UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An investigation of a vortex stabilized arc Neilson, John Bruce 1981

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1981_A1 N43.pdf [ 4.63MB ]
Metadata
JSON: 831-1.0085771.json
JSON-LD: 831-1.0085771-ld.json
RDF/XML (Pretty): 831-1.0085771-rdf.xml
RDF/JSON: 831-1.0085771-rdf.json
Turtle: 831-1.0085771-turtle.txt
N-Triples: 831-1.0085771-rdf-ntriples.txt
Original Record: 831-1.0085771-source.json
Full Text
831-1.0085771-fulltext.txt
Citation
831-1.0085771.ris

Full Text

AN INVESTIGATION OF A VORTEX STABILIZED ARC by JOHN BRUCE NEILSON B . S c , The U n i v e r s i t y Of B r i t i s h C o lumbia, 1975 M . S c , McMaster U n i v e r s i t y , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHYSICS) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d . THE UNIVERSITY OF BRITISH COLUMBIA September 1981 © J . Bruce N e i l s o n , 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agr e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . J. Bruce Neilson Department o f Physics The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date September 11, 1981 ( 9 /~7Q\ i i ABSTRACT The power b a l a n c e e q u a t i o n s f o r the column of a 500 A DC v o r t e x s t a b i l i z e d a r c i n argon have been s o l v e d n u m e r i c a l l y t o g i v e p r e d i c t e d v a l u e s f o r the temperature p r o f i l e , r a d i a t i v e power and heat t r a n s p o r t t o the w a l l . These q u a n t i t i e s have a l s o been measured u s i n g s p e c t r o s c o p y and c a l o r i m e t r y , and compared w i t h the t h e o r e t i c a l r e s u l t s so as t o r e f i n e our u n d e r s t a n d i n g of the heat t r a n s p o r t p r o c e s s e s i n the a r c . F u r t h e r study i s r e q u i r e d t o p r e d i c t c o n v e c t i v e e f f e c t s a c c u r a t e l y , e s p e c i a l l y a t low power l e v e l s , but the r e s u l t s show t h a t the b e h a v i o u r of the column i s p r e d i c t a b l e a t h i g h power l e v e l s . F o r c e d c o n v e c t i o n has been shown t o p l a y an im p o r t a n t r o l e i n the a r c column. i i i TABLE OF CONTENTS A b s t r a c t i i T a b l e Of C o n t e n t s i i i L i s t Of T a b l e s v L i s t Of F i g u r e s v i Acknowledgements i x Nomenclature x I . I n t r o d u c t i o n 1 I I . Theory Of The Arc Column 8 A. I n t r o d u c t i o n 8 B. The E l e n b a a s - H e l l e r E q u a t i o n 9 C. The Computer C a l c u l a t i o n s 11 D. Assumptions 20 E. C o n v e c t i v e T r a n s p o r t 24 I I I . E x p e r i m e n t a l A p p a r a t u s 37 A. I n t r o d u c t i o n 37 B. The Gas System 39 C. The V o r t e x 41 D. The C o o l i n g System 44 E. The E l e c t r i c a l System < 45 F. The Arc V e s s e l And E l e c t r o d e s 47 IV. D i a g n o s t i c Methods 52 A. I n t r o d u c t i o n 52 B. The O p t i c a l System 53 C. The Data H a n d l i n g System 56 D. AI L i n e I n t e n s i t y Measurements 57 E. A b s o l u t e I n t e n s i t y C a l i b r a t i o n 61 i v F. AI L i n e Width Measurements 65 G. Continuum I n t e n s i t y Measurements 66 V. R e s u l t s And A n a l y s i s 69 A. I n t r o d u c t i o n 69 B. C a l o r i m e t r y 70 C. Temperature P r o f i l e s 81 D. L i n e Width And Continuum Measurements 85 E. S y s t e m a t i c E r r o r s 86 F. Time V a r i a t i o n 89 G. C y l i n d r i c a l Symmetry 92 H. A x i a l V a r i a t i o n 92 V I . C o n c l u s i o n s 96 A. I n t r o d u c t i o n 96 B. C o n c l u s i o n s 97 C. O r i g i n a l C o n t r i b u t i o n s 99 D. Suggested F u t u r e Work 100 B i b l i o g r a p h y 103 Appendix A. Steady S t a t e C a l c u l a t i o n Programs 105 Appendix B. T r a n s i e n t H e a t i n g C a l c u l a t i o n Programs ....107 Appendix C. Data H a n d l i n g Programs 110 Appendix D. Gas Flow In The System 122 Appendix E. The V i s c o u s Decay Of A V o r t e x 127 V LIST OF TABLES Tab l e I I - 1 . Assumptions For The Steady S t a t e Model ....20 T a b l e I I - 2 . Models Of Argon M a t e r i a l F u n c t i o n s 25 T a b l e D-1. Gas Flow Rates And P r e s s u r e s 126 LIST OF FIGURES F i g u r e 1-1. The V o r t e x S t a b i l i z e d A rc 4 F i g u r e 11 -1 . Regions Of The Arc 8 F i g u r e I I - 2 . V a r i a t i o n Of P r o f i l e W i t h S t a r t i n g Temperature 12 F i g u r e I I - 3 . T vs R. P=1 Atm 14 F i g u r e I I - 4 . T vs R. P=3 Atm 15 F i g u r e I I - 5 . T vs R. P=5 Atm 16 F i g u r e I I - 6 . T vs R. P=7 Atm 17 F i g u r e I I - 7 . Core Temperature vs C u r r e n t 19 F i g u r e I I - 8 . Core R a d i u s vs C u r r e n t 19 F i g u r e I I - 9 . K vs T (T>7000K) 26 F i g u r e 1 1 - 1 0 . K vs T (T<7000K) 26 F i g u r e 1 1 - 1 1 . e vs T, P=1 Atm 27 F i g u r e 1 1 - 1 2 . Qrad vs T, P= 1 Atm 27 F i g u r e 1 1 - 1 3 . R a d i a l Isotherms vs P o s i t i o n 29 F i g u r e 1 1 - 1 4 . />Cp vs T 31 F i g u r e 1 1 - 1 5 . T vs R, T 0=10000K ...32 F i g u r e 1 1 - 1 6 . T vs R, T 0=11000K 33 F i g u r e 1 1 - 1 7 . 1000K Ra d i u s vs Time 34 F i g u r e 1 1 - 1 8 . Heat F l u x vs Time 34 F i g u r e 1 1 - 1 9 . P o s i t i o n Of The Gas D i s c C e n t e r Of Mass vs Time 35 F i g u r e 111-1. B l o c k Diagram Of The System 38 F i g u r e I I I - 2 . The Gas System 40 F i g u r e I I I - 3 . The C o o l i n g System 46 F i g u r e I I I - 4 . The E l e c t r i c a l System 46 F i g u r e I I I - 5 . T y p i c a l C u r r e n t And V o l t a g e Waveforms ...48 F i g u r e I I I - 6 . Drawing Of The Arc V e s s e l 51 F i g u r e I V - 1 . The D i a g n o s t i c System 55 F i g u r e IV-2. The AI 430 nm L i n e 59 F i g u r e I V - 3 . The C a l i b r a t i o n Setup 63 F i g u r e IV-4. Radiance vs B r i g h t n e s s Temperature 64 F i g u r e IV-5. N vs Radiance 64 F i g u r e IV-6. Temperature vs E l e c t r o n D e n s i t y 67 F i g u r e V-1. Power L o s s e s vs C u r r e n t , P=1.5 Atm 71 F i g u r e V-2. Power L o s s e s vs C u r r e n t , P=2.1 Atm 72 F i g u r e V-3. Power L o s s e s vs C u r r e n t , P=2.8 Atm 73 F i g u r e V-4. T o t a l Column L o s s e s vs C u r r e n t 75 F i g u r e V-5. R a d i a n t L o s s e s vs C u r r e n t 76 F i g u r e V-6. W a l l L o s s e s vs C u r r e n t 77 F i g u r e V-7. The M o d i f i e d W a l l L o a d i n g P r e d i c t i o n 80 F i g u r e V-8. T y p i c a l Temperature P r o f i l e s 82 F i g u r e V-9. Core Temperature vs C u r r e n t 83 F i g u r e V-10. Core R a d i u s vs C u r r e n t 84 F i g u r e V-11. Core Temperatures From Three Methods 87 F i g u r e V-12. C u r r e n t , Temperature And Ra d i u s vs Time ..91 F i g u r e V-13. T y p i c a l I n t e n s i t y P r o f i l e s 93 F i g u r e V-14. Temperature vs A x i a l P o s i t i o n 94 F i g u r e V-15. Core R a d i u s vs A x i a l P o s i t i o n 94 F i g u r e V I - 1 . The C o n v e c t i o n F r e e V o r t e x S t a b i l i z e d A rc 121 F i g u r e E-1. The F u n c t i o n J , ( x ) / x vs x 128 v i i i F i g u r e E-2. S o l i d Body Vortex 129 F i g u r e E-3. Annular J e t Vortex 129 i x ACKNOWLEDGEMENTS I would l i k e t o thank my s u p e r v i s o r Dr. Curzon f o r h i s support t hroughout t h i s work and f o r the many u s e f u l s u g g e s t i o n s he has made. I am a l s o g r a t e f u l t o the members of my s u p e r v i s o r y committee, D r s . A h l b o r n , Nodwell and White, f o r t h e i r h e l p i n the p r e p a r a t i o n of t h i s t h e s i s . In a d d i t i o n I have g a i n e d a g r e a t d e a l from many u s e f u l d i s c u s s i o n s w i t h a l l the members of the plasma group, p a r t i c u l a r l y L o m e G e t t e l and John Pearson. The t e c h n i c a l s t a f f have always been e x t r e m e l y h e l p f u l , and I am i n d e b t e d e s p e c i a l l y t o A l Cheuck, who f i x e d a l l my e l e c t r o n i c equipment as f a s t as I c o u l d damage i t w i t h RF p u l s e s , and t o E r n i e W i l l i a m s who pr e p a r e d dozens of q u a r t z t u b e s . I g r a t e f u l l y acknowledge f i n a n c i a l s u p p o r t from the N a t i o n a l S c i e n c e and E n g i n e e r i n g Research C o u n c i l and the U n i v e r s i t y of B.C. T h i s t h e s i s i s d e d i c a t e d t o R. A. Nordman, who f i r s t i n t r o d u c e d me t o the j o y s of P h y s i c s . NOMENCLATURE a r c v e s s e l r a d i u s t r a n s i t i o n p r o b a b i l i t y speed of l i g h t c o n s t a n t p r e s s u r e s p e c i f i c heat v e r t i c a l s p a c i n g between s p e c t r a e l e c t r i c f i e l d e x c i t e d s t a t e energy s t a t i s t i c a l weight P l a n c k c o n s t a n t c u r r e n t volume e m i s s i v i t y t h e r m a l c o n d u c t i v i t y Boltzmann c o n s t a n t a n g u l a r momentum argon atomic mass mass f l o w r a t e number of OMA c o u n t s n e u t r a l gas d e n s i t y e l e c t r o n d e n s i t y e x c i t e d s t a t e p o p u l a t i o n p r e s s u r e power r a d i a t i v e e m i s s i o n of argon power r e q u i r e d f o r gas h e a t i n g x i r a d i a t i v e column l o s s e s Q T • t o t a l i n p u t power Q w • power l o a d on v e s s e l w a l l r r a d i a l p o s i t i o n r i r a d i u s of 1000K i s o t h e r m r 1 0 - r a d i u s of 10000K i s o t h e r m R v i s c o u s f l o w r e s i s t a n c e Ro, r a d i a n c e per OMA count R« 3 0 r a d i a n c e a t 430 nm S i n t e g r a t e d t h e r m a l c o n d u c t i v i t y T t e m p e r a t u r e T 0 c e n t e r t e m p e r a t u r e V v e l o c i t y V v o l t a g e w OMA c h a n n e l w i d t h y v e r t i c a l p o s i t i o n 2 a x i a l p o s i t i o n V s p e c i f i c heat r a t i o X w a v e l e n g t h f v i s c o s i t y fi d e n s i t y a e l e c t r i c a l c o n d u c t i v i t y T decay time c o n s t a n t continuum e m i s s i o n f a c t o r U a n g u l a r v e l o c i t y 1 CHAPTER I INTRODUCTION For many y e a r s the e l e c t r i c a r c has been a mai n s t a y of e x p e r i m e n t a l plasma p h y s i c s , as w e l l as an e x t r e m e l y u s e f u l h i g h i n t e n s i t y l i g h t s o u r c e , both f o r g e n e r a l l i g h t i n g and f o r s p e c t r o s c o p i c i n v e s t i g a t i o n s . S i n c e the f i r s t a r c d i s c h a r g e was produced by Davy and R i t t e r i n 1808 between c h a r c o a l e l e c t r o d e s i n a i r , many d i f f e r e n t t y p e s of a r c have been s t u d i e d by many i n v e s t i g a t o r s . For a r e v i e w of the h i s t o r i c a l development and b a s i c p h y s i c s of a r c s the i n t e r e s t e d reader i s r e f e r r e d t o F i n k e l n b u r g and Maecker (1956) or Hoyaux (1968). B e f o r e g o i n g f u r t h e r we s h o u l d f i r s t c o n s i d e r the q u e s t i o n of what c o n s t i t u t e s an a r c . P r a c t i c a l l y s p e a k i n g , an a r c i s an e l e c t r i c d i s c h a r g e ( i . e . an e l e c t r i c c u r r e n t c a r r i e d by a gaseous c o n d u c t o r ) of l a r g e c u r r e n t (>1A) and s m a l l v o l t a g e (<100V). G e n e r a l l y t h r e e t y p e s of d i s c h a r g e are e n c o u n t e r e d ; a r c s , s p a r k s and glow d i s c h a r g e s , and d i f f e r e n t a u t h o r s ( F i n k e l n b u r g and Maecker 1956, Hoyaux 1968, Kesaev 1964) have d i f f e r e n t c r i t e r i a f o r s e p a r a t i n g them, a l t h o u g h they g e n e r a l l y g i v e the same g r o u p i n g s . Sparks a r e e a s i l y d i s t i n g u i s h e d as b e i n g of d u r a t i o n comparable t o the e q u i l i b r a t i o n time of the d i s c h a r g e ( 1 0 " 6 s ) , but the d i s t i n c t i o n between glow d i s c h a r g e s and some a r c s i s not so c l e a r - c u t . The most o b v i o u s d i f f e r e n c e i s the magnitude of the c u r r e n t d e n s i t y , which i s much lower 2 i n a glow d i s c h a r g e . I t a p p e a r s , however, t h a t the mechanism which s e t s t h i s d e n s i t y i s a f e a t u r e of the cathode r e g i o n . In an a r c , the v o l t a g e drop between the cathode and the a d j a c e n t plasma i s of the o r d e r of the i o n i z a t i o n p o t e n t i a l of the gas, and a cathode spot o c c u r s from which a v e r y l a r g e e l e c t r o n c u r r e n t d e n s i t y i s e m i t t e d by some c o m b i n a t i o n of t h e r m i o n i c and f i e l d e m i s s i o n . T h i s i s q u i t e d i f f e r e n t from the case i n a glow d i s c h a r g e , i n which the e l e c t r o n s a r e e m i t t e d w i t h a v e r y much lower c u r r e n t d e n s i t y , o f t e n from a l a r g e r s u r f a c e a r e a , and the cathode v o l t a g e i s on the o r d e r of ten times t h e i o n i z a t i o n p o t e n t i a l . The presence of a cathode s p o t , w i t h g r e a t l y enhanced e l e c t r o n e m i s s i o n a t low v o l t a g e , t h e r e f o r e d i s t i n g u i s h e s an a r c . One of the major problems t o be d e a l t w i t h i n m a i n t a i n i n g an a r c i s p o s i t i o n a l s t a b i l i t y ; l e f t t o i t s own d e v i c e s , an a r c i s o f t e n s e v e r e l y d i s r u p t e d by the n a t u r a l c o n v e c t i v e f o r c e which d r i v e s hot plasma upward i n the s u r r o u n d i n g c o l d gas (the f a m i l i a r 'Jacob's l a d d e r ' demonstrates t h i s b e h a v i o u r ) . S e v e r a l methods can be used t o c o n t r o l t h i s tendency, depending on what i s r e q u i r e d . For many purposes an a r c can be s t a b i l i z e d by s i m p l y keeping i t s h o r t so t h a t the attachment t o the e l e c t r o d e s i s s u f f i c i e n t f o r s t a b i l i z a t i o n . T h i s type of ' f r e e - b u r n i n g ' a r c i s f r e q u e n t l y used f o r a r c l i g h t i n g and sometimes f o r s p e c t r o s c o p y , and has the advantage of g r e a t s i m p l i c i t y . A n o t h e r , s l i g h t l y more demanding method of s t a b i l i z a t i o n i s 3 t o s u r r o u n d the a r c w i t h a w e l l - c o o l e d w a l l . T h i s w a l l must be a b l e t o a b s o r b a l a r g e heat f l u x w i t h o u t b e i n g d e s t r o y e d , and i s t h e r e f o r e u s u a l l y made of a s t a c k of water c o o l e d copper r i n g s , i s o l a t e d e l e c t r i c a l l y from one a n o t h e r . The o b v i o u s d i s a d v a n t a g e of t h i s scheme i s t h a t the w a l l i s opaque, and f o r t h i s reason w a l l s t a b i l i z e d or 'cascade' a r c s a r e used m a i n l y f o r r e s e a r c h , where they p r o v i d e u n i f o r m , s t e a d y and r e p r o d u c i b l e h i g h t e m p e r a t u r e plasmas f o r study (Maecker 1960, P r e s t o n 1977, B a e s s l e r 1980, Chen 1980) . A t h i r d method of s t a b i l i z a t i o n i s t o h o l d the a r c s t a t i o n a r y w i t h a gas f l o w , e i t h e r i n the form of a j e t or a v o r t e x f l o w . T h i s r e q u i r e s a more c o m p l i c a t e d system, but has the advantage t h a t i t i s c o m p a t i b l e w i t h a t r a n s p a r e n t w a l l and a l o n g a r c , and thus has g r e a t p o t e n t i a l f o r h i g h i n t e n s i t y l i g h t i n g a p p l i c a t i o n s . The b e h a v i o u r of an a r c i n a gas f l o w i s a l s o of i n t e r e s t f o r c i r c u i t b r e a k e r d e s i g n , and much work has been done on a r c s i n f l o w i n g gas f o r t h a t reason (Chien and Benenson 1980, P f e n d e r 1980). The a r c w i t h which t h i s t h e s i s i s concerned i s v o r t e x s t a b i l i z e d ; i n t h i s system the a r c column i s s t a b i l i z e d i n the c e n t e r of the q u a r t z v e s s e l by a gas v o r t e x c r e a t e d by i n j e c t i n g argon t a n g e n t i a l l y a t h i g h speed i n one end of the v e s s e l (see F i g u r e 1-1). The v o r t e x s t a b i l i z e d a r c , f i r s t used by Schoenherr (1909) i s b e i n g s t u d i e d by s e v e r a l groups (Tuchman 1967, Tarn 1972, Camm 1974, G e t t e l 1980) as a p o s s i b l e v e r y h i g h i n t e n s i t y l i g h t s ource f o r use i n l a r g e 4 Gas Vortex Figure 1-1. The vortex s t a b i l i z e d arc 5 ar e a l i g h t i n g , s e a r c h and r e s c u e , s o l a r s i m u l a t i o n , and o t h e r a p p l i c a t i o n s . V o r t e k I n d u s t r i e s i s now d e v e l o p i n g a commercial model of t h i s t ype of a r c . A s i n g l e v o r t e x s t a b i l i z e d a r c lamp may be a b l e t o use lOOkW of power t o produce l i g h t w i t h a n e a r l y s o l a r spectrum a t above 50% e f f i c i e n c y . G e t t e l (1980) has r e c e n t l y i n v e s t i g a t e d an AC v o r t e x s t a b i l i z e d a r c and found t h a t i t has some advantages over the DC a r c f o r l i g h t i n g a p p l i c a t i o n s . These advantages i n c l u d e improved e l e c t r o d e l i f e t i m e s and reduced power s u p p l y c o s t f o r an AC a r c . He s t u d i e d the e l e c t r o d e c o n d i t i o n s i n d e t a i l and d i d p a r a m e t r i c s t u d i e s of the e f f i c i e n c y of DC and AC a r c s , which r e v e a l e d some f e a t u r e s of the a r c t h a t a r e not c l e a r l y u n d e r s t o o d , and p r o v i d e d p a r t of the m o t i v a t i o n f o r the work d e s c r i b e d i n t h i s t h e s i s . In o r d e r t o improve our u n d e r s t a n d i n g of the b e h a v i o u r of the a r c , w i t h a view t o i n c r e a s i n g i t s e f f i c i e n c y and l o n g e v i t y , the temperature p r o f i l e s w i t h i n the a r c have been measured s p e c t r o s c o p i c a l l y and c a l c u l a t e d t h e o r e t i c a l l y from the power b a l a n c e e q u a t i o n s over a range of c u r r e n t s and p r e s s u r e s . The heat l o a d i n g on each component of the system has a l s o been c a l c u l a t e d t h e o r e t i c a l l y and measured u s i n g c a l o r i m e t r y . The measured and c a l c u l a t e d r e s u l t s have been compared, and some c o n c l u s i o n s drawn about the mechanisms i n the a r c , and the e f f e c t w hich the gas f l o w has on the p o s i t i v e column. The purpose of the m o d e l l i n g p r o c e s s has 6 not been s i m p l y t o p r e d i c t the b e h a v i o u r of the a r c ; the aim i s p r i m a r i l y t o t e s t and improve our u n d e r s t a n d i n g of the b a s i c p r o c e s s e s o c c u r r i n g i n the a r c column. Chapter I I c o n t a i n s the t h e o r y of the column which was used f o r the c a l c u l a t i o n s , t he assumptions made, and the r e s u l t s of tho s e c a l c u l a t i o n s . I t a l s o c o n t a i n s the s o l u t i o n of the c o n v e c t i v e heat l o s s problem which a r i s e s from the f a c t t h a t the gas f l o w c a r r i e s heat out of the a r c a x i a l l y . In Chapter I I I the e x p e r i m e n t a l a p p a r a t u s used f o r the e x p e r i m e n t a l work i s d e s c r i b e d , and the gas f l o w dynamics a r e examined i n d e t a i l . Some of the assumptions made i n Chapter I I a r e j u s t i f i e d i n g r e a t e r depth h e r e . Chapter IV c o n t a i n s a d e s c r i p t i o n of the t h e o r e t i c a l b a s i s of t h e d i a g n o s t i c t e c h n i q u e s used and the da t a a n a l y s i s r e q u i r e d t o e x t r a c t the temperature p r o f i l e s from the raw d a t a . In Chapter V the e x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d and c r i t i c a l l y examined. Some e x p e r i m e n t a l j u s t i f i c a t i o n i s p r e s e n t e d f o r a number of ass u m p t i o n s g i v e n i n Chapter I I . The r e s u l t s a r e here compared w i t h the t h e o r e t i c a l p r e d i c t i o n s , and the agreement i s examined and d i s c u s s e d . Chapter VI c o n t a i n s the c o n c l u s i o n s of the work, an e v a l u a t i o n of f u t u r e p r o s p e c t s and s u g g e s t i o n s f o r f u t u r e work. In t h i s c h a p t e r the a u t h o r ' s o r i g i n a l c o n t r i b u t i o n s t o t h i s f i e l d of study a re a l s o p r e s e n t e d . Appendices A, B and C l i s t the computer programs used i n t h i s t h e s i s f o r the temp e r a t u r e c a l c u l a t i o n s , c o n v e c t i v e 7 t r a n s p o r t c a l c u l a t i o n s and d a t a a n a l y s i s , r e s p e c t i v e l y . Appendix D c o n t a i n s the d e t a i l s of the gas f l o w i n the system and Appendix E c o n t a i n s the s o l u t i o n of the d i f f e r e n t i a l e q u a t i o n d e r i v e d i n Chapter I I I which governs the b e h a v i o u r of the gas v o r t e x . 8 CHAPTER I I THEORY OF THE ARC COLUMN A. INTRODUCTION An a r c can be d i v i d e d i n t o t h r e e d i s t i n c t r e g i o n s ; the cathode r e g i o n , the anode r e g i o n , and the a r c column (see F i g u r e 11 — 1 ). The e l e c t r o d e r e g i o n s a r e s t r o n g l y i n f l u e n c e d by the pr e s e n c e of the e l e c t r o d e s . Heat i s conducted t o the e l e c t r o d e s , r a d i a l e l e c t r i c f i e l d s a r e produced, and the a x i a l f i e l d i s enhanced near the e l e c t r o d e s , a l l of which g r e a t l y c o m p l i c a t e s the b e h a v i o u r of the s e r e g i o n s . C a t h o d e R e g i o n C a t h o d e C o l u m n A n o d e R e g i o n L 7 A n o d e F i g u r e I I - 1 . Regions of the a r c The c e n t r a l r e g i o n , or a r c column, i s more e a s i l y 9 a n a l y s e d , s i n c e the e l e c t r i c f i e l d i s u n i f o r m and a x i a l . The column makes up a l a r g e p a r t of the a r c , and i s t h u s of prime c o n c e r n f o r l i g h t i n g a p p l i c a t i o n s and i s the f o c a l p o i n t of t h i s t h e s i s . T h i s c h a p t e r d e s c r i b e s how the temperature p r o f i l e i n the a r c column i s c a l c u l a t e d a t v a r i o u s c u r r e n t s and p r e s s u r e s . The assumptions made f o r the a n a l y s i s a r e a l s o g i v e n , a l o n g w i t h the r e s u l t s of t h e s e c a l c u l a t i o n s . The e v a l u a t i o n of the r e s u l t s and comparison w i t h the e x p e r i m e n t a l measurements i s l e f t t o Chapter V. To c a l c u l a t e the temperature p r o f i l e the E l e n b a a s -H e l l e r e q u a t i o n , which i s d e s c r i b e d i n s e c t i o n B, i s s o l v e d n u m e r i c a l l y by computer. E x p e r i m e n t a l v a l u e s from the l i t e r a t u r e a r e used f o r the p r o p e r t i e s of argon ( c o n d u c t i v i t y , e t c . ) and a number of assumptions a r e made about the a r c column. The assumptions a r e g i v e n and j u s t i f i e d i n s e c t i o n C and the c a l c u l a t i o n s a r e d e s c r i b e d i n s e c t i o n D. S i n c e t h e computer model n e g l e c t s the e f f e c t s of c o n v e c t i v e heat t r a n s p o r t , c o r r e c t i o n s t o the model a r e r e q u i r e d . Another computer model was used t o p r e d i c t t h e s e c o r r e c t i o n s , and t h a t model i s d e s c r i b e d i n s e c t i o n E. B. THE ELENBAAS-HELLER EQUATION The n u m e r i c a l s o l u t i o n s f o r the a x i a l l y i n v a r i a n t , r a d i a l l y symmetric t e m p e r a t u r e p r o f i l e s a r e found by t h e method of c o n s e r v a t i o n of energy. In t h i s p a r t i c u l a r system, the c o n s e r v a t i o n of energy i s most e a s i l y e x p r e s s e d as 10 'power b a l a n c e ' , i . e . power i n e q u a l s power o u t . We c o n s i d e r a s m a l l volume of plasma, and c o n s t r u c t t h e e q u a t i o n t h a t e quates power i n t o power o u t . The assumptions made suggest the use of c y l i n d r i c a l c o o r d i n a t e s t o d e s c r i b e the plasma. The d i f f e r e n t i a l volume element i s rdrdzd© and we i n t e g r a t e dz and de t o get a c y l i n d r i c a l s h e l l of u n i t l e n g t h , a r e a 2 nr, and volume 2 i r r d r . The heat conducted t h r o u g h a s u r f a c e a t r a d i u s r i s 2irr»KdT/dr, and the net heat c o n d u c t i o n i n t o the s h e l l i s thus 2irdr«d/dr ( r K d T / d r ) , where K i s the t h e r m a l c o n d u c t i v i t y of t h e argon i n the s h e l l . There i s a l s o an ohmic h e a t i n g term w h i c h , f o r an a p p l i e d f i e l d E and e l e c t r i c a l c o n d u c t i v i t y c, i s 2irrdr«tfE2, and f i n a l l y t h e r e i s a r a d i a t i v e l o s s term g i v e n by the r a d i a t i v e l o s s f u n c t i o n °-rad ' w h i c h * s 2irrdr«Q_rad . The power b a l a n c e e q u a t i o n i s thu s 27Tdr ^ r(rKg) + 2 1rrdr-aE 2 = 2 T r r d r - Q r a d or I d ( r K d T = Q a E 2 ( H - D r dr dr x r a d E q u a t i o n 11— 1 , known i n the l i t e r a t u r e as the m o d i f i e d E l e n b a a s - H e l l e r e q u a t i o n , i s the e q u a t i o n t h a t f o l l o w s from power b a l a n c e or c o n s e r v a t i o n of energy. I f t h e parameters were c o n s t a n t , i t would be a r a t h e r s i m p l e second o r d e r d i f f e r e n t i a l e q u a t i o n , and c o u l d be s o l v e d a n a l y t i c a l l y , but i t i s c o m p l i c a t e d by t h e f a c t t h a t K, Q r a^ and a are s t r o n g 11 f u n c t i o n s of t e m p e r a t u r e and p r e s s u r e , which makes the problem t r a c t a b l e o n l y by n u m e r i c a l i n t e g r a t i o n . E q u a t i o n 11 — 1 can e a s i l y be i n t e g r a t e d t o the form ( I I - 2 ) and i s then amenable t o n u m e r i c a l i n t e g r a t i o n from r=0. The models used f o r K, Q r a (j and « are d e s c r i b e d i n s e c t i o n D. C. THE COMPUTER CALCULATIONS The c a l c u l a t i o n b e g i n s by s e t t i n g a p r e s s u r e and a f i e l d s t r e n g t h . A s t a r t i n g c o r e t e m p e r a t u r e T 0 i s chosen, and E q u a t i o n I I - 2 i s i n t e g r a t e d from t h e c e n t e r out t o r a d i u s a. Once the m a t e r i a l f u n c t i o n s and c o r e temperature T 0 a r e g i v e n , t h i s i n t e g r a t i o n i s s t r a i g h t f o r w a r d , as d e s c r i b e d i n Appendix A. The d i f f i c u l t p a r t of the c a l c u l a t i o n i s t o e n f o r c e the boundary c o n d i t i o n (T=600K) a t r=a. A f i r s t attempt a t ' n e g a t i v e feedback' was t o s u b t r a c t a s m a l l f r a c t i o n of t h e temperature e r r o r a t the w a l l from the c o r e t e m p e r a t u r e w i t h each i t e r a t i o n , but t h i s p r o c e d u r e d i d not converge w e l l , because the w a l l t e mperature i s such a s t r o n g f u n c t i o n of the c o r e temperature (or c o n v e r s e l y the c o r e t e m p e r a t u r e depends v e r y weakly on t h e w a l l t e m p e r a t u r e ) . For example, w i t h a g i v e n p r e s s u r e and e l e c t r i c f i e l d , c h a n g i n g the s t a r t i n g t e mperature by 1K c o u l d change the c a l c u l a t e d w a l l t e m p e r a t u r e from 300K t o 10000K (see F i g u r e I I - 2 ) . T h i s remarkable swing i s due t o the s t r o n g t emperature dependence of t h e r a d i a t i v e l o s s e s ; a 12 s m a l l change i n c o r e t e m p e r a t u r e can a l t e r the c o r e c o n d i t i o n s enough t h a t m a i n t a i n i n g t h a t c o r e temperature r e q u i r e s heat i n p u t from o u t s i d e the column. T h i s heat must be conducted i n , and so the s o l u t i o n f o r the p e r t u r b e d c o r e t e m p e r a t u r e r e q u i r e s t e m p e r a t u r e s which i n c r e a s e w i t h r a d i u s . T h i s i l l u s t r a t e s the f a c t t h a t the w a l l temperature T T 1=10001 K - " V \ r 10000K \ <- r r -~» F i g u r e I I - 2 . V a r i a t i o n of p r o f i l e w i t h s t a r t i n g temperature i s not c r i t i c a l t o the c a l c u l a t i o n , but i t makes i t more d i f f i c u l t t o e n f o r c e the boundary c o n d i t i o n . T h i s was e v e n t u a l l y done by a d e c r e a s i n g s t e p method, where a f i x e d c o r r e c t i o n was s u b t r a c t e d from the c o r e temperature r e p e a t e d l y u n t i l the c a l c u l a t e d w a l l t e m p e r a t u r e f e l l below the p r e s e t v a l u e (600K), whereupon the c o r r e c t i o n was d i v i d e d by 10 and the p r o c e s s c o n t i n u e d , u n t i l the c o r e 13 temperature converged on the c o r r e c t v a l u e from above. To i l l u s t r a t e t h i s , a sample approach t o , say, 8.46 would be the sequence (10,9,8.9,8.8,8.7,8.6,8.5,8.49,8.48,8.47,8.46). The c a l c u l a t i o n , as w e l l as p r o d u c i n g n u m e r i c a l r e s u l t s , g i v e s some i n t e r e s t i n g i n s i g h t s i n t o the a c t u a l a r c mechanisms. The convergence p r o c e s s r e f l e c t s the a c t u a l c o l l a p s e of an o v e r h e a t e d a r c column back t o e q u i l i b r i u m by r a d i a t i n g and c o n d u c t i n g away exce s s energy, and i n t h i s l i g h t the s t a b i l i t y problems e n c o u n t e r e d i n the c a l c u l a t i o n s can be seen t o be the same as the c o n d i t i o n s i n an a c t u a l a r c ; p r i m a r i l y they r e s u l t from the f a c t t h a t , once e x t i n g u i s h e d , an a r c w i l l not s p o n t a n e o u s l y r e s t a r t , a l t h o u g h i t i s ' l o c a l l y ' s t a b l e , i . e . s t a b l e over a moderate range of c o r e t e m p e r a t u r e s or c u r r e n t d e n s i t i e s , and i f kept w i t h i n t h i s range w i l l r e t u r n t o e q u i l i b r i u m . T h i s was i l l u s t r a t e d i n e a r l y a t t e m p t s a t c a l c u l a t i o n , when d u r i n g a s e r i e s of c a l c u l a t i o n s the c o n d i t i o n s would c r o s s a 'watershed' and b e g i n c o n v e r g i n g t o z e r o c u r r e n t . In f a c t , the z e r o c u r r e n t s o l u t i o n i s a p e r f e c t l y v a l i d , s t a b l e s o l u t i o n t o the g i v e n e q u a t i o n s , and t o a v o i d c o n v e r g i n g t o i t the method of a p p r o a c h i n g from above had t o be adopted. Once a s e l f - c o n s i s t e n t s o l u t i o n i s reached, the t o t a l c u r r e n t 1= 2irrtfEdr, which i s s e t by the v a l u e s of E and P, can be c a l c u l a t e d . Then the r a d i a t i v e and c o n d u c t i v e l o s s e s a r e c a l c u l a t e d and the r e s u l t s a r e p r i n t e d . R e p e a t i n g the e n t i r e c a l c u l a t i o n f o r d i f f e r e n t p r e s s u r e s and f i e l d F i g u r e I I - 3 . T vs r . P=1 Atm T e m p e r a t u r e ( K ) 1 2 0 0 0 1(A) .561 ,427 .300 .160 ,121 6 0 0 0 A 0 0 0 2 0 0 0 e — i — i 12 16 R a d i u s ( m m ) F i g u r e I I - 4 . T vs r . P=3 Atm F i g u r e I I - 5 . T vs r . P=5 Atm Temperature (K) F i g u r e I I - 6 . T vs r . P=7 Atm 18 s t r e n g t h s g e n e r a t e s a f a m i l y of t e m p e r a t u r e p r o f i l e s ( F i g u r e s I I - 3 t h r o u g h 6) which can be compared w i t h e x p e r i m e n t a l p r o f i l e s . F i g u r e s I I - 7 and 8 show p l o t s of c o r e t e m p e r a t u r e T 0 and c o r e r a d i u s r 1 0 vs p r e s s u r e and c u r r e n t . As t h e c u r r e n t i n c r e a s e s , the c o r e t e m p e r a t u r e and r a d i u s (which we d e f i n e as the r a d i u s a t which T=10000K) i n c r e a s e . T h i s a g r e e s w i t h what we might i n t u i t i v e l y e x p e c t . I n a d d i t i o n , a t h i g h c u r r e n t s and t e m p e r a t u r e s , the p r o f i l e s become much s q u a r e r as the r a d i a t i v e l o s s e s b e g i n t o dominate. The change i n shape of the c u r v e s i s due t o the c h a n g i n g mechanism of heat l o s s . At low temperature the r a d i a t i v e l o s s e s a r e s m a l l and the t e m p e r a t u r e p r o f i l e must have a l a r g e enough g r a d i e n t t o d r i v e the c o n d u c t i v e heat l o s s . At h i g h e r t e m p e r a t u r e s the r a d i a n t l o s s e s i n c r e a s e d r a m a t i c a l l y and s u f f i c i e n t heat can be r a d i a t e d w i t h o u t a l a r g e t e m p e r a t u r e g r a d i e n t . I n t h i s way, as the c o r e temperature i n c r e a s e s the p r o f i l e s become f l a t t e r i n t h e c e n t e r and s q u a r e r i n shape. The low temperature p r o f i l e s a r e q u i t e s i m i l a r i n shape t o t h o s e c a l c u l a t e d by M a e c k e r ( 1 9 5 9 ) , who m o d e l l e d * as a p i e c e w i s e l i n e a r f u n c t i o n of the i n t e g r a t e d t h e r m a l c o n d u c t i v i t y S=/KdT and then s o l v e d the n o n - r a d i a t i v e p r o f i l e a n a l y t i c a l l y . 3 A t m 20 D. ASSUMPTIONS In the p r e c e d i n g a n a l y s i s , a number of assumptions have been made about the a r c column, which w i l l now be g i v e n and j u s t i f i e d . The assumptions i n c l u d e a x i a l i n v a r i a n c e , r a d i a l symmetry, u n i f o r m p r e s s u r e , o p t i c a l t h i n n e s s , f i x e d w a l l t e m p e r a t u r e , and an assumed s e t of m a t e r i a l f u n c t i o n s ( i . e . c o n d u c t i v i t i e s and a t o t a l r a d i a t i v e e m i s s i o n c o e f f i c i e n t as f u n c t i o n s of temperature and p r e s s u r e ) . T a b l e 11 — 1 . Assumptions f o r the steady s t a t e model A x i a l I n v a r i a n c e R a d i a l Symmetry Un i f o r m P r e s s u r e O p t i c a l T h i n n e s s W a l l Temperature=600K M a t e r i a l F u n c t i o n s The a r c column or ' p o s i t i v e ' column r e f e r s t o t h a t p a r t of the a r c plasma which i s a x i a l l y i n v a r i a n t , i . e . not a f f e c t e d by the e l e c t r o d e s , and i t i s o n l y t h i s r e g i o n t o which the p r e s e n t a n a l y s i s i s a p p l i c a b l e . The e x p e r i m e n t a l measurements a r e made i n a d i s c of about 1 mm a x i a l t h i c k n e s s near the c e n t e r of a 100 mm l o n g a r c . In Chapter V r e s u l t s a r e g i v e n f o r measurements a t 10 mm i n t e r v a l s a l o n g the column which show t h a t the temperature p r o f i l e changes 21 v e r y l i t t l e o ver a t l e a s t the c e n t e r 30 mm of the a r c , and s i n c e the t o t a l c u r r e n t and d i a m e t e r a r e a l s o i n v a r i a n t , we can s a f e l y c o n c l u d e t h a t the a x i a l l y i n v a r i a n t a r c column makes up a l a r g e p a r t of t h e a r c , i n c l u d i n g the r e g i o n i n which the temperature measurements a r e made. When the a x i a l t emperature g r a d i e n t s a r e s m a l l the a x i a l gas f l o w has l i t t l e e f f e c t on the p r o f i l e , s i n c e we can move our c o o r d i n a t e system w i t h the gas. The assumption of r a d i a l symmetry has a l s o been v e r i f i e d e x p e r i m e n t a l l y (see Chapter V ) . The o n l y d e p a r t u r e s from symmetry a r e s m a l l f l u c t u a t i o n s i n the a r c d u r i n g the s c a n n i n g time which show up as f l u c t u a t i o n s i n the measured i n t e n s i t y p r o f i l e . These a r e d e a l t w i t h by a v e r a g i n g the two h a l v e s of t h e p r o f i l e t o g e t h e r and smoothing t h e r e s u l t i n g p r o f i l e p r i o r t o p r o c e s s i n g the d a t a , as d e s c r i b e d i n Chapter IV. R a d i a l symmetry s h o u l d be e x p e c t e d , s i n c e the v o r t e x a c c e l e r a t i o n i s much g r e a t e r than the asymmetric g r a v i t a t i o n a l buoyant f o r c e s . To show t h a t the assumption of u n i f o r m p r e s s u r e i s q u i t e j u s t i f i e d f o r the f l o w c o n d i t i o n s i n the e x p e r i m e n t a l a p p a r a t u s we must show t h a t both a x i a l and r a d i a l p r e s s u r e g r a d i e n t s a r e n e g l i g i b l e . I t i s easy t o c a l c u l a t e the a x i a l p r e s s u r e g r a d i e n t due t o the v i s c o u s drag of the a r c v e s s e l w a l l , and more d i f f i c u l t t o c a l c u l a t e t h e r a d i a l g r a d i e n t r e q u i r e d t o m a i n t a i n the c e n t r i p e t a l a c c e l e r a t i o n . These c a l c u l a t i o n s a r e c a r r i e d out i n d e t a i l i n Chapter I I I and i t i s shown t h a t the a x i a l p r e s s u r e v a r i a t i o n i s c o m p l e t e l y 22 n e g l i g i b l e (on the o r d e r of 1 p a r t i n 10 5) and t h a t the r a d i a l v a r i a t i o n i s about 2% i n gas a t room t e m p e r a t u r e . At h i g h e r t e m p e r a t u r e s the d e n s i t y i s reduced, and so when the a r c c o r e i s hot the r a d i a l p r e s s u r e v a r i a t i o n , which s c a l e s w i t h gas d e n s i t y , w i l l be on the o r d e r of 0.2%. None of the m a t e r i a l f u n c t i o n s v a r i e s s t r o n g l y w i t h p r e s s u r e , so such s m a l l v a r i a t i o n s a r e n e g l i g i b l e and the i s o b a r i c a p p r o x i m a t i o n i s indeed v a l i d . A plasma can be c o n s i d e r e d o p t i c a l l y t h i n i f i t does not a b s o r b a s i g n i f i c a n t f r a c t i o n of the r a d i a t i o n i t e m i t s . For the purposes of the model, o n l y average o p t i c a l t h i n n e s s i s r e q u i r e d , i . e . the t o t a l r e a b s o r b e d power s h o u l d be a n e g l i g i b l e f r a c t i o n of the r a d i a t e d power. In t h i s work power measurements a r e a c c u r a t e t o about 10%, so v a r i a t i o n s of l e s s than 5% can be n e g l e c t e d . At the p r e s s u r e s and t e m p e r a t u r e s under c o n s i d e r a t i o n t h i s c o n d i t i o n i s w e l l s a t i s f i e d . Evans and T a n k i n (1967) measured the t o t a l r a d i a t i o n from an argon plasma of d i a m e t e r about 10 mm a t a t m o s p h e r i c p r e s s u r e w i t h and w i t h o u t c o r r e c t i n g f o r s e l f -a b s o r p t i o n , and t h e i r r e s u l t s show a n e g l i g i b l e d i f f e r e n c e (<5%) a t t e m p e r a t u r e s below 12000K. I t i s g e n e r a l l y a c c e p t e d i n the l i t e r a t u r e (Hoyaux 1968, Evans 1967, O l s e n 1963) t h a t a t p r e s s u r e s below 10 Atmospheres and t e m p e r a t u r e s below 12000K, o n l y a few of the s t r o n g e s t atomic l i n e s can be o p t i c a l l y t h i c k . We w i l l r e t u r n t o t h i s q u e s t i o n i n Chapter IV where we must e s t a b l i s h the o p t i c a l t h i n n e s s of the e m i t t e d l i g h t used f o r d i a g n o s t i c s , but here a l l we need i s 23 t o show t h a t the r a d i a t i o n does not a f f e c t heat c o n d u c t i o n . The boundary c o n d i t i o n used i n t h e s e c a l c u l a t i o n s ( o n l y one i s r e q u i r e d because of the symmetry) i s the w a l l t e m p e r a t u r e . The a c t u a l w a l l t e m p e r a t u r e i n the experiment i s the te m p e r a t u r e of t h e i n s i d e s u r f a c e of the q u a r t z w a l l . The o u t s i d e s u r f a c e i s f i x e d a t t h e temp e r a t u r e of the c o o l i n g water (290±10K) and the i n s i d e t e m p e r a t u r e i s de t e r m i n e d by the t h e r m a l l o a d i n g . The s u r f a c e a r e a of 120 mm of 27 mm bore t u b i n g i s A=10" 2 m2, and the t o t a l heat f l u x i s about Q=3 kW or Q/A=300 kWm"2. T a k i n g the t h e r m a l c o n d u c t i v i t y of q u a r t z t o be K=1.4 Wm~1K~1, and the w a l l t h i c k n e s s as Ax=1.5 mm, the temp e r a t u r e d i f f e r e n c e w i l l be AT=QAx/KA=300K. T h i s c a l c u l a t i o n i s o b v i o u s l y o n l y an e s t i m a t e , but i t g i v e s the r e a s o n a b l e v a l u e of 600K f o r the w a l l t e m p e r a t u r e , w i t h an u n c e r t a i n t y of about 100K (30% i n AT). The n u m e r i c a l c a l c u l a t i o n s a r e i n any case v e r y i n s e n s i t i v e t o c o n d i t i o n s near the w a l l , as w i l l be shown l a t e r , so the s i m p l i f y i n g but not e n t i r e l y n e c e s s a r y assumption t h a t t h e w a l l t e mperature i s f i x e d a t 600K has been made. The f i n a l a s s u m p t i o n , or s e t of a s s u m p t i o n s , i s t h e s e t of models chosen f o r the t h e r m a l and e l e c t r i c a l c o n d u c t i v i t i e s K and e, and the t o t a l r a d i a t i v e e m i s s i v i t y °-rad °^ a r g o n . These p r o p e r t i e s do not have s i m p l e t e m p e r a t u r e and p r e s s u r e dependences, but have been m o d e l l e d a n a l y t i c a l l y u s i n g p u b l i s h e d e x p e r i m e n t a l d a t a . The models, shown i n F i g u r e s I I - 9 t h r o u g h 12, were c o n s t r u c t e d by 24 p l o t t i n g d a t a from the l i t e r a t u r e ( s o u r c e s a r e i n d i c a t e d on the a p p r o p r i a t e f i g u r e s ) and c h o o s i n g an a p p r o p r i a t e f u n c t i o n a l form f o r each c u r v e vs t e m p e r a t u r e , then f i t t i n g the parameters t o match the d a t a as c l o s e l y as p o s s i b l e a t each p r e s s u r e . P r e s s u r e s of 1,3,5 and 7 Atmospheres were used i n the c a l c u l a t i o n s . G e n e r a l l y the f i t i s good enough t o be w i t h i n the u n c e r t a i n t y of the o r i g i n a l source or the a c c u r a c y of r e a d i n g the graph from which the d a t a was t a k e n . The a n a l y t i c models which r e s u l t e d from t h i s p r o c e s s a r e l i s t e d i n T a b l e I I - 2 . The o n l y p a r t of t h e s e m a t e r i a l f u n c t i o n s which might d i f f e r s i g n i f i c a n t l y from the a n a l y t i c a p p r o x i m a t i o n s i s the th e r m a l c o n d u c t i v i t y K, which can be g r e a t l y i n f l u e n c e d by c o n v e c t i v e t r a n s p o r t such as t u r b u l e n t m i x i n g or f o r c e d c o n v e c t i o n . One of the o b j e c t s of t h i s t h e s i s , i n f a c t , i s to d e termine the importance of t u r b u l e n t m i x i n g i n the a r c . The i n f l u e n c e of f o r c e d c o n v e c t i o n i s s i g n i f i c a n t and w i l l be d e a l t w i t h s e p a r a t e l y , l a t e r i n t h i s c h a p t e r , when we c o n s i d e r a computer model of the t r a n s i e n t h e a t i n g of the argon gas i n the v o r t e x . E. CONVECTIVE TRANSPORT In the c o u r s e of t h i s r e s e a r c h , a s i g n i f i c a n t d i s c r e p a n c y appeared between the c a l c u l a t e d and measured v a l u e s of the w a l l l o a d i n g . T h i s was a l s o r e p o r t e d by G e t t e l (1980) who used a l e s s s o p h i s t i c a t e d model but found the same d i s c r e p a n c y , and su g g e s t e d i t might be due t o t u r b u l e n t T a b l e I I - 2 . Models of argon m a t e r i a l f u n c t i o n s P r e s s u r e ( A t m ) : *o(Ohm - 1nr 1 ) Ts(K) Q 0(Wm- 3) Tm(K) Tq(K) K 0(Wm- 1K- 1) Tk(K) K,(Wm"'K-1) A(Wm- 1K" 2) <r=«0/exp(Ts/T) 2 Q_ f l f 1=Qo/exp( (T-Tm)/Tq) 2 K=K 0«exp(T/Tk) (T>7000K) (T<7000K K=K 1+A»T 1 1 .5E4 1.27E4 5.24E9 1.65E4 3500 1.08E-2 2400 1.1E-2 2.7E-5 3 1.81E4 1.41E4 5.55E10 2.0E4 4400 5 2.07E4 1 .5E4 1 .02E1 2 2.5E4 5700 7 2.29E4 1.81E4 6.69E13 2.7E4 5800 26 Region of interest < F i gu re 11-9. K vs T (T>7000K) o J 1 1 I 20OO 4 0 0 0 T ( K ) 6 0 0 0 F i g u r e 11- 10. K vs T (T<7000K) 27 6 0 0 0 -0" (A1m1) 4000 2 0 0 0 --+- E m m o n s (1967) M o d e I Region of interest < 5 0 0 0 10000 T ( K ) F i g u r e 1 1 - 1 1 . c vs T, P=1 Atm 1 0 1 0 -l "rad (Wm"3) 10 10 6 Reg ion of i n t e res t < E m m o n s ( 1967) M o d e l 5 0 0 0 10000 T(K) F i g u r e 11-12. Qrad vs T, P=1 Atm 28 heat t r a n s p o r t . A s i g n i f i c a n t f e a t u r e of the v o r t e x s t a b i l i z e d a r c , however, i s the l a r g e gas f l o w t h r o u g h the column. In t h i s e x p e r i m e n t , the f l o w was about l O - 3 k g s _ 1 , and i f t h a t gas i s heated from 300 t o 5000K when i t e n t e r s the column, and c o o l e d a g a i n when i t l e a v e s , a c o n v e c t i v e power l o s s of about 3 kW o c c u r s ( a t 3 Atm). T h i s i s a s i g n i f i c a n t amount of power, comparable t o the c a l c u l a t e d w a l l l o a d i n g , and c o u l d w e l l be the source of the above-mentioned d i s c r e p a n c y . In the i n i t i a l n o n - e q u i l i b r i u m r e g i o n the power l o s s from the c o r e i s enhanced due t o the l a r g e r temperature g r a d i e n t s , thus i n c r e a s i n g the n o n - r a d i a t i v e l o s s e s w i t h o u t i n v o k i n g t u r b u l e n t mechanisms. At the same time t h i s c o n v e c t i v e t r a n s p o r t would have l i t t l e e f f e c t on t h e c o r e t e m p e r a t u r e , which we have seen does not depend s t r o n g l y on the o u t s i d e c o n d i t i o n s , a l t h o u g h i t w i l l of c o u r s e a f f e c t the o u t e r p a r t s of the temperature p r o f i l e , and might a f f e c t the c o r e r a d i u s . In o r d e r t o examine the importance of t h i s c o n v e c t i v e t r a n s p o r t , a c a l c u l a t i o n has been made of the t r a n s i e n t b e h a v i o u r of a c o l d gas f l o w down a 27 mm d i a m e t e r t u b e , w i t h a heat r e s e r v o i r of 13 mm d i a m e t e r a t 10000K i n i t s c e n t e r . F i g u r e 11-13 i s a s k e t c h i n d i c a t i n g the v a r i a t i o n i n the t emperature p r o f i l e w i t h a x i a l p o s i t i o n , showing s e v e r a l i s o t h e r m s i n the column. By moving w i t h the gas f l o w the problem i s made s i m p l e r , and we can d e a l w i t h the d i f f u s i o n of heat i n a u n i f o r m c y l i n d e r . To r e l a t e the t e m p o r a l t o the a x i a l v a r i a t i o n we use the mass f l o w r a t e of argon m t o 29 13 mm r Electro de 300K 5000K 10000K 27mm F i g u r e 11-13. R a d i a l isotherms vs p o s i t i o n c a l c u l a t e the v e l o c i t y which can be i n t e g r a t e d to g i v e the p o s i t i o n z=/vdt as a f u n c t i o n of time. The c a l c u l a t i o n was performed by m o d e l l i n g the thermal c o n d u c t i v i t y K, d e n s i t y p and heat c a p a c i t y C p of argon and n u m e r i c a l l y i n t e g r a t i n g the heat flow equation which can be w r i t t e n where T ( t , r ) i s the temperature as a f u n c t i o n of time and r a d i u s . The same c o n d u c t i v i t y model was used as f o r the temperature p r o f i l e c a l c u l a t i o n (see F i g u r e s II-9 and 10) and the q u a n t i t y pC p, the heat c a p a c i t y per u n i t volume at 9T = _1 3_ 3t ~ pC r 3r P ( I I - 4 ) 30 c o n s t a n t p r e s s u r e , was m o d e l l e d as shown i n F i g u r e 11-14. These c a l c u l a t i o n s were performed a t 1.44 Atm, s t a r t i n g w i t h a p r o f i l e made up of T=T 0 f o r r<6.6 mm and T=300K f o r r>6.6 mm. The boundary c o n d i t i o n s a p p l i e d i n the subsequent time e v o l u t i o n were T=T 0 a t r=6.6 mm and T=300K a t r=13.5 mm (the r a d i u s of the c o n f i n i n g t u b e ) . These boundary c o n d i t i o n s a re q u i t e s i m i l a r t o the c o n d i t i o n s imposed on c o l d argon f l o w i n g over a 6.4 mm r a d i u s e l e c t r o d e and e n t e r i n g the a r c column. The c a l c u l a t i o n s were performed f o r two v a l u e s of T 0, 10000K and 11000R. In F i g u r e s 11-15 and 16 the c a l c u l a t e d temperature p r o f i l e s a r e shown a t i n t e r v a l s of 5 m i l l i s e c o n d s as they e v o l v e , and i t can be seen t h a t the p r o f i l e s change smoothly from t h e i n i t i a l c o n d i t i o n s , r e a c h i n g s t e a d y s t a t e i n a time of about 20 m i l l i s e c o n d s . F i g u r e s 11-17 and 18 show some i m p o r t a n t f e a t u r e s of the s e p r o f i l e s , the r a d i u s R, a t which T=1000K and the power Q co n d u c t e d t o the w a l l , as f u n c t i o n s of t i m e . The r a d i u s changes smoothly from 6.6 t o 13.4 mm but t h e w a l l l o a d i n g undergoes a t r a n s i t i o n between t = 15 and 20 m i l l i s e c o n d s when the t emperature s t e p reaches the w a l l . T h i s sudden i n c r e a s e i n the w a l l l o a d i n g i s s i g n i f i c a n t when we c o n s i d e r the power b a l a n c e ; i n an a r c i n which the gas moves an a p p r e c i a b l e d i s t a n c e i n 20 m i l l i s e c o n d s the t h e r m a l l o a d on the w a l l s h o u l d be a s t r o n g f u n c t i o n of p o s i t i o n , i n c r e a s i n g q u i t e s t e e p l y a t the p o i n t where the heat wave reaches the w a l l . The a x i a l p o s i t i o n has been c a l c u l a t e d as a f u n c t i o n F i g u r e 11-14. pCp vs T F i gu re 11-15. T vs r, T 0=10000K T e m p e r a t u r e ( K ) 1 2 0 0 0 -I F i g u r e 11-16. T vs r, T 0 = 1 1 000K 15 H Vessel radius Rj (m m) 10 H — r 1 0 1— t (ms) — i — 2 0 Q (kWm1) 4 0 H F i g u r e 11-17. 1000K r a d i u s vs time 2 0 i F i g u r e 11-18. Heat f l u x vs time 35 F i g u r e 11-19. P o s i t i o n of the gas d i s c c e n t e r of mass vs time of time u s i n g E q u a t i o n 11-3 and i s p l o t t e d i n F i g u r e 11-19. I t can be seen from F i g u r e s I I - 1 8 and 1 9 t h a t the heat wave reaches the w a l l near z=50 mm. The c o n d u c t i v e l o a d i n g on the w a l l s h o u l d t h e r e f o r e o n l y be on the l a s t 50 mm of the column, which i s 100 mm l o n g . In a d d i t i o n t o t h i s c o n d u c t i v e l o a d i n g a s i g n i f i c a n t amount of power i s c a r r i e d away c o n v e c t i v e l y by the gas l e a v i n g the column, and t h i s must be i n c l u d e d i n t h e f i n a l model. These c a l c u l a t i o n s do not t a k e account of the f a c t t h a t the e l e c t r i c a l power i n p u t t o the column i s l i m i t e d , so they cannot be r e l i e d upon a t low power l e v e l s , i . e . when the power r e q u i r e d t o heat the gas i s more than h a l f the i n p u t power. At h i g h power l e v e l s the model i s v a l i d and i n d i c a t e s 36 t h a t the n o n - r a d i a t i v e l o s s e s s h o u l d i n c l u d e a x i a l c o n v e c t i o n as w e l l as the r a d i a l c o n d u c t i o n which o c c u r s i n 50 mm of column l e n g t h . The computer program "TRANSIT" which was used t o make the c a l c u l a t i o n s i s l i s t e d i n Appendix B. 37 /CHAPTER I I I EXPERIMENTAL APPARATUS A. INTRODUCTION The a p p a r a t u s used f o r the e x p e r i m e n t a l work c o n s i s t s of two p a r t s ; f i r s t l y t h e a r c v e s s e l , w i r i n g , gas and water systems and o t h e r equipment r e q u i r e d t o o p e r a t e the a r c , and s e c o n d l y the o p t i c a l d i a g n o s t i c s and d a t a p r o c e s s i n g equipment which were used t o make measurements on the a r c . The equipment used t o m a i n t a i n the a r c i s d e s c r i b e d i n d e t a i l i n t h i s c h a p t e r , and the o p t i c a l d i a g n o s t i c system i s d i s c u s s e d i n the next c h a p t e r . A v o r t e x s t a b i l i z e d a r c can be d e s c r i b e d i n terms of a v e s s e l i n which the a r c burns and t h r e e s e p a r a t e subsystems t o p r o v i d e e l e c t r i c a l power, c o o l i n g w a t e r , and a gas s u p p l y . B e f o r e d e s c r i b i n g each subsystem i n d e t a i l l e t us b r i e f l y o u t l i n e the o v e r a l l o p e r a t i o n . F i g u r e I I I - 1 shows a b l o c k diagram of the a p p a r a t u s w i t h the major components of each subsystem. The a r c v e s s e l i s the h e a r t of the system, and i t s main f u n c t i o n i s t o s u p p o r t the a r c s t a b l y w i t h o u t i n t e r f e r i n g w i t h the d i a g n o s t i c s . I t c o n s i s t s of a chamber w i t h double g l a s s w a l l s and c o o l i n g water between them, gas j e t s t o m a i n t a i n the argon v o r t e x , end p l a t e s t o c o n t a i n the p r e s s u r e of up t o 10 Atm, and water c o o l e d t u n g s t e n t i p p e d e l e c t r o d e s between which the a r c burns. The power s u p p l y W A T E R P U M P - < ( T A N K C O O L I N G F L O W M E T E R S T E M P E R A T U R E M E T E R S VESSEL • O a- K > H E A T F L O W M E T E R E X C H A N G E R G A S <E> P R E S S U R E M E T E R G A S B O T T L E & ( R E G U L A T O R POWER S T A R T I N G C IRCU I T P O W E R S U P P L Y Figure 1 1 1 — 1. Block diagram of the system 39 must p r o v i d e a d j u s t a b l e c u r r e n t s up t o 500A a t about 100V and a means of s t a r t i n g the a r c , i n t h i s case a s o u r c e of r a d i o f r e q u e n c y h i g h v o l t a g e (30kV). I t a l s o c o n t a i n s equipment t o monitor t h e a r c c u r r e n t and v o l t a g e . The c o o l i n g system c i r c u l a t e s c o o l i n g water t h r o u g h a l l the components, i n c l u d i n g a l i g h t a b s o r b i n g s h i e l d , and i s used t o measure the t h e r m a l l o a d i n g of each component by m o n i t o r i n g f l o w r a t e s and t e m p e r a t u r e s . The gas system c o n s i s t s of an argon s u p p l y w i t h p r e s s u r e meters and f lowmeters and a heat exchanger t o c o o l the exhaust gas. B. THE GAS SYSTEM The s i m p l e s t of the subsystems i s the gas system, which s u p p l i e s argon f o r the v o r t e x . P r e s s u r e s up t o about 8 Atmospheres ( a b s o l u t e ) and f l o w r a t e s of about 2•10~ 4 m 3s~ 1 are r e q u i r e d . The s u p p l y i s a s t a n d a r d gas b o t t l e and r e g u l a t o r w i t h a p r e s s u r e meter c a l i b r a t e d i n p s i g b u i l t i n t o the r e g u l a t o r . The gas used i s l a b o r a t o r y grade argon. The f l o w r a t e i s measured w i t h a Matheson 11670 r o tometer type flowmeter c a l i b r a t e d f o r a i r a t S.T.P. and c o r r e c t e d f o r gas d e n s i t y . A c c u r a t e knowledge of the chamber p r e s s u r e i s i m p o r t a n t f o r both the a r c m o d e l l i n g and the s p e c t r o s c o p i c a n a l y s i s , and the mass f l o w r a t e i s c r i t i c a l f o r the c o n v e c t i v e t r a n s p o r t c a l c u l a t i o n s , so the f l o w r a t e and p r e s s u r e drops must be c a l c u l a t e d . These c a l c u l a t i o n s and the measurements which c o n f i r m them a r e g i v e n i n d e t a i l i n Appendix D. 40 P R E S S U R E M E T E R R E G U L A T O R 12 -3-1 R= 2 . 0 * 1 0 N m s 1.18 m m 0 R = 9 . 4 « 1 0 7 N m 3 s 1 M A I N G A S B O T T L E ( A R G O N ) J R=1.6 -1 O 1 3 N m 3 s 1 P _ A ( P J _ j s t r i e r e s j S t a n c e due to v i s c o s i t y m F i g u r e I I I - 2 . The gas system 41 C. THE VORTEX L e t us now c o n s i d e r t h e a c t u a l gas v o r t e x . The gas i s e n c l o s e d i n a smooth c y l i n d e r of r a d i u s a, and i t e n t e r s t h r o u g h a t a n g e n t i a l n o z z l e a t the w a l l a t v e l o c i t y u a t one end, d r i f t s r e l a t i v e l y s l o w l y down the c y l i n d e r , and e x i t s t a n g e n t i a l l y from the o t h e r end. We assume t h e gas t o be i d e a l w i t h d e n s i t y p and v i s c o s i t y ». C o n s i d e r a d i s c of u n i t t h i c k n e s s a f t e r i t l e a v e s the v i c i n i t y of t h e i n l e t , w h i l e i t d r i f t s down the t u b e . I f the gas a t r a d i u s r has a n g u l a r v e l o c i t y o ( r ) , the s l i c e has t o t a l a n g u l a r momentum L=J o2v pur3dr . The r e t a r d i n g t o r q u e on the s l i c e i s p r o v i d e d by the v i s c o u s d r a g on the w a l l and has magnitude r = [ 2 * r 3 y | ^ ] L 3r Jr=a There i s a l s o t o r q u e due t o a x i a l v a r i a t i o n i n u , but i t i s p r o p o r t i o n a l t o -|jiL and i s much l e s s than the t o r q u e due t o r a d i a l g r a d i e n t s , a t l e a s t f o r the s l o w l y d e c a y i n g modes which a r e most i m p o r t a n t . C o n s e r v a t i o n of a n g u l a r momentum can then be a p p l i e d t o get N o t h i n g has so f a r r e s t r i c t e d the r a d i u s a t o be the s o l i d w a l l , so i f we l e t a be a r b i t r a r y and d i f f e r e n t i a t e we get 42 3 (VT3 3co-N 3_9w 3 r l p ar-1 at or 3w y 3 c 3 3un / T T T « \ 3r =-T3r"^ r 37^ ( I I I - D The s o l u t i o n t o t h i s e q u a t i o n i s o u t l i n e d i n Appendix E. For t h e boundary c o n d i t i o n s u(a)=0, |^rl 0= 0 t n e s o l u t i o n can be w r i t t e n as a B e s s e l f u n c t i o n e x p a n s i o n w(r,t) =1 — J j C - x n) e n n where t h e B e s s e l f u n c t i o n J , has z e r o e s a t ( x ^ ) and T = a 2 p / v = B . 8 0 s . The f i r s t few z e r o e s o c c u r a t 3.83, 7.02 and 10.17, and i f the a x i a l d r i f t v e l o c i t y i s about 0.4 ms~ 1 the ti m e decay i s such t h a t a f t e r 50 mm the f i r s t t h r e e components a r e reduced by f a c t o r s of 0.81, 0.50 and 0.23 r e s p e c t i v e l y , so the net e f f e c t i s t h a t the f i r s t harmonic i s the most i m p o r t a n t i n the r e g i o n of the a r c . Near the gas i n l e t t h e v o r t e x c o n t a i n s a m i x t u r e of harmonics (see Appendix E) but the h i g h e r harmonics of t h i s m otion r a p i d l y decay, l e a v i n g a f t e r a few c e n t i m e t e r s C r , t ) = w o F T J i ( x i r / a ) e " X i t / T 0) As the gas l e a v e s the chamber, the average t a n g e n t i a l v e l o c i t y near the o u t l e t ( w i t h i n 2 mm of t h e w a l l ) i s 0.005o 0a, or about 1 ms" 1, which i s n e g l i g i b l e compared t o the v e l o c i t y i n t h e o u t l e t t u b i n g . T h i s i s assumed t o be 43 t r u e i n Appendix D, and i s ind e e d the c a s e . W i t h i n the column, the r a d i a l p r e s s u r e g r a d i e n t i s j u s t s u f f i c i e n t t o p r o v i d e the c e n t r i p e t a l a c c e l e r a t i o n u 2 r , t h u s 8P 2 but s u b s t i t u t i n g f o r u we get AP = / pw 2rdr = 0 . 0 4 p u 2 a 2 e ~ 2 x l t / x  Jo o Now u 0 a i s the i n j e c t i o n v e l o c i t y U=180 ms" 1 and p=Pm/kT. The p r e s s u r e d i f f e r e n c e a t 293K i s thus f = 0.02 e - 2 x f t / T o which i s l e s s than 2 % , and a t 3000K i t i s l e s s than 0 . 2 % and need not be c o n s i d e r e d . H a v i n g s o l v e d t h e e q u a t i o n s of motion f o r the v o r t e x f l o w , we can a d d r e s s the problem of s t a b i l i t y i n the v o r t e x s t a b i l i z e d a r c . S t a b i l i t y r e q u i r e s some mechanism t o keep the a r c c e n t e r e d i n t h e v e s s e l , and i n t h i s case the mechanism i s t h a t of a c e n t r i f u g e . A u s e f u l a n a l o g y t o c o n s i d e r i s a system i n which a water and a i r m i x t u r e s p i n s i n s i d e a t u b e , and f o r t h a t system i t i s c l e a r t h a t when the a n g u l a r v e l o c i t y i s s u f f i c i e n t t he water w i l l s t a y near the w a l l and the a i r near the c e n t r e . The c e n t r i p e t a l a c c e l e r a t i o n o 2 r i s a measure of the s t r e n g t h of the c e n t r i f u g e and t o g e t h e r w i t h the d e n s i t y d i f f e r e n c e d e t e r m i n e s the amount of s e p a r a t i o n o b s e r v e d . In the a r c the system c o n s i s t s of hot and c o l d a r g o n , 44 w i t h a s u b s t a n t i a l d e n s i t y d i f f e r e n c e ( 3 0 : 1 ) . The c e n t r i p e t a l a c c e l e r a t i o n a t h a l f the tube r a d i u s i s o 2r=1.3«l0 5 m s - 2 , which p r o v i d e s a l a r g e buoyant f o r c e t o keep the h o t t e r , more c o n d u c t i v e gas a t the c e n t e r , and t h u s s t a b i l i z e s the a r c . The c e n t r i p e t a l a c c e l e r a t i o n i s a l s o l a r g e enough t o make the e f f e c t of the asymmetric g r a v i t a t i o n a l buoyant f o r c e n e g l i g i b l e and t h e r e f o r e keep the column symmetric. D. THE COOLING SYSTEM C o o l i n g i n the a r c i s a c h i e v e d by pumping c o l d water t h r o u g h a l l the components t o be c o o l e d . These i n c l u d e t h e e l e c t r o d e s , w a l l , r a d i a t i o n a b s o r b e r and some power s u p p l y components. The water i s not r e c y c l e d so no heat exchanger i s r e q u i r e d and the water i s s u p p l i e d by a t u r b i n e pump conne c t e d t o the mains t h r o u g h a r e s e r v o i r t a n k , and dumped t o the d r a i n . The water s u p p l y p r o v i d e s about 6«l0" < tm 3s" 1 (40 l i t r e s per minute) a t about 10 Atm p r e s s u r e . F i g u r e I I I -3 shows the c o o l i n g system s c h e m a t i c a l l y , and F i g u r e I I I - 6 shows how the v e s s e l i s c o o l e d . The water f l o w i s metered w i t h t h r e e 4.5 GPM rotometer type f l o w m e t e r s c a l i b r a t e d i n the l a b i n cm 3s~ 1 which measure the f l o w r a t e s i n each p a r t of the system. The i n l e t t e m p e r a t u r e ( T 0 ) a t the f l o w m e t e r s and the o u t l e t t e m p e r a t u r e s (T,-T(,) of the branch l i n e s a r e measured w i t h s o l i d s t a t e temperature t r a n s d u c e r s ( N a t i o n a l Semiconductor LM335). The tempe r a t u r e s a r e s u b t r a c t e d and d i g i t i z e d 45 e l e c t r o n i c a l l y and the d i f f e r e n c e s a re output d i g i t a l l y on a mon i t o r p a n e l . The mass f l o w r a t e m and the temperature change AT i s then known f o r each branch of the c o o l i n g system, so the heat l o a d i n g Q=CmAT can be c a l c u l a t e d , where C=4200 J k g ^ K " 1 i s the heat c a p a c i t y of water . E. THE ELECTRICAL SYSTEM The e l e c t r i c a l system p r o v i d e s power f o r s t a r t i n g and r u n n i n g the a r c and i n c l u d e s d e v i c e s t o m o n i t o r the c u r r e n t , v o l t a g e , and power i n t o the a r c . F i g u r e I I I - 4 i s a schematic diagram of the system. The s t a r t i n g c i r c u i t i n j e c t s a h i g h v o l t a g e p u l s e (30kV RMS a t 4 MHz) through a t r a n s f o r m e r i n s e r i e s w i t h the h i g h c u r r e n t s u p p l y , which i s p r o t e c t e d by a c a p a c i t o r r e t u r n (C1) t o ground. The RF p u l s e i s g e n e r a t e d by a d i s c h a r g e t h r ough a spark gap which forms p a r t of an LC c i r c u i t . The s t a r t i n g c i r c u i t , i n c l u d i n g r e t u r n c a p a c i t o r C1, was bought as a package ('750 A a r c lamp i g n i t e r ' ) from L.P. A s s o c i a t e s I n c . of B e v e r l y H i l l s , C a l i f o r n i a . When the a r c i s s t a r t e d the e l e c t r o d e s a r e moved t o about 10 mm s e p a r a t i o n and a 1 Ohm c u r r e n t l i m i t i n g r e s i s t o r (R1) i s i n the c i r c u i t t o l i m i t damage t o the e l e c t r o d e s w h i l e s t a r t i n g . The RF p u l s e breaks down the gap and the h i g h c u r r e n t s u p p l y p r o v i d e s about 200 A t o m a i n t a i n the a r c . The e l e c t r o d e s e p a r a t i o n i s then i n c r e a s e d t o 100 mm and the l i m i t i n g r e s i s t o r s h o r t e d out by a h i g h c u r r e n t SCR i n p a r a l l e l w i t h i t . The SCR i s a water c o o l e d assembly TURBINE PUMP 13 fcW CURRENT SHUNT A N O D E 3 k W CATHODE TANK V E S S E L WALL HEAT EXCHANGER r SWITCH S C R RADIATION ABSORBER 1 r~ \ 25 W 4 kW F i g u r e 111-3. The c o o l i n g system RUN VOLTAGE MONITOR CURRENT MONITOR F i g u r e I I I - 4 . The e l e c t r i c a l system 47 r a t e d a t 400 PRV and 550 A average (Westinghouse PTW7T7200455). The h i g h c u r r e n t s u p p l y i s q u i t e s i m p l e . I t c o n t a i n s an i s o l a t i o n t r a n s f o r m e r c o n n e c t e d t o 208 V RMS 60Hz mains, a 1mH i n d u c t a n c e f o r b a l l a s t and a s a t u r a b l e r e a c t o r ( v a r i a b l e i n d u c t a n c e about 0.5 t o 1 mH) t o c o n t r o l the c u r r e n t , a f u l l wave water c o o l e d d i o d e b r i d g e r e c t i f i e r and a 1mH f i l t e r choke. The o u t p u t of t h i s c i r c u i t depends on the c h a r a c t e r i s t i c s of the l o a d , and when o p e r a t i n g a t y p i c a l a r c the v o l t a g e and c u r r e n t have about 20% and 50% r i p p l e r e s p e c t i v e l y , a t 120 Hz. The c u r r e n t i s m o n i t o r e d by measuring the v o l t a g e a c r o s s a water c o o l e d copper tube of 10"" Ohms r e s i s t a n c e . The t u b i n g has an o u t s i d e d i a m e t e r of 9.5 mm, a w a l l t h i c k n e s s of 0.89 mm, and a l e n g t h of about 140 mm which can be a d j u s t e d f o r a c c u r a t e c a l i b r a t i o n . The v o l t a g e i s measured u s i n g a 10:1 r e s i s t i v e v o l t a g e d i v i d e r . The c u r r e n t and v o l t a g e a r e d i s p l a y e d on an o s c i l l o s c o p e and a l s o f e d i n t o an a n a l o g m u l t i p l y i n g / a v e r a g i n g a m p l i f i e r which measures the average c u r r e n t and power i n the a r c . T y p i c a l c u r r e n t and v o l t a g e waveforms a r e shown i n F i g u r e I I I - 5 . F. THE ARC VESSEL AND ELECTRODES The a u x i l i a r y systems converge i n the a r c v e s s e l , where the a r c burns i n a gas v o r t e x i n s i d e a 27 mm bore q u a r t z tube between t u n g s t e n t i p p e d e l e c t r o d e s , a l l water c o o l e d . The w a l l i s c o o l e d by c i r c u l a t i n g water between the q u a r t z 48 49 tube and a 43 mm bore pyrex tube o u t s i d e i t . The pyrex a l s o s e r v e s as a UV f i l t e r t o b l o c k p o t e n t i a l l y hazardous u l t r a v i o l e t l i g h t i n the 220 nm r e g i o n . The end p l a t e s of the v e s s e l , machined out of b r a s s , have o - r i n g s e a l s t o mate w i t h the tubes and t h r e a d e d f i t t i n g s f o r water c o n n e c t i o n s . They a l s o have 13 mm h o l e s t h r o u g h which the e l e c t r o d e s e n t e r . One end p l a t e c o n t a i n s the gas i n l e t (a 1.18 mm h o l e d r i l l e d t o i n t e r s e c t t h e i n s i d e s u r f a c e t a n g e n t i a l l y ) and the o t h e r end c o n t a i n s the o u t l e t , which i s v e n t e d t o the o u t s i d e t h r o u g h a heat exchanger c o n s i s t i n g of about 2 m of 2.4 mm bore copper t u b i n g immersed i n t h e c o o l i n g water coming from the double w a l l . I t s h o u l d be noted t h a t i t i s not p o s s i b l e w i t h t h i s v e s s e l t o d i s t i n g u i s h the c o n v e c t i v e heat l o s s e s from the c o n d u c t i v e l o s s e s i n the a r c column, but i n any case the gas would be c o o l e d s u b s t a n t i a l l y w h i l e l e a v i n g the r e g i o n of the column, and such a s e p a r a t i o n would be a l m o s t i m p o s s i b l e t o a c h i e v e w i t h o u t some d i r e c t measurement of the w a l l l o a d i n g a l o n g the column as a f u n c t i o n of p o s i t i o n , something which i s suggested i n Chapter VI as a p o s s i b l e t o p i c f o r f u r t h e r s t u d y . The e l e c t r o d e s a r e 12 mm d i a m e t e r b r a s s tubes capped w i t h t u n g s t e n t i p s c o n t a i n i n g 1% t h o r i u m . The t u n g s t e n t i p s a r e bonded t o copper d i s c s which a r e machined t o shape and s i l v e r s o l d e r e d t o b r a s s t u b e s . To o b t a i n good a r c s t a b i l i t y and l o n g e l e c t r o d e l i f e t i m e s the t i p g e o m e t r i e s were chosen based on p r e v i o u s work by G e t t e l (1980) i n t h i s l a b o r a t o r y . 50 The cathode has a c o n i c a l t i p w i t h a 90 degree v e r t e x a n g l e and the anode has a f l a t t i p w i t h chamfered edges. The t i p s a r e c o o l e d w i t h a water f l o w , u s i n g a s m a l l e r c o a x i a l tube t o f o r c e the f l o w a g a i n s t the t i p , and the e l e c t r i c a l c o n n e c t i o n i s made through the o u t e r b r a s s t u b e . The a r c v e s s e l i s drawn i n F i g u r e I I I - 6 . 51 t I .Water Cooling i 111 Water Cooling F i gu re 111-6. Drawing of the arc v e s s e l 52 CHAPTER IV DIAGNOSTIC METHODS A. INTRODUCTION One purpose of t h i s r e s e a r c h i s t o d e v e l o p , t e s t and e v a l u a t e a model f o r the b e h a v i o u r of t h e column of a v o r t e x s t a b i l i z e d a r c , and so, i n a d d i t i o n t o the c a l o r i m e t r i c d a t a g a t h e r e d by m o n i t o r i n g t h e c o o l i n g system, t h e a r c has been s t u d i e d o p t i c a l l y t o d e t e r m i n e the r a d i a l t e mperature p r o f i l e i n the column. Three independent methods have been used t o dete r m i n e the t e m p e r a t u r e s ; a b s o l u t e measurements of the t o t a l i n t e n s i t y i n the AI 430 nm atomic l i n e , a b s o l u t e measurements of the continuum i n t e n s i t y a t 431.4 nm, and measurements of the AI 430 nm l i n e w i d t h . These s p e c t r o s c o p i c measurements, w h i l e s i m p l e enough i n p r i n c i p l e , a r e c o m p l i c a t e d by the f a c t t h a t the column i s c y l i n d r i c a l . Data o b t a i n e d by v i e w i n g s i d e - o n must be i n v e r t e d t o get the a c t u a l r a d i a l p r o f i l e s . T h i s p r o c e d u r e , known as A b e l i n v e r s i o n , i s based on c y l i n d r i c a l symmetry and r e q u i r e s an o p t i c a l l y t h i n plasma. I f we t a k e a q u a n t i t y ( u s u a l l y a volume e m i s s i v i t y ) F ( r ) and i n t e g r a t e i t a l o n g a l i n e of s i g h t we get R u Y(y) = 2 / r r F ( r ) ( r 2 - y 2 ) ^dr which w i l l be a r a d i a n c e , or power e m i t t e d per u n i t a r e a . T h i s i n t e g r a l can be i n v e r t e d t o g i v e 53 which i s known as the A b e l i n t e g r a l ( F l e u r i e r 1974). In t h i s work the i n v e r s i o n has been c a r r i e d out by computer u s i n g s u b r o u t i n e "ABEL" , which i s l i s t e d i n Appendix C. T h i s p r o c e d u r e r e q u i r e s l a r g e q u a n t i t i e s of d a t a , so t h e d a t a c o l l e c t i o n and p r o c e s s i n g system has been l a r g e l y automated. Data a r e c o l l e c t e d w i t h an o p t i c a l m u l t i c h a n n e l a n a l y z e r (OMA), s t o r e d d i g i t a l l y on t a p e , and p r o c e s s e d by computer. A t y p i c a l s i n g l e r a d i a l p r o f i l e r e q u i r e s about 40 s p e c t r a , each c o n s i s t i n g of 500 i n t e g e r i n t e n s i t y v a l u e s . J u s t t o l i s t a l l the s p e c t r a l d a t a c o l l e c t e d f o r t h i s r e s e a r c h would f i l l 100 volumes the s i z e of t h i s t h e s i s , so t h e a u t o m a t i o n of the d a t a h a n d l i n g i s an i m p o r t a n t f e a t u r e of t h e system. T h i s c h a p t e r d i s c u s s e s the d i a g n o s t i c system and d e s c r i b e s the a b s o l u t e i n t e n s i t y c a l i b r a t i o n p r o c e d u r e s used, as w e l l as the t h e o r e t i c a l b a s i s on which the t e m p e r a t u r e p r o f i l e s a r e e x t r a c t e d from the raw d a t a . B. THE OPTICAL SYSTEM The purpose of the o p t i c a l system i s t o image a s m a l l spot i n t h e a r c column onto the e n t r a n c e s l i t of a s p e c t r o m e t e r , which then d i s p e r s e s t h a t l i g h t onto the OMA head where the d i s p e r s e d spectrum i s sensed and d i g i t i z e d . The spot from which the l i g h t o r i g i n a t e s i s swept a c r o s s the a r c column a t a steady speed w h i l e the d a t a a r e r e c o r d e d a t 54 f i x e d i n t e r v a l s by the OMA t o produce a r a d i a l p r o f i l e . The image of the a r c i s f i r s t r o t a t e d 90 degrees by a p a i r of m i r r o r s mounted a t 60 degrees t o one a n o t h e r so t h a t the column appears v e r t i c a l . The column i s then imaged onto the p l a n e of the s p e c t r o m e t e r e n t r a n c e s l i t w i t h a 400 mm f/16 l e n s mounted on a screw d r i v e which i s t u r n e d by a geared down synchronous motor t o sweep the image p o i n t a c r o s s the a r c a t a c o n s t a n t speed of about 0.16 nuns"' . The m a g n i f i c a t i o n i s 5 so the 0.05 mm by 2 mm s l i t sees an area of the column which i s 0.01 mm r a d i a l l y by 0.4 mm a x i a l l y . The s a m p l i n g r a t e i s such t h a t each spectrum i s averaged over 0.51 mm r a d i a l l y , which i s s u f f i c i e n t r e s o l u t i o n f o r the d a t a a n a l y s i s r e q u i r e m e n t s . The monochromator i s a Spex 1702 3/4 m C z e r n y - T u r n e r s c a n n i n g s p e c t r o m e t e r w i t h an a p e r t u r e of f/6.8 and a 1200 l i n e s per mm g r a t i n g b l a z e d a t 500 nm. The d i s p e r s i o n a t 430 nm i n f i r s t o r d e r i s about 11 nm/cm and w i t h the OMA a t t a c h e d t o t h e e x i t p o r t t h e 500 c h a n n e l o u t p u t c o v e r s a t o t a l range of about 12.4 nm. Each OMA c h a n n e l has a w i d t h w=0.0247 nm and the r e s o l u t i o n i s l i m i t e d by c r o s s t a l k between c h a n n e l s t o about 2 c h a n n e l s or 0.05 nm (fwhm) which i s adequate f o r t h i s work. A n e u t r a l d e n s i t y f i l t e r of 1.34% t r a n s m i s s i o n and a s t r o b e d i s c of 3.77% t r a n s m i s s i o n ( s y n c h r o n i z e d w i t h the s u p p l y v o l t a g e ) reduce the l i g h t i n t e n s i t y t o a v o i d s a t u r a t i n g the OMA. The s t r o b e , which c o n s i s t s of a d i s c w i t h a narrow s l o t c u t i n i t mounted on a synchronous motor, i s used t o sample the l i g h t a t a f i x e d 55 Arc Mirror Pair 400 mm f/16 Lens Variable Phase 120 Hz _ St robe" <—> Mot or Dri ve OM A Head Spex 1702 Spectrometer OMA Digital Console Tape Recorder F i g u r e IV-1 . The d i a g n o s t i c system 56 p o i n t i n the AC c y c l e so the r i p p l e i n the power s u p p l y does not a f f e c t the r e s u l t s . The synchronous motor on the s t r o b e i s mounted on a frame which can be r o t a t e d t o a l t e r the phase of the o b s e r v a t i o n s . C. THE DATA HANDLING SYSTEM As the column image i s scanned a c r o s s t h e s p e c t r o m e t e r s l i t , l i g h t p u l s e s of about 0.3 ms d u r a t i o n and 8 ms s e p a r a t i o n a r e passed by the s t r o b e d i s c t h rough the sp e c t r o m e t e r and onto the OMA head. The s p e c t r a l i n t e n s i t i e s a r e i n t e g r a t e d f o r about 3 seconds and the t o t a l i s s t o r e d d i g i t a l l y on magnetic t a p e . T h i s p r o c e s s i s r e p e a t e d about 50 times as the image sweeps a c r o s s the s l i t , g i v i n g 50 s p e c t r a spaced u n i f o r m l y a c r o s s the a r c column. A f i l e mark i s then w r i t t e n on the tape t o t e r m i n a t e the r u n , and anot h e r run can then be made w i t h d i f f e r e n t a r c parameters ( c u r r e n t , p r e s s u r e , e t c . ) . At the end of the s e s s i o n an end of tape marker (3 f i l e marks) i s w r i t t e n on the t a p e , which i s then rewound and removed from the tape d r i v e . The tape c o n t a i n i n g the da t a i s taken t o the UBC computing c e n t e r and mounted on a tape d r i v e connected t o the Amdahl computer. Program "CONV" reads the raw d a t a , w hich a r e coded i n a form not c o m p a t i b l e w i t h the Amdahl c o d i n g , and c o n v e r t s each spectrum t o an a r r a y of 500 h a l f w o r d (16 b i t ) i n t e g e r s . The program a l s o d i v i d e s each v a l u e by 2 t o ensure t h a t the OMA l i n e a r range w i l l not o v e r f l o w the h a l f w o r d i n t e g e r s and c a l l s s u b r o u t i n e "CAL" 57 which compensates f o r the v a r i a t i o n s i n response of d i f f e r e n t c h a n n e l s by s c a l i n g each c h a n n e l an a p p r o p r i a t e amount. Each spectrum ( a r r a y of 500 i n t e g e r s ) i s then w r i t t e n i n 16 b i t b i n a r y form on one l i n e of a d i s c f i l e , and a header l i n e i s g e n e r a t e d a t t h e b e g i n n i n g of each run which l a b e l s the run and g i v e s the number of s p e c t r a i t c o n t a i n s t o a l l o w the computer t o scan through q u i c k l y . G e n e r a l purpose program "SCAN" a l l o w s e x a m i n a t i o n and some m a n i p u l a t i o n of these d a t a f i l e s . The user can i n t e r a c t i v e l y d i s p l a y p l o t s of i n d i v i d u a l s p e c t r a or r a d i a l p r o f i l e s of any g i v e n c h a n n e l , smoothed or A b e l i n v e r t e d as d e s i r e d . T h i s program i s a l s o used t o i n s e r t a p p r o p r i a t e t i t l e s i n the header l i n e s , t o g e n e r a t e an index of d a t a , and t o o b t a i n h a r d copy ( p r i n t e d or p l o t t e d ) o utput such as the p l o t t e d o utput i n c l u d e d i n t h i s t h e s i s . These programs ar e l i s t e d i n Appendix C. D. AI LINE INTENSITY MEASUREMENTS S e v e r a l methods can be used t o c a l c u l a t e the tempe r a t u r e of an argon plasma from i t s l i g h t e m i s s i o n s . One method q u i t e f r e e from s y s t e m a t i c e r r o r s i s t o measure the r a t i o of the t o t a l i n t e n s i t i e s i n two (or more) s p e c t r a l l i n e s w i t h w i d e l y s e p a r a t e d upper l e v e l s , and assuming l o c a l t h e r m a l e q u i l i b r i u m (LTE) c a l c u l a t e the temperature which would y i e l d t h a t r a t i o . Another a c c u r a t e method depends on the e x i s t e n c e , f o r any l i n e , of a unique temperature a t which the t o t a l i n t e n s i t y i n t h a t l i n e i s a maximum ( s i n c e 58 at h i g h e r t e m p e r a t u r e s t h e p o p u l a t i o n of t h a t i o n i z a t i o n s t a g e i s d e p l e t e d ) . Both of t h e s e methods have been used t o g i v e a c c u r a t e t e m p e r a t u r e s f o r argon plasmas ( P r e s t o n 1977, L a r e n z 1951) but both r e q u i r e a f a i r l y l a r g e degree of i o n i z a t i o n t o make the A l l spectrum v i s i b l e above the continuum background, so t h e y can o n l y be a p p l i e d a t t e m p e r a t u r e s above about 12000K. The a r c which i s the s u b j e c t of t h i s t h e s i s has maximum t e m p e r a t u r e s i n the a r e a of 11000K, which i s t o o low f o r t h e A l l spectrum t o be u s a b l e . The methods which can be used when o n l y the AI spectrum i s a v a i l a b l e a r e q u i t e l i m i t e d . The r a t i o of l i n e i n t e n s i t i e s i s p r o p o r t i o n a l t o e E i " E 2 ^ T - where and E 2 a r e t he two upper l e v e l e n e r g i e s , which have v a l u e s r a n g i n g from 13 t o 15 eV f o r u s a b l e l i n e s i n the AI spectrum. E,-E 2 i s t h u s l i m i t e d t o about 2 eV maximum, and a 10% e r r o r i n the i n t e n s i t y r a t i o would t h u s i n t r o d u c e an e r r o r w hich c o r r e s p o n d s t o a te m p e r a t u r e e r r o r of a t l e a s t 500K a t T=10000K and i s u n a c c e p t a b l y l a r g e . The same problem o c c u r s f o r t h e r a t i o of AI l i n e t o continuum r a t i o s ; b o t h i n t e n s i t i e s a r e s t r o n g l y t e m p e r a t u r e dependent, but the r a t i o o n l y v a r i e s weakly w i t h t e m p e r a t u r e . The p r i m a r y method used t o det e r m i n e t e m p e r a t u r e s from the argon s p e c t r a was a b s o l u t e measurement of the t o t a l i n t e n s i t y of the AI l i n e a t 430 nm. A b s o l u t e i n t e n s i t y 59 1 1 1 1 1 1 1 1 1 1 1 1 i 4 2 5 4 3 0 . 4 3 5 A (nm) F i g u r e I V - 2 . The AI 430 nm l i n e measurements are g e n e r a l l y more d i f f i c u l t t o p e r f o r m ' t h a n r e l a t i v e measurements ( i . e . measurement of r a t i o s ) but t h i s p r o c e d u r e was judged t o be the be s t a v a i l a b l e a l t e r n a t i v e . The 430 nm l i n e was chosen because i t i s v i s i b l e and e a s i l y h a n d l e d o p t i c a l l y , many r e s e a r c h e r s have s t u d i e d i t and i t s s p e c t r o s c o p i c parameters a r e t h e r e f o r e a v a i l a b l e i n the l i t e r a t u r e , and i t i s w e l l s e p a r a t e d from o t h e r l i n e s t h a t might o v e r l a p and cause problems (see F i g u r e I V - 2 ) . T h i s l i n e i s a l s o o p t i c a l l y t h i n f o r the c o n d i t i o n s (T<12000K , P<10 Atm) o c c u r r i n g i n the a r c . T h i s c o n c l u s i o n i s support e d by numerous a u t h o r s (Olsen 1 9 6 3 , P r e s t o n 1 9 7 7 , R i c h t e r 1 9 6 5 , B a e s s l e r 1 9 8 0 ) , many of whom worked w i t h l o n g e r plasmas s i n c e they were l o o k i n g a x i a l l y t h r o u g h w a l l s t a b i l i z e d a r c s . The r a d i a t e d power per s t e r a d i a n per u n i t volume i n an atom i c l i n e of wavelength X i s 60 J = 5^A N 4TTA nm n where A ^ i s the t r a n s i t i o n p r o b a b i l i t y f o r an atom from the upper s t a t e n t o the lower s t a t e m and N i s the number d e n s i t y of atoms i n s t a t e n. I f the energy of the upper s t a t e i s E n and the n e u t r a l atom d e n s i t y i s N 0 , we have a t low t e m p e r a t u r e ( k T « E n ) XT XT ~ " E A T N = N g e n n o 5 n where g n i s the s t a t i s t i c a l weight of s t a t e n. Below 12000K the i o n i z a t i o n i s low enough (O l s e n 1963) t h a t we can use the i d e a l gas law N 0=P/kT and we get J = ] i ^ g A p _ e - V k T ( I V _ 1 ) 4TTX Bn"nm kT S p e c t r o s c o p i c measurements of t h e t r a n s i t i o n p r o b a b i l i t y f o r the AI 430nm l i n e have been made by many a u t h o r s ( O l s e n 1963, P r e s t o n 1977, R i c h t e r 1965) and r e c e n t l y B a e s s l e r and Kock (1980) summarized t h e s e r e s u l t s and e x p l a i n e d most of t h e d i s c r e p a n c i e s between e x p e r i m e n t s u s i n g the r e s u l t s of t h e i r i n t e r f e r o m e t r i c measurements. The mean c o r r e c t e d v a l u e from t h e i r work, A =3.12-10 s s _ 1 ± 7 % nm has been used h e r e . The upper l e v e l f o r the t r a n s i t i o n has energy E n=14.51 eV and s t a t i s t i c a l w eight g n=5 (Ols e n 1963). T a k i n g P 0=1 .01 • l 0 5 N m " 2 , T 0=10000K and J o = 0.70 W i r r ' s r " 1 ( t h e volume e m i s s i v i t y which c o r r e s p o n d s t o 1 count i n the f i n a l 61 p r o f i l e - see s e c t i o n E) E q u a t i o n IV-1 becomes T _ 1.68'105 K (IV-2) 24.81+ln(P/P o)-ln(T/T o)-ln(J/J o) E q u a t i o n IV-2 was used t o c a l c u l a t e the temperature p r o f i l e s from the A b e l i n v e r t e d i n t e n s i t y p r o f i l e s and produce the r e s u l t s which a r e g i v e n i n the next c h a p t e r . The s p e c t r a which have been c o l l e c t e d and f i l e d a r e c o n v e r t e d , u s i n g program "FT" , t o t e m p e r a t u r e p r o f i l e s . The program f i r s t i n t e g r a t e s the t o t a l i n t e n s i t y i n t h e AI430 nm l i n e f o r each spectrum, then smooths and A b e l u n f o l d s the l i n e i n t e n s i t y p r o f i l e , and f i n a l l y uses E q u a t i o n IV-2 t o c o n v e r t the volume e m i s s i v i t i e s i n t o t e m p e r a t u r e s . The temperature p r o f i l e s a r e then l i s t e d and p l o t t e d f o r i n s p e c t i o n . Program "FT" i s l i s t e d i n Appendix C. E. ABSOLUTE INTENSITY CALIBRATION In o r d e r t o use the method of a b s o l u t e l i n e i n t e n s i t y the o p t i c a l system must be c a l i b r a t e d f o r a b s o l u t e i n t e n s i t y , which i s c o m p l i c a t e d i n t h i s work by the f a c t t h a t the l i g h t passes t h r o u g h two c y l i n d r i c a l g l a s s w a l l s and the water j a c k e t between them. The most common c a l i b r a t i o n s o u r c e used i s the carbon cathode of a f r e e -b u r n i n g a r c , but i n o r d e r t o compensate f o r t h e a r c v e s s e l w a l l , the c a l i b r a t i o n of t h e system was done u s i n g a t u n g s t e n f i l a m e n t s o u r c e . The f i l a m e n t was mounted i n the c e n t e r of the v e s s e l (see F i g u r e I V - 3 ) , h e a t e d e l e c t r i c a l l y , 62 and p r o t e c t e d from o x i d a t i o n by f l o w i n g argon s l o w l y t h r o u g h the v e s s e l . The f i l a m e n t t e m p e r a t u r e was measured w i t h an o p t i c a l pyrometer and the r a d i a n c e a t 430 nm was c a l c u l a t e d , c o r r e c t i n g f o r t h e e m i s s i v i t y of t u n g s t e n as a f u n c t i o n of tem p e r a t u r e and w a v e l e n g t h . T h i s c a l c u l a t i o n , based on the t u n g s t e n e m i s s i v i t y d a t a i n the CRC Handbook (1973), r e s u l t s i n a c u r v e f o r the r a d i a n c e a t 430 nm ( R , 3 0 ) v e r s u s the b r i g h t n e s s t e m p e r a t u r e (T f c) as measured w i t h the o p t i c a l pyrometer a t 650 nm wa v e l e n g t h . T h i s f u n c t i o n i s p l o t t e d i n F i g u r e IV-4. The OMA system was then used ( w i t h o u t the s t r o b e ) t o measure the r a d i a n c e of the f i l a m e n t a t s e v e r a l d i f f e r e n t t e m p e r a t u r e s . The average number of c o u n t s per c h a n n e l N was measured, and knowing t h a t the s p e c t r a l w i d t h of each c h a n n e l i s w=0.0247 nm, the r a d i a n c e per count i s R o = w R » 3 o / N . I f the system i s l i n e a r , R 0 s h o u l d be c o n s t a n t , and as F i g u r e IV-5 shows, a l o g - l o g p l o t of N vs R, 3 0 a t v a r i o u s b r i g h t n e s s t e m p e r a t u r e s i s l i n e a r , and g i v e s a v a l u e of R 0=wR„ 3 0/N=2.69-10-°Wm- 2sr- 1. T h i s c o n s t a n t r e l a t e s the number of co u n t s t o the r a d i a n c e from t h e column; we must a l s o c o n s i d e r the e f f e c t s of the d a t a p r o c e s s i n g , A b e l i n v e r s i o n , and s t r o b e on the d a t a . The s t r o b e reduces the number of c o u n t s by a f a c t o r of 0.0377, and the A b e l i n v e r s i o n g i v e s a r e s u l t which must be d i v i d e d by 20d=l0.2 mm (d i s the s p a c i n g between s a m p l e s ) . The a c t u a l volume e m i s s i v i t y per count i s thus R j = _ ° _ = 0.70 Wm"3sr_1 (IV-3) T u n g s t e n R i b b o n F i l a m e n t F i gu re IV-3. The c a l i b r a t i o n setup F i g u r e IV-4. Radiance vs b r i g h t n e s s t emperature F i g u r e IV-5. N vs r a d i a n c e 65 and the a b s o l u t e c a l i b r a t i o n i s comp l e t e . F. AI LINE WIDTH MEASUREMENTS One of the methods used t o c o n f i r m the l i n e i n t e n s i t y t e m perature measurements was t o measure the w i d t h of the AI430 nm l i n e , which i s p r o p o r t i o n a l t o the e l e c t r o n d e n s i t y i n the r e g i o n of i n t e r e s t . F o r e l e c t r o n d e n s i t i e s between 1 0 2 1 and 2• 1 0 2 3 n r 3 the w i d t h of the l i n e i s p r e d i c t e d a c c u r a t e l y by l i n e a r S t a r k b r o a d e n i n g t h e o r y as A A = C -N - T 1 / 6 (IV-4) w e where AX. i s the l i n e w i d t h (fwhm), N. i s t h e e l e c t r o n d e n s i t y and C i s a c o n s t a n t w i t h the v a l u e J w 3.5•10" 1 6 m a K - l / 6 ± 1 u % ( R i c h t e r 1965). To c a l c u l a t e the e l e c t r o n d e n s i t y , the d a t a a r e f i r s t A b e l u n f o l d e d c h a n n e l by c h a n n e l , then the l i n e w i d t h i s found by i n t e r p o l a t i o n . The i n s t r u m e n t w i d t h was measured, u s i n g a He-Ne l a s e r , t o be 2.5 c h a n n e l s (0.062 nm±l0%). Assuming t h a t the l i n e w i d t h s add l i n e a r l y i n wavelength ( i . e . L o r e n t z i a n shapes) the i n s t r u m e n t w i d t h i s s u b t r a c t e d and E q u a t i o n IV-4 i s then used t o c a l c u l a t e t h e e l e c t r o n d e n s i t y N e , and the tem p e r a t u r e c o r r e s p o n d i n g t o t h a t d e n s i t y can be c a l c u l a t e d under the assumption of LTE. Olsen (1963) has c a l c u l a t e d a t a b l e of N a vs T a t 1.1 Atm p r e s s u r e , and h i s c a l c u l a t i o n s f i t a c u r v e N = N A e " E / k T e o 66 w i t h A=669 and E=8.96 eV. A g a i n s u b s t i t u t i n g N 0=P/kT and s e t t i n g P 0= 1 .01 • 1 0 5Nirr 2 and T 0 = 1 0000K we get T _ 1.04-105 K (IV-5) 1 " 61.45+ln(P/P )-ln(T/T )-ln(Nj O O G where N g i s i n n r 3 . F i g u r e IV-6 shows t h e c u r v e T vs N e a t P=1.1 Atm and the p o i n t s c a l c u l a t e d by O l s e n . E q u a t i o n IV-5 i s used t o c a l c u l a t e the t e m p e r a t u r e p r o f i l e from the e l e c t r o n d e n s i t y p r o f i l e , which p r o v i d e s an independent check of the c o r e t e m p e r a t u r e . G. CONTINUUM INTENSITY MEASUREMENTS The t h i r d method of o b t a i n i n g t e m p e r a t u r e s i s from the continuum i n t e n s i t y a t 431.4 nm, which i s p r o p o r t i o n a l t o the square of the e l e c t r o n d e n s i t y . The volume e m i s s i v i t y i n a wavelength i n t e r v a l AX. i s g i v e n by C N2 J = £ ( X , T ) A X (IV-6) X 2 T ^ where C c=1.62•10"• 3 K 1 / 2Wm f tsr- 1 and the continuum e m i s s i o n f a c t o r *(x,T), which i n c l u d e s t h e c o n t r i b u t i o n s of b o t h f r e e - f r e e and f r e e - b o u n d r a d i a t i o n , has a v a l u e of 1.67±10% ( P r e s t o n 1977). The temperature c a l c u l a t i o n from continuum i n t e n s i t y i s done by u n f o l d i n g the d a t a c h a n n e l by c h a n n e l , a v e r a g i n g over 5 c h a n n e l s and u s i n g the average number of c o u n t s per c h a n n e l (AX.=0.0247 nm) t o c a l c u l a t e the e l e c t r o n d e n s i t y N g from E q u a t i o n IV-6. The temperature i s then c a l c u l a t e d from N as d e s c r i b e d i n s e c t i o n F. Program "ASPEC" , which was 67 0 5 000 T(K) 10000 F i g u r e IV-6. Temperature vs e l e c t r o n d e n s i t y 68 used t o p e r f o r m the l i n e w i d t h and continuum i n t e n s i t y c a l c u l a t i o n s , i s l i s t e d i n Appendix C. 69 CHAPTER V RESULTS AND ANALYSIS A. INTRODUCTION In t h i s r e s e a r c h two t y p e s of measurement were made, c a l o r i m e t r i c and s p e c t r o s c o p i c . The c a l o r i m e t r i c measurements c o n s i s t of f l o w r a t e and temperature change measurements u s i n g the c o o l i n g system a l o n g w i t h c u r r e n t , v o l t a g e and average i n p u t power measurements u s i n g the e l e c t r i c a l system. These p r o v i d e the d e t a i l s of the o v e r a l l power b a l a n c e i n the a r c f o r comparison w i t h the p r e d i c t i o n s of the computer model, s i n c e an a c c u r a t e p r e d i c t i o n of the power b a l a n c e i s an i m p o r t a n t r e s u l t of a u s e f u l model of the v o r t e x s t a b i l i z e d a r c . The s p e c t r o s c o p i c measurements c o n s i s t of scanned sequences of s p e c t r a which have been p r o c e s s e d by computer t o g i v e r a d i a l t e mperature p r o f i l e s of the a r c column. These p r o f i l e s p r o v i d e a n other t e s t of the a c c u r a c y of the computer model, p a r t i c u l a r l y i n the hot c o r e , s i n c e the tem p e r a t u r e measurements a r e o n l y r e l i a b l e a t t e m p e r a t u r e s above about 9000K. I n t h i s c h a p t e r t h e d a t a which have been c o l l e c t e d a r e p r e s e n t e d and compared w i t h the p r e d i c t i o n s of the computer model. The e r r o r a n a l y s i s i s a l s o d e s c r i b e d , and some of the assumptions used f o r the s p e c t r o s c o p i c a n a l y s i s and the computer model a r e j u s t i f i e d on the b a s i s of e x p e r i m e n t a l e v i d e n c e . 70 B. CALORIMETRY Throughout the o p e r a t i n g range of the a r c , measurements of the power d i s s i p a t e d i n each component of the system have been made. The f o u r components a r e the c a t h o d e , the anode, the w a l l and the r a d i a t i o n a b s o r b e r . The w a l l l o a d i n g a l s o i n c l u d e s c o n v e c t i v e l o s s e s t o the f l o w i n g gas. S i n c e the computer model o n l y d e a l s w i t h the a r c column and n e g l e c t s e l e c t r o d e e f f e c t s we a r e p r i m a r i l y c o n cerned w i t h the r a d i a t i v e and w a l l l o s s e s , a l t h o u g h measurement of the e l e c t r o d e l o s s e s i s i m p o r t a n t t o ensure t h a t the sum of the l o s s e s e q u a l s the i n p u t power. The average e l e c t r i c a l power i n p u t was measured and when the temperature t r a n s d u c e r s were p r o p e r l y z e r o e d the power b a l a n c e d t o b e t t e r than 10%, which i s the a c c u r a c y w i t h which the flowm e t e r s can be re a d . There was some d i f f i c u l t y w i t h d r i f t i n the temperature measurements, s i n c e the RF s t a r t i n g p u l s e f r e q u e n t l y a f f e c t e d the t r a n s d u c e r s , e i t h e r c a u s i n g d r i f t i n g or complete f a i l u r e . A f t e r i m p r o v i n g t h e i s o l a t i o n , good r e s u l t s were o b t a i n e d , a l t h o u g h the t r a n s d u c e r s and e l e c t r o n i c s remained s u s c e p t i b l e t o RF damage and had t o be z e r o e d f r e q u e n t l y and r e p l a c e d f a i r l y o f t e n . F i g u r e s V-1 t o 3 show the f o u r power l o s s components vs c u r r e n t a t v a r i o u s p r e s s u r e s , w i t h the cathode upstream and downstream. These r e s u l t s a r e c o n s i s t e n t w i t h those of G e t t e l (1980) a l t h o u g h he used d i f f e r e n t a r c l e n g t h s , which c o n f u s e s the comparison. The beha v i o u r of t h e e l e c t r o d e 71 F i g u r e V-1 . Power l o s s e s vs c u r r e n t , P=1.5 Atm 72 F i g u r e V-2. Power l o s s e s vs c u r r e n t , P=2.1 Atm 73 F i g u r e V-3 . Power l o s s e s vs c u r r e n t , P=2.8 Atm 74 l o s s e s i s i n t e r e s t i n g , and c e r t a i n l y w a r r a n t s f u r t h e r s t u d y , but i n t h i s work we a r e c o n cerned p r i m a r i l y w i t h the p r o p e r t i e s of the column. F i g u r e s V-1 t o 3 i n d i c a t e t h a t c h a n g i n g th e f l o w d i r e c t i o n causes a s m a l l change i n the column l o s s e s , i . e . w i t h the cathode downstream the r a d i a t i v e and w a l l l o s s e s a r e about 10% g r e a t e r than w i t h the cathode upstream. T h i s would i n d i c a t e a s l i g h t l y i n c r e a s e d t o t a l power but no change i n the r a d i a t i v e e f f i c i e n c y , and seems t o i m p l y t h a t the gas f l o w a f f e c t s the column shape and c h a r a c t e r i s t i c s somewhat. The d i r e c t i o n of gas f l o w used f o r most of the c a l o r i m e t r y and a l l of the s p e c t r o s c o p y was from cathode t o anode, so the r e s u l t s which a r e used f o r comparison w i t h the model a r e the column l o s s e s QR and Qw when the cathode i s upstream. These a r e p l o t t e d i n f i g u r e s V-4 t o V-6 a l o n g w i t h t h e computer model p r e d i c t i o n s . These graphs show c l e a r l y t h a t w h i l e t h e s t e a d y s t a t e model a c c u r a t e l y p r e d i c t s the t o t a l column l o s s e s , or e q u i v a l e n t l y the e l e c t r i c f i e l d , the p r e d i c t e d w a l l l o a d i n g i s too s m a l l and the p r e d i c t e d r a d i a n t l o s s i s t o o l a r g e . In f a c t , the w a l l l o a d i n g i s the key p r e d i c t i o n , s i n c e the mechanism which d e t e r m i n e s the r a d i a n t l o s s i s e s s e n t i a l l y power b a l a n c e ; a s m a l l change i n the c o r e temperature s u b s t a n t i a l l y a l t e r s the r a d i a n t l o s s w i t h o u t c h a n g i n g a n y t h i n g e l s e s i g n i f i c a n t l y and t h u s e s t a b l i s h e s power b a l a n c e . The e f f e c t of an enhanced w a l l l o s s would thus be F i g u r e V-4. T o t a l column l o s s e s vs c u r r e n t F i g u r e V-5. R a d i a n t l o s s e s vs c u r r e n t 7 7 30 \ Q (kW) 20 10H 0 Experiment •+ 1.5 A tm OZ2 A t m x 2.9 A t m Computer Model Modified M odel 3 Atm 1 A tm 3 A t m 1 A t m Steady S ta te Model 100 300 500 KA) 700 F i g u r e V - 6 . W a l l l o s s e s vs c u r r e n t 78 t o reduce the r a d i a n t l o s s c o r r e s p o n d i n g l y , r a t h e r than i n c r e a s i n g the t o t a l column l o s s e s , s i n c e the r a d i a n t l o s s w i l l change t o accomodate the new w a l l l o s s w i t h o u t c h a n g i n g the t o t a l power much. T h i s has a l a r g e e f f e c t on the r a d i a t i v e e f f i c i e n c y and i s t h e r e f o r e of g r e a t i n t e r e s t f o r l i g h t i n g a p p l i c a t i o n s . The most o b v i o u s e x p l a n a t i o n f o r t h e enhanced w a l l l o a d i n g i n t h i s system i s the l a r g e c o n v e c t i v e l o s s a s s o c i a t e d w i t h the f l o w i n g gas - a l o s s which i s n e g l e c t e d i n the s t e a d y s t a t e model. The r e s u l t s of the t r a n s i e n t gas h e a t i n g c a l c u l a t i o n i n d i c a t e t h a t the gas s h o u l d r e a c h steady s t a t e about h a l f w a y down the column i f t h e r e i s s u f f i c i e n t e l e c t r i c a l power a v a i l a b l e i n the c o r e . U s i n g t h i s r e s u l t , a new p r e d i c t i o n of t h e w a l l l o a d i n g can be made by combining the s t e a d y s t a t e l o a d i n g i n the l a s t 50 mm of the column w i t h the heat c o n t e n t of the gas as i t l e a v e s the column. At low power l e v e l s the gas h e a t i n g i s l i m i t e d by the power i n p u t t o the a r c c o r e ; under these c o n d i t i o n s ( d i s t i n g u i s h e d by the f a c t t h a t the c a l c u l a t e d heat c o n t e n t of the gas exceeds the i n p u t power) i t i s t o be e x p e c t e d t h a t the c o n d u c t i v e l o s s e s t o the w a l l w i l l be n e g l i g i b l e . The heat c a r r i e d away by the gas i s d i f f i c u l t t o e s t i m a t e , but s h o u l d be g r e a t e r than the c a l c u l a t e d s t e a d y s t a t e w a l l l o a d i n g s i n c e the t h e r m a l g r a d i e n t s w i l l be s u b s t a n t i a l l y l a r g e r . For want of a b e t t e r e s t i m a t e , a v a l u e of t w i c e the s t e a d y s t a t e l o a d i n g has been assumed. The m o d i f i e d w a l l l o a d i n g can now be c a l c u l a t e d , 79 s e p a r a t i n g the a r c c o n d i t i o n s i n t o t h r e e r e gimes. At low power z e r o c o n d u c t i v e l o s s and a c o n v e c t i v e l o s s of t w i c e the p r e d i c t e d s t e a d y s t a t e l o s s a r e assumed. At h i g h power the c o n v e c t i v e l o s s can be c a l c u l a t e d assuming the gas reaches the s t e a d y s t a t e p r o f i l e and the c o n d u c t i v e l o s s i s assumed t o be z e r o f o r the f i r s t h a l f of the column and e q u a l t o t h e s t e a d y s t a t e v a l u e f o r the second h a l f , a f t e r the hot gas has reached the w a l l . I n the t r a n s i t i o n r e g i o n , where t h e power i s i n t e r m e d i a t e , the l o s s e s a r e assumed t o change l i n e a r l y w i t h c u r r e n t from the low power t o the h i g h power v a l u e s . The h i g h and low power r e g i o n s a r e d e l i m i t e d by Q T > 2 Q G and Q T < Q G r e s p e c t i v e l y . Here 0^, i s the t o t a l i n p u t power and Q G i s the c a l c u l a t e d power l o s s by c o n v e c t i o n i f the gas has reached the stea d y s t a t e p r o f i l e . F i g u r e V-7 i l l u s t r a t e s how the m o d i f i e d w a l l l o a d i n g was c a l c u l a t e d f o r P=3 Atm. T h i s m o d i f i e d w a l l l o a d i n g i s p l o t t e d i n F i g u r e V-6 w i t h the e x p e r i m e n t a l d a t a , and shows good agreement f o r such a crude model. I t would be u s e f u l t o c o n s t r u c t a more a c c u r a t e model of a c o n v e c t i o n dominated a r c u s i n g n u m e r i c a l m o d e l l i n g t e c h n i q u e s which c o u l d p r e d i c t the b e h a v i o u r of the a r c a t low power l e v e l s , and t h i s i s suggested as a p o s s i b l e f u t u r e r e s e a r c h t o p i c . The c a l o r i m e t r i c measurements have shown t h a t the stead y s t a t e model p r e d i c t s t h e t o t a l column l o s s e s a c c u r a t e l y but g i v e s poor r e s u l t s f o r the w a l l and r a d i a n t l o s s e s . T h i s can be u n d e r s t o o d i n view of t h e i m p o r t a n t r o l e 80 Modified Total Convective Conductive Maximum Convective Loss St eady S ta te C onductive Loss F i g u r e V-7. The m o d i f i e d w a l l l o a d i n g p r e d i c t i o n 81 p l a y e d by c o n v e c t i o n i n the power b a l a n c e of t h i s type of a r c , and a crude c a l c u l a t i o n shows t h a t the d i s c r e p a n c y c o u l d be a c c o u n t e d f o r by c o n v e c t i o n due t o the gas f l o w . I t i s p o s s i b l e , but by no means c e r t a i n t h a t t h i s i s t h e cause of the enhanced w a l l l o a d i n g , and more work w i l l be r e q u i r e d i n o r d e r t o c o n s t r u c t a model which p r e d i c t s the power b a l a n c e a c c u r a t e l y . C. TEMPERATURE PROFILES The temperature p r o f i l e s were measured p r i m a r i l y u s i n g t h e a b s o l u t e i n t e n s i t y of the AI430 nm l i n e , as d e s c r i b e d i n Chapter IV. The r e s u l t s were c o n f i r m e d i n d e p e n d e n t l y by measurements of the l i n e w i d t h and continuum i n t e n s i t y , which a r e d e s c r i b e d i n s e c t i o n D. Each e x p e r i m e n t a l run produces about 40 s p e c t r a spaced e v e n l y a c r o s s the a r c , and when the s p e c t r a are u n f o l d e d and a n a l y s e d a temperature p r o f i l e l i k e the ones shown i n F i g u r e V-8 i s o b t a i n e d . S i n c e p l o t t i n g f a m i l i e s of c u r v e s l i k e F i g u r e V-8 would be of l i m i t e d v a l u e i n u n d e r s t a n d i n g the a r c , the temperature p r o f i l e s have been c h a r a c t e r i z e d by the c o r e temperature T 0 and the 10000K r a d i u s R i 0 « Measured v a l u e s of T 0 and R 1 0 a r e p l o t t e d i n F i g u r e s V-9 and V-10 a l o n g w i t h the p r e d i c t e d v a l u e s from the s t e a d y s t a t e model. I t can be seen from t h e s e graphs t h a t the agreement between the s t e a d y s t a t e model and the measured v a l u e s i s good a t h i g h c u r r e n t s and f a i r a t low c u r r e n t s . At c u r r e n t s below 500A the a r c i s somewhat h o t t e r and narrower than the Temperature (K) 2000 H -I 1 1 1 ' 1 1 1 I 16 12 © 4 0 4 6 12 16 Radius(mm) F i g u r e V - 8 . T y p i c a l t e m p e r a t u r e p r o f i l e s 12 0 0 0 A T ( K ) F i g u r e V-9. Core t e m p e r a t u r e vs c u r r e n t 84 i i i 100 300 500 I (A) 700 F i gu re V - 1 0 . Core r a d i u s vs c u r r e n t 85 model p r e d i c t s , which c o u l d be another consequence of the s t r o n g l y c o n v e c t i v e n a t u r e of the a r c a t low power l e v e l s . In t h i s r e g i o n t h e r e i s s i m p l y i n s u f f i c i e n t power a v a i l a b l e i n the c o r e t o heat the gas up a l l the way t o the w a l l , so the a r c c h a n n e l remains narrow, and i n o r d e r t o c a r r y the r e q u i r e d amount of c u r r e n t the a r c c o r e must be h o t t e r than would be e x p e c t e d i f i t were of l a r g e r d i a m e t e r . T h i s i s c o n s i s t e n t w i t h the r e s u l t s of the c a l o r i m e t r y i n s e c t i o n B, and a g a i n p o i n t s out t h a t i t would be u s e f u l t o have a model f o r the b e h a v i o u r of a c o n v e c t i o n dominated a r c . D. LINE WIDTH AND CONTINUUM MEASUREMENTS In o r d e r t o c o n f i r m the te m p e r a t u r e s c a l c u l a t e d from any one s p e c t r a l q u a n t i t y i t i s d e s i r a b l e t o have measurements of some o t h e r independent q u a n t i t y as w e l l . There a r e many c o m p l i c a t e d p r o c e s s e s which a f f e c t the e m i s s i o n of l i g h t from a plasma, and such phenomena as d e v i a t i o n s from l o c a l t h e r m a l e q u i l i b r i u m , l o w e r i n g of the i o n i z a t i o n p o t e n t i a l , and i n a c c u r a t e t r a n s i t i o n p r o b a b i l i t i e s o f t e n plague the s p e c t r o s c o p i s t and produce e r r o r s . I n t h i s work the raw d a t a c o n s i s t of argon s p e c t r a from 424 t o 436 nm, and so the use of f e a t u r e s of t h a t spectrum f o r secondary t e m p e r a t u r e measurements i s p a r t i c u l a r l y s u i t a b l e . The continuum i n t e n s i t y and the w i d t h of the AI430 nm l i n e have been chosen as secondary f e a t u r e s , as d e s c r i b e d i n Chapter IV, and F i g u r e V-11 i l l u s t r a t e s the e x c e l l e n t agreement among the t e m p e r a t u r e s c a l c u l a t e d by the 86 t h r e e methods. The e r r o r bars shown i n F i g u r e V-11 r e p r e s e n t the r e p r o d u c i b i l i t y of the measurements from run t o run (1 s t a n d a r d d e v i a t i o n ) as measured i n a s e r i e s of runs a t 430 A and 1.5 Atm. The agreement among the t h r e e methods i s b e s t c o n s i d e r e d i n terms of the s y s t e m a t i c u n c e r t a i n t i e s i n t h e s e measurements, d e r i v e d i n s e c t i o n E and shown on F i g u r e V-11. E. SYSTEMATIC ERRORS The e r r o r a n a l y s i s of the c a l o r i m e t r i c system i s f a i r l y s i m p l e , and of the s p e c t r o s c o p i c system somewhat more c o m p l i c a t e d . L e t us f i r s t c o n s i d e r the c a l o r i m e t r y . The measured v a l u e s a r e water f l o w r a t e s and t e m p e r a t u r e s . The f l o w meters can o n l y be r e a d t o about 10% a c c u r a c y due t o v a r i a t i o n s i n the f l o w r a t e , a l t h o u g h the meters a r e c a l i b r a t e d t o w i t h i n 5%. The d i g i t a l t emperature measurements a r e a c c u r a t e t o ±0.1K which c o r r e s p o n d s t o about 50 W a t normal f l o w r a t e s , and i s not i m p o r t a n t . The l e n g t h of the a r c i s 100±2 mm which c o n t r i b u t e s about 2% u n c e r t a i n t y t o the column l o s s e s . In a d d i t i o n t h e r e i s u n c e r t a i n t y a s s o c i a t e d w i t h the e l e c t r o d e r e g i o n s of the a r c . These r e g i o n s a r e g e n e r a l l y h o t t e r than the column, but t h i s i s compensated by the heat and r a d i a t i o n absorbed by the e l e c t r o d e s . I t has been assumed t h a t t h e s e e f f e c t s c a n c e l out and t h a t the a r c column has an e f f e c t i v e l e n g t h of 100 mm. T h i s assumption c o u l d i n t r o d u c e an e r r o r of up t o 5% i n the c a l o r i m e t r y , but the e l e c t r o d e r e g i o n s a r e v e r y c o m p l i c a t e d (see f o r example P f e n d e r 1980) and an e x a c t 87 Figure V-11 . Core temperatures from three methods 88 a n a l y s i s i s beyond the scope of t h i s t h e s i s . The r a d i a t i v e l o s s e s a r e d i m i n i s h e d by s e v e r a l mechanisms; up t o 5% of the l i g h t i s l o s t from the ends of the a r c , and between 5 and 10% i s absorbed by the g l a s s w a l l s , water j a c k e t and end p l a t e s of the v e s s e l and added t o the measured w a l l l o a d i n g . These components i n c r e a s e s the measured w a l l l o a d i n g and d e c r e a s e the measured r a d i a t i v e l o s s e s , but not enough t o a l t e r the power b a l a n c e s u b s t a n t i a l l y . The measured power l o s s e s g e n e r a l l y l i e between 0 and 5% below the i n p u t power, which c o n f i r m s the e s t i m a t e of l i g h t l e a k a g e . The a c c u r a c y of the c a l o r i m e t r i c measurements i s thus l i m i t e d t o about 10% by f l o w measurements, but the o n l y i m p o r t a n t s y s t e m a t i c e r r o r i s t h e 5 t o 10% f r a c t i o n of the r a d i a t i o n e m i t t e d which i s absorbed by the v e s s e l w a l l s . The t h r e e temperature measurements have d i f f e r e n t s y s t e m a t i c e r r o r s a s s o c i a t e d w i t h them. For the continuum and l i n e i n t e n s i t y measurements one source of e r r o r s i s the a b s o l u t e c a l i b r a t i o n which has an u n c e r t a i n t y of about ±20% as i n d i c a t e d i n F i g u r e s IV-4 and 5. The AI t r a n s i t i o n p r o b a b i l i t y i s a c c u r a t e t o 7%, which i m p l i e s an o v e r a l l t emperature u n c e r t a i n t y of ±200K f o r the l i n e i n t e n s i t y method. The continuum f a c t o r ^ has a t h e o r e t i c a l u n c e r t a i n t y of ±30%, a l t h o u g h t h a t appears from e x p e r i m e n t a l e v i d e n c e t o be an o v e r e s t i m a t e ( P r e s t o n 1977) and might b e t t e r be e s t i m a t e d as ±10%. The continuum i n t e n s i t y t h e r e f o r e can be used t o c a l c u l a t e the e l e c t r o n d e n s i t y w i t h a s y s t e m a t i c 89 u n c e r t a i n t y of ±25% which p r o b a b l y o v e r e s t i m a t e s the e r r o r . The u n c e r t a i n t y i n the l i n e w i d t h measurement i s p a r t l y due t o the l a r g e i n s t r u m e n t w i d t h of the system (0.063 nm) compared t o the l i n e w i d t h (0.1 nm). A f t e r smoothing and A b e l i n v e r s i o n the l i n e w i d t h can o n l y be d e t e r m i n e d w i t h an a c c u r a c y of about ±0.01 nm or 10%. The l i n e w i d t h c o e f f i c i e n t i n t r o d u c e s an a d d i t i o n a l u n c e r t a i n t y of 10% ( R i c h t e r 1964) so the l i n e w i d t h method has an u n c e r t a i n t y of ±20% i n the e l e c t r o n d e n s i t y . Assuming t h a t l o c a l t h e r m a l e q u i l i b r i u m (LTE) e x i s t s , the temperature can be c a l c u l a t e d from the e l e c t r o n d e n s i t y as o u t l i n e d i n Chapter IV, and the u n c e r t a i n t i e s i n the temperature from the continuum and l i n e w i d t h methods a r e ±300K and ±250K r e s p e c t i v e l y . These u n c e r t a i n t i e s a re shown i n F i g u r e V-11 and the v a l u e s from the t h r e e methods c l e a r l y agree t o w e l l w i t h i n the s y s t e m a t i c u n c e r t a i n t i e s , which s u p p o r t s the assumption of LTE and i s e v i d e n c e of good e x p e r i m e n t a l t e c h n i q u e . F. TIME VARIATION The s p e c t r o s c o p y of the a r c was c o m p l i c a t e d by the l a r g e r i p p l e (50%) i n the c u r r e n t , so a synchronous d i s c s t r o b o s c o p e was used t o e l i m i n a t e the time dependence of the ar c column. The s t r o b e was s y n c h r o n i z e d w i t h the c u r r e n t maximum, but i t i s n e c e s s a r y t o examine the time e v o l u t i o n of the column i n o r d e r t o ensure t h a t we un d e r s t a n d i t s b e h a v i o u r . 90 F i g u r e V-12 shows the v a r i a t i o n t h rough the c y c l e (8.3 ms s i n c e the r i p p l e f r e q u e n c y i s 120 Hz) of the c u r r e n t , the c o r e temperature T 0, and the c o r e r a d i u s R i o * The s p e c t r o s c o p y i s done c o i n c i d e n t w i t h maximum c u r r e n t , 2.5 t o 3 ms a f t e r minimum c u r r e n t , and the a r c i s n e a r l y s t a t i o n a r y a t t h a t t i m e , which reduces s y n c h r o n i z a t i o n r e q u i r e m e n t s and p r e v e n t s time dependent e f f e c t s from c a u s i n g d i f f i c u l t y . I t can be seen t h a t the c o r e temperature and r a d i u s f o l l o w the c u r r e n t q u i t e c l o s e l y t h r o u g h the c y c l e , w i t h a l a g of l e s s than one m i l l i s e c o n d . There i s a c e r t a i n amount of time dependence i n the column, however, as e v i d e n c e d by the f a c t t h a t the c o r e c o n d i t i o n s measured 2 ms a f t e r maximum c u r r e n t do not c o r r e s p o n d t o the c u r r e n t o b s e r v e d a t t h a t t i m e , but t o the l a r g e r c u r r e n t f l o w i n g a s h o r t time b e f o r e . As the c u r r e n t d e c r e a s e s , t h e c o r e remains s l i g h t l y h o t t e r and l a r g e r than measurements a t the s t a t i o n a r y p o i n t would p r e d i c t . I t would be h e l p f u l t o e l i m i n a t e the r i p p l e e n t i r e l y , e s p e c i a l l y f o r c a l o r i m e t r i c measurements which cannot be s t r o b e d , but t h i s p r e s e n t s a major d i f f i c u l t y a t t h e s e h i g h power l e v e l s . A p o s s i b l e r e s e a r c h t o p i c t o be pursued i n the f u t u r e would be the c o n s t r u c t i o n of a lower power, r i p p l e f r e e a r c so t h a t the p r o p e r t i e s of a stea d y c o n v e c t i o n dominated a r c c o u l d be d e t e r m i n e d a c c u r a t e l y . 1000-1(A) v, • 1 1 1 1 0 2 4 6 t (ms) 8 F i g u r e V-12. C u r r e n t , temperature and r a d i u s vs t ime 92 G. CYLINDRICAL SYMMETRY One of the assumptions made i n both the computer c a l c u l a t i o n s and the s p e c t r o s c o p i c a n a l y s i s was c y l i n d r i c a l symmetry i n the a r c column. T h i s was e n f o r c e d i n the A b e l i n v e r s i o n by a v e r a g i n g the two h a l v e s of the p r o f i l e t o remove any asymmetry, but the raw d a t a show t h e symmetry of the column c l e a r l y . F i g u r e V-13 shows some t y p i c a l i n t e n s i t y p r o f i l e s b e f o r e the d a t a p r o c e s s i n g t a k e s p l a c e . These p r o f i l e s show the v a r i a t i o n of i n t e n s i t y ( a t the c e n t e r of the AI430 nm l i n e ) w i t h v e r t i c a l p o s i t i o n i n the a r c , and they have been superimposed on t h e i r r e f l e c t i o n s t o show the symmetry. S i n c e any asymmetry due t o g r a v i t a t i o n a l buoyancy would appear i n the v e r t i c a l p r o f i l e , t h i s p r o v i d e s c l e a r e v i d e n c e of c y l i n d r i c a l symmetry. H. AXIAL VARIATION The ste a d y s t a t e model i s a p p l i c a b l e t o an a x i a l l y i n v a r i a n t s e c t i o n of t h e a r c column, so t h e a x i a l v a r i a t i o n i n the column has been i n v e s t i g a t e d by moving the a r c r e l a t i v e t o the o p t i c a l system. T h i s r e v e a l s the changes i n the temperature p r o f i l e a l o n g the a r c a x i s , as shown i n f i g u r e s V-14 and 15. D i r e c t o b s e r v a t i o n s u g g e s t s t h a t the e l e c t r o d e s s t r o n g l y i n f l u e n c e the a r c f o r a d i s t a n c e of 10 t o 20 mm but the c e n t r a l 50 mm of the column appears u n i f o r m . F i g u r e s V-14 and 15 show t h a t the u n i f o r m r e g i o n i s 30 mm l o n g and F i g u r e V-13. T y p i c a l i n t e n s i t y p r o f i l e s 11000 T(K) 10800< — i -20 CATHODE —1— -10 Flow Direction » 0 10 z (mm) — i — 20 1 30 ANODE -> F i g u r e V-14. Temperature vs a x i a l p o s i t i o n io i R (mm) Flow Direction > -20 -10 • - C A T H O D E 0 - i 1 —r~ 10 20 *0 z (mm) ANODE H F i g u r e V-15. Core r a d i u s vs a x i a l p o s i t i o n 95 that outside that region the temperature begins to decrease and the radius to increase, due to a combination of electrode and gas flow e f f e c t s . Near the plane in which the spectroscopy was done (z=0 on the graphs) the a x i a l v a r i a t i o n is small enough to allow the assumption of a x i a l invariance. 9 6 CHAPTER VI CONCLUSIONS A. INTRODUCTION In t h i s t h e s i s the r e s u l t s a r e p r e s e n t e d from r e s e a r c h which was undertaken w i t h the u l t i m a t e g o a l of i m p r o v i n g the performance of the v o r t e x s t a b i l i z e d a r c lamp, p r i m a r i l y i n the a r e a s of l o n g e v i t y and r a d i a t i v e e f f i c i e n c y . T h i s g o a l has v been approached i n d i r e c t l y , on the p r i n c i p l e t h a t the s u r e s t road t o improvement i s u n d e r s t a n d i n g . The immediate aim of the r e s e a r c h has t h e r e f o r e been t o study the a r c e x p e r i m e n t a l l y and model i t t h e o r e t i c a l l y i n o r d e r t o expand our u n d e r s t a n d i n g of how i t behaves. A second i m p o r t a n t o b j e c t i v e has been t o de t e r m i n e what we do and do not un d e r s t a n d about the a r c , and thus t o suggest the d i r e c t i o n f u t u r e r e s e a r c h might most p r o f i t a b l y t a k e . In t h i s c h a p t e r the e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s a r e summarized, the c o n c l u s i o n s drawn from them a r e p r e s e n t e d , and the a u t h o r ' s o r i g i n a l c o n t r i b u t i o n s t o the work a r e d e t a i l e d . In a d d i t i o n , a number of r e s e a r c h t o p i c s which c o u l d h e l p t o answer many of the q u e s t i o n s s t i l l r e m a i n i n g about the a r c a r e su g g e s t e d . 97 B. CONCLUSIONS A st e a d y s t a t e model has been used t o p r e d i c t c o n d i t i o n s i n the a r c column, augmented by c a l c u l a t i o n s of the c o n v e c t i v e l o s s e s a s s o c i a t e d w i t h the gas f l o w . These c a l c u l a t i o n s suggest t h a t c o n v e c t i v e e f f e c t s a re im p o r t a n t a t h i g h c u r r e n t s and dominant a t low c u r r e n t s , and e x p e r i m e n t a l r e s u l t s support t h i s c o n c l u s i o n . D e s p i t e t h i s p e r t u r b a t i o n of stea d y s t a t e c o n d i t i o n s the model p r e d i c t s many f e a t u r e s of the a r c a d e q u a t e l y , which shows t h a t the g e n e r a l n a t u r e of the column i s w e l l u n d e r s t o o d , a l t h o u g h some d e t a i l s a r e s t i l l u n c l e a r . The s e l f - c o n s i s t e n t s o l u t i o n s of the steady s t a t e e q u a t i o n s produce temperature p r o f i l e s whose shapes agree w i t h both i n t u i t i v e e x p e c t a t i o n s and e x p e r i m e n t a l r e s u l t s . The c o r e t e m p e r a t u r e s and r a d i i e x t r a c t e d from t h e s e p r o f i l e s a r e i n good agreement w i t h e x p e r i m e n t a l v a l u e s a t c u r r e n t s above 500 A, but a t lower c u r r e n t s the measured p r o f i l e s a r e somewhat h o t t e r and narrower than the c a l c u l a t i o n s p r e d i c t . T h i s i s c o n s i s t e n t w i t h the r e s u l t s of the c o n v e c t i v e t r a n s p o r t c a l c u l a t i o n s , which i n d i c a t e t h a t under t h e s e c o n d i t i o n s the e l e c t r i c a l power i n p u t t o the column i s i n s u f f i c i e n t t o heat the gas up t o the stea d y s t a t e p r o f i l e . The t h e o r e t i c a l p r e d i c t i o n s of t h e power b a l a n c e i n t h e column a r e a l s o a c c u r a t e a t h i g h c u r r e n t s when they a r e c o r r e c t e d f o r c o n v e c t i v e e f f e c t s . Due t o l i m i t a t i o n s i n the power s u p p l y , c a l o r i m e t r i c r e s u l t s a r e o n l y a v a i l a b l e up t o 98 about 500 A, so most of the comparison of t h e o r y and experiment i s i n the lower power regime where s u b s t a n t i a l c o n v e c t i v e e f f e c t s o c c u r , but a t the h i g h e s t a v a i l a b l e power l e v e l s the p r e d i c t i o n s a r e a c c u r a t e . At low power the c o n v e c t i o n dominated a r c has c o n v e c t i v e l o s s e s about t w i c e as l a r g e as the p r e d i c t e d s t e a d y s t a t e l o s s e s , a r e s u l t which seems r e a s o n a b l e , a l t h o u g h i t has not been r i g o r o u s l y p r e d i c t e d . None of the s e r e s u l t s i s a f f e c t e d s u b s t a n t i a l l y by p r e s s u r e i n the range of 1 t o 3 Atmospheres, n e i t h e r i n the c a l c u l a t i o n s nor the e x p e r i m e n t a l r e s u l t s . T h i s work has a l s o shown t h a t the most i m p o r t a n t f e a t u r e of an a c c u r a t e model of the a r c i s the m o d e l l i n g of the n o n - r a d i a t i v e l o s s e s . The b e h a v i o u r of the a r c i s such t h a t the r a d i a t i v e l o s s e s change t o accomodate changes i n the o t h e r components of the power b a l a n c e , and so i n o r d e r t o p r e d i c t the r a d i a t i v e l o s s e s one must f i r s t a c c u r a t e l y p r e d i c t the o t h e r power l o s s e s i n the column. In summary, the c o n d i t i o n s i n the column of a v o r t e x s t a b i l i z e d a r c can and have been p r e d i c t e d by computer m o d e l l i n g . At h i g h power l e v e l s a steady s t a t e model, w i t h s u b s t a n t i a l c o r r e c t i o n s f o r c o n v e c t i v e e f f e c t s , produces a c c u r a t e r e s u l t s but a t lower power c o n v e c t i v e e f f e c t s a r e dominant and the s t e a d y s t a t e model of the a r c i s no l o n g e r a p p l i c a b l e . 99 C. ORIGINAL CONTRIBUTIONS The r e s e a r c h d e s c r i b e d i n t h i s t h e s i s f o l l o w s the t r a d i t i o n a l s c i e n t i f i c method of m o d e l l i n g a system, t e s t i n g the model e x p e r i m e n t a l l y , and then u s i n g the e x p e r i m e n t a l r e s u l t s t o r e f i n e the model. The m o d e l l i n g has two a s p e c t s ; the column m o d e l l i n g and the v o r t e x c a l c u l a t i o n . I c o n c e i v e d and wrote the computer programs used f o r the t r a n s i e n t h e a t i n g c a l c u l a t i o n and the steady s t a t e model, which p r e s e n t e d some d i f f i c u l t s t a b i l i t y problems and r e q u i r e d a good u n d e r s t a n d i n g of the c h a r a c t e r i s t i c s of the a r c column. In s t u d y i n g the problem of the gas v o r t e x I a n a l y t i c a l l y s o l v e d the problem of the v i s c o u s decay of a gas v o r t e x i n a c y l i n d r i c a l v e s s e l . E x p e r i m e n t a l l y , i n a d d i t i o n t o d e s i g n i n g and a s s e m b l i n g the a r c system and the d i a g n o s t i c system I made a number of i n n o v a t i o n s i n the a r c v e s s e l and support s t r u c t u r e . I changed the d e s i g n of the v e s s e l end caps t o a l l o w e a s i e r assembly and d i s m a n t l i n g of the v e s s e l and improve the gas f l o w . The improved gas f l o w has the advantage of a much smoother f l o w p a t t e r n due t o the t a n g e n t i a l i n j e c t i o n j e t . T h i s makes the flo w e a s i e r t o model, reduces the p o s s i b i l i t y of r e v e r s e d a x i a l f l o w and s t a g n a t i o n r e g i o n s , and s h o u l d reduce the amount of t u r b u l e n c e near the i n l e t . I a l s o i n c o r p o r a t e d an i n t e g r a l exhaust gas heat exchanger i n the downstream end cap. I d e s i g n e d and b u i l t the f i l a m e n t assembly f o r the i n s i t u a b s o l u t e c a l i b r a t i o n and r e d e s i g n e d the v e s s e l and e l e c t r o d e s u p p o r t s t r u c t u r e t o improve 100 a c c e s s i b i l i t y and r i g i d i t y . F i n a l l y I c o n c e i v e d and implemented the automated d a t a h a n d l i n g system which i s a ne c e s s a r y p a r t of the d i a g n o s t i c system i n an experiment such as t h i s . T h i s i n c l u d e d w r i t i n g a l a r g e number of computer programs f o r d a t a h a n d l i n g and s e t t i n g up the d i g i t a l equipment n e c e s s a r y f o r the d a t a c o l l e c t i o n . D. SUGGESTED FUTURE WORK The r e s e a r c h d e s c r i b e d i n t h i s t h e s i s has had two e f f e c t s ; i t has answered a number of q u e s t i o n s and i t has posed as many more. The g e n e r a l b e h a v i o u r of the a r c has been shown t o be p r e d i c t a b l e , but some of the d e t a i l s a r e s t i l l not c l e a r . The b e h a v i o u r of the c o n v e c t i o n dominated a r c , i . e . an a r c i n which the gas never reaches an a x i a l l y i n v a r i a n t temperature p r o f i l e , i s p o o r l y u n d e r s t o o d . I t i s suggested t h a t a computer model be c o n s t r u c t e d t o p r e d i c t the be h a v i o u r of such an a r c , and t h a t a w e l l f i l t e r e d power s u p p l y be b u i l t f o r use a t low c u r r e n t s (<200 A) t o study the power b a l a n c e i n a c o n v e c t i o n dominated a r c . T h i s r e s e a r c h has shown t h a t f o r c e d c o n v e c t i o n c o u l d be the mechanism which p e r t u r b s the stea d y s t a t e a r c c o n d i t i o n s . The p o s s i b i l i t y r e mains, however, t h a t much of the d e v i a t i o n c o u l d be due t o t u r b u l e n t heat t r a n s p o r t , a v e r y complex p r o c e s s which i s s t i l l not t h o r o u g h l y u n d e r s t o o d (Chien and Benenson 1980). To r e s o l v e t h i s q u e s t i o n i t i s e s s e n t i a l t h a t an experiment be d e s i g n e d t o 101 measure the w a l l l o a d i n g i n the a r c as a f u n c t i o n of a x i a l p o s i t i o n . T h i s w i l l p r o v i d e an e x c e l l e n t t e s t of the c o n c l u s i o n s which have been drawn r e g a r d i n g the n a t u r e and importance of the c o n v e c t i v e l o s s e s i n the a r c . F i n a l l y , i n keeping w i t h the a p p l i e d n a t u r e of t h i s r e s e a r c h , I would l i k e t o propose the c o n s t r u c t i o n of a system d e s i g n e d t o reduce the c o n v e c t i v e l o s s e s i n the a r c and t h u s improve the r a d i a t i v e e f f i c i e n c y . Such a system c o u l d c r e a t e a v o r t e x f l o w by i n j e c t i n g gas t a n g e n t i a l l y a t both end caps s i m u l t a n e o u s l y and a l l o w i n g the gas t o escape from both ends through gaps between the e l e c t r o d e s h a f t s and end caps (see F i g u r e V I - 1 ) . C a r e f u l b a l a n c i n g of the f l o w s s h o u l d reduce a x i a l v e l o c i t y t o a minimum and has the p o t e n t i a l of p r o d u c i n g improved e f f i c i e n c y . Whether t h a t p o t e n t i a l i s r e a l i z e d or n o t , the experiment s h o u l d p r o v i d e v a l u a b l e i n f o r m a t i o n about th e b e h a v i o u r of the v o r t e x s t a b i l i z e d a r c . 1 0 2 / Argon out Gas Vortex c 3 Argon i n Argon i n \ Argon out F i g u r e V I - 1 . The c o n v e c t i o n f r e e v o r t e x s t a b i l i z e d a r c 103 BIBLIOGRAPHY B a e s s l e r , P . and Kock,M., J o u r n a l of P h y s i c s BJ_3,1351 (1980) Camm,D.M. and Nodwell,R., US P a t e n t A p p l i c a t i o n #478872 (1974) Chen,D.M.,Hsu,K.C.,Liu,C.H. and P f e n d e r , E . , IEEE Trans on Plasma S c i e n c e . PS8.425 (1980) Chien,Y.K. and Benenson,D.M., IEEE Trans on Plasma S c i e n c e , PS8.411 (1980) , CRC Handbook of C h e m i s t r y and P h y s i c s , C h e m i c a l Rubber Company, C l e v e l a n d (1973) Emmons,H.W., P h y s i c s of F l u i d s J_0_, 1 1 25 ( 1 967) Evans,D.L. and Tankin,R.S., P h y s i c s of F l u i d s J_0,1137 (1967) F l e u r i e r , C . And C h a p e l l e , J . , Computer P h y s i c s Communications J_0,200 (1974) F i n k e l n b u r g , W . and Maecker,H., Handbuch der P h y s i k X X I I , S p r i n g e r - V e r l a g , B e r l i n (1956) G e t t e l , L . E . , A c o m p a r i t i v e study of DC and AC v o r t e x s t a b i l i z e d a r c s , Ph.D. T h e s i s , UBC (1980) Hoyaux.M.F., Arc P h y s i c s , S p r i n g e r - V e r l a g , New York (1968) Kesaev,I.G., Cathode P r o c e s s e s i n the Mercury A r c , C o n s u l t a n t s Bureau, New York (1964) Larenz,W., Z. P h y s i k 129,327 (1951) Maecker,H., Z. P h y s i k 157,1 (1959) Maecker,H., Z. P h v s i k 158.392 (1960) 01sen,H.N., J . Quant. Spec, and Rad. T r a n s . _3_, 305 ( 1963) P f e n d e r , E . , Pure and A p p l i e d C h e m i s t r y .52,1773 (1980) 104 P res ton , R . C . , J . Quant . Spec , and Rad. T r a n s . J_8,337 (1977) R i c h t e r , J . , Z. A s t r o p h y s i k 61.57 (1965) S c h o e n h e r r , 0 . , E l e c t r o t e c h n . Z e i t s c h r i f t (1909) Tam,S .Y .K . and G ibbs , B .W . , RCA Report #96208-2 (1972) Tuchman,A. and E n o s , G . , Avco C o r p o r a t i o n T e c h n i c a l Report j AVSSD-0043-67-RR (1967) 105 APPENDIX A STEADY STATE CALCULATION PROGRAMS PROGRAM TEMPROFILE; (* 22/V/80 *) VAR E,E2,STEP,EMAX,DR,DR2,T0,T1,T:REAL; CORR,ERR,COND,CURR,P,QWALL:REAL; Q,QRT,M,R9,SIGO,TS,KAP0,TK,QR0,TM,TQ:REAL; I,L:INTEGER;ERROR:BOOLEAN; TP:ARRAY(1..100) OF INTEGER-PROCEDURE TITLE; SLINKAGE 'TITLE'; PROCEDURE PLOT; SLINKAGE 'RP100'; FUNCTION KAPPA(T:REAL):REAL; BEGIN IF T>7000 THEN KAPPA:=KAP0*EXP(T/TK) ELSE KAPPA:=1.1E-4+2.7E-7*T; END; FUNCTION SIGMA(T:REAL):REAL; BEGIN IF T<2000 THEN SIGMA:=0 ELSE SIGMA:=SIG0/EXP(SQR(TS/T)); END; FUNCTION QRAD(T:REAL):REAL; VAR T1:REAL; BEGIN T1:=(TM-T)/TQ; IF T1>10 THEN QRAD:= 0 ELSE QRAD:=QR0/EXP(T1*T1); END; PROCEDURE INTEGRATE; VAR R,INTEGRAL:REAL; L:INTEGER; BEGIN R:=0;T:=T0;INTEGRAL:=0; FOR L:=1 TO 100 DO BEGIN IF T<100 THEN T:=0 ELSE BEGIN INTEGRAL:=INTEGRAL+(QRAD(T)-E2 * SIGMA(T))*L*DR2; T:=T+INTEGRAL/(L*KAPPA(T)); END; TP(L):=INTEGER(T); END; END; 106 BEGIN (*TEMPROFILE*) READ(INPUT,P,E,STEP,EMAX); T1:=600;DR:=135E-4;DR2:=DR*DR;ERR:=1E-3; KAP0:=108E-6;TK:=2400; IF P=1 THEN BEGIN SIG0:=150;TS:=12700;QRO : = 5240;TM: =16500;TQ:=35 ELSE IF P=3 THEN BEGIN SIG0:=181;TS:=14100;QRO:=55500;TM:=20E3;TQ:=44 ELSE IF P=5 THEN BEGIN SIG0:=207;TS:=15000;QR0:=102E4;TM:=25E3;TQ:=57 ELSE IF P=7 THEN BEGIN SIG0:=229;TS:=15600;QR0:=5.4E6;TM:=27E3;TQ:=58 ELSE ERROR:=TRUE; TITLE(6, 1***TEMPROFILE***',16); IF ERROR THEN WRITELNC INVALID DATA: P=',P:3:0) ELSE WHILE E<=EMAX DO BEGIN E2:=E*E;CORR:=1000;T0:=15E3; WHILE CORR>ERR DO BEGIN INTEGRATE; IF T>T1 THEN T0:=T0"CORR ELSE BEGIN T0:=T0+CORR; C0RR:=C0RR/10; END; END; COND:=0; FOR L:=1 TO 100 DO COND:=COND+SIGMA(TP(L))*L; COND:=COND*DR*DR* 6.283;CURR:=E*COND; QWALL:=3.11*KAPPA(TP(99))*(TP(98)-T); Q:=E*CURR/100;QRT:=Q-QWALL; M:=QRT/QWALL; L:=1; WHILE TP(L)>9000 DO L:=L+1; R9:=L*0.125; WRITELN('4 E=',E:5:2,'V/CM I=',CURR:4:0,'A, P:2:0,' ATM, Q=',Q:4:1,'KW',EOL,' QRAD=',Q 'KW, QWALL=',QWALL:5:2,'KW, M=',M:4:2,',R(9000)=' FOR I:=1 TO 100 DO IF I MOD 10=1 THEN WRITE(EOL,TP(I):13) ELSE WRITE(TP(I):7); WRITELN; PLOT(TP,CURR,P); E:=E+STEP; END; WRITELN(';'); END. 107 APPENDIX B TRANSIENT HEATING CALCULATION PROGRAMS PROGRAM TRANSIT; (* 7/IV/81 *) VAR K0,TK,K1,A,T,DELTA,DR,CON1:REAL; P,TMSEC,TIME,DT,TMAX,QOUT,QIN:REAL; T0,T1,REPS,NC,L,N,M:INTEGER; TP:ARRAY(1..100) OF REAL; QCORE,QWALL,R1:REAL; INITIAL T0=11000;DR=1.35E-4; (* INITIAL TEMP TO. RADIAL INCREMENT DR(M) *) K0=1.08E-2;TK=2400;K1=1.1E-2;A=2.7E-5; (* VALUES FOR FUNCTION KAPPA *) TIME=0;NC=51;P=1.44; (* NC IS THE CORE RADIUS. P IS PRESSURE(ATM) *) TP=( 11000:55,300:45 );QCORE=0;QWALL=0; (* INITIALIZE TEMPERATURE PROFILE *) PROCEDURE TITLE; SLINKAGE 'TITLE'; FUNCTION KAPPA(T:REAL):REAL; (* THERMAL CONDUCTIVITY *) BEGIN IF T>7000 THEN KAPPA:=K0*EXP(T/TK) ELSE KAPPA:=K1+A*T; END; FUNCTION KDT(T1,T2:REAL):REAL; VAR KDT1:REAL; BEGIN KDT1:=KAPPA((T1+T2)/2)*(T1-T2)/DR; IF KDT1>P*1E6 THEN KDT:=P*1E6 (* DIFFUSION LIMIT *) ELSE KDT:=KDT1; END ; FUNCTION CPR(T:REAL):REAL; (* SPECIFIC HEAT PER UNIT VOLUME * VAR VAL:ARRAY (4..12) OF REAL; INDEX:INTEGER; FR:REAL; INITIAL VAL=(63.3,50.5,42.5,38.3,39.5,49.5,71.5,104.4, 1 38. BEGIN INDEX:=INT(T/1000) ; FR:=T/1000-INDEX; IF INDEX<4 THEN CPR:=P*2.78E5/T ELSE IF INDEX>11 THEN CPR:=P*VAL(12) ELSE CPR:=P*(VAL(INDEX)*(1-FR)+VAL(INDEX+1)*FR); END; BEGIN (*TRANSIT*) 108 READ(INPUT,DT,REPS,TMAX);CON1:=DT/REPS/DR; (* PRINT INTERVAL DT HAS REPS ITERATIONS. T<TMAX *) TITLE(6,'***TRANSIT***',13); WRITELNC DT=',DT:6:4,' REPS=',REPS:3,EOL); R1:=NC*DR; WHILE TIME<=TMAX DO BEGIN WRITECO T=' ,TIME:7:4, ' SEC R1 = ' , R1 : 4 : 1 , ' MM'); WRITELNC QC=',QCORE:5:2,' QW=',QWALL:5:2); FOR L:=NC TO 100 DO IF L MOD 10=0 THEN WRITELN(TP(L):8:1) ELSE WRITE(TP(L):8:1); TIME:=TIME+DT; IF TIME<=TMAX THEN FOR L:=1 TO REPS DO BEGIN T:=TP(NC); QIN:=RDT(T0,T)*(NC-0.5); QC0RE:=QIN/1.18E6; FOR N:=NC TO 99 DO BEGIN T:=TP(N); QOUT:=KDT(T,TP(N+1))*(N+0.5); TP(N):=T+CON1/N/CPR(T)*(QIN-QOUT); QIN:=QOUT; IF T<300.1 THEN BEGIN M:=N; QIN:=0; N:=99; END; END ; WHILE TP(M)<1000 DO M:=M-1; T:=TP(M); R1:=(T-TP(M+1)); R1:=(T-1000)/R1; R1:=(M+R1)*DR*1000; QWALL:=QIN/1.18E6; END; END; WRITELNC;'); END. c  C PROGRAM TRPL c  DIMENSION TEMP(50), IY(100) 109 DATA IY/50*11000/ 11 FORMAT(14X,F 6.4) 12 FORMAT(7X,10F8.1 ) 13 FORMAT(' ',6X,F4.1,1016) 14 FORMATC ',6X,'T(MSEC) TEMP') CALL TITLE(6,'***TRPL***',10) WRITE(6,14) 100 READ(0,11,END= 990) TIME TMSEC=TIME*1000. READ(0,12) TEMP DO 150 1=1,50 150 IY(50+I)=INT(TEMP(I)) WRITE(6,13) TMSEC,(IY(10*1),I=1,10) CALL RP100(IY,TMSEC,1.0) GOTO 100 990 STOP END Note: s u b r o u t i n e "RP100" draws a p l o t of the r a d i a l t e mperature p r o f i l e . 110 APPENDIX C DATA HANDLING PROGRAMS C C PROGRAM SCAN (10/IX/80) c  IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR LOGICAL*1 COPY(24)/'$C -TSCAN TO -COPY(*L+1)'/,CMND(10) CALL FTNCMD('DEFAULT 7=-TSCAN',16) CALL FTNCMD('DEFAULT 1=*PRINT*',17) CALL FTNCMD('DEFAULT 0=DATA',14) CALL FTNCMD('DEFAULT 2=INDEX(*L)',19) 1 F0RMAT(I4,6X,I3) 2 FORMAT(' SCAN BEGINS. DATA CONTAINS RUNS ',14,' TO ',14 3 FORMAT('VALID SCAN COMMANDS ARE:', -/,'GET,LIST,STOP,$CMND,COPY,TITLE,FIX,NEWFILE,DISPLAY') 4 FORMAT('SCAN OFF.'/l4,' PLOTS IN -PLOT#') 5 FORMAT('RUN',15,' IS LOADED *) C INITIALIZE CALL SETPFX(']',1) 100 LNRUN=1000 INDEX=0 NPLOTS=0 LAST=0 NREC=0 110 READ(0'LNRUN,1,END=150) NRUN,LINC IF (LINC.EQ.0) GOTO 150 LAST=LAST+1 NDIR(LAST,1)=NRUN NDIR(LAST,2)=LNRUN NDIR(LAST,3)=LINC LNRUN=LNRUN+1000*(LINC+1) GOTO 110 150 IF (LAST.EQ.0) GOTO 600 PRINT 2,NDIR(1,1),NDIR(LAST,1) c GET COMMAND 200 CALL ASK(CMND,LEN) JMP=ITBLC(CMND(1),'GLS$CTFND#') GOTO (250,300,350,400,450,500,550,600,650),JMP IF (NUMGET(CMND).NE.0) CALL IGET(NUMGET(CMND),&200) IF (LEN.GT.0) PRINT 3 IF (INDEX.GT.0) PRINT 5,NDIR(INDEX,1) GOTO 200 C COMMANDS C GET 250 CALL GET(NUMGET(CMND),&200) 111 c L I S T 300 CALL LIST(NUMGET(CMND),&200) C STOP 350 PRINT 4,NPLOTS STOP C $CMND 400 CALL CMD(CMND,10) GOTO 200 C COPY 450 CALL CMD(COPY,24) GOTO 200 C TITLES 500 CALL TITLESU200) c F I X 550 CALL F I X ( 7 ) GOTO 200 C NEWFILE 600 CALL NEWFUl00,£200) c DISPLAY 650 CALL DlSPLY(NUMGET(CMND),&200) STOP END C SUBROUTINE DISPLY(N,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,HBUF(500 LOGICAL*1 CMND(10) 1 FORMAT('RUN ',14,' RECORD ',14,' IS LOADED.') 2 FORMAT('VALID DISPLAY COMMANDS ARE:'/ + 'DRAW,/MOD,STOP,PLOT,LI ST,RESCALE,ERASE,VIEW,GET,XHAIRS 3 FORMAT(* 1' ,12X,'LISTING OF RUN ',14,' RECORD ',14//) 4 FORMAT(* ',6X,1016) C INITIALIZE CALL SETPFX(')',1) IF (INDEX.EQ.0) CALL GET(NDIR(1,1)) IF (N.NE.0) CALL DRAW(N) C GET COMMAND 100 CALL ASK(CMND,LEN) NG=NUMGET(CMND) JMP=ITBLC(CMND(1),'D/SPLREVGX') GOTO (200,250,300,350,400,450,500,550,600,650),JMP IF (NG.NE.O) CALL DRAW(NREC+NG,& 100) IF (LEN.GT.0) PRINT 2 150 PRINT 1,NDIR(INDEX,1),NREC GOTO 100 c COMMANDS C DRAW 200 CALL DRAW(NG,&100) 1 1 2 C /MOD 250 CALL MOD(CMND,&100) c S T 0 P 300 CALL SETPFX(' ] ' , 1 ) RETURN1 C PLOT 350 IF (NG.NE.O) CALL ERASE(NG) CALL IGDRON('CALC') NPLOTS=NPLOTS+1 GOTO 100 c L I S T 400 IF (NG.NE.O) CALL DRAW(NG) IF (NREC.EQ.0) GOTO 150 WRITE(1,3) NDIR(INDEX,1),NREC WRITE(1,4) (HBUF(I),I=1,LASTR) GOTO 100 c RESCALE 450 CALL RSCL(CMND,&100) C ERASE 500 CALL ERASE(NG,& 100) c _ V I E W 550 IF (NG.NE.O) CALL GET(NG) CALL VIEWU100) c GET 600 IF (NG.NE.O) CALL GET(NG,& 100) CALL IGET(1, &100) c XHAIRS 650 CALL XHAIRS GOTO 100 END C SUBROUTINE GET(N,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,CBUF(100 1 FORMAT('THERE IS NO RUN ',15) 2 FORMAT('GOT RUN ',14,' CONTAINING ',14,' RECORDS') 1 = 1 50 IF (NDIR(I,1).EQ.N) GOTO 100 1 = 1 + 1 IF (I.LE.LAST) GOTO 50 C '• • NO MATCH PRINT 1,N RETURN1 C ENTRY IGET(N,*) C I=INDEX+N IF ((I.LT.1).OR.(I.GT.LAST)) 1=1 N=NDIR(I,1) c MATCH 100 INDEX=I 1 1 3 CALL EMPTYF(7,&990) NL00PS=NDIR(I,3)+1 CALL READ(CBUF,HL,2,NDIR{I,2),0,&990) DO 150 J=1,NLOOPS CALL WRITE(CBUF,HL,0,LNUM,7,&990) CALL READ(CBUF,HL,0,LNUM,0,& 150) 150 CONTINUE PRINT 2,N,NDIR(I,3) RETURN1 990 CALL ERROR('GET ') RETURN 1 END C SUBROUTINE LIST(NDEV,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,CBUF(32) 1 FORMAT(' 1 ' ,16X,'DATA LI ST:'//8X'RUN' ,5X,'#RECS' ,6X,'TIT 2 FORMAT(' ',6X,30A1) 3 FORMAT(';') IF (NDEV.EQ.0) NDEV=6 IF (NDEV.EQ.2) GOTO 20 CALL TITLE(NDEV,'***SCAN***',10) WRITE(NDEV,1) 20 DO 50 1=1,LAST CALL READ(CBUF,HL,2,NDIR(I,2),0,&990) 50 WRITE(NDEV,2)(CBUF(J),J=1,30) WRITE(NDEV,3) RETURN1 990 CALL ERRORCLIST ') RETURN 1 END C SUBROUTINE TITLES(*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON" I NDEX, LAST, NDI R ( 2 0 , 3 ) , NPLOTS , NREC , LASTR, CBUF (32) LOGICAL*1 CTITLE(10) EQUIVALENCE (CTITLE(1),CBUF(20)) 1 FORMAT('&',30A1,' ->') DO 50 1=1,LAST CALL READ(CBUF,HL,2,NDIR(I,2),0,&990) PRINT 1,(CBUF(J),J=1,30) CALL ASK(CTITLE,LEN) IF (LEN.NE.0) CALL WRITE(CBUF,HL,2,NDIR(I,2),0,&990) 50 CONTINUE RETURN 1 990 CALL ERROR('TITLES ') RETURN 1 1 1 4 END C SUBROUTINE NEWF(*,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) LOGICAL*1 CMND(19)/'ASSIGN 0=DATA '/,CFILE(10) EQUIVALENCE (CFILE(1),CMND(10)) 1 FORMAT('ENTER NEW DATAFILE NAME') PRINT 1 CALL ASK(CFILE,LEN) IF (LEN.EQ.O) RETURN2 CALL FTNCMD(CMND,19) RETURN1 END c  SUBROUTINE ERROR(CMSG) c  IMPLICIT LOGICAL*1(C) rINTEGER*2(H) DIMENSION CMSG(10) 1 FORMAT ('I/O ERROR OCURRED IN ROUTINE ',1fJAl) PRINT 1,(CMSG(I),1=1,10) RETURN END C SUBROUTINE DRAW(N,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,HBUF(500 1 FORMAT('THERE IS NO RECORD ',14) IF (N.LT.1) GOTO 990 LNUM=(N+1)*1000 CALL READ(HBUF,HL,2,LNUM,7,&990) NREC=N CALL VECTOR(500) CALL LABEL(NDIR(INDEX,1),'.',N) RETURN1 C BAD NUMBER 990 PRINT 1,N RETURN 1 END c  SUBROUTINE VECTOR(N) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,HBUF(500 LOGICAL*1 CFIRST/.FALSE./ REAL X(500),Y(500) 1 1 5 IF (CFIRST) GOTO 100 DO 50 1=1,500 50 X(I) = F L O A T ( l ) CALL START CFIRST=.TRUE. 100 DO 150 1=1,N 150 Y(I)=HBUF(I) CALL IGCTNS('SPEC) CALL IGVEC(N,X,Y) CALL IGENDS('SPEC') LASTR=N RETURN END c  SUBROUTINE LABEL(N1,CHAR,N2) c  IMPLICIT LOGICAL*1(C),INTEGER*2(H) CALL IGCTNS('CNUM') CALL IGFMTH(N1,'I') CALL IGFMTH(CHAR,'A') CALL IGFMTH(N2,'I') CALL IGTXTH(',<E>') CALL IGENDS('CNUM') CALL IGDRON('TERM') RETURN END c  SUBROUTINE MOD(CMND,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,HBUF(500 LOGICAL*1 CMND(10) DATA RATIO/.33/ 1 FORMAT('PEAK AT ',13,' WIDTH ',15,' HEIGHT ',15) IF (NREC.EQ.0) RETURN1 C GET COMMAND ITYPE=NUMGET(CMND) 100 JMP=ITBLC(CMND(2),'SRAPET####') GOTO (200,250,300,350,400,500),JMP CALL VECTOR(LASTR) CALL LABEL(NDIR(INDEX,1),'/',NREC) RETURN1 C /SMOOTH 200 CALL SMOOTH(HBUF,LASTR,ITYPE) CALL VECTOR(LASTR) CALL LABEL(NREC,'S',ITYPE) RETURN1 C /ROTATE 250 CALL ROTATE(HBUF,500,ITYPE) CALL VECT0R(NDIR(INDEX,3)) 1 16 NREC=ITYPE CALL TICS CALL LABEL(NDIR(INDEX,1),'R',ITYPE) RETURN 1 c /ABEL 300 CALL PEAK(HBUF,500,LASTR,NPK,NHT,NWD) CALL ABEL(HBUF,500,NPK,ITYPE) CALL VECTOR(NPK) CALL LABEL(NREC,'A',ITYPE) RETURN1 C /PEAK 350 I F (ITYPE.EQ.O) ITYPE=LASTR CALL PEAK(HBUF,500,1TYPE,NPK,NHT,NWD) PRINT 1,NPK,NWD,NHT RETURN1 c /ERASE 400 CALL ERASE(0) CMND(2)=CMND(3) GOTO 100 C /TEMP 500 CALL PEAK(HBUF,500,LASTR,NPK,IMAX,NWD) RMAX=FLOAT(IMAX) IF (ITYPE.NE.0) RATI0=100./ITYPE DO 550 1=1,LASTR 550 HBUF(I)=INT(11600./(I."RATIO*ZLOG(HBUF(I)/RMAX))) CALL VECTOR(LASTR) CALL TICS CALL LABEL(NDIR(INDEX,1),'T',NREC) RETURN1 END c  SUBROUTINE TICS C IMPLICIT LOGICAL*1(C),INTEGER*2(H) CALL IGCTNS('SPEC') CALL IGMA(0.,50.) DO 50 1=1,9 CALL IGMR(9.09,-1000.) CALL IGDR(0.,1000.) 50 CONTINUE CALL IGENDS('SPEC') RETURN END c  FUNCTION ZLOG(X) C ZLOG=1E5 IF (X.LE.0) RETURN ZLOG=ALOG(X) RETURN END C SUBROUTINE ERASE(N,*) C 1 17 IMPLICIT L0GICAL*1(C),INTEGER*2(H) CALL IGBGNS('SPEC') CALL IGENDS('SPEC) CALL IGBGNS('CNUM') CALL IGENDS('CNUM') IF (N.NE.O) CALL DRAW(N) RETURN 1 END c  SUBROUTINE RSCL(CMND,*) c  IMPLICIT LOGICAL*!(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR LOGICAL*1 CMND(10) N1=NUMGET(CMND) N2=NUMG2(CMND) IF (N2.NE.0) GOTO 50 IF (N1.NE.0) CALL SCL(NI ,& 100) N1 = 1 N2=LASTR 50 XR=N1+(N2-N1)/500. XL=2.*N1-XR CALL IGTRAN('XAXE','WIND',XL,XR,-1.,1.) 100 CALL IGDRON('TERM') RETURN 1 END C SUBROUTINE SCL(N,*) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) IF (N.GT.0) CALL IGTRAN('SPEC','WIND',-1.,1.,-10./N,10. IF (N.LT.0) CALL IGTRAN('SPEC ,'WIND' ,- 1 .,1.,-100., 100. RETURN 1 END C SUBROUTINE START C IMPLICIT LOGICAL*1(C),INTEGER*2(H) REAL X0(14)/0.,500.,500.,500.,400.,400.,300.,300.,200., +100.,100.,0.,0./ REAL Y0(14)/0.,0.,-10.,10.,-10.,10.,-10.,10.,-10.,10.,-+10.,-10.,10./ CALL SETUP C DRAW BACKGROUND CALL IGTRAN('MAIN','WIND',0.,500.,-50.,500.) CALL IGVEC(14,'(MD)',Y0,X0) CALL IGBGNS('XAXE') CALL IGVEC(14,'(MD)',X0,Y0) DO 70 1=1,4 CALL IGMA(100.*I,10.) CALL IGFMTH(100*1,'I *) 70 CONTINUE 118 CALL CALL CALL CALL CALL CALL IGBGNS( IGENDS( SPEC' ) SPEC' ) •INITIALIZE PICTURES IGENDS('XAXE') IGBGNS('CNUM') IGENDS('CNUM') IGTRAN('CNUM', MOVE',10. CALL SCL(- 1) RETURN END SUBROUTINE VIEW(*) , 4 9 0 . ) DEFAULT SCALE IMPLICIT LOGICAL*1(C),INTEGER*2(H) COMMON INDEX,LAST,NDIR(20,3),NPLOTS,NREC,LASTR,HBUF(500 DIMENSION CMND(10) CALL DRAW(1) CALL ERASE(0) CALL RSCL('R 1 50 ') NREC=1 DO 50 I=INDEX,LAST CALL GET(NDIR(I,1)) CALL MOD('/R 265 ') 5 CALL ASK(CMND,LEN) JMP=ITBLC(CMND(1),' RE/S#####') GOTO(50,10,20,30,60),JMP RETURN1 10 CALL RSCL(CMND,&5) 20 CALL ERASE(0,5.50) 30 CALL MOD(CMND,&5) 50 CONTINUE 60 RETURN 1 END c  SUBROUTINE ROTATE(HB1,LEN,N) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) DIMENSION HBUF(500),HB1(LEN) COMMON INDEX,LAST,NDIR(20,3) NREPS=NDIR(INDEX,3) REWIND 7 CALL READ(HBUF,HL,0,LNUM,7,&990) DO 50 1=1,NREPS CALL READ(HBUF,HL,0,LNUM,7,&990) 50 HB1(I)=HBUF(N) 990 RETURN END C SUBROUTINE FIX(NDEV) C IMPLICIT LOGICAL*1(C),INTEGER*2(H) 119 DIMENSION HBUF(500) REWIND NDEV 100 CALL READ(HBUF,HL,0,LNUM,NDEV,&990) IF (HL.NE.1000) GOTO 200 DO 150 1=1,500 N=HBUF(I) IF (N.LT.O) N=65536+N 150 HBUF(l)=N/2 200 CALL WRITE(HBUF,HL,2,LNUM,NDEV,&990) GOTO 100 990 RETURN END c  SUBROUTINE XHAIRS C LOGICAL*1 CHECK 1 FORMAT('CHANNEL #',F4.0,' HEIGHT IS ',F6.0/'&AGAIN?') CALL IGCTNS('SPEC') 50 CALL IGXYIN(X,Y,&100) PRINT 1,X,Y IF (CHECK('Y')) GOTO 50 100 CALL IGENDS('SPEC) RETURN END 1 20 c j. C PROGRAM ASPEC (23/V/81) c  IMPLICIT LOGICAL*1(C),INTEGER*2(H) CALL FTNCMD('DEFAULT 0=DATA',14) CALL FTNCMD('DEFAULT 1=*PRINT*;') DIMENSION CBUF(30),HBUF(500),HMAT(100,40),IT(3),EN(2) DATA T430/16.8/,TNE/l.04E5/,CHAR/'A'/ 1 FORMAT(' ' ,1 OX,3 0A1//1 OX,'EN(WIDTH) EN(CONT) T(WIDTH) +'T(CONT) T(430)'/) 2 FORMAT(' ',8X,2E10.3,3I10) 3 FORMAT(';') C NEXT RUN 100 CALL RUNIS(CBUF,NLINES,CHAR,&990) CALL TITLE(1,'***ASPEC***',11) WRITE(1,1) CBUF C CHAMBER PRESSURE P=1+NUMGET(CBUF(25))/49. ALOGP=ALOG(P) NCTR=250 C FIND THE LINE CALL READ(HBUF,HL,0,LNUM,0,&990) DO 150 1=220,280 150 IF (HBUF(I).GT.HBUF(NCTR)) NCTR=I NLOW=NCTR-2l DO 200 1=1,40 t 200 HMAT(1,I)=HBUF(NLOW+I) C READ AND STORE DATA DO 250 J=2,NLINES CALL READ(HBUF,HL,0,LNUM,0,&990) DO 250 1=1,40 250 HMAT(J,I)=HBUF(NLOW+I) C ABEL UNFOLD DO 300 1=1,40 300 CALL FSA(HMAT(1,1),NLINES,MAX) MAX=MAX-1 DO 450 J=1,MAX C BACKGROUND BACK=HMAT(J,1) DO 310 1=2,5 310 BACK=BACK+HMAT(J,I) DO 320 1=36,40 320 BACK=BACK+HMAT(J,I) BACK=BACK/10. C LINE AREA AREA=HMAT(J,6) MAXHT=HMAT(J,2 0) DO 350 1=7,35 IF (HMAT(J,I).GT.MAXHT) MAXHT=HMAT(J,I) 350 AREA=AREA+HMAT(J,I) AREA=AREA-BACK*30. 121 C LINE WIDTH WIDTH=1E-2 NHALF=INT((MAXHT+BACK ) /2 . ) IF (NHALF.GE.MAXHT) GOTO 400 1 = 10 380 1=1+1 IF (HMAT(J,I).LT.NHALF) GOTO 380 FIRST=I-(HMAT(J,I)-NHALF)*1./(HMAT(J,I)-HMAT(J,I-1)) 390 1=1+1 IF (HMAT(J,I).GT.NHALF) GOTO 390 WIDTH=I-FIRST -(NHALF-HMAT(J,I))*1./(HMAT(J,I)-HMAT(J,I-C INSTRUMENT WIDTH=2.5 WIDTH=WIDTH-2.5 C ZERO CHECKS; 400 IF (BACK.LE.0) BACK=1E~2 IF (AREA.LE.0) AREA=1E~2 IF (WIDTH.LE.0.) WIDTH=1E~2 C DO CALCULATIONS DEN3=24.81+ALOGP-ALOG(AREA) T=T430/DEN3 DEN3=DEN3-ALOG(T) T=T430/DEN3 IT(3)=INT(T*1E4) DEN1=61.45+ALOGP-ALOG(T) EN(1)=1.52E22*WIDTH/T**.1667 IT(1)=INT(TNE/(DEN1 -ALOG(EN(1)))) EN(2)=1.40E21*SQRT(BACK)*T**.25 IT(2)=INT(TNE/(DEN1-ALOG(EN(2)))) 450 WRITE(1,2) EN,IT GOTO 100 C EOF OR ERROR 990 WRITE(1,3) STOP END 122 APPENDIX D GAS FLOW IN THE SYSTEM In order to ensure that the gas flow i s completely understood, the v a r i o u s pressure drops through the system and the t o t a l mass flow have been c a l c u l a t e d f o r a room temperature gas flow with the a r c c u r r e n t o f f , and compared with measured v a l u e s . Two pre s s u r e l o s s mechanisms have been c o n s i d e r e d ; a d i a b a t i c a c c e l e r a t i o n s o c c u r r i n g when the gas i s f o r c e d q u i c k l y i n t o or out of a s m a l l a p e r t u r e , and v i s c o u s l o s s e s o c c u r r i n g when the gas flows along a s t a t i o n a r y w a l l . We assume that argon i s a v i s c o u s i d e a l gas with r=5/3, atomic mass m=39.95amu (6.63«10~ 2 6 kg) and v i s c o s i t y *« = 3.7 • 1 0~ 5 Nsnr 2 which i s c o n s t a n t . For an a d i a b a t i c a c c e l e r a t i o n i n a s t r a i g h t l i n e the c o n s e r v a t i o n equations are pvA = m (D-1) (D-2) (D-3) where the c r o s s - s e c t i o n a l area A i s a f u n c t i o n of p o s i t i o n ( x ) , the mass flow r a t e m i s constant and p,v and P are the gas d e n s i t y , v e l o c i t y and p r e s s u r e , and do not change with 123 t i m e . I f we s t a r t i n a p i p e of l a r g e a r e a a t p r e s s u r e P 0 and d e n s i t y p0 and move thr o u g h a n o z z l e of much s m a l l e r a r e a A e q u a t i o n s D-1 t o 3 can be s o l v e d t o g i v e ( f ) 1 ' 2 ^ ) 1 ' 6 " - ^ ( » - « > o o 5A 2P 2 o The f u n c t i o n f(x)=x 3-x° has a maximum v a l u e of 0.1055 a t x=0.75, which means t h a t t h e maximum mass f l o w p o s s i b l e o c c u r s a t (£. ) 0 , 4 = 0.75 or P=0.49P o, when the f l o w becomes s o n i c , and d e c r e a s i n g the o u t l e t p r e s s u r e w i l l not i n c r e a s e t h e f l o w r a t e . V i s c o u s d r a g i n the s u p p l y l i n e s and the heat exchanger t u b i n g causes a slow p r e s s u r e change which w i l l be c o n s i d e r e d i s o t h e r m a l . The v i s c o u s d r a g on gas f l o w i n g t h r o u g h a p i p e of r a d i u s r and l e n g t h 1 a t average v e l o c i t y v i s F=8rrnvl and must be overcome by a p r e s s u r e g r a d i e n t d P / d x = F / r r r 2 l , t h u s dP/dx=-8,»v/r 2 . Now v=m/trr2/> and />=P(m/kT) so s u b s t i t u t i n g we get pdP_ = -8pm kT dx u m or d rr>2-> -16ymkT ^ ( } = ^ T " ( D _ 5 ) TTT m and f o r a p i p e of r a d i u s r and l e n g t h 1 A(P 2) = ^ 1 lr" 4 m = Rm (D~6) The v i s c o u s d r a g f a c t o r s R f o r the s u p p l y t u b i n g , t h e heat exchanger and the a r c v e s s e l a r e g i v e n i n F i g u r e I I I - 2 , 124 which shows the gas system s c h e m a t i c a l l y . The p r e s s u r e drop through the heat exchanger i s s i g n i f i c a n t , t h r o u g h the s u p p l y t u b i n g s m a l l , and t h r o u g h the a r c chamber c o m p l e t e l y n e g l i g i b l e . The p r e s s u r e drops t h r o u g h the system can now be c a l c u l a t e d . The s u p p l y p r e s s u r e P 0 i s reduced by v i s c o u s drag t o P, a t the n o z z l e , and the a d i a b a t i c a c c e l e r a t i o n i n t o the n o z z l e reduces the p r e s s u r e t o about 0.49'P, and a c c e l e r a t e s the gas t o about 200 ms~ 1. When the j e t e n t e r s the chamber i t expands ( i f the chamber p r e s s u r e i s below the n o z z l e p r e s s u r e ) and i s v e r y q u i c k l y slowed down by v i s c o u s drag on the w a l l , as we see i n c h a p t e r I I I s e c t i o n C. The chamber p r e s s u r e i s thus l e s s than or e q u a l t o the n o z z l e p r e s s u r e and r i s e s u n t i l the f l o w r a t e out t h r o u g h the exhaust heat exchanger e q u a l s the f l o w r a t e i n t h r o u g h the i n l e t n o z z l e . The l o s s e s i n the heat exchanger a r e f i r s t an a d i a b a t i c a c c e l e r a t i o n i n t o the t u b i n g , then a v i s c o u s l o s s i n the t u b i n g and f i n a l l y a d e c e l e r a t i o n a t the e x i t . At h i g h p r e s s u r e s the f l o w becomes s o n i c a t the e x i t , which s e t s the p r e s s u r e t h e r e , and a t lower p r e s s u r e s the p r e s s u r e a t the e x i t i s c a l c u l a t e d by assuming t h a t the f l o w a f t e r the e x i t d e c e l e r a t e s a d i a b a t i c a l l y t o r e s t , and a p p l y i n g E q u a t i o n D-4. Given a measured s u p p l y p r e s s u r e we can c a l c u l a t e the mass f l o w r a t e through the system and the p r e s s u r e s a t d i f f e r e n t p o i n t s , and compare t h e s e w i t h measured v a l u e s . We b e g i n by assuming t h a t the f l o w t h r o u g h the i n l e t n o z z l e i s 125 s o n i c and c a l c u l a t i n g the mass f l o w t h r o u g h i t . T h i s g i v e s us a mass f l o w t o use i n c a l c u l a t i n g the r e m a i n i n g p r e s s u r e drops t h r o u g h the system, and we get the chamber p r e s s u r e and the s u p p l y l i n e l o s s e s f o r t h i s mass f l o w r a t e , then c a l c u l a t e t h e mass f l o w r a t e t h r o u g h the n o z z l e t a k i n g t h e s e i n t o a c c o u n t . The chamber p r e s s u r e has no e f f e c t except a t v e r y low s u p p l y p r e s s u r e s , so a t normal p r e s s u r e s o n l y the s m a l l c o r r e c t i o n f o r s u p p l y l i n e l o s s e s a p p l i e s , and the c a l c u l a t i o n converges v e r y q u i c k l y . At low p r e s s u r e s the i n l e t n o z z l e d e p a r t s s l i g h t l y from s o n i c , and 3 or 4 i t e r a t i o n s a r e r e q u i r e d b e f o r e the s o l u t i o n c o n v e r g e s . T a b l e D-1 l i s t s t he p r e d i c t e d and measured p r e s s u r e s and f l o w r a t e s a t d i f f e r e n t s u p p l y p r e s s u r e s . The f l o w r a t e s i n t h i s t a b l e were measured w i t h a v e n t u r i and manometer system, which a g r e e s w i t h but i s much more a c c u r a t e than the flowmeter i n the l i n e , and the p r e s s u r e s were measured w i t h a m e c h a n i c a l gauge. The n o z z l e and t u b i n g dimensions a r e i n d i c a t e d i n F i g u r e I I I - 2 and a r e o n l y a c c u r a t e t o about 5%, as a r e the p r e s s u r e and f l o w r a t e measurements. The dim e n s i o n s of the heat exchanger t u b i n g a r e v e r y approximate and so the v i s c o u s drag c o e f f i c i e n t was d e t e r m i n e d by b e s t f i t t o the measured chamber p r e s s u r e , and i t d i f f e r s by about 40% from the nominal v a l u e , which c o u l d be due t o a 10% e r r o r i n the e s t i m a t e d average r a d i u s . Agreement i s found i n the t a b l e t o w i t h i n 5%, as i t s h o u l d be. T a b l e D-1. Gas f l o w r a t e s and p r e s s u r e s P 0 ( k P a ) measured ±5 P,(kPa) c a l c u l a t e d m(gs" 1) c a l c u l a t e d measured ±0.03 P 2 ( k P a ) c a l c u l a t e d measured ±5 P 3 ( k P a ) c a l c u l a t e d 200 300 197 297 0.55 0.92 0.60 0.93 138 153 133 154 130 133 500 700 497 697 1.54 2.15 1.52 2.14 212 288 222 288 177 218 P„(kPa) 90 54 81 114 c a l c u l a t e d 127 APPENDIX E THE VISCOUS DECAY OF A VORTEX In s e c t i o n I I I - C i t i s shown that the v i s c o u s decay of vortex motion i n s i d e a tube of r a d i u s a leads to the p a r t i a l d i f f e r e n t i a l equation 22- = -U- M r 3 ^ ] (E-1 ) 3t 3 9 r i r 3r J with boundary c o n d i t i o n s u(a)=0, = 0. I f we assume 3r'o u ( r , t ) = R ( r ) f ( t ) with f ( t ) = e ' a t we get 3OJ y 3 o 3OK i t = " a W " — i r " C r 3 37^ pr 3 and we can d i v i d e out f ( t ) and expand the r a d i a l d e r i v a t i v e to get R + ^ i i d R + - J i d f R = 0 (E-2) a p r d r ap ^ 2 T h i s equation i s s a t i s f i e d by the f u n c t i o n R(r) = £ J^ iTS^) (E-3) The B e s s e l f u n c t i o n J , has zeroes at {* n}. so to s a t i s f y the boundary c o n d i t i o n u(a)=0, R(r) must a l s o be of the form R(r) = £ JjCx^/a) (E-4) Equations E-3 and E-4 together r e s t r i c t the allowed v a l u e s of o to a n = yx 2/pa 2 = x 2/x , 128 where T=pa 2 / i » . Now we have a set of s o l u t i o n s of the form o)(r,t) = i j 1 C x n r / a ) - e ' x n t / T and any l i n e a r combination of these, c o r r e s p o n d i n g t o some i n i t i a l d i s t r i b u t i o n of angular v e l o c i t y , i s a v a l i d s o l u t i o n t o Equation E-1. The f u n c t i o n J , ( x ) / x i s p l o t t e d i n F i g u r e E-1 over the range 0<x<20, which covers the f i r s t s i x zer o e s . These zeroes a r e : 3.83, 7.02, 10.17, 13.32, 16.47 and 19.62. F i g u r e E-1. The f u n c t i o n J , ( x ) / x vs x To study the e v o l u t i o n of an i n i t i a l v e l o c i t y d i s t r i b u t i o n we need t o express i t i n the form 1 2 9 130 U(r,0) = ^ i j ( x r / a ) n One l i k e l y d i s t r i b u t i o n i s a s o l i d body r o t a t i o n o(r)=o 0 which f a l l s o f f to zero very c l o s e to the w a l l . Another p o s s i b i l i t y i s an annular j e t type d i s t r i b u t i o n with u=u 0 near the w a l l and «=0 i n the center. I n i t i a l c o n d i t i o n s of both these types have been composed from the f i r s t s i x harmonics with the a i d of a computer. The i n i t i a l d i s t r i b u t i o n s are shown i n Figures E-2 and 3, along with the d i s t r i b u t i o n s a f t e r 0.125, 0.25 and 0.50 seconds have elapsed (corresponding to roughly 50, 100 and 200 mm of a x i a l t r a v e l ) . The values of the harmonic c o e f f i c i e n t s {A n} are i n d i c a t e d on each f i g u r e . I t can be seen that a f t e r 0.5 second has elapsed these d i s t r i b u t i o n s are very s i m i l a r , both having decayed nearly to the fundamental mode. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085771/manifest

Comment

Related Items