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UBC Theses and Dissertations

Electrical power extraction from a supersonic plasma flow Kwan, Wai-Ming Joe 1982

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ELECTRICAL POWER EXTRACTION FROM A SUPERSONIC PLASMA FLOW by WAI-MING JOE KWAN B . S c , C a l i f o r n i a S t a t e U n i v e r s i t y a t Long Beach, 1974 M.A., S t a t e U n i v e r s i t y of New York a t Stony Brook, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES ( P h y s i c s Department) 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 A p r i l 1982 c) Wai-ming Joe Kwan, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Wai-ming Joe Kwan Department of Physics The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Ap r i l 6, 1982 11 ABSTRACT The i n d u c e d p o t e n t i a l d i f f e r e n c e a c r o s s a s t a n d i n g shock f r o n t can be used t o d r i v e e l e c t r i c c u r r e n t t h r o u g h e x t e r n a l l o a d s . Such a " s t a n d i n g shock g e n e r a t o r " p r e s e n t s an a l t e r n a t i v e scheme t o the MHD g e n e r a t o r f o r d i r e c t c o n v e r s i o n of energy from h i g h temperature s u p e r s o n i c f l o w s . We make use of t h e s u p e r s o n i c f l o w b e h i n d a f r e e r u n n i n g shock i n a shock tube and l e t the f l o w impinge onto a c o n i c a l o b s t a c l e mounted i n the m i d d l e of the tube t o c r e a t e a s t a n d i n g shock. T h i s cone has a c o n d u c t i n g s u r f a c e which a c t s as the anode. The cathode i s mounted f l u s h w i t h the shock tube w a l l at a p o s i t i o n upstream of the s t a n d i n g shock. By v a r y i n g the l o a d r e s i s t a n c e , we have o b t a i n e d c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s f o r the s t a n d i n g shock g e n e r a t o r . These c h a r a c t e r i s t i c s a r e f a m i l i e s of p a r a l l e l s t r a i g h t l i n e s s i m i l a r t o the c h a r a c t e r i s t i c s of a b a t t e r y . The o p e n - c i r c u i t v o l t a g e i s found t o depend on the s t a n d i n g shock p o t e n t i a l and the work f u n c t i o n s of the e l e c t r o d e s . The i n t e r n a l r e s i s t a n c e ( e q u i v a l e n t t o the s l o p e of the c h a r a c t e r i s t i c ) i s found t o be dominated by the p l a s m a - w a l l i n t e r a c t i o n a t the s u r f a c e boundary l a y e r s . The maximum c u r r e n t c o l l e c t e d by the e l e c t r o d e i s t h e r e f o r e l i m i t e d t o i t s i o n s a t u r a t i o n c u r r e n t . A t y p i c a l r e s u l t of the s t a n d i n g shock g e n e r a t o r i n our experiment g i v e s 1 v o l t w i t h 0.5.amp f o r the. 30X10" 6 sec t e s t time d u r a t i o n . i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v i i LIST OF TABLES x i ACKNOWLEDGEMENTS x i i NOMENCLATURE x i i i CHAPTER 1. INTRODUCTION 1 1.1 A b r i e f s u r v ey on the p r e s e n t e n e r g y - c o n v e r s i o n t e c h n o l o g y 1 1.2 Power e x t r a c t i o n by the s t a n d i n g shock g e n e r a t o r 2 CHAPTER 2. PRINCIPLES OF THE STANDING SHOCK GENERATORS 5 2.1 I n t r o d u c t i o n t o the s t a n d i n g shock g e n e r a t o r .. 5 2.2 The l o a d i n g c h a r a c t e r i s t i c of the g e n e r a t o r ... 8 2.3 The s i z e of the induced e.m.f 10 i v 2.4 The gasdynamics of shock waves 11 2.5 O b l i q u e shock waves 16 2.6 Shock wave f r o n t s t r u c t u r e s 18 2.7 The s t a n d i n g shock g e n e r a t o r e.m.f. and i n t e r n a l r e s i s t a n c e 24 CHAPTER 3. THE EXPERIMENT 29 3.1 P r o d u c t i o n s of an i o n i z e d s u p e r s o n i c f l o w 29 3.2 The shock tube system 32 3.3 C a l i b r a t i o n of the shock wave speed 35 3.4 The t e s t f l o w p r o p e r t i e s and t h e i r d i a g n o s t i c s 39 3.5 The t e s t s e c t i o n and the e l e c t r i c a l measurements 49 3.6 The c o n i c a l s t a n d i n g shocks 53 CHAPTER 4. EXPERIMENTAL RESULTS AND INTERPRETATIONS . 57 4.1 V o l t a g e measurements of the s t a n d i n g shock g e n e r a t o r 57 4.2 The l o a d l i n e s 60 4.3 R e p r o d u c i b i l i t y of measurements 61 4.4 Comparison of the o p e n - c i r c u i t v o l t a g e measurement w i t h the s t a n d i n g shock g e n e r a t o r e.m.f 63 4.5 Comparison of the b u l k plasma r e s i s t a n c e t o the s t a n d i n g shock g e n e r a t o r i n t e r n a l r e s i s t a n c e .. 65 4.6 E f f e c t s of the s e p a r a t i o n between e l e c t r o d e s .. 68 4.7 E f f e c t s of the e l e c t r o d e m a t e r i a l s 68 V 4.8 Summary and p r e l i m i n a r y c o n c l u s i o n s 71 CHAPTER 5. THE BOUNDARY LAYERS 7 3 5.1 The concept of boundary l a y e r 73 5.2 Laminar and t u r b u l e n t t y p e s of f l o w 75 5.3 The boundary l a y e r s i n our experiment 76 5.4 F o r m u l a t i o n of the boundary l a y e r t h e o r y 78 5.5 Boundary c o n d i t i o n s of the boundary l a y e r f l o w 81 5.6 The v e l o c i t y and temperature p r o f i l e s i n the boundary l a y e r f l o w 82 5.7 The boundary l a y e r on the cone s u r f a c e 86 5.8 A summary of the boundary l a y e r study 87 CHAPTER 6. THE PLASMA-WALL INTERACTION AND RESULTS OF THE DOUBLE-PROBE MEASUREMENT 89 6.1 The charge boundary l a y e r 89 6.2 The double-probe system 95 6.3 E l e c t r i c a l probes i n plasma d i a g n o s t i c and t h e i r c l a s s i f i c a t i o n 100 6.4 The i o n and e l e c t r o n t e m p e r a t u r e s 103 6.5 The c u r r e n t s i n the sh e a t h r e g i o n 105 6.6 The s l o p e s of the l o a d i n g c h a r a c t e r i s t i c s 110 6.7 The double-probe experiment 112 6.8 Theory of the i o n s a t u r a t i o n c u r r e n t 115 6.9 Comparison of the l o a d i n g c h a r a c t e r i s t i c s l o p e 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 121 CHAPTER 7. THERMODYNAMICS OF THE STANDING SHOCK GENERATOR 125 7.1 Thermodynamics of shock compressions 125 7.2 A v a i l i b i l i t y and e f f e c t i v e n e s s 127 7.3 E f f e c t i v e n e s s of the s t a n d i n g shock g e n e r a t o r .128 7.4 The o v e r a l l system 131 7.5 C o n v e r s i o n e f f i c i e n c y of a s t a n d i n g shock g e n e r a t o r 132 CHAPTER 8. CONCLUSIONS 135 8.1 Summary and c o n c l u s i o n s 135 8.2 Important g e n e r a l r e s u l t s 137 8.3 S u g g e s t i o n s f o r f u t u r e work 138 BIBLIOGRAPHY ..140 APPENDIX A. SHOCK WAVE RELATIONS 144 APPENDIX B. THE ION SATURATION CURRENT AND THE AMBIPOLAR EQUATION 150 v i i LIST OF FIGURES 2.1 A c o n f i g u r a t i o n of the s t a n d i n g - s h o c k g e n e r a t o r . 7 2.2 The sch e m a t i c c i r c u i t of the g e n e r a t o r 9 2.3 The l o a d i n g c h a r a c t e r i s t i c of a b a t t e r y - l i k e g e n e r a t o r 9 2.4 A r u n n i n g and a s t a n d i n g shock wave 13 2.5 The f i n a l t o i n i t i a l d e n s i t y , p r e s s u r e and temperature r a t i o s of an i d e a l normal shock wave. 15 2.6 A p l a n a r o b l i q u e shock wave 17 2.7 O b l i q u e shock a n g l e as a f u n c t i o n of o b s t a c l e a n g l e and i n t a k e Mach number 19 2.8 Temperature, p r e s s u r e and d e n s i t y p r o f i l e s b e h i n d a s t r o n g shock wave 21 2.9 Temperature and degree of i o n i z a t i o n b e h i n d a Mach 16 argon shock 23 2.10 The s t r u c t u r e of a shock wave i n a p a r t i a l l y i o n i z e d plasma 25 2.11 The a m b i p o l a r d i f f u s i o n at the shock f r o n t 26 3.1 A schematic diagram of the shock tube system. .. 30 3.2 The d r i v e r s e c t i o n f o r the shock tube 31 3.3 An x - t diagram of the shock wave i n the shock tube 33 3.4 The p r e s s u r e d i s t r i b u t i o n i n the shock tube. ... 33 3.5 C i r c u i t diagram of the i g n i t i o n spark 34 v i i i 3.6 A smear camera photograph f o r the shock speed measurement 36 3.7 A f r a m i n g camera photograph f o r the shock speed measurement 37 3.8 A t y p i c a l o s c i l l o g r a m from the p r e s s u r e probe measurement 38 3.9 Shock speed c a l i b r a t i o n a g a i n s t o x y - a c e t y l e n e f i l l i n g p r e s s u r e a t 5 T o r r a r g o n . 40 3.10 E l e c t r o n temperature and number d e n s i t y b e h i n d the r u n n i n g argon shock wave 42 3.11 Degree of i o n i z a t i o n and t e s t f l o w Mach number be h i n d the r u n n i n g argon shock wave 42 3.12 A schematic diagram of the s p e c t r o s c o p i c d i a g n o s t i c a p p a r a t u s 44 3.13 S t a r k broadened l i n e shape of A r l 6032.13 46 3.14 E x p e r i m e n t a l r e s u l t s of e l e c t r o n d e n s i t y measurement by the S t a r k b r o a d e n i n g method 47 3.15 R e l a t i v e i n t e n s i t y r a t i o of A r l 6416 t o A r l 6965 as a f u n c t i o n of the LTE temperature 48 3.16 The t e s t s e c t i o n 50 3.17 A f r a m i n g camera photograph of the c o n i c a l shock. 52 3.18 E l e c t r o n d e n s i t y and temperature r a t i o s a c r o s s the s t a n d i n g shock as a f u n c t i o n of the f r e e r u n n i n g shock Mach number and cone h a l f - a n g l e . .. 54 3.19 D e n s i t y r a t i o and s t a n d i n g shock h a l f - a n g l e as a f u n c t i o n of the f r e e r u n n i n g shock Mach number and cone h a l f - a n g l e 55 i x 4.1 T y p i c a l examples of o s c i l l o g r a m s o b t a i n e d i n the v o l t a g e measurement. 58 4.2 E l e c t r i c p o t e n t i a l p o l a r i t i e s of the f r e e r u n n i n g shock and the s t a n d i n g shock 59 4.3 P l o t t i n g a l o a d l i n e on the c u r r e n t - v o l t a g e p l a n e 60 4.4 C a l c u l a t e d e.m.f. of the g e n e r a t o r at v a r i o u s cone h a l f - a n g l e and r u n n i n g shock Mach number. . 62 4.5 Measured e.m.f. of the g e n e r a t o r at v a r i o u s cone h a l f - a n g l e and r u n n i n g shock Mach number 62 4.6 Comparison of e x p e r i m e n t a l r e s u l t s t o the p r e d i c t e d e.m.f 64 4.7 E x p e r i m e n t a l r e s u l t s of the s t a n d i n g shock g e n e r a t o r i n t e r n a l r e s i s t a n c e 66 4.8 The s i m p l i f i e d e l e c t r o d e c o n f i g u r a t i o n f o r the i n t e g r a t i o n of the b u l k plasma r e s i s t a n c e 67 4.9 R e l a t i o n between the measured o p e n - c i r c u i t v o l t a g e , the shock p o t e n t i a l and the work f u n c t i o n s 70 5.1 S k e t c h of boundary l a y e r on a f l a t p l a t e i n p a r a l l e l f l o w a t z e r o i n c i d e n c e 74 5.2 The boundary l a y e r s a t the anode and cathode i n our experiment 76 5.3 The boundary l a y e r s b e h i n d a shock wave on the shock tube w a l l i n both l a b o r a t o r y and shock r e s t frames 77 X 5.4 P r o f i l e s of boundary l a y e r v e l o c i t y , temperature and e r r o r f u n c t i o n a g a i n s t the reduced d i s t a n c e . 84 5.5 P r o f i l e s of boundary l a y e r v e l o c i t y , temperature and e r r o r f u n c t i o n a g a i n s t the n o r m a l i z e d d i s t a n c e 84 5.6 The v e l o c i t y p r o f i l e of the boundary l a y e r on the cone s u r f a c e 87 6.1 Elements of the charge boundary l a y e r 92 6.2 A t y p i c a l c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of an i d e a l probe 94 6.3 The s t a n d i n g shock g e n e r a t o r e l e c t r i c a l c i r c u i t . 94 6.4a The schematic diagram of the c u r r e n t p a t h i n the s t a n d i n g shock g e n e r a t o r c i r c u i t 96 6.4b Regions of p o t e n t i a l d i f f e r e n c e i n the s t a n d i n g shock g e n e r a t o r c i r c u i t 96 6.5 The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of a d o u b l e -probe system 99 6.6 The double-probe e x p e r i m e n t a l s e t - u p 113 6.7 The s a t u r a t i o n c u r r e n t d e n s i t y measurements and t h e o r e t i c a l p r e d i c t i o n s 114 6.8 Comparison of the l o a d i n g c h a r a c t e r i s t i c s l o p e , i n t h e o r y and i n measurement 123 7.1 E f f e c t i v e n e s s of work e x t r a c t i o n from s t a n d i n g shock g e n e r a t o r 130 7.2 An a r r a y of s t a n d i n g shock g e n e r a t o r s 132 7.3 The energy c o n v e r s i o n of the s t a n d i n g shock g e n e r a t o r 1 33 XI LIST OF TABLES 4.1 O p e n - c i r c u i t v o l t a g e s f o r d i f f e r e n t c o m b i n a t i o n s of e l e c t r o d e m a t e r i a l s 69 5.1 V a l u e s of £ f o r common shock tube m a t e r i a l s . ... 81 5.2 V a l u e s of the boundary l a y e r t h i c k n e s s c o e f f -i c i e n t g 86 6.1 C l a s s i f i c a t i o n s of probe t h e o r i e s 101 6.2 Plasma parameters of the t e s t f l o w a t f r e e stream 102 6.3 Ion c u r r e n t s c o l l e c t e d a t the cone 122 XI 1 ACKNOWLEDGEMENTS I am g r e a t l y i n d e b t e d t o my s u p e r v i s o r , Dr. B. A h l b o r n , f o r h i s e n t h u s i a s t i c g u i d a n c e . H i s c o n s t a n t encouragement and h i s e v e r l a s t i n g p a t i e n c e a r e m o s t l y a p p r e c i a t e d . I w i s h t o e s p e c i a l l y thank Dr. F. Curzon f o r h i s many v a l u a b l e s u g g e s t i o n s . I would a l s o l i k e t o e x p r e s s my g r a t i t u d e t o the o t h e r two committee members, Dr. R. Nodwell and Dr. B. B e r g e r s e n , f o r t h e i r h e l p f u l d i s c u s s i o n s d u r i n g the c o u r s e of t h i s work. I am a l s o t h a n k f u l t o Dr. I . G a r t s h o r e and Dr. A . J . B a r n a r d , whose comments on t h i s t h e s i s were found t o be v e r y u s e f u l . I t has been my p l e a s u r e t o be a member of the plasma p h y s i c s group a t UBC. I have had many v a l u a b l e d i s c u s s i o n s w i t h Dr. W. L i e s e , Mr. K. Fong and Mr. J . Pea r s o n . The a s s i s t a n c e o b t a i n e d from Mr. A. Cheuck i n h a n d l i n g e l e c t r o n i c equipments must be g r a t e f u l l y acknowledged. F i n a l l y I am much o b l i g e d t o M i s s G. Kong f o r her h e l p i n t he p r e p a r a t i o n of t h i s document and a l s o f o r p u t t i n g up w i t h many d e s e r t e d weekends throughout the y e a r s . x i i i NOMENCLATURE Numbers denote the c h a p t e r ( s ) i n which the symbol has the d e f i n e d meaning. When the word ' a l l ' i s used, the d e f i n e d meaning i s v a l i d t h roughout the t h e s i s . a sound speed, 2, app.B A s u r f a c e a r e a of e l e c t r o d e s A a v a i l a b i l i t y , 7 AQ d i m e n s i o n l e s s parameter i n eq. ( B . 6 ) , app.B t> u n i t v e c t o r i n magnetic f i e l d d i r e c t i o n , 2 -»-B magnetic f i e l d , 2 c speed of l i g h t ( i n v o l v e d i n the d e f i n i t i o n of ), 3 c p s p e c i f i c heat a t c o n s t a n t p r e s s u r e , a l l C mass f r a c t i o n , use s u b s c r i p t s ' i ' or 'e' f o r i o n or e l e c t r o n , 6, app.B C i o n mass f r a c t i o n a t f r e e stream, 6, app.B D d i f f u s i o n c o e f f i c i e n t , use s u b s c r i p t s i , 'e' or 'a' f o r i o n , e l e c t r o n or a m b i p o l a r d i f f u s i o n , 6, app.B e e l e c t r o n c h a r g e , a l l E e l e c t r i c f i e l d , a l l E e x c i t a t i o n energy l e v e l f o r the r e l a t i v e l i n e i n t e n s i t y , 3 f o s c i l l a t o r s t r e n g t h , 3 F f o r c e on a charged p a r t i c l e , e q . ( 2 . l ) , 2 F stream f u n c t i o n , 5, app.B g s t a t i c a l w e i g h t , e q . ( 3 . 2 ) , 3 g boundary l a y e r t h i c k n e s s c o e f f i c i e n t , 5 g e n t h a l p y c o e f f i c i e n t d e f i n e d by e q . ( A . l O ) , 2, app.A h w i d t h of the cathode r i n g , 4 h e n t h a l p y f u n c t i o n , 2, 7 x i v s t a g n a t i o n e n t h a l p y , 7 random c u r r e n t as i n e g . ( 6 . 8 ) , 6 c u r r e n t , use s u b s c r i p t s ' i ' or 'e' f o r the i o n c u r r e n t or e l e c t r o n c u r r e n t , 6 , app.B i o n s a t u r a t i o n c u r r e n t , 6 , app.B e l e c t r o n s a t u r a t i o n c u r r e n t , 6 the net c u r r e n t i n the c i r c u i t , 6 c u r r e n t d e n s i t y , may use the same s u b s c r i p t s as I , 6 , app.B Boltzmann c o n s t a n t , a l l m o b i l i t y , use s u b s c r i p t ' i ' or 'e' f o r i o n or e l e c t r o n , 6 c u r r e n t p a t h i n e q . ( 2 . l 0 ) , 2 l i n e of s i g h t d i s t a n c e , 3 f l u i d - d y n a m i c c h a r a c t e r i s t i c l e n g t h , 6 p a r t i c l e mass, use s u b s c r i p t ' i ' or 'e' f o r i o n or e l e c t r o n , 'h' f o r heavy p a r t i c l e , a l l Mach number i n g e n e r a l , s u b s c r i p t s f o r shock wave, a l l Mach number of the f r e e r u n n i n g shock i n the shock t u b e , a l l number d e n s i t y , use s u b s c r i p t ' i ' o r 'e' f o r i o n or e l e c t r o n , s u b s c r i p t 'n' f o r n e u t r a l , a l l i o n number d e n s i t y a t s h e a t h edge, 6 , app.B i o n number d e n s i t y a t mean f r e e p a t h from the w a l l , 6 , app.B lower s t a t e p o p u l a t i o n of the e m i s s i o n t r a n s i t i o n , 3 number of c o l l i s i o n s i n the random walk ( s e c t i o n 6 . 4 ) , 6 p r e s s u r e , use s u b s c r i p t ' i ' or 'e' f o r i o n or e l e c t r o n p a r t i a l p r e s s u r e , a l l P r a n d t l number, a l l XV i n t e r n a l plasma b u l k r e s i s t a n c e , a l l gas c o n s t a n t , a l l Reynold's number, a l l i o n - s l i p f a c t o r , e q . ( 2 . 2 ) , 2 s p e c i f i c e n t r o p y , 2, 7 e n t r o p y , 2, 7 Schmidt number, app.B t i m e , a l l t e m p e r a t u r e , use s u b s c r i p t ' i ' or 'e' f o r i o n or e l e c t r o n , a l l temperature of the gas (heavy p a r t i c l e s ) , a l l temperature a t w a l l , 5, 6, app.B temperature a t f r e e stream, 5, 6, app.B i n i t i a l t e m perature ahead of the f r e e r u n n i n g shock, 2 s t a g n a t i o n t e m p e r a t u r e , 7 temperature of the dead s t a t e , 7 v e l o c i t y i n g e n e r a l , x-component v e l o c i t y i n c h a p t e r 5 v e l o c i t y of w a l l ( i n shock r e s t f r a m e ) , 5 v e l o c i t y of f r e e stream ( i n shock r e s t f r a m e ) , 5 y-component v e l o c i t y i n c h a p t e r 5 v e l o c i t y of shock, a l l average p a r t i c l e speed, 6 o p e n - c i r c u i t v o l t a g e , a l l measured v o l t a g e d i f f e r e n c e from anode t o ca t h o d e , a l l bow shock p o t e n t i a l , a l l « p o t e n t i a l drop a t the s h e a t h , 6 x v i p o t e n t i a l drop a t the a m b i p o l a r r e g i o n , 6 p o t e n t i a l drop due t o the plasma r e s i s t a n c e , 4, 6 -work f u n c t i o n , use s u b s c r i p t s 'a' f o r anode, 'c' f o r c a t h o d e , ' s t ' f o r s t e e l , ' a l ' f o r aluminum, and 'br' f o r b r a s s s , 3, 4, 6 heat r e l e a s e f u n c t i o n , 2, 7 work o u t p u t , 7 h o r i z o n t a l c o o r d i n a t e , a l l v e r t i c a l c o o r d i n a t e , a l l n o r m a l i z e d mass f r a c t i o n , 6, app.B degree of i o n i z a t i o n , a l l s t a r k b r o a d e n i n g parameter, 3 H a l l parameter, 2 i d e a l gas r a t i o of s p e c i f i c heat c o n s t a n t , a l l mass f l o w r a t e , 7 boundary l a y e r t h i c k n e s s , 5, 6, app.B i n t e r n a l energy, e q . ( a . 7 ) , app.B e f f e c t i v e n e s s , 7 a parameter d e f i n e d on page 8 3 , 5 n o r m a l i z e d t e m p e r a t u r e , app.B ef f i c i e n c y , 1 , 7 reduced d i s t a n c e , e q . ( 5 . 6 ) , 5, 6, app.B reduced d i s t a n c e a t v e l o c i t y boundary l a y e r t h i c k n e s s , 5 reduced d i s t a n c e a t temperature boundary l a y e r t h i c k n e s s , 5 nose cone h a l f - a n g l e , a l l t h e r m a l c o n d u c t i v i t y , 5 a b s o r p t i o n c o e f f i c i e n t , 3 x v i i X e m i s s i o n l i n e wave l e n g t h , 3 X c o l l i s i o n a l mean f r e e p a t h , s u b s c r i p t s 'en', ' i n ' , ' e i ' s p e c i f y the c o l l i s i o n p a r t n e r s , 6 X d Debye l e n g t h , a l l X ^ r a d i a t i o n mean f r e e p a t h , 6 y v i s c o s i t y c o e f f i c i e n t , a l l v k i n e m a t i c v i s c o s i t y c o e f f i c i e n t , a l l v e m i s s i o n l i n e f r e q u e n c y , 3 5 e x c i t a t i o n energy, e q . ( A . 9 ) , app.A 5 parameter used i n e q . ( 5 . l 0 ) , 5 p d e n s i t y of gas, a l l p r a d i u s of shock tube, 4 a c o n d u c t i v i t y , 2, 4 a bow shock h a l f - a n g l e , a l l T t e s t t i m e , a l l x f l u i d f r i c t i o n f o r c e , 5 $ p o t e n t i a l , a l l OJ S t a r k b r o a d e n i n g parameter, 3 a) a parameter d e f i n e d i n e q . ( 7 . 7 ) , 7 (1) i o n mass p r o d u c t i o n r a t e , 6, app.B i § e l e c t r o n 3-body r e c o m b i n a t i o n r a t e , app.B 1 CHAPTER 1. INTRODUCTION 1.1 A b r i e f s u r v e y on the p r e s e n t e n e r g y - c o n v e r s i o n  t e c h n o l o g y At p r e s e n t , the o n l y a v a i l a b l e energy c o n v e r s i o n scheme f o r l a r g e c e n t r a l - s t a t i o n power p r o d u c t i o n i s the steam t u r b o g e n e r a t o r which o p e r a t e s e c o n o m i c a l l y a t temp e r a t u r e not h i g h e r than 850°K. Meanwhile, the heat s o u r c e can be o b t a i n e d a t te m p e r a t u r e h i g h e r than 2500°K. A c c o r d i n g t o the thermodynamics of heat e n g i n e s , the e f f i c i e n c y a t maximum power o u t p u t , n, i s d e t e r m i n e d by t h e o p e r a t i n g t e m p e r a t u r e s of the g e n e r a t o r 1 8 : n - l -where T h and T c a r e the heat s o u r c e and heat s i n k t e m p e r a t u r e s r e s p e c t i v e l y . Thus i t i s most d e s i r a b l e t o o p e r a t e the g e n e r a t o r a t t h e h i g h e s t p o s s i b l e t emperature i n o r d e r t o o b t a i n h i g h e r e f f i c i e n c y . The gap between the a v a i l a b l e heat source temperature a t 2500°K and the much lower t u r b o g e n e r a t o r l i m i t a t 850°K i m p l i e s t h a t improvement i s needed i n our e n e r g y - c o n v e r s i o n t e c h n o l o g y . At 2500°K, the temperature i s h i g h enough t o p a r t i a l l y i o n i z e the w o r k i n g gas which can be seeded w i t h lower 2 i o n i z a t i o n p o t e n t i a l e l ements. Plasma d e v i c e s such as the magnetohydrodynamic g e n e r a t o r s and the t h e r m i o n i c g e n e r a t o r s are d e v e l o p e d t o e x t r a c t energy from h i g h temperature gases. They a r e o f t e n c a l l e d " D i r e c t Energy C o n v e r s i o n " (D.E.C.) d e v i c e because e l e c t r i c i t y i s produced d i r e c t l y from heat energy. There i s no i n t e r m e d i a t e s t e p t o generate k i n e t i c work. 1.2 Power e x t r a c t i o n by the s t a n d i n g shock g e n e r a t o r In t h i s t h e s i s , we d i s c u s s a new scheme of D.E.C. which makes use of the e x i s t e n c e of a s t a n d i n g shock wave i n a s u p e r s o n i c plasma f l o w . T h i s scheme i s g i v e n the name " s t a n d i n g shock g e n e r a t o r " . Comparing the s t a n d i n g shock g e n e r a t o r w i t h the MHD g e n e r a t o r , we f i n d t h a t b oth g e n e r a t o r s a r e based on the g e n e r a l i z e d Ohm's law a p p l i e d t o a plasma f l o w . The MHD g e n e r a t o r r e q u i r e s an a p p l i e d magnetic f i e l d , whereas the s t a n d i n g shock g e n e r a t o r r e q u i r e s an e l e c t r o n temperature and d e n s i t y g r a d i e n t . No magnetic f i e l d i s needed f o r the s t a n d i n g shock g e n e r a t o r . The p o t e n t i a l d i f f e r e n c e i n d uced by a shock wave was d i s c o v e r e d a t the time when i o n i z i n g shock wave s t r u c t u r e s were i n t e n s e l y s t u d i e d f o r the i n t e r e s t of space v e h i c l e r e -e n t r y p r o b l e m " 3 ' 8 . T h i s p o t e n t i a l d i f f e r e n c e , which i s caused by the e l e c t r o n p r e s s u r e g r a d i e n t , i s of the o r d e r of the e l e c t r o n temperature d i f f e r e n c e (measured i n eV) a c r o s s the shock wave. A plasma f l o w m a i n t a i n i n g a stea d y s t a n d i n g shock can u l t i l i z e t h i s p o t e n t i a l d i f f e r e n c e as the 3 e.m.f. t o d r i v e c u r r e n t through an e x t e r n a l l o a d . The b a s i c p r i n c i p l e s of the s t a n d i n g shock g e n e r a t o r and the gasdynamics of shock waves a r e d i s c u s s e d i n c h a p t e r two. Chapter t h r e e p r e s e n t s the e x p e r i m e n t . The purpose of the experiment was t o i n v e s t i g a t e the output performance of a s t a n d i n g shock g e n e r a t o r . The g e n e r a t o r was b u i l t i n s i d e a shock tube so as t o use the s u p e r s o n i c s h o c k - i n d u c e d t e s t f l o w b e h i n d a s t r o n g r u n n i n g shock wave. A c o n i c a l bow shock was produced as the t e s t f l o w impinged on a cone. The cone was mounted a l o n g the a x i s of the shock tube. T h e o r e t i c a l c o m p u t a t i o n s and e x p e r i m e n t a l d i a g n o s t i c s were done t o dete r m i n e the p r o p e r t i e s of the s u p e r s o n i c t e s t f l o w . C u r r e n t - v o l t a g e c h a r a c t e r i s t i c s (the l o a d l i n e s ) of the g e n e r a t o r o u t p u t were measured as f u n c t i o n s of the cone a n g l e and the t e s t f l o w c o n d i t i o n s . These l o a d l i n e s were found t o be l i n e a r , a b e h a v i o u r comparable t o t h a t of a dry c e l l . The e x p e r i m e n t a l r e s u l t s a r e d i s c u s s e d i n c h a p t e r f o u r . I t was found t h a t the measured e.m.f. agreed w i t h the s t a n d i n g shock p o t e n t i a l d i f f e r e n c e p r e d i c t i o n when the e l e c t r o d e work f u n c t i o n s were i n c l u d e d . The i n t e r n a l r e s i s t a n c e of the g e n e r a t o r , or e q u i v a l e n t l y the s l o p e of the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c , however, appeared t o be much l a r g e r than the plasma b u l k r e s i s t a n c e . T h i s l a r g e r e s i s t a n c e must be e x p l a i n e d i n terms of the plasma s h e a t h boundary l a y e r . Chapter f i v e c o n t a i n s the t h e o r e t i c a l study of the gas-4 dynamic ( v e l o c i t y and temperature) boundary l a y e r and c h a p t e r s i x d i s c r i b e s the p l a s m a - w a l l i n t e r a c t i o n of the f l o w i n g plasma. The gasdynamic boundary l a y e r reduces the t e m p e r a t u r e and v e l o c i t y of the f l o w a t the r e g i o n next t o the w a l l ( e l e c t r o d e ) . The plasma s h e a t h i n t h i s r e g i o n p r o v i d e s an e l e c t r i c f i e l d which d y n a m i c a l l y c o n t r o l s the amount of c u r r e n t a c c e p t e d by the e l e c t r o d e a c c o r d i n g t o the p o t e n t i a l between the e l e c t r o d e and the plasma. Theory of the double-probe plasma d i a g n o s t i c s was adopted t o e x p l a i n the r e s u l t s of our l o a d l i n e s measurements. The agreement was good. Boundary l a y e r s and p l a s m a - w a l l i n t e r a c t i o n s are i n t r i n s i c f e a t u r e s of the s t a n d i n g shock g e n e r a t o r s ; t h e i r e f f e c t s must be u n d e r s t o o d i n o r d e r t o f i n d ways of i m p r o v i n g the g e n e r a t o r ' s performance. F i n a l l y , i n c h a p t e r seven, the thermodynamics of the s t a n d i n g shock g e n e r a t o r a r e b r i e f l y d i s c u s s e d . Some p r e l i m i n a r y c o n c l u s i o n s are reached which may be h e l p f u l f o r d e s i g n i n g f u t u r e models of s t a n d i n g shock g e n e r a t o r s . N e v e r t h e l e s s , a steady f l o w c o n t i n u o u s output g e n e r a t o r i s needed i n o r d e r t o e v a l u a t e the o v e r a l l engine e f f i c i e n c y . B u i l d i n g a s t a n d i n g shock g e n e r a t o r of t h i s n a t u r e i s d e f i n i t e l y the next s t e p i n d e v e l o p i n g t h i s new concept f o r d i r e c t energy c o n v e r s i o n . 5 CHAPTER 2. PRINCIPLES OF THE STANDING SHOCK GENERATORS The s t a n d i n g shock g e n e r a t o r can be c o n s i d e r e d as a d i r e c t energy c o n v e r s i o n (D.E.C.) d e v i c e which g e n e r a t e s e l e c t r i c a l power d i r e c t l y from a s u p e r s o n i c plasma f l o w . I t i s i n p r i n c i p l e an a l t e r n a t i v e D.E.C. d e v i c e t o the b e t t e r known magneto-hydrodynamic (MHD) g e n e r a t o r . U n l i k e the MHD g e n e r a t o r , the s t a n d i n g shock g e n e r a t o r does not r e q u i r e any a p p l i e d magnetic f i e l d . The b a s i c p r i n c i p l e s u n d e r l y i n g the mechanism of energy c o n v e r s i o n by the g e n e r a t o r a r e d e s c r i b e d i n t h i s c h a p t e r . 2.1 I n t r o d u c t i o n t o the s t a n d i n g shock g e n e r a t o r The f o r c e e x e r t e d on a p a r t i c l e of charge q moving i n an e l e c t r o m a g n e t i c f i e l d i s c a l l e d the L o r e n t z f o r c e , F = q(E + vxB) , ( 2 . 1 ) where v i s the v e l o c i t y of the p a r t i c l e . In a plasma, t h e c o l l e c t i v e motion of charge d p a r t i c l e s i s de t e r m i n e d not o n l y by the L o r e n t z f o r c e but a l s o by the " f o r c e " of d i f f u s i o n , which i s p r o p o r t i o n a l t o the c o n c e n t r a t i o n g r a d i e n t of the charge d p a r t i c l e s . The t o t a l c harge c u r r e n t , J , t h a t f l o w s i n a p a r t i a l l y i o n i z e d plasma of a r b i t r a r y degree of i o n i z a t i o n ( b e i n g q u a s i - n e u t r a l w i t h N e = N i ) i s g i v e n a s * 0 6 m.,Vp - m Vp. J + B Jxb + sbx(JxB) i e e r i e (2.2) E + u x B + e(N m +N.m.) = 5 e e 1 1 Here it i s the average f l u i d v e l o c i t y , o i s the c o n d u c t i v i t y , £ i s a u n i t v e c t o r i n the d i r e c t i o n of the if f i e l d , m e i s the e l e c t r o n mass and m i s the i o n mass, and N e and p e are r e s p e c t i v e l y the e l e c t o n number d e n s i t y and the e l e c t r o n p a r t i a l p r e s s u r e . p e and N e a r e r e l a t e d t o each o t h e r by the p a r t i a l p r e s s u r e r e l a t i o n p e = N e k T e , where T e i s the t e m p e r a t u r e of the e l e c t r o n gas. L i k e w i s e , we have p i=N ikT i. S i n c e mi>>me, the g r a d ( p i ) term i s o f t e n n e g l e c t e d . The parameters 3 and s a r e c a l l e d t he H a l l parameter and the i o n - s l i p f a c t o r r e s p e c t i v e l y ; b oth of them a r e f u n c t i o n s of t h e magnetic f i e l d . In b r i e f , the H a l l parameter i s d e f i n e d as the r a t i o of the c y c l o t r o n f r e q u e n c y t o the c o l l i s i o n f r e q u e n c y . The i o n - s l i p f a c t o r has the d e f i n i t i o n ' •'• ir)'».'i • where p n and p a r e the n e u t r a l p a r t i c l e d e n s i t y and t o t a l d e n s i t y r e s p e c t i v e l y . We s h a l l not e l a b o r a t e f u r t h e r on th e s e two parameters because we a r e e v e n t u a l l y o n l y i n t e r e s t e d i n the c a s e of z e r o magnetic f i e l d - - a c o n d i t i o n i n which b o t h 3 and s v a n i s h . E q u a t i o n (2.2) i s g e n e r a l l y known as the g e n e r a l i z e d Ohm's law. I t i s apparent t h a t each term on the l e f t - h a n d s i d e of e q u a t i o n (2.2) r e p r e s e n t s a f o r c e c o n t r i b u t i n g t o the 7 F i g . 2.1 A c o n f i g u r a t i o n of the s t a n d i n g - s h o c k g e n e r a t o r . c u r r e n t f l o w i n a plasma. The e l e c t r i c and magnetic f o r c e s a r e f a m i l i a r ones whereas the f i n a l term t h a t o r i g i n a t e s from the e f f e c t of e l e c t r o n d i f f u s i o n , i s not n e g l i g i b l e when t h e r e a r e l a r g e g r a d i e n t s p r e s e n t i n the plasma, f o r i n s t a n c e i n the case of shock waves. W h i l e the MHD g e n e r a t o r u t i l i z e s the u x B term, the g r a d ( p e ) term may a l s o be used t o produce e l e c t r i c i t y i f the p r o p e r c o n d i t i o n s a r e p r o v i d e d . Shock waves can be c r e a t e d . i n a plasma f l o w . F i g u r e 2.1 shows one way of d o i n g i t . A s u p e r s o n i c f l o w i s produced by the use of a c o n v e r g e n t - d i v e r g e n t n o z z l e 1 * and t h e f l o w i s s u b s e q u e n t l y impinged onto an o b s t a c l e t h e r e b y f o r m i n g an o b l i q u e s t a n d i n g shock wave ( a l s o known as the 8 bow shock) i n f r o n t of the o b j e c t . E l e c t r o n c u r r e n t w i l l t e n d t o f l o w o p p o s i t e t o the d i r e c t i o n of the e l e c t r o n c o n c e n t r a t i o n g r a d i e n t a t the shock f r o n t . I f e l e c t r o d e s are p l a c e d on both s i d e s of the shock wave, c u r r e n t ( p o s i t i v e ) w i l l f l o w i n the d i r e c t i o n i n d i c a t e d i n f i g u r e 2.1. The s t a n d i n g - s h o c k g e n e r a t o r t h u s p o s s e s s e s n e a r l y a l l the advantages a t t a i n e d by the MHD g e n e r a t o r as a D.E.C. d e v i c e , and i n a d d i t i o n , i t does not r e q u i r e a magnetic f i e l d . B e s i d e s , the s t a n d i n g shock g e n e r a t o r may be u s e f u l i n i t s own r i g h t when s p e c i a l e n v i r o n m e n t a l c o n d i t i o n s a r e imposed e.g. on space v e h i c l e s t r a v e l i n g at hyper-speed i n the space plasma. 2.2 The l o a d i n g c h a r a c t e r i s t i c of the g e n e r a t o r The performance of any g e n e r a t o r , i n c l u d i n g the s t a n d i n g shock g e n e r a t o r , can be e v a l u a t e d a c c o r d i n g t o the amount of v o l t a g e s and c u r r e n t s i t s u p p l i e s . In p a r t i c u l a r , we s h a l l be i n t e r e s t e d t o f i n d the g e n e r a t o r ' s maximum e.m.f. and power o u t p u t . F i g u r e 2.2 shows a sc h e m a t i c c i r c u i t of the g e n e r a t o r c o n n e c t i n g t o an e x t e r n a l r e s i s t i v e l o a d , Rj^  . The t o t a l i n t e r n a l r e s i s t a n c e of the g e n e r a t o r i s r e p r e s e n t e d by the r e s i s t o r r i n the diagram. As the v a l u e of the l o a d i n g r e s i s t a n c e v a r i e s , the o u t p u t of the g e n e r a t o r r e a c t s a c c o r d i n g l y . The b e h a v i o r of the g e n e r a t o r ' s response can b e s t be un d e r s t o o d by means of the c u r r e n t - v o l t a g e r e l a t i o n s h i p shown i n f i g u r e 2.3. The c u r v e i n the graph i s u s u a l l y c a l l e d the l o a d i n g c h a r a c t e r i s t i c , 9 r-VWvV 1|— RL y 4> F i g . 2.2 The s c h e m a t i c c i r c u i t of the g e n e r a t o r . F i g . 2.3 The l o a d i n g c h a r a c t e r i s t i c of a b a t t e r y - l i k e g e n e r a t o r . 10 or s i m p l y , the l o a d l i n e . I t t u r n s out t h a t the s t a n d i n g shock g e n e r a t o r i n our experiment always g i v e s s t r a i g h t l o a d l i n e s . A common d e v i c e which a l s o e x h i b i t s s t r a i g h t l o a d l i n e s i s the d r y c e l l . S t r a i g h t l o a d l i n e s a r e r e l a t i v e l y s i m p l e because the e n t i r e l i n e i s c h a r a c t e r i z e d by two c o n s t a n t parameters namely the v o l t a g e i n t e r c e p t and the s l o p e . The v o l t a g e i n t e r c e p t i s e q u i v a l e n t t o the o p e n - c i r c u i t v o l t a g e , V 0, and the s l o p e i s e q u i v a l e n t t o t h e i n t e r n a l r e s i s t a n c e , r , of the g e n e r a t o r . Thus p a r t of our i n v e s t i g a t i o n on the s t a n d i n g shock g e n e r a t o r i s t o p r e d i c t t h e o r e t i c a l l y and t o det e r m i n e e x p e r i m e n t a l l y the v a l u e s of V 0 and r . 2.3 The s i z e of the i n d u c e d e.m.f. The g e n e r a l i z e d Ohm's law i n the absence of any magnetic f i e l d can be o b t a i n e d by d r o p p i n g a l l the terms i n e q u a t i o n (2.2) t h a t i n v o l v e B: E + l N - = o . (2.3) e In the o p e n - c i r c u i t s i t u a t i o n , a t stea d y s t a t e , the c u r r e n t i n the plasma must come t o r e s t . Thus the e l e c t r o n p r e s s u r e g r a d i e n t f o r c e i s c o u n t e r - b a l a n c e d by the in d u c e d e l e c t r i c f o r c e . W r i t i n g E =-v<s> where $ i s the e l e c t r i c a l p o t e n t i a l of the plasma, the above e q u a t i o n a t J = 0 becomes or V* = ^VT + -T V ( l n ( N )) . (2.4) e e e e e T h i s e q u a t i o n can be r e w r i t t e n i n the i n t e g r a l form f o r f u t u r e c o n v e n i e n c e : "2- »1 " 7 < * e 2 - T e l ) + l / i T e * f e ( l n ( N e ) ) d X • <2'5> When a p p l y i n g t h i s e q u a t i o n t o the s t a n d i n g shock g e n e r a t o r , the s u b s c r i p t 1 and 2 would r e f e r t o the e q u i l i b r i u m c o n d i t i o n s upstream and downstream of the shock wave r e s p e c t i v e l y . A l s o ( $ 2-$,) becomes the o p e n - c i r c u i t v o l t a g e , V 0 . In accordance w i t h the f o r e g o i n g e q u a t i o n , V 0 can be c a l c u l a t e d i f the d i s t r i b u t i o n s of the e l e c t r o n t e m p e r a t u r e and the e l e c t r o n d e n s i t y w i t h i n the shock t r a n s i t i o n zone i s known. 2.4 The gasdynamics of shock waves In o r d e r t o o b t a i n the s t a n d i n g shock e.m.f., the e s s e n t i a l f e a t u r e s of the shock t r a n s i t i o n must be u n d e r s t o o d f i r s t . Only t h o s e t o p i c s which a r e c l o s e l y r e l a t e d t o our i n t e r e s t w i l l be b r i e f l y d i s c u s s e d , o t h e r d e t a i l e d d e r i v a t i o n s and some u s e f u l formulae a r e t o be found i n appendix A. F u r t h e r m o r e , i n accordance w i t h the need of our e x p e r i m e n t s , the s p e c i a l case of shock waves i n argon gas w i l l be used f o r i l l u s t r a t i n g the shock f r o n t s t r u c t u r e . Argon was used t h r o u g h o u t a l l our e x p e r i m e n t s because i t i s a monatomic gas commonly chosen f o r shock wave 12 r e s e a r c h 2 * . I t i s r e l a t i v e l y easy t o i o n i z e and i t i s s a f e t o use i n the l a b o r a t o r y . N e v e r t h e l e s s , we b e l i e v e t h a t the phenomena i l l u s t r a t e d i n our example a r e a p p l i c a b l e t o o t h e r i o n i z e d w o r k i n g f l u i d s as w e l l . A shock wave may be c o n s i d e r e d i n the l a b o r a t o r y frame of r e f e r e n c e as b e i n g a l a r g e p r e s s u r e s t e p t h a t p r o p a g a t e s i n t o a medium w i t h a speed h i g h e r than the sound speed. The gas b e h i n d the shock f r o n t i s r a p i d l y h e ated and compressed t o a new e q u i l i b r i u m . On the o t h e r hand, one may choose t o observe the gasdynamics i n the r e s t frame of the shock. In t h i s frame, the shock f r o n t appears t o be s t a t i o n a r y . The c o l d gas e n t e r s the shock f r o n t a t a s u p e r s o n i c i n t a k e v e l o c i t y u,, and the heated gas l e a v e s the f r o n t a t a s u b s o n i c exhaust v e l o c i t y u 2 . As shown i n f i g u r e 2.4, the two r e f e r e n c e frames move w i t h r e s p e c t t o each o t h e r a t the shock speed, v g . I f v g i s a c o n s t a n t speed, then t r a n s f o r m a t i o n between the frames i s s i m p l y done by a d d i n g or s u b t r a c t i n g v g . a c c o r d i n g l y . The magnitude of the p r e s s u r e r a t i o , the te m p e r a t u r e r a t i o and o t h e r p h y s i c a l q u a n t i t i e s must be independent of the c h o i c e of the o b s e r v i n g frame. In r e a l l i f e , b o t h r u n n i n g and s t a n d i n g shock waves can o c c u r . For c o n v e n i e n c e , shock waves a r e g e n e r a l l y t r e a t e d i n t h e i r r e s t frames. The f l o w parameters of the upstream i o n i z a t i o n e q u i l i b r i u m s t a t e and the downstream i o n i z a t i o n e q u i l i b r i u m s t a t e of the shock wave a r e c o n n e c t e d t o each o t h e r by the f i n i t e d i f f e r e n c e e q u a t i o n s r e p r e s e n t i n g the c o n s e r v a t i o n 13 2 ) © u<vs u = o RUNNING SHOCK ® © v=o u2<u M=V s>a, STANDING SHOCK F i g . 2.4 A r u n n i n g and a s t a n d i n g shock wave, l a w s : mass P 1 V 1 = P 2 V 2 (2.6) momentum P i + P , v 1 '1 P 2 + P 2 v 2 (2.7) and energy 1 2 j . v W 2 1 1 p . v 1*1 1 2 2 2 (2.8) Here, h i s the s p e c i f i c e n t h a l p y f u n c t i o n , which can be w r i t t e n f o r an i d e a l gas as . T , P , and W i s the energy ( Y - 1 ) P f l u x l o s t t o the environment d u r i n g the shock t r a n s i t i o n . The c o n s e r v a t i o n e q u a t i o n s (2.6) t o (2.8) a r e s a t i s f i e d by 14 many d i f f e r e n t k i n d s of heat waves such as i n t e r s t e l l a r r a d i a t i o n f r o n t s , d e t o n a t i o n waves, d e f l a g r a t i o n waves and shock waves. A h l b o r n and L i e s e 1 have shown t h a t t h e s e heat waves a r e d i s t i n g u i s h a b l e by t h e i r unique heat r e l e a s e f u n c t i o n W = f ( h 2 ) . In t h i s way, a shock wave i s o n l y a s p e c i a l case of the heat wave t h a t has W=0. We note t h a t the s t a n d i n g - s h o c k g e n e r a t o r o n l y r e q u i r e s a l a r g e e l e c t r o n p r e s s u r e g r a d i e n t , t h e r e f o r e o t h e r k i n d s of i o n i z i n g heat waves p o s s e s s i n g t h i s p r o p e r t y w i l l produce the e.m.f. j u s t as w e l l as an i o n i z i n g shock wave does. One example i s the huge i n d u c e d e l e c t r i c and magnetic f i e l d s found i n the s o l i d l a s e r t a r g e t i m p l o s i o n e x p e r i m e n t s 5 1 . For a shock wave (W=0) a t a g i v e n speed i n an i d e a l gas, where p,, p, and v, a r e s p e c i f i e d , the c o n s e r v a t i o n e q u a t i o n s a r e s u f f i c i e n t t o de t e r m i n e p 2 , p 2 and v 2 c o m p l e t e l y . The t e m p e r a t u r e s T, and T 2 can be o b t a i n e d by u s i n g the p r o p e r e q u a t i o n of s t a t e 2 0 . F i g u r e 2.5 shows the p r e s s u r e , d e n s i t y and t e m p e r a t u r e r a t i o s of an i d e a l shock wave as f u n c t i o n s of the shock wave Mach number. The Mach number i s d e f i n e d as the r a t i o of a f l u i d speed t o the sound speed of the f l u i d . Thus the shock wave Mach number has the d e f i n i t i o n M E v s / a , where a, i s the sound speed of the upstream gas. G e n e r a l l y s p e a k i n g the shock Mach number i s a l s o a parameter i n d i c a t i n g t h e shock s t r e n g t h . S i n c e a shock wave always t r a v e l s a t a v e l o c i t y l a r g e r than or e q u a l t o t h e sound speed, the minimum v a l u e of M i s 1. A l a r g e r v a l u e of M i m p l i e s a s t r o n g e r shock, h i g h e r p r e s s u r e r a t i o 15 F i g . 2.5 The f i n a l t o i n i t i a l d e n s i t y , p r e s s u r e and temper-a t u r e r a t i o s of an i d e a l normal i d e a l shock wave, ( i n i t i a l t e m perature i s a t room te m p e r a t u r e ) 16 and h i g h e r compression r a t i o . M a t h e m a t i c a l e x p r e s s i o n s r e l a t i n g the f i n a l e q u i l i b r i u m s t a t e t o the i n i t i a l e q u i l i b r i u m s t a t e can be found i n appendix A. A l s o d i s c u s s e d i n the appendix are the n e c e s s a r y c o r r e c t i o n s f o r r e a l gas b e h a v i o r when the temperature of the shock-compressed gas becomes h i g h enough f o r e x c i t a t i o n and i o n i z a t i o n . By and l a r g e , both e x c i t a t i o n and i o n i z a t i o n has e f f e c t s such as l o w e r i n g the s p e c i f i c h e a t s r a t i o , i n c r e a s i n g the degree of freedom and adding more p a r t i c l e s t o the gas m i x t u r e . Shock compressions d e v i a t i n g from the i d e a l gas p r e d i c t i o n a t h i g h e r Mach number are shown i n f i g u r e 2.5 by the dashed c u r v e s . An i m p o r t a n t phenomenon o c c u r r i n g i n the shock t r a n s i t i o n i s the i n c r e a s e of e n t r o p y . I n t e r e s t i n g l y , the amount of i n c r e a s e can be d e t e r m i n e d from the c o n s e r v a t i o n e q u a t i o n s and the f l u i d thermodynamic p r o p e r t i e s ; i t i s e n t i r e l y independent of the a c t u a l d i s s i p a t i o n mechanisms t h a t causes t h i s i n c r e a s e . N e v e r t h e l e s s , d i f f u s i o n , cond-u c t i o n and the d i s s i p a t i o n mechanisms such as v i s c o u s f r i c t i o n and i o n i z a t i o n a re r e s p o n s i b l e f o r d e t e r m i n i n g the s t r u c t u r e of the shock t r a n s i t i o n z o n e 5 5 . 2.5 O b l i q u e shock waves The shock waves shown i n f i g u r e 2.4 were normal shock waves. The term "normal" means the d i r e c t i o n of the i n t a k e f l o w v e l o c i t y i s p e r p e n d i c u l a r t o the p l a n e of the shock 17 wave f r o n t . Bow shocks on the o t h e r hand, a r e " o b l i q u e " shock waves 3". As shown i n f i g u r e 2.6, the upstream v e l o c i t y can be r e s o l v e d i n t o components p a r a l l e l and normal t o the o b l i q u e shock f r o n t . The t a n g e n t i a l component i s u n a f f e c t e d by the o b l i q u e shock wave; the normal components t o g e t h e r w i t h the thermodynamic q u a n t i t i e s on both s i d e s of the o b l i q u e shock wave are a c t i n g e x a c t l y l i k e t h o s e of a normal shock wave. P h y s i c a l l y , the o b l i q u e shock wave t u r n s the s u p e r s o n i c f l o w by a d e f l e c t i o n a n g l e , 6 , a l o n g a new d i r e c t i o n i n c o m p l i a n c e w i t h the boundary c o n d i t i o n s . An o b s t a c l e of wedge shape w i l l g e n e r a t e a 2-d i m e n s i o n a l p l a n e o b l i q u e shock, whereas a cone w i l l g e n e r a t e a 3 - d i m e n s i o n a l c o n i c a l shock. In the l a t t e r c a s e , the downstream f l o w i s d i v e r g i n g t h e r e f o r e nonuniform. SHOCK F i g . 2.6 A p l a n a r o b l i q u e shock wave. 18 F o l l o w i n g the method o r i g i n a l l y proposed by T a y l o r 5 0 , both the p l a n a r and the c o n i c a l o b l i q u e shocks were n u m e r i c a l l y s o l v e d . The o b l i q u e shock h a l f - a n g l e , a , as a f u n c t i o n of the upstream Mach number, M,, and the o b s t a c l e h a l f - a n g l e , 6 , i s d e p i c t e d i n f i g u r e .2.7. I t i s i n t e r e s t i n g t o note t h a t f o r each g i v e n M, , t h e r e i s a maximum l i m i t t o the t u r n i n g of the f l o w . Beyond t h i s l i m i t , f u r t h e r i n c r e a s e of the o b s t a c l e a n g l e w i l l not r e s u l t i n a l a r g e r shock a n g l e . The o b l i q u e shock wave s i m p l y g e t s d etached. There w i l l be no f u r t h e r i n c r e a s e of the shock s t r e n g t h p 2 / P i • 2.6 Shock wave f r o n t s t r u c t u r e s The shock wave f r o n t , o f t e n c a l l e d the shock f r o n t f o r s h o r t , i s the r e g i o n i n which the t r a n s i t i o n from the i n i t i a l t o f i n a l thermodynamic steady s t a t e s t a k e s p l a c e . The s t r u c t u r e of t h i s shock f r o n t i s d e t e r m i n e d by the k i n e t i c s of the r e l a x a t i o n p r o c e s s e s . In a s i m p l e c a s e , a moderate shock wave i n a monatomic gas such as argon w i l l have a ve r y t h i n shock f r o n t : o n l y a few k i n e t i c mean f r e e p a t h s t h i c k . The o n l y r e l a x a t i o n p r o c e s s which i s o c c u r r i n g would be the one i n the t r a n s l a t i o n a l degree of freedom. T h i s p a r t i c u l a r r e l a x a t i o n i s a f a s t one; i t t a k e s no more than 3 t o 4 atomic c o l l i s i o n s t o t h e r m a l i z e the k i n e t i c energy c a r r i e d by the d i r e c t e d motion of the i n t a k e f l o w . Not a l l the r e l a x a t i o n mechanisms have the same r a t e ; 19 F i g . 2.7a P l a n e o b l i q u e shock h a l f - a n g l e as a f u n c t i o n of wedge h a l f - a n g l e and i n t a k e Mach number (Y=5/3). 80° 1.0 2.0 3D 4.0 5.0 ' 1 F i g . 2.7b C o n i c a l o b l i q u e shock h a l f - a n g l e as a f u n c t i o n of cone h a l f - a n g l e and i n t a k e Mach number. 20 the f a s t ones and the slow ones can d i f f e r by o r d e r s of m a g n i t u d e 9 1 . The r a t e of t r a n s l a t i o n a l r e l a x a t i o n mentioned above i s the f a s t e s t of a l l . I f o t h e r s l o w e r p r o c e s s e s such as d i s s o c i a t i o n , e x c i t a t i o n , i o n i z a t i o n , or v i b r a t i o n a l r e l a x a t i o n a r e i n v o l v e d , the shock f r o n t t h i c k n e s s w i l l be e n l a r g e d . In f a c t , by d e f i n i t i o n , the t h i c k n e s s of a shock f r o n t i s the r e l a x a t i o n l e n g t h of the s l o w e s t p r o c e s s i n v o l v e d i n the t r a n s i t i o n t o f i n a l e q u i l i b r i u m . The narrow r e g i o n a t the f r o n t p a r t of the shock dominated by t r a n s l a t i o n a l r e l a x a t i o n s i s d e s i g n a t e d as the "compression shock" t o be d i s t i n g u i s h e d from the wider " g e n e r a l r e l a x a t i o n l a y e r " . The p r o f i l e s of p r e s s u r e and temperature d i s t r i b u t i o n b e h i n d a s t r o n g argon shock wave (M > 10) a r e d e p i c t e d i n f i g u r e 2.8. By t r a n s l a t i o n a l c o l l i s i o n s , the gas i s f i r s t h e a t ed and compressed t o temperature T*, p r e s s u r e p' and d e n s i t y p ' a t the compression shock. As f u r t h e r i o n i z a t i o n of argon atoms c o n t i n u e s t o absorb energy, the gas g r a d u a l l y r e - e s t a b l i s h e s e q u i l i b r i u m a t a lower temperature and h i g h e r p r e s s u r e and d e n s i t y ( T 2 , p 2 , P 2 ) . I o n i z a t i o n of atoms can be due t o atom-atom i n e l a s t i c c o l l i s i o n or e l e c t r o n - a t o m i n e l a s t i c c o l l i s i o n . The l a t t e r p r o c e s s has a c r o s s - s e c t i o n much h i g h e r than the former one, thus i o n i z a t i o n i n a shock wave i s g e n e r a l l y dominated by e l e c t r o n - a t o m c o l l i s i o n except i n the s p e c i a l o c c a s i o n when e l e c t r o n s a r e s c a r c e e.g. a t the compre s s i o n shock r e g i o n . Moreover, the r a t e of e l e c t r o n - a t o m c o l l i s i o n depends on the 21 T 1, p.f F i g . 2.8 Temperature, p r e s s u r e and d e n s i t y p r o f i l e s b e h i n d a s t r o n g shock wave. e l e c t r o n number d e n s i t y , t h e r e f o r e i o n i z a t i o n i n c r e a s e s e x p o n e n t i a l l y w i t h t h e d i s t a n c e b e h i n d the shock f r o n t . As an example, f i g u r e 2.9 shows the d i s t r i b u t i o n s of the i o n and e l e c t r o n t e m p e r a t u r e s and the degree of i o n i z a t i o n b e h i n d a Mach 16 argon shock computed by Biberman and Yakubov 5. The degree of i o n i z a t i o n i s d e f i n e d as a: N e / ( N e + N n ) . In t h e f i g u r e X=0 i s the l o c a t i o n of the co m p r e s s i o n shock and the r e l a x a t i o n l e n g t h i s shown t o be more than 4cm l o n g , a l t h o u g h o n l y l e s s than a q u a r t e r of t h i s l e n g t h i s o c c u p i e d by the i o n i z a t i o n a v a l a n c h e r e g i o n . E a r l y s t u d i e s by P e t s c h e k and B y r o n " 3 and a l s o by H a r w e l l and J a h n 2 6 have a l r e a d y d i s c u s s e d t h a t " p r i m i n g 22 e l e c t r o n s " a r e needed t o s t a r t the i o n i z a t i o n a v a l a n c h e . These p r i m i n g e l e c t r o n s may o r i g i n a t e from p h o t o - i o n i z a t i o n , p h o t o - e m i s s i o n , (by l i g h t e m i t t e d from the shock-heated g a s ) , e l e c t r o n d i f f u s i o n or atom-atom c o l l i s i o n s . S h o r t e n i n g the p r i m i n g e l e c t r o n p r o d u c t i o n time w i l l s i m u l t a n e o u s l y s h o r t e n the r e l a x a t i o n l e n g t h . In f a c t i t i s found t h a t i m p u r i t i e s can enhance the p r o d u c t i o n of p r i m i n g e l e c t r o n s because they o f t e n have a lower i o n i z a t i o n p o t e n t i a l . I t was c o n f i r m e d e x p e r i m e n t a l l y by I g r a 3 0 t h a t i m p u r i t i e s c o u l d s h o r t e n the i o n i z a t i o n r e l a x a t i o n l e n g t h and t i m e , and enhance the l a t e r a l u n i f o r m i t y , but i m p u r i t i e s d i d not a l t e r the f i n a l e q u i l i b r i u m s t a t e . Thus the presence of some i m p u r i t i e s i n the shock tube may even be d e s i r a b l e because i t would ensure r a p i d i o n i z a t i o n t h e r e b y i n c r e a s i n g the t o t a l a v a i l a b l e t e s t time i n the ex p e r i m e n t . T h i s p o i n t w i l l become c l e a r as we d e s c r i b e the experiment i n the next c h a p t e r . In t he case of a shock wave p r o p a g a t i n g i n t o a plasma, the upstream gas i s a l r e a d y i o n i z e d ; t h e r e are p l e n t y of p r i m i n g e l e c t r o n s . E l e c t r o n impact i o n i z a t i o n b e g i n s i m m e d i a t e l y b e h i n d the compression shock. The s t r u c t u r e of a shock wave i n a p a r t i a l l y i o n i z e d plasma a c c o r d i n g t o J a f f r i n 3 1 i s shown i n f i g u r e 2.10. At the compression shock, b o t h the charge d e n s i t y and the i o n temperature a r e bo o s t e d . The e l e c t r o n t e m p e r a t u r e , however, t a k e s a smooth t r a n s i t i o n : i t b e g i n s t o r i s e g r a d u a l l y from a f a r and by the time i t g e t s t o the compression shock i t i s a l r e a d y near the F i g . 2.9 Temperature and degree of i o n i z a t i o n b e h i n d a Mach 16 argon shock, ( r e s u l t s computed by Biberman and Yakubov) 24 f i n a l e q u i l i b r i u m v a l u e . T h i s b e h a v i o u r of the e l e c t r o n t e m p e r a t u r e i s due t o the a b i l i t y of the e l e c t r o n s t o d i f f u s e and t o conduct h e a t . We s h a l l f i n d such e l e c t r o n t e m p e r a t u r e d i s t r i b u t i o n " h e l p f u l " when we come t o e v a l u a t e the s t a n d i n g shock g e n e r a t o r e.m.f. i n the next s e c t i o n . The e l e c t r o n s , b e i n g l i g h t and m o b i l e , can d i f f u s e b e t t e r than the i o n s or n e u t r a l s . Hence the e l e c t r o n s tend t o get ahead of the shock f a s t e r than the i o n s . N e v e r t h e l e s s , the e l e c t r o n s cannot run too f a r away from the i o n s , because any s i g n i f i c a n t charge s e p a r a t i o n would c r e a t e a l a r g e a t t r a c t i v e Coulomb f o r c e . T h i s phenomenon i s known as the a m b i p o l a r d i f f u s i o n . F i g u r e 2.11 i l l u s t r a t e s how a m b i p o l a r d i f f u s i o n u p s e t s the charge n e u t r a l i t y c o n d i t i o n . I t i n t r o d u c e s charge p o l a r i z a t i o n , e l e c t r i c f i e l d s and p o t e n t i a l s i n the plasma. Note t h a t t h i s i s the same p o t e n t i a l we have seen e a r l i e r when c o n s i d e r i n g the g e n e r a l i z e d Ohm's law. By and l a r g e , the e l e c t r o n and i o n d e n s i t i e s a re r i g i d l y c o u p l e d t h r o u g h the Coulomb i n t e r a c t i o n . Hence the p o s i t i v e and n e g a t i v e charge d e n s i t i e s a r e e q u a l t o the f i r s t o r d e r and the charge s e p a r a t i o n due t o the a m b i p o l a r d i f f u s i o n i s a second o r d e r ef f e c t . 2.7 The s t a n d i n g shock g e n e r a t o r e.m.f. and i n t e r n a l  r e s i s t a n c e L e t us now r e t u r n t o e q u a t i o n (2.5) t o e v a l u a t e the p o t e n t i a l d i f f e r e n c e a c r o s s a s t a n d i n g shock. For 25 UPSTREAM PLASMA DOWNSTREAM PLASMA Ti Ti 0 \ COMPRESSION SHOCK F i g . 2.10 The s t r u c t u r e of a shock wave i n a p a r t i a l l y i o n i z e d plasma. ( s c h e m a t i c diagram not draw t o s c a l e ) c o n v e n i e n c e , the e q u a t i o n i s r e p e a t e d here The second term on the r i g h t hand s i d e i n v o l v e s an i n t e g r a t i o n which r e q u i r e s t h a t t he the d i s t r i b u t i o n of N e and T e i n s i d e the r e g i o n between 1 and 2 be known. A c c o r d i n g t o f i g u r e 2.10, the v a l u e of T e has a l r e a d y come c l o s e t o i t s f i n a l e q u i l i b r i u m v a l u e a t the p l a c e where the degree of i o n i z a t i o n i s s i g n i f i c a n t l y v a r y i n g . Thus the i n t e g r a t i o n i s s t i l l a p p r o x i m a t e l y c o r r e c t i f T e i s assumed c o n s t a n t : - ~ ( T ,-T .) + - J 2 T - ^ - ( l n ( N ) ) d x 2 1 e e2 e l e i e dx e 26 e(Ni-Ne) F i g . 2.11 The arnbipolar d i f f u s i o n a t the shock f r o n t _ $ = — (T -T ,) + -T A ^ - ( l n ( N ) ) d x [ 1 e e2 e l e e 2 J j dx e (2.9) e2 " e l The f o r e g o i n g e q u a t i o n i m p l i e s t h a t the p o t e n t i a l d i f f e r e n c e 27 a c r o s s a plasma shock f r o n t i s d e t e r m i n e d o n l y by the end s t a t e s ; i t i s independent of the i n t e r m e d i a t e s t r u c t u r e s . Note t h a t t h i s e q u a t i o n i s not a p p l i c a b l e t o an i o n i z i n g shock t h a t has an n o n i o n i z e d upstream gas w i t h N e 1=0. In t h i s c a s e , more d e t a i l e d c o n s i d e r a t i o n s of the e l e c t r o n a m b i p o l a r d i f f u s i o n are n e c e s s a r y i n o r d e r t o c a l c u l a t e the p o t e n t i a l 5 3. In a c l o s e d c u r r e n t p a t h of the s t a n d i n g shock g e n e r a t o r , l i k e the one shown i n f i g u r e 2.1, c u r r e n t f l o w s from the anode through the e x t e r n a l l o a d i n t o t h e c a t h o d e , c o n t i n u e s a c r o s s the upstream plasma r e g i o n , then t h r o u g h the shock, the downstream plasma r e g i o n , and e v e n t u a l l y back t o the anode t o complete the l o o p . The v o l t a g e d i s t r i b u t i o n can be o b t a i n e d by i n t e g r a t i n g e q u a t i o n (2.3) from the cathode t o the anode: / c < " S * > - * * - • / ' £ • * * . < 2 - , 0 ) e On the l e f t hand s i d e , the f i r s t term i s j u s t the p o t e n t i a l d i f f e r e n c e , V 0, induced by t h e shock as d e s c r i b e d i n e q u a t i o n s (2.5) and ( 2 . 9 ) . The second term i s the p o t e n t i a l d r o p V r due t o the plasma b u l k r e s i s t a n c e . F i n a l l y , the r i g h t hand s i d e of the e q u a t i o n i s by d e f i n i t i o n e q u a l t o t h e p o t e n t i a l d i f f e r e n c e , V m , measured a c r o s s the e l e c t r o d e s . Assuming t h a t the s h o c k - i n d u c e d p o t e n t i a l i s u n a f f e c t e d by the amount of c u r r e n t f l o w , e q u a t i o n (2.10) can be r e w r i t t e n as 28 V =V -V (2.11) m o r which i s e q u i v a l e n t t o a p p l y i n g the K i r c h h o f f s v o l t a g e law t o the c i r c u i t shown i n f i g u r e 2.2. I f the t o t a l c u r r e n t f l o w i n g i n the c i r c u i t i s denoted by I and the t o t a l i n t e r n a l r e s i s t a n c e ( i . e . the s l o p e of the l o a d l i n e ) i s denoted by r , then we may w r i t e T h i s i n t e g r a t i o n depends on the a c t u a l geometry of the e l e c t r o d e s and the v a l u e of the plasma c o n d u c t i v i t y . Symmetry a p p r o x i m a t i o n s a r e o f t e n n e c e s s a r y i n o r d e r t o make the c a l c u l a t i o n p o s s i b l e . (2.12) 29 CHAPTER -3. THE EXPERIMENT In our ex p e r i m e n t , a s h o r t d u r a t i o n of p r e - i o n i z e d s u p e r s o n i c f l o w was produced i n a shock tube. We d i a g n o s e d t h i s plasma f l o w t o c o n f i r m i t s gasdynamic pa r a m e t e r s . T h i s s u p e r s o n i c plasma fl o w was a l l o w e d t o impinge onto an o b s t a c l e t o y i e l d a s t a n d i n g bow-shock. The e l e c t r i c a l l o a d i n g c h a r a c t e r i s t i c s of t h i s s t a n d i n g shock g e n e r a t o r were measured as f u n c t i o n s of the s u p e r s o n i c plasma f l o w c o n d i t i o n s and of the s t a n d i n g shock s t r e n g t h s . D i f f e r e n t e l e c t r o d e m a t e r i a l s and e l e c t r o d e l o c a t i o n s were t e s t e d t o observe t h e i r e f f e c t s on the g e n e r a t o r . 3.1 P r o d u c t i o n s of an i o n i z e d s u p e r s o n i c f l o w I o n i z e d and s u p e r s o n i c i n t a k e f l o w i s r e q u i r e d t o o p e r a t e the s t a n d i n g shock g e n e r a t o r . S i z a b l e i o n i z a t i o n can be a c h i e v e d a t tempe r a t u r e s above 2500°K by s e e d i n g the gas w i t h low i o n i z a t i o n p o t e n t i a l s u b s t a n c e s such as p o t a s s i u m or cesium compounds". Meanwhile, ste a d y f l o w s a t s u p e r s o n i c v e l o c i t i e s are commonly produced by means of c o n v e r g e n t - d i v e r g e n t n o z z l e s . U n f o r t u n a t e l y , t h i s k i n d of arrangement was not a v a i l a b l e i n our l a b o r a t o r y ; i n s t e a d we di d ' our experiment on an a p p a r a t u s which was most f a m i l i a r t o u s — t h e shock tube. Shock tubes a re used t o produce f r e e r u n n i n g shock 30 10 KV MIXING TANK vaccum pump ^shock tube I"diameter jJ~ 140 micron diaphragm oxygen acetylene DUMP TANK — argon vaccum pump F i g . 3.1 A schematic diagram of the shock tube system. waves. The shock wave s i m u l t a n e o u s l y h e a t s and a c c e l e r a t e s the p r e - f i l l e d t e s t gas. I f the shock wave i s a s t r o n g one, the h e a t i n g may be s u f f i c i e n t t o i o n i z e the t e s t gas. In the l a b o r a t o r y frame of r e f e r e n c e , the gas b e h i n d the f r e e r u n n i n g shock f r o n t moves i n the same d i r e c t i o n e x c e p t b e i n g s l o w e r than the shock wave. For a smooth s u r f a c e c o n s t a n t d i a m e t e r shock tube, a v e r y s t r o n g r u n n i n g shock wave ( w i t h Mach number > 10) w i l l be f o l l o w e d by a column of i o n i z e d , s u p e r s o n i c , steady and u n i f o r m f l o w which q u a l i f i e s t o be a 31 aluminum ring 10 KV» iucite-block F i g . 3.2 The d r i v e r s e c t i o n f o r the shock t u b e . t e s t f l o w i n our e x p e r i m e n t . P r o d u c i n g the t e s t f l o w by g e n e r a t i n g a s t r o n g shock wave i n a shock tube i s s i m p l e and r e p r o d u c i b l e . Shock t u b e s a r e known t o be r e l i a b l e . There a r e , however, two s e r i o u s drawbacks i n t h i s method. F i r s t , the t e s t f l o w produced t h i s way i s r a t h e r b r i e f , b e i n g l i m i t e d by the d r i v e r and the shock tube l e n g t h . In p r a c t i c e , the boundary l a y e r d r a g due t o the r e a l gas e f f e c t s such as v i s c o s i t y and heat c o n d u c t i o n would f u r t h e r reduce the a v a i l a b l e t e s t t i m e 2 2 . The t e s t time was about 30 m i c r o - s e c o n d l o n g i n our e x p e r i m e n t . Second, the f l o w parameters can not be i n d e p e n d e n t l y a d j u s t e d because they a r e c o u p l e d t o g e t h e r by the shock r e l a t i o n s . A change of the r u n n i n g shock Mach number w i l l a l t e r the e n t i r e s e t of f l o w parameters 32 s i m u l t a n e o u s l y . 3.2 The shock tube system The shock tube system employed i n our experiment i s d e p i c t e d i n f i g u r e 3.1. The system was composed of f i v e e s s e n t i a l u n i t s : the shock tube, the d r i v e r s e c t i o n , the t e s t s e c t i o n , the i g n i t i o n u n i t , and the m i x i n g t a n k . The shock tube was made of a Pyrex g l a s s p i p e , 25mm i n n e r d i ameter and 4mm w a l l t h i c k n e s s . At one meter l e n g t h from the d r i v e r end, i t was j o i n e d t o the t e s t s e c t i o n which was f o l l o w e d by another s i x i n c h l o n g Pyrex p i p e and a dump tank. The purpose of the dump tank was to p r o v i d e enough "room" f o r d i f f u s i n g the t e s t f l o w such t h a t no r e f l e c t i o n of the r u n n i n g shock wave c o u l d o c c u r . R e f l e c t i o n i s u n d e s i r a b l e because i t may reduce the a l l o w a b l e t e s t t i m e . The d r i v e r s e c t i o n , shown i n f i g u r e 3.2 was made up of two c o n i c a l cups and an i g n i t i o n spark gap. To l a u n c h a shock wave, the d r i v e r chamber would be e v a c u a t e d , then f i l l e d w i t h an c o n t r o l l e d amount of e x p l o s i v e gas m i x t u r e , and f o l l o w e d by i g n i t i n g a d e t o n a t i o n . The d e t o n a t i o n i s i n i t i a t e d a t the back end of the chamber by an i g n i t i o n s p a r k . The c o n i c a l geometry of the d r i v e r chamber was d e s i g n e d t o o b t a i n an o v e r - d r i v e n d e t o n a t i o n ; v e r y h i g h p r e s s u r e and temperature were reached as the d e t o n a t i o n converged a t the f r o n t e n d 2 9 . The d r i v e r s e c t i o n and the shock tube was i n i t i a l l y s e p a r a t e d by a 40 micron t h i c k a c e t a t e diaphragm p r i o r t o F i g . 3.3 An x - t diagram of the shock wave i n the shock t u b e . PRESSURE v r UrO x=0 c.s. g. 3.4 The p r e s s u r e d i s t r i b u t i o n i n the shock tube. 34 the i g n i t i o n . S h o r t l y a f t e r t h e d e t o n a t i o n wave was r e f l e c t e d from the diaphragm, the diaphragm b u r s t and a shock wave was produced. F i g u r e 3.3 and 3.4 show r e s p e c t i v e l y an i d e a l x - t d i a g r a m and the c o r r e s p o n d i n g p r e s s u r e d i s t r i b u t i o n of the shock tube system a t some time t ' a f t e r the l a u n c h i n g . In p ' r a c t i c e , the shock and c o n t a c t s u r f a c e may not m a i n t a i n c o n s t a n t speeds, thus they are not n e c e s s a r y s t r a i g h t l i n e s as showing i n the x - t diagram. O x y - a c e t y l e n e was chosen as the e x p l o s i v e gas m i x t u r e . The r a t i o of 2 p a r t s oxygen t o 1 p a r t a c e t y l e n e was found t o g i v e a n i c e c l e a n d e t o n a t i o n . Too much a c e t y l e n e would l e a v e carbon r e s i d u e i n the chamber and would a l s o darken the shock t u b e . The o x y - a c e t y l e n e was premixed i n the m i x i n g tank f o r over h a l f an hour b e f o r e used. P r i o r t o SCR PULSER 6 0 0 V trigger pulse trigger — spark tT-> • ignition spark 4mm dump switch 25 KV l.72>4f F i g . 3.5 C i r c u i t diagram of the i g n i t i o n s p a r k . 35 each i g n i t i o n , the shock tube was f i l l e d w i t h argon gas a t a p r e s s u r e of 5 T o r r . The argon was 99.996% pure coming out of the gas b o t t l e , but no e f f o r t was made t o keep the system i m p u r i t y f r e e f o r i t was b e l i e v e d t h a t a s m a l l amount of i m p u r i t i e s might enhence the r a t e of i o n i z a t i o n r e l a x t i o n w i t h o u t a f f e c t i n g the f i n a l e q u i l i b r i u m 3 0 . The c i r c u i t diagram of the i g n i t i o n spark i s shown i n f i g u r e 3.5. The spark gap d i s t a n c e was s e t a t 4mm and the c a p a c i t o r bank was charged up t o 10 t o I3kv. The spark o n l y s e r v e d t o s t a r t the i g n i t i o n , the a c t u a l gap w i d t h and d i s c h a r g e v o l t a g e were found t o have no e f f e c t on any e x p e r i m e n t a l r e s u l t . 3.3 C a l i b r a t i o n of the shock wave speed The speed of the f r e e r u n n i n g shock wave produced i n the shock tube i s an i m p o r t a n t parameter because i t d e t e r m i n e s the t e s t f l o w c o n d i t i o n s . Three s e p a r a t e i n s t r u m e n t s have been used t o d e t e c t the speed: smear camera, f r a m i n g camera and p r e s s u r e probes. A l t h o u g h the p r e s s u r e probe method was found t o g i v e the best a c c u r a c y , the o t h e r two methods were s t i l l v a l u a b l e i n terms of showing the s t r u c t u r e s of the t e s t f l o w . F i g u r e s 3.6 and 3.7 show t y p i c a l photographs o b t a i n e d from the smear camera and the f r a m i n g camera r e s p e c t i v e l y . The smear photograph i s s i m p l y a s t r e a k r e c o r d i n g of the l u m i n o s i t y of a h o r i z o n t a l s l i t a l o n g the shock tube ( w i t h time i n c r e a s i n g towards t o p of the page). I t c l o s e l y 36 A T I M E — *- x F i g . 3.6 A smear camera photograph f o r the shock speed measurement. (shock speed i s 3.4km/sec, d i s t a n c e between markers i s 5cm.) resembles the x - t diagram i n f i g u r e 3.3. The bottom edge of the l i g h t band i s the t r a c e of the shock f r o n t moving t o the r i g h t s i d e . The s l o p e of t h i s l i n e c o r r e s p o n d s t o the shock speed a t any p o s i t i o n ; t h u s the s t r a i g h t n e s s of t h i s l i n e i m p l i e s t h a t the shock wave has n e g l i g i b l e a t t e n u a t i o n w h i l e t r a v e l l i n g over t h i s p o r t i o n of the shock tube. The f r a m i n g camera photographs d i s p l a y s a sequence of 5 p i c t u r e s t aken at p r e - s p e c i f i e d time i n t e r v a l s (time i n c r e a s e s upward i n the f i g u r e ) . A g a i n the shock f r o n t appears as a s h a r p and b r i g h t " f r o n t " moving t o the r i g h t . 37 Both the smear and f r a m i n g photographs showed a "dark space" l o c a t e d i m m e d i a t e l y behind the shock f r o n t . We b e l i e v e d t h i s dark space was due t o the r a d i a t i o n a b s o r p t i o n d u r i n g t h e i o n i z a t i o n r e l a x a t i o n p e r i o d . The shock f r o n t was b r i g h t e r because of the h i g h e r t e m p e r a t u r e T' (see f i g u r e 2.8) a t the compression shock. From the l e n g t h of the luminous r e g i o n f o l l o w i n g the shock f r o n t ( a p p r o x i m a t e l y > x F i g . 3.7 A f r a m i n g camera photograph f o r the shock speed measurement. (shock speed i s 3.4km/sec and i n t e r - f r a m e time i n t e r v a l i s 5 m i c r o - s e c . ) 38 60mm), the t e s t f l o w time was det e r m i n e d t o be about 30 mic r o - s e c o n d l o n g . F i n d i n g the shock speed w i t h any camera r e q u i r e s a d i s t a n c e measurement on the photograph. Measurements of t h i s k i n d were t y p i c a l l y l i m i t e d t o 5% a c c u r a c y . The p r e s s u r e probe method, on the o t h e r hand, d i d much b e t t e r than t h a t . By d e t e c t i n g the time i n t e r v a l between the a r r i v a l of the compre s s i o n shock a t two p r e s s u r e probes s i t u a t e d a t an a c c u r a t e l y measured d i s t a n c e a p a r t , the shock speed c o u l d be measured up t o 2% or b e t t e r a c c u r a c y . A t y p i c a l p r e s s u r e probe s i g n a l i s p r e s e n t e d i n f i g u r e 3.8. The p r e s s u r e probes were LD-80 p i e z o - e l e c t r i c p r e s s u r e t r a n s d u c e r s made by C e l e s c o . We were unable t o e l i m i n a t e the 0.14 MHz r i n g i n g , s u pposedly due t o the n a t u r a l response F i g . 3.8 A t y p i c a l o s c i l l o g r a m from the p r e s s u r e probe measurement. ( s c a l e : 1 v o l t , 10 m i c r o - s e c per d i v . ) 39 of the c r y s t a l , p r e s e n t i n the p r e s s u r e probe s i g n a l s . The time i n t e r v a l measurement was made by an e l e c t r o n i c time c o u n t e r which was a c c u r a t e t o w i t h i n a m i c r o - s e c o n d . In acc o r d a n c e w i t h the camera measurements, the p r e s s u r e s i g n a l p r o f i l e s a l s o i n d i c a t e d an a c c e p t a b l e t e s t time of a p p r o x i m a t e l y 30 m i c r o - s e c o n d s . B e f o r e i g n i t i o n , the shock tube was always f i l l e d w i t h 5 T o r r of argon a t room t e m p e r a t u r e . The p r e s s u r e of the o x y - a c e t y l e n e m i x t u r e i n the d e t o n a t i o n chamber c o n t r o l l e d the speed of the r u n n i n g shock wave. C a l i b r a t i o n of the shock speed a g a i n s t the o x y - a c e t y l e n e f i l l i n g p r e s s u r e , done by measuring the speed w i t h the p r e s s u r e probe method, a r e d e p i c t e d i n f i g u r e 3.9. We o b s e r v e d t h a t w i t h i n our range of o p e r a t i o n (180 T o r r t o 250 T o r r o x y - a c e t y l e n e ) , the shock speed i n c r e a s e d almost l i n e a r l y . F u r t h e r m o r e , the r e p r o d u c i b i l i t y of shock waves a t the same speed was e x c e l l e n t . 3.4 The t e s t f l o w p r o p e r t i e s and t h e i r d i a g n o s t i c s From now on, we w i l l use s u b s c r i p t s 0, 1 and 2 t o i n d i c a t e the f l o w r e g i o n s i n f r o n t of the r u n n i n g shock, b e h i n d the r u n n i n g shock (the t e s t f l o w ) and b e h i n d the s t a n d i n g bow shock r e s p e c t i v e l y . The c o n d i t i o n s of r e g i o n 0 were f i x e d f o r a l l the e x p e r i m e n t s — a t 5 T o r r of argon and at room t e m p e r a t u r e . The f l o w p arameters i n r e g i o n 1 can be computed (assuming i d e a l shock c o m p r e ssion) as f u n c t i o n s of the r u n n i n g shock Mach number a c c o r d i n g t o the p r o c e d u r e s 4 0 MACH NUMBER 11.2 I ARGON PR ESSURE= 5 ' "orr j, \ -f-OXY-ACET PRE /LENE SSURE )CH 1 — ' 1 — 1 7 ' 180 200 220 240 260 . 270 F i g . 3.9 Shock speed c a l i b r a t i o n a g a i n s t o x y - a c e t y l e n e f i l l i n g p r e s s u r e a t 5 T o r r argon. 41 d e s c r i b e d i n appendix A. In the c a l c u l a t i o n , r a d i a t i o n and heat l o s s e s were n e g l e c t e d w h i l e i o n i z a t i o n e f f e c t s were i n c l u d e d by means of t h e Saha e q u a t i o n . The r e s u l t s , i n the range of the Mach number t h a t concerned us (M10 t o M12), are graphed i n f i g u r e s 3.10 and 3.11. T y p i c a l v a l u e s of the f l o w parameters a r e T ^ I O " degree K, N e 1 = 2x 1 0 2 2/m3, a,=2.5xl0" 2 and M,=1.5. Here M, i s the r a t i o of the t e s t f l o w v e l o c i t y (measured i n the l a b frame) t o the l o c a l sound v e l o c i t y w i t h i n the t e s t f l o w i n r e g i o n 1; M, i s a l s o the f l o w Mach number upstream of the s t a n d i n g shock. Our c o m p u t a t i o n r e s u l t s a g r e e d , t o w i t h i n 10%, t o those p u b l i s h e d by N e t t " 2 and a l s o by Horton and M e n a r d 2 8 which were e x p e r i m e n t a l l y v e r i f i e d on a c o n v e n t i o n a l shock t u b e . In o r d e r t o ensure t h a t the computed v a l u e s were r e l i a b l e t o d e s c r i b e the t e s t f l o w p r o p e r t i e s , d i a g n o s t i c measurements were performed t o c o n f i r m the f l o w p r o p e r t i e s . S i n c e the e l e c t r o n d e n s i t y and t e m p e r a t u r e were d i r e c t l y i n v o l v e d i n the p o t e n t i a l c a l c u l a t i o n i n e q u a t i o n ( 2 . 9 ) , t h e s e two parameters were measured. The s p e c t r o s c o p i c methods of l i n e b r o a d e n i n g was used t o measure the e l e c t r o n d e n s i t y . Other o p t i c a l methods commonly used i n shock tube r e s e a r c h such as i n t e r f e r o m e t r y , s c h l i e r e n or s h a d o w g r a p h 2 3 were not a p p l i c a b l e t o our t e s t f l o w c o n d i t i o n s , because i n t h i s t e mperature range the i o n i z a t i o n e f f e c t on the r e f r a c t i v e index of the plasma i s c o u n t e r -b a l a n c e d by those due t o the d e n s i t y i n c r e a s e . The e l e c t r o n t e m p e r a t u r e was measured by the r e l a t i v e l i n e i n t e n s i t y 42 TEMP.'K 11000 i Ne 10000 9000-8000. / ^y^ / / / - > i y y y ^y"^ yy"^ "y y y y y y y y S y ^y -"Ne, y^ y yy y y RUNNING SHOCK MACH NUMBER 10.0 10.5 11.0 x10 / m 3.0 2.0 1.0 11.5 12.0 F i g . 3.10 E l e c t r o n temperature and number d e n s i t y b e h i n d the ru n n i n g argon shock wave. a,x10 4.0 | Mi 3 0 2.0 ro y/ _ , -RUNNING NUMBER 10.0 10.5 20 •1.75 1.5 •1.25 •no 11.5 120 F i g . 3.11 Degree of i o n i z a t i o n and t e s t f l o w Mach number behi n d the r u n n i n g argon shock wave. 43 r a t i o method. Both s p e c t r o s c o p i c methods were independent of 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 and the a c t u a l t r a n s p a r e n c y of the shock tube g l a s s w a l l . For the t e s t flow plasma, the l i n e b r o a d e n i n g mechanisms a r e c o m p l e t e l y dominated by Coulomb i n t e r -a c t i o n s — t h e S t a r k e f f e c t . S t a r k b r o a d e n i n g g i v e s a L o r e n t z i a n l i n e shape. The f u l l w i d t h a t h a l f maximum (FWHM) of the s p e c t r a l l i n e i s o n l y weakly dependent on the t e m p e r a t u r e but h e a v i l y depends on the e l e c t r o n d e n s i t y . A c c o r d i n g t o G r i e m 2 5 , the h a l f - w i d t h i s a p p r o x i m a t e l y g i v e n as AX = [ l + 5 . 5 3 x l O " 6 N ^ a ( l - 6 . 8 x l O " 3 N ^ T " l s ) ] x 2 X 1 0 " 2 2 t o N (3.1) 6 e e e > where a and io are the S t a r k b r o a d e n i n g parameters t a b u l a t e d i n Griem's t e x t (be c a r e f u l not t o mix up t h i s a w i t h the degree of i o n i z a t i o n ) . E q u a t i o n (3.1) has AX e x p r e s s e d i n ft, N e i n m"3 and T e i n °K. Take note t h a t the f o r e g o i n g e q u a t i o n i s v a l i d r e g a r d l e s s of whether the plasma i s i n l o c a l t h e r m a l e q u i l i b r i u m (LTE) or n o t . At our plasma t e m p e r a t u r e , both the Doppler b r o a d e n i n g l i n e w i d t h and the n a t u r a l l i n e w i d t h a r e s e v e r a l o r d e r s of magnitude s m a l l e r than the S t a r k l i n e w i d t h , hence they need not be c o n s i d e r e d . S t r o n g e r l i n e s are p r e f e r r e d i n the d i a g n o s t i c . The r e a b s o r p t i o n of s t r o n g l i n e s i n the plasma can a f f e c t the f i n a l l i n e shape or a l t e r the r e l a t i v e i n t e n s i t y r a t i o of two l i n e s . For a l i n e h a v i n g w i d t h Av , c e n t e r e d on v , the 4 4 SHOCK TUBE J •WINDOW 5 mm LENS ENTRANCE SLITS .1mm 3/4 meter SPEX MONOCHROMATOR (CZERNY-TURNER TYPE) OMA F i g . 3.12 A schematic diagram of the s p e c t r o s c o p i c d i a g n o s t i c a p p a r a t u s . c o r r e c t i o n f o r r e a b s o r p t i o n can be w r i t t e n i n the f i r s t a p p r o x i m a t i o n a s 2 1 1 = 1 exp(-< 1) o v where 1 i s the l e n g t h of l i g h t p a t h t h rough the plasma and K v denotes the a b s o r p t i o n c o e f f i c i e n t which i s d e f i n e d as 45 Here m i s the e l e c t r o n mass, N j i s the p o p u l a t i o n of the lower s t a t e j , and f i s the a b s o r p t i o n o s c i l l a t o r s t r e n g t h . For a l l the argon l i n e s which have been used i n the ex p e r i m e n t , < v l was much l e s s than u n i t y . Thus r e a b s o r p t i o n was n e g l i g i b l e and the plasma i s c a l l e d o p t i c a l l y " t h i n " . F i g u r e 3.12 c o n t a i n s a sch e m a t i c diagram of the arrangement used t o measure the S t a r k b r o a d e n i n g of the A r l 6032.13^ l i n e . The 3/4 meter Spex monochromator r e s o l v e d the spectrum onto 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 s e r (OMA) which r e c o r d e d the spectrum i n t o 500 s e p a r a t e c h a n n e l s . S p e c t r a l l i n e s of l e s s than 1A can be obser v e d by t h i s method. Showing i n f i g u r e 3.13 i s a t y p i c a l p r i n t - o u t o b t a i n e d from the OMA. The A r l 6032.13 l i n e shown i n the f i g u r e had a h a l f - w i d t h of a p p r o x i m a t e l y 5A which c o r r e s p o n d t o 18 c h a n n e l s i n the OMA. E l e c t r o n d e n s i t i e s o b t a i n e d from the r e s u l t s of t h e s e measurements were compared w i t h the c o m p u t a t i o n a l p r e d i c t i o n s p r e v i o u s l y shown i n f i g u r e 3.10. As shown i n f i g u r e 3.14, the agreement between the d a t a and the c u r v e was q u i t e good. By assuming LTE, e l e c t r o n t emperature can be r e l a t e d t o the r e l a t i v e i n t e n s i t y r a t i o , I / I ' , of two s p e c t r a l l i n e s of the same element a c c o r d i n g t o the f o l l o w i n g e q u a t i o n 2 5 : 46 Ar l 6032.13 at Mach 11.15 P0=5torr FWHM 10&(36chan.) 50 COUNTS F i g . 3.13 S t a r k broadened l i n e shape of A r l 6032.13. where E, A, g, and f are r e s p e c t i v e l y the e x c i t a t i o n energy l e v e l , the w a v e l e n g t h , the s t a t i s t i c a l weight and the o s c i l l a t o r s t r e n g t h . T a b l e s of g and f can be found i n the l i t e r a t u r e 5 * . S i n c e we a r e not o n l y i n t e r e s t e d i n the magnitude of the l i n e i n t e n s i t y r a t i o but a l s o i n t e r e s t e d i n the magnitude of the change of t h i s r a t i o over the t e m p e r a t u r e range, i n t e n s i t y r a t i o of l i n e s from d i f f e r e n t i o n i z a t i o n l e v e l s would be more s e n s i t i v e . U n f o r t u n a t e l y , the t e s t f l o w plasma was not hot enough t o emit A r i l l i n e s . Amongst the a v a i l a b l e A r l l i n e s , f o r which we can f i n d the c o r r e s p o n d i n g g and f v a l u e s , the r a t i o of A r I 6 4 1 6 . 3 l A t o 47 F i g . 3.14 E x p e r i m e n t a l r e s u l t s of e l e c t r o n d e n s i t y measurement by, the S t a r k b r o a d e n i n g method. ArI6965.43A ) was our b e s t s e l e c t i o n . F i g u r e 3.15 d i s p l a y s the t h e o r e t i c a l v a l u e of the l i n e r a t i o as a f u n c t i o n of T e. I t ought t o be mentioned t h a t the t a b u l a t e d g and f v a l u e s were s a i d t o have poor a c c u r a c y ; the p r o d u c t gf has an u n c e r t a i n t y of about 25%. To do the e x p e r i m e n t , the OMA (see f i g u r e 3.12) was r e p l a c e d by a p h o t o - m u l t i p l i e r t o g e t h e r w i t h an 1 mm s l i t i n between the p h o t o - m u l t i p l i e r and the monochromator. The 48 p h o t o - m u l t i p l i e r used i n the experiment was a model C31034 GaAs p h o t o c a t h o d e , type 128, made by RCA. For each i n t e n s i t y r a t i o d a t a p o i n t , the shock tube had t o be i g n i t e d f o u r t i m e s : t w i c e f o r the l i n e s and t w i c e f o r the backgrounds. At Mach 10.7 and Mach 11.7, the i n t e n s i t y r a t i o s found were 0.11±0.05 and 0.12±0.03 r e s p e c t i v e l y . A c c o r d i n g t o the cu r v e i n f i g u r e 3.15, the s e d a t a i m p l i e d t h a t the e l e c t r o n t e m p e r a t u r e was almost t w i c e as much as the i d e a l shock c o m p u t a t i o n a l p r e d i c t i o n (as shown i n f i g u r e 3.10). Meanwhile, the 25% u n c e r t a i n t y of the measured I / I ' and the INTENSITY RATIO .14 .12 Mach 11.7 _S Mach 10.7 _ .10-.08 error limit / A r l 641631 i Arl 6965.43 | .06 ! .04 I .02 j I , i— , 1. 1—. :—I 6000 8000 10000 12000 14000 16000 18000 20000 LTE TEMPERATURE F i g . 3.15 R e l a t i v e i n t e n s i t y r a t i o of A r l 6416 t o A r l 6965 as a f u n c t i o n of the LTE te m p e r a t u r e . 4 9 25% u n c e r t a i n t y of the t a b u l a t e d gf v a l u e s can o n l y g i v e c r e d i t t o not more than 25% u n c e r t a i n t y of the v a l u e of T e. The o t h e r p o s s i b l e reason f o r the l a r g e d i s c r e p a n c y i s the inadequacy of e q u a t i o n ( 3 . 2 ) . The e l e c t r o n number d e n s i t y i n t h e t e s t f l o w ( a t - I 0 2 2 m " 3 ) was j u s t a t the edge of m eeting the LTE c r i t i c a l l i m i t 3 6 . A non-LTE plasma i s known t o g i v e e r r o n o u s r e s u l t s of T e measurement i n shock tube e x p e r i m e n t s 2 3. In summary, we found good agreement between the measured e l e c t r o n number d e n s i t y and the c a l c u l a t e d one. The e l e c t r o n temperature measurement d i d not agree so w e l l , we b e l i e v e d t h a t the disagreement was due t o the i n c o r r e c t a s sumption of LTE f o r e q u a t i o n ( 3 . 2 ) . F i n a l l y , we c o n c l u d e d t h a t the f l o w p r o p e r t i e s p r e d i c t e d by t h e i d e a l shock c o m p u t a t i o n were g e n e r a l l y c o r r e c t . These f l o w p r o p e r t i e s were s u b s e q u e n t l y used t o i n t e r p r e t the r e s u l t s of the s t a n d i n g shock g e n e r a t o r e x p e r i m e n t . 3.5 The t e s t s e c t i o n and the e l e c t r i c a l measurements The t e s t s e c t i o n was l o c a t e d a t a p p r o x i m a t e l y 1 meter away from the shock tube diaphragm. As i l l u s t r a t e d i n f i g u r e 3.16, i t c o m p r i s e d o n l y a m e t a l r i n g and a s o l i d cone. The i n n e r d i a m e t e r of the r i n g was f l u s h w i t h the shock tube w a l l so as t o m i n i m i z e the t e s t f l o w d i s t u r b a n c e . The r i n g was 1cm wide. The cones used i n our experiment were a l l machined out of 1/2 i n c h d i ameter m e t a l r o d s . The cone was mounted a l o n g the c e n t e r a x i s of the shock tube on trigger signal Tektronic 7 7 0 4 oscilloscope photo-di cone (anode) -bow shock •^ clomps "-ring (cathode) I cm. 15 cm W W W 1 7 cm J DUMP TANK y .2cm EXTENSION ROD T -lucite jacket ^spacer PUMP F i e . 3 . 1 6 The t e s t s e c t i o n , cn o 51 the t i p of an e x t e n s i o n metal r o d ( b r a s s ) p r o t r u d i n g from the back end of the dump ta n k . C u r r e n t can be conducted from the cone s u r f a c e t o the o u t s i d e a l o n g the e x t e n s i o n r o d . The extenson r o d and a l s o the back p a r t of the cone were i n s u l a t e d from the shock plasma by a l u c i t e j a c k e t . By a d d i n g s p a c e r s t o the end of the e x t e n s i o n r o d , the r e l a t i v e p o s i t i o n of the cone t o the r i n g c o u l d be a d j u s t e d . The e l e c t r i c a l c i r c u i t f o r measuring the s t a n d i n g shock g e n e r a t o r l o a d i n g c h a r a c t e r i s t i c i s a l s o i n c l u d e d i n f i g u r e 3.16. N o r m a l l y p o s i t i v e c u r r e n t f l o w e d from t h e cone (anode) t h r o u g h a r e s i s t o r and then t o the r i n g ( c a t h o d e ) . The v o l t a g e drop a c r o s s the r e s i s t o r was measured by a d i f f e r e n t i a l a m p l i f i e r i n an o s c i l l o s c o p e . The d i f f e r e n t i a l a m p l i f i e r a v o i d e d any ground l o o p c u r r e n t g o i n g t h r o u g h the shock plasma i n t o the i g n i t i o n ground. The o s c i l l o s c o p e was t r i g g e r e d by the s i g n a l o b t a i n e d from a ph o t o - d i o d e p l a c e d a t 4cm i n f r o n t of the t e s t s e c t i o n . The e l e c t r i c a l measurement i n v o l v e d d e t e r m i n i n g l o a d l i n e s f o r the s t a n d i n g shock g e n e r a t o r a t v a r i o u s t e s t f l o w c o n d i t i o n s . Mach number of the f r e e r u n n i n g shock was r a n g i n g from Mach 10.3 t o Mach 12.0 produced by u s i n g 180 T o r r t o 270 T o r r of o x y - a c e t y l e n e i n the d r i v e r and 5 T o r r of argon i n the shock t u b e . Cones w i t h h a l f - a n g l e (e) a t 15°, 20°, 25°, 30° and 45° were t e s t e d . In most of the e x p e r i m e n t s , the cone and the r i n g were both made of s t e e l , but o t h e r c o m b i n a t i o n s of m a t e r i a l s such as aluminum and b r a s s were a l s o used t o observe the e f f e c t s of d i f f e r e n t F i g . 3.17 A f r a m i n g camera (tube d i a m e t e r i s 1 i n c h p o s i t i o n s of r u n n i n g photograph of the c o n i c a l shock, and cone h a l f - a n g l e i s 30°, the shock are marked by a r r o w s . ) 53 e l e c t r o d e m a t e r i a l work f u n c t i o n s on the l o a d i n g c h a r a c t e r i s t i c . Due t o the r a t h e r s h o r t t e s t f l o w d u r a t i o n , i t was not p o s s i b l e t o o b t a i n the e n t i r e I-V p l o t of a l o a d l i n e w i t h i n a s i n g l e shock tube i g n i t i o n e.g. by r a p i d l y sweeping the v a l u e of the l o a d i n g r e s i s t a n c e . Many runs were r e q u i r e d t o determine one l o a d l i n e ; i n each run a d i s c r e t e l o a d i n g r e s i s t o r , r a n g i n g from 0.5 ohm t o 100K ohm, measured t o b e t t e r than 1% a c c u r a c y , was used. The e l e c t r o d e s u r f a c e c o n t a m i n a t i o n was t r o u b l e s o m e . D e p o s i t s of the d e t o n a t i o n p r o d u c t s (from the o x y - a c e t y l e n e and the e v a p o r a t e d diaphram m a t e r i a l ) were d i s c o v e r e d on the e l e c t r o d e s u r f a c e a f t e r each r u n . S i n c e t h e s e c o n t a m i n a t i o n s had been found t o a f f e c t the e l e c t r i c a l measurements, both e l e c t r o d e s had t o be c l e a n e d f o r each new run. The d e p o s i t s c o u l d be sanded o f f w i t h a grade 600 sandpaper, but f i r s t the t e s t s e c t i o n had t o be d i s m a n t l e d i n o r d e r t o get a t the e l e c t r o d e s u r f a c e s . The c l e a n i n g time and the pumping time t o g e t h e r r e q u i r e d a t l e a s t 30 minutes f o r each " i g n i t i o n c y c l e " . S i n c e many i g n i t i o n s were needed t o determine one l o a d l i n e , and many l o a d l i n e s were needed f o r each c o m b i n a t i o n of Mach number and cone a n g l e , the e l e c t r i c a l measurements were very time consuming. 3.6 The c o n i c a l s t a n d i n g shocks As soon as the t e s t f l o w a t t a c k e d the cone a t the t e s t s e c t i o n , a c o n i c a l s t a n d i n g shock was formed. F i g u r e 3.17 i s a t y p i c a l f r a m i n g camera photograph of the s t a n d i n g 54 TEMP. RATIO 1.201 Ne RATIO 1.16J 1.12H 1.08H 1.04-f 1 . 0 0 0.0 10.0 10.4 10.8 11.2 11.6 RUNNING SHOCK MACH NUMBER 12.0 F i g . 3.18 E l e c t r o n d e n s i t y and temperature r a t i o s a c r o s s the s t a n d i n g shock as a f u n c t i o n of the f r e e r u n n i n g shock Mach number and cone h a l f - a n g l e shock. For two r e a s o n s , no attempt was made t o compare the p r e d i c t e d s t a n d i n g shock a n g l e (see f i g u r e 2.7) w i t h the one seen on the photograph. F i r s t , the c y l i n d r i c a l shock tube w a l l t h i c k n e s s had o p t i c a l l y deformed the s t a n d i n g shock 55 DENSITY RATIO BOW SHOCK ANGLE e 7 0 10.0 10.4 10.8 11.2 11.6 RUNNING SHOCK MACH NUMBER 12.0 F i g . 3.19 D e n s i t y r a t i o P 2 / P I , and s t a n d i n g shock h a l f -a n g l e a, as f u n c t i o n s of the f r e e r u n n i n g shock Mach number and cone h a l f - a n g l e image when v i e w i n g from o u t s i d e . Second, the obse r v e d luminous f r o n t might not c o i n c i d e w i t h the a c t u a l c o m p r e s s i o n shock f r o n t . N e v e r t h e l e s s , the photographs were u s e f u l i n c o n f i r m i n g the t e s t f l o w d u r a t i o n . T h i s d u r a t i o n 56 was a l s o found t o be a p p r o x i m a t e l y 30 m i c r o - s e c o n d i n agreement w i t h the r e s u l t s o b t a i n e d through o t h e r means. The t h e o r y of c o n i c a l shocks has been d i s c u s s e d i n c h a p t e r 2 . The t e s t f l o w p r o p e r t i e s were adopted t o c a l c u l a t e the f l o w p r o p e r t i e s b e h i n d the c o n i c a l shock. U s i n g the r u n n i n g shock Mach number as the f r e e parameter, the computer program c a l c u l a t e d f i r s t the t e s t f l o w then the s t a n d i n g shock. R e s u l t s of the com p u t a t i o n such as the s t a n d i n g shock a n g l e , the e l e c t r o n d e n s i t y r a t i o and the tem p e r a t u r e r a t i o a c r o s s the s t a n d i n g shock as f u n c t i o n s of the r u n n i n g shock Mach number, M,, and the cone a n g l e , e, ar e shown i n f i g u r e s 3 . 1 8 and 3.19. These v a l u e s w i l l become u s e f u l i n p r e d i c t i n g the e.m.f. of the s t a n d i n g shock g e n e r a t o r . 57 CHAPTER 4. EXPERIMENTAL RESULTS AND INTERPRETATIONS 4.1 V o l t a g e measurements of the s t a n d i n g shock g e n e r a t o r The v o l t a g e a c r o s s the s t a n d i n g shock g e n e r a t o r e l e c t r o d e s (or e q u i v a l e n t l y a c r o s s the l o a d i n g r e s i s t o r R L) was measured by a d i f f e r e n t i a l a m p l i f i e r i n an o s c i l l o s c o p e . The shape of the v o l t a g e p r o f i l e v a r i e d c o n s i d e r a b l y depending on the v a l u e s of R L, cone a n g l e , e, and the r u n n i n g shock Mach number, M,. F i g u r e 4.1 c o n t a i n s a few t y p i c a l o s c i l l o g r a m s of the v o l t a g e measurements. By and l a r g e , l e s s f l u c t a t i o n on the s i g n a l was found f o r lower R L w h i l e l a r g e r cone a n g l e s c a l l e d f o r peaks t o o c c u r i n the s i g n a l p r o f i l e s . R i s e t i m e s of the s i g n a l s were l o n g e r f o r c a s e s of s m a l l e r e or s m a l l e r M,. In t h e s e measurements we adopted the c o n v e n t i o n t h a t a p o s i t i v e v o l t a g e i s o b t a i n e d from a p o s i t i v e c u r r e n t f l o w i n g from the cone (anode) t o the r i n g (cathode) through the e x t e r n a l l o a d . A s m a l l n e g a t i v e s h o r t p u l s e was always found ahead of the r i s i n g edge of the s i g n a l . T h i s p u l s e was b e l i e v e d t o be caused by the n e g a t i v e p o t e n t i a l i n f r o n t of the r u n n i n g shock wave. As i l l u s t r a t e d i n f i g u r e 4.2, the d i r e c t i o n of i n c r e a s i n g p o t e n t i a l of the r u n n i n g shock i s o p p o s i t e t o t h a t of the s t a n d i n g shock. Hence b e f o r e the r u n n i n g shock 58 VOLT 0.5 V/div. 0.5V/div. 0.5V/div. 0.5V/div. MACH 10.3 15°cone RL=100ja t.lOjjs/div. MACH 10.3 25° cone R L =100i l t ,10^s/div. MACH 10.3 30° cone l\=100il -> t.10A5/div MACH 10.3 45°cone R L=100_a - + J MM/] A MAH ^ t ,10^s/div . F i g . 4.1 T y p i c a l examples of o s c i l l o g r a m s o b t a i n e d i n the v o l t a g e measurement. 59 wave impinged on the cone, the e l e c t r o d e s were e x p e c t e d t o d e t e c t a n e g a t i v e s i g n a l . I t was n e c e s s a r y t o d e t e r m i n e a d e f i n i t e v o l t a g e v a l u e f o r each v o l t a g e p r o f i l e r e c o r d e d such t h a t the d a t a can be p l o t t e d i n t o a c u r r e n t - v o l t a g e p l a n e . The maximum v o l t a g e on the s i g n a l t r a c e w i t h i n the f i r s t 30 m i c r o - s e c o n d of t e s t f l o w d u r a t i o n was t a k e n . S m a l l f l u c t a t i o n s on the s i g n a l p r o f i l e s were averaged, but s i g n a l s which c o n t a i n e d v e r y s h a r p p e a k ( s ) were not e n t e r e d i n t o t h e I-V p l o t . About 1/5 of the d a t a was d i s c a r d e d t h i s way. F i g . 4.2 E l e c t r i c p o t e n t i a l p o l a r i t i e s of the f r e e r u n n i n g shock and the s t a n d i n g shock. 60 4.2 The l o a d l i n e s A f t e r d e t e r m i n i n g the v o l t a g e , the amount of c u r r e n t f l o w i n g out of the g e n e r a t o r c o u l d be c a l c u l a t e d by u s i n g the o r d i n a r y Ohm's law, I=V/R L, where the v a l u e of R L was a c c u r a t e l y known. F i g u r e 4.3 shows a t y p i c a l p l o t t i n g of the l o a d l i n e . The s t r a i g h t l i n e s f a n n i n g out from the o r i g i n a r e l i n e s of c o n s t a n t r e s i s t a n c e ; the s l o p e of each l i n e c o r r e s p o n d e s t o a r e s i s t a n c e v a l u e of R L. The e r r o r bar of a v o l t a g e d a t a t a k e n a t a s p e c i f i c R L c o u l d be p l o t t e d a l o n g the c o r r e s p o n d i n g c o n s t a n t r e s i s t a n c e l i n e . T h i s method of p l o t t i n g t h e d a t a e l i m i n a t e d the t r o u b l e of VOLTAGE (V.) CURRENT (AMP.) F i g . 4.3 P l o t t i n g a l o a d l i n e on the c u r r e n t - v o l t a g e p l a n e . 61 f i n d i n g e r r o r bars f o r the c u r r e n t . A l l our e x p e r i m e n t a l d a t a on the c u r r e n t - v o l t a g e measurement c o u l d be f i t t e d by s t r a i g h t l o a d l i n e s . In t h i s a p p r o x i m a t i o n , the s t a n d i n g shock g e n e r a t o r i s d e s c r i b e d by the v o l t a g e i n t e r c e p t and ' the s l o p e of the l o a d l i n e s . These two parameters are e q u i v a l e n t t o the e.m.f. and the i n t e r n a l r e s i s t a n c e of the g e n e r a t o r . 4.3 R e p r o d u c i b i l i t y of measurements We had e x p e r i e n c e d some d i f f i c u l t y i n the e a r l i e r days of the expe r i m e n t : the v o l t a g e measurements were not r e p r o d u c i b l e . M e asuring the o p e n - c i r c u i t v o l t a g e between the e l e c t r o d e s a t v a r i o u s Mach numbers and cone a n g l e s was the f i r s t t h i n g we d i d i n the expe r i m e n t . I t was found a t t h a t time t h a t the o p e n - c i r c u i t v o l t a g e o b t a i n e d would be d i f f e r e n t (beyond the l i m i t of experiment e r r o r ) on a day t o day b a s i s , d e s p i t e our e f f o r t t o keep a l l c o n t r o l l a b l e parameters c o n s t a n t . In a d d i t i o n , we l a t e r d i s c o v e r e d t h a t i n p l o t t i n g the l o a d l i n e , the d a t a would t e n d t o f o l l o w one l i n e i n a c e r t a i n day and then formed another l i n e i n the next day. S u r p r i s i n g l y , the l i n e s were n e a r l y p a r a l l e l ; o n l y the o p e n - c i r c u i t v o l t a g e s were s h i f t i n g . I t took us a l i t t l e w h i l e b e f o r e we found the anomalous b e h a v i o u r was due t o s u r f a c e e f f e c t s . Every time the e l e c t r o d e s u r f a c e was sanded d u r i n g c l e a n i n g , the o l d monatomic s u r f a c e l a y e r was r e p l a c e d by a new one. O x i d a t i o n and gas a b s o r p t i o n on t h i s new s u r f a c e depended 62 BOW SHOCK POTENTIAL (Vsh) 2.0 1.5 1.0 0.5 VOLT --__4tf___ 35* 30' - 8=25° 20° 15° 0 10 10.4 10.8 11.2 11.6 RUNNING SHOCK MACH NUMBER F i g . 4.4 C a l c u l a t e d e.m.f. of the g e n e r a t o r a t v a r i o u s cone h a l f - a n g l e and r u n n i n g shock Mach number. E.M.F. (Volt) • • A • • A A o A O X O X X • e=30* A 25' O 20° X 15° error bars typically ±0.07V 10.0 10.4 10.8 11.2 11.6 12.0 RUNNING SHOCK MACH NUMBER F i g . 4.5 Measured e.m.f. of the g e n e r a t o r a t v a r i o u s cone h a l f - a n g l e and r u n n i n g shock Mach number. 63 h e a v i l y on the a i r c o m p o s i t i o n and the exposure t i m e . M o i s t u r e i n the a i r , i n f a c t , c o u l d be a v e r y i m p o r t a n t parameter which had been n e g l e c t e d by us. These parameters were the hidden causes f o r the i r r e p r o d u c i b i l i t y . I t would not be easy t o c o n t r o l the a i r m o i s t u r e , but i t was. p o s s i b l e t o p r o v i d e p l e n t y of water by p u r p o s e l y m o i s t e n i n g the new s u r f a c e s f o r a few moments b e f o r e the e l e c t r o d e s were d r i e d and r e s t o r e d i n t o the shock tube. Our r e s u l t s became r e p r o d u c i b l e a f t e r t h i s p r o c e d u r e was adopted. 4.4 Comparison of the o p e n - c i r c u i t v o l t a g e measurement w i t h  the s t a n d i n g shock g e n e r a t o r e.m.f. The e.m.f. of the s t a n d i n g shock g e n e r a t o r can be c a l c u l a t e d a c c o r d i n g t o e q u a t i o n ( 2 . 9 ) w i t h the f l o w parameters g i v e n i n f i g u r e s 3.18 and 3 . 1 9 . The r e s u l t of t h i s c a l c u l a t i o n i s shown i n f i g u r e 4.4. C l e a r l y , the e.m.f. i s shown t o be o n l y weakly dependent on the r u n n i n g shock Mach number; i t i s d e t e r m i n e d m o s t l y by the h a l f - a n g l e of the cone. The e x p e r i m e n t a l measurement agreed w i t h t h i s p r e d i c t i o n . As shown i n f i g u r e 4.5, the open c i r c u i t v o l t a g e of the s t a n d i n g shock g e n e r a t o r was found t o be o n l y s e n s i t i v e t o the cone a n g l e and independent of the r u n n i n g shock Mach number. F i g u r e 4.6 d i s p l a y s 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 v a l u e s of the s t a n d i n g shock g e n e r a t o r e.m.f. as f u n c t i o n s of the cone h a l f - a n g l e . The s o l i d c u r v e was c a l c u l a t e d i n accordance w i t h the assumption t h a t the i o n i z a t i o n 64 e q u i l i b r i u m was a c h i e v e d a t the cone s u r f a c e . In p r a c t i c e , the c o n i c a l s t a n d i n g shock was found t o be a t t a c h e d t o the t i p of the cone. Hence t h e r e may not be s u f f i c i e n t r e l a x a t i o n d i s t a n c e b e h i n d the c o n i c a l shock f o r the i o n i z a t i o n t o a c h i e v e e q u i l i b r i u m a t the cone s u r f a c e . T h i s means the e l e c t r o n d e n s i t y r a t i o i n e q u a t i o n (2.9) would have i t s v a l u e e q u a l t o the n e u t r a l gas compre s s i o n r a t i o . E.m.f. c a l c u l a t e d a c c o r d i n g t o t h i s " f r o z e n i o n i z a t i o n l i m i t " i s shown as the dashed c u r v e i n f i g u r e 4.6. The e x p e r i m e n t a l r e s u l t s f e l l between the t h e o r e t i c a l upper and lower bounds. In f a c t , the d a t a appeared t o be c o r r e l a t i n g w i t h the dash c u r v e i f the e n t i r e c u r v e c o u l d be r a i s e d by 2J0I 1C E.M.F. VOLT UPPE o LIMIT / PRED CTION / . / \ • \ y / X ^ ~y SLOWER ^ y PREDICT y' LIMIT ON CONE HALF-ANGLE o^  io5 io7 3f7 40° F i g . 4.6 Comparison of e x p e r i m e n t a l r e s u l t s t o the p r e d i c t e d e.m.f. 65 a p p r o x i m a t e l y 0.3 v o l t . One p o s s i b l e s o u r c e of t h i s u p s h i f t i n g v o l t a g e c o u l d be the d i f f e r e n c e of work f u n c t i o n s of the anode and the cathode. The e f f e c t s of work f u n c t i o n can be t e s t e d by u s i n g d i f f e r e n t c o m b i n a t i o n s of m a t e r i a l s f o r the e l e c t r o d e s . The r e s u l t s of t h i s s e r i e s of experiment are d i s c u s s e d i n s e c t i o n 4.7. 4.5 Comparison of the b u l k plasma r e s i s t a n c e t o the  s t a n d i n g shock g e n e r a t o r i n t e r n a l r e s i s t a n c e The s l o p e of the l o a d l i n e which i s e q u i v a l e n t t o the i n t e r n a l r e s i s t a n c e of the s t a n d i n g shock g e n e r a t o r was e x p e r i m e n t a l l y found t o be independent of the cone a n g l e i n use. The s l o p e s d e c r e a s e d as the r u n n i n g shock Mach number i n c r e a s e d . The e x p e r i m e n t a l r e s u l t s are d e p i c t e d i n f i g u r e 4.7, t h e i r v a l u e s were r a n g i n g from 0.5 ohm t o 1 ohm. In o r d e r t o compare the s e r e s u l t s w i t h the s t a n d i n g shock g e n e r a t o r model, we must c a l c u l a t e the b u l k plasma r e s i s t a n c e between the cathode and the anode. F o l l o w i n g e q u a t i o n (2.12) i n s e c t i o n 2.8, which was w r i t t e n as IJ c a , t h i s e q u a t i o n i s t o be i n t e g r a t e d a c c o r d i n g t o the s t a n d i n g shock g e n e r a t o r geometry. S i n c e the i n t e g r a t i o n i s f i e l d -dependent, the e l e c t r o d e geometry must be s i m p l i f i e d . I f the c o n i c a l anode i s r e p r e s e n t e d by a c y l i n d e r i c a l rod which has a p p r o x i m a t e l y the same s u r f a c e a r e a and the same w i d t h as the o r i g i n a l cone (see f i g u r e 4.8), an i n t e g r a b l e 66 LOAD LINE SLOPE 12 OB OA V . / A M P . i — B — i • — f t O. 10.0 10.5 11.0 U 5 RUNNING SHOCK MACH NUMBER 120 F i g . 4.7 E x p e r i m e n t a l r e s u l t s of the s t a n d i n g shock g e n e r a t o r i n t e r n a l r e s i s t a n c e . c y l i n d e r i c a l symmetry i s o b t a i n e d . E q u a t i o n (2.12) becomes r = ^-r J — d P U.1) 2uh J Po o* P , where h=lOmm i s the w i d t h of the r i n g and p 0=l2.5mm i s the d i a m e t e r of the shock t u b e . S t r i c t l y s p e a k i n g , t h e s t a n d i n g shock i s i n between the p a t h from the anode t o the cathode. Due t o h i g h e r e l e c t r o n d e n s i t y , the c o n d u c t i v i t y i s h i g h e r i n the plasma on the anode s i d e . However, i n o r d e r t o get an upper bound v a l u e of r , and t o s i m p l i f y the c a l c u l a t i o n , 67 Y h—> F i g . 4.8 The s i m p l i f i e d e l e c t r o d e c o n f i g u r a t i o n f o r the i n t e g r a t i o n of the b u l k plasma r e s i s t a n c e . we assumed the c o n d u c t i v i t y a t o be c o n s t a n t : t a k i n g i t s l owest p o s s i b l e v a l u e . A c c o r d i n g t o the d e f i n i t i o n of conduct i v i t y 2 1 N e e 2 ° = m \T , (4.2) e en where \F e n i s the average e l e c t r o n - a t o m c o l l i s i o n f r e q u e n c y , the c o n d u c t i v i t y f o r a Mach 10 s h o c k - i n d u c e d plasma i s 2344 mho/m. The r e s u l t of the i n t e g r a t i o n g i v e s an upper l i m i t of r i n the o r d e r of 0.007 ohm. T h i s i s two o r d e r s of magnitude s m a l l e r than the measured r e s i s t a n c e . Thus we c o n c l u d e t h a t the b u l k plasma r e s i s t a n c e was not the dominant component of the i n t e r n a l r e s i s t a n c e of the s t a n d i n g shock g e n e r a t o r t e s t e d i n our e x p e r i m e n t . There must be o t h e r e f f e c t ( s ) which we have so f a r o v e r l o o k e d i n 68 our s i m p l e s t a n d i n g shock g e n e r a t o r model. 4.6 E f f e c t s of the s e p a r a t i o n between e l e c t r o d e s The r e l a t i v e d i s t a n c e between cathode and anode c o u l d be v a r i e d by moving the cone f o r w a r d or backward a l o n g the shock tube a x i s . However no e v i d e n c e was found t h a t e i t h e r the o p e n - c i r c u i t v o l t a g e s or the s l o p e s of l o a d l i n e s were a f f e c t e d by the e l e c t r o d e s e p a r a t i o n . T h i s f i n d i n g f u r t h e r s u p p o r t s the f a c t t h a t the b u l k plasma r e s i s t a n c e does not account f o r the h i g h i n t e r n a l r e s i s t a n c e measured i n the e x p e r i m e n t . 4.7 E f f e c t s of t h e . e l e c t r o d e m a t e r i a l s So f a r a l l the d a t a which have been d i s c u s s e d were ta k e n w i t h a p a i r of s t e e l e l e c t r o d e s : a s t e e l cone anode and a s t e e l r i n g c athode. E l e c t r o d e s made of o t h e r m e t a l s such as b r a s s and aluminum were a l s o used f o r d e m o n s t r a t i n g the e f f e c t of e l e c t r o d e m a t e r i a l s on the s t a n d i n g shock g e n e r a t o r . The r e s u l t s of t h i s s e r i e s of experiment c l e a r l y i n d i c a t e d t h a t work f u n c t i o n s were s i g n i f i c a n t i n d e t e r m i n i n g the e.m.f. A f i x e d s e t of f l o w c o n d i t i o n s was chosen t o do the experiment namely a Mach 11.15 r u n n i n g shock and a 30° h a l f - a n g l e cone. We r e p e a t e d l y measured the l o a d l i n e w i t h d i f f e r e n t c o m b i n a t i o n s of anode and cathode m a t e r i a l s . I t was found t h a t the s l o p e v a r i a t i o n on the l o a d l i n e was n e g l i g i b l y s m a l l but the o p e n - c i r c u i t v o l t a g e responded markedly. Even n e g a t i v e v o l t a g e was measured i n 69 the case of aluminum anode and s t e e l c a t h o d e. The r e s u l t s of the o p e n - c i r c u i t v o l t a g e measurement a r e t a b u l a t e d i n t a b l e 4.1. Tab l e 4.1 O p e n - c i r c u i t V o l t a g e For D i f f e r e n t C o mbinations Of E l e c t r o d e M a t e r i a l s . Anode (cone) Cathode ( r i n g ) v m ( v o l t ) V -V t, •m * sh ( v o l t ) s t e e l aluminum b r a s s s t e e l aluminum b r a s s 0.9 ±0.05 0.75±0.1 0.8 ±0.1 s t e e l aluminum aluminum s t e e l 1.75±0.2 -0.25±0. 1 0.93 -1 .07 s t e e l b r a s s b r a s s s t e e l 0.8 ±0.15 0.8 ±0.2 -0.02 -0.02 I f we assumed t h a t the measured o p e n - c i r c u i t v o l t a g e i s composed of the s t a n d i n g shock p o t e n t i a l , V s^, the anode work f u n c t i o n w a, and the cathode work f u n c t i o n w c, then we i can w r i t e V = V , + w - w m sh c a • (4.3) T h i s r e l a t i o n i s i l l u s t r a t e d i n f i g u r e 4.9. In the case when the anode and the cathode were both of the same m a t e r i a l , the two work f u n c t i o n s e x a c t l y c a n c e l l e d and V m became e q u i v a l e n t t o V s h ; hence the measured v a l u e of V m would be independent of the p a r t i c u l a r k i n d of m a t e r i a l b e i n g used. The d a t a e v i d e n t l y a greed w i t h t h i s p r e d i c t i o n . In f a c t , t he average v a l u e of V s h was found t o be 0.82±0.1 70 v o l t . When the anode and cathode m a t e r i a l s a r e d i f f e r e n t , e q u a t i o n (4.3) can be used t o c a l c u l a t e the r e l a t i v e d i f f e r e n c e between the anode and cathode work f u n c t i o n s . L e t w a i , w s t and t o be the work f u n c t i o n s f o r aluminum, s t e e l and b r a s s r e s p e c t i v e l y . A c c o r d i n g t o the d a t a i n t a b l e 4.1, we f i n d w a l - » s t = • 0 .93 v o l t w s t " w a l = = -1 .07 v o l t w b r " w s t = = -o .02 v o l t w s t " w b r = = -o .02 v o l t From th e s e f i g u r e s , we can i n f e r t h a t the work f u n c t i o n of aluminum i s about 1 v o l t l a r g e r than t h a t of s t e e l , whereas s t e e l and b r a s s have work f u n c t i o n s n e a r l y e q u a l t o each sh Wc V = V.+W -W m F i g . 4.9 R e l a t i o n between the measured o p e n - c i r c u i t v o l t a g e , the shock p o t e n t i a l and the work f u n c t i o n s . 71 o t h e r . We d i d not compare these r e s u l t s w i t h the a c c e p t e d v a l u e s because work f u n c t i o n s f o r a l l o y s such as s t e e l and b r a s s were not r e a d i l y V a v a i l a b l e . 4.8 Summary and p r e l i m i n a r y c o n c l u s i o n s In summary, we have found t h a t the s t a n d i n g shock g e n e r a t o r has a l i n e a r l o a d i n g c h a r a c t e r i s t i c . The e.m.f. of the g e n e r a t o r i s g i v e n by the s t a n d i n g shock p o t e n t i a l p l u s o t h e r p o t e n t i a l s a t the e l e c t r o d e s u r f a c e such as the work f u n c t i o n s . The i n t e r n a l r e s i s t a n c e of the g e n e r a t o r d i d not c o r r e s p o n d t o the b u l k plasma r e s i s t a n c e ; the l a t t e r one i s s m a l l e r by a t l e a s t two o r d e r s of magnitude. S i n c e the maximum output power of the g e n e r a t o r i s i n v e r s e l y p r o p o r t i o n a l t o the i n t e r n a l r e s i s t a n c e , i t i s v i t a l t o f i n d out the r e a l cause t o t h i s u n e x p e c t e d l y h i g h r e s i s t a n c e i n o r d e r t o improve the g e n e r a t o r ' s performance. In the next two c h a p t e r , the p l a s m a - w a l l i n t e r a c t i o n o c c u r r i n g i n the s t a n d i n g shock g e n e r a t o r w i l l be i n v e s t i g a t e d . I t i s found t h a t the plasma cannot be u n i f o r m a l l the way up t o the s o l i d w a l l boundary. A d j a c e n t t o the w a l l , t h e r e i s a t h i n boundary l a y e r which e f f e c t i v e l y s c r e e n s the plasma from the e l e c t r o d e s . The amount of c u r r e n t t h a t can f l o w from the e l e c t r o d e s i s t h e r e f o r e c o n t r o l l e d by the p r o p e r t i e s of t h i s boundary l a y e r . F l u i d -d y n a m i c a l l y , the boundary l a y e r reduces the v e l o c i t y and t e mperature of the f l o w i n g plasma so as t o match the boundary c o n d i t i o n s imposed by the s o l i d w a l l . 72 E l e c t r i c a l l y , the boundary l a y e r s e t s up an e l e c t r i c f i e l d which d y n a m i c a l l y b a l a n c e s the charge c u r r e n t i n a c c o r d a n c e w i t h the b i a s p o t e n t i a l between the e l e c t r o d e and the plasma. I t i s indeed t h i s boundary l a y e r t h a t g i v e s the s t a n d i n g shock g e n e r a t o r the u n d e s i r a b l e h i g h i n t e r n a l r e s i s t a n c e . 73 CHAPTER 5. THE BOUNDARY LAYERS 5.1 The concept of boundary l a y e r For a r e a l f l u i d i n which v i s c o s i t y e f f e c t a r e not n e g l i g i b l e , the f l u i d element next t o the w a l l must have z e r o v e l o c i t y w i t h r e s p e c t t o the w a l l ( t h e n o - s l i p c o n d i t i o n ) . Hence, the f l o w r e g i o n i m m e d i a t e l y a d j a c e n t t o the w a l l must c o n t a i n a s i g n i f i c a n t v e l o c i t y g r a d i e n t i n o r d e r t o meet the boundary c o n d i t i o n s . T h i s f l o w r e g i o n i s c a l l e d the boundary l a y e r . P h y s i c a l l y , t h e r e i s no d i v i d i n g l i n e s e p a r a t i n g the boundary l a y e r from the f r e e stream f l o w because the f l u i d v e l o c i t y approaches the f r e e stream v a l u e a s y m p t o t i c a l l y . The boundary l a y e r t h i c k n e s s , 6 , i s m a t h e m a t i c a l l y d e f i n e d a c c o r d i n g t o s p e c i f i c i n t e r e s t s ; one example i s t o use the d i s t a n c e a t which the v e l o c i t y has reached 99% of the f r e e stream v a l u e . Other examples u s i n g the momentum or energy i n s t e a d of the v e l o c i t y a r e a l s o commonly used. F i g u r e 5.1 shows the s t e a d y s t a t e boundary l a y e r on a f l a t p l a t e i n a p a r a l l e l u n i f o r m f l o w f i e l d . The e f f e c t of the boundary d r a g g r a d u a l l y " d i f f u s e s " outward i n an e f f o r t t o reduce the magnitude of the l o c a l v e l o c i t y g r a d i e n t . As a r e s u l t , t he t h i c k n e s s of the boundary l a y e r grows w i t h the 74 BOUNDARY LAYER THICKNESS "T" ' 5(x) 7 u ( x y ) x=o -X,U FLAT PLATE F i g . 5.1 S k e t c h of boundary l a y e r on a f l a t p l a t e i n a p a r a l l e l f l o w at z e r o i n c i d e n c e . d i s t a n c e x where x i s measured from the f r o n t edge of the f l a t p l a t e . We s h a l l show l a t e r t h a t the boundary l a y e r t h i c k n e s s i n c r e a s e s as the square r o o t of x. S i m i l a r l y , i f heat c o n d u c t i o n and d i s s i p a t i o n a re p r e s e n t i n the f l u i d , t h e r e i s a l s o a temperature boundary l a y e r superimposed on the v e l o c i t y boundary l a y e r . I f the d e n s i t y i s s i g n i f i c a n t l y a f f e c t e d by the v a r i a t i o n i n t e m p e r a t u r e or v e l o c i t y , the two boundary l a y e r s i n t e r a c t w i t h each o t h e r . The boundary l a y e r c o m p l i c a t e s the d i s t r i b u t i o n s of f l o w p a r a m e t e r s . In o r d e r t o determine i t s e f f e c t s on the plasma e l e c t r o d e i n t e r a c t i o n , i t i s n e c e s s a r y t o l o o k f o r the v e l o c i t y and temperature p r o f i l e s and a l s o the t h i c k n e s s of the boundary l a y e r f l o w . 75 5.2 Laminar and t u r b u l e n t t y p e s of f l o w In a l a m i n a r f l o w , f l u i d elements a r e moving a l o n g s t r e a m l i n e s . The tangent of a s t r e a m l i n e d e f i n e s the d i r e c t i o n of the l o c a l f l u i d v e l o c i t y . S t r e a m l i n e s never c r o s s each o t h e r s . Thus the v i s c o u s f r i c t i o n a l f o r c e between l a y e r s of f l u i d elements moving w i t h d i f f e r e n t v e l o c i t i e s a r e r e a d i l y g i v e n by Newton's law of f r i c t i o n : T = (5.1) 9 y where u i s the c o e f f i c i e n t of v i s c o s i t y . The t u r b u l e n t f l o w on the o t h e r hand i s more c o m p l i -c a t e d and l e s s o r g a n i z e d . The s t r e a m l i n e concept can not be used h e r e . M i x i n g of f l u i d i n the y - d i r e c t i o n p r o v i d e s a way t o p r e v e n t l a r g e v e l o c i t y and t e m p e r a t u r e g r a d i e n t s from d e v e l o p i n g i n the f l u i d . Due t o the m i x i n g e f f e c t , t u r b u l e n t f l o w s are g e n e r a l l y d e s c r i b e d by t i m e - a v e r a g e d p a r a m e t e r s . A l a m i n a r f l o w may t u r n t u r b u l e n t i f the d i m e n s i o n l e s s Reynolds number exceeds a c r i t i c a l l i m i t 3 " . The Reynolds number i s d e f i n e d as R e = (uL/v)'^ where v i s the k i n e m a t i c v i s c o s i t y c o e f f i c i e n t e q u a l t o n/p , and L i s the c h a r a c t e r i s t i c l e n g t h i n the f l o w . P h y s i c a l l y , the Reynolds number r e p r e s e n t s the r a t i o of i n e r t i a l f o r c e t o v i s c o u s f o r c e . Flows of e q u a l Reynolds number a r e s a i d t o be s i m i l a r because they can be d e s c r i b e d 76 by s i m i l a r d i m e n s i o n l e s s e q u a t i o n s . 5.3 The boundary l a y e r s i n our experiment We s h a l l o n l y be c o n s i d e r i n g l a m i n a r boundary l a y e r f l o w because the s h o c k - i n d u c e d f l o w (our t e s t f l o w ) remains l a m i n a r w i t h i n the t e s t time p e r i o d , i n s p i t e of the t r a n s i t i o n i n t o t u r b u l e n t f l o w a t some l a t e r t i m e 2 " . As shown i n f i g u r e 5.2, t h e r e a r e two s e p a r a t e boundary l a y e r s t o be examined, one on each e l e c t r o d e . The boundary l a y e r on the cone s u r f a c e w i l l l o o k somewhat s i m i l a r t o the one shown i n f i g u r e 5.1 except t h a t t h i s time the w a l l s u r f a c e i s c o n i c a l and 3 d i m e n s i o n a l . On the o t h e r hand, the o t h e r boundary l a y e r on the r i n g e l e c t r o d e , which i s f l u s h w i t h the shock tube w a l l , i s cathode ring running shock shock tube \rvall cone anode bow shock F i g . 5.2 The boundary l a y e r s a t the anode and cathode i n our e x p e r i m e n t . 77 © shock wave (Oj edge of boundary layers u r u f 6„6. U T illlllll/l wal l \wuiiminiimiiiiii/iiii/7uhi/i/ii t, o y i u =0 w F i g . 5.3a The v e l o c i t y and temp e r a t u r e boundary l a y e r s of a shock wave i n a shock tube i n the l a b o r a t o r y frame, © edge of boundary layers _ , \\ if wal l ftjr t iiiuuinuiu 1 u To 57777 7777 F i g . 5.3b The same boundary l a y e r s o bserved i n the shock r e s t frame. Note the w a l l moves at o r i g i n a l shock speed. 78 unsteady i n the l a b o r a t o r y frame of r e f e r e n c e because the e n t i r e boundary l a y e r p r o f i l e moves w i t h the f r e e r u n n i n g shock wave. N e v e r t h e l e s s , i f the shock v e l o c i t y i s c o n s t a n t , the boundary l a y e r can be c o n s i d e r e d as ste a d y i n the shock r e s t frame. F i g u r e 5.3 i l l u s t r a t e s the boundary l a y e r b e h i n d a r u n n i n g shock wave as observed i n both frames of r e f e r e n c e . N a t u r a l l y the r e s t frame s h o u l d be chosen f o r the a n a l y s i s because the mathematics i s s i m p l i e r i n the steady c a s e . We note t h a t i n the shock wave r e s t frame, the upstream gas and the shock tube w a l l a re moving t o g e t h e r t o the l e f t w i t h a speed which e q u a l s t o the o r i g i n a l shock speed, w h i l e the shock f r o n t i s a t r e s t ( a t x=0). In t h i s frame, v e l o c i t i e s appears t o be l a r g e r i n the boundary l a y e r than i n the f r e e stream. 5.4 F o r m u l a t i o n of the boundary l a y e r t h e o r y L i k e any f l o w problem, the boundary l a y e r f l o w i s governed by the c o n t i n u i t y e q u a t i o n , the momentum (or N a v i e r - S t o k e s ) e q u a t i o n and the energy e q u a t i o n . We assume t h a t the boundary l a y e r i s t h i n , the f r e e stream f l o w i s u n i f o r m , and the p r e s s u r e g r a d i e n t ( i n b o t h x and y d i r e c t i o n s ) can be n e g l e c t e d . L e t u and v t o be the x and y v e l o c i t y components of the f l u i d r e s p e c t i v e l y , the g o v e r n i n g e q u a t i o n s f o r boundary l a y e r f l o w become" 5 79 3 3 c o n t i n u i t y — ( p u ) + - — ( p v ) = 0 (5.2) d x d y , 3 u , 3 u 1 3 , 3u> / c Q \ momentum u r — + v-— = — — (y-—) V O . J ; dx 3y p 3y 3 y , energy P c p ( u — + v ^ ) - J ^ ^ y ) + v ( 9 y> (5.4) s t a t e P = pRT = c o n s t a n t , (5.5) where c p=h/T i s the s p e c i f i c heat a t c o n s t a n t p r e s s u r e and K i s the t h e r m a l c o n d u c t i v i t y . The momentum and energy e q u a t i o n s can be t r a n s f o r m e d i n t o d i m e n s i o n l e s s o r d i n a r y d i f f e r e n t i a l e q u a t i o n s by the i n t r o d u c t i o n of a d i m e n s i o n -l e s s stream f u n c t i o n , F, as an e x p l i c i t f u n c t i o n of an o t h e r d i m e n s i o n l e s s s i m i l a r i t y parameter n» where n i s d e f i n e d i n terms Of X and y as ( t h e H o w a r t h - D o r . o d n l t s y n t r a n s f o r m a t i o n ) n = i^-^f ^ d y (5.6) Here the s u b s c r i p t s f and w are used t o i n d i c a t e v a l u e s a t the f r e e stream and a t the w a l l r e s p e c t i v e l y . L a t e r on we s h a l l use s u p e r s c r i p t ' t o r e p r e s e n t d e r i v a t i v e s t aken w i t h r e s p e c t t o n. n i s c a l l e d t he "reduced d i s t a n c e " ; i t i s e q u a l t o z e r o a t the w a l l . Assuming t h a t the stream f u n c t i o n has the s p e c i a l 80 p r o p e r t y t h a t F'=u/u f, e q u a t i o n (5.2) can be used t o o b t a i n an e x p r e s s i o n f o r v / u f : - - " T 1 < 2 T T > , 5 - ( F - T ' F , ) ( 5 - 7 ) p 2 u^x With t h e s e d e f i n i t i o n s of u, v and n , the momentum and energy e q u a t i o n s can be i n d e p e n d e n t l y t r a n s f o r m e d i n t o two o r d i n a r y d i f f e r e n t i a l e q u a t i o n s : F' " + FF" (5.8) 2 V ^ " * V V " C p V * ' * (5.9) (£-)" + P r • F* (•=—) ' = - ^ ^ ( F " ) 2 where Pr i s the P r a n d t l number d e f i n e d as V / K . Pr has the v a l u e s 0.67 f o r argon and 0.72 f o r a i r ( v e r y weak temperature d e p e n d e n t ) 9 . E q u a t i o n s (5.8) and (5.9) were d e r i v e d from e q u a t i o n s (5.3) and (5.4) by assuming t h a t the pr o d u c t p u i s a c o n s t a n t . T h i s c o n d i t i o n may not always be t r u e f o r a c o m p r e s s i b l e f l u i d . In 1955, M i r e l s 3 7 suggested t h a t s i n c e y i s p r o p o r t i o n a l t o temperature t o the power N, where N i s some number between 0.5 t o 1 depending on the gas (the S u t h e r l a n d ' s l a w ) , and d e n s i t y i s i n v e r s e l y p r o p o r t i o n a l t o temperature ( f o r c o n s t a n t p r e s s u r e ) , the pr o d u c t py i s a p p r o x i m a t e l y c o n s t a n t . Hence the above two e q u a t i o n s a re s t i l l a p p l i c a b l e t o the boundary l a y e r i n a shock tube f l o w . 81 To check t h i s argument, he l a t e r r e p e a t e d h i s c a l c u l a t i o n w i t h o u t the assumption and found no s i g n i f i c a n t d i f f e r e n c e i n t he s o l u t i o n s 3 8 . 5.5 Boundary c o n d i t i o n s of the boundary l a y e r f l o w In the shock r e s t frame c o o r d i n a t e s , the boundary l a y e r on the shock tube w a l l has the f o l l o w i n g boundary c o n d i t i o n s : a t the w a l l , u/u,=u / u , , T/T =T /T. ; t w r f w f and a t the f r e e stream, u/Uf=1, T/Tf=1 . S i n c e Uw i s e q u i v a l e n t t o the upstream i n t a k e v e l o c i t y (or the shock "speed v s ) , the r a t i o u^/Uf i s i d e n t i c a l t o the d e n s i t y c o m p r e s s i o n r a t i o a c r o s s the shock. T w i s the tempe r a t u r e a t the w a l l . I t s v a l u e can be o b t a i n e d by b a l a n c i n g the heat t r a n s f e r from the gas i n t o the w a l l m a t e r i a l ; t he temperature must be c o n t i n u o u s a c r o s s the T a b l e 5.1 V a l u e s Of K For Common Shock Tube M a t e r i a l s . M a t e r i a l ( P C L K ) H i n M.K.S. S t e e l q u a r t z pyrex g l a s s a i r ( S T P ) argon(STP) 1.26x10* 1.529X10 3 1 . 3 9 6 X 1 0 3 1.159x10 3 3.878 5.586 boundary as shown i n f i g u r e 5.3. T h i s t e m p e r a t u r e i s im p o r t a n t t o us because i t i s a l s o the temp e r a t u r e a t the 82 e l e c t r o d e s u r f a c e . L e t T 0 t o be the i n i t i a l shock tube t e m p e r a t u r e , which i s the room temperature i n our c a s e , and 5 t o be d e f i n e d as is E= ( p C p < ) 2 , heat t r a n s f e r c o n s i d e r a t i o n can show t h a t the f o l l o w i n g e x p r e s s i o n i s a p p r o x i m a t e l y c o r r e c t 3 9 : T -T T- £ w o , i s g a s T — = (t^)-£ (5.10) o o S o l i d The v a l u e s of £ f o r v a r i o u s m a t e r i a l s are l i s t e d i n t a b l e 5.1. I t i s o b v i o u s t h a t even f o r a v e r y s t r o n g shock (e.g . T f / T o = 3 0 ) , the r i g h t hand s i d e of e q u a t i o n (5.10) i s always s m a l l . Thus f o r a l l p r a c t i c a l p u r p o s e s , T w can be c o n s i d e r e d e q u a l t o T 0. Such low e l e c t r o d e s u r f a c e temperature i m p l i e s t h a t t h e r m i o n i c e l e c t r o n e m i s s i o n can be n e g l e c t e d . 5.6 The v e l o c i t y and temperature p r o f i l e s i n the boundary  l a y e r f l o w E q u a t i o n s (5.8) and (5.9) have no a n a l y t i c a l s o l u t i o n s . The v e l o c i t y and t e m p e r a t u r e p r o f i l e s must be o b t a i n e d t h r o u g h n u m e r i c a l methods. A c c o r d i n g t o the k i n d of boundary c o n d i t i o n s p r o v i d e d i n the l a s t s e c t i o n , i t i s n e c e s s a r y t o i t e r a t e t h e v a l u e s of F"(0) and (T/T )'(0) u n t i l the p r o p e r p r o f i l e s , which a s y m p t o t i c a l l y approach the f r e e stream c o n d i t i o n s , are found. M i r e l s 3 9 has o b t a i n e d s o l u t i o n s a t v a r i o u s shock s t r e n g t h s and c o r r e l a t i o n f o r m u l a s were g i v e n f o r a few p a r t i c u l a r l y u s e f u l p a r a m e t e r s . D e n o t i n g n u t o be the reduced d i s t a n c e 8 3 c o r r e s p o n d i n g t o the v e l o c i t y boundary l a y e r t h i c k n e s s , and l e t eEu w/u f-1 where u w / u f i s the d e n s i t y compression r a t i o a c r o s s the shock wave, the f o l l o w i n g c o r r e l a t i o n f o r m u l a s can be a p p l i e d t o the boundary l a y e r f l o w : F " ( 0 ) = - E - O . 7 9 7 9 - ( 1 + 1 . 2 8 5 E + 0 . 3 8 2 7 E 2 ) ^ ( 0 . 1 % ) , ( ^ ) ( 0 ) = - F " ( 0 ) • ( 1 - ^ ) T r * ' <•*• + ^ — ^ ) ( 0 . 2 % ) , n.. = 2. 5 7 5 / ( 1 + 0 . 8030E+0. lObSz2)^ ( 0 . 6 % ) . The p e r c e n t a g e f i g u r e b e h i n d each f o r m u l a g i v e s the a c c u r a c y of the f o r m u l a as compared t o the a c t u a l n u m e r i c a l r e s u l t s . h For our i n t e r e s t , n i ~ n u / P r = 1.22n u. The above f o r m u l a s f o r F " ( 0 ) and ( T / T F ) ' ( 0 ) e n able us t o i n t e g r a t e e q u a t i o n s ( 5 . 8 ) and ( 5 . 9 ) n u m e r i c a l l y w i t h o u t the t r o u b l e of i t e r a t i o n . S i n c e we a r e e v e n t u a l l y i n t e r e s t e d i n the v e l o c i t y d i s t r i b u t i o n measured i n the l a b frame of r e f e r e n c e , the p r o p e r n o r m a l i z e d v e l o c i t y we s h a l l be l o o k i n g f o r i s (u-u w)/(u f-u„). L i k e w i s e , f o r the t e m p e r a t u r e , we have ( T - T w ) / ( T f - T w ) . P r o f i l e s of t h e s e n o r m a l i z e d v e l o c i t y and t e m p e r a t u r e as f u n c t i o n s of the reduced d i s t a n c e a r e shown i n f i g u r e 5 . 4 . A l s o shown i n the f i g u r e i s an e r r o r f u n c t i o n b e i n g a d j u s t e d t o the a p p r o p r i a t e t h i c k n e s s n u : BOUNDARY LAYER PROFILES u r u w / \ERF(1.8) m / /// REDUCED DISTANCE, 0 0.4 0.8 \2 1.6 2.0 F i g . 5.4 P r o f i l e s of boundary l a y e r v e l o c i t y , temperature and e r r o r f u n c t i o n a g a i n s t the reduced d i s t a n c e . BOUNDARY LAYER PROFILES U - U w T " T W . T f - T w ERF(1.87/>Ju) 1 i NORMALIZED DISTANCE, Y/<5 0 3 3 . 6 I I ~ . F i g . 5.5 P r o f i l e s of boundary l a y e r v e l o c i t y , temperature and e r r o r f u n c t i o n a g a i n s t the n o r m a l i z e d d i s t a n c e . 85 e r f ( ^ ) = h± j1'**'"* e x p ( _ t 2 ) d t where e r f ( 1 . 8 ) = 0 . 9 9 i s i n acco r d a n c e w i t h the d e f i n i t i o n of the boundary l a y e r t h i c k n e s s . I t can be seen t h a t the v e l o c i t y and temp e r a t u r e p r o f i l e s can be approx i m a t e d q u i t e w e l l by the e r r o r f u n c t i o n . The r e a l p h y s i c a l d i s t a n c e i n s i d e the boundary l a y e r f l o w can be r e l a t e d t o the reduced d i s t a n c e by i n v e r t i n g the d e f i n i t i o n g i v e n i n e q u a t i o n ( 5 . 6 ) : y = (— ) "T^ L < f > d T 1 ' ' ( 5 - 1 l ) u f p w Tw 0 T f U s i n g t h i s e q u a t i o n , the boundary l a y e r t h i c k n e s s (y=6) i s o b t a i n e d as n becomes n u ; the e q u a t i o n a l s o shows t h a t 6 i s a f u n c t i o n of the square r o o t of x. The v e o l o c i t y and tempe r a t u r e p r o f i l e s p l o t t e d a g a i n s t the n o r m a l i z e d d i s t a n c e , y/6 , a r e shown i n f i g u r e 5.5. I f we w r i t e 6=g(x) , then the c o e f f i c i e n t g i s found by i n t e g r a t i n g the temp e r a t u r e p r o f i l e . Assuming y w e q u a l 2 x l 0 " s P a s (M.K.S.) a t T w of 300°K, the v a l u e s of g f o r v a r i o u s r u n n i n g shock Mach number w i t h p 0=5 T o r r i s g i v e n i n t a b l e 5.2. The v a l u e g i s o b v i o u s l y v e r y i n s e n s i t i v e t o the r u n n i n g shock Mach number. By and l a r g e , the boundary l a y e r t h i c k n e s s i s e q u a l t o 9 .5x10" 3(x) 2m. For example, a t x=10cm be h i n d the r u n n i n g shock f r o n t , the c o r r e s p o n d i n g boundary l a y e r t h i c k n e s s i s 3 mm. 86 T a b l e 5.2 V a l u e s Of The Boundary Layer T h i c k n e s s C o e f f i c i e n t g. Mach number g (m** ) 10.0 9.80X10' 3 10.5 9.69X10" 3 11.0 9.53X10" 3 11.5 9.47x10- 3 12.0 9.15x10- 3 5.7 The boundary l a y e r on the cone s u r f a c e The boundary l a y e r on a c o n i c a l o b j e c t i s v e r y c o m p l i c a t e d because the f l o w o u t s i d e the o b j e c t i s 3-d i m e n s i o n a l and d i v e r g e n t , the v e l o c i t y and p r e s s u r e a r e non - u n i f o r m . However, i n the case of the f l o w b e h i n d a c o n i c a l s t a n d i n g shock, the p r e s s u r e , v e l o c i t y and d e n s i t y are c o n s t a n t over each c o a x i a l cone p a s s i n g t h r o u g h the same v e r t e x 5 0 . Thus f o r the c i r c u l a r cones which we used i n t h e ex p e r i m e n t , the boundary l a y e r a s sumptions ( u n i f o r m f r e e stream v e l o c i t y and p r e s s u r e ) might s t i l l be a p p r o x i m a t e l y t r u e . The same o r d i n a r y d i f f e r e n t i a l e q u a t i o n s (5.8) and (5.9) can be a p p l i e d , but a new s e t of boundary c o n d i t i o n s must be imposed: at the w a l l , u/u f=0 , T w/T f=0 ; and a t the f r e e stream, u/Uf=1, T/Tf=1 . The s o l u t i o n f o r u / u f (by n u m e r i c a l method a c c o r d i n g t o S c h l i c h t i n g " 5 ) i s shown i n f i g u r e 5.6. Once a g a i n the p r o f i l e i s comparable t o an a d j u s t e d e r r o r f u n c t i o n . 87 0.75 0.25 A AA / // // • / REDUCED DISTANCE, t\ O 1J0 2.0 3.0 4 .0 F i g . 5.6 The v e l o c i t y p r o f i l e of the boundary l a y e r on the cone s u r f a c e . S i m i l a r l y , t he boundary l a y e r t h i c k n e s s i s g i v e n by the f o r m u l a &=qxH where g i s a p p r o x i m a t e l y e q u a l t o 7.0x10" 3m and x i s the d i s t a n c e measured a l o n g the c o n i c a l s u r f a c e s t a r t i n g from the cone v e r t e x . 5.8 A summary of the boundary l a y e r study In summary, we have s t u d i e d the t h e o r y of f l u i d boundary l a y e r a d j a c e n t t o the e l e c t r o d e s . The v e l o c i t y and te m p e r a t u r e of the f l o w a r e found t o d e c r e a s e toward t h e w a l l , t h e i r p r o f i l e s a r e a p p r o x i m a t e l y d e s c r i b e d by e r r o r f u n c t i o n s . Moreover, the te m p e r a t u r e a t the s u r f a c e of the e l e c t r o d e i n the presence of the shock induced f l o w i s n e a r l y e q u a l t o the i n i t i a l shock tube t e m p e r a t u r e . Thus t h e r e i s no t h e r m i o n i c e m i s s i o n a t the e l e c t r o d e s u r f a c e 88 i n v o l v e d i n the e x p e r i m e n t . These r e s u l t s are u s e f u l i n c o n s i d e r i n g the charge boundary l a y e r t o be d i s c u s s e d i n the next c h a p t e r . The charge boundary l a y e r i s c o u p l e d t o the f l u i d boundary l a y e r t h r ough the heavy p a r t i c l e s v e l o c i t y and temperature f i e l d s . 89 CHAPTER 6. THE PLASMA-WALL INTERACTION AND RESULTS OF THE DOUBLE-PROBE MEASUREMENT 6.1 The charge boundary l a y e r The d e s c r i p t i o n of the s t a n d i n g shock g e n e r a t o r i s never completed u n l e s s the p l a s m a - w a l l i n t e r a t i o n s a r e i n c l u d e d i n the a n a l y s i s , t h i s i s because e l e c t r o d e s a re i n e v i t a b l y i n v o l v e d i n the c o n d u c t i o n of c u r r e n t from and t o the plasma. The p l a s m a - w a l l i n t e r a c t i o n i n our case was f u r t h e r c o m p l i c a t e d by the c o u p l i n g of boundary l a y e r v e l o c i t y and temperature f i e l d s i n t o the e l e c t r o s t a t i c space charge r e g i o n . The s u b j e c t of p l a s m a - w a l l i n t e r a c t i o n i s by no mean new i n plasma r e s e a r c h , we encounter i t i n many a r e a s of plasma p h y s i c s such as a r c and glow d i s c h a r g e , t h e r m i o n i c c o n v e r s i o n , plasma probe d i a g n o s t i c s and f u s i o n w a l l l o s s e s . The e l e c t r i c a l e f f e c t s a r e however as w i d e l y d i f f e r e n t from each o t h e r as t h e i r o r i g i n s . We must t h e r e f o r e r e s t r i c t our i n v e s t i g a t i o n t o the case t h a t i s c l o s e l y r e s e m b l i n g our e x p e r i m e n t a l c o n d i t i o n s . B a s i c a l l y , the p l a s m a - w a l l i n t e r a t i o n i s d e s c r i b e d by a se t of s i m u l t a n e o u s p a r t i a l d i f f e r e n t i a l e q u a t i o n s , some-what s i m i l a r i n n a t u r e t o those which we have seen i n 90 c h a p t e r 5. These g o v e r n i n g e q u a t i o n s s i m p l y r e p r e s e n t the fundamental c o n s e r v a t i o n laws (mass, momentum, and energy) a p p l i e d s e p a r a t e l y t o the e l e c t r o n s , the i o n s and the n e u t r a l s . On t o p of t h e s e laws t h e r e i s always one m o r e — P o i s s o n ' s e q u a t i o n . A l l the e q u a t i o n s a r e l i n k e d t o g e t h e r t h r o u g h parameters such as the number d e n s i t i e s , f l o w v e l o c i t i e s and the e l e c t r i c a l p o t e n t i a l . F u r t h e r m o r e , the v e l o c i t y d i s t r i b u t i o n f u n c t i o n s of the charge c a r r i e r s a r e v i t a l i n d e t e r m i n i n g the c u r r e n t s e n t e r i n g the e l e c t r o d e s . No g e n e r a l s o l u t i o n t o these e q u a t i o n s i s a v a i l a b l e , n u m e r i c a l methods a r e commonly used. A p p r o x i m a t i o n must be employed depending on the plasma c o n d i t i o n s and o f t e n the s o l u t i o n i s s e n s i t i v e t o the p a r t i c u l a r e l e c t r o d e geometry i n v o l v e d . L e t us f i r s t c o n s i d e r the s i m p l e s t c a s e : an e l e c t r i c a l l y i s o l a t e d c o l d w a l l i s immersed i n a u n i f o r m tenuous plasma w i t h the e l e c t r o n temperature e q u a l t o the i o n t e m p e r a t u r e . The w a l l s u r f a c e i s c o n s i d e r e d t o be f u l l y c a t a l y t i c : a t the s u r f a c e a l l p o s i t i v e i o n s i m m e d i a t e l y recombine w i t h e l e c t r o n s and a l l the e x c e s s e l e c t r o n s a r e a b s o r b e d . In e q u i l i b r i u m , the e l e c t r o n s and the i o n s have M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n , t h u s t h e i r average t h e r m a l v e l o c i t i e s a r e g i v e n a s : 8kT v irm ' v = i , e . v = ( .„ V )H „ = 4 p . (6.1) A c c o r d i n g t o t h i s e q u a t i o n , the e l e c t r o n t h e r m a l v e l o c i t y i s much l a r g e r than t h a t of the i o n s due t o the e x c e e d i n g l y 91 s m a l l e l e c t r o n mass. I n i t i a l l y , the e l e c t r o n and i o n c u r r e n t s e n t e r i n g the w a l l s u r f a c e a r e p r o p o r t i o n a l t o t h e i r r e s p e c t i v e t h e r m a l v e l o c i t i e s . Howevery-the ex c e s s amount of e l e c t r o n d r a i n w i l l c r e a t e a p o s i t i v e space charge c l o u d at the s u r f a c e . In the steady s t a t e the c u r r e n t s e v e n t u a l l y become space-charge l i m i t e d . S i n c e the w a l l i s e l e c t r i c a l l y i s o l a t e d , the t o t a l net c u r r e n t r e c e i v e d a t the s u r f a c e must v a n i s h . A d j a c e n t to the w a l l , t h e r e i s a space charge e l e c t r i c f i e l d ( i n accordance t o the P o i s s o n ' s e q u a t i o n ) ; the f i e l d a c c e l e r a t e s the slow moving i o n s towards the w a l l and s i m u l t a n e o u s l y r e t a r d s the f a s t e l e c t r o n s . We s h a l l c a l l t h i s net charge r e g i o n the " s h e a t h " . S i n c e the sh e a t h e f f e c t i v e l y s h i e l d s the plasma from the w a l l , i t i s o n l y n a t u r a l t o f i n d the sheath t h i c k n e s s r o u g h l y e q u a l t o the Debye s h i e l d i n g l e n g t h . The c o r r e s p o n d i n g n e g a t i v e p o t e n t i a l ( w i t h r e s p e c t t o the plasma) a c h i e v e d by the i s o l a t e d w a l l , i n the l i m i t of e q u a l e l e c t r o n and i o n c u r r e n t s , i s d e s i g n a t e d as the f l o a t i n g p o t e n t i a l . T h i s p o t e n t i a l i s of the o r d e r of the plasma e l e c t r o n temperature (measured i n eV). A l t h o u g h almost a l l of the f l o a t i n g p o t e n t i a l i s a t t a i n e d i n s i d e the s h e a t h , t h e r e i s s t i l l a weak e l e c t r i c f i e l d p r e s e n t o u t s i d e the sh e a t h r e g i o n . The q u a s i - n e u t r a l r e g i o n beyond the sh e a t h edge i n which the e l e c t r i c f i e l d i s s t i l l e f f e c t i v e i s c a l l e d the "ambipolar r e g i o n " . I t was shown by Bohm 7 t h a t i n o r d e r t o form a s h e a t h w i t h m o n o t o n i c a l l y d e c r e a s i n g p o t e n t i a l , the i o n s must e n t e r the 92 arbitrary scale i i Wall Sheath ed ge Boundary layer edge " Ambipofer region F i g . 6.1 Elements of the charge boundary l a y e r . s h e a t h edge w i t h a v e l o c i t y g r e a t e r than the a c o u s t i c v e l o c i t y . Thus we have the c o n d i t i o n i mi (6.2) E q u a t i o n (6.2) i s g e n e r a l l y known as the "Bohm sheath c r i t e r i o n " . The a m b i p o l a r r e g i o n t h e r e f o r e can be viewed as the i n t e r m e d i a t e t r a n s i t i o n r e g i o n between the she a t h and the u n d i s t u r b e d plasma, i n t h i s r e g i o n the i o n s a re b e i n g a c c e l e r a t e d t o meet the Bohm c r i t e r i o n . F i g u r e 6.1 shows the g e n e r a l b e h a v i o u r of the t e m p e r a t u r e s , the charge d e n s i t i e s and the e l e c t r i c p o t e n t i a l d i s t r i b u t i o n near the w a l l . Now suppose the w a l l i s no l o n g e r e l e c t r i c a l l y i s o l a t e d ; i n s t e a d i t i s a c o n d u c t i n g e l e c t r o d e b i a s e d a t a 93 chosen p o t e n t i a l w i t h r e s p e c t t o the plasma. C u r r e n t w i l l f l o w i n or out of the plasma a c c o r d i n g t o the p o t e n t i a l b i a s . The net c u r r e n t c o l l e c t e d by the e l e c t r o d e i s always the sum t o t a l of the i o n c u r r e n t and the e l e c t r o n c u r r e n t ; I t = I i + I e . In the l i m i t of l a r g e p o s i t i v e b i a s , a l l i o n c u r r e n t w i l l be r e j e c t e d and the t o t a l c u r r e n t c o l l e c t e d by the e l e c t r o d e i s c a l l e d the e l e c t r o n s a t u r a t i o n c u r r e n t . In the o p p o s i t e l i m i t of l a r g e n e g a t i v e b i a s , the i o n s a t u r a t i o n c u r r e n t i s o b t a i n e d . F i g u r e 6.2 shows a t y p i c a l c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of an i d e a l e l e c t r o d e . The c u r v e i n between the two s a t u r a t i o n l i m i t s i s d e t e r m i n e d by the e l e c t r o n v e l o c i t y d i s t r i b u t i o n : f o r a M a x w e l l i a n d i s t r i b u t i o n , the c u r v e w i l l be e x p o n e n t i a l . At p o s i t i o n P on the c u r v e ( t h e " k n e e " ) , the p o t e n t i a l a t t a i n e d by the e l e c t r o d e i s e q u a l t o the space p o t e n t i a l of the plasma. Hence the i o n s h e a t h d i s a p p e a r s and the e l e c t r o n c u r r e n t r eaches i t s s a t u r a t i o n l i m i t . At p o s i t i o n F, t h e net c u r r e n t i s z e r o and the e l e c t r o d e i s f l o a t i n g . S i n c e the s i z e of the i o n s a t u r a t i o n c u r r e n t i s much s m a l l e r than t h a t of the e l e c t r o n one, we f i n d the p o i n t F i s l o c a t e d c l o s e t o the i o n s a t u r a t i o n r e g i o n . For t h i s r e a s o n , i t i s g e n e r a l l y t r u e t h a t t h e f l o a t i n g p o t e n t i a l i s s u f f i c i e n t l y n e g a t i v e t o t h e plasma such t h a t the amount of the i o n c u r r e n t component c o l l e c t e d by a f l o a t i n g e l e c t r o d e i s c l o s e t o i t s s a t u r a t i o n l i m i t . 94 F i g . 6.2 A t y p i c a l c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of an i d e a l probe. cathode ring shock tube wall mm running shock 1 ^ _ J _ " 2 ^ / — ^ S H E A T H 'boundary l a y e r s ^ — sheath cone anode Y////A~f bow shock 5 VSAAAr F i g . 6.3 The s t a n d i n g shock g e n e r a t o r e l e c t r i c a l c i r c u i t , 95 6.2 The double-probe system F e a t u r e s of the s t a n d i n g shock g e n e r a t o r c i r c u i t a re shown i n f i g u r e 6.3. The anode cone, the cathode r i n g , the s t a n d i n g shock and the boundary l a y e r s have been d i s c u s s e d i n p r e v i o u s c h a p t e r s . In the l a s t s e c t i o n , we a l s o i n t r o d u c e d the s heath and the a m b i p o l a r r e g i o n , b oth are embedded i n s i d e the v i s c o u s boundary l a y e r . The l o a d i n g c h a r a c t e r i s t i c , measured by v a r y i n g the v a l u e of R L, produced a r e l a t i o n s h i p between the output v o l t a g e , V , and t h e c u r r e n t , I t . We found e x p e r i m e n t a l l y t h a t each c h a r a c t e r i s t i c was a l i n e a r l i n e w i t h an unique s l o p e and an o p e n - c i r c u i t voltage,' V 0 (see f i g u r e 4.3). The o p e n - c i r c u i t v o l t a g e was d e t e r m i n e d by the s t a n d i n g shock p o t e n t i a l and t h e e l e c t r o d e work f u n c t i o n s . I t i s the purpose of t h i s c h a p t e r t o e x p l a i n the s l o p e of the l o a d i n g c h a r a c t e r i s t i c . F i g u r e 6.4a i l l u s t r a t e s a s chematic diagram of the c u r r e n t p a t h i n the s t a n d i n g shock g e n e r a t o r c i r c u i t . As shown i n the diagram, each e l e c t r o d e r e c e i v e s an i o n c u r r e n t p l u s an e l e c t r o n c u r r e n t . The t o t a l c u r r e n t , however, must be c o n t i n u o u s t h r o u g h o u t the c i r c u i t , t h u s An e q u i v a l e n t amount of t o t a l c u r r e n t would a l s o d i f f u s e t h r o u g h the plasmas and the s t a n d i n g shock p r e v e n t i n g the plasmas from d i s c h a r g i n g . Between the anode and the c a t h o d e , t h e r e i s a p o t e n t i a l d i f f e r e n c e V m, where V m = I t R L . No g r o u n d i n g r e f e r e n c e i s r e q u i r e d i n the c i r c u i t . BOUNDARY SHOCK LAYER WAVE AMBIPOLAR SHEATH REGION x I anode g. 6.4a The schematic diagram of the c u r r e n t p a t h i n the bow shock g e n e r a t o r c i r c u i t . < LU X I V) < < < to < 8 X to CM < 10 < a.' tr o ° X i -COLLI < LU < cathode anode YrXh-tv* +\y-v a 2 -v5 2 +va, +vs, i g . 6.4b Regions of p o t e n t i a l d i f f e r e n c e i n the bow shock g e n e r a t o r c i r c u i t . 97 V s h + < V s l + V a l > " ( V s 2 + V a 2 > " V 1 + IT + 17> = ° ' ^ . 4 ) The c o r r e s p o n d i n g " p o t e n t i a l p a t h " i s shown i n f i g u r e 6.4b. V r , and V r 2 a r e the v o l t a g e drops due t o the c u r r e n t f l o w i n g i n the plasmas, they can be r e w r i t t e n as Vmry/RL and V m r 2 / R L r e s p e c t i v e l y . V s h i s the s t a n d i n g shock p o t e n t i a l , V s and V a a r e the magnitudes of the p o t e n t i a l drops of the s h e a t h and the a m b i p o l a r r e g i o n r e s p e c t i v e l y . We have l e f t out the work f u n c t i o n s by assuming t h a t both e l e c t r o d e s are made out of the same m a t e r i a l . From K i r c h h o f f ' s c i r c u i t r u l e the f o l l o w i n g e q u a t i o n can be d e r i v e d : r, i *L R L Under normal c o n d i t i o n s , r , , r 2 « , t h e r e f o r e the l a s t e q u a t i o n becomes V - V . + ( V . - V 0 ) + ( V . - V . ) . (c e\ m sh s i s2 a l a2 \o.o) O b v i o u s l y , when V s , i s not e q u a l t o V g 2 and V , not e q u a l t o V a 2 r the open c i r c u i t v o l t a g e measured from the s t a n d i n g shock g e n e r a t o r would be d i f f e r e n t from V . T h i s s h phenomenon p r o v i d e s a p o s s i b l e e x p l a n a t i o n t o the d i s c r e p a n c y found i n the o p e n - c i r c u i t v o l t a g e measurement as compared t o the c o m p u t a t i o n a l p r e d i c t i o n s (shown i n f i g u r e 4.5). The s i t u a t i o n p r e s e n t e d i n f i g u r e 6.4 i s r e m i n i s c e n t of d o u b l e - p r o b e s i n plasma d i a g n o s t i c . The double-probe system has a v e r y s p e c i a l p r o p e r t y : the two p r o b e s , a l t h o u g h b i a s e d w i t h r e s p e c t t o each o t h e r , a r e both a t p o t e n t i a l s more n e g a t i v e than the f l o a t i n g p o t e n t i a l w i t h r e s p e c t t o the 98 plasma such t h a t the i o n c u r r e n t s which they c o l l e c t e d a r e always near the i o n s a t u r a t i o n l i m i t 1 5 . The i n t e r - p r o b e b i a s , V m , manages o n l y t o a d j u s t the sheath p o t e n t i a l s V s 1 and V S 2 i n an e f f o r t t o a l l o w the p r o p e r amount of e l e c t r o n c u r r e n t t o f l o w i n t o each probe. V a i and V a 2 a r e g e n e r a l l y i n s e n s i t i v e t o t h i s a d j u s t m e n t . C o n s i d e r the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of a double-probe system ( t h e s o l i d c u r v e ) as shown i n f i g u r e 6.5, f o r V s h=0 and f o r i d e n t i c a l plasma 1 and 2, the t o t a l c u r r e n t i s z e r o when V m v a n i s h e s . In t h i s c a s e , the anode and the cathode are e s s e n t i a l l y s e p a r a t e l y i s o l a t e d , b o th e l e c t r o d e s a r e f l o a t i n g w i t h r e s p e c t t o the plasma. As V m becomes l a r g e r than z e r o , the p o t e n t i a l of the anode i s e l e v a t e d and t h a t of the cathode i s d e p r e s s e d . C o n s e q u e n t l y , more e l e c t r o n s w i l l f l o w t o the anode and l e s s t o the c a t h o d e . E v e n t u a l l y f o r l a r g e p o s i t i v e V , no e l e c t r o n can re a c h the c a t h o d e : the c u r r e n t a t t a i n s a s a t u r a t i o n l i m i t w i t h I =—1. 1 o Q t . . L i k e w i s e , I =1.,. 2 o „ «_ i s a t t a i n e d a t the o p p o s i t e l i m i t of l a r g e n e g a t i v e V . The r e v e r s e p o l a r i t y of V m and I t shown i n f i g u r e 6.5 may appear t o be i n c o n t r a d i c t i o n t o the p r e v i o u s d e f i n i t i o n of v m = I t R L ' The e x p l a n a t i o n t o t h i s paradox i s t h a t i n the c o n v e n t i o n a l o p e r a t i o n of probe d i a g n o s t i c , V"m i s an e x t e r n a l l y a p p l i e d v o l t a g e source i n c o n t r a s t t o b e i n g an IR drop i n our c a s e . Hence the p r e v i o u s d e f i n i t i o n of the d i r e c t i o n of I t (shown i n f i g . 6.4a) i s " u n c o n v e n t i o n a l " . We may a l s o l o o k a t t h i s paradox i n a more u s e f u l way: i n 9 9 accordance w i t h e q u a t i o n ( 6 . 5 ) , p o s i t i v e V m and p o s i t i v e I t can o c c u r t o g e t h e r o n l y i f V S h > 0 ( V s 2 i s u s u a l l y > V s , f o r T e 2 > T e l ) , To get V m and I t both > 0, the c h a r a c t e r i s t i c w i l l have t o s h i f t t o the r i g h t as shown by the d o t t e d c u r v e i n the f i g u r e . The p a r t of the d o t t e d c u r v e w i t h both V m and I t > 0 i s a c t u a l l y the l o a d i n g c h a r a c t e r i s t i c t h a t we were l o o k i n g f o r . I f we assume V s h t o be independent of I t , from e q u a t i o n (6.5) the s l o p e of the c h a r a c t e r i s t i c can be w r i t t e n as I t I; i2,sat loading characteristic V, m i1,sat F i g . 6.5 The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of a double-probe system. 1 0 0 dV m dV . s 1 i -o d I t d V s 2 i -o d I t t t t (6.6) 1 = 0 We note t h a t the s o l i d c u r v e i n f i g u r e 6.5 i s assymmetric about the o r i g i n because the two probes may not have e q u a l i o n s a t u r a t i o n c u r r e n t l i m i t s . Only i f the probes have e q u a l a r e a s and o n l y i f they a r e both e n c o u n t e r i n g an i d e n t i c a l plasma w i l l t he c u r v e become symmetric. As a matter of f a c t , l a t e r on we s h a l l show t h a t I i 2 S a t > : > ^ i s a t * n o u r e x p e r i m e n t . 6.3 E l e c t r i c a l probes i n plasma d i a g n o s t i c and t h e i r  c l a s s i f i c a t i o n S i n c e the s t a n d i n g shock g e n e r a t o r i s t o be a n a l y z e d i n terms of the e l e c t r o s t a t i c probe t h e o r y , we s h a l l make use of t h e p r e s e n t l y a v a i l a b l e probe d i a g n o s t i c t h e o r y t o study the e l e c t r i c a l response of our s t a n d i n g shock g e n e r a t o r . The o b j e c t i v e of a l l probe t h e o r i e s i s t o p r e d i c t the r e l a t i o n s h i p between the probe c u r r e n t and the v o l t a g e b i a s on t h e probe. These t h e o r i e s , however, a r e d i v e r s i f i e d . For each domain of probe o p e r a t i o n , a d i f f e r e n t s e t of a p p r o x i m a t i o n l i m i t s must be a p p l i e d i n o r d e r t o s o l v e the g o v e r n i n g e q u a t i o n s . A common way of h a n d l i n g the a p p r o x i m a t i o n i s t o r e w r i t e the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s i n t o n o r m a l i z e d d i m e n s i o n l e s s form; a l l terms which have t h e i r magnitudes much s m a l l e r then u n i t y can be d i s c a r d e d . Three d i f f e r e n t s c a l e l e n g t h s a r e i n v o l v e d i n 101 T a b l e 6.1 C l a s s i f i c a t i o n s Of Probe T h e o r i e s . Group C o n d i t i o n C l a s s i f i c a t i o n C l a s s i c a l Lang-muir l i m i t A >> L A » L »X d X » X d » L A d » X » L C o l l i s i o n l e s s c o n v e n t i o n a l t h i n s h e a t h C o l l i s i o n l e s s o r b i t a l - l i m i t t h i c k s h e a t h C o l l i s i o n a l t h i c k s h e a t h Continuum l i m i t L » A X d » L »X L » A d » A L » X » A d C o l l i s i o n a l t h i c k s h e a t h C o l l i s i o n a l t h i n s h e a t h C o l l i s i o n l e s s t h i n s h e a t h (dense case) the n o r m a l i z a t i o n and by means of comparing t h e i r r e l a t i v e s i z e s , we d i v i d e the t h e o r i e s i n t o v a r i o u s regimes. These s c a l e l e n g t h s a re not h a r d t o guess; they a r e the Debye l e n g t h , x d (or the i s h e a t h t h i c k n e s s ) , the c o l l i s o n a l mean f r e e p a t h , A , and the s m a l l e s t f l u i d - d y n a m i c c h a r a c t e r i s t i c l e n g t h , L, e.g. the v e l o c i t y boundary l a y e r t h i c k n e s s . When the mean f r e e p a t h i s r e l a t i v e l y s m a l l , c o l l i s i o n a l e f f e c t s a r e i m p o r t a n t . The continuum l i m i t i m p l i e s t h a t the charge t r a n s p o r t mechanism can be d e s c r i b e d by charged p a r t i c l e s d i f f u s i o n s i n an e l e c t r i c f i e l d . In the o p p o s i t e l i m i t of l a r g e A , the c o l l i s i o n l e s s c o n d i t i o n r e q u i r e s the charged p a r t i c l e t o obey a f r e e - f a l l motion or o r b i t a l motion i n s i d e the e l e c t r o s t a t i c f i e l d . T a b l e 6.1 g i v e s the c l a s s i f i c a i o n 102 of probe o p e r a t i o n regimes a c c o r d i n g t o C h u n g 1 6 ; and t a b l e 6.2 c o n t a i n s the approximate v a l u e s of the f r e e stream T a b l e 6.2 Plasma Parameters Of The Te s t Flow At Fr e e Stream. Q u a n t i t y Symbol Approx. v a l u e (SI) e l e c t r o n d e n s i t y Ne 1 . 5 x l 0 2 2 / m 3 n e u t r a l d e n s i t y N n 8 . 2 x l 0 2 3 / m 3 degree of i o n i z a t i o n a 1 .8 X10" 2 f r e e - s t r e a m temperature T f 1.0x10" °K w a l l s u r f a c e t emperature T w 300.°K Debye l e n g t h Ad 5.6xl0"8m e—n mean f r e e p a t h ^ en 3.9xl0"6m i — n mean f r e e p a t h ^ i n 3.9xl0"6m e — i mean f r e e p a t h X e i 1.1x10" 5m probe dimension h 1.0x10" 2m boundary l a y e r t h i c k n e s s 6 1.0x10" 3m shock tube d i a m e t e r D 2.5xl0" 2m plasma parameters i n our e x p e r i m e n t . With L b e i n g i n the o r d e r of I0~ 3m, x i n I0" 6m and x d i n I0' 8m, we c o n c l u d e t h a t the continuum l i m i t , c o l l i s i o n l e s s t h i n s h e a t h regime i s the most s u i t a b l e d e s c r i p t i o n t o t h e p l a s m a - w a l l i n t e r a c t i o n i n our e x p e r i m e n t . N e v e r t h e l e s s , s i n c e the heavy p a r t i c l e t e m p e r a t u r e i s low near the w a l l , the i n c r e a s e i n n e u t r a l number d e n s i t y and the d e c r e a s e i n the degree of i o n i z a t i o n 103 would c o n s e q u e n t l y r a i s e t h e v a l u e of the Debye l e n g t h and reduce t h a t of the mean f r e e p a t h . Hence i t i s not s u r p r i s i n g t o f i n d the c o n d i t i o n of c o l l i s i o n a l t h i n s h e a t h case (L >> X d >>A ) b e i n g s a t i s f i e d near the w a l l r e g i o n . We s h a l l be c o n s i d e r i n g b o t h p o s s i b i l i t i e s as we c o n t i n u e . 6.4 The i o n and e l e c t r o n t e m p e r a t u r e s In c h a p t e r 5, we have o b t a i n e d the gas temperature p r o f i l e of the boundary l a y e r (see f i g u r e 5.4). T h i s p r o f i l e d e s c r i b e d the f a l l i n g of the heavy p a r t i c l e s t e m p e r a t u r e ( n e u t r a l s and i o n s ) from the f r e e stream v a l u e t o the w a l l t e m p e r a t u r e . The e l e c t r o n s , however, may have a tem p e r a t u r e d i f f e r e n t from t h a t of the heavy p a r t i c l e s due t o the i n e f f i c i e n t exchange of k i n e t i c energy i n the e l e c t r o n - h e a v y p a r t i c l e e l a s t i c c o l l i s o n . S t r i c t l y s p e a k i n g , i n o r d e r t o f i n d the e l e c t r o n t e m p e r a t u r e d i s t r i b u t i o n , we must s o l v e the energy e q u a t i o n f o r the e l e c t r o n g a s 1 6 . However, a s i m p l e random walk model as suggested by Smy 4 6 can q u i c k l y p r o v i d e an e s t i m a t e of the e l e c t r o n c o o l i n g e f f e c t . A c c o r d i n g t o t h i s model, s u b s t a n t i a l e l e c t r o n c o o l i n g can oc c u r o n l y i f L 2 m e <r> > > 1 A m, n where me and i % a re the e l e c t r o n and heavy p a r t i c l e masses r e s p e c t i v e l y . For L=l0" 3m and x = 4 x l 0 _ 6 m (see t a b l e 6.2), the l e f t hand s i d e of the i n e q u a l i t y amounts t o 0.8; thus no e l e c t r o n c o o l i n g i s a n t i c i p a t e d . 104 To be more s p e c i f i c , we used the random walk concept t o c a l c u l a t e the e l e c t r o n t e mperature d e p r e s s i o n due t o energy l o s s i n e l a s t i c c o l l i s i o n s . For each c o l l i s i o n , the f r a c t i o n a l l o s s of energy i s p r o p o r t i o n a l t o t h e i r mass r a t i o . Hence d(T ) m T m, » (6.7) e n where N i s the t o t a l number of c o l l i s i o n s s u f f e r e d by an e l e c t r o n i n moving from the f r e e stream t o the w a l l . N i s a l s o r e l a t e d t o the d i s t a n c e of t r a v e l by the random walk model as x (N)1* - y or dN - dy . (6.8) S u b s t i t u t i n g the l a s t e q u a t i o n i n t o e q u a t i o n ( 6 . 7 ) , and i n t e g r a t i n g y t h r o u g h the boundary l a y e r t h i c k n e s s , 6 , we found t h a t the e l e c t r o n temperature a t the w a l l s u r f a c e was l a r g e r than 80% of i t s f r e e stream v a l u e even i n the most u n f a v o r a b l e l i m i t ( u s i n g the s h o r t e s t \ ) . For a l l f u t u r e i n t e r e s t , we s h a l l assume T e t o be f r o z e n a t the f r e e stream v a l u e everywhere i n s i d e the boundary l a y e r . ' 105 6.5 The c u r r e n t s i n the s h e a t h r e g i o n The d i f f u s i o n of charg e d p a r t i c l e s f o r a weakly i o n i z e d c o l l i s i o n - d o m i n a t e d gas a r e d e t e r m i n e d by both the e l e c t r i c f i e l d and the p r e s s u r e g r a d i e n t s . The d i f f u s i o n v e l o c i t i e s f o r i o n s and e l e c t r o n s may be w r i t t e n a s " 0 Z. = K.f - D 4 ^ ± (6.9) 1 1 i P ± and u = -K E - D — - (6.10) e e e P e where p i = N i k T i and p e = N e k T e a r e the i o n p r e s s u r e and the e l e c t r o n p r e s s u r e r e s p e c t i v e l y . Here K ±, K e, D i and D e a r e the c o r r e s p o n d i n g c h a r g e d p a r t i c l e t r a n s p o r t c o e f f i c i e n t s of m o b i l i t y and d i f f u s i o n . Making use of the s e two e q u a t i o n s , we can w r i t e the d e f i n i t i o n of i o n c u r r e n t and e l e c t r o n c u r r e n t as J 5 -eN u = eD N — - - eK N E (6.11) 1 1 1 p i 1 1 • and 3 = eN ou = -eD N - eK N E (6.12) e e e e e p g e e We note t h a t the s i g n c o n v e n t i o n f o r J± and J g i s w r i t t e n i n such a way t h a t an i o n c u r r e n t d i r e c t e d towards the w a l l i s p o s i t i v e . S i n c e we a r e o n l y i n t e r e s t e d i n c u r r e n t s f l o w i n g i n t he y - d i r e c t i o n , we r e p l a c e the v e c t o r q u a n t i t i e s by t h e i r y-components (but s t i l l use the same s y m b o l s ) . I n a f l o w i n g plasma, the p l a s m a - w a l l i n t e r a t i o n would 106 be c o u p l e d w i t h the hydrodynamic v i s c o u s m o t i o n s , which means the s i z e of the a m b i p o l a r r e g i o n i s comparable t o the boundary l a y e r t h i c k n e s s , 6 . N e v e r t h e l e s s the e f f e c t of c o n v e c t i o n i s not t o be found a t r e g i o n s v e r y c l o s e t o the w a l l (y<< 6 ) , f o r example i n s i d e the t h i n s h e a t h . F u r t h e r -more i f the r e c o m b i n a t i o n e f f e c t i s s m a l l , t h a t i s A r » A ( j , the i o n c u r r e n t and the e l e c t r o n c u r r e n t s must be i n d i v i d u a l l y c o n s e r v e d w i t h i n t h i s n o n - c o n v e c t i v e zone: thus we may w r i t e 3 J ± 3 J The f a c t t h a t J i and J e a r e b o t h i n v a r i a n t p r o v i d e s the c o n v e n i e n c e of b e i n g a b l e t o e v a l u a t e them a t l o c a t i o n s away from the w a l l . The c o n d i t i o n A r > > X d i s found t o be v a l i d f o r our plasma u s i n g the i o n i z a t i o n - r e c o m b i n a t i o n r a t e s g i v e n by Hinnov and H i r s c h b e r g 2 7 . For the reasons mentioned i n s e c t i o n 6.2, we are i n t e r e s t e d i n the s i t u a t i o n where J e i s not l a r g e r than . L e t us re-examine e q u a t i o n s (6.11) and ( 6 . 1 2 ) . S i n c e the e l e c t r o n s a r e much more m o b i l e than the i o n s , we always have De >>Di and K e>>K 1. C o n s e q u e n t l y , the c o n d i t i o n J e= J± has the i m p l i c a t i o n t h a t J e i s the r e s u l t of t a k i n g the a l g e b r i a c d i f f e r e n c e of two l a r g e but almost e q u a l terms. In t h e f i r s t o r d e r a p p r o x i m a t i o n , J e can be s e t t o z e r o i n e q u a t i o n ( 6 . 1 2 ) ; thus we o b t a i n an e x p r e s s i o n f o r the e l e c t r i c f i e l d : 107 E S " K 0 P 0 9 y ( p e ) . (6.14) t o the E i n s t e i n (6.15) t o become kT -— = - — ( I n N ) 3y e 3y e' Here, we have t r e a t e d T e as a c o n s t a n t parameter. Upon i n t e g r a t i n g the above r e l a t i o n , we can w r i t e the e l e c t r o n d e n s i t y as a f u n c t i o n of the e l e c t r i c a l p o t e n t i a l i n the f o l l o w i n g way: N = N exp [T-|-(*-* ) ] » i a i c \ e e s ^ l k T v s J (6.16) e where N e s and $ s denote e l e c t r o n d e n s i t y and p o t e n t i a l a t the sheath edge. A l t h o u g h the f o r e g o i n g e q u a t i o n was d e r i v e d w i t h the c o l l i s i o n a l t h i n s h e a t h a s s u m p t i o n , the same e q u a t i o n i s s t i l l v a l i d i n the c o l l i s i o n l e s s t h i n s h e a t h regime because e q u a t i o n (6.16) was merely a statement of the f a c t t h a t the e l e c t r o n s a r e a t t a i n i n g s t a t i c e q u i l i b r i u m w i t h the e l e c t r i c p o t e n t i a l . The e l e c t r o n c u r r e n t does not e n t i r e l y v a n i s h even though i t was ap p r o x i m a t e d t o be z e r o i n the f i r s t o r d e r . The a c t u a l e l e c t r o n c u r r e n t c o l l e c t e d by the probe s u r f a c e can be e v a l u a t e d by making use of e q u a t i o n ( 6 . 1 6 ) . Suppose T h i s e q u a t i o n can be s i m p l i f i e d a c c o r d i n g r e l a t i o n - -D kT 108 the e l e c t r o n d e n s i t y i s N g A a t a d i s t a n c e one mean f r e e p a t h away from the w a l l , and the v e l o c i t y d i s t r i b u t i o n i s i s o t r o p i c w i t h a random average speed of v e . A v e r a g i n g the d i r e c t i o n c o s i n e over a hemisphere would g i v e the average v e l o c i t y i n the y - d i r e c t i o n t o be v e / 2 . Moreover, o n l y h a l f of the p o p u l a t i o n w i l l be t r a v e l i n g towards the w a l l . As a r e s u l t , t he e l e c t r o n c u r r e n t d e n s i t y ( per u n i t a r e a of probe s u r f a c e ) i s g i v e n by e 4 eX e The r i g h t hand s i d e of t h e above e q u a t i o n i s v e r y o f t e n d i v i d e d by an e x t r a f a c t o r c a l l e d the " s c r e e n i n g f a c t o r " which has a v a l u e between one and o n e - h a l f 9 . T h i s s c r e e n i n g f a c t o r i s due t o the n o n - i s o t r o p i c v e l o c i t y d i s t r i b u t i o n as y becomes v e r y c l o s e t o t h e w a l l ( e . g . a t y <x ). i n p a r t i c u l a r , no e l e c t r o n can come out of the w a l l , the s c r e e n i n g f a c t o r a t p o s i t i o n y=0 i s e x a c t l y 1/2. Hence we o b t a i n J = --UN v (6.17) e 2 ew e S u b s t i t u t i n g N g w from e q u a t i o n ( 6 . 1 6 ) , t h e e l e c t r o n c u r r e n t can be w r i t t e n as I = _ i r e x p [ _ | _ ( $ w _ * ) ] 8 e 109 1 8 k T Jc where * = o-e(- 6 ) N A e » and A i s the e f f e c t i v e s u r f a c e a r e a of the e l e c t r o d e . S i n c e V s = $ s - $ w (see f i g u r e 6.4b), we r e w r i t e e q u a t i o n (6.18) as I = - i e x p ( - ^ ) (6.19) e r v kT e We ought t o acknowledge here t h a t the e l e c t r o n c u r r e n t c o l l e c t e d by the probe i s m a i n l y due t o those e l e c t r o n s a t the Boltzmann d i s t r i b u t i o n t a i l which have enough energy t o conquer the r e t a r d i n g e l e c t r i c f i e l d a t the s h e a t h . In view of t h i s poor s a m p l i n g l o c a t i o n , the e l e c t r o n c u r r e n t g i v e n i n e q u a t i o n (6.19) i s not an a c c u r a t e measure of the e l e c t r o n t e m p e r a t u r e , e s p e c i a l l y f o r a non-LTE plasma. Now l e t us t u r n our a t t e n t i o n t o the i o n c u r r e n t . I f we combine e q u a t i o n (6.11) f o r the i o n c u r r e n t and e q u a t i o n (6.14) f o r the e l e c t r i c f i e l d , we get , - 1- , K i D e N i 9 J i ' 6 P ± 9 y P i + e K e p e 3 ? e x J e J At t h e s h e a t h edge, we may c l a i m H± t o be e q u a l t o N e , then the above e q u a t i o n i s reduced t o J i = K i W [ N i k ( T i + T e ) ] y=y s . (6.20) 110 I t s h o u l d be emphasised t h a t e q u a t i o n (6.20) was d e r i v e d f o r the case of s m a l l J e , t h e r e f o r e i t i s t r u l y the p r o p e r e x p r e s s i o n f o r the i o n c u r r e n t i n t h e s a t u r a t i o n l i m i t . Moreover, the e q u a t i o n remains u s e f u l even i n the c o l l i s i o n l e s s s h eath regime, except i n t h i s case the d e r i v a t i v e must be e v a l u a t e d a t a p o s i t i o n y > x i n s t e a d of a t y s i n o r d e r t o be m e a n i n g f u l . 6.6 The s l o p e s of the l o a d i n g c h a r a c t e r i s t i c s We a r e now ready t o d e r i v e an e x p r e s s i o n which d e s c r i b e s the s l o p e s of the l o a d i n g c h a r a c t e r i s t i c s . F i r s t , l e t us r e c a l l two p r e v i o u s l y d e f i n e d r e l a t i o n s namely e q u a t i o n s (6.3) and ( 6 . 6 ) : X i 2 + Xe2 dV m d l dV s i It-0 d l dV s2 i t - o d I t It-0 I f we s u b s t i t u t e i n t o e q u a t i o n (6.3) the c o r r e s p o n d i n g e l e c t r o n c u r r e n t as g i v e n i n e q u a t i o n (6.19) and i f we a l s o assume t h a t the i o n c u r r e n t s a r e s a t u r a t e d , then we can r e w r i t e e q u a t i o n (6.3) as I - 1. 0 = - i , e x p ( - f — t 12,sat r2 k T „ e2 111 V a n d - T t - ^ l . s a t ' ^ r l ^ P ^ t ^ e 1 T a k i n g the l o g on both s i d e s and r e - a r r a n g i n g the terms, we f i n d V s 2 = - I T e 2 t l n ( I i 2 , s a t - I t ) " l n ( i r 2 ^ ' (6.21) a n d V s l = - I T e l I l n ( I i l , B a t + I t ) " ^ r l ' J ' ( 6 ' 2 2 ) Here i r , , i r 2 f I ± i s a t , and I j L 2 s a t a r e a 1 1 supposed t o be c o n s t a n t s , not b e i n g a f f e c t e d by the v a r i a t i o n of the t o t a l c u r r e n t I t . Combining the l a s t two e q u a t i o n s w i t h e q u a t i o n ( 6 . 6 ) , we o b t a i n the s l o p e of the l o a d i n g c h a r a c t e r i s t i c , dV d I t e e l1 ! . , +I> e e 2 l L , I =0 I t - 0 e ' i l . s a t A t c " i i 2 , s a t * t 1 I t T T e ^ i l . s a t I12. s a t ' (6-23) We note t h a t the s l o p e i s n e g a t i v e which i s i n agreement w i t h the s l o p e of the l o a d i n g c h a r a c t e r i s t i c shown i n f i g u r e 6.5. E q u a t i o n (6.23) i s an e x c e e d i n g l y s i m p l e r e s u l t . I t says the s l o p e of the l o a d i n g c h a r a c t e r i s t i c i s g i v e n as a s i m p l e f u n c t i o n of o n l y the e l e c t r o n t e m p e r a t u r e s and the s a t u r a t i o n i o n c u r r e n t s a t the e l e c t r o d e s . W h i l e the 1 1 2 e l e c t r o n t e m p e r a t u r e s may be p r e d i c t e d from the shock wave t h e o r y and the d i a g n o s t i c s , the v a l u e s of i o n s a t u r a t i o n c u r r e n t s can be o b t a i n e d e i t h e r by p e r f o r m i n g a d d i t i o n a l independent e x p e r i m e n t a l measurements or by t h e o r e t i c a l l y e v a l u a t i n g the io'n c u r r e n t from e q u a t i o n ( 6 . 2 0 ) . These two methods are d e s c r i b e d i n the f o l l o w i n g s e c t i o n s . 6.7 The double-probe experiment The method of measuring the i o n s a t u r a t i o n c u r r e n t by a p a i r of double-probe i s s t r a i g h t f o r w a r d . We have a l r e a d y d i s c u s s e d the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of such a system as shown i n f i g u r e 6.5. Our purpose i n t h i s e x periment i s t o f i n d the v a l u e a t which the c u r r e n t a t t a i n s s a t u r a t i o n . Upon r e a c h i n g t h i s l i m i t , f u r t h e r i n c r e a s e of the b i a s v o l t a g e w i l l not produce any change on the c u r r e n t r e c e i v e d by the probe. In the e x p e r i m e n t , we have used a p a i r of symmetric p r o b e s , both made out of the same 1/8-inch d i a m e t e r s t e e l r o d . The probes were mounted f l u s h w i t h the shock tube w a l l a t the same p o s i t i o n where the r i n g - c a t h o d e was p r e v i o u s l y l o c a t e d (see f i g u r e 6.6). The s e p a r a t i o n between the two probes was found t o have no i n f l u e n c e on the d a t a . The b i a s - v o l t a g e a p p l i e d between the probes was r a n g i n g from 0 t o 50V. C u r r e n t s a t u r a t i o n was u s u a l l y reached a t a b i a s of about 10V t o 15V. O c c a s i o n a l l y , a t beyond 45V b i a s , we o b s e r v e d l a r g e c u r r e n t p u l s e s which a r e e v i d e n c e s of a r c -d i s c h a r g i n g between the p r o b e s . The c i r c u i t diagram of t h i s 1 13 V)v=5onxi BIAS F i g . 6.6 The double-probe e x p e r i m e n t a l s e t - u p . experiment i s a l s o i n c l u d e d i n f i g u r e 6.6. The s a t u r a t i o n c u r r e n t which we o b t a i n e d i n t h i s e x periment i s o b v i o u s l y a f u n c t i o n of the probe s u r f a c e a r e a , one might t h e r e f o r e q u e s t i o n why we d i d n ' t use a probe of the same s u r f a c e a r e a as the r i n g - c a t h o d e used i n the s t a n d i n g shock g e n e r a t o r e x p e r i m e n t . There were two r e a s o n s . F i r s t , the a r e a of the r i n g was too l a r g e , the c o r r e s p o n d i n g s a t u r a t i o n c u r r e n t (about 2 amps, net) would g r e a t l y d i s t u r b the weak plasma. Second, i t i s p h y s i c a l l y i m p o s s i b l e t o p l a c e two e l e c t r o d e s of t h i s s i z e on the shock 1 14 tube w a l l and s t i l l be a b l e t o keep good s p a t i a l r e s o l u t i o n (see e.g. f i g u r e 6.3). B e s i d e s , s i n c e the boundary l a y e r t h i c k n e s s grew b e h i n d the i n c i d e n t shock f r o n t as a f u n c t i o n of the square r o o t of d i s t a n c e , the two probes must be l o c a t e d a t the same normal p l a n e i n o r d e r t o ensure t h a t they a re " s e e i n g " the same plasma c o n d i t i o n s . T h i s arrangement i s o n l y p o s s i b l e f o r a p a i r of s m a l l p r o b e s . In f a c t , we s h a l l show i n the next s e c t i o n t h a t the i o n s a t u r a t i o n c u r r e n t d e n s i t y i s i n v e r s e l y p r o p o r t i o n a l t o the boundary l a y e r t h i c k n e s s ( i . e . - ( x 0 ) _ J s ) . S i n c e the probe SATURATION CURRENT DENSITY e 0 0 C * | a m p / m * 6 0 0 0 , 4 0 0 0 . 2 0 0 0 , 10.0 105 11.0 11.5 RUNNING SHOCK MACH NUMBER 12.0 F i g . 6.7 The s a t u r a t i o n c u r r e n t d e n s i t y measurements and t h e o r e t i c a l p r e d i c t i o n s . 115 had a dia m e t e r of o n l y 3.2mm, the v a r i a t i o n of J i over the s u r f a c e a r e a would not be more than 5% d i f f e r e n c e i f x 0 was l a r g e r than 32mm (measured from the back of the r u n n i n g shock f r o n t ) . T h i s v a l u e of x 0 c o r r e s p o n d s t o the time a t 10 m i c r o - s e c o n d a f t e r the i n c i d e n t shock a r r i v a l . Measurements of J i ) S a t a s a f u n c t i o n of the i n c i d e n t shock Mach number are shown i n f i g u r e 6.7. We d i d not attempt t o measure the s a t u r a t i o n c u r r e n t on the anode s i d e b e h i n d the s t a n d i n g shock. T h i s was because the experiment would be ha r d t o do and the r e s u l t s were not go i n g t o be v e r y r e w a r d i n g . In s e c t i o n 6.9, an e s t i m a t e i s g i v e n which demonstrates t h a t the i o n s a t u r a t i o n c u r r e n t e n t e r i n g the cone-anode i s much g r e a t e r than the one g o i n g t o t he r i n g - c a t h o d e , i . e . I i 2 S a t > > I i i s a f W h i l e T e 2 i s b i g g e r than T e, by o n l y a s m a l l f r a c t i o n , the second term on the r i g h t hand s i d e of e q u a t i o n (6.23) becomes i n s i g n i -f i c a n t . Thus the s l o p e of the l o a d i n g c h a r a c t e r i s t i c i s det e r m i n e d o n l y by T e 1 and I ±, s a t . T h i s r e s u l t agrees w e l l w i t h the e x p e r i m e n t a l o b s e r v a t i o n s t a t e d i n s e c t i o n 4.5 t h a t the measured s l o p e i s independent of the cone a n g l e , but i t depends on the i n c i d e n t shock Mach number ( i . e . the s t a n d i n g shock upstream c o n d i t i o n s ) . 6.8 Theory of the i o n s a t u r a t i o n c u r r e n t At the end of s e c t i o n 6.5, we have a r r i v e d a t an e x p r e s s i o n f o r e v a l u a t i n g the i o n s a t u r a t i o n c u r r e n t , namely: 116 J i - K i f 7 [ N i k ( T i + T e > 3 y=y s where K.j_ i s the i o n m o b i l i t y and y s i s the p o s i t i o n of the shea t h edge. T h i s r e l a t i o n was o r i g i n a l l y d e r i v e d f o r the she a t h r e g i o n , but s i n c e i t was r e q u i r e d t o be e v a l u a t e d a t the s h e a t h edge, the e q u a t i o n i s e q u a l l y a p p l i c a b l e t o the bottom p a r t of the a m b i p o l a r r e g i o n as w e l l . A f t e r a l l , the two r e g i o n s must be c o n t i n u o u s l y matched a t the she a t h edge. The a m b i p o l a r r e g i o n i s q u a s i - n e u t r a l , t h u s N i=N e. The e l e c t r o n t e m p e r a t u r e i s assumed t o be f r o z e n a t the f r e e -s tream t e m p e r a t u r e , T f, whereas the i o n and n e u t r a l t e m p e r a t u r e s a r e supposedly e q u a l t o the gas te m p e r a t u r e , T , which i s p r e d i c t e d by the boundary l a y e r t h e o r y i n c h a p t e r 5. A c c o r d i n g t o t h i s t h e o r y , the temp e r a t u r e of the w a l l , T , i s n e a r l y e q u a l t o the room t e m p e r a t u r e . For the con v e n i e n c e of f o r m u l a t i n g the g o v e r n i n g e q u a t i o n s , we i n t r o d u c e the f o l l o w i n g symbols: C±s m iN i /p i s the i o n mass f r a c t i o n , C e = m e N e / p i s t n e e l e c t r o n mass f r a c t i o n , C i f i s the v a l u e of C at f r e e - s t r e a m , Z= c±/^±f * s t n e n o r m a l i z e d mass f r a c t i o n and p g= P R T ^ / I I K i s the gas p r e s s u r e . The d e n s i t y , p , i s i n v e r s e l y p r o p o r t i o n a l t o T whereas p i s kept c o n s t a n t everywhere i n accordance w i t h the boundary l a y e r t h e o r y a s s u m p t i o n s . For a s i n g l y i o n i z e d gas, the d e f i n i t i o n of i s i d e n t i c a l t o the d e f i n i t i o n of the l o c a l degree of i o n i z a t i o n N i / ( N n + N i ) . The g o v e r n i n g e q u a t i o n s f o r the charge f l o w a r e 1 17 o b t a i n e d from s e p a r a t e l y t a k i n g the i o n mass moment and the e l e c t r o n mass moment of the Boltzmann e q u a t i o n 1 2 . They a re w r i t t e n as 3C, 3C 3C p u i l T + P V 3 3 T " fy-tpDi 5 7 - p K i c i 7 ^ = w i ; ( 6 ' 2 4 ) 3 C 3 C T T P U T - 1 +Pvr-^ " ITIPD t(C +PK C V»] 3x — 3 y 3 y - e T g 3y ^ e T g' " a e ' e , ( 6 > 2 5 ) where <Li and (Le are the mass p r o d u c t i o n r a t e s f o r the i o n s and the e l e c t r o n s r e s p e c t i v e l y . In a d d i t i o n t o these two e q u a t i o n s , the P o i s s o n ' s e q u a t i o n must be s a t i s f i e d . In a s i n g l y i o n i z e d gas w i t h e q u a l e l e c t r o n and i o n d e n s i t i e s , i t has the s i m p l e form: V 2* = 0 . , (6.26) E q u a t i o n (6.24) and (6.25) a r e r e a d i l y c o m b i n e d — b y m u l t i p l y i n g t he former one by K e and the l a t t e r one by Kimi/m e and then a d d i n g both e q u a t i o n s t o g e t h e r . The r e s u l t a n t e q u a t i o n i s a l r e a d y d e c o u p l e d from the P o i s s o n ' s e q u a t i o n and i t has the form: 9 C i *Ci 3 p D l K e Ifi . p u 3 l T + p V3lT " ST^+K, 3y pD K. T m C T e 1 B 1 l _ ( _ e _ e - ) l = a)j K,+K T in 3 y v T J i i e e e g T h i s e q u a t i o n can be f u r t h e r s i m p l i f i e d i n terms of the n o r m a l i z e d mass f r a c t i o n Z t o y i e l d 1 1 8 e g ' g i , f where D a denotes the a m b i p o l a r d i f f u s i o n c o e f f i c i e n t 1 1 . I t i s d e f i n e d as D K +D K T D a " V + K 6 * V 1 + TT> . (6.28) l e i We can a l s o r e w r i t e the i o n s a t u r a t i o n c u r r e n t f o r m u l a i n terms of t h e s e new pa r a m e t e r s : 6 6 y=y s = 1£c D r— + Z I m. i , f a L 3 y 1+T /T 3 y v T ; 1  1 e g ' g y=y ' (6.29) s Comparing e q u a t i o n s (6.27) w i t h the f o r e g o i n g e x p r e s s i o n , we observe t h a t the i o n s a t u r a t i o n c u r r e n t i s t o be found by s o l v i n g e q u a t i o n (6.27) w i t h the known p r o f i l e s of u, v and T (from c h a p t e r 5 ) , and a l s o w i t h the a p p r o p r i a t e boundary o c o n d i t i o n s . The p r o c e d u r e s of s o l v i n g t h i s e q u a t i o n i n c l u d e the f o l l o w i n g s t e p s : 1) t r a n s f o r m i n g the e q u a t i o n i n t o an o r d i n a r y d i f f e r e n t i a l e q u a t i o n u s i n g the independent v a r i a b l e n ( d e f i n e d as the reduced d i s t a n c e i n boundary l a y e r f l o w ) , 2) f i n d i n g an a p p r o x i m a t i o n f o r the n o n - e q u i l i b r i u m i o n mass p r o d u c t i o n term, 3) d e t e r m i n i n g the sh e a t h edge boundary c o n d i t i o n (by 119 r e l a t i n g Z t o dZ/dy), 4) i t e r a t i n g the n u m e r i c a l i n t e g r a t i o n u n t i l one f i n d s t h e s o l u t i o n i n which Z a s y m p t o t i c a l l y approaches 1 f o r l a r g e n. The m a t h e m a t i c a l m a n i p u l a t i o n s a r e r a t h e r l e n g t h y and u n i n t e r e s t i n g . The d e t a i l s can be found i n appendix B. In t h e r e m a i n i n g p a r t of t h i s s e c t i o n , we a r e g o i n g t h r o u g h a n o n - r i g o r o u s argument which l e a d s t o a r e s u l t t h a t i s v e r y c l o s e t o the f o r m a l s o l u t i o n , and y e t p r o v i d e s an even b e t t e r p h y s i c a l i n s i g h t than t h e d a z z l i n g mathematics. L e t us s t a r t from e q u a t i o n ( 6 . 2 9 ) . For p h y s i c a l r e a s o n , a t y=y s- 0, must almost v a n i s h . Thus the second term on the r i g h t hand s i d e of t h i s e q u a t i o n can be abandoned. Meanwhile, f o r t h e f i r s t term, t h e d e r i v a t i v e may be a p p r o x i m a t e d by _ i . __1 , -LL± (6.30) 3y Ay 6 » where 6 i s the boundary l a y e r t h i c k n e s s g i v e n i n c h a p t e r 5 by e q u a t i o n ( 5 . 1 4 ) . T h i s a p p r o x i m a t i o n s i m p l y means the degree of i o n i z a t i o n d e c r e a s e s l i n e a r l y from the f r e e - s t r e a m v a l u e t o z e r o v a l u e a t the w a l l over the t r a n s i t i o n zone which has a s i z e e q u a l t o the hydrodynamic boundary l a y e r . Combining e q u a t i o n s (6.29) and ( 6 . 3 0 ) , the i o n s a t u r a t i o n c u r r e n t f o r m u l a i s s i m p l i f i e d t o J i • •ci"g<1 + h V 6 w A c c o r d i n g t o B i o n d i & C h a n i n 6 and D a l g a r n o 1 9 , the i o n 120 m o b i l i t y has the v a l u e K, * 3.362x10 -3.1 (U 0.432 (M.K.S.) S u b s t i t u t i n g K ± and 6, and l e t T =300°K, we o b t a i n the f o l l o w i n g f o r m u l a : where x 0 i s the d i s t a n c e measured from the s t a r t i n g p o i n t of the boundary l a y e r ( i . e . the shock f r o n t ) . A l l q u a n t i t i e s a r e e x p r e s s e d i n M.K.S. u n i t s . T h i s r e s u l t d i f f e r s from the more r i g o r o u s s o l u t i o n o n l y by the magnitude of the n u m e r i c a l c o n s t a n t i n the f r o n t , a l l o t h e r parameters a r e i d e n t i c a l . B e s i d e s , i n view of the many u n c e r t a i n t i e s i n v o l v e d i n the probe t h e o r i e s , i n the boundary l a y e r t h e o r y , and i n the v a l u e s of the plasma p a r a m e t e r s , we ought not p l a c e t o o much emphasis on t h e a c c u r a c y of t h i s n u m e r i c a l c o n s t a n t . 1 E q u a t i o n (6.31) c o n t a i n s an ( x 0 ) " ^ dependence. In the r e a l e x p e r i m e n t a l s i t u a t i o n , the v a l u e of ( x 0 ) 2 o n l y v a r i e s from 0.2 t o 0.25 d u r i n g the time i n t e r v a l from 10 m i c r o -second t o 20 mi c r o - s e c o n d a f t e r the shock a r r i v a l . T h i s range of v a l u e s c o v e r s the a c t u a l o b s e r v a t i o n time f o r a l l Mach numbers of i n t e r e s t . From now on, the v a l u e of ( x 0 ) 2 i s f i x e d a t 0.25m 2. The v a l u e s of the i o n c u r r e n t d e n s i t y i n a c c o r d a n c e w i t h e q u a t i o n (6.31) were p l o t t e d i n f i g u r e 6.7. The comparison between the e x p e r i m e n t a l d a t a and the t h e o r e t i c a l c u r v e i s s a t i s f a c t o r y . i , s a t « 4.8C (6.31 ) 121 6.9 Comparison of the l o a d i n g c h a r a c t e r i s t i c s l o p e 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 The i o n c u r r e n t c o l l e c t e d by the r i n g - c a t h o d e i s c a l c u l a t e d by i n t e g r a t i n g J i j S a t over the e l e c t r o d e s u r f a c e a r e a . Hence I . = 4.8C. (p uf)h° ( * d ) J X ° ( x ) _ i s d x 1 , 1 g x -h o where d and h r e p r e s e n t the di a m e t e r and the w i d t h of the r i n g r e s p e c t i v e l y . For s m a l l h / x 0 f which i s t r u e f o r our c a s e , the r e s u l t of t h i s i n t e g r a t i o n i s eq u a l t o h / ( x 0 ) ' s . I f A i s the ar e a of the r i n g , t he t o t a l i o n c u r r e n t i s I±l = A ' J i l , s a t ' (6.32) F o r h=8.8xl0 _ 3m and d=2.54x10" 2m, A i s 7.0xl0-"m 2. Take a Mach 11.0 r u n n i n g shock f o r example, one then f i n d s by e v a l u a t i n g e q u a t i o n (6.31) t h a t 1 s a t = 3450amp/m2 and , i s t h e r e f o r e e q u a l t o 2.4amps. The i o n c u r r e n t c o l l e c t e d by the cone anode can be o b t a i n e d by a s i m i l a r method. A d o p t i n g the approximate boundary l a y e r d e s c r i b e d i n s e c t i o n 5.7, we have f o r the i o n c u r r e n t , I i 2 = 4 . 8 C i ) f ( P g U f ) l 5 . 2 7 T s i n e J*(x)hdx where 0 i s the h a l f a n g l e of the cone and s i s the l e n g t h of the s l a n t e d edge measured from the v e r t e x . For a cone w i t h a base r a d i u s of 1/4 i n c h , we s i m p l i f y the e x p r e s s i o n t o become 122 12 -2 C i f ( p g U f V = 1.0x10 Z  / ( s i n 6 ) T a b l e 6.3 shows some t y p i c a l v a l u e s of the i o n c u r r e n t s at the cone f o r v a r i o u s cone a n g l e s and Mach numbers. Again t a k e the Mach 11.0 case f o r c o m p a r i s o n , i t i s o b v i o u s t h a t I±2 » I n -T a b l e 6.3 Ion C u r r e n t s C o l l e c t e d At The Cone, ( c u r r e n t s a r e e x p r e s s e d i n ampere) Mach 15° 20° 25° 30° 10 4.0 4.9 6.3 8.3 1 1 12.0 13.3 15.5 18.5 12 25.3 27.2 30.6 35.0 S i n c e the t e m p e r a t u r e o n l y i n c r e a s e s by about 10% a c r o s s the s t a n d i n g shock, the c o n d i t i o n l±2 » I±i i m p l i e s t h a t the term Te2/l±2 does not c o n t r i b u t e s i g n i f i c a n t l y t o the magnitude of the l o a d i n g c h a r a c t e r i s t i c s l o p e . Thus by n e g l e c t i n g t h i s term, the s l o p e may be o b t a i n e d from e q u a t i o n s (6.23) and (6.32) as dV d l -kT I t - 0 eAJ i 1 , s a t -3 = - 6 . 4 x l 0 " J T e / [ C i > f ( p g u f ) ] 123 LOAD LINE SLOPE 0.8 V./AMP. r • \ o , r ' l J 1 • r s. O J 1 i r ' I J ' > 1 i Measured Slope 0 0.123 T e measured 6.4x10"3T e ^ O ^ ( '-• " S 1 • 10.0 105 11.0 11.5 120 RUNNING SHOCK MACH NUMBER F i g . 6.8 Comparison of the l o a d i n g c h a r a c t e r i s t i c s l o p e , i n t h e o r y and i n measurement. F i g u r e 6.8 compares t h i s p r e d i c t i o n w i t h the e x p e r i m e n t a l d a t a of the l o a d i n g c h a r a c t e r i s t i c s l o p e measurement. A l s o shown i n the f i g u r e a r e the v a l u e s of the e x p r e s s i o n - k T e / ( e A J 0 ) where J 0 i s the s a t u r a t i o n c u r r e n t d e n s i t y measured i n the double-probe experiment a t f i v e d i f f e r e n t Mach numbers. D e s p i t e the f a c t t h a t the c u r v e 124 l i e s o u t s i d e the e r r o r b a r s , the d a t a and the t h e o r y a re undoubtedly i n q u a l i t a t i v e agreement. The d i s c r e p a n c y i s most l i k e l y due t o the e l e c t r o n p a r t i c l e v e l o c i t i e s b e i n g n o n - M a x w e l l i a n d i s t r i b u t e d . We a l r e a d y d i s c u s s e d t h i s p roblem i n s e c t i o n 6.5 i n c o n n e c t i o n w i t h the r e l i a b i l i t y of e q u a t i o n ( 6 . 1 9 ) . Another cause might be the edge e f f e c t on the e l e c t r o d e s u r f a c e a r e a . T h i s phenomenon has been known to c r e a t e t r o u b l e when u s i n g p l a n e probes i n plasma d i a g n o s t i c s 1 5 . F i n a l l y , the u n c e r t a i n t i e s on the hydrodynamic boundary l a y e r or even p o s s i b l y the e a r l y o c c u r r e n c e of t u r b u l e n c e i n the plasma f l o w may become s o u r c e s of e r r o r s . In c o n c l u s i o n t o t h i s c h a p t e r , we s t a t e t h a t the s l o p e s of the s t a n d i n g shock g e n e r a t o r l o a d i n g c h a r a c t e r i s t i c s , which cannot be e x p l a i n e d by the plasma b u l k r e s i s t a n c e , a r e due t o the e f f e c t s of the p l a s m a - w a l l boundary l a y e r . T h i s boundary l a y e r e f f e c t i s common t o a l l plasma d e v i c e s (such as the MHD g e n e r a t o r and the t h e r m i o n i c c o n v e r t o r ) which i n v o l v e the f l o w of e l e c t r i c c u r r e n t t h r o u g h plasma-e l e c t r o d e boundary. The i n t e r n a l r e s i s t a n c e of our d e v i c e i s found t o be d i r e c t l y p r o p o r t i o n a l t o the plasma e l e c t r o n temperature and i n v e r s e l y p r o p o r t i o n a l t o the degree of i o n i z a t i o n . Hence, i n o r d e r t o o b t a i n s m a l l i n t e r n a l r e s i s t a n c e , i t i s d e s i r a b l e t o maximize the i o n s a t u r a t i o n c u r r e n t by i n c r e a s i n g the cathode s u r f a c e a r e a , the charge d e n s i t y or the v e l o c i t y and p r e s s u r e of the f r e e stream f l o w . 125 CHAPTER 7. THERMODYNAMICS OF THE STANDING SHOCK GENERATOR At p r e s e n t , e l e c t r i c i t y i s m a i n l y produced by t u r b o g e n e r a t o r s . A t u r b o g e n e r a t o r i s however o n l y c a p a b l e of o p e r a t i n g a t a maximum temperature of 800°K w h i l e heat s o u r c e t e m p e r a t u r e s a r e a l r e a d y a v a i l a b l e a t 2500°K or h i g h e r . I t i s un d e r s t o o d t h a t the maximum e f f i c i e n c y of the system i s l i m i t e d by the temperature employed i n the energy c o n v e r s i o n d e v i c e . C l e a r l y t h e r e i s a temperature gap i n our energy c o n v e r s i o n t e c h n o l o g y . The s t a n d i n g shock g e n e r a t o r may be u t i l i z e d t o f i l l t h i s gap as a t o p p i n g d e v i c e t o the t u r b o g e n e r a t o r . In o r d e r t o e v a l u a t e the u s e f u l n e s s of the s t a n d i n g shock g e n e r a t o r , the thermodynamics of the s t a n d i n g shock g e n e r a t o r w i l l be i n v e s t i g a t e d i n t h i s c h a p t e r . 7.1 Thermodynamics of shock c o m p r e s s i o n s Let us f i r s t c o n s i d e r the thermodynamics of shock co m p r e s s i o n s because the e s s e n t i a l element i n the s t a n d i n g shock g e n e r a t o r i s the s t a n d i n g shock i t s e l f . The shock t r a n s i t i o n can be d e s c r i b e d by the t h r e e g o v e r n i n g e q u a t i o n s which had been s t a t e d e a r l i e r i n c h a p t e r 2 as e q u a t i o n s (2.6) t o ( 2 . 8 ) . Amongst these e q u a t i o n s , the energy c o n s e r v a t i o n e q u a t i o n i s w r i t t e n as 126 h l + 2 " " W v 2 - h 2 + - (7.1 ) where h i s t h e e n t h a l p y , v i s t h e f l u i d v e l o c i t y and W i s the energy l o s s t o the environment (per u n i t a r e a and t i m e ) . S u b s c r i p t s 1 and 2 a r e used t o i n d i c a t e the upstream and downstream r e g i o n s r e s p e c t i v e l y . For c o n v e n i e n c e , we d e f i n e a s t a g n a t i o n s t a t e ( i n d i c a t e d by s u b s c r i p t o) which i s the s t a t e a t t a i n e d by the f l u i d when the f l u i d f l o w i s brought t o r e s t i s e n t r o p i c a l l y . The e n t h a l p y a t t h i s s t a g n a t i o n s t a t e i s r e l a t e d t o the o r i g i n a l s t a t e e n t h a l p y and k i n e t i c energy by d e f i n i t i o n s , the energy c o n s e r v a t i o n e q u a t i o n i m p l i e s t h a t the s t a g n a t i o n e n t h a l p y and s t a g n a t i o n t e m p e r a t u r e i n c r o s s i n g t h e shock t r a n s i t i o n a r e lowered by W/p^, and W/p^Cp r e s p e c t i v e l y . For the case of no energy l o s s (W=0), the s t a g n a t i o n e n t h a l p y remains c o n s t a n t t h r o u g h o u t . A l t h o u g h the energy remains c o n s e r v e d a c r o s s the shock, some of the k i n e t i c energy from the upstream f l o w i s c o n v e r t e d i n t o heat so as t o a t t a i n a h i g h e r t e m p e r a t u r e a t downstream. Thermodynamically t h i s i s an i r r e v e r s i b l e p r o c e s s because d i s s i p a t i o n has o c c u r r e d . The s p e c i f i c e n t r o p y i n c r e a s e , As, between the f i n a l and the i n i t i a l e q u i l i b r i u m s t a t e s ( n e g l e c t i n g the e f f e c t s of i o n i z a t i o n ) i s 2 (7.2) For an i d e a l gas, h=c pT, hence To=h 0/c p. A c c o r d i n g t o t h e s e 127 g i v e n as 5 5 6 (7.3) where y i s the r a t i o of s p e c i f i c h e a t s f o r an i d e a l gas. 7.2 A v a i l i b i l i t y and e f f e c t i v e n e s s In thermodynamics, the maximum amount of u s e f u l work which can be o b t a i n e d from a stea d y f l o w i s known as the a v a i l i b i l i t y a s s o c i a t e d w i t h the f l o w , i t can be w r i t t e n where s t a t e d i s the dead s t a t e of the system ( e.g. N.T.P.). The l o s s o f a v a i l i b i l i t y f o r a st e a d y f l o w system i s t h e r e f o r e e q u a l t o T h i s i s the work energy t h a t one can o b t a i n i f a r e v e r s i b l e e n g i ne i s used t o o p e r a t e between t h e i n t a k e and the output f l o w s . A g a i n f o r an i d e a l shock, both AH 0 and W a r e z e r o , and AS i s g i v e n by e q u a t i o n ( 7 . 3 ) . Hence the e n t r o p y p r o d u c t i o n a t shock t r a n s i t i o n r e s u l t s i n the d e g r a d a t i o n of the energy s t o r e d i n the f l o w even i f no energy was e x t r a c t e d d u r i n g the p r o c e s s . In an o t h e r words, a l e s s e r as 3 5 A - ( H o - T dS) - ( H o > ( J - T d S d ) , A. - A 0 • T.AS - AH 1 2 d o (7.4) 128 p o r t i o n of the energy c o n t e n t i n t h e f l o w can become u s e f u l work a f t e r the t r a n s i t i o n has t a k e n p l a c e . In f a c t , the e f f e c t i v e n e s s of a d e v i c e i s d e f i n e d as the r a t i o of the a c t u a l work o u t p u t , W0 , t o the maximum a v a i l a b l e o utput (as o b t a i n e d by a r e v e r s i b l e e n g i n e ) . Thus we have W o E " ( A j - A ^ '-Tw^risy . < 7 - 5 ) a Here W0 would be the e l e c t r i c a l power g e n e r a t e d by the s t a n d i n g shock g e n e r a t o r and W i s the t o t a l power l o s s which i n c l u d e s W0 and o t h e r r a d i a t i o n or c o n d u c t i o n heat l o s s e s . 7.3 E f f e c t i v e n e s s of the s t a n d i n g shock g e n e r a t o r To get some i d e a s about the e f f e c t i v e n e s s of the s t a n d i n g shock g e n e r a t o r , we assume W=W0 by n e g l e c t i n g the heat l o s s e s . Hence combining e q u a t i o n (7.3) and (7.5) p r o v i d e s an e s t i m a t i o n of the e f f e c t i v e n e s s as w. W 1+T.AS/W d T J c . p 9 p. = 1/U + -S-j-P-lImC-i) + l n ( - i ) ] } (7.6) " T v Y ^2 ' where r i s the mass f l o w r a t e t h r o u g h the shock wave. The p r e s s u r e r a t i o and the compr e s s i o n ( d e n s i t y ) r a t i o a c r o s s 129 the shock wave can be c a l c u l a t e d , a c c o r d i n g t o the method d e s c r i b e d i n appendix A, i n terms of the s t a n d i n g shock Mach number, M s, ( t h e energy l o s s , W / ( p , v 1 h 1 ) , i s s m a l l enough t o be n e g l e c t e d f o r our p u r p o s e ) . For s i m p l i c i t y , the shock i s a l s o assumed t o be weak such t h a t y and c remain c o n s t a n t . P L e t us d e f i n e the d i m e n s i o n l e s s parameter d p d ' where A T 0 i s the change i n s t a g n a t i o n t e m p e r a t u r e as d i s c u s s e d i n s e c t i o n 7.1. F i g u r e 7.1 d i s p l a y s the e f f e c t i v e n e s s a g a i n s t the s t a n d i n g shock Mach number f o r v a r i o u s v a l u e s of tn. G e n e r a l l y s p e a k i n g , the e f f e c t i v e n e s s of work energy e x t r a c t i o n from a " l e a k i n g " s t a n d i n g shock wave tends t o be l a r g e r f o r l a r g e r W 0 and f o r s m a l l e r Mach numbers. I t i s i n t e r e s t i n g t o f i n d out the e f f e c t i v e n e s s of the p r i m i t i v e s t a n d i n g shock g e n e r a t o r i n our p r e s e n t e x p e r i m e n t . From f i g u r e s 4.5 and 4.6, we f i n d the maximum power output from the g e n e r a t o r t o be about 0.5 watt ( w o E V 0 2 / 4 r ) . Meanwhile the t e s t f l o w has a mass f l o w r a t e of 0.07kg/sec and c p=500joule/kg-°K. The t o t a l change i n s t a g n a t i o n t e m p e r a t u r e ( A T 0 = W / P , v , C p ) i s t h e r e f o r e 1.5X10~ 2 degree. A c c o r d i n g t o e q u a t i o n ( 7 . 7 ) , the v a l u e of w o b t a i n e d f o r the s t a n d i n g shock g e n e r a t o r a t T d=300°K i s about 5 x 1 0 " 5 . On the o t h e r hand, the c o n i c a l shock b e i n g a t an a n g l e of around 50°, would have a s t a n d i n g shock Mach number near 1.2. At CJ = 5 X 1 0 " 5 and M s = 1.2, the e f f e c t i v e n e s s 130 10 cn to UJ UJ 10 <_J UJ UJ i d 3 -A 10 a;=io"3 1 .0 STANDING1 SHOCK MRCH NUMBER8(Ms) 2 ' ° F i g . 7.1 E f f e c t i v e n e s s of work e x t r a c t i o n from s t a n d i n g shock g e n e r a t o r . i s about 0.2 t o 0.3% ( i n d i c a t e d i n f i g u r e 7.1 by the shaded r e g i o n ) . C l e a r l y , the s m a l l e f f e c t i v e n e s s i s due t o t h e s m a l l v a l u e of W0 ( o r w) which we o b t a i n e d i n the e x p e r i m e n t . In f a c t , the AT 0 a s s o c i a t e d w i t h t h i s W0 i s u n r e a l i s t i c a l l y s m a l l . E f f o r t s must be made t o promote w by about two o r d e r s of magnitude i n o r d e r t o p l a c e the s t a n d i n g shock g e n e r a t o r a t a more e f f e c t i v e r e g i o n . These e f f o r t s w i l l i n c l u d e l o w e r i n g the work f u n c t i o n of the anode t o g i v e a 131 h i g h e r V 0 , c o n t r o l l i n g the boundary l a y e r s t o g i v e a s m a l l e r r , o p t i m i z i n g the e l e c t r o d e geometry and keeping the Mach number of the flow low. F i n a l l y , we must bear i n mind t h a t the e l e c t r i c a l work i s the f r i n g e b e n e f i t of u s i n g s t a n d i n g shock g e n e r a t o r s to reduce the s t a g n a t i o n temperature of the f l o w as opposed t o u s i n g o t h e r means of r e d u c i n g the s t a g n a t i o n t e m p e r a t u r e . Other schemes such as m i x i n g the w o r k i n g gas w i t h another c o l d gas would a l s o r e s u l t i n the l o s s of a v a i l i b i l i t y but g i v e no u s e f u l work done at a l l . 7.4 The o v e r a l l system U s i n g the concept of the s t a n d i n g shock g e n e r a t o r as a t o p p i n g d e v i c e t o the t u r b o g e n e r a t o r , we can c o n c e i v e the f o l l o w i n g system. The gas a t the heat source temperature of above 2500°K can be seeded w i t h low i o n i z a t i o n p o t e n t i a l compounds (such as p o t a s s i u m or cesium) t o enhance the charge d e n s i t y . The gas i s a l l o w e d t o f l o w through a c o n v e r g i n g - d i v e r g i n g n o z z l e t o a t t a i n s u p e r s o n i c speed. The e f f i c i e n c i e s of n o z z l e s and d i f f u s e r s are g e n e r a l l y v e r y h i g h (above 95%) f o r i s e n t r o p i c c o mpressions or e x p a n s i o n s 1 4 . S t a n d i n g shock g e n e r a t o r u n i t s a r e p o s i t i o n e d i n the s u p e r s o n i c f l o w t o e x t r a c t e l e c t r i c i t y . T h i s c o m b i n a t i o n of n o z z l e s and s t a n d i n g shock g e n e r a t o r s may be r e p e a t e d u n t i l the s t a g n a t i o n temperature has dropped s u b s t a n t i a l l y . F i g u r e 7.2 shows a p o s s i b l e c o n f i g u r a t i o n of an a r r a y of s t a n d i n g shock g e n e r a t o r s ; each u n i t i s a wedge i n s e r t e d 132 F i g . 7.2 An a r r a y of s t a n d i n g shock g e n e r a t o r s . i n t o the s u p e r s o n i c f l o w . C o n n e c t i n g the u n i t s i n s e r i e s t o a c h i e v e h i g h e r v o l t a g e may cause problem s i n c e the plasma i s q u i t e u n w i l l i n g t o su p p o r t a l a r g e g r a d i e n t of e l e c t r i c p o t e n t i a l . However, c o n n e c t i n g the u n i t s i n p a r a l l e l i s most l i k e l y s u i t a b l e as a l o w - v o l t a g e c u r r e n t s o u r c e . 7.5 C o n v e r s i o n e f f i c i e n c y of a s t a n d i n g shock g e n e r a t o r O b v i o u s l y the most i n t e r e s t i n g f i g u r e of e f f i c i e n c y i s the o v e r a l l engine e f f i c i e n c y . At t i m e s , the e f f i c i e n c y would a l s o be a f u n c t i o n of the ou t p u t power; i t i s t h e r e f o r e the e f f i c i e n c y a t maximum power t h a t we a r e most co n c e r n e d . U n f o r t u n a t e l y , t h e s e f i g u r e s a r e hard t o o b t a i n 133 because they r e q u i r e a complete a n a l y s i s of an e x i s t i n g system which i s i n o p e r a t i o n . The s t a n d i n g shock g e n e r a t o r i s o n l y c o n s i d e r e d as a t o p p i n g u n i t , i t i s not e x p e c t e d t o have a v e r y h i g h e f f i c i e n c y . The p r e v i o u s l y d e f i n e d e f f e c t i v e n e s s i s s t i l l the b e s t f i g u r e of m e r i t which can be used t o e v a l u a t e the system. N e v e r t h e l e s s , t h e r e i s one k i n d of e f f i c i e n c y which d e s e r v e s some a t t e n t i o n because i t i n v o l v e s the fundamental p h y s i c s o c c u r r i n g i n the c o n v e r s i o n p r o c e s s . T h i s i s the i d e a l e f f i c i e n c y of the c o n v e r s i o n of t h e r m a l energy i n t o e l e c t r i c a l work. As shown i n f i g u r e 7.3, the e l e c t r i c a l work done i s IV f f i. The s t e a d y f l o w and t h e ' shock wave steady; flew • standing shock T, Q 5 V / e cathode anode F i g . 7.3 The energy c o n v e r s i o n of the s t a n d i n g shock g e n e r a t o r . 134 m a i n t a i n the "heat b a t h s " a t T, and T 2; a s i t u a t i o n somewhat s i m i l a r t o the t h e r m i o n i c g e n e r a t o r 4 7 . The heat l o s s by t h i s T 2 b a t h t o the c o n v e r s i o n p r o c e s s i s the sum of t h e energy used t o l i b e r a t e the e l e c t r o n s from the anode s u r f a c e i n t o the plasma and the energy r e q u i r e d t o t h e r m a l i z e t h e s e e l e c t r o n s . These e l e c t r o n s e v e n t u a l l y 'leave the T 2 bath and e n t e r the T, b a t h , s u b s e q u e n t l y heat i s g i v e n t o the T, b a t h as the e l e c t r o n s "condense" onto the cathode s u r f a c e . The t h e r m a l e f f i c i e n c y of t h i s p r o c e s s can be w r i t t e n as IV n = i ( w _ + 1.5kT 2 /e) , ( 7 * 8 ) a where w a i s the work f u n c t i o n of the anode. I f the s t a n d i n g shock g e n e r a t o r has a l i n e a r l o a d l i n e , the maximum power o u t p u t i s a t V m=V 0/2 where V 0 i s the o p e n - c i r c u i t v o l t a g e which we may approximate ( f o r w a = w c ) t o be the s t a n d i n g shock p o t e n t i a l , v s h ' (see e q u a t i o n 2.9). F u r t h e r m o r e , i f one assumes t h a t the anode m a t e r i a l has a n e g l i g i b l e work f u n c t i o n , then the e f f i c i e n c y of c o n v e r s i o n becomes k AT 2e e 2e e2 T 2 - T l 3T 2 . ( 7 . 9 ) I t i s i n t e r e s t i n g t o see t h a t the e x p r e s s i o n o n l y d i f f e r s from the Car n o t e f f i c i e n c y by the f a c t o r 1/3. 1 35 CHAPTER 8. CONCLUSIONS 8.1 Summary and c o n c l u s i o n s In t h i s t h e s i s , a new concept of d i r e c t energy c o n v e r s i o n (D.E.C.) from a s u p e r s o n i c plasma f l o w i s p r o posed. T h i s method makes use of the p o t e n t i a l d i f f e r e n c e i n d u c e d by a s t a n d i n g shock wave i n a s u p e r s o n i c plasma; hence a d e v i c e which i s based on t h i s p r i n c i p l e i s c a l l e d " s t a n d i n g shock g e n e r a t o r " . The a u t h o r has found t h r o u g h t h e o r e t i c a l and e x p e r i m e n t a l s t u d i e s t h a t the e.m.f. of the g e n e r a t o r i s of the o r d e r of the temperature d i f f e r e n c e (measured i n eV) a c r o s s the shock. The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of the g e n e r a t o r i s found t o be a s t r a i g h t l i n e , s i m i l a r t o the one which belongs t o a b a t t e r y . Each c h a r a c t e r i s t i c i s t h e r e f o r e d e s c r i b e d by o n l y two p a r a m e t e r s , namely the open-c i r c u i t v o l t a g e and the c o n s t a n t s l o p e , the l a t t e r r e p r e s e n t s the i n t e r n a l r e s i s t a n c e of the g e n e r a t o r . E x p e r i m e n t a l r e s u l t s of the s t a n d i n g shock g e n e r a t o r c u r r e n t - v o l t a g e c h a r a c t e r i s t i c measurement a r e s u c c e s s f u l l y e x p l a i n e d by c o n s i d e r i n g the anode and cathode of the g e n e r a t o r as a p a i r of Langmuir doubl e - p r o b e s i n a s u p e r s o n i c plasma. The o p e n - c i r c u i t v o l t a g e i s shown t o be 136 an a l g e b r a i c sum of the bow shock e.m.f., the plasma sheath p o t e n t i a l s and the work f u n c t i o n s of both e l e c t r o d e s . The s l o p e i s found t o be a f u n c t i o n of the e l e c t r o n t e m p e r a t u r e s and i o n c u r r e n t s a t the e l e c t r o d e s . T h i s v a l u e of the i n t e r n a l r e s i s t a n c e i s two o r d e r s of magnitude l a r g e r than the bulk plasma r e s i s t a n c e . Hence the s c r e e n i n g of the boundary l a y e r s a t the e l e c t r o d e s u r f a c e s f o r b i d s the g e n e r a t o r from r e a c h i n g i t s maximum o u t p u t power l i m i t . Due t o the presence of boundary l a y e r s , the b e h a v i o u r of the s t a n d i n g shock g e n e r a t o r i s determined by the g e o m e t r i c c o n f i g u r a t i o n s of the e l e c t r o d e s . The f o l l o w i n g c o n d i t i o n s a r e b e l i e v e d t o be f a v o u r a b l e to the s t a n d i n g shock g e n e r a t o r : ( i ) l a r g e e l e c t r o d e s u r f a c e s , ( i i ) work f u n c t i o n b e i n g s m a l l f o r the anode but l a r g e f o r the ca t h o d e , ( i i i ) l a r g e i o n s a t u r a t i o n c u r r e n t s , ( i v ) s m a l l boundary l a y e r t h i c k n e s s , and (v) plasma f l o w b e i n g a t h i g h p r e s s u r e , h i g h charge d e n s i t y and h i g h v e l o c i t y . At p r e s e n t , the power output of the p r i m i t i v e s t a n d i n g shock g e n e r a t o r i s v e r y low due t o the presence of h i g h i n t e r n a l r e s i s t a n c e . Thermodynamic study done by the aut h o r has shown t h a t i t i s n e c e s s a r y t o improve the e f f e c t i v e n e s s of the g e n e r a t o r by a t l e a s t one o r d e r of magnitude i n o r d e r f o r i t t o become an a t t r a c t i v e t o p p i n g d e v i c e . 1 37 8,2 Important g e n e r a l r e s u l t s The s t a n d i n g shock g e n e r a t o r i s o p e r a t i n g l i k e a p a i r of d o u b l e - p r o b e s ; and the e l e c t r i c a l measurements i n the experiment have become more or l e s s l i k e those of plasma probe d i a g n o s t i c s . I t was i n t e r e s t i n g t o observe t h a t the g e n e r a t o r ' s l o a d l i n e was merely a s e c t i o n of the g e n e r a l double-probe c h a r a c t e r i s t i c b e i n g v o l t a g e s h i f t e d i n t o the p o s i t i v e v o l t a g e and p o s i t i v e c u r r e n t quadrant by the shock p o t e n t i a l . The a u t h o r found t h a t the work f u n c t i o n s of the e l e c t r o d e s p l a y e d an i m p o r t a n t r o l e i n d e t e r m i n i n g the measured v o l t a g e . T h i s f a c t i s o f t e n o v e r l o o k e d i n probe d i a g n o s t i c s . The work f u n c t i o n of the e l e c t r o d e can be d e t e r m i n e d by the method suggested i n s e c t i o n 4.7. L i k e w i s e , the presence of bow shock p o t e n t i a l must be c o n s i d e r e d when probe measurements a r e done i n s u p e r s o n i c plasma f l o w . E q u a t i o n (2.9) i s u s e f u l i n e v a l u a t i n g t h i s p o t e n t i a l . A g r e a t d e a l of e f f o r t has been put i n t o e v a l u a t i n g the i o n s a t u r a t i o n c u r r e n t s f l o w i n g i n t o the e l e c t r o d e s . The a u t h o r a p p l i e d both t h e o r e t i c a l and e x p e r i m e n t a l methods and the two r e s u l t s have e x c e l l e n t agreement. The e x p r e s s i o n f o r the i o n s a t u r a t i o n c u r r e n t ( e q u a t i o n 6.31), o r i g i n a l l y d e v e l o p e d by the a u t h o r by combining the t h i n s heath probe t h e o r y w i t h the s u p e r s o n i c v e l o c i t y and t h e r m a l boundary l a y e r s , s h o u l d be g e n e r a l l y u s e f u l f o r probe d i a g n o s t i c of a w e a k l y - i o n i z e d , s u p e r s o n i c plasma f l o w . 1 38 A major d i f f i c u l t y was e x p e r i e n c e d i n measuring the e l e c t r i c a l output from the g e n e r a t o r — t h e problem of i r r e p r o d u c i b i l i t y . T h i s i s i n f a c t a common problem i n d o i n g e l e c t r i c a l probe d i a g n o s t i c s . The author has found a way t o overcome t h i s d i f f i c u l t y i n the expe r i m e n t . R e p r o d u c i b l e r e s u l t s were o b t a i n e d by m o i s t e n i n g the newly formed e l e c t r o d e s u r f a c e w i t h water f o r a minute ( a f t e r s a n d i n g o f f the o l d one d u r i n g c l e a n i n g ) . In f a c t , w i t h o u t t h i s t e c h n i q u e , i t would have been i m p o s s i b l e t o c o n t i n u e the experiment i n the i n v e s t i g a t i o n of the s t a n d i n g shock g e n e r a t o r c h a r a c t e r i s t i c . I t i s a n t i c i p a t e d t h a t t h i s d i s c o v e r y w i l l advance the s t a t e of the a r t i n probe d i a g n o s t i c measurements. 8.3 S u g g e s t i o n s f o r f u t u r e work The b i g g e s t s h o r t c o m i n g of our experiment was the r e l a t i v e l y s h o r t d u r a t i o n (about 30 micro-second) of the a v a i l a b l e t e s t time p r o v i d e d by the shock induced s u p e r s o n i c f l o w . In r e a l l i f e , the s t a n d i n g shock g e n e r a t o r must o p e r a t e c o n t i n u o u s l y , t h u s f u t u r e e x p e r i m e n t a l s t u d i e s ought t o be con d u c t e d on a c o n t i n u o u s s t e a d y f l o w a p p a r a t u s such as the s u p e r s o n i c wind t u n n e l . A l o n g e r t e s t d u r a t i o n would a l s o make the experiment e a s i e r and l e s s time consuming: by c o n t i n u o u s l y v a r y i n g the e x t e r n a l l o a d i n g r e s i s t a n c e , the e n t i r e l o a d l i n e can be o b t a i n e d q u i c k l y whereas p r e s e n t l y the shock tube must be f i r e d many t i m e s i n o r d e r t o p l o t a s i n g l e l o a d l i n e . 1 39 In a ste a d y plasma f l o w , the e l e c t r o d e s u r f a c e s would be heated t o h i g h t e m p e r a t u r e . The t h e r m i o n i c e m i s s i o n s or secondary e m i s s i o n s of e l e c t r o n s from the s u r f a c e may become s i g n i f i c a n t . Hence, i t would be i n t e r e s t i n g t o extend the t h e o r y of the s t a n d i n g shock g e n e r a t o r t o i n c l u d e the e f f e c t of e l e c t r o n e m i s s i o n s . Moreover, the t e c h n o l o g y which i s a l r e a d y d e v e l o p e d f o r h a n d l i n g the e l e c t r o d e s u r f a c e s ( e . g . heat r e s i s t a n c e and work f u n c t i o n s ) i n t h e r m i o n i c g e n e r a t o r would a l s o be a p p l i c a b l e t o the s t a n d i n g shock g e n e r a t o r . F i n a l l y , no matter what the e l e c t r o d e geometry i s , the boundar l a y e r s and the plasma sheaths must be u n d e r s t o o d because t h e s e e f f e c t s o c c u r i n a l l plasma d e v i c e s which use e l e c t r o d e s t o conduct c u r r e n t s away from the plasma t o the e x t e r n a l w o r l d (e.g. the MHD and the t h e r m i o n i c g e n e r a t o r s ) . 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Mag. 36, 507-531, (1893). 49 S w i f t , J.D. and Schwar, M.J.R., ' E l e c t r i c Probes f o r Plasma D i a g n o s t i c s ' , I L I F F E Books L t d . , London, (1970) . 50 T a y l o r , G.I. and M a c c o l l , J.W., P r o c . Roy. Soc. A (London) 219, 278-297, (1933). 51 Tidman, D.A., Phys. F l u i d s U3, 1454-1459, (1975). 52 Tseng, R.C. and T a l b o t , L., AlAA J o u r n a l 9, 1365-1372, (1971) . 53 Weymann, H.D., Phys. F l u i d s 3, 545-548, ( i 9 6 0 ) . 54 Wiese, W.L. et a l , 'Atomic T r a n s i t i o n P r o b a b i l i t i e s ' , U.S. Dept. Of Commerce, N.B.S., NSRDS-NBS 22 V o l I I , (1966). 55 Z e l ' d o v i c h Ya.B. and R a i z e r Yu.P., ' P h y s i c s of Shock Waves and High-Temperature Hydrodynamic Phenomena', Academic P r e s s , New York, (1966). 56 B i e n k o w s k i , G.K., Phys. F l u i d s J_0, 381-391, (1967). 144 APPENDIX A. SHOCK WAVE RELATIONS The b a s i c g o v e r n i n g e q u a t i o n s f o r the two e q u i l i b r i u m s t a t e s a c r o s s a wave f r o n t a re the c o n s e r v a t i o n e q u a t i o n s f o r mass, f o r momentum and f o r energy. These e q u a t i o n s can be w r i t t e n as p.v - p v , 1 1 2 2 » .(A. I ) P i + p l v ! = p 2 + p 2 V 2 , (A.2) 1 2 . u W 1 2 _L u 2 V 1 + V - 2 V 2 + h 2 . ( A.3) where p, p, v and h a r e r e s p e c t i v e l y the p r e s s u r e , d e n s i t y , v e l o c i t y and s p e c i f i c e n t h a l p y . S u b s c r i p t 1 and 2 a r e used t o i n d i c a t e r e g i o n s upstream and downstream of the shock wave r e s p e c t i v e l y . The f u n c t i o n W r e p r e s e n t s the energy l o s s o r g a i n d u r i n g t h e t r a n s i t i o n t o new e q u i l i b r i u m . When W=0, e q u a t i o n s (A.1) t o (A.3) d e s c r i b e a normal i d e a l shock. I f a l l upstream f l o w parameters a r e s p e c i f i e d and g i v e n W=0, we a r e l e f t w i t h 3 e q u a t i o n s and 4 v a r i a b l e s ; which means the down stream f l o w parameters can be s o l v e d w i t h an a d d i t i o n a l e q u a t i o n of s t a t e . For example, we may use 145 h = c T = £ , (A.4) p y-l P f o r an i d e a l gas, where y i s the s p e c i f i c heat r a t i o . I f e q u a t i o n (A.4) i s used t o s u b s t i t u t e h, and h 2 i n e q u a t i o n ( A . 3 ) , and i f the upstream f l o w parameters a r e not s p e c i f i e d , we can e l i m i n a t e v 0 and v, w i t h e q u a t i o n s (A.1) and (A.2) t o o b t a i n the f o l l o w i n g e q u a t i o n P 2 (Y + D.P2 + ( Y - D P J P7 = ( Y - D P 2 + ( Y + 1 ) P J P 2 n p 2 / P j + 1 or Pl ~ P2^pl + ^ ' (A.5) where n= . T h i s e q u a t i o n i m p l i e s t h a t the r a t i o of d e n s i t y a c r o s s the shock can be d e t e r m i n e d once th e p r e s s u r e r a t i o i s known. Base on e q u a t i o n ( A . 5 ) , we can o b t a i n r e l a t i o n s amongst o t h e r r a t i o s such as T 2/T, or v , / a , ( t h e Mach number), where a 1 i s the upstream sound speed d e f i n e d as a = (YV2 P These shock wave r e l a t i o n s may be found f o r example i n r e f e r e n c e 1 0 . I n t e r e s t i n g l y , e q u a t i o n (A.5) shows t h a t the c o m p r e s s i o n r a t i o i s l i m i t e d t o n a l t h o u g h the p r e s s u r e r a t i o may get v e r y l a r g e f o r a s t r o n g shock. For a monatomic gas, n i s o n l y 4. The c a l c u l a t i o n of f l o w parameters becomes c o m p l i c a t e d 146 when e x c i t a t i o n or i o n i z a t i o n a r e not n e g l i g i b l e (W i s s t i l l e q u a l t o 0 ) . F i r s t of a l l , the v a l u e of c p i s no l o n g e r c o n s t a n t and we must l o o k f o r a more s u i t a b l e e q u a t i o n of s t a t e . Then we f i n d the gas c o n t a i n s many d i f f e r e n t s p e c i e s of p a r t i c l e s . Hence the t o t a l p r e s s u r e i s a sum of the p a r t i a l p r e s s u r e s , and the t o t a l d e n s i t y i s a sum of the p a r t i a l d e n s i t i e s . The temp e r a t u r e may be d i f f e r e n t f o r each s p e c i e s ; hence, f o r s i m p l i c i t y , t h e r m a l e q u i l i b r i u m i s o f t e n assumed. E q u a t i o n (A.4) i s o n l y v a l i d f o r an i d e a l gas. The o r i g i n a l d e f i n i t i o n of s p e c i f i c e n t h a l p y i s h 5 p + e (A.6) where e i s the s p e c i f i c i n t e r n a l energy. Thus the new e q u a t i o n of s t a t e can be found i f z can be e x p r e s s e d as a f u n c t i o n of the f l o w parameters p, p or T. T h i s i s not an easy s t e p i n g e n e r a l , but f o r a weakly s i n g l y i o n i z e d gas, can be ap p r o x i m a t e d by e . 3 £ + V (A.7) 2 p p Here, the f i r s t term on the r i g h t s i d e i s due t o the t r a n s l a t i o n a l k i n e t i c energy ( f o r monatomic gas w i t h 3 degrees of freedom). The second term i s t h e i o n i z a t i o n energy where H± i s the i o n d e n s i t y and I i s t h e i o n i z a t i o n p o t e n t i a l (=18350°K). Combining e q u a t i o n (A.6) w i t h e q u a t i o n (A.7) and w r i t e H± i n terms of the degree of - i o n i z a t i o n , a = Nj[/(N i+N n) = mali±/p, where ma i s the atomic 147 mass, we o b t a i n h - H + a < - l h (A. 8) a T h i s a can be r e l a t e d t o the f l o w parameters p and T through the Saha e q u a t i o n . N e t t * 2 has found the e x p r e s s i o n f o r a weakly i o n i z e d argon plasma: ° * { 0 - 7 5 [ - 5 7 2  2 1 l e x p ( - + (A 9) where £ i s the energy of the f i r s t e x c i t e d s t a t e of the s i n g l y i o n i z e d argon i o n (2062°K above the i o n ground l e v e l ) . E q u a t i o n s (A.8) and (A.9) t o g e t h e r become the new e q u a t i o n ( s ) of s t a t e i n s o l v i n g the shock r e l a t i o n s . In view of the c o m p l i c a t e d form of e q u a t i o n ( A . 9 ) , no a l g e b r a i c s o l u t i o n was found, but n u m e r i c a l i t e r a t i o n method on a computer i s s i m p l e . A d i f f e r e n t approach i s suggested f o r the case when W i n e q u a t i o n (A.3) i s not n e g l i g i b l e . By and l a r g e W i s caused by r a d i a t i o n l o s s , o r a b s o r p t i o n , or the d i f f u s i o n l o s s as i n our s t a n d i n g shock g e n e r a t o r . Hence, at the c o n d i t i o n when W i s s i g n i f i c a n t , the gas i s u s u a l l y a t h i g h temperature such t h a t i o n i z a t i o n must be c o n s i d e r e d . To s i m p l i f y the c a l c u l a t i o n , we keep the format of the e q u a t i o n of s t a t e t o the one shown i n e q u a t i o n (A.4) but s u b s t i t u t e by a v a r i a b l e e n t h a l p y c o e f f i c i e n t g. T h e r e f o r e we f i n d 2 h = _E_ £ (A.10a) g-1 P 1 48 or 8 E hT^p- = 7 (A.10b) O b v i o u s l y , a t low t e m p e r a t u r e , g=y. Combining e q u a t i o n s (A.1) t o (A.3) w i t h e q u a t i o n (A.10), a f t e r some l e n g t h y a l g e b r a i c s t e p s , we found the f o l l o w i n g e x p r e s s i o n s f o r the p r e s s u r e and d e n s i t y r a t i o s : " 1 + Tft - I T V ' 1 1 {14,,(»,-»(«2-»,»1>-PI'II'I> 1 (A.11a) 2 g 2 M 2 ( g 2 - l ) w -hrr^- u + e , ( g 1 - n ( g 2 - g 1 « i > ' p i v i h i ( A . l i b ) = 2 ( g 2 + 1 ) ( g r g 2 ) M i g i where E ( g j - 1 ) ( g ^ g j M * ) * and 1 1 7 g i P i These e x p r e s s i o n s remain a p p l i c a b l e f o r as l o n g as we can f i n d the a p p r o p r i a t e v a l u e of g, and g 2 f o r the upstream and downstream gas. Ag a i n one may chose an i t e r a t i v e p r o c e d u r e t o s o l v e the shock r e l a t i o n s ( t h i s time w i t h W not e q u a l 0 ) 3 . The s t e p s a r e the f o l l o w i n g : i ) assume g 2 e q u a l g,, a p p l y e q u a t i o n (A.11) t o 1 49 f i n d approximate s o l u t i o n s of the downstream f l o w parameters; i i ) d e termine a b e t t e r v a l u e of g 2 from these new parameters i n accordance w i t h the e q u i l i b r i u m c a l c u l a t i o n or w i t h e x p e r i m e n t a l accepted. d a t a 3 2 ; i i i ) use the a d j u s t g 2 v a l u e and a p p l y e q u a t i o n (A.11) a g a i n ; i v ) i t e r a t e u n t i l the s o l u t i o n s are s e l f - c o n s i s t e n t . We have found t h i s method of shock wave c a l c u l a t i o n t o be q u i t e e f f i c i e n t ; and the s o l u t i o n s a r e a l s o c o m p a t i b l e w i t h the more c o m p l i c a t e d model of c a l c u l a t i o n p u b l i s h e d i n the 1 i t e r a t u r e 2 8 . 1 50 APPENDIX B. THE I ON SATURATION CURRENT AND  THE AMBIPOLAR EQUATION In c h a p t e r 6, we a p p l i e d the t h e o r y of e l e c t r i c probe d i a g n o s t i c i n a f l o w i n g plasma t o e x p l a i n the c u r r e n t -v o l t a g e c h a r a c t e r i s t i c found i n the bow shock g e n e r a t o r . Due t o the s p e c i a l c o n f i g u r a t i o n of the e l e c t r o d e s , we c o n c e n t r a t e d our s t u d i e s on the double-probe t h e o r y . In a d d i t i o n , the t e s t f l o w plasma c o n d i t i o n s r e q u i r e s the t h e o r y t o be r e s t r i c t e d t o the continuum l i m i t c o l l i s i o n l e s s t h i n s h e a t h regime. E q u a t i o n (6.23) d e r i v e d i n s e c t i o n 6.6 f o r the s l o p e of the I-V c h a r a c t e r i s t i c was found t o be g e n e r a l l y a p p l i c a b l e under the assumptions of d o u b l e - p r o b e , f r o z e n e l e c t r o n t emperature and M a x w e l l i a n d i s t r i b u t i o n of e l e c t r o n v e l o c i t y . T h i s e q u a t i o n was i n f a c t i n good agreement w i t h the r e s u l t s found by C h u n g 1 6 f o r a p a i r of p a r a l l e l p l a t e s d o u b l e - p r o b e . E q u a t i o n (6.23) c a l l s f o r the e v a l u a t i o n of the i o n s a t u r a t i o n c u r r e n t . To t h i s end, we have d e r i v e d two i m p o r t a n t e q u a t i o n s i n c h a p t e r 6, e q u a t i o n (6.27) and (6 . 2 9 ) : P U 9 ^ + p v a 7 " 97 { p Da [ 97 + i+TE/TG 5 7 (T^ ] } " c ^ f , ( 6 > 2 7 ) 151 i i , T 3 y vT J i > J a (6.29) E q u a t i o n ( 6 . 2 9 ) , which o r i g i n a t e s from e q u a t i o n ( 6 . 2 0 ) , r e l a t e s the i o n s a t u r a t i o n c u r r e n t t o the g r a d i e n t of i o n d e n s i t y at the s h e a t h edge f o r a c o m p r e s s i b l e f l o w . A s i m i l a r e x p r e s s i o n had been o b t a i n e d by Tseng and T a l b o t 5 2 . F urthermore i f the f l o w can be c o n s i d e r e d as i n c o m p r e s s i b l e , the i o n s a t u r a t i o n f o r m u l a reduces t o those g i v e n by Lam 3 3 and by Smy" 6. The i o n d e n s i t y g r a d i e n t must be found by s o l v i n g the a m b i p o l a r e q u a t i o n (6.27) f o r the i o n d e n s i t y d i s t r i b u t i o n i n s i d e the a m b i p o l a r r e g i o n . In s e c t i o n 6.8, we have a v o i d e d s o l v i n g the a m b i p o l a r e q u a t i o n by making an a p p r o x i m a t i o n t h a t the i o n d e n s i t y mass f r a c t i o n reduced l i n e a r l y from the f r e e stream v a l u e t o z e r o a t the w a l l over a d i s t a n c e e q u a l t o the f l u i d - d y n a m i c boundary l a y e r t h i c k n e s s . The r e s u l t o b t a i n e d from t h i s a p p r o x i m a t i o n was v e r y c l o s e t o t h a t o b t a i n e d from the f o r m a l a n a l y s i s , d i f f e r i n g o n l y by a n u m e r i c a l c o n s t a n t . In p a r t i c u l a r , the i o n c u r r e n t d e n s i t y was found t o have the 1 / ( X ) ^ dependence which was i n accordance w i t h the e x p e r i m e n t a l o b s e r v a t i o n s o b t a i n e d by B u r k e 1 3 and by Scharfman and B r e d f e l d t 4 4 . In the r e m a i n i n g p a r t of t h i s a p p e n d i x , we s h a l l d i s c u s s the s o l u t i o n t o the a m b i p o l a r e q u a t i o n f o r the s h o c k - i n d u c e d boundary l a y e r f l o w i n our e x p e r i m e n t . The i n t e r a c t i o n between the a m b i p o l a r r e g i o n and the 1 52 f l u i d - d y n a m i c boundary l a y e r i s c l e a r l y i n d i c a t e d by the s i m i l a r i t y of the a m b i p o l a r e q u a t i o n t o the boundary l a y e r g o v e r n i n g e q u a t i o n s s t a t e d p r e v i o u s l y a s ' e q u a t i o n s (5.3) and ( 5 . 4 ) . As a matter of f a c t , the a m b i p o l a r e q u a t i o n i s i n e f f e c t the f i f t h g o v e r n i n g e q u a t i o n f o r the i o n i z e d boundary l a y e r f l o w . Thus we l e t the a m b i p o l a r e q u a t i o n t o take the same t r a n s f o r m a t i o n as the one a p p l i e d t o e q u a t i o n s (5.3) and (5.4) making use of the reduced d i s t a n c e n and the stream f u n c t i o n F. A f t e r t r a n s f o r m i n g , the f i r s t two terms on the l e f t hand s i d e of e q u a t i o n (6.27) become 8Z 3Z p U f r i 7, p u r — + p v r — = 7T- FZ' ox 8 y 2x , where the s u p e r s c r i p t ' stands f o r t a k i n g d e r i v a t i v e w i t h r e s p e c t t o n. The next ' l o n g ' term on the l e f t hand s i d e i s t r a n s f o r m e d i n t o p u f kZ' p u f k z r ' • *-2x ; V S } 2^x ; l S (1+?) 5J c c ' where c i s the n o r m a l i z e d t e m p e r a t u r e d e f i n e d as ?=T g/Tf, S & i s the Schmidt number d e f i n e d as S 3v/D 0 (where v i s the v i s c o s i t y c o e f f i c i e n t ) and k i s the parameter e q u a l t o ( y p ) / ( y w P w ) which i s n o r m a l l y e q u a l t o 1 a c c o r d i n g t o M i r e l 3 8 . The new a m b i p o l a r e q u a t i o n can be w r i t t e n as 1 53 r ' ( C ' ) 2 (1+2 c) z» + [ F S c - ( i n s c ) ' - T ^ z y l z ' - U" - U ( ) , + ^ ) 2x x „ u i (B. 1 'f - i f • Now l e t us examine the mass p r o d u c t i o n term on the r i g h t hand s i d e of t h i s e q u a t i o n . By d e f i n i t i o n u) iHm iN i, the r a t e of change of the i o n number d e n s i t y i s f i x e d by the i o n i z a t i o n - r e c o m b i n a t i o n r a t e of the n o n - e q u i l i b r i u m f l o w i n g plasma. Assuming Nj^Ng, t h i s r a t e can be w r i t t e n as' , « 0 *± = ^ VN^* " N P • (B'2) n where § i s the e l e c t r o n 3-body r e c o m b i n a t i o n c o e f f i c i e n t g i v e n f o r example as i n r e f e r e n c e 27. The term (N^ /N ) i s the Saha e q u a t i o n e q u i l i b r i u m r a t i o which i s a f u n c t i o n of o n l y the e l e c t r o n t e m p e r a t u r e . A l s o b e i n g a f u n c t i o n of the e l e c t r o n temperature i s the r e c o m b i n a t i o n c o e f f i c i e n t § . S i n c e we had been assuming f r o z e n e l e c t r o n t emperature i n the boundary l a y e r (see s e c t i o n 6.4), i t i s r e l a t i v e l y s i m p l e t o det e r m i n e t h e s e q u a n t i t i e s . W r i t i n g e q u a t i o n (B.2) i n terms of our n o t a t i o n s , we f i n d p p J C ^ m . P f Hence the r i g h t hand s i d e of e q u a t i o n ( B . l ) becomes -2S c[(^-)(§N. f)](l - |i) ^ (B.4) We note t h a t i f we d e f i n e x/uf as the flo w c h a r a c t e r i s t i c time and l/(§N i f) as the r e c o m b i n a t i o n c h a r a c t e r i s t i c t i m e , then the e x p r e s s i o n i n s i d e the b r a c k e t s i n e q u a t i o n (B.4) i s 1 54 s i m p l y the r a t i o of t h e s e two c h a r a c t e r i s t i c t i m e s i n the n o n - e q u i l i b r i u m f l o w plasma. When the f l o w c h a r a c t e r i s t i c time i s l a r g e i n comparison w i t h the r e c o m b i n a t i o n c h a r a c t e r i s t i c t i m e , the r e c o m b i n a t i o n e f f e c t i s i m p o r t a n t i n d e t e r m i n i n g the i o n number d e n s i t y d i s t r i b u t i o n i n the a m b i p o l a r r e g i o n , and when t h i s time r a t i o i s s m a l l , the e f f e c t i s n e g l i g i b l e . S u b s t i t u t i n g e x p r e s s i o n (B.4) i n t o e q u a t i o n ( B . 1 ) , the a m b i p o l a r e q u a t i o n i s ready t o be s o l v e d s i m u l t a n e o u s l y w i t h the boundary l a y e r e q u a t i o n s (5.3) and (5.4) as soon as we f i x the boundary c o n d i t i o n s . The b e h a v i o u r of Z on n i s e x p e c t e d l y s i m i l a r t o the v e l o c i t y (or temperature) p r o f i l e showing i n c h a p t e r 5, t h a t i s , we l i k e t o see Z approach 1 a s y m p t o t i c a l l y . We do not know b e f o r e hand the v a l u e of Z or Z' a t n=0 because i t i s a c t u a l l y our u l t i m a t e o b j e c t i v e t o l o o k f o r the v a l u e of Z'(0). N e v e r t h e l e s s we may f i n d an a p p r o x i m a t i o n f o r Z(0) by c o n s i d e r i n g the charge f l u x a t the s h e a t h edge. S i n c e we a r e d e a l i n g w i t h a t h i n s h e a t h i n our problem, the s c a l e l e n g t h of the a m b i p o l a r r e g i o n i s much l a r g e r than t h a t of the s h e a t h . T h e r e f o r e , i t i s v a l i d t o assume Z(0)=Z s or Z'(0)=Z' s. A c c o r d i n g t o the Bohm's she a t h c r i t e r i o n , e q u a t i o n ( 6 . 2 ) , the i o n c u r r e n t can be r e l a t e d t o the s h e a t h edge i o n d e n s i t y , thus J k eN . ( —) = — Z ( 0 ) ( — ( B . 5 ) i i s m. s m m. , x i i and J± can be e x p r e s s e d i n terms of Z'(0) as i n e q u a t i o n 1 55 (6. 2 9 ) . A more r i g o r o u s t r e a t m e n t on the c o l l i s i o n l e s s s h e ath due t o B i e n k o w s k i 5 6 p r o v i d e d a s i m i l a r boundary c o n d i t i o n : 2kT i T. J i - ^ i s ^ J ( 1 + T ^ s A ° , (B.6) i e ' where A 0 i s some number about e q u a l t o 1, and T^/T * s almost z e r o i n our c a s e . These boundary c o n d i t i o n s imply t h a t e q u a t i o n (B.1) must be s o l v e d by n u m e r i c a l i t e r a t i o n : f i r s t choose a v a l u e of Z ' ( 0 ) , determine Z(0) from e q u a t i o n ( B . 6 ) , then n u m e r i c a l l y i n t e g r a t e the a m b i p o l a r e q u a t i o n w i t h the boundary l a y e r e q u a t i o n s , the p r o p e r v a l u e of Z'(0) i s o b t a i n e d when Z i s found t o approach 1 a s y m p t o t i c a l l y at l a r g e rl. We had o b t a i n e d r e s u l t s which e v i d e n t l y showed t h a t Z(0) was a c t u a l l y s m a l l enough t o be taken as z e r o at a l l time w i t h o u t a f f e c t i n g the f i n a l s o l u t i o n . Z'(0) was found t o be about 0.1 f o r the t e s t f l o w behind a shock wave w i t h shock speed r a n g i n g from Mach 10 t o Mach 12 ( o n l y v e r y s m a l l change was found w i t h i n t h i s r a n g e ) . T u r n i n g our a t t e n t i o n t o e q u a t i o n (6.29) which d e s c r i b e s the i o n c u r r e n t d e n s i t y c o l l e c t e d by the probe, we found the second term i n s i d e the b r a c k e t i n the e q u a t i o n i s always s m a l l (even more so as we take Z 0 ) . The i o n c u r r e n t i s t h e r e f o r e s i m p l i f i e d as e C i f _ 3Z J i = p D a 37 y=0 . <B'7> 8 Z In s e c t i o n 6.8, we assumed = w n e r e i s t n e boundary 1 56 l a y e r t h i c k n e s s . Here we may f i n d U _ D T 9_z = 3_n „, = (_J__V)hz, 3y 9y  K2\i x J 3 3 w y = 0 U s i n g the d e f i n i t i o n of 6 ( i n e q u a t i o n 5.11) t o e x p r e s s the c o e f f i c i e n t i n f r o n t of Z', we o b t a i n 3_Z = K_5 1±„% 8y 6 T w n ( E - 8 y=0 For the v a l u e s of T f/T w=30 and Z'(0)=0.1 a c c o r d i n g t o the c o m p u t a t i o n , the r e s u l t of e q u a t i o n (B.7) becomes a f a c t o r of 4 l a r g e r than t h a t g i v e n by e q u a t i o n (6.31) i n c h a p t e r 6. However, except the n u m e r i c a l c o n s t a n t , the parameters i n both e q u a t i o n s are i d e n t i c a l . 

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