.03497 T ) x 10-8 (3.10) and a curve fit of the therm al conductivity d a ta yields kth = 84.60 + 77.76 exp (-7 .9 9 5 x 10~‘r ) + 149.67 exp (-5 .0 2 6 x 10_3r ) . (3.11) The linear resistivity can be substituted for th e constant value of pe in Eqs. (3.8) and (3.9). T his change can significantly affect the results for currents g reater than 1000 A, as seen by com paring Fig. (3.2) and Fig. (3.7). T he tem perature dependent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 2.0-1 O solid pe data gj ----- solid p, fit □ © liquid pe data lb ----- liquid pe fit G J □ solid kth data solid kth fit 19 liquid kth data • “ ■ liquid kth fit X r T T T T T TT T r p T f t j t i 1 1 | i r r r i T i ' n ^ 9 §. c n < [ v* 3 * 2000 3000 Temperature (K) 4000 5000 Figure 3.6: Therm al conductivity and electrical resistivity for tungsten as a function of tem perature. therm al conductivity has a sm aller but still significant effect. T here is a significant effect on the tip heat flux when the tem perature dependent resistivity is used as seen in Fig. (3.8). Recall th a t the surface heat flux and current density for th e near- cathode plasm a model are sensitive to the cathode tem perature. Therefore, this shift in the relationship between tip heat flux and tip tem perature in the therm al model will greatly alter the overall model solution. It is therefore im portant to use the tem p eratu re dependent properties for high-current cases, while th e constant properties m ay be used for low-current cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 1 3500 | 3000 £ 8. I 2500 H u ■ P 5 2000 C Q u 1500 *100 -80 -60 -40 -2 0 0 Distance from Tip (mm) Figure 3.7: Cylindrical cathode tem perature distribution w ith tem perature depen dent electrical resistivity. 0.6 0.4 " l 0 2 | 0.0 * - 0.2 X E -0.4 n X *0.6 o. P -0.8 - 1.0 2000 2500 3000 3500 4000 Tip Temperature (K) Figure 3.8: Heat flux as a function of tip surface tem perature for a cylindrical cathode w ith and w ithout tem perature dependent m aterial properties. 1 = 3000 A Cylindrical U p e = 0.5 Tinf= 500 K Tbue = 1500 K th = M T ). pc = ptCD kth = const, Pe = Pe(T) k u > = kih(T), ft = const kth = const, = const Cylindrical Tip I = 3000 A . e = 0.5 / " \ 1 iaf * - 3 w t\ / Tb M «= 1500K / / / / / / / / V / / / / / / h / - - ” ------------------ ------kih = consL, pe = const -----k u > = km(T), (v = pt(I) ----- kth = const, p. = pbCD ------kth==kth(T). pe = const Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 2 W hile these m odels are adequate for sim ple cathode geom etries and constant value boundary conditions, they are not sufficient to fully model the cathode char acteristics in electric thrusters. It is expected th a t the plasm a characteristics may change significantly along the cathode, and therefore the characteristics of the near cathode plasm a model will be significantly different. Also, observations a t JP L have shown severe m elting for conical-tip cathode and negligible m elting w ith spherical- tip cathodes at sim ilar and higher power levels [78]. A model capable of modeling a spherical-tip cathode is also necessary. 3.2 Two-Dim ensional Axisym m etric Therm al M odel To fully predict the therm al characteristics of the cathode, a two-dimensional model is probably required. Experim ental evidence shows th at th e cathode centerline tem peratures significantly exceed the outer tem peratures for some operating conditions. T he cathode tip craters, Fig. (1.11), indicate th a t m elting was greatest on the cen terline. If th e m aterial tem perature exceeds the melting point, th e m aterial will melt internally. T he voids in the S tu ttg art cathode, shown in Figs. (1.15) and (1.16), probably formed this way. Experim ental d a ta suggest th a t for M PD thrusters the arc attachm ent may be over a significant portion of the exposed surface area [100]. Therefore, the therm al model m ust be capable of modeling Ohmic heating and arc attach m en t over a range Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 of axial positions as well as the tip. Tem perature-dependent m aterial properties are also required due to th e large range of tem peratures experienced in the cathode. T he total current is determ ined by integrating the current density values over th e em itting surface. For conditions where the arc is attached over a m ajor portion of the cathode surface, and therefore a large range of cathode tem peratures exists, th e surface heat flux can vary significantly. T h a t is, because th e cathode surface tem p eratu re will change significantly, so will th e heat flux and th e current density to the surface. For this study, a two-dimensional finite volume model was developed. This model is capable of variable geom etries, tem perature-dependent m aterial properties, and m ultim ode heat transfer (arc attachm ent plus radiative and convective transport) a t any axial location. Energy and current balances are perform ed on each cell for th e two-dimensional model using a finite volume technique. T he energy balance for each cell, as shown in Fig. (3.9), is given by: 9 i - i j + 9 .+ ij + 9 .J - 1 + Qij+i + q'ijVtj = 0- (3-12) T he heat generated per unit volume for each cell due to Ohmic heating is given by (3I3) where th e cell resistance, R , is given by R \j = P e,ijdzi/A c,ij (3.14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 T T u * ' i.j+1 »+1 j H J i+1 j Figure 3.9: Energy in p u ts for the two-dimensional therm al model. and I is the total cell current. T he axial heat load, calculated a t th e cell wall is given by 9 i+ lj = 4 i+ ljA ’ + lJ = fr<A,i+lj'4i+lj— ^ (3.15) where the tem perature gradient has been discretized using the central differencing technique and is of order ( d z /2)2 [119). Since the cell wall tem perature is not needed to calculate the heat flux, this technique effectively doubles th e num ber of grid cells used. T h a t is, half of the num ber of cells are required to achieve the same spatial resolution as a direct discretization of the differential equations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 Using Eq. (3.15) and parallel equations for the other three wall heat fluxes, Eq. (3.12) can then be rew ritten as fcffc.i-ljA '-lj'ff, , ^M,i+l J ^ i+ l J rr , ^ Tz Ti- '* + Tz Tr Ti* - 1 + — J+l TiJ+ 1 - C T tj + f t V i j = 0 (3.16) where Q _ kth ,i- l j A ' - l j kfhj+1 jA j+ i J f^th.ij+l • A j J + l dz dz dr dr For th e m ethod of Successive Over Relaxation (SO R), Eq. (3.16) is rew ritten as rr *+1 _ i* * . u ( k t h .i - \ ,j A i - \ j rr kth .i+ ijA i+ ij ^ T,< - T tJ + 7 i_ jj + ---------j z-------- Ti+i j ) , ( k t h ,ij- iA ij- i ^ kth,ij+ \A ij+ i \ + ( Tr + Tr TiJ+1) ~ ( C T i j + t f j V i J ) . (3.18) T he tem p eratu re value a t the “k th ” tim e-step is 7)*, 7)fc+ 1 is the value a t th e next tim e-step, and u is the relaxation coefficient. If u is less than one the calculation is “under-relaxed” while if w is greater than one the calculation is “over-relaxed” [119]. An iteration is performed until th e tem perature change between consecutive tim e steps is w ithin the specified tolerance and the global energy balance is w ithin a specified tolerance. Over-relaxation will generally make the solution converge more quickly by taking larger steps b u t may also make the calculation diverge. The com puter program is setup to autom atically decrease the relaxation coefficient if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10G the calculation diverges more th an a specified num ber of consecutive steps. This technique has proven to be very useful in m aintaining code stability. C urrent conservation is considered for the group of cells a t each axial position. The total current for each group is given by cells toward th e cathode base. T he current is distributed am ongst the cells within the group by considering the cells as a parallel resistor network where th e resistance for each cell is given by Eq. (3.14). T he voltage drop is determ ined by, V = /totf2tot where J2tot is th e total resistance for th e parallel network. T he current for each cell is given by /,• = V / R\ where V is th e voltage drop across the group. This current value is used to calculated the Ohm ic heating within th e cell. For cells w ith radial current in p u ts, this will over-predict the am ount of Ohmic heating. However, for a sufficiently fine grid spacing, this effect is negligible. T he nonlinearities of the therm al equations produce larger tem perature gradi ents in th e tip region than in th e base region. To increase th e accuracy of the discretization w ithout increasing th e num ber of cells and correspondingly increasing the solution tim e, the cells are “packed” towards the tip. An algebraic grid generator th a t allows th e m inimum grid spacing to be placed anywhere along the cathode was used. This transform ation consists of two third-order polynomials and is described n (3.19) where n is th e num ber of radial cells and I tot is the current leaving th e group of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 in detail in A ppendix B [120]. T he cells could also be “packed” radially towards the centerline. 3.3 Two-Dim ensional Tip Approximation Model There is a severe lim itation w ith the arc attachm ent a t the cathode tip in the quasi- two-dimensional models. P ast models using the quasi-two-dimensional therm al m od els have assum ed th a t the arc attachm ent area covered the entire tip [86,101,102,113], which presents tw o problems. F irst, as th e surface tem perature is changed, the cur rent density will significantly change and therefore th e total current will change. For high-current cases this change in th e total current will significantly affect the therm al model since the dom inant heating mechanism is from Ohmic heating. It also makes com paring the effects of th e different param eters a t constant current difficult since each case produces a different total current. Second, this assum ption does not correctly account for operation where the attachm ent area is only over a small portion of th e tip, for exam ple see Fig. (1.10). For this stu d y the two-dimensional finite-volume model was used to develop the tip approxim ation model for the spot heating effect a t the cathode tip. This model was then com bined with the quasi-two-dimensional therm al model. T he geom etry considered is shown in Fig. (3.10). T he variables were normalized such th at T = T*vt/T c, R = R e/R a p , and q = g/(*thT»vei?tip) where T»ve is the area weighted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 insulated ^ / / / / / / / / / / / / / / / / A y 'a v e Figure 3.10: Two-dimensional tip approxim ation geometry. average surface tem p eratu re, Te is th e tem perature of th e arc spot assum ed to be uniform , R^ is th e arc spot radius, R ^p is the tip radius, and q is the heat load. A series of cases were run to determ ine T as a function of R and q. T he natural logarithm s of th e results were fit to a 16-param eter (constant through cubic products) surface shown in Fig. (3.11). A b e tte r fit was obtained in natural log space than in linear space. T h e curve fit has a m axim um relative error of 1.7 percent and a maximum absolute error of 0.9 percent when transform ed back to linear space. W ith this fit the spot tem perature, Te, from the plasm a model is related to the average tip tem p eratu re Tavf, used for th e therm al model for a given heat load. The attachm ent area is determ ined from the current density calculated from the plasm a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 0 ■ 1 ln(T) 2 ln(R) _4 -1 Figure 3.11: Two-dimensional tip approxim ation surface fit. model and the known to tal current. T he heat load is then determ ined from th e heat flux calculated from th e plasm a model and th e calculated attachm ent area. The final solution is obtained through an iterative process. Comparisons w ith two-dimensional solutions reveal th a t this approach captures m ost of the two-dim ensional therm al effects. T he m ajor advantage is th at an ana lytic expression can be used instead of a numerical solution in th e overall model. For a typical case the full two-dimensional solution can take 8-40 hours on a 486-33MHz P C com pared to about one m inute using the quasi-two-dimensional therm al model w ith the two-dimensional tip approxim ation model. However, th e simple solution assum es th a t all of the arc attachm ent conditions are the same. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 0 C hapter 4 C om bined M od el S olu tion s T he near-cathode plasm a model and th e cathode therm al model are combined to form an overall model of the cathode-plasm a interaction. Two configurations were used for th e integration of the plasm a model w ith the quasi-two-dimensional ther mal m odel. T h e first configuration, which was sim ilar to previously developed m od els [86,98], predicted excessively large cathode tem peratures. A second model was developed to correct this problem and it has yielded reasonable comparisons with experim ental d ata. T he near-cathode plasm a model was also combined with the two-dimensional therm al model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill 4.1 Quasi-Two-Dimensional Model Two configurations were used for the integration of the plasm a model w ith the quasi- two-dimensional therm al model. The first configuration was sim ilar to previously developed models [86,98]. T he first model produced reasonable results for the low- pressure cases b u t predicted cathode tip tem peratures th a t were unreasonably large for high-pressure operation. For exam ple, for arcjet th ru ster operation a t about 100 kP a on argon th e cathode tip tem peratures usually exceeded 4000 K which is well above the m elting point of tungsten (TrkU = 3660 K). To help correct this problem th e Saha equation, describing equilibrium ionization/recom bination, was added to the ionization region of the plasm a model. A brief description of the first model is presented here for com parison. A more detailed description of this plasm a model is presented in Ref. [101], and the de tails of combining this plasm a model w ith the quasi-two-dimensional model are presented in Ref. [102]. In this configuration the plasma model did not include ion ization/recom bination equilibrium , but instead used the energy balance in the ion ization region to determ ine the num ber of ions produced. Also, only singly-charged ions were considered. Figure (4.1) shows th e heat flux and surface tem perature re lations for both the first near-cathode plasm a model and the simple heat transfer model. Solutions exist a t three points; namely, at the origin (trivial solution) and th e two nonzero solutions where the curves intersect. Of these, only th e high tem Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 2 2.0 Plasma Model - - Thermal Model V e= 10 T«= 1.0 eV P = 1.0 kPa 4 = 3 5 eV Ar = 60 A/Kfcm B u £ X _ _ ■ = 0 .5 - E 0.0 -0.5 2000 2200 2400 2 6 0 0 2800 3000 3200 Cathode Tip Temperature (K) Figure 4.1: H eat flux as a function of surface tem perature w ith therm al model solutions. perature, or fully ionized, solution is stable. Therefore, the value of the pressure is im portant for this type of discharge and the energy balance in the ionization region is not needed. For the special case where the arc only attaches at the tip, such as for the arc- je t, a series of reasonable solutions can be determ ined. For a gjven set of therm al characteristics for the quasi-two-dimensional m odel, and given th e plasm a proper ties, one can solve for th e intercept point of th e one-dimensional therm al model and near-cathode plasm a model for the fully ionized (stable) case. T h e model param eters can then be adjusted to examine their effects on the solution. Generally, for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sheath Voltage Figure 4.2: Peak and zero point tem peratures as functions of norm alized sheath voltage w ith pressure as a param eter for first model version. cases w ithout internal heat generation (quasi-two-dimensional therm al m odel), the tip tem perature solution will be bounded by th e point of m axim um or peak heat flux and the zero-heat-flux solution. This range of solutions can be lim ited for cer tain com binations of param eters, for exam ple, sm all sheath voltages. T h a t is, the difference between the zero-heat-flux tem perature and the peak tem p eratu re may be sm all. An exam ple of the lim iting tip tem perature solutions are shown in Fig. (4.2) as a functions of the sheath voltage. Note th a t two different sheath voltages can produce the same cathode tip tem perature and th a t a tem perature m inim um exists. T he decreasing tem perature w ith decreasing voltage for the large voltage range is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 a result of decreasing ion energies. For small voltages, the plasm a electron heating becomes significant and dom inates for very small voltages (< 4 volts). A minimum sheath voltage exists a t the point where T p ^ and Tiero are the sam e. This m inimum is not the sam e as the minimum discussed previously, although the voltage values for both may be sim ilar. There are also different attachm ent areas corresponding to the two different voltage solutions for a given tem perature. T he addition of the Saha equation relaxes the need for a fully ionized plasm a for a stable operating point therefore allowing solutions for lower cathode tem peratures (and lower therm ionic emission current) for a given sheath voltage. Recall th a t the prim ary energy source to the ionization region for ion production is a result of therm ionic electrons being accelerated through the sheath. T he second version of the plasm a model was used for all of the analyses presented here. T he two- dimensional tip approxim ation was also added to the therm al model for the second version of th e combined model. Figure (4.3) shows the heat flux as a function of surface tem perature for both the second near-cathode plasm a model and the quasi- two-dimensional therm al model. Again, solutions exist for the combined model at the intercept points of the plasm a and therm al models. A series of investigations was done to determ ine the effects of changes in the operating param eters, < f > , P , J and Te, on the cathode operation. A flat tipped, cylindrical cathode with a diam eter of 9.5 mm and a length of 75 mm was assum ed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 2.0 Plasma Model Pressure (kPa) * - Thermal Model 1.0 - 2 0 .5 : -0.5 3000 3200 3400 2600 2800 Cathode Temperature (K) Figure 4.3: H eat flux as a function of cathode surface tem p eratu re w ith therm al model solutions for second model version. T he boundary conditions for the quasi-two-dimensional therm al model were Tb**, = 1500 K, c = 0.5, and T|» = 500 K. T he therm al model was used to calculate th e tip heat load, qtip, as a function of the tip tem perature, Ttjp. These solutions were then curve flt and this fit was used for the therm al evaluation for each of th e param eter com binations. Com parisons between this simplified technique and direct therm al model solutions a t each condition showed negligible differences. By using this simplification the solution tim e for each set of param eters was reduced from about one hour to about 30 seconds. For situations where the attachm ent area is less than the tip area, th e two-dimensional tip approxim ation was used Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to relate the attachm ent spot tem perature, Te, to the area weighted average tip tem perature, Ttip, and the spot heat load, qe to the tip average heat load, gtip- For conditions where the attachm ent area is larger than the tip area, it is assum ed th a t the cathode has an enlarged tip with a uniform tem perature (Tc = Ttip). T h a t is, as the attachm ent area increases, the cathode correspondingly grows longer, and has a constant tem perature where the arc is attached. T his simplification allows th e sam e set of therm al model solutions to be used for each com parison, which significantly reduces th e com putational tim e. D orodnov, et al. [88] have also had reasonable results w ith this approxim ation. As long as the enlarged tip is small relative to th e total cathode surface area (22.4 cm2), the results will be reasonable. Also, m any of the experim ents have shown th a t variations in the cathode tem perature w ithin the attachm ent area are small relative to the tem perature changes outside of the attachm ent area. The baseline values of the param eters, P , 0, and / , were selected as values th a t would be representative of an M PD thruster. T he param eter ranges were selected to illustrate the effects over a wide operating envelope. T he effect of changes in the pressure are shown in Figs. (4.4) through (4.8). This wide pressure range extends from the low-pressure regim e expected for M PD thrusters (1-10 kP a) to high-pressure operation more typical of arcjet thrusters (100 kP a). For this cathode geom etry, only cases w ith attachm ent areas less th an 10 cm2 (Tc a 2560 K) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 100 - Pressure (kPa) argon I = 1000 A 6 = 3.5 eV S 10- - - 5 0 — 100 C o u n J < 2200 2 4 0 0 2600 2 8 0 0 3000 3200 3400 3600 Cathode Tip Temperature (K) Figure 4.4: A ttachm ent area as a function of cathode tip tem p eratu re with pressure as a param eter. 2.0 Pressure (kPa) - - 5 0 — 100 1.4 : s 0 . 8 : argon I = 1000 A $ = 3.5 eV 0.6 3600 3200 2400 2800 Cathode Tip Temperature (K) Figure 4.5: Electron tem perature as a function of cathode tip tem perature w ith pressure as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 Pressure (kPa) 00 2 o > 5 (S i : c/5 - - 5 0 — 100 2800 3200 3600 2400 Cathode Tip Temperature (K) Figure 4.6: Sheath voltage as a function of cathode tip tem p eratu re with pressure as a param eter. 3.5 & 3.4 g 3.3J Pressure (kPa) 3 .2 - --50 — 100 3.0 2200 2400 2600 2800 3000 3200 3400 3600 Cathode Tip Temperature (K) Figure 4.7: Effective work function as a function of cathode tip tem perature with pressure as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 1 0 0 argon 1= 1000 A $ = 3.5 cV 10 1 Pressure * > (kPa) O 3 o .i - - 5 0 — 100 0.01 6 8 10 12 4 Sheath Voltage Figure 4.8: A ttachm ent area as a function of sheath voltage w ith pressure as a param eter. should be considered reasonable bu t all are shown to indicate th e trends. For a given cathode tem perature the pressure has only a small effect on th e attachm ent area. A s expected, sim ilar trends are observed for the electron tem perature and the sheath voltage plots. Recall th a t th e electron tem perature is determ ined from the energy balance in the ionization region and th a t the prim ary energy source for this region is the acceleration of the therm ionic electrons through th e sheath region. T he values for different pressures are sim ilar a t the low cathode tem peratures. As the cathode tem perature is increased, the values increase slowly until the plasm a begins to reach full ionization. At this corner point both the sheath voltage and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 0 the electron tem perature increase rapidly. T he slight bend about halfway up the vertical portion of the electron tem perature curves is the full ionization point. As the cathode tem perature increases beyond this point the doubly-charged ions begin to become significant. The curves are term inated a t a point where th e large num ber of doubly-charged ions created convergence difficulties in the com puter program . T he effective work function includes th e m aterial work function, 3.5 eV in this case, and the work function lowering from th e Schottky effect. T he pressure appears to have a small effect on the effective work function except for conditions near full ionization of the plasm a. Some previous works [91,113,114] have used a m inim um in the voltage to determ ine the operating point for th e discharge. Figure (4.8) shows the relationship between the attachm ent area and the sheath voltage. For th e near cathode portion of the discharge no m inim um exists and therefore the voltage drop in the arc column and the near-anode region would need to be included for this technique. T he effect of changes in th e m aterial work function is shown in Figs. (4.9) through (4.13). As expected, changes in the work function have large effects on the operating characteristics. T he trends appear sim ilar for each work function value bu t the curves are shifted to a different cathode tem perature. For exam ple, th e com er point for each curve is shifted to higher cathode tem peratures for correspondingly higher work function values. For the work function comparisons only, the lowering of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 1 100 Work Function (cV) — 3.0 3.5 — 4.0 - - 4 .5 S C 3 H < w c w 1 o 2 < 0.1 4 0 0 0 3500 2500 3000 Cathode Tip Temperature (K) Figure 4.9: A ttachm ent area as a function of cathode tip tem perature w ith work function as a param eter. work function is presented in Fig. (4.12) rather than the effective work function for clarity. T he effect of changes in the current are shown in Figs. (4.14) through (4.18). For this geom etry the 100 A curves are significantly shifted from the others. Unlike all of the o th er conditions, a local minimum of about 10.29 exists in th e sheath voltage for th e 100 A case, an attachm ent area of about 0.0715 cm2 and a cathode tem perature of about 2870 K. A local m inim um exists in the effective work function. It is m ost easily seen in Fig. (4.17). This local m inimum corresponds to the maximum electron num ber Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 2 Work Function (cV) 3.0 3.5 4.0 4.5 2 .0 - c 1.0- 0.5 4 0 0 0 3500 2500 3000 Cathode Tip Temperature (K) Figure 4.10: Electron tem perature as a function of cathode tip tem p eratu re with work function as a param eter. Work Function (cV) 3.0 3.5 — 4.0 --4.5 10- o to 3 "o > ■ S ra u je C O a r g o n Is 1000 A P= lOkPa 4 0 0 0 3500 3000 2500 Cathode Tip Temperature (K) Figure 4.11: Sheath voltage as a function of cathode tip tem perature w ith work function as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 0 .3 0 - argon 1= 1000A P = lOkPa c 0.20 E 0.10 5 0.05 Work Function (eV) 3.0 3 5 4.0 45 2500 3 0 0 0 ^ ' 3500 Cathode Tip Temperature (K) 4000 Figure 4.12: W ork function lowering as a function of cathode tip tem perature w ith work function as a param eter. 100 Work Function (eV) 3.0 argon I =1000 A P = lOkPa < 9 a < e u < 12 8 10 6 4 Sheath Voltage Figure 4.13: A ttachm ent area as a function of sheath voltage with work function as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. density. This m inimum is also near the corner point in the Vc and Te curves. This m inim um point may be a “preferred” operating point for the discharge. O peration a t high cathode tem peratures requires large increases in th e sheath voltage and th e corresponding electron tem perature. O peration a t lower cathode tem peratures increases the attachm ent area. In some cases this point also corresponds to a local m axim um in the current density, depending on the relative rates of change of the cathode tem perature and the effective work function. Also, comparisons between these types of curves and the experim ental d ata, discussed in C hapter E ight, suggest th a t the system tends to operate near this local minimum. T he local maximum located ju st right of the local minimum corresponds to a local m inimum in the surface electric field. T he effect of changes in the gas type are shown in Figs. (4.19) through (4.23). W ith the exception of lithium , a variation in the gas type has only a small effect on th e attachm ent area for a gjven cathode tem perature. The gas type does, however, significantly affect th e electron tem perature and the corresponding sheath voltage as shown in Figs. (4.20) and (4.21). The sheath voltages and electron tem perature values for thq slowly increasing portion of the curves are related to the first ionization energy of the gas. T he rates of increase in the electron tem perature and the sheath voltage after the corner point are prim arily determ ined by th e second ionization energy for each gas. Lithium , for example, has the lowest first ionization energy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 1000 j Current (A) 100 ..— 5 0 0 1000 5000 argon P= lOkPa 6 = 3.5 cV 100 E u < 2200 2400 2600 2800 3000 3200 Cathode Tip Temperature (K) Figure 4.14: A ttachm ent area as a function of cathode tip tem p eratu re with current as a param eter. Current (A) 100 500 1000 5000 argon P= lOkPa 6 = 3.5 eV > 2.0 0.5- 2800 3000 3200 2400 2600 2200 Cathode Tip Temperature (K) Figure 4.15: Electron tem perature as a function of cathode tip tem perature with current as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 Current (A) 100 500 1000 5000 argon P= lOkPa 0 = 3.5 eV « > 1 2 . eo 1 ^ 2 o > 10J € « o 8 co o - 2400 2200 2600 2800 3000 3200 Cathode Tip Temperature (K) Figure 4.16: Sheath voltage as a function of cathode tip tem perature w ith current as a param eter. 3.5 0 Current (A) 100 500 1000 - - • 5 0 0 0 £ 3.45 2 3.40 3.35J 3 .3 0 : argon P = lOkPa < > = 3.5 eV 3 .2 5 : 3 .20 3000 3200 2400 2600 2800 2200 Cathode Tip Temperature (K) Figure 4.17: Effective work function as a function of cathode tip tem p eratu re with current as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 1000 h Current (A) 100 500 1000 5000 100 - M £i 1 0 - < 5 i t < 4 6 8 10 12 14 16 Sheath Voltage Figure 4.18: A ttachm ent area as a function of sheath voltage w ith current as a param eter. (5.39 eV) and th e largest second ionization energy (75.6 eV). Therefore, th e lithium curve sta rts out lower than the others b u t climbs more quickly once the double ions become significant. For hydrogen, the ra te of increase is the m ost rapid because no double ions exist. All of the effective work function curves show the local m inim a discussed previously. This local m inim um is most pronounced for the lithium curve due to the large difference between the first and second ionization energies. For lithium , th e local m inim um in the effective work function also corresponds to a local m aximum in th e current density. If the discharge does tend to operate near this point then lithium would have the lowest operating cathode tem perature and hydrogen Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 Gas Type I =1000 A P = lOkPa 4 > = 3.5 eV 100 B o W - - N --- Ar 8 < ’V " u 2 < 22 0 0 2400 2600 2800 3000 3200 3400 3600 Cathode Tip Temperature (K) Figure 4.19: A ttachm ent area as a function of cathode tip tem perature for different gases. th e highest. Usually helium is expected to have the highest cathode tem perature due to its large first ionization energy, and therefore the surface heating each ion is larger. However, for these conditions, the model predicts hydrogen to have a slightly higher cathode tem perature. 4.2 Two-Dim ensional M odel For the two-dim ensional therm al model the near-cathode plasm a model is solved for each surface elem ent. T h at is, for given values of the work function, sheath voltage and tem p eratu re, the surface heat flux and current density are calculated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 > 4 o M e 3 & B £ 2 e e u . « i w Gas Type ----- H I =1000 A / ...... He P= lOkPa / / / .......Li ♦ = 3.5 eV / V ----- N 0 / / ----- At 0 / 1 0 ft • I I * II * 11 0 II / / / / • ' J \ / -------- J ~ — .......— — “ " * 0 2200 24 0 0 2600 2800 30 0 0 3200 3400 3600 Cathode Tip Temperature (K) Figure 4.20: Electron tem perature as a function of cathode tip tem perature for different gases. 16- Gas Type H He I = 1000 A P = lOkPa 4 > = 3.5 eV a to n - - N 12: o > 10 - •S A O JC 2200 24 0 0 2600 2800 3000 3200 3400 3600 Cathode Tip Temperature (K) Figure 4.21: Sheath voltage as a function of cathode tip tem perature for different gases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 3.50 Gas Type H — He •2 3.40 - - N A t 3.35 3.30 1= 1000 A P = lOkPa 6 = 35 eV g 3.25 3.20 2200 2400 2600 2800 3000 3200 3400 3600 Cathode Tip Temperature (K) Figure 4.22: Effective work function as a function of cathode tip tem perature for different gases. 100 Gas Type 1= 1000 A P = lOkPa 6 = 3.5 eV He - - N 0 3 t < 1 u c a < 0.1 10 12 14 16 6 8 4 Sheath Voltage Figure 4.23: A ttachm ent area as a function of sheath voltage for different gases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 for each elem ent. T h e total current is then calculated by sum m ing the elemental currents. T he total current can be controlled by adjusting the attachm ent area a n d /o r the sheath voltage. In practice the sheath voltage can only be adjusted over a small range (typically 3-5 volts for argon) and still m aintain sheath model solutions. A djustm ents to the attachm ent area are more effective. T he com putational tim es required for th e two-dimensional solutions were signif icantly longer than for the quasi-two-dimensional model. T he solution tim e on a 486-33 MHz PC for a given operating condition increased from ab o u t one m inute to about 8-40 hours dependent upon the initial values for the cathode tem pera tu re distribution. Also, th e two-dimensional model solutions were not significantly different from the quasi-two-dimensional model solutions. A com parison between th e two models using th e first version of the plasm a model are given in Ref. [121]. Therefore, the two-dimensional model was only used for select com parisons w ith the experim ental data, which are discussed in C hapters Six and Seven. This two-dimensional m odel, used w ith the first version of the plasm a model to investigate the effect o f the m agnetic pressure on the radial cathode tem perature distribution for a flat-tipped cathode. A com parison of the radial tip tem perature profiles is shown in Fig. (4.24). For the static pressure cases, the tem perature does not change significantly for different current levels or radially across th e attachm ent area. O utside the attach m en t area the tem perature decreases more rapidly due to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 2800 2 7 5 0 - 2 7 0 0 1 J 2 6 5 0 - Current (A) Magnetic 2 5 5 0 - Radial Position (cm) Figure 4.24: Radial C athode tip tem perature distributions. radial cooling. T he addition of th e m agnetic pressure effect results in an increase in the tem perature and a corresponding decrease in the attachm ent area for a specific current level. T he larger radial decreases in th e tem perature profiles using the m agnetic pressure are prim arily due to the radial change in pressure. T he heat flux and current density are sensitive to the pressure values in the plasm a model for the fully ionized solution [101]. T h e peak tem perature for these plasm a conditions is about 2600 K. The tem peratures w ithin the attachm ent area for all of these cases are above this value and, therefore, all of these solutions are on the fully ionized side of the curve [101,102]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 C hapter 5 E xp erim en tal F acility and D iagn ostics Experim ents are required to verify or correct the modeling assum ptions and to iden tify the n ature of th e cathode discharge. Also, testing often reveals phenom ena th a t were not anticipated and therefore m ust be incorporated into th e models. A series of experim ents were perform ed to characterize the cathode operation. Since the cath ode tem perature has a significant effect on th e cathode operating characteristics, th e m ajority of the effort was focused on this area. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 IMAGING PYROMETER CATHODE FIXTURE TO PUMPING PLANT VACUUM CHAMBER WATER-COOLED VACUUM CHAMBER DOOR ON RAILS (DOOR OPEN) Figure 5.1: Diagram of th e cathode test facility. 5.1 Cathode Test Facility A facility was constructed specifically for testing cathodes used in arcjet and M PD thrusters [122,123,124]. A diagram of the cathode test facility is shown in Fig. (5.1). The stainless steel vacuum cham ber is 0.5 m in diam eter and 2.4 m long and is com posed of 4 water-cooled cylindrical segments. In addition, a water-cooled copper liner has been inserted in the middle two segm ents to perm it long-duration oper ation. As the schem atic in Fig. (5.2) shows, the first segment forms the discharge cham ber. A water-cooled, ring-shaped copper anode w ith a diam eter of 7.6 cm is m ounted on a flange located between and electrically isolated from the first two tank segments. T h e cathode fixture m ounted on th e vacuum cham ber door is composed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 TANK TANEDOO* CATHODE Figure 5.2: Schematic of the electrode configuration. of two coaxial tubes electrically isolated from each other and the door w ith m icarta rings. T he inner tu b e serves as the cathode current feed and has a w ater-cooled cap on the end to which the cathode is clamped. T he outer tu b e is electrically floating and has a water-cooled copper disk m ounted on th e end w ith an ap ertu re through which the cathode protrudes. T he propellant gas is injected between the two tubes and flows in to the discharge cham ber through an annulus around the base of the cathode. T h e interelectrode gap is set by the thickness of a spacer in th e cathode assembly. T h e cathodes used in this investigation were rods of 2 percent thoriated tungsten 76 mm long and 9.5 mm in diam eter w ith hemispherical tips. T h e last tank segm ent contains a heat exchanger m ade of w ater-coded, finned copper tubing to cool the exhaust before it enters the pum ping system . T he tank has a num ber of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 ports which provide optical access to the discharge cham ber as well as the plum e. In addition, the cathode and the discharge can be viewed along the tank axis through a window at th e rear of the tank. This facility is similar to the one a t the University of S tu ttg a rt [51]. T he main difference between the two facilities is th a t the S tu ttg a rt system consisted of a sim ulated th ru ster with no sides operating within a larger vacuum cham ber. T h at is, th e discharge cham ber is not completely enclosed. Therefore, the m ass flow through the anode could not be determ ined and the pressures within the plenum cham ber and downstream of th e anode were the sam e. Since the first cham ber section of the JP L system acts as the plenum cham ber and is sealed, all of the propellant entering the cham ber m ust pass through th e anode. The system also allows for choking conditions w ithin the flow resulting from th e pressure drop across the anode orifice. This configuration should provide a m ore accurate sim ulation of thruster operation. T he vacuum cham ber is pum ped by a 610 1/s Roots blower backed by a 140 1/s Stokes mechanical pump. The system is capable of achieving a vacuum of less than 0.13 P a with no propellant flow and approxim ately 80 P a with an argon flow rate of 0.75 g /s. The pressure within the discharge/plenum cham ber can be controlled by adjusting the gas flow rate into the cham ber and an adjustable gate valve located between the vacuum tank and the pum ps. The cooling for th e cathode base, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 anode, the buffer electrode, and each of the three tank sections can be controlled by adjusting the w ater flow rate to each section. Therm ocouples are placed in the inlet and outlet cooling lines to each com ponent. A flow m eter and a valve are used to m onitor and control the cooling w ater flow rate. Thus calorim etry may be performed on each com ponent to determ ine their heat loads. Higher am bient gas pressures are achieved by thro ttlin g the pum ping speed with a valve on a bypass around the main vacuum valve. T he am bient pressure can be controlled to w ithin approxim ately ± 30-70 Pa. The arc is powered by two M iller welding power supplies, each of which can provide 1500 A a t a load voltage of 60 V continuously or 2000 A a t 50 V for up to 20 m inutes. T h e initial arc breakdown is accomplished w ith a 4 A, 850 V sta rt supply. 5.2 D iagnostics T he factory shunts in the Miller welders have been replaced w ith precision shunts th a t are used to m onitor the arc current. The term inal voltage is m easured a t the current feedthroughs into the vacuum tank. T he propellant flow rate is m easured with a Sierra Instrum ents Side-Trak Model 830 flow m eter and a M icromotion Model D6 flow m eter and controlled with a th rottling valve located ju s t upstream of the inlet to the cathode fixture. T he flow m eter outp u t w f as calibrated by m easuring the m ass loss from an argon bottle as a function of tim e. T hree MKS B aratron Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. capacitance m anom eters with ranges of 0-133 Pa, 0-1.3 x 10* P a, and 0-1.3 x 105 P a are used to m onitor pressures. The three transducers are m ounted in a single m anifold with three input tubes. One line measures the tank pressure through a feedthrough on the cham ber door. A special cathode was fabricated w ith two 1 mm diam eter holes. One hole was along the centerline and th e o th er offset from the centerline about 3.5 mm with an opening on th e side of th e cathode near th e tip. These holes serve as pressure taps to m easure th e pressure a t th e cathode tip and on th e cathode side. These pressure taps were connected to th e o th er manifold inlets. T he pressures could be read independently by valving off th e o th er inputs. These param eters and a num ber of facility tem peratures are recorded w ith a M acintosh com puter system utilizing Lab View software and Opto-22 d a ta acquisition hardw are. A C ID TEC 2250-D Charge Injection Device (CID) cam era was chosen as an optical pyrom etric sensor to m easure the two-dimensional tem p eratu re field on the cathode. T he system optics are composed of two interference filters with a 10 nm bandpass centered at 632.8 nm and a long pass filter w ith a cutoff wavelength of 570 nm . T he cam era lens aperture is fixed a t a relatively small value of / /4 and neutral density filters are used to control the image intensity. The im aging array has 512 x 512 CID detectors which are read out at a m axim um rate of thirty tim es per second. These values are converted to an analog signal, which is then fu rth er processed and o u tp u t as a normal video signal by th e cam era electronics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 T he video signal is digitized by a D ata Translation DT-28G2 8-bit fram e-grabber board, which yields a final value between 0 and 255 corresponding to the incident power. T he cam era outp u t was calibrated as a function of incident radiance using a tungsten ribbon lam p. T he calibration procedure and a detailed error analysis for th e tem p eratu re m easurem ents are discussed in Appendix C and Ref. [122]. In th e experim ents th e cam era and optics were m ounted outside th e cham ber ab o u t 39.5 cm from the cathode. T he video o u tp u t from th e cam era was digitized to provide real-tim e m onitoring of the tem perature distribution. O ne line in video m em ory chosen to correspond to the axis of th e cathode was sam pled from each fram e. A given num ber of lines were averaged, displayed in real tim e, and periodi cally stored on disk. T h e surface em ittance was measured using a special cathode fabricated with radial cavities located a t axial locations 4.5,10 and 15 mm from the cathode tip. The 1 mm diam eter, 4 mm deep cavities were formed by electric-discharge machining. Experim ents and theoretical calculations yield an em ittance o f about 0.95 for cavities w ith this length-to-diam eter ratio and rough walls [125,126]. The em ittance of the surrounding cathode surface was calculated by com paring th e radiance of the cavities w ith the surface radiance. T he cam era was also used to study the extent of the arc attachm ent region. Two interference filters w ith a 10 nm bandpass centered a t 488 nm were used to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 select radiation from an intense argon ion line. E ntire images of the cathode and near-cathode discharge region were then captured w ith the frame grabber board and analyzed to yield th e lateral intensity distribution. These images were then Abel inverted to give the radial emissivity distributions and the resulting profiles were used to estim ate th e arc attachm ent area. Emission spectroscopy was also used to determ ine the election tem p eratu re near the cathode tip and dow nstream of the tip. T he system was set up to form an image of the cathode on a screen w ith a lens m ounted outside the vacuum cham ber. A length of fiber optic cable was installed w ith the 100 micron diam eter inlet located a t the center of th e screen and flush with the im age plane. T he screen and fiber inlet were m ounted on a m icrom eter-operated X-Y translation stage so th a t the inlet could be positioned a t any desired point in the image. W ith this technique, light-gathering with high spatial resolution from any image point could be achieved. The light em erging from the fiber exit was focused on the entrance slit of a one- m eter M cPherson m onochrom ator using an optical system designed to m atch the fiber num erical ap ertu re w ith th a t of the m onochrom ator. Variable m onochrom ator slits were set a t 30-50 microns and a 1200 groove p er mm grating was used to disperse th e light, providing a potential resolution in first order of 0.10 Angstrom s. The o u tp u t of a H am am atsu R928 photom ultiplier tube was filtered (30 Hz corner frequency w ith 12 d B /o c t rolloff) and stored on a com puter. T he fiber inlet was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. moved vertically by a motorized stage and the axial translation was done manually. Scans were perform ed a t several axial locations moving away from th e cathode tip, and then some of these points were repeated moving back tow ards th e tip. The vertical intensity scans v/ere then Abel inverted to produce radial em ittance profiles [127]. A dditional filtering of the d a ta was done w ithin th e Abel inversion routine using a FIR Blackman windowed filter. For each case, tw o vertical scans were averaged and centered. The Abel inversion routine then folds th e d a ta set for a total of four averaged half-profiles, and perform s the Abel inversion [111,127]. T he surface m icrostructure and elemental com position of cathodes after opera tion for various lengths of tim e were analyzed using a scanning electron microscope (SEM ) and energy-dispersive spectroscopy (ED S). These m easurem ents were used to characterize changes in th e surface finish and th e distribution of thorium m etal on the surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 C hapter 6 P ure T u n gsten C ath od e E xp erim en ts A series of experim ents were performed using a pure tungsten cathode to elim inate the spatial and tem poral variations in the surface work function th a t were observed with the thoriated tungsten cathodes discussed in the next chapter. A lthough most of the thoriated tungsten tests were performed first, th e pure tungsten d a ta had b etter repeatability and are therefore presented first. T he tests were perform ed using argon gas a t pressure levels of 1.5 and 3.0 kP a, and for currents of 600, 1000 and 1400 A. T he pressure was lim ited to 3.0 kP a and below because th e model predicted th a t cathode m elting would occur for th e higher pressures and the high cathode tip tem peratures measured during the tests confirmed this. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 initial f i n a l E E W -50 -40 -30 -20 -10 0 10 Distance From Initial Cathode Tip (mm) Figure 6.1: Pure tungsten cathode profiles. 6.1 Axial Temperature D istributions T he cathode initially had a hemispherically shaped tip th a t becam e more “bullet shaped” as the tests proceeded. The initial and final profile shapes are shown in Fig. (6.1). Some of the m aterial th a t evaporated from the sides was redeposited on the tip. T he initial tip growth rate, determ ined from th e cam era im ages, was approxim ately 0.4 m m /h r but the rate had slowed to about 0.1 m m /h r at th e end of the first series o f tests. C athode tem perature and electron tem perature scans were performed during each of the first series of tests, which lasted about three to six hours each. Negligible growth was observed during the second and third Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 4000 0.40 0.50 0.60 3500 2 3 £ 2 I 3000 < L > ■ g a 2500 (9 u argon P = 3.0 kPa 1=1400 A pure tungsten 2000 0 -10 -40 -30 -20 Distance From Cathode Tip (mm) Figure 6.2: Pure tungsten cathode axial tem perature profiles for a current of 1400 A w ith surface em issivity as a param eter. series of tests. T he second and third series of tests lasted about one hour each, ju st long enough to obtain cathode tem perature profiles. T he pure tungsten cathode reached therm al equilibrium (> 90 percent of final value) in two to three cam era scans (30 seconds between scans) com pared to about one hour for the thoriated tungsten cathodes. T h e long equilibration tim e for the thoriated tungsten cathodes is probably due to thorium m igration on the surface and will be discussed in m ore detail in the next chapter. T he effect of using different values of the surface em ittance is shown in Fig. (6.2). T he values range from 0.4 for a polished tungsten surface to 0.6 for a rough surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 At th e higher tem peratures, this range of emissivity produces a 200 K range in tem perature. The pure tungsten cathode surface roughness appeared to fall between these extrem es, so a value of 0.5 was used for all of the tem perature m easurem ents. A ctual emissivity m easurem ents were not m ade because the larger erosion rates make the use of cavities, like those used for th e thoriated tungsten te st, im practical. T he tem perature profiles, shown in Figs. (6.3) and (6.4), for th e second and third series of tests indicate th a t there is good repeatability in the cathode tem perature m easurem ents. The tem perature profiles from th e first series of tests were not used because of both the changing tip shape and a small shift in the transm ittance of the window used for the im aging cam era. T he transm ittance shift was probably due to tungsten evaporated from the cathode depositing on th e window because cleaning th e window restored the transm ittance to its original value. No change in the window transm ittance was observed during th e cam era calibrations th a t followed the second and third series of tests. All of the axial tem perature profiles decrease m onatonically as one moves from the cathode tip towards the base. T his shape was also seen for the 4.5 and 6.0 kP a cases w ith thoriated tungsten cathodes, which are discussed in .th e next chapter, but for the 1.5 and 3.0 kP a cases a m axim um in the tem perature profiles existed a short distance from the tip. As shown in Fig. (6.3) and Fig. (6.4) increasing the current causes an increase in the cathode tem peratures away from th e tip b u t does not significantly affect the tip tem perature. T h a t is, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 4000 /■S £ E 3500 3 E & | 3000 H o ■g f 2500 U 2000 Figure 6.3: Pure tungsten cathode axial tem perature profiles for a tank pressure of 1.5 k P a w ith current as a param eter. attach m en t area increases m ore th an the cathode tem perature, or current density. Tests were lim ited to pressures of 3.0 kP a and less because for higher pressures the model predicted th at the cathode tip would melt. T he m arkers shown w ith error bars in the cathode tem perature plots are the m easurem ents m ade w ith the Leeds and N orthrup disappearing filam ent optical pyrom eter. T here is excellent agreem ent between the disappearing filam ent pyrom eter m easurem ents and th e im age cam era m easurem ents. T his confirms th a t the cam era is accurately determ ining the cathode tem perature. A t the cathode tip the disappearing filament pyrom eter m easurem ents are consistently low because these Current Run Run Pyrometer (A) No. No. Value 600 — - 113 ----- 119 □ 600 — * 121a V 1000 .... ... j |4 ----- 120 O 1400 — -• 115 ----- 121 A argon P = 15 kPa pure tungsten c = 050 Distance From Cathode Tip (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 4 0 0 0 -,--------------------------------------------------------------------------------------- Current Run Run Pyrometer (A) No. No. Value 2 600 ------- 116 ------ 122 A w 35 0 0 - 1000 ------ 117 ------ 123 O T r 2 • 1400 ------ 118 — 124 □ : | T ? L '' - ■ - & 3 0 0 0 - > • < r‘ . f * I 1 O _ > f I A r* / *8 / y * . i t*v^ 5 2 5 0 0 - ^ . c - ' > F ^ argon 3 P = 3.0kPa : . " pure tungsten v¥ S w ^ - e = o^o 2 0 0 0 - ' ^ r " i » r t ■ >■»»•» | i i i i | i n i | i i i i | i i i i -4 0 -30 -20 -1 0 0 Distance From Cathode Tip (mm) Figure 6.4: P u re tungsten cathode axial tem perature profiles for a tan k pressure of 3.0 k P a w ith current as a param eter. tem peratures are near th e upper end of its m easurem ent range where it is less accu rate. T he spatial error bars are conservative estim ates for the instrum ent pointing. T he m easurem ents could be accurately repeated for the locations near th e cathode tip and th e two furthest from the tip because m arks on the cathode m ade them easy to locate. T he spatial error bars are based on the second and third points which were m ore difficult to locate. T he tem perature error bars are based on a conservative estim ate of 5 percent accuracy [128,129]. T he error bars for the cam era tem p eratu re profiles, th a t are located on the left side of the plot, were sized at ± 3 percent based on the error analysis in Appendix C. A spatial erro r of plus or Current Run (A) No. 600 116 - 1000 ------ 117 - 1400 ------ 118 — Run Pyrometer No. Value — 122 A - 1 2 3 O 124 □ L J L J J * n r j i j p , ^ > f 1 X r* / S argon r ~ - ' _ „ P = 3.0kPa X v r . v - : t . v V S l ' pure tungsten '■ ° - 50 I I I 11 1 -10 4 0 ■ i • -20 Distance From Cathode Tip (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 3600 U 2200 1 3000 2 §• 2800 £ „ 2600 *8 S 2400 e d 3400 argon 1 = 600 A Pure Tungsten e =0.50 Run Pressure (kPa) 119 1.5 122 3.0 2000 -25 -20 -15 -10 -5 Distance From Cathode Tip (mm) 0 Figure 6.5: P u re tungsten cathode axial tem perature profiles for a current of 600 A with tank pressure as a param eter. minus one pixel (0.298 mm) was used for all of the cam era im ages. Both in stru m ents and th eir respective vacuum tan k windows were calibrated using th e sam e black body calibration source. T he effect of changes in the tank pressure on the tem perature profiles is shown in Figs. (6.5) through (6.7). Increasing the tank pressure raises th e cathode tip tem perature slightly and lowers the tem peratures further from the tip. T he same effect was observed in the thoriated tungsten cathode tests which are discussed in the next chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 3600 3400 < * “s. a: 3200 e | 3000 §• 2800 £ „ 2600 ■ g £ 2400 (3 2200 2000 Figure 6.6: P u re tungsten cathode axial tem perature profiles for a current of 1000 A w ith tank pressure as a param eter. 3600 ~ 3400 U £ 3200 a 1 3000 2 | 2800 £ „ 2600 ■ g •5 2400 ra U 2200 2000 Figure 6.7: P ure tungsten cathode axial tem perature profiles for a current of 1400 A w ith tank pressure as a param eter. argon I =1400 A Pure Tungsten e = 0.50 Run Pressure (kPa) 121a 1.5 124 3.0 ' ] ' » i | f 1 *■ r »' j"» v ? v j i v f r j f i i i i | i r t f j ■!"»" v 4 0 -30 -20 -10 0 Distance From Cathode Tip (mm) argon 1= 1000 A Pure Tungsten e = 0.50 -30 -2 0 -10 Distance From Cathode Tip (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 6.2 The 488 nm Argon II Line Emission T h e extent of the arc attachm ent zone is an im portant param eter in the model because it determ ines w hat fraction of th e boundary is subjected to the arc heat in p u ts and it is used com putationally to lim it the total current to the desired value. It is, of course, very difficult to m easure directly the surface current density or th e heat fluxes. In these experim ents filter photography and emission spectroscopy were used to resolve th e regions of most intense argon ion line emission and an estim ate of the attachm ent area on the cathode surface. A typical image of the cathode and discharge region captured w ith the CID cam era using two 488 nm interference filters is shown in Fig. (6 .8 ) w ith a contour plot of the 488 nm Ar II line intensity distribution shown in Fig. (6.9). T h e contours represent lines of constant cam era response in gray levels and start from 2 0 on the outside, increasing inward in increm ents of 2 0 . This example shows the features common to all of the pure tungsten cathode im ages of the arc attachm ent zone. These im ages show a more diffuse arc attachm ent th an the ones for th e thoriated tungsten operation th a t are presented later. The tip of the cathode appears to be covered w ith a thin bright layer th a t may be the ionization region located outside of the sheath. Most of the im ages had a darker region ju st dow nstream of the cathode tip . This dark region also appears as a depression in the Abel inverted contours presented below indicating th a t there is a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 Figure 6 .8 : Photograph of th e 488 nm A r II line intensity distribution a t 1000 A and 1.5 kP a for a pure tungsten cathode. Figure 6.9: Contours of the 488 nm Ar II line intensity distribution a t 1000 A and 1.5 k P a for a pure tungsten cathode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 decrease in the 488 nm emission in this region. T his effect is m ore dram atic with thoriated tungsten cathodes. A sim ilar depression in the discharge was observed by H aidar and Farm er for a 200 A argon discharge a t 100 kP a [130]. T he current, as indicated by the lum inous region, appears to leave the cathode in an annulus. Tw o experim ents were perform ed to verify th a t these brightness peaks near the tip are not due to cathode surface luminosity. F irst, the intensity in this region was m onitored during arc extinction, which revealed a decay transient lasting less than 1/30 of a second, th e m inim um tem poral resolution of the cam era system , which is much shorter than the cathode therm al transients. Second, a m onochrom ator was used to gain finer spectral resolution of the intensity peak than the filter photog raphy could provide. In axial scans from the cathode shaft into the plasm a plum e dow nstream of the cathode a strong peak in the intensity a t 488 nm sim ilar to th a t in th e photographs was found, b u t no increase in lum inosity was observed in the continuum near the 488 nm line. These two experim ents confirm th a t the observed peak is due to intense argon ion line emission. T he contour plots of the 488 nm brightness for the pure tungsten cathode tests are shown in Figs. (6.10) through (6.15). T he contour levels represent the same brightness value for all of these plots. T he first five contour levels are shown with higher values in increm ents of five units. T he largest gradients for both cathode m aterials tested are on the side of the cathode near the tip. However, the gradients Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 Figure 6.10: D istribution of th e 488 nm A r II line intensity distribution a t 600 A and 1.5 kP a for a pure tungsten cathode. Figure 6.11: D istribution of th e 488 nm Ar II line intensity distribution at 1000 A and 1.5 kP a for a pure tungsten cathode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 Figure 6.12: D istribution of the 488 nm Ar II line intensity distribution at 1400 A and 1.5 kP a for a pure tungsten cathode. Figure 6.13: D istribution of the 488 nm A r II line intensity distribution a t 600 A and 3.0 kP a for a pure tungsten cathode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 Figure 6.14: D istribution of the 488 nm A r II line intensity distribution at 1000 A and 3.0 kP a for a pure tungsten cathode. Figure 6.15: D istribution of the 488 nm A r II line intensity distribution at 1400 A and 3.0 kPa for a pure tungsten cathode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for th e pure tungsten cathode are not as large o r as concentrated as in the thoriated tungsten tests. T his indicates th a t the arc attachm ent is not as concentrated with pure tungsten as is expected from its larger work function. Also, increasing the current causes contours of any specific value to move closer to the base. T h at is, the arc attachm ent area increases w ith increasing current. T he m axim um contour values only change slightly, indicating th a t the m axim um ion brightness does not change significantly w ith current. If one can assum e th a t the brightness corresponds to ion density, then the local ion densities would not significantly change with current. Since th e cathode tip tem perature did not change significantly w ith current, the local current density is probably sim ilar between current levels. Also, since similar conditions exist, th e sheath voltages should be similar. Therefore, since the ions are prim arily produced from energy obtained by therm ionic electrons passing through th e sheath, equal to current density tim es th e sheath voltage for a given length of tim e, one would expect a correlation between the cathode tem perature and the ion brightness contours. Similarly, the expansion of the contours tow ards the cathode base w ith increasing current also agrees well w ith the cathode tem perature results. T h a t is, one would expect the ion brightness a t a specific axial location to increase w ith an increase in th e cathode tem perature at th at location. Similarly, the cathode tem peratures and the brightness contours expand closer to the cathode base for the 1.5 k P a tests th an for th e 3.0 kPa tests as one would expect. However, extrem e care Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 must be used in these assum ptions because only one energy-level transition is being considered, and the gas may not be in local therm odynam ic equilibrium . Also, the ion num ber density is not a linear function of th e current density. 6.3 Electron Temperature M easurem ents Electron tem perature m easurem ents were m ade near the cathode tip by emission spectroscopy to help characterize the plasm a for verifying the model predictions. Scans perpendicular to th e cathode axis were taken at approxim ately ten axial locations dow nstream of th e cathode tip. These lateral scans were Abel inverted to determ ine the radial emissivity profiles. T he scans were repeated for each Ar II transition of interest and a plot such as Fig. (6.16) was created for each spacial location. T he electron tem perature was calculated by plotting the logarithm ic term of Eq. (6.1) versus the excited state energy. T he slope of the line fit to these points is proportional to 1 JTt . k is B oltzm ann’s constant, Te is the electron tem perature, En is the energy of level V , Anm is the wavelength for the transition between levels “m ” and “n ", A m„ is the transition probability, (mn is the m easured em ittance, and gn is th e degeneracy + const (6.1) where (6.2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 2 + tfala fit to 4 highest points 401.4 nm 1 407.2 nm 410.4 nm 0 488.0 nm 1 397.9 nm -2 405.3 nm argon P = 1.5 kPa 1= 1000 A 3 372.9 nm 4 19 21 20 25 22 23 24 Excited State Energy (eV) Figure 6.16: B oltzm ann fit to electron tem perature d ata. of level “n” [111,131]. T he m easured em ittance, cmn, is determ ined from the Abel inversion of the lateral spectrom eter scans, Ix, for each spectral line. An earlier a ttem p t to use lower energy levels in this study revealed a nonequi librium condition w ithin th e plasm a [124]. T his plot is shown in Fig. (6.16) for a point on th e centerline located 0.25 mm dow nstream of the cathode tip. Clearly a single line will not fit the d a ta correctly, indicating th at nonequilibrium is present. A line plotted through the three left points, excluding the 488.0 nm poin t, yields electron tem peratures from 1.2-2.5 eV, which is larger than is expected. Tem pera tures calculated using a line fit through the right four points yields a substantially lower tem peratures (typically around 0.3 to 0.7 eV). Also shown in Fig. (6.16) is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 the 488.0 nm point for com parison. To correct this problem a higher energy line was added to the scans (372.9 nm ). T he electron tem perature was then determ ined using a line fit to the points for the A r II 410.4, 397.9, 405.3, and 372.9 nm lines, shown in Fig. (6.16). An a ttem p t to use the 394.6 nm lines was unsuccessful because they consistently fell far from the fit line indicating th a t this level is over-populated, perhaps due to a m etastable condition. T he radial electron tem perature profiles from the pure tungsten cathode test are shown in Figs. (6.17) through (6.22). In general the electron tem peratures were higher for these tests th an for th e thoriated tungsten tests and ranged from 0.58- 0.73 eV. T he radial profiles show sim ilar shapes except th a t the 1.5 kP a profiles show a depression in the tem perature near the centerline and the cathode tip . It is particularly noticeable for th e 1000 and 1400 A cases. T his effect does not appear in the 3.0 k P a plots or any of th e thoriated tungsten cathode tests. T he depression occurs in all of the near cathode scans, indicating th a t it is a real effect and not a com putational artifact. T he tem perature is more uniform in the scans further from th e tip, indicating th a t the depression is localized near the cathode tip and not along th e entire centerline. It is not d e a r w hat is causing this depression. T h e ion brightness contour plots for these two conditions do not appear any different from the others. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 0.74 ^ 0 . 7 2 0 a 0.70 s e 1 0.68 £ 1 0.66 w U E 0.64 0.62 Figure 6.17: Electron tem perature as a function cf radius for a pressure of 1.5 kP a and a current level of 600 A with axial position as a param eter (pure tungsten). T he oscillations in th e curves m ay be from a com bination of the Abel inversion and th e curve fit techniques. Small variations in the inversion of each line scan, such as those used for the 488 nm contours, are amplified by the line fitting technique. T h at is, the oscillations are small for each Abel inverted im age but the oscillations becom e large when the scans are combined. T he prim ary reason for the magnifica tion is th a t th e peaks and valleys of the oscillations are a t slightly different radial locations for each line scan. T his, in tu rn , causes the slope changes in the line fit resulting in th e electron tem perature oscillations. Axial Position (mm) 0.13 • 0.25 — - 0.25 - - 0.25 —- 0.89 - - - 1.52 — - 2.79 - - 2.79 - 4.06 - - 5.33 a r g o n P= 1.5 kPa 1 = 600 A Radial Position (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 0.72 ? 0.70 N m " * e I 0 68 8. E H 0.66 e e j£ 0.64 w 0.62 Figure 6.18: Electron tem perature as a function of radius for a pressure of 1.5 kP a and a current of 1000 A w ith axial position as a param eter (pure tungsten). 6.4 Calculated R esults T he cathode emission characteristics for each operating condition were calculated from the m easured cathode tem perature profiles. First, an average effective work function, presented in Table (6.1), was calculated for each of the operating cases. T he therm al im age of each cathode was used to determ ine th e radial edge locations. These radial values were then used to com pute the surface area for each axial location using a trapezoidal m ethod. T he area of the element at the tip includes both the side area and the tip surface area. Using the measured tem perature profile, th e surface areas and the total current, the average effective work function was calculated. This Axial Position (mm) — 0.0 0.13 0.13 0.13 0.76 1.40 2.67 - - 2.67 P = 1.5 kPa — 3 .9 4 1=1000 A — - 5.21 4 6 Radial Position (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 2 0.74 Axial Position (mm) — 0.0 0.13 — - 0.13 - - 0.13 - - 0.76 - - - 1.40 — - 2.67 - - 2.67 - 3.94 - - 5.21 argon P = 1.5 kPa 1= 1400 A 0.72 0 .68- / ' / 0.66 0 2 4 6 8 Radial Position (mm) Figure 6.19: Electron tem perature as a function of radius for a pressure of 1.5 kPa and a current of 1400 A w ith axial position as a param eter (pure tungsten). Pressure Current (A) (kPa) 600 1 0 0 0 1400 1.5 4.1694 4.1896 4.2045 3.0 4.1166 4.1334 4.1632 Table 6.1: Experim ental effective work function values for pure tungsten cathode operation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 0.68 > 0.66 a | 0.64 8. E H 0.62 c s « 0.6 0 u 0.58 Figure 6.20: Electron tem perature as a function of radius for a pressure of 3.0 kPa and a current level of 600 A w ith axial position as a param eter (pure tungsten). calculated work function includes both the m aterial work function and any lowering conditions such as the Schottky effect. T he average effective work function increases with increasing current and decreasing pressure. Since the m aterial work function is not expected to change for pure tungsten, these changes are probably due to changes in the electric field which in tu rn produces changes in the effective work function by the Schottky effect. Using the calculated effective work function, the attachm ent areas corresponding to 25, 50, 75 and 98 percent of the total current were calculated. T h e calculated results and the corresponding 488 nm emissivities for different percentages of the Axial Position (mm) 0.13 0.25 0.25 - - - 0.25 0.89 * 1.52 2.79 - - 2.79 — 4.06 — - 5.33 argon P = 3.0 kPa I = 600 A Radial Position (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 2 4 6 Radial Position (mm) 8 Figure 6.21: Electron tem perature as a function of radius for a pressure of 3.0 kPa and a current level of 1000 A w ith axial position as a param eter (pure tungsten). enclosed current are presented in Tables (6.2) through (6.7). T h e enclosed cur rent fractions are not exactly 25, 50, 75 and 98 percent because of th e discrete experim ental tem perature values. The closest percentage values were used. The attachm ent areas for the different enclosed currents and both pressures are shown in Fig. (6.23). All of the areas increase approxim ately exponentially w ith current. For each attachm ent area and current com bination, the cathode tem p eratu re, 7 ) ^ , assum ing th a t all of the current is from therm ionic electron em ission, was calcu lated using the Richardson equation Eq. (2.3). This tem perature is th e value the cathode would have if the attachm ent area had a uniform tem perature. Therefore Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 0.68 argon P = 3.0 kPa I = 1400 A 1.40 2.67 — - 2.67 — 3.94 — - 5.21 0.13 - - - 0.13 Position (mm) 0.0 0.13 Axial 0.76 0 2 4 6 8 Radial Position (mm) Figure 6.22: Electron tem perature as a function of radius for a pressure of 3.0 kPa and a current level of 1400 A w ith axial position as a param eter (pure tungsten). these tem p eratu re and area values can be used to com pare the experim ental results with th e quasi-two-dimensional cathode model th a t are presented in C hapter Eight. Also listed is the value of the 488 nm Argon II line emissivity corresponding to the edge of the attachm ent area. T he 488 nm brightness for each enclosed current value are sim ilar for each current level at a specific pressure and the values increase with pressure for a specific current level. A 488 nm emission value of 5 ± 1 corresponds to the 98 percent enclosed current value for all of the tests indicating th a t this value may be useful for determ ining the arc attachm ent area. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 Enclosed C urrent A ttachm ent Thich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K ) (A rbitrary U nits) 23.8 142.8 0.3452 3379 34.6 51.2 307.1 0.7308 3382 30.4 74.6 447.3 1.3920 3327 21.2 97.9 587.8 3.386 3208 5.68 Table 6.2: E xperim ental values for pure tungsten cathode operation a t 1.5 k P a and 600 A. Enclosed C urrent A ttachm ent Thieh 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K ) (A rbitrary Units) 25.1 251.4 0.4024 3481 40.0 50.4 504.2 0.9450 3447 31.0 75.4 754.5 1.8112 3393 20.6 98.0 979.8 4.564 3262 5.66 Table 6.3: Experim ental values for tungsten pure cathode operation a t 1.5 k P a and 1000 A. Enclosed Current A ttachm ent TfUch 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K ) (A rbitrary Units) ' 24.6 343.6 0.6749 3448 38.0 49.5 693.6 1.5526 3420 27.8 74.7 1045.2 3.041 3366 20.4 98.0 1372.0 7.237 3248 6.16 Table 6.4: Experim ental values for pure tungsten cathode operation a t 1.5 kP a and 1400 A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 Enclosed Current A ttachm ent 7'm ch 488 nm C urrent Fraction Area Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 23.2 139.0 0.2060 3444 50.0 50.3 301.8 0.4199 3458 35.9 76.3 457.6 0.8127 3406 24.0 98.0 588.2 2.192 3255 4.53 Table 6.5: E xperim ental values for pure tungsten cathode operation at 3.0 k P a and 600 A. Enclosed Current A ttachm ent 7 * * 488 nm Current Fraction Area Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 25.8 258.5 0.2783 3528 49.6 50.3 502.8 0.6290 3494 38.4 74.5 744.9 1.2512 3430 26.6 98.0 979.6 3.5819 3270 5.30 Table 6.6: E xperim ental values for pure tungsten cathode operation a t 3.0 k P a and 1000 A. Enclosed Current A ttachm ent Trad, 488 nm C urrent Fraction Area Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 23.8 333.8 0.4047 3523 47.1 51.2 716.4 0.9181 3510 35.0 74.8 1047.8 1.9276 3432 21.8 98.1 1372.9 5.2469 3282 5.39 Table 6.7: Experim ental values for pure tungsten cathode operation a t 3.0 k P a and 1400 A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 8 Pressure Current Enclosed (kPa) (percent) - e - 1.5 - e - 1. 5 - A - 3.0 H K 3.0 - * - 3 . 0 - • - 3 . 0 0.1 600 800 1000 1200 1400 1600 1800 Current (A) Figure 6.23: Arc attachm ent areas for pure tungsten cathode operation. T he correlation between the 488 nm Argon II line em issivity brightness and the cathode current densities for all of the pure tungsten tests is shown in Fig. (6.24). T he current densities were calculated using th e m easured cathode tem peratures and the effective work functions calculated above and given in Table (6.1). The 488 nm brightness values were those one pixel above the surface. T he peaks on the right side of the plot are from th e large brightness values observed near the tip. T he curvature on the left side may be a cam era effect. T he brightness values for both the 488 nm images and the cathode tem perature images for the left portion of the graph are near the detection lim it of th e cam era and therefore may contain significant errors. All of the values tend to fall on one characteristic curve. Since the m ajor energy source Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 P = 1.5 kPa Run Current No. (A) - 113 600 - 114 1000 * 115 1400 P = 3.0 kPa Run Current No. (A) ■ 116 600 117 1000 118 1400 Data Fit argon pure tungsten 0.1 1 10 100 1000 Current Density (A/cm ) Figure 6.24: Brightness of the 488 nm Argon II line em issivity as a function of current density for the pure tungsten tests. for th e ionization region is from therm ionic electrons being accelerated through the sheath, one would expect there to be a correlation between these values. However, th e exact correlation can not be determ ined because both the sheath voltage and ion densities are unknown. A curve fit to this d ata yields In (C 48s) = 0.6330 + 0.248781n (j6) - 0.03680 [In (jt )]2 (6.3) or In (jb) = -2 .4 1 8 + 3.8412In (£4 8 8) - 0.40278 [In (f48s)]2 . (6.4) T his correlation will be used to estim ate the local work function for th e thoriated tungsten cases in the next chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 C hapter 7 T h oriated T u n gsten C ath od e E xp erim en ts T he m ajority of the experim ents were performed with two percent thoriated tung sten cathodes. The lower operating tem peratures for this m aterial significantly reduce th e erosion rate m aking it a more practical m aterial for th ru sters. However, the distribution of thorium along the surface can significantly change the local work function and therefore th e discharge characteristics making fundam ental observa tions m ore difficult. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 7.1 Cathode Surface Em ittance Knowledge of the em ittance is not only of im portance in predicting the radiant heat flux as a boundary condition on the therm al models, it is also required to convert the radiance m easured directly by the im aging pyrom eter to surface tem peratures. T he thoriated tungsten cathode w ith the three cavities was imaged in a series of experim ents a t 1000 A with tank pressures of 1.5-3.0 kP a and argon m ass flow rates of 0.25-0.75 g /s. T he radiance from the back walls of th e cavities and from spots on th e cathode surface in the same axial locations and approxim ately 1 mm from th e centerline of the cylindrical cavities was m easured. T h e surface em ittance calculated from the ratio of th e radiances and a cavity em ittance of 0.95 is shown in Fig. (7.1). T he em ittance is essentially uniform along this part of th e cathode and ranges from 0.53-0.66. T his is higher than em ittances m easured by de Vos on polished tungsten ribbons [132] and in sim ilar experim ents by Myers [133] in a low-power M PD thruster, b u t agrees well with later m easurem ents by Fillm ore on a sim ilar th ru ster [126]. Subsequent elemental analysis of the area surrounding the cavities using Energy Dispersive Spectroscopy (EDS) showed only pure tungsten, b u t Scanning Electron Microscope (SEM ) exam inations revealed a very complex m icrostructure. The observed increase in em ittance is consistent w ith m easurem ents for tungsten w ith a characteristic surface roughness of 1-3 m icrons [134]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 1.0 0.8 8 0.6 I 1 0.4 0.2 - 0.0 30 20 25 0 5 10 15 Distance from Tip (mm) Figure 7.1: C athode surface em ittance m easured a t three axial locations. W hen p lotted as a function of the local tem perature in Fig. (7.2), the m easure m ents indicate a slight decrease in em ittance with increasing tem perature. This behavior was also observed by de Vos [132] and Fillmore [126]. A mean value of about 0.57 was chosen for all subsequent therm al d ata analysis. 7.2 Axial Temperature D istributions The focus of these experim ents was th e development of a database of tem perature m easurem ents for use in validating th e models. T he axial tem perature distribution in the first 40-60 mm of the cathode, m easured from the tip, was determ ined using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 1.0 0.8 8 0.6 S3 J 0.4 0.2 0.0 2000 2200 2400 2600 2800 Temperature (K) Figure 7.2: C athode surface em ittance as a function of surface tem perature. the im aging pyrom eter for current levels of 600, 1000 and 1400 A, tank pressures of 1.5, 3.0, 4.5 and 6.0 k P a and an argon m ass flow rate of 0.275 g /s. T h e results from tests using different mass flow rates are presented in a later section. Also, additional d ata from this study are presented in Refs. [122] and [123]. T he effects of current level for each pressure are shown in Figs. (7.3) through (7.6) and the effect of pressure for each current level are shown in Figs. (7.7) through (7.9). Two distinctly different tem perature profiles were observed. A t lower pressures and higher currents a tem perature peak is located on the shaft of the cathode (convex shape), while a t higher pressures th e peak is located on the tip (concave shape). T he tem perature peaks located on th e shaft were observed to move towards o © n % O % , Q ° ° 0 6 ° UO O q i >iif11 11 11 11i 1111 11 11 | 11 i ■11 11 11 i 11 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 3400 3200 * 3000 a 2 2800 2 | 2600 £ 2400 M 2200 cs U 2000 1800 -60 -5 0 -40 -30 -20 -10 0 Distance From Tip (mm) Figure 7.3: Axial cathode tem perature distribution for a tank pressure of 1.5 kP a w ith current as a param eter. 3500 X 'w ' K 3000 e & 0 £ 2500 u •a o •S < a u 2000 -50 -4 0 -30 -20 -10 0 Distance From U p (mm) Figure 7.4: Axial cathode tem perature distribution for a tank pressure of 3.0 kPa with current as a param eter. Run Current No. (A) 96 600 argon P * * 3.0 kPa ~ 128 600 93 1000 - - - 102 1000 129 1000 97 1400 — 130 1400 1 * 1 I | I I I I I I I I I | I I I I I I I I I Run Current Run Current No. (A) No. (A) aigon ------94 600 ----- 126 1000 P = 1.5 kPa ....... 125 600 ------ 95 1400 ....... 91 1000 ----- 127 1400 - - - 8 4 1000 ' V ■— * r ' — .— • - / V * * / ./ .* X 5-' 4 -*' -** j f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 3400 3200 > ■ * - % * 3000 2 2 2800 2 | 2600 £ « 2400 ■ 5 5 2200 A u 2000 1800 Figure 7-5: Axial cathode tem perature distribution for a tank pressure of 4.5 kPa with current as a param eter. 3400 3200 * 3000 2 2 2800 .2 | 2600 « 2400 T 3 I 2200 < 3 U 2000 1800 Figure 7.6: Axial cathode tem perature distribution for a tank pressure of 6.0 kPa with current as a param eter. Run Current No. (A) 100 600 Run Current No. (A) 135 1000 ■ 101 1400 argon P = 6.0 kPa 1000 t t * t -30 40 -20 -10 0 Distance From Tip (mm) Run Current No. (A) 98 600 131 600 90 1000 85 1000 132 1000 99 1400 133 1400 argon P = 4.5 kPa 40 20 Distance From Tip (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 3200 1 = 600 A £ 24 0 0 1 2200 Distance From Tip (mm) Figure 7.7: Axial cathode tem p eratu re distribution for a current of 600 A w ith tank pressure as a param eter. 4 0 0 0 £ 3500 g 3 | 3000 £ £ •S 2 5 0 0 o JS w C Q U 2000 -40 -30 -20 -10 0 Distance From Up (mm) Run Pressure No. (kPa) - 9 1 1.5 ■ - 8 4 1.5 ■ •• 126 15 • - 102 3.0 - 130 3.0 Run Pressure No. (kPa) - - 9 0 — 132 - 89 - 135 4.5 4.5 6.0 6.0 argon I =1000 A .____ - ' j ............................................... '<5 - - * ■ JrW *?' I I I I I I I I I I I I I ■ I I I I I I -TT- I i i r r r P H r Figure 7.8: Axial cathode tem perature distribution for a current of 1000 A with tank pressure as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 3600- 3400- s X g 3200 3 2 3000 & | 2800 1 2600 Run Pressure No. (kPa) — 95 1.5 127 1.5 — * 97 3.0 Run Pressure No. (kPa) 99 4.5 133 4.5 101 6.0 argon I =1400 A 130 3.0 - - 136 6.0 Y ' ' ' & - ^ * ,.»•* / ^ _ 2400: - - Z - •• 2200- -40 -30 -20 -10 Distance From Up (mm) Figure 7.9: Axial cathode tem perature distribution for a current of 1400 A w ith tank pressure as a param eter. the cathode base providing a flatter profile during th e longer tests. This can be seen by com paring the profiles for tests 84 (240 m inutes) and 126 (100 m inutes), and tests 95 (280 minutes) and 127 (70 m inutes) in Fig. (7.3). T he three profiles for 1000 A in Fig. (7.4) also show this trend. The profile for test 129 (50 m inutes) has a concave shape while the profiles for tests 102 (180 m inutes) and test 93 (300 m inutes) are convex and become flatter w ith increasing operation tim e. At the tip, the tem perature increases weakly w ith current for all pressures. However, a m ore dram atic increase in tem perature w ith current is observed on the cathode shaft. It appears th a t increasing the current prim arily results in an increase in the attachm ent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. area w ith only a small increase in the cathode tip tem perature. The higher pressure a n d /o r lower current profiles (concave) did not change much during the longer tests. Some of these profiles becam e flatter with tim e while others became m ore curved. These trends were also observed during testing w ith currents levels of 600, 800, and 1000 A b u t a t a higher flow rate of 0.75 g /s [122,123]. Some caution m ust be exercised when com paring the curves for different tests, since some of th e profiles were not repeatable. T h a t is, the tem perature a t a given location may vary as much as 200 K between two tests a t similar conditions. This is discussed in m ore detail in the m ass flow ra te effects section below. Also, several of the profiles contain a small “dip” near th e tip th a t corresponds to a visible dark band. M aterials analysis, which is discussed below, revealed th a t th e dark band was prim arily tungsten w ith a rough texture. T he tem perature “dip” is therefore a result of a drop in th e surface brightness an d not a real tem perature effect. The intensity peak a t th e tip may contain some contribution from plasm a rad i ation, either the continuum in the 10 nm bandpass of the 632 nm interference filter o r the integrated effect of plasm a radiation collected in the wings of th e blocking filters. However, m easurem ents of the plasm a intensity off the cathode surface indi cate th a t this contribution is small. In addition, th e timescale for decay of the tip intensity peak when the arc is extinguished is much longer than the plasm a decay timescales, proving th a t the peak is due to surface luminosity. The erro r for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 3000 Cathode Tip 2800- * 2600- e I 2400- 8. I 2200- H Cavity No. 3 (4.5 mm from Tip) Cavity No. 2 (15 mm from Tip) 2000- Cavity No. 1 (25 mm from Tip) 1800 300 100 250 0 50 150 200 Elapsed Time (min) Figure 7.10: V ariation in cathode tem perature w ith operating tim e. cam era tem peratures was ± 3 percent, which was previously discussed for th e pure tungsten cathode tests. T he evolution of th e tem perature profile during a five hour run a t 1000 A, 3.0 kP a and 0.75 g /s is sum m arized in Fig. (7.10), which displays the tip tem pera ture calculated assum ing an em ittance of 0.57 and the tem perature a t each of the three cavities calculated w ith an em ittance of 0.95. T he dotted lines are interpo lated values in regions where the tip tem perature and cavity tem peratures were not available because the m easurem ent line was moved off of the axis to m onitor the radiance outside the cavities. T h e discontinuous increase in the tem peratures at about 1 0 m inutes elapsed tim e occurred when the current was increased from the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 start value of 800 A to the 1000 A operating point. A fter a transient lasting only several m inutes, the tip tem perature becomes relatively constant a t 2750-2780 K. However, after about two hours the tem perature on the shaft started to climb. The high-tem perature zone was effectively creeping back on th e cathode shaft. T h e fact th a t the cavity radiance shows th is effect suggests th a t it is a tru e increase in tem perature, not ju st an increase in em ittance. These transients are m ost likely due to an increase in the length of the attachm ent zone precipitated by changing surface conditions, as discussed below. 7.3 M ass Flow Hate Effects The focus of this series of experim ents was to determ ine w hat effect, if any, th e mass flow rate had on the axial cathode tem perature profiles. The axial tem p eratu re distribution in the first 40-50 m m of the cathode (m easured from the tip ) was determ ined using the im aging pyrom eter for current level of 1000 A, tank pressures of 1.5, 3.0, 4.5 and 6.0 kP a and an argon mass flow rates of 0.074-0.878 g /s. T he results for each pressure are shown in Figs. (7.11) through (7.16). A dditional d ata from this study are presented in Refs. [122] and [123]. Again, th e test num bers are included to show the order in which the experim ents were performed. T he profiles dem onstrate the degree of irreproducibility in the cathode tem p era ture for the sam e operating point. W hile the tem perature profile shape is generally Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. quite repeatable, the tem perature m agnitudes can vary by as much as 200 K from one trial to the next. The irreproducibility is greater for th e lower pressures. For example, all of th e profiles shown in Fig. (7.13) were perform ed under sim ilar con ditions. T he highest tem perature profile in the 1.5 kP a plots is from test num ber 126 while the profile from test num ber 87 is one of th e lowest. Curiously, b oth of these were a t m ass flow rates of 0.290 g /s but they have different tem perature values. Similarly, in Fig. (7.14), test num bers 83 and 129 were performed a t the same flow rate, and they tend to bound th e d ata for the 3.0 kP a tests. Figure (7.11) shows all of the tests performed for 1.5 kP a and Fig. (7.12) shows selected tests for b etter clarity. All of the profiles fall in a band between tests 87 and 126. W here each profile falls in the band docs not appear to be correlated with the mass flow rate. T he spread in the profiles suggests th a t additional heating is occurring on the cathode shaft which pulls the distribution to higher tem peratures. Some of the irreproducibility can be attrib u ted to varying run durations, because the tem pera tu re distribution appears to evolve w ith tim e. However, it is m ore likely a result of variations in the surface work function resulting from thorium m igration which is discussed in a later section. T he thorium surface coverage seems to be determ ined by the length of the test and the conditions of the preceding tests. The profiles of Fig. (7.15) from the 4.5 kP a tests show the two profile shapes observed for this pressure. All of th e tests started with tem perature profiles sim ilar to those of tests Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 81 and 85 (concave). However, in later tests, a shift to the profiles like those of tests 90 and 132 (convex) was observed. Typically, the profile shift occurred over a period of about 10 m inutes after about 60-80 minutes of testing. However, this shift is not strictly a tim e dependent effect since test num ber 132 was actually shorter than both tests 81 and 85. It appears th a t 4.5 kP a is near the pressure threshold th a t divides the two types of profile shapes. At 6.0 kP a th e spread in the profiles is much sm aller th an for the other pressures. At this pressure, alm ost all of th e current attachm ent is a t th e tip, and therefore, thorium m igration on the sides, which effects the tem perature profile of the cathode, would have little effect. T he variations in the tip tem peratures probably result from differences in the thorium coverage a t the tip. 7.4 The 488 nm Argon II Line Emission A typical im age of a thoriated tungsten cathode and discharge region captured w ith the CID cam era using two 488 nm interference filters is shown in Fig. (7.17) w ith a contour plot of th e 488 nm A r II line intensity distribution shown in Fig. (7.18). T he contours represent lines of constant cam era response in gray levels and sta rt from 20 on the outside, increasing inward in increm ents of 20. T his exam ple shows the features common to all of the images of the arc attachm ent zone. There is a strong axial gradient in intensity dow nstream of the cathode which is probably due Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 £ 3 2 & e H u 1 (3 4000- 3500- 3000- 2500 Run Flow Rate Run Flow Rate Run Flow Rate N o. (g/s) - 75 0.074 * 57 0.156 - 62 0.156 - 76 0.156 - 55 0290 - 56 0290 N o. (g/s) • 61 0.290 74 0.290 79 0.290 - 80 0.290 ■ 84 0.290 ■87 0290 N o. (g/s) •■■91 0.290 - 126 0.290 - 53 0.775 • • 54 0.878 2000 argon P= 15 kPa 1 = 1000 A ■ 1 1 1 1 1 1 1 -50 -40 -30 -20 -10 Distance From Cathode Tip (mm) Figure 7.11: Axial cathode tem perature distribution for a tan k pressure of 1.5 kPa, a current of 1000 A, and all mass flow rates. 3400 3200 * 3000 E > 3 2800 2 | 2600 & 2400 0 1 2200 a U 2000 1800 -50 -40 -30 -20 -10 0 Distance From Cathode Tip (mm) Figure 7.12: Axial cathode tem perature distribution for a tan k pressure of 1.5 kPa, a current of 1000 A, and selected m ass flow rates. Flow Rate (g/s) 0.074 - - • 76 0 .1 5 6 ----- 80 0.156 91 Run Flow R ate Run Flow Rate No. (g/s) No. (g/s) 0.156 - - 126 0290 0.290 — 53 0.775 0290 - • 54 0.878 argon P= 1.5 kPa 1 = 1000 A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 3000 ~ 2800 A 'w' | 2600 2 I* 2400 £ ■ g 2200 j= u 2000 1800 -50 -40 -30 -20 -10 0 Distance From Cathode Tip (mm) Figure 7.13: Axial cathode tem perature distribution for a tan k pressure of 1.5 kPa, a current of 1000 A , and a mass flow rate of 0.290 g /s. 3400 3200 * 3000 a 3 2800 2 | 2600 £ „ 2400 •O | 2200 « s U 2000 1800 -40 -30 -20 -10 0 Distance From Cathode Tip (mm) Figure 7.14: Axial cathode tem perature distribution for a tank pressure of 3.0 kPa, a current of 1000 A, and all mass flow rates. Run Flow Rate Run Flow Rate No. (g/s) No. (g/s) 72 0.074 93 0.290 i i i i i i r r n ' i i t » " i i t"» ..r n * r 'T it j t t f r i r -r - r T argon P = 1.5 kPa 1= 1000 A mass llow = 0290 g/s x s ' Run Run No. No. - 5 5 -------80 ■ 5 6 -------84 - 61 - - 87 74 — 91 7 9 ------- 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 3000- £ 2 8 0 0 - W | 2 6 0 0 - e I 2 4 0 0 £ | 2200 I U 2000J Run No. — 81 85 — 90 - - 132 argon P = 4.5 kPa 1 = 1000 A flow rate = 0.290 g/s 1800 -40 Distance From Cathode Tip (mm) Figure 7.15: Axial cathode tem perature distribution for a tank pressure of 4.5 kPa, a current of 1000 A, and all mass flow rates. 3000- ~ 2 8 0 0 - m Run Flow Rate Run Flow Rate No. (g/s) No. (g/s) ----- 71 0.074 ------8 8 0.290 -----7 0 0.156 - * - 8 9 0.290 .......69 0.290 - - 135 0.290 ------78 0.290 — 6 8 0.545 ------82 0.290 ----- 67 0.775 ------8 6 0.290 .§ 2200 - o l! U 2000- 1800- argon P = 6.0 kPa 1= 1000 A -30 -20 -10 Distance From Cathode Tip (mm) Figure 7.16: Axial cathode tem perature distribution for a tank pressure of 6.0 kPa, a current of 1000 A, and all mass flow rates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.17: Photograph of th e 488 nm A r II line intensity distribution a t 1000 A and 1.5 k P a for a thoriated tungsten cathode. 5 Figure 7.18: C ontours of the 488 nm Ar II line intensity distribution a t 1000 A and 1.5 kP a for a thoriated tungsten cathode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to a decrease in current density as the current lines diverge toward the anode, and a drop in density and tem perature as the flow accelerates. T here is also a strong radial gradient th a t reflects a true increase in density and an increase in integrated lateral luminosity due to a lengthening optical path length through the cylindrical plasm a column. All of th e profiles show a strong peak in intensity along the hem ispherical tip of the cathode. The large radial extent of the intensity plateau a t this axial location also suggests th at it is not ju st due to the longer optical path along the center of the im age. This bright plasm a cloud on the tip of the cathode may be the ionization zone located above the sheath, m arking the p art of the attachm ent zone with the highest current density. Contour plots of the Abel inverted images of the discharge region captured w ith the CID cam era sim ilar to those for the pure tungsten tests are shown in Figs. (7.19) through (7.30). T h e contours represent lines of constant cam era response in gray levels and sta rt from five on th e outside, increasing inward in increm ents of five. All of the contours presented here have been adjusted to the sam e relative em issivity so th a t they can be com pared to each other and with the pure tungsten cathode results. Comparison of the images reveals th at the size of the attachm ent zone increases w ith current and decreases with tank pressure, as predicted by the models (121). This behavior was also observed by Hugel (135). The distributions for a pressure of 1.5 kP a show much m ore plasm a intensity along the cathode shaft, mirroring the behavior Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 8 Figure 7.19: D istribution of th e 488 nm A r II line em issivity distribution at 600 A and 1.5 kPa. Figure 7.20: D istribution of the 488 nm Ar II line em issivity distribution at 1000 A and 1.5 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 Figure 7.21: D istribution of th e 488 nm A r II line emissivity distribution a t 1400 A and 1.5 kPa. Figure 7.22: D istribution of th e 488 nm Ar II line emissivity distribution at 600 A and 3.0 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 Figure 7.23: D istribution of th e 488 nm A r II line emissivity distribution a t 1000 A and 3.0 kPa. Figure 7.24: D istribution of th e 488 nm A r II line emissivity distribution a t 1400 A and 3.0 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 Figure 7.25: D istribution of th e 488 nm A r II line emissivity distribution at 600 A and 4.5 kPa. Figure 7.26: D istribution of the 488 nm A r II line emissivity distribution a t 1000 A and 4.5 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 Figure 7.27: D istribution of the 488 nm A r II line emissivity distribution a t 1400 A and 4.5 kPa. Figure 7.28: D istribution of the 488 nm A r II line emissivity distribution a t 600 A and 6.0 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 C50S Figure 7.29: D istribution of the 488 nm A r II line emissivity distribution a t 1000 A and 6.0 kPa. Figure 7.30: D istribution of the 488 nm A r II line emissivity distribution a t 1400 A and 6.0 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. observed in the tem p eratu re m easurem ents. A very interesting characteristic is observed in all of the cases. The largest gradients in the ion intensity are found near the corner of the cathode indicating th a t the m ajority of the arc attachm ent is in a thin annulus around th e tip, and not on the centerline as w ith arcjet th ru ster cathodes. This current concentration moves tow ards the centerline as it moves away from the tip to form a conical current envelope. In addition, there is a depression in th e intensity values in the center beginning a few m illim eters from the tip. T h a t is, th e intensity increases w ith radius and then decreases as one moves from the inside to the outside of this current envelope. This effect can be seen m ost strongly in the higher pressure cases. F urther away, th e maxim um intensity is on the centerline. Also, th e intensity profiles are flat in the radial direction near the tip , particularly in th e two low-pressure cases, indicating th a t radial gradients are small. If the intensity levels represent num ber density levels, then th e radial num ber density profiles would also be expected to be fiat in this region. However, a m eans of calibrating the brightness levels is needed for any quantitative results. 7.5 Electron Temperature M easurem ents T he radial tem perature distributions dow nstream of the cathode tip are shown in Figs. (7.31) through (7.42) for the thoriated tungsten cathode tests. T he radial profiles show sim ilar shapes to the pure tungsten ones. T he tem perature profiles are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 0.70 a r g o n P = 1.5 kPa 1 = 600 A Position (mm) Axial - - 0.13 -- 1.40 — 2.67 - 2.67 3.94 - 5.21 0.76 0.0 0.13 0.13 0.58 0 1 2 3 4 5 6 Radial Position (mm) Figure 7.31: Electron tem perature as a function o f radius for a pressure of 1.5 kPa and a current level of 600 A w ith axial position as a param eter. flatter radially as th e pressure is decreased. Recall, th a t the 488 nm brightness values were also flat radially for th e low-pressure cases. As the pressure is increased the electron tem peratures begin to increase radially, suggesting th a t the arc attachm ent is concentrated in an annulus near the edge of th e cathode. A dditional inform ation discussed below will su b stan tiate this idea. T he d a ta were less repeatable at the higher pressures. T he axial tem p eratu re distributions are shown in Figs. (7.43) through (7.45). T he axial tem perature distributions were obtained by curve fitting a line to the portion of the radial curves near the centerline. In general the tem peratures decrease Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 196 0.66 5 0.64 w a I 0.62 6 E £ 0.6 0 M 0.58 ID 0.56 Figure 7.32: Electron tem perature as a function o f radius for a pressure of 3.0 kP a and a current level of 600 A w ith axial position as a param eter. 0.66 5 0.64 W e I 0.62 w 8. E £ 0.60 " 0.58 tu 0.56 Figure 7.33: Electron tem perature as a function of radius for a pressure of 4.5 kPa and a current level of 600 A w ith axial position as a param eter. / / //*i ; \ ' \ ,r ( I ? ' argon P = 4.5kPa 1 = 600 A Axial Position (mm) 0.13 0.25 0.25 - - - 0.25 0.89 - - - 1.52 2.79 - - 2.79 - 4.06 - - 5.33 -r 2 T 3 — r- 4 T Radial Position (mm) A Axial Position — o.i3 - 025 ■ > ■ ■ / argon 0.25 - - - 0.25 0.89 - - - 1.52 2.79 2.79 P = 3.0 kPa — 4 06 1 = 600 A 5 3 3 T 0 2 3 4 Radial Position (mm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 0.62 Axial Position (mm) —- 0.25 — - - 0.25 — - 0.89 * 1.52 •2.79 — - 2.79 — 4.06 — - 5.33 f / / v ? > - argon P = 6.0 kPa 1 = 600 A Radial Position (mm) Figure 7.34: Electron tem p eratu re as a function of radius for a pressure of 6.0 kPa and a current level of 600 A w ith axial position as a param eter. 0.68 jo 0.66 a I 0.64 & E £ 0.62 c B " 0.60 u 0.58 . \ Axial ■ v _ ✓ '■ Position ' /C*C (mm) v ' " a / ^ - ' r ' — 00 ^ -* • ~ su 0.13 / / .......0.13 ^ ' ---0.13 . - 0.76 ^ ' / ----------------- ----- 1.40 ^ . ' N _ ---- 2.67 r * — * ■ " argon - - 2 6 7 P=1.5kPa _ 3 9 4 1=1000 A _ . 5' 2 l — r 2 I I I I I I I I I I I I I 0 1 Radial Position (mm) Figure 7.35: Electron tem perature as a function of radius for a pressure of 1.5 kPa and a current level of 1000 A w ith axial position as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 0.64 > 0 . 6 2 a | 0.60 8. E £ 0.58 I « 0.56 u 0.54 0 1 2 3 4 5 6 Radial Position (mm) Figure 7.36: Electron tem perature as a function of radius for a pressure of 3.0 kP a and a current level of 1000 A w ith axial position as a param eter. 0.62 ? 0.60 £ | 0.58 & E £ 0.56 c e 1 0.54 u 0.52 Figure 7.37: Electron tem perature as a function of radius for a pressure of 4.5 kP a and a current level of 1000 A with axial position as a param eter. ~ r ~ r T Axial Position (mm) — 0.25 argon P = 4.5 kPa 1 = 1000 A 2 3 4 Radial Position (mm) \ * argon Axial Position (mm) 0.13 -..... 0.25 0.25 0.25 0.89 - - - 1.52 — - 2.79 - - 2.79 P = 3.0kPa — 4.06 I * 1000 A 5 .3 3 f 'V I * f t t | V I I T V I V I 1 I" I 1 'f I V I I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 1 2 3 4 5 6 Radial Position (mm) Figure 7.38: Electron tem perature as a function of radius for a pressure of 6.0 kP a and a current level of 1000 A w ith axial position as a param eter. 0.66 £ 0.64 Posiuon g 0.62 £ 0.60 « 0.58 _ / (mm) ' _ 0.13 0.25 / ^ S ' J ..........0.25 / ' - - - 0.25 f - ' ........ 0.89 - * - 1.52 — - 2.79 argon ------- 2.79 P = 1.5 kPa — 4 0 6 I = 1400 A ---- 5 3 3 Radial Position (mm) Figure 7.39: Electron tem perature as a function of radius for a pressure of 1.5 kP a and a current level of 1400 A with axial position as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 0 0.64 - " ■ • S . > 0 0.62 K 3 « ■ » 1 0.60 E £ 5 0.58 § 1 0 5 6 0.54 0 1 2 3 4 5 6 Radial Position (mm) Figure 7.40: Electron tem perature as a function of radius for a pressure of 3.0 kP a and a current level of 1400 A w ith axial position as a param eter. 0.64 > 0.62 a 1 0.60 2 1 E £ 0.58 C 2 “ 0.56 u 0.54 Figure 7.41: Electron tem perature as a function of radius for a pressure of 4.5 kPa and a current level of 1400 A w ith axial position as a param eter. ■ ?V : ....' v / v - ^ — - / i / \ r argon Axial Position (mm) 0.13 0.25 0.25 0.25 0.89 - * - 1.52 2.79 - - 2.79 P = 4.5 kPa — 4.06 I = 1400 A — - 5.33 — r 4 — r 5 0 1 T 2 T 3 Radial Position (mm) •4 Axial ■ \ _ / v' - ^ / 7 ' Position ' / f ’ (mm) \ / ' \ ' / / V * ' V argon 0.13 0.25 0.25 - - - 0.25 - - 0.89 - - - 1.52 — • 2.79 - - 2.79 T P = 3.0 kPa — 4.06 I = 1400 A 5.33 V I I I I I I f f * f I V 1 I I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 1 0.64 Position ■ * 0.25 - * - 1.52 2.79 - - 2.79 argon P « 6.0 kPa 1= 1400 A ' I 1 ' ' 2 3 Radial Position (mm) Figure 7.42: Electron tem p eratu re as a function of radius for a pressure of 6.0 kP a and a current level of 1400 A w ith axial position as a param eter. approxim ately exponentially w ith increasing distance from the cathode tip and all of the values tend to fall between 0.55 and 0.70 eV. Electron tem peratures near 1.0 eV are expected near th e cathode for argon discharges but were not observed here [114,131]. It is also expected th a t the maximum tem peratures would be in th e ionization region near the cathode surface and fall as one moved radially and axially outw ard. This trend was observed in all of th e scans. T he tem perature profiles become flatter radially as th e pressure is decreased. T he tem peratures for 4.5 and 6.0 k P a begin to rise again around 4 mm from th e tip. Also, th e electron tem perature decreases w ith increasing pressure, as expected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 2 ~ 0.68 > 0 3 2 0.64 8. 1 0.62 1 0.60 8 53 0.58 M 1 0.56 E 0 u 0.54 0 1 2 3 4 5 6 Axial Distance From Cathode T ip (mm) Figure 7.43: Electron tem perature as a function of axial position from the cathode tip for a current level of 600 A w ith pressure as a param eter. ~ 0.68 % 0.66 3 2 0.64 & 1 0.62 E- c g 0.60 U J U " 0.58 0 c 1 0.56 3 0.54 0 1 2 3 4 5 6 Axial Distance From Cathode Tip (mm) Figure 7.44: Electron tem perature as a function of axial position from the cathode tip for a current level of 1000 A w ith pressure as a param eter. .1 + argon Pressure (kPa) 1 = 1000 A □ 1.5 3.0 + + O 4.5 m A 6 . 0 A □ + C0 O + A O A o ad e o r ' T ' T I I I I 1 I I I I I r I I I I I I ■ T " T T " 1 I I I I I $ + argon Pressure (kPa) + 1 = 600 A + 1 .5 z E o o D a a ° □ . o □ o □ 3.0 O 4.5 A 6.0 °o A A A | I I I " f 'r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 0.66 Pressure (kPa) + 1.5 □ 3.0 O 4.5 A 6.0 £ > 0.64 argon I = 1400 A 8 , 0.62 0.60 0.5 8 - 0 .5 6 - 0.54 0 1 2 3 4 5 6 Axial Distance From Cathode Tip (mm) Figure 7.45: Electron tem perature as a function of axial position from the cathode tip for a current level of 1400 A w ith pressure as a param eter. A ttem pts to m easure the electron tem perature upstream of the cathode tip yielded poor results. T he d ata were found to have poor repeatability and oscillations in the d a ta obscured th e details of the distributions. For scans which included the cathode, th e intensities m ust be doubled for the portion of th e scan that includes th e cathode. T h at is, the recorded image is only half of the actual value because the emission from the far side of the cathode is not visible. This doubling of th e center d a ta further amplified the problems with oscillations because of th e large step th at was created in the im age. The filtering techniques used in the Abel inversion routine sm ooth discontinuities, which results in oscillations. Different types of digital filters Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 -1 > u W R Z 12 8. E H e 8 G o E 1.2 - 1.0 0 .8- 0.6 0 / \F U a r g o n P= 1.5 kPa 1= 1400 A A xial Position (m m ) — 0.13 — -0.13 — - -0.13 ■ — -038 — -0.64 ■ - - -0.89 •— -1.14 Cathode Edge T V V f I V I I I | I I I I | I Radial Position (mm) Figure 7.46: Electron tem perature as a function of radius for a tan k pressure of 1.5 kP a and a current level of 1400 A upstream of the cathode tip w ith axial position as a param eter. were tried , but no im provem ent in the results was observed. Also, removing the filter resulted in poor results for all of the scans. The best of the upstream scans are presented in Fig. (7.46). T he values a t the edge of the cathode w ’hich are shown with m arkers have values near 1.0 eV but are not repeatable. T he large oscillations, however, account for a significant error in the m agnitudes. Because of these errors, the d a ta from the upstream scans was considered unusable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 Pressure C urrent (A) (kPa) 600 1 0 0 0 1400 1.5 3.2166 3.2559 3.3498 3.0 2.8996 3.5330 3.7776 4.5 3.0869 3.2730 3.5505 3.0 3.1621 3.2903 3.6302 Table 7.1: Experim ental effective work function values for T horiated tungsten cath ode operation. 7.6 Calculated R esults T he average effective work functions calculated for each of th e operating cases are presented in Table (7.1). As expected both th e effective work functions and the average tem peratures, Tfud,, are significantly lower for the thoriated tungsten cath odes. T he corresponding attachm ent areas were also much larger for these tests. T he range in effective work functions for the different operating conditions is also much larger due to the effect of the thorium m igration. T he effective work functions increase w ith increasing current sim ilar to the pure tungsten tests. However, the work functions in these tests increase with increasing pressure w ith th e exception 600 A and 3.0 kPa. T he value for the 600 A and 3.0 kP a test is significantly lower th an the other tests due to its lower cathode tem perature profile. T he 488 nm brightnesses are not as consistent for each enclosed current value as they were for th e pure tungsten tests. Again, this is probably due to the variations in work func Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tion and th e corresponding variations in local current density. For these cases the 98 percent enclosed current contour was in the range of about 5-16. T he contour values th a t are not listed were outside of the cam era im age range. T he cathode emission characteristics for each operating condition were calculated from the m easured cathode tem perature profiles by the sam e m ethods described in the previous chapter. T he results for different fractions of the total current are presented in Tables (7.2) through (7.13). Enclosed C urrent A ttachm ent TfUch 488 nm C urrent F!raction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 26.2 157.01 0.6573 2602 77.0 49.2 295.2 1.7622 2546 46.7 74.6 447.5 3.191 2520 28.7 98.0 587.8 6.404 2457 11.5 Table 7.2: Experim ental values for thoriated tungsten cathode operation a t 1.5 kPa and 600 A. Enclosed C urrent A ttachm ent Tkich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 24.5 245.2 2.203 2512 65.0 50.3 503.7 3.721 2541 63.0 '75.1 750.8 5.417 2545 53.8 98.0 980.5 9.167 2506 16.9 Table 7.3: Experim ental values for thoriated tungsten cathode operation a t 1.5 kPa and 1000 A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 Enclosed Current A ttachm ent Tmch 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary U nits) 25.2 353.0 4.494 2524 6 6 . 8 50.0 700.6 6.904 2562 69.0 74.6 1045.1 9.404 2576 97.9 1371.3 13.510 2562 Table 7.4: E xperim ental values for thoriated tungsten cathode operation a t 1.5 kP a and 1400 A. Enclosed C urrent A ttachm ent TRich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary U nits) 24.1 144.37 0.2964 2485 98.0 50.6 303.7 0.5420 2507 76.0 74.1 494.4 1.1712 2445 40.5 98.0 588.1 4.386 2293 7.85 Table 7.5: Experim ental values for thoriated tungsten cathode operation a t 3.0 kP a and 600 A. Enclosed C urrent A ttachm ent TtUeh 488 nm C urrent Fraction Area Emissivity (percent) (A) (cm2) (K) (A rbitrary U nits) 24.6 245.9 1.1609 2805 75.2 50.7 507.2 2.262 2814 59.7 74.8 747.9 3.691 2797 52.0 98.0 978.0 7.083 2735 2 2 . 0 Table 7.6: Experim ental values for thoriated tungsten cathode operation a t 3.0 kPa and 1000 A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 Enclosed Current A ttachm ent Twch 488 nm C urrent Fraction Area Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 24.1 337.8 1.4092 2997 74.9 50.5 707.5 2.659 3016 70.6 75.7 1059.2 4.087 3011 64.8 98.0 1371.7 10.783 2887 Table 7.7: Experim ental values for thoriated tungsten cathode operation a t 3.0 kP a and 1400 A. Enclosed C urrent A ttachm ent Tmdi 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 25.5 152.96 0.2969 2634 114.0 47.9 287.3 0.7053 2594 97.0 75.6 453.2 1.4586 2551 37.4 98.0 588.3 3.780 2445 8.29 Table 7.8: Experim ental values for thoriated tungsten cathode operation a t 4.5 kP a and 600 A. Enclosed Current A ttachm ent Thich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 24.1 241.0 0.6657 2711 87.0 48.7 487.2 1.4693 2696 48.2 75.1 750.8 2.469 2681 38.6 98.0 980.3 5-950 2583 7.60 Table 7.9: Experim ental values for thoriated tungsten cathode operation a t 4.5 kP a and 1000 A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 Enclosed C urrent A ttachm ent Tkich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm 2) (K) (A rbitrary Units) 25.5 356.4 1.1607 2881 59.5 49.4 691.2 2.250 2881 54.7 75.0 1050.4 3.589 2873 42.6 98.0 1371.6 9.659 2751 Table 7.10: Experim ental values for thoriated tungsten cathode operation a t 4.5 kP a and 1400 A. Enclosed C urrent A ttachm ent Tkich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 27.4 164.54 0.19436 2777 50.0 50.5 303.0 0.4496 2736 58.1 75.6 453.9 1.2719 2628 16.2 98.0 588.1 6.378 2421 4.83 Table 7.11: Experim ental values for thoriated tungsten cathode operation a t 6.0 kP a and 600 A. Enclosed C urrent A ttachm ent Thich 488 nm C urrent Fraction A rea Emissivity (percent) (A) (cm2) (K) (A rbitrary Units) 22.5 225.4 0.2385 2896 70.0 50.8 508.4 0.5551 2890 70.1 74.0 740.1 1.1788 2820 28.6 98.0 980.1 3.078 2701 7.27 Table 7.12: Experim ental values for thoriated tungsten cathode operation at 6.0 kPa and 1000 A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1 0 Enclosed C urrent A ttachm ent Tkich 488 nm C urrent Fraction Area Emissivity (percent) (A) (cm2) (K) (A rbitrary U nits) 24.0 . 335.5 0.6703 3028 55.9 50.4 705.5 1.6078 3004 26.7 74.4 1041.0 2.501 2994 22.4 98.0 1372.1 7.768 2842 4.76 Tabic 7.13: Experim ental values for thoriated tungsten cathode operation at 6.0 kPa and 1400 A. T h e relationship between th e 488 nm Ar II emissivity brightness and the local current density for the thoriated tungsten cathode test is shown in Fig. (7.47). Also shown in this figure is the curve fit from the pure tungsten tests for comparison. Most of th e d a ta from the thoriated tungsten test fall above this fit indicating th at these brightness values are significantly larger than those from the pure tungsten tests a t a given current density. Since this relationship is for current density and not tem perature, the expected work function effects should be m inor. A b etter correlation is seen between the 6.0 kP a tests and the pure tungsten fit results as seen in Fig. (7.48). T he relationship between the 488 nm emission brightness and the local current density determ ined from the pure tungsten d ata was used in an a tte m p t to estim ate the work function distribution on the thoriated tungsten cathodes. T he curve fit, Eq. (6.4), from the d a ta displayed in Fig. (6.24) and the 488 nm brightness data were used to estim ate the local current density. This current density distribution and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1 1 1= 1000 A Run Pressure No. (kPa) 1= 1400 A Run Pressure No. (kPa) i ' > i/i I U 102 3.0 99 4.5 - 101 6.0 Pure Tungsten Data Fit 00 00 1 0 ' i 0.01 0.1 1 10 100 1000 Current Density (A / cm 2 ) Figure 7.47: Brightness of the 488 nm Argon II line emissivity as a function of current density for the thoriated tungsten tests. the m easured cathode tem perature profile were used to calculate th e local effective work function. The results for the two extrem e pressure cases, 1.5 and 6.0 kPa are shown in Figs. (7.49) and (7.50). T he values for the 1.5 kP a cases are much lower th an would be expected and the values from the 6.0 kP a tests show a wide range in values. To check the validity of this approach, the total current for each case was calculated using the m easured tem perature profiles and areas, and the estim ated work functions. For the 6.0 kPa cases the calculated currents for the 600, 1000, and 1400 A cases were 801, 1082, and 1459 A, respectively. T his is excellent agreem ent considering th e roughness of the estim ate and th e sensitivity of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1 2 Run Current No. (A) 8 8 600 100 1000 101 1400 100- Pure Tungsten data fit B ID Urn < 10- oo oo T 0.1 10 Current Density (A / cm 1 ) 1 ,100 1000 Figure 7.48: Brightness of the 488 nm Argon II line em issivity as a function of current density for th e thoriated tungsten tests at 6.0 kPa. 3.2 Run Current No. (A) 94 600 91 1000 95 1400 ft 3.0 ■ o 2.8 K k ■ * * » W 2 .7 : 2.6 -25 -30 -20 -15 Distance From Cathode Tip (mm) -10 Figure 7.49: Calculated work function for the 1.5 kP a tests. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 4.0 Run Current No. (A) ■ 88 600 100 1000 1 0 1 1400 a r g o n P = 6.0 kPa o 3.8 3.6 3.4- S 3.2- 3.0- 2.8 -30 -25 -20 -10 5 0 -15 Distance From Cathode Tip (mm) Figure 7.50: C alculated work function for the 6.0 kP a tests. th e current to errors in th e work function. T he results for the 1.5 k P a case were quite poor. T he calculated currents for the 600, 1000, and 1400 A cases were 1422, 4602, and 5136 A , respectively. The results from the 3.0 and 4.5 kP a cases were sim ilar to those of the 1.5 k P a cases. In all of the cases except the 6.0 kP a cases, the calculated currents using this technique were a factor of 2 to 6 too large, indicating th a t this technique is not appropriate for these cases. W hether or not the good agreem ent from the 6.0 kP a cases are simply fortuitous is not known. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 7.7 Cathode Tip Pressure M easurem ents T he near-cathode plasm a models have shown th a t the surface heat flux is quite sensitive to the gas pressure near the surface, which may be different from the tank pressure. The interaction of the arc current w ith the induced m agnetic field will produce a radial pressure gradient as the plasm a is “squeezed” tow ards the centerline. Assuming a uniform current density given by / (0(/jrr^ , w here I tot is the total current, and integrating this force over an arc column of radius re yields a pressure difference between th e am bient pressure and the centerline pressure of Pamb ~ Ptip = ~ g j.2 ^ 2 where fi0 is the perm eability of vacuum. Estim ates based on this relationship using the current levels and arc colum n radii from the experim ents suggest th a t a pressure increase of several hundred Pascals over the am bient pressure should be observed on the cathode tip. Such overpressures have been experim entally confirmed in a self-field device under sim ilar operating conditions [41]. Experim ents performed over a current range of 600-1400 A and a pressure range of 1.5 to 6.0 kP a produced th e opposite result. A cathode containing two pressure taps, as shown in Fig. (7.51), was used for these tests. T he pressure ta p holes were approxim ately 1 mm in diam eter. As shown in Figs. (7.52) through (7.59), the pressures measured w ith the pressure taps on the side and centerline of the cathode were substantially lower th an th e m easured tank pressure for all of the tests. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 97 N om inal D im e n sio n s in m m Figure 7.51: Schem atic diagram of the cathode containing the pressure taps. A num ber of experim ents were performed to test the validity of the pressure tap m easurem ents. Because the sam e pressure transducer was used to m easure both th e tank pressure and the cathode pressure taps, sensor effects such as electrical noise from th e arc can be ruled out. A return to the low pressure tap readings after pressurizing and isolating th e pressure tap lines showed th a t the pressure tap s in th e cathode were not blocked during arc operation. Also, the pressure m easured through th e pressure tap s recovered to the tank pressure level when the arc was extinguished and again dropped to th e low value again when the arc restarted. The d a ta taken during a 3 hour run at 1000 A is shown in Fig. (7.52). Following th e test the centerline pressure tap had a reduced diam eter of about 0.35 m m while Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the side pressure tap was unaffected. Even with the substantial reduction in area, the centerline pressure tap was still working correctly. T h at is, th e m easured pressure returned to the tan k pressure when th e arc was extinguished. T he pressure rise, however, took much longer due to the reduced area. This figure shows the transients for each m easurem ent as the pressure sensor was switched between the different pressure taps. T he tank pressure quickly reached its equilibrium value because it had a much larger tube diam eter. In the following plots, only the equilibrium pressure d a ta are presented for simplicity. The tests were repeatable as seen by the comparison between these d a ta and d a ta taken in the short duration test shown in Fig. (7.53). T he short test was th e first test of the series and the long test was the last one. T here is only a slight difference in the side pressure tap m easurem ent th at may be due to changes in the arc attachm ent resulting from thorium m igration during the tests. Tests were also performed a t 6.0 k P a but asym m etry in th e arc attachm ent m ade the d ata unusable. As the test progressed, the arc moved tow ards the side pressure tap which was initially outside of the visible arc. During this transition the pressure drop between the side tap and the tan k increased from zero to 0.655 kPa. During the same period the tip pressure drop decreased from 1.104 to 0.994 kP a as the arc began to uncover this tap. This trend was common to all of the tests. T h a t is, the pressure drop increased as the arc attachm ent moved tow ards the cathode base. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 a 0.8 3 p 0.6 0.4 (ip 0 .2 : -Argon 1 = 1000 A 0.0 100 120 140 160 2 0 40 Elapsed Time (min) Figure 7.52: Pressure m easurem ents during long duration test a t a current level of 1000 A w ith th e pressure tap cathode. W hen the side tap was outside of the visible arc, its pressure was the sam e as the tan k pressure, indicating th at there is a negligible change in th e pressure due to the expanding gas a t the cathode base. T his result was expected since th e gas velocities in this region are much smaller than in the hot, accelerating region. Following this test a dark spot was seen approxim ately midway between th e pressure taps. T his dark spot is a result of thorium and tungsten migration and was observed for all of the 6.0 kP a tests. T he pressure difference between th e tip pressure ta p and the tank changed slightly w ith increasing current, as shown in Figs. (7.54) through (7.56). T he scat- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 1.6 1.4 1.2 I 1 0 £ > 0.8 1 0.6 C u 0.4 0.2 0.0 0 2 0 4 0 60 80 100 120 140 160 Elapsed Time (min) Figure 7.53: Pressure m easurem ents for operation at 1000 A and 1.5 kP a for short and long durations tests. ter in the d a ta a t each point is the variation in each value with tim e. The values range from 1.06-1.16, 1.25-1.63, and 2.30-2.90 kP a for tank pressures of 1.5, 3.0 and 4.5 kPa, respectively. The pressure drops for the side ta p were 0.887-0.650, 0.123-0.777, and 0.160-0.967 kPa for 1.5, 3.0 and 4.5 kPa, respectively. A series of previous tests in this study, using a cathode w ith only a centerline pressure tap, produced sim ilar results [122]. The values for each tap are consistent for each sam pling although a slow drift can be seen in some of the m easurem ents, particularly for the larger current tests. These drifts are probably a result of the arc attachm ent area changing as a result of the work function changes discussed previously. gfgon Long Test Short Test I = 1 0 0 0 A A P u B k A P u n k P = 1.5 kPa D p* “ * ■ p* > « « O P i ip ® P tfp I I I I I I I I I I 1 1 I 1 I I I r p*TT Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 219 2.0 argon P = lJ k P a A P u n k D P«ide /•S C 3 52 g 1.0. I t/i , a C u 0.5- 0.0 600 800 1000 1200 1400 Current (A) Figure 7.54: Pressure m easurem ents for operation a t 1.5 kP a w ith current as a param eter. T h e effect of current on the pressure m easurem ents are shown in Figs. (7.57) through (7.59). T he pressure measured a t th e side tap increases linearly w ith in creasing pressure, while the pressure a t the tip appears to exhibit asym ptotic be havior. This effect is m ost clearly seen in Fig. (7.59). It is not clear why these tests consistently produce an under-pressure at the cathode while m easurem ents with thrusters produce an over-pressure consistent with electrom agnetic pinching [41]. The main difference between the two experim ents is the anode/nozzle diam eter. In the th ruster experim ents th e flow is constrained into a 1 .2 cm diam eter nozzle while in the experim ents here the tank wall inner diam eter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 0 4 3 2 1 A P u n k O P iid c 0 600 800 1000 1200 1400 Current (A) Figure 7.55: Pressure m easurem ents for operation a t 3.0 kP a with current as a param eter. is 49.5 cm and th e anode inner diam eter is 7.62 cm. T he maximum diam eter of the visible arc in these tests was approxim ately th e sam e as the anode inner diam eter. Therefore in these experim ents the gas is essentially unconstrained. As th e gas passes into th e arc its tem perature should rise from abo u t 1000-1500 K corresponding to the cathode base tem perature where the gas is injected to about 10,000-15,000 K in the arc. This rapid heating of the gas will cause th e gas to expand perhaps creating the m easured low-pressure conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 1 6 5 4 3 2 1 A Punk □ Pfidc 0 600 800 1000 1200 1400 Current (A) Figure 7.56: Pressure m easurem ents for operation at 4.5 kP a with current as a param eter. 7.8 Surface M icrostructure and Chemical State The surface finish and chemical state of the cathode are im portant because they de term ine to a large extent the therm al radiation and electron emission properties of the surface1. In the models the radiant heat flux varies directly with the em ittance and the current density depends exponentially on the work function. T horium oxide is added to tungsten during the sintering process, and a t high operating tem pera tures is reduced by the tungsten, forming thorium m etal which diffuses to th e surface 'M ajor portions of this section were taken directly from Reis. [124] and [123], and were provided by Jay Polk as part of the JPL CTF project. Used here with permission Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 2 5 argon 1 = 600 A g A P u n k Q P«id« 3 - O P ^ & 0 8 i 8 0 T TX'r 'T'i 1 f | T ,•, i , i,m ,i 1 2 3 Tank Pressure (kPa) 4 5 Figure 7.57: Pressure m easurem ents for operation a t 600 A w ith tank pressure as a param eter. and form s an adsorbate layer. T he electric dipole layer form ed by this electropositive atom on tungsten lowers the cathode work function, facilitating the escape of elec trons from the surface and allowing a lower operating tem perature. The variation in the work function with thorium coverage is shown in Fig. (7.60) [136]. T he coverage / is defined by <r/<T 0, where a is the surface density of adsorbed thorium atom s and < r„ represents the num ber density a t th e minimum work function. At low coverage the work function approaches th a t of pure tungsten, about 4.5 eV, while a t high coverage the work function is close to th a t of bulk thorium m etal, about 3.27 eV. At high tem peratures the thorium m etal evaporates from the surface. The equilibrium Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 223 argon I = 1000 A C Q 22 a 2 X /i £ 1 2 3 4 5 Tank Pressure (kPa) Figure 7.58: Pressure m easurem ents for operation at 1000 A w ith tank pressure as a param eter. surface coverage, which determ ines the lowering of the work function, is dependent on the relative rates of supply by diffusion from the cathode interior and loss by evaporation and mass tran sp o rt through the surrounding plasm a [122]. Because of th e extrem e model sensitivity to this param eter, it is essential to characterize the extent of thorium coverage on the cathode surface. In this stu d y six separate exam inations of cathodes using energy-dispersive spec troscopy (ED S) in an electron microscope were perform ed. T h e first cathode had been operated for a total of approxim ately 2 0 hrs and the final run had lasted about 1 hr. T he sam e cathode was then polished and exam ined, and then tested for about Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 5 1400 A 4 Punk P«ide 3 2 1 0 1 2 3 5 4 Tank Pressure (kPa) Figure 7.59: Pressure m easurem ents for operation at 1400 A w ith tan k pressure as a param eter. 4.5 Work Function Richardson Coefficient | 4 .0 - I £ * 3 .5- ! , C S JZ U S 60 cl 4 0 n 2.5 0.0 0.5 1.0 1.5 2.0 Surface Coverage, f Figure 7.60: Variation of work function w ith thorium coverage on tungsten. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 12 s and studied again. T he pressure tap cathode, with a cum ulative operating tim e of 13.3 hrs was examined after a final run of 6.5 min. T hen the cathode with the cavities was exam ined after a final runtim e of 5 hrs, 13 min and a total operating tim e of 8 hours. Finally, a new cathode th a t accum ulated m ore th an 80 hours of operation a t 1000 A and was last operated a t 6 kPA and 1000 A was exam ined. T he exam ination of the polished cathode showed a pure tungsten surface scored by sandpaper with no appreciable thorium coverage. After the 12 s run the surface m icrostructure was dom inated by th e surface cratering which occurs during the sta rt phase. On start-u p , th e cathode surface is too cold to su p p o rt therm ionic emission, so current continuity is m aintained by a num ber of highly m obile hot spots. A lthough th e bulk tem perature is low, th e local tem perature in these emission sites is well above the boiling point of the m etal, so vigorous vaporization and m elting occurs. T his extrem ely destructive process forms small craters w ith dimensions ranging from less than a micron to several hundred microns. T he emission sites heat th e cathode, and after several seconds the bulk cathode tem p eratu re becomes sufficiently high and the attachm ent transitions to the therm ionic m ode discussed above. However, in the process the original surface becomes obliterated by the cratering from thousands of tiny emission sites. T he crater size increases with bulk cathode tem perature, so th e largest craters are formed in th e highest tem perature region ju s t before transition to the therm ionic m ode. The SEM exam ination revealed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 Figure 7.61: Photom icrograph of thorium deposits on floor of crater found after 12 s of operation. th a t a num ber of these large craters had deposits of thorium in them . Figure (7.61) is a photom icrograph showing a representative crater w ith a diam eter of ab o u t 1 0 0 microns found on the tip of this cathode. T he lighter sm ooth regions and globules are thorium m etal. T he cratering process apparently serves to excavate deposits of thorium from the cathode bulk. T he cathode tested for 6 m inutes was sim ilar in appearance to th e cathode tested for 12 s. T he m icrostructure on the shaft of th e cathode th a t had been operated for about 1 h r was also dom inated by the initial cratering process; however, significant surface restructuring and redistribution of thorium had occurred on th e tip . A large Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 Figure 7.62: Photom icrograph of sm ooth tungsten crystals w ith deposits of thorium found after 1 hour of operation. circular region on th e tip was composed of a porous tungsten structure th a t looked like a coral reef. T his was surrounded by a ring composed of tiny rectangular crystals of pure tungsten, as shown in Fig. (7.62). M any of the large craters were filled to the rim w ith porous accum ulations of these crystals, as shown in Fig. (7.63). These structures are probably formed by recrystallization of tungsten at high operating tem peratures or by vapor deposition of tungsten. T he m ost astonishing feature of this cathode, however, was the enorm ous concentration of thorium on th e tip. The rough, lighter m aterial smeared am ongst the crystalline structures in Fig. (7.62) is thorium m etal. Such deposits were found all over the tip , dem onstrating clearly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 Figure 7.63: Photom icrograph of craters clogged w ith tungsten crystals after 1 hour of operation. th a t some process, perhaps ionized thorium being drawn to th e surface by the arc, results in the concentration of thorium a t the cathode tip. Exam ination of th e shaft of the cathode th a t had been run for over five hours revealed thorium deposits on the floor of large craters about 15 m m from the tip. T he shaft dow nstream of this point appeared to be depleted of thorium . Substantial restructuring of the tip had also occurred on this cathode, with th e characteristic tungsten crystals, coral-like structures and clogged craters. However, on the extrem e tip was an extraordinary new feature. A circular region approxim ately 1.6 mm in diam eter was com posed of structures th a t looked like fern leaves, as shown in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 Figure 7.64: Photom icrograph of tungsten and thorium structures found on the cathode tip after 5 hours of operation. Fig. (7.64). T he stem and ribs of the structures are pure tungsten, and th e m aterial th a t appears to have flowed between them is pure thorium . A magnified image of one of the leaves is shown in Fig. (7.65). EDS analysis reveals th a t thorium metal has accum ulated between each of the ribs shown in this photom icrograph. In other areas on the tip the surface was composed of scalloped or wave-like structures as shown in Fig. (7.66). Each of these pure tungsten depressions is filled with a tiny lake of thorium . T he cathode tip tem perature exceeds the m elting tem perature of thorium , so the cathode tip m ust operate with pools of molten thorium m etal which collect in the depressions between the tungsten structures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 Figure 7.65: Photom icrograph of the fern-like structures. Figure 7.66: Photom icrograph of tiny tungsten depressions filled w ith thorium m etal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 Figure 7.67: Photom icrograph of cathode tip after two hours of operation a t 1000 A and 6 kPa. A photom icrograph of the cathode tip after operation a t 6 kPA and 1000 A is shown in Fig. (7.67). T he different gray bands are a result of different thorium coverage fractions and therefore different work functions. These characteristics are discussed in m ore detail below. T he changes in work function can significantly affect the current density values [101]. Interestingly, the inner edge of the high 488 nm argon II brightness contours corresponds to the edge of th e w hite area in the figure. This cathode had been run for a total of 80 hours a t a current level of 1000 A , pres sures ranging from 1.5-6.0 kPa and argon flow rates of 0.060-0.878 g /s. T h e final two hour run was a t a current level of 1000 A, a pressure of 6.0 kP a and a flow rate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of 0.276 g /s. O ver the course of these tests a shiny protuberance with a diam eter of about 2 m m and a height of approxim ately 1 / 2 mm developed on the tip of the cathode and a dark band formed around this region. T his photograph shows th a t the tip region can be divided into four zones; the bright w hite protuberance, the patchy w hite ring surrounding it, th e dark band encircling the white areas and the lighter gray region on the outside. T his photograph was taken using backscattered electrons, so th e response is sensitive to the elemental com position of the surface. EDS analysis confirmed th a t the bright white regions are enriched in thorium while the dark areas are essentially pure tungsten. In the central raised area th e thorium response was alm ost as strong as th e tungsten response, indicating nearly equal pro portions. H igher m agnification revealed fern-like structures sim ilar to those found in earlier experim ents on the tip of a cathode operated at lower pressures [123]. Again, the ribs of th e structures were found to be pure tungsten, while th e surrounding areas were pure thorium m etal. T h e patchy region surrounding this was composed of small tungsten crystals and pools of thorium m etal. T he dark band consisted of the tungsten crystals with no significant traces of thorium , while the outerm ost region was pure tungsten w ith a m uch sm oother texture. T he dendritic structures found in the central region are consistent w ith the existence of a th in m olten layer of thorium and tungsten on th e tip which solidified at shutdow n. T he feathery or fern-like structures are formed when the two immiscible Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. com ponents segregate upon cooling. A lthough the measured tip tem perature was only on the order of 2750 K, th e phase diagram for this system shows th a t thorium reduces the freezing point of tungsten to a minimum of 1968 K a t the eutectic point [137]. T he tip of the cathode exists in a molten state only because of the high concentration of thorium m etal there. T he desorption rate of thorium from tungsten a t the m easured tip tem perature greatly exceeds the diffusion ra te from the interior, and im probably low gas diffusion rates are required to m aintain significant surface coverage [76]. However, the surface structures observed in this zone strongly suggest th a t thorium vapor condenses on th e tip during operation, providing further evidence for a mass tran sp o rt process first proposed in [123]. A t these am bient gas pressures th e arc attachm ent is concentrated on the tip, so an arc column w ith a high electron density and tem perature form s downstream of this region. Evidently m etal vapor from th e cathode is ionized in the arc column and drawn back to the surface by th e electric field. T he sheath in these devices is collisionless, so ionization m ust occur outside th e sheath. T he weak electric field in the column is apparently sufficient to slow th e gas diffusion rate significantly. This also provides a m echanism for mass tran sp o rt from other regions of th e cathode to the region where the arc attaches. Paradoxically, the volatile species accum ulate in the hottest zone. The effect is to delay depletion of the thorium a t the tip, because it lowers the effective gas diffusion rate and allows access to a much larger supply of thorium . T he thorium Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 which can accum ulate on the tip is not lim ited to th a t available locally; thorium th a t evaporates from any p a rt of the cathode can conceivably be deposited on the tip if it is first convected in to the arc column dow nstream of the tip. T he tungsten crystals in the dark band were probably formed in a sim ilar pro cess. Tungsten evaporated from the cathode is ionized in the arc column and drawn back to th e surface, creating small crystals by vapor deposition. T he patchy region between the thorium -rich central area and the ring of tungsten crystals represents a transition zone in which tungsten and thorium vapor are deposited on the sur face, b u t the concentration of thorium is not sufficiently high to lower the m elting tem perature below the tip tem perature. The thorium pools appeared to have been m olten during operation, b u t the tungsten crystals in this region clearly were not. Because there is a surface concentration gradient, thorium may also be supplied to the transition zone from th e central region by surface diffusion. T he outerm ost region is evidently a high-tem perature zone with little vapor deposition of tungsten or thorium . The craters formed during startu p have been sm oothed ou t by therm al recrystallization over the course of the final two hour run. These observations are consistent with cathode surface structures observed at lower pressures and higher discharge currents [51], suggesting th a t this is represen tative of cathode operation. T he surface analysis indicates th a t th e entire range of work functions previously discussed exists on the cathode during operation. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 central region where thorium m etal accum ulates is probably characterized by a work function close to th at of bulk thorium , while the outerm ost region of pure tungsten would have the highest work function. The interm ediate zone represents a transi tion betw een these two extrem es. T he minimum work function and th e peak current density therefore occur not on th e cathode centerline, b u t a t an interm ediate radius on the hemispherical tip. As shown above, this conclusion is consistent w ith m ea surem ents of th e argon ion line intensity distribution on th e tip. T horium can apparently be supplied to the surface from a thin layer by diffusion along grain boundaries or pores [76], and a t th e tem peratures observed in these experim ents th e desorption ra te of thorium greatly exceeds the diffusion rate from the interior. T he discovery of large quantities of thorium on the hottest p art of th e cathode proves the existence of an additional mass transport step which lim its the rate of thorium loss. Zimin has proposed th a t thorium is ionized in the arc and draw n back to the surface by the electric field near the cathode [138]. This recycling of thorium m etal could explain the observed results. T he cold cathode cratering process exposes new deposits of thorium during startu p . D uring steady state operation the thorium evaporates rapidly in th e tip and further upstream on the shaft. T he thorium vapor diifuses into the intense discharge zone near the cathode tip and is ionized and drawn back to th e surface. T he supply rate by this mechanism is evidently sufficiently high compared to th e evaporation rate to result Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 236 in bulk condensation on the tip. T his is a powerful mechanism for the transport of thorium from the cooler p arts of th e shaft to the hottest region, and it relies completely on th e existence of th e arc to function. T his process results in the depletion of th e thorium supply on th e shaft, so the work function in this region will increase as th e coverage decreases. T he tip work function m ay also increase initially as the coverage passes through th e value / = 1 and forms bulk thorium lakes on th e surface. T h e electron emission on th e tip may be dom inated by emission from the bulk thorium sites or from thoriated tungsten surfaces between the bulk deposits, so it is difficult to conclude w hat th e work function would be. This physical picture may explain th e therm al transient behavior observed in th e experim ents. The initial increase in tip tem perature m ay be associated w ith the buildup of thorium , after which th e effective work function becomes stable a t some value between th e m inimum and th a t of bulk thorium . T he tip tem perature then becomes stable. However, depletion of thorium still occurs along the shaft, requiring increasing tem peratures to m aintain the emission com ponent from this part of the attachm ent. T his process obviously cannot continue indefinitely. G radual loss of thorium , even rate-lim ited by th e gas tran sp o rt processes, will eventually deplete the resources uncovered in the cratering process or available from thin surface layers. T he m icrostructural changes th a t occur during operation may also affect the radiant heat flux and the tem perature m easurem ents. Surface roughening by for- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m ation of the porous structures can increase the surface em ittance, while subsequent therm al polishing can cause it to decrease. Further characterization of long-term m aterial behavior is required to resolve this issue. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 238 C hapter 8 C om parison o f M od el w ith E xp erim en ts Model com parisons are presented for two different operating m odes. F irst, com par isons are m ade for high-pressure operation w ith pressures around 100 kPa. Existing d a ta for arcjet th ru sters are compared w ith model predictions. Second, comparisons are m ade between th e low-pressure discharges discussed in the previous two chapters w ith both th e quasi-two-dimensional and th e two-dimensional m odels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 239 8.1 Arcjet Thruster Comparisons and M odel Predic tions The plasm a model and th e quasi-two-dimensional therm al model are combined to form th e overall solution. Solutions are found where the heat flux curves for the two solutions intersect. In general there m ay be four possible solutions, namely, the trivial solution, two low -tem perature solutions, and a high-tem perature solution (typically for a fully ionized plasm a as w ith th e previous models). T h e second low- tem perature point is a result of adding th e two-dimensional tip approxim ation to the therm al model described previously. T h a t is, the therm al m odel solution for constant current case has m ore curvature th an th e constant attachm ent area case, enabling an additional intersection point. T his effect can be seen in Fig. (8.1). The two intercept points shown are the two possible low -tem perature points but only the lower tem perature point on the left is stable. T h a t is, tem perature perturbations from this point will restore the solution while for the higher tem p eratu re point a perturbation will cause th e solution to move away from the point. Therefore the addition of the new approxim ation to the therm al model has enabled the overall model to have a stable solution on the partially ionized portion of th e curve. The solution point near full ionization still grossly over-predicts the cathode tem perature and th e current density. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 am m onia P = 85 kPa I = 100 A 4 > = 4.5eV V c= 11.8 V N I £ JX 10 - K 3 E 3 X Plasma model Therm al m odel 3300 3400 3500 3600 3700 3800 3900 4000 Cathode Temperature (K) Figure 8.1: H eat flux as a function of cathode tem perature for a Vc value of 11.8 V. Com parisons between th e model and 10 kW and 25 kVV am m onia arcjet long duration te st d a ta w ith thoriated tungsten cathodes [72,70] are presented. A plot of th e sheath voltage as a function of cathode tem perature for both cases is shown in Fig. (8.2) and predicted electron tem peratures are shown in Fig. (8.3). T here are two possible solutions for a given sheath voltage corresponding to the two intercepts in Fig. (8.1). T he peak value occurs when only one intercept point exists and it is m arginally stable. T he stable points are to the left of the m axim um point. T he slight step in the curve left of the m axim um point occurs a t the point where th e attach m en t area equals the tip area, and is a result of lim itations of the two- dimensional tip approxim ation surface fit a t this point. For cases left of this point, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 241 18- 16- — 10 kW, P = 85 kPa, I = 100 A, e = 0.5 *— 10 kW, P = 85 kPa, I = 100 A, e = 0.57 — • 25 kW, P = 157 kPa. 1 = 215 A .e = 0.5 — 25 kW. P = 157 kPa. 1 = 215 A .e = 0.57 — * - 25 kW. P = 157 kPa, I = 215 A. c = 0.57. water-cooled am m onia $ = 4.5 eV 3500 3600 37 0 0 3800 Cathode Temperature (K) 4000 Figure 8.2: Sheath voltage as a function of cathode tem perature for 10 kW and 25 kW am m onia arcjets. 10 kW , P = 85 kPa, I =100 A. t = 10 kW . P = 85 kPa, I =100 A. e = 25 kW , P = 157 kPa, I = 215 A.e — 25 kW , P = 157 kPa, I = 215 A.e --25kW,P= 157 kPa, I = 215 A. e w ater-cooled = 0.57 = 0.57, am m onia $ = 4.5 eV 1 ' I 3500 I I I I | I I 1 3600 3700 3800 Cathode Temperature (K) 4000 Figure 8.3: Electron tem perature as a function of cathode tem perature for 10 kW and 25 kW am m onia arcjets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 the attachm ent area will be larger than the tip area. Recall th a t the model assumes for these cases th a t an “enlarged” tip area equal to the attachm ent area exists a t a uniform cathode tem perature, Tc. The attachm ent area as a function of cathode tem perature is shown in Fig. (8.4) for the 10 kW case and in Fig. (8.5) for the 25 kW case. T he horizontal lines represent physical areas for each cathode tip. For the 10 kW case, the lower line represents the size of the oval spot shown in Fig. (1.10) [70]. M aterials analysis of this cathode after testing showed th a t it was m olten prior to shut-down [70] so th a t th e actual operating tem perature would be around 3660 K. A surface work function of 4.5 eV for pure tungsten is used in the model since analysis of the molten spot revealed no traces of thorium . T he model predicts a cathode tem perature of 3670 K for an attachm ent area equal to th a t of the m olten spot. Although the model appears to agree well w ith the experim ental d ata, this point falls on the unstable side of the voltage curve. T he closest stable point would be near the peak voltage point. T he tem perature of the peak voltage point or m axim um stable point is about 3585 K for the 10 kW case and about 3790 K for the 25 kW case. For the 25 kW case it is less obvious from the experim ental d ata w hat size the attachm ent area was a t shut-dow n. A cross section of the cathode tip after testing is shown in Fig. (1.11) [71,72]. T he severe erosion for the 25 kW case indicates th a t a significant portion of the tip was m olten. T he horizontal line in Fig. (8.5) represent the areas Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 243 l o . i 3300 3400 3500 3600 3700 3800 3900 4 0 0 0 Cathode U p Temperature (K) Figure 8.4: A ttachm ent area as a function of cathode tem perature for a 10 kW am m onia arcjet. associated w ith th e entire hem ispherical surface ( 2 rri2c3) and the cross-sectional area a t the opening (trRc2) where is the crater radius. T h e actual attachm ent area probably is between these two extrem es. For this case th e m olten tem perature point falls on the stable side of the curve and with an area ju st larger than the cross-sectional area suggesting th a t the model agrees well w ith the d ata. Exam ples of the sensitivity of the model to the cathode therm al radiation pa ram eters are also shown in Figs. (8.2) through (8.5). Tw o values of th e surface emissivity representing the expected extrem es are considered for each power level. A rough surface is characterized by a value of 0.57 while th e value of 0.40 is more M odel attachm ent area, e = 050 M odel attachm ent area, e = 057 Total tip area M olten spot area ammonia 10 kW I = 100 A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 1 6 ^ . 1 <0 a < o .o i 3300 3400 3500 3600 3700 3800 39 0 0 40 0 0 Cathode Temperature (K) Figure 8.5: A ttachm ent area as a function of cathode tem p eratu re for a 25 kW am m onia arcjet. typical for a polished surface. In addition, a water-cooled th ru ste r configuration is considered for the 25 kW case. T h a t is, an environmental tem p eratu re of 300 K is used instead of 2000 K. For the 25 kW case the shift of the em issivity from 0.50 to 0.57 shifts the cathode tip tem perature for the intercept of th e model area with th e cross-sectional area from 3668.6 K to 3666.6 K. The corresponding shifts in the sheath voltage and th e electron tem perature are 9.37 to 10.0 volts and 0.732 to 0.735 eV, respectively. For the water-cooled case the tip tem p eratu re drops from 3666.6 K to 3659.4 K w ith corresponding shifts of 10.0 to 9.5 volts in sheath voltage and 0.735 to 0.734 eV in electron tem perature. T he shifts for th e 10 kW case are am m onia $ = 4.5 eV 25 kW P = 157 kPa 1 = 215 A ---- M odel attachm ent area ---- Spherical area -----C ross-sectional area -----25 kW , P = 157 kPa, I = 215 A, e = 0.57 -----25 kW . P = 157 kPa, 1 = 215 A. e=0.57. w ater-cooled I I J I I I I | V I I V | I I V I | I I I I I V II Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 245 sim ilarly small. These sm all changes indicate the tip operating tem p eratu re is in sensitive to the cathode therm al conditions. Therefore, actively cooling the cathode base will have very little effect on th e cathode tem perature and therefore little effect on th e cathode erosion. T h e effect of different propellant types is shown in Figs. ( 8 .6 ) through (8 .8 ) for operation at 1 0 0 A. W ith the exception of helium, there is little change in the sheath voltage for the various gas types. As expected, the results for am m onia and hydrazine are sim ilar to those for hydrogen. T h e solution values corresponding to an attachm ent area of 0.15 cm2, the estim ated area value from th e 10 kW am m onia test, are given in Table (8.1). All of these points fall on th e stable aide (left of m axim um ) of the sheath voltage curve. For this area argon produces the lowest cathode tem perature and helium the highest. An additional model of th e arc column is necessary to relate the attachm ent area and th e sheath voltage to determ ine the specific operating conditions. Gas Type H He N Ar N1I3 N2 H4 r c (K ) 3488 3528 3457 3447 3483 3482 v e 12.38 13.69 12.33 12.37 12.36 12.37 T ' (eV) 0.694 1 . 2 1 0 0.780 0.838 0.707 0.713 Table 8.1: Model solutions for arcjet th ru ster configuration w ith different propel lants. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 246 14 13 12 11 He 1 0 4 0 0 0 3800 3200 3400 3600 Cathode Temperature (K) Figure 8 .6 : Sheath voltage as a function of cathode tem perature for 100 A arcjets. 1.4 Ar 1.2 E 0.8 0.6 3600 3800 4 0 0 0 3400 3000 3200 Cathode Temperature (K) Figure 8.7: Electron tem perature as a function of cathode tem p eratu re for 100 A arcjets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 247 H He I = 100 A P = 100 kPa $ = 4.5 eV NHj —-N,H« — — Tip area V! 0.1 'v? 0.01 3800 4000 3400 3600 3000 3200 Cathode Temperature (K) Figure 8 .8 : A ttachm ent area as a function of cathode tem perature for 100 A arcjets. T he eifect of different operating pressures is shown in Figs. (8.9) through (8 .1 2 ), again for 100 A operation. Only the solution values from the stable side of the voltage m axim um are shown. T he values corresponding to an attachm ent area of 0.15 cm2 are given in Table (8.2). Increasing th e pressure for a given attachm ent area produces only small increases in the cathode tem perature, the sheath voltage, and the electron tem perature. For these high-pressure conditions, no local minimum in the effective work function is seen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 248 Pressure (kPa) 50 1 0 0 150 2 0 0 T c{ K) 3483 3483 3483 3484 v c 12.32 12.34 12.37 12.42 T< (eV) 0.684 0.690 0.707 0.731 Table 8.2: Model solutions for arcjet th ru ster configuration a t different pressures. ammonia 1 = 100 A $ = 4 .5 eV e =0.50 12.4 12.0 Pressure (kPa) 50 100 150 200 11 .8 - oo 11.6: 11.4: 11.2 3400 3450 3500 3550 3300 3350 Cathode Temperature (K) Figure 8.9: S heath voltage as a function of cathode tem perature for 100 A am m onia arcjets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 249 0.80 Pressure (kPa) 0.75 — 100 — 150 - - 200 S. 0.70 0.65- 0.60 3300 3350 3400 3450 3500 3550 Cathode Temperature (K) Figure 8.10: Electron tem perature as a function of cathode tem perature for 100 A am m onia arcjets. 0.5 Pressure (kPa) 50 0.4 100 150 200 Tip Area < 0 .3 O 0.2 0.1 3350 3400 3450 3500 3550 3300 Cathode Temperature (K) Figure 8.11: A ttachm ent area as a function of cathode tem perature for 100 A am m onia arcjets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 4.40 Pressure (kPa) — 150 - - 200 4.36 | 4.34 ammonia I = 100 A $ = 4.5 eV e = 0.50 I 4.32. 4.3 0 3300 3350 3400 3450 3500 3550 Cathode Temperature (K) Figure 8.12: Effective work function as a function of cathode tem perature for 100 A am m onia arcjets. 8.2 Pure Tungsten Experim ents 8.2.1 Quasi-Two-Dimensional Comparisons T he d a ta from the low-pressure experim ents with the pure tungsten cathode, dis cussed in C hapter Six, com pare well with th e model predictions. T he model predic tions for a cathode w ith th e sam e geom etry as the experim ental ones and a work function of 4.35 eV are shown in Figs. (8.13) through (8.16). T he work function reduction predicted by th e model for these operating conditions is about 0.15 eV. If the m aterial work function is about 4.35 eV , corresponding to som e of the crys tal faces for tungsten [103], instead of 4.5 eV , the cathode tem peratures predicted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by the model are d ose to w hat was observed in the experim ents. Two types of comparisons were m ade. T he results for both com parisons are presented in Table (8.3). F irst, the attachm ent area for the experim ental d a ta was m atched to the model attachm ent area value corresponding to th e local m inim um in the effective work function. T he value of was then determ ined using th e experim ental area and th e experim ental effective work function. Recall from C hapters Five and Six th a t th e experim ental effective work function is determ ined from th e cathode tem p eratu re distribution and th e total current. T he relative differences for the two tem peratures are 0.21, 0.15 and 0.27 percent for 600, 1000 and 1400 A , respectively. T he differences in the effective work functions were 0.49, 0.10 and 0.20 percent, re- spectively. T he model electron tem peratures and sheath voltages are also consistent w ith m easured electron tem peratures and arc voltages. T he excellent agreem ent between the two effective work functions indicates th a t the model is properly de term ining the surface electric field for the Schottky effect. T h e agreem ent between the tem peratures indicates th a t the Richardson equation is properly determ ining th e therm ionic emission current density and th at th e m ajority of th e total current is carried by the therm ionic electrons. However, this m ethod does not produce a repeatable am ount of enclosed current. T he enclosed current values fell between 70 and 85 percent for all of th e pure tungsten experim ents, b u t there is no apparent correlation between enclosed current percentage and either total current or pres- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 252 Cunent (A) 600 - 1000 1400 argon P= 13 kPa $ = 4.35 eV co 6 - 2800 2900 3000 3100 3200 3300 3400 3500 Cathode Temperature (K) Figure 8.13: Sheath Voltage as a function of cathode tem perature for a pure tungsten cathode and a pressure of 1.5 kP a with current as a param eter. sure. T his m ethod is capable of predicting cathode operational characteristics since no additional inform ation is needed. For th e second com parison, the experim ental attachm ent area corresponding to 98 percent of the enclosed current was determ ined. Then th e cathode tem perature in the model was adjusted to give a com parable attachm ent area. T he relative differences for the two tem peratures are 0.68, 0.89 and 0.94 percent for 600, 1000 and 1400 A, respectively. T h e differences in the effective work functions were 1.14, 0.63 and 0.46 percent, respectively. A lthough th e differences are slightly larger for th is com parison they are still typically less than one percent. It was expected th at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 253 2.0 Current (A) 600 , o o o 1400 W 0.8- 0.6 2 8 0 0 2900 3000 3100 3200 3300 3400 3500 Cathode Temperature (K) Figure 8.14: Electron tem perature as a function of cathode tem perature for a pure tungsten cathode and a pressure of 1.5 kP a w ith current as a param eter. Current (A) 600 1000 140 0 100 B o C S , fc i z < 2 10 - g 8: B 6 JZ U C 9 < M < 28 0 0 2900 3000 3100 3200 3300 3400 3500 Cathode Temperature (K) Figure 8.15: A ttachm ent area as a function of cathode tem perature for a pure tungsten cathode and a pressure of 1.5 kPa w ith current as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 254 4.30 Current (A) 600 1000 1400 4.28 £ 4.26 4.24 o 4.22 > 4.20. § 4.18 4.16 2 8 0 0 2900 3000 3100 3200 3300 3400 3500 Cathode Temperature (K) Figure 8.16: Effective work function as a function of cathode tem perature for a pure tungsten cathode and a pressure o f 1.5 k P a with current as a param eter. <& nin Model Soln. E xpt. D ata Aot Te Ve Te Attach 4> tn fan 7flich fend (A) (K) (eV) (cm 2) (eV) (cm2) (eV) (K) (%) 600 3321 8.450 1.1490 1.5479 4.190 1.5426 4.169 3328 80 1 0 0 0 3355 7.605 1.1393 2.331 4.194 2.427 4.190 3360 85 1400 3374 7.201 1.1360 3.086 4.196 3.041 4.204 3365 75 A rea M atch Model Soln. E xpt. D ata 600 3186 7.757 0.8942 3.391 4.217 3.386 4.169 3208 98 1 0 0 0 3233 7.047 0.9103 4.556 4.216 4.564 4.190 3262 98 1400 3218 6.619 0.8876 7.234 4.229 7.237 4.204 3248 98 Table 8.3: Model solutions and experim ental d ata for th e pure tungsten cathode configuration a t 1.5 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 255 the differences w ith this m ethod would be lower than for the first m ethod since th e cathode conditions for 98 percent of th e enclosed current should be close to those for the entire cathode (100 percent of the current). T he problem w ith this technique, however, is th a t the attachm ent area corresponding to 98 percent of enclosed current is not known a-priori. For predicting operational characteristics, an additional m odel, such as an arc column m odel, would be needed to determ ine th e attachm ent area. T he same two m ethods were used for comparisons w ith the tests a t a pressure of 3.0 kPa. The m odel predictions are shown in Figs. (8.17) through (8.20) and the comparisons are presented in Table (8.4). For the minimum work function com par isons, the relative differences between the model cathode tem perature and Trudi are 0.84,1.28 and 1.09 percent for 600,1000 and 1400 A, respectively. T he differences in th e effective work functions were 1.04, 0.76 and 0.112 percent, respectively. For the 98 percent enclosed current comparison th e relative differences between the model cathode tem perature and 7)u * are 0.043, 0.061 and 0.34 percent for 600, 1000 and 1400 A , respectively. T he differences in th e effective work functions were 2.0, 1.76 and 1.18 percent, respectively. Again, the model electron tem peratures and sheath voltages are also reasonable for both com parisons. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 256 Current (A) 600 1000 1400 oc < 9 C O 6 - 2800 3000 3200 3400 3600 Cathode Temperature (K) Figure 8.17: Sheath Voltage as a function of cathode tem perature for a pure tungsten cathode and a pressure of 3.0 k P a with current as a param eter. 2.2 Current (A) 600 1000 1400 2.0 > o w a 3 1.6 E 1 1.4 c H 1.2 1 1.0 5 3 0.8 0.6 3000 3200 3400 3600 2800 Cathode Temperature (K) Figure 8.18: Electron tem p eratu re as a function of cathode tem perature for a pure tungsten cathode and a pressure of 3.0 kP a w ith current as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 257 Current (A) 600 • 1000 1400 argon P = 3.0 kPa $ = 435 eV 100 e <4 K < 1 0 , - 81 c 6; u . 1 4: < 2800 3000 3200 3400 3600 Cathode Temperature (K) Figure 8.19: A ttachm ent area as a function of cathode tem perature for a pure tungsten cathode and a pressure of 3.0 kP a with current as a param eter. 4 .3 0 Current (A) 600 1000 1400 £ > 4 .2 8 - c 4 2 6 ' 14 - 2 ^ i 4 -2 2 ; $ 4.20. o | 4.18. u 4.16. 4.14 2800 3000 3200 3400 3600 Cathode Temperature (K) Figure 8.20: Effective work function as a function of cathode tem perature for a pure tungsten cathode and a pressure of 3.0 kP a with current as a param eter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 258 min Model Soln. Expt. D ata Aot Tc Vc Tt Aattach 4 > ett Aattach Tkich ■ fend (A) (K ) (eV) (cm2) (eV) (cm2) (eV) (K ) (%) 600 3434 8.696 1.1847 0.8050 4.160 0.8127 4.117 3405 75 1 0 0 0 3474 7.793 1.1844 1.2035 4.165 1.2512 4.133 3430 75 1400 3497 7.369 1.1864 1.5845 4.168 1.5987 4.163 3459 70 A rea M atch Model Soln. E xpt. D ata 600 3256 7.859 0.8975 2.192 4.201 2.192 4.117 3255 98 1 0 0 0 3272 7.033 0 . 8 8 8 8 3.582 4.208 3.582 4.133 3270 98 1400 3271 6.627 0.8783 5.243 4.213 5.247 4.163 3282 98 Table 8.4: Model solutions and experim ental d a ta for the pure tungsten cathode configuration a t 3.0 kPa. 8.2.2 Two-Dimensional Model Results The results from the two-dimensional model were not significantly different from those of th e quasi-two-dimensional model for th e operating conditions considered here. T h e cathode centerline and surface tem perature for two exam ple cases are shown in Fig. (8.21). As expected the radial tem perature variations are small com pared to th e axial ones. For all of th e model solutions the cathode tem peratures were uniform in the arc attachm ent area. T he tem perature variation was usually less th an 50 K. N ear the tip , th e surface tem perature is higher than the centerline value because of the surface heating from the arc attachm ent. At the tip the two tem peratures are closer because of the arc attachm ent. U pstream of the attachm ent area th e centerline tem perature is higher than the surface tem perature because of the radial cooling due to radiation. T he surface tem peratures for different sheath Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 259 e 3 e a E £ o a 3600 3400 3200 3000 2800 2600 2400 2200 2000 Radius (mm) Vc Auu<* Current 0.0 4.75 (cm ) (A) 10.3 1.42 1193 11.0 1.68 1859 -30 -25 -20 -15 -10 -5 Distance From Cathode Tip (mm) Figure 8.21: C athode surface and centerline tem peratures as a function of axial position for a pure tungsten cathode and a pressure of 1.5 kPa. voltages and two attachm ent areas are shown in Fig. (8.22). Increasing the attach m ent area produced a small increase in the cathode tip tem perature because of the increased heat load to th e tip area. T he increase in th e cathode tem peratures to wards th e base with increasing sheath voltage is prim arily due to the increase in the Ohmic heating. Again, the cathode tip tem peratures were uniform and changed slightly for this range of conditions. A com parison between the experim ental d ata and two model solutions are shown in Fig. (8.23). Again, the value of th e base tem perature was selected to give a reasonable m atch to the experim ental d a ta near the cathode base. T he model was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 260 3500 s? w H 3000 3 e S . g 2500 H 0 1 H 2000 U 1500 -60 -50 -40 -3 0 -20 -10 0 Distance From Cathode Tip (mm) Figure 8.22: C athode surface tem perature as a function of axial position for a pure tungsten cathode and a pressure of 1.5 kPa. not sensitive to these adjustm ents, and values of 1500 and 2000 K produced sim ilar results. T he cathode tip tem peratures agree quite well even though the model currents were larger. As w ith the previous tem perature profile com parisons, the model predicts a faster decrease in the cathode tem perature as one moves towards the cathode base. A lthough the model agrees well with th e experim ental d ata, it is still not capable of accurately predicting behavior at o th er conditions because the attachm ent area and the sheath voltage are not known. argon pure tungsten P = lJ k P a $o b 4.5 eV e = 0.5 Tbuc- 1500 K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 261 4000- 3500- £ > 2 E 8 . E £ C T 3 u 3000- 2500- 2000- Exp. Data Current Run (A) No. 600 116 1000 ------ 117 1400 ------ 118 Model Data (A*uc* = 1.42 cm ) Vc Tbau Current (K) (A) — 8.7 2000 1533 • - 8 .8 1500 1567 a rg o n P = 3.0 kPa pure tungsten E = 0.50 -30 -20 -10 Distance From Cathode Tip (mm) Figure 8.23: C athode surface tem perature as a function of axial position for a pure tungsten cathode and a pressure of 3.0 kPa. 8.3 Thoriated Tungsten Experim ents Comparisons between the experim ents and the model predictions are m ore difficult for the thoriated tungsten tests because of the thorium m igration effect on the work function. This effect is seen in th e experim ents as variance in the cathode tem perature d ata for sim ilar operating conditions. For the m odel, there is no m eans of determ ining the m aterial work function a-priori. An additional model is needed to calculate the work function as a function of the surface tem p eratu re and the characteristics of the surrounding plasm a. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Experim ental and predicted tem peratures as a function of axial distance from th e tip are displayed in Figs. (8.24) and (8.25). These tests were perform ed at cur rent levels ranging from 600 to 1400 A , an argon mass flow ra te of 0.75 g /s and a discharge cham ber pressure of 2.8 kPa. For th e comparison w ith m easured profiles, th e upstream tem perature boundary condition was set equal to th e m easured tem peratu re on the shaft near the base. T he tip tem perature for th e model profiles was selected to be close to th e experim ental values. Therefore, only th e tem perature profiles can be com pared and not the end points. T he experim ental values for these cases were calculated by the same m ethods as the d a ta presented in C hapters Six and Seven. T he experim ental values are given in Table (8.5) and model predictions are given in Table ( 8 .6 ) For the model predictions, th e cathode tem perature was set to th e Trjd, value. T h e work function was then varied to m atch the model a t tachm ent area to th e attachm ent area calculated from the experim ental d a ta for the 98 percent enclosed current condition. T h e work function was varied for these com parisons instead of th e tem perature, as was done in the pure tungsten cathode com parisons, because there was greater uncertainty in the work function values than in the tem perature values for this d ata. T he predicted tem perature profiles agree rem arkably well with th e measured distributions for currents of 600 and 800 A , indicating th a t in this range th e tem p eratu re can be adequately described by tip attachm ent and radiative cooling. For Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 263 2 8 0 0 . * 2600 § 2400 H 2200 S 2000 Measured Temperature Predicted Temperature 800 A T ’ -5 Distance from Cathode Tip (mm) Figure 8.24: M easured and predicted axial tem perature profiles for low currents. 2800 * 2600 Measured Temperature Predicted Temperature 2000- 1400 A 1200 A 1000 A Distance from Cathode Tip (mm) Figure 8.25: M easured and predicted axial tem perature profiles for high currents. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. higher current levels, however, the calculated profiles do not m atch th e measured profiles as well. W hile th e calculated tem perature profiles agree well w ith the mea sured profiles near the tip (w ithin 1 0 m m ) for the high-current cases, th e profiles are clearly different closer to the cathode base. T he model over-predicts th e radial cooling from radiation along the cathode. T he tem perature profiles for high-current cases calculated w ith an em ittance of 0.4 agreed b etter w ith the m easured values. T his indicates th a t the em ittance may not be uniform along the cathode. T he m ea sured tem perature profiles have linear shapes tow ards th e base indicating th a t the heat transfer is prim arily conductive and th a t the cooling from radiation ju st bal ances the Ohmic heating. T he large tem perature drops near the tip are prim arily from the radiation coding. A nother possible explanation for the profile variations is th e large difference in th e attachm ent areas between the low-current cases and the high-current cases. T he attachm ent areas enclosing 98 percent of the current for the 600 and 800 A cases includes the hemispherical tip (tip area « 1.42 cm2) plus 0.89 and 1.17 cm of the shaft, respectively. In contrast, for the 1200 and 1400 A cases the attachm ent areas include the tip plus the first 2.9 and 4.1 cm of th e shaft, respectively. T h a t is, the attachm ent areas for the low-current cases are prim arily at th e tip , while for the higher current cases a significant portion of the cathode shaft is also included. Note th a t for the 1200 and 1400 A cases the attachm ent areas for 75 percent of the enclosed current include th e tip plus the first 0.37 and 1.7 cm of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 265 the shaft, respectively. Therefore, the last 25 percent of the current is spread over a large area of the cathode due to the lower tem peratures (and current densities) away from the cathode tip. Even though the current densities are low on th e shaft, there may be sufficient heating to make the m easured tem peratures higher th an the calculated ones. Recall th a t th e quasi-two-dimensional model assum es th a t all of the arc attachm ent is a t th e tip . Therefore, one would expect the predictions to become less accurate as the arc attachm ent moves further towards th e cathode base as is shown here. P aram eter Case 1 Case 2 Case 3 Case 4 Case 5 C urrent (A) 600 800 1 0 0 0 1 2 0 0 1400 T ip Tem perature (K) 2618 2654 2694 2699 2725 Effective Work Function (eV) 3.116 3.125 3.138 3.191 3.278 A ttachm ent A rea (0.25) (cm5) 0.2932 0.2932 0.2932 0.4555 0.7130 Tw a (0-25) (K ) 2654 2710 2760 2754 2766 A ttachm ent A rea (0.50) (cm 2) 0.6124 0.7130 0.7130 1.0459 2.344 Tfua (0-50) (K ) 2647 2676 2725 2729 2782 A ttachm ent A rea (0.75) (cm 1 5 ) 1.1430 1.4827 1.5734 2.515 6.538 Trja (0.75) (K ) 2610 2621 2658 2649 2582 A ttachm ent A rea (0.98) (cm 2) 4.056 4.912 6.709 10.13 13.81 Trja (0.98) (K ) 2455 2475 2471 2474 2510 Table 8.5: Experim ental d ata for tem perature profile comparisons. In general, comparisons between the m easured cathode tem perature distribu tions and those calculated w ith th e models were reasonable for the concave shaped profiles, such as those for the 6.0 kP a tests, but produced poor results for th e convex shaped profiles. Also, th e quasi-two-dimensional and the two-dimensional models Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 266 Param eter Case 1 Case 2 Case 3 Case 4 Case 5 C urrent (A) 600 800 1 0 0 0 1 2 0 0 1400 Tem perature (0.98) (K) 2455 2475 2471 2474 2510 Work Function (eV) 3.264 3.270 3.275 3.324 3.406 A ttachm ent A rea (cm2) 4.065 4.920 6.708 1 0 . 1 2 13.82 Electron Tem perature (eV) 0.8112 0.8162 0.8055 0.7875 0.7753 Sheath Voltage 5.602 5.426 5.195 5.145 5.079 Effective Work Function (eV) 3.154 3.159 3.169 3.223 3.310 Table 8 .6 : Model predictions for tem perature profile com parisons using 98 percent enclosed current area m atch m ethod. produced sim ilar tem perature profiles. The main difference between the two m od els was th a t for the two-dimensional model, the tem peratures near the cathode tip where th e arc was attached had a slight curvature. T h e m axim um tem perature could be a t either the tip or further back on the cathode shaft, depending prim ar ily on th e arc attachm ent area. In general these tem perature variations w ithin the arc attachm ent area were typically less than two percent of the cathode tem per ature. Recall th a t for the quasi-two-dimensional model all of the arc attachm ent area is assum ed to be a t one surface tem perature. T he com parisons between the two models showed this assum ption is reasonable. Also, calculations for the M PD th ru ster a t the University of S tu ttg a rt have shown sim ilarly flat tem perature profiles near the cathode tip [51,139]. T he University of S tu ttg a rt model couples th eir flow model [140] with the cathode surface heating model developed here. However, m ost of the experim ental tem perature profiles have a convex (peaked) shape sim ilar to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 267 low-pressure test presented here. Also, the cathode tem peratures in th e attachm ent region ranged from about 2600-2850 K, which are similar to the thoriated tungsten d a ta presented here. These tests were perform ed in the ZT3 th ru ster which uses a thoriated tungsten cathode, for a argon m ass flow rate of 2 g /s and a current range of 4-10 kA. T he local variations in th e m aterial work function make com parisons between the model and the experim ental d a ta difficult for th e thoriated tungsten cathode tests. T he technique of m atching the model attachm ent area th a t encloses 98 percent of th e current to the experim ental value was used for these com parisons. Comparisons using th e minimum effective work function in the model d ata were not performed since there is no accurate m ethod for estim ating the m aterial work function. For th e pure tungsten cathode tests, a constant m aterial work function of 4.35 eV was used. T he cathode tem perature, T ^ch, was then varied until th e 98 percent areas were sim ilar. However, for the thoriated tungsten cathode com parisons, the work function is not known, and one estim ated value would not work for all of the tests. Therefore, for these com parisons, the cathode tem perature from th e experim ental d a ta was used in the model and the work function was varied until th e attachm ent areas w f ere similar. The comparison results are presented in Tables (8.7) through (8.10). T he relative differences in the effective work functions between the model and th e experim ental d a ta were typically 4-5 percent. As expected, this difference Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 268 is larger th an th e differences for the pure tungsten cathode results b u t are still reasonable. T he model d ata include the m aterial work function used to m atch the attachm ent areas. T he m aterial work function ranged from 3.03-3.94 eV which again is reasonable. Recall from C hapter Seven th a t the work function for thoriated tungsten can vary from 2.9-4.5 eV depending on th e thorium coverage. A rea M atch Model Soln. E xpt. D ata /to t Tc V c Tt A *tU ch 4 > o 4> tf! A a tta d i dwr (A) (K ) (eV) (cm 2) (eV) (eV) (cm 2) (eV) 600 2457 5.708 0.8017 6.404 3.359 3.260 6.404 3.217 1 0 0 0 2506 5.358 0.8086 9.167 3.394 3.294 9.167 3.256 1400 2562 5.241 0.8028 13.510 3.485 3.288 13.510 3.350 Table 8.7: Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 1.5 kPa. A rea M atch Model Soln. E xpt. D ata /to t Te Ve Te A attach < t > o 4> ta A attach (A) ( K ) (eV) (cm 2) (eV) (eV) (cm 2) (eV) 600 2293 5.208 0.7943 4.386 3.036 2.933 4.386 2.900 1 0 0 0 2735 5.801 0.8069 7.083 3.695 3.584 7.083 3.533 1400 2887 5.856 0.8016 10.783 3.942 3.833 10.783 3.778 Table 8 .8 : Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 3.0 kPa. T he d a ta from both the pure and thoriated tungsten cathode tests were used to estim ate th e correlations between th e cathode tem perature, th e effective work function and the current density. T he current density is shown as a function of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 269 A rea M atch Model Soln. Expt. D ata /to t Te Ve Te A a tu c h < t > o A a tta d i 4>tft (A) ( K ) (eV) (cm2) (eV) (eV) (cm2) (eV) 600 2445 5.552 0.7952 3.780 3.240 3.129 3.780 3.087 1 0 0 0 2583 5.951 0.7978 5.950 3.426 3.315 5.950 3.273 1400 2751 5.520 0.7890 9.659 3.708 3.598 9.659 3.550 Table 8.9: Model solutions and experim ental d a ta for thoriated tungsten cathode configuration a t 4.5 kPa. A rea M atch Model Soln. E xpt. D ata A ot Tc Vc Te ^ tlU c h 4 > o < t > e l f A a tta d i 4>ts (A) ( K ) (eV) (cm2) (eV) (ev) (cm 2) (eV) 600 2421 5.510 0.7480 6.378 3.304 3.206 3.162 3.162 1 0 0 0 2701 5.595 0.8414 3.078 3.466 3.334 3.290 3.078 1400 2842 5.669 0.7963 7.768 3.799 3.682 7.768 3.630 Table 8.10: Model solutions and experim ental d ata for thoriated tungsten cathode configuration a t 6.0 kPa. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 270 1000 c T 800 S 600 * > * 1 £ 400 «•* 1 | 200 0 2200 2400 2600 2800 3000 3200 3400 Cathode Temperature (K) Figure 8.26: C urrent density as a function of cathode tem perature. the cathode tem p eratu re in Fig. (8.26). T he continuous lines show th e current density predicted from the Richardson equation, Eq. (2.3), for different effective work functions. T h e symbols indicate the d a ta from all of the tests presented in this section. T he cathodes for this range of operating conditions appear to o perate such th a t th e current density slowly increases w ith cathode tem perature. This trend is significantly different from w hat is predicted using the Richardson equation. The pure tungsten d a ta have a shape sim ilar to the Richardson equation lines over a small range, b u t the thoriated tungsten d a ta do not. T he effective work function is shown as a function of the cathode tem p eratu re in Fig. (8.27). T he experim ental d a ta and the model predictions using the 98 percent E ffective Experimental Data W oik T hW Pure W Function (eV ) Press. (kPa) Press. (lc P a ) ---- 3.0 O 13 © 13 ......... 35 □ 3.0 B 3.0 ......4.0 A 43 ..... Exp. fit ---- 43 V 6.0 (all dam ) / » / i / * □ / A / / k / . / -A..... V” >L n * ~ w — — * ~ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 271 T h-W pure W M odel M o d e l A ll Exp ~*~r— ■ " 2800 3200 3400 Cathode Temperature (K) Figure 8.27: Effective work function as a function of cathode tem perature. enclosed current area are shown w ith the symbols, and agree well. A linear fit to all of the experim ental d a ta is also shown. It appears th a t when all of the operating param eters, such as attachm ent area, sheath voltage, electron tem perature and surface electric field are considered, th at there is a linear relationship between the effective work function and the cathode tem perature. T he curved lines in Fig. (8.27) are predictions from the model using the m inim um effective work function point discussed previously. Although these model predictions have the same shape and sim ilar slopes, they predict higher cathode tem peratures for a given effective work function value than was observed in the experim ents. T his further confirms th a t the minimum effective work function point may represent an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 272 upper bound on th e operation, but does not represent th e average values. T h at is, for some of the operating points, the conditions near the tip where the cathode tem p eratu re is the largest, may be represented by the m inimum effective work function point. Therefore, th is m ethod may be useful for predicting operation a t the hottest o r highest erosion location of the cathode. Model com parisons using the two-dimensional model were not inform ative for th e thoriated tungsten tests. Any of the tip tem peratures could be m atched by th e proper selection of th e attachm ent area, th e sheath voltage and the m aterial work function. T he tem p eratu re profiles for all of the solutions had sim ilar shapes to those presented previously for the pure tungsten tests. To reproduce th e peaked tem perature profiles observed in the experim ents an axial variation in both the work function and th e sheath voltage are probably necessary. However, additional inform ation is needed to determ ine these distributions, so those solutions should be included in future work. 8.4 Cathode Evaporation T h e prim ary erosion mechanisms for steady-state cathode operation are dependent on the cathode operating environm ent. For low discharge pressures where the gas tran sp o rt processes can be neglected, the evaporation ra te can approach th e vacuum evaporation rate. As th e pressure is increased, the gas tran sp o rt process become Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 273 increasingly im p o rtan t and the erosion process is diffusion-limited. T he presence of oxygen, even in small quantities, can significantly increase the erosion rate through the form ation of volatile tungsten oxides. Tests performed by Polk [51] measured predicted by the vacuum evaporation ra te indicating th a t diffusion processes were present. These tests also revealed an order of m agnitude increase in erosion when poorer gas purity was used. T he vacuum evaporation rate was used as an upper bound prediction for these experim ents. T he evaporation rate is given by, where P fq is the equilibrium vapor pressure, which depends only on th e surface tem perature of the condensed phase and a is the sticking probability. For the vacuum evaporation rate, P, is assumed to be zero. A value of unity for o which is consistent with the sticking probability for m etals [141,142]. An analytical expression for the vapor pressure (in T orr) from Eucken [143] based on d ata from Langm uir [144] and Zwikker [145] is given in Eq. (8.2). T he vacuum evaporation rate for tungsten is strongly dependent on the cathode tem perature is shown in Fig. (8.28). For the thoriated tungsten cathode tests the erosion rates th a t were approxim ately an order of m agnitude less th a t the rates (8.1) p e q _ jq [(^ 2 2 )+ 0 .1 4 6 1 n (T t ) - 0.164x10 JTc +9.84] (8.2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 274 100 - 50.01 0.001 2600 2800 3000 3200 3400 3600 Cathode Temperature (K) Figure 8.28: M aximum evaporation rate for tungsten. cathode tem peratures were typically 2600-2800 K which correspond to erosion rates of 0.0090-0.113 /ig/cm 2s while th e pure tungsten cathode tip tem peratures were typ ically 3400-3600 K which correspond to erosion rates of 81-167 /ig /cm 2 s. Therefore, a factor of 1.3 increase in cathode tem perature produces roughly a factor of 1500 increase in erosion rate. The vacuum evaporation rates for the pure tungsten cath odes are shown in Fig. (8.29). For each tem perature profile, th e local evaporation rates were calculated using Eq. (8 . 1 ) and then combined w ith th e local surface areas to predict the total cathode erosion. The total evaporation rates increase w ith cur rent b u t do not ap p ear to be pressure dependent as seen in Fig. (8.29). Recall th at increasing the current a n d /o r decreasing the pressure resulted in an increase in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 275 agon pure tungsten 100 - u ea C C c 0 1 & R u 3 £ Pressure (kPa) - e - 1-5 ~E3- 3.0 1000 Current (A) 600 800 1200 1400 Figure 8.29: C athode evaporation rates for pure tungsten cathode tests. arc attachm ent area and an increase in the overall cathode tem perature. For the pressure increases, however, th e tip tem perature decreased slightly. T h e increase in th e tem p eratu re and erosion rates towards the cathode base appear to be offset by th e cathode tem perature and erosion rate decreases at the cathode tip . Although th e cathode tip area is sm aller, the tem peratures were the greatest and therefore tem p eratu re changes here would have a larger effect on the erosion rate. T he total m ass lost during the pure tungsten cathode tests was 2.098 g in 19.12 hours with an average current of 961.3 A for an average m ass loss rate of 110 m g /h r. T h e vacuum evaporation rate prediction for this operation was 4.841 g which is a factor of 2.31 higher th an the actual mass loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 276 argon thoriated tungsten 1 0.1 '3 Pressure (kPa) - S - 1.5 -O - 3.0 ••Q* 4.5 - A - 6.0 0.01 0.001 600 800 1000 1200 1400 Current (A) Figure 8.30: C athode evaporation rates as a function of current for thoriated tung sten cathode tests with pressure as a param eter. T h e erosion rates for the thoriated tungsten cathode tests were much lower than for th e pure tungsten tests, as expected. For one segment of th e tests a cathode weight loss of 0.8697 g was observed for 23.70 hours of operation w ith an average current of 1062 A. The predicted m ass loss for this test segment is 0.1123 g which is a factor of 7.74 lower than th e actual mass loss. After a later series of tests, a small w ater leak was detected in the system and, therefore, the larger actual mass loss m ay be due to the form ation of volatile tungsten oxides. T h e factor of 7.74 increase is consistent w ith th e order of m agnitude increase observed by University of S tu ttg a rt where oxygen contam ination was present [79]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 277 Current (A) - B - 600 1000 -£}- 1400 1 < 0 0.1 0.01 argon thoriated tungsten 0.001 Pressure (kPa) Figure 8.31: C athode evaporation rates as a function of pressure for thoriated tung sten cathode tests with current as a param eter. T he vacuum evaporation rates for the thoriated tungsten cathodes are shown in Fig. (8.30) as a function of current and in Fig. (8.31) as a function of pressure. T here is a m ore noticeable increase in the erosion rates with increasing pressure and a strong increase with increasing current for these tests. The irreproducibility of th e cathode tem perature profiles for these tests complicates the erosion estim ates because of th e strong dependence of the vacuum evaporation ra te on the cathode tem perature. T his may explain th e different trends seen in Fig. (8.31) for the 3.0 kPa values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 278 C hapter 9 C onclusions and R ecom m en d ation s The objective of this work was to develop a model capable of describing the in ter action of the cathodes w ith the surrounding plasm a in electric propulsion thrusters. This model is capable of providing inform ation to help explain experim ental ob servations as well as predicting th e effect of thruster operational changes on the cathode operation and erosion, and th e near-cathode plasm a. This model also pro vides th e cathode boundary conditions for models of the main plasm a. C athode operation and erosion have been shown to depend strongly on the cathode tem per ature. Therefore, the main focus of this study was to provide a simple m eans of predicting the cathode tem perature for various thruster operating conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 279 T he cathode model consists of two p arts, namely a near-cathode plasm a model and a therm al m odel of the cathode. T he near-cathode plasm a model connects the properties of the m ain plasm a with th e cathode. Specifically, given the plasm a prop erties w ithin an ionization m ean-free-path of the surface, the near-cathode model predicts the heat flux and current density to th e cathode surface. W ith these bound ary conditions and th e traditional therm al transport mechanisms, th e therm al model can predict the tem perature distribution w ithin the cathode. Because of the inter dependency of th e tw o models, they m ust be solved simultaneously. T he near-cathode plasm a is considered as a series of layers w ithin a mean-free- path of th e surface. Specifically, th e cathode surface, sheath, presheath and ion ization regions are m odeled. The sheath region is assumed to contain collisionless particles w ith co nstant total energy (potential plus kinetic) since th e Debye length is much larger th an th e collisional m ean free paths. Six species are considered; monoen- ergetic therm ionic (or beam ) electrons, singly- and doubly-charged monoenergetic ions for two m onatom ic gases, and Maxwellian electrons originating in the plasm a. Furtherm ore, th e sheath thickness is assum ed to be much less than the Larm or radii of the particles, and therefore, m agnetic field effects on the particle trajectories are negligible. D oubly-charged ions were included because the cathode heating from doubly-charged ions may be significant a t low pressures for high-current discharges. T he gases are assum ed to be m onatom ic and nonreacting. For this model, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 280 presheath region is combined with the ionization region by requiring th a t ions leave the ionization region w ith the Bohm energy. T he ionization region generates the required num ber of ion and electron pairs to m atch the sheath and m ain plasm a body values. T he energy balance w ithin th e ionization region is used to determ ine the electron tem perature and then equilibrium ionization/recom bination is used to determ ine the species densities. C athode therm al m odels with different levels of sophistication were developed to describe the tem perature distributions w ithin th e cathode. T he large cathode tem p eratu re variations require the inclusion of tem perature dependent m aterial proper ties, and the high tem peratures require th e inclusion of radiation cooling from the surface. T he heat loads to the surface from th e arc attachm ent are also strongly tem perature dependent. These three nonlinear effects significantly com plicate the therm al model solution. Simple models were developed to provide insights into the operation and to provide initial tem perature distributions for the m ore com plicated m odels. T he quasi-two-dimensional m odel, which considers axial conduction with radial boundary heat fluxes, was found to be th e most useful. Its solutions were not significantly different from those of the two-dimensional model for th e conditions tested, and it could be solved in m inutes ra th e r than hours on a 486-33 MHz PC com puter. Ohmic heating w ithin the cathode is the dom inant heating mechanism for large current discharges (> 3000 A) and heating from th e arc attachm ent dom Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 281 inated for the low currents. T he arc attachm ent can either heat or cool the surface depending on the plasm a characteristics, and th e cathode tem perature and work function. A database of axial tem perature distributions on a cylindrical, tw o percent th o riated tungsten cathode has been collected for current levels of 600-1400 A for argon m ass flow rates of 0.074 to 0.878 g /s and am bient gas pressures ranging from 1.5- 6.0 kPa. At the higher pressures the cathode tem perature increases monotonically from th e base to the tip while a t the lower pressures the maximum tem p eratu re was located further back on th e cathode. The peak tem peratures for these profiles were observed to move tow ards th e cathode base over periods of several hours. These slow variations are probably a result of thorium m igration on the cathode surface chang ing the local m aterial work function. Changes in the mass flow ra te were found to have no significant effect on the axial cathode tem peratures b u t tem perature variations resulting from work function changes between tests m ay have obscured this effect. Electron tem p eratu re m easurem ents were m ade utilizing th e m ethod of relative line intensity ratios. The radial tem perature profiles are flat for the low- pressure case and increased radially for the high-pressure cases. T h e variation of th e attachm ent area w ith current and pressure was characterized by m easuring the intensity distribution of an argon ion line near th e cathode surface. For all of the pressures considered th e arc is attached in an annular ring on the cathode tip and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not on th e centerline. M aterials analysis of the cathode following a test a t 1000 A and 6.0 kP a for two hours revealed th a t the thorium tends to accum ulate at the tip, is depleted on the shaft, and a transition occurs in between. A m inimum in the work function therefore occurs in an annulus around th e cathode tip. Tests performed with a cathode containing tw o pressure taps revealed th a t th e pressure w ithin the arc attach m en t area can be significantly lower th an the am bient pressure. Addi tional tests were perform ed over a pressure range of 1.5-3.0 k P a and current levels of 600-1400 A w ith a pure tungsten cathode which had more repeatable tem pera ture profiles. T he pure tungsten cathode tem peratures reached equilibrium w ithin a m inute and did not drift as with the thoriated tungsten cathodes. For both sets of experim ents, increases in th e operating current a n d /o r decreases in th e pressure resulted prim arily in increases in the arc attachm ent area with small increases in the cathode tip tem perature. Increasing the current tended to increase the attachm ent area m ore th an the current density. Excellent agreem ent between the model and high-power long-duration am m onia arcjet d a ta was achieved. P ost-test analysis of these cathodes indicates th a t portions of the cathode tips were m olten during operation. T he model predicts tem peratures near th e m elting point of tungsten for attachm ent areas consistent w ith the areas observed in the experim ents. In addition, model predictions revealed th a t the use of a water-cooled anode o r additional cathode base cooling had little effect on the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cathode tip tem perature. Excellent agreem ent was also seen between the model and th e pure tungsten tests. T he comparisons with the thoriated tungsten tests were com plicated by the irreproducibility of the cathode tem perature between tests. The m easured m ass loss for the pure tungsten cathode was also consistent with the model predictions. Excellent comparisons were seen between th e model predictions and th e experim ents for both high-pressure (100 kPA), and low-pressure (1 -6 kPa) operation. Tw o additional m odels are required to complete the description of cathode oper ation in these thrusters. F irst, a model describing the work function distribution on thoriated tungsten cathodes resulting from the m igration of thorium on th e surface is necessary. Because this model will need to include th e evaporation and diffusion, and possibly ionization of the thorium , it m ust include a description of th e bound ary layer characteristics which are not well understood for these devices. Observing th e tem p eratu re variations of a cathode a t one specific operating condition over hundreds of hours w ith periodic elemental analyses may prove useful in establishing these relationships. Second, a model describing the relationship between th e arc a t tachm ent area and the sheath voltage, or total arc voltage, is needed to com plete the model developed here. U nderstanding the work function characteristics and the arc a ttach m en t are the rem aining key issues to predicting thoriated tungsten cathode lifetim es. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 284 A ppend ix A List o f V ariables w ith T ypical U n its a constant A area (m2) or transition probability (1/eV nro) A attack arc attachm ent area (m 2) A r Richardson coefficient (A /m 2/K 2) C constant d ’ cathode diam eter or ionization region thickness (m ) c electron charge (C) E electric field (V /m ) E n energy of atom level “N” (V /m ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ei'O Normalized therm ionic electron therm al energy / surface coverage F particle flux (particle/s) Ft th ru st (N) g energy level degeneracy g0 acceleration of gravity on E arth (m /s2) or ground sta te degeneracy h Planck constant (J s) hconv therm al convection coefficient (W /m 2/K ) I total current (A) I * specific impulse (s) j current density (A /m 2) Jb normalized therm ionic current density k Boltzm ann constant (J /K ) K reaction rate k,h therm al conductivity (W /m /K ) L characteristic length (m ) m particle mass (kg) M l fraction of im pregnate mass loss n num ber density (particle/m 3) n particle generation rate (p a rtid e /m 3s) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 286 N norm alized num ber density P pressure (Pa) P* th ru ste r power (W ) p t , equilibrium vapor pressure (P a or Torr) 9 heat flux (W /m 2) r radius (m) rcyl cylidrical radius (m ) Rc radius of arc attachm ent sp o t or crater radius (m ) Rz ratio of maximum to m inim um cell size 9 heat generated per unit volum e (W /m 3) *£ lifetim e (hr) T tem perature (K) T i insert tem perature (K) To, environmental tem perature (K ) V velocity (m /s) V voltage (V) or volume (m3) X position from cathode surface (m ) Y mole fraction z position from cathode tip (m ) Z norm alized position from cathode tip, species charge, or partition function Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 287 Z, norm alized position from cathode tip to conical base a ionization fraction, conical half-angle, or sticking probability a ratio of partial pressures e norm alized electric field or em issivity f. ionization energy for singly-charged ion (eV) c» ionization energy for doubly-charged ion (eV) f o perm itivity of free space (C 2/N /m 2) €r em issivity V norm alized voltage V B norm alized Bohm energy V ' th ru ster efficiency Vi norm alized ionization energy for singly-charged ion Vii norm alized ionization energy for doubly-charged ion n r th ru ster efficiency T e.net evaporation rate (p g /cm 2s) A wavelength (nm ) Ho perm eability of free space (N /A 2) Vi norm alized ion density a t sheath edge for singly-charged ion Vii norm alized ion density a t sheath edge for doubly-charged ion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 288 u relaxation coefficient pc charge density (C /m 3) pt electrical resistivity (ft - m) < t > work function (eV) Ot electrical conductivity (m ho - m Or Stefen-Boltzm ann constant (W /m s /K 4) e electron to heavy tem perature ratio or normalized tem p eratu re £ normalized position S ubscripts am b am bient b therm ionic (beam ) electron base cathode base B Bohm energy’ value c cathode surface or arc spot C cross-sectional con conical cyl cylindrical D Debye e plasm a electron Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 289 eff effective ei electron-ion collision enc, e n d endosed h heavy (neutral or ion) i singly-charged ion ii doubly-charged ion iii triply-charged ion inf or oo infinity max m axim um min m inim um n neutral P main plasm a o reference value ri inside radius Rich Richardson ro outside radius s gas spedes type s sh sheath edge surf surface tip cathode tip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 290 to t total 1 gas spedes type 1 2 gas spedes type 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 291 A ppend ix B G rid T ransform ation T he nonlinearities of th e therm al equations produce larger tem perature gradients in the cathode tip region th an in th e base region. To increase the accuracy of the discretization w ithout increasing th e num ber of cells and correspondingly increasing th e solution tim e, th e cells are “packed” towards the tip. An algebraic grid generator th a t allows th e m inim um grid spacing to be placed anyw here along the cathode (cell num ber £mm a t position 2m,n )- T h e transform ation consists of two third-order polynom ials given by Lmin,baM e = 'm in ~ Zbate ( B . l ) f'ftp .m in = Ztip ~~ Zjnin (B.2) F = l min ~ ] (B.3) ( m a r 1 1 n - 1 Gr = - i - (B.4) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 292 H = f " ,n ~ 2 (B.5) s m o i “ * r j _ ________________ G ♦ ( H z — l)f«m in,6a»e________________ /t> g \ F[G (G - F)(G - 2F) + R t H(h - F){H - 2F)\ 1 ' ’ C = - 3 F D (B.7) B = LminMte/ F - F C - F 2D (B.8) x _ | a ; e - 1 ( B g ) ( m a t * z = *ia« + B X + C X 2 + D X 3 (B .1 0 ) for the first polynomial (base to m inim um ) where Rt is the ratio of the maximum cell size to the m inimum cell size given by Rt = ( B .ll) and m = B + 2 C F + ZF2D (B.12) (B.13) n f lip ,m in 1 F) ~ (1 - F)3 C - - Z F D (B.14) B — m - 2F C - 3F 2D (B.15) A = zmin - F B - F 2C - FaD (B.16) X = f'* 1 " ~ 1 (B.17) Sm ox * Z = A + B X + C X 2 + D X 3 (B.18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for th e second polynom ial. T he above polynomials transform the uniform ly spaced £ positions to the nonuniform A' and X positions. The values of A' and A' are used to determ ine the cell centers and then the cell wall is set to th e m id-point between two cell centers. T his produces a cell where th e “center” is slightly shifted towards th e wall on th e m inim um side. T his provides th e same < is value for both sides of the wall for th e central differencing approxim ation of the tem perature gradient. If the cells are arranged such th a t cell “center” is tru ly centered, then the spacing between th e cell walls and th e centers is not uniform for each cell wall and a coordinate transform ation is required for the central differencing approxim ation. T h e sam e axial grid transform ation is used for the quasi-two-dimensional model and th e two-dimensional model. T he radial grid spacing in the two-dimensional model only uses one of th e polynomials and th e minimum spacing set a t the cen terline/inside radius. Equations (B .l) and (B .3) through (B.10) are used replacing rfca*. w ith ra, and z„,in w ith r,. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 294 A ppend ix C T h e T em perature M easurem ent S y stem T he operation and calibration of the cathode tem perature m easurem ents are de scribed in this section. C .l The Im aging Pyrom eter A C ID T E C 2550-D Charge-Injection Device (C ID ) cam era was chosen as an op tical pyrom etric sensor to m easure the two-dimensional tem perature field on the cathode1. Figure (C .l) shows a diagram of the system . An area A on th e source ’M ajor portions of this appendix were taken directly from Refs. [51] and [122], and were provided by Jay Polk as part of the JPL C TF project. Used here with permission Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 295 CU D O m e n Im *fc S o n a r B e a ro o ia VCR D ig jn a r j f i t o B iter Figure C .l: Diagram of im aging pyrom etry system. em its a spectral radiance of € (0 ,< j> ,\,T )L t\(\,T ), where is the surface em ittance a t wavelength A a t an angle 0 relative to the surface norm al and < f > from some reference line on th e surface, and L b \(X ,T ) is th e radiance of a black body a t the sam e tem perature T . This area is viewed by th e system optics through a window w ith a transm ittance of t&(A) and one or m ore neutral density filters with a combined transm ittance of T 2 (A). T he system optics are composed of tw o inter ference filters w ith a 10 nm bandpass centered a t 632.8 nm and a long-pass filter with a cutoff wavelength of 570 nm , all three with a combined transm ittance rj(A ); a normal cam era lens w ith a transm ittance T 3 (A); and a protective coating on the sensor array w ith a transm ittance of r 4 (A). The optics intercept radiation em itted by the area into a solid angle Q and focus it onto the CID sensors in the cam era array. T he detector o u tp u t Vd for an input power W is defined as the responsivity of the sensor, R q. The im aging array has 512 x 512 CID detectors which are read out at a m axim um rate of thirty tim es per second. These values are converted to an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 296 analog signal, which is then further processed and outp u t as a normal video signal by the cam era electronics, which have a gain of Go. T he video signal is digitized by a D ata lY anslation DT-2862 8 -bit fram e-grabber board w ith a gain of G \, which yields a final value between 0 and 255 corresponding to th e incident power. C.2 Calibration o f the Pyrom eter T he system enclosed in dashed lines in Fig. (C .l), which includes the cam era and associated electronics, all optical com ponents except the neutral density filters, and the detector-source geometry, was calibrated as a unit. T he neutral density filters are chosen to properly m oderate th e input radiance and may be varied from experim ent to experim ent depending on th e source intensity. T heir transm ittance is therefore calibrated separately and used to scale the calibration for the subsystem shown inside th e dashed lines. T he incident power is given by radiance of the source transm itted by the var ious intervening m edia integrated over the spectrum , th e em itting area, and the intercepted solid angle, W = r f f L ix (\,T )((0 ,< t> ,\,T ) x T T T i(A )cos0d,4dA dn. (C .l) Jo J a J n ~_q T he blackbody radiance Ltx is described by P lanck’s spectral radiance distribution: i as -h c/\',k T , q 2 ) LbX ~ A f, e W A .,* T _ i ~ A ?, C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 297 where h is Planck’s constant, k is B oltzm ann’s constant, and c is the velocity of light. T he second expression is W ien’s Law, an approxim ation th a t is valid for the tem per ature range of interest. Assuming th a t the directional em ittance c(0 ,^ ,A ,T ) does not vary significantly over the solid angle fi and th a t th e interference filter blocks sufficiently well in regions outside of a narrow band AA a t a central wavelength of A, 7 , the integral can be simplified to W = A c o s0 ftL kA ( T ) c ( f l ,^ ,T ) n r , AA \x=Xt, (C .3) 1=0 where A c o sS is th e projected area in the viewing direction. T he responsivity, R ", which relates th e system o u tp u t Vt to the source radiance a t th e interference filter wavelength through the neutral density filters L \, is defined as « • = f t C . C . W a j f t r . A A - r ) r o = % . (C .4 , This is the desired calibration relation which contains th e responsivities of the elec tronic com ponents, the source-detector geometry, and the optica] com ponent param eters. If the gains of the electronic com ponents are constant, the system responsivity R * should have th e sam e functional form as the cam era responsivity R q, which is constant up to a point in the range of 50-80 percent of the saturation value, after which it decreases. Equation (C.4) also emphasizes th a t each optical setup and source-detector geom etry m ust be individually calibrated. The value R * m ust be empirically determ ined using a reference source with a known tem perature and em it tance at the interference filter wavelength, accounting for the transm ittances tq of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 298 th e m edia between the source and the cam era optics. This m easured R ' can then be used to determ ine the tem perature of another source with known em ittance viewed through filters with known transm ittances. T h e subsystem responsivity R ’ was m easured with an Eppley tungsten ribbon lam p and verified independently w ith a blackbody calibration source. The radiance in the center portion of th e tungsten ribbon lam p was determ ined by comparison w ith an Optronics M odel 550 calibration lam p traceable to N IST standards using an O ptronics Model 746-D spectroradiom eter, and is accurate to w ithin ± 3 percent. T he lam p was operated over a radiance range of 0.035 to 28.4 m W //im cm2 sr and th e center p art of the ribbon was im aged on th e detector array. T he response of 5 pixels illum inated by th e uniform part of the image was m easured. These d ata were also used to estim ate th e variation in response from pixel to pixel. T he vari ation am ong the 5 detectors did not exceed 1 gray level up to 150 gray levels and n o t m ore than 1.5 gray levels a t saturation. For b etter laboratory calibrations, the voltage from a calibrated current shunt was used rather th an using lam p current directly. T he calibration between the cam era response (gray level) and the lam p shunt voltage was regularly performed in the cathode test facility. These two cal ibrations, radiance verses shunt voltage and cam era response verses shunt voltage, were then combined to determ ine the subsystem responsivityfZ*. T he regular in-situ calibrations provided regular checks for system atic errors. For exam ple, a shift in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 299 tem p eratu re m easurem ents resulting from a decrease in the tank window transm is sivity from contam ination could be easily quantified. T h e M ikron model M300 blackbody source used to verify the cam era calibration had an em issivity of 0.999±0.005. T he tem perature of the spherical cavity was m easured to w ithin ± 1°C w ith a T ype S therm ocouple em bedded in the wall. T he radiance L \ of th e source was calculated from the m easured tem perature using th e Planck distribution in E q. (C.2). T he possible error in the calculated radiance due to uncertainties in th e tem perature is less than ±2.5 percent. T he source was operated over a tem perature range of 900 to 1200°C, corresponding to a radiance range of 0.44 to 23 m W //im cm 3 sr. T he aperture of the cavity was im aged on the array w ith the sam e detector-source geom etry used in the experim ents, bu t with no neutral density filters. T he tem perature was set and allowed to stabilize to w ithin ± 1 °C before m easuring the system response. T he radiance was uniform across the ap ertu re w ith about three pixels illum inated. These m easurem ents were also used to determ ine the intrinsic variability in pixel response. Experim ents perform ed with the source held a t a constant tem perature over five hours show no drift in pixel o u tp u t and random fluctuations on the order of ± 2 gray levels. T h e uncertainty associated with these fluctuations can be reduced to a negligible level by averaging a num ber of frames. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 300 250 O Blackbody Source A Tungsten Ribbon Lamp — Linear-Quadratic Regression I 200- >* E g 150- I J S 3 O S E « 50- i o.i Radiance (mW / pm cm 2 sr) Figure C.2: CID TEC cam era calibration. Figure (C.2) shows th e final calibration curve relating the cam era o u tp u t Vt to the incident radiance L \ using th e tungsten ribbon lam p. D ata points from the two blackbody verifications are also shown. The excellent agreem ent indicates th a t the lam p calibration technique correctly relates the cam era response to th e radiance. The slope of this curve is th e responsivity defined in Eq. (C.4) and, as expected, is constant until about gray level 117 and then drops off. T he d a ta can be well represented by the function V, = < h + < h L \ + H o - L l ) 2 (C.5) where a is th e position of th e knee, and the subscript minus sign a t th e end of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 301 last term signals th a t it is to be included only when L \ is greater th an a . The param eters f a , fa , fa , and a were estim ated using a non-linear fitting program . T he transm ittance of the neutral density filters was m easured using a tungsten coil lam p and th e spectroradiom eter with an uncertainty of approxim ately ± 3 per cent. C.3 Application in the Experim ents In the experim ents the cam era and optics were m ounted outside the cham ber with about 49.2 cm between th e cathode and the cam era lens. T he video o u tp u t from the cam era was digitized to provide real-tim e m onitoring of th e tem p eratu re distri bution. O ne line in video m em ory chosen to correspond to the axis of th e cathode was sam pled from each fram e. A given num ber of lines were averaged, displayed in real tim e, and periodically stored on disk. Figure (C .3) shows a typical cathode axial profile obtained by averaging 20 frames. The axial position is m easured in mm from th e cathode tip and is defined to be positive in the dow nstream direction. A small signal due to radiation from the plasm a plum e is visisble dow nstream of the cathode. T he intensity varies with the pressure and the current level, b u t is typically 7 to 15 gray levels. T he inverse of the calibration relation given in Eq. (C .5) was used to determ ine the radiance incident on th e system from the m easured response. T he tem perature Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 302 200 O ^ 150 e p . | 100. & (A « ■ > c* 1 2 5 0 - o U -50 -40 -30 0 10 -20 •10 Distance from Cathode Tip (mm) Figure C.3: C athode axial intensity distribution. is determ ined from Planck’s distribution, Eq. (C.2), the cathode em ittance and neutral density filter transm ittance, and the measured radiance, he T = A ,/* C.4 Uncertainty in the Temperature M easurem ents The uncertainty in the tem perature m easurem ent is given approxim ately by the expression 4 = (*£*r\3 \d + 5* + fk + (5 _ _i^Y T 2 \ he ) c2 r 2 + I * A 2 + XijkTJ 4 „ Xh (C .7) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 303 The sensitivity to the independent param eters is small because the m ultiplying factor (X ijk T /h c ) 2 is approxim ately equal to 0.01-0.02 [51]. T he interference filter is assumed to block perfectly outside a narrow region, so the last term is negligible. U ncertainties in th e neutral density filter transm ittance contribute approxim ately 1 percent to th e stan d ard error. T he prim ary contributors to th e uncertainty are the relatively large and, to a certain ex ten t, unquantifiable errors in th e em ittance of the cathode surface and the proper m easurem ent of th e incident radiance. The em ittance of the cathode surface was assum ed to be independent of th e viewing angles 0 and < f > , which is tru e for diffuse em itters, and rough surfaces often approach this behavior. In addition, since th e cathode axial tem perature profiles were taken parallel to the surface norm al, m easurem ents for th e normal em ittance of tungsten were used in the analysis. A curve fit perform ed by Pon [146] to em ittance d ata for tungsten ribbon lam ps m easured by DeVos [132] for the wavelength 632.8 nm yields em ittances ranging from about 0.44 a t 2000 K down to 0.42 at 3400 K. The errors in th e m easurem ents and the curve fit are quoted to be about 2 percent. To simplify th e analysis an interm ediate value of 0.43 corresponding to the em ittance a t 2800 K was used in Eq. (C .6 ) for all tem peratures. T his approxim ation represents an error in em ittance of about 2.5 percent a t 2000 K, b u t a very small error in the tem perature range of interest. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, this d a ta is not necessarily representative of the cathode surface be cause em ittance is extrem ely sensitive to surface topology and chemical composition. Roughness can substantially increase the em ittance, and lim ited d a ta from [134] in dicate th a t the em ittance for a tungsten surface with a characteristic roughness of 1-3 pm could be as high as 0.6 in the wavelength range of interest. Since th e energy radiating from a surface is characteristic of th e m aterial in a thin layer less than 1 0 0 0 A thick, relatively thin oxide layers can have a significant im pact. Because it is virtually im possible to adequately characterize a surface o r quantify th e effect of surface irregularities and com position, th e em ittance should ideally be m easured in the experim ent. M easurem ents of em ittance using cathodes containing small, high-em ittance cavities [126,133] or unspecified m ethods [147] are inconclusive. The m easured values range from about 0.4 to as high as 0.8. T he uncertainty in th e estim ate of incident radiance L \ depends on three factors - the uncertainty in th e system output determ ination, uncertainties arising from the calibration curve fit, and system atic errors associated w ith inapplicability of the calibration relation. As discussed above, the variation in sensor o u tp u t can be reduced below one gray level by averaging a sufficient num ber of m easurem ents. In all analyses 2 0 sam ples were averaged, giving a random error in th e determ ination of V, under the inherent digitizer resolution, less then 0.5 percent. T he uncertainty in the fit, which reflects the random errors in the calibration m easurem ents, is on Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 305 th e order of 0.04 m W //im cm2, or less than 0.5 percent a t the higher tem peratures. However, there could be a system atic error in the calibration equal to th e uncertainty in th e radiance values of th e calibration sources, which is about 3 percent. T h e applicability of th e calibration rests on th e validity of the assum ptions lead ing to Eq. (C.4). It is assum ed th a t the gains of the electronic com ponents are constant, and th a t the d etector responsivity Ro is the sam e for all sensors in the array, because the sensors which receive the cathode im age in the experim ents are n o t necessarily those used in the system calibration. T he tests w ith th e blackbody source indicated th a t there is very little drift in the system gain. In addition, the variability among im age pixel outputs is only 0.5 to 1.5 gray levels, and th e effect of th is uncertainty on tem perature m easurem ents can be minimized by averaging several adjacent pixels. T he calibration relation also contains the source-detector geom etry in the term s A c o sB , ft, so errors in these factors produce an uncertainty in the calculated radi ance. For the focal length and source-detector distance used in these tests an error in th e setup distance of as much as 25 mm produces less than a 1 percent deviation from th e responsivity m easured in the calibration. For the calibration relation to be valid the system optics m ust have the same transm ittances as those used for the calibration. T his requires th a t th e sam e optics b e used, th a t the surfaces be kept clean, and th a t all adjustable ap ertu re settings Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 306 be repeatable. T he m axim um aperture of the lens was chosen because interm edi ate settings of the diaphram proved to be irreproducible. Dlumination levels were controlled with neutral density filters. T he final assum ption used in the calibration is th a t the observed radiance is em itted by the cathode surface, and not other sources such as plasm a radiation. T he interference filter was chosen because it minimized the contribution from the intense plasm a. T he prim e contributor is apparently continuum radiation a t the wavelength of the interference filters. The plum e intensity typically observed several mm in front of the cathode tip corresponds to a bias of only about 20 K, assum ing th a t it is representative of th e plasm a signal intensity over th e cathode surface. Because th e plasm a layer over th e cathode surface im aged by the cam era is thinner than th e je t in front of th e tip , th e effect may be even less significant. In conclusion, random errors in the tem perature calculation associated with un certainties in the transm ittance of the neutral density filters, em ittance of th e sur face and th e m easurem ent of the incident radiance are under 1 percent. System atic errors from variations in source-detector geom etry and optical param eters can be minimized by careful atten tio n to the experim ental setup. However, underestim a tion of th e tru e em ittance because of the effects of roughness or oxide layers can lead to errors as high as 9 percent, and plasm a radiation perhaps as high as 1 percent. These tw o effects are difficult to quantify, and both overestim ate the tem perature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 307 R eferen ces [1] J . R. Brophy. Com et and A steroid Rendezvous Missions Using Solar Electric Propulsion Based on th e E L IT E Spacecraft. JP L IOM 353EP-92-048, Jet Propulsion Laboratory / California In stitu te of Technology, P asadena, CA, May 1992. [2] C. E. V aughan and R . J . Cassady. An U pdated Assessment of Electric Propul sion Technology for N ear-E arth Space Missions. In 28‘* Joint Propulsion Conference, Nashville, T N , 1992. AIAA 92-3202. [3] W . D. D eininger and R. J . Vondra. Electric Propulsion for C onstellation De ploym ent and Spacecraft M aneuvering. In 25,A Jo in t Propulsion Conference, Boston, M A, 1988. AIAA 88-2833. [4] A. S B ober, V. P. Kim, A. S. Koroteev, L. A. Latyshev, A. I. Morozov, G. A. Popov, Y . P. Rylov, and V . V. Zhurin. S tate of W ork on Electrical T hrusters in USSR. In 22n < i International Electric Propulsion Conference, Viareggio, Italy, 1991. IE PC 91-003. [5] J . W . B arn ett. A Review of Soviet Plasm a Engine Development. In 21J( International Electric Propulsion Conference, O rlando, FL, 1990. AIAA 90- 2600. [6 ] J . R. Brophy, J . W . B arn ett, J . M. 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Creator Goodfellow, Keith David (author)

Core Title A theoretical and experimental investigation of cathode processes in electric thrusters

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, aerospace,engineering, electronics and electrical,OAI-PMH Harvest,Physics, Fluid and Plasma

Language English

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Advisor Erwin, David (committee chair), Kunc, Joe (committee member), McCurdy, Alan (committee member), Muntz, Phillip (committee member)

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