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Forces on tubes immersed in a fluidized bed Hosny, Nasr. M. 1982

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FORCES ON TUBES IMMERSED IN A FLUIDIZED BED by NASR M. HOSNY M.Eng., Ca r l e ton U n i v e r s i t y , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Chemical Eng ineer ing) We accept t h i s t h e s i s as conforming to the r equ i r ed s tandard THE UNIVERSITY OF BRITISH COLUMBIA August 1982 © NASR M. HOSNY, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or pu b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Che^YhiCaJl E^-njjj-vifo rinj The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Da t e j-W). nn3 1982 o • /new - i i -SUMMARY Heat t rans fe r tubes and other surfaces f ixed ins ide f l u i d i z e d beds are subjected to bu f fe t t i ng forces due to s o l i d s and gas motion. Coupled with erosion and c o r r o s i o n , these forces can lead to tube f a i l u r e . The ob jec t i ve of the present work was to inves t igate the nature and o r i g i n of the forces act ing on tubes immersed h o r i z o n t a l l y in gas f l u i d i z e d beds, and to provide information to be used in s t ruc tura l design of heat exchanger tube bundles. Experiments were c a r r i e d out in a column o f c r o s s - s e c t i o n 215 x 200 mm and height 1.5 m. The f l u i d i z i n g gas was a i r and the bed was operat ing at ambient temperature. The parameters var ied in the experiments were s u p e r f i c i a l gas v e l o c i t y (from U ^ to Um^ + 1.4 m/s), s t a t i c bed height (from 0.30 m to 0.45 m), mean p a r t i c l e s i z e (from 185 to 430 ym), p a r t i c l e dens i ty (from 920 to 4100 kg/m 3 ), tube diameter (from 15 to 32 mm) and tube shape (f inned v s . unf inned) . A l l tubes were mounted h o r i z o n t a l l y at a height of 0.30 m above the o r i f i c e p late d i s t r i b u t o r . Both v e r t i c a l and hor izontal forces were measured on tubes in i s o l a t i o n and on tubes within arrays of d i f f e r e n t con f igura t ions . For some runs , loca l pressure v a r i a t i o n s at the surface of the tes t tubes were a lso measured simultaneously with the f o r c e s . The r e l a t i o n between force c h a r a c t e r i s t i c s and bubble propert ies was examined by separate experiments invo lv ing i n j e c t i o n of s ing le bubbles. S t a t i s t i c a l parameters o f the measured fo rces have been c a l c u l a t e d . Instantaneous fo rces on each tube w i t h i n a f r e e l y bubbl ing bed cons i s t ed of a s e r i e s o f pu lses whose magnitudes, dura t ions and ra te o f occurrence depend on the opera t ing parameters. The magnitude of the fo rces i s s t r ong l y i n f l uenced by s u p e r f i c i a l gas v e l o c i t y , s l i g h t l y dependent on p a r t i c l e s i z e , and moderate ly a f f e c t ed by bed depth and p a r t i c l e d en s i t y . Hor i zon ta l fo rces are gene ra l l y o f s i g n i f i c a n t l y lower magnitude than v e r t i c a l components and o s c i l l a t e from s ide to s i de w i th a zero mean. Forces can be cha ra c t e r i z ed as con ta in i ng both p e r i o d i c and random components and are s t a t i s t i c a l l y s t a t i o n a r y . The pr imary frequency content of the fo rces i s i n the range 0-20 HZ, w i th the major f requenc ies being almost independent of the opera t ing cond i t i on s and always below 10 HZ at the he ight o f measurement, 300 mm above the d i s t r i b u t o r . The trends are c ons i s t en t w i th the hydro-dynamics o f bubbl ing f l u i d i z e d beds. Force pulses have been found to be r e l a t e d to bubble mot i on , and pu lse maxima correspond to a r r i v a l o f bubbles at the tube su r f a ce . The i n t e n s i t y of fo rces on i n d i v i d u a l tubes w i t h i n an a r ray have been found to depend on the p o s i t i o n of the tube i n the a r r a y . The i n t e n s i t y o f f o r ces can be reduced s i g n i f i c a n t l y by us ing reasonab ly t i g h t i n t e r - t ube spacings and by reduc ing the l e ve l of the tube-bundle above the gas d i s t r i b u t o r . ACKNOWLEDGEMENT The r e s e a r c h was superv ised by Dr. J . R . Grace and the author i s g r a t e f u l to him f o r h i s c o n t i n u e d a d v i c e and encouragement. I would l i k e to express my a p p r e c i a t i o n to Mr. B i l l Hoogendorn, Chemical E n g i n e e r i n g Department, M c G i l l U n i v e r s i t y , f o r s u p p l y i n g the t i m e r used i n the s i n g l e - b u b b l e exper iments . Thanks are a l s o due to the N a t u r a l Sc ience and E n g i n e e r i n g Research C o u n c i l o f Canada f o r a grant i n a i d o f t h i s work. - V -TABLE OF CONTENTS Page SUMMARY i i ACKNOWLEDGEMENT i v TABLE OF CONTENTS v LIST OF FIGURES i x LIST OF TABLES x x CHAPTER 1 NATURE OF THIS STUDY 1 1.1 General Introduction 1 1.2 Objectives of th i s Work 3 1.3 Previous Studies and the Present Study 4 1.4 Outline of th i s Thesis 6 CHAPTER 2 HYDRODYNAMIC CONSIDERATIONS 8 2.1 Introduction 8 2.2 Hydrodynamic Behaviour of Gas F lu id ized Beds 8 2.2.1 Character is t ics of Single bubbles r i s i ng in f l u i d i zed beds 9 2.2.1.1 General nature of bubbles 9 2.2.1.2 Bubble pressure f i e l d and associated pressure f luctuat ions 12 2.2.2 Gross behaviour of f ree ly bubbling beds 18 2.3 Hydrodynamic Effect of Hor izonta l ly Immersed Tubes in a Gas F lu id ized Bed 24 2.3.1 Flow pattern in the v i c i n i t y of immersed tubes 24 2.3.2 Interact ion of immersed tubes with r i s i ng bubbles 27 - v i -Paje CHAPTER 3 EXPERIMENTAL EQUIPMENT AND PROCEDURE , 32 3.1 General Requirements 32 3.2 The F l u i d i z a t i o n Column and A u x i l i a r y Equipment 33 3.3 S o l i d P a r t i c l e s 37 3.4 Force Measurements 39 3.4.1 Force t ransducer 39 3.4.2 Test c y l i n d e r s 42 3.4.3 Inst rumentat ion 47 3.4.4 Exper imental procedure 50 I . P r e l im i na r y t e s t s 50 I I . C a l i b r a t i o n 54 I I I . Force measurements 56 3.5 Pressure Measurements 58 3.6 S i ng l e Bubble Experiments 61 CHAPTER 4 METHODS OF DATA PROCESSING AND ANALYSIS 65 4.1 General Requirements 65 4.2 General Cons idera t ions i n Data Sampling and Process ing 66 4.2.1 Data d i g i t i z a t i o n and sampling cons i de ra t i ons •• 66 4.2.2 Data p r e - ana l y s i s cons i de ra t i ons (general c h a r a c t e r i s t i c s o f the data) 68 4.3 Data Ana l y s i s Procedures 82 CHAPTER 5 SINGLE TUBE EXPERIMENTS: RESULTS AND DISCUSSION 92 5.1 In t r oduc t i on ............... 92 5.2 E f f e c t s o f F lu id-Bed Parameters on the Forces on an Immersed Tube 92 5.2.1 E f f e c t s o f s u p e r f i c i a l gas v e l o c i t y 93 5.2.2 E f f e c t s o f s t a t i c bed height 110 5.2.3 E f f e c t s o f p a r t i c l e s i z e 118 5.2.4 E f f e c t s o f p a r t i c l e type and dens i t y 128 - v i i -Page 5.3 E f f e c t s o f S i ze and Shape o f Immersed Tubes on the Forces 138 5.3.1 E f f e c t s o f tube s i z e 138 5.3.2 E f f e c t s o f tube shape ( f i nned v s . unf inned) 149 5.4 Causes o f the Forces 151 5.5 General P rope r t i e s o f the V e r t i c a l and Ho r i zon ta l Components o f Force 164 5.5.1 Magnitude, du ra t i on and frequency o f the fo rce pulses 164 5.5.2 S t a t i s t i c a l p r ope r t i e s 170 5.5.3 P e r i o d i c i t y and s t a t i o n a r i t y p rope r t i e s 172 CHAPTER 6 TUBES WITHIN ARRAYS: RESULTS AND DISCUSSION 174 6.1 In t roduc t i on 174 6.2 Exper imental Forces on a Tube at the Centre o f an Ar ray o f F ive Tubes 174 6.3 Exper imental Forces on a Tube w i t h i n an Ar ray o f Three Tubes i n Two Con f i gu ra t i ons 190 6.4 General D i scuss ion 196 CHAPTER 7 GENERAL DISCUSSION, CONCLUSIONS AND RECOMMEN-DATIONS , 201 7.1 General D i scuss ion 201 7.2 Conc lus ions 203 7.3 Recommendations 204 NOMENCLATURE 205 REFERENCES 207 - v i i i -Page APPENDIX A. NATURAL FREQUENCIES OF VIBRATION OF THE TEST TUBES 213 APPENDIX B. COMPUTER PROGRAM FOR CALCULATION OF THE RMS, STANDARD DEVIATION AND MEAN VALUES OF THE DATA 216 - i x -LIST OF FIGURES F igure Page 2-1 Pressure d i s t r i b u t i o n along the ax i s o f symmetry o f the bubble, re fe rence (Reuter , 1966) 13 2-2 Dimensionless pressure d i f f e r en c e between po in t s on the v e r t i c a l bubble ax i s and po in ts i n the same ho r i z on t a l plane we l l removed from the bubb le- three-d imens iona l bubble, re fe rence (Stewar t , 1968) 15 2-3 Pressure-Time curve recorded dur ing the r i s e o f a s i n g l e i n j e c t e d bubble i n t o i n c i p i e n t l y f l u i d i z e d bed, re fe rence (L i t tman & Homolka, 1970) 15 2-4 Pressure-Time t races fo r 25 mm diam. tube at four d i f f e r e n t angular p o s i t i o n s . Time 0 corresponds to a r r i v a l o f the f r on t o f the bubble a t the measuring po i n t , re fe rence (Nguyen & Grace, 1978) 17 2-5 Comparison o f Pressure-Time t races f o r 6.3 mm diam. tube and 6 = 0 w i th t h e o r e t i c a l pressure p r o f i l e s and wi th exper imenta l va lues measured by L i t tman and Homolka, re fe rence (Nguyen & Grace, 1978) 17 2-6 Bubble s p a t i a l - d i s t r i b u t i o n across the bed f o r d i f f e r e n t he ights (Copper powder f l u i d i z e d i n the 0.2 m d i a . bed, U = 0.084 m/s), re fe rence (Werther, 1973) 22 2-7 Development o f l o c a l f l u i d i z a t i o n around a ho r i z on t a l tube , re ference (Ginoux et a l ., 1 974) 25 2-8 Sur face pressure v a r i a t i o n at a po int on the c y l i n d e r dur ing t r a n s i t o f an i n j e c t e d bubble i n t o two-dimensional f l u i d i z e d bed, re ference (Ginoux et a l . , 1974) 29 - x -F igure Page 2- 9 Typ ica l v a r i a t i o n in sur face p a r t i c a i v e l o c i t y w i th time f o r the cen t ra l tubes , re ference (Pee le r & Whitehead, 1982) 2 9 3- 1 General set-up o f the exper imental apparatus 34 3-2 Pr imary fea tures o f the f l u i d i z a t i o n column and i t s a u x i l i a r i e s 35 3-3 V e r t i c a l s e c t i on through the f l u i d i z a t i o n column 36 3-4 General dimensions o f the f o r ce t ransducer 40 3-5 Balanced Wheatstone br idge and add i t i o na l compensation r e s i s t o r s 41 3-6 Dimensions o f the c r o s s - s e c t i o n a l area o f the t e s t c y l i n d e r s 43 3-7 S ide p l a t e o f column f o r tube a r ray assembly 44 3-8 A ho r i z on ta l c r o s s - s e c t i o n at l e ve l 300 mm above the gas d i s t r i b u t o r showing the tube-gage assembly, w i th the gages set to measure the v e r t i c a l fo rces 46 3-9 Inst rumentat ion ( record mode) f o r force measurement 48 3-10-a Output s i gna l ( a f t e r a m p l i f i c a t i o n and w i thout f i l t e r i n g ) f o r one o f the t r a n s -ducers w i th no bed i n t e r a c t i n g wi th the t e s t cy l i nder 53 3-10-b Output s i gna l ( a f t e r amp l i c a t i on ) f o r : (1) r i g h t hand t ransducer w i th f i l t e r i n g , (2) l e f t hand t ransducer wi thout f i l t e r i n g 53 3-11 Sample gage fo rce c a l i b r a t i o n 55 3-12 Sec t i ona l drawing o f the 51D20 low pressure t ransducer 59 3-13 Inst rumentat ion (record mode) o f pressure measurement 62 - x i -F igure Page 3- 14 General set-up of s i n g l e i n j e c t ed bubble experiment 63 4- 1 Histograms cons t ruc ted from two d i f f e r e n t segments o f one sample data r e c o r d . The record i s f o r the measured v e r t i c a l f o rce on a 32 mm tube . The sample was recorded at bed c o n d i t i o n s : (U-U m f ) = 1.2 m/s, H 0 = 0.3 m and bed m a t e r i a l : Ottawa sand (dp = 430 um) 72 4-2 Power spe c t r a l p l o t o f a sample o f t o t a l v e r t i c a l f o r ces on a 32 mm tube . (U-Umf) = 0.1 m/s; bed m a t e r i a l : Ottawa sand; d p = 430 ym; H Q = 0.3 m 75 4-3 P r o b a b i l i t y dens i t y p l o t o f a sample o f t o t a l v e r t i c a l f o rces on a 32 mm tube. (U-Urnf) = 0-1 m/s; bed m a t e r i a l : Ottawa sand; dp = 430 ym; H 0 = 0.3 m 77 4-4 Autocovar iance p l o t o f a sample o f t o t a l v e r t i c a l fo rces on a 32 mm tube. (U-Umf) = 0 .1 ; bed m a t e r i a l : Ottawa sand; <3p = 430 ym; H 0 = 0.3 m 78 4-5 Power spec t r a l p l o t o f a sample of t o t a l v e r t i c a l fo rces on a 32 mm tube. (U-Umf) = 1 - 4 m / § ; bed m a t e r i a l : Ottawa sand; d = 430 ym; H Q = 0.3 m : 79 4-6 P r o b a b i l i t y dens i t y p l o t o f a sample o f t o t a l v e r t i c a l f o rces on a 32 mm tube. (U-U m f ) = 1.4 m/s; bed m a t e r i a l : Ottawa sand; dp = 430 ym; Ho = 0.3 m 80 4-7 Autocovar iance p l o t o f a sample of t o t a l v e r t i c a l forces on a 32 mm tube. (U -U m f ) = 1.4 m/s; bed m a t e r i a l : Ottawa sand; d p = 430 ym; H 0 = 0.3 m 81 5-1-a V e r t i c a l f o rces at oppos i te ends o f the t e s t c y l i n d e r recorded at (U-U m f ) = 0.05 m/s. Bed m a t e r i a l : sand; d p = 430 ym; H 0 = 0.3 m 94 5-1-b Power spe c t r a l est imates of t o t a l v e r t i c a l fo rces f o r the same exper imenta l cond i t i ons as 5-1-a 94 - x i i -F igure Page 5-2-a V e r t i c a l f o rces at oppos i te ends o f the t e s t c y l i n d e r recorded at (U-Umf) = 0 . 3 m/s. Bed m a t e r i a l : sand; dp = 430 ym; H 0 = 0.3 m 96 5-2-b Power spec t r a l es t imates o f t o t a l v e r t i c a l forces f o r the same exper imental cond i t i ons as 5-2-a 96 5-3-a V e r t i c a l forces at oppos i te ends of the t e s t c y l i n d e r recorded at (U-U m f ) = 1.4 m/s. Bed m a t e r i a l : sand; dp = 430 ym; H 0 = 0.3 m 97 5-3-b Power spe c t r a l es t imates o f t o t a l v e r t i c a l f o rces f o r the same exper imenta l cond i t i ons as 5-3-a 97 5-4 V a r i a t i o n of the v e r t i c a l f o rce major frequency w i th excess s u p e r f i c i a l v e l o c i t y . Bed m a t e r i a l : sand; d p = 430 ym; H 0 = 0.3 m . 99 5-5 V a r i a t i o n of RMS v e r t i c a l fo rce wi th excess s u p e r f i c i a l gas v e l o c i t y ( U - U m f ) . Bed m a t e r i a l : sand; d p = 430 ym; H 0 = 0.3 m 100 5-6 V a r i a t i o n o f mean value and standard dev i a t i on o f v e r t i c a l fo rce wi th excess s u p e r f i c i a l gas v e l o c i t y . Bed m a t e r i a l : sand; d p = 430 ym; H Q = 0.3 m 1° 2 5-7-a Ho r i zon ta l fo rces a t oppos i te ends o f the t e s t c y l i n d e r recorded at (U-U m f ) = 0.3 m/s. Bed m a t e r i a l : sand; d p = 430 ym; H Q = 0.3 m/s 1° 4 5-7-b Power spe c t r a l est imates o f t o t a l ho r i zon ta l fo rces f o r the same exper imenta l cond i t i ons as 5-7-a 104 5-8-a Ho r i zon ta l fo rces at oppos i te ends o f the t e s t c y l i n d e r recorded at (U-U m f ) = 0.8 m/s. Bed m a t e r i a l : sand; d p = 430 ym; H Q = 0.3 m 1° 5 5-8-b Power spe c t r a l est imates o f t o t a l ho r i z on t a l f o r ces f o r the same exper imenta l c ond i t i on s as 5-8-a 1° 5 5-9-a Ho r i zon ta l fo rces at oppos i te ends o f the t e s t c y l i n d e r recorded at (U-U m f ) = 1.4 m/s. Bed m a t e r i a l : sand; d p = 430 ym; H 0 = 0.3 m 1° 6 - x i i i -Fi gure Page 5-9-b Power spec t r a l es t imates o f t o t a l ho r i zon ta l fo rces f o r the same exper imenta l cond i t i ons as i n 5-9-a 106 5-10 V a r i a t i o n o f the ho r i z on ta l fo rce major frequency wi th excess s u p e r f i c i a l v e l o c i t y . Bed m a t e r i a l : sand; d p = 430 pm; H Q = 0.3 m 108 5-11 V a r i a t i o n o f RMS and mean va lue o f ho r i zon ta l components o f force w i th excess s u p e r f i c i a l v e l o c i t y , and i n comparison w i th RMS and mean value o f v e r t i c a l components o f the f o r ce 109 5-12-a V e r t i c a l fo rces on the t e s t c y l i n d e r measured i n a bed o f s t a t i c depth 0.45 m. Bed m a t e r i a l : sand; d p = 430 ym; (U -U m f ) = 1 .2 m/s I l l 5-12-b Power spe c t r a l est imates o f t o t a l v e r t i c a l fo rces f o r the same exper imenta l cond i t i ons as in 5-12-a I l l 5-13-a V e r t i c a l fo rces on the t e s t c y l i n d e r measured i n a bed o f s t a t i c depth 0.30m. Bed m a t e r i a l : , , ? sand; a p = 430 ym; (U -U m f ) = 1 .2 m/s 5-13-b Power spec t r a l es t imates o f t o t a l v e r t i c a l fo rces f o r the same exper imental cond i t i ons as i n 5-13-a 112 5-14 Va r i a t i o n o f the major frequency wi th excess s u p e r f i c i a l v e l o c i t y ( U - U m f ) , f o r two d i f f e r e n t s t a t i c bed he i gh t s : H 0 = 0.45 and 0.30 m 114 5-15 RMS v e r t i c a l f o rce v s . excess s u p e r f i c i a l gas v e l o c i t y (U-U m f ) f o r two d i f f e r e n t bed he i gh t s : H 0 = 0.45 and 0.30 m 116 5-16 Mean va lues o f v e r t i c a l f o rce vs . excess s u p e r f i c i a l gas v e l o c i t y (U -U m f ) 117 5-17 Hor i zon ta l fo rces on the t e s t c y l i n d e r measured in a bed o f _ s t a t i c depth 0.45 m. Bed m a t e r i a l : sand; d p = 430 ym; (U-U m f ) = 1 .2 m/s !, 119 - x iv -Fi gure Page 5-18 Hor i zon ta l fo rces on the t e s t c y l i n d e r measured i n a bed o f s t a t i c depth 0.30 m. Bed ma te r i a l : sand; d p = 430 ym; (U-U~mf) = 1 .2 m/s 119 5-19 RMS and mean va lues o f h o r i z on t a l f o r ce v s . excess s u p e r f i c i a l v e l o c i t y ( U - U m f ) f o r two cases o f d i f f e r e n t bed he i gh t s : H 0 = 0.45 and 0.3 m 120 5-20-a V e r t i c a l f o rces on_the t e s t c y l i n d e r measured in a bed o f sand, d n = 280 ym; ( U - U m J = 0.3 m/s; H 0 = 0.3 m ? 122 5-20-b Power spec t r a l es t imates o f t o t a l v e r t i c a l f o rces fo r the same exper imental cond i t i ons as 5-20-a 122 5-21-a V e r t i c a l forces on the t e s t c y l i n d e r measured i n a bed o f sand; d n = 185 ym; ( U - U m f ) = 0.3 m/s; H 0 = 0.3 m 123 5-21-b Power spec t r a l est imates o f t o t a l v e r t i c a l f o r ces f o r the same exper imenta l cond i t i ons as 5-21-a 123 5-22 E f f e c t o f p a r t i c l e s i z e on the major frequency .. 124 5-23 E f f e c t o f p a r t i c l e s i z e on RMS o f v e r t i c a l f o r ce at d i f f e r e n t va lues o f excess super-f i c i a l v e l o c i t y , ( U - U m f ) 126 5-24 E f f e c t o f p a r t i c l e s i z e on RMS and mean va lues o f ho r i z on t a l f o rce at d i f f e r e n t va lues o f excess gas v e l o c i t y , (U-Umf) 127 5-25-a V e r t i c a l fo rces on the t e s t c y l i n d e r measured i n a bed o f alundum, d p = 295 ym; ( U - U m f ) = 0.05 m/s; H 0 = 0.3 m 129 5-25-b Power spe c t r a l es t imates o f t o t a l v e r t i c a l f o rces f o r the same exper imental cond i t i ons as 5-25-a 129 5-26-a V e r t i c a l f o r ces on the t e s t c y l i n d e r measured i n a bed o f sand, dp = 280 ym; ( U - U m f ) = 0.05 m/s; H Q = 0.3 m 130 5-26-b Power spe c t r a l es t imates o f t o t a l v e r t i c a l f o rces f o r the same exper imental cond i t i ons as 5-26-a 130 - xv -F igure Page 5-27-a V e r t i c a l f o rces on the t e s t c y l i n d e r measured i n a bed o f po lye thy lene powder, d p = 280 um; (U-Umf) = 0-05 m/s; H 0 = 0.3 m 132 5-27-b Power s pe c t r a l es t imates o f t o t a l v e r t i c a l f o rces f o r the same exper imenta l cond i t i ons cts 5 _ 2 7~& ••••••••••••••••«•••••••••••••••••••••• 132 5-28 E f f e c t o f p a r t i c l e dens i t y on standard d ev i a -t i o n o f v e r t i c a l fo rce at d i f f e r e n t values o f excess s u p e r f i c i a l v e l o c i t y , (U-U m f ) 133 5-29 E f f e c t of p a r t i c l e dens i t y on mean values o f v e r t i c a l f o rce at d i f f e r e n t va lues o f excess gas v e l o c i t y , (U-U m f ) 134 5-30 E f f e c t o f p a r t i c l e dens i t y on RMS o f v e r t i c a l fo rce at d i f f e r e n t values o f excess gas v e l o c i t y , (U-U m f ) 135 5-31 E f f e c t o f p a r t i c l e dens i t y on RMS and mean values o f ho r i z on t a l f o r ce at d i f f e r e n t va lues o f excess gas v e l o c i t y , (U-U m f ) 137 5-32 E f f e c t of tube s i z e on RMS v e r t i c a l fo rce a t d i f f e r e n t va lues of excess gas v e l o c i t y , (U-Umf) 142 5-33 V e r t i c a l f o rces on a 15 mm diameter tube recorded at (U-Umf) = 0.1 m/s. Bed m a t e r i a l : sand; 3 p = 430 ym; H 0 = 0.3 m 143 5-34 V e r t i c a l f o rces on a 32 mm diameter tube recorded at (U-Umf) = 0.1 m/s. Bed m a t e r i a l : sand; d p = 430 ym; H Q = 0.3 m 143 5-35-a V e r t i c a l f o rces on a 15 mm diameter tube recorded at (U-Umf] = 0 - 3 m/s. Bed m a t e r i a l : sand; d p = 430 ym; H Q = 0.3 m 144 5-35-b Power s pe c t r a l es t imates o f t o t a l v e r t i c a l fo rces f o r exper imenta l cond i t i ons as in 5-35-a 144 5-36-a V e r t i c a l fo rces on a 32 mm diameter tube recorded at (U-Umf.) = 0.3 m/s. Bed m a t e r i a l : sand; 3 p = 430 ym; H 0 = 0.3 m 145 5-36-b Power spe c t r a l es t imates o f t o t a l v e r t i c a l fo rces fo r same exper imental cond i t i ons as 5-35-a 145 - x v i -F igure 5-37 5-38 5-39 5-40 5-41 5-42 5-43 5-44 5-45 Page E f f e c t o f tube s i z e on the major frequency 146 E f f e c t o f tube s i z e on RMS and mean va lues o f ho r i z on t a l fo rce at d i f f e r e n t va lues o f excess gas v e l o c i t y , (U-Umf) 148 E f f e c t o f tube shape ( f i nned v s . unf inned) on RMS v e r t i c a l fo rce a t d i f f e r e n t values o f excess gas v e l o c i t y , (U-U m f ) 150 Power spe c t r a l est imates o f t o t a l v e r t i c a l f o r ce on a bare tube o f outer d iameter 32 mm, recorded at (U-U m f ) = 0 . 8 m/s. Bed m a t e r i a l : sand, dp = 430 ym; H 0 = 0.3 m 152 Power s pe c t r a l est imates o f t o t a l v e r t i c a l f o r ce on a f inned tube o f ou ts ide diameter 32 mm, recorded at (U-Umf) = 0 . 8 m/s. Bed m a t e r i a l : sand; dp = 430 ym; H 0 = 0.3 m 152 E f f e c t o f tube shape ( f i nned v s . unf inned) on RMS and mean va lues o f ho r i z on t a l f o rce at d i f f e r e n t va lues o f excess gas v e l o c i t y , (U-Umf) 1 5 3 Simultaneous fo rce and pressure s i g n a l s . The fo rce s i gna l i s the ins tantaneous t o t a l f o r ce on a 32 mm diameter tube; the pressure s i gna l i s the pressure v a r i a t i o n at the bottom o f the tube, recorded at (U-U m f ) = 0 . 1 m/s; H 0 = 0.3 m 157 Simulatneous fo rce and pressure s i g n a l s . The fo rce s i gna l i s the instantaneous t o t a l fo rce on a 32 mm diameter tube; the pressure s i gna l i s the pressure v a r i a t i o n a t the bottom o f the tube , recorded at (U-Umf) = I - 2 m / s > H 0 = 0.3 m 158 V e r t i c a l f o r ce - t ime curve dur ing the r i s e o f a bubble i n j e c t e d i n to a t h ree -dimensional i n c i p i e n t l y f l u i d i z e d bed o f 430 ym sand p a r t i c l e s . I n j e c t i o n was d i r e c t l y below the centre o f the tube 100 mm above the d i s t r i b u t o r 160 - xvi i -Figure Page 5-46 Vert ica l force-time and pressure-time curves recorded simultaneously during the r i se of a bubble injected into an i nc ip i en t l y f l u id i zed bed of 430 ym sand pa r t i c l e s . Inject ion was d i r e c t l y below the centre of the tube 100 mm above the d i s t r i bu to r 163 5-47 Upward and downward magnitudes of the force pulse vs . bubble_equivalent diameter. Bed mater ia l : sand; d = 430 ym; bed depth = 0.3 m T 166 5-48 Upward and downward magnitudes of the force pulse vs . horizontal distance between the axis of the tes t cy l inder and the point of in ject ion of s ingle bubbles 167 5- 49 Pulse duration vs . bubble equivalent diameter. Bed mater ia l : sand; d = 430 ym; bed depth = 0.3 m 7 168 6- 1 Array of f i ve tubes, each of outer diameter 32 mm, in an equ i la te ra l - t r iangu lar p i t ch . The instrumented tube is at the centre of the array. A l l dimensions are in mm .. . . 175 6-2-a Vert ica l forces at opposite ends of a 32 mm tube at the centre of an array of f i ve tubes, recorded at (U-U m f) = 0.8 m/s 177 6-2-b Power spectral estimates of the tota l ve r t i ca l forces on the tube at the centre of the f ive tube array. (U-Umf) = 0.8 m/s 177 6-3-a Vert i ca l forces at opposite ends of a s ingle 32 mm tube in i s o l a t i on , recorded at (U-U m f) = 0.8 m/s 178 6-3-b Power spectral estimates o f tota l ve r t i ca l forces on the iso lated tube 178 6-4-a Vert ica l forces on a 32 mm tube at the centre of an array of f ive tubes, recorded at (U-U m f) = 0.2 m/s 180 6-4-b Power spectral estimates of to ta l ve r t i ca l forces on the tube at the centre of the array 180 - x v i i i -Figure 6-5-a 6-5-b 6-6 6-7 6-8 6-9-a 6-9-b 6-10 6-11 6-12 6-13 6-14 Page V e r t i c a l forces on a s i n g l e 32 mm diameter tube i n i s o l a t i o n , recorded at ( U - U m f ) = 0.3 m/s . . 181 Power s p e c t r a l estimates o f t o t a l v e r t i c a l forces for the tube in i s o l a t i o n 181 V a r i a t i o n o f the major frequency with excess s u p e r f i c i a l gas v e l o c i t y , ( U - U m f ) , for two d i f f e r e n t tube arrangements ( s i n g l e tube i n i s o l a t i o n and tube at the centre o f an a r r a y o f f i v e tubes) 182 RMS v e r t i c a l force v s . excess s u p e r f i c i a l gas v e l o c i t y , ( U - U m f ) 183 Standard d e v i a t i o n o f v e r t i c a l force v s . excess s u p e r f i c i a l gas v e l o c i t y , ( U - U m f ) 185 H o r i z o n t a l forces at opposite ends o f a 32 mm diameter tube at the centre o f an a r r a y o f f i v e t u b e s , recorded at ( U - U m f ) = 0.8 m/s m ! 186 Power s p e c t r a l estimates o f t o t a l h o r i -zontal forces 186 H o r i z o n t a l forces at opposite ends o f a s i n g l e 32 mm diameter tube i n i s o l a t i o n , recorded at ( U - U m f ) = 0.8 m/s 187 RMS and mean values o f h o r i z o n t a l force v s . excess s u p e r f i c i a l gas v e l o c i t y , (U-U m f ) 189 Array o f three tubes, each o f outer diameter 32 mm, in an e q u i l a t e r a l -t r i a n g u l a r p i t c h ; instrumented tube i n downstream o f the other tubes 191 Array o f three tubes, each o f outer diameter 32 mm, in an e q u i l a t e r a l -t r i a n g u l a r p i t c h ; instrumented tube i s upstream o f the two other tubes 191 V e r t i c a l forces at opposide ends o f the instrumented tube o f the tube-a r r a y shown i n F i g . 6-12, recorded at (U-Umf) = 0.8 m/s 192 x i x _ F igure Page 6-15 RMS v e r t i c a l f o rce v s . excess s u p e r f i c i a l gas v e l o c i t y ( U - U m f ) , f o r two d i f f e r e n t tube arrangements 194 6-16 V e r t i c a l f o r ces a t oppos i te ends o f the instrumented tube o f the tube-ar ray shown i n F i g . 6-13, recorded at (U -U m f ) = 0.8 m/s 195 6-17 V e r t i c a l f o rce v s . excess s u p e r f i c i a l gas v e l o c i t y ( U - U m f ) , f o r two d i f f e r e n t tube arrangements 197 6-18 RMS v e r t i c a l fo rce v s . excess s u p e r f i c i a l gas v e l o c i t y ( U - U m f ) , f o r d i f f e r e n t tube c on f i gu r a t i on s 198 XX -L I S T OF T A B L E S Table Page 3.1 Principal properties of the particles used in the experiments 38 3.2 Natural frequency of the tube-gage assemblies in air and in bed at minimum fluidization 51 3.3 Operating conditions for measurements of vertical and horizontal force components for isolated tubes 57 4.1 Sample of properties of data determined from different segments, i l lustrat ing stationarity of the data 71 5.1 RMS vertical forces (in Newtons) measured with the three sizes of tube. H 0 = 0.3 m; material: 430 ym sand 139 5.2 Ratio between RMS values of the vertical force at different gas velocities to that for 32 mm dia . tube. Conditions are the same as in Table 5.1 "141 - 1 -CHAPTER 1 NATURE OF T H I S STUDY 1 .1 General I n t r o d u c t i o n Heat t r a n s f e r i n f l u i d i z e d beds cont inues to be a t o p i c o f great t e c h n i c a l i n t e r e s t . Tubes are commonly employed i n f l u i d i z e d beds f o r heat t r a n s f e r purposes. In p a r t i c u l a r , some o f the more r e c e n t l y proposed a p p l i c a t i o n s o f f l u i d i z e d beds, such as i n - b e d combust ion, as w e l l as many t r a d i t i o n a l f l u i d bed p r o c e s s e s , i n v o l v e heat t r a n s f e r to immersed tubes and tube bundles . These tubes are most often h o r i z o n t a l , a l though v e r t i c a l tubes are a l s o q u i t e common. Most tubes are c i r c u l a r i n c r o s s - s e c t i o n , but e x t e r n a l l y f inned tubes are i n c r e a s i n g l y under i n v e s t i g a t i o n . Tubes and o t h e r o b j e c t s immersed i n f l u i d i z e d beds are subjected to b u f f e t t i n g f o r c e s due to s o l i d s and gas m o t i o n , e s p e c i a l l y those a s s o c i a t e d w i t h b u b b l i n g . Coupled w i t h e r o s i o n due to motion o f a b r a s i v e p a r t i c l e s over the tube s u r f a c e , these forces can l e a d to tube f a i l u r e . In order to permit safe and a p p r o p r i a t e design o f tubes and t h e i r s u p p o r t s , i t i s important to c h a r a c t e r i z e these forces and to i n v e s t i g a t e t h e i r causes . - 2 -Forces exer ted on a tube immersed i n a f l u i d i z e d bed r e s u l t main ly from a combinat ion o f : 1) F l u c t ua t i n g pressure fo rces caused by bubbles as they pass the tube. 2) Va r i ab l e momentum forces caused by the impact of s o l i d s , a s soc i a t ed w i th bubble mot ion, aga ins t the tube. 3) Bed buoyancy fo rce on the tube. I f one cou ld p r ed i c t the frequency composi t ion of the f o r c e s , the tubes may be designed in such a way that natura l and f o r c i ng frequencies are s a f e l y a pa r t . In format ion on the magnitude o f the forces i s a l so needed in order to p r ed i c t the f a t i gue l i f e o f components subjected to v a r i a b l e l o ad s . The general p rope r t i e s o f the forces i n t h e i r r e l a t i o n to hydrodynamics o f f l u i d i z e d beds must a l so be i n v e s t i g a t ed t o a s s i s t our understanding o f the nature and o r i g i n o f the f o r c e s . There has been l i t t l e work i n t h i s area u n t i l t h i s t ime. In t h i s i n v e s t i g a t i o n , the experiments were performed i n a column of c r o s s - s e c t i o n 215 mm x 200 mm and o f he ight 1.5 m. Hor i zon ta l c y l i n d r i c a l tubes o f outer d iameter 15, 25 and 32 mm, and one e x t e r n a l l y f inned tube , were mounted in i s o l a t i o n across the bed and supported on i d e n t i c a l f o rce t r ansduce r s , one at each end. Both the v e r t i c a l and ho r i z on t a l instantaneous fo rce components were measured. Force measurements were a l so made on tubes w i t h i n tube-ar rays o f d i f f e r e n t c o n f i g u r a t i o n s . The f l u i d - b ed opera t ing parameters va r i ed in the experiments were s u p e r f i c i a l gas v e l o c i t y , s t a t i c bed he i gh t , p a r t i c l e s i z e and d en s i t y . In some o f these exper iments , l o c a l pressure v a r i a t i o n s a t the su r face o f the t e s t tubes were measured s imu l taneous ly - 3 -with the forces. In addition, single bubble experiments were carried out in order to confirm the relation between force characteristics and bubble properties and to have insight into the mechanism by which bubbles induce forces. Measured forces are presented in this thesis by showing represen-tative samples of time-varying force records. Because the forces are nondeterministic time series, the data are also described in terms of their statistical parameters such as RMS forces and power spectra. Some representative mean values of the forces, standard deviations, probability density functions and autocovariance functions are also presented. 1 .2 Objectives of this work The fundamental objectives of the present work are: 1. To measure and characterize the forces exerted on single horizontal tubes in isolation and on tubes within arrays in a gas fluidized bed. 2. To correlate these forces with the bed operating parameters (superficial gas velocity, bed depth, particle size and particle density) and to determine how these forces depend on other factors such as tube size, tube shape (finned vs. unfinned), tube-array configuration and tube position within an array. 3. To investigate the causes of the forces and to relate these forces to hydrodynamic conditions, including bubble properties, prevailing within the bed. - 4 -4. To make recommendations regard ing the forces to guide des igners o f f l u i d i z e d bed equipment in the design of tube systems. 1 .3 Prev ious s tud ies and the present study In t h i s t h e s i s we are concerned wi th the i n t e r a c t i o n between a f l u i d i z e d bed and immersed tubes . While there has been a cons iderab le amount o f work on heat t r a n s f e r between f l u i d i z e d beds and immersed tubes and a number o f papers de s c r i b i ng the i n f l uences o f tubes on bubbles and o ther bed p r o p e r t i e s , there have been s u r p r i s i n g l y few s tud ies on the i nve r se problem, i . e . on the i n f l u ences o f the bed on the tube. Some o f the re l evan t hydrodynamic s tud ies showing the e f f e c t s o f tubes on the bed behavior are reviewed in Chapter 2. In t h i s chapter we cons ide r on ly those s tud i e s where fo rces on immersed ob jec t s have been measured. L i v s h i t s et a l . (1978) conducted fo rce measurements on a spher i ca l ob s t a c l e immersed i n gas f l u i d i z e d beds o f d iameter 0.3 and 0.7 m. Almost no d e t a i l was given on the method o f mounting the obs tac l e and fo rce t r ansduce r , an important cons i de ra t i on s ince f l u i d i z a t i o n c ond i t i o n s can be e a s i l y d i s t u r b e d . The fo r ces were found to i nc rease w i th he ight above the gas d i s t r i b u t o r and s u p e r f i c i a l gas v e l o c i t y . The fo rce maxima were assumed to occur when wakes o f bubbles passed the o b s t a c l e . Throughout the i n v e s t i g a t i o n the bed buoyance fo rce was not cons i de red . Nguyen and Grace (1978) est imated the net pressure fo rces on tubes - 5 -o f outer d iameter 6.3 and 25 mm immersed in an i n c i p i e n t l y two-dimensional f l u i d i z e d bed, by i n t e g r a t i n g the instantaneous pressure p r o f i l e s measured around the tube caused by passage o f s i n g l e bubbles. They found the tube encounters an up- thrus t as the bubble approaches the tube and then a downward fo r ce "as the bubble r i s e s away from the tube. The magnitude o f the net fo rces was found to depend on the s i z e o f the tube and on the r e l a t i v e p o s i t i o n o f the bubbles as they pass the tube. Separate measurements by the same workers showed tha t the buoyancy f o r ce on immersed ob jec t s could be p red i c ted from the immersed volume m u l t i p l i e d by the bed dens i t y under minimum f l u i d i z a t i o n c o n d i t i o n s . Kennedy et a l . (1981) measured fo r ces on tubes o f va r i ous lengths immersed in f l u i d i z e d beds o f c r o s s - s e c t i o n 0.3 x 0.3 m, 0.91 x 0.91 m and 2.4 x 0.3 m. C i r c u l a r tubes o f d iameter 50 mm were used in the exper iments . The l eng th o f the tube i n a l l measurements was s l i g h t l y l e s s than the bed width in which the tube was f i x e d . Force measurements were conducted by suppor t ing each end o f the t e s t tube w i th s t r a i n gage load c e l l s . A l though the system was designed to measure the v e r t i c a l and ho r i z on t a l components of fo rce s imu l taneous ly and to minimize c r o s s -t a l k between the two components, i t i s imposs ib le to e l im i na t e t h i s c r o s s - t a l k a l t o ge t he r (see repor t o f EPRI CS-1542, 1980) , and t h i s may have inc reased the e r r o r i n the measurements. The parameters va r i ed in the exper iments , as ide from tube l e ng t h , were s u p e r f i c i a l gas v e l o c i t y and tube a r r ay he ight above the gas d i s t r i b u t o r . Th is study prov ides use fu l i n fo rmat i on i n des ign o f i n t e r n a l tubes f o r f l u i d i z e d beds. However, i n a dd i t i o n to being l i m i t e d to a s i n g l e tube d iameter , the measurements were obta ined f o r on ly one p a r t i c l e type ( l a rge p a r t i c l e s o f sand o f d iameter 0.8 mm) and high s u p e r f i c i a l - 6 -a i r v e l o c i t i e s (> 1.5 m/s) . The present work i s a comprehensive study to measure and c ha r a c t e r i z e the fo rces a c t i n g on tubes o f d i f f e r e n t s i z e and shape, i n i s o l a t i o n and w i th a r r a y s , over a wide range o f f l u i d - bed ope ra t i ng parameters. The parameters v a r i e d in the experiments were s u p e r f i c i a l gas v e l o c i t y , s t a t i c bed he i gh t , p a r t i c l e dens i t y and t ype , i n a dd i t i o n to tube s i z e and shape, tube p o s i t i o n w i t h i n an a r ray and tube-ar ray c o n f i g u r a t i o n . Most o f these parameters have been va r i ed over a wide range in order to cover the ranges i n which most f l u i d i z e d beds operate . The fo rce t rends and p r ope r t i e s are d i scussed in r e l a t i o n to the hydrodynamics o f bubb l ing f l u i d i z e d bed. Exper imental set-up i n the present work was designed in a way to overcome the de fec ts a s soc i a t ed w i th the fo rce measurements in the prev ious s t u d i e s . The fo rce t ransducers were l oca ted outs ide the f l u i d i z a t i o n column, con t ra ry to the set-up o f Kennedy et a l . (1981). Transducers on the i n s i d e have d isadvantages as d i scussed i n Chapter 3. A l s o , t ransducers have been designed i n the present work to measure the v e r t i c a l and ho r i z on t a l fo rce components wi thout c r o s s - t a l k . 1 .4 Ou t l i ne o f t h i s t h e s i s Th is t r e a t i s e i s d i v i d ed i n to seven chap te r s . The hydrodynamic cons i de ra t i ons necessary to enable us to understand the nature and o r i g i n o f the f o r ces under study are cons idered i n Chapter 2. Chapter 3 desc r ibes the exper imenta l set-up used i n t h i s work as we l l as the techniques o f ob ta i n i ng fo rce and pressure measurements. Chapter 4 desc r ibes methods o f data process ing and ana l y s i s and i l l u s t r a t e s some important p r ope r t i e s e x h i b i t e d by the measured f o r c e s . Resu l ts o f - 7 -single tube experiments are presented and discussed in Chapter 5, while corresponding results for tubes within arrays are treated in Chapter 6. A general discussion and conclusions appear in Chapter 7. - 8 -CHAPTER 2 H Y D R O D Y N A M I C C O N S I D E R A T I O N S 2.1 I n t r o d u c t i o n I n o r d e r t o u n d e r s t a n d t h e n a t u r e a n d o r i g i n o f e x t e r n a l f o r c e s on a n o b j e c t , i t i s n e c e s s a r y t o h a v e i n s i g h t i n t o t h e s u r r o u n d i n g s o f t h a t o b j e c t . T u b e s i m m e r s e d i n g a s f l u i d i z e d b e d s u s u a l l y e n c o u n t e r v i g o r o u s l y b u b b l i n g c o n d i t i o n s . H e n c e , i t i s n e c e s s a r y t o d e s c r i b e b r i e f l y t h e f u n d a m e n t a l h y d r o d y n a m i c b e h a v i o r o f b u b b l i n g g a s f l u i d i z e d b e d s . T h i s i s d i s c u s s e d i n S e c t i o n 2 . 2 . I m m e r s i o n o f an o b j e c t w i t h i n a bed c o n s i d e r a b l y i n f l u e n c e s t h e f l o w p a t t e r n a n d b u b b l e b e h a v i o r n e a r t h e i m m e r s e d s u r f a c e . T h i s i s d i s c u s s e d , f o r t h e c a s e o f h o r i z o n t a l l y i m m e r s e d t u b e s , i n S e c t i o n 2 . 3 . 2 . 2 H y d r o d y n a m i c B e h a v i o r o f Gas F l u i d i z e d B e d s M o s t g a s f l u i d i z e d b e d s o p e r a t e i n t h e b u b b l i n g r e g i m e . W i t h an i n c r e a s e i n g a s f l o w r a t e b e y o n d t h a t r e q u i r e d f o r m i n i m u m f l u i d i z a t i o n , g a s p o c k e t s o r v o i d s f o r m a t o r n e a r t h e g a s d i s t r i b u t o r a n d g r o w i n s i z e , m o s t l y by c o a l e s c e n c e , a s t h e y r i s e . T h e s e g a s p o c k e t s a r e c a l l e d b u b b l e s b e c a u s e o f t h e a n a l o g i e s b e t w e e n t h e m a n d l a r g e b u b b l e s i n r e a l l i q u i d s . B u b b l e s i n g a s f l u i d i z e d b e d s a r e v e r y i m p o r t a n t f o r - 9 -t h e y a r e r e s p o n s i b l e f o r most o f t h e f e a t u r e s t h a t d i f f e r e n t i a t e a packed f rom a f l u i d i z e d b e d . For e x a m p l e , t h e y a f f e c t g a s - s o l i d c o n t a c t i n g and gas f l o w t h r o u g h t he sy s t em and cause t he p a r t i c l e movement wh i ch i s r e s p o n s i b l e f o r r a p i d heat t r a n s f e r and s o l i d s m i x i n g . In t h i s s e c t i o n t h e c h a r a c t e r i s t i c s o f s i n g l e bubb l e s r i s i n g i n f l u i d i z e d beds i s t r e a t e d f i r s t . The g r o s s b e h a v i o r o f f r e e l y b u b b l i n g beds i n c l u d i n g the r e l a t i o n s h i p s between key v a r i a b l e s i s t r e a t e d s e c o n d . 2 .2 .1 C h a r a c t e r i s t i c s o f s i n g l e bubb l e s r i s i n g i n f l u i d i z e d beds 2 . 2 . 1 . 1 Gene ra l n a t u r e o f bubb l e s The r e a son f o r t h e f o r m a t i o n o f b ubb l e s i n gas f l u i d i z e d beds i s not e n t i r e l y u n d e r s t o o d , but t h e on s e t o f b u b b l i n g appea r s t o be a s s o -c i a t e d w i t h i n s t a b i l i t y i n t h e u n i f o r m s t a t e o f f l u i d i z a t i o n ( G r a c e , 1 9 8 2 ) . A number o f wo r k e r s have a t t emp t ed t o a p p l y the p r i n c i p l e s o f c on t i nuum mechan i c s o v e r zones l a r g e compared w i t h p a r t i c l e s p a c i n g bu t s m a l l compared w i t h t o t a l s y s t e m . The q u a n t i t a t i v e c o n c l u s i o n s v a r y f rom s t u d y t o s t u d y , but q u a l i t a t i v e c o n c l u s i o n s a r e c o n s i s t e n t show ing t h a t t h e f l u i d i z e d sys tem i s a lways u n s t a b l e t o s m a l l p e r t u r -ba t i on s o f v o i d a g e . The o n s e t o f b ubb l e f o r m a t i o n depends on t h e r a t e o f d i s t u r b a n c e g rowth w i t h i n t h e bed , and t h i s v a r i e s f rom one sys tem t o a n o t h e r , d epend i ng on t h e p h y s i c a l c h a r a c t e r i s t i c s o f t h e s y s t e m . S i n g l e bubb l e s r i s i n g i n f l u i d i z e d beds a r e commonly r e p r e s e n t e d as s p h e r i c a l l y - c a p p e d v o i d s w i t h concave i n d e n t a t i o n s a t t h e i r bases ( D a v i d s o n and H a r r i s o n , 1 9 6 3 ) , a l t h o u g h the t r u e shape i s o f t e n c l o s e r t o an e l l i p s o i d a l cap ( C l i f t e t a l . , 1 9 7 8 ) . The m o t i o n o f a v o i d i n a gas f l u i d i z e d bed i s a na l o gou s t o t h e mo t i o n o f a gas bubb l e t h r o u g h - 10 -a rea l l i q u i d o f v i s c o s i t y 2-10 po i s e . There are however, s i g n i f i c a n t d i f f e r en ce s between bubbles i n rea l f l u i d s and . in f l u i d i z e d beds . (Davidson and Ha r r i s on , 1963). Surface tens ion i s pronounced at the i n t e r f a c e between gases and t rue l i q u i d s , whereas the sur face tens ion i s gene ra l l y taken to be zero at the i n t e r f a c e between a gas bubble and the surrounding dense phase o f a f l u i d i z e d bed. Moreover, the boundary o f bubbles in f l u i d i z e d beds i s permeable l ead ing to s ub s t an t i a l f low through the su r f a c e . The v e l o c i t y o f r i s e o f an i s o l a t e d bubble i n a f l u i d i z e d bed has been i n ve s t i g a t ed by a number of workers . Davidson et a l . (1959) found expe r imen ta l l y that the r i s e v e l o c i t y of a s i n g l e bubble i n a f l u i d i z e d bed i s r e l a t e d to the bubble d iameter i n a s i m i l a r manner to tha t g iven by Davies and Tay lo r equat ion f o r the r i s e o f gas bubbles in a t rue l i q u i d , i . e . U b 0 = k ( g r b P (2.1) where r^  i s the rad ius o f curvature of the l ead ing edge o f the bubble; k i s a cons tan t . The express ion i s o f ten w r i t t e n (Rowe, 1971): U 5 0 = k ' (g d t , ) 5 * (2.2) where d^ i s the diameter o f sphere having the same volume as the bubble; k' has a va lue o f 0.711 f o r s phe r i c a l - c ap bubbles in a t rue low v i s c o s i t y l i q u i d , wh i l e the exper imental va lue o f k' f o r bubbles i n f l u i d i z e d beds was found to vary between 0.57 and 0.85 ( K u n i i , 1969). Rowe and Pa r t r i dge (1965), and Romero et a l . (1965) found tha t the constant - 11 -k 1 depends on the s i z e and type o f p a r t i c l e s . In beds o f f i n e p a r t i c l e s , bubbles r i s e f a s t e r . Masson and Jo t t rand (1978) have r e cen t l y presented an exper imenta l i n v e s t i g a t i o n o f bubble p rope r t i e s i n a f l u i d i z e d bed i n which they proposed a l i n e a r r e l a t i o n between the v e l o c i t y and the s i z e o f the bubb les . However, the equat ion U b 0 = 0.711 (g d b ) 2 i s gene ra l l y accepted and w i l l be used here. The pa t te rn o f the s o l i d s motion w i th respect to the r i s i n g bubble i s , to a f i r s t approx imat ion , analogous to po t en t i a l f low past a sphere (Reuter , 1966). The p a r t i c l e s around the bubble con t i nuous l y stream around the s i des o f the bubble. Behind the bubble there i s a wake reg ion i n which p a r t i c l e s are c a r r i e d upward at the bubble v e l o c i t y (Rowe and P a r t r i d g e , 1965). The wake f r a c t i o n , de f ined as the r a t i o between the wake volume and the bubble volume, i s about 0.1-0.4 depending on s o l i d p r o p e r t i e s . Small and rounded p a r t i c l e s g ive la rger wakes than coarse or angular p a r t i c l e s . In add i t i o n to the t r anspo r t o f s o l i d s i n the bubble wake, a r i s i n g bubble causes d i s p l a c e -ment o f s o l i d s i n a d r i f t p r o f i l e which resembles d r i f t caused by the motion o f a sphere through an i n v i s c i d f l u i d (Rowe, 1971). Gas motion i n the v i c i n i t y o f a r i s i n g bubble i n f l u i d i z e d beds d i f f e r from that around gas bubbles i n rea l l i q u i d s . In a f l u i d i z e d bed, there i s an interchange o f gas between bubbles and cont inuous phase i n f l u i d i z e d beds through the permeable bubble boundary. Gas enters a bubble at i t s base and leaves through the r oo f . The f low pa t t e rn near and w i t h i n a s i n g l e r i s i n g bubble depends on the r e l a t i v e bubble v e l o c i t y , u"b, and on the remote i n t e n s t i t i a l emulsion phase v e l o c i t y , u f = U mf / e mf ' W h e n t h e r a t l ° ( u b emf^ Umf^ > 1 ' a g a s " c l o u c l " o c cu r s . - 12 -The best-known t h e o r e t i c a l models o f gas motion i n and around a r i s i n g bubble i n a f l u i d i z e d bed are due to Davidson (1961), Jackson (1963) and Murray (1965). De t a i l s o f these t h e o r e t i c a l treatments o f gas motion are prov ided in r e f e r ences : Jackson (1971), Kun i i and Levenspie l (1969), and Grace (1982). 2 .2 .1 .2 Bubble pressure f i e l d and a s soc i a t ed pressure f l u c t u a t i o n s In f l u i d i z e d beds, pressure f l u c t u a t i o n s caused by r i s i n g bubbles are l i k e l y to have a s t rong i n f l uence on the f o r c i ng func t i on caus ing tube v i b r a t i o n s . This b r i e f review on the bubble pressure f i e l d g ives i n s i g h t i n t o the c h a r a c t e r i s t i c s and the causes o f the forces under i n v e s t i g a t i o n . In a p a r t i c u l a t e f l u i d i z e d bed, the pressure decreases l i n e a r l y w i th he i gh t , g i v i n g a h yd r o s t a t i c pressure f i e l d as in a t rue l i q u i d . Each bubble i s a vo i d whose r e s i s t an ce to the f low o f f l u i d i z i n g f l u i d i s a lmost zero compared w i th the r e s i s t an ce o f the p a r t i c u l a t e ( s o l i d -gas) phase around the bubble. Thus, the pressure g rad ien t i n the bed i s d i s t u rbed by the presence o f bubbles. Reuter (1966) measured the pressure d i s t r i b u t i o n around a r i s i n g bubble in a three-d imens iona l f l u i d i z e d bed by means o f a s e n s i t i v e pressure t r ansduce r . F igure 2-1 shows the pressure d i s t r i b u t i o n along the ax i s o f symmetry of a r i s i n g bubble. The pressure w i t h i n the bubble i s approx imate ly cons tan t . At the top of the bubble the pressure was found to be h igher than pressure in the und is turbed f l u i d i z e d bed at the same l e v e l . Converse ly , i n the lower pa r t and behind the bubble the pressure i s lower than i n the und is turbed f l u i d i z e d bed at the same l e v e l . As one moves f u r t he r - 13 -0-3 ox\ 0.1 - 0 3 ; -0.2 - 0 . 3 \ symrnefry ' v . Pressure } \ c e n t e r level. \ \ -100 -loo lOOmmofu/fltcr pressure p F i g . 2-1. P r e s s u r e d i s t r i b u t i o n a l o n g t h e a x i s o f symmetry o f t h e b u b b l e , r e f e r e n c e ( R e u t e r , 1966) - 14 -away from the bubble, the pressure va r i a t ion caused by the bubble was found to progress ive ly d isappear . Theoret ica l pred ic t ions o f the loca l pressure changes due to the presence o f an i so l a t ed bubble in f l u i d i z e d beds are summarized by Stewart (1968), for two and three-dimensional bubbles. His summary i s based on the theore t i ca l p red ic t ions o f Davidson (1961), Jackson (1963) and Murray (1965). Figure 2-2 shows the dimensionless pressure f l uc tua t ions caused by a three-dimensional bubble as predicted by the theore t i ca l ana lyses , and compared with experimental data o f Reuter (1963). Davidson's theo re t i ca l p red i c t ion appears to give the best representat ion of pressure. The pressure change caused by the presence of a three-dimensional bubble can therefore be predicted by the equation (Grace, 1982): 3 g rb P - P 0 = P S (1 - e m f ) — - - cos e (r >_ r 5 ) (2.3) where r^  i s the radius o f the bubble, r and 6 are polar coordinates measured from the instantaneous bubble cent ro id and the v e r t i c a l axis o f the bubble, p Q is the mean hydrostat ic pressure at the measuring leve l in the absence o f bubbles. Littman and Homolka (1970) measured the pressure f i e l d along the v e r t i c a l axis o f symmetry o f a two-dimensional bubble. The pressure-time curve recorded during the r i s e of a s ing le bubble in jec ted into an i n c i p i e n t l y f l u i d i z e d bed, is shown in Figure 2-3. The curve has two maxima (points 1 and 2) and two minima (points 3 and 4 ) . The f i r s t maximum corresponds to the time when the bubble was in jec ted into the bed; the second maximum occurred when the roof of the bubble reached - 15 -Curve 1: Davidson Curve2: Jackson Vb = 1o,/(ga) Curve 3: Jickso-n Ub - Zft^ga) Curve 4: Murray Curve + : Reuters ZxperrmaTits F i g . 2 -2 . D i m e n s i o n l e s s p r e s s u r e d i f f e r e n c e between o o i n t s on the v e r t i c a l bubb le a x i s and p o i n t s i n the same h o r i z o n t a l p l a w e l l removed from the bubb 1 e — t h r e e - d i mens i ona 1 b u b b l e , r e f . ( S t e w a r t , 1968) T 1 1 r 1 —r 3 -I 1 1 1 I 1 L t 2 0 -Z -•* -6 -8 DlSTONce OF BUBBLE FROM THE PRESSURE PROBE IN BUBBLE RflDlI F i g . 2 -3 . P r e s su r e - T ime cu rve r e c o r ded d u r i n g the r i s e o f a s i n g l e i n j e c t e d bubb le i n t o i n c i p i e n t l y f l u i d i z e d b e d , L i t t m a n and Homolka, r e f e r e n c e ( L i t t m a n & Homolka, 1970) - 16 -the pressure t ransducer t a p . The f i r s t maximum was sa i d to correspond to the i n s t an t when the rear o f the bubble reached the pressure t r an s -ducer t ap ; wh i l e the second minimum occurred when the bubble began to d i s t o r t the upper su r face o f the bed. Nguyen and Grace (1978) c a r r i e d out experiments to measure the l o c a l pressure v a r i a t i o n s caused by the r i s e o f bubbles past ho r i z on ta l c y l i n d r i c a l tubes i n a two-dimensional f l u i d i z e d bed. Widely separated bubbles having the same s i z e were i n j e c t e d i n t o the column. The l o c a l pressure around the sur face o f the tube was then recorded at d i f f e r e n t angles us ing a s e n s i t i v e pressure t r ansduce r . Tests were conducted f o r three d i f f e r e n t tube s i z e s . F igure 2-4 shows pressure- t ime t races f o r a 0.63 cm diameter at four d i f f e r e n t angu lar p o s i t i o n s . The f i r s t maximum was observed when the bubble was i n j e c t ed i n t o the bed, but the second maximum occurred when the f r on t o f the bubble a r r i v e d at the measuring po in t and the minimum when the back o f the bubble a r r i v e d a t the measuring p o i n t . F igure 2-5 shows a comparison of the p ressure-time t r a c e , measured at 6 = 0 on the sur face o f tha t tube , w i th the t h e o r e t i c a l pressure p r o f i l e s caused by a two-dimensional bubble and w i th the exper imenta l data o f L i t tman (1973). In summary, an immersed ob jec t i n a f l u i d i z e d bed i s sub jec t to a f l u c t u a t i n g pressure fo rce due to the pressure g rad ien t o f the bubbles r i s i n g i n the bed. The fo r ce i s i n an upward d i r e c t i o n wh i l e the bubble i s approaching the ob jec t i n the lower po r t i on of the bed, and i n downward d i r e c t i o n wh i l e the bubble i s t r a v e l l i n g away from the ob jec t i n upper po r t i on of the bed. In a f r e e l y bubbl ing bed, where many bubbles are present a t one t ime and where bubble coa lescence and s p l i t t i n g take p lace n a t u r a l l y and con t i nuous l y , the s i t u a t i o n w i l l - 17 -F i g . 2 -4 . P r e s su r e - T ime t r a c e s f o r 2.5 cm d i am. tube a t f o u r d i f f e r e n t a n g u l a r p o s i t i o n s . Time 0 c o r r e sponds to a r r i v a l o f the f r o n t o f the bubb le a t the measu r i ng p o i n t , r e f e r e n c e (Nguyen & G ra ce , 1 978 ) /I J1 , />/ / — — ' / • 5 1 •'"TX^ O i T n e n s i O M l e t s fressur^p^^go .10 D a v i d s o n Theory \ C o l t t n s \ l \ ^ ^ N j U Y c n a n e f Srace(o.63CmTube,9-0) - U i — B u b b l e r\ r \ o i m c n s i o n t e s s oista-nctpt/a ? \\ U ^ — ' 0 - E ' _ ^ i _ J - - = : ^ _ _ 3 L — \ / o L i t t - m o n a n t i H o T T i o V f c a 0 o U o E n p e r i n i e - n t s -1.0 V o ° F i g . 2 -5 . Compar ison o f P r e s su r e - T ime t r a c e s f o r 0.63 cm d i am. tube and 8=0 w i t h t h e o r e t i c a l p r e s s u r e p r o f i l e s and w i t h exp . v a l u e s measured by L i t t m a n and Homolka, r e f e r e n c e (Nguyen & G ra ce , 1978) - 18 -be more comp l i ca ted . The fo rces w i l l c l e a r l y depend on the s i z e , frequency and s p a t i a l d i s t r i b u t i o n of the bubb les , and these w i l l i n turn be i n f l uenced by ope ra t i ng and equipment v a r i a b l e s , i n c l u d i ng the presence o f o ther f i x ed tubes or o b s t a c l e s . 2.2.2 Gross behavior o f f r e e l y bubbl ing beds In an aggregat ive f l u i d i z e d bed there are regions o f very low s o l i d s dens i t y de f ined as bubble phase and reg ions o f h igher s o l i d dens i t y c a l l e d emulsion or p a r t i c u l a t e phase. As a f i r s t approx imat ion , a l l gas i n excess o f tha t needed to j u s t f l u i d i z e the bed passes through the bed as bubbles, wh i l e the emulsion phase remains at minimum f l u i d i z i n g c ond i t i on s w i th a voidage f r a c t i o n o f and i n t e r s t i t i a l v e l o c i t y U^/e^. This pos tu l a te i s known as the "two-phase theory o f f l u i d i z a t i o n . " Many s tud i e s have been conducted i n o rd ina ry bubb l ing beds to f i n d the ra te o f bubbles growth and coa lescence , and the bubble s i z e and f requency. The average bubble s i z e i s found to inc rease r a p i d l y w i th he ight and w i th an inc rease in gas f low r a t e , main ly as a r e s u l t o f coa lescence . A number o f c o r r e l a t i o n s have been proposed to est imate the mean bubble s i z e at a c e r t a i n l e ve l i n a f l u i d i z e d bed. The best known pub l i shed c o r r e l a t i o n s are those of Darton et a l . (1977), Rowe (1976) , Werther (1976) , Ge ldar t (1972) , Mori and Wen (1975), Yasui and Johanson (1958), Kato and Wen (1969), and Bar-Cohen et a l . (1980). In a l l o f these c o r r e l a t i o n s , the average bubble diameter i s a f unc t i on of the gas f low ra te and the height above the gas d i s t r i b u t o r . In some o f them, the e f f e c t o f bed s c a l e , c h a r a c t e r i s t i c s o f the gas - 19 -d i s t r i b u t o r , and t h e powder p r o p e r t i e s a r e a l s o c o n s i d e r e d . The c o r r e l a t i o n o f Rowe i s a s e m i - e m p i r i c a l c o r r e l a t i o n : 311* d b = (U - U m / 2 (h + h Q ) /Vs (2.4) where (U - U m f ) i s t h e e x c e s s s u p e r f i c i a l v e l o c i t y , h i s the h e i g h t above t he d i s t r i b u t o r and h Q i s a c o n s t a n t c h a r a c t e r i z i n g the gas d i s t r i b u t o r based on t h e d a t a o f Whitehead ( 1 9 6 7 ) , Rowe found t h a t h Q i n c r e a s e s l i n e a r l y w i t h t o t a l bed h e i g h t . Whitehead (197 9 ) , p o i n t e d o u t t h a t t h e i n c r e a s e i n bed depth and gas f l o w r a t e enhances the speed o f bubble c o a l e s c e n c e i n the immediate v i c i n i t y o f the d i s t r i b u t o r due t o t h e development o f " g u l f - s t r e a m i n g " , and t h i s i s what causes the dependence between t o t a l bed depth and bubble s i z e a t a p a r t i c u l a r l e v e l . The c o r r e l a t i o n o f Darton e t a l . (1977) i s an a n a l y t i c a l c o r r e l a t i o n based on l a t e r a l bubble c o a l e s c e n c e . The proposed e q u a t i o n i s d b = 0.54(U - U m f ) ° - 4 (h + 4>/A o-)°- 8/g°- 2 (2.5) where A i s t h e catchment a r e a a t the d i s t r i b u t o r , i . e . the are a o f o d i s t r i b u t o r p l a t e per o r i f i c e . The above e q u a t i o n a g r e e s w e l l w i t h most l i t e r a t u r e d a t a , p r o v i d i n g t h a t n e i t h e r a maximum s t a b l e bubble s i z e nor s l u g g i n g i s a c h i e v e d . Bar-Cohen e t a l . (1980) m o d i f i e d the c o n s t a n t s i n the s e m i - e m p i r i c a l c o r r e l a t i o n o f Darton e t a l . t o o f f e r b e t t e r agreement w i t h the da t a o f a number o f r e c e n t w o r k e r s . The m o d i f i e d e q u a t i o n i s : - 20 -d b = 0.45(U - U m f ) 0 - 4 (h + 4.63v^)°-8/g0-2 (2.6) S e v e r a l f u n c t i o n s have been proposed by a number o f workers t o d e s c r i b e e m p i r i c a l l y t h e d i s t r i b u t i o n o f bubble s i z e . Functions i n c l u d e t h e normal d i s t r i b u t i o n (Park e t a l . , 1969), Gamma d i s t r i b u t i o n (Rowe and Yacono, 1975), and log-normal d i s t r i b u t i o n ( W e r t h e r , 1974). However, t h e d i s t r i b u t i o n o f b u b b l e s i z e i s d i f f e r e n t a t eac h l e v e l , and t h e r e i s no g e n e r a l means o f p r e d i c t i n g the d i s t r i b u t i o n parameters o t h e r t h a n t h e mean. The f r e q u e n c y o f bubbles p a s s i n g a g i v e n l e v e l i n a gas f l u i d i z e d bed has been d e t e r m i n e d i n a number o f s t u d i e s . K u n i i and L e v e n s p i e l (1969), i n summarizing the e x p e r i m e n t a l e v i d e n c e , s u g g e s t e d t h a t the f r e q u e n c y o f b u b b l e s p a s s i n g a p o i n t 10-15 cm o r h i g h e r above the d i s t r i b u t o r i s a p p r o x i m a t e l y the same no m a t t e r what the gas f l o w r a t e . C o n s e q u e n t l y , t h e p r i m a r y e f f e c t o f gas f l o w r a t e i s t o change t h e s i z e o f b u b b l e s p a s s i n g a p o i n t , not t h e f r e q u e n c y . C o n t r a d i c t o r y to t h e above c o n c l u s i o n , Haines e t a l . (1 972) found t h a t f r e q u e n c y o f b u b b l e s a t a g i v e n l e v e l depends on the gas f l o w r a t e , e s p e c i a l l y when the bed i s composed o f f i n e m a t e r i a l . G e l d a r t (1970) c a r r i e d out e x p e r i m e n t a l work t o measure t h e s i z e and f r e q u e n c y o f bubbles i n a f l u i d i z e d bed. The r e s u l t o f t h i s i n v e s t i g a t i o n i n d i c a t e d t h a t t h e p o i n t bubble f r e q u e n c y i s r e l a t i v e l y i n s e n s i t i v e t o i n c r e a s e s i n gas f l o w r a t e , i n agreement w i t h the f i n d i n g noted above, w h i l e t h e b u b b l e f r e q u e n c y d e c r e a s e s s t e a d i l y w i t h i n c r e a s i n g bed h e i g h t . The mean bub b l e f r e q u e n c y a t any l e v e l can be e s t i m a t e d ( G r a c e , 1982) - 21 -from: V /A b (ir/6) d b 3 where d b i s the a v e r a g e bubble s i z e a t t h a t l e v e l i n the bed and V^/A i s t he gas f l o w c a r r i e d as b u b b l e s . A c c o r d i n g t o t h e two-phase t h e o r y o f f l u i d i z a t i o n V b/A i s equal t o (U - U f ) . The f r e q u e n c y o f gas b u b b l e s has been found a l s o t o i n c r e a s e and t h e i r d i m e n s i o n s t o d e c r e a s e as the p a r t i c l e s i z e d e c r e a s e s ( G e l p e r i n e t a l . , 1968). Werther (1977) used m i n i a t u r i z e d c a p a c i t a n c e probes t o d e t e r m i n e t h e f r e q u e n c y and s i z e o f b u b b l e s a t d i f f e r e n t p o s i t i o n s i n s i d e t h r e e - d i m e n s i o n a l f l u i d i z e d beds, t h e b u b b l e f r e q u e n c y was found t o e x h i b i t superimposed random and p e r i o d i c c h a r a c t e r i s t i c s . S i t n a i e t a l . (1981) d e m o n s t r a t e d s i m i l a r b ubble b e h a v i o r i n a l a r g e - s c a l e f l u i d i z e d bed u s i n g d i f f e r e n t i a l p r e s s u r e probes t o d e t e c t the p r e s s u r e f i e l d o f r i s i n g b u b b l e s . S t a t i s t i c a l a n a l y s i s o f the d a t a i n d i c a t e d t h a t t h e b u b b l i n g p a t t e r n i n a f l u i d i z e d bed i s s t a t i o n a r y w i t h a marked p e r i o d i c i t y s u perimposed on random b u b b l i n g . The s p a t i a l d i s t r i b u t i o n s o f b u b b l e s i n f l u i d i z e d beds have been i n v e s t i g a t e d by a number o f w o r k e r s . Among t h o s e i n v e s t i g a t o r s are Grace and H a r r i s o n (1968) and Werther ( 1 9 7 3 ) . Both i n v e s t i g a t i o n s showed a c h a r a c t e r i s t i c s b ubble f l o w p r o f i l e d evelopment, F i g u r e 2-6. Near t h e gas d i s t r i b u t o r more b u b b l e s were found near the bed w a l l than a t t h e bed c e n t r e . The zone o f i n c r e a s e d bubble o c c u r r e n c e moves towards t h e v e s s e l c e n t r e - l i n e w i t h i n c r e a s i n g h e i g h t above t h e d i s t r i b u t o r . The m e r g i n g o f t h e a n n u l a r zone r e p r e s e n t s t h e s t a r t o f the t r a n s i t i o n o f t h e bed t o t h e s l u g g i n g r e g i m e . T h i s development a r i s e s n a t u r a l l y - 22 -F i g . 2 - 6 . B u b b l e s p a t i a l - d i s t r i b u t i o n a c r o s s t h e b e d f o r d i f f e r e n t h e i g h t s h ( c o p p e r p o w d e r f l u i d i z e d i n t h e 0.2m d i a . b e d , U=0.084 m/s ) , r e f e r e n c e ( W e r t h e r , 1 9 7 3 ) - 23 -from bubble c o a l e s c e n c e , even f o r a u n i f o r m s p a t i a l d i s t r i b u t i o n a t the gas d i s t r i b u t o r . The a b s o l u t e r i s e v e l o c i t y o f bubbles a t a p a r t i c u l a r h e i g h t i n f r e e l y b u b b l i n g beds can be e s t i m a t e d (Davidson and H a r r i s o n , 1963) from U b = U b 0 + < u " u m f > ( 2' 8 ) where i s the v e l o c i t y o f t h e bubble i n i s o l a t i o n , e s t i m a t e d from e q u a t i o n ( 2 . 2 ) . The above e q u a t i o n i m p l i e s t h a t t h e mean bubble v e l o c i t y i n c r e a s e s w i t h d i s t a n c e from the d i s t r i b u t o r and w i t h the gas v e l o c i t y . T h i s e q u a t i o n has been w i d e l y u s e d , a l t h o u g h i t i s s u b j e c t t o t h e o r e t i c a l o b j e c t i o n s ( G r a c e , 1982). P a r t i c l e movement and s o l i d c i r c u l a t i o n i n gas f l u i d i z e d beds a r e caused p r i m a r i l y by the motion and d i s t u r b a n c e o f gas bubbles p a s s i n g t h r o u g h t h e bed. R i s i n g b u b b l e s cause t r a n s p o r t o f s o l i d s by two mechanisms, f i r s t by c a r r y i n g s o l i d s i n t h e i r wakes and second by drawing up t h e s o l i d s i n a d r i f t p r o f i l e b e h i n d t h e b u b b l e . O u t s i d e b u b b l e paths s o l i d s move downward i n the bed t o r e p l a c e the s o l i d s b r ought t o t h e s u r f a c e . The no n u n i f o r m s p a t i a l d i s t r i b u t i o n o f the r i s i n g b ubbles tends t o enhance p a r t i c l e movement and t o e s t a b l i s h s o l i d c i r c u l a t i o n p a t t e r n s d e p e n d i n g on t h e bed d e p t h . The p r e v i o u s d i s c u s s i o n a p p l i e s when the e f f e c t o f the v e s s e l d i m e n s i o n s i s n e g l i g i b l e . I f t h e s i z e o f t h e r i s i n g bubble approaches the bed d i a m e t e r , t h e bubble tends to e l o n g a t e , t he r i s e v e l o c i t y i s r e t a r d e d by t h e c o n t a i n i n g w a l l , and t h e bubble wake i s s m a l l e r . When d h i s about D/2 o r l a r g e r t h e s l u g f l o w regime has been a c h i e v e d . For - 24 -d i s c u s s i o n o f t h i s regime see Hovmand and Davidson (1971). 2.3 Hydrodynamic E f f e c t o f H o r i z o n t a l l y Immersed Tubes i n a Gas  F l u i d i z e d Bed T h i s s e c t i o n f i r s t t r e a t s the f l o w p a t t e r n i n t h e v i c i n i t y o f s i n g l e h o r i z o n t a l tubes and a r r a y s o f h o r i z o n t a l c y l i n d r i c a l t u b es immersed i n two- and t h r e e - d i m e n s i o n a l f l u i d i z e d beds. The i n t e r a c t i o n between t h e immersed t u b e s and r i s i n g b u b b l e s i s a l s o d i s c u s s e d . 2.3.1 Flow p a t t e r n i n t h e v i c i n i t y o f immersed tubes A number o f i n v e s t i g a t o r s have u t i l i z e d p h o t o g r a p h i c t e c h n i q u e s t o s t u d y t h e p a r t i c l e and f l u i d b e h a v i o r near immersed tubes i n two-and t h r e e - d i m e n s i o n a l f l u i d i z e d beds. The r e s u l t o f t h e s t u d i e s i n a t w o - d i m e n s i o n a l column ( G l a s s and H a r r i s o n , 1964; Ginoux e t a l . , 1974; Loew e t a l . , 1979; Fakhimi and H a r r i s o n , 1980; Hager e t a l . , 1976) can be summarized as f o l l o w s : (a) U < U m f The p r e s e n c e o f a h o r i z o n t a l t u b e i n a gas f l u i d i z e d bed a l t e r s the f l o w f i e l d o f the gas and i n i t i a t e s b u b b l i n g a t f l o w r a t e s lower than t h a t r e q u i r e d f o r i n c i p i e n t f l u i d i z a t i o n . When t h e s u p e r f i c i a l v e l o c i t y U i s w e l l below U ,^ l o c a l f l u i d i z a t i o n around t he o b s t a c l e i s o b s e r v e d , Ginoux e t a l . ( 1 9 7 4 ) , as shown i n F i g u r e 2-7. F i r s t a zone o f i n f l u e n c e i s formed on each s i d e . Small b u b b l e s a r e formed and move a l o n g the o b s t a c l e s u r f a c e . They then d i s a p p e a r a t the upper boundary o f t h e f l u i d i z e d r e g i o n . As U i s i n c r e a s e d , t he two - 25 -F i g . 2 - 7 . D e v e l o p m e n t o f l o c a l f 1 u i d i z a t i o n a r o u n d a h o r i z o n t a l t u b e , r e f e r e n c e ( G i n o u x e t a l . , 1 9 7 4 ) - 26 -zones expand and merge a t t h e upstream p a r t o f the t u b e . Bubbles d e t a c h from the tube and move upwards. Above the t u b e , t h e r e remains a dead zone w i t h no p a r t i c l e movements. (b) u > u m f When U i s equal o r j u s t above U m ^ the f l o w regime i s s i m i l a r t o t h a t shown i n F i g u r e 2-7-d. The b u b b l e s formed a t the s i d e s o f t e n a l t e r n a t e , one a t one s i d e , then a n o t h e r a t the o p p o s i t e s i d e , c a u s i n g p e r i o d i c sweeping o f t h e p a r t i c l e s a t the s i d e s o f t h e t u b e s . For a i r f l o w r a t e s up t o 2-3 t i m e s U ^ , t h r e e d i s t i n c t f l o w regimes a r e o b s e r v e d i n t h e n e i g h b o r h o o d o f the t u b e , G l a s s and H a r r i s o n (1964). Below the t u b e a t the upstream f a c e t h e r e i s a t h i n gas f i l m ; above the tube t h e r e e x i s t s a d e f l u i d i z e d cap whose e x t e n t and permanence depend on t h e f l u i d i z i n g v e l o c i t y ; t h e t h i r d r e g i o n , a t e i t h e r s i d e o f t h e t u b e , i s c h a r a c t e r i z e d by gas from the f i l m under the tube f o r m i n g i n t o i r r e g u l a r c h a i n s o f b u b b l e s . In g e n e r a l , the s o l i d p a r t i c l e s c i r c u l a t e upwards near the s i d e s o f t h e tube and downwards f u r t h e r away. However, t h e c i r c u l a t i o n i s e r a t i c and d i s c o n t i n u o u s because o f bubble f o r m a t i o n . In a t h r e e - d i m e n s i o n a l bed, Rooney and H a r r i s o n (1976) i n v e s t i -g a t e d t h e f l o w p a t t e r n i n the v i c i n i t y o f a s i n g l e h o r i z o n t a l l y immersed t u b e u s i n g a c a p a c i t a n c e probe and p h o t o g r a p h i c t e c h n i q u e . The d e f l u i d i z e d cap and upstream gas f i l m were n e i t h e r as l a r g e nor as permanent as t h o s e o b s e r v e d i n t w o - d i m e n s i o n a l beds. At s u p e r f i c i a l gas v e l o c i t i e s g r e a t e r than U ^ , bubbles o c c a s i o n a l l y swept a c r o s s t h e t o p o f t h e tube d i s l o d g i n g the d e f l u i d i z e d c a p . Replacement o f t h i s cap became more f r e q u e n t a t h i g h e r gas v e l o c i t i e s . Rooney and H a r r i s o n (1976) a l s o c a r r i e d out e x p e r i m e n t s w i t h an a r r a y - 27 -o f seven t u b e s . The main d i f f e r e n c e s between the s i n g l e tube and the a r r a y was a t low f l u i d i z i n g v e l o c i t i e s ( i n t h e range o f f l u i d i z i n g a i r f l o w - r a t e s up t o 4 U m f ) . The tube a r r a y appeared t o d e c r e a s e the s t a b i l i t y o f the d e f l u i d i z e d caps and t o r e d u c e p a r t i c l e c o n t a c t w i t h the t h e s i d e s o f the t u b e . At h i g h gas v e l o c i t i e s , t h e p r e s e n c e o f t u b e a r r a y s can cause l o c a l changes from one f l u i d i z a t i o n regime t o a n o t h e r because o f i n c r e a s e d gas v e l o c i t i e s i n r e g i o n s o f reduced c r o s s - s e c t i o n a l a r e a . Staub and Canada (1978) have o b s e r v e d r e g i o n s o f t u r b u l e n t f l u i d i z a t i o n c o e x i s t i n g w i t h b u b b l i n g r e g i o n s w i t h i n the tube banks. At h i g h e r gas v e l o c i t i e s , f u r t h e r m o r e , the f l o w regime i n a t u b e bank i s p r e d o m i n a n t l y o f a q u a s i - s t e a d y s t a t e c o u n t e r - c u r r e n t f l o w n a t u r e w i t h a l a r g e r number o f s m a l l l o c a l c i r c u l a t i n g e d d i e s than a t lower v e l o c i t i e s . 2.3.2 I n t e r a c t i o n o f immersed t u b e s w i t h r i s i n g b ubbles In b u b b l i n g f l u i d i z e d beds, t h e r e i s mutual i n t e r a c t i o n between r i s i n g b u b b l e s and immersed t u b e s . The r i s i n g b u b b l e s impose v a r i a b l e p r e s s u r e f o r c e s on the immersed t u b e , as d i s c u s s e d i n S e c t i o n 2.2.1.2. In a d d i t i o n t o the p r e s s u r e f o r c e , the immersed tubes are s u b j e c t t o dynamic f o r c e s as a r e s u l t o f momentum t r a n s f e r r e d from s o l i d p a r t i c l e s a s s o c i a t e d w i t h the r i s i n g b u b b l e . F r i c t i o n a l d r a g due t o gas and p a r t i c l e s p a s s i n g the tube w i l l a l s o be p r e s e n t t o some e x t e n t . At t h e same t i m e as tubes a r e a f f e c t e d by p a s s i n g b u b b l e s , t h e tubes i n f l u e n c e the g e n e r a l b e h a v i o r o f the r i s i n g b u b b l e s . When a bubble a r r i v e s i n the v i c i n i t y o f a tube t h e i r f l o w - f i e l d s i n t e r a c t and the b u b b l e may d e v i a t e from i t s v e r t i c a l p a t h ; when i t makes c o n t a c t w i t h the tube t h e bubble may s p l i t up i n t o two p a r t s (seldom more than two) - 28 -depending on t h e r e l a t i v e s i z e o f the bubble and tube and d i s p l a c e m e n t between the a x i s o f t h e tube and t r a j e c t o r y o f the bubble c e n t r o i d . Ginoux e t a l . (1974) s t u d i e d t h e i n t e r a c t i o n o f s i n g l e i n j e c t e d b u b b l e s w i t h a h o r i z o n t a l l y immersed c y l i n d e r i n a t w o - d i m e n s i o n a l column u s i n g a p h o t o g r a p h i c t e c h n i q u e and a s e n s i t i v e p r e s s u r e t r a n s d u c e r . S i n g l e b u b b l e s o f d i f f e r e n t s i z e s were i n j e c t e d i n t o i n c i p i e n t l y f l u i d i z e d beds. Bubbles o f d i f f e r e n t s i z e s were found t o behave i n d i f f e r e n t manners. The s m a l l e r ones s p l i t up i n t o two p a r t s o f a l m o s t equal s i z e . These two d a u g h t e r b u b b l e s e v e n t u a l l y r e c o a l e s c e d above t h e c y l i n d e r . L a r g e r b u b b l e s , on the o t h e r hand, tended t o pass around one s i d e o r t h e o t h e r w i t h o u t s p l i t t i n g . Measurements o f p r e s s u r e v a r i a t i o n s a t a p o i n t on the c y l i n d e r i n d i c a t e d t h a t l o c a l f l u i d i z a t i o n a t the s u r f a c e was always p r e s e n t e x c e p t when t h e bubble was i n c o n t a c t w i t h the c y l i n d e r , F i g u r e 2-8. Hager and Thomson(1973) used X-rays t o s t u d y bubble b e h a v i o r around i n c l i n e d and h o r i z o n t a l t u b e s i n a t h r e e - d i m e n s i o n a l f l u i d i z e d bed. A h o r i z o n t a l tube was o b s e r v e d t o d i s r u p t bubble growth by d i v i d i n g r i s i n g b u b b l e s i n t o a l a r g e r number o f s m a l l d i a m e t e r b u b b l e s , but r e c o m b i n a t i o n ( c o a l e s c e n c e ) u s u a l l y o c c u r s above the t u b e . Thus, the number o f the r i s i n g b ubbles a t a s h o r t d i s t a n c e above t h e tube was found t o be t h e same o r l e s s than below the t u b e . Hager and Thomson a l s o c a r r i e d o u t a n o t h e r s e r i e s o f e x p e r i m e n t s u s i n g d i r e c t o b s e r v a t i o n o f bubble b e h a v i o r around t h e o b s t a c l e s i n a t w o - d i m e n s i o n a l bed. B e h a v i o r was q u a l i t a t i v e l y s i m i l a r t o the t h r e e - d i m e n s i o n a l column. A p p r o x i m a t e l y h a l f t h e b u b b l e s i m p i n g i n g upon t h e h o r i z o n t a l tube d i v i d e d i n t o p a i r s o f s m a l l e r b u b b l e s as t h e y e n v e l o p e d t h e t u b e ; the p a i r s r a p i d l y c o a l e s c e d above t h e t u b e . The r e m a i n d e r o f the bubbles - 29 -locaL fluidisotion F i g . 2 - 8 . S u r f a c e p r e s s u r e v a r i a t i o n a t a p o i n t on t h e c y l i n d e r d u r i n g t r a n s i t o f an i n j e c t e d b u b b l e i n t o 2-d i m e n s i o n a l f l u i d i z e d b e d , r e f e r e n c e ( G i n o u x e t a l . , 1 9 7 4 ) T I M E ( S E C S ) F i g . 2 - 9 . T y p i c a l v a r i a t i o n i n s u r f a c e p a r t i c a l v e l o c i t y w i t h t i m e f o r t h e c e n t r a l t u b e s , r e f e r e n c e ( P e e l e r & W h i t e h e a d , 1 9 8 2 ) - 30 -passed u n d i s t u r b e d . G l a s s (1967), and Nguyen (1976) o b s e r v e d s i m i l a r b u b b l e b e h a v i o r around h o r i z o n t a l o b s t a n c l e s i n t w o - d i m e n s i o n a l columns. The e f f e c t o f h o r i z o n t a l tube b u n d l e s on the p r o p e r t i e s o f r i s i n g b u b b l e s ( b u b b l e s i z e and d i s t r i b u t i o n ) has been s t u d i e d by a number o f workers ( G l a s s , 1 967; Loew e t a l . , 1 979; Newby and K e a i r n s , 1978; Nguyen e t a l . , 1979). From t h e s e i n v e s t i g a t i o n s i t i s a p p a r e n t t h a t h o r i z o n t a l tube b u n d l e s i n d u c e s p l i t t i n g o f r i s i n g b ubbles c a u s i n g a marked r e d u c t i o n i n bubble s i z e and a more u n i f o r m bubble s i z e d i s t r i -b u t i o n and a much more u n i f o r m s p a t i a l d i s t r i b u t i o n a c r o s s the bed c r o s s - s e c t i o n . The a r r a y i s most e f f e c t i v e i f t h e t u b e - b u n d l e i s compact, i . e . w i t h a tube p i t c h ( c e n t r e - t o - c e n t r e d i s t a n c e ) o f 2-2.5 tube d i a m e t e r s . However, i n o r d e r t o a l l o w s o l i d s t o c i r c u l a t e f r e e l y , the gap between the tubes s h o u l d be 20-30 p a r t i c l e diameters o r more ( G r a c e , 1982) . S o l i d s m otion a t h o r i z o n t a l tube s u r f a c e s i n d u c e d by r i s i n g b ubbles i n a gas f l u i d i z e d bed was i n v e s t i g a t e d by P e e l e r and Whitehead (1982). They used a p h o t o g r a p h i c t e c h n i q u e t o s t u d y the motion o f f l u i d i z e d p a r t i c l e s ( s i l i c a s and, d p = 700 ym) a t t h e s u r f a c e o f 38 mm d i a m e t e r h o r i z o n t a l t u bes immersed i n a 1 .2 m square column. The s u p e r f i c i a l gas v e l o c i t y a t which t h i s i n v e s t i g a t i o n was c a r r i e d o u t was 0.9 m/s. The s t u d y showed t h a t s o l i d motion a t the s u r f a c e o f a h o r i z o n t a l tube i s dependent on t u b e l o c a t i o n . Tubes l o c a t e d i n t h e main body o f t h e bed e x p e r i e n c e d a c y c l i c b e h a v i o r p a t t e r n as t h e y e n c o u n t e r e d each r i s i n g b u b b l e . T h i s c y c l i c b e h a v i o r i s e v i d e n t from the v a r i a t i o n o f s o l i d c o n c e n t r a t i o n around the t u b e and t h e v a r i a t i o n i n t h e s u r f a c e p a r t i c l e v e l o c i t y as t h e b u b b l e p a s s e s t h e t u b e . F i g u r e 2-9 shows the change d u r i n g t r a n s i t o f a r i s i n g bubble p a s t t h a t t u b e . The - 31 -surface p a r t i c l e velocity can be divided into 4 d i s t i n c t stages: I. Dense phase movement preceding the bubble a r r i v a l . I I . Bubble a r r i v a l at the tube. During this stage particles move in an upward d i r e c t i o n , and later in a random motion with a velocity much higher than the bubble rise velocity. I I I . Bubble envelopes the tube. IV. Bubble leaves the tube and the dense phase reappears at the surface. As discussed in Chapter 5 this behavior i s very important in understandi the dynamic force induced by s o l i d particles associated with the motion of a r i s i n g bubble past a horizontal tube. Tubes adjacent to the vessel w a l l , experienced a " s t i c k - s l i p " solids descent pattern. Peeler and Whitehead also observed that the defluidized cap on the central tube existed only intermittently and was always swept away during bubble t r a n s i t . - 32 -C H A P T E R 3 E X P E R I M E N T A L EQUIPMENT AND PROCEDURE 3.1 General Requirements The p r i n c i p a l o b j e c t i v e o f t h i s work was to measure the v e r t i c a l and h o r i z o n t a l t i m e - v a r y i n g f o r c e s on i n d i v i d u a l tubes i n i s o l a t i o n and w i t h i n an a r r a y i n v i g o r o u s l y f l u i d i z e d beds. C o r r e l a t i n g t h e s e f o r c e s w i t h the hydrodynamic c o n d i t i o n s p r e v a i l i n g w i t h i n the bed was a n o t h e r major o b j e c t i v e o f t h i s t h e s i s . In o r d e r t o u n d e r s t a n d t he n a t u r e and o r i g i n o f t h e s e f o r c e s , p r e s s u r e f l u c t u a t i o n s around t he immersed tubes have a l s o been measured. Moreover, s i n g l e bubble e x p e r i m e n t s have been c a r r i e d out t o have i n s i g h t i n t o the mechanism by which i n d i v i d u a l b u b b l e s i n d u c e f o r c e s on the immersed o b s t a c l e s , and t o i n v e s t i g a t e t h e e f f e c t o f bubble s i z e and p o s i t i o n on the i n d u c e d f o r c e s . The r e q u i r e m e n t s have been met by mounting a h o r i z o n t a l t e s t c y l i n d e r a c r o s s t h e f l u i d i z i n g bed. Force t r a n s d u c e r s have been used to measure the f l u c t u a t i n g f o r c e components. A p r e s s u r e t a p p i n g on th e c y l i n d e r p e r i p h e r y c o n n e c t e d t o a d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r , has been used t o o b t a i n f l u c t u a t i n g p r e s s u r e measurements. In s i n g l e bubble e x p e r i m e n t s , i n d i v i d u a l bubbles o f d i f f e r e n t s i z e were i n j e c t e d - 33 -into an i n c i p i en t l y f l u i d i zed bed, and induced force and pressure signals were recorded. The general set-up of the experimental apparatus i s shown in Figure 3-1. A detai led descr ipt ion of the test f a c i l i t i e s , instrumen-ta t i on , and data co l l ec t ion system is given below. 3.2 The F lu id i za t ion Column and Aux i l i a ry Equipment The f l u id i zed bed i s contained in a three-dimensional column, 215 mm x 200 mm in cross-sect ion and 1500 mm high. This s ize of column allows f l e x i b i l i t y for i n s t a l l a t i on of a su f f i c i en t number of tubes (3 rows with 1-2 tubes in each row) while the avai lable blower capacity was su f f i c i en t to give a reasonable maximum bed super f i c ia l ve loc i ty (1.7 m/s). Figure 3-2 shows the primary features of the f l u i d i z a t i on column and i t s aux i l i a r i e s including i n l e t and out let ducts and the cycl one. Figure 3-3 shows a ve r t i ca l section through the column, arranged for experiments with a s ingle test cy l i nder . The test cy l inder i s centered 300 mm above the gas d i s t r i bu to r . The f l u i d i z i ng column could be modified so that the test cy l inder was at the centre of an array of ident ica l tubes, arranged in a t r iangular pitch configuration as described in Section 3.3.2 below. The bed was constructed ent i re ly from mild steel with a thickness of 5 mm to achieve wall r i g i d i t y . The a i r d i s t r i bu to r was a mu l t i -o r i f i c e p la te , 5 mm thick with o r i f i c e s of diameter 3 mm and a spacing between o r i f i c e s of 15 mm. The plate was covered by a steel wire screen (150 Tyler mesh) to prevent so l i d par t i c les from dropping through the F i g . 3 - 1 . G e n e r a l s e t - u p o f t h e e x p e r i m e n t a l a p p a r a t u s . ( 1 ) a i r b l o w e r , ( 2 ) s i l e n c e r , ( 3 ) f l o w m e a s u r e m e n t o r i f i c e , ( 4 ) m a n o m e t e r , ( 5 ) f l u i d i z a t i o n c o l u m n , ( 6 ) c y c l o n e , ( 7 ) s i l e n c e r . - 35 -F i g . 3 - 2. P r i m a r y F e a t u r e s o f t h e F l u i d i z a t i o n C o l u m n a n d i t s A u x i l i a r i e s ( d i m e n s i o n s a r e i n mm) . ( 1 ) c y c l o n e , ( 2 ) t u b e p a n e l , ( 3 ) t e s t c y l i n d e r p o s i t i o n , ( 4 ) f l e x i b l e j o i n t , ( 5 ) t o p g l a s s w i n d o w , ( 6 ) r i g h t - s i d e g l a s s w i n d o w . - 36 1 F i g . 3-3. V e r t i c a l section through the f l u i d i z a t i o n column ( a l l dimensions are i n mm) . (1) top glass window, (2) test cylinder, (3) tube panel (4) gas d i s t r i b u t o r , (5) plenum. - 37 -ho l e s . Three g lass windows were used to a l l ow v iewing and f i l m i n g of the bed behav iour . Seven pressure taps cou ld be used to measure pressure drops across d i f f e r e n t he ight i n t e r v a l s above the d i s t r i b u t o r . The bed was connected to the a i r - s u p p l y l i n e through a f l e x i b l e bel lows j o i n t to minimize t r ansm i t t ed v i b r a t i o n s from upstream. F l u i d i z i n g a i r was provided by a SUTORBILT b lower, Model 7 HV, 3 wi th a maximum capac i t y o f 5 m /min . at 67 KPa. The a i r f low ra te was monitored wi th an o r i f i c e meter designed and const ruc ted f o r t h i s purpose. S o l i d p a r t i c l e s could be d ischarged through a 20 mm diameter hole d r i l l e d j u s t above the d i s t r i b u t o r . 3.3 S o l i d P a r t i c l e s Three grades o f Ottawa sand, one o f alundum and one of po lyethy lene powder were used f o r the exper iments . A l l were of narrow s i z e d i s t r i -bu t i ons , obta ined by s i e v i n g . P rope r t i e s o f the p a r t i c l e s are given in Table 3.1. The l i s t e d values o f dp (the sur face- to-vo lume mean) are obta ined from dp = 1/z(x^/dp^), where x^  i s the weight f r a c t i o n c o l l e c t e d between s i eves o f mean aper ture dp^. The U ^ values were obta ined from the standard pressure drop versus s u p e r f i c i a l v e l o c i t y p l o t method. The va lues of p n and e m f were found expe r imen t a l l y . - 38 -T a b l e 3.1 P r i n c i p l e p r o p e r t i e s o f t h e p a r t i c l e s used i n the e x p e r i m e n t s Type Shape d p P g U m f (um) (kg/m 3) (m/s) Ottawa sand n e a r l y s p h e r i c a l 430 2600 0, .15 0. .41 Ottawa sand n e a r l y s p h e r i c a l 280 2600 0. .078 0. .42 Ottawa sand n e a r l y s p h e r i c a l 185 2600 0, .039 0. .44 Alundum i r r e g u l a r 295 4100 0, .14 0. .52 P o l y e t h y l e n e i r r e g u l a r 280 920 0. .029 0. .49 - 39 -3.4 F o r c e Measurement 3.4.1 F o r c e t r a n s d u c e r Two i d e n t i c a l f o r c e t r a n s d u c e r s , one a t each end o f t h e t e s t c y l i n d e r , were used t o measure the i n s t a n t a n e o u s i n - l i n e ( v e r t i c a l ) and t r a n s v e r s e ( h o r i z o n t a l ) f o r c e s . The b a s i c t r a n s d u c e r was purchased from BLH E l e c t r o n i c s I n c . under the t r a d e name o f Type METRIC-LBP1 , c a t a l o g u e No. 403. The i m p o r t a n t d i m e n s i o n s o f the t r a n s d u c e r are shown i n F i g u r e 3-4. Each gage had a c a p a c i t y o f 10 Kg f o r c e (98 N), w i t h an o v e r l o a d c a p a c i t y o f 200%. The d e f l e c t i o n o f the gage f o r h i g h e s t c a p a c i t y was l e s s than 0.25 mm. S t r a i n gages (SR-4) were bonded t o a h i g h - s t r e n g t h s t e e l e l e m e n t . These s t r a i n gages were e l e c t r i c a l l y c o n n e c t e d to form a b a l a n c e d Wheatstone b r i d g e . A d d i t i o n a l compensation r e s i s t o r s were added t o t h e c i r c u i t t o m a i n t a i n t h e a c c u r a c y o f t h e b r i d g e o v e r a wide range o f t e m p e r a t u r e , F i g u r e 3-5. D e f l e c t i o n o f t h e l o a d element changed the s t r a i n gage r e s i s t a n c e , t h e r e b y u n b a l a n c i n g the b r i d g e . As a r e s u l t , f o r a g i v e n i n p u t v o l t a g e , the o u t p u t v o l t a g e o f the b r i d g e v a r i e d w i t h the l o a d ( d e f l e c t i o n ) . The gages were a l i g n e d so t h a t t h e y measured e i t h e r o n l y the i n - l i n e f o r c e ( a c t i n g i n the v e r t i c a l d i r e c t i o n ) o r t h e t r a n s v e r s e f o r c e ( a c t i n g i n t h e h o r i z o n t a l d i r e c t i o n at r i g h t a n g l e s t o the c y l i n d e r a x i s ) . A s p e c i a l h o u s i n g was b u i l t f o r each gage so t h a t i t c o u l d be mounted on the o u t s i d e bed w a l l and r o t a t e d t o measure the v e r t i c a l f o r c e o r the h o r i z o n t a l f o r c e , d e p e n d i n g on the p o s i t i o n i n which i t was s e t , w i t h o u t c r o s s - t a l k between the two components. - 40 -g. 3-4. G e n e r a l d i m e n s i o n s o f t h e f o r c e t r a n s d u c e r . - 41 -F i g . 3-5. B a l a n c e d W h e a t s t o n e b r i d g e and a d d i t i o n a l c o m p e n s a t i o n r e s i s t o r s . - 42 -Each gage was s e p a r a t e l y c a l i b r a t e d , as shown l a t e r . Repeated c a i i b r a t i o n s have shown t h a t : (a) The gages were l i n e a r up t o the maximum a p p l i e d f o r c e ; t y p i c a l c a l i b r a t i o n v a l u e s a r e shown i n S e c t i o n 3.4.4. (b) The gages y i e l d e d the same s i g n a l f o r l o a d s a p p l i e d upward o r downward. 3.4.2 T e s t c y l i n d e r s E x p e r i m e n t s were c a r r i e d out w i t h c y l i n d r i c a l tubes o f t h r e e d i f f e r e n t o u t s i d e d i a m e t e r s , 32 mm, 25 mm and 15 mm. In a d d i t i o n , one e x t e r n a l l y f i n n e d tube was a l s o u s e d . The d i m e n s i o n s o f the c r o s s - s e c t i o n o f each tube a r e shown i n F i g u r e 3-6. The f i n n e d tube had a maximum d i a m e t e r ( t o the o u t s i d e o f t h e f i n s ) o f 32 mm and a minimum e x t e r n a l d i a m e t e r o f 15 mm, a l l o w i n g ready comparison w i t h the bare t u b e s . The tubes were chosen t o meet the a s s u m p t i o n o f tube r i g i d i t y , and t o have n a t u r a l f r e q u e n c i e s high enough t o e n s u r e the absence o f r e s o n a n c e e f f e c t s , see Appendix ( A ) . The tubes were made from m i l d s t e e l and were d e s i g n e d t o have r e s o n a n t f r e q u e n c i e s g r e a t e r than 500 HZ. The i n s i d e o f the tube i n each case c o n t a i n e d a i r . A l l tubes can be c o n s i d e r e d smooth r e l a t i v e t o the s i z e o f the p a r t i c l e s used. F o r c e measurements were c a r r i e d out w i t h each o f the f o u r tubes i n i s o l a t i o n and w i t h the 32 mm bare tube mounted a t the c e n t r e o f an a r r a y o f 5 t u b e s , i n an e q u i l a t e r a l p i t c h w i t h c e n t r e - t o - c e n t r e tube s p a c i n g o f 64 mm. The c o n f i g u r a t i o n f o r the tube a r r a y i s shown i n F i g u r e 3-7. - 43 -a D=32 mm d = l 9 mm C D = l 5 mm d=8 mm F i g . 3 - 6 . D i m e n s i o n s o f o f t h e t e s t c y l i n d e r s . b D D=25 mm d = l 5 mm d D D=3 2 mm d = l 5 mm c r o s s - s e c t i o n a l a r e a - 44 -*. 1 J F i g . 3 - 7 . S i d e P l a t e o f Co lumn f o r Tube A r r a y A s s e m b l y , ( a l l d i m e n s i o n s i n mm ) . ( 1 ) i n s t r u m e n t e d t u b e , ( 2 ) u n i n s t r u m e n t e d c y l i n d e r s , ( 3 ) h o u s i n g o f t h e f o r c e t r a n s d u c e r , . (4) h o l e i n t h e bed w a l l . - 45 -F o r c e s were measured w i t h t h e t e s t c y l i n d e r mounted a c r o s s the bed, and clamped r i g i d l y a t each end t o one o f t h e gages. The a x i s o f the i n s t r u m e n t e d tube was 300 mm above the gas d i s t r i b u t o r i n each c a s e . The tube-gage assembly i s shown i n F i g u r e 3-8 f o r the case where the two gages were s e t t o measure the v e r t i c a l components o f f o r c e . The e n t i r e assembly i s r o t a t e d t h r o u g h 90° t o measure t h e h o r i z o n t a l components. A f o r c e on t h e tube i s t r a n s m i t t e d t o the l o a d beam ( a t each end o f the t u b e ) , which a c t s l i k e a c a n t i l e v e r beam s u p p o r t i n g the t u b e . The s t r a i n caused by the d e f l e c t i o n o f the l o a d beam i s measured . From F i g u r e 3-8, the e f f e c t i v e l e n g t h o f the t e s t c y l i n d e r i n s i d e t h e bed i s 200 mm. The t u b e - t o - c o l u m n w a l l j u n c t i o n was s e a l e d w i t h a n n u l a r f l e x i b l e t h i n p i e c e s o f r u b b e r a t both s i d e t o p r e v e n t escape o f p a r t i c l e s and a i r . The two r u b b e r s e a l s had a n e g l i g i b l e r e s i s t a n c e , i n a l l d i r e c t i o n s , up t o the maximum measured f o r c e . T h i s was shown by a p p l y i n g s t a t i c f o r c e s on the t e s t tube w i t h and w i t h o u t the r u b b e r s e a l s i n p l a c e and w i t h the column empty o f p a r t i c l e s . I t was found t h a t t h e r u b b e r s e a l s had no m e a s u r a b l e e f f e c t on the r e c o r d e d f o r c e s up t o the maximum a p p l i e d f o r c e . The f l e x i b l e s e a l s a l l o w e d t h e f o r c e t r a n s d u c e r s t o be l o c a t e d o u t s i d e t h e f l u i d i z a t i o n column. T h i s i s c o n t r a r y t o the s e t - u p o f Kennedy e t a l . (1981) who put t h e i r t r a n s d u c e r s on the i n s i d e . T r a n s d u c e r s on the i n s i d e have t h e f o l l o w i n g d i s a d v a n t a g e s : (a) I n t e r f e r e n c e w i t h the f l u i d i z e d bed. (b) F a i l u r e t o measure t h e f o r c e s o v e r the e n t i r e tube l e n g t h . T h e r e a r e s u b s t a n t i a l r a d i a l o r t r a n s v e r s e g r a d i e n t s i n f l u i d i z e d beds (Grace & H a r r i s o n , 1968; Werther, 1973; F i g . 3 - 8 . A H o r i z o n t a l C r o s s - S e c t i o n a t L e v e l 3 0 0 mm A b o v e t h e Gas D i s t r i b u t o r S h o w i n g t h e T u b e - G a g e A s s e m b l y , w i t h t h e G a g e s S e t t o M e a s u r e t h e V e r t i c a l F o r e es . ( 1 ) f o r c e t r a n s d u c e r , ( 2 ) h o u s i n g o f t h e f o r c e t r a n s d u c e r , ( 3 ) t e s t c y l i n d e r , ( 4 ) t h i n f l e x i b l e r u b b e r w i t h n e g l i g i b l e t e n s i o n r e s i s t a n c e . - 47 -P e e l e r & Whitehead 1982),so t h a t the regimes a d j a c e n t t o t h e w a l l s a r e not w e l l r e p r e s e n t e d by what happens i n t h e c o r e o f t h e bed. ( c ) E r o s i o n o f and i n t e r f e r e n c e w i t h the s t r a i n gages by the f l u i d i z e d p a r t i c l e s . Our s e t - u p a l l o w e d t h e s e d i s a d v a n t a g e s t o be overcome. 3.4.3 I n s t r u m e n t a t i o n The system f o r r e c o r d i n g the f o r c e s i s i n d i c a t e d i n the b l o c k diagram shown i n F i g u r e 3-9. Two F o r c e T r a n s d u c e r s were used as d e s c r i b e d above. The changes o f r e s i s t a n c e o f the s t r a i n gages, bonded t o the l o a d beam, produced a s i g n a l p r o p o r t i o n a l t o the a p p l i e d f o r c e s as a r e s u l t o f l o a d beam d e f l e c t i o n . T h i s s i g n a l was a m p l i f i e d . Each t r a n s d u c e r was c o n n e c t e d w i t h a v a r i a b l e r e s i s t a n c e t o a d j u s t the o u t p u t s i g n a l t o the z e r o l e v e l when t h e r e was no l o a d p r e s e n t . The gages were used i n c o n j u n c t i o n w i t h a p a i r o f a s i n g l e - c h a n n e l DC a m p l i f i e r s w i t h c o n t r o l s f o r v a r y i n g g a i n t o a d j u s t the o u t p u t l e v e l s t o t h e range o f ±5V b e f o r e r e c o r d i n g . Too l a r g e an a m p l i f i c a t i o n would r e s u l t i n a v o l t a g e c u t - o f f by the tape r e c o r d e r , w h i l e too low an a m p l i f i c a t i o n would r e s u l t i n an u n n e c e s s a r i l y low s e n s i t i v i t y . The two t r a n s d u c e r s were c o n n e c t e d t o a s i n g l e power s o u r c e , an AC-DC t r a n s f o r m e r , NAAIEK E l e c t r o n i c s L t d . , Model 50-IS. A d u a l - c h a n n e l ROCKLAND Analogue F i l t e r , Model 442, was used t o e l i m i n a t e the h i g h f r e q u e n c y c o n t e n t o f the s i g n a l s r e s u l t i n g from e l e c t r i c a l c i r c u i t n o i s e . The c u t - o f f f r e q u e n c y was a d j u s t a b l e i n t h e range 10 HZ t o 1.1 MHZ. In the p r e s e n t work the c u t - o f f f r e q u e n c y FORCE TRANSDUCER(1) ANALOGUE FILTER CHANNEL(1) FORCE TRANSDUCER(2) A M D 1 i C T r n M n r L l r l t K ANALOGUE FILTER CHANNEL(2) TAPE RECORDER F i g . 3-9. I n s t r u m e n t a t i o n ( r e c o r d mode ) f o r Fo rce Measurement - 49 -was s e t a t 50 HZ as d i s c u s s e d i n S e c t i o n 3.4.4. The s i g n a l s from t h e f i l t e r s were r e c o r d e d on magnetic tape and c o u l d s u b s e q u e n t l y be r e p l a y e d as r e q u i r e d . A magnetic t a p e r e c o r d e r , TANDBERG INSTRUMENTATION, Model TIR 115, was used. T h i s r e c o r d e r has 4 c h a n n e l s and 3 tape speeds (24, 95, 380 mm/s). The r e c o r d i n g was c a r r i e d o u t a t a c o n s t a n t speed o f 95 mm/s. The tape r e c o r d e r was a d j u s t e d t o r e c o r d s i g n a l s w i t h i n the range ±5V c o r r e s p o n d i n g to the range o f a (LSI-IT) computer used f o r a n a l o g u e - t o - d i g i t a l c o n v e r s i o n . S i g n a l s s t o r e d on magnetic t a p e were l a t e r r e p l a y e d i n t o the a n a l o g u e - t o - d i g i t a l c o n v e r s i o n u n i t f o r a n a l y s i s on the U.B.C. Computer, AMDAHL 470 V/6 Model 11 . The r e c o r d e d s i g n a l s were a l s o r e p l a y e d i n t o o t h e r i n s t r u m e n t a t i o n used t o examine and measure t h e s i g n a l s : 1. A Brush Mark 220 Reco r d e r Model 15-6327-50 (two c h a n n e l , a n a l o g u e r e c o r d e r ) was c o n n e c t e d i n p a r a l l e l t o the magnetic t a p e r e c o r d e r t o o b t a i n permanent r e c o r d s o f f o r c e - t i m e h i s t o r y o f r e c o r d e d d a t a . The r e c o r d i n g paper speed c o u l d be v a r i e d from 5 t o 125 mm/s. 2. A K e i t h l e y D i g i t a l V o l t m e t e r , Model 179 DMM, was used f o r c a l i b r a t i o n purposes and d i s p l a y e d the measured v o l t a g e t o examine the l e v e l o f the o u t p u t s i g n a l s from time t o time d u r i n g the e x p e r i m e n t s . 3. A T e l e q u i p m e n t o s c i l l o s c o p e , t y p e DM53A, was used t o examine v i s u a l l y t h e c h a r a c t e r i s t i c s o f t h e s i g n a l s . - 50 -3.4.4 E x p e r i m e n t a l p r o c e d u r e I. P r e l i m i n a r y t e s t s : P r e l i m i n a r y t e s t s were c a r r i e d out w i t h a t e s t c y l i n d e r i n p o s i t i o n i n o r d e r t o check f o r p o s s i b l e e r r o r s a r i s i n g from the method o f measurement, and t o examine the g e n e r a l c h a r a c t e r i s t i c s o f the measured f o r c e s . The n a t u r a l f r e q u e n c y o f the tube-gage assembly was measured e x p e r i m e n t a l l y t o e n s u r e t h a t i t was h i g h enough to have no a d v e r s e e f f e c t . T h i s was a c c o m p l i s h e d by a p p l y i n g l i g h t i m p u l s e s t o the t u b e -gage assembly. The f r e q u e n c y c o n t e n t o f t h e r e s u l t i n g s i g n a l s was then examined w i t h a s t o r a g e o s c i l l o s c o p e . When the t e s t c y l i n d e r was immersed i n an i n c i p i e n t l y f l u i d i z e d bed - the case c o r r e s p o n d i n g to the l o w e s t n a t u r a l f r e q u e n c y - the r e s o n a n t f r e q u e n c y was found t o exceed 100 HZ f o r a l l tubes t e s t e d . The n a t u r a l f r e q u e n c y o f v i b r a t i o n o f the same assembly i n a i r , was found t o be i n e x c e s s o f 125 HZ f o r a l l t u b e s . The e x p e r i m e n t a l v a l u e s o f the n a t u r a l f r e q u e n c i e s o f t h e tube-gage assembly a r e t a b u l a t e d i n T a b l e 3.2. S i n c e the e f f e c t i v e bed v i s c o s i t y i s e x p e c t e d t o d e c r e a s e w i t h i n c r e a s i n g s u p e r f i c i a l gas v e l o c i t y f o r U > U ^ ( S c h i i g e r l , 1971), the n a t u r a l f r e q u e n c y was e x p e c t e d t o be a t l e a s t i n the range 100-125 HZ f o r a l l t e s t c o n d i t i o n s employed i n t h i s work. T h i s f r e q u e n c y i s a t l e a s t an o r d e r o f magnitude l a r g e r than the l a r g e s t e x p e c t e d f o r c i n g f r e q u e n c y a s s o c i a t e d w i t h b u b b l e s . A f u r t h e r t e s t was c a r r i e d out t o e n s u r e the absence o f low f r e q u e n c y v i b r a t i o n s t r a n s m i t t e d t h r o u g h the f l o o r and p i p e l i n e from - 51 -T a b l e 3.2 N a t u r a l f r e q u e n c y o f the tube-gage a s s e m b l i e s i n a i r and i n bed o f 430 ym sand a t minimum f 1 u i d i z a t i o n S i z e o f t h e N a t u r a l f r e q u e n c y o f the tube-gage assembly, t e s t c y l i n d e r — — — ( o u t s i d e d i a m e t e r T e s t c y l i n d e r i n an i n mm) i n c i p i e n t l y f l u i d i z e d T e s t c y l i n d e r i n a i r bed 32 220 275 25 136 170 15 102 126 - 52 -the b l o w e r . With the t e s t c y l i n d e r i n p o s i t i o n and no p a r t i c l e s i n the column, the a i r blower was s w i t c h e d on. With no bed i n t e r a c t i n g w i t h t h e t u b e , t h e s i g n a l s produced by t h e f o r c e gages were examined. The f o r c e s measured were much s m a l l e r than t h o s e a s s o c i a t e d w i t h the bed-tube i n t e r a c t i o n s . Hence i t was c o n f i r m e d t h a t any low f r e q u e n c y i n t e r f e r e n c e r e a c h i n g the t e s t c y l i n d e r c o u l d be n e g l e c t e d . The gage o u t p u t s i g n a l , a f t e r a m p l i f i c a t i o n , c o n t a i n e d c o n s i d e r a b l e h i g h - f r e q u e n c y n o i s e . V i s u a l o b s e r v a t i o n s o f s e v e r a l s i g n a l r e c o r d s as w e l l as i n i t i a l a n a l y s i s o f t h e s e r e c o r d s i n the frequency-domain (as d i s c u s s e d i n Chapter 4) i n d i c a t e d t h a t the p r i m a r y f r e q u e n c y c o m p o s i t i o n o f t h e measured f o r c e s was i n t h e range below 50 HZ and t h a t the high f r e q u e n c y c o n t e n t o f t h e s i g n a l s r e s u l t e d from e l e c t r i c a l c i r c u i t n o i s e . The a n a l o g u e f i l t e r was used i n the low-pass mode to s u p p r e s s the h i g h f r e q u e n c y c o n t e n t o f t h e s i g n a l s i n o r d e r to f a c i l i t a t e r e c o r d i n g and m e a s u r i n g o f t h e f o r c e s a s s o c i a t e d w i t h the b u b b l i n g phenomena. The c u t - o f f f r e q u e n c y was a d j u s t e d t o 50 HZ. F i g u r e 3-10-a shows t h e o u t p u t s i g n a l o f one o f the t r a n s d u c e r s , a f t e r a m p l i f i c a t i o n and p r i o r t o f i l t e r i n g , r e c o r d e d w i t h no bed i n t e r a c t i n g w i t h the t e s t c y l i n d e r . F i g u r e 3-10-b shows the a m p l i f i e d s i g n a l o f one o f the t r a n s d u c e r s a f t e r f i l t e r i n g , and t h e a m p l i f i e d s i g n a l o f the o t h e r t r a n s d u c e r w i t h t h e f i l t e r b y p a s s e d ; both s i g n a l s were r e c o r d e d s i m u l t a n e o u s l y . Note the " t h i c k n e s s " o f t h e u n f i l t e r e d s i g n a l s i n d i c a t i n g h i g h - f r e q u e n c y n o i s e . - 53 -u 5 £ o j i i ! i ; i j |. 1 ! i 1 ; j | i ! i i i ' i ! ! ! '• ! : ! '• ! 1 ] J ; ! | j " ; [ ; • i — i — j | ! ! i 1 i i ! i j : | ! 1 ! : 1 i i ! ! i ! ..A i i i I i 1 i i 5 10i 4 TIME (S) 5 F i g . 3-1O-a. Output sig n a l (after amplification and without f i l t e r i n g ) for one of the transducers with no bed intera c t i n g with the test cylinder. ^ + 6 10 w ^ 5 O fc 0 5 10 3 TIME (S) 4 F i g . 3-10-b. Output signal (after amplification) for: (1) r i g h t hand transducer with f i l t e r i n g , (2) l e f t hand transducer without f i l t e r i n g . Both were recorded simultaneously at ( U - U j r j f ^ O ^ m/s. H=0.3 m, and bed material was sand (dp=430iam). - 54 -I I . C a l i b r a t i o n : A s t a t i c c a l i b r a t i o n was c a r r i e d out by h a n g i n g l o a d s from t h e m i d d l e o f the tube a f t e r s e t t i n g both gages t o sense o n l y the v e r t i c a l f o r c e components w i t h no p a r t i c l e s i n t h e column. The magnitudes o f th e r e s u l t i n g v o l t a g e s i g n a l s , c o r r e s p o n d i n g to p a r t i c u l a r a m p l i f i e r g a i n s , were then measured u s i n g t he d i g i t a l v o l t m e t e r . The d i r e c t i o n a l s e n s i t i v i t y o f the gages was a l s o checked by a p p l y i n g i d e n t i c a l l o a d s upwards on t h e tube w i t h the h e l p o f a l i g h t f i s h i n g l i n e run over a p u l l e y and a t t a c h e d t o the same w e i g h t s . A t y p i c a l c a l i b r a t i o n o f the f o r c e gages i s shown i n F i g u r e 3-11. The f o r c e shown i s the sum o f th e two s e p a r a t e c h a n n e l s . These c a l i b r a t i o n s showed t h a t : - The two gages gave s i g n a l s f o r the same l o a d which were v i r t u a l l y i d e n t i c a l ( w i t h i n 0.8% o f each o t h e r ) . - The gages were l i n e a r up t o the maximum a p p l i e d f o r c e . - The gages y i e l d e d t he same s i g n a l f o r l o a d s a p p l i e d upward o r downward. - The gages were found t o be c a p a b l e o f s e n s i n g l o a d s as small as 0.01 N. The c a l i b r a t i o n s a r e v a l i d a l s o f o r dynamic l o a d s p r o v i d e d t h a t t h e n a t u r a l f r e q u e n c y o f t h e system i s s u f f i c i e n t l y h i g h r e l a t i v e t o t h e f r e q u e n c y o f a p p l i e d f o r c e s (Kennedy e t a l . , 1981). F u r t h e r c a l i b r a t i o n checks were c a r r i e d out r e g u l a r l y , and no s i g n i f i c a n t s h i f t i n the c u r v e was n o t i c e d . The d i g i t a l o u t p u t o f the (LS1-11) Computer was a l s o c a l i b r a t e d by a p p l y i n g a s e r i e s o f s t a t i c known l o a d s to t he t e s t c y l i n d e r ; t h e d i g i t a l o u t p u t s o f the computer were then e v a l u a t e d a f t e r p r o c e s s i n g the c a l i b r a t i o n d a t a t h r o u g h the e n t i r e F i g . 3-11. Sample gage force c a l i b r a t i o n - 56 -system d e s c r i b e d i n S e c t i o n 3.4.3. I I I . F o r c e measurement p r o c e d u r e : In o r d e r t o c h a r a c t e r i z e and c o r r e l a t e the f o r c e a c t i n g on the tubes w i t h the bed o p e r a t i n g v a r i a b l e s , s e v e r a l t e s t runs were c a r r i e d out a t d i f f e r e n t bed o p e r a t i n g c o n d i t i o n s w i t h a 32 mm smooth tube a l o n e and i n an a r r a y o f t u b e s i n t r i a n g u l a r p i t c h a r r a n g e m e n t s . F u r t h e r e x p e r i m e n t s were c a r r i e d o u t u s i n g the 25 mm and 15 mm smooth tubes as w e l l as the f i n n e d t u b e . The e x p e r i m e n t a l runs were performed i n the f o l l o w i n g sequence: (1) Measurement o f t h e v e r t i c a l components o f f o r c e on an i s o l a t e d t u b e : The t e s t c y l i n d e r was mounted i n a h o r i z o n t a l p o s i t i o n a c r o s s the bed 0.3 m above the d i s t r i b u t o r p l a t e and s u p p o r t e d r i g i d l y a t each end by the two f o r c e gages, as shown i n F i g u r e 3-8. The o u t p u t s i g n a l was a d j u s t e d t o t h e z e r o l e v e l w i t h the tube under no e x t e r n a l l o a d . Force measurements were c a r r i e d o u t on t h e s p e c i f i e d tubes w i t h each o f the f i v e bed m a t e r i a l s d e s c r i b e d i n T a b l e 3.1. The o p e r a t i n g c o n d i t i o n s a r e s p e c i f i e d i n T a b l e 3.3. The h i g h e s t v a l u e s o f ( U - U m f ) c o u l d not be used f o r t h e p o l y e t h y l e n e and f o r t h e f i n e s t sand because o f e x c e s s i v e c a r r y o v e r from the bed under t h e s e c o n d i t i o n s . (2) Measurement o f the h o r i z o n t a l components o f f o r c e on an i s o l a t e d t u b e : The gages were r o t a t e d u n t i l t he v e r t i c a l f o r c e components became z e r o . The gages were h e l d i n t h i s new p o s i t i o n , o r t h o g o n a l t o the f i r s t p o s i t i o n . The p r e v i o u s e x p e r i m e n t a l p r o c e d u r e was r e p e a t e d under i d e n t i c a l - 57 -T a b l e 3.3 O p e r a t i n g c o n d i t i o n s f o r measurements o f v e r t i c a l and h o r i z o n t a l f o r c e components f o r i s o l a t e d t u b e s . P a r t i c l e s Tubes S t a t i c bed h e i g h t (m) U - U, (m/sf 430 ym sand 15, 25 and 32 mm and f i n n e d 0.30 and 0.45 0.05, 0.10, 0.20, 0.30, 0.50, 0.80, 1 .2 and 1 .4 280 ym sand 185 ym sand alundum p o l y e t h y l e n e 32 mm 32 mm 32 mm 32 mm 0.30 and 0.45 0.30 0.30 0.30 0.05, 0.10, 0.20, 0.30, 0.50, 0.80, 1 .2 and 1 .4 0.05, 0.10, 0.20, 0.30, 0.50 and 0.80 0.05, 0.1 0, 0.20, 0.30, 0.50, 0.80, 1 .2 and 1.4 0.05, 0.10, 0.20, 0.30, 0.50 and 0.80 - 58 -o p e r a t i n g c o n d i t i o n s . (3) Measurements o f the v e r t i c a l and h o r i z o n t a l components o f f o r c e on a tube w i t h i n an a r r a y : The i n s t r u m e n t e d 32 mm tube was p l a c e d a t the c e n t r e o f an a r r a y as d e s c r i b e d i n S e c t i o n 3.4.2. Con t i n u o u s measurements o f the v e r t i c a l and h o r i z o n t a l components o f f o r c e s were c a r r i e d out f o r the 430 ym sand, s t a t i c bed h e i g h t 0.45 m, and f o r the ( U - U m f ) v a l u e s l i s t e d i n T a b l e 3.3. F u r t h e r e x p e r i m e n t s were c a r r i e d out w i t h 3 tubes i n the a r r a y i n two d i f f e r e n t c o n f i g u r a t i o n s , as d e s c r i b e d i n Cha p t e r 6. In a l l o f t h e s e e x p e r i m e n t a l r u n s , f o r c e components were r e c o r d e d u s i n g t h e system d i s c u s s e d i n S e c t i o n 3.4.3 and i n d i c a t e d i n F i g u r e 3-9. R e c o r d i n g s were made a t a tape speed o f 95 mm/s. In o r d e r t o a s s u r e s u f f i c i e n t d a t a f o r l a t e r a n a l y s i s , a r e c o r d l e n g t h o f 300 s was made f o r each s i g n a l . At a l a t e r s t a g e the s i g n a l s were r e p l a y e d a t a tape speed o f 95 mm/s i n t o t h e a n a l o g u e - t o - d i g i t a l c o n v e r s i o n u n i t . 3 .5 P r e s s u r e Measurement P r e s s u r e measurements around t h e immersed tube were employed t o d e t e c t t h e p r e s e n c e o f bubbles near t h e t u b e , and t o measure the p r e s s u r e f l u c t u a t i o n s caused by the b u b b l e s . The v a r i a t i o n o f p r e s s u r e a t a p o i n t on the tube p e r i p h e r y was most c o n v e n i e n t l y measured as a v o l t a g e produced by a p r e s s u r e t r a n s d u c e r and a s s o c i a t e d i n s t r u m e n t a t i o n . A l o w - p r e s s u r e t r a n s d u c e r , DISA Type 51D20, was us e d . The t r a n s d u c e r i s o f c a p a c i t i v e t y p e w i t h the p r e s s u r e t o be measured c a u s i n g v a r i a t i o n i n the c a p a c i t a n c e o f a c a p a c i t o r . The s e t - u p i s shown i n F i g u r e 3-12. The c a p a c i t o r i s composed o f a diaphragm ( 1 ) , - 59 -F i g . 3-12. S e c t i o n a l d rawing o f the 51D20 low p r e s s u r e t r a n s d u c e r . (1) diaphragm, (2) e l e c t r o d e , (3) i n s u l a t i o n d i s c , (4) and (5) p r e s s u r e c h a n n e l s . - 60 -a t z e r o p o t e n t i a l , and an e l e c t r o d e (2) which i s i n s u l a t e d from the diaphragm by an aluminium o x i d e i n s u l a t i o n d i s c ( 3 ) . The p r e s s u r e d i f f e r e n c e a c r o s s the diaphragm causes i t t o d e f l e c t , t h u s c h a n g i n g the c a p a c i t a n c e between t h e diaphragm and t h e e l e c t r o d e . The p r e s s u r e was conveyed t o t h e t r a n s d u c e r t h r o u g h p r e s s u r e c h a n n e l s (4) and ( 5 ) . The s p e c i f i c a t i o n s o f t h e t r a n s d u c e r used i n t h e p r e s s u r e measurements were: Diaphragm t h i c k n e s s , 0.14 mm 2 P r e s s u r e r a n g e , 4.9 KN/m 2 Maximum o v e r l o a d , 14.7 MN/m Diaphragm re s o n a n c e i n a i r , 4 KHZ T r a n s d u c e r r e s o n a n c e i n a i r , 1.6 KHZ The p r e s s u r e t r a n s d u c e r was used i n c o n j u n c t i o n w i t h a t u n i n g p l u g (Type 51E03), an o s c i l l a t o r (Type 51E02) which d e l i v e r s a h i g h -f r e q u e n c y s i g n a l whose f r e q u e n c y i s i n v e r s e l y p r o p o r t i o n a l t o the s t i m u l u s a p p l i e d t o t h e a s s o c i a t e d t r a n s d u c e r , and a r e a c t a n c e c o n v e r t e r (Type 51E01) which a m p l i f i e s the o s c i l l a t o r s i g n a l and produces a DC v o l t a g e p r o p o r t i o n a l t o the change o f c a p a c i t a n c e . In o r d e r t o measure t h e p r e s s u r e v a r i a t i o n s a t a p o i n t on t h e s u r f a c e o f the immersed t u b e , d i f f e r e n t i a l p r e s s u r e measurements (between t h i s p o i n t and t h e p r e s s u r e o f t h e dense phase a t the same l e v e l ) were c a r r i e d o u t . T h i s was a c h i e v e d by c o n n e c t i n g one o f t h e two p r e s s u r e c h a n n e l s i n t h e body o f t h e t r a n s d u c e r t o t h e p o i n t o f measurement on the tu b e s u r f a c e , w h i l e t he o t h e r channel was c o n n e c t e d t o a chamber w i t h a r e f e r e n c e p r e s s u r e which was s e t t o match the p r e s s u r e o f the dense phase a t the same l e v e l w i t h the s p e c i f i e d p o i n t . The t e s t c y l i n d e r was a 32 mm o u t s i d e d i a m e t e r tube s i m i l a r - 61 -t o t h a t shown i n F i g u r e 3-6. A p r e s s u r e t a p o f d i a m e t e r 2 mm was d r i l l e d t h r o u g h the w a l l a t t h e m i d - s e c t i o n o f t h e tube and c o v e r e d by a s c r e e n t o p r e v e n t p a r t i c l e s from e n t e r i n g the p r e s s u r e c h a n n e l . The p r e s s u r e channel passed t h r o u g h the h o l l o w c e n t r e o f the t u b e , c o n n e c t i n g the t r a n s d u c e r w i t h the p r e s s u r e t a p . The tube c o u l d be t u r n e d t o measure the p r e s s u r e a t a n g l e s o f 0° and 180° t o the v e r t i c a l . A n o t h e r p r e s s u r e t a p i d e n t i c a l t o the f i r s t but p e r p e n d i c u l a r , was d r i l l e d t h r o u g h the tube w a l l t o measure the p r e s s u r e a t a n g l e s o f 90° and 270° to the v e r t i c a i . The system f o r r e c o r d i n g the p r e s s u r e f l u c t u a t i o n s i s i n d i c a t e d i n t h e b l o c k diagram shown i n F i g u r e 3-13. The s i g n a l s produced by the two f o r c e gages and by the p r e s s u r e t r a n s d u c e r were r e c o r d e d s i m u l t a n e o u s l y , i n o r d e r t o match the f o r c e p u l s e s w i t h the p r e s s u r e f l u c t u a t i o n s . Repeated c a l i b r a t i o n s o f the p r e s s u r e t r a n s d u c e r and the a s s o c i a t e d system were c a r r i e d out from time t o time t o e n s u r e a c c u r a c y and r e p r o d u c i b i l i t y o f t h e measured p r e s s u r e . 3.6 S i n g l e Bubble E x p e r i m e n t The experiment was c a r r i e d out i n the f l u i d i z a t i o n column d e s c r i b e d i n S e c t i o n 3.2. S i n g l e bubbles were i n j e c t e d i n t o an i n c i p i e n t l y f l u i d i z e d bed o f sand, by means o f a bubble i n j e c t o r d e s c r i b e d i n F i g u r e 3-14. The i n j e c t o r c o n s i s t s o f a s i n g l e tube h a v i n g an i n j e c t i o n p o r t o f d i a m e t e r 4 mm a t a l e v e l 100 mm above the gas d i s t r i b u t o r . The t u b e was c o n n e c t e d t o a p r e s s u r i z e d v e s s e l t h r o u g h a s o l e n o i d v a l v e ( e l e c t r o m a g n e t i c ) . The p r e s s u r e i n t h e p r e s s u r i z e d FORCE S I G N A L ( 1 ) P R E S S U R E TRANSDUCER TUNING PLUG O S C I L L A T O R CONVERTER FORCE S I G N A L ( 2 ) F i g . 3 - 1 3 . I n s t r u m e n t a t i o n ( r e c o r d mode ) o f P r e s s u r e M e a s u r e m e n t F i g . 3-!4 . General set-up of single injected bubble experiment. 14 " l e n o i d ^ r ^ r w ^ r ^ ^ ^ r " " 3 ^ 0 ^ 5 ' »>*»J«tion port, ^ p r e s s u r e gage^'<9) c o n t " i v a l ? e V e S S S l ' < 7 , a l r - 64 -v e s s e l was measured by a p r e s s u r e gage, and the a i r was f e d from an a i r c y l i n d e r . An e l e c t r i c t i m e r was c o n n e c t e d t o the s o l e n o i d v a l v e t o c o n t r o l the f r e q u e n c y and d u r a t i o n o f gas i n j e c t i o n s . Force and p r e s s u r e s i g n a l s , i n d u c e d by the passage o f i n j e c t e d b u b b l e s w i t h i n the bed, were r e c o r d e d u s i n g the Brush r e c o r d e r d e s c r i b e d i n S e c t i o n 3.4.3. Two d i f f e r e n t e x p e r i m e n t a l runs were c a r r i e d out i n the f o l l o w i n g manner: (1) S i n g l e b u b b l e s o f d i f f e r e n t s i z e were i n j e c t e d a t the c e n t r e o f the column i n o r d e r t o s t u d y the r e l a t i o n between c h a r a c t e r i s t i c s (magnitude and d u r a t i o n ) o f i n d u c e d f o r c e and bubble s i z e . (2) S i n g l e bubbles o f t h e same s i z e were i n j e c t e d a t d i f f e r e n t h o r i z o n t a l p o s i t i o n s (0, 30, 60 and 90 mm from the c e n t r e - l i n e o f t h e column i n a p e r p e n d i c u l a r d i r e c t i o n t o t h e t e s t c y l i n d e r ) a t a l e v e l 100 mm above the gas d i s t r i b u t o r , i n o r d e r to s t u d y the e f f e c t o f bubble p o s i t i o n on c h a r a c t e r i s t i c s o f measured f o r c e . These e x p e r i m e n t s were c a r r i e d out t o c l a r i f y the mechanism by which i n d i v i d u a l bubbles i n d u c e f o r c e s on an immersed t u b e . - 65 -C H A P T E R 4 METHODS OF DATA P R O C E S S I N G AND A N A L Y S I S 4.1 G e n e r a l Requirements Data a c q u i s i t i o n and p r o c e s s i n g may be d i v i d e d i n t o two p r i m a r y o p e r a t i o n s : (a) d a t a c o l l e c t i o n and r e c o r d i n g , and (b) data p r e p a r a t i o n and a n a l y s i s . The f i r s t o p e r a t i o n was d i s c u s s e d i n d e t a i l i n the p r e c e d i n g c h a p t e r . The second o p e r a t i o n i s t h e s u b j e c t o f t h i s c h a p t e r . D i g i t a l a n a l y s i s was employed i n t h e p r e s e n t work. T h i s r e q u i r e d d i g i t a l d a t a p r e p a r a t i o n . B a s i c c o n s i d e r a t i o n s a s s o c i a t e d w i t h the data s a m p l i n g and d i g i t i z a t i o n p r o c e s s a r e d i s c u s s e d i n S e c t i o n 4.2.1. A n a l y s i s i s f a c i l i t a t e d where the d a t a e x h i b i t s t a t i s t i c a l r e g u l a r i t y . Data p o s s e s s i n g such c h a r a c t e r i s t i c s may be d e f i n e d as s t a t i o n a r y e r g o d i c . The p r o p e r t i e s o f s t a t i o n a r y ( e r g o d i c ) d a t a can be d e t e r m i n e d from t i m e - a v e r a g e s o f i n d i v i d u a l sample r e c o r d s . These c h a r a c t e r i s t i c s and o t h e r g e n e r a l p r o p e r t i e s o f the data under i n v e s t i g a t i o n a r e d i s c u s s e d i n S e c t i o n 4.2.2. The t i m e - h i s t o r y r e c o r d s o f the measured f o r c e s under s t u d y do not e x h i b i t g r e a t r e g u l a r i t y and t h e r e f o r e s h o u l d be t r e a t e d as a n o n d e t e r m i n i s t i c (random) time s e r i e s . In such c a s e s i t i s more f e a s i b l e t o d e s c r i b e the d a t a i n terms o f s t a t i s t i c a l parameters - 66 -t h a n t o seek a d e t e r m i n i s t i c d e s c r i p t i o n . The b a s i c m a t h e m a t i c a l methods employed i n d a t a a n a l y s i s a r e p r e s e n t e d i n S e c t i o n 4.3. The d i s c u s s i o n i n c l u d e s the g e n e r a l i n t e r p r e t a t i o n o f the main s t a t i s t i c a l f u n c t i o n s used t o d e s c r i b e the b a s i c p r o p e r t i e s o f the d a t a . 4.2 General C o n s i d e r a t i o n s i n Data Sampling and P r o c e s s i n g 4.2.1 Data d i g i t i z a t i o n and s a m p l i n g c o n s i d e r a t i o n s A d i g i t a l computer was used i n the d a t a a n a l y s i s . T h i s r e q u i r e d a n a l o g u e - t o - d i g i t a l c o n v e r s i o n o f t h e measured f o r c e r e c o r d s . The parameters r e q u i r i n g c o n s i d e r a t i o n a r e t h e s a m p l i n g i n t e r v a l At and t h e r e c o r d l e n g t h T. F i n d i n g an a p p r o p r i a t e s a m p l i n g i n t e r v a l At i n v o l v e s a compromise. Sampling a t p o i n t s which a r e too c l o s e t o g e t h e r w i l l y i e l d c o r r e l a t e d and h i g h l y redundant d a t a , and t h u s u n n c e s s a r i l y i n c r e a s e the time and c o s t o f c a l c u l a t i o n s . On the o t h e r hand, sa m p l i n g a t p o i n t s which a r e too f a r a p a r t w i l l l e a d t o a l i a s i n g p r o b l e m s , i . e . t o c o n f u s i o n between t h e low and h i g h f r e q u e n c y compo-nen t s i n the o r i g i n a l d a t a . Care must be taken t o ensure t h a t a h i g h enough s a m p l i n g f r e q u e n c y i s chosen so t h a t a l i a s i n g can be a v o i d e d . T h i s m a t t e r i s t r e a t e d i n d e t a i l i n r e f e r e n c e s : Bendat ( 1 9 7 1 ) , Bendat ( 1 9 8 0 ) , and J e n k i n s ( 1 9 6 8 ) . The h i g h e s t f r e q u e n c y which can be d e t e c t e d w i t h d a t a sampled at i n t e r v a l s o f At i s f c = l / ( 2 A t ) (4.1) c a l l e d t h e N y q u i s t f r e q u e n c y o r f o l d i n g f r e q u e n c y . To a v o i d a l i a s i n g , t h e s a m p l i n g f r e q u e n c y must be chosen t o have a v a l u e a t l e a s t equal to - 67 -the h i g h e s t f r e q u e n c y a n t i c i p a t e d i n the d a t a . As d i s c u s s e d i n Cha p t e r 3, frequency-domain (PSD) a n a l y s i s o f a few a r b i t r a r y chosen f o r c e r e c o r d s , d i g i t i z e d a t the maximum p o s s i b l e r a t e (10,000 samples per second) p r i o r t o f i l t e r i n g , has i n d i c a t e d t h a t t h e p r i m a r y f r e q u e n c y c o m p o s i t i o n o f the measured f o r c e was i n the range below 50 Hz. S t u d i e s o f bubble f r e q u e n c i e s i n f l u i d i z e d beds ( e . g . K u n i i e t a l . , 1967; G e l d a r t , 1970/71) a l s o i n d i c a t e f r e q u e n c i e s w e l l below t h i s v a l u e . Hence, an analogue f i l t e r was used i n the low-pass mode, a d j u s t e d a t a c u t - o f f f r e q u e n c y o f 50 HZ, t o s u p p r e s s the h i g h f r e q u e n c y c o n t e n t o f t h e s i g n a l s , p r o b a b l y due t o e l e c t r i c a l n o i s e . As a r e s u l t , i n f o r m a t i o n above t he maximum f r e q u e n c y o f i n t e r e s t was no l o n g e r c o n t a i n e d i n t h e f i l t e r e d d a t a . Thus, f c was chosen equal t o t h e maximum f r e q u e n c y o f i n t e r e s t (50 H z ) , g i v i n g At = 10 ms. The r e l a t i o n between the r e c o r d l e n g t h , T, and the number, N, o f d a t a samples per r e c o r d i s : T = N At (4.2) The sample s i z e N s h o u l d be s e l e c t e d on the b a s i s o f t h e d e s i r e d a c c u r a c y f o r e s t i m a t i n g v a r i o u s c h a r a c t e r i z a t i o n v a l u e s , and t h i s i s d i s c u s s e d i n S e c t i o n 4.3. A b a l a n c e must be a c h i e v e d between the c o s t o f computing time and t h e a c c u r a c y o f c a l c u l a t i o n . At the same t i m e , t h e sample r e c o r d must be l o n g enough t o p e r m i t n o n s t a t i o n a r y t r e n d s t o be d i f f e r e n t i a t e d from the random f l u c t u a t i o n s o f the f o r c e - t i m e h i s t o r y . As d i s c u s s e d below, t h i s t e s t was done on d i f f e r e n t r e c o r d l e n g t h o f d a t a . I t was found t h a t t h e da t a a r e s t a t i o n a r y , and t h a t a r e c o r d l e n g t h o f 30 seconds ( c o r r e s p o n d i n g to 3000 d a t a samples) i s s u f f i c i e n t t o v e r i f y t h i s p r o p e r t y . - 68 -In t h e d a t a d i g i t i z a t i o n , machine language was used t o sample d a t a a t t h e s p e c i f i c t i m e i n t e r v a l s . The d i g i t i z e d d a t a were s t o r e d on m a g n e t i c tape f o r p r o c e s s i n g on the UBC d i g i t a l computer. The a n a l y s i s was made from r e c o r d s 5 minutes i n d u r a t i o n i n each c a s e . Each r e c o r d was broken i n t o 10 segments f o r a v e r a g i n g . 4.2.2 Data p r e - a n a l y s i s c o n s i d e r a t i o n s ( g e n e r a l c h a r a c t e r i s t i c s o f  the d a t a ) The c o r r e c t p r o c e d u r e s f o r a n a l y z i n g random d a t a , as w e l l as i n t e r p r e t i n g t h e a n a l y z e d r e s u l t s , a r e i n f l u e n c e d by c e r t a i n b a s i c c h a r a c t e r i s t i c s which may o r may not be e x h i b i t e d by t h e d a t a . For t h e t i m e - v a r y i n g f o r c e d a t a under i n v e s t i g a t i o n , t h e two most i m p o r t a n t o f t h e s e c h a r a c t e r i s t i c s a r e t h e s t a t i o n a r i t y o f t h e d a t a , and t h e p r e s e n c e o f p e r i o d i c i t i e s i n the d a t a . Q u a l i f i c a t i o n o f t h e sampled d a t a i n terms o f t h e s e c h a r a c t e r i s t i c s was performed p r i o r t o d e t a i l e d d a t a a n a l y s i s . I . S t a t i o n a r y n a t u r e o f t h e measured f o r c e s : The s t a t i o n a r i t y ( e r g o d i c i t y ) p r o p e r t y o f the data i s o f c o n c e r n because i t makes a n a l y s i s much e a s i e r than would be the c a s e f o r n o n s t a t i o n a r y d a t a , and i t e n a b l e s us t o d e r i v e the r e q u i r e d s t a t i s t i c a l i n f o r m a t i o n from a p p r o p r i a t e a n a l y s i s o f a s i n g l e a r b i t r a r y time h i s t o r y r e c o r d . A random p r o c e s s i s c o n s i d e r e d s t a t i o n a r y i f the ensemble a v e r a g e s o v e r c o l l e c t i o n o f sample f u n c t i o n s a t t i m e , t-j , are c o n s t a n t and i n d e p e n d e n t o f t-j . The ensemble a v e r a g e s a r e s t a t i s t i c a l p r o p e r t i e s a v e r a g e d o v e r a l a r g e number o f sample r e c o r d s a t any i n s t a n t - 69 -o f t i m e . The e r g o d i c p r o c e s s i s a s t a t i o n a r y p r o c e s s i n which t he p r o p e r t i e s computed from t i m e - a v e r a g e s o v e r i n d i v i d u a l r e c o r d s o f the ensemble a r e the same from one r e c o r d t o t h e n e x t and equal t he c o r r e s p o n d i n g p r o p e r t i e s computed from an ensemble a v e r a g e . In some r e f e r e n c e s ( e . g . Bendat, 1971; J e n k i n s , 1968) t h e word s t a t i o n a r y i s used t o i m p l y both p r o p e r t i e s t o g e t h e r , s t a t i o n a r y and e r g o d i c , and i t w i l l be used here i n the same s e n s e . The s t a t i o n a r y p r o p e r t y o f random data can be t e s t e d by i n v e s t i -g a t i n g a s i n g l e time h i s t o r y r e c o r d , as r e p o r t e d i n Bendat (1971), i n th e f o l l o w i n g manner: 1. d i v i d e t he sample r e c o r d i n t o N equal time i n t e r v a l s . 2. c a l c u l a t e a mean squa r e v a l u e o r mean v a l u e and v a r i a n c e f o r each i n t e r v a l . I f the computed v a l u e s do not show v a r i a n c e o r t r e n d s o t h e r than t h o s e due t o e x p e c t e d s a m p l i n g v a r i a n c e , then the sampled data e x h i b i t s t a t i o n a r i t y . A n o t h e r s i m p l e method, which has been r e p o r t e d i n J e n k i n s (1968), i n v o l v e s c o n s t r u c t i n g a s e p a r a t e h i s t o g r a m f o r each h a l f o f an a r b i t r a r i l y chosen sample r e c o r d . I f the two h i s t o g r a m s a r e c o n s i s t e n t , t h e assumption o f a c o n s t a n t p r o b a b i l i t y d e n s i t y f u n c t i o n i s p r o b a b l y j u s t i f i e d , and t h e sampled d a t a may be c o n s i d e r e d t o be s t a t i o n a r y . In o r d e r t o demonstrate t h e s t a t i o n a r i t y c h a r a c t e r i s t i c s o f the measured f o r c e under i n v e s t i g a t i o n i n the p r e s e n t work, a n a l y s e s o f a r b i t r a r y chosen f o r c e - t i m e h i s t o r y r e c o r d s were c a r r i e d o u t . The root-mean squ a r e v a l u e s , mean v a l u e s and s t a n d a r d d e v i a t i o n s were d e t e r m i n e d by the methods d e s c r i b e d i n the coming s e c t i o n , from - 70 -d i f f e r e n t segments o f each r e c o r d . The c a l c u l a t e d v a l u e s were always i n agreement w i t h each o t h e r w i t h i n ± 3 % . By way o f i l l u s t r a t i o n , T a b l e 4.1 shows t h e c a l c u l a t e d v a l u e s and p e r c e n t a g e o f d e v i a t i o n o f the measured v e r t i c a l f o r c e components on a 32 mm tube i n i s o l a t i o n under the f o l l o w i n g o p e r a t i n g c o n d i t i o n s : Bed m a t e r i a l : Ottawa sand; dp = 430 ym; s t a t i c bed h e i g h t = 0.3 m; (U - U m f ) = 0.1, 0.5 and 1 .2 m/s. Each segment o f d a t a c o r r e s p o n d e d t o 30 s (3000 d a t a p o i n t s ) . In each c a s e , t h r e e segments were chosen a r b i t r a r i l y from the 10 segments o f data f o r t h e s e c a l c u l a t i o n s . I t i s o b v i o u s from T a b l e 4.1 t h a t the p e r c e n t a g e d e v i a t i o n between the v a l u e s c a l c u l a t e d from d i f f e r e n t segments o f one sample r e c o r d i s i n . t h e range o f ± 3 % . T h i s d e m o n s t r a t e s t h e s t a t i o n a r i t y p r o p e r t y o f the data a c c o r d i n g t o the f i r s t method d e s c r i bed. The o t h e r way o f d e m o n s t r a t i n g the s t a t i o n a r i t y p r o p e r t y o f the d a t a was performed by c o n s t r u c t i n g two s e p a r a t e h i s t o g r a m s , one f o r each segment o f t h e same sample r e c o r d . The work was r e p e a t e d on a few sample r e c o r d s c o l l e c t e d under d i f f e r e n t o p e r a t i n g c o n d i t i o n s . In a l l c a s e s , any two h i s t o g r a m s c o n s t r u c t e d from the same i n d i v i d u a l r e c o r d , but from two d i f f e r e n t segments, were c o n s i s t e n t w i t h each o t h e r , w i t h no s i g n i f i c a n t d i f f e r e n c e between them. As an i l l u s t r a t i o n , F i g u r e 4.1 shows two h i s t o g r a m s c a l c u l a t e d from two d i f f e r e n t segments o f one sample data r e c o r d ( r e c o r d e d a t the s p e c i f i e d o p e r a t i n g c o n d i t i o n s ) . The s o l i d l i n e can be c o n s i d e r e d as the P r o b a b i l i t y D e n s i t y F u n c t i o n o f t h i s s e t o f d a t a . The methods used i n t h i s c a l c u l a t i o n are d i s c u s s e d i n S e c t i o n 4.3. D e t a i l e d a n a l y s i s o f a l l d a t a s e t s , p r e s e n t e d below, a r e i n good agreement w i t h t h e s e r e s u l t s . Thus the f o r c e s under i n v e s t i g a t i o n can T a b l e 4.1 Sample o f p r o p e r t i e s o f data d e t e r m i n e d from d i f f e r e n t segments, i l l u s t r a t i n g s t a t i o n a r i t y o f t h e d a t a (U - U m f ) , Segment m/s a 0.1 b The Mean Value o f the Fo r c e V a l u e , N Per c e n t a g e D e v i a t i o n % 2.096 -0.05 2.097 0.00 2.098 +0.05 S t a n d a r d D e v i a t i o n V a l u e , N P e r c e n t a g e D e v i a t i o n % 1.132 +2.1 1.076 -2.8 1.118 +0.8 Root-Mean Square (RMS) V a l u e , N P e r c e n t a g e D e v i a t i o n % 2.384 +0.52 2.354 -0.75 2.377 +0.20 a 4.600 +1.5 3.609 -0.18 5.846 +0.84 b 4.531 -0.02 3.730 +3.0 5.869 +1.2 c 4.466 -1.4 3.507 -2.9 5.677 -2.07 a 6.414 -1.3 5.833 -0.01 8.669 -1.17 b 6.532 +0.43 6.01 +2.1 8.877 +1.2 c 6.566 +0.95 5.814 -1.2 8.769 -0.03 72. 24.0 (N) o U -co z UJ o CQ < m o or 16.0 20.0 24 0 8.0 12.0 FORCE (N) Fig. 4-1. Histograms constructed from two different segments of one sample data record. The solid line can be considered as the probability density function. The record is for the measured vertical force on q 32 mm tube. The sample was recorded at bed conditions :(U-Umf)= 1.2 m/s, H0= 0.3 m and bed material' Ottawa sand (d_=430ttmY - 73 -be t r e a t e d as s t a t i o n a r y and r e p r o d u c i b l e . I I . P r e s e n c e o f p e r i o d i c i t y i n t h e measured f o r c e s : I t was d e s i r a b l e t o i d e n t i f y the p r e s e n c e o f p e r i o d i c components i n the measured f o r c e s so t h a t t h e y can be r e l a t e d t o the o t h e r p e r i o d i c ( o r a l m o s t p e r i o d i c ) b u b b l i n g phenomena and t h e a s s o c i a t e d p r e s s u r e f l u c t u a t i o n s p r e v a i l i n g i n the f l u i d i z e d bed. At t h e same t i m e , i t was b e n e f i c i a l t o d e t e c t the p r e s e n c e o f p e r i o d i c i t i e s i n the data p r i o r t o d e t a i l e d a n a l y s i s o f a l l sample r e c o r d t o a v o i d e r r o n e o u s i n t e r -p r e t a t i o n o f l a t e r r e s u l t s . The p r e s e n c e o f p e r i o d i c and/or q u a s i -p e r i o d i c components i n the o t h e r w i s e random da t a may o f t e n be d e t e c t e d by v i s u a l i n s p e c t i o n o f a power s p e c t r a l f u n c t i o n , p r o b a b i l i t y d e n s i t y f u n c t i o n , o r a u t o c o r r e l a t i o n f u n c t i o n c a l c u l a t e d from s t a t i o n a r y data ( B e n d a t , 1971). The power spectrum o f sample d a t a r e v e a l s p e r i o d i c components i n the d a t a as s h a r p peaks. A p e r i o d i c component i n o t h e r w i s e random da t a may a l s o be d e t e c t e d from the p r o b a b i l i t y d e n s i t y f u n c t i o n o f the d a t a . A p r o b a b i l i t y d e n s i t y p l o t o f a s i n u s o i d a l wave mixed w i t h random da t a has a s a d d l e shape about the mean v a l u e o f the s i g n a l s . T h i s shape r e s u l t s from a c o m b i n a t i o n from a b e l l - s h a p e d p l o t (P.D.F. o f random d a t a ) and a d i s h - s h a p e d p l o t (P.D.F. o f a s i n e wave). The p r e s e n c e o f more t h a n one superimposed s i n u s o i d a l wave i s d i f f i c u l t t o d i s t i n g u i s h from random d a t a by the p r o b a b i l i t y d e n s i t y p l o t (Bendat, 1971). The t h i r d method which may be used t o d e t e c t the p r e s e n c e o f p e r i o d i c components i n t h e data i s by p r e s e n t i n g an a u t o c o r r e l o g r a m o f t h e data sample. The a u t o c o r r e l a t i o n f u n c t i o n o f a p e r i o d i c wave w i t h i n random da t a i s c h a r a c t e r i z e d by an o b v i o u s o s c i l l a t i o n o f the - 74 -f u n c t i o n . In the s p e c i a l case where the p e r i o d i c wave i s a s i n e wave or c o l l e c t i o n o f s i n e waves the a u t o c o r r e l a t i o n f u n c t i o n w i l l c o n t i n u e t o o s c i l l a t e , r e g a r d l e s s o f the time d i s p l a c e m e n t . On the o t h e r hand, the a u t o c o r r e l a t i o n f u n c t i o n o f random d a t a w i t h o u t p e r i o d i c i t i e s w i l l always a p p r o a c h a v a l u e equal t o the s q u a r e o f the mean v a l u e as the time d i s p l a c e m e n t becomes l a r g e (Bendat, 1971). In t h i s c a s e the a u t o -c o r r e l a t i o n f u n c t i o n , w i l l a p p r oach a v a l u e o f z e r o as the time d i s p l a c e -ment becomes l a r g e . On t h e b a s i s o f t h e s e c o n s i d e r a t i o n s , we can have i n s i g h t i n t o c o m p o s i t i o n o f the measured f o r c e . I n i t i a l a n a l y s i s o f a few samples o f d a t a i n terms o f power s p e c t r a l , p r o b a b i l i t y d e n s i t y , and a u t o -c o v a r i a n c e f u n c t i o n s have shown the c h a r a c t e r i s t i c t r e n d s o f both p e r i o d i c ( o r q u a s i - p e r i o d i c ) and random s i g n a l s . Both are p r e s e n t i n a l l a n a l y z e d samples o f d a t a . The form and i n t e n s i t y o f p e r i o d i c i t i e s , i n t h e o t h e r w i s e random d a t a , d i f f e r e d from one sample t o the n e x t , depending on the bed o p e r a t i n g c o n d i t i o n s . Data a n a l y s e s c a r r i e d out a t l a t e r s t a g e s and d i s c u s s e d i n the next c h a p t e r s , demonstrate the same r e s u l t . Here, a n a l y s i s o f o n l y two d i f f e r e n t samples o f d a t a i s p r e s e n t e d as an i l l u s t r a t i o n . The f i r s t sample i s a r e c o r d o f the measured v e r t i c a l f o r c e component on a 32 mm tube i n i s o l a t i o n r e c o r d e d under t h e f o l l o w i n g c o n d i t i o n s : dp = 430 ym (Ottawa s a n d ) ; s t a t i c bed h e i g h t = 0.3 m; (U - U m f ) = 0.1 m/s. The sample was a n a l y z e d i n the f r e q u e n c y domain, the a m p l i t u d e domain, and the time domain by the methods d e s c r i b e d i n S e c t i o n 4.3. F i g u r e 4-2 shows the power s p e c t r a l d e n s i t y o f t h e sample. The main f e a t u r e o f t h i s spectrum i s t h a t t h e r e i s a s h a r p peak a t a f r e q u e n c y o f 3 Hz which c o u l d e x p l a i n the p e r i o d i c i t y C L I O T I T 3 =3 -J-I I CU • -P" OO -pa oo r>o i o rs j 3 . •c 3 3 T J <-+ O c S i n c r fD o n> - 5 P O W E R S P E C T R A L E S T I M A T E S ( N / H Z ) o.o 0.02 0.04 I I o -CO T 3 - IT) • <= O OO I r + d "S 3 3 Cu T 3 I I — 1 O O r + 3 Cu to u • CO CU c r 3 ft) T 3 Q. — • fD 3 Cu o r + -h fD - S r + _ J . o Cu r + O < r + (D r + - 5 Cu r + S -<• c u O Cu to —' Cu 3 -h CL O »• -s O ro CO - 9Z -- 76 -o f t he d a t a . F i g u r e 4-3 g i v e s the p r o b a b i l i t y d e n s i t y p l o t o f the same s e t o f d a t a . The f i g u r e shows a s a d d l e about the mean v a l u e i n d i c a t i n g the p r e s e n c e o f p e r i o d i c components, most p r o b a b l y s i n u s o i d a l waves, i n o t h e r w i s e random d a t a . The c o r r e s p o n d i n g a u t o c o v a r i a n c e f u n c t i o n , F i g u r e 4-4, i s c o n s i s t e n t w i t h t he o t h e r two f u n c t i o n s , s i n c e i t e x h i b i t s the c h a r a c t e r i s t i c t r e n d s o f both p e r i o d i c and random s i g n a l s . ( T h e maximum v a l u e o f t h e f u n c t i o n i s a t z e r o l a g ; as the time d i s p l a c e m e n t becomes l a r g e i t shows c o n t i n u o u s f l u c t u a t i o n s ) . The second sample o f data i s a r e c o r d o f the measured v e r t i c a l f o r c e on a 32 mm tube r e c o r d e d f o r the same e x p e r i m e n t a l c o n d i t i o n s e x c e p t t h a t the s u p e r f i c i a l a i r v e l o c i t y was h i g h e r g i v i n g (U - U ^ ) = 1.4 m/s. (U - U m f ) has a s t r o n g e f f e c t on the c h a r a c t e r i s -t i c s o f t h e measured f o r c e as shown i n Chapter 5. The c o r r e s p o n d i n g p l o t s o f the power s p e c t r a l , t he p r o b a b i l i t y d e n s i t y and a u t o c o v a r i a n c e f u n c t i o n s a r e shown i n F i g u r e s 4-5, 4-6 and 4-7. The power spectrum p l o t i s c h a r a c t e r i z e d by a w e l l - d e f i n e d peak w i t h a width o f l e s s than 1 Hz. T h i s i n d i c a t e s the p r e s e n c e o f p e r i o d i c components w i t h i n the random d a t a . F i g u r e s 4-6 and 4-7 c o n f i r m t h e e x i s t e n c e o f the p e r i o d i c components. The o n l y d i f f e r e n c e between t h e data c o m p o s i t i o n o f the f i r s t and second samples i s i n t h e i n t e n s i t y and t y p e o f p e r i o d i c i t i e s i n c l u d e d i n t h e d a t a . While the p e r i o d i c components i n the f i r s t sample a r e most p r o b a b l y s i n e wave o r a c o l l e c t i o n o f s i n e waves, t h i s may not be the case f o r the second sample. The f i n d i n g t h a t f o r c e s on tubes e x h i b i t both p e r i o d i c and random components i s e n t i r e l y c o n s i s t e n t w i t h e x p e r i m e n t a l measurements o f bubble p r o p e r t i e s i n f l u i d i z e d beds o b t a i n e d by Werther (19 7 7 ) . - 77 -FORCE (N ) Fig. 4 - 3 . Probability density plot of a sample of total vertical forces on a 32 mm tube. (U - Um f) = O.I m/s ; bed material 1 Ottawa sand ; d p * 430 /im ; H 0 • 0.3 m. OD 0.0 0.2 0.4 LRG A u t o c o v a r i a n c e p l o t a 32 mm t u b e . ( U-U F i g . 4-4 . f o r c e s _ o n s a n d ; dp = 4 3 0 pm; H 0.3 m mf ( S E C O N D ) 6 o f a s a m p l e o f t o t a l v e r t i c a l ) = 0 . 1 ; b e d m a t e r i a l : O t t a w a C L I O T I TJ 3 - . . t O II CU . -P» OJ -P» o o r o i O o n T J ( T O C S : cr ro ro -s P O W E R S P E C T R A L E S T I M A T E S ( N / H Z ) 0.0 0.5 ] .0 II CO O C f D I o O J C I + 3 - 5 3 -h Cu II T 3 — 1 O • r + t o c u co c r Cu fD 3 C L " O 3 CD CU r + O fD -h - 5 r + Cu O ~ c+ 01 o —• r + c + < Cu f D S -J CU <T _j. t o O CU CU Z3 —» C L -+) O -s o fD CO - 6Z -- 80 -0.4 FORCE(N) Fig. 4 - 6 . Probability density plot of a sample of total vertical forces on a 32 mm tube. (U-U m f) s 1.4 m/s ; bed material : Ottawa sand d p = 430 fi m H„ s 0.3 m. o i n F i g . 4 - 7 . A u t o c o v a r i a n c e p l o t o f a s a m p l e o f t o t a l v e r t i c a l f o r c e s o n a 32 mm t u b e . ( U - U m f ) = 1.4 m/s; b e d m a t e r i a l : O t t a w a s a n d ; dp = 4 3 0 pm; H 0 = 0.3 m. - 82 -4.3 Data a n a l y s i s p r o c e d u r e The s t a t i s t i c a l p r o p e r t i e s o f t h e da t a c a l c u l a t e d i n c l u d e t h e mean v a l u e , mean s q u a r e v a l u e , and power s p e c t r a l d e n s i t y f u n c t i o n o f . t h e measured f o r c e s . Some r e p r e s e n t a t i v e a u t o c o r r e l a t i o n ( a u t o c o v a r i a n c e ) and p r o b a b i l i t y d e n s i t y f u n c t i o n s were a l s o c a l c u l a t e d . The methods o f c a l c u l a t i o n assume t h a t t h e d a t a a r e s t a t i o n a r y ( e r g o d i c ) , as de m o n s t r a t e d i n the preceedi.ng s e c t i o n . Hence the p r o p e r t i e s o f the da t a c o u l d be d e t e r m i n e d from t i m e - a v e r a g e s o f i n d i v i d u a l sample r e c o r d s . 1. Mean and mean s q u a r e v a l u e s o f t h e d a t a : (a) Mean v a l u e : The mean v a l u e i s s i m p l y t h e ave r a g e o f a l l d a t a v a l u e s , and can be r e g a r d e d as t h e s t a t i c component o f t h e f o r c e ( t i m e - i n v a r i a n t component). Hence, t h e mean v a l u e p r o v i d e s an approx i m a t e measure o f t h e bed buoyance f o r c e a t low l e v e l s o f t h e gas s u p e r f i c i a l v e l o c i t y c l o s e t o minimum f l u i d i z a t i o n (where t h e buoyancy f o r c e i s d o m i n a n t ) . The f o r c e mean v a l u e ( y x ) o f a sample t i m e - f o r c e h i s t o r y r e c o r d x ( t ) , i s g i v e n by: T u x = l i m y / x ( t ) d t (4.3) T-*°° o I f x ( t ) i s a s i n g l e t i m e h i s t o r y r e c o r d from a s t a t i o n a r y ( e r g o d i c ) random p r o c e s s , t h e mean v a l u e o f x ( t ) can be e s t i m a t e d by t i m e - a v e r a g i n g o v e r a f i n i t e t i m e i n t e r v a l T, g i v i n g an u n b i a s e d e s t i m a t e o f t h e t r u e v a l u e , a s : - 83 -1 T y = j j x(t) dt (4.4) x 1 o I f x(t) is a transformed record (sampled and d ig i t i zed at discrete points as discussed in Section 4.2.1) , the sample mean value i s given by: 1 N x = M I x. (4.5) n i=l 1 where, N is the number of data points over the record length T, and x.j are the corresponding data values. (b) Mean square value: The mean square value of the data provides a measure of the in tens i ty of the forces. However, knowledge of th is parameter has special importance in the design of the elements subjected to varying forces. The mean square value ( i ^ 2 ) of a sample time history record x (t) i s : 1 T Vx - l im I J x 2 ( t ) dt (4.6) Since the process is stat ionary (ergodic), an unbiased estimate of the mean square value (^  ) can be evaluated from a f i n i t e single time history record of x(t) by th i s equation: *x = T ^ x 2 ( t ) d t ( 4 - 7 ) o - 84 -The p o s i t i v e s q u a r e r o o t o f the mean square v a l u e i s c a l l e d the r o o t -mean-square (RMS). The sample RMS v a l u e c a l c u l a t e d from a t r a n s f o r m e d t i m e - h i s t o r y r e c o r d i s g i v e n by: RMS N I i = l (4.8) The v a r i a n c e o f t h e data v a l u e s i s the mean square v a l u e o f th e d e v i a t i o n about t he mean, and can be c o n s i d e r e d as a measure o f the data v a r i a b i l i t y . An u n b i a s e d e s t i m a t e o f the v a r i a n c e (a ) X e v a l u a t e d from a f i n i t e t ime h i s t o r y r e c o r d x ( t ) i s g i v e n by: ~2 1 Y J [x(t) - y x r dt (4.9) From e q u a t i o n ( 4 . 4 ) , (4.7) and (4.9) we see t h a t cs - \b - y X y X X I f t h e mean v a l u e o f t h e da t a (y ) = 0, t h e n , a2 = \\>2 . T h i s i s the X X X c a s e f o r t h e h o r i z o n t a l component o f the f o r c e s under s t u d y i n the p r e s e n t work, as we s h a l l see i n t h e next c h a p t e r . The sample v a r i a n c e ( s 2 ) c a l c u l a t e d from the t r a n s f o r m e d t i m e h i s t o r y r e c o r d i s : -.2 -N v L i = l (x, - x ) 1 (4.10) - 85 -where x i s the sample mean c a l c u l a t e d from e q u a t i o n ( 4 . 5 ) . The v a l u e o f t h e sample v a r i a n c e e v a l u a t e d from e q u a t i o n (4.10) i s an u n b i a s e d e s t i m a t e o f t h e e x a c t v a r i a n c e ( B e n d a t , 1971; L i p s o n , 1973). The s q u a r e - r o o t o f t h e v a r i a n c e , d e f i n e d as the s t a n d a r d d e v i a t i o n , can be r e g a r d e d as a measure o f the dynamic component o f t h e f o r c e . S i n c e t h e f l u c t u a t i n g (dynamic) component o f t h e measured f o r c e s a r e caused m a i n l y by bubbles m o t i o n and the a s s o c i a t e d s o l i d c i r c u l a t i o n t h r o u g h o u t t h e f l u i d i z e d bed, t h e s t a n d a r d d e v i a t i o n o f t h e measured f o r c e can be used as a measure o f the i n t e n s i t y o f b u b b l i n g . A computer programme was w r i t t e n by t h e a u t h o r t o c a l c u l a t e the mean v a l u e , RMS v a l u e and t h e s t a n d a r d d e v i a t i o n ( s ) o f the da t a samples e v a l u a t e d from e q u a t i o n s ( 4 . 5 ) , (4.8) and (4.10) r e s p e c t i v e l y , see Appendix ( B ) . 2. A u t o c o r r e l a t i o n ( a u t o c o v a r i a n c e f u n c t i o n ) : The a u t o c o r r e l a t i o n f u n c t i o n d e s c r i b e s the dependence o f the v a l u e s o f t h e d a t a a t one t i m e on t h e v a l u e s a t a n o t h e r t i m e . From a n o t h e r p o i n t o f view, i t i s a measure o f how f u t u r e v a l u e s o f the da t a can be p r e d i c t e d based on p a s t o b s e r v a t i o n . The a u t o c o r r e l a t i o n f u n c t i o n and t h e power s p e c t r a l d e n s i t y f u n c t i o n a r e F o u r i e r t r a n s f o r m p a i r s and knowledge o f e i t h e r one i s e q u i v a l e n t t o knowledge o f the o t h e r . In t h e a n a l y s i s o f t h e measured f o r c e s i n the p r e s e n t work, t h e spectrum i s o f d i r e c t p h y s i c a l i n t e r e s t . T h e r e f o r e , t he a u t o c o r r e l a t i o n ( a u t o c o v a r i a n c e ) f u n c t i o n was n o t used e x t e n s i v e l y i n o u r d a t a a n a l y s i s . I n s t e a d , power s p e c t r a l d e n s i t y f u n c t i o n was employed, w i t h t he a u t o c o r r e l a t i o n f u n c t i o n used m a i n l y as an i n t e r m e d i a t e s t e p i n the - 86 -e s t i m a t i o n o f the power s p e c t r a l d e n s i t y f u n c t i o n . The a u t o c o r r e l a t i o n f u n c t i o n o f a sample time h i s t o r y r e c o r d x ( t ) i s d e f i n e d a s : R ( T ) = l i m | / T x ( t ) . x ( t + T ) d t (4.11) T-*» o where, x ( t ) i s the v a l u e o f t h e sample r e c o r d a t time t , x ( t + x ) i s th e v a l u e o f t h e r e c o r d a t t i m e ( t + x ) , and x i s t h e l a g time between them. In some c a s e s , t h e a u t o c o v a r i a n c e f u n c t i o n i s used i n p l a c e o f t h e a u t o c o r r e l a t i o n f u n c t i o n i n data a n a l y s i s . The a u t o c o v a r i a n c e f u n c t i o n i s d e f i n e d a s : C x ( T ) = l i m 1 / { X ( t ) - u H x ( t + x ) - y x ) d t (4.12) T->°° o where, y x i s the mean v a l u e o f the da t a c a l c u l a t e d from e q u a t i o n ( 4 . 3 ) . From e q u a t i o n (4.11) and (4.12) C x ( x ) = R X ( T ) - P X Thus, t h e a u t o c o v a r i a n c e f u n c t i o n i s i d e n t i c a l w i t h the a u t o c o r r e l a t i o n f u n c t i o n when t h e mean v a l u e o f t h e d a t a i s z e r o . I t t u r n s o u t t h a t whenever y f 0, i t i s more c o n v e n i e n t t o work w i t h C ( x ) than R v ( x ) . An u n b i a s e d e s t i m a t e o f t h e a u t o c o v a r i a n c e f u n c t i o n o f a s t a t i o n a r y ( e r g o d i c ) random p r o c e s s based on a s i n g l e sample t i m e - h i s t o r y r e c o r d - 87 -x ( t ) i s given by: C ( T ) = j - ! — n J (x ( t ) -y }{x(t + T) - y }dt (0 < i < T) X I - x O x X " (4.13) If x ( t ) i s a transformed record (sampled at time interval A t ) , then the estimated a u t o v a r i a n c e function at lag time, r = r A t is defined by: N-r C ¥ ( T ) = C (rat) = ^~ I < xi " * ) ( x i + r " x ) r = 0,1,2,....m x v x N-r i=1 (4.14) where, C (T) i s the estimate of the true value of the autocovariance function. N i s the number of data points in the record x(t) r i s the lag number = x / A t m i s the maximum lag number = x / A t a max x is the data mean value calculated from equation (4.5) The subroutine BMD02T from the UBC Library was used to calculate the autocovariance function of the data samples. 3. Power spectral density function: The power spectral density function describes the frequency composition of the data in terms of the spectral density of i t s mean square v a l u e . Hence, the data spectrum provides a measure of the - 88 -f r e q u e n c y d i s t r i b u t i o n o f t h e mean squa r e v a l u e o f the d a t a . A l s o , the power spectrum p l o t can be used i n the d e t e c t i o n o f p e r i o d i c i t i e s i n t h e d a t a as d e s c r i b e d i n S e c t i o n 4.2.2. The a u t o s p e c t r u m i s d e f i n e d as the F o u r i e r t r a n s f o r m o f t h e a u t o c o r r e l a t i o n f u n c t i o n ( B e n d a t , 1971; Bendat, 1980; M e i r o v i t c h , 1975). Hence, the a u t o s p e c t r u m o f t h e t i m e -h i s t o r y r e c o r d x ( t ) i s : co . S x ( f ) = / R x(x) e " j 2 * f T d T ( - c c < f < c c ) (4.15) — c o The s p e c t r a l d e n s i t y f u n c t i o n i n e q u a t i o n (4.15) i s d e f i n e d o v e r a l l f r e q u e n c i e s , both p o s i t i v e and n e g a t i v e , and i s r e f e r r e d t o as a two s i d e d s p e c t r a . However, s i n c e t h e power spectrum i s an even f u n c t i o n , e q u a t i o n (4.15) can be w r i t t e n as G x ( f ) = 2 S x ( f ) = 2 / R x(x) e " j 2 7 T f x dx (0 < f < -) (4.16) — co where G ( f ) i s the o n e - s i d e d s p e c t r a l d e n s i t y f u n c t i o n . The a u t o -c o r r e l a t i o n f u n c t i o n , R (T), i s an even f u n c t i o n o f x; hence the a u t o s p e c t r a i n e q u a t i o n (4.16) a r e g i v e n by the r e a l p a r t o f the F o u r i e r t r a n s f o r m , i . e . co G v ( f ) = 4 J R v(x) cos 2^fx dx (0 < f ) (4.17) A A o In some r e f e r e n c e s ( e . g . J e n k i n s , 1968), t h e a u t o s p e c t r u m d e n s i t y f u n c t i o n i s d e f i n e d as the F o u r i e r t r a n s f o r m o f the a u t o c o v a r i a n c e f u n c t i o n , i . e . - 89 -S x ( f ) = J C x(x) e " J ^ T T dx (-00 < f < 00) (4.18) When t h e mean v a l u e o f t h e d a t a i s z e r o , i . e . y = 0, t h e two d e f i n i t i o n s X g i v e n by e q u a t i o n s (4.15) and (4.18) a r e a c t u a l l y the same, s i n c e R V ( T ) = C V ( T ) . However, i f y * 0, e q u a t i o n (4.15) can be w r i t t e n A A X i n t he form: co S x ( f ) = J C x(x) e " j 2 l T f T dx + y 2 6 ( f ) ( - » < f < « , ) -co X (4.19) The d i f f e r e n c e between e q u a t i o n s (4.18) and (4.19) i s i n the f i n a l 2 term, y^ 6 ( f ) , where 6 ( f ) i s the d e l t a f u n c t i o n . A nonzero mean v a l u e appears i n the s p e c t r a l d e n s i t y f u n c t i o n as a d e l t a f u n c t i o n a t f = 0. T h e r e f o r e , whether t h e power spectrum f u n c t i o n i s d e f i n e d a c c o r d i n g t o e q u a t i o n (4.15) o r e q u a t i o n ( 4 . 1 8 ) , the g e n e r a l shape o f the power spectrum o f the d a t a i s the same, e x c e p t a t f = 0 and then o n l y i f the mean v a l u e o f t h e data does not equal z e r o . The same subprogramme (BMD02T) used i n t h e c a l c u l a t i o n o f t h e a u t o c o v a r i a n c e f u n c t i o n o f the d a t a samples, was used t o c a l c u l a t e t h e power a u t o s p e c t r u m e s t i m a t e s o f t h e d a t a r e c o r d s . The s u b r o u t i n e i s based on e q u a t i o n (4.18) a f t e r the a u t o c o v a r i a n c e f u n c t i o n i s e v a l u a t e d u s i n g e q u a t i o n ( 4 . 1 4 ) . The a r e a under the a u t o s p e c t r u m p l o t between any two f r e q u e n c y l i m i t s f-| and fr, g i v e s the mean squa r e v a l u e o f t h e d a t a w i t h i n t h a t - 90 -f r e q u e n c y range. In e q u a t i o n form t h i s can be w r i t t e n as (4.20) 1 4. P r o b a b i l i t y d e n s i t y f u n c t i o n : The p r o b a b i l i t y d e n s i t y f u n c t i o n p r o v i d e s a d e s c r i p t i o n o f t h e p r o p e r t i e s o f the data i n the a m p l i t u d e domain, and can a l s o be used t o d i s t i n g u i s h between p e r i o d i c and random d a t a as d e s c r i b e d i n S e c t i o n 4.2.2. C o n s i d e r a sample time h i s t o r y r e c o r d x ( t ) . The p r o b a b i l i t y t h a t x ( t ) assumes p a r t i c u l a r a m p l i t u d e v a l u e s between (x - z/2) and (x + z/2) d u r i n g a t i m e i n t e r v a l T may be w r i t t e n as where, z i s the w i d t h o f the narrow i n t e r v a l c e n t e r e d a t x and T i s the t o t a l amount o f time t h a t x ( t ) f a l l s i n s i d e the range [ ( x - z / 2 ) , (x + z / 2 ) ] d u r i n g an o b s e r v a t i o n time T. I f x ( t ) i s a s i n g l e sample t i m e h i s t o r y r e c o r d from a s t a t i o n a r y ( e r g o d i c ) random p r o c e s s , then an u n b i a s e d e s t i m a t e o f the t r u e p r o b a b i l i t y may be o b t a i n e d (Bendat, 1971) from: T Prob. [ (x - z/2) < x ( t ) <_ (x + z/2) ] = l i m (4.21) x T P(x,z) = Prob. [ (x - z/2) < x ( t ) < (x + z/2) ] = ^ (4.22) The p r o b a b i l i t y d e n s i t y f u n c t i o n p(x) i s d e f i n e d (Bendat, 1971) a s : - 91 -Hence, p ( x ) = l i m P l i L s i i ( 4 2 3 ) z+o p ( X ) = = ! x _ ( 4 2 4 ) I f x ( t ) i s a t r a n s f o r m e d r e c o r d w i t h N d a t a v a l u e s , t h e n t h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f x ( t ) may be e s t i m a t e d v i a : N where, N x i s the number o f d a t a v a l u e s which f a l l w i t h i n t he range (x ± z/2). The subprogram FREQ from the UBC L i b r a r y was used i n c a l c u l a t i n g the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r s e l e c t e d sample data r e c o r d s . The h i s t o g r a m o f t h e c a l c u l a t e d d i s t r i b u t i o n was t h e n p l o t t e d . - 92 -C H A P T E R 5 S I N G L E T U B E E X P E R I M E N T S : R E S U L T S AND D I S C U S S I O N 5.1 I n t r o d u c t i o n In t h i s c h a p t e r , e x p e r i m e n t a l r e s u l t s o f f o r c e measurements on a s i n g l e tube i n i s o l a t i o n a r e p r e s e n t e d and d i s c u s s e d i n r e l a t i o n t o hydrodynamic c o n d i t i o n s p r e v a i l i n g w i t h i n the bed. The e f f e c t s o f the f l u i d - b e d o p e r a t i n g parameters on the c h a r a c t e r i s t i c s o f the f o r c e s a r e d e m o n s t r a t e d i n S e c t i o n 5.2. The e f f e c t s o f u s i n g tubes o f d i f f e r e n t s i z e and shape on t h e c h a r a c t e r i s t i c s o f t h e f o r c e s are shown i n S e c t i o n 5.3. The causes o f the f o r c e s a r e i d e n t i f i e d and d i s c u s s e d i n S e c t i o n 5.4. F i n a l l y , t h e g e n e r a l p r o p e r t i e s o f t h e f o r c e s under s t u d y are d i s c u s s e d w i t h r e s p e c t t o the hydrodynamics o f b u b b l i n g f l u i d i z e d beds i n S e c t i o n 5.5. 5.2 E f f e c t s o f F l u i d - B e d Parameters on t h e F o r c e s on an Immersed Tube In o r d e r t o s t u d y the n a t u r e o f t h e e x t e r n a l f o r c e s on an immersed tube w i t h i n a f l u i d i z e d bed and t h e i r r e l a t i o n t o t h e hydrodynamics o f the bed, e x p e r i m e n t s were con d u c t e d t o examine the e f f e c t s o f the f l u i d - b e d parameters on the c h a r a c t e r i s t i c s o f the measured f o r c e s . - 93 -Force measurements were made on a tube o f o u t e r d i a m e t e r 32 mm w i t h t h e a x i s o f t h e tube 300 mm above the gas d i s t r i b u t o r . E x p e r i m e n t a l arrangements and p r o c e d u r e s are o u t l i n e d i n Chapter 3. The parameters v a r i e d i n the e x p e r i m e n t s were s u p e r f i c i a l gas v e l o c i t y , s t a t i c bed h e i g h t , p a r t i c l e s i z e and p a r t i c l e d e n s i t y . The e x p e r i m e n t a l d a t a a r e p r e s e n t e d i n terms o f t h e i r s t a t i s t i c a l p a r a m e t e r s . However, some r e p r e s e n t a t i v e samples o f f o r c e - t i m e r e c o r d s a r e a l s o p r e s e n t e d . As d i s c u s s e d i n C h a p t e r 4, s t a t i s t i c a l p arameters were c a l c u l a t e d from d a t a r e c o r d s o f 9000 d a t a p o i n t s e a c h , which m i n i m i z e d s t a t i s t i c a l e r r o r s . 5.2.1 E f f e c t s o f s u p e r f i c i a l gas v e l o c i t y V e r t i c a l components o f f o r c e : The v e r t i c a l components o f f o r c e on t h e t e s t c y l i n d e r were measured a t e x c e s s s u p e r f i c i a l gas v e l o c i t i e s , (U - U m f ) > °f 0.05, 0.1, 0.2, 0.3, 0.5, 0.8, 1.2 and 1.4 m/s w h i l e t h e o t h e r bed parameters (bed m a t e r i a l : sand, d p = 430 pm, p p = 2600 Kg/m , H o = 0.3 m) remained unchanged. V e r t i c a l f o r c e d a t a (upward d i r e c t i o n t a k e n as p o s i t i v e ) from t h e f o r c e t r a n s d u c e r s a t t h e o p p o s i t e ends o f the t u b e , r e c o r d e d f o r (U - U f ) = 0.05, 0.3 and 1.4 m/s, a r e shown i n F i g u r e s 5-1-a, 5-2-a and 5-3-a r e s p e c t i v e l y . In F i g u r e 5-1-a t h e f o r c e - t i m e h i s t o r y c o n s i s t s o f a s e r i e s o f p u l s e s , 2-3 p u l s e s per s e c o n d . Most o f the p u l s e s are between 1 and 2 N i n magnitude, w i t h a d u r a t i o n ( t i m e e l a p s e d d u r i n g o c c u r r e n c e o f a p u l s e ) o f 0.2 t o 0.4 S. The p u l s e s o c c u r a t each end s i m u l t a n e o u s l y . F ig. 5-1-a. V e r t i c a l f o r c e s at o p p o s i t e ends of the t e s t c y l i n d e r r e co rded at (U-Umf) = 0.05 m/s. Bed m a t e r i a l : sand; dp = 430 ym; H 0 = 0.3 m. o 4.0 F R E Q U E N C Y 32.0 (HZ) 3G.0 20.0 F i g . 5-1-b. Power spectral estimates of tota l ve r t i ca l forces for the same experimental conditions as 5-1-a. - 95 -The magnitudes a r e g e n e r a l l y o f t h e same o r d e r , but d i f f e r somewhat because n o t a l l r i s i n g b u b b l e s pass t h e t u b e near t h e m i d d l e . The t o t a l v e r t i c a l f o r c e a r e o b t a i n e d by summing the v e r t i c a l f o r c e s i g n a l from each gage. The power s p e c t r a l e s t i m a t e s o f the t o t a l v e r t i c a l f o r c e were c a l c u l a t e d and a r e p l o t t e d i n F i g u r e 5-1-b. The power s p e c t r a l e s t i m a t e s p l o t has a peak a t 2.5 HZ. In F i g u r e 5-2-a, where (U - U ^) has been i n c r e a s e d t o 0.3 m/s, the f o r c e a g a i n appears as p u l s e s a t a r a t e o f 2-3 p u l s e s per second but the magnitudes a r e now between 4 and 7 N, s e v e r a l t i m e s l a r g e r than t h e s e measured a t (U - U .^) = 0.05 m/s ( c f . F i g u r e 5-1 - a ) . The d u r a t i o n o f t h e p u l s e s i s about 0.08-0.25 s, somewhat s h o r t e r than the d u r a t i o n a t t h e p r e v i o u s s u p e r f i c i a l gas v e l o c i t y , presumably because o f t h e h i g h e r b u b b l e s v e l o c i t y a t h i g h e r gas f l o w r a t e . The power s p e c t r a l e s t i m a t e s o f t h e t o t a l v e r t i c a l f o r c e o f t h i s r e c o r d appear i n F i g u r e 5-2-b. The c u r v e has a peak a t 2 HZ which c o r r e s p o n d s to t h e r a t e a t which p u l s e s o c c u r i n t h i s sample o f d a t a . There i s some s i m i l a r i t y between the f o r c e - t i m e r e c o r d shown i n F i g u r e 5-3-a and t h e two p r e v i o u s f o r c e r e c o r d s . The f o r c e a g a i n a p p e a r s as p u l s e s o c c u r r i n g a t r a t e 2-3 per s e c o n d . However, the magnitudes o f t h e p u l s e s a r e t y p i c a l l y 10-22 N, about t h r e e times t h o s e measured a t (U - U ^) = 0.3 m/s, and t h e d u r a t i o n o f the p u l s e s i s o n l y 0.04-0.12 s. The power s p e c t r a l e s t i m a t e s , shown i n F i g u r e 5-3 i n d i c a t e a major f r e q u e n c y a t 2.5 HZ. These t h r e e samples o f d a t a d e m o n s t r a t e t h a t t h e v e r t i c a l f o r c e component appea r s as a s e r i e s o f p u l s e s . The magnitudes and d u r a t i o n o f t h e s e p u l s e s a r e s t r o n g l y i n f l u e n c e d by the s u p e r f i c i a l gas v e l o c i t y F i g . 5 - 2 - a . V e r t i c a l f o r c e s a t o p p o s i t e e n d s o f t h e t e s t c y l i n d e r r e c o r d e d a t ( U - U m f ) = 0.3 m/s. B e d m a t e r i a l : s a n d ; dp = 4 3 0 ym; H 0 = 0.3 m. 0.0 4.0 8.0 12.0 ]6.0 20.0 FREQUENCY (HZ) F i g . 5 - 2 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 2 - a . F i g . 5 - 3 - a . V e r t i c a l f o r c e s a t o p p o s i t e e n d s o f t h e t e s t c y l i n d e r r e c o r d e d a t ( U - U m f ) = 1.4 m/s. Bed m a t e r i a l : s a n d ; dp = 43 0 ym; H 0 = 0.3 m. - 93 -H i g h e r gas v e l o c i t i e s l e d t o l a r g e r a m p l i t u d e s and s h o r t e r p u l s e d u r a t i o : However t h e f r e q u e n c y o f p u l s e o c c u r r e n c e was a p p r o x i m a t e l y c o n s t a n t . The i n s e n s i t i v i t y o f the peak f r e q u e n c y t o v a r i a t i o n s i n (U - U m^) i s even more e v i d e n t i n F i g u r e 5 - 4 . I t i s o b v i o u s t h a t t h e e f f e c t o f the e x c e s s s u p e r f i c i a l gas v e l o c i t y on t h e c h a r a c t e r i s t i c s o f the measured f o r c e s i s s i m i l a r t o t h a t on bubble p r o p e r t i e s d i s c u s s e d i n Chapter 2 . The c h a r a c t e r i s t i c s o f t h e f o r c e t h e r e f o r e appear t o r e f l e c t bubble p r o p e r t i e s . Hence, t h e measured f o r c e s and r i s i n g b u b b l e s a r e c l o s e l y r e l a t e d , and t h e c h a r a c t e r i s t i c s o f the f o r c e can o n l y be i n t e r p r e t e d i n terms o f bubble b e h a v i o u r . More e x p e r i m e n t a l and a n a l y t i c a l work w i l l be p r e s e n t e d l a t e r i n t h i s c h a p t e r t o c o n f i r m t h i s r e l a t i o n s h i p . The root-mean-square (RMS) v a l u e s o f the measured f o r c e s are p l o t t e d a g a i n s t e x c e s s gas v e l o c i t y , (U - U ^ ) , i n F i g u r e 5 - 5 . The RMS v a l u e i n c r e a s e s l i n e a r l y w i t h i n c r e a s i n g gas v e l o c i t y a t s m a l l v a l u e s o f (U - U ^ ) , but l e v e l s o f f a t h i g h e r v a l u e s o f (U - U ^ ) . T h i s i n d i c a t e s t h a t t h e v e r t i c a l f o r c e s l e v e l o f f a t h i g h gas v e l o c i t i e s . As the gas v e l o c i t y i n c r e a s e s , the bed expands f u r t h e r and f u r t h e r . Hence, r i s i n g b u b b l e s t r a v e l l o n g e r d i s t a n c e s above the tube i n the upper p o r t i o n o f the bed toward t h e expanded bed s u r f a c e . As a r e s u l t , a t h i g h gas v e l o c i t i e s t he downward component o f the v e r t i c a l f o r c e ( c a u s e d p a r t i a l l y by the a c t i o n o f t h e bubbles p r e s s u r e f i e l d d u r i n g t h e i r t r a v e l l i n g i n the second p o r t i o n o f the bed) i n c r e a s e s a t a h i g h e r r a t e than t h a t o f t h e upward component o f f o r c e . The v e c t o r i a l summation o f the upward and downward components a t any moment r e p r e s e n t s the i n s t a n t a n e o u s t o t a l v e r t i c a l f o r c e . T h e r e f o r e , the i n t e n s i t y o f the v e r t i c a l f o r c e s l e v e l s o f f o r even d e c l i n e s w i t h an i n c r e a s e i n (U - U f ) a t h i g h gas v e l o c i t i e s . 0 0.5 1.0 1.5 ( U - U m f ),m/s Fig. 5 - 4 Variation of the vertical force major frequency with excess superficial velocity. Bed material1 sand, dp = 430 jtim , H0= 0.3 m. - 100 -UJ o CL O < o I-cr UJ > CO 0 0.5 1.0 1.5 (U-Umf), m/s Fig. 5 - 5 . Variation of RMS vertical force with excess superficial gas velocity (U - U m f ) . Bed material' sand, d p = 430 /*m, H 0 = 0.3 m - 101 -Other f a c t o r s may a l s o c o n t r i b u t e t o the e v e n t u a l d e c l i n e o f the i n t e n s i t y o f f o r c e s w i t h i n c r e a s i n g s u p e r f i c i a l gas v e l o c i t y . The o n s e t o f t r a n s i t i o n t o t u r b u l e n t f l u i d i z a t i o n , and i n c r e a s i n g e n t r a p -ment and hold-up o f s o l i d p a r t i c l e s i n the f r e e - b o a r d o f the f l u i d i z i n g column a r e two such f a c t o r s . These causes have a moderate r o l e i n the p r e s e n t c a s e , but they become more i m p o r t a n t w i t h s m a l l e r and l i g h t e r s o l i d p a r t i c l e s . T h i s i s i n v e s t i g a t e d and d i s c u s s e d f u r t h e r below. The mean v a l u e o f t h e measured f o r c e s i s p l o t t e d v e r s u s (U - U ^) i n F i g u r e 5-6. The mean v a l u e c a n be r e g a r d e d as t h e s t a t i c component o f f o r c e . At low l e v e l s o f s u p e r f i c i a l gas v e l o c i t y , c l o s e t o t h a t o f minimum f l u i d i z a t i o n where the buoyancy f o r c e o f the bed r e p r e s e n t s the b i g g e s t c o n t r i b u t i o n t o the t o t a l v e r t i c a l f o r c e on the t u b e , the mean v a l u e i s a p p r o x i m a t e l y equal t o the d i f f e r e n c e between the buoyancy f o r c e and t h e w e i g h t o f s o l i d p a r t i c l e s s u p p o r t e d i n the d e f l u i d i z e d cap above t h e t u b e . E x t r a p o l a t i o n o f the mean v a l u e p l o t shows a net f o r c e o f v a l u e 1 .6 N at (U - U ^) = 0. At t h e s e c o n d i t i o n s , the e s t i m a t e d v a l u e o f t h e buoyancy f o r c e on the tube i s 2.5 N ( i n the upward d i r e c t i o n ) ; w h i l e t h e volume o f the d e f l u i d i z e d cap above the t u b e c o u l d be e s t i m a t e d (by means o f v i s u a l o b s e r v a t i o n and by t h e h e l p o f i n f o r m a t i o n o b t a i n e d from e a r l i e r work ( e . g . Hager and S c h r a g , 1976) t o g i v e an e s t i m a t e d v a l u e o f the w e i g h t o f the s o l i d p a r t i c l e s i n t h e s t a g n a n t cap o f 1 N (downward). T h e r e f o r e , the e s t i m a t e d net s t a t i c f o r c e on t h e tube i s 1.5 N i n the upward d i r e c t i o n , i n good agreement w i t h t h e v a l u e 1.6 N o b t a i n e d by e x t r a p o l a t i n g the e x p e r i m e n t a l d a t a p l o t . The mean v a l u e o f f o r c e l e v e l s o f f a t high (U - U m f ) f o r the same r e a s o n s as the RMS v a l u e s , d i s c u s s e d above. 102 -10 UJ Q o o or o < o I-or ui > Lu O 5 > O CO Q < LU _J 2 < LU I - Mean values 2 - S t a n d a r d deviation 0.5 1.0 (U-U m f ) , m/s Fig. 5-6. Variation of mean value and standard deviation of vertical force with excess superficial gas velocity. Bed material: sand, d~p = 430 pm , H0 » 0.3 m. 1.5 - 103 -The s t a n d a r d d e v i a t i o n o f the f o r c e s , which p r o v i d e s a measure o f the dynamic components, i s p l o t t e d a g a i n s t (U - U ^) i n F i g u r e 5-6. The dynamic component o f t h e measured f o r c e i s a v a r i a b l e l o a d b e l i e v e d t o be caused m a i n l y by bubbles motion and a s s o c i a t e d s o l i d c i r c u l a t i o n . At minimum f l u i d i z a t i o n , where r i s i n g b u b b l e s a r e small i n s i z e and r i s e r e l a t i v e l y s l o w l y , the dynamic component i s e x p e c t e d t o be small . E x t r a p o l a t i n g the p l o t i n F i g u r e 5-6 r e s u l t s a p p r o x i m a t e l y i n a z e r o v a l u e o f t h e s t a n d a r d d e v i a t i o n a t minimum f l u i d i z a t i o n . As the gas v e l o c i t y i n c r e a s e s bubbles become l a r g e r and r i s e more q u i c k l y . Hence, the bubbles and a s s o c i a t e d s o l i d s become a b l e t o t r a n s f e r h i g h e r momentum t o the t u b e . Hence the dynamic component o f f o r c e i n c r e a s e s c o n t i n u o u s l y as (U - U ^ ) i n c r e a s e s . H o r i z o n t a l components o f f o r c e s : The h o r i z o n t a l components o f f o r c e on t h e t e s t c y l i n d e r were measured a t t h e same bed c o n d i t i o n s as i n the v e r t i c a l f o r c e measure-ments. The e x c e s s s u p e r f i c i a l gas v e l o c i t y was a g a i n v a r i e d from 0.05 t o 1.4 m/s. Three r e p r e s e n t a t i v e samples o f h o r i z o n t a l f o r c e -time h i s t o r i e s a r e shown i n F i g u r e s 5-7-a, 5-8-a and 5-9-a; t h e s e samples were c o l l e c t e d a t (U - U m^) = 0.3, 0.8 and 1.4 m/s r e s p e c t i v e l y . These t h r e e samples i n d i c a t e t h a t t h e h o r i z o n t a l f o r c e o s c i l l a t e s from s i d e t o s i d e i n p u l s e s , w i t h a mean v a l u e near z e r o . The magnitudes and d u r a t i o n s o f t h e p u l s e s depend on s u p e r f i c i a l gas v e l o c i t y . The magnitudes a r e l a r g e r and t h e d u r a t i o n s s h o r t e r as U i n c r e a s e s , but t h e changes become l e s s s i g n i f i c a n t a t h i g h l e v e l s o f (U - U ^ ) . The m agnitudes o f h o r i z o n t a l f o r c e p u l s e s a r e t y p i c a l l y about o n e - t h i r d o f t h o s e o f the c o r r e s p o n d i n g v e r t i c a l f o r c e . T y p i c a l examples F i g . 5 - 7 - a . H o r i z o n t a l f o r c e s a t o p p o s i t e e n d s o f t h e t e s t c y l i n d e r r e c o r d e d a t ( U - U m f ) = 0.3 m/s. B e d m a t e r i a l : s a n d ; dp = 4 3 0 y m ; H 0 = 0 . 3 m . o ,—,o 0.0 4.0 8.0 32.0 ]6.0 20.0 FREQUENCY (HZ) F i g . 5 - 7 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l h o r i z o n t a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 7 - a . F i g . 5 - 8 - a . H o r i z o n t a l f o r c e s a t o p p o s i t e e n d s o f t h e t e s t c y l i n d e r r e c o r d e d a t ( L ) - L ) m f ) = 0.8 m/s. B e d m a t e r i a l : s a n d ; dp = 4 3 0 ym; H 0 = 0.3 m. F i g f o r . 5 - 8 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l h o r i z o n t a l f o r c e s t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 8 - a . - 106 35 4d r—\ z w u O u. 3 TIME CSJ F i g . 5 - 9 - a . H o r i z o n t a l f o r c e s a t o p p o s i t e e n d s o f t h e t e s t c y l i n d e r r e c o r d e d a t ( U - L ) m f ) = 1.4 m/s. Bed m a t e r i a l : s a n d ; dp = 4 3 0 ym; H 0 = 0.3 m. ,—>o IE 0.0 4.0 8.0 ]2.0 ]6.0 20.0 F R E Q U E N C Y ( H Z ) F i g . 5 - 9 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l h o r i z o n t a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s i n - 5 - 9 - , a . - 107 -a r e p r o v i d e d by comparing the h o r i z o n t a l f o r c e data i n F i g u r e 5-7-a w i t h the v e r t i c a l f o r c e d a t a i n F i g u r e 5-2-a and t h e data i n F i g u r e 5-8-a w i t h t h o s e i n F i g u r e 5-3-a. The power s p e c t r a l e s t i m a t e s o f the t h r e e r e p r e s e n t a t i v e samples appear i n F i g u r e s 5-7-b, 5-8-b and 5-9-b. The f r e q u e n c y c o n t e n t o f h o r i z o n t a l f o r c e s i s i n g e n e r a l s i m i l a r t o t h a t o f v e r t i c a l f o r c e s . However, t h e major f r e q u e n c i e s a r e between 0.25 and 1 HZ as shown i n F i g u r e 5-10. T h i s c o r r e s p o n d s to t h e r a t e a t w h ich p u l s e s o f s i g n i f i c a n t magnitude o c c u r i n the f o r c e - t i m e h i s t o r i e s . The root-mean-square (RMS) and mean v a l u e s o f the h o r i z o n t a l components o f the measured f o r c e a r e p l o t t e d a g a i n s t (U - U ^) i n F i g u r e 5-11. The RMS and mean v a l u e s o f t h e v e r t i c a l components o f t h e same f o r c e a r e shown i n t h e same f i g u r e f o r c o m p a r i s o n . The RMS h o r i z o n t a l f o r c e i s g e n e r a l l y about o n e - q u a r t e r o f the RMS v e r t i c a l f o r c e . As shown i n t h e f i g u r e , t h e RMS h o r i z o n t a l f o r c e s t a r t e d t o l e v e l o f f a t lower v a l u e s o f (U - U ,-) than t h a t o f v e r t i c a l f o r c e s . mf T h i s might be e x p l a i n e d i n terms o f bubble b e h a v i o r around an immersed tu b e . As (U - U ^) i n c r e a s e s bubbles grow i n s i z e c a u s i n g h i g h e r and more i n t e n s i v e v e r t i c a l f o r c e s as e x p l a i n e d b e f o r e , but l a r g e bubbles u s u a l l y behave around t h e tube i n a d i f f e r e n t way than s m a l l e r ones (Nguyen e t a l . , 1978; G r a c e , 1982; H a r r i s o n and G r a c e , 1971; Hager and Thomson, 1973). As b u b b l e s become l a r g e r than the t u b e , t h i s i n c r e a s e s the p o s s i b i l i t y o f b u b b l e s e n v e l o p i n g the tube o r s p l i t t i n g i n t o two p a r t s o f a l m o s t equal s i z e . ' T h i s b e h a v i o r may l e a d t o n e a r l y b a l a n c i n g h o r i z o n t a l f o r c e s on the two s i d e s o f the t u b e . P r o b a b l y , o n l y bubbles p a s s i n g the t u b e a t one s i d e o r a t t h e o t h e r t h e n c a u s e h o r i z o n t a l f o r c e p u l s e s o f l a r g e a m p l i t u d e . T h i s may a l s o h e l p t o e x p l a i n why the major f r e q u e n c i e s o f h o r i z o n t a l f o r c e s a r e l e s s than t h a t o f v e r t i c a l 4 or o 3 I o o 0 0.5 1.0 1.5 ( U - U m f ) , m/s Fig. 5-10. Variation of the horizontal force major frequency with excess superficial velocity. Bed material •  sand, d p = 430 /trn, H0= 0.3 m. - 1 09 -0.5 1.0 1.5 (U-U m , ) ,m/s Fig. 5-11. Variation of RMS and mean value of horizontal components of force with excess superficial velocity, and in comparison with RMS and mean value of vertical components of the force. - n o -f o r c e s . E x t r a p o l a t i o n o f t h e RMS p l o t l e d t o a v a l u e v e r y c l o s e t o z e r o a t (U - U ^) = 0. T h i s i s e x p e c t e d s i n c e t h e s t a t i c f o r c e c o n t r i -b u t i o n t o t h e h o r i z o n t a l f o r c e components i s n u l l and t h e dynamic f o r c e c o n t r i b u t i o n (caused by b u b b l e s and a s s o c i a t e d s o l i d motion) s h o u l d be v e r y s m a l l a t minimum f l u i d i z a t i o n . T h i s r e s u l t a g a i n i n d i c a t e s t h a t dynamic f o r c e s on an immersed tube a r e c a used m a i n l y by b u b b l e s and a s s o c i a t e d s o l i d s m o t i o n . As e x p e c t e d , t h e time-mean v a l u e o f h o r i z o n t a l f o r c e components i s a p p r o x i m a t e l y z e r o . T h i s i s shown i n F i g u r e 5-11 and can be o b s e r v e d a l s o from the f o r c e - t i m e r e c o r d s ( F i g u r e s 5-7-a, 5-8-a and 5-9-a). 5.2.2 E f f e c t s o f s t a t i c bed h e i g h t The e f f e c t s o f s t a t i c bed h e i g h t s on the c h a r a c t e r i s t i c s o f t h e measured f o r c e s were examined by c a r r y i n g out measurements o f the v e r t i c a l and h o r i z o n t a l components o f f o r c e w i t h two d i f f e r e n t bed depths (H = 0.30 and 0.45 m) o v e r a wide range o f e x c e s s s u p e r f i c i a l gas v e l o c i t y (0.05 - 1.4 m/s), w h i l e the o t h e r bed v a r i a b l e s were m a i n t a i n e d unchanged (bed m a t e r i a l : sand, d p = 430 ym). The h e i g h t o f the t e s t c y l i n d e r above t h e d i s t r i b u t o r p l a t e was always the same, 0.3 m above the d i s t r i b u t o r . V e r t i c a l components o f f o r c e s : F i g u r e s 5-12-a and 5-13-a show two samples o f the measured v e r t i c a l f o r c e d a t a r e c o r d e d w h i l e t h e bed depth ( s t a t i c ) was 0.45 and 0.30 m r e s p e c t i v e l y . The e x c e s s gas v e l o c i t y was 1.2 m/s i n both c a s e s . In both f i g u r e s t h e f o r c e - t i m e r e c o r d i s a s e r i e s o f p u l s e s . A l t h o u g h , t h e g e n e r a l c h a r a c t e r i s t i c s o f t h e f o r c e s i g n a l s are s i m i l a r , t h e F i g . 5 - 1 2 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s t a t i c d e p t h 0.45 m. B e d m a t e r i a l : s a n d ; dp = 430 u ( U - U m f ) = 1.2 m/s. H P F i g f o r . 5 - 1 2 - b . t h e same P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l e x p e r i m e n t a l c o n d i t i o n s a s i n 5 - 1 2 - a . f o r c e s F i g . 5 - 1 3 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s t a t i c d e p t h 0.30 m. B e d m a t e r i a l : s a n d ; dp = 4 3 0 um; ( U - U m f ) = 1.2 m/s . F i g f o r . 5 - 1 3 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s t h e same e x p e r i m e n t a l c o n d i t i o n s a s i n 5 - 1 3 - a . - 113 -m a g n i t u d e s , d u r a t i o n s , and r a t e o f p u l s e o c c u r r e n c e a r e somewhat d i f f e r e n t . In F i g u r e 5-12-a most o f t h e p u l s e s a r e between 10-20 N i n magnitude, w i t h a d u r a t i o n o f 0.12-0.32 s, w h i l e t h e p u l s e s appear a t 1-2 per s e c o n d ; i n F i g u r e 5-13-a most o f the p u l s e s a r e between 9-18 N i n magnitude w i t h a d u r a t i o n o f 0.05-0.16 s and a t a r a t e o f 2-3 per s e c o n d . These show t h a t i n the deeper bed the magnitude o f the f o r c e p u l s e s i s l a r g e r and t h e r a t e o f p u l s e o c c u r r e n c e l e s s . These t r e n d s may be i n t e r p r e t e d i n t h e l i g h t o f the d i f f e r e n t hydrodynamic e v e n t s w i t h each bed d e p t h . F i r s t , as p o i n t e d out by Whitehead ( 1 9 7 9 ) , an i n c r e a s e i n bed depth enhances the speed o f b u b b l e s c o a l e s c e n c e i n the immediate v i c i n i t y o f t h e d i s t r i b u t o r due t o the development o f g u l f - s t r e a m i n g . Hence, bub b l e s a r e e x p e c t e d t o be l a r g e r i n s i z e and l e s s f r e q u e n t i n the deeper bed, l e a d i n g t o l e s s f r e q u e n t and l a r g e r a m p l i t u d e f o r c e p u l s e s . The second c o n t r i b u t i n g f a c t o r may be t h a t r i s i n g bubbles have more chance o f c a t c h i n g each o t h e r around the tube i n t he deeper bed t o form b i g g e r b u b b l e s c a u s i n g l a r g e r p u l s e a m p l i t u d e s . T h i s a l s o may be the r e a s o n why some o f the f o r c e p u l s e s show two peaks d u r i n g one p u l s e as i n F i g u r e 5-12-a. F i g u r e s 5-12-b and 5-13-b a r e the s p e c t r a l e s t i m a t e p l o t s o f the c o r r e s p o n d i n g d a t a samples. The f r e q u e n c y c o n t e n t s o f both samples a r e s i m i l a r , but the major f r e q u e n c y o f t h e f i r s t sample i s 1.5 HZ, w h i l e i t i s 2 HZ f o r the second.The v a r i a t i o n s o f the major f r e q u e n c y w i t h e x c e s s s u p e r f i c i a l v e l o c i t y f o r t h e two bed depths appear i n F i g u r e 5-14. The major f r e q u e n c i e s o f t h e f o r c e s measured i n t h e deeper bed a r e seen t o be g e n e r a l l y l e s s than f o r the s h a l l o w bed f o r t h e same re a s o n s as d e s c r i b e d above. 8 N X >-o UJ a u i or o 6 L 4 U 2 L - O H 0 = 0.45 m H 0 = 0.30m 0.5 .0 1.5 (U -U m f ) , m/s Fig. 5-14. Variation of the major frequency with excess superficial velocity (U-U m f ) , for two different static bed heights = H0= 0.45 and 0.30 m. - 115 -The RMS v a l u e s o f the measured v e r t i c a l f o r c e a r e p l o t t e d a g a i n s t (U - U m f ) f o r t h e two bed depths i n F i g u r e 5-15. The RMS f o r c e i n the deeper bed i s always h i g h e r than t h a t i n the s h a l l o w bed because o f t h e l a r g e r a m p l i t u d e o f f o r c e p u l s e s . E x t r a p o l a t i o n o f the two p l o t s shown i n F i g u r e 5-15 appears t o l e a d t o a p p r o x i m a t e l y the same v a l u e a t (U - U m^) = 0 i n d i c a t i n g t h a t t h e s t a t i c f o r c e s a r e a p p r o x i -m a t e l y i n d e p e n d e n t o f bed depth as e x p e c t e d . In the deep bed and a t h i g h gas v e l o c i t i e s , the i n t e n s i t y o f f o r c e s l e v e l s o f f o r even d e c l i n e s (as d i s c u s s e d i n S e c t i o n 5.2.1). The v e r t i c a l f o r c e i n the downward d i r e c t i o n c a u s e d by c o l l a p s e o f t h e bed upper s u r f a c e a f t e r bubble e r u p t i o n i n c r e a s e s ( c a u s i n g d e c r e a s e i n the t o t a l v e r t i c a l f o r c e ) as s u p e r f i c i a l v e l o c i t y i n c r e a s e s . T h i s downward component may be more s i g n i f i c a n t i n deep beds because b u b b l e s e s c a p i n g the upper s u r f a c e a r e much l a r g e r t h a n i n s h a l l o w beds. Hence, d e c l i n e o f RMS o f f o r c e s o c c u r s i n t h e deep bed a t somewhat lower l e v e l s o f (U - U f ) than i n t h e shal1ow bed . The mean v a l u e s o f the f o r c e s a r e shown i n F i g u r e 5-16. The p l o t s f o r t h e two d i f f e r e n t bed depths a r e c o n s i s t e n t w i t h the RMS p l o t s and show t h e same t r e n d s . The mean v a l u e s o f f o r c e i n the deeper bed f a l l b e hind t h e mean v a l u e s i n t h e s h a l l o w e r bed a t h i g h v a l u e s o f (U - U ^ ) . The r e a s o n s are b e l i e v e d t o be the same as p o s t u l a t e d above. In s u p p o r t o f t h i s r e a s o n i n g note t h a t i n F i g u r e 5-13-a (H = 0.30 m) t h e f o r c e s i g n a l i s always p o s i t i v e , w h i l e i t shows n e g a t i v e v a l u e s i n the deeper bed, F i g u r e 5-12-a. - 116 -10 9 h 8 o S 5 < o cr LU > - O Static bed height (H 0) = 0.45 m - • - Static bed height (H 0) =0.30m 0.5 1.0 1.5 (U-U m f ) ,m/s Fig. 5-15. RMS vertical force vs. excess superficial gas velocity (U-U m f ) for two different bed heights1 H0 = 0.45 and 0.30 m. - 117 -I I 0 0.5 1.0 (U-U r af), m/s Fig. 5 - 16. Mean values of vertical force vs. excess superficial gas velocity ( U - U m f ) . - 118 -H o r i z o n t a l components o f f o r c e s : Samples o f h o r i z o n t a l f o r c e t r a c e s , r e c o r d e d a t bed depths o f 0.45 and 0.30 m, a r e shown i n F i g u r e s 5-17 and 5-18 r e s p e c t i v e l y . The samples show some t r e n d s s i m i l a r t o t h o s e o f the v e r t i c a l f o r c e samples. In p a r t i c u l a r , t h e magnitude o f f o r c e p u l s e s i n c r e a s e d as the bed depth i n c r e a s e d . O s c i l l a t i o n o f p u l s e s from one s i d e t o the o t h e r around an approxi m a t e z e r o mean v a l u e i s a c h a r a c t e r o f the measured f o r c e i n both samples. F i g u r e 5-19 p l o t s the RMS and mean v a l u e s o f the h o r i z o n t a l f o r c e measured i n the deep and s h a l l o w bed. The RMS v a l u e s a re c o n s i s t e n t w i t h t h e f o r c e - t i m e r e c o r d s , e x h i b i t i n g t he same t r e n d s as the RMS v e r t i c a l f o r c e w i t h the change i n bed d e p t h . E x t r a p o l a t i o n o f both RMS p l o t s i n F i g u r e 5-19 l e a d s to a p p r o x i m a t e l y z e r o f o r c e s at (U - U f ) = 0> i n d i c a t i n g t h a t t he s t a t i c h o r i z o n t a l f o r c e i s z e r o whatever t h e bed d e p t h . The mean v a l u e s o f t h e h o r i z o n t a l f o r c e a r e a p p r o x i m a t e l y z e r o , i n d e p e n d e n t o f bed depth and s u p e r f i c i a l v e l o c i t y . T h i s can a l s o be o b s e r v e d from the f o r c e - t i m e r e c o r d s o f d a t a . 5.2.3 E f f e c t s o f p a r t i c l e s i z e The e f f e c t o f u s i n g d i f f e r e n t p a r t i c l e s i z e was examined by c o n d u c t i n g f o r c e measurements w i t h t h r e e grades o f Ottawa sand: dp = 430, 280 and 185 ym. These t e s t s were r e p e a t e d a t d i f f e r e n t gas v e l o c i t i e s as s p e c i f i e d i n T a b l e 3.3, w i t h the s t a t i c bed h e i g h t c o n s t a n t a t 0.3 m. F i g . 5 - 1 7 . H o r i z o n t a l f o r c e s o n t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s t a t i c d e p t h 0.45 m. B e d m a t e r i a l : s a n d ; dp = 4 3 0 ym; ( U - U m f ) = 1 .2 m/s. F i g . 5 - 1 8 . H o r i z o n t a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s t a t i c d e p t h 0.30m. B e d m a t e r i a l : s a n d ; dp = 4 3 0 ym; ( U - U m f ) = 1 .2 m/s. - 120 -UJ o or o u. 0.5 1.0 1.5 ( U - U m f ) , m/s Fig.5-19. RMS and mean values of horizontal force vs. excess superficial velocity ( U - U m f ) for two cases of different bed heights • H0=0.45 and 0.3 m. - 121 -V e r t i c a l components o f f o r c e : Three r e p r e s e n t a t i v e samples o f f o r c e - t i m e h i s t o r y c o l l e c t e d f o r dp = 430, 280 and 185 ym a r e shown i n F i g u r e s 5-2-a, 5-20-a, and 5-21-a. A l l t h r e e samples were o b t a i n e d a t t h e same e x c e s s gas v e l o c i t y o f 0.3 m/s. As p a r t i c l e s i z e i s r e d u c e d , f o r c e p u l s e s became more f r e q u e n t , l a r g e r i n a m p l i t u d e and s h o r t e r i n d u r a t i o n . These t r e n d s can be i n t e r p r e t e d as a r e s u l t o f d i f f e r e n t bubble b e h a v i o r w i t h d i f f e r e n t p a r t i c l e s i z e s . G e l ' p e r i n e t a l . (1968) found t h a t the f r e q u e n c y o f gas b u b b l e s i n c r e a s e d and t h e i r d i m e n s i o n s d e c r e a s e d as dp i s r e d u c e d . An i n c r e a s e i n the b u b b l e s f r e q u e n c y would l e a d t o an i n c r e a s e i n the f r e q u e n c y o f f o r c e p u l s e s . A r e d u c t i o n i n bubble s i z e would be e x p e c t e d t o l e a d t o f o r c e s o f s m a l l e r m a g n i t u d e s , but t h i s d i d not o c c u r here because t h e b u b b l e s r i s e f a s t e r i n a bed o f f i n e p a r t i c l e s t h a n i n medium o r c o a r s e p a r t i c l e s (Rowe and P a r t r i d g e , 1965; Romero and S m i t h , 1965). F a s t e r r i s i n g b u b b l e s cause f o r c e p u l s e s l a r g e r i n magnitude and s h o r t e r i n d u r a t i o n . These t r e n d s do not h o l d o v e r the whole t e s t range o f gas s u p e r f i c i a l v e l o c i t y ; i n s t e a d t h e y a r e a f u n c t i o n o f (U - U ^ ) . Power s p e c t r a l e s t i m a t e s o f the r e l e v a n t samples a r e p l o t t e d i n F i g u r e s 5-2-b, 5-20-b and 5-21 -b. The f r e q u e n c y c o n t e n t s o f the f o r c e s i g n a l s a r e s i m i l a r , but t h e major f r e q u e n c i e s (which a p p r o x i m a t e l y c o r r e s p o n d t o the r a t e o f p u l s e o c c u r r e n c e ) d i f f e r . The major f r e q u e n c y i s shown as a f u n c t i o n o f p a r t i c l e s i z e i n F i g u r e 5-22. T h i s f i g u r e i n d i c a t e s t h a t the major f r e q u e n c y depends on p a r t i c l e s i z e , and t h a t i t i s more s e n s i t i v e t o s u p e r f i c i a l gas v e l o c i t y as the p a r t i c l e s i z e i s r e d u c e d . Haines e t a l . (1 972) found t h a t the - 122 -3 TIME CS) 4 F i g . 5 - 2 0 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s a n d ; dp = 2 80 ym; ( U - U m f ) = 0.3 m/s; H 0 = 0.3 m. i.O 4.0 FREQUENCY 12.0 (HZ) 16.0 20.0 F i g . 5 - 2 0 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 2 0 - a . F i g . 5 - 2 1 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s a n d ; dp = 185 ym; ( U - U m f ) = 0.3 m/s; H 0 = 0.3 m. MAJOR FREQUENCY (HZ) - PZl -- 125 -f r e q u e n c y o f bubbles a t a g i v e n l e v e l i n a f l u i d i z e d bed has a g r e a t e r dependence on gas f l o w r a t e when the bed i s composed o f f i n e p a r t i c l e s . Our f i n d i n g s a p p e a r t o be i n agreement w i t h t h i s t r e n d . The e f f e c t o f p a r t i c l e s i z e on the RMS measured f o r c e i s demon-s t r a t e d i n F i g u r e 5-23. A t e x c e s s gas v e l o c i t i e s between 0.1 and 0.5 m/s, t h e RMS v a l u e s a r e c o n s i s t e n t w i t h the t h r e e samples o f f o r c e - t i m e h i s t o r y d i s c u s s e d above: l a r g e r magnitudes o f f o r c e p u l s e s l e a d t o h i g h e r v a l u e s o f the RMS v e r t i c a l f o r c e . However, a t h i g h e r l e v e l s o f e x c e s s gas v e l o c i t y , the RMS p l o t s s t a r t e d t o behave i n a d i f f e r e n t way. As p a r t i c l e s i z e d e c r e a s e d the i n t e n s i t y o f f o r c e s s t a r t e d t o l e v e l o f f and t o f a l l o f f a t lower gas v e l o c i t i e s . T h i s may w e l l r e s u l t from i n c r e a s e d e n t r a i n m e n t and h o l d up o f the s m a l l e r s o l i d p a r t i c l e s i n t h e f r e e b o a r d o f the column. ( E n t r a i n m e n t i s 5 -2 a p p r o x i m a t e l y p r o p o r t i o n a l t o U d p and r e s u l t s i n a r e d u c e d dense bed h e i g h t because o f s o l i d s h o l d up i n the f r e e b o a r d and c y c l o n e . ) T r a n s i t i o n t o the t u r b u l e n t regime may be a n o t h e r r e a s o n f o r t h e d e c l i n e o f f o r c e s , s i n c e the t r a n s i t i o n gas v e l o c i t y depends on p a r t i c l e s i z e ( Y e r u s h a l m i and C a n k u r t , 1979; G r a c e , 1982). A t (U - U f ) < 0.1 m/s, t h e d e c r e a s e i n t h e i n t e n s i t y o f f o r c e s as d p i s r e d u c e d p r o b a b l y r e s u l t s from d i f f e r e n t buoyancy f o r c e s , s i n c e buoyancy depends on the dense phase v o i d a g e which tends t o i n c r e a s e f o r s m a l l e r p a r t i c l e s . H o r i z o n t a l components o f f o r c e : The e f f e c t o f p a r t i c l e s i z e on the RMS and mean v a l u e s o f the h o r i z o n t a l f o r c e i s shown i n F i g u r e 5-24. The RMS v a l u e s e x h i b i t the same t r e n d s as the v e r t i c a l components, e s p e c i a l l y a t h i g h l e v e l s o f gas v e l o c i t y . As p a r t i c l e s i z e i s ' r e d u c e d , the i n t e n s i t y o f f o r c e s I I 0 0.5 1.0 1.5 ( U - U m f ) , m / s Fig. 5-23. Effect of particle size on RMS of vertical force at different values of excess superficial velocity, ( U - U m f ) . - 127 -0.5 1.0 1.5 ( U - U m f ), m/s Fig. 5-24. Effect of particle size on RMS and mean values of horizontal force at different values of excess gas velocity, (U-Um f). - 128 -r e a c h e d a maximum a t e a r l i e r v a l u e s o f (U - I L ^ ) , f o r the same re a s o n d i s c u s s e d f o r t h e v e r t i c a l f o r c e s . As e x p e c t e d , the mean v a l u e s a r e a l m o s t t h e same and e q u a l t o z e r o , whatever t h e p a r t i c l e s i z e and gas v e l o c i t y . 5.2.4 E f f e c t s o f p a r t i c l e t y p e and d e n s i t y The e f f e c t s o f p a r t i c l e d e n s i t y on the c h a r a c t e r i s t i c s o f the measured f o r c e s were i n v e s t i g a t e d by c a r r y i n g out e x p e r i m e n t s u s i n g , 3 3 as bed m a t e r i a l , alundum (p = 4100 Kg/m ) , Ottawa sand (p $ = 2600 Kg/m ) and p o l y e t h y l e n e powder (p = 920 Kg/m ). The t h r e e powders had a p p r o x i m a t e l y the same mean p a r t i c l e s i z e (dp = 280-295 ym). The sand and alundum p a r t i c l e s b e l o n g t o group B o f G e l d a r t ' s c l a s s i f i c a -t i o n ( G e l d a r t , 1973), w h i l e t h e p o l e t h y l e n e powder b e l o n g s t o group A. V e r t i c a l components o f f o r c e s : F i g u r e s 5-25-a and 5-26-a p r e s e n t two samples o f d a t a measured i n t h e alundum bed and i n t h e sand bed r e s p e c t i v e l y . At the time o f measurement, the s u p e r f i c i a l gas v e l o c i t y was 0.05 m/s and the s t a t i c bed h e i g h t was 0.3 m. F o r c e p u l s e s i n the two samples appear s i m i l a r , but f o r c e s i g n a l s i n F i g u r e 5-25-a f l u c t u a t e around a mean o f a p p r o x i -m a t e l y +1.2 N, w h i l e i n F i g u r e 5-26-a the f l u c t u a t i o n s a r e around a mean v a l u e o f a p p r o x i m a t e l y 0.8 N. T h i s i n d i c a t e s t h a t i n the two samples, dynamic ( f l u c t u a t i n g ) f o r c e s are s i m i l a r o r a p p r o x i m a t e l y the same, but s t a t i c f o r c e s a r e d i f f e r e n t . F i g u r e s 5-25-b and 5-26-b g i v e t h e c o r r e s p o n d i n g s p e c t r a l e s t i m a t e p l o t s . F o r c e s a r e d i s t r i b u t e d o v e r t h e f r e q u e n c y range i n a s i m i l a r manner i n both c a s e s , and the major 3 TIME (S) 4 F i g . 5 - 2 5 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f a l u n d u m ; dp = 295 ym; ( U - U m f ) = 0.05 m/s; H 0 = 0.3 m o Q _ 0.0 4.0 8.0 J2.0 FREQUENCY (HZ) ]6.0 20.0 F i g . 5 - 2 5 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 2 5 - a . F i g . 5 - 2 6 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f s a n d ; dp = 2 8 0 ym; (U-U f ) = 0.05 m/s; H n 0.3 m . o o 0.0 CL. 4.0 16.0 l — i r e.o ]2 . o FREQUENCY (HZ) F i g . 5 - 2 6 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 2 6 - a . - 131 -f r e q u e n c i e s a r e i d e n t i c a l . As d i s c u s s e d i n C hapter 2, dynamic components o f f o r c e on t h e tube r e p r e s e n t c o n t r i b u t i o n s m a i n l y from p r e s s u r e f o r c e s i n d u c e d by bubble p r e s s u r e f i e l d s and momentum t r a n s m i t t e d by moving s o l i d p a r t i c l e s a s s o c i a t e d w i t h bubble movement. Hence s i m i l a r i t y o f dynamic f o r c e s i n t h e two m a t e r i a l s i m p l i e s t h a t bubble c h a r a c t e r i s -t i c s a r e s i m i l a r and t h a t i n e r t i a f o r c e s o f s o l i d p a r t i c l e s a r e not a p p r e c i a b l e a t t h i s l e v e l o f gas v e l o c i t y . S t a t i c f o r c e s a r e d i f f e r e n t b ecause buoyancy f o r c e s i n a bed o f alundum a r e h i g h e r than i n a bed o f sand, s i n c e t h e buoyancy f o r c e i s p r o p o r t i o n a l t o p a r t i c l e d e n s i t y (Nguyen and Grace, 1978). F i g u r e 5-27-a p r o v i d e s r e p r e s e n t a t i v e t r a c e s o f i n s t a n t a n e o u s f o r c e s measured i n the p o l y e t h y l e n e powder bed a t t h e same gas v e l o c i t y and bed h e i g h t as F i g u r e s 5-25-a and 5-25-a. T h i s sample i s q u i t e d i f f e r e n t t h a n t h e two p r e v i o u s samples. S t a t i c f o r c e s seem to be dominant, w i t h p u l s e s a l m o s t a b s e n t . T h i s i s because o n l y a few small b u b b l e s were p r e s e n t under t h e s e c o n d i t i o n s . T h i s b e h a v i o u r i s no doubt a s s o c i a t e d w i t h the f a c t t h a t s o l i d s i n group B (sand and alundum p a r t i c l e s ) behave i n a d i f f e r e n t way than s o l i d s i n group A ( p o l y -e t h y l e n e powder) ( G e l d a r t , 1 973). These t r e n d s e x h i b i t e d by the f o r c e - t i m e r e c o r d s a r e a g a i n d e m onstrated i n F i g u r e s 5-28, 5-29 and 5-30, a t low l e v e l s o f e x c e s s gas v e l o c i t y . In F i g u r e 5-28 t h e s t a n d a r d d e v i a t i o n p l o t s , which p r o v i d e a measure o f dynamic components o f f o r c e , show t h a t dynamic f o r c e s measured i n t h e beds o f alundum and sand a r e a l m o s t the same at low (U - U m f ) . As gas v e l o c i t y i n c r e a s e s , b u b b l e s grow i n s i z e , t h e i r r i s i n g v e l o c i t i e s i n c r e a s e , and v e l o c i t i e s o f a s s o c i a t e d s o l i d p a r t i c l e s a r e c o n s i d e r a b l y h i g h e r . Hence, i n e r t i a f o r c e s o f the s o l i d - 132 -i j j' ! ! ; j 1 j j i i 1 I | | i 1 I ! T i l ! i I i i J ; •. ,! 1 ! : • ! i i i i M j 1 j ! ; 1 -1 i i ! 1 ! 1 ! i ! i 1 1 ! ! i i i i i i j I i ! i 1 j I i i ! u + ot £ 5 i j i i i 1 1 ! i ! i ! i j ; i i 1 : i t 1 i 1 1 | I i ; i t ! ! 1 1 1 i | i ] ! ! i 1 1 1 ! \ .. i i i ' . 1 . 1 , • . J 1 ! ! i I '< 1 1 ! i ; | 1 1 1 ; i l l ! i 1 1 1 1 1 1 i i 1 j ! j i i I I i i ! M M ! i i j i 1 ! 1 i i 1 ! i • : : 1 i j i i i J : 1 . i 1 I ! i : : | ! 1 j i ; 1 i 1 1 1 1 1 ! 1 ! ! ! 1 1 ! i | ! i : ! 3 TIME (S) 4 F i g . 5 - 2 7 - a . V e r t i c a l f o r c e s on t h e t e s t c y l i n d e r m e a s u r e d i n a b e d o f p o l y e t h y l e n e p o w d e r ; dp = 280 ym; ( U - U m f ) = 0.05 m/s; H 0 = 0.3 m. a- FREQUENCY (HZ) F i g . 5 - 2 7 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s f o r t h e same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 2 7 - a . - 1 3 3 -10 8 -•- Alundum, p = 4IOOkg/m O- Ottawa sand, p= 2600kg/m T^T Polyethylene, ps= 920kg/m 0 0.5 1.0 1.5 (U -U m f ) , m/s Fig. 5-28. Effect of particle density on standard deviation of vertical force at different values of excess superficial velocity ( U - U m f ) - 134 -10 -9 -0 0.5 1.0 1.5 ( U - U m f ) ,m/s Fig. 5 - 2 9 . Effect of particle density on mean values of vertical force at different values of excess gas velocity ( U - U m f ) . - 135 -12 10 8 A l u n d u m , p= 4 IOOkg/m - O - Ottawa s a n d , /0=2600kg/m -A- Polyethylene, P s=920kg/irT 0.5 1.0 1.5 (U-Umf),m/s Fig .5 -30 . Effect of particle density on RMS of vertical force at different values of excess gas velocity ( U - U m f ) . - 136 -p a r t i c l e s i n the alundum bed exceed t h o s e i n the sand bed. The i n c r e a s -i n g d i f f e r e n c e between p a r t i c l e i n e r t i a f o r c e s i n t h e two beds causes the c u r v e s t o d i v e r g e w i t h i n c r e a s i n g gas v e l o c i t y . At h i g h e r gas v e l o c i t y , o t h e r f a c t o r s ( d i s c u s s e d p r e v i o u s l y ) maximize the d i f f e r e n c e between the two p l o t s . These causes i n c l u d e t h e i n c r e a s i n g e n t r a i n -ment and hol d - u p o f s o l i d p a r t i c l e s i n the f r e e b o a r d and the o n s e t o f t r a n s i t i o n o f the t u r b u l e n t regime f o r the l i g h t e r p a r t i c l e s . The s t a n d a r d d e v i a t i o n o f t h e f o r c e s i n the p o l y e t h y l e n e powder bed i s c o n s i d e r a b l y lower t h a n f o r the o t h e r two m a t e r i a l s , not o n l y because t h e powder d e n s i t y i s d i f f e r e n t , but a l s o because c h a r a c t e r i s t i c s o f r i s i n g b u b b l e s a r e d i f f e r e n t , G e l d a r t ( 1 9 7 3 ) . The d o t t e d p a r t o f t h e p l o t r e f e r s t o i n s t a b i l i t y i n the system d u r i n g t h e measurements where th e s o l i d m a t e r i a l i n t h e bed d e c r e a s e d c o n t i n u o u s l y w i t h t i m e : E l e c t r o s t a t i c f o r c e s l e d t o c o n t i n u o u s a c c u m u l a t i o n o f the e n t r a i n e d p o l y e t h y l e n e p a r t i c l e s i n t h e c y c l o n e and on t h e i n s i d e w a l l o f the column i n the f r e e b o a r d r e g i o n . The mean v a l u e p l o t s , shown i n F i g u r e 5-29, p r o v i d e an approximate measure o f s t a t i c f o r c e s by e x t r a p o l a t i n g the p l o t s t o (U - U m^) = 0. At gas v e l o c i t i e s up t o about 0.05 m/s, t h e r a t i o between mean v a l u e s i s a p p r o x i m a t e l y equal t o the d e n s i t y r a t i o . The RMS p l o t , F i g u r e 5-30, i s c o n s i s t e n t w i t h the two p r e v i o u s p l o t s , and demonstrates t he same t r e n d s d i s c u s s e d above. I t i n d i c a t e s t h e i n t e n s i t y o f the t o t a l f o r c e (dynamic and s t a t i c components). H o r i z o n t a l components o f f o r c e : The e f f e c t o f p a r t i c l e d e n s i t y on t h e RMS and mean v a l u e s o f the measured h o r i z o n t a l f o r c e i s dem o n s t r a t e d i n F i g u r e 5-31 f o r sand and - 137 -( U - U m f ) , m / s Fig. 5-31. Effect of particle density on RMS and mean values of horizontal force at different values of excess gas velocity ( U - U m f ) . - 138 -alundum. The RMS p l o t s e x h i b i t the same t r e n d s p r e v i o u s l y d i s c u s s e d w i t h v e r t i c a l f o r c e ( c f . F i g u r e 5-30), but t h e r e i s an o b v i o u s d i f f e r e n c e . At low l e v e l s o f gas v e l o c i t y , p a r t i c l e d e n s i t y has a minor e f f e c t on t h e RMS h o r i z o n t a l f o r c e i n comparison w i t h t h a t o f v e r t i c a l f o r c e . T h i s i s because s t a t i c f o r c e s a r e n e u t r a l o r equal t o z e r o i n the h o r i z o n t a l d i r e c t i o n s ; s i n c e p a r t i c l e i n e r t i a f o r c e s a r e not a p p r e c i a b l e a t t h i s l e v e l o f gas v e l o c i t y , dynamic f o r c e s are a p p r o x i m a t e l y the same f o r t h e two d i f f e r e n t p a r t i c l e d e n s i t i e s . As (U - U ^) i n c r e a s e s , the d i f f e r e n c e between the two p l o t s o f RMS ( i n F i g u r e 5-31) i n c r e a s e s f o r t h e same r e a s o n s as d e s c r i b e d b e f o r e . The mean v a l u e s o f h o r i z o n t a l f o r c e shown i n F i g u r e 5-31 a r e a p p r o x i m a t e l y equal t o z e r o f o r both m a t e r i a l s as e x p e c t e d . 5.3 E f f e c t s o f S i z e and Shape o f Immersed Tubes on the Forces 5.3.1 E f f e c t s o f t u b e s i z e Smooth tubes o f o u t e r d i a m e t e r 15, 25 and 32 mm were used i n t h i s i n v e s t i g a t i o n t o examine the i n f l u e n c e o f tube s i z e on the c h a r a c t e r i s t i c s o f t h e e x t e r n a l l y a p p l i e d f o r c e s . Each tube was mounted 0.3 m above the gas d i s t r i b u t o r as d e s c r i b e d i n C hapter 3. The s u p e r f i c i a l gas v e l o c i t y was v a r i e d between 0.05 m/s and 1.2 m/s, w h i l e the o t h e r bed parameters remained c o n s t a n t (H = 0.3 m, bed m a t e r i a l : Ottawa sand, d = 430 ym). P V e r t i c a l components o f f o r c e s : The RMS v a l u e s o f t h e measured f o r c e s a r e p r e s e n t e d i n T a b l e 5.1. The r a t i o between t h e RMS f o r c e s a t each s p e c i f i c gas v e l o c i t y , - 139 -T a b l e 5.1 RMS v e r t i c a l f o r c e s ( i n Newtons) measured w i t h t he t h r e e s i z e o f t u b e s . H 0 = 0.3 m; bed m a t e r i a l : 430 ym sand ( U - U m f ) , m/s 0.05 0.1 0.2 0.3 0.5 0.8 1.2 Tube d i a . 32 mm 2.04 2.37 3.40 4.38 5.80 7.43 8.77 25mm 1.42 1.70 2.54 3.30 4.42 5.70 6.84 15mm 0.55 0.73 1.42 1.98 2.70 3.48 4.11 - 140 -c a l c u l a t e d from T a b l e 5.1, a ppear i n T a b l e 5.2. I f we c o n s i d e r t h a t the r a t i o between the immersed volumes o f the t h r e e t e s t t u bes i s 1:0.61:0.22, and the r a t i o between the d i a m e t e r s o f the t u bes i s 1:0.78:0.47, the t a b u l a t e d v a l u e s i n T a b l e 5.2 show t h a t the RMS r a t i o a t each gas v e l o c i t y i n v e s t i g a t e d always l i e w i t h i n t h e s e l i m i t s . The RMS f o r c e r a t i o has a v a l u e c l o s e t o t h e tube volume r a t i o a t low gas v e l o c i t i e s , and i n c r e a s e s g r a d u a l l y t o approach t h e tube d i a m e t e r r a t i o as gas v e l o c i t y i n c r e a s e s . T h i s i s c o n s i s t e n t w i t h the p i c t u r e t h a t , a t gas v e l o c i t i e s c l o s e t o minimum f l u i d i z a t i o n , buoyancy f o r c e s , p r o p o r t i o n a l t o tube volume, a r e dominant, w h i l e a t h i g h gas v e l o c i t i e s dynamic f o r c e s , p r o p o r t i o n a l t o the t u b e a r e a i n p l a n view (and hence t o t u b e d i a m e t e r ) , a r e dominant. The e f f e c t o f tube s i z e on the root-mean-square o f t h e measured f o r c e s i s p l o t t e d i n F i g . 5-32. The e f f e c t o f t u b e s i z e on t h e c h a r a c t e r i s t i c s o f t h e f o r c e s i s d e m onstrated f u r t h e r by r e p r e s e n t a t i v e data samples. F i g u r e s 5-33 and 5-34 p r e s e n t two samples o f f o r c e - t i m e r e c o r d measured f o r the 15 mm and 32 mm t u b e , r e s p e c t i v e l y . Both samples were r e c o r d e d a t (U - U ^) = 0.1 m/s. Comparison o f t h e two samples shows t h a t the f r e q u e n c y and d u r a t i o n o f f o r c e p u l s e s a r e s i m i l a r , but p u l s e magni-tudes are s i g n i f i c a n t l y d i f f e r e n t , p u l s e magnitudes i n F i g . 5-33 b e i n g about o n e - t h i r d o f t h o s e i n F i g . 5-34. F i g u r e s 5-35-a and 5-36-a p r e s e n t two o t h e r samples o f d a t a , but r e c o r d e d a t an e x c e s s gas v e l o c i t y o f 0.3 m/s. F o r c e - t i m e r e c o r d s a r e a g a i n s i m i l a r e x c e p t t h a t p u l s e magnitudes f o r the 15 mm tube ( F i g . 5-35-a) a r e between 40 and 50% o f t h o s e i n F i g . 5-36-a. These samples o f d a t a not o n l y demonstrate t h e dependence o f f o r c e magnitudes on tube s i z e , but a l s o i n d i c a t e t h a t the i m p o r t a n c e o f tube d i a m e t e r v a r i e s w i t h a change i n gas v e l o c i t y . - 141 -T a b l e 5.2 The r a t i o between RMS v a l u e s o f t h e v e r t i c a l f o r c e a t d i f f e r e n t gas v e l o c i t i e s t o t h a t f o r the 32 mm d i a . t u b e . C o n d i t i o n s are the same as i n T a b l e 5.1 ( U - U m f ) , m/s 0.05 0.1 0.2 0.3 ' 0.5 0.8 1.2 Tube d i a . 32 mm 1 .0 1.0 1.0 1.0 1.0 1.0 1 .0 25- mm 0.70 0.72 0.75 0.75 0.76 0.77 0.78 15 mm 0.29 0.31 0.42 0.45 0.46 0.47 0.47 10 - 142 -0 0.5 1.0 15 (U-U m f ) ,m/s Fig. 5-32. Effect of tube size on RMS vertical force at different values of excess gas velocity, (U - U m f ) . F i g . 5 - 3 3 . V e r t i c a l f o r c e s o n a 15 mm d i a m e t e r t u b e r e c o r d e d a t ( U - U m f ) = 0.1 m/s. Bed m a t e r i a l : s a n d ; dp = 4 3 0 ym; H Q = 0.3 m. F i g . 5 - 3 4 . V e r t i c a l f o r c e s o n a 32 mm d i a m e t e r t u b e r e c o r d e d a t ( U - U m f ) = 0.1 m/s. B e d m a t e r i a l : s a n d ; dp = 430 ym; H 0 = 0.3 m. - 144 -F i g . 5 - 3 5 - a . V e r t i c a l f o r c e s on a 15 mm d i a m e t e r t u b e r e c o r d e d a t ( U - U m f ) = 0.3 m/s. B e d m a t e r i a l : s a n d ; dp = 430 um; H 0 = 0.3 m . - 145 -10 J ! I ! ! 1 ; I ! I I ! j ! L _ i 1 2 3 TIME CS) 4 F i g . 5 - 3 6 - a . V e r t i c a l f o r c e s on a 32 mm d i a m e t e r t u b e r e c o r d e d a t ( U - U m f ) = 0.3 m/s. B e d m a t e r i a l : s a n d ; dp = 4 3 0 pm; H Q = 0.3 m. o i 1 1 1 1 1 1 1 i i i 0.0 4.0 8.0 12.0 36.0 20.0 FREQUENCY (HZ) F i g . 5 - 3 6 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s f o r same e x p e r i m e n t a l c o n d i t i o n s a s 5 - 3 6 - a . MAJOR FREQUENCY (HZ ) — ro CM J> cn 01 T H C _ CD Ol B ro o ro 01 O Ol r- p o o oo N « « N — 6 - 9 H -- 147 -Power s p e c t r a l e s t i m a t e s f o r the l a s t two c a s e s are shown i n F i g u r e s 5-35-b and 5-36-b. The two p l o t s a r e a l m o s t i d e n t i c a l , i n d i c a t i n g t h a t f r e q u e n c y c o n t e n t s i n t h e samples a r e v e r y s i m i l a r ; t h e major f r e q u e n c i e s a r e a l s o e x a c t l y t h e same. T h i s i s p r o b a b l y because t h e d i f f e r e n c e between the s i z e s o f tubes i s small r e l a t i v e t o t h e bubble s i z e a t t h i s gas v e l o c i t y . D i f f e r e n c e s would be e x p e c t e d f o r tubes whose d i a m e t e r a r e s i g n i f i c a n t l y d i f f e r e n t r e l a t i v e t o the s i z e o f b u b b l e s . The e f f e c t o f tube s i z e on the major f r e q u e n c y i s shown i n F i g . 37. In g e n e r a l , the major f r e q u e n c y has a s h i f t dependence on tube s i z e . I t appears t o be more s e n s i t i v e t o tube s i z e a t low gas v e l o c i t i e s where bub b l e s a r e r e l a t i v e l y s m a l l . H o r i z o n t a l components o f f o r c e s : RMS and mean h o r i z o n t a l f o r c e s measured on the 25 mm and 32 mm tubes a r e shown i n F i g . 5-38. The RMS v a l u e i s a p p r o x i m a t e l y p r o p o r -t i o n a l t o the tube d i a m e t e r a t a l l l e v e l s o f gas v e l o c i t y . As d i s c u s s e d above, t h e h o r i z o n t a l f o r c e i s composed o n l y o f dynamic components, and dynamic components o f f o r c e are p r o p o r t i o n a l t o tube d i a m e t e r . Note a l s o from t h e f i g u r e t h a t the mean h o r i z o n t a l f o r c e i s a p p r o x i m a t e l y z e r o w i t h both s i z e s o f t u b e . - 148 --2 I I | I 0.5 1.0 1.5 ( U - U m f ),m/s Fig. 5-38. Effect of tube size on RMS and mean values of horizontal force at different values of excess gas velocity ( U - U m f ). - 149 -5.3.2 E f f e c t s o f tube shape ( f i n n e d v s . u n f i n n e d ) The e f f e c t s o f tube shape were examined by c o n d u c t i n g measurements on a l o n g i t u d i n a l l y f i n n e d t u b e . The f i n peak-to-peak d i a m e t e r was 32 mm w h i l e t h e v a l 1 e y - t o - v a l l e y d i a m e t e r was 15 mm (see F i g . 3-6-d). For comparison p u r p o s e s , f o r c e measurements were a l s o made on bare tubes o f o u t e r d i a m e t e r 32 mm and 15 mm under t he same o p e r a t i n g c o n d i t i o n s . V e r t i c a l components o f f o r c e : The RMS v a l u e s o f the v e r t i c a l f o r c e s measured on the tubes a re shown i n F i g . 5-39. The RMS f o r c e s f o r the f i n n e d t u b e i s seen t o always l i e between t h e f o r c e s f o r t h e c i r c u m s c r i b i n g and " i n s c r i b e d " t u b e s . V a l u e s a r e near t h o s e f o r the 15 mm tube a t low gas v e l o c i t i e s , and e x t r a p o l a t i o n o f the two p l o t s l e a d s t o alm o s t equal v a l u e s a t U = U T h i s i s because net s t a t i c f o r c e s ( v e c t o r i a l summation o f buoyancy f o r c e and weight o f d e f l u i d i z e d cap above t he tube) are a p p r o x i m a t e l y e q u a l ; s i n c e t h e s t a t i c f o r c e s a r e dominant a t t h i s l e v e l o f . g a s v e l o c i t y , t h e t o t a l f o r c e i n the v e r t i c a l d i r e c t i o n on both tubes a r e alm o s t e q u a l . As s u p e r f i c i a l gas v e l o c i t y i n c r e a s e s , dynamic components o f f o r c e i n c r e a s e , but a t a h i g h e r r a t e f o r the f i n n e d t u b e , making the two p l o t s d i v e r g e s h a r p l y . In comparison w i t h t he p l o t f o r the 32 mm bare t u b e , t he RMS o f t he f o r c e s on t h e f i n n e d tube i s always l e s s than t h a t on the bare t u b e , but t h e d i f f e r e n c e between them i s a p p r o x i m a t e l y c o n s t a n t o v e r t he e n t i r e range o f gas v e l o c i t y . The main r e a s o n i s p r o b a b l y t h a t t h e buoyancy f o r c e on the f i n n e d tube i s much l e s s than t h a t on - 150 -0 0.5 1.0 1.5 ( U - U m f ),m/s Fig. 5-39. Effect of tube shape (finned vs. unfinned) on RMS vertical force at different values of excess gas velocity, ( U - U m f ) . - 151 -t h e u n f i n n e d t u b e , w h i l e t h e s i z e o f t h e d e f l u i d i z e d cap above the f i n n e d t u b e i s l a r g e r and more s t a b l e . Dynamic f o r c e s on both tubes seem t o be s i m i l a r , p o s s i b l y because most bubbles do not p e n e t r a t e i n t o t h e f i n s t r u c t u r e (Hger and Thomson, 1973). Power spectrum p l o t s , shown i n F i g u r e s 5-40 and 5-41, g i v e f u r t h e r i n d i c a t i o n o f s i m i l a r i t y between dynamic f o r c e s on t h e f i n n e d and c i r c u m s c r i b i n g bare t u b e . Hence, we may c o n c l u d e t h a t the dynamic component o f f o r c e i s p r o p o r t i o n a l t o t h e tube o u t e r d i a m e t e r . H o r i z o n t a l components o f f o r c e : F i g u r e 5-42 shows the RMS h o r i z o n t a l f o r c e measured on t h e f i n n e d t u b e and on t h e bare tube o f o u t e r d i a m e t e r 32 mm. The RMS f o r c e s f o r t he two c a s e s a r e a l m o s t i d e n t i c a l . T h i s i n d i c a t e s t h a t dynamic f o r c e s a r e a l m o s t e q u a l , s i n c e s t a t i c f o r c e s do not c o n t r i b u t e t o the t o t a l h o r i z o n t a l f o r c e . 5.4 Causes o f t h e F o r c e s The r e s u l t s p r e s e n t e d i n t h i s c h a p t e r have i n d i c a t e d t h a t the t o t a l i n s t a n t a n e o u s f o r c e on an immersed tube i n a f l u i d i z e d bed i s composed o f a s t a t i c component and a dynamic ( f l u c t u a t i n g ) component. I . S t a t i c component o f f o r c e : S t a t i c f o r c e s on immersed tubes i n a f l u i d i z e d bed a r e caused m a i n l y by the buoyancy f o r c e a c t i n g i n an upward d i r e c t i o n , and the weight o f s o l i d p a r t i c l e s i n the d e f l u i d i z e d cap above the tube a c t i n g downward. The d r a g o f t h e f l u i d i z i n g gas on t h e o b j e c t i s n e g l i g i b l e , F i g . 5 - 4 0 . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e o n a b a r e t u b e o f o u t e r _ d i a m e t e r 32 mm; r e c o r d e d a t ( U - U m f ) = 0.8 m/s. Bed m a t e r i a l : s a n d ; dp = 4 3 0 ym; H 0 = 0.3 m. 0.0 4 .0 8.0 12.0 FREQUENCY XHZ) 16.0 20.0 F i g . 5 - 4 1 . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e o n a f i n n e d t u b e o f o u t s i d e d i a m e t e r 32_mm, r e c o r d e d a t ( U - U m f ) = 0.8 m/s. B e d m a t e r i a l : s a n d ; dp = 4 3 0 ym; H 0 = 0.3 m - 153 -4 h 3 h -A- Finned Tube ( f i n t i p - t o - t i p 32mm and root-to root 15mm) Unfinned Tube 32 mm outer dia. M E A N A 0.5 1.0 1.5 ( U - U m f ) , m/s Fig. 5-42. Effect of tube shape (finned vs. unfinned) on RMS and mean values of horizontal force at different values of excess gas velocity, ( U - U m f ) . - 154 -p r o v i d e d t h a t t h e s c a l e o f t h e o b j e c t i s much l a r g e r than t h a t o f the f l u i d i z e d p a r t i c l e s (Nguyen and Grace, 1978). A c c o r d i n g t o Archimedes' p r i n c i p l e , t h e bed buoyancy f o r c e can be w r i t t e n : F B = V x P f i f f x g where V i s the volume o f the immersed o b j e c t , p ^ the e f f e c t i v e d e n s i t y o f the bed dense phase, and g the g r a v i t a t i o n a l a c c e l e r a t i o n . Nguyen and Grace (1978) found t h a t 9eff = P m^> i . e . the bed d e n s i t y a t minimum f l u i d i z a t i o n g i v e n by p g ( l - P m f ) • Hence a t minimum f l u i d i z a -t i o n t h e bed buoyance f o r c e i s : FB = V X p s ( 1 " ( 5" 1 } At gas v e l o c i t i e s h i g h e r than t h a t o f minimum f l u i d i z a t i o n , t ^ i n the above e q u a t i o n s h o u l d be r e p l a c e d by the o v e r a l l bed v o i d a g e . S i n c e t h e o v e r a l l bed v o i d a g e i n c r e a s e s as U i n c r e a s e s , t h e bed buoyancy f o r c e d e c r e a s e s . The bed buoyancy f o r c e i s t h e dominant component o f f o r c e at minimum f l u i d i z a t i o n . Above t h e t u b e , t h e r e i s a d e f l u i d i z e d cap whose e x t e n t and permanence depend on t h e f l u i d i z i n g v e l o c i t y . At minimum f l u i d i z a t i o n t h e s o l i d p a r t i c l e s i n t h e d e f l u i d i z e d cap a r e a l m o s t s t a g n a n t ( H a r r i s o n and Grace, 1971; Hager and Thomson, 1973; Ginoux e t a l . , 1974; Loew e t a l . , 1979). Thus, t h e w e i g h t o f s o l i d p a r t i c l e s h e l d up by the tube can be c o n s i d e r e d t o p r o v i d e the s t a t i c f o r c e i n a - 155 -downward d i r e c t i o n . The r a t i o o f the d e f l u i d i z e d cap h e i g h t t o the o b j e c t d i a m e t e r was found t o be 0.8 i n a t w o - d i m e n s i o n a l bed (Hager and S c h r a g , 1976). In t h r e e - d i m e n s i o n a l beds t h i s r a t i o i s e x p e c t e d t o be lower (Rooney and H a r r i s o n , 1976). At s u p e r f i c i a l gas v e l o c i t i e s g r e a t e r than U ,^ b u b b l e s o c c a s i o n a l l y sweep a c r o s s the t o p o f the t u b e , d i s l o d g i n g the d e f l u i d i z e d cap. The n e t s t a t i c component o f f o r c e i n an upward d i r e c t i o n may be e s t i m a t e d from: where Wc i s the weight o f s o l i d p a r t i c l e s i n t h e d e f l u i d i z e d cap. 11. Dynamic ( f l u c t u a t i n g ) component o f f o r c e : The d a t a p r e s e n t e d have demonstrated t h a t the c h a r a c t e r i s t i c s o f t h e f l u c t u a t i n g component o f t h e f o r c e i s c l o s e l y r e l a t e d t o bubble p r o p e r t i e s . F o r c e - t i m e r e c o r d s c o n s i s t o f s e r i e s o f p u l s e s whose magnitudes a r e r e l a t e d t o t h e bubble s i z e ; p u l s e d u r a t i o n s appear t o be i n v e r s e l y p r o p o r t i o n a l t o bubble r i s i n g v e l o c i t y , and p u l s e f r e q u e n c y appears t o be s t r o n g l y r e l a t e d t o the bubble f r e q u e n c y . The c h a r a c t e r i s t i c s o f t h e f o r c e t h e r e f o r e r e f l e c t bubble p r o p e r t i e s . In a d d i t i o n , t h e bed parameters have de m o n s t r a t e d s i m i l a r e f f e c t s on the f o r c e c h a r a c t e r i s t i c s as on bubble p r o p e r t i e s . A l l o f t h e s e p r o v i d e s t r o n g e v i d e n c e t h a t t h e f l u c t u a t i n g f o r c e i s produced by t h e passage o f b u b b l e s p a s t t h e t u b e . The p l o t s o f power spectrum p r e s e n t e d are c h a r a c t e r i z e d by a - 156 -w e l l i d e n t i f i e d peak, i l l u s t r a t i n g t h a t the measured f o r c e s e x h i b i t both p e r i o d i c and random components (see Chapter 4 ) . Bubble f r e q u e n c y a l s o e x h i b i t s superimposed random and p e r i o d i c c h a r a c t e r i s t i c s , Werther ( 1 9 7 7 ) . T h i s p r o v i d e s a n o t h e r i n d i c a t i o n t h a t t h e f o r c e s measured i n t h e p r e s e n t work a r e c l o s e l y r e l a t e d t o the bubble p r o p e r t i e s i n t h e bed. S e v e r a l e x p e r i m e n t s were c a r r i e d out to c o n f i r m t h i s r e l a t i o n , and t o have i n s i g h t i n t o t h e mechanism by which i n d i v i d u a l bubbles i n d u c e f o r c e s on immersed t u b e s . S i m u l t a n e o u s p r e s s u r e and f o r c e measurements were c o n d u c t e d as d e s c r i b e d i n Chapter 3. F i g u r e 5-43 shows a r e c o r d o f s i m u l t a n e o u s f o r c e and p r e s s u r e s i g n a l s . The p r e s s u r e s i g n a l i s t h e measure o f p r e s s u r e v a r i a t i o n a t t h e bottom o f t h e t u b e , w h i l e t he f o r c e s i g n a l i s the i n s t a n t a n e o u s t o t a l f o r c e e x e r t e d on t h e t u b e . The e x c e s s gas s u p e r f i c i a l v e l o c i t y was 0.1 m/s, and the s t a t i c bed h e i g h t was 0.3 m. The f i g u r e i n d i c a t e s t h a t p r e s s u r e p u l s e s and f o r c e p u l s e s o c c u r s i m u l t a n e o u s l y , and a t the same r a t e , i . e . about 3 p u l s e s per s e c o n d . F i g u r e 5-44 shows a n o t h e r sample o f f o r c e and p r e s s u r e measurements a t a h i g h gas v e l o c i t y , (U - U m f ) = 1.2 m/s. In t h i s f i g u r e , as i n t h e p r e v i o u s c a s e , t he f o r c e and p r e s s u r e p u l s e s o c c u r s i m u l t a n e o u s l y . T h i s not o n l y c o n f i r m s the r e l a t i o n between the f o r c e p u l s e s and bubble movement, but a l s o i n d i c a t e s t h a t t h e f o r c e maxima c o r r e s p o n d t o the a r r i v a l o f bubb l e s at t h e tube s u r f a c e , s i n c e p r e s s u r e p u l s e maxima c o r r e s p o n d t o t h e a r r i v a l o f b u b b l e s a t t h e p o i n t o f p r e s s u r e measurement ( L i t t m a n and Homolka, 1970; Ginoux e t a l . , 1974; Nguyen and G r a c e , 1978). U J o O u. © Z U J BS U J CC 0-3 TIME (S) 4 F i g . 5 - 4 3 . S i m u l t a n e o u s f o r c e a n d p r e s s u r e s i g n a l s . T h e f o r c e s i g n a l i s t h e i n s t a n t a n e o u s t o t a l f o r c e o n a 32 mm d i a m e t e r t u b e ; t h e p r e s s u r e s i g n a l i s t h e p r e s s u r e v a r i a t i o n a t t h e b o t t o m o f t h e t u b e , r e c o r d e d a t (U-U f ) = 0.1 m/s; H = 0.3 m. r + c + t o - n 3 - 3 - - • • - • • fD ro i n i n 3 • <-+ cu c -j —• cn o- ro i ro to -p» PRESSURE (N/m2)(X10) FORCE IN) -J -J ro ro o 0 to -5 -•• C L I O ro 3 C L OJ r + - ' • LO C Z r t -1 zr d ro 3 -h "a •~— -s ro II t o t o — 1 c . -s ro ro 3 < \ Cu t o - 5 * . —J. CU ^: <-+ o -•• o II 3 O Cu . r + C O 3 " CO ro -•• 3 — • t o c + r + Cu CU 3 3 ro r + O cu cr 3 t o ro o -h c t o o -5 o r + ro o r + CU CU 3 —> Q --+»T3 O -5 -s ro O (/> ro to SZ O -5 3 ro cu t o C O L O ro 3 cu 3 —' 3 to fD C L C T CU — | O 3 3 " r + ro ro o ro -b 3 - J O -5 o <-+ o - h c ro cr ro - 89L -- 159 -In a n o t h e r e x p e r i m e n t , l o c a l p r e s s u r e v a r i a t i o n s a t a d i s t a n c e 0.05 m below the tube were measured s i m u l t a n e o u s l y w i t h t he f o r c e on the t u b e . The p r e s s u r e d i s t u r b a n c e p r o d u c i n g the f o r c e p u l s e was found t o pr o p a g a t e upward toward t h e tube a t a v e l o c i t y between 0.6 and 1.0 m/s; w h i l e t he average r i s e v e l o c i t y o f bubbles a p p r o a c h i n g the t u b e , c a l c u l a t e d from e q u a t i o n ( 2 - 8 ) , was 0.8 m/s. Hence, t h e upward t r a v e l l i n g v e l o c i t y o f t h e p r e s s u r e d i s t u r b a n c e , c a u s i n g t h e f o r c e p u l s e , c o r r e s p o n d s t o the bubble r i s e v e l o c i t y . The f o r c e - t i m e c u r v e r e c o r d e d d u r i n g the r i s e o f a s i n g l e bubble i n j e c t e d i n t o an i n c i p i e n t l y f l u i d i z e d bed, d i r e c t l y below the t u b e , i s shown i n F i g . 5-45. I t was found t h a t 1-2 o f t h e c u r v e c o r r e s p o n d e d t o t he b u b b l e i n j e c t i o n t i m e ; p o i n t 3 o c c u r r e d when the f r o n t o f the bubble a r r i v e d a t the s u r f a c e o f t h e t u b e ; 3-4 c o r r e s p o n d e d t o t h e p e r i o d when the bubble was p a s s i n g the t u b e ; p o i n t 5 p r o b a b l y o c c u r r e d when s o l i d s i n t h e wake o f t h e bubble impacted a g a i n s t t h e t u b e ; p o i n t 6 was found t o c o r r e s p o n d t o a r r i v a l o f the bubble a t the upper s u r f a c e o f t h e bed; t h e minimum p o i n t 7 was the r e s u l t o f the bed upper s u r f a c e c o l l a p s e and r e c i r c u l a t i o n o f s o l i d p a r t i c l e s i n t h e downward d i r e c t i o n . The sm a l l a m p l i t u d e f l u c t u a t i o n s f o l l o w i n g p o i n t 7 ar e b e l i e v e d t o r e s u l t from bed i n s t a b i l i t y (up and down) f o l l o w i n g the c o l l a p s e o f the bed s u r f a c e . T h i s e x p e r i m e n t i n d i c a t e d t h a t t h e peak o f t h e f o r c e p u l s e ( p o i n t 3) c o r r e s p o n d e d t o t h e a r r i v a l o f t h e bubble a t the tube s u r f a c e , which i s c o n s i s t e n t w i t h our r e s u l t i n t h e f r e e l y b u b b l i n g bed. Kennedy e t a l . (1981) assumed t h a t t h e f o r c e p u l s e s r e s u l t from s o l i d s i m p a c t i n g a g a i n s t the tube a f t e r t h e passage o f b u b b l e s . However, s o l i d s i n the wake o f the F i g . 5 - 4 5 . V e r t i c a l f o r c e - t i m e c u r v e d u r i n g t h e r i s e o f a b u b b l e i n j e c t e d i n t o a t h r e e - d i m e n s i o n a l i n c i p i e n t l y f l u i d i z e d b e d o f 430 ym s a n d p a r t i c l e s . I n j e c t i o n was d i r e c t l y b e l o w t h e c e n t r e o f t h e t u b e 100 mm a b o v e t h e d i s t r i b u t o r . S t a t i c b e d h e i g h t 0.45 m. - 161 -bubble were found i n t h e p r e s e n t work t o have a minor e f f e c t i n c a u s i n g t h e f o r c e on t h e t u b e . The mechanism by which i n d i v i d u a l b u b b l e s i n d u c e f o r c e s on the t u b e can be c l a r i f i e d f u r t h e r by b r e a k i n g t h e f l u c t u a t i n g f o r c e i n t o i t s p r i n c i p a l c o n t r i b u t i n g components: 1) F l u c t u a t i n g p r e s s u r e f o r c e c a used by bubble p r e s s u r e f i e l d : The t u b e e n c o u n t e r s an i n c r e a s i n g upward p r e s s u r e f o r c e as a r i s i n g b u bble approaches t he tube and then a downward p r e s s u r e f o r c e as the bubble r i s e s away from the tube ( d i s c u s s e d i n Chapter 2 ) . The p r e s s u r e f o r c e i s a maximum a t the i n s t a n t when the bubble a r r i v e s a t the tube s u r f a c e and a minimum when t h e r e a r o f t h e bubble passes t he tube (Nguyen and Grace, 1978). 2) C y c l i c momentum f o r c e caused by t h e impact o f s o l i d p a r t i c l e s ( a s s o c i a t e d w i t h bubbles motion) a g a i n s t t he tube: Tubes l o c a t e d i n t h e main body o f a f l u i d i z e d bed e x p e r i e n c e v a r i a t i o n i n s o l i d s c o n c e n t r a t i o n around t h e tube and v a r i a t i o n i n the s u r f a c e p a r t i c l e v e l o c i t y as t h e y e n c o u n t e r each r i s i n g bubble ( P e e l e r e t a l ., 1 982). These v a r i a t i o n s show a c y c l i c b e h a v i o u r as the bubble passes the tube (see C h a p t e r 2 ) . As the bubble approaches t he t u b e , the p a r t i c l e v e l o c i t y i n c r e a s e s , m o s t l y i n an upwards d i r e c t i o n , r e a c h i n g a maximum when t h e bubble a r r i v e s a t t h e tube s u r f a c e ( s e e F i g . 2 - 9 ) . The maximum p a r t i c l e v e l o c i t y c o r r e s p o n d s a p p r o x i m a t e l y to the p r e s s u r e f o r c e peak. T h i s e x p l a i n s why t h e r e i s o n l y one major peak i n the f o r c e - t i m e c u r v e o f s i n g l e bubble (see F i g . 5-45). In the p a r t i c l e v e l o c i t y - t i m e p l o t , t h e r e i s a n o t h e r peak c o r r e s p o n d i n g t o wake a p p e a r a n c e . T h i s peak a l s o c o r r e s p o n d s a p p r o x i m a t e l y to t h e minimum p r e s s u r e f o r c e and - 162 -causes a sm a l l s e c o n d a r y peak o f the r e s u l t a n t f o r c e , as shown by p o i n t 5 i n t h e f o r c e - t i m e c u r v e . 3) S e m i - p e r i o d i c downward t h r u s t caused by c o l l a p s e o f the bed upper s u r f a c e : T h i s downwards f o r c e a r i s e s from downwards f l o w o f s o l i d s a f t e r bubble e r u p t i o n a t t h e bed s u r f a c e . F i g u r e 5-46 shows a f o r c e - t i m e h i s t o r y r e c o r d e d s i m u l t a n e o u s l y w i t h a p r e s s u r e - t i m e h i s t o r y d u r i n g the r i s e o f a s i n g l e bubble i n j e c t e d c e n t r a l l y i n t o an i n c i p i e n t l y f l u i d i z e d bed o f sand. The f i g u r e s u p p o r t s the above d e s c r i p t i o n o f the b u b b l e - i n d u c e d f o r c e mechanism. The f o r c e p u l s e peak ( p o i n t 3) c o r r e s p o n d s t o t h e p r e s s u r e p u l s e peak. T h i s i n d i c a t e s t h a t t h e p r e s s u r e f o r c e maximum and the momentum f o r c e maximum o c c u r s i m u l t a n e o u s l y a t the i n s t a n t when the r o o f o f the bubble r e a c h e s t h e bottom o f the tube ( t h e p o i n t o f p r e s s u r e measurement). The f i r s t minimum o f t h e f o r c e - t i m e c u r v e ( p o i n t 4) c o r r e s p o n d s t o t h e minimum o f t h e p r e s s u r e - t i m e c u r v e and t o the i n s t a n t when t h e f l o o r o f t h e bubble r e a c h e d t h e bottom o f the tu b e . Immediately a f t e r t he f l o o r o f the bubble r e a c h e d the bottom o f the t u b e , s o l i d p a r t i c l e s i n t h e wake o f t h e bubble impacted a g a i n s t the tube c a u s i n g t h i s minor peak ( p o i n t 5) o f t h e f o r c e - t i m e c u r v e . The bub b l e c o n t i n u e d t o t r a v e l upward, and t h e p r e s s u r e - t i m e c u r v e c o n t i n u e d t o r e c o v e r . C o l l a p s e o f the bed upper s u r f a c e and r e c i r -c u l a t i o n o f s o l i d s i n downward d i r e c t i o n , f o l l o w i n g t he bubble b u r s t , l e d t o a second minimum ( p o i n t 7) o f the f o r c e - t i m e c u r v e , w i t h much l e s s i n f l u e n c e on t h e p r e s s u r e - t i m e c u r v e . F i g . 5 - 4 6 . V e r t i c a l f o r c e - t i m e a n d p r e s s u r e - t i m e c u r v e s r e c o r d e d s i m u l t a n e o u s l y d u r i n g t h e r i s e o f a b u b b l e i n j e c t e d i n t o an i n c i p i e n t l y f l u i d i z e d b e d o f 430 um s a n d p a r t i c l e s . I n j e c t i o n was d i r e c t l y b e l o w t h e c e n t r e o f t h e t u b e 100 mm a b o v e t h e d i s t r i b u t o r . S t a t i c b e d d e p t h 0.45 m. - 164 -5.5 G e n e r a l P r o p e r t i e s o f t h e V e r t i c a l and H o r i z o n t a l Components o f  F o r c e The f o r c e under s t u d y can be d e s c r i b e d i n the time-domain i n terms o f the m a gnitude, f r e q u e n c y and d u r a t i o n o f t h e f o r c e p u l s e s . However, s i n c e t h e f o r c e - t i m e r e c o r d s were found t o be n o n d e t e r m i n i s t i c time s e r i e s , i t was more f e a s i b l e t o d e s c r i b e the f o r c e i n terms o f s t a t i s t i c a l p a r a m e t e r s . In a d d i t i o n , i t was f e l t n e c e s s a r y to d e t e r -mine whether o r not the f o r c e d a t a e x h i b i t p e r i o d i c and s t a t i o n a r y p r o p e r t i e s t o a s s i s t our u n d e r s t a n d i n g o f the n a t u r e o f the f o r c e . 5.5.1 M agnitude, d u r a t i o n and f r e q u e n c y o f the f o r c e p u l s e s As p r e v i o u s l y d e m o n s t r a t e d , t h e f o r c e - t i m e h i s t o r y c o n s i s t s o f s e r i e s o f p u l s e s whose magnitudes, d u r a t i o n s and f r e q u e n c y a r e dependent on the bed o p e r a t i n g v a r i a b l e s and r e l a t e d t o bubble p r o p e r t i e s . I f t h e bed i s o p e r a t i n g i n the b u b b l i n g regime and t h e e f f e c t o f s o l i d s e n t r a i n m e n t i s m i n o r , p u l s e magnitudes are s t r o n g l y i n f l u e n c e d by the e x c e s s s u p e r f i c i a l gas v e l o c i t y ; s l i g h t l y dependent on p a r t i c l e s i z e ; and m o d e r a t e l y a f f e c t e d by bed depth and p a r t i c l e d e n s i t y . However, p a r t i c l e d e n s i t y has a s t r o n g i n f l u e n c e a t h i g h gas v e l o c i t i e s . The bed v a r i a b l e s e x h i b i t t h e same t r e n d s o f i n f l u e n c e on magnitudes o f t h e f o r c e p u l s e s as on s i z e o f p r e v a i l i n g b u b b l e s . In o r d e r t o c o n f i r m the r e l a t i o n between the magnitude o f t h e f o r c e p u l s e and bubble s i z e , s i n g l e b u b b l e s o f d i f f e r e n t s i z e were i n j e c t e d i n t o an i n c i p i e n t l y f l u i d i z e d bed a t the c e n t r e o f t h e column as d e s c r i b e d i n C h a p t e r 3. The upward and downward - 165 -magnitudes o f t h e r e s u l t i n g f o r c e p u l s e a r e p l o t t e d v e r s u s bubble e q u i v a l e n t d i a m e t e r i n F i g . 5-47. Each p o i n t i s o b t a i n e d from a v e r a g i n g 300 i n d i v i d u a l v a l u e s a l t h o u g h t h e s e v a l u e s were a l m o s t t h e same. From the f i g u r e , t h e upward and downward magnitudes o f the f o r c e p u l s e appear t o be l i n e a r l y p r o p o r t i o n a l to bubble s i z e . T h i s i s c o n s i s t e n t w i t h our r e s u l t i n the f r e e l y b u b b l i n g bed i f t h e bubble s i z e i n a f r e e l y b u b b l i n g bed i s c o n s i d e r e d t o be l i n e a r l y p r o p o r t i o n a l t o the e x c e s s gas v e l o c i t y ( S i t n a i e t a l . , 1981), and t h e measured f o r c e i s p r o p o r t i o n a l t o t h e e x c e s s gas v e l o c i t y as found e a r l i e r i n t h e p r e s e n t work. A l s o , magnitudes o f the f o r c e p u l s e s a r e r e l a t e d t o t h e t r a j e c t o r i e s o f t h e b u b b l e s r e l a t i v e t o t h e t u b e . T h i s r e l a t i o n was examined by i n j e c t i n g s i n g l e bubbles o f the same s i z e a t d i f f e r e n t h o r i z o n t a l d i s t a n c e s from the a x i s o f the tube i n t o an i n c i p i e n t l y f l u i d i z e d bed. The p r o c e d u r e i s d e s c r i b e d i n Chapter 3. The measured upward and downward magnitudes o f the f o r c e p u l s e s are p l o t t e d a g a i n s t t h e h o r i z o n t a l d i s t a n c e o f t h e i n j e c t i o n p o r t from the bed c e n t r e i n F i g . 5-48. The f i g u r e shows t h a t the magnitudes o f t h e f o r c e p u l s e v a r y i n v e r s e l y w i t h t h e d i s t a n c e o f bubble c e n t r e from the t e s t c y l i n d e r a x i s . As b u b b l e s grow i n s i z e t h e y r i s e more q u i c k l y . As a r e s u l t , d u r a t i o n s o f the f o r c e p u l s e s ( d e f i n e d i n S e c t i o n 5.2) become s h o r t e r . T h i s i s a p p a r e n t i n t h e d a t a samples r e c o r d e d a t d i f f e r e n t gas v e l o c i t i e s , as i n F i g u r e s 5-1-a, 5-2-a and 5-3-a. D u r a t i o n s o f f o r c e p u l s e s , i n d u c e d by i n d i v i d u a l b u b b l e s o f d i f f e r e n t s i z e i n j e c t e d i n t o an i n c i p i e n t l y f l u i d i z e d bed, were a l s o measured. The measured v a l u e s a r e p l o t t e d v e r s u s bubble e q u i v a l e n t d i a m e t e r i n F i g . 5-49. - 166 -+ 6 . £ h Downward force -2 --3 -I I I I ; I I ; 50 60 70 8 0 90 100 110 120 BUBBLE EQUIVALENT DIAMETER, mm Fig. 5 - 47. Upward and downward magnitudes of the force pulse vs. bubble equivalent diameter. Bed material : sand, dp= 430pm ; bed depth =0.3 m. - 167 -+ 2.0 1.6 1.2 0.8 ~ 0.4 -0.4 -0.8 -1.2 k -1.6 h UPWARD FORCE DOWNWARD FORCE 0 20 40 60 80 100 DISTANCE FROM THE BED CENTRE,mm Fig. 5-48. Upward and downward magnitudes of the force pulse vs. horizontal distance between the axis of the test cylinder and the point of injection of single bubbles. Fig. 5 - 4 9 . Pulse duration vs. bubble equivalent diameter. Bed material* sand, dp • 430 pm ; bed depth • 0.3 m. - 169 -The p l o t shows an i n v e r s e r e l a t i o n s h i p between the p u l s e d u r a t i o n s and bubble s i z e , t he p u l s e d u r a t i o n s d e c r e a s i n g w i t h bubble d i a m e t e r . However, c a r e f u l e x a m i n a t i o n o f f o r c e - t i m e r e c o r d s o f s i n g l e bubbles i n d i c a t e s t h a t b u b b l e s spend l o n g e r than e x p e c t e d i n the v i c i n i t y o f t h e t u b e , i . e . t h e i r movement i s i n t e r r u p t e d and r e t a r d e d by the e x i s t a n c e o f the t u b e . T h i s i s a l s o c l e a r i n the data measured i n t h e f r e e l y b u b b l i n g bed, e s p e c i a l l y a t low gas v e l o c i t i e s . F requency a n a l y s i s ( s p e c t r a l a n a l y s i s ) o f a l l d a t a r e c o r d s o f the measured f o r c e have i n d i c a t e d t h a t t h e p r i m a r y f r e q u e n c y c o n t e n t i s always i n t h e range 0-20 HZ. F o r c e s a r e d i s t r i b u t e d o v e r t h i s range o f f r e q u e n c y i n a s i m i l a r manner f o r a l l c a s e s . However, t h e f r e q u e n c y o f t h e spectrum peak v a r i e d between 1 and 5 HZ, depending on t h e bed o p e r a t i n g v a r i a b l e s . I t was found t o be s l i g h t l y dependent on e x c e s s gas v e l o c i t y , e x h i b i t e d the same dependence on p a r t i c l e s i z e as bubble f r e q u e n c y , d e c r e a s e d w i t h an i n c r e a s e i n bed depth and showed no s i g n i f i c a n t dependence on p a r t i c l e d e n s i t y . The above r e s u l t s s h o u l d be used w i t h c a u t i o n f o r bed m a t e r i a l s o t h e r than group B o f G e l d a r t ' s c l a s s i f i c a t i o n , as d i s c u s s e d i n S e c t i o n 5.2. F o r c e - t i m e r e c o r d s o f t h e h o r i z o n t a l component o f the f o r c e c o n s i s t o f p u l s e s o s c i l l a t i n g from s i d e t o s i d e around an approximate v a l u e o f z e r o . The magnitude , d u r a t i o n and f r e q u e n c y o f the f o r c e p u l s e s a r e r e l a t e d t o b u b b l e p r o p e r t i e s and bubble m o t i o n . As i n t h e c a s e o f the v e r t i c a l f o r c e s , t h e s e parameters a r e dependent on the bed o p e r a t i n g v a r i a b l e s . The magnitudes o f the h o r i z o n t a l f o r c e p u l s e s a r e g e n e r a l l y about o n e - t h i r d o f t h e v e r t i c a l p u l s e s . I f the - 170 -bed i s o p e r a t i n g i n the b u b b l i n g regime w i t h o u t s i g n i f i c a n t s o l i d e n t r a i n m e n t , p u l s e magnitudes i n c r e a s e w i t h i n c r e a s i n g s u p e r f i c i a l gas v e l o c i t y , p a r t i c l e d e n s i t y and bed depth and a r e o n l y s l i g h t l y a f f e c t e d by p a r t i c l e s i z e . The h o r i z o n t a l f o r c e p u l s e s t h e r e f o r e e x h i b i t t h e same t r e n d s w i t h the o p e r a t i n g v a r i a b l e s as f o r the v e r t i c a l f o r c e . However, t h e r e i s no s t a t i c p a r t f o r the h o r i z o n t a l component o f f o r c e . F r equency c o m p o s i t i o n o f t h e h o r i z o n t a l f o r c e i s s i m i l a r to t h a t o f t h e v e r t i c a l f o r c e , but the major f r e q u e n c y i s u s u a l l y l e s s . T h i s might be because the p r o b a b i l i t y o f a r i s i n g bubble s t r i k i n g one s i d e o f t h e tube and then the o t h e r i s l i m i t e d . A l s o , f o r c e - t i m e h i s t o r i e s o f the h o r i z o n t a l f o r c e a r e l e s s r e g u l a r . As i n the c a s e o f t h e v e r t i c a l f o r c e , p u l s e d u r a t i o n s are r e l a t e d t o bubble r i s e v e l o c i t y and bubbles b e h a v i o u r around the immersed o b j e c t . P u l s e d u r a t i o n s become s h o r t e r as bubbles r i s e more q u i c k l y . 5.5.2 S t a t i s t i c a l p r o p e r t i e s S t a t i s t i c a l p r o p e r t i e s o f t h e measured f o r c e have been d e s c r i b e d m o s t l y i n terms o f root-mean-squares (RMS), a r i t h m e t i c mean v a l u e , and s t a n d a r d d e v i a t i o n o f t h e f o r c e . The RMS v a l u e p r o v i d e s a measure o f the i n t e n s i t y o f t h e f o r c e s . Hence, i t was used e x t e n s i v e l y i n t h e data a n a l y s i s and i s i m p o r t a n t i n d e s i g n c o n s i d e r a t i o n s . As d i s c u s s e d e a r l i e r , i n s t a n t a n e o u s v e r t i c a l f o r c e s on t h e t e s t t u b e s , which c o n s t i t u t e the RMS v a l u e , a r e composed o f s t a t i c and dynamic components. The s t a t i c component - 171 -i s dominant a t low s u p e r f i c i a l gas v e l o c i t i e s ( c l o s e t o U m f ) , but i t d i m i n i s h e s as U i s i n c r e a s e d . The dynamic component, caused m a i n l y by bubble motion and a s s o c i a t e d s o l i d s c i r c u l a t i o n , i n c r e a s e s r a p i d l y as t h e b u b b l i n g becomes more v i g o r o u s . As a r e s u l t , t h e i n t e n s i t y (RMS) o f t h e f o r c e s i n c r e a s e s as b u b b l i n g i n t e n s i t y i n c r e a s e s , i . e . as U i n c r e a s e s . P a r t i c l e d e n s i t y and bed depth a r e a l s o i m p o r t n a t . The e f f e c t o f p a r t i c l e s i z e depends on t h e l e v e l o f gas v e l o c i t y , as d i s c u s s e d i n S e c t i o n 5.2. High e n t r a i n m e n t and s o l i d s hold-up i n the f r e e b o a r d , t r a n s i t i o n t o t u r b u l e n t f l u i d i z a t i o n , and an i n c r e a s e i n the downward component o f t h e f o r c e p u l s e s due t o s l u g g i n g above the tube a l l c a u s e t h e i n t e n s i t y o f t h e f o r c e s t o l e v e l o f f o r even d e c l i n e a t h i g h gas v e l o c i t i e s . The l i n e a r mean v a l u e o f the f o r c e s p r o v i d e s an approximate measure o f t h e s t a t i c component, e s p e c i a l l y c l o s e t o minimum f l u i d i -z a t i o n . At h i g h gas v e l o c i t i e s t he mean v a l u e s t a r t s t o d e c l i n e e a r l i e r i n a deep bed t h a n i n a s h a l l o w bed, because t h e downward component o f t h e f o r c e p u l s e s grows f a s t e r i n t h e deep bed t h a n i n t h e s h a l l o w one due t o l o n g e r bubble t r a j e c t o r i e s above the t u b e , and l a t e r , due t o s l u g s near t h e t o p o f the bed. The s t a n d a r d d e v i a t i o n o f the f o r c e p r o v i d e s an a p p r o x i m a t e measure o f t h e dynamic component. E x t r a p o l a t i o n o f the s t a n d a r d d e v i a t i o n p l o t l e a d s t o an a p p r o x i m a t e l y z e r o v a l u e a t minimum f l u i d i z a t i o n ; as b u b b l i n g i n t e n s i t y i n c r e a s e s , t h e s t a n d a r d d e v i a t i o n i n c r e a s e s i n the same way as the RMS f o r c e . I n s t a n t a n e o u s h o r i z o n t a l f o r c e s a r e composed o f the dynamic component o n l y , because s t a t i c f o r c e s i n the h o r i z o n t a l d i r e c t i o n are - 172 -a b s e n t . The RMS o f t h e h o r i z o n t a l f o r c e s has a v a l u e c l o s e t o z e r o a t minimum f l u i d i z a t i o n , where t h e dynamic component o f the f o r c e i s v e r y s m a l l , and i n c r e a s e s as the i n t e n s i t y o f b u b b l i n g i n c r e a s e s , e x h i b i t i n g the same t r e n d s w i t h the bed o p e r a t i n g v a r i a b l e s as the v e r t i c a l f o r c e . However, t h e RMS o f t h e h o r i z o n t a l f o r c e seems t o be l e s s s e n s i t i v e t o (U - U m^) a t h i g h gas v e l o c i t i e s f o r the r e a s o n s d i s c u s s e d i n S e c t i o n 5.2. In t h e p r e s e n t work, the mean v a l u e o f the h o r i z o n t a l f o r c e was c l o s e t o z e r o under a l l bed o p e r a t i n g c o n d i t i o n s . T h i s i s because the b u b b l i n g p r o f i l e was s y m m e t r i c a l around the t u b e , w i t h equal p r o b a b i l i t y f o r b u b b l e s to s t r i k e each s i d e . Non-zero m e a n " h o r i z o n t a l f o r c e s would l i k e l y a r i s e , however, f o r tubes c l o s e t o the w a l l o f the bed due t o t h e n a t u r a l development o f n o n - u n i f o r m s p a t i a l bubble d i s t r i b u t i o n s ( d i s c u s s e d i n C hapter 2 ) . 5.5.3 P e r i o d i c i t y and s t a t i o n a r i t y p r o p e r t i e s I. P e r i o d i c i t y : As d i s c u s s e d i n C hapter 4, t h e p r e s e n c e o f p e r i o d i c components i n o t h e r w i s e random da t a can be d e t e c t e d by v i s u a l i n s p e c t i o n o f the power s e p c t r a l f u n c t i o n , p r o b a b i l i t y d e n s i t y f u n c t i o n , o r a u t o c o r r e l a t i o n f u n c t i o n o f t h e d a t a . A n a l y s e s o f two a r b i t r a r y chosen samples o f the measured f o r c e , ( i n terms o f power spectrum, p r o b a b i l i t y d e n s i t y and a u t o c o v a r i a n c e f u n c t i o n s ) , were p r e s e n t e d and d i s c u s s e d i n C h a p t e r 4. These a n a l y s e s have i n d i c a t e d t h e e x i s t e n c e o f p e r i o d i c components w i t h i n random d a t a . However, the form and i n t e n s i t y o f p e r i o d i c i t i e s d i f f e r e d from one sample t o the next depending on the bed o p e r a t i n g c o n d i t i o n s . - 173 -Power s p e c t r a p r e s e n t e d e a r l i e r i n t h i s c h a p t e r a r e a l l c h a r a c t e r i z e d by a w e l l - d e f i n e d peak, w i t h a narrow w i d t h o f l e s s than 1 HZ. T h i s i n d i c a t e s t h a t the f o r c e s c o n t a i n b o t h p e r i o d i c and random components. T h i s c h a r a c t e r i s o f a g r e a t i m p o r t a n c e i n d e s i g n c o n s i d e r a t i o n s o f tube s y s t e m s . S i n c e f a t i g u e d e s i g n under random l o a d i n g i s a d i f f i c u l t t a s k (Kennedy, 1981), the f o r c e - t i m e r e c o r d may be a p p r o x i m a t e d by d e t e r m i n i s t i c p e r i o d i c f o r c i n g f u n c t i o n w i t h f r e q u e n c y equal t o the major f r e q u e n c y ( t h e spectrum peak f r e q u e n c y ) . The major f r e q u e n c y o f v e r t i c a l f o r c e s i s i n t h e range 1-5 HZ d epending on t h e bed o p e r a t i n g v a r i a b l e s , as d e m o n s t r a t e d b e f o r e . I I . S t a t i o n a r i t y : The s t a t i o n a r i t y p r o p e r t y o f t h e measured f o r c e s was d i s c u s s e d i n C h a p t e r 4. I t was d e m o n s t r a t e d t h e r e t h a t the f o r c e s a r e s t a t i o n a r y ( e x h i b i t s t a t i s t i c a l r e g u l a r i t y ) and t h a t t h e i r s t a t i s t i c a l p a rameters a r e r e p r o d u c i b l e . T h i s i s an e x p e c t e d f i n d i n g s i n c e causes o f the f o r c e f l u c t u a t i o n s , s p e c i f i c a l l y b u b b l e p r o p e r t i e s , a r e t h e m s e l v e s s t a t i o n a r y ( S i t n a i , 1981). - 174 -C H A P T E R 6 TUBES WITHIN ARRAYS: R E S U L T S AND D I S C U S S I O N 6.1 I n t r o d u c t i o n In t h i s c h a p t e r , e x p e r i m e n t a l f o r c e measurements on tubes w i t h i n v a r i o u s t u b e - a r r a y s are p r e s e n t e d and d i s c u s s e d . Force measurements on a t u b e a t the c e n t r e o f an a r r a y o f f i v e t u bes a r e p r e s e n t e d f i r s t . F o rce measurements on a tube w i t h i n a r r a y s o f t h r e e tubes i n two d i f f e r e n t c o n f i g u r a t i o n s are p r e s e n t e d n e x t . The measured f o r c e s a r e then r e l a t e d t o the hydrodynamics o f f l u i d i z e d beds. The r e s u l t s are f i n a l l y d i s c u s s e d as a g u i d e i n the d e s i g n o f f l u i d i z e d bed heat exchangers tube b u n d l e s . Throughout t h e e x p e r i m e n t s p r e s e n t e d i n t h i s c h a p t e r , bed m a t e r i a l was Ottawa sand; d p = 430 ym; bed depth ( s t a t i c ) = 0.45 m. 6.2 E x p e r i m e n t a l F o r c e s on a Tube a t t h e C e n t r e o f an A r r a y o f F i v e  Tubes E x p e r i m e n t s were c a r r i e d out w i t h an a r r a y o f f i v e 32 mm d i a m e t e r tubes i n an e q u i l a t e r a l - t r i a n g u l a r p i t c h a r r a y w i t h c e n t r e - t o - c e n t r e tube s p a c i n g o f 64 mm, as shown i n F i g . 6-1. The c e n t r e tube was i n s t r u m e n t e d . The c e n t r e - l i n e o f t h i s tube was 0.3 m above the gas - 175 -instrumented tube Fig. 6-1. Array of five tubes, each in an equ i l a t e r a l - t r i a n g u l a r pitch, is at the centre of the array. All of outer diameter The instrumented dimensions are in 3 2 mm , tube mm . - 176 -d i s t r i b u t o r . E x p e r i m e n t a l d e t a i l s and p r o c e d u r e s a r e d e s c r i b e d i n Chapter 3. In t h e s e e x p e r i m e n t s , t h e e x c e s s gas v e l o c i t y was v a r i e d from 0.05 t o 1.4 m/s. V e r t i c a l components o f f o r c e s : F i g u r e 6-2-a i s a f o r c e - t i m e r e c o r d o f the v e r t i c a l f o r c e s a t o p p o s i t e ends o f the c e n t r e tube o f t h e a r r a y ; r e c o r d e d a t an e x c e s s s u p e r f i c i a l gas v e l o c i t y o f 0.8 m/s. The f o r c e - t i m e r e c o r d ( a t each end o f t h e tube) c o n s i s t s o f p u l s e s , 3-4 p u l s e s per second w i t h d u r a t i o n s o f 0.08 t o 0.16 s. The magnitudes o f t h e p u l s e s a re between 6 and 12 N, and t he p u l s e s o c c u r a t each end s i m u l t a n e o u s l y . The magnitudes a r e g e n e r a l l y o f the same o r d e r , but to some e x t e n t a re d i f f e r e n t because i n s t a n t f o r c e s a r e not u n i f o r m l y d i s t r i b u t e d o v e r t he tube l e n g t h when bubbl e s a r e a t one end o r t h e o t h e r . F i g u r e 6-3-a p r e s e n t s a f o r c e -time r e c o r d o f t h e v e r t i c a l f o r c e s a t o p p o s i t e ends o f the same tube w i t h t h e tube i n i s o l a t i o n , r e c o r d e d under o p e r a t i n g c o n d i t i o n s i d e n t i c a l t o t h e above d a t a sample. In t h e l a t e r c a s e , magnitudes o f t h e f o r c e p u l s e s a r e between 6-18 N, about 1% ti m e s t h o s e o f the former sample. The p u l s e d u r a t i o n s a r e a l s o l o n g e r ; and the p u l s e s o c c u r a t a r a t e o f about 2 s"^ , about h a l f t h e f r e q u e n c y o b s e r v e d when the tube was i n t h e a r r a y . These d i f f e r e n c e s a r e demonstrated f u r t h e r by t h e spectrum p l o t s shown i n F i g u r e s 6-2-b and 6-3-b. These d i f f e r e n c e s appear t o a r i s e because t h e t u b e - a r r a y s i n d u c e s p l i t t i n g o f r i s i n g b u b b l e s , t h e r e b y i n h i b i t i n g f o r m a t i o n o f l a r g e bubbles (Newby and K e a i r n s , 1978; Nguyen e t a l . , 1979; G r a c e , 1982) which produce l a r g e f o r c e p u l s e s . F i g . 6 - 2 - a . V e r t i c a l f o r c e s a t o p p o s i t e a t t h e c e n t r e o f a n a r r a y o f f i v e t u b e s ( U - U m f ) = 0.8 m/s. e n d s o f a r e c o r d e d 32 a t mm t u b e 4.Q FREQUENCY J2J3 (HZJ 36.0 20.0 F i g . 6 - 2 - b . P o w e r s p e c t r a l e s t i m a t e s o f t h e t o t a l v e r t i c a l f o r c e s on t h e t u b e a t t h e c e n t r e o f t h e f i v e t u b e a r r a y . ( U - U m f ) = 0.8 m/s. - 178 -F i g . 6 - 3 - a . V e r t i c a i f o r c e s a t o p p o s i t e e n d s o f a s i n g l e 32 mm t u b e i n i s o l a t i o n , r e c o r d e d a t ( U - U m f ) = 0.8 m/s. - 179 -F i g u r e s 6-4 and 6-5 p r e s e n t data samples and t h e i r spectrum a n a l y s i s , r e c o r d e d a t t h e same e x c e s s gas v e l o c i t y (0.2 m/s) f o r t h e tube i n t h e a r r a y and f o r t h e same tube i n i s o l a t i o n , r e s p e c t i v e l y . A l t h o u g h , the f o r c e p u l s e s i n the two samples d i f f e r t o some e x t e n t i n magnitude, r a t e o f p u l s e o c c u r r e n c e and u n i f o r m i t y o f t h e p u l s e s , the d i f f e r e n c e s between them a r e not as s i g n i f i c a n t as i n the two p r e v i o u s samples. T h i s i n d i c a t e s t h a t t h e n e i g h b o u r i n g tubes have most i n f l u e n c e a t h i g h gas v e l o c i t i e s where bubbles ( i n the absence o f t u b e s ) a r e l a r g e r e l a t i v e t o i n t e r - t u b e s p a c i n g s . The f r e q u e n c y c o m p o s i t i o n o f the f o r c e s on the c e n t r e - t u b e o f t h e a r r a y i s i n t h e range 0-20 HZ, as f o r an i s o l a t e d t u b e . However, th e major f r e q u e n c y ( s p e c t r u m peak) i s h i g h e r than t h a t f o r an i s o l a t e d tube o v e r t h e e n t i r e range o f gas v e l o c i t y i n v e s t i g a t e d , as shown i n F i g u r e 6-6. As r i s i n g b ubbles pass t h r o u g h an a r r a y o f t u b e s , t h e y tend t o break up i n t o a l a r g e r number o f b u b b l e s as the r e s u l t o f d i r e c t c o l l i s i o n s w i t h i n d i v i d u a l elements o f t h e a r r a y , e s p e c i a l l y t h o s e t u bes a t the bottom o f t h e a r r a y . T h i s i n c r e a s e s the r a t e o f p u l s e o c c u r r e n c e i n t h e f o r c e - t i m e h i s t o r i e s . The f i g u r e a l s o shows t h a t the a r r a y i s most e f f e c t i v e i n i n c r e a s i n g the major f r e q u e n c y a t h i g h gas v e l o c i t i e s , where bubble s i z e i s r e l a t i v e l y l a r g e compared t o i n t e r - t u b e s p a c i n g s . The root-mean s q u a r e (RMS) v e r t i c a l f o r c e s a r e p l o t t e d a g a i n s t (U - U m^) i n F i g u r e 6-7, f o r the s i n g l e tube and t u b e - i n - a r r a y . The f i g u r e shows t h a t t h e i n t e n s i t y o f t h e f o r c e s on a tube w i t h i n the a r r a y , does not i n c r e a s e as q u i c k l y as f o r the tube i n i s o l a t i o n . I n s t e a d , t h e c o n t r o l o f bubble s i z e and bubble d i s t r i -b u t i o n by t h e n e i g h b o u r i n g t u b e s p l a y s a d r a m a t i c r o l e i n l i m i t i n g the i n t e n s i t y o f f o r c e s . At h i g h gas v e l o c i t y , the i n t e n s i t y o f F i g . 6 - 4 - a . V e r t i c a l f o r c e s o n a 3 2 mm t u b e a t t h e c e n t r e o f a n a r r a y o f f i v e t u b e s , r e c o r d e d a t ( U - U m f ) = 0.2 m/s. 0.0 4.0 8.0 32.0 36.0 20.0 FREQUENCY (HZ) F i g . 6 - 4 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l v e r t i c a l f o r c e s o n t h e t u b e a t t h e c e n t r e o f t h e a r r a y . Fig. 6-5-a. Vertical forces on a single 32 mm diameter tube in i s o l a t i o n , recorded at (U-U f) = 0.2 m/s. o i i i 1 1 1 1 i i i —I 0.0 4.0 6.0 ]2.0 36.0 20.0 FREQUENCY (HZ) Fig. 6-5-b. Power spectral estimates of total v e r t i c a l forces for the tube in i s o l a t i o n . 8 N X > o z LU ID O LU CC Lu CC O -a < 2 2 r-o • O 0.5 Single tube in isolation -O- Tube at the centre of an array of 5 tubes. 1.0 1.5 ( U - U m f ) , m / s Fig. 6 - 6 . Variation of the major frequency with excess superficial gas velocity, ( U - U m f ) , for two different tube arrangements (single tube in isolation and tube at the centre of an array of five tubes). CO ro - 183 -10 8 UJ g 6 p o 0 I-UJ > CO 4 L 3 r 2 L - • - Single tube in isolation - O Tube at the centre of an array of 5 tubes. 0.5 1.0 ( U - U m f ) , m / s 1.5 Fig. 6 -7 . RMS vertical force vs. excess superficial gas velocity, ( U - U m f ) . - 184 -t h e f o r c e s a p p e a r s t o l e v e l o f f a f t e r b e i n g maximum. T h i s o c c u r s not o n l y because o f t u b e - a r r a y - i n d u c e d s p l i t t i n g o f b u b b l e s , b u t , i n a d d i t i o n , r e g i o n s o f t u r b u l e n t f l u i d i z a t i o n may be c o e x i s t i n g w i t h b u b b l i n g r e g i o n s w i t h i n the tube a r r a y (Staub and Canada, 1978). At low (U - U ^ ) , below 0.1 m/s, t h e tube a r r a y appeared t o i n c r e a s e s l i g h t l y the f o r c e s on the c e n t r e - t u b e o f t h e a r r a y . At such a low gas v e l o c i t y , the t u b e - a r r a y has l i t t l e i n f l u e n c e on bubble s i z e , but d e c r e a s e s t h e s i z e and s t a b i l i t y o f t h e d e f l u i d i z e d cap above the c e n t r e - t u b e o f the a r r a y (Rooney and H a r r i s o n , 1976). F i g u r e 6-8 shows the v a r i a t i o n i n the s t a n d a r d d e v i a t i o n o f the v e r t i c a l f o r c e s f o r the two tube a r r a n g e m e n t s . As d i s c u s s e d p r e v i o u s l y , the s t a n d a r d d e v i a t i o n can be r e g a r d e d as a measure o f t h e dynamic component o f the f o r c e s . The f i g u r e i n d i c a t e s t h a t the t u b e - a r r a y s u p p r e s s e s the dynamic component o f the f o r c e s , e s p e c i a l l y as gas v e l o c i t y i n c r e a s e s . T h i s a g a i n can be a t t r i b u t e d t o break-up o f b u b b l e s to produce bubbles o f much more unif o r m s i z e , which produce f o r c e s o f more u n i f o r m magnitudes. H o r i z o n t a l components o f f o r c e : A f o r c e - t i m e r e c o r d o f the h o r i z o n t a l f o r c e s on the c e n t r e tube o f t h e a r r a y i s shown i n F i g u r e 6-9-a, r e c o r d e d at an e x c e s s gas v e l o c i t y o f 0.8 m/s. The f o r c e o s c i l l a t e s from s i d e to s i d e i n p u l s e s around mean v a l u e o f z e r o . The g e n e r a l c h a r a c t e r o f t h e f o r c e i s s i m i l a r t o t h a t o f the h o r i z o n t a l f o r c e on a s i n g l e tube i n i s o l a t i o n , shown i n F i g u r e 6-10, but the magnitudes o f t h e peaks are much s m a l l e r . The r e a s o n f o r t h i s i s the same as f o r the v e r t i c a l f o r c e s , i . e . t h e t u b e - a r r a y i n d u c e s s p l i t t i n g o f b u b b l e s and p r e v e n t s f o r m a t i o n o f 10 - 185 -0 0 . 5 1.0 1.5 (U-Umf),m/s Fig. 6 -8 . Standard deviation of vertical force vs. excess superficial gas velocity, ( U - Umf). F i g . 6 - 9 - a . H o r i z o n t a 1 ' f o r c e s a t o p p o s i t e e n d s o f a 32 mm d i a m e t e r t u b e a t t h e c e n t r e o f an a r r a y o f f i v e t u b e s r e c o r d e d a t (U-U f ) = 0.8 m/s. F i g . 6 - 9 - b . P o w e r s p e c t r a l e s t i m a t e s o f t o t a l h o r i z o n t a l f o r c e s 5 - — — j ^ — - — — -0 nj" V. — — — — N\ L — \ V 5 — - — — - l" — - — — -: 1 1 i I \ I 1 ( 1 1 1 ' I I I + 5 — — — — - . o - . fjrc V — - i — - — - -L A - ~ 7 t \ 4 H 5 — * V — — - - t V V t - - — - -- 2 T IME (S) 4 J > ( F i g . 6 - 1 0 . H o r i z o n t a l 32 mm d i a m e t e r t u b e i n f o r c e s a t o p p o s i t e e n d s o f a s i n g l e i s o l a t i o n , r e c o r d e d a t ( U - U m f ) = 0.8 m/s . - 188 -l a r g e b u b b l e s which produce p u l s e s o f l a r g e magnitude. In comparing the h o r i z o n t a l f o r c e t r a c e i n F i g u r e 6-9-a w i t h t h a t o f t h e v e r t i c a l f o r c e i n F i g u r e 6-2-a, we f i n d t h a t the magnitude o f t h e h o r i z o n t a l f o r c e p u l s e s i s about 1/4 o f t h a t o f t h e v e r t i c a l f o r c e . Power s p e c t r a l a n a l y s i s o f t h e h o r i z o n t a l f o r c e d a t a samples a g a i n i n d i c a t e t h a t t h e p r i m a r y f r e q u e n c y c o m p o s i t i o n o f the f o r c e l i e s i n th e range 0-20 HZ. However, t h e major f r e q u e n c i e s o f t h e h o r i z o n t a l f o r c e a r e much lower t h a n t h o s e o f t h e v e r t i c a l f o r c e . The RMS and mean v a l u e s o f t h e h o r i z o n t a l f o r c e s a r e p l o t t e d v e r s u s e x c e s s s u p e r f i c i a l gas v e l o c i t y i n F i g u r e 6-11 f o r both tube a r r a n g e m e n t s . The f i g u r e shows t h a t t h e t u b e - a r r a y appeared t o have a s t r o n g e f f e c t i n s u p p r e s s i n g t h e i n t e n s i t y o f f o r c e s on the c e n t r e -t u b e o f t h e a r r a y . T h i s e f f e c t i s e s p e c i a l l y i m p o r t a n t a t h i g h s u p e r f i c i a l gas v e l o c i t i e s as w i t h the RMS p l o t o f t h e v e r t i c a l f o r c e s ( c f . F i g u r e 6-7). As d i s c u s s e d i n Chapter 5, h o r i z o n t a l components o f f o r c e a r e composed o f dynamic c o n t r i b u t i o n s caused by m o t i o n o f b u b b l e s and a s s o c i a t e d s o l i d s . Hence, the c o n t r o l o f bubble s i z e and bubble d i s t r i b u t i o n by n e i g h b o u r i n g tubes reduces the i n t e n s i t y o f the h o r i z o n t a l f o r c e s , as w i t h t h e v e r t i c a l f o r c e s . The tube a r r a y i s b e l i e v e d t o have s h i e l d i n g e f f e c t s on both s i d e s o f the c e n t r e - t u b e . The mean v a l u e o f the h o r i z o n t a l f o r c e i s v e r y c l o s e to z e r o f o r a l l c a s e s as shown i n t h e f i g u r e . - 189 -Single tube in isolation - O Tube at the centre of an array of 5 tubes . RMS M E A N — • — % % 8 8 g _ i i 0.5 1.0 1.5 (U-U m f ) ,m/s Fig. 6-11. RMS and mean values of horizontal force vs. excess superficial gas velocity, ( U - U m f ) . - 190 -6.3 E x p e r i m e n t a l F o r c e s on a Tube w i t h i n an A r r a y o f Three Tubes  i n Two C o n f i g u r a t i o n s The r e s u l t s p r e s e n t e d above have demonstrated t h a t n e i g h b o u r i n g t u b e s have a s t r o n g i n f l u e n c e on t h e i n t e n s i t y and f r e q u e n c y o f f o r c e s on i n t e r n a l t u b e s o f a r r a y s , e s p e c i a l l y when the gas f l o w r a t e i s r e l a t i v e l y l a r g e . In o r d e r t o g a i n i n s i g h t i n t o the mechanism o f t h e e f f e c t o f upstream tubes and o f downstream t u b e s , e x p e r i m e n t s were co n d u c t e d w i t h two d i f f e r e n t c o n f i g u r a t i o n s , t h r e e tubes each, shown i n F i g u r e s 6-12 and 6-13. The e l e v a t i o n o f t h e i n s t r u m e n t e d t u b e above the gas d i s t r i b u t o r was 0.3 m i n t h e both c a s e s . E x p e r i m e n t a l d e t a i l s and p r o c e d u r e a r e d e s c r i b e d i n C h a p t e r 3. 1. E x p e r i m e n t s w i t h t u b e - a r r a y shown i n F i g u r e 6-12: F i g u r e 6-14 shows a f o r c e - t i m e r e c o r d o f t h e measured v e r t i c a l f o r c e s a t o p p o s i t e ends o f the i n s t r u m e n t e d t u b e , r e c o r d e d at an excess gas v e l o c i t y o f 0.8 m/s. Comparison o f t h i s data sample w i t h F i g u r e 6-2-a, ( f i v e - t u b e a r r a y a t same gas v e l o c i t y ) r e v e a l s t h a t t h e t r a c e s a r e s i m i l a r w i t h two d i f f e r e n c e s . In F i g u r e 6-14 some p u l s e s have n e g a t i v e magnitudes; such p u l s e s do not e x i s t i n t h e o t h e r sample. These p u l s e s o f n e g a t i v e magnitudes a r e c a u s e d , as d i s c u s s e d i n C h a p t e r 5, by down-flow o f s o l i d s t r e a m s , e s p e c i a l l y t h o s e f o l l o w i n g c o l l a p s e o f the upper bed s u r f a c e , and by the n e g a t i v e p r e s s u r e f i e l d o f bubbles t r a v e l l i n g above the t u b e . T h i s i n d i c a t e s t h a t the h i g h e s t row i n t h e f i v e - t u b e a r r a y shown i n F i g u r e 6-1 s h e l t e r e d the lower t u b e , d e c r e a s i n g the downward t h r u s t by s p l i t t i n g d e s c e n d i n g s o l i d streams i n t o s m a l l e r ones and r e t a r d i n g the downward motion (Nguyen - 191 -instrumented tube F i g . 6 - 1 2 . A r r a y o f t h r e e t u b e s , e a c h o f o u t e r d i a m e t e r 32 mm, i n an e q u i l a t e r a l - t r i a n g u l a r p i t c h ; i n s t r u m e n t e d t u b e i n d o w n s t r e a m o f t h e o t h e r t u b e s . instrumented tube F i g . 6 - 1 3 . A r r a y o f t h r e e t u b e s , e a c h o f o u t e r d i a m e t e r 32 mm, i n a n e q u i l a t e r a l - t r i a n g u l a r p i t c h ; i n s t r u m e n t e d t u b e i s u p s t r e a m o f t h e two o t h e r t u b e s . F i g . 6 - 14. V e r t i c a l f o r c e s a t o p p o s i t e e n d s o f t h e i n s t r u m e n t e d t u b e o f t h e t u b e - a r r a y s h o wn i n F i g . 6 - 1 2 , r e c o r d e d a t ( U - U m f ) = 0.8 m/s. - 193 -et a l . , 1979). In a d d i t i o n , the upper row o f tubes p r e v e n t s o r d e l a y s r e c o a l e s c e n c e o f d i v i d e d b u b b l e s , d e c r e a s i n g t h e e f f e c t o f t h e n e g a t i v e p r e s s u r e f i e l d . Some p u l s e s o f n e g a t i v e magnitudes may be c a n c e l l e d by p u l s e s o f p o s i t i v e magnitude r e s u l t i n g i n s m a l l d i s t o r t e d p u l s e s . T h i s may e x p l a i n why t h e magnitudes o f t h e f o r c e p u l s e s i n F i g u r e 6-14 a r e on a v e r a g e l e s s than t h o s e i n F i g u r e 6-2-a, but t h i s d i f f e r e n c e i s not v e r y s i g n i f i c a n t . Except f o r t h e s e two d i f f e r e n c e s , the two samples a r e s i m i l a r , b u t t h e y d i f f e r s i g n i f i c a n t l y from the c o r r e s p o n d i n g sample f o r t h e s i n g l e tube i n i s o l a t i o n shown i n F i g u r e 6-3-a. Hence, the l o w e s t row o f an a r r a y o f t u b e s , w i t h t i g h t i n t e r - t u b e s p a c i n g , appears to have t h e major r o l e i n c h a n g i n g the c h a r a c t e r i s t i c s o f the f o r c e s . The above o b s e r v a t i o n s , o b t a i n e d by v i s u a l e x a m i n a t i o n o f the f o r c e - t i m e r e c o r d s , a r e d e m o n s t r a t e d f u r t h e r by t h e RMS p l o t shown i n F i g u r e 6-15. The f i g u r e shows t h a t t h e upstream tubes have a v e r y s i g n i f i c a n t e f f e c t i n c o n t r o l l i n g the i n t e n s i t y o f f o r c e s on the downstream, e s p e c i a l l y a t high gas v e l o c i t i e s . 2. E x p e r i m e n t s w i t h t u b e - a r r a y shown i n F i g u r e 6-13: A f o r c e - t i m e r e c o r d o f the measured v e r t i c a l f o r c e s i s shown i n F i g u r e 6-16 f o r an e x c e s s gas v e l o c i t y o f 0.8 m/s. T h i s r e c o r d i s s i m i l a r t o F i g u r e 6-3-a f o r a s i n g l e tube i n i s o l a t i o n a t the same gas v e l o c i t y , e x c e p t t h a t f o r c e p u l s e s o f n e g a t i v e magnitude were a b s e n t i n the p r e s e n t sample. As mentioned above, t h e downstream row o f t h e a r r a y r e t a r d s the m o t i o n o f d e s c e n d i n g s o l i d streams and a l s o d e c r e a s e s the e f f e c t o f t h e n e g a t i v e p r e s s u r e f i e l d o f b u b b l e s on t h e lower t u b e . T h i s d e c r e a s e i n the downward components i n c r e a s e s - 194 -10 8 3 h 2 h I h O Single tube in isolation ••• Tube within an array of 3 tubes . s h o w n in F i g . 6-12 0.5 1.0 1.5 ( U - U m f ) , m/s Fig. 6-15. RMS vertical force vs. excess superficial gas velocity ( U - U m f ) , for two differrent tube arrangements. F i g . 6 -16. V e r t i c a l f o r c e s t u b e o f t h e t u b e - a r r a y s h o ( U - l i f ) = 0.8 m/s. a t o p p o s i t e e n d s o f t h e i n s t r u m e n t e d wn i n F i g . 6 - 1 3 , r e c o r d e d a t - 196 -the i n s t a n t a n e o u s r e s u l t a n t f o r c e s ( i n t h e upward d i r e c t i o n ) on the upstream t u b e . The same t r e n d i s e x h i b i t e d by the RMS p l o t shown i n F i g u r e 6-17. The major r o l e o f the downstream tubes o f an a r r a y i s t h e r e f o r e t o d e c r e a s e t h e downward components o f f o r c e on the upstream t u b e s , hence i n c r e a s i n g the t o t a l f o r c e i n upward d i r e c t i o n . T h i s s u g g e s t s t h a t the h i g h e s t row o f an a r r a y s h o u l d e x p e r i e n c e t h e minimum i n t e n s i t y o f f o r c e s among the tubes i n an a r r a y . 6.4 G e n e r a l D i s c u s s i o n The d a t a p r e s e n t e d above have shown the e f f e c t s o f n e i g h b o u r i n g tubes on the g e n e r a l c h a r a c t e r i s t i c s o f f o r c e s under s t e a d y o p e r a t i n g c o n d i t i o n s . In a d d i t i o n , t h e y have e n a b l e d us to u n d e r s t a n d the i n f l u e n c e o f upstream and downstream rows on the o t h e r tubes i n an a r r a y . As shown i n F i g u r e 6-18, the l o w e s t row i n an a r r a y reduces s i g n i f i c a n t l y the i n t e n s i t y o f f o r c e s on the h i g h e r tubes by c o n t r o l l i n g bubble s i z e and p r o b a b l y by r e t a r d i n g the motion o f bubbles as they pass t h r o u g h t h e s e t u b e s . T h i s i n f l u e n c e o f upstream tubes no doubt depends on the i n t e r - t u b e s p a c i n g r e l a t i v e t o bubble s i z e at the bottom o f the a r r a y . The e f f e c t o f n e i g h b o u r i n g tubes i s e x p e c t e d t o d i m i n i s h as the i n t e r - t u b e s p a c i n g becomes l a r g e r e l a t i v e to bubble s i z e . For a f i x e d i n t e r - t u b e s p a c i n g , the e f f e c t o f tubes a t the bottom o f t h e bundle i n c r e a s e s w i t h i n c r e a s i n g s u p e r f i c i a l gas v e l o c i t y , as shown i n F i g u r e 6-18, and w i t h i n c r e a s i n g h e i g h t o f t h e a r r a y above t h e gas d i s t r i b u t o r . The h i g h e r t h e b u n d l e , the l a r g e r t h e r i s i n g b u bbles grow b e f o r e b e i n g broken up, so t h a t l a r g e r f o r c e s a r e t r a n s m i t t e d t o t h o s e tubes a t the bottom o f t h e a r r a y . T h i s means - 197 -10 8 U J o tr 5 o ° < £ 4 - • - Single tube in isolation - + - Tube within an array of 3 tubes, shown in Fig. 6-13 0.5 1.0 1.5 ( U - l W . m / s Fig. 6-17. RMS vertical force vs. excess superficial gas velocity (U-Umf), for two different tube arrangements. - 198 -10 8 UJ o cc £ 5 _ l < o i= 4 oc Ul > CO - • - Single tube in isolation - O Tube at the centre of an array of 5 tubes. -•- Tube within arrangement of tubes shown in Fig. 6-12 - + - Tube within arrangement of tubes shown in Fig.6-13 1.5 0 0.5 1.0 (U-Umf ), m/s Fjg. 6-18. RMS vertical force vs. excess superficial gas velocity, (U-Umf), for different tube configurations. - 199 -t h a t tubes i n t h e l o w e s t row i n a bundle o f t i g h t l y spaced tubes a r e e x p e c t e d t o be t h e most s e v e r e l y l o a d e d t u b e s i n t h e a r r a y ; however, t h e i n t e n s i t y o f the f o r c e s on t h e s e tubes can be reduced by l o w e r i n g t h e i r l e v e l above t h e gas d i s t r i b u t o r . Large i n t e r - t u b e s p a c i n g s a l l o w bubbles t o grow f u r t h e r as they pass t h r o u g h the b u n d l e , i n c r e a s i n g the i n t e n s i t y o f f o r c e s on tubes i n t h e m i d d l e and near the t o p o f t h e a r r a y . F i g u r e 6-18 a l s o shows t h a t t h e e f f e c t o f the h i g h e s t row i n a b u n d l e i s t o i n c r e a s e t h e i n t e n s i t y o f f o r c e s on the lower tubes i n t he a r r a y by r e d u c i n g the downward components o f the f o r c e s on the l a t e r . The r e a s o n s f o r t h i s e f f e c t have been d e s c r i b e d i n the p r e v i o u s s e c t i o n . The f o r c e s on tubes o f t h e topmost row are e x p e c t e d t o be a minimum i n comparison w i t h the f o r c e s on the o t h e r tubes i n t h e b u n d l e . I t becomes c l e a r t h a t d i s t r i b u t i o n o f f o r c e s on i n d i v i d u a l t u bes w i t h i n an a r r a y depends on t h e i n t e r - t u b e s p a c i n g s and h e i g h t o f t h e t u b e bundle above the gas d i s t r i b u t o r . Tubes a t the bottom o f the a r r a y a r e e x p e c t e d t o be t h e most s e v e r e l y l o a d e d t u b e s , w i t h the l o a d s e v e r i t y d i m i n i s h i n g w i t h i n c r e a s i n g tube h e i g h t and r e a c h i n g a minimum f o r t h e t o p t u b e s o f t h e a r r a y . T h i s i s i n agreement w i t h f i n d i n g o f Kennedy e t a l . ( 1 9 8 1 ) . At h i g h gas v e l o c i t i e s , t he e f f e c t o f t r a n s i t i o n t o t u r b u l e n t f l u i d i z a t i o n i n t h e t u b e - a r r a y r e g i o n s h o u l d a l s o be c o n s i d e r e d . I f t u b e s a r e i n a s q u a r e - p i t c h a r r a y , i n s t e a d o f a t r i a n g u l a r - p i t c h , t h e t u b e s o f t h e a r r a y a r e e x p e c t e d t o have an even g r e a t e r s h i e l d i n g e f f e c t on the h i g h e r tubes which may l e a d t o a f u r t h e r d e c r e a s e i n the i n t e n s i t y o f v e r t i c a l f o r c e s on the h i g h e r t u b e s . - 200 -F o r c e s on tubes c l o s e t o the w a l l o f t h e bed, e s p e c i a l l y i n deep beds, a r e e x p e c t e d to be p r e d o m i n a n t l y downward, s i n c e s o l i d p a r t i c l e s i n t h i s zone f l o w i n downward d i r e c t i o n i n " s t i c k - s l i p - f l o w " ( P e t e r and Whitehead, 1982). However, t h e s e f o r c e s a r e not e x p e c t e d t o be as s e v e r e as t h o s e on tubes i n t h e main body o f the bed because the v e l o c i t y o f d e s c e n d i n g s o l i d s i s much lower than t h a t o f s o l i d s a s s o c i a t e d w i t h r i s i n g b u b b l e s which o c c u r m a i n l y c l o s e t o the bed c e n t r e i n deep beds ( P e e l e r and Whitehead, 1982). In c o n c l u s i o n , a p p r o p r i a t e d e s i g n o f a t u b e - b u n d l e immersed i n a gas f l u i d i z e d bed c o u l d be a c h i e v e d by: 1) U s i n g r e a s o n a b l y t i g h t i n t e r - t u b e s p a c i n g s . F a c e - t o - f a c e tube s p a c i n g s h o u l d be chosen as s m a l l as p o s s i b e , but not l e s s than the n e c e s s a r y space r e q u i r e d t o a l l o w s o l i d s t o c i r c u l a t e f r e e l y , which i s 20-30 p a r t i c l e d i a m e t e r s o r more ( G r a c e , 1982). C o n s t r u c t i n g t u b e - a r r a y s w i t h t i g h t i n t e r - t u b e s p a c i n g s p r o v i d e s a mean o f c o n t r o l l i n g the i n t e n s i t y o f f o r c e s on i n d i v i d u a l t u bes f o r m i n g the a r r a y , e x c e p t on t h e bottom t u b e s . Compact t u b e - b u n d l e s have o t h e r a d v a n t a g e s . They improve the q u a l i t y o f f l u i d i z a t i o n , reduce a m p l i t u d e o f bed p r e s s u r e f l u c t u a t i o n s , and reduce e l u t r i a t i o n . Moreover, the c o n t r o l o f bubble s i z e and d i s t r i b u t i o n promoted by compact t u b e - b u n d l e s a s s i s t s s i m u l a t i o n o f commerical f l u i d i z e d beds (Newby and K e a i r n s , 1978). On t h e o t h e r hand, c l o s e l y spaced tubes may h i n d e r s o l i d s m i x i n g and p r e v e n t smooth c i r c u l a t i o n o f s o l i d s , a l l o w i n g temperature g r a d i e n t s t o d e v e l o p i n the bed and i n c r e a s i n g the p r e s s u r e drop s i g n i f i c a n t l y . 2) The t u b e - a r r a y s h o u l d be as c l o s e as p o s s i b l e t o the gas d i s t r i b u t o r , but f a r enough from g r i d j e t s t o m i n i m i z e tube e r o s i o n and p a r t i c l e a t t r i t i o n . - 201 -C H A P T E R 7 GENERAL D I S C U S S I O N , C O N C L U S I O N S AND RECOMMENDATIONS 7.1 G e n e r a l D i s c u s s i o n The " s c a l e - u p " o f f l u i d i z e d beds has l o n g been r e c o g n i z e d as a major problem because the dynamics o f f l u i d i z e d systems are not s u f f i c i e n t l y u n d e r s t o o d ( B o t t e r i l l , 1975). There are a number o f t e c h n i q u e s f o r s c a l i n g - u p p r o c e s s equipment ( J o h n s t o n e and T h r i n g , 1957) based on s i m i l a r i t y p r i n c i p l e s . Foremost among t h e s e i s d i m e n s i o n a l a n a l y s i s , but i n the case o f f l u i d i z e d systems t h e r e a r e so many p o s s i b l e r a t i o s t h a t i t i s not p r a c t i c a l t o c o n s i d e r them a l l (Rowe, 1963). A more u s e f u l a p p roach i s to t r y t o u n d e r s t a n d why performance o f f l u i d i z i n g systems change w i t h i n c r e a s i n g s c a l e ( G r a c e , 1980). There a r e t h r e e i m p o r t a n t d i f f e r e n c e s between l a r g e and sm a l l f l u i d i z e d beds ( D a v i d s o n , 1973): (1) In sma l l columns, w a l l e f f e c t s i n f l u e n c e r i s i n g b ubbles e s p e c i a l l y a t h i g h s u p e r f i c i a l v e l o c i t i e s c a u s i n g many l a b o r a t o r y - s c a l e f l u i d i z e d beds t o o p e r a t e i n the s l u g f l o w r e g i m e . (2) Large bubbles have an o p p o r t u n i t y t o form i n l a r g e d i a m e t e r beds. (3) G u l f - s t r e a m c i r c u l a t i o n i s l i k e l y t o be more v i g o r o u s i n l a r g e beds. In a d d i t i o n , f o r r e l a t i v e l y l a r g e - 202 -beds, bubbles have a t e n d e n c y to r i s e i n c h a i n s o r " t r a c k s " r a t h e r than randomly i n space (Werther, 1977, Whitehead and Young, 1967). The t r a n s i t i o n t o t u r b u l e n t f l u i d i z a t i o n may a l s o be i n f l u e n c e d by s c a l e , a l t h o u g h t h e r e i s i n s u f f i c i e n t e x p e r i e n c e t o i n d i c a t e whether o r not t h i s o c c u r s . These d i f f e r e n c e s and u n c e r t a i n t i e s as to the n a t u r e o f f l u i d i z e d systems s t i l l d i c t a t e c a u t i o n i n s c a l i n g - u p . An i m p o r t a n t r e s u l t i n the p r e s e n t work which may be b e n e f i c i a l i n the d e s i g n o f tube bundle heat exchangers f o r l a r g e s c a l e f l u i d i z e d beds i s t h a t f o r c e s on immersed tubes a r e d i r e c t l y r e l a t e d t o bubble c h a r a c t e r i s t i c s . Hence the i n t e n s i t y o f t h e f o r c e s can be s i g n i f i -c a n t l y reduced by c o n t r o l l i n g bubble s i z e . T h i s can be a c h i e v e d , as d i s c u s s e d i n C hapter 6, by u s i n g r e a s o n a b l y t i g h t i n t e r - t u b e s p a c i n g s and by l o w e r i n g the l e v e l o f b u n d l e s above the gas d i s t r i b u t o r . As d i s c u s s e d i n C h a p t e r s 4 and 5, f o r c e s on immersed t u b e s i n a f l u i d i z e d bed can be c h a r a c t e r i z e d as c o n t a i n i n g both p e r i o d i c and random components. T h i s b e h a v i o u r i s r e l a t e d t o the b u b b l i n g p a t t e r n , which e x h i b i t s superimposed random and p e r i o d i c c h a r a c t e r i s t i c s . The p e r i o d i c components o f f o r c e a r e p r o b a b l y c a u s e d by what i s s o - c a l l e d bubble " t r a c k s " . As noted above, b u b b l e s have a g r e a t e r t e n dency to r i s e i n c h a i n s i n l a r g e f l u i d i z e d beds so t h a t p e r i o d i c components o f f o r c e on immersed t u b e s i n l a r g e f l u i d i z e d beds are e x p e c t e d t o be more i n t e n s e . The a v e r a g e w a i t i n g p e r i o d between s u c c e s s i v e c h a i n s r i s i n g w i t h i n a r e l a t i v e l y l a r g e bed (1 m d i a m e t e r ) has been found t o be o f t h e o r d e r o f seconds (Werther, 1977). Hence, f r e q u e n c i e s o f t h o s e p e r i o d i c components o f f o r c e on t u bes i n a l a r g e - s c a l e d f l u i d i z e d bed a r e e x p e c t e d t o be low (below 10 HZ). The n a t u r a l f r e q u e n c i e s o f t h e s t r u c t u r e can t h e r e f o r e be a r r a n g e d s a f e l y t o a v o i d r e s o n a n c e . - 203 -7.2 C o n c l u s i o n s o f t h i s Work The main c o n c l u s i o n s o f t h i s work a r e : 1) F o r c e measurements on i n d i v i d u a l tubes show t h a t the v e r t i c a l component c o n s i s t s o f a s e r i e s o f p u l s e s whose m a g n i t u d e s , d u r a t i o n s and r a t e o f o c c u r r e n c e depend on the bed o p e r a t i n g p a r a m e t e r s , i n a d d i t i o n t o the s i z e and shape o f the t u b e . The h o r i z o n t a l component o f the f o r c e c o n s i s t s o f p u l s e s o s c i l l a t i n g from s i d e t o s i d e around a v a l u e o f z e r o , and g e n e r a l l y e x h i b i t i n g t h e same t r e n d s w i t h t h e v a r i a b l e s as t h e v e r t i c a l component. H o r i z o n t a l f o r c e s a r e g e n e r a l l y o f s i g n i f i c a n t l y lower magnitude than v e r t i c a l . 2) F o r c e s e x h i b i t superimposed p e r i o d i c and random c h a r a c t e r i s t i c s , and p o s s e s s s t a t i o n a r i t y p r o p e r t i e s . 3) Power s p e c t r a l a n a l y s i s o f the t i m e s e r i e s i n d i c a t e d t h a t the p r i m a r y f r e q u e n c y c o m p o s i t i o n o f the f o r c e s i s i n the range 0-20 HZ, w i t h major f r e q u e n c i e s below 10 HZ and i n s e n s i t i v e t o the o p e r a t i n g c o n d i t i o n s . 4) F o r c e s a r e r e l a t e d t o hydrodynamic c o n d i t i o n s , e s p e c i a l l y bubble p r o p e r t i e s , p r e v a i l i n g w i t h i n t h e bed. F o r c e p u l s e s a r e c a used by b u b b l e s and a s s o c i a t e d s o l i d m o t i o n . Force maxima c o r r e s p o n d t o a r r i v a l o f bubbles at t h e tube s u r f a c e . 5) The i n t e n s i t y and d i s t r i b u t i o n o f f o r c e s on t u bes w i t h i n an a r r a y depend on the p o s i t i o n o f t h e tube w i t h i n the a r r a y , i n t e r - t u b e s p a c i n g and t h e h e i g h t o f the t u b e - b u n d l e above the - 204 -gas d i s t r i b u t o r . Tubes a t t h e bottom o f an a r r a y a re the most s e v e r e l y l o a d e d t u b e s , w i t h the l o a d s e v e r i t y d i m i n i s h i n g w i t h i n c r e a s i n g tube h e i g h t . S t r e s s on t u b e - b u n d l e s can be reduced by c h o o s i n g a r e a s o n a b l y t i g h t i n t e r - t u b e s p a c i n g and l o w e r i n g t h e i r l e v e l above the gas d i s t r i b u t o r , w h i l e r e t a i n i n g enough d i s t a n c e above g r i d j e t s . 3 Recommendations f o r F u r t h e r Work In o r d e r t o g e n e r a l i z e t h e r e s u l t s i n the p r e s e n t work, s i m i l a r e x p e r i m e n t s s h o u l d be con d u c t e d i n a l a r g e r s c a l e f l u i d i z e d bed. The work o f Kennedy ejt a l _ . (1981) provides some g u i d a n c e . More s t u d i e s a r e r e q u i r e d on t h e hydrodynamic b e h a v i o u r o f l a r g e s c a l e f l u i d i z e d beds, e s p e c i a l l y t h o s e e q u i p p e d w i t h o b s t a c l e s , i n o r d e r t o p r e d i c t s u c c e s s f u l l y the f o r c e s on immersed o b j e c t s . The work p r e s e n t e d i n t h i s s t u d y has been l i m i t e d t o a c o l d f l u i d i z e d bed. On t h e o t h e r hand, most i n d u s t r i a l f l u i d i z e d bed systems o p e r a t e w e l l above ambient t e m p e r a t u r e . Thus, f o r c e measurements s h o u l d be c o n d u c t e d i n a hot model, a l t h o u g h no s i g n i f i c a n t d i f f e r e n c e s i n the c h a r a c t e r o f t h e f o r c e s a r e e x p e c t e d . - 205 -NOMENCLATURE A c r o s s - s e c t i o n a l a r e a o f bed A 0 a r e a o f d i s t r i b u t o r p l a t e per o r i f i c e D bed d i a m e t e r d b bubble d i a m e t e r 5 b mean bubble d i a m e t e r p a r t i c l e d i a m e t e r % mean p a r t i c l e d i a m e t e r d T tube d i a m e t e r FB buoyance f o r c e f b bubble f r e q u e n c y f c N y q u i s t f r e q u e n c y ( f o l d i n g f r e q u e n c y ) g a c c e l e r a t i o n due t o g r a v i t y H expanded bed h e i g h t H o s t a t i c bed h e i g h t Hmf h e i g h t o f f l u i d i z e d bed a t minimum f l u i d i z a t i o n h v e r t i c a l d i s t a n c e from bottom o f f l u i d i z e d bed h o c o n s t a n t c h a r a c t e r i z i n g gas d i s t r i b u t o r , e q u a t i o n (2.4) P l o c a l p r e s s u r e Po l o c a l p r e s s u r e i n t h e absence o f bubbles N sample s i z e , T/At r b radius o f c u r v a t u r e o f l e a d i n g edge o f bubble T r e c o r d l e n g t h t time - 206 -U s u p e r f i c i a l gas v e l o c i t y U B bubble v e l o c i t y U^ Q bubble v e l o c i t y i n i s o l a t i o n i n t e r s t i t i a l gas v e l o c i t y a t minimum f l u i d i z a t i o n , Umf/ Emf U M^ s u p e r f i c i a l v e l o c i t y a t minimum f l u i d i z a t i o n v o l u m e t r i c f l o w r a t e due t o bubble d i s p l a c e m e n t Greek N o t a t i o n Ap p r e s s u r e drop At s a m p l i n g i n t e r v a l e t o t a l v o i d f r a c t i o n volume f r a c t i o n o f bed o c c u p i e d by bubbles e ^ v o i d a g e a t minimum f l u i d i z a t i o n 9 a n g l e Peff e f f e c t i v e d e n s i t y o f bed dense phase p ^ bed d e n s i t y a t minimum f l u i d i z a t i o n PS s o l i d s d e n s i t y - 207 -REFERENCES Bar-Cohen, A., G l i c k s m a n , L.R., and Hughes, R.W. 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E n g r s . , 52^, 149-159. Werther, J . (1976) B u b b l e growth i n l a r g e d i a m e t e r f l u i d i z e d beds, i n " F l u i d i z a t i o n T e c h n o l o g y " , e d . D.L. K e a i r n s , V o l . 1, Hemisphere P u b l i s h i n g , Washington, pp. 215-235. We r t h e r , J . (1977) B u b b l e c h a i n s i n l a r g e d i a m e t e r gas f l u i d i z e d beds, I n t . J . M u l t i p h a s e Flow, 3, 367-381. - 212 -Whitehead, A.B., and Young, A.D. (1967) F l u i d i z a t i o n performance i n l a r g e s c a l e equipment: P a r t I , P r o c . I n t . Symp. on F l u i d i z a t i o n , ed. A.A.H. D r i n k e n b u r g , N e t h e r l a n d s , U n i v . P r e s s , Amsterdam, pp. 284-293. Whitehead, A.B. (1971) Some problems i n l a r g e - s c a l e f l u i d i z e d beds, i n " F l u i d i z a t i o n " , e d. J . F . Davidson and D. H a r r i s o n , Academic P r e s s , New York. Whitehead, A.B. (1979) P r e d i c t i o n o f bubble s i z e i n a g a s - f l u i d i z e d bed, Chem. Eng. S c i . 34_, p. 751. Y a s u i , G., and Johanson, L.N. (1 958) C h a r a c t e r i s t i c s o f gas p o c k e t s i n f l u i d i z e d beds, AIChE J . 4, p. 445. Y e r u s h a l m i , J . , and C a n k u r t , N.T. (1979) F u r t h e r s t u d i e s o f t h e regimes o f f l u i d i z a t i o n , Powder Techno!. 24, 187-205. - 213 -APPENDIX A NATURAL F R E Q U E N C I E S OF V I B R A T I O N OF THE T E S T T U B E S The n a t u r a l f r e q u e n c i e s o f v i b r a t i o n o f a u n i f o r m beam can be c a l c u l a t e d by u s i n g the E u l e r e q u a t i o n f o r the beam (Thomson, 1965). oi = n 2 /gEI/W where the number n depends on the boundary c o n d i t i o n s o f the problem and: E = modulus o f e l a s t i c i t y I = moment o f i n e r t i a o f c r o s s - s e c t i o n a l a r e a W = weig h t o f t h e beam per u n i t l e n g t h . For clamped-clamped beam (Thomson, 1965): (nl)2 = 22.4 ( F o r fundamental mode) .'. The n a t u r a l f r e q u e n c y o f the s p e c i f i c beam i s : ui = 22.4 /gEI /J> 4W = 22.4 /TTJsJm - 214 -Where, m i s the t o t a l mass o f the beam and a i s the l e n g t h o f t h e beam. 1 . The n a t u r a l f r e q u e n c y o f t h e l a r g e s t s i z e o f the tubes (shown i n F i g u r e 3-6-a): E = 20.3 x 1 0 1 0 N/m2 i - u <°4 - d4» = 4.339 x 10" 8 m 4 9 9 m = j (D -d ) £.p , where I = 0.3 m m = 1 .1851 kg co = 22.4 [8.809 x 1 0 3 / ( 0 . 3 3 x 1.1851 ) ] 1 = 11753 ra d / s f ( n a t u r a l f r e q u e n c y ) = = 1870 HZ 2. The n a t u r a l f r e q u e n c y o f the medium s i z e o f the tubes (shown i n F i g u r e 3-6-b) : E = 20.3 x 1 0 1 0 N/m2 I = 1.668 x 1 0 " 8 m 4 m = 0.7348 kg co = 7286 ra d / s f = 1160 HZ - 215 -3. The n a t u r a l f r e q u e n c y o f the s m a l l e s t s i z e o f tubes (shown i n F i g u r e 3 - 6 - c ) : E = 20.3 x i o 1 0 N/m2 I = 2.28 x I O " 9 J m m = .295 kg to = 5392 r a d / s f = 850 HZ - 216 -APPENDIX B COMPUTER PROGRAM FOR C A L C U L A T I O N OF THE RMS, STANDARD D E V I A T I O N AND MEAN V A L U E S OF THE DATA DIMENSION J (3000) N=3000 READ(3, 1 0 ) ( J ( I ) , I = 1 , N ) 10 FORMAT ( I X , 16) IA1=0 DO 20 1 = 1,N 20 IA1=IA1+J(I) IA=IA1/N WRITE(6,10)IA VA=0.0 DO 30 1 = 1 ,N 30 VA=VA+(J(I)-IA)**2 SD=SQRT(VA/(N-1)) WRITE(6,35)SD 35 F0RMAT(IX, 'S.D.=',F6.0) RS=0.0 DO 40 1 = 1 ,N 40 RS=RS+(J(I))**2 S=SQRT(RS/N) WRITE(6,50)S 50 F0RMAT(1X,'R.M.S.=',F6.0) STOP END 

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