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A study of the fluid-dynamic characteristics of turbulent gas-liquid bubble plumes Castillejos-Escobar, Alfonso Humberto 1986

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A STUDY OF THE FLUID-DYNAMIC CHARACTERISTICS OF TURBULENT GAS-LIQUID BUBBLE PLUMES by ALFONSO H. CASTILLEJOS E. M e t a l l . Eng., U n i v e r s i d a d Nacional Autonoma de Mexico, 1 9 7 7 DIC, M.Sc., Imperial C o l l e g e of Science and Technology, 1 9 7 9 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of M e t a l l u r g i c a l Engineering) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1986 • A l f o n s o H. C a s t i l l e j o s E. 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of M E T A L L U R G I C A L E N G I N E E R I N G The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date J U L Y 1 5 t h , 1 9 8 6 . DE -6 (3/81) ABSTRACT The p h y s i c a l behaviour of air-water plumes during upward i n j e c t i o n i n ladle-shaped v e s s e l s has been I n v e s t i g a t e d . The study i n v o l v e d the experimental determination of the s p a t i a l d i s t r i b u t i o n of the flow parameters c h a r a c t e r i z i n g the behaviour of the gas phase : gas f r a c t i o n , bubble frequency, mean bubble v e l o c i t y and p i e r c e d length, and the spectrum of the bubble v e l o c i t y and p i e r c e d l e n g t h . A computer-aided e 1 e c t r o r e s i s t i v i t y sensor was developed to determine simultaneously the s e v e r a l l o c a l parameters. In p a r t i c u l a r the accurate measurement of the bubble r i s e v e l o c i t y and bubble p i e r c e d length, under the t u r b u l e n t c o n d i t i o n s s t u d i e d , n e c e s s i t a t e d an instrument capable of ensuring that s e q u e n t i a l voltage pulses o r i g i n a t i n g at the probe t i p s corresponded to a s i n g l e bubble t r a v e l l i n g a x i a l l y and undisturbed between the t i p c o n t a c t s . To achieve t h i s , s p e c i a l e l e c t r o n i c i n s t r u m e n t a t i o n and software was b u i l t to analyse, i n r e a l time, the s i g n a l s produced by the contact of the bubbles with the sensor. The extensive e v a l u a t i o n of the measuring system i n d i c a t e d a high accuracy and r e p r o d u c i b i l i t y of the r e s u l t s . The plumes were i n v e s t i g a t e d under v a r i o u s c o n d i t i o n s of a i r flow r a t e , o r i f i c e diameter and bath depth. The measurements i n d i c a t e that the r a d i a l gas f r a c t i o n p r o f i l e s , at d i f f e r e n t 11I a x i a l p o s i t i o n s i n the plume, e x h i b i t s i m i l a r i t y . The reduced gas f r a c t i o n p r o f i l e s can be approximated by a s i n g l e Gaussian d i s t r i b u t i o n f o r a l l the c o n d i t i o n s s t u d i e d . Thus a f u l l d e s c r i p t i o n of the s p a t i a l d i s t r i b u t i o n of gas could be obtained through the c o r r e l a t i o n of the a x i a l gas f r a c t i o n and h a l f - v a l u e r a d i u s with the modified Froude number. The development of flow i n g a s - l i q u i d plumes i s evidenced by changes in the bubble frequency, mean bubble v e l o c i t y and mean p i e r c e d length d i s t r i b u t i o n s . In the region c l o s e to the i n j e c t i o n p o i n t , there Is a steep change r a d i a l l y i n bubble v e l o c i t y and the motion of the bubbles i s s t r o n g l y a f f e c t e d by the gas i n j e c t i o n v e l o c i t y . Measurements of bubble frequency and p i e r c e d length i n d i c a t e that bubble break-up occurs i n t h i s zone before a dynamic process of break-up and coalescence e s t a b l i s h e s a n e a r l y constant bubble s i z e d i s t r i b u t i o n . In the region of f u l l y developed flow i n the plume, the mean bubble v e l o c i t y and the standard d e v i a t i o n of the bubble v e l o c i t y spectrum e x h i b i t r e l a t i v e l y f l a t r a d i a l p r o f i l e s and the bubbles a f f e c t the flow only through buoyancy. The s p e c t r a of bubble p i e r c e d length and diameter i n t h i s zone can be f i t t e d to a log-normal d i s t r i b u t i o n . I n j e c t i o n c o n d i t i o n s have only a s l i g h t Influence i n determining the s i z e of the bubbles i n t h i s r e g i o n . Close to the bath s u r f a c e a t h i r d zone i s i d e n t i f i e d i n which bubble v e l o c i t y decreases more r a p i d l y as l i q u i d begins to flow r a d i a l l y outward from the plume. A mathematical model proposed i n the l i t e r a t u r e f o r bubble plumes has been used f o r comparison with the experimental r e s u l t s . IV TABLE OF CONTENTS Page ABSTRACT . II TABLE OF CONTENTS IV LIST OF FIGURES : VI 11 LIST OF TABLES . . XV TABLE OF NOMENCLATURE . . XVI ACKNOWLEDGEMENTS XX CHAPTER 1 INTRODUCTION 1 1.1 Submerged Gas Je t s i n M e t a l l u r g i c a l Processes 1 CHAPTER 2 LITERATURE REVIEW 5 2.1 Bubbling and J e t t i n g Phenomena i n Submerged Gas I n j e c t i o n 5 2.2 Si z e of Gas Bubbles at Detachment 8 2.3 Si z e of Gas Bubbles a f t e r Detachment 13 2.4 Shape and D i s t r i b u t i o n of Flow Parameters i n Gas-Liquid Bubble J e t s 16 2.5 F l u i d Motion i n the Gas-L i q u i d Region 22 2.6 Techniques f o r Measuring Loc a l Gas-Liquid Bubble Flow Parameters 29 2.6.1 D e f i n i t i o n of Some Fundamental Q u a n t i t i e s D e s c r i b i n g Two-Phase Flow 31 2.6.2 E l e c t r o r e s i s t i v i t y Probes 37 2.6.3 O p t i c a l Probes 42 V 2.6.4 I s o k i n e t i c , D i f f e r e n t i a l Pressure and Impact Probes 44 2.6.5 Hot Film / Wire Anemometry 46 2.6.6 Laser-Doppler Anemometry 49 2.6.7 Miscellaneous Methods 51 CHAPTER 3 OBJECTIVES OF THE PRESENT WORK 53 3.1 Summary of Previous Work 53 3.2 O b j e c t i v e s 57 CHAPTER 4 EXPERIMENTAL APPARATUS AND CONDITIONS 58 4.1 Experimental Apparatus 58 4.1.1 The P h y s i c a l Model 60 4.1.2 The Ladle-Shaped V e s s e l 62 4.1.3 Nozzle and Gas D e l i v e r y System 63 4.1.4 The E1 ect r or es i s t i v i t y Probe 66 4.1.5 T r a v e r s i n g Mechanism 69 4.1.6 C o n d i t i o n i n g and Logic C i r c u i t 70 4.1.7 Computer, Counter/Timer and P a r a l l e l Input/Output I n t e r f a c e 75 4.1.8 Mi s c e l l a n e o u s Equipment 76 4.2 Co n d i t i o n s f o r the Tests and General Procedure 76 CHAPTER 5 SIGNAL ANALYSIS AND EVALUATION OF INSTRUMENT PERFORMANCE 80 VI 5.1 S i g n a l A n a l y s i s and D e f i n i t i o n of Measured Parameters 80 5.1.1 Local Gas F r a c t i o n 82 5.1.2 Local Bubble Frequency 82 5.1.3 Local Bubble Rise V e l o c i t y 83 5.1.4 P i e r c e d Length of Bubbles 92 5.2 Data A c q u i s i t i o n and Data Reduction 93 5.2.1. Program f o r Data A c q u i s i t i o n (GLJET700) 96 5.2.1.1 Frequency of Occurrence of the Accepted P a t t e r n Classes 97 5.2.2 Program f o r Data P r o c e s s i n g and Reduction (DATPRC) 99 Q 5.2.2.1 D i s c r i m i n a t i o n of Time-Delay (t ) According to P i e r c e d Length....? 100 5.2.3 Program f o r Data A c q u i s i t i o n (GLJET100) 106 5.2.4 Sample Size 107 5.2.5 Measurement Locations 109 5.3 C h a r a c t e r i s t i c s of the Response of the Sensor and the C o n d i t i o n i n g C i r c u i t 110 5.3.1 E f f e c t of Probe C h a r a c t e r i s t i c s on the Measurements I l l 5.3.2 Threshold Level 114 5.4 E v a l u a t i o n of Measuring System 116 5.4.1 Measurement of Rise V e l o c i t y of I n d i v i d u a l S pherical-Cap Bubbles 116 5.4.2 Gas Phase Volume Balance 119 CHAPTER 6 PRESENTATION AND DISCUSSION OF RESULTS 123 6.1 P r o f i l e s of Void F r a c t i o n 123 VI I 6.1.1 C o r r e l a t i o n s f o r the A x i a l Gas F r a c t i o n and the Half-Value Radius 137 6.2 P r o f i l e s of the Bubble Frequency 147 6.3 Bubble Rise V e l o c i t y and i t s Spectrum 154 6.3.1 A x i a l P r o f i l e s of Bubble Rise V e l o c i t y -Influence of I n j e c t i o n C onditions 165 6.3.2 Comparison of the Experimental Results with the P r e d i c t i o n s of the Model of Tacke et a l . 5 6 169 6.4 D i s t r i b u t i o n of Bubble P i e r c e d Length 172 6.4.1 D i s t r i b u t i o n of Bubble Diameters 179 CHAPTER 7 SUMMARY AND CONCLUSIONS 185 REFERENCES 190 APPENDIX I Speed of Displacement of a R i s i n g S p h e r i c a l Bubble 198 APPENDIX II Cond i t i o n s of the Experiments 201 APPENDIX III Data A c q u i s i t i o n and Data Reduction Programs 202 APPENDIX IV C a l c u l a t i o n of Gas Volume Flow Rate and Area Averaged Gas F r a c t i o n 231 APPENDIX V Model of Tacke et a l . f o r the Zone of F u l l y Developed Buoyant Flow 233 VIII LIST OF FIGURES Figure 2.1 Schematic diagram of homogeneous j e t showing constant v e l o c i t y core and v e l o c i t y p r o f i l e s at v a r i o u s d i s t a n c e s from the o r i f i c e 17 Figure 4.1 Schematic diagram of experimental f a c i l i t y 59 Figure 4.2 Photograph of the ladle-shaped model and a n c i l l a r y apparatus used i n the t e s t s 64 Fi g u r e 4.3 Nozzle holder and n o z z l e s , dimensions i n mm 65 Fi g u r e 4.4 D e t a i l s of e l e c t r o r e s i s t i v i t y probe f o r simultaneous gas f r a c t i o n and v e l o c i t y measurement, dimensions i n mm 67 Figure 4.5(a) Wiring diagram of e l e c t r o n i c c i r c u i t f o r e l e c t r o r e s i s t i v i t y probe measuring system 71 Figure 4.5(b) C o n t i n u a t i o n of w i r i n g diagram of e l e c t r o n i c c i r c u i t f o r e l e c t r o -r e s i s t i v i t y probe system 72 Figure 4.6 Schematic diagram of bubble sensor and s i g n a l s obtained 73 Figure 5.1 Voltage t r a c e s showing the modified d i g i t a l s i g n a l s generated by bubbles i n t e r c e p t i n g the sensor contacts 81 F i g u r e 5.2 Schematic diagram of the events o c c u r r i n g at the probe t i p s 85 Fi g u r e 5.3 Timing diagram of the d i f f e r e n t s i g n a l p a t t e r n s generated by the i n t e r a c t i o n of bubbles with the sensor under t u r b u l e n t c o n d i t i o n s 86 Fi g u r e 5.4 D i g i t a l s i g n a l p a t t e r n s generated under a c t u a l i n j e c t i o n c o n d i t i o n s , IX s h o w i n g t h e c o r r e s p o n d a n c e t o t h e c l a s s e s g i v e n i n F i g u r e 5.3 91 F i g u r e 5.5 B l o c k d i a g r a m o f t h e m e a s u r i n g p r o c e s s e s . . . 94 F i g u r e 5.6 F l o w d i a g r a m o f c o m p u t e r p r o g r a m f o r d a t a a c q u i s i t i o n and d a t a r e d u c t i o n p r o c e s s e s 95 F i g u r e 5.7 H i s t o g r a m s o f t h e p a t t e r n c l a s s e s g e n e r a t e d by t h e b u b b l e s a t d i f f e r e n t l o c a t i o n s i n t h e plume and f o r d i f f e r e n t gas f l o w r a t e c o n d i t i o n s 98 F i g u r e 5 . 8 ( a ) T y p i c a l b u b b l e v e l o c i t y d i s t r i b u t i o n s i n d i c a t i n g t h e v e l o c i t i e s a s s o c i a t e d t o p i e r c e d l e n g t h s l a r g e r t h a n t h e maximum a c c e p t e d 101 F i g u r e 5 . 8 ( b ) T y p i c a l b u b b l e v e l o c i t y d i s t r i b u t i o n s i n d i c a t i n g t h e v e l o c i t i e s a s s o c i a t e d t o p i e r c e d l e n g t h s l a r g e r t h a n t h e maximum a c c e p t e d 102 F i g u r e 5.9 L a r g e b u b b l e s o c c u r r i n g a t d i f f e r e n t p o s i t i o n s i n t h e plume, (Q = 876 Ncm / s , h^ = 400 mm, d = 6.35 mm, p r o b e a t c e n t r e l i n e z =°110 mm f o r a and b, z = 200 mm f o r c and d p h o t o g r a p h s ) 104 F i g u r e 5.10 P e r c e n t a g e o f a c c e p t e d t i m e d e l a y s as f u n c t i o n o f p o s i t i o n ; empty and f u l l s y m b o l s c o r r e s p o n d t o c e n t r e l i n e and plume b o u n d a r y l o c a t i o n s , r e s p e c t i v e l y 107 F i g u r e 5.11 E s t a b l i s h m e n t o f s a m p l e s i z e f o r s t a t i s t i c a l l y m e a n i n g f u l m e a s u r e m e n t s by e 1 e c t r o r e s i s t i v i t y p r o b e i n t u r b u l e n t g a s - l i q u i d p l u m e s 110 F i g u r e 5.12 C o m p a r i s o n o f s i g n a l s p r o d u c e d ( a ) i n t h e p r e s e n c e and (b) a b s e n c e o f l i q u i d b r i d g i n g b e t w e e n t h e s e n s o r c o n t a c t s 112 F i g u r e 5.13 I n f l u e n c e o f v o l t a g e s i g n a l a m p l i t u d e d i f f e r e n c e on t h e d e t e c t e d t i m e d e l a y , t C 113 g F i g u r e 5.14 O s c i l l o g r a p h s s h o w i n g s i g n a l s p r o d u c e d by b u b b l e s i n t e r c e p t i n g t h e l o w e r and up p e r c o n t a c t s o f t h e p r o b e , (Q = 876 Ncm / s , z = 65 mm, r = 0 mm) 115 X F i g u r e 5.15 V a r i a t i o n of i n d i c a t e d f l o w p a r a m e t e r s w i t h t h r e s h o l d l e v e l f o r d o u b l e - c o n t a c t e l e c t r o r e s i s t i v i t y s e n s o r 1 1 7 F i g u r e 5.16 F i g u r e 5.17 F i g u r e 5.18 B u b b l e v e l o c i t i e s r e p o r t e d by t h e probe (O) f o r s i n g l e s p h e r i c a l cap b u b b l e s , compared w i t h t h o s e measured from h i g h speed f i l m (•) and 1 3 4 c a l c u l a t e d from D a v i e s - T a y l o r r e l a t i o n s h i p 118 D i s c r e p a n c i e s between In p u t A i r R a t e s and A i r R a t e s o b t a i n e d from E q u a t i o n (5.17) 120 B u b b l e r i s e v e l o c i t y and p i e r c e d l e n g t h d i s t r i b u t i o n s measured n i n e days a p a r t . . . 122 F i g u r e 6.1 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r - w a t e r plume 124 F i g u r e 6.2 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r - w a t e r plume. ....... 125 F i g u r e 6.3 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r - w a t e r plume 126 F i g u r e 6.4 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r - w a t e r plume 127 F i g u r e 6.5 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r - w a t e r plume 128 F i g u r e 6.6 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r - w a t e r plume 129 F i g u r e 6.1(a) D i m e n s i o n l e s s r a d i a l gas f r a c t i o n p r o f i l e s a t d i f f e r e n t a x i a l d i s t a n c e s from th e n o z z l e i n an a i r - w a t e r plume ; symbols b e l o n g t o t h e t r a n s -v e r s e s e c t i o n s i n the c o r r e s p o n d i n g p r e v i o u s f i g u r e 131 F i g u r e 6.2(a) D i m e n s i o n l e s s r a d i a l gas f r a c t i o n p r o f i l e s a t d i f f e r e n t a x i a l d i s t a n c e s from th e n o z z l e i n an a i r - w a t e r plume ; symbols b e l o n g t o t h e t r a n s -v e r s e s e c t i o n s i n the c o r r e s p o n d i n g p r e v i o u s f i g u r e 132 XI Figure 6.3(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s -verse s e c t i o n s i n the corresponding previous f i g u r e 133 Figure 6.4(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s -verse s e c t i o n s i n the corresponding previous f i g u r e 134 Figure 6.5(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s -verse s e c t i o n s i n the corresponding previous f i g u r e 135 Figure 6.6(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s -verse s e c t i o n s i n the corresponding previous f i g u r e 136 Figure 6.7(a) Gas f r a c t i o n maps f o r d i f f e r e n t a i r -water bubble plumes 138 Figure 6.7(b) Gas f r a c t i o n maps f o r d i f f e r e n t a i r -water bubble plumes 139 Figure 6.7(c) Gas f r a c t i o n maps f o r d i f f e r e n t a i r -water bubble plumes 140 Figure 6.8 V a r i a t i o n of a x i a l gas f r a c t i o n with dimensionless d i s t a n c e from the nozzle f o r d i f f e r e n t a i r - w a t e r plumes 141 Figu r e 6.9 V a r i a t i o n of dimensionless h a l f - v a l u e r a d i u s with dimensionless d i s t a n c e from the nozzle f o r d i f f e r e n t a ir-water plumes 142 Figure 6.10 I n t e r c e p t s of the p a r a l l e l p o r t i o n of the l i n e s i n Figure 6.8 as f u n c t i o n of the modified Froude number 143 Figure 6.11 In t e r c e p t s of the l i n e s i n Figure 6.9 as f u n c t i o n of the modified Froude number 144 X I I Figure 6.12 C o r r e l a t i o n f o r the v a r i a t i o n of a x i a l gas f r a c t i o n with d i s t a n c e from the nozzle, i n a i r water-plumes 145 Figure 6.13 C o r r e l a t i o n f o r the v a r i a t i o n of the h a l f - v a l u e radius with d i s t a n c e from the n o z z l e , i n a i r water-plumes 146 Figure 6.14 Bubble frequency p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air- w a t e r plume 148 Figure 6.15 Bubble frequency p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air-water plume... 149 Figure 6.16 Bubble frequency p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air-water plume 150 Figure 6.17 Bubble frequency p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air-water plume 151 Figure 6.18 Bubble frequency p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air-water plume 152 Figure 6.19 Bubble frequency p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air-water plume 153 Figure 6.20 Bubble v e l o c i t y s p e c t r a at the plume c e n t r e l i n e 155 Figure 6.21 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r -water plume 157 Figure 6.22 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r -water plume 158 Figure 6.23 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r -water plume 159 Figu r e 6.24 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r -water plume 160 Figure 6.25 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r -water plume 161 Figu r e 6.26 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an a i r -water plume 162 XII I Figure 6.27 Local standard d e v i a t i o n of bubble v e l o c i t y spectrum f o r d i f f e r e n t gas flow rate c o n d i t i o n s 164 Figure 6.28 Mean bubble v e l o c i t y p r o f i l e s at the c e n t r e l i n e and boundary of plumes to i l l u s t r a t e the e f f e c t of gas i n j e c t i o n v e l o c i t y at two gas flow r a t e s 166 Fig u r e 6.29 Mean bubble v e l o c i t y p r o f i l e s at the c e n t r e l i n e and boundary of plumes formed under d i f f e r e n t gas flow rate c o n d i t i o n s 167 Figure 6.30 Mean bubble v e l o c i t y p r o f i l e s at the c e n t r e l i n e and boundary of plumes f o r two bath depth c o n d i t i o n s . . . . 167 F i g u r e 6.31 Comparison between experimental and p r e d i c t e d v a r i a t i o n of a x i a l gas f r a c t i o n h a l f - v a l u e r a d i u s and bubble v e l o c i t y , with d i s t a n c e from the i n j e c t i o n pointy P r e d i c t i o n s from model of Tacke et a l 171 Figure 6.32 Comparison between experimental and p r e d i c t e d v a r i a t i o n of a x i a l gas f r a c t i o n h a l f - v a l u e r a d i u s and bubble v e l o c i t y , with d i s t a n c e from the i n j e c t i o n poin^g P r e d i c t i o n s from model of Tacke et a l 173 Figure 6.33 Comparison between experimental and p r e d i c t e d v a r i a t i o n of a x i a l gas f r a c t i o n h a l f - v a l u e r a d i u s and bubble v e l o c i t y , with d i s t a n c e from the i n j e c t i o n pointy P r e d i c t i o n s from model of Tacke et a l 174 Figure 6.34 V a r i a t i o n of mean p i e r c e d length of bubbles with d i s t a n c e from the nozzle along the plume c e n t r e l i n e , f o r d i f f e r e n t i n j e c t i o n c o n d i t i o n s 175 Figure 6.35 Fast speed p i c t u r e s taken at 1500 frames/s of the breakup of bubbles i n the v i c i n i t y of the nozzle, (Q = 371Ncm /s, h b = 400mm, d = 6.35mm, probe at c e n t r e l i n e , z = 110 mm) 177 Figure 6.36 Fast speed p i c t u r e s of bubbles i n the region of developed buoyant plume (Q = 371Ncm /s, h = 400mm, d = 6.35mm. probe at c e n t r e l i n e , z = 200mm) 178 Figure 6.37 Log-normal p r o b a b i l i t y p l o t of bubble p i e r c e d length f o r d i f f e r e n t c e n t r e l i n e XIV p o s i t i o n s i n the region of developed flow of air-water plumes 180 Figure 6.38 Log-normal p r o b a b i l i t y p l o t of bubble diameter at the c e n t r e l i n e of the region of developed flow of air - w a t e r plumes 183 Figure 6.39 Geometric mean bubble diameter at the c e n t r e l i n e of the region of developed flow versus Reynolds number 184 Figure 6.40 Local geometric mean bubble diameter i n the region of developed flow of an a i r -water plume 184 Figure 1.1 Sketch of a bubble moving toward a double-contact sensor 198 XV LIST OF TABLES Table 4.1 Truth t a b l e f o r l o g i c c i r c u i t ... 74 Table 4.2 Experimental c o n d i t i o n s of the study 77 Table 5.1 Comparison of mean bubble v e l o c i t y and standard d e v i a t i o n obtained by GLJET700 and GLJET100 108 Table 5.2 V a r i a t i o n of i n d i c a t e d p r o p e r t i e s with probe t i p A length 112 Table II.1 Experimental c o n d i t i o n s of the study 201 XVI TABLE OF NOMENCLATURE 2 A r e a ; c r o s s - s e c t i o n a 1 a r e a of the plume, mm Lower t i p of e l e c t r o r e s i s t i v i t y s e n s o r , c o e f f i c i e n t i n E q . ( 2 . 3 0 ) . Nominal plume w i d t h w i t h r e s p e c t t o gas f r a c t i o n ; n o m i n a l plume w i d t h w i t h r e s p e c t to l i q u i d v e l o c i t y , mm. Upper t i p of e l e c t r o r e s i s t i v i t y s e n s o r , c o e f f i c i e n t i n E q . ( 2 . 3 0 ) . D e l a y p u l s e c h a n n e l . Added mass c o e f f i c i e n t . D r a g c o e f f i c i e n t . I n t e r c e p t s of t h e p a r a l l e l l i n e s a p p e a r i n g i n F i g u r e s 6.8 and 6.9 r e s p e c t i v e l y . L o c a l b u b b l e d i a m e t e r ; g e o m e t r i c mean l o c a l b u b b l e d i a m e t e r , mm. V e r t i c a l s e p a r a t i o n between l o w e r and upper c o n t a c t s of the p r o b e , mm. O r i f i c e d i a m e t e r , mm. Anemometer v o l t a g e , v. L o c a l b u b b l e t r a n s i t f r e q u e n c y , s D o p p l e r f r e q u e n c y , s F i e l d v a r i a b l e a s s o c i a t e d w i t h phase k. Drag f o r c e , N. 2 5 M o d i f i e d F r o u d e number = Q p /gd ( P , - p ) o go o 1 go 2 A c c e l e r a t i o n due t o g r a v i t y , (9.81 ra/s ). A t m o s p h e r i c p r e s s u r e ; B a t h d e p t h , mm. Phase i n d e x . XV11 Local bubble p i e r c e d length ; a r i t h m e t i c mean of l o c a l p i e r c e d lengtih ; geometric mean of l o c a l p i e r c e d length, mm. Mass of l i q u i d bath, kg. Slope of f a l l i n g edge of s i g n a l s , v s Subindex i n d i c a t i n g model. Space dimension, exponent i n Eq. (2.30), subindex i n Eq. (5.15) i n d i c a t i n g c l a s s i n t e r v a l with the l a r g e s t c e n t r a l value i n bubble v e l o c i t y histogram. Number of p i e r c e d lengths i n histogram i n t e r v a l i . Number of bubble v e l o c i t i e s c o l l e c t e d i n the experiment. Number of bubbles with diameter i n the i n t e r v a l iA - A/2 to iA + A/2, whose mean diameter i s i A. Subindex i n d i c a t i n g prototype. Atmospheric pressure, Pa. Instantaneous volume f l u x of phase k_j time averaged volume f l u x of phase k, m s 3 Gas flow rate at STP Ncm /s ; gas flow rate at given c o n d i t i o n s (atmospheric pressure plus s t a t i c head of water and 20 C), cm /s. Rad i a l c o o r d i n a t e across the plume, s t a t i s t i c d e f i n e d i n Eq.(5.15) ; radiu s of plume ; h a l f -value r a d i u s , mm. I n t e r n a l r a d i u s of o r i f i c e ; r a d i u s of bubble at detachment, mm. Radius of the v e s s e l , mm. V e l o c i t y of expansion of bubble s u r f a c e , ms 1 . C r o s s - c o r r e l a t i o n c o e f f i c i e n t . Reynolds number = 4Q p / TT d y . o go o V e r t i c a l d i s t a n c e of centre of bubble from centre of o r i f i c e , mm. XVI11 Standard d e v i a t i o n of bubble v e l o c i t y spectrum, ms t, t Time ; residence time of bubble at probe contacts , s u p e r s c r i p t s A and B stand for lower and upper contacts r e s p e c t i v e l y , s. C C t , t Time delay ; time delay d e f i n e d i n Fig.5.13, a * t Time r a t i o d e f i n e d i n Eq. (5.9). T, T^ Time of d u r a t i o n of experiment ; cummulative residence time of phase k, s. U, U K Z . , U q Flow v e l o c i t y ; flow v e l o c i t y of phase k i n d i r e c t i o n z ; l i q u i d v e l o c i t y , s u p e r f i c i a l v e l o c i t y of the gas at the o r i f i c e , ms U K u u> u u Local bubble v e l o c i t y ; average of l o c a l bubble b, b b max . .. , , , ° , v e l o c i t y ; average l o c a l bubble v e l o c i t y at plume c e n t r e l i n e , m/s. U^^, Bubble t e r m i n a l v e l o c i t y ; bubble t r a n s p o r t v e l o c i t y , ms V^, V ^ Volume of bubble; volume of bubble at detachment, mm V , V j , V Voltage l e v e l f o r gas ; v o l t a g e l e v e l f o r l i q u i d ; t h r e s h o l d voltage l e v e l , s u p e r s c r i p t s A and B represent lower and upper contacts r e s p e c t i v e l y , v . We Weber number = P L U 2 / o . x P o s i t i o n . X^(x,t) Phase d e n s i t y f u n c t i o n f o r phase k, dependent on p o s i t i o n and time. Y ( l ) , Y(2) Functions d e f i n e d i n Eqs. (V.10), (V.9). z V e r t i c a l c o ordinate along the plume, mm. a, a , <a> Loc a l gas f r a c t i o n ; l o c a l gas f r a c t i o n at plume c e n t r e l i n e ; area-averaged gas f r a c t i o n . g Bubbling f a c t o r . e Entrainment c o e f f i c i e n t . e^.e^ S p e c i f i c buoyancy power ; s p e c i f i c k i n e t i c power, Watt Kg Length of histogram i n t e r v a l . 2 Pressure drop, N/m . Wavelength of l i g h t , m, constant d e f i n e d i n Eq V. 6 Angle between i n c i d e n t and r e f l e c t i v e beam, angle d e f i n e d i n F i g . 1.1 ; j e t cone angle. Angle d e f i n e d i n F i g . 1.1 . Density of l i q u i d ; d e n s i t y of gas at o r i f i c e , g/cm . _ 3 Mean d e n s i t y of the two phase mixture, kg m V i s c o s i t y of a i r , cP. 2 -1 Kinematic v i s c o c i t y , cm s Time i n t e r v a l at which R j 2 * s a B i a x i m u m < s-Time averaging operator over T. Time averaging operator over T^. Space averaging operator over e n t i r e space domain. Space averaging operator over domain of phase k . XX ACKNOWLEDGEMENTS 1 would l i k e t o e x p r e s s my s i n c e r e g r a t i t u d e t o my s u p e r v i s o r , P r o f e s s o r J.K. Brimacombe f o r h i s c o n t i n u o u s a s s i s t a n c e and g u i d a n c e . I would a l s o l i k e t o thank P r o f e s s o r J.R. G r a c e f o r many h o u r s of u s e f u l d i s c u s s i o n . My s i n c e r e t h a n k s t o Mr. Andre K i n d s v a t e r f o r h i s f r i e n d s h i p and h i s s u p p o r t i n b u i l d i n g the e l e c t r o n i c i n s t r u m e n t a t i o n . My a p p r e c i a t i o n _ t o t h o s e f e l l o w s t u d e n t s who h e l p e d me i n many ways i n my work as w e l l as to M e s s r s . P. Wenman and E. K l a s s e n . The g e n e r o u s s u p p o r t o f t h e U n i v e r s i d a d N a c i o n a l Autonoma de M e x i c o , C o n s e j o N a c i o n a l de C i e n c i a y T e c n o l o g i a and t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada i s g r a t e f u l l y a c k n o w l e d g e d . F i n a l l y I want t o e x p r e s s my g r a t i t u d e t o my w i f e Lynda; her h e l p and u n d e r s t a n d i n g c o n s t i t u t e d the s t r o n g e s t s t i m u l u s d u r i n g my work. 1 CHAPTER 1 INTRODUCTION 1. 1 Submerged Gas Jets In M e t a l l u r g i c a l Processes Submerged gas i n j e c t i o n techniques are ubi q u i t o u s i n the smel t i n g and r e f i n i n g of metals both i n the ferro u s and nonferrous i n d u s t r i e s . The i n j e c t i o n of a r e a c t i v e gas has been used f o r a very long time to remove i m p u r i t i e s through chemical r e a c t i o n s o c c u r r i n g at gas/melt i n t e r f a c e s . More r e c e n t l y , the i n j e c t i o n of an i n e r t gas has been used to s t i r up the melt to produce homogeneous c o n d i t i o n s w i t h i n the bath and to promote slag-metal r e a c t i o n s . V a r i a t i o n s i n submerged gas i n j e c t i o n techniques i n v o l v e d i f f e r e n c e s i n gas blowing r a t e s (0.01 to 5 3 Nm /min tonne), d i r e c t i o n of i n j e c t i o n (bottom, top, l a t e r a l and combined i n j e c t i o n ) , i n j e c t i o n method (tuyere, porous plug and lance) and m a t e r i a l s i n j e c t e d ( r e a c t i v e gas, i n e r t gas and gas-powder mi x t u r e s ) . In the f e r r o u s i n d u s t r y there has been a very r a p i d growth 1 2 in the u t i l i z a t i o n of submerged gas i n j e c t i o n . The Q-BOP ' process f o r the r e f i n i n g of p i g i r o n r e v i v e d bottom blown i n j e c t i o n i n steelmaking. T h i s process uses c o n c e n t r i c tuyeres, placed at the bottom of the converte r , to i n j e c t oxygen, lime and hydrocarbons. The Q-BOP o f f e r s improved i n t e r a c t i o n between s l a g and metal and higher r a t e s of d e c a r b u r i z a t i o n than the BOF process. Thus r e c e n t l y , a number of processes have emerged that r e t a i n the BOF top blown c o n f i g u r a t i o n and advantages e.g. e a r l y 2 s l a g formation and high scrap to metal r a t i o u t i l i z a t i o n , but also allow f o r bottom i n j e c t i o n of i n e r t (LD-KG, LBE) or o x i d i z i n g (LD-OB, LD-OTB) g a s e s . 3 - 6 The A O D 7 p r o c e s s f o r the p r o duction of s t a i n l e s s s t e e l u t i l i z e s c o n c e n t r i c tuyeres, set at the s i d e s of the r e a c t o r , to i n j e c t argon and oxygen. Secondary steelmaking has developed i n t o a wide v a r i e t y of 9-11 l a d l e m e t a l l u r g i c a l treatments. The treatments extend from simple p r a c t i c e s to improve d e o x i d a t i o n and a l l o y a d d i t i o n , to s o p h i s t i c a t e d l a d l e processes which can supply heat, i n j e c t m a t e r i a l s and vacuum degas. The non-ferrous i n d u s t r y a l s o makes extensive use of submerged gas j e t s . Today, c o n v e r t i n g of copper mattes i s almost u n i v e r s a l l y c a r r i e d out i n Pierce-Smith c o n v e r t e r s . The P i e r c e -Smith converter i s a h o r i z o n t a l c y l i n d r i c a l v e s s e l with a row of h o r i z o n t a l l y d i r e c t e d tuyeres that blow a i r i n t o matte to produce 1 2 b l i s t e r copper The r e s u l t i n g b l i s t e r copper i s f u r t h e r r e f i n e d by the i n j e c t i o n of o x i d i z i n g and reducing gases i n an anode furnace. The slag-fuming furnace i s another process that u t i l i z e s submerged i n j e c t i o n of a i r and carbon to recover z i n c and t i n from t h e i r r e s p e c t i v e s l a g s . The z i n c fuming furnace i s a r e c t a n g u l a r shaped v e s s e l with two rows of h o r i z o n t a l l y d i r e c t e d 1 3 tuyeres, one on e i t h e r s i d e of the v e s s e l This review of i n d u s t r i a l submerged processes has been n e c e s s a r i l y b r i e f but serves to o u t l i n e the v a r i a t i o n s that e x i s t and t h e i r widespread a p p l i c a t i o n i n the metal p r o c e s s i n g 3 i n d u s t r y . A more complete d e s c r i p t i o n may be found i n the papers by F e l s k i et a l 1 4 , Upadhya 1 5 and Themelis et a l . 1 6 . Submerged gas i n j e c t i o n techniques have been adopted widely f o r the f o l l o w i n g reasons e f f i c i e n t mixing of the l i q u i d bulk, l a r g e s urface area produced f o r r e a c t i o n , easy c o n t r o l of process v a r i a b l e s such as gas flow r a t e and r a t e of a d d i t i o n , r e l a t i v e s i m p l i c i t y and low c a p i t a l c o s t s . However, a number of problems a l s o can be found such as: severe bath s l o p p i n g , poor u t i l i z a t i o n of c e r t a i n a d d i t i o n s , tuyere e r o s i o n and blockage and r e f r a c t o r y wear. The o r i g i n of most of the advantages and problems of gas i n j e c t i o n systems can be traced back to t h e i r f l u i d flow behaviour induced by the buoyancy of the d i s c h a r g i n g gas and to the behaviour of the j e t i t s e l f , as S z e k e l y 1 and Brimacombe have pointed out. In a s i m p l i f i e d r e p r e s e n t a t i o n , a g a s - s t i r r e d melt can be considered to c o n s i s t of two regions : a l i q u i d r e c i r c u l a t i n g zone and a gas d i s p e r s e d - 1 i q u i d zone. A review of the l i t e r a t u r e shows that most of the t h e o r e t i c a l and experimental work has focussed on the l i q u i d zone and on the zone of g a s - l i q u i d c l o s e to the i n j e c t i o n p o i n t . The g a s - l i q u i d bubble zone has r e c e i v e d l e s s a t t e n t i o n due to the a n a l y t i c a l and experimental d i f f i c u l t i e s that the study of two-phase flow systems e n t a i l s 1 7 , 1 9 . Two-phase flow i s not yet an area i n which t h e o r e t i c a l p r e d i c t i o n of flow parameters i s g e n e r a l l y p o s s i b l e . Thus, the r o l e of experiments and measurements of flow parameters i s 4 p a r t i c u l a r l y important. Measurement of time-averaged and space-averaged v o i d f r a c t i o n and v e l o c i t y of the phases i s r e q u i r e d to confirm current modeling e f f o r t s . Measurements are a l s o r e q u i r e d to v e r i f y hypotheses r e g a r d i n g the shape and i n t e r r e l a t i o n of the p r o f i l e s of vo i d f r a c t i o n , bubble passage frequency and v e l o c i t y p r o f i l e s . Measurements a l s o may provide u s e f u l dimensionless c o r r e l a t i o n s e x p r e s s i n g the v a r i a t i o n of vo i d f r a c t i o n , v e l o c i t y and turbulence with energy input. The experimental i n f o r m a t i o n produced i n t h i s work represents an e f f o r t i n these d i r e c t i o n s . The general aims of t h i s i n v e s t i g a t i o n were to develop s i g n a l i n t e r p r e t a t i o n techniques that allow r e l i a b l e measurement of two-phase flow parameters i n g a s - l i q u i d bubble j e t s and to provide i n f o r m a t i o n of the kind mentioned i n the previous paragraph. The parameters measured were l o c a l time-averaged q u a n t i t i e s : gas f r a c t i o n , bubble frequency, bubble v e l o c i t y and p i e r c e d l e n g t h , as well as the spectrum of bubble v e l o c i t y and p i e r c e d length. The s p e c i f i c o b j e c t i v e s of t h i s work are presented i n Chapter 3. CHAPTER 2 LITERATURE REVIEW 2.1 Bubbling and J e t t i n g Phenomena in Submerged Gas I n j e c t i o n The gas i n j e c t e d i n t o a molten bath may discharge i n the form of d i s c r e t e bubbles or i n the form of a j e t depending on the k i n e t i c energy of the gas and on the p h y s i c a l p r o p e r t i e s of the gas and l i q u i d . In the metal i n d u s t r y , both regimes of discharge are important. In steelmaking processes, using tuyeres, gas i s i n j e c t e d at v e l o c i t i e s of 200-300 m/s and discharges i n the 18 16 Ifi 20 j e t t i n g regime, e.g. Q-BOP , STB , K-BOP , AOD . On the other hand, processes using porous plugs f o r i n j e c t i o n have gas discharge v e l o c i t i e s of only lm/s and operate i n the bubbling 16 10 21 regime, e.g. LBE , LF , VOD . In the non-ferrous i n d u s t r y , processes normally i n v o l v e i n j e c t i o n i n the bubbling regime e.g. 18 13 Pierce-Smith converter , s l a g fuming furnace and anode 20 furnace . Thus any s u c c e s s f u l comprehension of these processes r e q u i r e s an understanding of the mechanism of bubble formation and i t s t r a n s i t i o n i n t o long gas envelopes or j e t s . As a consequence, much in f o r m a t i o n has been generated on the phenomena o c c u r r i n g c l o s e to the point of i n j e c t i o n . Three d i f f e r e n t flow regimes have been i d e n t i f i e d as a 2 2-27 f u n c t i o n of gas flow r a t e At very low gas flow r a t e s (Re < 500), the process of bubble formation i s a s t a t i c one. The bubble grows u n t i l the buoyancy exceeds the s u r f a c e t e n s i o n f o r c e 6 2 8 h o l d i n g i t to the nozzle The bubbling frequency i s p r o p o r t i o n a l to the gas flow r a t e and the bubble s i z e i s almost constant. At higher gas flow r a t e s (500 < Re < 2100) the process of bubble formation becomes dynamic. The s i z e of the bubbles i s determined by buoyancy and i n e r t i a l f o r c e s that r e s u l t from the displacement of the surrounding f l u i d . The bubble volume in c r e a s e s with gas flow r a t e while the frequency of formation 2 2 remains almost constant With i n c r e a s i n g gas flow r a t e , the forming bubble becomes a f f e c t e d by the wake of the preceding 2 4 2 9 30 bubble ' ' and l a r g e and small bubbles s t a r t to form i n p a i r s . As the flow rate f u r t h e r i n c r e a s e s , coalescence takes place more f r e q u e n t l y and c l o s e r to the o r i f i c e . F i n a l l y at very high flow rates the j e t t i n g regime appears i n which a continuous 18 gas channel i s formed c l o s e to the nozzle During bubble formation, the pressure i n s i d e the bubble decreases due to the upward movement of i t s c e n t r o i d and t h e r e f o r e the gas flow r a t e may vary with time. If there i s a r e s t r i c t i o n that produce a high pressure drop between the gas r e s e r v o i r and the o r i f i c e , the pressure f l u c t u a t i o n s due to forming bubbles w i l l not have an e f f e c t on the pressure drop. Then, the gas flow r a t e d u r i n g bubble formation can be considered as constant. If the volume of the r e s e r v o i r i s l a r g e compared with the volume of the bubbles being formed, the pressure i n the chamber w i l l not s i g n i f i c a n t l y change. This corresponds to the case of bubble formation under constant pressure c o n d i t i o n s . 7 A large number of models for p r e d i c t i n g bubble formation under d i f f e r e n t c o n d i t i o n s have evolved. They have been reviewed 31 32 by C l i f t , Grace and Weber and Kumar and Kuloor . A l l the models are mechanistic and depend on a f o r c e balance to d e s c r i b e the e v o l u t i o n of bubble growth. The t r a n s i t i o n from bubbling to j e t t i n g has been 1 8 i n v e s t i g a t e d by Hoefele and Brimacombe They s t u d i e d the discharge of gas through a h o r i z o n t a l nozzle i n t o water, a z i n c c h l o r i d e s o l u t i o n and mercury. Based on the i n t e r p r e t a t i o n of tuyere pressure t r a c e s and high speed f i l m s , they were able to represent the behaviour of the d i s c h a r g i n g gas i n a flow p a t t e r n map. The flow regimes - bubbling, t r a n s i t i o n and j e t t i n g - were determined by the modified Froude number and the r a t i o of the d e n s i t y of the gas to that of the l i q u i d . The j e t regime was d e f i n e d as that c h a r a c t e r i z e d by the continuous presence of a short s t a b l e gas envelope at the nozzle t i p . For systems with a low g a s - l i q u i d d e n s i t y r a t i o , choked flow i n the tuyere i s necessary f o r steady j e t t i n g ; f o r systems with a g r e a t e r gas-l i q u i d d e n s i t y r a t i o , steady j e t t i n g can be achieved under c o n d i t i o n s i n v o l v i n g smaller gas v e l o c i t i e s i n the tuyere. Mori 33 34 et a l . ' found that the t r a n s i t i o n from bubbling to j e t t i n g was achieved once the v e l o c i t y of the gas at the nozzle exceeded the s o n i c v e l o c i t y i r r e s p e c t i v e of the p h y s i c a l p r o p e r t i e s of the 3 5 l i q u i d and o r i f i c e diameter. McNallan and King have l a t e r proposed that the j e t regime i s e s t a b l i s h e d when the mass f l u x of 2 the gas i s g r e a t e r than 40 g/cm s 8 As w i l l be d e s c r i b e d l a t e r , s h o r t l y a f t e r i n j e c t i o n , both j e t s and large bubbles break up i n t o a multitude of small bubbles. These bubbles e n t r a i n l i q u i d and create a two-phase bubble flow which r i s e s to the s u r f a c e , to produce a r e c i r c u l a t i n g motion i n the bath. 2 . 2 S i z e of Gas Bubbles at Detachment Many models have been proposed to p r e d i c t the s i z e of the bubbles that detach from a gas source. The f i r s t f u l l y t h e o r e t i c a l models, f o r a vi s c o u s l i q u i d and l a t e r f o r an 3 6 3 7 i n v i s c i d l i q u i d , were developed by Davidson and Schuler ' . In one of t h e i r models, which has r e c e i v e d wide a t t e n t i o n , they assumed constant flow, an i n v i s c i d l i q u i d , and that the bubble behaves as a r i g i d expanding sphere growing from a point source. The v e r t i c a l motion of the bubble was considered to be determined by a balance between the buoyancy fo r c e and the a c c e l e r a t i o n of the f l u i d around the bubble ; the mass of t h i s l i q u i d swept by the moving bubble i s considered through an added mass c o e f f i c i e n t . N e g l e c t i n g the mass of the bubble, the balance may be w r i t t e n as Pj Vfe g = d_ ( C a Pj Vfe ds) ( 2 . 1 ) C o n s i d e r i n g that detachment occurs when the bubble centre has t r a v e l l e d a d i s t a n c e equal to i t s r a d i u s , i . e . s = r b f i n t e g r a t i o n of Equation ( 2 . 1 ) produces a f i n a l bubble volume of 9 V 6g -3/5 6/5 (2.2) b.f For a s p h e r i c a l bubble forming at an o r i f i c e i n a f l a t p l a t e , the added mass c o e f f i c i e n t has a value of 11/16 ; then the bubbling c o e f f i c i e n t i n Equation (2.2) becomes 1 . 3 7 8 3 6 , 3 7 . This value produced good agreement with experiments i n aqueous systems with i n j e c t i o n r a t e s i n the range 1.5 to 20 cm /s. However, the agreement becomes i n c r e a s i n g l y poor f o r higher flow r a t e s as the observed bubble volumes are smaller than p r e d i c t e d . This was i n t e r p r e t e d as being due to the upward current induced by previous bubbles and to the deformation of the bubble forming at the o r i f i c e . An e m p i r i c a l expression s i m i l a r to Equation (2.2) with 3 = 1.725 had been p r e v i o u s l y developed by Van Krevelen and H o f t i j z e r 3 8 . 3 7 The Davidson-and-Schuler model was adapted by Walters and 39 Davidson to represent bubble formation i n a f r e e - s t a n d i n g n o z z l e . This was achieved by using an added mass c o e f f i c i e n t of 1/2, which corresponds to the motion of a sphere i n an i n f i n i t e i d e a l f l u i d . For t h i s case Equation (2.2) with 3 = 1.138 gave good agreement with experiments, f o r flow r a t e s i n the range of 10 to 10000 cm 3/s. 4 0 Wraith s t u d i e d bubble formation during v e r t i c a l downward i n j e c t i o n . He developed a model which reduced to that of Walters 3 9 and Davidson when the r a d i u s of the lance i s small or the gas 4 0 flow rate i s l a r g e . Wraith pointed out that as the gas flow rate i n c r e a s e s , the bubble continues to grow a f t e r i t s base has l e f t the gas source. The a d d i t i o n a l growth i s due to a 10 connecting stem of gas. The f i n a l volume i s t h e r e f o r e l a r g e r than the value p r e d i c t e d by Equation (2.2) with 3 = 1.138 The c o n d i t i o n of gas c u t - o f f to a forming bubble was f u r t h e r 41 i n v e s t i g a t e d by Wraith and Kakutani They found that the s e v e r i n g of the connecting stem occurred a f t e r the bubble has r i s e n to a c e r t a i n d i s t a n c e and the pressure behind the bubble has r i s e n s h a r p l y . They i n c o r p o r a t e d these o b s e r v a t i o n s i n t o a simple one-stage model and found 3 = 1.39 f o r bubble formation i n a f r e e - s t a n d i n g nozzle and 3 = 1.54 f o r formation at a p l a t e 4 2 32 o r i f i c e . Wraith and Kumar and Kuloor a l s o have proposed two-stage models f o r bubble formation at p l a t e o r i f i c e s . The models c o n s i s t of a f i r s t stage dominated by i n e r t i a l f o r c e s and a second stage dominated by buoyancy. The c o n s i d e r a t i o n s regarding bubble shape e v o l u t i o n are d i f f e r e n t i n the two models. The 4 2 3 2 Wraith model y i e l d s B = 1.09 and the Kumar and Kuloor model g i v e s , a f t e r some s i m p l i f i c a t i o n s , 6 = 0.976. Hoefele and 1 8 Brimacombe , working on a one tuyere model of a copper conver t e r , i n d i c a t e d that bubble formation can be adequately d e s c r i b e d by Equation (2.2) with 3 = 1.57 or 0.88 i n mercury and aqueous baths r e s p e c t i v e l y . It i s i n t e r e s t i n g to observe that the v a r i o u s equations mentioned p r e v i o u s l y d i f f e r only i n the value of the c o e f f i c i e n t , 3. d e s p i t e the f a c t that they have been based on d i f f e r e n t mechanisms or that they have been adj u s t e d to apply to systems with d i f f e r e n t p h y s i c a l p r o p e r t i e s . A f a c t o r that becomes important at high i n j e c t i o n v e l o c i t i e s i s the i n e r t i a of the gas i s s u i n g from the n o z z l e . Davidson and 3 6 Schuler proposed that the net e f f e c t of the gas momentum was to 11 exert an a d d i t i o n a l f o r c e f o r bubble detachment. Nilmani and 4 3 Robertson i n c l u d e d the gas momentum i n the equation of motion given i n Equation (2.1). They found that t h e i r model agreed with the Davidson-and-Schuler model f o r a small g a s - l i q u i d d e n s i t y r a t i o . However, f o r an air-water system the p r e d i c t i o n s d i f f e r e d 2 9 c o n s i d e r a b l y . McCann and P r i n c e have pointed out that at high flow r a t e s , wake e f f e c t s and bulk c i r c u l a t i o n a f f e c t bubble growth ; the bubble that forms i s elongated and grows so r a p i d l y that i t coalesces with the previous bubble at some d i s t a n c e from the o r i f i c e . As the flow ra t e i n c r e a s e s the coalescence point moves c l o s e r to the o r i f i c e . Equation (2.2) contains no dependence upon o r i f i c e diameter s i n c e the gas was assumed to issue from a point source. The e f f e c t of a f i n i t e o r i f i c e diameter was considered by Davidson 3 6 and Schuler i n t h e i r model, they assumed that a r e s i d u a l sphere of gas was l e f t a f t e r the r e l e a s e of a bubble and that the detachment occurred when s = r. _ + r . I n t h i s case the model b , f o has no simple a n a l y t i c s o l u t i o n although i t does p r e d i c t that the bubble volume w i l l i n c r e a s e with i n c r e a s i n g o r i f i c e diameter. 2 2 Davidson and Amick proposed the f o l l o w i n g e m p i r i c a l r e l a t i o n f o r bubble s i z e V. . = 0.11 (Q o P 0.5 0.867 b, f o o 3 4 4 f o r flow r a t e s up to 250 cm /s i n water. Sano et a l . , during an i n v e s t i g a t i o n of bubble formation i n mercury and l i q u i d s i l v e r , 12 found that Equation (2.3) was a p p l i c a b l e i f the i n t e r n a l o r i f i c e r a d i u s was r e p l a c e d by the e x t e r n a l r a d i u s . For an aqueous system, where the l i q u i d wets the nozzle, the bubble forms around the edge of the o r i f i c e w h i l s t i n a m e t a l l i c non-wetting system, the bubbles grow beyond the edge of the o r i f i c e . Irons and G u t h r i e 2 4 , 2 5 a l s o observed t h i s phenomenon and f u r t h e r pointed out the pronounced e f f e c t that the antechamber volume has on bubble s i z e , p a r t i c u l a r l y at low gas flow r a t e s . 4 5 A n d r e i n i et a l . discharged argon at low flow r a t e s i n t o copper, lead and t i n baths. They found that the bubble s i z e depended on the gas flow r a t e , o r i f i c e diameter, l i q u i d s u r face t e n s i o n , and gas d e n s i t y . T h e i r r e s u l t s could be c o r r e l a t e d as a f u n c t i o n of the o r i f i c e Froude and Weber numbers ; t h e r e f o r e they concluded that e m p i r i c a l c o r r e l a t i o n s cannot be e x t r a p o l a t e d from aqueous s o l u t i o n s with a very d i f f e r e n t s u r f a c e t e n s i o n . However, as i n aqueous systems, a constant volume and a constant frequency range are i d e n t i f i e d . S i m i l a r r e s u l t s have been reported by d ft Berdnikov et a l . . 4 7 Guthrie developed a mathematical model to p r e d i c t the shape and s i z e of bubbles growing at an o r i f i c e . The model was a p p l i e d to bubble formation i n water and molten i r o n . It p r e d i c t e d that argon bubbles forming i n i r o n are about three times l a r g e r than a i r bubbles forming i n water f o r the same gas flow rate and o r i f i c e diameter. Another numerical m o d e l 4 8 was developed to p r e d i c t bubble formation i n a copper c o n v e r t e r . The model c o n s i d e r s the e f f e c t of heat t r a n s f e r , chemical r e a c t i o n 13 and bath motion on bubble formation. It was found that heat t r a n s f e r acts to decrease bubble frequency while bath c i r c u l a t i o n i n c r e a s e s i t . 2.3 Size of Gas Bubbles a f t e r Detachment It was p r e v i o u s l y mentioned that s h o r t l y a f t e r i n j e c t i o n both j e t s and larg e bubbles break up i n t o a multitude of small bubbles. These bubbles w i l l , with continuous coalescence and d i s i n t e g r a t i o n , tend to e s t a b l i s h a dynamic range of s t a b l e 3 9 s i z e s . Walters and Davidson have i n v e s t i g a t e d , t h e o r e t i c a l l y and e x p e r i m e n t a l l y , the changing shape of growing and a c c e l e r a t i n g f r e e bubbles. T h e i r work d e a l t with the a n a l y s i s of the i n i t i a l motion of a s p h e r i c a l bubble s t a r t i n g from r e s t . T h e i r theory p r e d i c t e d that the pressure f i e l d generated around the a c c e l e r a t i n g bubble i s accompanied by the d i s t o r t i o n of the bubble i n t o the form of a mushroom. As t h i s happens the base of the bubble i s depressed to develop u l t i m a t e l y i n t o a r i s i n g tongue of l i q u i d which reaches i n t o the bubble i n t e r i o r and r e s u l t s i n i t s d i s i n t e g r a t i o n . The i n i t i a l motion theory was extended to deal with the problem of a growing bubble. Walters 39 and Davidson found that the expansion of the bubble causes the 40 42 49 change i n shape to occur much more slowly. Wraith et a l in agreement with t h i s r e s u l t , have observed that growing bubbles a c c e l e r a t e and deform more slowly than detached bubbles. T h e i r o b s e r v a t i o n s i n v e r t i c a l upward and downward i n j e c t i o n i n d i c a t e that bubble d i s i n t e g r a t i o n i s r e t a r d e d u n t i l the bubble i s severed from the source. 14 2 3 Leibson et a l . reported, i n t h e i r study with water, that an increase i n the Reynolds number produces an i n c r e a s i n g l y w r i n k ly appearance of the bubble s u r f a c e . P o r t i o n s of the a i r bubble s u r f a c e are pinched or torn o f f to form very minute bubbles that remain o s c i l l a t i n g near the gas source. These i n v e s t i g a t o r s a l s o have pointed out that the formation of bubbles having nonuniform s i z e was g r e a t l y a c c e l e r a t e d by the e x p l o s i o n of large i r r e g u l a r bubbles at a point approximately 10 cm above the o r i f i c e . The bubbles thus produced had a s i z e d i s t r i b u t i o n that could be f i t t e d by a log-normal p r o b a b i l i t y d i s t r i b u t i o n . The volume-surface mean diameter was approximately 4 mm i n the range of 10000 < Re < 50000 ; the volume su r f a c e mean diameter i s the diameter of a h y p o t h e t i c a l bubble whose r a t i o of volume to su r f a c e area i s e q u i v a l e n t to that of the e n t i r e bubble s i z e d i s t r i b u t i o n . O r y a l l and Brimacombe 5 0 and Fruehan et a l . 5 1 used an e l e c t r o r e s i t i v i t y probe to study g a s - l i q u i d j e t s . They found a very l a r g e bubble frequency, r e l a t i v e to that of formation, w i t h i n 0.5 cm of the gas source. This i n d i c a t e s that the gas stream d i s c h a r g i n g from the tuyere d i s i n t e g r a t e s i n t o bubbles 5 2 very r a p i d l y . Fruehan rep o r t e d that the volume of argon bubbles emerging at the s u r f a c e of copper and s i l v e r baths were much sma l l e r that the expected volumes f o r a frequency of formation of 22 s 1 . T h e r e f o r e , i t was obvious that the lar g e bubbles formed at the o r i f i c e were breaking up i n t o smaller bubbles. The gas 3 flow r a t e s were r e l a t i v e l y low "19 cm /s. 15 5 3 Grace et a l . have reported, on the basi s of p e r t u r b a t i o n theory, that i t i s p o s s i b l e to set a lower l i m i t on the maximum s t a b l e bubble diameter. T h e i r p r e d i c t e d maximum s t a b l e diameter for a i r bubbles i n water and mercury are 4.9 cm and 3.4 cm 3 1 r e s p e c t i v e l y . Other models have been proposed to des c r i b e bubble break up. They have been reviewed comprehensively by 3 1 C l i f t , Grace and Weber Some of these models have been a p p l i e d by Sano et a l . 3 0 , 5 4 to o b t a i n an e s t i m a t i o n of bubble s i z e i n 54 bubble swarms. Sano et a l . used the d i s t r i b u t i o n of gas pulse length, generated with an e 1 e c t r o r e s i s t i v i t y probe, as an es t i m a t i o n of the bubble s i z e d i s t r i b u t i o n . They i n j e c t e d n i t r o g e n upward i n mercury through a nozzle, and found that the gas pulse lengths were widely d i s t r i b u t e d and independent of the r a d i a l p o s i t i o n and nozzle diameter. The frequency of la r g e wave lengths i n c r e a s e d with gas flow r a t e s . P i e r c e d lengths were c a l c u l a t e d assuming an average bubble r i s e v e l o c i t y of 0.66m/s and were i n the range of 7 to 75 mm. In c o n t r a s t to Leibson et 2 3 30 a l . , Sano et a l . found that the volume-area mean diameter in c r e a s e d with Reynolds number. The discrepancy was a t t r i b u t e d to the method of measurement and to the presence of coalescence. 5 5 Kawakami et a l . found s i m i l a r r e s u l t s i n t h e i r study of n i t r o g e n and argon i n j e c t i o n i n t o l i q u i d i r o n . 5 6 In a recent i n v e s t i g a t i o n , Tacke et a l . de f i n e d two q u a n t i t i e s r e l a t e d to the s i z e of the bubbles a f t e r detachment. These s i z e parameters i n d i c a t e d a decrease i n bubble s i z e toward the p e r i p h e r y of the j e t and a l s o a tendency f o r the bubble s i z e to decrease with the o r i f i c e Reynolds number. In t h i s 16 i n v e s t i g a t i o n , however, no d i r e c t measurement of bubble s i z e was undertaken. 2.4 Shape and D i s t r i b u t i o n of Flow Parameters i n Gas-Liquid  Bubble Jets Some general s i m i l a r i t i e s e x i s t i n the behaviour of homogeneous and heterogeneous two-phase j e t s with respect to the shape and d i s t r i b u t i o n of flow parameters. The theory d e s c r i b i n g the flow of a t u r b u l e n t j e t in t o a stagnant bath of the same f l u i d (homogeneous j e t ) has been t r e a t e d i n d e t a i l by 5 7 5 8 Abramovich and S c h l i c h t i n g Figure 2.1 i l l u s t r a t e s the s i t u a t i o n i n which a c i r c u l a r j e t of f l u i d d i s c h a r g e s , with uniform v e l o c i t y , i n t o a larg e body of the same f l u i d at r e s t . The v e l o c i t y d i f f e r e n c e produces a t u r b u l e n t axisymmetric shear l a y e r around the constant v e l o c i t y core. The t h i c k e n i n g of the boundary l a y e r due to turbulence leads to the eventual e x t i n c t i o n of t h i s core. This f i r s t r e g i o n i s c a l l e d the flow development r e g i o n . At some d i s t a n c e downstream from t h i s region the j e t appears to is s u e from a point source. The j e t continues to spread as i t e n t r a i n s the surrounding f l u i d and the maximum a x i a l v e l o c i t y , o c c u r r i n g on the ax i s of the j e t , decreases c o n t i n u o u s l y . In t h i s new region the flow i s s a i d to be f u l l y developed. 5 7 Abramovich has pointed out some important p r o p e r t i e s of homogeneous j e t s 17 (a) The pressure i s constant throughout the j e t . This i m p l i e s that the a x i a l momentum flow i s constant. (b) The j e t grows rather slowly, that i s , the thickness of any s e c t i o n i s small compared to i t s d i s t a n c e from the nozzl e . (c) The t r a n s v e r s e v e l o c i t y component i s much smaller than the a x i a l v e l o c i t y component. (d) The r a d i a l p r o f i l e s of the a x i a l v e l o c i t y at d i f f e r e n t s e c t i o n s of the f u l l y developed region have the same shape. Furthermore, the p r o f i l e s can be non-dimensional i z e d so thus they are d e s c r i b e d by one general curve. Thus i t i s s a i d that s i m i l a r i t y e x i s t s at a l l s e c t i o n s p e r p e n d i c u l a r to the main flow d i r e c t i o n . The d i s t r i b u t i o n s of c o n c e n t r a t i o n and temperature e x h i b i t t h i s property as w e l l . F igure 2.1 Schematic diagram of homogeneous j e t showing constant v e l o c i t y core and v e l o c i t y p r o f i l e s at v a r i o u s d i s t a n c e s from the o r i f i c e . 18 The j e t expansion angle i s a v i t a l parameter i n c h a r a c t e r i s i n g the behaviour of a submerged j e t , since i t determines the degree to which the j e t i n t e r a c t s with the surrounding f l u i d . The j e t angle i s def i n e d as the angle between the outer edges of the t u r b u l e n t boundary l a y e r i n the region of developed flow. L a a t s 5 9 , H e t s r o n i et a l . 6 0 and Popper et a l . 6 1 found that a c i r c u l a r j e t of a i r d i s c h a r g i n g i n t o a i r , both at the same temperature, expanded with a cone angle of 18° while 6 2 C o r r s i n and Uberoi determined, by blowing hot a i r i n t o cool a i r , that a r e d u c t i o n i n the d e n s i t y of the j e t r e l a t i v e to the r e c e i v i n g medium i n c r e a s e s the r a t e of spread. They rep o r t e d an o 63 angle of 21 Bin n i e found that a c i r c u l a r water j e t i n water expanded with a cone angle of 14°. He t s r o n i et a l . 6 0 and Popper 61 o et a l . measured a cone angle of 16 f o r an a i r - d r o p l e t s j e t 59 o with a small l o a d i n g r a t i o . Laats measured an angle of 16 f o r 64 a dusty a i r j e t , with a small l o a d i n g r a t i o . Engh reported a cone angle of 3° f o r a polyethene-loaded a i r j e t i n j e c t e d i n t o a i r . 6 5 Donald and Singer performed a s e r i e s of experiments with homogeneous and nonhomogeneous j e t s and proposed the f o l l o w i n g e m p i r i c a l r e l a t i o n s h i p tan ( 6 / 2) = 0.238 , j U i 0 0 (2.4) c This r e l a t i o n s h i p i m p l i e s that the expansion of the j e t i s determined s o l e l y by the p h y s i c a l p r o p e r t i e s of the i n j e c t e d f l u i d . Further c o n f i r m a t i o n was provided by Themelis et a l . who 19 measured a cone angle of 20° f o r a i r j e t s i n water. O r y a l l and 2 6 5 0 Brimacombe ' i n j e c t e d a i r h o r i z o n t a l l y i n t o mercury and measured the j e t t r a j e c t o r y using an e l e c t r o r e s i s t i v i t y probe. T h e i r measurements rev e a l e d that the j e t s expanded extremely r a p i d l y upon discharge from the nozzle with an i n i t i a l expansion angle of 155°. This value i n d i c a t e s that the p h y s i c a l p r o p e r t i e s of the l i q u i d exert c o n s i d e r a b l e i n f l u e n c e on the behaviour of 51 gas j e t s i n l i q u i d s . Fruehan et a l . a l s o used an e l e c t r o r e s i s t i v i t y probe to i n v e s t i g a t e the shape of a i r j e t s i n j e c t e d v e r t i c a l l y upward and h o r i z o n t a l l y i n t o water and g l y c e r o l . They found that the j e t s e x h i b i t a r a p i d expansion near the tuyere and then r i s e almost v e r t i c a l l y . They a l s o reported that the expansion of the j e t s i n c r e a s e d with the gas flow r a t e , as w e l l as with the d e n s i t y and v i s c o s i t y of the l i q u i d . In a 1 8 l a t e r i n v e s t i g a t i o n Hoefele and Brimacombe showed that f o r the 5 0 nozzle v e l o c i t i e s used by O r y a l l and Brimacombe the gas emerged as bubbles, implying that the concept of cone angle was not s t r i c t l y a p p l i c a b l e . 6 7 Szekely et a l . i n j e c t e d argon at the bottom of p i l o t and i n d u s t r i a l s c a l e l a d l e s c o n t a i n i n g s t e e l . They observed that the d i s t u r b e d r e g i o n on the s u r f a c e of the melt i n c r e a s e d with the depth of the bath and the gas flow r a t e . T h i s could give an 21 i n d i c a t i o n that the j e t has a c o n i c a l shape. Hsiao et a l . , on the b a s i s of t h e i r experiments with argon i n j e c t i o n i n s t e e l , have found that the g a s - l i q u i d flow zone could be represented adequately as a buoyant plume having a c o n i c a l shape. 20 6 6 Themelis et a l . a p p l i e d an I n t e g r a l - p r o f i 1 e method to develop a model for p r e d i c t i n g the t r a j e c t o r y of an a i r j e t i n j e c t e d h o r i z o n t a l l y i n t o water and l i q u i d metals. C o n s i d e r a t i o n of the c o n s e r v a t i o n equations of mass as well as of v e r t i c a l and h o r i z o n t a l momentum r e s u l t e d i n an equation f o r the j e t t r a j e c t o r y . They assumed that the j e t expansion was p r o p o r t i o n a l to the h o r i z o n t a l d i s t a n c e from the o r i g i n of the cone, with an angle of 20° f o r a l l systems. The gas c o n c e n t r a t i o n and v e l o c i t y d i s t r i b u t i o n t r a n s v e r s e to the j e t a x i s were assumed constant and to be a f u n c t i o n only of a x i a l d i s t a n c e from the o r i f i c e ; the i n f l u e n c e of buoyancy was then a p p l i e d . The model gave very good agreement with experiments on a i r - w a t e r systems but f a i l e d to 6 8 represent j e t t r a j e c t o r i e s i n gas-metal systems. Engh et a l . 6 7 introduced some m o d i f i c a t i o n s to the model of Themelis et a l . but the p r e d i c t i o n s from both models were s i m i l a r . The model of Themelis et a l . produced good r e s u l t s f o r m e t a l l i c systems when an angle of expansion of 155° was c o n s i d e r e d 5 ^ . However, 6 9 M c K e l l i g e t et a l . demonstrated that the p r e d i c t i o n was f o r t u i t o u s and that the model breaks down when the radius of curvature of the j e t a x i s becomes equal to the r a d i u s of the j e t . 50 O r y a l l and Brimacombe generated d e t a i l e d contour maps of the gas f r a c t i o n and bubble frequency d i s t r i b u t i o n of h o r i z o n t a l l y i n j e c t e d a i r j e t s i n mercury. T h e i r measurements i n d i c a t e d that c l o s e to the nozzle the j e t c o n s i s t e d of a core of high gas c o n c e n t r a t i o n which g r a d u a l l y decreased toward the edge of the j e t . They found that the d i s t r i b u t i o n of gas i s a f f e c t e d by the Froude number and the nozzle diameter ; both 21 f a c t o r s have the e f f e c t of i n c r e a s i n g the forward p e n e t r a t i o n of the j e t and a l s o the gas f r a c t i o n i n the core of the j e t . However, they do not a l t e r the columnar shape of the plume nor 5 1 i t s back p e n e t r a t i o n . Fruehan et a l . reported that the s i z e of the g a s - r i c h zone inc r e a s e s with the gas flow rate and that s t r o n g r a d i a l v a r i a t i o n s i n gas c o n c e n t r a t i o n are present. Recently Tacke et a l . used an e l e c t r o r e s i s t i v i t y probe to measure the r a d i a l p r o f i l e s of gas c o n c e n t r a t i o n and bubble frequency d u r i n g upward v e r t i c a l i n j e c t i o n . They found that the r a d i a l p r o f i l e s of gas f r a c t i o n and bubble frequency are s i m i l a r and could be represented by a s i n g l e Gaussian curve. Using the modified Froude number they presented c o r r e l a t i o n s f o r the a x i a l gas f r a c t i o n and h a l f - v a l u e r a d i u s to d e s c r i b e the d i s t r i b u t i o n of gas w i t h i n the plume, i n water and mercury systems. Kawakami 5 5 et a l . , i n j e c t i n g argon i n t o p i g i r o n , found that the p r o f i l e s of bubble frequency and gas f r a c t i o n e x h i b i t e d d i f f e r e n t shapes -s i n g l e peak, double peak and combination of the two - depending on the p o s i t i o n along the j e t . Single-peak p r o f i l e s could be represented by a Gaussian curve. S e v e r a l r e s e a r c h e r s have developed mathematical models to represent f l u i d flow i n t u r b u l e n t bubble-driven r e c i r c u l a t i n g 70-74 flows. In the more recent models the g a s - l i q u i d plume has been i n c l u d e d . For the purpose of modelling, the plume has been t r e a t e d as a homogeneous medium of v a r i a b l e d e n s i t y . M c K e l l i g e t 71 et a l . have r e p o r t e d the a x i a l p r o f i l e s of the gas f r a c t i o n f o r two d i f f e r e n t e x i t gas v e l o c i t i e s . The model p r e d i c t s that the 22 gas f r a c t i o n decays at a steady rate which i s v i r t u a l l y the same fo r both j e t s . i 2.5 F l u i d Motion i n the Gas-Liquid Region The g a s - l i q u i d r e gion formed during submerged i n j e c t i o n i s c h a r a c t e r i z e d by a suspension of bubbles moving i n a continuous l i q u i d . Owing to i t s reduced d e n s i t y , t h i s g a s - l i q u i d plume provides a r e c i r c u l a t i n g flow p a t t e r n , with upward flow c l o s e to i t and downward flow near the w a l l s of the bath c o n t a i n i n g v e s s e l . In regard to f l u i d motion i n bubble-driven r e c i r c u l a t i n g flows, i t has been found that the system i s s t r o n g l y non-7 5 uniform The r e g i o n of high v e l o c i t i e s and high values of t u r b u l e n t k i n e t i c energy i s conf i n e d to the j e t plume and to the v i c i n i t y of the f r e e s u r f a c e , w h i l s t the remainder of the system i s r e l a t i v e l y q uiescent. F l u i d motion i n the g a s - l i q u i d zone i s h i g h l y important as the determining f a c t o r of t u r b u l e n t mixing. The motion of the gas envelope produced by a point source of gas l o c a t e d i n a body of i d e a l l i q u i d of lar g e s i z e has been much di s c u s s e d by many i n v e s t i g a t o r s as was seen i n Sec t i o n s 2.1 and 3 7 2.2 In the Davidson and Schuler model, f o r the growth of bubble i n an i n v i s c i d l i q u i d and under constant flow, a gas-f i l l e d envelope, assumed r i g i d , expands r a d i a l l y from the source and t r a n s l a t e s upward under buoyancy. As the bubble grows and the buoyancy f o r c e begins to act s t r o n g l y , pushing the center of the bubble upward, the bottom surface of the bubble i s brought to r e s t when i t s downward v e l o c i t y r e l a t i v e to the centre of the 23 bubble i s equal to that of the bubble center. T h e r e a f t e r the bubble base has a net upward v e l o c i t y and detachment of the bubble occurs when the base reaches the point source. With an e f f e c t i v e i n e r t i a of l/2p.V. , the theory of Davidson and Schuler 1 0 p r e d i c t s the i n i t i a l r i s e v e l o c i t y of the bubble at detachment to be ds _ 1.138 g 2 / 5 Q 1 / 5 (2.5) dt 0 41 Wraith and Kakutani have observed that f u l l y formed bubbles detach from the o r i f i c e to form a lengthening stem and then continue to grow as they r i s e . The i n i t i a l t h e o r e t i c a l r i s e 2 1/5 v e l o c i t y of t h i s bubble when the stem i s severed i s 1.6(Q g ) and i t s a c c e l e r a t i o n i s about 1.8g. This la r g e severed gas bubble d i s i n t e g r a t e s to produce a t u r b u l e n t column of smaller bubbles. 4 5 A n d r e i n i et a l . measured the r i s e v e l o c i t y of gas bubbles in l i q u i d metals under laminar flow c o n d i t i o n s at the o r i f i c e . The v e l o c i t i e s were c a l c u l a t e d by the r a t i o of the bath depth to the r esidence time of the bubbles i n the melt. They found that bubble r i s e v e l o c i t i e s were gre a t e r than those corresponding to the r i s i n g of s i n g l e bubbles. This was b e l i e v e d to be caused by the p r o x i m i t y and wake e f f e c t s of the bubbles as they r i s e under dynamic c o n d i t i o n s . The v e l o c i t y was observed to in c r e a s e with i n c r e a s i n g bubble s i z e and d e c r e a s i n g bubble p r o x i m i t y . 51 Fruehan et a l . determined the average gas v e l o c i t i e s at d i f f e r e n t l e v e l s of an a i r j e t i n water and w a t e r - g l y c e r o l s o l u t i o n s . The average v e l o c i t i e s of the gas were c a l c u l a t e d from 24 i n t e g r a t i o n of the gas f r a c t i o n over a small volume and from the known gas flow r a t e . The r e s u l t s of t h e i r c a l c u l a t i o n s show that the gas v e l o c i t y decreased very r a p i d l y from the s u p e r f i c i a l v e l o c i t y at the o r i f i c e to a l i m i t i n g value of about 1.2 m/s at a d i s t a n c e of 5 cm above the tuyere, f o r a l l the c o n d i t i o n s 5 5 s t u d i e d . Kawakami et a l . used a double-contact e l e c t r o -r e s i s t i v i t y sensor to measure d i r e c t l y the r i s e v e l o c i t y of n i t r o g e n bubbles i n p i g i r o n . T h e i r r e s u l t s i n d i c a t e d that at low flow rates the r a d i a l bubble v e l o c i t y p r o f i l e s were very f l a t and d i d not change with v e r t i c a l p o s i t i o n , the p r o f i l e s became steeper with i n c r e a s i n g gas flow r a t e . The average bubble r i s e v e l o c i t i e s were i n the range of 0.5 to 3.5 m/s. 7 6 Haida and Brimacombe used an e l e c t r o c h e m i c a l technique to measure the l i q u i d v e l o c i t y i n n i t r o g e n and helium-water plumes. They r e p o r t e d that i n c r e a s i n g gas k i n e t i c energy r e s u l t s i n higher l i q u i d v e l o c i t i e s i n the v i c i n i t y of the tuyere t i p . However, at higher l e v e l s above the tuyere, channeling, due to high gas k i n e t i c energy reduces the l i q u i d v e l o c i t y . They concluded that the l i q u i d v e l o c i t y i n the plume decreases with v e r t i c a l d i s t a n c e from the tuyere when the gas k i n e t i c energy exceeds the buoyancy energy by a f a c t o r of about 10. 21 Hsiao et a l . s t u d i e d gas i n j e c t i o n i n t o a water model and In l a d l e s , 6 and 60 tonnes, s t i r r e d with argon. They c a r r i e d out measurements of the l i q u i d v e l o c i t y i n the g a s - l i q u i d plume with the a i d of a drag probe. T h e i r measurements show that the r a d i a l v e l o c i t y p r o f i l e s could be represented by a Gaussian curve. The 25 c e n t r e l i n e v e l o c i t y v a r i e s with the gas flow rat e to the power of 0.23 and i s almost independent of the height i n the plume, except i n the i n i t i a l r e g i o n . This region was found to be about one tenth of the t o t a l height of the bath. They concluded that the b u b b l e - l i q u i d flow above the o r i f i c e i s d r i v e n by buoyancy except i n the i n i t i a l r e g i o n , and that a plume model i s more adequate than a j e t model to represent the g a s - l i q u i d r e g i o n . 7 7 F i g u e i r a and Szekely r e c e n t l y r e p o r t e d measurements of f l u i d flow and turbulence i n a water model of an AOD v e s s e l . Laser Doppler v e l o c i m e t r y was used to determine time-smoothed v e l o c i t i e s and rms of the f l u c t u a t i n g v e l o c i t y components i n s i d e the g a s - l i q u i d zone. The r a d i a l v e l o c i t y p r o f i l e s were normalized with respect to the maximum v e l o c i t y and the t o t a l t h i c k n e s s of the j e t . The p r o f i l e s showed reasonable s i m i l a r i t y , with the maximum d i s p l a c e d toward the no z z l e . The r e l a t i v e rms of the f l u c t u a t i n g v e l o c i t y i n d i c a t e d that turbulence o u t s i d e the j e t i s f a i r l y i s o t r o p i c and high. A comparative study concluded that f o r the AOD model, both the v e l o c i t y f i e l d and the s p a t i a l d i s t r i b u t i o n of t u r b u l e n t k i n e t i c energy seem reasonably uniform o u t s i d e the two-phase r e g i o n , i n c o n t r a s t to the model of the a r g o n - s t i r r e d l a d l e , where the v e l o c i t i e s and turbulence l e v e l s are very high near the j e t cone and i n the f r e e s u r f a c e but are much lower elsewhere. Recently s e v e r a l macroscopic models based on the co n s e r v a t i o n of mass, energy and momentum have been 56 78 79 proposed ' ' to represent the f l u i d motion i n the plume. This 26 research has been motivated by the need to d e r i v e some simple mathematical expressions that r e l a t e l i q u i d c i r c u l a t i n g flow r a t e and mixing time to v e s s e l dimensions, gas flow rates and plume c h a r a c t e r i s t i c s . 7 9 Sano et a l . have proposed a simple model to c h a r a c t e r i z e f l u i d flow i n a l i q u i d bath a g i t a t e d by bubbles. The model p o s t u l a t e s that the bath c o n s i s t s of two zones : a c e n t r a l bubble plume moving upward and a l i q u i d annular zone moving downward. The a n a l y s i s i s based on a s t e a d y - s t a t e energy balance f o r the l i q u i d plume, which e s t a b l i s h e s that the rate of energy d i s s i p a t i o n due to l i q u i d c i r c u l a t i o n i s equal to the r a t e of energy input. These authors assumed that the rate of energy d i s s i p a t i o n i s equal to the d i f f e r e n c e between the rate of k i n e t i c energy a s s o c i a t e d with the l i q u i d moving upward and that of the l i q u i d moving downward plus an energy d i s s i p a t i o n due to the bubble s l i p . The equations thus d e r i v e d allow the c a l c u l a t i o n of the l i q u i d v e l o c i t y i n the plume zone, the l i q u i d c i r c u l a t i o n flow rate and the mixing time as a f u n c t i o n of gas flow r a t e , l i q u i d depth and cross s e c t i o n a l area of the plume. The model p r e d i c t s that the l i q u i d v e l o c i t y i n the plume i s d i r e c t l y p r o p o r t i o n a l to the gas flow r a t e to the power of o n e - t h i r d . 7 8 Sahai and Guthrie a l s o have proposed a model to analyze the i n t e r a c t i o n of a submerged gas j e t with l i q u i d s to cause 7 9 s t i r r i n g . Like Sano et a l . they considered v e r t i c a l upward i n j e c t i o n at the center of a c y l i n d r i c a l v e s s e l and a l s o conducted an energy balance. The balance e s t a b l i s h e s that once 27 steady s t a t e i s reached i n the bath the rate of energy input by the r i s i n g bubbles counter-balances the ra t e of t u r b u l e n t energy d i s s i p a t i o n w i t h i n the bulk of the l i q u i d . The equation r e s u l t i n g from the a n a l y s i s p r e d i c t s that U a(Q 1 / 3 h h 1 / 4 ) / R 1 / 3 (2.6) p o b In t h i s model i t i s assumed that gas and l i q u i d i n the plume 8 0 move at the same v e l o c i t y . More r e c e n t l y the model was modified to represent v e r t i c a l downward i n j e c t i o n with a lance. 5 6 Tacke et a l . a p p l i e d an i n t e g r a l - p r o f i l e technique to study v e r t i c a l gas i n j e c t i o n i n water and mercury. This i n v o l v e d c o n s i d e r a t i o n of the c o n s e r v a t i o n equations of mass f o r the gas and l i q u i d together with the equation of c o n s e r v a t i o n f o r the v e r t i c a l momentum i n i n t e g r a t e d form. The p r o f i l e s of gas c o n c e n t r a t i o n and l i q u i d v e l o c i t y were taken to be Gaussian and the v e l o c i t y d i f f e r e n c e between the r i s i n g bubbles and the l i q u i d was introduced. The group of d i f f e r e n t i a l equations that r e s u l t e d from the a n a l y s i s was solved f o r a r e g i o n separated from the nozzle where the maximum gas f r a c t i o n had dropped to f i f t y percent, s i n c e i n t h i s r e g i o n the assumptions of the model were considered to be c l o s e l y approached. Reasonable p r e d i c t i o n s of a x i a l bubble v e l o c i t y p r o f i l e s were obtained. Most of the work on i n t e g r a l - p r o f i l e methods a p p l i e d to g a s - l i q u i d plumes has been d i r e c t e d to environmental problems, such as i n h i b i t i o n of i c e formation, b a r r i e r s a g a i n s t i n t r u s i o n of s a l t water to locks and water a e r a t i o n . 3 1 8 4 ' Here l a r g e q u a n t i t i e s of gas are i n j e c t e d i n t o l a r g e volumes of water, and w h i l s t these c o n d i t i o n s do not 28 r e l a t e d i r e c t l y to bath s t i r r e d i n v e s s e l s , some common behaviour can be observed. For example the v e l o c i t y and gas c o n c e n t r a t i o n p r o f i l e s are Gaussian and the mean r i s i n g speed of the bubble 84 stream i n c r e a s e s with the gas flow r a t e to the 1/6 power In recent years major advances have been made i n the development of mathematical models f o r t u r b u l e n t r e c i r c u l a t i n g buoyancy-driven flows . In these models the researchers have recognized the importance of the plume i n generating buoyancy flow and have proposed computational schemes wherein the gas-l i q u i d mixture contained i n the j e t region i s represented by a f l u i d of v a r i a b l e d e n s i t y . Reasonably good agreement between computed and observed flow patterns o u t s i d e the g a s - l i q u i d zone have been achieved. However, computed r e s u l t s on the plume c h a r a c t e r i s t i c s (voidage, v e l o c i t y of the phases and so f o r t h ) 7 1 have s c a r c e l y been v e r i f i e d or repo r t e d . M c K e l l i g e t et a l . have s t u d i e d the i n f l u e n c e of the gas i n j e c t i o n v e l o c i t y on the c e n t e r l i n e v e l o c i t y of the plume. They found that the r e l a t i v e e f f e c t of buoyancy on the induced plume motion was i n v e r s e l y p r o p o r t i o n a l to the square of the i n j e c t i o n v e l o c i t y . For an i n j e c t i o n v e l o c i t y of 10 m/s the a x i a l v e l o c i t y along the plume f e l l c o n t i n u o u s l y as the surrounding f l u i d was e n t r a i n e d , while f o r lower s u p e r f i c i a l v e l o c i t i e s the a x i a l v e l o c i t y p r o f i l e e x h i b i t e d a lower r e g i o n where the plume was a c c e l e r a t e d by a comparatively s t r o n g buoyancy f o r c e u n t i l i t reached a maximum v e l o c i t y . The v e l o c i t y then decreased as the e f f e c t of buoyancy decreased as more l i q u i d was being e n t r a i n e d and the gas hold-up dimi n i s h e d . 29 2.6 Techniques f o r Measuring Local Gas-Liquid Bubble Flow  Parameters The r e a l i z a t i o n that two-phase flows i n general are not homogeneous and that more a t t e n t i o n has to be paid to the d e t a i l e d d i s c r e t e s t r u c t u r e of these flows, has r e s u l t e d i n the development of many experimental devices and techniques. Two-phase flow i n s t r u m e n t a t i o n has most of the problems and c h a r a c t e r i s t i c s found i n single-phase i n s t r u m e n t a t i o n , plus a d d i t i o n a l problems inherent to the presence of a second phase. 8 5 Hinze has l i s t e d the requirements that a measuring instrument must s a t i s f y before l o c a l measurements can be undertaken i n t u r b u l e n t one-phase flows. Jones and Delhaye have l i s t e d the d i f f i c u l t i e s encountered i n two-phase flow measurements, due to the presence of the second phase. In both s i n g l e and two-phase flow systems an instrument must have the f o l l o w i n g c h a r a c t e r i s t i c s to produce r e l i a b l e r e s u l t s : (a) The d e t e c t i n g element introduced i n t o the f l o w i n g f i e l d must have an adequate shape and be so small that i t causes only a minimum adm i s s i b l e d i s t u r b a n c e of the flow p a t t e r n and of the p a s s i n g i n t e r f a c e s . (b) The d e t e c t i n g element must be s m a l l e r than the dimen-sions of the flow element to be detected. (c) The i n e r t i a of the instrument must be low, so that i t can respond to f l u c t u a t i o n s i n the measured parameter. (d) The instrument must have s e n s i t i v i t y to small v a r i a t i o n s i n the measured parameter. 30 (e) The instrument must be p h y s i c a l l y and c h e m i c a l l y s t a b l e , so that no n o t i c e a b l e changes i n response occur to f i x e d c o n d i t i o n s . ( f ) The instrument must be s u f f i c i e n t l y s trong and r i g i d to withstand v i b r a t i o n s or motions caused by the t u r b u l e n t f i e l d . The a d d i t i o n a l c h a r a c t e r i s t i c s r e q u i r e d i n two-phase inst r u m e n t a t i o n are (a) The instrument must e x h i b i t a s u b s t a n t i a l l y d i f f e r e n t response to the passing phases. (b) The instrument must have an adequate d i s c r i m i n a t i o n l e v e l to be able to separate the c o n t r i b u t i o n of each phase to the s i g n a l . (c) The instrument must be provided with a s u i t a b l e s i g n a l i n t e r p r e t a t i o n technique to produce i n f o r m a t i o n r e l a t e d to each phase. Some of the instruments used to measure l o c a l parameters i n two-phase bubble flows are (a) L o c a l v o i d f r a c t i o n - e l e c t r o r e s i s t i v i t y probes - o p t i c a l probes - i s o k i n e t i c probes - hot wire anemometers - phase d e t e c t i n g microthermocouples 3 1 (b) Gas v e l o c i t y - double r e s i s t i v i t y probes - double o p t i c a l probes (c) L i q u i d v e l o c i t y - hot-wire anemometers - h o t - f i l m anemometers - i s o k i n e t i c probes - impact probes - l a s e r Doppler anemometers (d) Temperature of each phase - phase d e t e c t i n g microthermocouples. In t h i s review these methods are d e s c r i b e d , a few of which have been a p p l i e d to the study of g a s - l i q u i d j e t s i n i n j e c t i o n m e t a l l urgy. E x c e l l e n t reviews on the general subject of two-phase bubble flow i n s t r u m e n t a t i o n are given i n the works of Jones and 86 8 7 8 8 Delhaye , Hewitt and i n the work e d i t e d by Hetsroni and by 8 9 LeTourneau and Bergles 2.6.1 D e f i n i t i o n of Some Fundamental Q u a n t i t i e s D e s c r i b i n g  Two-Phase Flows Given the f l u c t u a t i n g c h a r a c t e r of two-phase flows, averaging operators have to be introduced. The d e f i n i t i o n and a p p l i c a t i o n of these operators have been reviewed by Delhaye and go 9 1 Q 2 93 Achard , Delhaye * and I s h i i . Some of the operator 32 d e f i n i t i o n s w i l l be presented here owing to t h e i r relevance to the present work. (a) Phase d e n s i t y f u n c t i o n The e x i s t a n c e of a phase k at a given p o i n t , x, and time, t, can be expressed by a phase d e n s i t y f u n c t i o n X k ( x , t ) , 1 i f point x at time t p e r t a i n s to k X k ( x , t ) = (2.7) 0 i f point x at time t does not p e r t a i n to k (b) L o c a l time-averaging If we c o n s i d e r a given p o i n t , x, i n a two-phase flow and phase k passes t h i s p o i n t i n t e r m i t t e n t l y , then a f i e l d v a r i a b l e f k ( x , t ) a s s o c i a t e d with phase k w i l l be a piece-wise continuous f u n c t i o n . Two time-averaging operators can be d e f i n e d f o r the phase d e n s i t y f u n c t i o n dt (2.8) T T which averages over the whole time i n t e r v a l T, and - x T k ( x , t ) dt (2.9) T k ( x , t ) which averages over the cumulative resi d e n c e time of phase k, T R ( x , t ) . The l o c a l time-averaged k - f r a c t i o n , a^, i s d e f i n e d as the average over T of the phase d e n s i t y f u n c t i o n X^(x,t) 33 ak ( x , t ) = X k ( x , t ) 1 T X. dt k 1 T dt (2.10) a (2.11) (c) Instantaneous space averaging An instantaneous f i e l d v a r i a b l e a s s o c i a t e d with phase k may be averaged over a space M of dimension n. Two space-averaging operators can be d e f i n e d 1 M dM (2.12) M over the whole flow space, and < > k J dM (2.13) over the space p e r t a i n i n g to phase k The instantaneous space-averaged k - f r a c t i o n , R^n, i s d e f i n e d as the average over M of the phase d e n s i t y f u n c t i o n X (x,t) K R R = < X k ( x , t ) > 1 M X R dM M 1 M dM M, (2.14) M k R k " — K M (2.15) (d) Commutativity of averaging operators 34 C o n s i d e r i n g t h e t i m e - a v e r a g i n g o f a s p a c e - a v e r a g e d f u n c t i o n , t h e f o l l o w i n g can be w r i t t e n < X k f k > 1 T X k f k dM j d t ( 2 . 1 6 ) M and s i n c e b o t h i n t e r v a l s o f i n t e g r a t i o n a r e f i n i t e , t h e y can be commutated s u c h t h a t * X k f k > 1 M X R f R d t j dM ( 2 . 1 7 ) < X k f k > " < X k f k > ( 2 . 1 8 ) F o r t h e s p e c i a l c a s e where f = 1 kn < a. > k n ( 2 . 1 9 ) T h a t i s , t h e t i m e - a v e r a g e d and s p a c e - a v e r a g e d v o i d f r a c t i o n s a r e r e l a t e d t h r o u g h t h e c o m m u t a t i v i t y o f t h e a v e r a g i n g o p e r a t o r s . (e) Time and s p a c e - a v e r a g e d v olume f l u x The i n s t a n t a n e o u s v olume f l u x r a t e o f p h a s e k t h r o u g h an a r e a A, may be g i v e n as 1 A X k U k z d A ( 2 . 2 0 ) A v e r a g i n g E q u a t i o n ( 2 . 2 0 ) w i t h r e s p e c t t o t i m e and c o m muting o p e r a t o r s 35 1_ A 1 A if X. U. dt n dA k kz ] U k z d t ] d A 1 A T k "kz d A (2.21) (2.22) \ = < \ U k z > (2.23) 90 =—= A p p l i c a t i o n of Reynolds r u l e s leads to f g Equation (2.23) can be w r i t t e n as \ = < ° k " U k z > f g . Thus (2.24) (f) T r a n s i t v e l o c i t y of a moving i n t e r f a c e If a f l u c t u a t i o n t r a v e l s with the flow at the flow v e l o c i t y , the f l u c t u a t i o n can be t r e a t e d as a t r a c e r . For example, when a probe i s used to sense a l o c a l parameter such as v o i d f r a c t i o n , and another s i m i l a r probe i s placed downstream, two f l u c t u a t i n g curves w i l l r e s u l t . The curves w i l l be s i m i l a r but w i l l e x h i b i t time s h i f t . The time s h i f t r e presents the time r e q u i r e d f o r the f l u c t u a t i o n to t r a v e l from one probe l o c a t i o n to the other. Double-contact probes have been used to determine l o c a l bubble m i g r a t i o n v e l o c i t i e s by two methods : two-state s i g n a l method and c r o s s - c o r r e l a t i o n method. In the two-state s i g n a l method, an i n d i v i d u a l p a i r of consecutive s i g n a l s i s considered and t h e i r time d i f f e r e n c e i s measured. For a d e f i n e d and a c c u r a t e l y known probe s e p a r a t i o n 36 d U b = ^_ (2. 2 5 ) t C g T h i s method o f m e a s u r i n g g i v e s t h e mean b u b b l e v e l o c i t y t o g e t h e r w i t h t h e b u b b l e t r a n s p o r t v e l o c i t y s p e c t r u m . T h i s t e c h n i q u e p r e s e n t s t h e d i f f i c u l t y o f h a v i n g t o r e l a t e s e q u e n t i a l s i g n a l s i n the upper and lower c h a n n e l s to the i n d i v i d u a l e v e n t s o c c u r r i n g a t t h e probe t i p s . In t h e c r o s s - c o r r e l a t i o n method t h e whole s i g n a l s p e c t r u m , o b t a i n e d o v e r a l o n g t i m e , i s used to c a l c u l a t e a c r o s s -c o r r e l a t i o n c o e f f i c i e n t R 1 2 < T > = J T f ^ t ) f 2 ( t + T ) dt ( 2 . 2 6 ) where f j ( t ) i s t h e measured q u a n t i t y a t one p o i n t as a f u n c t i o n o f time and f ( t + T ) i s t h e q u a n t i t y measured a t a downstream p o i n t a t time t + T . The v a l u e of R 1 2 ( T ) r e a c h e s a maximum a t T = T M > where " T M " i s t h e most p r o b a b l e t r a n s i t t i m e of t h e d i s t u r b a n c e between the p r o b e s . The most p r o b a b l e v e l o c i t y o f the b u b b l e s i s t h e n g i v e n by "v, = -B. ( 2 . 2 7 ) T M One major p r o b l e m w i t h t h i s t e c h n i q u e i s t h a t i t measures o n l y 8 6 one v e l o c i t y . J o n e s and D e l h a y e have p o i n t e d out t h a t c a l c u l a t i n g b u b b l e s i z e from the v e l o c i t y o b t a i n e d by c r o s s -37 c o r r e l a t i o n may neglect the c o r r e l a t i v e e f f e c t s of bubble s i z e on v e l o c i t y . 9 0 Delhaye et a l . have pointed out that by using a double-contact probe, a bubble v e l o c i t y can be measured but one has to be c a r e f u l when a s s i g n i n g a p h y s i c a l s i g n i f i c a n c e to t h i s bubble v e l o c i t y . For a more d e t a i l e d d i s c u s s i o n on t h i s subject see Appendix I . 2.6.2 E l e c t r o r e s i s t i v i t y Probes The e l e c t r o r e s i s t i v i t y probe detects the presence of a phase based on the d i f f e r e n c e i n e l e c t r i c a l c o n d u c t i v i t y between the phases. The d i f f e r e n c e i n e l e c t r i c a l c o n d u c t i v i t y must be la r g e and the conductive phase must be continuous. The probe array c o n s i s t s of one small measuring e l e c t r o d e , made up of a wire i n s u l a t e d except at i t s t i p , and a ref e r e n c e e l e c t r o d e with a much l a r g e r s u r f a c e area. The ref e r e n c e e l e c t r o d e may be the m e t a l l i c p r o t e c t i n g case of the probe, a second wire adjacent to the a c t i v e probe and welded to i t s case or the general ground of the t e s t s e c t i o n . The two e l e c t r o d e s are g e n e r a l l y connected by a generator and a r e s i s t a n c e i n s e r i e s . Current flows w h i l s t the measuring e l e c t r o d e i s r e s i d e n t i n the conducting phase but i t i s i n t e r r u p t e d as the gas surrounds the e l e c t r o d e , and thus develops a voltage pulses across the load r e s i s t o r . S u i t a b l e p r o c e s s i n g of the s i g n a l allows the measurement of the l o c a l v o i d f r a c t i o n and the a r r i v a l frequency of the bubbles 4 i * 4 5 0 , 5 1 , 5 6 , 9 4 - 9 6 . . . . . at a given l o c a t i o n The a n a l y s i s of the s i g n a l s 38 may r e s u l t i n s t a t i s t i c a l i n f o r m a t i o n c h a r a c t e r i z i n g the flow 9 6 p a t t e r n Bubble migrat i o n v e l o c i t i e s and c h a r a c t e r i s t i c bubble dimensions have been measured through the use of double-contact sensors.55,56,97,98,106,107 The main f e a t u r e that d i f f e r e n t i a t e s e l e c t r o r e s i s t i v i t y c i r c u i t s i s the type of power supply. D i r e c t - c u r r e n t supply systems r e q u i r e low v o l t a g e s to minimize e l e c t r o c h e m i c a l phenomena at the sensor, when used i n an i o n i c l i q u i d conductor. In a d d i t i o n r e s u l t a n t e l e c t r o n i c s may prove troublesome as a m p l i f i c a t i o n of the s i g n a l s may be r e q u i r e d . Some authors have repor t e d problems when working with a d i r e c t - c u r r e n t supply i n 56 95 50 97 99 aqueous media ' , w h i l s t others have not ' ' . D i r e c t -c u r r e n t supply has been widely used i n l i q u i d m e t a l l i c systems , . 30,50,56,95 , 55 such as mercury ' ' and l i q u i d i r o n A l t e r n a t i n g - c u r r e n t supply has been used by s e v e r a l i n v e s t i g a t o r s 5 6 , 9 4 , 1 0 0 , 1 0 1 to suppress problems a s s o c i a t e d with e l e c t r o c h e m i c a l phenomena at the sensor t i p . The current frequency has to be s u b s t a n t i a l l y d i f f e r e n t from the frequency of the phenomena measured. Frequencies h i g h e r 5 6 , 9 4 and l o w e r 1 0 1 i n r e l a t i o n to that of the phenomena measured have been used. Lower f r e q u e n c i e s provide pseudo-direct c u r r e n t o p e r a t i o n . Depending on the manner i n which the sensor i s energized, the i d e a l output s i g n a l of a r e s i s t i v e probe i s e i t h e r a binary wave sequence, or a sequence of bursts of constant amplitude o s c i l l a t i o n s separated by zero v o l t a g e areas. In a c t u a l p r a c t i c e , however, the response of the e l e c t r o r e s i s t i v i t y probes and other 39 n e e d l e c o n t a c t p r o b e s e.g. o p t i c a l p r o b e s , i s m i s s h a p e n w i t h r e s p e c t t o the i d e a l s i g n a l s . T h i s i s due t o h y s t e r e s i s i n the 102 c o n t a c t of the phases w i t h the p r o b e . D e l h a y e e t a l . have c o n d u c t e d a d e t a i l e d i n v e s t i g a t i o n o f the r e s p o n s e o f r e s i s t i v i t y p r o b e s t o l o c a l v o i d f r a c t i o n f l u c t u a t i o n s . The tim e l a p s e f o r the r e moval of the l i q u i d ^ f i l m from the probe t i p has been f o u n d to be a f f e c t e d by t h e geometry and s i z e o f t h e probe t i p , the v e l o c i t y of the b u b b l e s 9 8 , 1 0 2 and t h e t y p e of l i q u i d . 5 6 , 9 5 . The probe p e r f o r m a n c e i s improved by u s i n g s m a l l s h a r p t i p p e d p r o b e s at h i g h f l o w r a t e s and i n n o n - w e t t i n g l i q u i d s . G a s - l i q u i d v o l t a g e t r a n s i t i o n s o c c u r more r a p i d l y t h a n l i q u i d - g a s v o l t a g e ... 99,103 t r a n s i t i o n s The t r u e s i g n a l i s g e n e r a l l y t r a n s f o r m e d t o a sequence of s q u a r e p u l s e s w i t h the h e l p of a s i n g l e t h r e s h o l d l e v e l n e a r the l i q u i d v o l t a g e . I t has been f o u n d , t h a t i f the l o c a l v o i d f r a c t i o n i s p l o t t e d v e r s u s t h e t r i g g e r l e v e l , an S-shaped c u r v e i s o b t a i n e d w i t h a s l o w l y c h a n g i n g p l a t e a u o v e r a c e r t a i n t r i g g e r l e v e l r a n g e . H e r r i n g e e t a l . 9 4 and I i d a e t a l . 1 0 0 chose a t r i g g e r l e v e l c o r r e s p o n d i n g t o t h i s p l a t e a u . The l e v e l a d j u s t m e n t a l s o has been b a s e d upon c o m p a r i s o n o f t h e l i n e - a v e r a g e d gas f r a c t i o n w i t h t h e l i n e v o i d f r a c t i o n measured d i r e c t l y w i t h Y - r a y 104 a b s o r p t i o n In o t h e r c a s e s t h e t h r e s h o l d l e v e l s i m p l y has been 56 97 99 s e l e c t e d c l o s e t o t h e l i q u i d v o l t a g e l e v e l . ' ' 105 G a l a u p f o u n d t h a t t h e o p t i c a l , t h e anemometer and the e l e c t r o r e s i s t i v i t y probe g i v e c o m p a r a b l e r e s u l t s r e g a r d i n g the measurement of l o c a l gas f r a c t i o n . In a n o t h e r s t u d y H e r r i n g e e t 40 94 a l . compared the q u a l i t y o f r e s p o n s e of h o t - w i r e and h o t - f i l m anemometer p r o b e s , e l e c t r o r e s i s t i v i t y p r o b e s and i n f r a r e d a b s o r p t i o n p r o b e s . They f o u n d t h a t t h e most s u i t a b l e s y s t e m f o r phase d e t e c t i o n was the e l e c t r o r e s i s t i v i t y p r o b e , f o l l o w e d by the h o t - w i r e anemometer. Double c o n t a c t p r o b e s have been used t o d e t e r m i n e l o c a l b u b b l e v e l o c i t y by two methods : t w o - s t a t e s i g n a l method and c r o s s - c o r r e l a t i o n method. 99 C a l d e r b a n k e t a l . d e v e l o p e d a t r i d i m e n s i o n a l r e s i s t i v i t y p r o b e , c o n s i s t i n g o f f i v e m e a s u r i n g c o n t a c t s , to d e t e r m i n e t h e v e l o c i t y , s i z e and shape o f b u b b l e s r i s i n g i n a v e r t i c a l t u b e . The a r r a n g e m e n t o f the c o n t a c t s and t h e s i g n a l a n a l y s i s p e r f o r m e d by a computer a l l o w e d r e s o l u t i o n o f t h e l o c a t i o n a t w h i c h the b u b b l e s were s t r u c k by the probe r e l a t i v e to t h e i r r e s p e c t i v e c e n t e r s . The t e c h n i q u e e l i m i n a t e d most of t h e u n c e r t a i n t i e s a s s o c i a t e d w i t h t h e v a r y i n g and random p o s i t i o n s and s e q u e n c e s a t w h i c h t h e b u b b l e c o n t a c t e d the p r o b e . The t e c h n i q u e p r o d u c e d v e r y r e l i a b l e i n f o r m a t i o n . However, the probe a r r a n g e m e n t was b u l k y and t h e s t r i c t d i s c r i m i n a t i n g l o g i c c o u l d r e n d e r the t e c h n i q u e d i f f i c u l t t o use i n two-phase s y s t e m s h a v i n g a h i g h l y t u r b u l e n t and c o n c e n t r a t e d d i s p e r s i o n o f b u b b l e s . O t h e r m u l t i p l e -e l e c t r o d e s y s t e m s have been r e p o r t e d 1 0 8 , 1 0 9 . 97 S e r i z a w a e t a l . used a d o u b l e - c o n t a c t s e n s o r t o s t u d y a i r -w a t e r b u b b l e f l o w i n a v e r t i c a l column. They measured the b u b b l e t r a n s p o r t v e l o c i t y u s i n g the t w o - s t a t e s i g n a l method and the c r o s s - c o r r e l a t i o n method. The mean and the most p r o b a b l e b u b b l e 41 v e l o c i t y , o b t a i n e d from each method compared r e a s o n a b l y w e l l . U n c e r t a i n time d e l a y s , which appear d u r i n g the use o f t h e two-s t a t e s i g n a l method e.g. due to the a r r i v a l of d i f f e r e n t b u b b l e s a t t h e c o n t a c t s or t o t h e b u b b l e t r a n s v e r s e m o t i o n , were c o n s i d e r e d to be homogeneously d i s t r i b u t e d and to have a s m a l l p r o b a b i l i t y of o c c u r r e n c e . Hence, t h e a v e r a g e b u b b l e v e l o c i t y was o b t a i n e d by s i m p l y n e g l e c t i n g t h e v e l o c i t i e s a s s o c i a t e d w i t h t h o s e t i m e s . A more r i g o r o u s d i s c r i m i n a t i n g a n a l y s i s of the 10 6 s i g n a l s c a r r i e d out by Lewis et a l . i n d i c a t e d t h a t t y p i c a l l y 40 p e r c e n t o f the b u b b l e s i n t e r c e p t e d c o u l d not be c o n s i d e r e d i n the measurement of the v e l o c i t y , s i n c e t h e s e b u b b l e s d i d not p r o d u c e c o n s e c u t i v e s i g n a l s h a v i n g good c o r r e l a t i o n . Gunn et 10 7 a l . In t h e i r s t u d y of b u b b l e s f l o w s i n f l u i d i z e d bed c o n s i d e r e d as p h y s i c a l l y p o s s i b l e o n l y t h o s e b u b b l e s v e l o c i t i e s w i t h i n t h e range 0.05 t o 2 m/s. V e l o c i t i e s o u t s i d e t h i s range were c o n s i d e r e d as o u t l i e r s and were n e g l e c t e d d u r i n g the p r o c e s s i n g of the d a t a . 9 8 H e r r i n g e e t a l . used a d o u b l e - c o n t a c t e l e c t r o r e s i s t i v i t y p r o b e to measure the m i g r a t i o n v e l o c i t y of b u b b l e s i n a i r - w a t e r m i x t u r e s f l o w i n g v e r t i c a l l y . They used t h e c r o s s - c o r r e l a t i o n method to measure the b u b b l e v e l o c i t y . The c h o r d l e n g t h s of the b u b b l e s were c a l c u l a t e d from t h e r e s i d e n c e t i m e of t h e b u b b l e s at the t i p and t h e v e l o c i t y o b t a i n e d from c r o s s - c o r r e l a t i o n of the s i g n a l s . S t a t i s t i c a l a n a l y s i s a l l o w e d them t o c a l c u l a t e the b u b b l e s i z e d i s t r i b u t i o n from the c h o r d l e n g t h d i s t r i b u t i o n . 42 Tacke et a l . measured gas f r a c t i o n and b u b b l e f r e q u e n c y d i s t r i b u t i o n s i n submerged gas j e t s i n water and mercury. They a l s o measured a r e d u c e d number of b u b b l e t r a n s i t v e l o c i t i e s u s i n g a d o u b l e - c o n t a c t s e n s o r . The t r a n s i t v e l o c i t i e s were measured on an o s c i l l o s c o p e f o r o n l y t h o s e b u b b l e s t h a t p r o d u c e d c o n s e c u t i v e 5 5 s i m i l a r s i g n a l s . Kawakami e t a l . r e p o r t e d , i n a r e c e n t i n v e s t i g a t i o n , the measurement of gas f r a c t i o n , b u b b l e f r e q u e n c y and b u b b l e t r a n s p o r t v e l o c i t y i n a n i t r o g e n j e t i n p i g i r o n . They used a probe made of a g r a p h i t e r o d of 1.5 ram i n d i a m e t e r c o n n e c t e d t o a n i c k e l w i r e . The g r a p h i t e r o d was e l e c t r i c a l l y i n s u l a t e d w i t h a m u l l i t e tube e n c a s e d i n a s i l i c a and a s t a i n l e s s s t e e l t u b e . The c r o s s - c o r r e l a t i o n method was used t o measure the b u b b l e t r a n s p o r t v e l o c i t y . 2.6.3 O p t i c a l P r o b e s An o p t i c a l p r o be i s a d e v i c e s e n s i t i v e t o the change i n t h e r e f r a c t i v e i n d e x of the s u r r o u n d i n g medium and i s t h u s r e s p o n s i v e t o the p a s s a g e of i n t e r f a c e s . T h i s e n a b l e s t h e measurement of l o c a l v o i d f r a c t i o n and i n t e r f a c e p a s s a g e f r e q u e n c y . The use of two s e n s o r s can a l l o w t h e measurement of t h e b u b b l e t r a n s i t v e l o c i t y . The o p t i c a l p r o be c o n s i s t s of a two-way g l a s s l i g h t g u i d e " j o i n e d " t o a r i g h t - a n g l e d p r i s m w h i c h r e f l e c t s l i g h t back i f s u r r o u n d e d by a i r but not i f s u r r o u n d e d by w a t e r . The l i g h t i s p r o v i d e d by a lamp at one end of the l i g h t g u i d e and i t i s d e t e c t e d a t t h e o t h e r end by a p h o t o t r a n s i s t o r . F o r a g i v e n 43 a n g l e , t e m p e r a t u r e , r e f r a c t i v e i n d e x of the probe m a t e r i a l and wave l e n g t h of l i g h t , the amount o f r e f r a c t i o n depends on the r e f r a c t i v e i n d e x of the s u r r o u n d i n g phase. A i r - w a t e r b u b b l e f l o w s can be s t u d i e d a d e q u a t e l y u s i n g a g l a s s f i b e r p r o be h a v i n g a t i p w i t h a h a l f a n g l e of 4 5 ° 8 5 . A U-shaped f i b e r p r o be a l l o w s a g r e a t e r d e g r e e o f m i n i a t u r i z a t i o n t h a n g l a s s - r o d or f i b e r - b u n d l e . 110, 111 p r o b e s A t h o r o u g h i n v e s t i g a t i o n of t h e r e s p o n s e of t h e o p t i c a l p r o b e f o r l o c a l v o i d f r a c t i o n measurement has been r e p o r t e d by 112 J o n e s e t a l . They f o u n d t h a t t h e dynamic r e s p o n s e o f t h e probe i s a f f e c t e d by the a d h e r e n c e o f l i q u i d t o t h e probe t i p . Thus 112 the o u t p u t d e p a r t s from the i d e a l b i n a r y form. J o n e s e t a l . f o u n d t h a t the p e r f o r m a n c e o f the probe d e c r e a s e d w i t h an i n c r e a s e i n b u b b l e v e l o c i t y . T h i s b e h a v i o u r was e x p l a i n e d by p r o p o s i n g t h a t the water f i l m t h i c k n e s s l e f t on the probe i n c r e a s e s w i t h t h e b u b b l e v e l o c i t y . T h i s r e s p o n s e o f the o p t i c a l p robe to the v e l o c i t y of the b u b b l e seems t o be o p p o s i t e to the r e s p o n s e o f the e l e c t r i c a l p r o b e s d e s c r i b e d i n S e c t i o n 2.6.2 S i g n a l d i s c r i m i n a t i o n i n o p t i c a l p r o b e s has been a c c o m p l i s h e d t h r o u g h t h e use of two a d j u s t a b l e t h r e s h o l d l e v e l s , w h i c h e n a b l e t h e t r a n s f o r m a t i o n of the t r u e s i g n a l i n t o a s q u a r e wave s i g n a l 1 1 1 . The t h r e s h o l d v o l t a g e s e l e c t i o n p r e s e n t s t h e same p r o b l e m s as d e s c r i b e d f o r r e s i s t i v i t y p r o b e s , and t h e r e s e a r c h e r s d e c i s i o n s f o r the s e l e c t i o n have a l s o been th e same. 94 H e r r i n g e et a l . r e p o r t e d a n o t h e r t y p e o f o p t i c a l p r o b e , an i n f r a r e d a b s o r p t i o n p r o b e . The s y s t e m was c o m p r i s e d o f an 44 e m i t t i n g and a d e t e c t i n g diode approximately 1.5 mm apart. An i n d i c a t i o n of the presence or absence of l i q u i d between the two elements r e l i e s upon the strong a b s o r p t i o n of i n f r a r e d r a d i a t i o n by l i q u i d s i n comparison with gases. S i g n a l i n t e r p r e t a t i o n i n t h i s case was c a r r i e d out by a n a l y s i s of the amplitude p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n of the s i g n a l s . The data produced by the probe had a wide s c a t t e r because i t was subjected to r e f l e c t i o n and r e f r a c t i o n of i n f r a r e d r a d i a t i o n by small bubbles . 2.6.4 I s o k i n e t i c , D i f f e r e n t i a l Pressure and Impact Probes I s o k i n e t i c probes belong to the c l a s s of flow s e p a r a t i o n methods. Here a two-phase sample i s withdrawn through a probe at such a rate that the pressure j u s t i n s i d e the probe o r i f i c e i s equal to the l o c a l s t a t i c pressure i n the stream. That i s to say, the flow i n t o the probe would i d e a l l y take place at the v e l o c i t y p r i o r to the i n s e r t i o n of the probe. Where there i s s l i p between 8 6 113 the two phases, the term " v e l o c i t y " i s ambiguous ' S h i r e s et 114 a l . , In an extensive study of two-phase bubble flow, have demonstrated t h e o r e t i c a l l y that f o r g a s - l i q u i d mixtures where p g/ Pj i s s m a l l , the s u p e r f i c i a l v e l o c i t y of the l i q u i d on e n t e r i n g the probe i s equal to the l o c a l value i n the stream, i . e . the probe samples i s o k i n e t i c a l l y i n terms of the s u p e r f i c i a l water v e l o c i t y , and the voidage of the sample i s equal to the l o c a l voidage i n the stream. Both c h a r a c t e r i s t i c s were confirmed e x p e r i m e n t a l l y i n a i r - w a t e r flows i n v e r t i c a l tubes. 45 I f the s a m p l i n g l i n e of the i s o k i n e t i c probe i s c l o s e d , the p r o be w i l l a c t as a P i t o t probe m e a s u r i n g th e l o c a l impact p r e s s u r e , but a g a i n , the i n t e r p r e t a t i o n of the r e s u l t s i s somewhat d i f f i c u l t when the two p hases move a t d i f f e r e n t v e l o c i t y . One a p p r o a c h which commonly has been u s e d i s to assume t h a t t h e two-phase f l o w behaves as a homogeneous m i x t u r e w i t h no s l i p v e l o c i t y . Under t h e s e c o n d i t i o n s the v e l o c i t y measured by t h e p r obe i s g i v e n by u _ (2 AP 1 / 2 (2.28) D where the homogeneous d e n s i t y must be known. T h i s a p p r o a c h has p r o v e n to be e f f e c t i v e i n c o n s i d e r i n g h i g h mass v e l o c i t y f l o w s 1 1 4 and d r o p - l a d e n f l o w s i n a gas c o n t i n u u m . S a h a i and G u t h r i e have used a P i t o t tube to d e t e r m i n e mean (homogeneous) plume 1 1 6 v e l o c i t i e s , i n g a s - l i q u i d plumes. H s i a o et a l . measured the impact p r e s s u r e a l o n g the a x i s of v a r i o u s g a s - w a t e r j e t s . O b v i o u s l y t h e main p r o b l e m w i t h the use o f P i t o t p r o b e s t o measure l o c a l v e l o c i t y i s t h a t t h e y can be a p p l i e d o n l y i n s i t u a t i o n s were the f l o w i s l i k e l y t o be homogeneous w i t h no s l i p between the gas and the l i q u i d . O t h e r d i f f i c u l t i e s a r i s e i n c o n t r o l l i n g t h e k i n d of f l u i d p r e s e n t i n t h e t a p p i n g l i n e s , w hich 1 1 7 i s g e n e r a l l y d i f f i c u l t i n two-phase f l o w measurement. 46 Drag body probes of d i f f e r e n t shapes have been used to measured the momentum flow of the f l u i d . In a s i n g l e l i q u i d - p h a s e flow the v e l o c i t y of the flow can be c a l c u l a t e d as 2 F . U = ( — ) 1 / 2 (2.29) P l CD A Drag form probes have been a p p l i e d widely i n two-phase flow s t u d i e s , but the i n t e r p r e t a t i o n of the recorded drag f o r c e i s d i f f i c u l t and v a r i o u s models have been adopted to i n t e r p r e t the 118 21 r e s u l t s Hsiao et a l . used a drag body probe to measure r a d i a l p r o f i l e s of an average plume v e l o c i t y . The formula given above was used with a c a l i b r a t e d drag c o e f f i c i e n t . 2.6.5 Hot Film / Wire Anemometry Anemometer measurements are based on the response of the e l e c t r i c a l r e s i s t a n c e of a d e t e c t i n g element to the flow v e l o c i t y of the f l u i d p assing the sensor. The d e t e c t i n g element i n a hot-wire anemometer i s a very short metal wire. H o t - f i l m anemometry uses an i n e r t s u b s t r a t e , such as g l a s s , p l a t e d with a t h i n m e t a l l i c f i l m . The geometry of the probe may be c y l i n d r i c a l or c o n i c a l . The r e s i s t a n c e of the metal i s a f u n c t i o n of i t s temperature ; as the sensor i s cooled by the f l o w i n g f l u i d , i t s e l e c t r i c a l r e s i s t a n c e d i m i n i s h e s . Wire c o o l i n g i s c o n t r o l l e d by f o r c e d c o n v e c t i o n . 47 Hot-wire and h o t - f i l m anemometers can operate i n the constant current mode and i n the constant temperature mode. Today the l a t t e r mode i s the most commonly used. The response of hot-wire and h o t - f i l m probes u n f o r t u n a t e l y i s very n o n - l i n e a r . The c a l i b r a t i o n curve f o r e i t h e r probe type u s u a l l y can be f i t t e d over workable v e l o c i t y ranges by a r e l a t i o n of the type e = A + B U n (2.30) Owing to the d i f f i c u l t i e s a s s o c i a t e d with the n o n l i n e a r i t y of the hot-wire / - f i l m probe c a l i b r a t i o n , l i n e a r i z e r s are a v a i l a b l e to give an output voltage p r o p o r t i o n a l to v e l o c i t y over a narrow 8 5 range. Hinze has given a d e t a i l e d d i s c u s s i o n of the p r i n c i p l e s of hot-wire and h o t - f i l m anemometry. Hot-wire and h o t - f i l m anemometers have been used widely f o r measuring v e l o c i t i e s and t u r b u l e n t f l u c t u a t i o n s i n gases over larg e v e l o c i t y ranges. The a p p l i c a t i o n to l i q u i d s has been more recent and r e s t r i c t e d to low v e l o c i t i e s . Various researchers have used these instruments i n two-phase ai r - w a t e r bubble flows to determine the instantaneous v e l o c i t y and t u r b u l e n t i n t e n s i t y of 9 7 119 the water phase ' and a l s o to measure the l o c a l gas f r a c t i o n and a r r i v a l frequency of the b u b b l e s 1 1 0 , 1 2 0 D e l h a y e 1 1 0 , In h i s study of air - w a t e r bubble flows, used a c o n i c a l quartz coated f i l m probe working i n the constant-temperature node. He found that the c o n i c a l geometry had the advantages of almost completely e l i m i n a t i n g the d e p o s i t i o n of small p a r t i c l e s on the sensor and of having l i t t l e i n f l u e n c e on 48 the t r a j e c t o r y and shape of the bubbles. The constant temperature anemometer, with a small superheat, a l s o avoided degassing on the sensor. In order to determine the v o i d f r a c t i o n and v e l o c i t y of the l i q u i d , the gas and l i q u i d s i g n a l s had to be separated. Delhaye did t h i s by o b t a i n i n g the amplitude p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n of the output s i g n a l . This d i s t r i b u t i o n i n d i c a t e s the p r o b a b i l i t y that the voltage i s near a given l e v e l (gas or l i q u i d ) , and i t d i s t i n c t l y shows high p r o b a b i l i t i e s of the v oltage s i g n a l near the a i r and water l e v e l . To a f i r s t approximation the l o c a l void f r a c t i o n was c a l c u l a t e d as the r a t i o of the area a s s o c i a t e d to the gas phase to the t o t a l area below the p r o b a b i l i t y d e n s i t y curve. To v e r i f y the s i g n a l a n a l y s i s procedure, the l o c a l gas f r a c t i o n was averaged over the diameter of the t e s t s e c t i o n and compared with the averaged g a s - f r a c t i o n on the same diameter obtained with a y _ r a y a b s o r p t i o n method. The l i q u i d time-averaged v e l o c i t y and the l i q u i d t u r b u l e n t i n t e n s i t y were c a l c u l a t e d from the l i q u i d - r e l a t e d area of the amplitude p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n and the c a l i b r a t i o n curve of the probe immersed i n l i q u i d . 12 0 Jones and Zuber used a c y l i n d r i c a l h o t - f i l m anemometer to measure v o i d f r a c t i o n and v e l o c i t y p r o f i l e s i n a v e r t i c a l channel. The sensor experienced problems at v e l o c i t i e s much gre a t e r than 1.5 m/s. The problems were zero d r i f t , c a l i b r a t i o n change, and degassing on the sensor and o r i g i n a t e d from the c r a c k i n g of the quartz c o a t i n g due to f a t i g u e and o v e r s t r e s s in moderate and high v e l o c i t y water flow. They used voltage l e v e l d i s c r i m i n a t o r s to produce a b i n a r y s i g n a l and to measure voi d 49 f r a c t i o n and bubble frequency. E r r o r s in the void f r a c t i o n were encountered when c a l i b r a t e d against X-ray v o i d measurements. The e r r o r s were a t t r i b u t e d to f i n i t e dewetting times. The l o c a l volume f l u x a l s o was measured ; t h i s property i s def i n e d as the time average of the s i g n a l which i s equal to the phase v e l o c i t y when the phase i s at a p o i n t , and equal to zero when the phase i s not at a p o i n t . The l i q u i d v e l o c i t y then was obtained by pointwise d i v i s i o n of the measured l i q u i d f l u x by the measured gas f r a c t i o n . This procedure f o l l o w s from the d e f i n i t i o n of the averaging operators given i n S e c t i o n 2.6.1 9 7 Serizawa et a l . used the method and s i g n a l procedure recommended by D e l h a y e 1 1 0 to measure water v e l o c i t y and t u r b u l e n t i n t e n s i t y i n bubble and s l u g flow i n a v e r t i c a l tube. The l i q u i d flow rate c a l c u l a t e d from i n t e g r a t i o n of the p r o f i l e s of the product of l o c a l l i q u i d v e l o c i t y and l o c a l l i q u i d f r a c t i o n was i n good agreement with that measured d i r e c t l y by a t u r b i n e flowmeter. Thus, the authors concluded that the c a l i b r a t i o n of the anemometer obtained i n water flows i s v a l i d a l s o f o r bubble flow as long as the temperature of the f l u i d and the overheat r a t i o of the probe are kept constant. 2.6.6 Laser-Doppler Anemometry The l a s e r - D o p p l e r anemometer (LDA) makes use of the frequency s h i f t of a wave which occurs when the wave i s t r a n s m i t t e d by a moving source or r e c e i v e d by a moving r e c e i v e r . The LDA uses a l a s e r to provide i n c i d e n t l i g h t , which i s 50 s c a t t e r e d by moving p a r t i c l e s i n the f l u i d under study. The p a r t i c l e s s c a t t e r the l i g h t and t h e r e f o r e can be considered as moving t r a n s m i t t e r s . The r e l a t i o n s h i p between the frequency s h i f t of the l i g h t i . e . Doppler frequency, and the f l u i d v e l o c i t y i s U = (2.31) 2 sin ( 6/2) The LDA technique i s i d e a l to measure v e l o c i t y and turbulence i n transparent f l u i d s because i t i s n o n - i n t r u s i v e , i t s reponse i s l i n e a r , and i t i s c a l i b r a t i o n f r e e . The l a s e r - D o p p l e r anemometer c o n s i s t s of the l a s e r , a beam s p l i t t e r , l enses, photodetector and e l e c t r o n i c s which may be arranged i n s e v e r a l forms of o p e r a t i o n . Two of the most common arrangements are : the reference-beam mode and the f r i n g e mode. Recently some re s e a r c h e r s have used LDA to study two-pha'se 121 flow systems. Ohba employed a l a s e r - D o p p l e r anemometer i n the reference-beam mode to measure vo i d f r a c t i o n and l i q u i d v e l o c i t y 12 2 and turbulence i n a r e c t a n g u l a r duct. S u l l i v a n et a l . used a l a s e r - D o p p l e r anemometer in the f r i n g e mode to measure l i q u i d v e l o c i t y and turbulence i n a bubble flow contained i n a v e r t i c a l c y l i n d r i c a l pipe. S i n g l e bubble experiments were done to v e r i f y that the l i g h t s c a t t e r e d by bubble i n t e r f a c e s was d i s c r i m i n a t e d and not i n t e r p r e t e d as a v a l i d l i q u i d v e l o c i t y measurement. The a x i a l t u r b u l e n t i n t e n s i t i e s from the LDA were approximately twice 97 the values obtained by Serizawa et a l . using a h o t - f i l m anemometer. The reason f o r t h i s d iscrepancy i s not known. 51 7 7 F i g u e i r a and Szekely used a LDA op e r a t i n g i n the f r i n g e mode, with a n a l y s i s of the fo r w a r d - s c a t t e r e d l i g h t , to measure the v e l o c i t y and turbulence of the water i n an air-water j e t . The problems of s i g n a l i n t e r p r e t a t i o n produced by the r e f r a c t i o n of the l a s e r by the bubbles were overcome by f i l t e r i n g the measurements using a microcomputer. 119 Recently Boerner et a l . explored the r e l a t i v e merits of las e r - D o p p l e r and h o t - f i l m anemometry techniques i n a study of buoyancy-induced bubble two-phase flow, i n a r e c t a n g u l a r column. The study i n v o l v e d the determination of the l o c a l mean l i q u i d v e l o c i t i e s and the root-mean-square values of the l i q u i d v e l o c i t y f l u c t u a t i o n s . The gas component of the s i g n a l , i n both techniques, was i d e n t i f i e d and d i s c r i m i n a t e d . The authors found that i n the case of mean v e l o c i t i e s , the two experimental techniques agreed very well with one another. On the other hand, the rms-values of the l i q u i d v e l o c i t y f l u c t u a t i o n s were s l i g h t l y l a r g e r with the l a s e r - D o p p l e r than with the h o t - f i l m anemometer. This d i f f e r e n c e was exp l a i n e d as a tendency of las e r - D o p p l e r v e l o c i m e t e r s to s t a t i s t i c a l l y overemphasize f a s t motions. The r e s u l t of the i n v e s t i g a t i o n a l s o showed that the sampling time r e q u i r e d to o b t a i n s t a t i s t i c a l l y meaningful averages increased with both the width of the flow f i e l d and the voi d f r a c t i o n , l i m i t i n g the p r a c t i c a l u t i l i t y of LDV i n the inst a n c e of a 120 mm wide bubble column, to a voi d f r a c t i o n < 2 * This problem was a s s o c i a t e d with the frequent d i s r u p t i o n of the measuring volume by bubbles. 52 2.6.7 Miscellaneous Methods 12 3 Chang et a l . have employed an u l t r a s o n i c technique to measure l i q u i d f i l m t h ickness and to c h a r a c t e r i z e flow regimes i n gas-water and g a s - l i q u i d metal flows. The use of t h i s method to o b t a i n v o i d i n f o r m a t i o n i s d i f f i c u l t i n geometries where r e f l e c t i o n s occur or at moderate and high v o i d l e v e l s , > 20 * . 124 Davis et a l . used X-ray cinematography to make observa t i o n s of bubble formation i n l i q u i d metals. The r e s u l t s confirmed that bubbles i n . l i q u i d metals are formed at the outer circumference of a noz z l e . Microthermocouples have been used to i n v e s t i g a t e the temperature and phase d i s t r i b u t i o n i n two-phase systems of i n t e r e s t i n nuclear e n g i n e e r i n g . The a n a l y s i s of the temperature histograms has allowed the s e p a r a t i o n of the temperature of the l i q u i d phase from the temperature of the gas phase. This instrument gives s t a t i s t i c a l p r o p e r t i e s of the temperature in 12 5 each phase as well as the l o c a l v o i d f r a c t i o n 53 CHAPTER 3 OBJECTIVES OF THE PRESENT WORK 3.1 Summary of Previous Works As d e s c r i b e d i n the l a s t chapter, c e r t a i n aspects of the f l u i d dynamic behaviour of gas i n j e c t i o n systems have r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n the m e t a l l u r g i c a l l i t e r a t u r e . These aspects are the process of bubble formation, the regimes of discharge of the gas and the motion of l i q u i d that surrounds the g a s - l i q u i d plume. The f o r c e s that determine the bubble volume at detachment, under d i f f e r e n t bubbling regimes, have been i d e n t i f i e d . Also models have been proposed to p r e d i c t , with s u f f i c i e n t accuracy, the bubble volume f o r gas i n j e c t i o n i n t o l i q u i d metals, p a r t i c u l a r l y under low gas flow r a t e s and non-r e a c t i v e c o n d i t i o n s . Regarding the behaviour of the d i s c h a r g i n g gas, bubbling and j e t t i n g regimes have been i d e n t i f i e d depending on the d r i v i n g pressure of the gas. Several t r a n s i t i o n c r i t e r i a , which allow the p r e d i c t i o n of the discharge regime under d i f f e r e n t s i t u a t i o n s , have been presented based on v i s u a l assessment of the flow. The d i f f i c u l t i e s i n o b t a i n i n g a general t r a n s i t i o n c r i t e r i o n undoubtedly I n d i c a t e the need f o r more res e a r c h i n t o the nature of the two-phase bubble plume. Exte n s i v e experimental and t h e o r e t i c a l work has been c a r r i e d out on the v e l o c i t y and turbulence f i e l d s i n the l i q u i d surrounding g a s - l i q u i d plumes ; and a f a i r l y good understanding 54 of mixing and r e a c t i o n k i n e t i c s i n g a s - s t i r r e d l a d l e systems has r e s u l t e d . A major f a c t o r which complicates the study of the o v e r a l l f l u i d dynamic behaviour of these systems i s the gas-l i q u i d plume which introduces a d d i t i o n a l v a r i a b l e s , i n a d d i t i o n to the problems of t u r b u l e n t motion which are common to both single-phase and two-phase flows. These e x t r a v a r i a b l e s i n c l u d e the d i s t r i b u t i o n of the phases, the v e l o c i t y of the two phases, the e x i s t e n c e of i n t e r f a c i a l and g r a v i t a t i o n a l f o r c e s , the expansion of the gas phase and the geometry of the boundary between the plume and the surrounding l i q u i d . Very l i t t l e experimental evidence e x i s t s on the r o l e of these v a r i a b l e s as a f u n c t i o n of the gas i n l e t c o n d i t i o n s and the p h y s i c a l p r o p e r t i e s of the system. Work has been reported on the i n f l u e n c e of the l i q u i d p h y s i c a l p r o p e r t i e s ( , U , a) and the o r i f i c e Reynolds number on the mean s i z e of bubbles r i s i n g i n g a s - l i q u i d d i s p e r s i o n s . G e n e r a l l y , i n v e s t i g a t o r s have concluded that an ' e q u i l i b r i u m ' bubble s i z e i s e s t a b l i s h e d a f t e r the bubbles have t r a v e l l e d a c e r t a i n d i s t a n c e from the n o z z l e . However very l i t t l e experimental evidence of the e x i s t a n c e of such an ' e q u i l i b r i u m ' c o n d i t i o n i s a v a i l a b l e . There i s only s c a t t e r e d i n f o r m a t i o n on the s i z e d i s t r i b u t i o n of the bubbles and the d i s t a n c e over which the d i s i n t e g r a t i o n of the i n i t i a l gas envelopes occurs i n systems of m e t a l l u r g i c a l i n t e r e s t . The s i z e of bubbles a f t e r detachment has been determined mainly by d i r e c t photography or e l e c t r o -r e s i s t i v i t y probe techniques. The photographic technique, d e s p i t e the tedious e f f o r t i n v o l v e d , u s u a l l y does not provide very high 5 5 accuracy. S i m i l a r l y , the determination of bubble s i z e by c r o s s -c o r r e l a t i o n of the s i g n a l s from the e l e c t r o r e s i s t i v i t y probes has severe l i m i t a t i o n s because i t assumes that a l l the bubbles move at the same v e l o c i t y . The r a d i a l p r o f i l e s of gas c o n c e n t r a t i o n and bubble frequency i n g a s - l i q u i d bubble j e t s have been measured using r e s i s t i v i t y probes. It has been found that these p r o f i l e s can be represented well by a Gaussian f u n c t i o n and that they e x h i b i t good s i m i l a r i t y along the j e t . The gas d i s t r i b u t i o n and the geometry of the Jet have been reported to depend on the modified Froude number and the p h y s i c a l p r o p e r t i e s of the l i q u i d . The l i q u i d v e l o c i t i e s i n the plume have been measured using d i f f e r e n t techniques. It has been rep o r t e d that these v e l o c i t i e s decrease only s l i g h t l y with p o s i t i o n along the j e t and that t h e i r r a d i a l p r o f i l e s are Gaussian. A few measurements of the mean bubble r i s e v e l o c i t y have been done i n g a s - l i q u i d plumes, using a double-contact e l e c t r o r e s i s t i v i t y sensor. The measurements have i n v o l v e d the c r o s s - c o r r e l a t i o n of recorded s i g n a l s or the d i r e c t measurement from an o s c i l l o s c o p e . The n a t u r a l l i m i t a t i o n s of these techniques precludes determination of the bubble v e l o c i t y f l u c t u a t i o n . There has been no attempt to assess the v a l i d i t y of the experimental techniques under the c o n d i t i o n s i n which they have been a p p l i e d . Several macroscopic hydrodynamic models of submerged gas i n j e c t i o n have been proposed to represent the gas-l i q u i d plume and i t s e f f e c t on l i q u i d s t i r r i n g . 56 Numerous instruments have been developed to o b t a i n i n f o r m a t i o n of g a s - l i q u i d bubble flows. While no s i n g l e instrument can give a complete d i a g n o s i s of the flow s t r u c t u r e , the e l e c t r i c a l probes have been the most widely used since they can be adapted to measure the g r e a t e s t number of parameters under d i f f e r e n t c o n d i t i o n s . Studies of v a r i o u s phase d e t e c t i o n techniques have i n d i c a t e d that the r e s i s i t i v i t y probe gives comparable r e s u l t s to o p t i c a l and anemometer probes and i n some cases has proven to be the most s u i t a b l e method f o r measuring l o c a l instantaneous phase changes i n g a s - l i q u i d systems. The measurement of bubble v e l o c i t y by a phase d e t e c t i n g probe, e.g. double-contact e l e c t r o r e s i s t i v i t y sensor, has commonly i n v o l v e d the use of the c r o s s - c o r r e 1 a t i o n technique. Severe l i m i t a t i o n s of t h i s technique are that the p a r t i c u l a r events o c c u r r i n g at the probe t i p s are ignored. Under the best c o n d i t i o n s t h i s technique allows the measurement of only the most probable bubble v e l o c i t y , p r o v i d i n g no d i r e c t i n f o r m a t i o n on the p r o b a b i l i t y d i s t r i b u t i o n of the v e l o c i t i e s and hence of t h e i r f l u c t u a t i o n s . In more s o p h i s t i c a t e d m u l t i p l e e l e c t r o d e systems, e f f o r t s have been made to measure the spectrum of bubble v e l o c i t y and p i e r c e d length through the i n d i v i d u a l a n a l y s i s of s e q u e n t i a l v o l t a g e p u l s e s . However, the l a r g e s i z e of the probes used i n these works means that only bubbles l a r g e r than "3 mm can be detected, which may under some c o n d i t i o n s cause a s e r i o u s b i a s i n the measuring process. These i n v e s t i g a t i o n s have shown that i n t u r b u l e n t gas-l i q u i d mixtures a s u c c e s s f u l bubble encounter with the probe i s a r e l a t i v e l y r a r e event. 57 3.2 Ob j e c t i v e s In view of the l i t e r a t u r e f i n d i n g s j u s t mentioned, the present work was undertaken to increase our understanding of the g a s - l i q u i d bubble plumes formed under upward gas i n j e c t i o n i n t o v e r t i c a l c y l i n d r i c a l v e s s e l s . In s t r i v i n g toward t h i s u l t i m a t e goal, t h i s r e s e a r c h p r o j e c t pursued the f o l l o w i n g s p e c i f i c o b j e c t i v e s : (a) To develop, c o n s t r u c t and v e r i f y an accurate computer-aided e l e c t r o r e s i s t i v i t y sensor f o r measuring the p r o p e r t i e s of the s t r u c t u r e of t u r b u l e n t g a s - l i q u i d plumes . (b) To determine the s t r u c t u r a l development of air-water bubble plumes by d e t a i l e d measurement of the l o c a l p r o p e r t i e s c h a r a c t e r i z i n g the behaviour of the gas phase, i n c l u d i n g v o i d f r a c t i o n , bubble frequency, mean bubble v e l o c i t y and p i e r c e d l e n g t h , and the s p e c t r a of bubble v e l o c i t y and p i e r c e d length. (c) To c l a r i f y the e f f e c t that i n j e c t i o n c o n d i t i o n s have on the s p a t i a l development of g a s - l i q u i d bubble plumes. (d) To compare the p r e d i c t i o n s of some of the mathematical models, proposed i n the l i t e r a t u r e , r e p r e s e n t i n g gas-l i q u i d plumes to the experimental r e s u l t s . 58 CHAPTER 4 EXPERIMENTAL APPARATUS AND CONDITIONS As d i s c u s s e d e a r l i e r , i t was e s s e n t i a l i n t h i s work to develop d e t a i l e d i n f o r m a t i o n on the s t r u c t u r a l development of v e r t i c a l g a s - l i q u i d plumes. To accomplish t h i s o b j e c t i v e a microcomputer c o n t r o l l e d e l e c t r o r e s i s t i v i t y probe system was developed to measure simultaneously two-phase flow parameters under a wide range of i n j e c t i o n c o n d i t i o n s . The f o l l o w i n g s e c t i o n s d e s c r i b e the experimental system under i n v e s t i g a t i o n and the measuring instrument. 4 . 1 Experimental Apparatus The apparatus employed i n the study of ai r - w a t e r plumes i s i l l u s t r a t e d s c h e m a t i c a l l y i n Figure 4.1 . It c o n s i s t e d of a ladle-shaped v e s s e l c o n t a i n i n g d e - i o n i z e d water which was s l i g h t l y a c i d i f i e d , an a i r d e l i v e r y system to supply c l e a n a i r to the nozzle and the newly developed measuring Instrument. The measuring system was based on a double-contact e l e c t r o -r e s i s t i v i t y sensor coupled to a microcomputer through a c o n d i t i o n i n g and l o g i c c i r c u i t and a counter-timer p a r a l l e l i nput/output i n t e r f a c e . The system was able to analyse i n r e a l time the e l e c t r i c a l s i g n a l s o r i g i n a t i n g from the i n t e r a c t i o n of the bubbles with the sensor t i p s . Thus, i t produced the data r e q u i r e d to determine the l o c a l gas f r a c t i o n , bubble frequency, 59 bubble v e l o c i t y and p i e r c e d length as well as the spectrum of the l a s t two q u a n t i t i e s . An o s c i l l o s c o p e , a d i g i t a l analyzer, a f u n c t i o n generator and a high-speed camera were used as supp o r t i n g equipment. Oscilloscope Anoiog cignols Digital onoiyzer Stgnol |j condition-! er module Reference electrode Digital signots r — z troversing mechonism Wire broces L. "Boffle •-Flowmeters Nozzle comero Inlet gos Solenoid volve Doto bus Function C o o n 1 er/timer 8 generator p o r o u e | I/O interface L -/—I Printer 16 bit microcomputer Figure 4.1 Schematic diagram of experimental f a c i l i t y 60 4.1.1 The P h y s i c a l Model P h y s i c a l models commonly have been used to measure v e l o c i t i e s and turbulence f i e l d s because of the inherent c o m p l i c a t i o n s that these measurements present p a r t i c u l a r l y i n g a s - l i q u i d bubble flows. In p h y s i c a l modelling work on gas i n j e c t i o n systems, the l i q u i d metal u s u a l l y i s r e p l a c e d by water or by other l i q u i d s which are e a s i e r to handle and allow a more d e t a i l e d i n v e s t i g a t i o n of the flow. However, i t i s necessary to recognize the i m p l i c a t i o n s of t h i s s u b s t i t u t i o n . A major d i f f i c u l t y encountered i n the p h y s i c a l modelling of a g a s - l i q u i d metal system l i e s i n the number of p h y s i c a l parameters i n v o l v e d i n the process. In order that the f l u i d flow i n the p h y s i c a l model does represent that i n the a c t u a l system, 12 6 c e r t a i n q u i t e s t r i c t c o n d i t i o n s have to be observed, namely : geometric, dynamic, kinematic, and thermal s i m i l a r i t i e s . Thermal s i m i l a r i t y r e q u i r e s that the dimensionless numbers i n v o l v i n g heat t r a n s f e r or t h e r m ally induced flow be equal i n the model and the p rototype. Thermal s i m i l a r i t y appears to be unimportant i n m odelling the o v e r a l l f l u i d motion i n gas s t i r r e d l a d l e systems si n c e the c o n t r i b u t i o n of thermal g r a d i e n t s to t h i s motion i s small compared to that of other f o r c e s . Close to the i n j e c t i o n p o i n t , however, thermal e f f e c t s are important s i n c e they determine the expansion of the d i s c h a r g i n g gas and the growth of a c c r e t i o n s . Kinematic s i m i l a r i t y i s ensured i n a model that conforms to geometric and dynamic s i m i l a r i t y . 6 1 In the case of l a d l e m e t a l l u r g i c a l processes, geometric s i m i l a r i t y i s e a s i l y and p e r f e c t l y obtained i n the water model. For dynamic s i m i l a r i t y the p r i n c i p a l f o r c e s to be considered in two-phase systems are : i n e r t i a l , g r a v i t a t i o n a l , v i s c o u s and s u r f a c e t e n s i o n f o r c e s . Thus, i n p r i n c i p l e , s i m i l a r i t y would be p o s s i b l e through simultaneous adjustment of the modified Froude number, Reynolds number, and Weber number which r e l a t e i n e r t i a l and g r a v i t a t i o n a l f o r c e s , i n e r t i a l and v i s c o u s f o r c e s and i n e r t i a l and surface t e n s i o n f o r c e s r e s p e c t i v e l y . However, in p r a c t i c e , t h i s i s qu i t e d i f f i c u l t owing to major d i f f e r e n c e s i n sur f a c e t e n s i o n and d e n s i t y between aqueous and molten metal system. Regarding bubble formation at the o r i f i c e , surface t e n s i o n dominates at low gas flow r a t e s . At higher gas flow rates and for l i q u i d s of low v i s c o s i t y , i n e r t i a l and buoyancy f o r c e s play the dominant r o l e i n determining bubble s i z e . Also i t has been 18 found that during gas i n j e c t i o n i n t o a l i q u i d the t r a n s i t i o n from bubbling to j e t t i n g depends on the modified Froude number as well as on the g a s - l i q u i d d e n s i t y r a t i o . In model experiments designed to simulate the bath motion dur i n g submerged i n j e c t i o n 1 2 7 i n a teeming l a d l e , u s e f u l r e s u l t s concerning mixing have been obtained by choosing modelling c o n d i t i o n s a c c o r d i n g to the 12 8 modified Froude number. In a recent i n v e s t i g a t i o n by Bustos concerning bath movement and s l o p p i n g i n copper c o n v e r t e r s , the r a t i o n a l i z a t i o n of the observ a t i o n s according to the modified Froude number proved to be s u c c e s s f u l i n t r a n s f e r r i n g l a b o r a t o r y i n f o r m a t i o n to i n d u s t r i a l systems. 62 Therefore s i m i l a r i t y based on the modified Froude number and on g a s - l i q u i d d e n s i t y r a t i o i s e s s e n t i a l to simulate gas i n j e c t i o n processes. This implies P P ( ) p = ( ) M (4.1) and P l P l " u = U n- ( — ) 1 / 2 ( 4 - 2 ) oM oP doP Then u s e f u l i n f o r m a t i o n concerning the behaviour of the discharged gas i n a gas-metal system can be obtained by i n j e c t i n g helium i n water. U n f o r t u n a t e l y economic c o n s i d e r a t i o n s owing to the high consumption of gas r e q u i r e d i n t h i s study made the use of helium p r o h i b i t i v e . Therefore i t was decided to i n j e c t a i r through the o r i f i c e of an isothermal water model and i n v e s t i g a t e the behaviour of g a s - l i q u i d j e t s under a wide range of experimental c o n d i t i o n s . 4.1.2 The Ladle-Shaped Vessel A l / 6 t h s c a l e model of a 150 tonne teeming l a d l e was b u i l t . Gas was i n j e c t e d through a nozzle l o c a t e d at the center of the bottom of the v e s s e l which contained d e i o n i z e d water that was s l i g h t l y a c i d i f i e d with n i t r i c a c i d , approximately 0.01 % by volume. The v e s s e l was made from 9.5 mm P l e x i g l a s s p l a t e and had an I n t e r n a l diameter of 500 mm and a height 900 mm. A b a f f l e c o n s i s t i n g of a P l e x i g l a s s r i n g with an e x t e r n a l diameter of 495 mm and an i n t e r n a l diameter of 300 mm was mounted 63 near the bath surface to minimize s l o p p i n g . The b a f f l e was held "3 mm above the quiescent bath s u r f a c e . This arrangement allowed the f r e e r i s i n g and r e l e a s e of bubbles from the bath while p r o v i d i n g a damping a c t i o n of the s u r f a c e waves. Excessive s u r f a c e waves produced i n s t a b i l i t y of the i n i t i a l l y v e r t i c a l r i s i n g path of the plume which r e s u l t e d i n a p e r i o d i c swaying motion r e l a t i v e to the v e s s e l a x i s . It i s p o s s i b l e that t h i s kind of motion of the j e t produced the double peaks that Kawakami et 5 5 a l . observed i n the p r o f i l e s of bubble frequency and gas hold-up. The damping a c t i o n of the b a f f l e on the waves c o r r e c t e d the j e t t r a j e c t o r y and permitted r e p r o d u c i b l e experiments. Figure 4.2 shows a photograph of the ladle-shaped v e s s e l employed i n t h i s work. 4.1.3 Nozzle and Gas D e l i v e r y System The nozzles used were c o n s t r u c t e d from s t r a i g h t - b o r e P l e x i g l a s s pipe having i n t e r n a l diameters 4.10 mm and 6.35 mm. The nozzles were l o c a t e d i n a nozz 1 e-ho 1 der such that the gas discharge was l e v e l with the bottom of the v e s s e l , Figure 4.3 The lower end of the n o z z l e - h o l d e r , made of s t a i n l e s s s t e e l , was connected to the gas supply through a needle valve used to c o n t r o l the flow. This was done to achieve constant flow rate to the n o z z l e by means of a high pressure drop across the needle v a l v e and the small sub-nozzle volume. The pressure drop across the n o z z l e was measured with a Bourdon-type pressure gauge connected c l o s e to the n o z z l e . 64 6 5 20 F i l l mm Nozzle (6.3 5 or 4.1 f ) Nozzle holder Stainless steel pipe 635 Figure 4.3 Nozzle holder and no z z l e s , dimensions i n mm A i r was s u p p l i e d to the nozzle by an a i r compressor, with a 3 c a p a c i t y of 0 . 0 2 m /s (47 CFM) at a gauge pressure of 1 0 3 . 4 kPA ( 1 5 p s i g ) . The gas flow rate was monitored using a rotameter with a s p h e r i c a l f l o a t of tungsten and c a l i b r a t e d with a dry gas meter. The rotameter was flanked by a needle valve at the top and a Bourdon-type pressure gauge at the bottom. The pressure i n s i d e the rotameter was held constant at 6 2 . 0 5 kPa ( 9 psig) i n a l l the experiments. The gas flow r a t e at the o r i f i c e was c a l c u l a t e d 66 c o n s i d e r i n g both the atmospheric pressure and the s t a t i c head of water. An a u x i l i a r y rotameter having s m a l l e r c a p a c i t y was used to prevent l i q u i d from d r a i n i n g i n t o the nozzle while the measuring system was being prepared. The switch from one rotameter to the other was made through the a c t i o n of two s o l e n o i d valves, as shown i n Figure 4.1 . 4.1.4 The E l e c t r o r e s i s t i v i t y Probe A double-contact e l e c t r o r e s i s t i v i t y probe was used since i n c o n j u n c t i o n with a s u i t a b l e s i g n a l a n a l y s i s procedure i t allows the simultaneous measurement of a larg e number of f l u i d flow parameters. The e l e c t r o r e s i s t i v i t y probe used i n t h i s work i s i l l u s t r a t e d i n Figure 4.4 . The probe sensing elements were s t a i n l e s s s t e e l needles s o l d e r e d to a length of i n s u l a t e d copper. The lower needle was c a r e f u l l y bent to be i n v e r t i c a l alignment with the upper needle. The e n t i r e s u r f a c e of the needles was coated with a t h i n l a y e r of red G l y p t o l (a General E l e c t r i c i n s u l a t i n g f i n i s h ) and baked at 400 K f o r 2 hours ; t h i s procedure was repeated s e v e r a l times u n t i l a hard coherent c o a t i n g developed. The G l y p t o l c o a t i n g was d i s s o l v e d from the needle t i p s u s i n g S t r i p - X (a s o l v e n t marketed by G.C. E l e c t r o n i c s ) l e a v i n g a bare t i p "0.130 mm long and "0.100 mm at i t s l a r g e s t diameter. The wires were then passed through the holes of a ceramic i n s u l a t i n g tube which was housed i n a s t a i n l e s s s t e e l support tube. The top and bottom of the s t a i n l e s s s t e e l case were se a l e d with s i l i c o n e s e a l a n t to prevent any 67 900 Stainless steel tube Ceramic tube 30 11 Copper wire Solder Silicone sealant -Upper contact 0.31 s.s. needle Lower contact 0.3 4 *.s. needle Figure 4 . 4 D e t a i l s of e l e c t r o r e s i s t i v i t y probe f o r simultaneous gas f r a c t i o n and v e l o c i t y measurement, dimensions i n mm. 68 p e n e t r a t i o n of water. The wires emerging from the top of the tube were encased i n heat-shrink tubing to secure complete i n s u l a t i o n . The geometry of the needle t i p s was found to be c r i t i c a l as they a f f e c t e d the q u a l i t y of the s i g n a l s and consequently the value of the measured parameters. A t r a v e l l i n g microscope was used to measure the s i z e of the t i p s , the v e r t i c a l s e p a r a t i o n between the t i p s and the l a t e r a l s e p a r a t i o n between the needles. These d i s t a n c e s were measured once the probe was assembled. The probes used i n the experiments were s e l e c t e d from s e v e r a l c o n s t r u c t e d on the b a s i s of these d i s t a n c e s and t h e i r performance as f o l l o w s : (a) The s i z e of the two t i p s was kept i n the range of 0.130 ± 0.015 mm. As d i s c u s s e d i n S e c t i o n 5.3.1 , both t i p s must be small and have a narrow s i z e d i f f e r e n c e to ensure r e l i a b l e measurements. (b) The v e r t i c a l s e p a r a t i o n between the contacts was kept in the range 1.9 - 0.2 mm i n a l l the probes used. The exact d i s t a n c e f o r a p a r t i c u l a r probe was measured to an accuracy of microns. This s e p a r a t i o n had to be small to ensure l o c a l v e l o c i t y measurements. (c) The c l o s e s t d i s t a n c e between the needles l a t e r a l l y was "1.5 mm, F i g u r e 4.4 It was found that c l o s e r s pacing produced a r e l a t i v e l y s t a b l e water bridge between the needles which had a very d e t r i m e n t a l e f f e c t on the s i g n a l q u a l i t y of the upper c o n t a c t . S i g n a l s from the 6 9 upper contact tended to have a very long f a l l i n g time, p a r t i c u l a r l y at low flows r a t e s and at p o s i t i o n s c l o s e to the j e t edge. The s t r e a m l i n i n g and small s i z e of the sensors prevented t h e i r i n t e r f e r e n c e with the t r a j e c t o r y of the bubbles as was v e r i f i e d by f i l m i n g the bubbles passi n g the probe under the experimental i n j e c t i o n c o n d i t i o n s . Also frequent v e r i f i c a t i o n of the t i p dimensions i n d i c a t e d that they d i d not change during weeks of experiments ; and micros c o p i c examination of t h e i r s u r f a c e s showed no a l t e r a t i o n . The r e f e r e n c e e l e c t r o d e common to both t i p s c o n s i s t e d of a 450 X 100 mm s t a i n l e s s s t e e l sheet 0.2 mm t h i c k , that was sol d e r e d to a copper wire. The wire was passed through a ceramic tube which was then encased i n t o a s t a i n l e s s s t e e l tube f o r support. The e l e c t r o d e was curved to f i t snugly against the v e s s e l w a l l . The ref e r e n c e e l e c t r o d e was placed on one side of the v e s s e l p a r a l l e l to the r a d i a l motion of the measuring probe as shown i n Fi g u r e 4.2 A l l the o b j e c t s i n contact with the a c i d i f i e d water were made of P l e x i g l a s or s t a i n l e s s s t e e l to avoid s o l u t i o n contamination that would a f f e c t the probe performance. 4.1.5 T r a v e r s i n g Mechanism A t r a v e r s i n g mechanism was b u i l t to l o c a t e the probe t i p s a c c u r a t e l y at d i f f e r e n t p o s i t i o n s w i t h i n the j e t , Figure 4.2 . To 7 0 take f u l l e r advantage of the symmetry of the flows s t u d i e d , the t r a v e r s i n g mechanism was b a s i c a l l y two-dimensional. The p o s i t i o n and alignment of the probe t i p s was determined using s c a l e s and l e v e l s which were placed on the v e r t i c a l and h o r i z o n t a l arms of the t r a v e r s i n g mechanism. The p o s i t i o n was measured with respect to the v e s s e l center-line and bottom. Wire braces were use to make the long probe s u f f i c i e n t l y s t rong and r i g i d to exclude v i b r a t i o n s or motions caused by the tu r b u l e n t flow and to allow i t s accurate placement. The wire braces worked very well and permitted the use of a small probe case . 4.1.6 C o n d i t i o n i n g and Logic C i r c u i t The c o n d i t i o n i n g and l o g i c c i r c u i t that was designed and b u i l t f o r the present purposes c o n s i s t e d b a s i c a l l y of three parts : (a) Voltage d e t e c t i o n c i r c u i t (b) Level d e t e c t i o n c i r c u i t (c) Pulse delay generation c i r c u i t The d e t a i l e d c i r c u i t r y i s shown i n Figures 4.5(a) and 4.5(b) The v o l t a g e d e t e c t i o n c i r c u i t r e a c t e d to the changes i n phase around the measuring e l e c t r o d e . The change i n r e s i s t a n c e between the measuring and the re f e r e n c e e l e c t r o d e was a s s o c i a t e d with the volt a g e accross the load r e s i s t o r , Figure 4.6 . The volta g e was maximum when the needle t i p s were i n the l i q u i d phase Figure 4.5(a) Wiring diagram of e l e c t r o n i c c i r c u i t f o r e l e c t r o r e s i s t i v i t y probe measuring system. 72 Figure 4.5(b) C o n t i n u a t i o n of w i r i n g diagram of e l e c t r o n i c c i r c u i t f o r e l e c t r o r e s i s t i v i t y probe system. and approached zero when the t i p s were surrounded by the gas phase. For the aqueous s o l u t i o n used, the vo l t a g e d e t e c t i o n c i r c u i t produced s i g n a l s with an amplitude of " 3 . 3 v o l t s . A v o l t a g e a m p l i f i e r with v a r i a b l e gain was used to compensate f o r the e f f e c t s that d i f f e r e n c e s i n t i p s i z e had on the amplitude of the s i g n a l s from contact A and B, F i g u r e 4 . 6 . In a l l the 7 3 5f-T TP Probe B • i 11 o Probe A Reference electrode * i o I I I Delay Channel Channel pulse channel C Digital signals F i g u r e 4.6 Schematic diagram of bubble sensor and s i g n a l s obtained. 7 4 experiments c a r r i e d out i n t h i s work the voltage amplitude a f t e r a m p l i f i c a t i o n was 3.52 ± 0.01 v o l t s . Close examination of the s i g n a l s on an o s c i l l o s c o p e v e r i f i e d that the a m p l i f i c a t i o n d i d not d i s t o r t the width of the p u l s e s . The phase d e t e c t i o n c i r c u i t c o n s i s t e d of a comparator to which a r e f e r e n c e v o l t a g e was a p p l i e d . When the incoming s i g n a l was above the t h r e s h o l d v o l t a g e i t was considered that the probe was i n the l i q u i d phase and v i c e v e r s a , Figure 4.6 . The t h r e s h o l d v o l t a g e was f i x e d , a f t e r examining i t s e f f e c t on the measurements, at 2.84 ± 0.01 v o l t s , i . e . 20 % below the l i q u i d l e v e l v o l t a g e . The b i n a r y s i g n a l r e s u l t i n g from t h i s step was i n a s u i t a b l e form to be analyzed by the l o g i c c i r c u i t . The delay pulse g e n e r a t i n g c i r c u i t provided a f i r s t step i n the s i g n a l a n a l y s i s procedure. This c i r c u i t c o n s i s t e d of a set of Boolean devices that s a t i s f i e d s p e c i f i e d input/output r e l a t i o n s necessary to determine the v e l o c i t y of the bubbles. The l o g i c of the c i r c u i t produced the t r u t h t a b l e given i n Table 4.1 Table 4.1 Truth t a b l e f o r l o g i c c i r c u i t A B C 0 0 0 1 0 1 0 1 0 1 1 0 7 5 The t r u t h t a b l e i n d i c a t e s that when at a c e r t a i n i n s t a n t a t r a n s i t i o n from 0 to 1 occurs at the lower contact i . e . bubble surrounds t i p A, and at that same i n s t a n t t i p B i s i n the l i q u i d phase, the l o g i c c i r c u i t s t a r t s to produce a delay pulse C. Pulse C remains high f o r as long as A i s high and B i s low. As shown i n Figu r e 4.6 , the width of pulse C corresponds to the time spent by the bubble t r a v e l l i n g from contact A to B. This time i s i n v e r s e l y p r o p o r t i o n a l to the bubble r i s e v e l o c i t y i f the bubble t r a v e l s c o l i n e a r to the probe. 4.1.7 Computer, Counter/Timer and P a r a l l e l Input/Output  I n t e r f a c e The computer system was a 16-Bit Proteus microcomputer from Innovative E l e c t r o n i c s Technology. The computer was programmed in 12 9 Z-8000 assembler language to r e s e t , i n i t i a l i z e and configure the I/O board to d r i v e the s i g n a l a n a l y s i s and data a c q u i s i t i o n i n r e a l time. The counter/timer and p a r a l l e l input/output i n t e r f a c e was a l s o b u i l t by Innovative E l e c t r o n i c Technology f o r i t s microcomputer. The board c o n s i s t e d of d i r e c t memory access, i n t e r r u p t c o n t r o l l e r s and counter/timers and p a r a l l e l input/output d e v i c e s . The counter/timer and p a r a l l e l I/O devices 130 were a p a i r of Z8036 Z-CIO chips (a r e g i s t e r e d trade mark of Z i l o g C o r p o r a t i o n ) . These devices allowed the p r e c i s e timing and counting of the s i g n a l s generated by the probe. 76 The ports i n the Z - C I O chips were c o n f i g u r e d as b i t ports f o r e x t e r n a l input access and were enabled f o r p a t t e r n A C r e c o g n i t i o n . Times t and t were measured by the two timers i n e g g one of the chips and time t by a timer i n the second chip. The timers had a r e s o l u t i o n of 1ys. A counter i n the f i r s t chip was used to r e g i s t e r the number of s i g n a l s coming from contact A. 4.1.8 M i s c e l l a n e o u s Equipment A n c i l l i a r y equipment c o n s i s t e d of : (a) T e k t r o n i c 564 storage o s c i l l o s c o p e (b) Sony/Tektronic 308 data a n a l i z e r (c) Exact 123A f u n c t i o n generator (d) Hycam K2054E high-speed camera. The f i r s t three instruments helped i n debugging and o p t i m i z i n g the probe, the c o n d i t i o n i n g and l o g i c c i r c u i t and the assembler program. The high-speed camera was used at 400 frames per second to measure the v e l o c i t y of i n d i v i d u a l s p h e r i c a l cap bubbles used to assess the performance of the probe and at speeds of 1000 to 3000 frames per second to record the events under a c t u a l experimental c o n d i t i o n s . 4.2 C o n d i t i o n s f o r the Tests and General Procedure The f o l l o w i n g parameters were s t u d i e d i n the air-water experiments : (a) Gas flow rate 77 (b) Nozzle diameter (c) Bath depth Table 4.2 summarizes the c o n d i t i o n s of the experiments. Appendix II l i s t s the modified Froude numbers, Reynolds numbers and s p e c i f i c k i n e t i c and buoyancy powers corresponding to each of the t e s t s . D e t a i l e d i n v e s t i g a t i o n of the flow s t r u c t u r e of the g a s - l i q u i d j e t was c a r r i e d out i n each of the t e s t s . Table 4.2 Experimental c o n d i t i o n s of the study O r i f i c e diameter d (mm) 0 Gas f l o w 3 r a t e Q (Ncm /s) Bath depth h. (mm) b 371 876 1257 6 . 35 o v O Y 400 600 4.10 • A 4 00 The procedure o u t l i n e d below was followed throughout p r e l i m i n a r y and f i n a l experiments. (a) The e l e c t r o r e s i s t i v i t y probe was placed on the center of the nozzle to e s t a b l i s h the r e f e r e n c e p o s i t i o n i n the t r a v e r s i n g mechanism and to make connections between the e l e c t r o d e s and the measuring system. The sensor was then moved to the center of the s e l e c t e d c r o s s - s e c t i o n where measurements were to s t a r t . 78 The compressor was turned on and a i r i n j e c t i o n was commenced to d e l i v e r the d e s i r e d a i r discharge. Time was allowed f o r flow establishment and p r e p a r a t i o n of the measuring equipment. This p r e p a r a t i o n c o n s i s t e d of s e t t i n g the proper t h r e s h o l d voltage and s i g n a l a m p l i f i c a t i o n . The l i q u i d - v o l t a g e l e v e l was measured i n s i d e and out s i d e the j e t and the gain of the a m p l i f i e r was adjusted u n t i l the l e v e l was s t a b l e and independent of p o s i t i o n . The s i g n a l s were monitored on an o s c i l l o s c o p e to detect and c o r r e c t any s i g n a l d i s t o r t i o n due to a m p l i f i c a t i o n or other o p e r a t i o n a l problem. Measurements s t a r t e d with the execution of program GLJET700 which c a r r i e d out s i g n a l a n a l y s i s and data a c q u i s i t i o n . The sampling time could take from ~90 to 5400 seconds depending on the l o c a t i o n of the sensor and the gas flow r a t e . At the end of the measuring p e r i o d A C the computer had c o l l e c t e d times t and t ' the number g g of bubbles that passed contact A and the time of d u r a t i o n of the experiment. Program DATSTRDK t r a n s f e r r e d t h i s i n f o r m a t i o n together with the experimental c o n d i t i o n s from computer memory to fl o p p y d i s k f o r l a t e r data r e d u c t i o n and a n a l y s i s . Programs GLJET700 and DATSTRDK are d i s c u s s e d i n S e c t i o n 5.2 . The measurements were repeated at symmetric (-r and +r) r a d i a l p o s i t i o n s . Before s t a r t i n g any experiment at new l o c a t i o n s , the s i g n a l s were observed on the o s c i l l o s c o p e 79 fo r c o n t r o l , e.g. v e r i f y the constancy of the l i q u i d v o l t a g e l e v e l . The l o c a t i o n s of the measuring s t a t i o n s are d i s c u s s e d f u r t h e r i n S e c t i o n 5.2.5 . (d) A f t e r the d e s i r e d t r a v e r s e was completed, program DATPRC was used to ca r r y out data r e d u c t i o n to obt a i n values of the measured flow parameters. Program DATPRC i s dis c u s s e d i n S e c t i o n 5.2 . The e n t i r e p r e p a r a t i o n , measurement and data r e d u c t i o n f o r a s e l e c t e d l e v e l l a s t e d about 8 to 10 hours. For each experimental c o n d i t i o n 9 l e v e l s were s e l e c t e d i n v o l v i n g approximately 170 measurements. The aqueous s o l u t i o n i n the tank was p e r i o d i c a l l y changed. Proper c o n t r o l of the c l e a n l i n e s s of the s o l u t i o n and of the a i r as well as adequate probe c o n s t r u c t i o n permitted maintenance of the amplitude of the s i g n a l s at 3.52 ±0.01 v o l t s and the t h r e s h o l d l e v e l at 2.84 ± 0.01 v o l t s i n a l l the experiments. 80 CHAPTER 5 SIGNAL ANALYSIS AND EVALUATION OF INSTRUMENT PERFORMANCE This chapter d e s c r i b e s the s i g n a l a n a l y s i s procedure developed to i n t e r p r e t the voltage pulses produced by the i n t e r a c t i o n of the bubbles with the sensor. The c o n s i d e r a t i o n given to the c h a r a c t e r i s t i c s of the e l e c t r o r e s i s t i v i t y sensor and the c o n d i t i o n i n g c i r c u i t , to ensure t h e i r quick and r e p r o d u c i b l e response to the passing of bubbles i s al s o d i s c u s s e d . At the end of the chapter i t i s demonstrated that c a r e f u l a t t e n t i o n to a l l the aspects mentioned produced h i g h l y r e l i a b l e measurements. 5.1 S i g n a l A n a l y s i s and D e f i n i t i o n of Measured Parameters Figure 5.1 shows t y p i c a l t r a c e s of the d i g i t a l form of the s i g n a l s generated by the contact of the bubbles with the sensor under a c t u a l experimental c o n d i t i o n s . The voltage t r a c e s reveal that the bubbles a r r i v e at the contacts at c l o s e i n t e r v a l s and i n random f a s h i o n . This makes i t necessary to examine the st a t e of the contacts c o n t i n u o u s l y by a computer to e x t r a c t i n f o r m a t i o n on the bubble behaviour i n g a s - l i q u i d plumes, as d e s c r i b e d below. D i g i t a l p r o c e s s i n g r e q u i r e s conversion of the analogue s i g n a l p r i o r to s i g n a l a n a l y s i s . The conversion may be e i t h e r an o n - l i n e or a delayed process. In t h i s i n v e s t i g a t i o n the d i g i t i z a t i o n of the response of the e l e c t r o r e s i s t i v i t y probe and the d i g i t a l 81 PRL TIMING. <BIN> SHPL. POST. POS ma pos=ao+i55 woRia-eeoeeiei (b) F i g u r e 5.1 Voltage t r a c e s showing the modified d i g i t a l s i g n a l s generated by bubbles i n t e r c e p t i n g the sensor c o n t a c t s . 82 a n a l y s i s of the s i g n a l s , based on the d e f i n i t i o n of the measured parameters, were c a r r i e d out i n r e a l time. 5.1.1 Local Gas F r a c t i o n The l o c a l gas f r a c t i o n i s d e f i n e d as the p r o b a b i l i t y that a p o i n t , probe t i p A i n Figure 4.6, i s i n the gas phase ; from Equation (2.11) the l o c a l time-averaged gas f r a c t i o n f o r steady flow i s given as T A a = Z t fl / T (5.1) As shown in S e c t i o n 4.1.6 and Figure 5.1 the response from the probe system c o n s i s t e d of a two-state s i g n a l i n d i c a t i n g the presence of gas or l i q u i d at probe t i p A. Then to measure the l o c a l time-averaged gas f r a c t i o n , the computer was enabled to f o l l o w every voltage t r a n s i t i o n o c c u r r i n g at contact A and a timer recorded each i n d i v i d u a l bubble residence time over the time of the experiment. The area-averaged gas f r a c t i o n s f o r t r a n s v e r s e s e c t i o n s of the plume could be obtained from i n t e g r a t i o n of the l o c a l g a s - f r a c t i o n p r o f i l e s over the areas, Appendix IV. 5.1.2 Local Bubble Frequency The l o c a l bubble frequency a l s o was measured u t i l i z i n g only the s i g n a l s from the lower t i p . The number of modified d i g i t a l s i g n a l s , produced by b u b b l e s . a r r i v i n g at Contact A, Figure 4.6 , f o r a given d u r a t i o n of the measuring p e r i o d , was determined by a 83 d i g i t a l counter. The sampling time depended on the time r e q u i r e d to acquire a predetermined number of delay pulses as explained below. These times were s u f f i c i e n t l y long to ensure good s t a t i s t i c a l s i g n i f i c a n c e of the l o c a l gas f r a c t i o n and bubble frequency measurements. 5.1 .3 L o c a l Bubble Rise V e l o c i t y From what has been s a i d i t i s obvious that the s i g n a l a n a l y s i s i n v o l v e d i n the measurement of the l o c a l gas f r a c t i o n and bubble frequency r e q u i r e s only the r e c o g n i t i o n , on the part of the computer, of the s t a t e (low or high) of the s i g n a l s generated at the lower contact. The computer must record the r e q u i r e d data u n c o n d i t i o n a l l y and independently of the events o c c u r r i n g at the upper contact. On the other hand, the measurement of the bubble v e l o c i t y and p i e r c e d length r e q u i r e s the simultaneous a n a l y s i s of the s t a t e of the s i g n a l s from both the lower and the upper c o n t a c t s . Figure 4 . 6 shows a schematic diagram of the voltage pulse sequence which i s generated when s i n g l e bubbles approach the assembly of c o n t a c t s . Under such ra t h e r i d e a l c o n d i t i o n s , the consecutive pulses at the lower and upper contact and the corresponding delay p u l s e s , Channels A, B and C r e s p e c t i v e l y , are uniquely a s s o c i a t e d with a s i n g l e u ndisturbed bubble t r a v e l l i n g a x i a l l y from the lower to the upper contact of the sensor, such that t A t B (5.2) g g 84 for a d e f i n e d and a c c u r a t e l y known probe s e p a r a t i o n , the bubble r i s e v e l o c i t y , Equation (2.25), i s given by d Under a c t u a l i n j e c t i o n c o n d i t i o n s the passage of bubbles through the sensor does not always generate consecutive s i g n a l s of s i m i l a r width, F i g u r e 5.1(b) and 5.1(c). This r e s u l t s from the f a c t that under t u r b u l e n t c o n d i t i o n s the bubble motion through the sensor present s e v e r a l p o s s i b i l i t i e s , F igure 5.2 (a) Bubbles moving p a r a l l e l to the probe axis (b) Bubbles moving o b l i q u e l y to the probe axis (c) D i f f e r e n t bubbles a r r i v i n g at the contacts (d) Bubble coalescence o c c u r r i n g during the probe i n t e r c e p t p e r i o d (e) Bubble break-up o c c u r r i n g during the probe i n t e r c e p t p e r i o d ( f ) Bubbles t r a v e l l i n g very c l o s e together The v a r i e t y of these events and the randomness and speed at which they occur make i t necessary to examine the s t a t e of channels A, B and C c o n t i n u o u s l y by computer. The purpose i s to p ensure that the time t used to determine the bubble r i s e g v e l o c i t y i s uniquely r e l a t e d to a s i g n a l sequence corresponding to a s i n g l e bubble t r a v e l l i n g undisturbed i n a d i r e c t i o n p a r a l l e l to the probe. The approach followed to determine the v e l o c i t y of i n d i v i d u a l bubbles was to e l i m i n a t e the u n c e r t a i n t y i n the o r i g i n 85 ' • Ol Jp '-o UP o o o o o Upper contoct Lower contoct (o) (b) (c) (d) (e) (f) F i g u r e 5.2 Schematic diagram of the events o c c u r r i n g at the probe t i p s . of time t by using p a t t e r n r e c o g n i t i o n methods. The object of pa t t e r n r e c o g n i t i o n i s to i d e n t i f y p a t t e r n s that share some 131 common p r o p e r t i e s and as s i g n them to well d e f i n e d c l a s s e s In t h i s study these methods allowed the sequence and d u r a t i o n of the s i g n a l s to be r e l a t e d to the most probable bubble behaviour at t h e p r o b e t i p s . F i g u r e 5.3 shows a l l the p o s s i b l e p a t t e r n c l a s s e s that can be produced when the bubbles i n t e r a c t with the sensor t i p s . The l o g i c used by the computer to analyse the s i g n a l s , i n order to measure the bubble r i s e v e l o c i t y and the p i e r c e d length i n t u r b u l e n t g a s - l i q u i d plumes, was based on t h i s f i g u r e . In the experiments the s i g n a l s from the probes are t r e a t e d by the 86 f e e i s to (2) (3) e _r s (4) c j -V > Timt (ol) 'JI J L J — I T L <o2)Jl , o 3 , JL (b)_n_ _n_ ItJ - t J / t J l > Q I L J — L Figure 5 . 3 Timing diagram of the d i f f e r e n t s i g n a l patterns generated by the i n t e r a c t i o n of bubbles with the sensor under t u r b u l e n t c o n d i t i o n s . 8 7 computer as f o l l o w s . The computer, i n i t i a l l y i n the ready s t a t e , waits u n t i l which i n d i c a t e s that the approaching bubble i n t e r f a c e has enveloped t i p A. At the i n s t a n t that t h i s c o n d i t i o n occurs, the computer examines the s t a t e of the delay pulse channel. If the condi t ion i s met at the i n s t a n t Equation ( 5 . 4 ) i s s a t i s f i e d , then the upper channel i s a l r e a d y i n the gas phase when the lower probe was str u c k by the bubble. It i s d i f f i c u l t under these c o n d i t i o n s to determine the bubble v e l o c i t y s i n c e the s i g n a l sequence was generated by a bubble moving in a d i r e c t i o n not p a r a l l e l to the probe or by bubbles which were t r a v e l l i n g very cl o s e together. For t h i s case, p a t t e r n c l a s s ( 3 ) i n Figure 5 . 3 , the l o g i c i n the c o n d i t i o n i n g c i r c u i t does not produce a delay pulse and the a n a l y s i s to determine bubble v e l o c i t y i s abandoned immediately. The computer waits u n t i l the pulse from the lower contact ends to measure i t s d u r a t i o n , before r e t u r n i n g to the ready s t a t e . If c o n d i t i o n V A ( t ) 1 ( 5 . 4 ) V C ( t ) = 0 ( 5 . 5 ) 0 ( 5 . 6 ) 88 at the i n s t a n t C o n d i t i o n (5.4) occurs, then the c o n d i t i o n i n g c i r c u i t s t a r t s generating the delay V c ( t ) = 1 ( 5 . 7 ) C o n d i t i o n ( 5 . 7 ) l a s t s f o r as long as the bubble remains in contact with the lower t i p and the l i q u i d stays i n the upper t i p . This c o n d i t i o n i s represented by p a t t e r n c l a s s (4) in Figure 5 . 3 . Depending on the r e l a t i o n among times t A , t B and g g t , p a t t e r n c l a s s (4) may or may not be r e l a t e d to a bubble t r a v e l l i n g p a r a l l e l and undisturbed from the lower to the upper contact. P a t t e r n c l a s s ( c ) , Figure 5 . 3 , i s such that ( 5 . 8 ) T h i s p a t t e r n presents i n t e r p r e t a t i o n problems f o r the determination of the bubble v e l o c i t y , s i n c e i t i s generated by bubbles t r a v e l l i n g o b l i q u e l y to the sensor or by bubbles sm a l l e r than the s e p a r a t i o n between the probe c o n t a c t s . For t h i s case the A C computer accepts time t and r e j e c t s time t , before r e t u r n i n g g g to the ready s t a t e . P a t t e r n c l a s s (b) , F i g u r e 5 . 3 , represents bubble sequences where delay pulses are generated but the t r a n s i t i o n of V (t) from A high to low l e v e l occurs before that of V (t) The delay pulse generated under these c o n d i t i o n s i s not s a t i s f a c t o r y f o r the i s s a t i s f i e d l o g i c i n the pulse, 89 determination of the bubble v e l o c i t y s i n c e i t can be the r e s u l t of bubbles moving l a t e r a l l y , of bubbles breaking up or of the A encounter of the probe with d i f f e r e n t bubbles. The time t i s g measured and s t o r e d i n memory while the accompaning time t i s r e j e c t e d . The computer then returns to r e a d y - s t a t e . P a t t e r n c l a s s (a) , Figure 5.3 , c o n s i s t s of sequences where the voltage from contact B remains high a f t e r the accompanying previous pulse from contact A has dropped to the low l e v e l . This p a t t e r n gives r i s e to a s e r i e s of c l a s s e s : ( a l ) , (a2a) , (a2b) , (a3a) and (a3b) The c a t e g o r i z a t i o n of patterns i n t o these c l a s s e s i s based on the r e l a t i v e d u r a t i o n of the voltage pulses from channels A and B and on whether or not a second bubble a r r i v e s at the lower probe before the pulse from contact B r drops. The f i n a l d e c i s i o n on the acceptance of time t depends A B on how s i m i l a r time t and t are. For an undisturbed bubble g g t r a v e l l i n g p a r a l l e l to the probe a x i s , t * = | (t A - t B ) / t A | = 0 (5.9) However, i n a t u r b u l e n t d i s p e r s i o n a p e r f e c t h i t i s a rare event and i t i s b e t t e r to d e f i n e l i m i t i n g c o n d i t i o n s f o r the time d i f f e r e n c e r a t i o to achieve a s a t i s f a c t o r y sampling time. In the present study t was allowed to f a l l i n the range t* < 0.1 (5.10) 9 0 The e f f e c t that a d i f f e r e n t choice of t has on the measurements i s d i s c u s s e d l a t e r i n the chapter. P a t t e r n c l a s s e s (a2a) and (a3a) represent the cases where the c o n d i t i o n s f o r the a c c e p t a b i l i t y c r i t e r i o n , C o n d i t i o n (5 .10), f o r t i s met, while p a t t e r n c l a s s e s ( a l ) , (a2b) and (a3b) represent s i t u a t i o n s where i t i s not p o s s i b l e to determine the v e l o c i t y of the pulses u n e q u i v o c a l l y s i n c e they could have been generated by bubbles t r a v e l l i n g o b l i q u e l y to the sensor, by bubbles undergoing coalescence or break-up during t h e i r i n t e r c e p t i o n by the probe, or by the encounter of the probe t i p s with d i f f e r e n t bubbles. It should be s t r e s s e d that Figure 5.3 shows a l l the p o s s i b l e combinations of s i g n a l p a t t e r n s that may occur when bubbles i n a t u r b u l e n t g a s - l i q u i d d i s p e r s i o n i n t e r a c t with the e l e c t r o -r e s i s t i v i t y probe t i p s . To confirm that a l l these patterns a c t u a l l y e x i s t , under the c o n d i t i o n s of the present experiments, an extensive i n v e s t i g a t i o n of the s i g n a l sequences was c a r r i e d out with a d i g i t a l a n a l y z e r . Figure 5.4 shows the d i g i t a l p a t t e r n s observed i n d i c a t i n g t h e i r correspondance to F i g u r e 5.3 The v e l o c i t y of the bubbles producing s u c c e s s f u l encounters with the probe was c a l c u l a t e d from Equation (5.3) It i s assumed in measuring the c h a r a c t e r i s t i c l o c a l v e l o c i t y d i s t r i b u t i o n that the bubbles i n t e r c e p t the probe i n a random f a s h i o n and that the measurements are t h e r e f o r e c h a r a c t e r i s t i c of an unbiased sampling. In t h i s work the ensemble average of the bubble r i s e v e l o c i t y measurements was given by the a r i t h m e t i c mean ( c ) ( a 3 a ) ( a 2 b ) ( a 2 a ) ( c ) ( 3 ) ( a 2 a ) ( a 3 b ) F i g u r e 5.4 D i g i t a l s i g n a l p a t t e r n s generated under a c t u a l i n j e c t i o n c o n d i t i o n s , showing the correspondance to the c l a s s e s given i n F i g u r e 5.3 . 92 "U U b = Z U b / N u (5.11) and the standard d e v i a t i o n of the bubble r i s e v e l o c i t y spectrum, i . e . the r.m.s. of the f l u c t u a t i n g component of the bubble v e l o c i t y , was ( z ( u b - u b ) 2 / N [ J ) 1/2 (5.12) 5.1.4 P i e r c e d Length of Bubbles For each bubble accepted by d i s c u s s e d i n the previous s e c t i o n , bubbles was c a l c u l a t e d as b b g The a r i t h m e t i c mean was used as the best estimator of the c e n t r a l tendency of the p i e r c e d length spectrum, NU l b = Z l b / (5.14) The values of t h i s s t a t i s t i c are compared with i t s geometric e q u i v a l e n t i n S e c t i o n 6.4 . Provided that the flow of the bubbles i s l o c a l l y homogeneous, the sensor has equal p r o b a b i l i t y of p i e r c i n g the the d i s c r i m i n a t i n g l o g i c the p i e r c e d length of the (5.13) 93 b u b b l e s a t any p o i n t of t h e i r p r o j e c t e d f r o n t a l a r e a , so t h a t the m e a s u r i n g p i e r c e d l e n g t h can v a r y from z e r o to the b u b b l e d i a m e t e r . Thus i t i s p o s s i b l e , under c e r t a i n c o n d i t i o n s , as d i s c u s s e d i n S e c t i o n 6.4.1 , t o o b t a i n i n f o r m a t i o n on the l o c a l d i s t r i b u t i o n of b u b b l e d i a m e t e r s from the d i s t r i b u t i o n of p i e r c e d l e n g t h s . 5.2. Data A c q u i s i t i o n and Dat a R e d u c t i o n The p r o c e s s e s of d a t a a c q u i s i t i o n and d a t a r e d u c t i o n c o n s i s t e d of the r e c o g n i t i o n and c l a s s i f i c a t i o n of the s i g n a l p a t t e r n s a c c o r d i n g to the c r i t e r i a and d e f i n i t i o n s d i s c u s s e d p r e v i o u s l y . F i g u r e 5.5 shows t h e f l o w c h a r t r e p r e s e n t a t i o n of t h e s e p r o c e s s e s . A l l the programs r e q u i r e d f o r s i g n a l a n a l y s i s and d a t a a c q u i s i t i o n were w r i t t e n i n a s s e m b l e r Z8000 12 9 13 0 l a n g u a g e ' s i n c e i t o f f e r e d t h e f l e x i b i l i t y and speed r e q u i r e d to c a r r y out t h e s e p r o c e s s e s i n r e a l t i m e . In a d d i t i o n t o t h i s s o f t w a r e , o t h e r programs were w r i t t e n i n B a s i c t o c a r r y out d a t a r e d u c t i o n and a n a l y s i s o f t h e i n f o r m a t i o n . F i g u r e 5.6 shows a f l o w s h e e t of t h e d a t a a c q u i s i t i o n and d a t a r e d u c t i o n p r o g r a m s . s An i m p o r t a n t f a c t o r i n any d a t a a c q u i s i t i o n p r o c e s s i s t h e d e t e r m i n a t i o n o f the amount of d a t a needed t o r e p r e s e n t t h e phenomena s t u d i e d . I n a d e q u a t e amounts of d a t a can l e a d t o i n a c c u r a t e r e s u l t s , but the c o s t o f the d a t a g a t h e r i n g and p r o c e s s i n g f o r c e s one t o l i m i t the amount t o a minimum w h i c h w i l l a s s u r e s t a t i s t i c a l l y m e a n i n g f u l r e s u l t s . In t h i s s t u d y two 9 4 C O M P U T E R C O N D I T I O N I N G A N D L O G I c C i R C U I T - * | P R I N T E R \~ TlMER 2 I N 2-CIO 1 I /O B O A R D , J L O C A L G A S ' | F " R * C T I O N C O M P U T E R C O N D IT I O N I N G A N D L O G I C C I R C U I T P R I N T E R C O U N T E R 3 I N Z-ClO 1 I/O B O A R D C O M P U T E R C O N D I T I O N I N G A N D L O G I C C I R C U I T P R I N T E R T I M E R 1 I N 2 -CIO 1 (C) T I M E R 2 1 N Z -CIO 1 ( A ) T I M E R 2 1 N 2 -CIO 2 ( B ) L O C A L B U B B L E Y F R E Q U E N C Y B U B B L E V E L O C I T Y A N D P I E R C E D L E N G T H S P E C T R U M A V E R A G E B U B B L E V E L O C I T Y A N D P I E R C E D L E N G T H S T A N D A R D D E V I A T I O N O F B U B B L E V E L O C I T Y A N D P I E R C E D L E N G T H Figure 5 . 5 Block diagram of the measuring processes aspects had to be considered i ) the s i z e of the s i g n a l sample r e q u i r e d f o r measurement of the air-phase flow parameters and, i i ) the number of t e s t p o i n t s and t h e i r l o c a t i o n . 95 ( S T A R T ) I N I T I A L I Z E Z 8 0 3 6 ( Z - C 1 0 ) 1 A N D 2 9519 I N T . C N T R . 8 2 3 7 D W , V R £ A D S T A R T I N G T I M ,0f E X P E R I M E N T . I D E N T I F Y P U L S E T I M I N G S E Q U E N C E B Y R E A D I N G I N P U T S AT P O R T D A T A R E G I S T E R B I T S A N D F L A G S S E T B Y P A T T E R N R E C O G N I T I O N ! L O G I C . D I D T H E S E Q U I N A N D D U R A T I O N OF S I G N A L S C O R R E S P O N D TO A ' B U B B L E T R A V E L L I L Y T O P R C 8 C , A S S E K I Y E S NO T H E N T I M I N G S E Q J E N C E S a 2 a . a3a. STORE O N C O N T I G L O J S M E -MORY L O C A T I O N S , THE C U -R A T I O ; OF LOWER A N D D E -L A Y P U L S E S . D E C R E A S E D E L A Y T I M E P U L S E C O U N T E R B Y 1 . C O N V E R T T I M E R C O - N T S TO T I M E . SORT D E L A Y T I M E S INTO D E C R E A S I N G O R D E R OF M A G N I T U D E K E E P I N G T H E I R C O R R E S P O N D I N G LOWER P J L S E T I M E . T H E N T I M I N C 3, b, c, al S E Q U E N C E S , a l b , «3b. STORE O N C O N T I G L O J S M E -MORY L O C A T I O N S , THE D U -RAT I O N O F T H E LOWER P U L S E A N D 0 F O R THE D C -L A Y P U L S E . E X P E R I M E N T C O N D I T I O N S • C O N T A C T S E P A R A T I O N • G A S F L O * R A T E . N O Z Z L E D I A M E T E R . B A T H D E P T H • P R O B E P O S I T I O N • E X P E R I M E N T D A T E . CCmfTE T H E F O L L O W I N G I N F O F -MAT ION : • L O C A L G A S F R A C T I O N • L O C A L B U B B L E F R E Q U E N C Y • B U B B L E V E L O C I T Y S P E C T R U N B U B B L E P I E R C E D L E N G T H S P E C T R L M • A V E R A G E B L B B L E V E L O C I T Y • A V E R A G E B L B B L E P I E R C E L " L E N G T H S T A N D A R D D E V I A T I O N ' OF B U B B L E V E L O C I T Y S T A N D A R D D E V I A T I O N O F B U B B L E P I E R C E D L E N G T H . C PS ) Figure 5.6 Flow diagram of a c q u i s i t i o n and computer program f o r data data r e d u c t i o n processes. 96 5.2.1. Program for Data A c q u i s i t i o n (GLJET700) Program GLJET700 was w r i t t e n i n assembler Z8000 language to ca r r y out the p a t t e r n r e c o g n i t i o n process a c c u r a t e l y i n r e a l time. The program r e s e t , i n i t i a l i z e d and co n f i g u r e d the d i f f e r e n t components of the I/O i n t e r f a c e . The Z8036 (Z-CIO) devices were c o n f i g u r e d to recognize the s t a t e of the ports to which channels A, B and C where connected. The pattern r e c o g n i t i o n l o g i c of these devices was a c t i v a t e d whenever the voltage l e v e l i n channel A s u f f e r e d a t r a n s i t i o n from 0 to 1. In t h i s a c t i v e c o n d i t i o n the computer c o n t i n u o u s l y checked the s t a t e of channels A, B and C to measure the times t A and f o r the S r acceptable s i t u a t i o n , time t a l s o . A f t e r completing these measurements the computer went back to the ready-state where i t waited f o r the a r r i v a l of another bubble at the lower contact. In the f i n a l experiments, the process of data a c q u i s i t i o n stopped Q once 800 values of t were c o l l e c t e d . When t h i s c o n d i t i o n was g A met, the computer had stored i n memory a l l the values of t Q generated, the predetermined number of t times, the number of bubbles passing contact A, the time d u r a t i o n of the experiment, and the l o c a t i o n of the measurement, Figure 5.6 . A program, DATSTRDK, w r i t t e n i n Basic t r a n f e r r e d these data from computer memory to fl o p p y d i s k . Programs GLJET700 and DATSTRDK are l i s t e d i n d e t a i l i n Appendix I I I . To ensure that program GLJET700 worked adequately i n counting and timing the events o c c u r r i n g at channels A, B and C, an e x t e n s i v e t e s t i n g of the program was undertaken using the 97 d i g i t a l a n a l y z e r . This instrument provided an independent form of a s s e s s i n g program performance i n regard to s i g n a l a n a l y s i s l o g i c and c o n f i g u r a t i o n of the I/O board. In the form that program GLJET700 was w r i t t e n , i t could lose i n f o r m a t i o n only i f consecutive pulses at channel A occurred during a time i n t e r v a l s m a l l e r than 11 u s , and t h i s was confirmed using a f u n c t i o n generator. This c o n d i t i o n occurred very r a r e l y i n t h i s work. 5.2.1.1 Frequency of Occurrence of the Accepted P a t t e r n Classes During the many experiments that were conducted, i t was found that t y p i c a l l y 25 to 35 % of the bubbles i n t e r c e p t e d by contact A were accepted, f o r e x t r a c t i n g i n f o r m a t i o n concerning bubble r i s e v e l o c i t y , although t h i s p r o p o r t i o n was smaller at c e r t a i n l o c a t i o n s . As di s c u s s e d p r e v i o u s l y , the accepted bubbles belonged to p a t t e r n c l a s s e s (a2a) and (a3a) f o r which t £ 0.1 Experiments c a r r i e d out by the r e l e a s e of i n d i v i d u a l s p h e r i c a l cap bubbles of d i f f e r e n t s i z e showed that under these flow c o n d i t i o n s a l l the bubbles i n t e r c e p t e d by the probe were accepted. The r e j e c t i o n of bubbles under a c t u a l experimental c o n d i t i o n s then was c l e a r l y a s s o c i a t e d with the i r r e g u l a r and random motion of the bubbles around the sensor. A v e r s i o n of program GLJET700 was w r i t t e n to count the frequency of occurrence of the d i f f e r e n t p a t t e r n c l a s s e s shown i n Figure 5.3 . Some r e s u l t s of these t e s t s are shown i n the form of histograms i n Figure 5.7 It can be seen that the accepted 98 hfc •400mm d o «635 " Z • 240 " r « 0 " 0 -I257 Ncm3 s"' n JZL ( 3 ) (c) (b) ( a l ) <a3b) <a2b) (o2o) + <c3o) (3) (c) (b) (a l ) (o2b) ( o 2 o ) + ( o 3 a ) (a3b) Pattern classes 60 50 / 40 £ 30 • £ 20 10 0 0 -371 Ncm3*'1 h b « 400mm d o •6.35 mm z • 20mm r « 0 mm f l n n n n. Q «l2 57NcmSs"1 n n n (3) ( c ) ( b ) (o l )(o2b) ( o 2 o ) + ( o 3 o ) (3) ( c ) (b) ( o l ) ( o 2 b ) ( o 2 o ) + ( o 3 o ) (o3b) (o3b) Pattern C I U M S F i g u r e 5.7 H i s t o g r a m s o f t h e p a t t e r n c l a s s e s g e n e r a t e d by t h e b u b b l e s a t d i f f e r e n t l o c a t i o n s i n t h e plume and f o r d i f f e r e n t gas f l o w r a t e c o n d i t i o n s . 99 p a t t e r n c l a s s e s (a2a and a3a) have a f r e q u e n c y of o c c u r r e n c e t h a t i s , i n g e n e r a l , l a r g e r t h a n or e q u a l to t h e o t h e r c l a s s e s . T h i s i n d i c a t e s t h a t a s u c c e s s f u l b u b b l e - p r o b e e n c o u n t e r had a s a t i s f a c t o r y p r o b a b i l i t y o f o c c u r r e n c e . P a t t e r n s (3) and (c) a r e a l s o p r e s e n t i n r e l a t i v e l y h i g h f r e q u e n c y . From t h e s e and o t h e r r e s u l t s a t d i f f e r e n t gas f l o w r a t e s and v e r t i c a l p o s i t i o n s , i t i s p o s s i b l e to say t h a t p a t t e r n c l a s s (3) was the r e s u l t o f the c l o s e p r o x i m i t y of the b u b b l e s , w h i l e c l a s s ( c ) was most p r o b a b l y the r e s u l t o f the o b l i q u e m o t i o n of some b u b b l e s w i t h r e s p e c t to t h e p robe a x i s or of b u b b l e s t h a t were s m a l l e r t h a n the t i p s e p a r a t i o n . The r e l a x a t i o n on t h e c r i t e r i o n of a c c e p t a b i l i t y b ased on t , i . e . e n l a r g i n g the range of a c c e p t a b i l i t y on t , had the e f f e c t of i n c r e a s i n g the a v e r a g e and s t a n d a r d d e v i a t i o n o f the b u b b l e v e l o c i t y and c o n s e q u e n t l y of t h e p i e r c e d l e n g t h . T h i s was most p r o b a b l y due to the i n c r e a s e d a c c e p t a n c e o f b u b b l e s which s t r i k e t h e p r o b e t i p s o b l i q u e l y . As d i s c u s s e d i n A p p e n d i x I, t h e s e k i n d s o f e n c o u n t e r s between the b u b b l e s and t h e s e n s o r * g e n e r a t e l a r g e " v e l o c i t i e s " . A t i m e d i f f e r e n c e r a t i o t £ 0 . 1 was a d o p t e d s i n c e i t p r o d u c e d the b e s t measurements, w h i l e i t k e p t t h e d u r a t i o n o f t h e a c q u i s i t i o n p r o c e s s w i t h i n r e a s o n a b l e l i m i t s . 5.2.2 Program f o r D a t a P r o c e s s i n g and R e d u c t i o n (DATPRC) The program DATPRC c o n s i s t i n g of a main program and s e v e r a l s u b r o u t i n e s , used the d a t a measured by GLJET700 to c a l c u l a t e 100 i n d i v i d u a l bubble v e l o c i t i e s and p i e r c e d lengths as well as the l o c a l gas f r a c t i o n , bubble frequency and the mean and standard d e v i a t i o n of the bubble v e l o c i t i e s and p i e r c e d length, Figure 5.6 These q u a n t i t i e s were c a l c u l a t e d a c c o r d i n g to the d e f i n i t i o n s given i n S e c t i o n 5.1 A l l the i n f o r m a t i o n was organized i n t o f i l e s c o n t a i n i n g : frequency d i s t r i b u t i o n of bubble v e l o c i t y , frequency d i s t r i b u t i o n of p i e r c e d length, and reduced i n f o r m a t i o n of a l l the measured parameters. A d e t a i l e d l i s t i n g of DATPRC i s given i n Appendix I I I . As d i s c u s s e d below, Q the values of time t c o l l e c t e d by GLJET700 had to be f i l t e r e d g by c o n s i d e r a t i o n of p h y s i c a l l y p o s s i b l e and t h e r e f o r e acceptable p i e r c e d l e n g t h . Program DATPRC contained i n s t r u c t i o n s to carry out t h i s f i l t e r i n g process. rj 5.2.2.1 D i s c r i m i n a t i o n of Time-Delay (tg ) According to  P i e r c e d Length The f i l t e r i n g of the s i g n a l s based on the c r i t e r i a d i s c u s s e d i n S e c t i o n 5.1.3. e l i m i n a t e s most of the u n c e r t a i n t i e s a s s o c i a t e d with the d i v e r s e o r i g i n of the delay p u l s e s . However, u n c e r t a i n t y s t i l l e x i s t s owing to the i m p o s s i b i l i t y of a c c u r a t e l y determining the d i r e c t i o n of movement of the bubble with r e s p e c t to the probe using a double-contact sensor. This problem manifested i t s e l f as the presence of a small number of unreasonably high v e l o c i t i e s i n the bubble v e l o c i t y spectrum, as i s shown i n Figures 5.8(a) and 5.8(b). Despite the very small frequency of occurrence of these apparent v e l o c i t i e s , t h e i r l a r g e magnitude had an e x c e s s i v e 101 9 V m 12 10 e 6 4 2 0 0.3 0.2 0.1 0 12 I 0 e 6 4 2 0 1 r (o) i — r l — i — r ' T T Somple size • 795 U b -2.58 m/s S g k » 2 3 7 m / » Q - l 2 5 7 N c m 3 / s h b • 4 0 0 m m d 0 » 635 mm z« 40mm, r»0mm 1 1 '1 • I — 1 ' ' + 4 (b) U b producing I b >l50mm Somple size • 15 I 1 'l (c) Somple size * 7 8 0 •2.30 m/s s " b » 1.24 m/s mini., 1 , 1 . 1 »J—L 8 10 12 14 16 18 20 30 Bubble rise velocity U b ,m/s F i g u r e 5 . 8 ( a ) T y p i c a l b u b b l e t h e v e l o c i t i e s l a r g e r t h a n t h e v e l o c i t y d i s t r i b u t i o n s i n d i c a t i n g a s s o c i a t e d t o p i e r c e d l e n g t h s maximum a c c e p t e d . 102 12 10 6 6 4 2 O 0 3 2 0.2 O c 3 CT 0.1 0 I 2 10 6 6 4 2 0 1 1 1 1 1 1 1 1 J Sample size - 8 0 0 Ob-l.79m/s i—T S U b » 1.46 m/s (a) 0- l257NcmVs h b» 400mm d 0« 6.35mm z • 190 mm r « 0 " (b) U b producing l b >7 0mm Sample size -13 T U J L H — I 1 1—H \ (c) Sample size • 787 U_ • 1.66 m/s D S U b « 0.72 m/s ± ± ± H—P—h— I — H — i — i — i — + i—I—4 ± 2 4 6 8 10 12 14 16 18 2 0 22 Bubble rise velocity U b ,m/s F i g u r e 5 . 8 ( b ) T y p i c a l b u b b l e t h e v e l o c i t i e s l a r g e r t h a n the v e l o c i t y d i s t r i b u t i o n s i n d i c a t i n g a s s o c i a t e d t o p i e r c e d l e n g t h s maximum a c c e p t e d . 103 weight on the value of the standard d e v i a t i o n of the bubble v e l o c i t y spectrum. The f i r s t step i n the f i l t e r i n g of the bubble v e l o c i t i e s c o n s i s t e d of n e g l e c t i n g the v e l o c i t i e s which exceeded the v e l o c i t y of the gas at the o r i f i c e . The number of these events was very small and occurred mainly at large gas flow r a t e s and at p o s i t i o n s c l o s e to the nozzle where c o n d i t i o n s are very t u r b u l e n t . A second r a t i o n a l form of n e g l e c t i n g the l a r g e " v e l o c i t i e s " was found by examining t h e i r a s s o c i a t e d p i e r c e d l e n g t h . V i s u a l o b s e r v a t i o n of the bubbles from recorded f i l m s with a high-speed camera permitted the determination of reasonable upper l i m i t s f o r the p i e r c e d l e n g t h . These l i m i t s then were used as the acceptable maximum i n the f i l t e r i n g of data v e l o c i t i e s . Figure 5.9 shows some examples of the l a r g e bubbles that e x i s t e d i n the plumes. Depending on the l o c a t i o n i n s i d e the j e t two maximum acceptable p i e r c e d lengths were s e l e c t e d : one of 150 mm f o r p o s i t i o n s below 100 mm from the nozzle and the other of 70 mm at higher p o s i t i o n s . These two l i m i t s were f i x e d i n the treatment of a l l the data c o l l e c t e d . Figures 5.8(a) and 5.8(b) show that the large bubble " v e l o c i t i e s " measured were i n general r e l a t e d to p i e r c e d lengths l a r g e r than those observed and that t h e i r p r o b a b i l i t y of occurrence was very s m a l l . Then i t was j u s t i f i a b l e to neg l e c t these v e l o c i t i e s without unduly i n t r o d u c i n g b i a s i n the data treatment . 104 (a) (b) Figure 5.9 Large bubbles occurring at d i f f e r e n t positions in the plume, (Q - 876 Ncm /s, h - 400 am, d = 6.35 mm, probe at centreline z = 110 mm for (a) and (b), z * 200 mm for (c) and (d) photographs). 105 As F i g u r e s 5.8(a) and 5.8(b) show, a few v e l o c i t y v a l u e s r e m a i n e d t h a t were m a r k e d l y d i f f e r e n t from the b u l k of t h e d a t a . 13 2 J o h n s o n and Leone have d i s c u s s e d a number r o f t e s t s of s i g n i f i c a n c e t o d e c i d e whether i s o l a t e d o b s e r v a t i o n s , c a l l e d o u t l i e r s , can be r e g a r d e d as b e i n g t o o d o u b t f u l t o be a c c e p t e d as p a r t o f a sample. They i n d i c a t e d t h a t such t e s t s a r e m a i n l y h e u r i s t i c and m o s t l y r e l a t e t o samples o f normal p o p u l a t i o n . In 13 3 t h i s work the f o l l o w i n g s t a t i s t i c g i v e n by D i x o n and Massey f o r l a r g e samples was a p p l i e d , (5.15) U b n-2 - U b 1 The o u t l i e r was n e g l e c t e d , i . e . , removed from the sample, i f t h e t e s t c r i t e r i o n was s i g n i f i c a n t . In t h i s work, from th e t a b l e of c r i t i c a l v a l u e s f o r t e s t i n g o u t l i e r s g i v e n by D i x o n and 13 3 Massey , i t can be s a i d t h a t t h e t e s t was s i g n i f i c a n t t o 95 % i f t h e v a l u e f o r , r , e x c e e d e d 0.40. T h i s v a l u e a d o p t e d was o n l y 13 2 a p p r o x i m a t e . As J o h n s o n and Leone have recommended, the t e s t i n g p r o c e d u r e was a p p l i e d s e q u e n t i a l l y r e m o v i n g one o u t l i e r a t a t i m e u n t i l no f u r t h e r s i g n i f i c a n t v a l u e s o f , r , r e s u l t e d . I t i s c o n s i d e r e d t h a t t h i s t e s t g i v e s enough i n f o r m a t i o n t o e l i m i n a t e , by t h i s o b j e c t i v e method, the o u t l i e r s . I t i s i m p o r t a n t to n o t e t h a t the c o n s i d e r a t i o n s d e s c r i b e d i n t h i s s e c t i o n , r e g a r d i n g t h e a c c e p t a n c e o f i n d i v i d u a l b u b b l e v e l o c i t y d a t a , had t h e i r l a r g e s t i n f l u e n c e on the v a l u e o f t h e c a l c u l a t e d s t a n d a r d d e v i a t i o n and had a much s m a l l e r i n f l u e n c e on t h e v a l u e s 106 of the mean bubble v e l o c i t y . This i s r e f l e c t e d i n the t y p i c a l r e s u l t s f o r U b and shown i n Figures 5.8(a) and 5.8(b) . b The average percent of accepted time delays, with respect to the t o t a l c o l l e c t e d by GLJET700, was t y p i c a l l y 97 * . Figure 5.10 shows how t h i s percentage behaved i n the experiments. It i s seen that the maximum r e j e c t i o n occurred i n the r e g i o n c l o s e to the nozzle and at the j e t edge, most probably due to the strong r a d i a l motion of the bubbles i n t h i s r e g i o n . 5.2.3 Program f o r Data A c q u i s i t i o n (GLJET100) A P Program GLJET100 was w r i t t e n to c o l l e c t times t and t g g without a p p l y i n g any of the d i s c r i m i n a t i n g c r i t e r i a based on p a t t e r n r e c o g n i t i o n . With t h i s program, the computer accepted and timed a l l the delay pulses that a r r i v e d from the c o n d i t i o n i n g c i r c u i t . A s e r i e s of experiments using t h i s program was performed to demonstrate the need f o r the r i g o r o u s s i g n a l a n a l y s i s based on the r e c o g n i t i o n of the pa t t e r n s shown i n Figure 5.3 The mean and the standard d e v i a t i o n of the bubble v e l o c i t y obtained from such t e s t s were compared with those obtained from GLJET700. The comparison showed that the data acquired by GLJET100 were c o n s i s t e n t l y much higher than those obtained from GLJET700. The ac q u i r e d sets of data from both programs were t r e a t e d i n the same form by DATPRC. The r e s u l t s from the t e s t s are shown i n Table 5.1 . It i s obvious from these r e s u l t s that i n order to o b t a i n meaningful i n f o r m a t i o n on the v e l o c i t i e s of bubbles i n t u r b u l e n t g a s - l i q u i d plumes i t i s necessary to c a r r y out the 107 s i g n a l a n a l y s i s p r o c e d u r e d i s c u s s e d i n S e c t i o n 5.1.3 o\00 E 96 8 { i { | - A - _ • * * 1 * E Averoge T3 f 92 0> u o ° 88 o c if 84 0. 4 • - A - A • d.(mm) Q (Ncm 3 s*1) h b (mm) 371 876 1257 6.35 om v • o• Y A 4 00 6 00 4.1 0 • • A A 400 1 100 200 300 400 Vertical position z. mm 500 F i g u r e 5.10 P e r c e n t a g e of a c c e p t e d t i m e d e l a y s as f u n c t i o n of p o s i t i o n ; empty and f u l l symbols c o r r e s p o n d t o c e n t r e l i n e and plume boundary l o c a t i o n s , r e s p e c t i v e l y . 5.2.4 Sample S i z e D e t e r m i n a t i o n o f mean v a l u e s of s t a t i o n a r y random p r o c e s s e s r e q u i r e s a n a l y s i s o f samples which a r e s u f f i c i e n t l y l a r g e to e n s u r e t h e i r s t a t i s t i c a l s i g n i f i c a n c e . To d e t e r m i n e the a p p r o p r i a t e a v e r a g i n g sample s i z e , p r e l i m i n a r y e x p e r i m e n t s were done under c o n d i t i o n s c h o s e n t o r e p r e s e n t t h e f i n a l e x p e r i m e n t s . S i n c e t h e number o f d e l a y p u l s e s c o l l e c t e d by the computer o v e r th e d u r a t i o n o f t h e e x p e r i m e n t was s m a l l e r t h a n t h e t o t a l number " 1 0 8 Table 5.1 Comparison of mean bubble v e l o c i t y and standard d e v i a t i o n obtained by GLJET700 and GLJET100. h, = 400 mm d, = 6.35 mm b b z (mm) Q = 371 Ncm3/s o GLJET100 GLJET700 °b SUb U b sub 20 65 240 1.87 1.58 1.22 0.80 1.65 1.08 1.57 0.60 1.38 1.25 1.03 0.42 z (mm ) Q = 1257 Ncm3/s o GLJET100 GLJET700 U b SUb u b s u b ( m / s ) 20 65 240 6.97 7.16 2.36 1.14 5.51 5.39 2.21 1.17 2.23 1.87 1.49 0.59 109 of s i g n a l s generated by contact A, i t was decided to base the sample s i z e on the number of the delay pulses necessary to ensure the convergence of the measured q u a n t i t i e s . R e s u l t s of these t e s t s f o r gas f r a c t i o n , bubble frequency and mean bubble v e l o c i t y are summarized i n Figure 5.11 The f i n a l e l e c t r o r e s i s t i v i t y probe measurements were based on data samples c o n t a i n i n g 800 time de l a y s . This large number r e q u i r e d measuring times between 90 to 5400 seconds depending on the l o c a t i o n i n the plume, but ensured r e p r o d u c i b l e r e s u l t s . 5.2.5 Measurement Lo c a t i o n s The accuracy i n the determination of the s p a t i a l d i s t r i b u t i o n of flow parameters depends upon the degree of s p a t i a l u n i f o r m i t y and improves as the number of measurement point s i s i n c r e a s e d . C o n s i d e r i n g that changes i n the measured q u a n t i t i e s i n the d i r e c t i o n of the main flow are slower than those i n the tra n s v e r s e d i r e c t i o n , together with the e f f o r t r e q u i r e d i n data h a n d l i n g and the advantage of the symmetry i n the flows, the measurement l o c a t i o n s given i n Figures 6.1 to 6.6 were adopted. The l o c a t i o n s of the measurements were referenced with respect to the nozzle center and the flow width was estimated based on experience from p r e l i m i n a r y experiments; moving outward r a d i a l l y measurements were taken u n t i l the bubble frequency was between 1 and 3 s 1 f o r a p a r t i c u l a r c r o s s - s e c t i o n . Together with the s i g n a l a n a l y s i s , the d e t a i l of the measurements allowed to determine an i n t e g r a l p i c t u r e of the flow behaviour of the plumes f o r the f i r s t time. 110 •« I u • 2 §• *- o i ? e o o I 2 « 3 5 0 mm r « 0 " + 0« 876 N cm 3 /s "b « 400 mm d o * 635mm z • 140mm r-0 " z•20mm r • 0 » 4 0 0 800 1200 Sample size (number of t£ times) 0 4 0 0.20 0 0.4 0 0 0.40 - 0.20 1600 c o • » o c a CT O I -a. • < o o n ~ 7 o 0.2 0 CI -o ^ » c m 9 ~> o 3 o rt o Figure 5.11 Establishment of sample s i z e f o r s t a t i s t i c a l l y meaningful measurements by e l e c t r o r e s i t i v i t y probe i n t u r b u l e n t g a s - l i q u i d plumes. 5.3 C h a r a c t e r i s t i c s of the Response of the Sensor and the  C o n d i t i o n i n g C i r c u i t T h i s s e c t i o n presents the r e s u l t s of p r e l i m i n a r y experiments c a r r i e d out e s p e c i a l l y to examine the e f f e c t of the probe and c i r c u i t c h a r a c t e r i s t i c s on the q u a l i t y of the s i g n a l s generated and on the measurements. I l l 5.3.1 E f f e c t of Probe C h a r a c t e r i s t i c s on the Measurements Probe s t r e a m l i n i n g and c o n t r o l of v e r t i c a l s e p a r a t i o n of the t i p s , as well as of the l a t e r a l s e p a r a t i o n between the needles of the sensor and the s i z e of the probe t i p s , S e c t i o n 4.1.4 , were necessary to ensure a f a s t and matched response from both elements. Figure 5.12 compares the s i g n a l s from two d i f f e r e n t probes. It i s seen that with c l o s e needle s e p a r a t i o n , l i q u i d b r i d g i n g r e s u l t s i n a very slow response from the upper contact. T h i s phenomenon was p a r t i c u l a r l y n o t i c e a b l e at low gas flow rate and at p o s i t i o n s c l o s e to the plume edge. This problem was c o r r e c t e d f o r the f i n a l experiments ; the t i p s were kept v e r t i c a l l y a l i g n e d to measure the bubble v e l o c i t y c o r r e c t l y and at the same time l i q u i d b r i d g i n g was avoided by c o n t r o l l i n g the l a t e r a l s e p a r a t i o n between the needles, S e c t i o n 4.1.4 . Table 5.2 shows the e f f e c t of probe t i p length on gas f r a c t i o n and bubble frequency. The smaller values of these q u a n t i t i e s , recorded by the probe with the longer t i p , were r e l a t e d to the longer times r e q u i r e d f o r the v o l t a g e t r a n s i t i o n s to occur. T h i s e f f e c t i s p a r t i c u l a r l y d e t r i m e n t a l at high bubble f r e q u e n c i e s because the f a l l i n g and r i s i n g times of the s i g n a l s are i n many cases longer than the time between the a r r i v a l of bubbles. For the f i n a l experiments the s i z e of the sensor t i p s was kept very s m a l l , S e c t i o n 4.1.4 The sensor t i p s i z e was approximately a twentieth of the s m a l l e s t p i e r c e d length measured. (a) 1 v/div ; 10 ms/div (b) 1 v/div ; 2 ms/div Figure 5.12 Comparison of s i g n a l s produced (a) i n the presence and (b) absence of l i q u i d b r i d g i n g between the sensor c o n t a c t s . Table 5.2 V a r i a t i o n of i n d i c a t e d p r o p e r t i e s with probe t i p A length. Q Q = 1257 Ncm^/s h b - 400 mm d. -b 6.35 mm z (mm) Exposed Ti p Length (mm) 0 a(\) 191 V 8 " 0 . 488 a ( * ) V 8 " 1 ) 65 72 . 6 103 6 61 . 3 58 . 6 290 14 . 4 25 11.3 21 . 4 113 A n o t h e r i m p o r t a n t f a c t o r c o n c e r n i n g the c h a r a c t e r i s t i c s of the probe i s the d i f f e r e n c e i n s i z e between t h e upper and lower t i p s , s i n c e i t r e s u l t s i n d i f f e r e n c e s i n the a m p l i t u d e of the s i g n a l s from the two c o n t a c t s . Under t h e s e c o n d i t i o n s , the d e f i n i t i o n of a s i n g l e t h r e s h o l d v o l t a g e becomes ambiguous. The e l i m i n a t i o n of a m p l i t u d e d i f f e r e n c e s i s a l s o i m p o r t a n t owing t o t h e i r e f f e c t s on t h e measured tim e d e l a y , as g i v e n by t C = t C - 1 ( V . B - V . A ) (5.16) g go — 1 1 v a s s u m i n g t h a t the f a l l i n g edges of the lower and upper s i g n a l s have the same s l o p e . F i g u r e 5.13 shows how t h e measured time d e l a y t depends on t h e r e l a t i v e a m p l i t u d e o f s i g n a l s A and B, when the r e s p e c t i v e f a l l i n g v o l t a g e edges a r e p a r a l l e l . In t h i s F i g u r e 5.13 I n f l u e n c e o f v o l t a g e s i g n a l a m p l i t u d e d i f f e r e n c e on the d e t e c t e d time d e l a y , t g 114 work the problems a s s o c i a t e d with d i f f e r e n c e s in s i g n a l amplitudes were i n i t i a l l y present but were c o r r e c t e d by f i n e adjustment using a m p l i f i e r s with v a r i a b l e gain in the s i g n a l c o n d i t i o n i n g c i r c u i t . F i g u r e 5.14 shows examples of analogue s i g n a l s generated duri n g the f i n a l experiments. The s i g n a l s e x h i b i t a very short f a l l i n g time of the order of approximately 600 us ; the s i g n a l s from both contacts a l s o have the same amplitude and t h e i r f a l l i n g edges were c l o s e l y p a r a l l e l . Therefore both probes had very s i m i l a r d e t e c t i o n times and consequently the bubble v e l o c i t y measurements were v i r t u a l l y independent of the t h r e s h o l d v o l t a g e . 5.3.2 Threshold Level The d i g i t a l i d e n t i f i c a t i o n of bubbles r e q u i r e s s p e c i f i c a t i o n of the t h r e s h o l d l e v e l V . Thus, there i s a need f o r a c o n s i s t e n t s e l e c t i o n of t h i s l e v e l owing to i t s e f f e c t s on the measured gas-phase parameters. As mentioned e a r l i e r , the l i q u i d v o l t a g e l e v e l was independent of the experimental c o n d i t i o n s (2.53 ± 0.01 v o l t s ) during the whole study. Then, the t h r e s h o l d l e v e l s could be s e l e c t e d as f r a c t i o n s of the peak voltage to study t h e i r e f f e c t on the measured gas phase c h a r a c t e r i s t i c s . In the s e v e r a l t e s t s c a r r i e d out i t was found that the values of the measured parameters v a r i e d only s l i g h t l y i n the range of t h r e s h o l d v o l t a g e s between 0.7Vj< V < 0.9V . The t h r e s h o l d l e v e l s e l e c t e d i n t h i s work was = 0.8W^ = 2.84 ± 0.01 v o l t s . This value was adequate s i n c e : 115 m s / d i v v / d i v F i g u r e 5.14 O s c i l l o g r a p h s showing s i g n a l s produced by bubbles i n t e r c e p t i n g the lower and upper c o n t a c t s of the probe, (Q - 876 Nca /a, z = 65 am, r - 0 mm) 116 (a) the time r e q u i r e d to reach the t h r e s h o l d l e v e l was of the order of 200 Us. or s h o r t e r , a l l o w i n g f o r an accurate determination of the bubble residence time. (b) most of the f a l l i n g and r i s i n g edges of the s i g n a l reached t h i s l e v e l thereby a l l o w i n g the i d e n t i f i c a t i o n of i n d i v i d u a l bubbles. (c) as mentioned e a r l i e r , around t h i s v o l t a g e l e v e l , the f a l l i n g edges of s i g n a l s A and B were p a r a l l e l . F i gure 5.15 shows the i n f l u e n c e of the t h r e s h o l d l e v e l on the value of the measured parameters. As d i s c u s s e d i n the next s e c t i o n , the computation of the gas flow rate from the c r o s s -s e c t i o n a l d i s t r i b u t i o n of mean bubble v e l o c i t i e s and gas f r a c t i o n support the adequacy of the s e l e c t e d t h r e s h o l d l e v e l . 5 . 4 E v a l u a t i o n of Measuring System The system d e s c r i b e d i n t h i s work was designed to c a r r y out s i g n a l a n a l y s i s i n r e a l time and generate i n f o r m a t i o n on the gr e a t e s t number of gas-phase parameters i n bubble plumes. In t h i s s e c t i o n , an attempt w i l l be made to evaluate the r e l i a b i l i t y of the system f o r measuring v o i d f r a c t i o n and bubble v e l o c i t y . 5.4.1 Measurement of Rise V e l o c i t y of I n d i v i d u a l Spherical-Cap  Bubbles One method employed to was to measure the v e l o c i t y i n water. S i n g l e s p h e r i c a l t e s t the performance of the sensor of s i n g l e bubbles r i s i n g v e r t i c a l l y cap bubbles were generated using a 117 0.5 Fraction of voltage amplitude 0.6 0.7 0.8 0.9 1.0 i — r 2.3 J5 2.1 1.9 0 »876 Ncm 5 s" 1 h b « 6 0 0 m m dp '6.35 " O 1.7 4 2 38 34 — 30 1.6 2 * 100mm r = 0 • — o O' 3 a E o 9 O o > ZD 2.4 2£ 3.2 3.6 Threshold voltage V t , volts - 20 18 -o — 16 14 72 68 64 4.0 > < cr c er er 3 3 w c c 3 O Figure 5.15 V a r i a t i o n of i n d i c a t e d flow parameters with t h r e s h o l d l e v e l f o r double-contact e l e c t r o -r e s i s t i v i t y sensor. 118 s t a i n l e s s s t e e l scoop to which the d e s i r e d volume of gas was i n j e c t e d from a s y r i n g e . Bubble r e l e a s e was accomplished by r o t a t i n g the scoop. The v e l o c i t y of the bubbles was measured by the probe and from f i l m s recorded by a high-speed camera. Figure 5.16 shows the r e s u l t s of the t e s t s f o r bubbles of d i f f e r e n t volume. The measurements are compared with values c a l c u l a t e d from 134 the D a v i e s - T a y l o r r e l a t i o n s h i p . C o n s i d e r i n g that t h i s formula has been t e s t e d e x t e n s i v e l y , the r e s u l t s obtained by the probe show very high accuracy. The small discrepancy f o r the l a r g e s t bubbles was due to the s l i g h t break-up that these bubbles s u f f e r e d during t h e i r r e l e a s e . Bubble volume V b,cm 3 F i g u r e 5.16 Bubble v e l o c i t i e s reported by the probe ( O ) f o r s i n g l e s p h e r i c a l cap bubbles, compared with those measured from h i g h ^ g e e d f i l m (•) and c a l c u l a t e d from D avies-Taylor r e l a t i o n s h i p . 119 5.4.2 Gas Phase Volume B a l a n c e S i m u l t a n e o u s measurement of t h e v o i d f r a c t i o n and b u b b l e v e l o c i t y d i s t r i b u t i o n s p e r m i t s a n o t h e r form o f e v a l u a t i o n of the r e l i a b i l i t y of t h e r e s i s t i v i t y probe f o r m e a s u r i n g t h e s e p a r a m e t e r s . From E q u a t i o n (2.23) e q u a t i n g the mean l o c a l gas phase v e l o c i t y t o t h e measured mean b u b b l e r i s e v e l o c i t y , the gas volume f l o w r a t e a t a g i v e n c r o s s - s e c t i o n can be c a l c u l a t e d as t Q o " <* U b dA (5.17) A P T h i s i n t e g r a t i o n was c a r r i e d out o v e r a l l the c r o s s - s e c t i o n s and c o n d i t i o n s s t u d i e d . The i n t e g r a t e d gas f l o w r a t e s were c o r r e c t e d f o r the t o t a l s t a t i c p r e s s u r e . D e t a i l s o f the i n t e g r a t i o n p r o c e d u r e a r e g i v e n i n A p p e n d i x IV. The v a l u e s o b t a i n e d from i n t e g r a t i o n of the l o c a l f l o w p r o p e r t i e s a r e g i v e n i n F i g u r e 5.17 i n t h e form of d i s c r e p a n c i e s from t h e i n p u t a i r volume f l o w r a t e , D i s c r e p a n c y _ i n t e g r a t e d a i r r a t e - i n p u t a i r r a t e x 100 (5.18) i n p u t a i r r a t e Most o f t h e c a l c u l a t e d gas volume f l o w r a t e s a r e w i t h i n ± 10 % of the v a l u e s o b t a i n e d from f l o w r a t e measurements and a r e g e n e r a l l y l o w e r . In a l l c a s e s t h e d i s c r e p a n c y i s l e s s t h a n 17 % e x c e p t f o r t h o s e c r o s s - s e c t i o n s a t 20 mm from t h e n o z z l e , w h i c h were c l o s e t o 25 % . From t h e s c a t t e r o f t h e d a t a a p p e a r i n g i n t h e f i g u r e , a c l e a r t r e n d i s a p p a r e n t from p o s i t i v e s c a t t e r i n the r e g i o n near the n o z z l e t o n e g a t i v e s c a t t e r a t h i g h e r p o s i t i o n s . The n e g a t i v e 120 >s U c o C L s> u 18 I 2r-6 0 - 6 -12 -18 • • V 8 O T o • o A o -9-do t mm Q. (Ncm3/ •) 371 876 1257 635 O V O Y 400 600 410 • A 400 9 A Y o A o A o o • o • o 1 1 200 400 Vertical position z.mm 600 Figure 5.17 D i s c r e p a n c i e s between Input A i r Rates and A i r Rates obtained from Equation (5.17) discrepancy can be explained p a r t i a l l y by the use of a t h r e s h o l d l e v e l and the p o s s i b l e d e f l e c t i o n of the s m a l l e s t bubbles. The p o s i t i v e d i s c r e p a n c i e s are more d i f f i c u l t to e x p l a i n but i t i s p o s s i b l e that they occur because the c a l c u l a t i o n of the gas volume flow rate d i d not consider dynamic pressure e f f e c t s l i k e l y to be important c l o s e to the nozzle where the bubbles a c c e l e r a t e . 2 9 McCann et a l . , i n t h e i r theory of "doublet" formation, 121 suggested that the pressure i n the wake of preceding bubbles can cause up to 20 % i n c r e a s e i n the volume of f o l l o w i n g bubbles. An important i n d i c a t i o n of the accuracy of any measurement technique i s the r e p r o d u c i b i l i t y of the r e s u l t s obtained. Figure 5.18 compares the bubble v e l o c i t y and p i e r c e d length d i s t r i b u t i o n taken nine days apart under i d e n t i c a l c o n d i t i o n s . There i s very good agreement between the d i s t r i b u t i o n s . The r e s u l t s r e p o r t e d i n Chapter 6 r e f l e c t a l s o the high degree of r e p r o d u c i b i l i t y obtained i n a l l the parameters measured. The very good agreement that i n general e x i s t e d between i n t e g r a t e d and input a i r rates give support to the measuring instrument and to the c o n s i d e r a t i o n s adopted i n the s i g n a l a n a l y s i s procedure, and i n d i c a t e that the r e s u l t s obtained i n t h i s work present a c o r r e c t p h y s i c a l p i c t u r e of the behaviour of the bubbles in t u r b u l e n t g a s - l i q u i d plumes. (0) FU • i H-UK I 0 * 876 N crrfVs h b •400mm do • 6.35 mm z « 100mm, r«Omm Bubb le r ise velocity Ub,m/s 20 40 60 80 Bubble pierced length lb,mm 100 Figure 5.18 Bubble rise velocity and pierced length di s t r i b u t i o n s measured nine days apart. 123 CHAPTER 6 PRESENTATION AND DISCUSSION OF RESULTS The previous chapter contains an assessment of the computer c o n t r o l l e d experimental technique a p p l i e d to measure l o c a l p r o p e r t i e s of g a s - l i q u i d plumes i n v e r t i c a l l y i n j e c t e d a i r j e t s i n water. This chapter r e p o r t s and d i s c u s s e s the f o l l o w i n g : the r e s u l t s of the experimental measurements of flow parameters i n such plumes, the i n f l u e n c e of d i f f e r e n t i n j e c t i o n c o n d i t i o n s on the flow behaviour of the bubbles w i t h i n the plume, the plume geometry, the r e s u l t s of some c o r r e l a t i o n s f o r the d i s t r i b u t i o n of gas i n s i d e the plume and a comparison of the model p r e d i c t i o n s 5 6 of Tacke et a l . with some of the experimental r e s u l t s . Measurements were performed under c o n d i t i o n s i n v o l v i n g d i f f e r e n t gas flow r a t e s , bath depths and nozzle diameters. In e f f e c t s i x d i f f e r e n t i n j e c t i o n c o n d i t i o n s were examined as shown in Table 4.2 The data obtained i n a l l the t e s t s c o n s i s t of : the time-averaged gas f r a c t i o n and bubble frequency, the mean bubble v e l o c i t y and p i e r c e d length and the spectrum of bubble v e l o c i t y and p i e r c e d length, measured at s e l e c t e d p o i n t s . 6.1 P r o f i l e s of Void F r a c t i o n The r a d i a l gas f r a c t i o n d i s t r i b u t i o n , at nine l e v e l s i n the plume, was measured f o r each of the s i x experimental c o n d i t i o n s s t u d i e d . The r e s u l t s are p l o t t e d i n Figures 6.1 to 6.6. The gas 124 i I I I i — l — i — I — i — i — i — i — i — i — r z,mm <ct>,% r 20 26.03 — O 40 19.61 -80 -60 -40 -20 0 20 40 60 80 Radial position r, mm Figure 6.1 Gas fraction p r o f i l e s in different cross-sections of an air-water plume. ri 2 5 1 I I I i i I i i I—i—i—r 9 0 0 = 876 N cmVs h f c=400 mm d e=6.35 " 2 ,mm < • >,% V 2 0 34.95 O 4 0 26.90 A 6 5 2 3.47 O 100 1 3.37 • 140 10.40 0 190 8.32 ~~ O 2 4 0 6.05 2 9 0 5.08 A 3 5 0 3.89 -- 2 0 0 2 0 4 0 Radiol position r,mm 8 0 Figure 6.2 Gas fraction p r o f i l e s in different cross-sections of an air-water plume. 126 3* O | E 100 90 80 70 60 50 t o o o u o 40 30 20 10 i — i — i — r Q = I257N cmVs hb = 400 mm do =6.35 " i—i—r z , mm <*>.% r 20 36.82 — O 40 30.68 65 28.00 o 100 1 7.74 _ • 140 13.57 0 190 8.50 o 240 6JB6 V 290 6.19 — X 350 4.84 -80 - 6 0 - 4 0 -20 0 20 Radial position r, mm 40 60 80 Figure 6.3 Gas fraction p r o f i l e s in different cross-sections of an air-water plume. 127 90 ~ i i i i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r ~ z 80-70 • 60' £ 50 u o in o o u o 40-30 — 20 — 10 Q = 876Ncm 3/s = 600 mm do = 6.35 " z,mm < e O . % Y 20 30.90 — o 40 25.43 A 65 23.19 O 100 13.78 — a 140 10.09 0 190 8.65 o 290 5.00 V 450 2.70 A 550 2.33 -100 -80 -60 -40 -20 0 20 40 Radial position rt mm 100 Figure 6.4 Gas f r a c t i o n p r o f i l e s i n d i f f e r e n t c r o s s - s e c t i o n s of an air- w a t e r plume. 1 2 8 100 9 0 Q= 371 Ncm' / s 8 0 — = 4 0 0 mm do = 4.10 " o c o u ^ 50 ¥> O o 4 0 o i i i i i—i—i—i—i—i—i—i—i—i—r 2 , m m V 2 0 29.92"" O 4 0 25.34 A 6 5 17.21 O 100 9 9 6 _ • 140 7.2 3 0 1 9 0 5.6 7 o 2 4 0 3.92 V 2 9 0 3.42 ~~ 3 5 0 2.44 - 2 0 0 2 0 4 0 Radial position r,mm 6 0 8 0 Figure 6 . 5 Gas f rac t ion p r o f i l e s in d i f fe rent c ross-sect ions of an air-water plume. 129 lOOl 90 8 0 7 0 6 0 c o I 50} ~i i i i i—i i—i—i—i—i—i—i—i—r D O U o 4 0 3 0 2 0 0 0 * = 0 = 8 7 6 N c m 8 / s h b = 4 0 0 mm d e=4.IO " z.mm <*>,•/. V 2 0 21.68 _ O 4 0 22.50 6 5 20.89 o 100 13.12 • 1 4 0 9.59 -0 1 9 0 6.40 o 2 4 0 553 2 9 0 5.08 3 5 0 359 ~ W /9J i • • - 8 0 - 6 0 -4 0 - 2 0 0 2 0 4 0 Radial position r,mm 6 0 8 0 Figure 6.6 Gas fraction p r o f i l e s in different cross-sections of an air-water plume. 130 f r a c t i o n p r o f i l e s are seen to be symmetrical and each e x h i b i t s a s i m i l a r b e l l - s h a p e . The measurements r e v e a l the continuous spread of the gas w i t h i n the plume, as l i q u i d i s e n t r a i n e d by the r i s i n g bubbles ; the p r o f i l e s become f l a t t e r and wider downstream from the i n j e c t i o n p o i n t . Viewed i n terms of area-averaged, the gas f r a c t i o n seen i n the f i g u r e s decreases with d i s t a n c e from the o r i f i c e . The s i m i l a r i t y of the gas f r a c t i o n p r o f i l e s i n s u c c e s s i v e s e c t i o n s of the j e t becomes evident when the p r o f i l e s are normalized by d i v i s i o n of the l o c a l gas f r a c t i o n by a and are e * max r e l a t e d to the dimensionless d i s t a n c e s obtained with a length s c a l e r r t . . This d i s t a n c e c a l l e d the h a l f - v a l u e r a d i u s i s the amax/2 di s t a n c e from the a x i s of symmetry at which the gas f r a c t i o n i s h a l f the maximum value. The normalized gas f r a c t i o n p r o f i l e s at the nine l e v e l s considered are shown i n Figures 6.1a to 6.6a. The s i m i l a r i t y of the c r o s s - s e c t i o n a l p r o f i l e s i s observed over the e n t i r e plume length. Under a l l the c o n d i t i o n s s t u d i e d , the experimental reduced gas f r a c t i o n d i s t r i b u t i o n s can be approximated by the f o l l o w i n g Gaussian curve 2 _a _ exp (-0.7 r ) (6.1) araax r,*max/2 This equation produces very good agreement with the experimental data p r a c t i c a l l y across the e n t i r e plume. However, values that are s l i g h t l y below the measurements are obtained near the apex of 131 t c o E e c o o o o o o o o c t> E 0.8 1.6 2.4 Dimensionless radial position r/ ro W O J ( / 2 F i g u r e 6.1(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s v e r s e s e c t i o n s i n the corresponding previous f i g u r e . 132 Figure 6.2(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an a i r - w a t e r plume ; symbols belong to the t r a n s v e r s e s e c t i o n s i n the corresponding previous f i g u r e . 133 x o £ o c o o o D o> O O O in V) a> c o c a> E Q = l 2 5 7 N c m 3 s 1 h b = 4 0 0 m m d 0 =6.35 mm 0 0.8 1.6 2 .4 Dimensionless rad ia l position r / T a f n Q X / 2 Figure 6.3(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s v e r s e s e c t i o n s i n the corresponding previous f i g u r e . 134 Figure 6 . 4 ( a ) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an a i r - w a t e r plume ; symbols belong to the t r a n s v e r s e s e c t i o n s i n the corresponding previous f i g u r e . 135 Figure 6.5(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an air-water plume ; symbols belong to the t r a n s v e r s e s e c t i o n s i n the corresponding previous f i g u r e . 136 Figure 6.6(a) Dimensionless r a d i a l gas f r a c t i o n p r o f i l e s at d i f f e r e n t a x i a l d i s t a n c e s from the nozzle i n an a i r - w a t e r plume ; symbols belong to the t r a n s v e r s e s e c t i o n s i n the corresponding previous f i g u r e . 137 the gas f r a c t i o n d i s t r i b u t i o n ; at the edge of the plume the values from Equation (6.1) are s l i g h t l y too high. Figures 6.7(a) to 6.7(c) show maps of the voi d f r a c t i o n i n a v e r t i c a l plane pas s i n g through the c e n t r e l i n e of the plume. The maps c o n s i s t of i s o v o i d f r a c t i o n curves which are numbered to i n d i c a t e the gas f r a c t i o n expressed as a percentage. From these f i g u r e s i t i s c l e a r that i n an axisymmetrical a i r - j e t i s s u i n g v e r t i c a l l y upward, the g a s - l i q u i d plume has the form of a re g u l a r cone with i t s o r i g i n some d i s t a n c e upstream of the nozzle mouth. The angle subtended by the cone v a r i e d between 18° and 22° and incr e a s e d with the gas flow r a t e . 6.1.1 C o r r e l a t i o n s f o r the A x i a l Gas F r a c t i o n and the Half-Value  Radius Since the l a b o r a t o r y work rev e a l e d s i m i l a r i t y i n the d i s t r i b u t i o n s of gas f r a c t i o n , i t was d e s i r a b l e to o b t a i n c o r r e l a t i o n s f o r a and rn as a f u n c t i o n of p o s i t i o n and max u max/2 K system v a r i a b l e s , so that combined with Equation (6.1) the gas d i s t r i b u t i o n throughout the plume could be c h a r a c t e r i z e d . F i g u res 6.8 and 6.9 show the l o g - l o g p l o t s of the a x i a l gas f r a c t i o n and dimensionless h a l f - v a l u e r a d i u s as a f u n c t i o n of dimensionless v e r t i c a l p o s i t i o n , r e s p e c t i v e l y . It i s observed that the f i t t e d l i n e s f o r the a x i a l gas f r a c t i o n are almost p a r a l l e l when a i s below approximately 70 * and that the max dimensionless h a l f - v a l u e r a d i u s l i n e s are c l o s e l y p a r a l l e l over the e n t i r e plume len g t h . The f i t t e d equations have c o r r e l a t i o n 138 80 40 0 80 «0 0 Radial position r, mm Radial position r, mm Figure 6 . 7(a) Gas fractio n maps for different air-water bubble piumes. Vertical position z, mm co 140 Figure 6 . 7 ( c ) Gas fraction maps for different air-water bubble plumes. 141 c o e f f i c i e n t s i n the range 0.96 - 0.99 and are expressed as "max - C a ( f - ) " ° " 3 < 6 2 > o r a C , z , 0 ' 4 8 (6.3) "max/2 = r (-j—) o The i n t e r c e p t s of the l o g a r i t h m i c l i n e s i n Figures 6.8 and 6.9 at z/d Q = l , denoted by C a and C p r e s p e c t i v e l y , c o r r e l a t e d well with 14 2 Figure 6.9 V a r i a t i o n of dimensionless h a l f - v a l u e r a d i u s with dimensionless d i s t a n c e from the nozzle f o r d i f f e r e n t a i r - w a t e r plumes. the o r i f i c e m odified Froude number, as can be seen i n Figures 6.10 and 6.11. Thus the f o l l o w i n g expressions f o r the a x i a l gas f r a c t i o n and h a l f - v a l u e r a d i u s were obtained * d o 5 <Pl " P „ > 0 2 6 9 Z 0 9 9 3 _ 1 "max = 2 9 3 7 7 " ° 2 ( ~ ) ] Q P d o go o N >4 (6.4) 143 4000 2000 1000 200 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 de(mm) 0 (Ncm 5 » ' ) 371 876 1257 € 3 5 O v 0 Y 4 0 0 6 0 0 4.10 D 6 4 0 0 ^ — " * A O 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 6 Fr • 10 20 60 100 Figure 6.10 In t e r c e p t s of the p a r a l l e l p o r t i o n of the l i n e s i n F igure 6.8 as f u n c t i o n of the modified Froude number . o x o go z 0 , 4 8 ( — ) ] (6.5) d Equations (6.4) and (6.5) are p l o t t e d i n Figures (6.12) and (6.13) together with the experimental data ; a c l o s e f i t with l i t t l e s c a t t e r i n the data i s observed. The v a r i a b l e N d e f i n i n g the range of a p p l i c a b i l i t y of Equation (6.4) i s def i n e d as 144 0.20: 0.10 00.6 1 I I I M i l l 1 1 1 1 1 1 1 L d* (mm) Q f N c m3 »"') h f c (mm) -371 876 1257 — 6.3 5 O v O Y 400 600 — — 4.1 0 0 6 400 — " Y_ — — — 1 I I I M i l l 1 1 1 1 i 11 r Fr « 10 QIP, 20 60 100 Figure 6.11 I n t e r c e p t s of the l i n e s i n Figure 6.9 as f u n c t i o n of the modified Froude number. g d D 5 (p. - P ) 0 2 6 9 z ° " 3  Q o Pgo d o For the region of the plume where a > 70 % the f o l l o w i n g max expres s i o n holds f o r the v a r i a t i o n of the a x i a l gas f r a c t i o n with p o s i t i o n and modified Froude number g d 5 (p. - p ) 0 2 6 9 z ° " 3 " ° - 2 2 ° w - 1 0 0 t < — — < - > J Q P d v o Kgo o N < 4 (6.7) 145 I 00r=r 60 40 o i 20 c o u tn O 9 X < I 1—I I MMI Eq.(§.7) 1 1 I I I M Li Eq.(M) d0(mm) Q( Ncm 3 s"') hjmm) D 371 876 1257 6.35 O ? O Y 400 6 0 0 4.10 a A 4 0 0 J I I M I I I I J I M i l l i o 20 60 100 ( 0 r Q 0 Figure 6.12 C o r r e l a t i o n f o r the v a r i a t i o n of a x i a l gas f r a c t i o n with d i s t a n c e from the noz z l e , i n air- w a t e r plumes. This equation i s al s o shown i n Figure (6.12). Equations (6.4) and (6.5) are s i m i l a r i n form to expressions reported r e c e n t l y by 5 6 Tacke et a l . , but the values of the c o e f f i c i e n t s and exponents are d i f f e r e n t . This d i s s i m i l a r i t y may a r i s e from d i f f e r e n c e s i n the Froude numbers employed and i n the speed of response of the measuring system to l i q u i d - g a s t r a n s i t i o n s . Tacke et a l . worked with modified Froude numbers above 200 and reported t r a n s i t i o n 146 O —' v. M o IO 3 •o O k_ I t) _ 3 O > J . 0.6 o to c o C E 0.2 0.1 1 1 1 1 INI 1 1 1 1 1 1 1 1 dg(mm) 0(Ncm3$*') h^( fr» m) 371 876 1257 6.35 O V O Y 400 600 y -4.1 0 • & 400 — — f — r / — 1 1 1 1 M i l 1 1 1 11111 0.2 0.6 I 10 ( g d 0 9 ( ^ , - / ' g o ) ) a , 8 4 ( z / d 0 ) 0 - 4 8 ,2 "90 0 0 Pc Figure 6.13 C o r r e l a t i o n f o r the v a r i a t i o n of the h a l f - v a l u e r a d i u s with d i s t a n c e from the nozzle, i n a i r -water plumes. times of 2 ms , while i n t h i s study the modified Froude numbers were below 100 and the t r a n s i t i o n times of the s i g n a l s were i n the order of 600 ys ; the times to reach the t h r e s h o l d l e v e l , i n d i c a t i n g the presence of gas at the probe c o n t a c t s , were even s h o r t e r . Equations (6.1), (6.4), (6.5) and (6.7) d e s c r i b e the gas d i s t r i b u t i o n i n plumes, under the c o n d i t i o n s i n v e s t i g a t e d , very w e l l . 147 6.2 P r o f i l e s of the Bubble Frequency The l o c a l bubble frequency, l i k e the gas f r a c t i o n , was measured by means of the lower contact of the e l e c t r o r e s i s t i v i t y probe. The r a d i a l p r o f i l e s of bubble frequency at d i f f e r e n t a x i a l d i s t a n c e s from the nozzles are shown i n Figures 6.14 to 6.19, f o r the d i f f e r e n t experimental c o n d i t i o n s i n v e s t i g a t e d . S i m i l a r to the v o i d f r a c t i o n d i s t r i b u t i o n s , the bubble frequency d i s t r i b u t i o n s are c l o s e l y symmetrical and e x h i b i t a b e l l - s h a p e . However, an important d i f f e r e n c e r e s i d e s i n the f a c t that the continuous f l a t t e n i n g observed i n the gas f r a c t i o n d i s t r i b u t i o n curves, with d i s t a n c e from the i n j e c t i o n p o i n t , does not e x i s t i n the d i s t r i b u t i o n of bubble frequency. Instead, what i s observed i s an i n c r e a s e i n the bubble frequency near the nozzle, i . e . over the t r a n s v e r s e s e c t i o n s at a x i a l d i s t a n c e s below 100 mm from the o r i f i c e . Over t h i s region of the plume the bubble frequency p r o f i l e s become g e n e r a l l y steeper as d i s t a n c e from the nozzle in c r e a s e s ; t h i s tendency i s p a r t i c u l a r l y s trong at higher gas flow r a t e s , F i g u r e s 6.15 and 6.16 . As w i l l be seen l a t e r , these r e s u l t s are c o n s i s t e n t with the occurrence of breakup, i n the v i c i n i t y of the n o z z l e , of the bubbles forming at the o r i f i c e . The f i g u r e s suggest that the sudden i n c r e a s e i n the number of bubbles i s not compensated by the expansion undergone by the plume as r e s u l t of l i q u i d entrainment and hence the l o c a l bubble frequency i n c r e a s e s . However, once a s t a b l e bubble s i z e d i s t r i b u t i o n i s e s t a b l i s h e d beyond 100 mm, the continued expansion of the j e t produces bubble frequency p r o f i l e s which 60 i i I i I i I—I—i—i—i—i—i—i—r Figure 6.14 Bubble frequency p r o f i l e s in different cross sections of an air-water plume. 149 Figure 6.15 Bubble frequency p r o f i l e s in different cross-sections of an air-water plume. 150 Figure 6.16 Bubble frequency p r o f i l e s in different cross-sections of an air-water plume. 151 Figure 6.17 Bubble frequency p r o f i l e s in different cross-sections of an air-water plume. 152 Figure 6.18 Bubble frequency p r o f i l e s in different cross-sections of an air-water plume. 153 Radiol position r.mm Figure 6.19 Bubble frequency p r o f i l e s in different cross-sections of an air-water plume. 154 become lower and wider with i n c r e a s i n g d i s t a n c e from the begining of the plume. 6.3 Bubble Rise V e l o c i t y and i t s Spectrum The l o c a l bubble v e l o c i t y spectrum was measured s i m u l t a -neously with the other parameters, through the a n a l y s i s of the s i g n a l s from the lower and upper contacts of the e l e c t r o -r e s i s t i v i t y probe. Figure 6.20 shows t y p i c a l bubble v e l o c i t y s p e c t r a f o r s e v e r a l a x i a l p o s i t i o n s along the c e n t r e l i n e of a plume. Thus i t i s seen that the shape of the bubble v e l o c i t y s p e c t r a changes with p o s i t i o n . The s p e c t r a become l e s s skewed to the l a r g e v e l o c i t y values i . e . the d i s t r i b u t i o n becomes more symmetric, with i n c r e a s i n g d i s t a n c e from the noz z l e . It i s i n t e r e s t i n g to note that l a r g e r v e l o c i t i e s , along the c e n t r e l i n e , become more probable i n moving from z = 40 ram to z = 100 mm ; beyond t h i s p o s i t i o n the tendency reverses to decreasing v e l o c i t i e s . This behaviour can be e x p l a i n e d by noting that c l o s e to the nozzle the probe i s measuring, to a lar g e degree, the upward v e l o c i t y of bubbles at detachment and the speed of displacement of the growing i n t e r f a c e s . C o n s i d e r i n g to a f i r s t 3 9 approximation Equation (2.7) given by Walters and Davidson f o r the bubble v e l o c i t y at detachment, a value of 0.70 m/s i s c a l c u l a t e d f o r the c o n d i t i o n s r e p o r t e d i n Figure 6.20. This v e l o c i t y corresponds reasonably well to the f i r s t peak of the v e l o c i t y spectrum at z = 40 mm. The l a r g e s t v e l o c i t i e s of the spectrum would correspond to those bubbles a c c e l e r a t i n g i n the wake of the l e a d i n g bubbles. Beyond the region where gas 1 5 5 16 A 1 Z •350 mm 0 16 1 1 I Q«876Ncm s /s h b «400mm c90 '635 mm r • 0mm A ill 5 16 Z * 240 mm z • 100 mm 0 16 f - * - h " f l 2 • 40 mm _L 2 4 6 8 Bubble rise velocity UB , m/s Figure 6 . 2 0 Bubble velocity spectra at the plume centreline 156 discharge has an i n f l u e n c e , the bubble v e l o c i t i e s decrease as l i q u i d i s e n t r a i n e d and the bubbles d i s s i p a t e t h e i r energy. Changes i n the motion of the bubbles with d i s t a n c e from the i n j e c t i o n point are f u r t h e r d i s c u s s e d i n the next s e c t i o n . The spread of the bubble v e l o c i t y s p e c t r a c l e a r l y r e v e a l s the inadequacy of the assumption made by some i n v e s t i g a t o r s , when c a l c u l a t i n g the p i e r c e d length of the bubbles, that a l l the bubbles move at the same v e l o c i t y . The l o c a l mean bubble v e l o c i t y was evaluated from the bubble v e l o c i t y spectrum a c c o r d i n g to Equation (5.11) F i g u r e s 6.21 to 6.26 show the r a d i a l p r o f i l e s of the mean bubble r i s e v e l o c i t y at d i f f e r e n t l e v e l s i n the g a s - l i q u i d plumes. It i s evident that the j e t d i s s i p a t e s i t s momentum almost immediately upon discharge due to the sudden expansion of the gas. Close to the i n j e c t i o n p o i n t , steep v e l o c i t y p r o f i l e s are observed and these become g r a d u a l l y l e s s pronounced downstream as more l i q u i d i s e n t r a i n e d i n the plume. S i m i l a r to the r a d i a l v e l o c i t y p r o f i l e s i n homogeneous j e t s , the bubble r i s e v e l o c i t y d i s t r i b u t i o n s can be approximated by a Gaussian curve. The maximum bubble r i s e v e l o c i t i e s are l o c a t e d along the c e n t r e l i n e of the j e t as were the maximum v o i d f r a c t i o n and bubble frequency. As mentioned p r e v i o u s l y , the gas f r a c t i o n and the mean bubble r i s e v e l o c i t y d i s t r i b u t i o n s appearing i n Figures 6.1 to 6.6 and Figures 6.21 to 6.26, r e s p e c t i v e l y , were used to assess the r e l i a b i l i t y of the measurements. Since the gas volume flow r a t e s , obtained by i n t e g r a t i o n of the product of the gas f r a c t i o n 157 M E -o >. u o > 15 0.5 1.5 0 5 1.5 0 5 1.5 Q5 1.5 u 0 5 a £> _ O S « > < 1.5 05 1.5 Q5 l.5|— i i—i—i—i—i—r T — r 1 1—i r-i - 3 5 0 m m ~ B — S Z 2 2 — Z a ~ H 1 h -t h 2 9 0 m m _ H 1 r - j g — c — s — o — a — a — z ~ H I I h H r-240mm _ H 1 r t j - g — s — a — s — s — n — z r ^ — i — I — i — i — h " 3 2 — Z ZS~ —\ 1 1 H i l 1 9 0 m m — -a—z—z a a s~a—s—s—z~s~ i 1 1 1 1 1 1 1 1 h -B-B-z- a " ^ I A * t - E S s -\ 1 1 1 I 1 1 1 — I 140 mm — I 1-100mm --\ 1 r Q-371 NcmVs ht • 400 mm d 0 » 635 mm 65 mm — - r — r -20 mm — I I L 05 -80 - 6 0 - 4 0 -20 0 20 Radial position r,mm 40 60 80 Figure 6 .21 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s -s e c t i o n s of an air-water plume. 158 13 o _o a> > a> _0J X) a> o> o L. a> > < 2D 1.0 1.5 0.5 1.5 0.5 1.5 0.5 1.5 0.5 2.0 2.0 1.0 2 0 1.0 2.0 1 .0 -i—i—r — i * • l—i—i—i—i—r -H 8 B S Z S-H 1 1 ~l—I—I-z = 350mm H 1 1 \ 1 1 r-~z—s—= » a—z—a—-g- 290mm -H 1 1 r--a A ~I—s » a — I T H—I—I—I—h H—I h Q = 8 7 6 N c m 3 / s - h b =400mm - do = 6.35 mm -f-H r—r-I T H 1 r-J L_l L J I I L H 1 h 240mm 190mm -•i 1 I h 140mm ~ 100mm 65mm 4 0 mm 20mm - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 Radial position r,mm 4 0 6 0 8 0 Figure 6.22 Mean bubble v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s -s e c t i o n s of an air- w a t e r plume. 159 1.5 i—i—r 10 1 3 o > *> in X J o > < 0.5 2.0 1.0 2.0 1.0 2 0 1.0 2.0 1.0 2.0 1.0 2.0 1.0 i I i i r -a—z—a—s—z~ i — r ~ i — l — r -z= 350mm 240mm -—I 1-190mm -H 1 1 1 1 1 h 140mm H 1 h H 1 h H 1 h 100mm Q = l 2 5 7 N c m 3 / s h b =400mm do = 6.35 " 40mm J I I I I L i 1 1 1- —I h-20mm J I I L - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 Radial position r, mm 6 0 8 0 F i g u r e 6.23 Mean b u b b l e v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s -s e c t i o n s of an a i r - w a t e r plume. 160 2j0 I—i—r i—i—i—r~r "i—i—i—i—i—i—i—i—i—i—r z s 5 5 0 m m 1.0-H — I — I — I — r --o s -B n~ H — I — I I I I I — I — I — h 1.5 05 15 e 0.5 £ 1.5 1 3 0.5 | 1.5 to I 0.5 42 20 I 1 I I 1 i 1 h 450mm -I I 1 I -\ 1 1 1 1 1 1 h 290mm 140mm ~ 1.0 JO JO •3 JO «, 20 a | 1.0 0 = 8 7 6 N c m ' / s h b = 600mm d 0 = 6.3 5mm 1 I I I I I I I I I 100mm _ 6 5mm — I I I I 20 1.0 2 0 1.0 H—I— 1 —I—H 40 mm H 1 1 1—\—I 1 1 1 1 1 1 1 h J I I I I L J I I I I I I i l i L 20 mm 100 - 8 0 - 6 0 - 4 0 - 2 0 0 20 40 60 80 100 Radial position r, mm F i g u r e 6.24 Mean b u b b l e v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s -s e c t i o n s o f an a i r - w a t e r plume. 161 2.0 I'.O 1.5 v, 0.5 E 1.5 T — i — i — i — i — i — i — i — i — r - * S a B B B B E H-1 3 ~B B Z S B 2 2j j -t s — a — B — B — s s S B s — 2 — g — j -n — r z =350mrrr 2 9 0 m m -240mm -- 8 0 - 6 0 - 4 0 -20 0 2 0 4 0 Radial position r, mm 6 0 8 0 F i g u r e 6.25 Mean b u b b l e v e l o c i t y p r o f i l e s i n d i f f e r e n t c r o s s -s e c t i o n s of an a i r - w a t e r plume. 162 6 «» JB >. "o o 4> > 10 JD JO 3 « o w-> < 2.0 1.0 1.5 0.5 1.5 0.5 1.5 0.5 1.5 0.5 2.0 1.0 2.0 1.0 2.5 1.5 i—i—i—i—i—r~i—i—i—r . ft S B B S Z S B~ n — i — r ~ z = 350mm H 1 1 1 1 1 1 r--a * s B B j -2 9 0 m m -•1 1 1 H TS g B S B B- 240mm -190mm -140mm -H—I 1 — h 100mm -65mm H 1 1 h Q=876Ncm 3 / s h b =400mm d o =4.10 mm I 1 1 1 r H 1 1 1 h 4 0 mm H 1 1 h 3.0 2.0 1 .0 -20mm J I I I I I I L - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 Radial position r,mm F i g u r e 6.26 Mean bubble v e l o c i t y p r o f i l e s i n s e c t i o n s of an air- w a t e r plume. d i f f e r e n t c r o s s -163 and mean bubble v e l o c i t y over the flow c r o s s - s e c t i o n , were g e n e r a l l y w i t h i n ±10 * of the i n j e c t e d gas volume flow, i t appears that the measured bubble v e l o c i t y s p e c t r a adequately r e p r e s e n t the motion of the bubbles w i t h i n the plume. The standard d e v i a t i o n of the bubble v e l o c i t y spectrum or i n t e n s i t y of turbulence were c a l c u l a t e d a c cording to Equation (5.12) . The r a d i a l v a r i a t i o n of the standard d e v i a t i o n f o r d i f f e r e n t a x i a l p o s i t i o n s i s p l o t t e d i n F i g u r e 6.27, under those c o n d i t i o n s i n v o l v i n g the lowest and highest gas flow r a t e s s t u d i e d . The f i g u r e shows that the standard d e v i a t i o n of the bubble v e l o c i t y spectrum has a f a i r l y uniform d i s t r i b u t i o n over most of the flow areas except c l o s e to the i n j e c t i o n point where i t decreases s h a r p l y toward the plume edge. This c h a r a c t e r i s t i c of the p r o f i l e s c l o s e to the nozzle may be explained by the suppression of the f l u c t u a t i n g motions of the bubbles due to the i n e r t i a of a r e l a t i v e l y quiescent l i q u i d at the bottom of the v e s s e l near the plume boundary. Also from the same f i g u r e i t i s seen that the standard d e v i a t i o n of the bubble v e l o c i t y spectrum i n c r e a s e s with the gas flow r a t e . For the low gas flow r a t e i t appears that the d i s s i p a t i o n of t u r b u l e n t motion occurs over a short d i s t a n c e a f t e r which the turbulence i n the d i r e c t i o n of flow becomes n e a r l y homogeneous. For the high gas flow r a t e the decay of the t u r b u l e n t motion of the bubbles i s slower. T h i s may be due to the i n t e r a c t i o n of the bubbles with a l i q u i d which i s more throughly mixed by a more buoyant plume and p o s s i b l y by a more e f f e c t i v e i n t e r a c t i o n among the bubbles themselves. 164 Figure 6.27 Local standard deviation of for different gas flow rate bubble velocity spectrum conditions. 165 6.3.1 A x i a l P r o f i l e s of Bubble Rise V e l o c i t y - Influence of  I n j e c t i o n C onditions Some other i n t e r e s t i n g c h a r a c t e r i s t i c s i n the development of g a s - l i q u i d plumes, and the e f f e c t that i n j e c t i o n v a r i a b l e s have on the bubble motion, can be observed from the a x i a l p r o f i l e s of the mean bubble r i s e v e l o c i t y along the plume c e n t r e l i n e and the plume edge. The p r o f i l e s are shown i n F i gures 6.28 to 6.30 f o r the v a r i o u s i n j e c t i o n c o n d i t i o n s i n v e s t i g a t e d . The a x i a l bubble v e l o c i t y d i s t r i b u t i o n s r e v e a l three c h a r a c t e r i s t i c flow r e g i o n s . In the f i r s t zone of developing flow the shape of the c e n t r e l i n e p r o f i l e i s s t r o n g l y a f f e c t e d by the i n j e c t i o n v e l o c i t y . The bubble v e l o c i t y at the j e t axis i n c r e a s e s with height for low gas flow r a t e s ; t h i s i n c r e a s e becomes smaller as the i n j e c t i o n v e l o c i t y i s r a i s e d u n t i l above ~41.2 m/s (Q = 1257Ncm 3/s, d o 6.35 ram), Figure 6.29, the bubble v e l o c i t y begins to show a decrease with height. These c h a r a c t e r i s t i c s of the v e l o c i t y p r o f i l e s seem to be i n t i m a t e l y r e l a t e d to the phenomena of bubble formation at the o r i f i c e . At low gas flow r a t e s , i n d i v i d u a l bubbles form and d u r i n g t h e i r i n i t i a l r i s e exert strong wake e f f e c t s on t r a i l i n g bubbles, thus i n c r e a s i n g the v e l o c i t y of the l a t t e r . With i n c r e a s i n g i n j e c t i o n v e l o c i t y , severance of the bubbles from the o r i f i c e becomes l e s s s u c c e s s f u l and elongated gas envelopes r e s u l t to form i r r e g u l a r and d i s c ontinuous j e t s which lose t h e i r momentum as they penetrate i n t o the l i q u i d . In t h i s r e g i o n of the plume, the bubble r i s e v e l o c i t y at the plume boundary decreases to an approximately constant value, f o r given i n j e c t i o n c o n d i t i o n s . 166 Vertical position z , mm 0 100 200 300 400 Vertical position z ,mm F i g u r e 6.28 Mean b u b b l e v e l o c i t y p r o f i l e s a t the c e n t r e l i n e and boundary of plumes t o I l l u s t r a t e the e f f e c t o f gas i n j e c t i o n v e l o c i t y at two gas f l o w r a t e s . 167 Vertical position z,mm Figure 6.29 Mean bubble v e l o c i t y p r o f i l e s at the c e n t r e l i n e and boundary of plumes formed under d i f f e r e n t gas flow rate c o n d i t i o n s . 2.5- U.- 28.68m/s V • hb*400mm U.-28.15 h b*600 Q =876 Ncm5/s do - 6.35 mm 0.5 1 100 200 300 400 Vertical position z , mm 500 600 Figure 6.30 Mean bubble v e l o c i t y p r o f i l e s and boundary of plumes f o r c o n d i t i o n s . at the c e n t r e l i n e two bath depth 168 In the second region of developed flow, the c e n t r e l i n e p r o f i l e s show a slow decrease i n the bubble r i s e v e l o c i t y as the bubbles continue to d i s s i p a t e t h e i r energy and more l i q u i d i s e n t r a i n e d i n t o the plume. In t h i s r e g i o n of the plume the bubbles a f f e c t the flow mainly (or only) through the buoyancy fo r c e they induce. As mentioned p r e v i o u s l y , the bubble v e l o c i t y at the edge of the plume in t h i s r e gion remains a p p r e c i a b l y constant with heig h t . This can be e x p l a i n e d i f the bubble v e l o c i t y depends on both l i q u i d v e l o c i t y and ter m i n a l bubble v e l o c i t y as "b = U l + U b t ( 6 - 8 ) The i n c r e a s e i n the v e l o c i t y of the l i q u i d with height, along the plume boundary, would compensate f o r the decrease i n the bubble t e r m i n a l v e l o c i t y a s s o c i a t e d with the s l i g h t decrease i n the s i z e of the bubbles as they r i s e . F i n a l l y i n the t h i r d r e g i o n , the bath surface comes i n t o play and causes the bubble r i s e v e l o c i t y to decrease more r a p i d l y as the d i r e c t i o n of l i q u i d c i r c u l a t i o n changes from upward to r a d i a l l y outward. Figure 6.28 i l l u s t r a t e s the e f f e c t of i n j e c t i o n v e l o c i t y on the a x i a l mean bubble v e l o c i t y d i s t r i b u t i o n . From the p l o t s i t i s seen that the v e l o c i t y of the bubbles i n the region of developed flow i s independent of the i n j e c t i o n v e l o c i t y . Thus the plumes i n t h i s r e gion a f f e c t the motion of the surrounding l i q u i d mainly 5 7 through t h e i r buoyancy. In agreement with Abramovich the r e s u l t s i n d i c a t e that the k i n e t i c energy of the i n j e c t e d gas has 169 o n l y a l o c a l i z e d e f f e c t upon the m o t i o n of the b u b b l e s s i n c e t h i s e n e r g y i s d i s s i p a t e d r a p i d l y upon i n j e c t i o n . The r e s u l t s a l s o 13 5 a g r e e w i t h the m i x i n g s t u d i e s of H a i d a and Brimacombe who f o u n d t h a t the k i n e t i c power o f the i n j e c t e d gas has o n l y a s m a l l i n f l u e n c e on m i x i n g as compared t o i t s buoyancy power. In F i g u r e 6.29 the a x i a l b u b b l e r i s e v e l o c i t y p r o f i l e s a r e p l o t t e d f o r d i f f e r e n t gas f l o w r a t e s . I t i s s e e n t h a t t h e mean v e l o c i t y of the b u b b l e s i n the plume i n c r e a s e s w i t h the gas f l o w r a t e owing p r i m a r i l y t o an i n c r e a s e i n t h e s p e c i f i c buoyancy power o f the i n j e c t e d gas. The b a t h d e p t h , on t h e o t h e r hand, d i d not have an e f f e c t on t h e r i s e v e l o c i t y of the b u b b l e s , as shown i n F i g u r e 6.30, l i k e l y b e c a u s e the power i n p u t per u n i t mass f o r the two b a t h s under c o n s i d e r a t i o n was v e r y s i m i l a r , A p p e n d i x I I . In a d d i t i o n , the gas d i s t r i b u t i o n and the plume s p r e a d i n the two b a t h s were n e a r l y the same ; t h u s from t h e gas mass c o n s e r v a t i o n e q u a t i o n i t s h o u l d be e x p e c t e d t h a t the b u b b l e v e l o c i t y d i s t r i b u t i o n i n b o t h c a s e s s h o u l d be c l o s e l y s i m i l a r . 6.3.2 C o m p a r i s o n of the E x p e r i m e n t a l R e s u l t s w i t h the 5 6 P r e d i c t i o n s o f the Model of Tacke e t a l . 5 6 Tacke e t a l . r e c e n t l y p r o p o s e d a model t o e s t i m a t e t h e a x i a l v a r i a t i o n of gas f r a c t i o n , h a l f - v a l u e r a d i u s and b u b b l e r i s e v e l o c i t y i n g a s - l i q u i d plumes. The model was of t h e i n t e g r a l - p r o f i l e t y p e and c o n s i s t e d of the f o r m u l a t i o n of the c o n t i n u i t y e q u a t i o n s f o r the gas and t h e l i q u i d i n t h e plume and t h e c o n s e r v a t i o n of v e r t i c a l momentum of the plume. The r a d i a l 170 p r o f i l e s of gas f r a c t i o n and l i q u i d v e l o c i t y were assumed G a u s s i a n . These p r o f i l e s were r e l a t e d t h r o u g h the r a t i o of t h e i r w i d t h , w h i l e the l i q u i d and gas v e l o c i t i e s were c o n n e c t e d by the s l i p v e l o c i t y o f the b u b b l e s which was assumed e q u a l t o t h e b u b b l e t e r m i n a l v e l o c i t y . The model e q u a t i o n s and a s s u m p t i o n s a d o p t e d by t h e a u t h o r s a r e p r e s e n t e d i n A p p e n d i x V, t o g e t h e r w i t h the method o f s o l u t i o n employed i n t h i s work. 5 6 Tacke et a l . d e f i n e d t h e i n i t i a l c o n d i t i o n s f o r the model, i . e . the c o n d i t i o n s s t a t i n g the s t a r t of the d e v e l o p e d buoyant r e g i o n of t h e plume, t o be the p o s i t i o n c o r r e s p o n d i n g t o a c e n t r e l i n e gas f r a c t i o n of 50 % . However, i n the p r e s e n t work i t was f o u n d t h a t t h i s gas f r a c t i o n , p a r t i c u l a r l y at low gas f l o w r a t e s , i s w i t h i n the r e g i o n of d e v e l o p i n g f l o w c l o s e to the n o z z l e . T h e r e f o r e the c o n d i t i o n a = 50 % does not s p e c i f y the max s t a r t of the f u l l y d e v e l o p e d b u o y a n t r e g i o n of the plume f o r w h i c h th e model was p r o p o s e d . The i n i t i a l c o n d i t i o n s used i n t h i s i n v e s t i g a t i o n c o r r e s p o n d e d to the p o s i t i o n a t which the s e c o n d zone, t h e f u l l y d e v e l o p e d buoyant zone, commences based on t h e measured c e n t r e l i n e v e l o c i t y p r o f i l e s , as d i s c u s s e d i n the p r e v i o u s s e c t i o n . The e x p e r i m e n t a l gas f r a c t i o n and h a l f - v a l u e r a d i u s a t t h i s p o s i t i o n were used to s t a r t t h e model. F i g u r e 6.31 g i v e s the p r e d i c t i o n s o f the model f o r a g i v e n s e t of e x p e r i m e n t a l c o n d i t i o n s . The p l o t s show the s e n s i t i v i t y o f t h e s o l u t i o n t o t h e e n t r a i n m e n t c o e f f i c i e n t , w hich f o r p r a c t i c a l p u r p o s e s can be c o n s i d e r e d a f i t t i n g p a r a m e t e r . To c a l c u l a t e the b u b b l e r i s e v e l o c i t y , t h e e n t r a i n m e n t c o e f f i c i e n t , e , was 171 too f o L I I I I I | 1 Z 0-371 N cm* i" 40 20 I 0 6 -4 40 ' I I I I I 100 60 100 200 400 600 Vertical position z ,mm 40 20 -1 1 1111 1 i - 11 r - ^ -0' 371 Ncm'/ • hb» 400mm d 0 • 6.35 mm II 1 1 O.I00 0.120 0.135 -l i n n 1 1 40 60 100 200 400 600 Vertical position z,mm 10.0 o - 1 1 1 111 i 1 1-— Q«37l NcmVi — - hb • 400 mm — — d0 « 6J5 mm — — QI00 — 0.120 0.135 -o -o — i & ^ S ? ° o o — — -1 1 1 11 1 1 1 f 40 60 100 200 400 600 Vertical position i , mm Figure 6.31 Comparison between experimental and p r e d i c t e d v a r i a t i o n of a x i a l gas f r a c t i o n , h a l f - v a l u e radius and bubble v e l o c i t y with d i s t a n c e from the i n j e c t i o n p o i n t . P r e d i c t i o n s from model of Tacke et a l . 5 6 172 adjusted to produce the best f i t of the a x i a l gas f r a c t i o n and h a l f - v a l u e r a d i u s . From the f i g u r e i t i s seen that the c a l c u l a t e d gas f r a c t i o n and h a l f - v a l u e radius vary i n opposite d i r e c t i o n s with the entrainment c o e f f i c i e n t , and t h e r e f o r e cannot be f u l l y a d j u s t e d . F i g u r e s 6.32 and 6.33 show the r e s u l t s of the c a l c u l a t i o n s together with the experimental data f o r the other two gas flow r a t e s s t u d i e d . It may be considered that the p r e d i c t i o n s of the model are s e m i q u a n t i t a t i v e 1 y s a t i s f a c t o r y , but again i t should be remembered that the model p r e d i c t i o n s depend on the entrainment c o e f f i c i e n t to f i t the r e s u l t s . 6.4 D i s t r i b u t i o n of Bubble P i e r c e d Length As d i s c u s s e d i n S e c t i o n 5.1.4 , the residence time of the bubbles at the lower t i p of the probe can be used i n co n j u n c t i o n with t h e i r i n d i v i d u a l t r a n s i t v e l o c i t i e s to determine the p i e r c e d length d i s t r i b u t i o n of the detected bubbles. A probe assembly of the type used here w i l l p i e r c e the bubble at any point of the p r o j e c t e d f r o n t a l area of the bubble, and t h e r e f o r e any c a l c u l a t i o n of the bubble diameter d i s t r i b u t i o n r e q u i r e s the adoption of c e r t a i n assumptions re g a r d i n g bubble shape. In F i g u r e 6.34, the a r i t h m e t i c mean of the p i e r c e d length of the bubbles along the c e n t r e l i n e i s p l o t t e d as f u n c t i o n of d i s t a n c e from the no z z l e . From t h i s f i g u r e i t i s c l e a r that the i n j e c t i o n c o n d i t i o n s , i n p a r t i c u l a r gas flow r a t e and nozzle diameter, i n i t i a l l y produce d i f f e r e n t - s i z e d bubbles ; but moving 173 " 0 1 0 0 4 0 0 4 0 6 0 100 2 0 0 4 0 0 Ver.icol position z, mm Vertical position z .mm 10.0 i 4.0 o u o a o < 2.0 1.0 0.6-0.4 -1 1 1111 1 1 1 _ — — Q • 876Ncms/t hb • 400mm - ds " 6.35mm « • 0.125 -— o 1 1 1111 1 1 4 0 6 0 100 2 0 0 4 0 0 Vertical position z .mm F i g u r e 6.32 C o m p a r i s o n between e x p e r i m e n t a l and p r e d i c t e d v a r i a t i o n o f a x i a l gas f r a c t i o n . , h a l f - v a l u e r a d i u s and b u b b l e v e l o c i t y w i t h d i s t a n c e from the i n j e c t i o n p o i n t . P r e d i c t i o n s from model of Tacke et a l . 174 100 U I I I I I | t a 40 • 20 ' I I I I I I I ' ! , 0 •l257Ncm5/» ht' 400mm d„ » 6.35 mm _ « « 0.14 5 J I L 40 60 100 200 400 600 V e r t l c o l p o i l l i o n j . m m 100 r-l I I I II 40 f o 20 6 — 0 « 1257 N cmS/» hb • 400 mm Oo • 6.35 mm < • 0.14 5 J I I I 40 60 100 200 400 600 V e r t i c a l p o s i t i o n z,mm 100 40 20 10 0.4 - 1 1 1111 ] 1 1 1-—  — — 0 •l257Ncm ,/t h, • 400mm d 0 • 6.35 mm € • 0.14 5 — o — 0 ° o — o III 1 1 1 111 1 1 1 40 60 100 200 400 Vlrtlcol pocltlon t , mm 600 F i g u r e 6 . 3 3 C o m p a r i s o n between e x p e r i m e n t a l and p r e d i c t e d v a r i a t i o n of a x i a l gas f r a c t i o n , h a l f - v a l u e r a d i u s and b u b b l e v e l o c i t y w i t h d i s t a n c e from the i n j e c t i o n p o i n t . P r e d i c t i o n s from model of Tacke e t a l . 5 6 175 100 200 300 Verticol position z , mm 4 00 F i g u r e 6.34 V a r i a t i o n o f mean p i e r c e d l e n g t h o f b u b b l e s w i t h d i s t a n c e f r o m t h e n o z z l e a l o n g t h e p l u m e c e n t r e -l i n e , f o r d i f f e r e n t i n j e c t i o n c o n d i t i o n s . u p w a r d f r o m t h e n o z z l e , f l o w d e v e l o p m e n t i s a c c o m p a n i e d by a r e d u c t i o n i n b u b b l e s i z e u n t i l a s t a b l e d i s t r i b u t i o n o f s m a l l -s i z e d b u b b l e s i s e s t a b l i s h e d . T h e " e q u i l i b r i u m " p i e r c e d - l e n g t h i s p r a c t i c a l l y i n d e p e n d e n t o f i n j e c t i o n c o n d i t i o n s . F r o m t h e m e a s u r e m e n t s o f t h e b u b b l e p i e r c e d - l e n g t h c l o s e t o t h e n o z z l e i t i s c l e a r t h a t t h e e f f e c t o f t h e g a s momentum i s t o d i s t o r t t h e g a s e n v e l o p e s i n t h e v e r t i c a l d i r e c t i o n , g i v i n g t h e b u b b l e s a j e t - l i k e a p p e a r a n c e a l t h o u g h i r r e g u l a r a n d d i s c o n t i n u o u s . O b s e r v a t i o n s o f t h e b u b b l e s w e r e made f r o m h i g h s p e e d m o t i o n p i c t u r e s t a k e n a t d i f f e r e n t i n j e c t i o n c o n d i t i o n s a n d l o c a t i o n s i n 176 the j e t . F i g u r e 6.35 shows a sequence of e v e n t s o c c u r r i n g c l o s e t o t h e n o z z l e under c o n d i t i o n s of low gas f l o w r a t e . Under t h e s e c o n d i t i o n s the b r e a k u p o f the b u b b l e s was most c l e a r l y o b s e r v e d . The g r o w i n g b u b b l e , F i g u r e 6 . 3 5 ( a ) , d e v e l o p e d a d e p r e s s i o n i n i t s base as i n f l o w i n g gas p e n e t r a t e d t h e b u b b l e d r a g g i n g a column of l i q u i d t h a t r e a c h e d th e t o p s u r f a c e of t h e b u b b l e . The change i n shape o f the b u b b l e and the d i s t o r t i o n of i t s s u r f a c e became more r a p i d as t h e b u b b l e d e t a c h e d from the o r i f i c e , F i g u r e 6 . 3 5 ( b ) , and the b a s a l p e n e t r a t i o n became more p r o n o u n c e d . The f i n a l d i s r u p t i o n and d i s i n t e g r a t i o n of t h e l e a d i n g b u b b l e was b r o u g h t about as r e s u l t of b i n a r y c o a l e s c e n c e , i . e whereby the g r o w i n g b u b b l e becomes e l o n g a t e d and c o a l e s c e s w i t h t h e p r e v i o u s l y r e l e a s e d b u b b l e r e e s t a b l i s h i n g the f l o w of gas i n t o t h i s b u b b l e , F i g u r e 6 . 3 5 ( c ) . The outcome of a l l t h e s e p r o c e s s e s was a c l o u d of d i f f e r e n t s i z e d - b u b b 1 es t r a v e l l i n g c l o s e l y p a cked as shown i n F i g u r e 6 . 3 5 ( d ) . The sequence of e v e n t s o b s e r v e d i n t h e s e p h o t o g r a p h s a g r e e s v e r y w e l l w i t h the t r e n d s r e v e a l e d by the b u b b l e f r e q u e n c y and b u b b l e p i e r c e d l e n g t h measurements, i n t h e r e g i o n of d e v e l o p i n g f l o w . F i g u r e 6.36 i l l u s t r a t e s the s i t u a t i o n t h a t e x i s t s h i g h e r i n the b a t h , where i t i s seen t h a t b u b b l e s o f d i f f e r e n t s i z e s a r e r andomly d i s t r i b u t e d i n the plume. In F i g u r e 6.37 t h e p r o b a b i l i t y d i s t r i b u t i o n of p i e r c e d l e n g t h s , f o r the r e g i o n where the a r i t h m e t i c mean of p i e r c e d l e n g t h s i s c o n s t a n t , i s seen t o f o l l o w a l o g - n o r m a l d i s t r i b u t i o n . The g e o m e t r i c mean, 1, , shown i n F i g u r e 6.34, i s t h u s a b e t t e r bg 136 e s t i m a t o r of the c e n t r a l t e n d e n c y o f the d i s t r i b u t i o n and as e x p e c t e d t h i s q u a n t i t y i s s m a l l e r t h a n the a r i t h m e t i c mean. The 177 ( c ) 61 ms ( d ) 100 ms F i g u r e 6.35 F a s t speed p i c t u r e s taken at 1500 f r a m e s / s of the breakup of b u b b l e s i n the v i c i n i t y of the n o z z l e , (Q • 371Nc« / s , h = 400 mm, d = 6.35 mm, probe at c e n t r e l i n e z - 110 • • ) . ° 30 mm Figure 6.36 Fast speed pictures of bubbles in tr^e region of developed buoyant plume, (Q • 371Ncm /s, h. = 400 m m , d 200 m m ) . 6.35 mm, probe at centreline z = 179 d i s t r i b u t i o n s of pierced length, as shown in Figure 6.37, are very close for different axial positions but show that smaller bubbles become s l i g h t l y more frequent downstream in the plume. The different results concerning axial variations of gas fr a c t i o n , bubble frequency, mean bubble velocity and pierced length show ch a r a c t e r i s t i c changes with position. In p a r t i c u l a r , i t should be noticed that the regions of developing flow and bubble break-up extend over the same distance of the plume and that the region of developed flow is characterized, apart from being mostly a buoyant flow region, by a p r a c t i c a l l y constant bubble size d i s t r i b u t i o n . Provided that the flow is l o c a l l y homogeneous, the probe has equal probability of piercing the bubbles at any point on the projected frontal area and the measured chord length may vary from zero to the largest v e r t i c a l dimension of the bubble, e.g. diameter for a spherical bubble. Considering this to be the case in the region of developed flow, the d i s t r i b u t i o n of bubble diameters can be obtained from the d i s t r i b u t i o n of pierced lengths. 6.4.1 D i s t r i b u t i o n of Bubble Diameters 13 7 Different methods are found in the l i t e r a t u r e that can be used to determine the bubble size d i s t r i b u t i o n from the d i s t r i b u t i o n of pierced lenghts of the detected bubbles. The most commonly applied methods require that 1) a l l bubbles have spherical shape, 2) spheres of various sizes are randomly 1 8 0 ( E E e O Q . X* 3 0.1 0.5 I 2 Q-1257 N cm 3 /s h b » 4 0 0 m m d 0 "6.35 mm z(mm) 140 190 290 O 10 50 Percent larger than l D 99 80 i muni—i i i i i i i—r E E 9 C a> T3 a> u w a X) x> CD 4 0 - 20 10 4 0.1 AGTV AD O Q .371 Ncm 3 /s h^* 400 mm d 0 • 6.35 mm Z(mm) 100 190 290 O A O 'bg M i l l I X . 10 50 90 99 Percent larger than I , Figure 6.37 Log-normal p r o b a b i l i t y p l o t s of bubble p i e r c e d length f o r d i f f e r e n t c e n t r e l i n e p o s i t i o n s i n the region of developed flow of air- w a t e r plumes. 181 distributed over a l l the region under study and 3) the size of the sample of pierced lengths is s t a t i s t i c a l l y meaningful. For the bubble swarms studied in this work the f i r s t two requirements would be most closely approximated in the region where the "equilibrium" bubble size d i s t r i b u t i o n has been established, while the t h i r d condition is met as demonstrated in Section 5.2.4 . Although the bubbles are more generally not spherical, the use of a method involving this assumption may be j u s t i f i e d as a f i r s t approximation since many different bubble shapes exist in turbulent gas-liquid plumes, and only one bubble size parameter has been measured i.e. bubble pierced length. 13 7 The method of Spektor which is discussed by Underwood has been applied in this work to determine the distributions of bubble diameters from the probability density function of the pierced lengths of detected bubbles. This technique is based upon consideration of the penetration of a polydispersed system of spheres randomly distributed in space. Then for a discrete d i s t r i b u t i o n of pierced lengths, the number of bubbles with a diameter equal to the pierced length characterizing an interval of the pierced length histogram is given as 4 N lb (i) N l b (1 + 1 ) N ( i ) = v TT A 2 [ ] (6.9) 2i - 1 2i + 1 The derivation and use of this expression is explained in d e t a i l 13 7 by Underwood . To overcome the problem that Equation (6.9) can predict negative numbers, the measured distr i b u t i o n s of pierced 182 length were arranged in i n t e r v a l s of 5 mm. This equation magnifies d i f f e r e n c e s i n the d i s t r i b u t i o n s of p i e r c e d length. Figure 6.38 shows the d i s t r i b u t i o n s of bubble diameter corresponding to the d i s t r i b u t i o n s of p i e r c e d length, along the c e n t r e l i n e of the plume, given i n Figures 6.37 . The l i n e s r e p r e s e n t i n g the d i s t r i b u t i o n of bubble diameters were c a l c u l a t e d from the d i s t r i b u t i o n s of p i e r c e d length denoted by the l i n e s in Fig u r e s 6.37, which hold reasonably well along the e n t i r e plume c e n t r e l i n e . The data p o i n t s on the bubble diameter d i s t r i b u t i o n p l o t s correspond s p e c i f i c a l l y to the axis of the plume 190 mm downstream of the nozzl e . It i s evident that the bubble s i z e d i s t r i b u t i o n i s well represented by a log-normal d i s t r i b u t i o n . Figure 6.39 shows the geometric mean of the bubbles t r a v e l l i n g along the c e n t r e l i n e of plumes f o r a l l the c o n d i t i o n s s t u d i e d . It i s seen that i n general the bubble s i z e decreases with an inc r e a s e i n the o r i f i c e Reynolds number ; the geometric mean diameter ranged between 5 and 8 mm. It i s important to note that Figure 6.39 presents the geometric mean diameter of the bubbles and not the volume-surface mean diameter which was the q u a n t i t y 2 3 reported by Leibson et a l . . Figure 6.40 shows an example of the d i s t r i b u t i o n of bubble diameter across s e v e r a l s e c t i o n s of the plume. It i s c l e a r that, as p r e v i o u s l y mentioned, the d i f f e r e n c e s i n bubble s i z e over the plume radi u s are s m a l l , i n d i c a t i n g that bubbles of d i f f e r e n t s i z e are p r a c t i c a l l y homogeneously d i s t r i b u t e d i n the region of developed flow. 183 O J 0 5 Q 5 2 1 0 5 0 9 0 9 9 Percent larger than db Percent larger than d Figure 6.38 Log-normal p r o b a b i l i t y p l o t s of bubble diameter at the c e n t r e l i n e of the region of developed flow of ai r - w a t e r plumes. 184 | 30 9 r o c D V E o L . a> E o o o> « J x> X) 3 CD 10 1 1 I N I l | 1 1 1 1 1 1 d0(mm) 0 (Ncm5/s) 371 876 1257 6.35 0 v o V 4 0 0 4.1 0 • A 6 0 0 — ° C 1 7 — — 1 1 M M 11 1 1 1 1 1 1 2000 5000 10000 20000 Orifice Reynolds number Re 80000 Figure 6.39 Geometric mean bubble diameter at the c e n t r e l i n e of the r e g i o n of developed flow versus Reynolds number. E E « E o •> X> 3 O E o 14 12 10 8 6 I I I 0 -371 Ncm 3/« h 6 »400mm dc •635mm T 0 o o o o o w o o o i (mm) o 100 o 140 o 190 O 2 40 V 290 x 350 o o ° o vo » ° o O O O o • - X X ° o * © o 80 60 40 20 0 20 40 Radial position r.mm 60 80 Figure 6.40 Local geometric mean bubble diameter in the region of developed flow of an air-water plume. 185 CHAPTER 7 SUMMARY AND CONCLUSIONS The present work has sought to shed new l i g h t on the complex behaviour of t u r b u l e n t g a s - l i q u i d bubble plumes, i n v e r t i c a l l y i n j e c t e d j e t s , through the experimental determination of the fl u i d - d y n a m i c c h a r a c t e r i s t i c s of the gas phase : gas f r a c t i o n , bubble frequency, mean bubble v e l o c i t y and p i e r c e d length, and the s p e c t r a of the bubble v e l o c i t y and p i e r c e d length. This has r e q u i r e d the development of a unique measuring system c o n s i s t i n g of a two-element probe to sense bubbles i n the plume and a new s i g n a l a n a l y s i s procedure. The l a t t e r has i n v o l v e d the assembly of hardware and software to implement p a t t e r n r e c o g n i t i o n l o g i c in r e a l time. The a n a l y s i s ensures that the measured delay times of s i g n a l s from the two probe contacts are uniquely r e l a t e d to the t r a n s i t of bubbles t r a v e l l i n g a x i a l l y from the lower to the upper e l e c t r o d e . This f e a t u r e i s v i t a l f o r the e f f e c t i v e d etermination of the bubble r i s e v e l o c i t y and p i e r c e d l e n g t h . The measuring system a l s o allows simultaneous a c q u i s i t i o n of the data necessary to evaluate a l l the parameters mentioned p r e v i o u s l y . Tests on the accuracy and r e p r o d u c i b i l i t y of the measuring system have r e v e a l e d the f o l l o w i n g (1) The dimensions, geometry and alignment of the probe contacts are c r i t i c a l to the r e l i a b i l i t y and r e p r o d u c i b i l i t y of the measurements and t h e r e f o r e must be c a r e f u l l y c o n t r o l l e d . 186 Probe e f f e c t s are manifested i n the speed of the s i g n a l t r a n s i t i o n s and i n the degree of p a r a l l e l i s m between the f a l l i n g edges of s i g n a l s from the two c o n t a c t s . (2) R e l i a b l e bubble v e l o c i t y measurements r e q u i r e that the s i g n a l s from both contacts have a common l i q u i d voltage l e v e l . This permits the use of a s i n g l e t h r e s h o l d l e v e l and ensures that both s i g n a l s have the same d e t e c t i o n time. (3) Under the t u r b u l e n t c o n d i t i o n s of the plumes s t u d i e d , t y p i c a l l y about 25 to 35 per cent of the bubbles i n t e r c e p t e d could be accepted to e x t r a c t i n f o r m a t i o n on bubble v e l o c i t y based on the p a t t e r n r e c o g n i t i o n l o g i c . There i s s t r o n g evidence that the s i g n a l a n a l y s i s does not unduly bia s the r e s u l t s . The r e p r o d u c i b i l i t y of the r e s u l t s , t h e i r i n t e r n a l c o n s i s t e n c y , and the f a c t that the bulk of data produced i n t e g r a t e d a i r flow rates w i t h i n ± 10 per cent of the input r a t e s are i n d i c a t i o n s of the s u i t a b i l i t y of t h i s technique and the v a l i d i t y of the r e s u l t s . With t h i s measuring system, the f o l l o w i n g r e s u l t s were obtained : (1) The r a d i a l gas f r a c t i o n d i s t r i b u t i o n s across the plume are symmetric and can be approximated by Gaussian curves. It has been found that the p r o f i l e s e x h i b i t s i m i l a r i t y along the e n t i r e plume len g t h . (2) A set of experimental c o r r e l a t i o n s have been formulated to express the a x i a l v a r i a t i o n of the gas f r a c t i o n and h a l f value r a d i u s with i n j e c t i o n c o n d i t i o n s , as represented by 187 t h e m o d i f i e d F r o u d e number. A v e r y good r e p r e s e n t a t i o n of t h e gas d i s p e r s i o n was o b t a i n e d under the c o n d i t i o n s s t u d i e d . D u r i n g v e r t i c a l upward i n j e c t i o n , a i r - w a t e r plumes expand as a cone w i t h an a n g l e between 18° and 22° ; t h e a n g l e i n c r e a s e s w i t h the gas f l o w r a t e . F o r t h e f i r s t t i m e , b u b b l e f r e q u e n c y measurements have r e v e a l e d the i n c r e a s e i n the number of b u b b l e s t h a t o c c u r s o v e r the r e g i o n of t h e plume w i t h i n 100 mm from th e n o z z l e . These r e s u l t s show t h a t s m a l l e r b u b b l e s a r e c o n t i n u o u s l y p r o d u c e d by t h e s h a t t e r i n g of l a r g e b u b b l e s o v e r t h i s r e g i o n . Downstream o f t h i s zone the b u b b l e f r e q u e n c y p r o f i l e s become i n c r e a s i n g l y f l a t t e r and w i d e r . I t has been f o u n d t h a t the b u b b l e v e l o c i t y s p e c t r a a r e skewed. The measurements a l s o i n d i c a t e t h a t the s t a n d a r d d e v i a t i o n o f the b u b b l e v e l o c i t y s p e c t r u m i n c r e a s e s w i t h the gas f l o w r a t e , t h u s r e v e a l i n g the i n c r e a s e d t u r b u l e n t n a t u r e o f the f l o w . The r a d i a l mean b u b b l e v e l o c i t y p r o f i l e s a r e s y mmetric and can be a p p r o x i m a t e d by a G a u s s i a n c u r v e . The mean b u b b l e v e l o c i t y g r a d i e n t s a r e l a r g e c l o s e t o the i n j e c t i o n p o i n t . A x i a l b u b b l e v e l o c i t y p r o f i l e s c l e a r l y r e v e a l t h r e e r e g i o n s o f b u b b l e m o t i o n b e h a v i o u r : a r e g i o n of n o z z l e i n f l u e n c e or d e v e l o p i n g f l o w , a r e g i o n of f u l l y d e v e l o p e d buoyant f l o w and a r e g i o n of s u r f a c e i n f l u e n c e . 188 (8) The a x i a l mean bubble v e l o c i t y p r o f i l e s i n the region of developing flow have d i f f e r e n t shapes depending on the i n j e c t i o n v e l o c i t y . At low i n j e c t i o n v e l o c i t i e s the p r o f i l e s r e v e a l an in c r e a s e i n the mean bubble v e l o c i t y downstream from the i n j e c t i o n p o i n t , while at i n j e c t i o n v e l o c i t i e s above "42 m/s the p r o f i l e s e x h i b i t a decrease with heig h t . High speed f i l m o b s e r v a t i o n s suggest that t h i s e f f e c t of the i n j e c t i o n v e l o c i t y i s r e l a t e d to the nature of gas discharge, i . e . i f the gas discharge produces s i n g l e bubbles or short j e t s . (9) The mathematical model of Tacke et a l . proposed to represent the behaviour of the developed buoyant region of the plume compares reasonably well with the experimental r e s u l t s . However, the usefulness of the model to make p r e d i c t i o n s on plume behaviour i s l i m i t e d s i n c e i t depends s e n s i t i v e l y on an entrainment c o e f f i c i e n t to f i t the r e s u l t s . (10) The p i e r c e d length measurements i n the region of developing flow, i n agreement with the bubble frequency measurements, i n d i c a t e that the la r g e bubbles forming at the o r i f i c e become r a p i d l y unstable a f t e r detachment from the o r i f i c e . The bubbles break c o n t i n u o u s l y u n t i l an " e q u i l i b r i u m " bubble s i z e d i s t r i b u t i o n i s e s t a b l i s h e d . (11) In the region of developed flow the bubble s i z e d i s t r i b u t i o n i s maintained The s p e c t r a of bubble p i e r c e d length dynamically s t a b l e reasonably constant, and bubble diameters 189 are represented by a log-norraal d i s t r i b u t i o n . I n j e c t i o n c o n d i t i o n s have only a s l i g h t e f f e c t on the s i z e of the bubbles e x i s t i n g i n t h i s r e g i o n . Suggestions f o r Further Work It i s b e l i e v e d that the present t h e s i s has extended the knowledge on many of the aspects concerning the p h y s i c a l c h a r a c t e r i s t i c s of g a s - l i q u i d plumes of i n t e r e s t i n l a d l e m etallurgy. However much work remains to be done. The f o l l o w i n g emerge as d e s i r a b l e extensions of the present work (1) In the l a b o r a t o r y work i t i s necessary to use the experimental techniques developed here to conduct a d d i t i o n a l experiments to cover more f u l l y the e f f e c t of s e v e r a l i n j e c t i o n v a r i a b l e s on plume behaviour. This must be pursued through the study of l a r g e r s c a l e water models and non-isothermal l i q u i d metal systems. 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H a i d a 0. and Brimacombe J.K., i n 3 r d I n t . Conf. on R e f i n i n g o f I r o n and S t e e l by Powder I n j e c t i o n , L u l e a , Sweden, 1983, pp.5:1-5:17. 136. B r o d k e y R.S. The Phenomena of F l u i d M o t i o n s , A d d i s o n -Wesley P u b l . Co., R e a d i n g , M a s s a c h u s e t t s , 1967. 137. Underwood E.E. i n Q u a n t i t a t i v e M i c r o s c o p y by DeHoff R.T. and R h i n e s F.N., M c G r a w - H i l l , N.Y., 1974. 138. C a rnahan B., L u t h e r H.A. and W i l k e s J.O., A p p l i e d N u m e r i c a l Methods, John W i l e y & Sons, I n c . , N.Y., 1969. 198 APPENDIX I Speed of Displacement of a R i s i n g S p h e r i c a l Bubble Consider a s p h e r i c a l bubble and l e t U be the v e l o c i t y of c i t s center and R be the v e l o c i t y of expansion o f the s u r f a c e . Then the v e l o c i t y , " n , p e r p e n d i c u l a r to a s u r f a c e element dS, Figure I.1 , i s Un = U c 0 0 8 Y + * ( I D F i g . I . 1 Sketch of a bubble moving toward a double-contact sensor. 199 From the f i g u r e i t i s seen a l s o that the r a d i a l displacement, dr, of a p o i n t on the s u r f a c e i s a s s o c i a t e d to a v e r t i c a l displacement, dz, of the i n t e r f a c e , given a p p r o x i m a t e l y 9 0 by dz = dr (1.2) cos9 The r a d i a l displacement can be expressed as dr = U dt (1.3) Thus the t r a n s p o r t v e l o c i t y , , detected by a v e r t i c a l l y a l i g n e d sensor, i s given as U cos Y + R U = S (1.4) cos 0 T h i s equation i n d i c a t e s that i f R = 0 and the bubble r i s e s along the v e r t i c a l a x i s , i . e . Y = 9 , then U t = U c (1.5) in which case the v e l o c i t y measured by the sensor w i l l be equal to the v e l o c i t y of the gas. The t r a n s i t v e l o c i t y , U^, can be i n t e r p r e t e d as measuring the speed of displacement of a v e r t i c a l l y r i s i n g and expanding bubble only i f 0 = 0 ° . Equation (1.4) a l s o show that a sensor with a small t i p s e p a r a t i o n w i l l r e g i s t e r an i n c r e a s i n g l y l a r g e r t r a n s p o r t v e l o c i t y , of a l a t e r a l l y moving and/or expanding bubble, as the 200 intersection with the bubble occurs closer to the equatorial plane. 201 APPENDIX II Con d i t i o n s of the Experiments The experimental c o n d i t i o n s d e f i n e d i n terms of Fr Q 2 P o Kgo o 1 go (II .1) Re 4 Q„ p o go TT d y o g (II.2) 2 Q n p . o a , a 1 b . In ( ) a (II.3) 1/2 P U Q g o x o (II .4) are given i n Table II.1 , where symbols correspond to those c o n d i t i o n s i n Table 4.2 Table II.1 Experimental c o n d i t i o n s of the study Experiments Fr Re e. X 10 2 z X 10 4 D K Watt K g - 1 Watt K g - 1 O V O Y • 1 8 5304 3 8 4 5 10 1 12542 8 9 59 0 20 9 18003 12 8 174 6 9 9 12307 8 7 37 9 16 1 8215 3 8 25 7 90 2 19425 8 9 339 6 202 APPENDIX III Data Acquisition and Data Reduction Programs ORG "X88000000 BASE_9519 CI01_BASE CI02_BASE DMA_BASE i S_GETTIME S_PUTCH S_PUTS S_PUTN j N_GDBUBL i CONTINUE: I * * * * O f f s e t * * * * C h i p ' s a d d r e s s e s * * * * EQU ~X79 19519 ADDR EQU "X7B !1STCIO ADDR EQU ~X7F I2NDCIO ADDR EQU ~X7E !DMA PGM 8237 ADDR ** * * S y s t m e 1 s c a l l s * * * * EQU ~X05 !RTRN TIME FROM REAL TIME CLOCK EQU ~X29 !PRNT CHARACTER MESS TO SCREEN EQU ~X2B !PRNT STRING MESS TO SCREEN EQU ~X2C !PRNT NUMBER MESS TO SCREEN * * * * E n d i n g c o n d i t i o n * * * * EQU ~X0320 !PGRM END AFTER 0320(16) GB * * * * O f f s e t 0 -- i n i t i a l i z e c h i p s * * * * CALR INIT_CHIP LDAR RR2.MESS_INDN !LOAD ADDR OF INIT DONE MESS CALR PUTMESS DW "X7F00 e n a b l e s c o u n t e r / t i m e r s c a r r y i n g out d a t a a c q u i s i t i o n * * * * ! !PNT TO START TIME BUF ADDR ! LDAR RR6,STARTTIME SC #S_GETTIME LD R6,#~X8A00 LD R7,#~X0000 SUB R12.R12 LD R13,#N_GDBUBL MULT RR12,#~X4 LD R12,#"X8A00 SUBB RL4.RL4 SUB R9.R9 SUBL RR10,RR10 LD R11,#~XFFFF LDB RL1,#~XF4 I1STCI0 STR DATA IN SEG A ... •STARTING AT OFFSET 0, GB !CALCULATE DATA. . . !OFFSET . . . !FOR NGB I1STCI0 STR DATA IN SEG A ... !STARTING AT OFFSET N_GDBUBL*4, NGB !CLR FLAG !CLR GB CNTR !CLR DIVIDEND 165535(10) IN R l l !NBL PA, PB, C / T l , C/T2, C/T3 BY .. to o w "X7B03,RL1 RL1 , #~X06 ~X7B19,RL1 "X7B17,RL1 ~X7B15,RL1 RL1 , 'X7B1B RL1,#0 Z,WAIT_START RL1 , "X7B13 RL1,#5 NZ,PARNBL_7F RL1 , "X7B1B RL1,#0 NZ,CHPA_IN_LC1 RH2,"X7B25 RL2,"X7B27 §RR12,R2 R13,#2 R2 , R2 §RR12,R2 DBL_JR_3 RL1,#"XA4 "X7F03,RL1 RL1 . 4TX26 ~X7F17,RL1 RL1 , "X7B1B RL1,#1 NZ,CHPA_IN_UC1 RL1,#0 Z,DBL_JR_1 RL1 , "X7F13 RL1,#5 NZ,DBL_JR_4 RL1 , "X7B1B RL1,#0 NZ,CHPA_IN_LC2 RH2,"X7B25 RL2.~X7B27 !F4(16) TO MCCR 1STCI0 ! 6(16) PUT 1 IN ... !TCB, GCB IN CSR C/T3 ... !C/T2 . . . IC/T1 ! READ PA'sDR !CH (PA 0 ) , BUBBLE IN LWR C.NTCT? !N0 - WAIT UNTIL IT COMES !YES - READ PB CSR, SEE IF A GB I IS IPB SET? ( E I . PATTERN MATCHED?) !YES - GO APPARENTLY A GB !N0 - CH (PA 0) !BUBBL PASSED LWR CNTCT? !N0 - WAIT UNTIL IT PASS !YES - READ C/T2 CCR - MSB !...'- LSB 1ST JUMP NBL PB, PA, C/T2 BY ... A4(16) TO MCCR 2NDCI0 LOAD 26(16) TO ... CLR IPB & IUSB, 1 TO TCB GCB C/T2 CSR READ PA'sDR CH (PA 1 ) , BUBL ARRIVED UP CNTCT? NO - WAIT UNTIL IT ARRIVES YES - BUBL PASSED LWR CNTCT TOO? YES - 1ST JUMP NO- READ PB CSR SEE IF A GB IS IPB SET? ( E I . PATTERN MATCHED?) YES - 1ST JUMP NO - READ PA'sDR CH (PA 0 ) , HAS BUBL PASSED LWR CNTCT? NO - WAIT UNTIL IT PASS YES - READ C/T2 CCR - MSB . . . - LSB RDY TMRS CHPAIN ULC1 CHPAIN ULC2 DBL 7F 1 DIF1 LD R3 , R2 LD R5 , R2 INB RH2,"X7B21 INB RL2 , "X7B23 LD R8 , R2 LDB RL1 ,#~X20 OUTB ~X7B13,RL1 LDB RL1 ,#~X26 OUTB ~X7B17,RL1 OUTB "X7B15,RL1 SUB Rl1,R3 DIV RR10 ,#~XOOOA INB RL1,"X7B1B B I TB RL1 , #0 JR NZ,CHPAIN_ULC2 INB RL 1 ,~X7F13 BI TB RL1,#5 JR Z , CHPAINJJLCl INB RH2,"X7F25 INB RL2,"X7F27 JR DBL_7F_1 INB RL1,"X7B1B B I TB RL1,#0 JR Z,DBL_7F_2 INB RL1,~X7F13 B I TB RL1,#5 JR Z,CHPAIN_ULC2 INB RH2,"X7F25 INB RL 2,~X7F27 LDB RL4,#1 LDB RL1,#~X00 OUTB ~X7F03,RL1 LDB RL1,#~X20 OUTB ~ X 7 F 1 3 , R L 1 CP R3.R2 JR ULT,DIF2 SUB R3 , R2 CP R 3 , R 1 1 JR UGE.Nl 1 !KEEP C/T2 CLK CNTS 1STCI0 MSB READ C / T l CCR ... - LSB KEEP C / T l CLK_CNTS 1STCI0 LOAD 20(16) TO CLR IPB & IUSB PB CSR LOAD 26(16) TO ... CLR IPB & IUSB, 1 TO TCB GCB C/T2 CSR ... C / T l CSR LOAD DIVIDEND IN RR10 ~ 1/10 OF C/T2 CLK_CNTS IN R l l READ PA'sDR CH (PA 0 ) . BUBL IN LWR CNTCT? YES - GO TO TAKE FURTHER DESITIONS NO - READ PB CSR 2NDCI0 IS IPB SET? ( E I . PATTERN MATCHED?) NO - KEEP CH YES - READ C/T2 CCR - MSB ... - LSB JUMP, WHILE CH UPR NO BUBL LWR CNTCT READ PA'sDR CH (PA 0 ) , BUBL S T I L L IN LWR CNTCT? NO - IT PASSED NGLCT C / T l CLK_CNTS YES - READ PB CSR 2NDCI0 IS IPB SET? ( E I . PATTERN MATCHED?) NO - KEEP CH YES - READ C/T2 CCR - MSB ... - LSB ISET FLAG !LOAD 00(16 ) TO ... !CLR MCR 2NDCI0 !LOAD 20(16) TO . . . !CLR IPB & IUSB PB CSR !C/T2 CLK_CNTS 1STCI0 < ... 2NDCI0? !YES - GO DIF2 !N0 - C/T2 CLK_CNTS 1STCI0 - ... 2NDCI0 IIS DIF >= QUTNT? !YES - NGLCT C / T l CLK CNTS DBL_JR_1 DBL JR 2 DBL_JR_3: DBL_JR_4: DI F 2 : Nl 1 : CH FLAG CHPA IN LC3 LD §RR6,R5 INC R7 , #2 LD @RR6,R8 INC R9 , #1 JR CH_FLAG JR DBL_7F_4 SUBB RL4,RL4 SUBL RR10,RR10 LD R 1 1 , # ~ X F F F F JR WAIT_START INC R13,#2 JR CH_CT3 JR DBL_7F_3 SUB R2 , R3 CP R2 , Rl1 JR U G E , N l _ l LD @RR6,R5 ] N n R7 , #2 LD @RR6,R8 INC R9 , #1 JR CH_FLAG LD @RR12,R5 INC R13,#2 SUB R8 , R8 LD §RR12,R8 INC R13,#2 BITB RI.4 , #0 JR Z,CH_CT3_S JR CHPA_IN_LC3 INC R7 , #2 BITB RL4,#0 JR Z,CH_CT3_S INB RL1,'X7B1B BITB RL1,#0 JR NZ,CHPA_IN_LC3 INB RH2,~X7B25 INB RL2,"X7B27 LD @RR12 , R2 INC Rl3.#2 !N0 - INC !G0 TO CH !2ND JUMP !CLR FLAG !CLR DIVIDEND GB CNTR FLAG - GO WRITE C/T2 NO C / T l CI.K !2NI) JUMP RESTART LOGIC !2ND JUMP - CH DOWN-COUNT !2ND JUMP _ GO WRITE C/T2 NO C / T l CLK IC/T2 CLK_CNTS 2NDCI0 - ... 1STCI0 IIS DIF >= QUTNT? 1VF.S - NGLCT C / T l CLK CNTS !N0 - INC GB CNTR !G0 TO CH FLAG !HAS FLAG BEEN SET? !N0 - CH DOWN COUNT !YES - GO CH LWR CNTCT !HAS FLAG BEEN SET? !N0 - CH DOWN-COUNT !YES - READ PA'sDR !CH (PA 0 ) , BUBL S T I L L IN LWR CNTCT? !YES - KEEP CH !N0 - READ C/T2 CCR - MSB ! . . . - LSB DBL 7F 2: N-1 2 : DBL 7F 3: CHPA IN LC4 DBL 7 F 4: READ T2 Nl SUB R2 , R2 LD §RR12,R2 INC R13,#2 JR CH_CT3 !CH DOWN-COUNT LDB R L l ,#~XOO ! LOAD 00(16) TO ... OUTB "X7F03,RLl !CLR MCCR 2NDCI0 LDB R L l ,#"X20 !LOAD 20(16) TO ... OUTB "X7F13,RLl !CLR IPB & IUSB PB CSR INB RH2,~X7B25 ! READ C/T2 CCR - MSB INB RL2 , "X7B27 ! . . . - LSB LD §RR12,R5 INC R13 , #2 SUB R8 , R8 LD @RR12,R8 INC R13,#2 LD @RR12,R2 INC R13,#2 SUB R2.R2 LD §RR12 , R2 INC R13,#2 JR CH_CT3 !CH DOWN_COUNT LDB RLl,#"XOO !LOAD 00(16 ) TO . . . OUTB "X7F03,RLl !CLR MCCR 2NDCI0 LDB RLl,#"X20 !LOAD 20(16) TO ... OUTB ~X7F13,RLl !CLR IPB & IUSB PB CSR INB R L l ,'X7B1B !READ PA'sDR BI TB R L l , #0 !CH (PA 0 ) , HAS BUBL PASSED LWR CNTCT JR NZ,CHPA_IN_LC4 !N0 - WAIT UNTIL IT PASS JR READ_T2_N1 !YES - GO READ C/T2 NGLCT C / T l CLK_CNTS LDB RLl,#~X00 !LOAD 00(16) TO ... OUTB ~X7F03,RLl !CLR MCCR 2NDCI0 LDB RL1,#"X20 !LOAD 20(16) TO ... OUTB ~ X 7 F 1 3 , R L 1 !CLR IPB & IUSB PB CSR INB RH2,~X7B25 ! READ C/T2 CCR - MSB INB RL2,~X7B27 ! . . . - LSB LD @RR12,R2 INC R13,#2 INB RH2,~X7B21 ! READ C / T l CCR - MSB INB RL2."X7B23 ! . . . - LSB CH CT3: CH CT3 S ENDACQ: PR I NF SUB R2 , R2 LD §RR12,R2 INC R13,#2 LDB RL1,#~X20 !LOAD 20(16) TO ... OUTB "X7B13,RL1 !CLR IPB & IUSB PB CSR LDB RL1 , #~X26 ! LOAD 26(16) TO ... OUTB ~X7B17,RL1 !CLR IPB & IUSB. 1 TO TCB GCB C/T2 CSR OUTB "X7B15.RL1 !... C / T l CSR INB RL1 , ~X7B19 !READ C/T3 CSR BITB RL1,#5 !HAS IPB BEEN SET? JR NZ,END_ACQ !YES - ACQUISITION FINISH CP R9,#N_GDBUBL !N0 - HAS ENDING CONDITION BEEN MET? JR NZ,DBL_JR_2 !N0 - 1ST JUMP LDB RL1,#~X08 !YES - LOAD 8 ( 1 6 ) TO ... OUTB "X7B19,RL1 ISTOP C/T3 BY 1 TO RCCB C/T3 CSR LD R9 , R13 !STR DATA OFFSET LDAR RR6,ENDTIME !PNT TO END TIME BUF ADDR SC #S_GETTIME LDAR RR2,MESS_FIN !LOAD ADDR OF FINISH MESS CALR PUTMESS LDAR RR2,MESS_ST !LOAD ADDR OF START TIME MESS CALR PUTMESS LDAR RR12,STARTTIME !PNT TO START TIME BUF ( E I . YEAR) CALR PR I NTT I ME LDAR RR2,MESS_NDT !LOAD ADDR OF END TIME MESS CALR PUTMESS LDAR RR12,ENDTIME !PNT TO END TIME BUF ( E I . YEAR) CALR PR I NTT IME LDL RR6,#STRING1 !PNT TO No. OF TIME SZ MESS SC #S_PUTS SUB R6 , R6 LD R7 , R9 ! RTRN DATA OFFSET DIV RR6,#"X4 !CALCULATE No. OF TIME SZ IN SEG A LD R9 , R7 !KEEP LENGTH OF TIME SZ LIST LD R5,#~D10 LD R4,#~D4 LD R3 , # ' SC #S_PUTN LDL RR6 , #STRING3 to o co SC #S_PUTS XOR R6.R6 !FAST CLR OF R6 LD R7,#~XFFFF INB RH2,~X7B29 !READ C/T3 CCR - MSB INB RL2,"X7B2B !... - LSB SUB R7.R2 !CALCULATE No. OF BUBL THAT PASSED ADD R7,#"X0001 !LWR CNTCT LD R5,#~D10 LD R4,#~D4 LD R3,#* SC #S_PUTN LDL RR6,#STRING3 SC #S_PUTS CALR CNTS_TIME CALR GAS_TIME LDL RR6,#STRING4 !PRNT ... SC #S_PUTS LDL RR6.RR4 LD R5,#~D10 LD R4,#~D4 LD R3,#' SC #S_PUTN ISUMTIME SZ LDL RR6,#STRING3 SC #S_PUTS CALR SORT_TVEL LDAR RR2,MESS_NDPRC CALR PUTMESS LDL RR6,#STRING5 !PRNT ... SC #S_PUTS SUB R6.R6 LD R7.R9 MULT RR6,#~X4 SUB R7,#*X1 LD R5,#"D16 LD R4,#'D4 LD R3,#' ' SC #S_PUTN !DATA OFFSET IN SEG A LDL RR6,#STRING3 SC f S PUTS S C # 0 INIT_CHIP: j * * * * I n i t i a l i z a t i o n of the 9519, 1STCI0 LDB RH1 ,#BASE_9519 LD R2,#INIT_SIZE_9519 LDAR RR4,TB_9519 CALR INIT LDB RH1,#CI 01_BASE LD R2,#CI01_INIT_SIZE LDAR RR4,CI01_TB CALR INIT LDB RH1,#CI02BASE LD R2,#CI02_INIT_SIZE LDAR R R 4 . C I 0 2 T B CALR INIT INIT SKIP : TB 9519 LDB LD LDAR CALR RET LDB INC ORB JR OUTIB JR RET INC DEC JR RET DW DW DW RH1,#DMA_BASE R2 , #DMA_INIT_SIZE RR4,DMA_TB INIT RLl,@RR4 R5 , #1 R L l , R L l Z,SKIP @R1,@RR4,R2 NOV,INIT R5 , #1 R2 , #1 NZ, INIT X0380 X03B0 XOl FF , 2NDCIO and DMA c h i p s * * * * !RH1= 79 !SZ OF INIT TABLE !R4, R5 HAVE ADDR OF TB_9519 ! INIT 9519 !RH1= 7B ! R H 1 = 7 F !RH1= 7E !GET LWR HALF OF PORT No. !R5= R5+1 --> INCR ADDR IN TABLE !CH END OF LWR HALF PORT No. ! IF END - SKIP !NO END - SHIP OFF CNTRL BYTE !REPEAT UNTIL R2= 0 !INCR ADDR IN TABLE !DECR CNTR ILOOP BACK IF COUNT <> 0 IACTIVE LOW, COMMON VECTOR !PRESELECT IMR ! ALL MASK OFF (NO INTERRUPTS AL -CI01 TB DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW DW X0340 X03C0 X0100 X03A9 XOOOO XOOOO XO l O l XOIOO X390C 'X1504 X2D00 X2F00 ' X3B0C 'X1704 X3100 X3300 X5102 X57FF X5B88 X5D80 X5F88 X3D20 X35FF X37FF XODOF X4140 X47FF XOOOO XOOOO XOOOO CI02 TB DW DW DW DW DW XO l O l XOIOO ' X3B0C X1 704 xrn no !L0W) CHANGE T O 01DF TO T U R N O N !CLR 9 519 !PRESELECT AUTO CLR REGISTER !NO AUTO CLR ICHIP ARMED, IREQ REG RESET CI01 CLR RESET C / T l : EGEB, REB MSR C / T l : GCB CSR C / T l : INIT TCR - MSB ... - LSB C/T2: EGEB, REB MSR C/T2: GCB CSR C/T2: INIT TCR - MSB ... - LSB PB BTPORT PMSB's (AND) MSR !PB - IN DIRECTION DDR IPB PPR 10001000 1 BIT3 PB PTR 10000000 AND PB PMR 10001000 0-->l BIT7 C/T3 C/T3 C/T3 PC -PA -PA -ECEB (CNTR) MSR 6 5 5 3 5 ( E I . F F F F ) TO TCR 0..3 IN DIRECTION INPUT PORT IN DIRECTION DDR IRESET CI02 ICLR RESET I C/T2 I C/T2 ! 0 / T 2 EGEB, REB MSR GCB CSR I N I T ren M S M DW DW DW DW DW DW DW DW DW DW DMA_TB: DW DW DW DW DW DW DW DW DW DW END_DMA: DS INIT_SIZE_9519 EQU C I 0 1 _ I N I T _ S I Z E EQU C I 0 2 _ I N I T _ S I Z E EQU DMA_INIT_SIZE EQU PUTMESS: i NEXT_CHAR: LDB ORB RET SC INC JR MESS INDN: DB ~X3300 !.. "X5102 !PB "X57FF !PB "X5B00 !PB ~X5D08 IPB "X5F88 IPB "X4140 !PA "X47FF !PA "XOOOO ~XOOOO - LSB BTPORT PMSB's (AND) MSR - IN DIRECTION DDR PPR OOOOOOOO l - > 0 BIT3 PTR 00001000 AND PMR 10001000 0 BIT7 - INPUT PORT - IN DIRECTION DDR "X1B00 !MASTER CLR 'X1900 !CLR BYTE PTR OFF ~X1140 !CMD REG, FIXED PRNT, DABL DMA ~X1715 !PRGRM MODE REGISTER "X0500 !DMA WRITE ADDR - INIT AT 2000 'X0520 ~ X07FF !DMA TC INIT AT IK "X0703 ! ( E I . IK-1= 3FF BYTES "XOOOO "XOOOO 0 (CI01_TB-TB_9519)/2 (CI02_TB-CI01_TB)/2 (DMA_TB-CI02_TB)/2 (END_DMA-DMA_TB)/2 * * * P r i n t messages RL1,@RR2 RL1 , RL1 Z #S_PUTCH R3 , #1 NEXT CHAR on the s c r e e n * * * * !RL1 HAS ONE CHARACTER !CH FOR END OF MESS !IF END RTRN !IF NO END - PRNT CHARACTER IMOVE TO NEXT CHARACTER 1L00P BACK " r \ n INIT DONE GO TO 08.000A TO CONTINUE" , 1 3 , 1 0 MESS_FIN: MESS_ST: MESS_NDT: MESS NDPRC DB DB DB DB . EVEN " r \ n ACQUISITION ENDED ... DOWN COUNTER= 0",13,10, "\r\nTIME AT WHICH EXPERIMENT STARTED :",13,10,0 " \r\nTIME AT WHICH EXPERIMENT ENDED :",13,10,0 "\r\nCONVERTION, SUM AND SORTING FINI SHED",13,10,0 PR INTT IME i * * * P r i n t s t a r t and end times of the e x p e r i m e n t * * * * SUBL RR6,RR6 !CLR RR6 LDB RL7,§RR12 !RR6 CONTAIN TIME UNIT TO PRNT LD R5,#~D16 !REMEMBER TIME UNIT IS IN BCD LD R4,#~D2 !R4 AND R3 TAKE CARE OF ... LD R3 , #' ' ! PRINTING FORMAT SC #S_PUTN LDL RR6.#STRING2 !PNT TO STRING2 SC #S PUTS INC R13,#1 !PNT TO NEXT TIME UNIT - M LD R5,#"D16 LD R4,#"D2 LD R3,# 1 ' SUBL RR6,RR6 LDB RL7.@RR12 SC #S_PUTN LDL RR6.#STRING2 SC #S_PUTS INC R13,#1 !PNT TO NEXT TIME UNIT - DM LD R5,#~D16 LD R4,#"D2 LD R3,#• 1 SUBL RR6,RR6 LDB RL7,@RR12 SC #S_PUTN LDL RR6,#STRING2 SC #S_PUTS INC R13,#l !PNT TO NEXT TIME UNIT - DW LD R5,#"D16 LD R4,#~D2 LD R3,#' ' SUBL RR6.RRB LDB RL7,@RR12 SC #S_PUTN LDL RR6,#STRING2 SC #S_PUTS INC R13,#l LD R5,#~D16 LD R4,#~D2 LD R3,#' ' SUBL RR6.RR6 LDB RL7,@RR12 SC #S_PUTN LDL RR6,#STRING2 SC #S_PUTS INC R13,#1 LD R5,#~D16 LD R4,#~[)2 LD R3,#' ' SUBL RR6.RR6 LDB RL7,@RR12 SC #S_PUTN LDL RR6,#STRING2 SC #S_PUTS INC R13,#l LD R5,#~D16 LD R4,#"D2 LD R3,#' ' SUBL RR6.RR6 LDB RL7,@RR12 SC #S_PUTN LDL RR6,#STRING3 SC #S_PUTS RET !PNT TO NEXT TIME UNIT !PNT TO NEXT TIME UNIT - MIN !PNT TO NEXT TIME UNIT SEC STARTTIME ENDTIME STRING1 STRING 2 STRING3 STRING4 DS DS DB DB DB DB 7 7 " \n\rNUMBER OF TIME-SIZES COLLECTED : \ n \ r \ 0 " "\ / \0" " \ n \ r \ 0 " "\n\rSUM TIME SIZE : \ n \ r \ 0 " t o STRING5: CNTS_TIME i NEXT DATA UPDT : DONE CONV GAS. i TIME [add SUM32 DB . EVEN \n\rDATA OFFSET IN SEGMENT A :\n \ r \ 0 ' * * * * C o n v e r t from c l o c k - c o u n t s t o m i c r o s e c o n d s * * * * LD R6,#*X8A00 ! RE INIT ADDR . . . LD R7,#~X0000 ! DATA PNTR LD Rl , R9 !LENGTH OF CNTS SZ LIST IN INC R l DEC Rl !No. LOOPS = No. ELEMENTS JR Z,DONE_CONV LD R2,#~XFFFF SUB R2,§RR6 ICONV CLK_CNTS SZ TO MCRSC LD @RR6,R2 !TI ME IN SAME MEM LOCATION INC R7 , #2 TEST @RR6 !IS CLK_CNTS VEL = 0? JR Z.UPDT !YES - SEE NEXT CNT LD R2,#~XFFFF SUB R2,@RR6 !N0 - CONV CLK_CNTS VEL TO LD @RR6,R2 INC R7 , #2 JR NEXT_DATA RET t i m e s c o r r e s p o n d i n g to gas p r e c e n s e a t lwr c n t c t * * * * LD R6,#'X8A00 ! REINIT ADDR . . . LD R7, #~X0000 ! DATA PNTR CLR R2 CLR R5 CLR R4 LD R1.R9 LD R3,@RR6 ADDL RR4,RR2 INC R7 , #4 DJNZ Rl ,SUM3 2 RET SORT TVEL:' j * * * * s o r t time v e l i n d e c r e a s i n g or PASS STEP DONE SORT CLR RO LD Rl,#N_GDBUBL DEC Rl JR LE,DONE_SORT LD R4,#"X8A00 LD R5,#~XOOOO LD R6,#"X8A00 LD R7,#~X0002 LD R3,@RR6 LD R2,@RR4 INC R7 , #4 INC R5 , #4 CP R3,@RR6 JR UGE,STEP LDK RO , #1 EX R3,@RR6 DEC R7 , #4 EX R3,@RR6 INC R7 , #4 EX R2,@RR4 DEC R5 , #4 EX R2,§RR4 INC R5 , #4 DJNZ Rl , PASS DEC RO JR Z , S O R T T V E L RET END k e e p i n g c o r r e s p o n d i n g time s i z e * * * * !CLR INTERCHANGE FLAG BEFORE PASS ! LENGTH OF CNTS SZ LIST IN R l !ONE LESS PAIR THAN ELEMENTS !CATCH 0 OR 1 ELEMENTS CASES ! TAKE ARRAY TIME VEL ELEMENT ! TAKE ARRAY TIME SZ ELEMENT !.IS 1ST LESS THAN 2ND IN PAIR? !NO - NO INTERCHANGE NECESSARY !YES - SET INTERCHANGE FLAG ! INTERCHANGE TIME VEL DATAS ! INTERCHANGE TIME SZ DATAS !WAS THERE ANY INTERCHANGE? !YES - GO THROUGTH IT AGAIN !NO - SORTING FINISHED 100 'THIS PROGRAM TRANSFER MEMFILE TO A FLOPPY DISK F I L E WHERE 103 'IT WAITS FOR FURTHER PROCESSING. 106 ' 109 GOSUB 352 112 OPEN "R", #1, "#2:DATA", 128 115 FIELD #1, 128 AS DT$ 118 OPEN "R", #2, "EXPDATA100", 128 121 FIELD #2, 2 AS NE$ , 2 AS DM$,2 AS DD$,2 AS DY$ , 4 AS TE$,4 AS PS$,4 AS NB$,4 ~§AS NV$,4 AS STS$,4 AS DP$,4 AS PZ$,4 AS PR$,4 AS PAS,4 AS R0$,4 AR QVG$ 124 FIELD #2 , 128 AS ED$ 127 GOSUB 313 130 PUT #2, 1 133 NRMEM= INT(NBEX! * 4/128) + 1 136 NRCLR= NRMEM +1 139 J= 1 142 J= J + l 145 1= 1+1 148 GET #1, NRCLR 151 GET # 1 , 1 154 LSET ED$= DT$ 157 PUT #2, J 160 IF LOC(1)<=NRCLR THEN GOTO 142 163 GOSUB 175 166 END 169 ' 172 1 175 PRINT CHR$(26) 178 PRINT TAB(19); 181 GOSUB 226 184 IF ANSWER$= "Y" OR ANSWER$= "y" 187 J= 1 : GET #2, J 190 GOSUB 241 193 J= J+l 196 K= 1 199 GET #2, J 202 EXDT!= CVI(MID$(ED$,K,2)) 205 IF EXDT! < 0 THEN EXDT!= 65536! : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT "DO YOU WANT TO SEE THE ACQUIRED DATA F I L E ? " THEN GOTO 187 ELSE GOTO 220 EXDT ! CO ~3 208 211 214 217 220 223 226 229 232 235 238 241 244 247 250 253 256 259 262 265 268 271 274 277 280 283 286 289 292 295 298 301 304 307 310 313 316 PRINT USING "########"; EXDT! K= K + 2 IF K<128 GOTO 202 IF L0C(2) <= NRMEM GOTO 193 RETURN PRINT : PRINT PRINT TAB(18); "TYPE ANSWER$= INPUT$(1) RETURN OR IF YES ANYTHING ELSE IF NOT' NEX% = EDM* = TEX ! = STSEX! PPZ ! = ROEX!= PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT RETURN CVI(NE$) CVI(DM$) : EUD% = CVI(DD$) : EDY%= CVI(DY$) CVS(TE$) : PSEX!= CVS(PS$) : NBEX ! = CVS(NB$) = CVS(STS$) : DPEX!= CVS(DP$) CVS(PZ$) : PPR!= CVS(PR$) : PPA!= CVS(PA$) CVS(RO$) : QVGEX!= CVS(QVG$) CHR$(26) : PRINT : PRINT : PRINT "EXPERIMENT'S NUMBER"; TAB(45); NEX% "EXPERIMENT'S DATE"; TAB(45); EDM*; CHR$(47) "DURATION OF THE EXPERIMENT";TAB(45 ) ; TEX!; "DISTANCE BETWEEN THE CONTACTS"; T A B ( 4 5 ) ; "No. OF BUBBLES THAT PASSED LOWER CONTACT NVEX!= CVS(NV$) EDD* ; ; " s e c " PSEX!; "m ; TAB(45) CHR$(47); EDY% NBEX 'No. OF MEASURED VELOCITIES"; TAB ( 4 5 ) ; NVEX! 'SUM TIME SIZE"; TAB(45); STSEX!; " s e c " 'DEPTH OF THE BATH"; TAB(45); DPEX! ; "m" "POSITION OF THE PROBE" PPR!; "m"; SP C ( 3 ) ; PPA! "ORIFE RADIUS"; TAB(45) "GAS VOLUME FLOW AT ORIFICE CONDITIONS' : PRINT : PRINT TAB(45) "deg" ROEX!; PPZ ! m SPC(3) T A B ( 4 5 ) ; QVGEX!; "m3/sec" LSET NE$= MKI$(NEX%) LSET DM$= MKI$(EDM*) : LSET I) D $ - MKI$(EDI)%) : LSET DY$= MK I $ ( ET)Y% ) 319 322 325 328 331 334 337 340 343 346 349 352 355 358 361 364 367 370 373 376 379 382 385 388 391 394 397 400 LSET TE$= MKS$(TEX! ) LSET PS$= MKS$(PSEX! ) LSET NB$= MKS$(NBEX!) : LSET NV$ = MKS$(NVEX!) LSET STS$= MKS$(STSEX!) LSET DP$= MKS$(DPEX!) LSET PZ$= MKS$(PPZ!) : LSET PR$= MKS$(PPR!) : LSET LSET R0$= MKS$(R0EX!) : LSET QVG$= MKS$(QVGEX!) RETURN PA$= MKS$(PPA!) PRINT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT INPUT RETURN CHR$(26) : PRINT : PRINT : PRINT "EXPERIMENT'S NUMBER ", nex% "EXPERIMENT'S MONTH ", EDM* "EXPERIMENT'S DAY ", edd% "EXPERIMENT'S YEAR ", EDY* "DURATION OF THE EXPERIMENT ", "DISTANCE BETWEEN CONTACTS "No. OF BUBBLES THAT PASSED "No. OF MEASURED VELOCITIES "SUM TIME SIZE ", STSEX! "DEPTH OF THE BATH ", DPEX! TEX! ", PSEX! LOWER CONTACT ", NVEX! NBEX ! PROBE 1S PROBE 1S PROBE'S ORIFICE AXIAL POSITION ", PPZ! RADIAL POSITION ", PPR! ANGULAR POSITION ", PPA! S RADIUS ", ROEX! GAS VOLUME FLOW AT ORIFICE CONDITIONS QVGEX 100 'THIS PROGRAM PROCESS THE DATA DISK F I L E TO PRODUCE FI L E S 103 'OF THE VELOCITIES AND SIZES OF BUBBLES AND OF THE 106 'REDUCED INFORMATION 10 9 ' 112 OPEN "R", #2, "EXPDATA100", 128 115 FIELD #2, 2 AS NE$,2 AS DM$,2 AS DD$ , 2 AS DY$ , 4 AS T E $,4 AS PS$,4 AS N B $,4 ~@AS NV$,4 AS STS$,4 AS DP$,4 AS PZ$,4 AS PR$,4 AS PA$,4 AS R0$,4 AS QVG$ 118 FIELD #2, 128 AS ED$ 121 J= 1 : GET #2, J 124 GOSUB 1231 127 OPEN "R", #3, "EXPVELSZ100", 64 130 FIELD #3, 2 AS NE$,2 AS DM$,2 AS DD$,2 AS DY$,4 AS TE$,4 AS PS$,4 AS NB$,4 ~@AS NV$,4 AS STS$,4 AS DP$,4 AS PZ$,4 AS PR$,4 AS PA$,4 AS R0$,4 AS QVG$ 133 FIELD #3, 4 AS V l $ , 4 AS Sl$',4 AS V2$ , 4 AS S2$,4 AS V3$,4 AS S3$ , 4 AS V4$ , 4 "§AS S4$,4 AS V5$,4 AS S5$,4 AS V6$,4 AS S6$,4 AS V7$.4 AS S7$,4 AS V8$,4 AS S8 136 GOSUB 1306 139 PUT #3, 1 142 OPEN "R", #4, "EXPSZ100", 64 145 FIELD #4, 2 AS NE$,2 AS DM$,2 AS DD$,2 AS DY$,4 AS TE$,4 AS PS$,4 AS NB$,4 ~§AS NV$,4 AS STS$,4 AS DP$,4 AS PZ$,4 AS PR$,4 AS PA$,4 AS R0$,4 AS QVG$ 148 FIELD #4, 4 AS Z l $ , 4 AS Z2$,4 AS Z3$,4 AS Z4$,4 AS Z5$,4 AS Z6$,4 AS Z7$,4 "§AS Z8$,4 AS Z9$,4 AS Z10$,4 AS Z l l $ ( 4 AS Z12$,4 AS Z13$,4 AS Z14$,4 AS Z15$,4 ~@AS Z16$ 151 GOSUB 1306 154 PUT #4, 1 157 FLAG= 0 160 PRINT CHR$(26) : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT 163 INPUT "MAXIMUM ALLOWED SIZE ", SNR! 166 PRINT CHR$(26) : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT 169 PRINT TAB(12); "DO YOU WANT TO SUPPLY THE MAXIMUM ALLOWED VELOCITY, TOO?" 172 GOSUB 1216 175 IF ANSWER$= "Y" OR ANSWER$= "y" THEN FLAG= 1 178 OPEN "R", #5, "EXPRI100", 64 181 FIELD #5, 2 AS NE$,2 AS DM$,2 AS DD$,2 AS DY$,4 AS TE$,4 AS PS$,4 AS NB$,4 "§AS NV$,4 AS STS$,4 AS DP$,4 AS PZ$,4 AS PR$,4 AS PA$,4 AS R0$,4 AS QVG$ 184 FIELD #5, 4 AS BF$, 4 AS BHLP$ , 4 AS AVSZ$, 4 AS SDS$, 4 AS AVV$, ~§4 AS SDV$, 4 AS NBVS$, 4 AS NBVSC$, 4 AS V$, 4 AS SL$, 4 AS V0$ 187 ON FLAG +1 GOTO 190, 199 190 FIELD #5, 4 AS AVSZS$,4 AS SDSS$,4 AS AVVS$,4 AS SDVS$ 193 FIELD #5, 4 AS NS$, 4 AS VS$ 196 GOTO 211 199 FIELD #5, 4 AS AVSZS$,4 AS SDSS$,4 AS AVSZSC$,4 AS SDSSC$,4 AS AVVS$, ~@4 AS SDVS$,4 AS AVVSC$,4 AS SDVSC$ 202 FIELD #5, 4 AS NS$, 4 AS VS$, 4 AS NSC$, 4 AS VSC$ 205 PRINT CHR$(26) : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT : PRINT 208 INPUT "MAXIMUM ALLOWED VELOCITY ", VNR! 211 GOSUB 1306 214 PUT #5, 1 217 NRMEM= INT(NBEX! * 4/128) + 1 : NRMEM1= INT(NVEX! * 4/128) + 1 220 NRCLR= NRMEM +1 223 GOSUB 829 226 ARY.SIZE= 850 229 DIM BS(ARY.SIZE) , BV(ARY.SIZE) , BSS(ARY.SIZE ) , BVS(ARY.SIZE) , S9(30,2) 232 SUM.EXBS!= 0 : SUM.EXBV!= 0 : SUM.SQS!= 0 : SUM.SQV!= 0 235 1= 0 : J= 1 238 PSEXM!= PSEX! * 1E+06 241 J= J + l : K= 1 244 GET #2, NRCLR 247 GET #2, J 250 EXBTS!= CVI(MID$(EDS,K , 2 ) ) 253 IF EXBTS! < 0 THEN EXBTS!= 65536! + EXBTS! 256 K= K+2 259 EXBTV!= CVI(MID$(EDS,K,2)) 262 IF EXBTV! < 0 THEN EXBTV!= 65536! + EXBTV! 265 IF EXBTV! <> 0 GOTO 274 268 K= K+2 271 IF K<128 THEN GOTO 250 ELSE GOTO 295 274 EXBV!= (PSEXM!/EXBTV! ) 277 EXBS!= EXBTS! * EXBV! * .000001 280 1= 1+1 283 BS(I ) = EXBS! 286 BV(I)= EXBV! 289 K= K+2 292 IF K<128 THEN GOTO 250 295 IF L0C(2) <= NRMEM1 THEN GOTO 241 298 N!= I 301 GOSUB 379 304 GOSUB 640 307 BFRC!= NBEX!/TEX! 310 GHUP!= (STSEX!/TEX! ) 313 LSET BF$= MKS$(BFRC!) : LSET BHLP$= MKS$(GHUP!) 316 LSET AVSZ$= MKS$(ABSZ!) : LSET SDS$= MKS$(SIGMA.S!) 319 LSET AVV$= MKS$(ABV!) : LSET SDV$= MKS$(SIGMA.V!) 322 LSET NBVS$= MKS$(N!) : LSET NBVSC$= MKS$(NC!) 325 LSET V$= MKS$(VL!) : LSET SL$= MKS$(SNR!) : LSET V0$= MKS$(VOE!) 328 PUT #5, 2 331 LSET AVSZS$= MKS$(ABSZS1!) : LSET SDSS$= MKS$(SIGMA.SS1!) 334 IF FLAG= 0 GOTO 340 337 LSET AVSZSC$= MKS$(ABSZS2! ) : LSET SDSSC$= MKS$(SIGMA.SS2! ) 340 LSET AVVS$= MKS$(ABVS1!) : LSET SDVS$= MKS$(SIGMA.VS 1 ! ) 343 IF FLAG = 0 GOTO 349 346 LSET AVVSC$= MKS$(ABVS2!) : LSET SOVSC$= MKS$(SIGMA.VS2 ! ) 349 PUT #5, 3 352 LSET NS$= MKS$(NS1!) : LSET VS$- MKS$(VS1!) 355 IF FLAG= 0 GOTO 361 358 LSET NSC$= MKS$(NS2!) : LSET VSC$= MKS$(VS2!) 361 PUT #5, 4 364 GOSUB 1009 367 END 370 ' 373 1 376 ' 379 ARO= 3.1416 * R0EX!"2 : VOE!= QVGEXI/ARO! : 1= N 382 IF BV(I) < VOE! GOTO 388 385 1= 1-1 : GOTO 382 388 NC= I : VL= BV(NC) 391 J= 0 394 FOR 1= 1 TO NC 397 IF BS(I) > SNR! THEN GOTO 403 ELSE J= J + l 400 BSS(J)= B S ( I ) : BVS(J)= BV(I) 403 NEXT I 406 NS1= J : VS1= BVS(NSl) 409 IF FLAG= 0 GOTO 430 412 1= NS1 415 IF BVS(I) < VNR GOTO 421 418 1= 1-1 : GOTO 415 421 424 427 430 433 436 439 442 445 448 451 454 457 460 463 466 469 472 475 478 481 484 487 490 493 496 499 502 505 508 511 514 517 520 523 526 529 532 5 3 f> SUM.EXBV!/NC! * ABSZ!~2))/NC) * ABV!'2))/NC) SUM.SQS!= 0 : SUM A= I : B= 1-1 : DIFV= BVS(A)-BVS(B ) IF DIFV <= .6 THEN NS2= A ELSE NS2= B VS2! = BVS(NS2) FOR 1= 1 TO NC SUM.EXBS!= SUM.EXBS! + BS(I) SUM.EXBV!= SUM.EXBV! + BV(I) SQS!= BS(I ) * BS(I ) SUM.SQS!= SUM.SQS! + SQS! SQV!= B V ( I ) * B V ( I ) SUM.SQV!= SUM.SQV! + SQV! NEXT I ABSZ!= SUM.EXBS!/NC! : ABV!= SIGMA.S!= SQR((SUM.SQS - (NC SIGMA.V!= SQR((SUM.SQV - (NC SUM.EXBS!= 0 : SUM.EXBV!= 0 : .SQV!= 0 FOR 1= 1 TO NS1 SUM.EXBS!= SUM.EXBS! + BSS(I) SUM.EXBV!= SUM.EXBV! + BVS(I) SQS ! = BSS( I ) * BSS ( I ) SUM.SQS!= SUM.SQS! + SQS! SQV!= BVS(I) * BVS(I) SUM.SQV!= SUM.SQV! + SQV! NEXT I ABSZS1! = SUM.EXBS!/NS1 : ABVS1!= SUM.EXBV!/NS1! SIGMA.SS1!= SQR((SUM.SQS - (NS1 * ABSZS 1 ! "2 ) )/NS 1 ) SIGMA.VS1!= SQR((SUM.SQV - (NS1 * ABVS1 ! ~2 ) )/NS1) IF FLAG = 0 GOTO 538 0 : SUM.EXBV!= 0 : SUM.SQS! NS 2 + BSS(I) + BVS(I) SUM FOR SUM SUM EXBS!= 1 = 1 TO EXBS ! = EXBV!= SUM.SQV!= 0 SUM SUM EXBS ! EXBV ! SQS!= B S S ( I ) * BSS(I) SUM.SQS!= SUM.SQS! + SQS! SQV!= BVS(I) * BVS(I) SUM.SQV!= SUM.SQV! + SQV! NEXT I ABSZS2!= SUM.EXBS!/NS2! : ABVS2!= SIGMA.SS2!= SQR((SUM.SQS - (NS 2 * SIGMA.VS2!= SQR((SUM.SQV ( N S 2 * SUM.EXBV!/NS 2! ABSZS2!"2) ) /NS2) A R V S 2 ! ~ 2 ) ) / N S 2 ) co CO 538 J= -1 : 1= 0 : 1= I + l 541 Ml= 2*N*4 : M01= Ml 544 IF M01= 0 THEN NR3= 547 J= J + l 550 VEL1!= BV(I) : SZ1! = 553 1= I+l : IF I>N THEN 556 1= I+l : IF I>N THEN 559 1= I+l : IF I>N THEN 562 1= I+l : IF I>N THEN 565 1= I+l : IF I>N THEN 568 1= I+l : IF I>N THEN 571 1= I+l : IF I>N THEN 625 628 631 634 637 640 643 646 649 652 MOD 64 Ml/64 ELSE NR3= I N T ( ( M l / 6 4 ) +1) BS( I ) GOTO 577 GOTO 580 GOTO 583 GOTO 586 GOTO 589 GOTO 592 GOTO 595 ELSE ELSE ELSE ELSE ELSE ELSE ELSE VEL2 ! VEL3 ! VEL4 ! VEL5 ! VEL6 ! VEL7 ! VEL8 ! BV( I BV( I BV( I BV( I BV( I BV( I BV( I 574 GOTO 598 577 VEL2 ! = 0 SZ 2!= 0 580 VEL3 ! = 0 SZ3!= 0 583 VEL4 ! = 0 SZ4!= 0 586 VEL5 ! = 0 SZ5!= 0 589 VEL6 ! = 0 SZ6!= 0 592 VEL7 ! = 0 SZ7!= 0 595 VEL8 ! = 0 SZ8!= 0 598 LSET V l $ = MKS$(VEL1!) LSET S l $ = MKS$(SZ1 ! ) 601 LSET V2$ = MKS$(VEL2!) LSET S2$ = MKS$(SZ2 ! ) 604 LSET V3$ = MKS$(VEL3!) LSET S3$ = MKS$(SZ3! ) 607 LSET V4$ = MKS$(VEL4! ) LSET S4$ = MKS$(SZ4 ! ) 610 LSET V5$ = MKS$(VEL5 ! ) LSET S5$ = MKS$(SZ5 ! ) 613 LSET V6$ = MKS$(VEL6!) LSET S6$ = MKS$(SZ6! ) 616 LSET V7$ = MKS$(VEL7 ! ) LSET S7$ = MKS$(SZ7!) 619 LSET V8$ = MKS$(VEL8 ! ) LSET S8$ = MKS$(SZ8 ! ) 622 PUT #3, J : 1= I + l : IF K=N THEN GOTO 547 GOSUB 883 RETURN i 11= 1 : J l = NC 1= II : J= J l : S= -1 IF BS(I)<=BS(J) THEN 655 T= B S ( I ) : BS(I) = B S ( J ) S= SGN('-S) SZ2 ! SZ3 ! SZ4 ! SZ5 ! SZ6 ! SZ7 ! SZ8 ! BS ( I BS ( I BS ( I BS ( I BS ( I BS ( I BS ( I BS(J ) = T to to 655 658 661 664 667 670 673 676 679 682 685 688 691 694 697 700 703 706 709 712 715 718 721 724 727 730 733 736 739 742 745 748 751 754 757 760 763 766 769 S9(P,2)= J I IF S= 1 THEN 1= 1+1 ELSE J= J - l IF K J THEN 646 IF I+1>=J1 THEN 667 P= P+l : S9(P,1)= 1+1 J l = 1-1 IF I K J 1 THEN 643 IF P= 0 THEN 682 11= S9(P,1) : J l = S9(P GOTO 643 J= 1 : 1= 0 : 1= 1+1 M2= NC*4 : M02= M2 MOD 64 IF M02= 0 THEN NR4 = M2/64 ELSE NR4= INT((M2/64) + J= J + l SZ1 ! = B S ( I ) 2) P= P - l 1 = 1 + 1 IF I>NC THEN GOTO 745 ELSE SZ2 ! = BS ( I ) I = 1 + 1 IF I>NC THEN GOTO 748 ELSE SZ3 ! = BS ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 751 ELSE SZ4 ! = BS ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 754 ELSE SZ5 ! = BS ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 757 ELSE SZ6 ! = BS ( I ) I = I + 1 IF I>NC THEN GOTO 760 ELSE SZ7 ! = BS ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 763 ELSE SZ8 ! = BS ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 766 ELSE SZ-9 ! = BS ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 769 ELSE SZ10 ! = B S ( I ) I = I + 1 IF I>NC THEN GOTO 772 ELSE SZ11 ! = B S ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 775 ELSE SZ12 ! = B S ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 778 ELSE SZ13 ! = B S ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 781 ELSE SZ14 ! = B S ( I ) 1 = I + 1 IF I>NC THEN GOTO 784 ELSE S Z 1 5 ! = B S ( I ) 1 = 1 + 1 IF I>NC THEN GOTO 787 ELSE SZ16 ! = B S ( I ) GOTO 790 SZ2 ! = 0 S Z 3 ! = 0 S Z 4 ! = 0 SZ5 ! = 0 SZ6 ! = 0 SZ 7 ! = 0 SZ8 ! = 0 SZ9 ! = 0 SZ10 ! = 0 772 775 778 781 784 787 790 793 796 799 802 805 808 81 1 814 817 820 823 826 829 832 83 5 838 841 844 847 850 853 856 859 862 865 868 871 874 877 880 883 886 SZ1 1 ! = SZ12!= SZ13!= SZ14!= SZ15 ! = SZ16 ! = LSET LSET LSET LSET LSET LSET LSET LSET PUT 0 0 0 0 0 0 Z l $ = Z3$ = Z5$ = Z7$ = Z9$ = Z l 1$ = Z13$ = Z15$ = #4 , J GOSUB 946 RETURN MKS$(SZ1 ! ) MKS$(SZ3 ! ) MKS$(SZ5 ! ) MKS$(SZ7 ! ) MKS$(SZ9! ) MKS$(SZ11 = MKS$(SZ13 = MKS$(SZ15 : 1= I+l : ) ) ) IF LSET Z2$= MKS$(SZ2 ! ) LSET Z4$= MKS$(SZ4! ) LSET Z6$= MKS$(SZ6 ! ) LSET Z8$= MKS$(SZ8 ! ) LSET Z10$= MKS$(SZ10! ) LSET Z12$= MKS$(SZ12 ! ) LSET Z14$= MKS$(SZ14!) LSET Z16$= MKS$(SZ16! ) K = NC THEN GOTO 691 "Y" OR ANSWER$= PRINT CHR$(26) PRINT TAB(19); GOSUB 1216 IF ANSWERS J= 1 GOSUB 1252 J= J + l K= 1 GET #2, J EXDT!= CVI(MID$(ED$,K,2)) IF EXDT! < 0 THEN EXDT!= 65536! PRINT USING "########"; EXDT!; K= K + 2 IF K<128 GOTO 856 IF L0C(2) <= NRMEM GOTO 847 RETURN PRINT : PRINT : PRINT : PRINT : PRINT : PRINT DO YOU WANT TO SEE THE ACQUIRED DATA F I L E ? " PRINT THEN GOTO 841 ELSE GOTO 874 + EXDT! PRINT CHR$(26) PRINT TAB(17); PRINT : PRINT '1)0 YOU WANT T O : PRINT : PRINT : PRINT : PRINT SEE THE VELOCITY AND SIZE F I L E ? ' PRINT t o t o 889 892 895 898 901 904 907 910 913 916 919 922 925 928 931 934 937 940 943 946 949 952 955 958 961 964 967 970 973 976 979 982 985 988 991 994 997 1000 1003 GOSUB 1216 IF ANSWER$= "Y" J= 1 : GET #3, GOSUB 1231 GOSUB 1252 J= J + l GET #3, J VEL1!= VEL3!= VEL5!= VEL7!= PRINT OR ANSWER$= THEN GOTO 895 ELSE GOTO 937 CVS(V1$) CVS(V3$) CVS(V5$) CVS(V7$) USING "##. PRINT USING "##. PRINT USING "##. PRINT USING IF L0C(3) <= NR3 RETURN PRINT CHR$(26) PRINT TAB(24); GOSUB 1216 IF ANSWER$= "Y J= 1 : GET #4, GOSUB 1231 GOSUB 1252 J = J + l GET #4, J SZ1!= CVS(Z1$) SZ5!= CVS(Z5$) SZ9!= CVS(Z9$) SZ13!= CVS(Z13$) PRINT USING "##. PRINT USING "##. PRINT USING "##. PRINT USING "##. IF L0C(4) <= NR4 RETURN SZ1!= CVS(S1$) : VEL2!= SZ3!= CVS(S3$) : VEL4!= SZ5!= CVS(S5$) : VEL6!= SZ7!= CVS(S7$) : VEL8!= ##### " ; V E L l ! ; SZ1! ##### •• ; VEL3 ! ; SZ3 ! ##### " ; VEL5!; SZ5! ##### » ; VEL7 ! ; SZ7 ! THEN GOTO 904 CVS(V2$) CVS(V4$) CVS(V6$) CVS(V8$) VEL2 ! VEL4 ! VEL6 ! VEL8 ! SZ2 ! SZ4 ! SZ6 ! SZ8 ! SZ2 ! SZ4 ! SZ6 ! SZ8 ! CVS(S2$) CVS(S4$) CVS(S6$) CVS(S8$) PRINT : PRINT : PRINT : PRINT : PRINT DO YOU WNAT TO SEE THE SIZE F I L E ? " OR ANSWER$= PRINT PRINT SZ2!= CVS(Z2$) SZ6!= CVS(Z6$) SZ10!= CVS(ZIOS) : : SZ14!= CVS(Z14$) ##### ##### ##### ##### THEN GOTO SZ3 ! - CVS(Z3$) : SZ4!= CVS(Z4$) SZ7!= CVS(Z7$) : SZ8!= CVS(Z8$) : SZ11!= C V S ( Z l l S ) : SZ12!= CVS(Z12$) SZ15!= CVS(Z15$) : SZ16!= CVS(Z16$) SZ1 ! ; SZ 2! ; SZ3! ; SZ4! S Z 5 ! ; S Z 6 ! ; S Z 7 ! ; S Z 8 ! SZ9 ! ; SZ10! ; SZ11 ! ; SZ12! SZ13 ! : SZ14! ; SZ15! ; SZ16! 967 1006 1009 1012 1015 1018 1021 1024 1027 1030 1033 1036 1039 1042 1045 1048 1051 1054 1057 1060 1063 1066 1069 1072 1075 1078 1081 1084 1087 1090 1093 1096 1099 1102 1105 1108 1111 1114 1117 1120 "Y" OR ANSWER$= PRINT CHR$(26) : PRINT : PRINT PRINT TAB(16); "DO YOU WANT TO GOSUB 1216 IF ANSWER$= GET #5, 1 GOSUB 1231 GOSUB 1252 GET #5, 2 BFRC!= CVS(BF$) : ABSZ!= CVS(AVSZ$) ABV!= CVS(AVV$) : NC!= CVS(NBVSC$) GHUP!= GHUP! * 100 GET #5, 3 ABSZS1!= CVS(AVSZS$) IF FLAG = 0 GOTO 1060 ABSZS2!= CVS(AVSZSC$) ABVS1!= CVS(AVVS$) : IF FLAG= 0 GOTO 1069 ABVS2!= CVS(AVVSC$) : GET #5, 4 NS1!= CVS(NS$) : VS1!= CVS(VS$) IF FLAG= 0 GOTO 1081 NS2!= CVS(NSC$) : VS2!= CVS(VSC$) : PRINT : PRINT : PRINT : PRINT : PRINT SEE THE REDUCED INFORMATION F I L E ? " THEN GOTO 1021 ELSE GOTO 1210 GHUP!= CVS(BHLP$) : SIGMA.S!= CVS(SDS$) SIGMA.V!= CVS(SDV$) : N!= CVS(NBVS$) VL!= CVS(V$) : SNR!= CVS(SL$) : VOE!= CVS(VO$) SIGMA.SS1!= CVS(SDSS$) : SIGMA.SS2= CVS(SDSSC$) SIGMA.VS1!= CVS(SDVS$) SIGMA.VS2!= CVS(SDVSC$) LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT IF FLAG "EXPERIMENT'S No."; TAB(30); USING "###"; NEX% "GAS HOLD-UP"; TAB(30); USING "###.###"; GHUP! "BUBL FREC ( b b l / s e c ) " ; TAB(30) USING "###.###"; "No. OF MEASURED USING "###"; N! "No. OF V-S DATA = 1 GOTO 1117 BFRC ! V & S"; TAB(30); USED IN IR"; TAB(30) LPRINT USING "### GOTO 1120 LPRINT USING "### LPRINT "AVRG BUBL S (m) NC! , NS1 ! NC! , NS1 ! TAB(30); NS 2 1123 1126 1129 1132 1135 1138 1141 1144 1147 1150 1153 1156 1159 1162 1165 1168 1171 1174 1177 1180 1183 1186 1189 1192 1195 1198 1201 1204 1207 1210 1213 1216 1219 1222 1225 1228 1231 1234 1237 IF FLAG = 1 GOTO 1132 LPRINT USING "###.### "; ABSZ!, ABSZS1! GOTO 1135 LPRINT USING "###.### "; ABSZ!, ABSZS1!, ABSZS2! LPRINT "SD OF BUBBLE SIZE (m)"; TAB(30); IF FLAG= 1 GOTO 1147 LPRINT USING "###.### "; SIGMA.S!, SIGMA.SSI! GOTO 1150 LPRINT USING "###.### "; SIGMA.S!, SIGMA.SSI! LPRINT "AVRG BUBL V (m/sec)"; TAB(30); IF FLAG= 1 GOTO 1162 LPRINT USING "###.### "; ABV!, ABVS1! GOTO 1165 LPRINT USING "###.### "; ABV!, ABVS1!, ABVS2! LPRINT "I OF TURB (m/sec)"; TAB(30); IF FLAG= 1 GOTO 1177 LPRINT USING "###.### "; SIGMA.V!, SIGMA.VS1! GOTO 1180 LPRINT USING "###.### "; SIGMA.V!, SIGMA.VS1! LPRINT "LRGST V CNSDR (m/sec)"; TAB(30); IF FLAG= 1 GOTO 1192 SIGMA.SS2 ! SIGMA.VS2! LPRINT USING "###.### GOTO 1201 LPRINT USING "###.### LPRINT "LRGST S CNSDR (m LPRINT USING "###.### LPRINT "GVOC (m/sec)"; TAB(30) LPRINT USING "###.###"; VOE! LPRINT : LPRINT : LPRINT RETURN VL!, VS1! VL! , VS1 ! ; TAB(30) SNR ! VS2 ! PRINT : PRINT PRINT TAB(18 ) ; "TYPE Y ANSWER$= INPUT$(1) RETURN OR IF YES ANYTHING ELSE IF NOT" NEX* = CVI(NE$) EDM%= XVI(DM$) : EDD*= CVI(DD$) : EDY%= CVI(DY$) TEX!= CVS(TE$) : PSEX!= CVS(PS$) : NBEX!= CVS(NB$) : NVEX!= CVS(NV$) t o t o co 1240 1243 1246 1249 1252 1255 1258 1261 1264 1267 1270 1273 1276 1279 1282 1285 1288 1291 1294 12 97, 1300 1303 1306 1309 1312 1315 1318 1321 1324 1327 1330 1333 STSEX! PPZ ! = ROEX!= RETURN PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT RETURN = CVS(STS$) : DPEX!- CVS(0P$) CVS(PZ$) : PPR!= CVS(PR$) : PPA! CVS(R0$) : QVGEX!= CVS(QVG$) CVS(PA$ ) CHR$(26) : PRINT : PRINT : PRINT "EXPERIMENT'S NUMBER"; TAB(48); NEX* "EXPERIMENT'S DATE"; TAB(48); EDM*; CHR$(47); EDD*; CHR$(47); EDY* "DURATION OF THE EXPERIMENT"; T A B ( 4 8 ) ; TEX!; " s e c " "DISTANCE BETWEEN THE CONTACTS"; T A B ( 4 8 ) ; PSEX!; "m" "No. OF BUBBLES THAT PASSED LOWER CONTACT"; T A B ( 4 8 ) ; NBEX! "No. OF MEASURED VELOCITIES"; T A B ( 4 8 ) ; NVEX! "SUM TIME SIZE"; TAB(48); STSEX!; " s e c " "DEPTH OF THE BATH"; TAB(48); DPEX!; "m" "POSITION OF THE PROBE"; TAB(48); PPZ!; "m"; S P C ( 3 ) ; PPR!; "m"; S P C ( 3 ) ; PPA!; "deg" "ORIFICE RADIOUS"; TAB(48); ROEX!; "m" "GAS VOLUME FLOW AT ORIFICE CONDITIONS"; T A B ( 4 8 ) ; QVGEX! ; "m3/sec" : PRINT : PRINT LSET NE$ = MKI$(NEX*) LSET DM$ = MKI$(EDM*) : LSET DD$= MKI $(EDD*) : LSET LSET TE$ = MKS$(TEX! ) LSET PS$ = MKS$(PSEX ! ) LSET NB$ = MKS$(NBEX!) : LSET NV$= MKS$(NVEX!) LSET STS$ = MKS$(STSEX! ) LSET DP$ = MKS$(DPEX! ) LSET PZ$ = MKS$(PPZ ! ) : LSET PR$= MKS$(PPR ! ) : LSET LSET R0$ = MKS$(ROEX! ) : LSET QVG$ = MKS$(QVGEX!) RETURN DY$= MKI $(EDY*) PA$= MKS$(PPA!) 231 APPENDIX IV C a l c u l a t i o n of Gas Volume Flow Rate and Area Averaged Gas  F r a c t i o n By equating the l o c a l gas phase v e l o c i t y to the l o c a l mean bubble r i s e v e l o c i t y , the gas volume flow rate across tr a n s v e r s e s e c t i o n s of the plume were obtained by numerical i n t e g r a t i o n as A n _ Q E U. . a . A. ( I V . l ) bi 1 1 where A. , A„ , A., .... A are the areas of a c i r c l e with 5 mm 1 2 3 n radius and that of r i n g s with 5 - 10, 10 - 15 r , - r p-1 P r a d i u s . The values of U, . and a . are the l o c a l values b l I corresponding to the r a d i a l p o s i t i o n s 0.25, 0.75, 1.25 r + 0.25 mm The radiu s of the plume, r , corresponded to a p-1 p gas f r a c t i o n of 0 * The i n t e g r a t i o n r e s u l t s are p l o t t e d i n Figure 5.17 i n the form of d i s c r e p a n c i e s from the input a i r r a t e . The input gas flow rate used i n the comparison was that f o r a temperature of 20°C and a pressure h & + (h^ - z) The area-averaged gas f r a c t i o n was c a l c u l a t e d from the l o c a l values as A n < a > _ i Z a . A . (IV.2) A 1 1 232 The i n t e g r a t i o n i n t e r v a l was s u b d i v i d e d i n the form m e n t i o n e d p r e v i o u s l y . R e s u l t s of t h e i n t e g r a t i o n a r e p r e s e n t e d i n F i g u r e s 6.1 to 6.6 . 233 APPENDIX V 56 Model of Tacke et a l . f o r the Zone of F u l l y Developed Buoyant  Flow In order to c a l c u l a t e the a x i a l v a r i a t i o n of the gas f r a c t i o n , h a l f - v a l u e r a d i u s and v e l o c i t y of the bubbles i n the re g i o n of developed buoyant flow, i n t u r b u l e n t g a s - l i q u i d plumes, Tacke et a l . formulated a model based on the c o n s e r v a t i o n of mass of gas and l i q u i d i n the plume and i n the c o n s e r v a t i o n of v e r t i c a l momentum, given r e s p e c t i v e l y as h - + h w a b oo aU. r d r D (V.l) d f 2 dz 3 d a ) d r ] 2 TT b ue U x (1 - a ) max (V.2) d f 2 TT P dz 1 °°(1 - a ) U 2 r dr ] 2 TT g( P, p ) a r dr g (V.3) The p r o f i l e s of gas f r a c t i o n and l i q u i d v e l o c i t y were assumed Gaussian and represented as °<max exp ( - — ) b a (V.4) Uj exp ( -max b u (V.5) 234 where t>u and b Q (b = r ^ 2 ^ " / l n 2 f o r G a u s s i a n d i s t r i b u t i o n s ) a r e the n o m i n a l plume w i d t h s a t U, / U, = e 1 . and a/ a = e 1 lmax max r e s p e c t i v e l y , and were r e l a t e d by a c o n s t a n t w i d t h r a t i o b a A = (V.6) b u The v e l o c i t y of the b u b b l e s i n the plume was g i v e n as U b = U x + U b t (V.7) I n t e g r a t i n g E q u a t i o n s ( V . l ) to (V.3) c o n s i d e r i n g E q u a t i o n s (V.4) t o ( V . 7 ) , t h e f o l l o w i n g e x p r e s s i o n s a r e o b t a i n e d 3 . 2 r 1 fll 3. X „ , . _. ^ . T = TT a m a v b [ + U ] (V.8) 1 v. A. v, r, m x O , ^ , 2 bt h a + h f a - z 1 + \ dY(2) d , U. bT ,1 - _^a_max, . i r ^ - - d T [ l m a x a < - — 7 2 > ] 1 + A 2EAb U. , 1 - a max . (V.9) a lmax ( TPk^ dY( 1 ) d r b 2 U 2 . 1 amax , . d z = Ii" 1 a l m a x ( —T" 2 1 ]. a z 2A^ 1 + 2 A 2 ga b 2 (V.10) max a The i n i t i a l c o n d i t i o n s r e q u i r e d f o r syst e m of e q u a t i o n s were s e l e c t e d a t the the s o l u t i o n of the above p o s i t i o n i n d i c a t i n g t h e 235 s t a r t of the d e v e l o p e d buoyant plume as g i v e n by the e x p e r i m e n t a l a x i a l mean b u b b l e v e l o c i t y p r o f i l e s . T h i s s p e c i f i c a t i o n of the 5 6 i n i t i a l c o n d i t i o n s d i f f e r s from t h a t g i v e n by Tacke e t a l . . In t h a t s t u d y the i n i t i a l c o n d i t i o n s c o r r e s p o n d e d to a p o s i t i o n a t wh i c h a 50 * , but as seen i n t h i s i n v e s t i g a t i o n t h i s max l o c a t i o n i s i n no form r e l a t e d to the o n s e t of f u l l y buoyant f l o w f o r w h ich t h e model was p r o p o s e d . V a l u e s ^ of X = 0.7 and U.. = 0.25 m/s were used i n t h e s o l u t i o n o f the e q u a t i o n s , bt To c a l c u l a t e t h e v a r i a t i o n of a . T r y ,„ and U, w i t h max umax/2 bmax p o s i t i o n , E q u a t i o n (V.9) and (V.10) were s o l v e d by a f o u r t h - o r d e r 13 8 R u n g e - K u t t a method f o r Y ( l ) and Y ( 2 ) . The sys t e m of n o n - l i n e a r a l g e b r a i c e q u a t i o n s t h a t r e s u l t e d max I a i ( U l m a x i + U b t 1 + X' h + h u a b z . I ( V . l l ) IT H 2 1 u, . b„, . . lmax l oil ( A 2ct max l 1 + A' Y(2) (V.12) 2 2 b a i U l m a x i ( 2X *max l 1 + 2X' ) - Y ( 1 ) i (V.13) 138 was t h e n s o l v e d u s i n g the Newton-Raphson method t o o b t a i n t h e v a l u e s of the d e s i r e d p a r a m e t e r s a t p o s i t i o n i . T h i s p r o c e d u r e was r e p e a t e d t o c o v e r t h e e n t i r e l e n g t h of the plume. D e t a i l e d l i s t i n g of the program c o n s t r u c t e d f o r the s o l u t i o n of the model 236 i s g i v e n at the end of the a p p e n d i x . I t i s i m p o r t a n t to note t h a t the r e s u l t s o f t h e model needed t o be f i t t e d t o t h e measured q u a n t i t i e s by s e l e c t i n g v a l u e s o f the e n t r a i n m e n t c o e f f i c i e n t , £. 237 C .. THIS PROGRAM SOLVES TACKE ET AL. MODEL. IT IS IDENTICAL C . . T O VEL2.F77 EXCEPT THAT IT IS LINKED USING 80287 .. IMPLICIT REAL*8 (A-H.O-Z) IMPLICIT INTEGER ( I , J , K, L, M. N) INTEGER RUNGE REAL * 8 LAMBDA REAL * 8 F. Y, PHI, SAVEY, XOLD, XINC, A REAL * 8 DUX, F l , Q, HA, H, Z, PI, G REAL * 8 ZO, HD, ALPHA, EPS1, EPS 2 REAL * 8 BE, BEPSO , EMAXO, ULMAXO, UGMAXO, RIO REAL * 8 BEPS, EMAX, ULMAX, ZMAX, UGMAX, R l REAL * 8 RD1, RD2, RD3 REAL * 8 DEXP, DATAN, DLOG, DSQRT DIMENSION F ( 2 ) , Y ( 2 ) , P H I ( 5 0 ) , SAVEY(50), X0LD(21) DIMENSION X I N C ( 2 1 ) , A(21,21) COMMON LAMBDA, DUX, F l , Q, HA, H, Z, PI, G, Y C ... READ AND PRINT DATA . . . 0PEN(UNIT = 10,FILE='NUM2 ' ) READ (10,100) ZO, R l , H, Q, HD, ALPHA 0PEN(UNIT=11,FILE='NUM3') R E A D ( l l . l O l ) ITMAX, IPRINT, N, EPS1, EPS2 OPEN (UNIT=12,FILE='DAV ' ) PI = 4.0*DATAN(DBLE(1 . 0 ) ) G = 9 . 810 BE = DLOG(DBLE(2 . ) ) BEPSO = (Rl/DSQRT(BE ) ) EMAXO = 0.50283 LAMBDA = 0 . 7 DUX = 0.250 F l = 1 . HA = 1.02347E1 WRITE(6,200) ZO, R l , H, Q, HD, ALPHA WRITE (6,201) ITMAX, IPRINT, N, EPS1, EPS2 WRITE ( 12,300) Q, H, ALPHA, HD, EPS1, EPS 2 C C ... I N I T I A L I Z E Z, Y ( l ) , Y ( 2) AND ULMAX ... BEPS = BEPSO EMAX = EMAXO ZMAX = H Z = ZO ULMAXO = (1. + LAMBDA**2)*( Q/(PI *BEPS* * 2 *EMAX)* + (HA/(HA + H - ZO)) - DUX*F1) ULMAX = ULMAXO UGMAXO= ULMAXO + 0 . 2 5 R10= BEPSO * DSQRT(BE) WRITE(12,301) ZO, EMAXO, RIO, UGMAXO WRITE( 12 , 302 ) Y ( l ) = BEPS**2*ULMAX**2*(1./(2.*LAMBDA**2) - EMAX/ + (1.+ 2.* LAMBDA* * 2) ) Y(2) = BEPS**2*ULMAX*(1. - LAMBDA** 2 *EMAX/ 238 + (1.+ LAMBDA* * 2) ) C GO TO 4 WRITE(6,202 ) READ(5,102) ICONT IF (ICONT .EQ. 0) STOP 4 WRITE(6,203) Z, Y ( 1 ) , Y ( 2 ) , ULMAX C C ... CALL ON THE FOURTH-ORDER RUNGE-KUTTA FUNCTION ... 11 K = RUNGE(2,Y,F,Z,HD) IF ( K . N E . l ) WRITE (6,203) Z, Y ( l ) , Y ( 2 ) , ULMAX C ... WHENEVER K=l, COMPUTE DERIVATIVE VALUES IF (K.NE . 1 ) GO TO 13 F ( l ) = BEPS**2*EMAX*G F ( 2 ) = 2.*ALPHA*LAMBDA*BEPS*ULMAX*(1.-(EMAX/ + DEXP(DBLE(1./LAMBDA* * 2 ) ) ) ) GO TO 11 C ... IF Z EXCEEDS ZMAX, TERMINATES INTEGRATION 13 IF (Z.LE.ZMAX) GO TO 1 STOP C .... CALCULATE ESTIMATES FOR X O L D ( I ) , 1=1,..,3 .... 1 X0LD(1)= DEXP(DBLE(-2.5565 + 0.5480*DL0G(Z)))/DSQRT(BE) X0LD(2)= DEXP(DBLE(-3.3224 - 1 .1307*DL0G(Z) ) ) X0LD(3)= DEXP(DBLE(-0.1821 - 253.6236*DLOG(Z) + 126.600* + D L 0 G ( Z * * 2 ) ) ) - 0.250 2 T l = X O L D ( l ) T2= X0LD(2) T3= X0LD(3) CALL SIMNLE (ITMAX, IPRINT, N, EPS1, EPS 2 , XOLD. ICONV, ITER) IF (ICONV.EQ.0) GO TO 2 WRITE(6,204) ITER, ( X O L D ( I ) , 1=1,N) BEPS = X O L D ( l ) EMAX = X0LD(2) ULMAX = X0LD(3) UGMAX= ULMAX + 0 . 2 5 Rl= BEPS * DSQRT(BE) RD1= -(XOLD(1)**2*X0LD(3 ) **2*(1 ./(2 . *LAMBDA* * 2) + - X 0 L D ( 2 ) / ( 1 . + 2. * LAMBDA* * 2) ) - Y ( l ) ) RD2= - ( X O L D ( 1 ) * * 2 * X 0 L D ( 3 ) * ( 1 . - (LAMBDA** 2*XOLD(2) )/ + (1. + LAMBDA* * 2 ) ) - Y(2) ) RD3= - ( P I * X O L D ( 1 ) * * 2 * X 0 L D ( 2 ) * ( X O L D ( 3 )/(1 . + LAMBDA* * 2) + + DUX*F1) - Q*HA/(HA t-H-Z ) ) WRITE(12,303) T l , T2, T3 WRITE(12 , 304) RD1, RD2, RD3 WRITE( 12 , 305 ) Z, R l , EMAX, UGMAX C .... UPDATE RUNGE'S FUNCTIONS .... Y ( l ) = BEPS**2*ULMAX**2*(1./(2.*LAMBDA**2) - EMAX/ + (1. + 2.*LAMBDA**2)) Y(2) = BEPS**2*ULMAX*(1. - LAMBDA** 2 *EMAX/ + (1. + LAMBDA* * 2) ) 239 100 101 102 200 201 202 203 204 300 301 302 303 304 305 C C C GO TO 4 FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT 2X. F18 FORMAT 3 ( 2X , F9 FORMAT 2H M / 7H EPS1 FORMAT 8X , 6H 4H M/S FORMAT 5X, 35H FORMAT FORMAT FORMAT STOP END (9X, 3(F9.6,9X) / 9X, (10X, 13, 17X,11, 19X, 13 ( 5 X , I 2 ) (9X, 3(F9.6,9X) / 9X, 3 ( F 9 . 6 , 9 X ) ) / 10X,E7.1,13X,E7 . 1 ) 3{F9.6,9X ) ) (10X,13,17X,II,19X,13 / 10X,E7.1,13X,E7.1) (5X, 'TO STOP PROGRAM INPUT 0 ' / 8H0 ** / ) / 10X,F10.5,5X,2F18.8 = , 13 / (5X,'Z, Y ( l ) , Y ( 2 ) , ULMAX : • 8) ( 24H0SUCCESSFUL CONVERGENCE / 10H0ITER • 6) ) (5X, 4H Q= , 4X, F9.6, 5H M3/S, 3X, 4H H= , 5X, 8H ALPHA= , F9.6, 8X, 5H HD = , 2X, F9.6 8X, 7H EPS2= , E9.2 / ) 3X, F9.6, 2H M / 5X, 8H EMAXO = , IX, E9.2 (5X, 5H Z0= 3X, F9.6, 2H M / 5X = , F9.6, R10= , IX, F9.6, 2H M, 5X, 9H UGMAXO= , F9.6, / ) (5X, 42H ALTERNATED ROWS OF ESTIMATED AND SOLUTION / PARAMETERS AS FUNCTION OF POSITION //) (19X, 3 ( 5 X , F 9 . 6 ) ) ( 19X , 3(5X,F9.6) ) (5X, 4(5X,F9.6 ) ) FUNCTION RUNGE(N,Y,F,X,HD) .. THE FUNTION RUNGE USES THE 4TH ORDER RUNGE-KUTTA METHOD IMPLICIT REAL*8(A-H, 0-Z) REAL*8 Y, F, PHI, SAVEY REAL * 8 X, HD INTEGER RUNGE DIMENSION P H I ( 5 0 ) , SAVEY(50), Y ( N ) , F(N) DATA M/0/ C C C C 22 M = M + 1 GO TO ( 1 , 2 , 3 , 4 , 5 ) , M .... PASS 1 .... RUNGE = 1 RETURN .... PASS 2 DO 22 J = 1 , N SAVEY(J) = Y ( J ) P H I ( J ) = F ( J ) Y ( J ) = SAVEY(J) CONTINUE X = X + 0.5*HD + 0.5*HD*F(J) 240 RUNGE = 1 RETURN C C .... PASS 3 .... 3 DO 33 J = 1, N P H I ( J ) = P H I ( J ) + 2 . 0 * F ( J ) Y ( J ) = SAVEY(J) + 0.5*HD*F(J) 33 CONTINUE RUNGE = 1 RETURN C . C PASS 4 .... 4 DO 44 J = 1 , N P H I ( J ) = PHI ( J) + 2.0 * F ( J ) Y ( J ) = SAVEY(J) + HD*F(J) 44 CONTINUE X = X + 0 . 5 * H D RUNGE = 1 RETURN C C .... PASS 5 .... 5 DO 55 J = 1, N Y ( J ) = SAVEY(J) + ( P H I ( J ) + F ( J ) ) * H D / 6 . 0 55 CONTINUE M = 0 RUNGE = 0 RETURN C END C C SUBROUTINE SIMNLE(ITMAX,IPRINT,N,EPS1 IEPS2,X0LD,ICONV,ITER) C THIS PROGRAM SOLVES N SIMULTANEOUS NON-LINEAR EQUATIONS C IN N UNKNOWNS BY THE NEWTON-RAPHSON ITERATIVE PROCEDURE. C INITIAL GUESSES FOR VALUES OF THE UNKNOWNS ARE READ INTO C XOLD(1)...XOLD(N ) . THE PROGRAM FIRST CALLS ON THE C CALCN TO COMPUTE THE ELEMENTS OF A, THE AUGMENTED MATRIX C OF PARTIAL DERIVATIVES, THEN ON FUNCTION SIMUL TO SOLVE C THE GENERATED SET OF LINEAR EQUATIONS FOR THE CHANGES IN C THE SOLUTION VALUES XINC(1) . . .XINC(N) . DETER IS THE C JACOBIAN COMPUTED BY SIMUL. THE SOLUTION ARE UPDATED C AND THE PROCESS CONTINUED UNTIL ITER, THE NUMBER OF C ITERATIONS, EXCEEDS ITMAX OR UNTIL THE CHANGE IN EACH C OF THE N VARIABLES IS SMALLER IN MAGNITUDE THAN EPS2 C (ITCON = 1 UNDER THESE CONDITIONS). EPS1 IS THE MINIMUM C PIVOT MAGNITUDE PERMITTED IN SIMUL. WHEN IPRINT = 1, C INTERMEDIATE SOLUTION VALUES ARE PRINTED AFTER EACH C ITERATION. C IMPLICIT INTEGER(I, J , K, L, M, N) 241 IMPLICIT REAL*8(A-H, 0-Z) REAL * 8 XOLD, XINC, A REAL * 8 EPS 2 , EPS1, DETER DIMENSION XOLD(21), X I N C ( 2 1 ) , A(21,21) L= 0 C C NEWTON-RAPHSON ITERATION DO 9 ITER = 1, ITMAX C C CALL ON CALCN TO SET UP A MATRIX CALL CALCN( XOLD, A, 21 ) C C .. CALL SIMUL TO COMPUTE JACOBIAN AND CORRECTIONS IN XINC . DETER = SIMUL(N, A, XINC, EPS1, 1, 21 ) L= L + l IF ( L . L T . 5 ) GO TO 84 WRITE (6 , * ) (X I N C ( I ) , 1 = 1 ,N) WRITE (6,83) 83 FORMAT (' THIS ARE X I N C ( I ) VALUES ') 84 IF ( DETER.NE.0. ) GO TO 3 WRITE (6,201) ICONV = 0 RETURN C C ... CHECK FOR CONVERGENCE AND UPDATE XOLD VALUES ... 3 ITCON = 1 ICONV = 0 DO 5 I = 1 , N IF ( A B S ( X I N C ( I ) ) . G T . E P S 2 ) ITCON = 0 XOLD(I) = XOLD(I) + X I N C ( I ) 5 CONTINUE IF (L . LT.5) GO TO 85 IF ( IPRINT.EQ.l ) WRITE (6,202 ) ITER,DETER, (XOLD(I) ,I = 1 , N ) L= 0 85 IF ( ITCON.EQ.O ) GO TO 9 ICONV = 1 WRITE (6,203) ITER,(XOLD(I),I=1,N) RETURN 9 CONTINUE C WRITE (6,204) RETURN C C FORMATS FOR INPUT AND OUTPUT STATEMENTS C 201 FORMAT ( 38H0MATRIX IS ILL-CONDITIONED OR SINGULAR ) 202 FORMAT ( 10H0ITER = ,18/ 10H DETER = , E18.5 / + 4(2X, E15.8) / 3(2X, E15.8)) 203 FORMAT ( 24HOSUCCESSFUL CONVERGENCE / 10H0ITER = ,13 / + 4(2X, E10.4) / 3(2X, E10.4)) 242 204 FORMAT ( 15H NO CONVERGENCE ) RETURN END C C SIMUL FUNCTION SIMUL( N, A, X, EPS, INDIC, NRC ) IMPLICIT REAL*8(A-H, 0,Z) IMPLICIT INTEGER(I, J , K, L, M, N) REAL * 8 Y, A, X REAL * 8 EPS, SIMUL, DABS, PIVOT, DETER, AIJCK DIMENSION IR0W(50), J C 0 L ( 5 0 ) , JORD(50), Y ( 5 0 ) , A(N R C,N R C) DIMENSION X(N) C MAX = N IF ( INDIC.GE.O ) MAX = N + 1 C C IS N LARGER THAN 50 IF ( N.LE.50 ) GO TO 5 WRITE (6,200) SIMUL = 0. RETURN C C BEGIN ELIMINATION PROCEDURE 5 DETER = 1. DO 18 K = 1, N KM1 = K - 1 C C SEARCH FOR THE PIVOT ELEMENT PIVOT = 0. DO 11 I = 1, N DO 11 J = 1, N C .. SCAN IROW AND JCOL ARRAYS FOR INVALID PIVOT SUBSCRIPTS IF ( K.EQ.1 ) GO TO 9 DO 8 ISCAN = 1, KM1 DO 8 JSCAN = 1 , KM 1 IF ( I . EQ. IROW(ISCAN) ) GO TO 11 IF ( J.EQ.JCOL(JSCAN)) GO TO 11 8 CONTINUE 9 IF ( D A B S ( A ( I , J ) ) . L E . D A B S ( P I V O T ) ) GO TO 11 PIVOT = A ( I , J ) IROW(K) = I JCOL(K) = J 11 CONTINUE C C ..... INSURE THAT SELECTED PIVOT IS LARGER THAN EPS IF ( DABS(PIVOT).GT.EPS ) GO TO 13 SIMUL = 0. RETURN C C UPDATE THE DETERMINANT VALUE IROWK = IROW(K) JCOLK = JCOL(K) DETER = DETER*PIVOT NORMALIZE PIVOT ROW ELEMENTS DO 14 J = 1, MAX A(IROWK,J) = A(IROWK,J)/PIVOT CONTINUE CARRY OUT ELIMINATION AND DEVELOP INVERSE A(IROWK,JCOLK) = 1./PIVOT DO 18 I = 1, N A l J C K = A(I,JCOLK) IF ( I.EQ.IROWK ) GO TO 18 A(I , J C O L K ) = - AIJCK/PIVOT DO 17 J = 1, MAX IF ( J.NE.JCOLK ) A ( I , J ) = A ( I , J ) - AIJCK*A(IROWK,J) CONTINUE CONTINUE .... ORDER SOLUTION VALUES ( I F ANY) AND CREATE JORD ARRAY DO 20 I = 1, N IROWI= IROW(I) JCOLI= J C O L ( I ) JORD(IROWI) = JCOLI IF ( INDIC.GE.O ) X ( J C O L I ) = A(IROWI,MAX) CONTINUE ADJUST SIGN OF DETERMINANT INTCH = 0 NM1 = N - 1 DO 22 I = 1, NM1 IP1 = 1 + 1 DO 2 2 J= I PI, N IF ( J O R D ( J ) . G E . J O R D ( I ) ) GO TO 22 JTEMP = JORD(J) JORD(J) = JORD(I) JORD(I) = JTEMP INTCH = INTCH + 1 CONTINUE IF ( INTCH/2*2.NE.INTCH ) DETER= - DETER IF INDIC IS POSITIVE RETURN WITH RESULTS IF ( INDIC.LE.0 ) GO TO 26 SIMUL = DETER RETURN ... IF INDIC IS NEGATIVE OR ZERO, UNSCRAMBLE THE INVERSE FIRST BY ROWS . . . DO 28 J = 1, N 244 DO 27 I = 1, N IROWI = IROW(I) JCOLI = J C O L ( I ) Y ( J C O L I ) = A(IROWI,J) 27 CONTINUE DO 28 I = 1, N A ( I , J ) = Y ( I ) 28 CONTINUE C THEN BY COLUMNS DO 30 I = 1, N • DO 29 J = 1, N IROWJ = IROW(J) JCOLJ = J C O L ( J ) Y(IROWJ) = A ( I , J C O L J ) 29 CONTINUE DO 30 J = 1 , N A ( I , J ) = Y ( J ) 30 CONTINUE C C RETURN FOR INDIC NEGATIVE OR ZERO SIMUL = DETER RETURN C C FORMAT FOR O U T P U T STATEMENT 200 FORMAT(1OHON TOO B I G ) END C C C SUBROUTINE CALCN( DXOLD, A, NRC ) IMPLICIT R E A L * 8 ( A H , O-Z) IMPLICIT INTEGER ( I, J , K, L, M, N) REAL*8 LAMBDA REAL * 8 Y REAL* 8 DUX, F l , Q, HA, H, Z, P I , G DIMENSION X0LD(20), DXOLD(NRC), A(NRC,NRC), Y(2) COMMON LAMBDA, DUX, F l , Q, HA, H, Z, P I , G, Y C C SHIFT ELEMENTS OF DXOLD TO XOLD AND CLEAR A ARRAY DO 1 1 = 1 , 3 XOLD(I) = DXOLD(I) DO 1 J = 1, 4 A ( I , J ) = 0 . 1 CONTINUE C C COMPUTE NON-ZERO ELEMENTS OF A A ( l , l ) = 2 . * X 0 L D ( 3 ) * * 2 * ( l . / ( 2 . * LAMBDA* * 2) - X 0 L D ( 2 ) / + ( 1. + 2.* LAMBDA * * 2 ) ) A ( l , 2 ) = - XOLD(1)**2*X0LD(3 ) **2/( 1. + 2.* LAMBDA** 2) A ( l , 3 ) = 2 . * X 0 L D ( l ) * * 2 * ( l . / ( 2 . * L A M B D A * * 2) -245 + X 0 L D ( 2 ) / ( 1. + 2.* LAMBDA* * 2) ) A ( l , 4 ) = - ( X O L D ( l ) * * 2 * X O L D ( 3 ) * * 2 * ( l . / ( 2 . * L A M B D A * * 2 ) + - X 0 L D ( 2 ) / ( 1 . + 2.*LAMBDA**2)) - Y ( l ) ) A ( 2 , l ) = 2. * X0LD(3)*(1.-(LAMBDA**2*XOLD(2))/ + (1. + LAMBDA* * 2)*XOLD(1)) A(2.2) = - (X0LD(1)**2*X0LD(3)*LAMBDA**2)/ + (1. + LAMBDA* * 2) A(2,3) = X O L D ( 1 ) * * 2 * ( 1 . - (LAMBDA**2*XOLD(2))/ + (1. + LAMBDA * * 2 ) ) A(2,4) = - ( X 0 L D ( 1 ) * * 2 * X 0 L D ( 3 ) * ( 1 . - (LAMBDA** 2*XOLD(2 ) ) / + ( 1 . + LAMBDA* * 2 ) ) - Y(2) ) A ( 3 , l ) = 2 * P I * X 0 L D ( 2 ) * ( X 0 L D ( 3 ) / ( 1 . + LAMBDA* * 2) + + DUX*F1) A(3,2) = P I * X 0 L D ( 1 ) * * 2 * ( X 0 L D ( 3 ) / ( 1 . + LAMBDA* * 2) + + DUX*F1) A(3,3) = P I * X 0 L D ( 1 ) * * 2 * X 0 L D ( 2 ) / ( 1. + LAMBDA* * 2) A(3,4) = - ( P I * X O L D ( 1 ) * * 2 * X 0 L D ( 2 ) * ( X O L D ( 3 ) / ( 1 . + LAMBDA**2) + + DUX*F1) - Q*HA/(HA + H - Z)) RETURN END 

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