THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE BEHAVIOR OF SLUGGING GAS-FLUIDIZED SOLIDS Robert H. Kadlec A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan Department of Chemical and Metallurgical Engineering 1961 September, 1961 IP-532

Doctoral Committee: Professor G. Brymer Williams, Chairman Doctor Gordon Atkinson Assistant Professor Dale Rudd Associate Professor M. R. Tek Professor J. Louis York iii

ACKNOWLEDGMENTS I gratefully acknowledge the assistance of Professors Go B, Williams, Jo Lo York, Do R. Rudd, Go Atkinson and M. Ro Tek in all phases of this worko Thanks are also due to the University of Michigan Industry Program and my wife Kathleen for great assistance in the preparation of the manuscripto I wish also to thank the Nati6nal Science F'oudation fr three year.s supporto iv

TABLE OF CONTENTS ABSTRACT ii ACKNOWLEDGEMENTS iv LIST QF FIGURES vii LIST OF TABLES xii NOMENCLATURE xiii INTRODUCTION 1 PRELIMINARY WORK 3 Io Microscopic Approach 3 11o Macroscopic Approach 10 PROPOSED MODEL FOR SLUGGING FLUID BEDS 23 BED PRESSURE DROPS AT INCIPIENT FLUIDIZATION 28 SOLIDS DOWNFLOW RATES 50 Io Bin Flow Measurements 50 II, Pressure Profile Measurements 57 IIIo Effect of Operating Variables on Solids 64 Downflow Rate BED CONFIGURATION 79 SLUG GENERATION 96 USE AND VERIFICATION OF PROPOSED MODEL 98 Io Prediction of Bed Configuration 98 IIo Period of Fluctuations 103 IIIo Pressure Profiles 113 EXTENSION OF MODEL TO BUBBLING SYSTEMS 132 SUMMARY 135 APPENDIX A. Details of Development of Average 138 Collision Frequency APPENDIX B. Solution of the Mass and Momentum 144 Balances for a Slug APPENDIX Co Calibrations 153 V

TABLE OF CONTENTS (continued) APPENDIX D. Response of the Pressure Measuring 162 System and Pickup Characteristics BIBLIOGRAPHY 171 vi

LIST OF FIGURES 1o Physical description of terms used in discussing 11 slugging fluidizationo 2, Four individual cases of slugs which may exist in 18 a slugging fluid bedo 3. Experimental equipment for measuring total bed 30 pressure drop, 44 Total bed pressure' drop versus gas velocity for 32 air-sand A in a 2" by 74o0 cm bed, 5a Total bed pressure drop versus gas velocity for 35 helium-sand A in a 2" by 72o0 cm bedo 6o Total bed pressure drop versus gas velocity for 37 air-sand A in a 1" by 21,2 cm bedo 7o Total bed pressure drop versus gas velocity for 39 air-sand A in a l"by 5800 cm bedo 80 Total bed pressure drop versus gas velocity for 41 CO02sand A in a 1" by 42 0 cm bedo 9. Total bed pressure drop versus gas velocity for 43 heliiim-sand A in a 1"1 by 42o0 cm bedo 10o Total bed pressure drop versus gas velocity for 45 air-sand B in a 1" by 33o0 cm bedo 11o Total bed pressure drop at incipient fluidization 48 versus bed height for sand Ao 12. Apparatus for measuring solids downflow rateso 52 13. Typical oscillographic recording of falling solids 53 optical density for solids downflow rate measurement. 14o Solids downflow rate versus gas velocity. 55 Bin flow techniqueo 15o Equipment used for measuring solids downflow via 58 pressure drop in a slugging fluid bed0 16O Oscillograph of pressure variation between taps 60 15o8 cm apart near the top of a slugging bedo vii

ISTC OF FIGURES (continued) 17. Solids downflow rate versus gas velocity. 63 Pressure measurement technique. 18o Equipment for measuring solids downflow rate in a 66 lo0 cm glass tube, 19. Equipment for measuring solids downflow in a 67 lo0 inch plexiglas tube. 20. Determination of particle terminal velocity. 69 Typical oscillograph. 21. Solids downflow rate versus solids density. 72 22. Effect of particle Reynolds number at incipient 74 fluidization on solids downflow rate. 23. Effect of bed diameter on solids downflow rate. 71 24. Level curves for air-sand A in a 2.00" by 74.0 81 cm bed. 25. Level curves for helium-sand A in a 2 00 by 82 72,0 cm bed. 26. Level curves for air-sand A in a 1.00" by 83 21o2 cm bed, 27. Level curves for air-sand A in a 1.00" by 84 58o0 cm bed, 28. Level curves for air-sand B in a O1600" by 85 33.0 cm bed. 29, Level curves for helium-sand A in a lo00" by 86 42.0 cm bed. 30. Level curves for C02-sand A in a lo00" by 87 42o0 cm bed. 31. Level curves for air-sand A in a 1.00" by 88 30,0 cm beds 32. Level curves for air-molecular sieve cylinders 89 in a 1,128 cm by 23,0 cm bed, viii

LIST OF FIGURES 33. Illustrstion of bed configuration versus time, 91 Physical meaning of HT, HM, HBg 34o Reduced HT curve slope versus bed diameter. 93 35. Reduced HM curve minimum versus gas velocity 95 at the minimum, 36, Graphical solution for bed configuration. 101 37, Period of upper bed fluctuations versus gas 105 velocity for air-sand A in a 2.00" by 74.0 cm bed. 38, Same for helium-sand A in a 2.00" by 72.0 cm bed. 106 39. Same for air-sand A in a 1,00" by 58.Oi1cm bed. 107 40, Same for air-sand A in a 1.00" by 21.2 cm bed. 108 41o. Same for helium-sand A in a 1.00" by 42.0 cm bed. 109 42. Same for C02-sand A in a 1.00" by 42.0 cm bed. 110 43, Same for air-sand B in a 1^.0" by 33.0 cm bed, 111 44. Same for air-sand A in a 1.00 cm by 30.0 cm bed0 112 45~ Predicted and observed pressure-time curves for 117 air-sand A in a 2.00" by 74.0 cm bed. High taps. Air velocity 7.00 cm/seco 46, Predicted and observed pressure time curves for 118 air-sand A in a 2~00" by 74,0 cm bed, Middle taps. Air velocity 7.00 cm/seco 47, Observed pressure-time curve for the same system 119 for low taps, 48. Observed pressure-time curves for the same system~ 120 High taps at velocities of 10.0 and 13.0 cm/seca 49~ Pressure-time records for three elevations in a 121 helium-sand A system in a 2,00" by 72.0 cm bed. Gas velocity is 30.2 cm/seco ix

LIST OF FIGURES (continued) 50, Pressure-time records for the same system at gas 122 velocities of 12o7 and 16.4 cm/seco Middle tap elevation, 51o Same for gas velocities of 18o3 and 21o4 cm/sec. 123 52, Pressure-time record for middle elevation in an 125 air-sand A system in a lo00" by 21o2 cm bed. Air velocity is 8.7 cm/sec. 530 Same for a velocity cf 24.0 cm/sec. 126 54. Same for a velocity of 36,4 cm/seco 127 55, Pressure-time record for helium-sand A in a lo00" 130 by 42.0 cm bed at a gas veldbctty of 43.7 cm/sec. Middle taps. 56. Predicted and observed bubble velocities versus 134 bubble diameter, 570 Calibration curve for pressure gage C2-174o 154 58. Calibration curve for pressure pickup 2172. 155 59. Calibration curve for pressure pickup 2171. 156 60. Calibration curve for pressure pickup 2150O 157 61. Calibration curve for pressure pickup 2144. 158 62. Calibration curve for pressure pickup 2133o 159 63. Calibration curve for pressure pickup 2100, 160 64o Calibration curve for rotameter 9145 with 161 glass float, 65, Apparatus for applying step and ramp pressure 163 impulses to the pressure measuring system. 66, Response of the measuring system to a step 165 pressure input, x

LIUST OF FIGURES (continued) 67& Response of the measuring system to a ramp 166 pressure input. 68 Apparatus for applying an approximately sinusoidal 168 pressure input to the pressure measuring system, 69, Response of the measuring system to a sinusoidal 169 input. xi

LIST OF TABLES I, Total bed pressure drop versus gas 31 velocity for air-sand A in a 2" by 74 cm bedIIo Total bed pressure drop versus gas 33 velocity for helium-sand A in a 2" by 72 cm bed. III, Total bed pressure drop versus gas 36 velocity for air-sand A in a 1" by 21.2 cm bed, IVo Total bed pressure drop versus gas 38 velocity for air-sand A in a 1" by 58,0 cm bed, V, Total bed pressure drop versus gas 40 velocity for C02-sand A in a 1" by 42,0 cm bed. VI, Total bed pressure drop versus gas 42 velocity for heium-sand A in a 1" by 42^0 cm bed, VIIo Total bed pressure drop versus gas 44 velocity for air-sand B in a 1" by 33.0 cm bed, VIII. Total bed pressure drop at incipi- 47 ent fluidization versus bed height for sand A, IX. Incipient fluidization data for 49 systems studied, X. Solids downflow rate versus gas ve- 54 locity, Bin flow technique. XI. Solids downflow rate versus gas ve- 62 locity, Pressure technique. XII, Summary of observed solid4l down- 71 flow rates. XIII, Solids downflow rate versus particle 73 Reynolds number at incipient fluidization. XIV, Effect of bed diameter on solids down- 76 flow rates, XVo Summary of results of pressure profile 128 measurements. xii

NOMENCLATURE A constant; bed cross sectional area Ap projected area of a particle C,0C constants Cd drag coefficient c particle speed D constant, diameter dp collision particle diameter E constant e base for natural logarithm F force f friction factor g acceleration of gravity gc gravitational constant H height I integral K constant L static bed length m mass Np particle number density n counting index P distribution function, pressure R reading Re Reynolds number xiii

NOMENCLATURE (continued t time u velocity v velocity, volt v relative velocity w nonrandom component of particle velocity; solids downflow rate Wa solids downflow rate x cartesian coordinate; coordinate of the top of a slug y cartesian coordinate; length of a slug za cartesian coordinate; coordinate of the bottom of a slug Zc collision frequency < P constants, variables: defined as required f ~dimensionless nonrandom particle speed 08 Dirac delta function 6 porosity, dimensionless particle speed 6 polar coordinate, measure of solids downflow rate P density a' standard deviation T* period,~ polar coordinate function symbol xiv

NOMENCLATURE (continued) Superscripts first time derivative second time derivative derivative, prime T t second derivative, double prime average Subscripts B bottom. b bubble o collision C,.Go. center.. of gravity D drag g gas fb fixed bed gmf gas, minimum fluidization M middle 0 initial' condition P pressure, particle, projected S solid 0slip T top t tube ts true, solid.x X direction ' 'XV'

NOMENCLATURE (continued) Subscripts xs excess y Y direction Iz Z direction xvi

INTRODUCTION A -wealth of information has been obtained in the field of fluidization in recent years. The most studied area has been that of gas fluidized solids, principally because of applications in the petroleum industry. Necessity has demanded that the majority of studies be correlative in nature. It is the goal of this work to select a specific type of gas solid fluidization, slugging, and develop an understanding of the basic concepts which determine the behaviour of such a system. A slugging fluid bed is a system in which slugs of high solids density alternate with regions of low solids density. This study is concerned with linear systems, that is, systems in which there are practically no variations in the directions perpendicular to direction of gas flow. All systems will be batch systems, i.e., beds which have a constant solids inventory. The variables to be studied are bed diameter, bed height, gas and gas velocity. The effect of solids variables is not studied because solids characterizations is a tough problem in itself and would unnecessarily complicate the problem of interest. When one attempts to develop a technique for prediction, or mathematical model, there are always several alternative routes to investigate. It is fortunate if the first attempt succeeds, but discarded approaches can be quite valuable because of the information gained in the attempt. In this work three approaches were tried; the third was successful. This report will begin with a discussion of the two abandoned approaches because each contains information of value. 1

2 The remainder of the discussio will be devoted to the construction, verification and use of a technique for predicting the behaviour of certain fluidized systems o

PRELIMINARY WORK Part Ao Microscopic Approach. One possible approach to the ibehviour of fluid systems, which was not pursued in this investigation^ is to consider a single particle and its immediate fsrroundingso If one ould predict the motion of one particle frm a. kowledge of the: pripertiese of fluid surrounding it and of the macroscopic properties of porosity or solids concentration and bed geoetry in its viainity, a firm basis for a nilcrosaopic scale model would be established. At least ion su h attempt has been made. Bowman attempted to describe fluitation by considering events occurring on the microscopic seale and applying statistical methods. Details of this work are apparently not available (2) The principle of statistieal mechanics are unfortunately not applicable to the Motion of a collection of particles which move at ordinary velocitieso The number of particles involved is not great enough nor is there any resemblance of a fluidized system to an equilibrium systemo Further, the Ergodie hypothesis is not satisfied, that is, the point in phase spaee representing the system does not pass through all possible states in a. short period of timeo A lIgical procede would seem to be to write. a force balance on an individual particleo The forces to be considered are gravity, buoyancy, impact due t:o ollisi:on, viscOus drag, and wall friction in' the case of a moving plug p agsectien Such a balanc will yield a second order differential equation Bonda.ry onditio can most likely be written at. the bed support, the upper bed surfae., nd the walls of the bed. This equation, coupled with a continuity equation for the gas, some sort of material balance equation for the solid,, and the. pressure drop equations for the 3

4 gas should permit utn f solution for sids velocity and concentration; gas velocity and densityo These are four variables four equations have been listed. The force balance on an individual particle will be considered first One of the most importfnt questions to be ansered before any microscopic approach can succeed is the problem of the drag on a particle under conditions varying from isolation to dense packingo The two extreme cases have been treated in detail0 The existing correlations for drag on isolated particles are lacking in only two respects: shape is a variable which still cannot be accurately treated- and drag under accelerative conditions still has not been thoroughly investigated. The question of shape is of concern in all cases where one deals with particles of irregular shape; unfortunatelyg this is most often the case with fluidized solidso The question of drag on particles in a fixed bed has. probably been answered more satisfactorily from an engineering viewpointo The problem of the effect of shape still exists, but the pro:blem of the effect of acceleration no longer existso One can, therefore, eliminate a major objection to the use f drag correlations by restricting the materials used to spherical partiele0so The weakest arta f drag theory i the region of hindered settlingo This term is here intended to encompass the entire range of porosity from zero for isolated particles to some value less than one for the packed bedo These are oSnditions under which an individual particle is partially restricted in its motion by the influence of neighboring particleso However, sufficient inforation is availabl te o ble to treat this topic quantitative lyo

5 The following would be feasible correlations to use for determining drag forces, or pressure drops which may be converted to average drag force per partie:le: il Isolated sphere. C: =. O) (1) D a C f A Vr = CD(o f Vr Plot of D veUs:B.< 2. Hindered settling (Cg 0.0 to Cfb) (2) FD j Ci f Dp Va Plot of (Re/CD) /3 versus (DRe.2)1/3 (A emprehenive plot of t tnre appear on page 235 of Zenz an Othber- ) 3. Fixed bed: (C b) (3) P- afe v/gDp Plot of f versus Re at ~ parameters (A comprehensive plot of this nture appears on page 181 of Zenz and Othmerv2)) These account for the drag force on a particle nder all conceivab le tonditionso A relativ.e veloity- vR, between partiEle and fluid, must be used in computing FD, Re and AP/L. One for.e upon a particle required for writing a foarce ba.lne is tbus determindo Onre might next inqui about the i ct force upon a partile when it cllides with a neighboro If one further asumes hard particles or in other words prfejtly aelastic. olisioi., tn e ftr om one such lliision might be repsted by a Di;e delta ifuntion at follows t (4) Fip.ct = (Amv) 8 [ (tx y, 2) where (Anv) represents the m entwa exchanged between the colliding particles

6 A moment's consideration sh.B tht t he frm of should be as follows: (5) + = t, l/ c Y) (t,^ yv where ~c (t X) ) y ) ) eollision freq.ency This indicates that oe must obtain the collision frequency in order to determine the impact force upon a particle0 It is appatret ttht collision frequeney will not depend explicitly on time and p.sitiono It will be some explicit functio of particle motion and particle concentration or in other wo-rs an implicit function of time and positionai One an imgine th.e _elceity of a particle to eonist of a random ormponent, with eqal mean cartesian etoponents, and a non-rando mnt n e t d t g boynt, and gravitatiomal forces The rando cam ponent would arise f rim two soures,. turbulent vartiees of the f'luidizing mium and Olliions.o A simplfied vsrsion of this problem nam ely tht 'wa here o ra m compoents exist iS the problem treated by t.e kinetic- thery of gaseso This theory produces the collision frequency for this simplified ase(4) (6) c(txyy))= Np (tx y ) rDp (\' +V/ v ) where N (txy,29z) mnmber of parteles per itt volume - average elocity f:olliding particle nmber1 T2 = aT age -veloeity of colliding particle number2 If V =V or for the cae of simiIar particles: (7 2 t,xy, )=- p(tx,y 2)\/: T D" v However, a prti-e in ' flid; bed d efinitely possesses a nonrandom component of vlsoity. This:nses gret.ofmplia.tion of the meehanics of obtaining ia avt e cllision frequenyo One must begin

7 with atn "ag ned stattitial distributiQn of individal particle vrsleXitieE o A convenient a.d reafson le 'ast mption is that the two horizontal cmpe Knt of partile le velo~ity are nomlly distributed about a zero men and that the vertial Comptnent is noma.lly di - trihbuted about a non-z ero mWAn whiLh is a function of time and positio.n These distri bution funetion arre: (8) P(vx) ev (9) P(vy) e ~ v v- w(X, y, ) (10) '. p(v, = eT where w(x,yz) is the non-random verticl velocity componento If one further as. -nes that ax = My cr- c one may obtaln the distribution of parti le sepeds; w(xi~ or-e l sinh (o Details of the deelO f off E atin (11) are included in Appendix A Geaomtrital eonsidtrationse how thbt 't. number of aelliseion between a particle amoving with speed e rwith statifonry partteles of:ocent 'tion Np per -nit -To l: (12) c= Np(t,X)y))Tt DP) Detail of the deveFlopment of Euation (2) ae inluded in Appendix AO If -on is dealing -with a set of particresn:rally ditributed abot a man diaeter Dp wth a standard deiatian; an immediate average o er

particle diameters yields: (13) lC N(tXYT c) +C /d4 Details of the development of Equation (13) are included in Appendix Ao Thus if one defined dp by (i4) dp - pL +a/4] then (15) p(tx, y, )CTT dp Equation (15) holds for a particle moving at a fixed speed c through stationary particleso What is desired is an expression for collision frequency of partiele whose speed is distributed according to Equation (1I) with particles whose speeds are distributed according to Equation (11). Averaging Ze over the speeds of the particles,, one obtains: -[a - S ) (~t ) d erf - + + where - (dimensionless) and Y (dimensionless) Details of the development of Equation (16) are included in Appendix Ao

9 Eqmtion (16) holds for a particle moving at fixed speed e through particles whose speeds are distributed according to Equation (1)o Averaging Z over the speeds of the particle under consideration, one C obtains a5 (17) ~c;46i~~pdd T~c 1 + 4r + ) erf( ) + e 3 where again Details of the development of Equation (17) are included in Appendix Ao The average collision frequency is thus seen to be a function of the non-mrandom velocity component, the particle concentration, and the standard deviation of the distribution of random particle velocity components. At this point it becomes apparent that the terms involved in setting up just a force balance on a single particle are complex and nonlinear to such an extent that there can be no hope of an analytical solutiono Nuinrical solutions would at best be very difficulto The micrssopic approach was abandoned for these reasonso

PRELIMINARY WORK Part Bo Macroscopic Approach In the foregoing discussion it was not necessary to limit in any way the type of fluidization under considerationo Appropriate equations and boundary conditions would conceivably handle all imaginable ases0 The maeroscopic approach does, however, necessitate the formulation of certain assumptions concerning the geometry and operation of the fluidized system to be describedo The remainder of this work will be concerned with systems that: lo Consist of only two phases. gas and solido 2a Are batch systems, that is, systems with a constant solids inventory0 30 Operate in the slugging regime of fluidizationo 4.. Are vertical in orientationo A slugging fluid bed is one in which regions of low solids concentration c:mpletely fill the bed cross section and alternate with regions of high solids concentration which also completely fill the bed cross section0 A slug is a region of solids concentration in the range from the solids concentration of a bed at incipient fluidization to the fixed bed solids concentrationo A void space is a region of lower solids concentrationo A bubbling bed is a fluid bed in which void spaces exist but do not fill th eentire bed cross sectiono Let us now consider a system consisting of one slugo Figure 1 will be of assistance in visualizing the variables to be discussedo There is some ruperficial gas velocity in the system, vgo The length 10

TUBE CROSS SECTION,A SOUDS INFLOW- W,, GM/SEC VELOCITY- v,, CM/SEC CENTER OF GRAVITY y SYSTEM I I 1 X SOoS OUTFLOW- W2 GM/SEC VELOCITY- v, CM/SEC Z GAS lNFOW VELOCITY v, CM/SEC Figure 1. Physical Description of Terms Used in Discussing Slugging Fluidization. 11

12 of the slug is yo The elevation of the upper surface is x with respect to some arbitrary reference leve0o The elevation of the lower surface is z, so by definition: (18) X = y + E The solids density in the slug is assumed to be constant and given by gso Consequently the mass of the slug is: (19) m = s Ay If there is a mass flow into the top of the slug of wl and a mass flow from the bottom of w2. the mass balance for the system is: (20) = w,-w A 0) it I -S - The momentum balance for the system will be considered nexto Its form is: 21) d(m V)s external forces + IY1 m \/o dt time t ime For purposes of a momentum balance, the velocity of the system is the velocity of its center of mass: (22) v =d(x-sy)- x- y The gravitational force on the system is: (23) 1( = s^ ys The frictional force exerted by the wall should be negligible if the slug is at incipient fluidization, for at that point drag renders the

13 particles weightless and hence they can exert no normal force on the wall, The momentum carried in by solids inflow is: (24) Rate of momentum in = ^, v, The momentum transported out of the system by solids outflow presents a small problem, When a solid particle leaves the lower surface of the slug, it leaves at the slip velocity of the slugo The slip velocity of the slug is the velocity of a particle within the slug relative to the bed wallo From the moment the particles leave, they begin accelerating until they reach terminal velocity or encounter another slugo The slip velocity of the slug may be defined in two ways: (25) \d C l X"/a (25) Vg S X - -—, _ ~ / (osA pA Therefore the momentum transported out with the effluent solids is. (26) Rate of momentum out Wa (\-,)} W fBuoyancy will of course contribute- a term to the momentum balan-ceo (27) Fb- Ay(l-E)~ This term is small and may be neglected in comparison to gravitational forces The most important external force is the pressure differential across the slugo This pressure drop is the sole agent tending to cause the slug to slip in the upward directiono The question is that of predicting the prerssutr drop through the slug for varying gas velocities.

14 If the slug is asstumed to be at the state of incipient fluidization, this pressure drop is a known constant quantityo This is trae because the prssuxre drop at incipient fluidization is a ehrarcteristic and well defined value. For batch beds, one may writet (28) -CM=_P)M ~ = d g P 9 IK L A /CP. FL. L 9 This is merely a mathematical statement of the fact that at incipient fluidization, pressure forces balance gravitational forceso If the velocity of the fluidizing gas is increased beyond the point of incipient fluidization no- further increase in the bed pressure drop is observed0 This indicates that the same AP prevails from the incipient fluidization velocity to some higher velocity, presumably near the velocity where dilute phase fluidization beginsO At gas velocities below incipient fluidization, the pressure drop relations for packed beds apply. It is conceivable that a moving slug could experience a gas velocity, relative to the moving slug, less than that rqtuired for incipient fluidizationo Since the fixed bed pressure drop:eorrelitions are not simple in nature, a linear approximation will be made for purposes of simplicity0 Thus approximation will be valid if physical conditions do not stray too far below the point of incipient fluidization The approximation is the following: (29) - (v-K Vs V 7-Vgmf Thus the pressure foree exerted on a slug is given by (30) Fp KAy.vg- vmf KAy (Vq-Vs)/VMif V> Vq - W.f

15 All elements necessary for a momentum balance have been consideredo The momentum balance is: (31) SAy(x- Y)]= [si Ay + WUA Vw:u,-w, '-1 ~sgd r I I(2)- 2A j Ay(XXYJ LL fJ p sAA S > VgV —mf The term r,- in Equation (31) should be equal to zero, according to Equation (28), if theory were precisely upheld, but this is not necessarily the case. It will, therefore be left in Equation (31)o Since(O and A are constant. the momentm balEanee may be reduced to:.w. w, W,/ i A (o'A) ] YA gL sAA sA (34) y(6-Vy)+ (-s V-SA Vs t> V-Mf

16 These two balances, mass and momentum, suffice to describe the movements of the slugo An energy balance may also be written, but it is not independent of the mass and momentum balanceso One further assumption may be made at this point with no experimental justificationo The solids flow into and out of a slug may be considered to be equal and the same for all slugs in the systemo This seems reasonable if the rate controlling event is the rate at which particles can disengage from the slugo Changes in gas velocity relative to the disengaging particles should have but little effect on disengaging rate since for all velocities in excess of the minimum fluidization velocity, particles are exposed to interstitial velo.cities in excess of the terminal particle velocityo The particles therefore do not fall until disengaged from the bulk of the slug0 Theorore let: (35) G- = W S A s A Equations (33) and (34) then become. upon substitution of Equations (35): ~36) ( X-;\'X e1 e( e)/ (37) y (; s 8) y 2 lfX [ 3 Vs^ -v~nI f The mass balance, Equation (20), then becomes, for the case of solids inflow and solids outflowt

17 (38) y - 0 There are four separate cases to be consideredo Each of these cases could conceivably exist in a slugging fluid bedO Figure 2 shows each case schematically o The above equations have been written for case 4~ In all four cases, a set of boundary conditions is required for the solution for slug motion. These conditions are specified as: (59) y when t t yO {(o) X - Xo ',hen t = to=r o a) X= Xa w/hen t -t QO (41) Wx o The velocityR, u, is assumed to be the velocity of the particles with respect to the bed wall and hence will vary with gas velocity. However, if one remains in a region where particle terminal velocity is far greater than gas velocity., a good approximation is that u is constant and equal to the average terminal velocity of the particles in the stagnant gaso With these asstmptions, all four cases can be solved. Case 1 No1 slids nflow or o u tflow The equations to be solved are: The mass. balane t (42) y = The two cases of the momentum balance: (43) y =( [sY f)y Vs Y-V Mf (44) yy S 5 V ~ " - LCs W Mf

CASE I CASE 2 NO SOLIDS INFLOW OR OUTFLOW SOLIDS OUTFLOW CASE 3 CASE 4 SOLIDS INFLOW SOLIDS INFLOW AND OUTFLOW Figure 2. Four Individual Cases of Slugs Which May Exist in a Slugging Fluid Bed. 18

19 For simplicity, let f K ~ The solutions of thee eequations yields (45) y = yo (46) X= Xo.+ Xota 4 t g^f (47) <g X V6e A Ae 3 (e A*-&I,) where A= _ _ The details of- solution are given in Appendix B, Case 2o Solids outf low only, In this case tth mass balance is: (48) y - The two forms of the momentum balanee are, upon simplification: (s), )(a -r1 (50) X= CS e; 9 (4- vX~ ^ 5V yV V f Vs v -v Mf The solution of these equations is more difficult than in Case 1, the solutions being: (51) y= - t

20 ' -V^9 Vgmf (5) X_ XO++ t9A+76l XI -I I I.' * The details of these solutionxs are shown in Appendix Bo Case o0 Solids inflow onlyo The mass.balance beeomes: 55) X C + - -: VS v \-g f 56) I The solution of these oequations detailed in Appendix B, is: (58) X 6 = Mfo- _ - yo(M^7e)7. ^ (5) (yyo, (y - y ) YV Vg -Vmf _ _ _ _ _ C A V X + 6 4

21 A (59) X- X~ - +e -L] n7 A) /') D Ln —I L yo Vs V -Vmf wherA K/rseVAf D:A A(v~+)- 9 E= ( + e) Oase 4, Solidd inflow and soids outflWo The mas. balanee in this oase is given by Equation (38). The two -ases of the mom.enturm blance are gien in Equations (36) and (37). Again,9 ust the final so-fltion of thesae equation will be presented the details to be found in Appendix Bo Solutioon of the mass balancee (60) y =y Solution of the momentim b.lanc: (62) = c + _ -Ie — e y o Yo where Y G ^ + 9 9 (3 A (v9+o)-9 _ ^t eu, V;

22 These equations should describe the slug in question under all circumstalnes0o They are to be applied to a system of n slugs in a slugging fluid bedo Howeverv the extreme complexity of these equations is discouraging, since there would be 2n of themo Furthers in order to couple the equations some assumptions must be made as to the time delay involved in the transfer of material by the raining process from a slug to its next lowest neighbor- A further complication would be the disappearance and appearance f slugs in the system; the number of slugs is observed to change with timeo And, as time progresse, the equations which describe a slug:ca. ch e n in two ways The first is a change from one momentum balance to another within the same casea Also, the case which appliB to a given s-lg may chang Thesi re-jtonS make: it extremly doubtfUl that any realistic and ac-curate reStults can be obtained frm this particular mathematical model0 Any results wold be obtained numerically at the expense of a great deal of efforto Certainly some simpler approach would be indicated if at all possibleo.@Henei although this approach is not necessarily the wrong fone, it was abandonedo

PROPOSED MODEL FOR SLUGGING FLUID BEDS The slug models in the unsteady state case just discussed appear unattractive for only one reason: the complexity of the mathematical descriptiono The term unsteady state is here intended to mean that slugs accelerate and decelerateo Therefore, before becoming lost in the intricacies of mathematical analysis, it was deemed wise to begin with the most highly simplified version of the macroscopic slug model and discard simplifying assumptions as they proved invalido A possible set of assumptions is the following: lo There are no radial variations in solids concentration0 2o The system consists of well defined slugs separated by "void" spaceso 30 These slugs do not accelerate or decelerate for any-finite period of timeo 4o There is a known process of slug generation0 5. There is a unique solids flow rate through the "void" spaces when such a flow existso 60 Material falls through a "void" space at a velocity which is very large compared to all other veloities in the systemo 7o. Only one system at a time can be dealt with; that is, one bed, one type.of solid, and one gaso 8. There is no friction between slug and wallo 90 There is a negligible gas pressure drop in a.void"l spaceO lO B Gas flows through a slug at the velocity for minimum fluidizationo These assumptions will be supported at appropriate points in the following discussiono Some can be supported a posteriori onlyo 23

24 Let us now examine the implications of these assumptionso First of all, if the superficial (based on tube cross section) gas velocity through a slug is the minimum fluidization velocity, the pressure drop per unit length of that slug must be the pressure drop per unit length of bed at incipient fluidizationo This variable will therefore be of interest in any discussion of pressure fluctuations within the bedo Second, if there is a unique solids flow for a given system, it will be desirable to know it, a priori if possibleo This variable will be of paramoun ntnterest in examining fluctuation frequencies and bed configurationso Third, if any or all of the four possible cases of slugs discussed in the preceeding section are present in a system, one must know which case exists when and at what locationo Finally, some sort of description of slug generation must be formulatedo There must exist some mechanism by which slugs come into being at the bottom of the bed3 otherwise the slugging regime could not exist in a batch systemO These then are four points which must be investigated experimentally to complement the description of the behaviour of the slugs in the systemo Each will be treated later in turno However, the kernel of the overall model is the set of equations describing slug behaviouro Therefore, these equations will be developed before proceeding to examine these complementary areaso

25 The equations governing the motion of a slug of any of the four types mentioned proceed very easily from the assumptionso If one define s \ = — total -uiperficial gas velocity /gf= superficial gas Teloeity at incipient fluidization then one may define an excess velocityT Vxr5 \/S V superficlal gas velocity in excess of that required for incipient fluidizationo This excess velocity is the component of the total gas velocity which moves through the bed as void spaces or, in large diameter beds, as bubbleso It follows that a void space which occupies the entire tube diameter must move with this excess gas velocityg provided that there is no solids flow through the voido Consequently, the solid slug above it will also slip with that linear veloeity if there is no solids flow into or:ot of the slugo If one now adds a solifds flow through the void Space, the velocity of the void space will increase proportionally. but its volume will remain the se The volume lost by the upper bounding slug will immediately be gained by thea ower bounding slug Howevers the slug above the void will continue to slip at Vxs o The upper and lower boundaries of the slug may have an additional eomponent of velocity proportional t the solids downflow rate, These concepts ma be formulated matheatically as followso Let w = solids downflow rate, gm/Sec A = tube cross section, C m = - density of a slug Qm /Cm3

26 Define WS 0 -_ Recalling Figures 1 and 2, let us now describe each slug caseo Case lo No solids inflow or outflowo The mass balance as previously, is: (63) y -O or y=yo The velocity of the various surfaces are given by: (64) - /S OIr X- (65) Z =Vxs or = -o = Vys(t-to Oase 2o Solids outfiow onlo The 'mass balance is' (66) Y= - or o Y-r(t 0) The velocities of the upper and lower surfaces are: (67) X Vxs o r X- X VX5 ( t-to) (68) Vxs' or - (Vx v-V)("s- ) Case 30 Solids inflow onlyo The mass balance is. (69) y or 7-yo= (t- t ) The velocities of the bounding surfaces are- (70) Vxs or X- 5 + )(~-~) (71) i = V or ' -V = VS ( &)i

27 Case e4 Solids inflow and solids outflfow The mass balance iss (72) y =- or y-\/y O The veloeities involved are: (73) — Vxste o x- X - (V-e)(t-to) (74) V VSe or Z- = (Xt-t) The:se equations epresent a complete description of the configuration of each type slug as a function of timeo Their ultimate use will be postponed until the four complementary areas mentioned above have been investigated0 The success or failure of the model will, as has already been mentioned, be judged on the basis of how accurately it can predict two quantities: the pressure profile of the bed as a function of time and the fluctuation frequencies of the bedo With this goal in mind, the four areas requiring investigation were studiedo

BED PRESSURE DROPS AT INCIPIENT FLUIDIZATION Ai Equipment The determination of the bed pressure drop at incipient fluidization is a relatively easy quantity to measure, The only requirements are two pressure taps, one below the bed support and one above the bed, a means of measuring gas velocity, and a manometer. The rotameters ied to measure gas flow were Manostat "Predictability" flowmeterso The calibrations of these instruments is discussed in Appendix Co The manometer used was filled with tetrabromoethane and was connected differentially between the two taps mentioned aboveo All flow rates were measured upstream from a sonic flow orifice (needle valve) so as to eliminate any fluctuations in the level of the rotameter float which might arise from fluctuations in the bedo This means all rotameter readings were taken at rotameter pressures at least twice the downstream pressure at the sonic flow orificeo Rotameter pressures were read from pressure gauge C2-1l74, the calibration of which appears in Appendix Co The various tubes used were all preceded by a packed calming sectiono The intent was to distribute gas flow to the bed support as evenly as possibleo The diameter of the calming section was, in all cases, three inches0 The packing consisted of a 10 inch lower section of 3/4" crushed stone separated by a coarse screen from a two inch deep upper section packed with 6mm glass beadso Two inches of free space remained between the top of the packed section and the bed supporto 28

29 The bed was supported by a wire cloth with 30 mesh openingso This support was oriented as horizontally as possible0 The tube was aligned as close to vertical as was possibleo An expansion head topped the tube in 'order to prevent any solids carry over0 A diagram of the apparatus appars in Figure 30 B0 Procaedureo The bed as reparep d by pouring the solid into the tube through the expansion heado Bed height was measured with the solid in the retnlting loose packed stateo The tube was then tapped until no further settling of the solid was observedo The change of bed height with tapping was less than 5% for all solids usedo The bed height at minimum fluidization also varied less than 5% from the poured heighto The remainder -of the procedure is completely straightforwardo The rotameter was set at a given reading and a period of a few minutes was allowed for transients to disappearo The manometer was then reado CO Results 0 It was assued at the outset that the bed support contributed a negligible pressrae drop to the masurd bed pressure drop. This assumption was later verifiado Figures 4 through 10 show the experimentally detrnined mrues of total bed pressure drop versus linear

30 EXPANSON HEAD AIR 1NE _ BOTTLE GAS LINE MANOMETER PRESSURE < /"-SONIC FLOW ORIFICE PRESSURE GAGE 2 ROTAMETER I -SUPPORT PACKED CALMING SECTION Figure 3. Experimental Equipment for Measuring Total Bed Pressure Drop.

31 TABLE I Total bed pressure drop versus linear gas velocity. System: lo Air-sand A (100-150 mesh) 2. 2" diameter bed. 3. 74.0 cm. bed height. Linear gas velocity Total bed pressure at latm and 60OF. drop, inches of tbe.* cm/sec 0o 00 000 0.90 2.3 1.30 5.4 1.85 8o.6 2.40 11.0 3.60 14.7 4.80 15.3 6.00 15.5 7.45 15.8 10.20 16.2 *tbe. tetrabromoethane. One inch tbe.=0.107 psig.

S2 - _...-. ----.- - ___ X, L- w --- — -- _IOT a N I: __._ _ ___. ~ = _..'..,' 111_IL- = ' 3aeN.i'dO, 3nss3,d 038 ivIoi ' dv 32

33 TABLE II Total bed pressure drop versus linear gas velocity. System: l1 Helium-Sand A. (100-150 mesh) 2. 2" diameter bedo 3. 72.0 cmo bed height. Linear gas velocity Total bed pressure at latm and 600F. drop, inches of tbe. cm/sec 0000 0.00 0.30 1.2.o10 307.240 9.3 3.90 13.6 5o60 14.4 7030 14.8 9.10 15.0

-34 -BLANK PAGE

: __= o = __ o CY I I N _ t Ii ', 2 \lS STTC\ S j t) L ---- - _I_ - - -..- I _ 1 NL....::............ 1 1.... 1 I... I.e I91 N1 1 i 11 1 1 i 1 1 3 '391*Nl ' dU 3SS3U 38 5t"101 ' dV 35;

36 TABLE III Total bed pressure drop versus linear gas velocity. System: 1. Air-Sand A (100-150 mesh) 2. 1" diameter bed. 3. 21o2 cm. bed height. Linear gas velocity Total bed pressure at latm and 600Fo drop, inches of tbe. cm/sec 0.00 0.00 0,19 19 0.19 0.38 0.30 0.83 0044 136 0.59 1.91 0.77 2.53 1.00 3.25 1.28 3.92 1.58 3.82 1.73 3.92 2.06 4.08 2.31 4.10 2.64 4.14 3,04 4.20 3.44 4./26

! I 1 I:1 1_! _I l/ l 1 e f. H b 88^ 1o 1 1 < r __I. 'I I:=: 1 1:= N I I ~~~~__ \~~ -~~~4 1 __......" —.. _ 'bu. (0 b to NM - o cq w 10 K) ~ 3891 'Nl t dO0 3dfnSS3ld 03' 1V".L01' dV 37

38 TABLE IV Total bed pressure drop versus linear gas velocity. System: 1, Air-Sand A (100-150 mesh) 2. 1" diameter bed. 3. 58o0 cmo bed height Linear gas velocity Total bed pressure at latm and 600~F drop, inches of tbeo cm/sec 0000 0.00 O092 2.55 1.60 8.10 2.42 13.48 3.50 11.88 4,83 12.60 6.04 13lo10

__o — 1zzzzz1- ~ L I - -I - - 8 1 I;; > 0 ED CD 0 U0 3 e itC "MO i i ii i I I I II -L _ O,, _ _ ) & *3EiNiM'dOWa 3&falSS3d e ao 101' dv 39

40 TABLE V Total bed pressure drop versus linear gas velocity. System: lo C02-Sand Ao (100-150 mesh) 2o 1" diameter bedo 3o 42.0 cmo bed heighto Linear gas velocity Total bed pressure at latm and 60~Fo drop, inches of tbeo cm/sec 0000 000 0O72 2l1 1.26 6,0 1.91 8,7 4.65 8.9 8.85 902

Z 11 en WIg 4 0 '1_ __ _ _ _ _ _!XH 0.47W1 @ so r t KCo O O0D 1~- In. V '3gl'NI dOlC 3luSS31d Ca8 1V.LO1. dV 41

42 TABLE VI Total bed pressure drop versus linear gas velocity, System: 1o Helium-Sand A. (100-150 mesh) 2, 1." diameter bedo 3o 42 0 cm. bed height Linear gas velocity Total bed pressure at latm and 600~F drop, inches of tbeo cm/sec 0000 000 2008 1.7 3.664 4o05 5.53 6085 7094 8060 13.48 9o50 19o5 9080 25.6 9080

(0 - w) \1 5o B 2: N 00~ ~ ~ ~ 0 -- l --- {o dni -- -_ l -I -- -a S Z 2 a (0 ---- —,.._ __ _____ *'381 'NI ' dOla 3Inss3~d 038 v1' dV 43

44 TABLE VII Total bed pressure drop versus linear gas velocity. System: 1. Air-Sand B. (monazite) 2o 1" diameter bed 3. 33,0 cm. bed height Linear gas velocity Total bed pressure at latm and 60~F. drop, inches of tbeo cm/sec 0.00 0.00 1.04 2066 lo55 4.65 2. 14 7 55 3,36 11.80 5o73 12.00 O832 12 55 10.92 12.65

2i l li 0 N -- - - -- - - - R =S - — iQ i i Io L -- - - 0)_- - - E e O- oC _ 0I.J 3el No -dowa (Mnss~sd a~s 5v10 'o S P45 N 1 *3ei'NI 'dOg 3NnSS3Md 38 IVlOI 'dV 45

46 gas velocity. The incipient fluidization velocity is quite apparent on each of these curveso A simple check on the accuracy of these curves and at the same time on the assumption of zero pressure drop across the bed support can be madeo The pressure drop at incipient fluidization should equal the weight of the bed per unit bed cross sectional areao This should be true of all gaseso If one plots pressure drop at incipient fluidization versus bed height, the intercept at zero bed height will be the pressure drop due to the bed supporto The slope of such a curve should be(SO/QC~ in appropriate units. Figure 11 shows the data and theoretical curve for sand A (100-150 Mesh) and zero support resistanceo It is apparent that the assumption of zero support resistance is valido Table IX lists all pressure drops per unit bed length at incipient fluidization and minimum fluidizing velocities for all systemso

47 TABLE VIII Total Bed Pressure Drop at Incipient Fluidization versus Bed Height for 100-150 Mesh Sand. Gas Bed Diameter Bed Height Pressure Drop in o cm. in..te. Air lo00 21.2 3.95 Helium 1o00 420.o 50 002 1.00 42.00 8o60 Air 1.00 58.0 10.92 Helium 2.00 72.0 13.80 Air 2.00 74.0 15.00

48......\ e,r a. 0! 1! S ISa 4 I \ I \ I - I 4,a I ~C H 61 — ~ _ Na 3 0iW\ N I \ 5ti 3 N Z t ) ~N3EII.I dOHO:~lsS& 03 " \~Ol 'dV uj V-a duuo u d 039 iIOi'dv

49 TABLE IX Incipient Fluidization Data for Systems Studied. Gas Bed Bed Solid Vgmf, Pressure Diameter Height, B im/ed Drop per in. mdi. unit length, psi/cm Air 1.00 21.2 Sand A 1.10 0.0199 Helium 1.00 42.0 Sand A 6.60 0.0217 002 1.00 42.0 Sand A 1.52 0.0219 Air 1.00 58.0 Sand A 1.82 0.0202 Helium 2.00 72.0 Sand A 3.60 0.0206 Air 2.00 74.0 Sand A 3.14 0.0217 Air 1.00 33.0 Sand B 2.95 0.0383

SOLIDS DOWNFLOW RATES The rate at which solids flow through void spaces is the most important variable in the characterization of slugging gas-solid fluid bedso Considerable effort was spent in determining precisely what conditions prevail in such systemso The first guess was that the solids downflow rate would vary with gas velocity; this was shown to be incorrect. Because slugs move at the excess velocity of the gasp v,xs the lower surface of the slug is always exposed to a relative velocity equal to that at incipient fluidizationo Io Bin Flow Measurementso Ao Equipment The first measurements were taken in the two inch diameter bedo The only additions to the equipment shown in Figure 3 were a slide valve at the expansion head and a photoelectric sensing device coupled to an oscillographic recordero The slide valve was designed to divert the gas flowing up the column out of the side of the column when closed to confine the solids to the expansion heado When the slide valve was opened, the gas was diverted up through the expansion heado This presumably exposed the solids at the throat of the expansion head to the velocities encountered by solid slugs in the tubeo The photocell measured the incident light from a projection lamp after two traverses of the tube via a mirroro The electrical output of the photocell was continuously recorded on a Sanborn 50

51 Twin-Viso oscillographic recorder. The output of photocell was not calibrated to measure solids density accurately; the only desired information was the time of fall of a known quantity of solids contained in the expansion head. A diagram of this equipment is shown in Figure 12. Bo Procedure The desired gas flow was set, the flow being diverted by the slide valve to the atmosphere. The expansion head was filled with solids sufficient to fill the tube to some point below the photocell arrangement. The oscillograph chart was started and the slide valve opened. The falling solids changed the transmittance characteristics of the tube and contents and provided a measure, qualitative since no calibrations were made, of the solids density of the tube. Figure 13 shows a trace of light transmitted versus time for a typical run. Co Results The results of these measurements for a 100-150 mesh sand in a two inch tube are shown in Figure 14. The solids downflow at zero gas velocity in similar systems has been investigated (7,8) (8) Kelly reports that the flow of solids through 2 inch orifices follows the equation: (75) W = 0.156D2 84

52 SLIDE VALVE DETAIL SOLID (SAND) SLIDE VALVE PHOTOCELL LAMP- 0- PH cLL MIRROR SUPPORT OSCILLOGRAPH Figure 12. Apparatus for Measuring Solids Downflow Rates.

53 1" 0 -O-P p4i _I ~~~I "" i.1.= A a 0 i " \ ' *H. 0 r 30NviiiMS~vyjL — ^ 77 N.l~II~SNVW. -.

54 TABLE X Solids Downflow Rate versus Gas Velocity. System: 1. Air-Sand A (100-150 Mesh) 2. 2" Diameter Tube, 3o Bin flow technique. Gas Velocity, Solids Downflow Rate cm/sec gm/cm2 sec 0,0 20.0 0,0 19.6 0.0 20.8 1.0 18.9 2.0 19.6 3~25 2602 402 27.2 5.7 24.4 6.0 29.1 6.0 2908 7.4 23.5 8.2 2004 8.7 21.0 11.5 15.3

55 0.............. --.... 0 i --- -— 0 _ ~o w Ca _co wt 'w B a oe 0 0 --- —--------- co / H..... _ _,a J w 0 in 0, 0 i" 0............. ~m~~~~~~~~~~~~~~V....,...........,,,..............U i i iii - iii L i ii ii ii i m i ii - 0~r 0 ~ 0 ~ 0 ~ 0~~~~~ o~:ma slv m~.~Ao an 'v

56 where D is in inches and W is in pounds per secondo This yields, for the two inch orifice used, a flow rate of 244o5 gm/cm2 seCo This compares favorably with the observed value of 20.5 gm/cm2seCo The discrepancy probably lies in the fact that Equation (75) is written for bead or pelletized catalyst, which may have differed significantly from the sand used in this studyo As the gas flow rate is increased from zero to the velocity for minimum fluidization, the solids downflow rate drops, most likely due to increased drag on particles as they separate from the bulko At the velocity of minimum fluidization there is a marked increase in solids downflow rate. Further increases in gas velocity cause a decrease in solids flowo The two data points in Figure 14 at gas velocities of 6~0 cm/sec are noted to be considerably higher than all other points~ The reason for this high solids flow rate is that some grease from the slide valve found its way into the solids and caused agglomeration of solidso The solids used were replaced whenever this occurred after the first incident. Do Conclusions Two important points may be made as a result of these bin flow testso The first is that solids flow tends to decrease with increasing gas velocity both above and below incipient fluidization. The second is that a sizable increase in solids flow occurs at the velocity of incipient fluidization. This increase is due, in all probability, to the

57 increase in the freedom of particle motion at this velocity. A particle which is free fromforces exerted by its neighbors is able to disengage from the bulk more easily. The important result is the value of w at the maximum, as will be seen. II. Pressure Profile Measurements. A logical question arises following the completion of these bin flow measurements. What relation is there between the solids downflow rates measured with bin and slide valve to the actual solids flow rates which prevail in a slugging fluid bed? The answer to this question was sought before proceeding with more bin flow tests. Pressure-time records were used to determine solids flow rates in slugging beds. Ao Equipment The apparatus used for these measurements is that shown in Figure 15. Two pressure taps were placed in the tube at heights of 60.0 and 75.8 cm. above the bed support. Each tap consisted of a '-" pipe to l" copper tubing tee with a medium mesh screen across the tube side leg. One tube leg, the downward oriented leg, was used as a trap to collect solids which entered the tap through the screen. This leg was emptied periodically. The upper leg led to a tee which permitted two pressure connections to one tap. One connection was made through a pet cock to one side of a differential pressure pickup. The other side of the pickup was connected via

58 EXPANSION HEAD TAP DETAL SECTION OUTPUT ) REFE RENCE I SED ANBORN RECORDER Figure 15. Equipment Used for Measuring Solids Downflow Via Pressure Drop in Slugging Fluid Bed.

59 rubber tubing with two pinch clamps to one connection on the lower tapo For these measurements, the extra pressure connection at each tap was closed and unusedo In other applications, these extra connections led to other pickupso The strain gage pressure pickups used were CEC model 4-315,~ 1 psi differential transducers. The pickup used in these measurements was number 2172, the calibration of which appears in Appendix Co Appendix C also contains other pertinent information on this pressure pickup such as zero shift with temperature, linearity, hysteresis, etc. A constant 5 volt Do C. excitation produced, through the transducer, an electrical output which was recorded on the Sanborn TwinViso recorder. Bo Procedure The proper valves were opened to permit operation of the transducero The gas flow rate was adjusted to the desired value and the bed allowed to reach steady state operation (5 minutes). Then the pickup excitation source was switched on and a one minute recording of the pickup output was madeo Co Results A sample oscillographic recording is shown in Figure 16. Since the pressure drop measured was at all times in a zone of solids downflow, the slopes of the observed pressure curve yield, under the assumptions of the postulated model, a measure of solids downflowo Consider first a decreasing pressure differential. This will

EMPTY CHANNEL I, I........,,, I. I I I I I ' 0 ~ -...I T. I I I I, I II t SEC TIME, SECONDS - Figure 16. Oscillograph of Pressure Variation Between Two Taps 15.8 cm Apart Near the Top of a Slugging Fluid Bed. Solid is 100 -150 Mesh Sand. Gas: Helium. Gas Velocity: 10.9 cm/sec in 2.0 Diameter Tube. Lines at the Right are for Slope Measurement. 60

61 be due to the upward movement of a slug-over-void interface through the zone between the pressure taps. As the void moves upward, the amount of solids between the taps decreases. If the total pressure difference is attributed to a slug at conditions of incipient fluidization, then the pressure difference. isL-a measure of the slug length between taps. If the top of the slug is above the top tap, then the rate of decrease of pressure is proportional to the rate of upward motion of the interface. But the proposed model gives this rate as: (76) i = +0 -The excess gas velocity is easily calculated from (77) Vx = - V f Thus S may be determined and converted, by use of solids density, to w,> solids downflow rate. Next consider an increasing pressure differential. This is due to a void-over-slug interface moving into the zone between taps. The model gives this rate: (78) X = VxS 0+ Therefore, the analysis of increasing pressure differentials is similar, except for sign, to the analysis of decreasing pressure differentials. Figure 17 shows the results of these measurements for 100-150 mesh sand in a 2.0 inch diameter tube for air and helium. It appears

62 TABLE XI Solids Downflow Rate versus Gas Velocity. Operational Pressure Drop Technique Gas Gas Velocity Downflow Rate cm/sec gm/cm2 sec Air 700 31,5 Air 10,0 3109 Air 13.0 35.1 Helium 9.1 3208 Helium l 109 3701 Helium 14.7 32.1 Helium 18 0 34. 6

63 - I I II __ __ _______ i _ a_ _ _ __ i4~ - 14;1 1 — ' - -- -' ---- o caa %1k I ---T-t/" 9 ---1-3 1^0 — ^SI~p!) mm 8 8? 8o o8 o 3as-^i/IO/e".v MO'UBNO0 sano s' %

64 that there is no change in wa with gas velocity. For this reason all subsequent measurements of wa or 0 were made at only one gas velocity, the velocity at minimum fluidizationo The data at this point are inconclusive as to whether or not the gas used affects solids downflowo Certainly the variation between air and helium is slighto Do Conclusions These results lead to several important conclusionso There are: lo Slugs slip with respect to the tube wall at the excess gas velocity vxsIo This is true necessarily in the bed zone studiedo 2. No acceleration or deceleration of slugs is presento The linearity of the pressure curves points this outo Assumption 3 of the proposed model is thus verified for the bed section studiedo 3o There is a unique solids downflow rate0 This verifies assumption 5 of the proposed model for the bed section studiedo 4o There is a negligible pressure drop in void spaceso This is true because the pressure trace returns to the base line when only falling solids occupy the zone between taps (see Figure 16)o Again this only holds at present for the bed section studiedo This verifies assumption 9 of the model under considerationo 5o The agreement between bin flow measurements and pressure drop measurements of wa indicates that the lower interface of a slug, and hence the slug itself, is exposed to a relative velocity the same as the velocity for incipient fluidizationo Assumption 10 of the model is thus verified for the bed section studiedo IIIo Effect of Operating Variables on Solids Downflow Rateo With gas velocity ruled out as a variable, the remaining variables of the gas used and tube diameter were investigated brieflyO Carbon dioxide, helium and air were used to fluidize 10Q-150 mesh

65 sand in a loO-inch-diameter Plexiglas tube. Air and this same sand were also checked inca loO-cmo-diameter glass tube. Finally, the system air-monazite sand was studied in a lo0-inch Plexiglas tube. In every case, except the loO-cm. tube rune, both bin-flow and pressure-curve techniques were used to obtain solids downflow rateso Ao Equipment 1o The apparatus for pressure measurement is the same as that previously discussed, with one exception. The screen over the pressure tap was replaced with a very thin glass-wool plug for measurements in the 1.0-inch-diameter bed. 2o The apparatus for photoelectric measurements was altered slightly to yield more information, Two light beams in these experiments made only one traverse each of the tube. The photocells and their respective sources were located near the bottom of the bed and were separated by a distance of 17.0 cm. Thus the solids optical density was measurable at two separate locations. The slide-valve arrangement already described was used for binflow measurements for the one-inch bed. A section of 1-cm, rubber tube and pinch clamp was used for controlling solids flow in the 1-cm, bed. Several pin holes in the rubber tube below the pinch clamp diverted gas flow to the atmosphere until the clamp was openedo The bins used were changed for convenience, A section of 3: inch Plexiglas pipe served as the hopper for measurements in the oneinch bed; a glass funnel performed the same function for the one centimeter diameter bedo The bed supports in both instances were changed to porous glass wool plugso No calming section was usedo Figures 18 and 19 show these arrangements o

SANDV FUNNEL l- 9 -PINCH CLAMP -.0 CM. GLASS TUBE LIGHT SOURCE PHOTOCELL SANBORN GLASS WOOL 2 CHANNEL PLUG RECORDER GAS INLET Figure 18. Equipment for Measuring Solids Downflow in a 1.0 cm. Glass Tube. 66

A^' ^SOLIDS BIN SAND SLIDE VALVE (SEE FIGURE 12) -1.0 IN. PLEXIGLASS TUBE LIGHT SOURCE PHOTOCELL GLASS WOOL PLUG W SANBORN 2 CHANNEL f*^ RECORDER GAS INLET- Figure 19. Equipment for Measuring Solids Downflow in a 1.0 inch Plexiglass Tube. 67

68 Bo Procedure The procedure for determining solids downflow rate from pressure measurements had already been outlined, The bin flow procedure operated as followso When the solids were released, falling solids interrupted both light beams to a certain extento Fluctuations in the light transmitted by the column of falling solids were observedo When the tube filled with solids in the light beam, these fluctuations ceasedo Therefore, the time delay between the cessation of transmittance fluctuations in the two photoelectric measuring points will give the value of solids downflowo The distance between beams divided by this time delay is exactly the value Q, which may be converted to w, Co Results The results of bin-flow experiments in the loO-cmo glass tube will be discussed firsto The solids, upon release from the bin, fall first through the upper light and then through the lower light beamo Before reaching the upper beam, the solids have fallen over 200 cmo, and therefore should be close to terminal velocity. The distance between beams divided by the time delay between initial beam interruptions is therefore a good measure of particle terminal velocityo A typical oscillograph showing this delay is shown in Figure 20. The value of terminal velocity determined in this way was 120 cm/seco This compares quite favorably with the value calculated from drag correlations for single spheres of 134 cm/seco The average mesh size was taken as the diameter for purposes of this calculation,

z lI.I — I — -' — --- -DELAY =0.142 SEC ".,.iSEC I I I I, I, I, i,, TIME, SECONDS -- Figure 20. Determination of Particle Terminal Velocity. System: 100-150 Mesh Sand in Air in a 1.0 cm Glass Tube. Distance Between Photo Beams: 17.00 cm. 69

70 Solids downflow rates determined by both procedures for nearly all tube diameters, gases, and solids are shown in Table XIIo Three variables are present which may affect the solids downflow rateo These are: lo A variable characterizing the solido 2o A variable characterizing the gaso 3. A variable characterizing the tubeo Since this study was not intended to examine the complex problems of solids characterization, only a brief excursion was made in this direction. The two solids considered were 100-150 mesh sand and 100-150 mesh monazite sando The former, sand A, had a bulk density of lo44 gms/cm3, the latter, sand B, 2.98 gm/cm3o These solids appeared identical under a microscopeo The particles were smooth and fairly regularo Figure 21 shows a plot of wa versus apparent solid density, the only solid variable of consequenceo Normally two data points are insufficient, but the plot shows, as expected, that the origin is also a point on the curveo The variable which best characterizes the gas is the particle Reynolds number at incipient fluidizationo The velocity for minimum fluidization is used because it has been shown that this is the velocity to which the disengaging particles at the bottom of a slug are exposedo The resultant effect of particle Reynolds number on Wa is shown in Figure 220 As expected, the increase of particle Reynolds number decreases the solids downflow rate Wa, although very slightlyo This plot also demonstrates the effect of the most important variable, bed diametero

71 TABLE XII Summary of Observed Solids Downflow Rates. Bed Diameter, Gas Solid w Technique cmo gm/cm.sec. 5,08 Air Sand A 29.5 Bin flow 5o08 Air Sand A 32.8 Pressure 5.08 Helium Sand A 34.1 Pressure 2,54 Air Sand A 26.4 Bin flow 2.54 Air Sand A.24.7 Pressure 2.54 Helium Sand A 23.5 Bin flow 2.54 Helium Sand A 22.8 Pressure 2.54 CO2 Sand A 18.4 Bin flow 2054 CO2 Sand A 21e4 Pressure 2o54 Air Sand B 53.4 Pressure o00 Air Sand A 12.2 Bin flow

'''' ' ''1 ' ' -..e, F~ 400 Xd 1MW01 1 b *as A1 #0A Os 8 s e 2 -~ ~~S -:Bot:/F ~~~' 3 F4 --— _-_. —. ~, o S 8 t~o 88 0 0 03S-^3/LO 31V A N SO f OS' M 72

73 TABLE XIII Solids Downflow Rate versus Particle Reynolds Number at Incipient Fluidization, Bed Diameter, Gas Rep w in. gm/cm sec. 1.0 Helium 0.0665 23.2 1.0 Air -.0 o0875 25.5 1.0 Air 0.145 25.5 1.0 002 0.238 19.9 2.0 Helium 0.0361 34o1 2.0 Air 0.249 31.2 Solid used: 100-150 Mesh Sand A. (S 1.o44 gm/cm3 bulk.

ii U lI~ 'I 11 I i D,i PP w 1 A 5Q V~~~~~~~~~~~~~~~~~~~~~~~) rd3 z I0z * (2 I 1 l TZ.. J 0 o IJ (U k 0 ~~~~~~II *^~~~~~~~~~~~~~~~~!I~ _ 0000 0~0 o3s- /W'aLvy cn1Moa sonos < m 74

75 Figure 23 shows the effect of bed diameter on solids downflow rateo This curve exhibits a tendency to level off at larger bed diameterso This curve must of necessity pass through w equal to a. zero when the bed diameter equals particle diameter. Since the diameter of a particle is very small, this curve appears to pass through the origino Do Conclusions Since the terminal velocity of a particle is much greater than all other system velocities, assumption 6 of the proposed model is verified. The largest impressed gas velocity was of the order of 40 cm/sec, which is much smaller than the observed terminal velocity for Sand A of 120 cm/seco This conclusion permits one to assume that solids transfer from slug to slug is very fasto The bin flow technique for measuring solids downflow is seen to be a reasonably accurate means of predicting wae Pressure- measurements give nearly the same values. This indicates that one need not operate a slugging system in order to know wa it can be measured with auxiliary equipment. The results of Figure 21 lead to the conclusion that solids downflow is directly proportional to solids density. The data available is not great in number, but the results are not to be deniedo This curve definitely passes through the origin, showing solid density to be a very significant variableo On the other hand, the properties of the gas -used seem to have but a weak effect on wa. The slight

76 TABLE XIV Effect of Bed Diameter on Solids Downflow Rate, Gas Bed Diameter w. cm. gm/cm sec Air 5.08 31o2 Helium 5.08 3401 Air 2~54 25.5 Helium 2 54 23.2 C02 2.54 19.9 Air 1.00 12.2

77 a). j }, I4I J PA 0 z H 5V SIy) (1) o3S-/9 3UNMO sOcOS'= s

78 decrease in wa with increasing particle Reynolds number can be attributed to increased drag on the disengaging particleso Bed diameter has a strong effect on w a indicating that bed geometry plays an important role in small beds. It is quite probable that there is some critical diameter above which w is not affected by a changes in bed diameter. The large equipment required to explore larger diameters made such a study impracticalo These results point to an expression for Wa of the form. (79) WA Ap (ep)'s (Dt) 3 (SoLID 5HAPE) The character of fl and f2 has already been demonstratedo The function f3 has always been a very difficult one, and, as stated earlier, beyond the scope of this worko The manner in which solid density enters if interesting because it means that the previously defined quantity, Q, is independent of solid density: (.o) e- A - f'fF Two conclusions regarding the study of solids downflow are worthy of further emphasiso They are: lo There is a unique solids downflow rate for a given systemo 2, This downflow rate may be determined without operating the actual system; ioeo, it may be determined with auxiliary equipment0

BED CONFIGURATION This section deals with the type of slugs which exist at various elevations in a slugging fluid bed. There are the four possibilities listed in the proposed model0 Certainly the top slug in any system must be a case 2 slug, that is a slug with material outflow from the bottom. The remainder of the slugs, however, may in theory fall into any of the four cases described. A. Equipment The equipment has already been described in preceding sections; no instrumentation was used0 B0 Procedure There were three events of note which recurred periodically at three statistically varying elevations0 The first of these was the maximum elevation of the top of the bed. The top surface of the slug rose upward until the top slug disappeared because of solids outflowo The.second was the stabilization of the lower interface of the slug below the top slug0 The term stabilization is here intended to mean the cessation of solids outflow from this interface0 The location of this interface at the moment of stabilization is the quantity of interest. The third quantity was the elevation at which bubbles, originating at the distributor, grew to occupy the whole bed cross section, thus creating a slug and a void space0 As will be seen these three elevations serve to characterize the bed and completely determine the case of all slugs in the system. 79

80 The procedure therefore consisted of measuring these three elevations as functions of gas velocityo Because statistical variations were present, approximately 30 measurements of each level at each velocity were taken for each system. The average value is reportedo A glass marking wax crayon was used to mark the elevation of the event. The average of the readings was thus visually apparento Resultso The three levels mentioned above are given by: T = maximum height of the top of the bed HM = height of the lower interface of the slug below the top slug when solids cease to fall from this interfaceo HB height at which bubbles formed at the distributor grow to ocupy the entire tu;be All heights are measured from the bed support. Figures 24 to 32 show the results of these measurementso These three "level curves". HT, HM, and HB versus velocity serve to characterize the bed configurationo The region between the stabilized interface and the top of the bed always contains two slugso The top slug always has outflow onlyo The second slug is a case 41 slug when at lower bed elevationso When the lower interface reaches H, it becomes a case 3 slug-one with inflow onlyo An occasional particle was observed to fall from this "stable" lower interfaee, but this:"dripping-" was of no consequence0 There existed in every system a babbling zone at the bottom of the bedo These bubbles grew to the size of the bed at the elevation

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160 _ 150 SYSTEM: 140 100-150 MESH SAND 1.0 IN. BED DIAMETER H 130 GAS: CARBON DIOXIDE DISTRIBUTOR WIRE CLOTH 120 120 il-j ----- --------- l --- —---— ___ — ---- 110 -V____ __ GAS VELCIYCMSE __ 100 - 90 0 40 H 0 5 0 15 20.25 vg, GAS VELOCITY, CM/SEC Figure 30. Level Curves for 100-150 Mesh Sand. -87 -

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90 designated HB, thus creating a void space and a slugo If HM was not too low, one or more slugs existed between the bubbling zone and the stable interfaceo The third slug from the top exists as a case 2 slug throughout most of its history0 All slugs between the third and the bubbling zone were case 4 slugs, Figure 33 show pictorially the situations discussed aboveo The shapes of the level curves for the systems studied are all similar, The curves in Figure 24 are the only ones which deviate in any respect In this case a bubbling region preceded the slugging region as velocity was increased0 The result is that the curves are translated to the righto Conclusions The bubbling zone will be covered in the following section0 The results of this section show that the bubbling zone occupies a small fraction of the total bed heighto The stabilization of the lower interface of the second slug can be explained in terms of drag forces 'Nearly all drag correlations are of the form (81) FD = A V Ani:entirely reasonable assumption for gases is that the product (OgV is constant throughout the bed. It will vary from void space to slugo Hence FD depends on gas velocity, which varies from point to point in the bedo The stable interface would then occur

WT MeaniA o 91.H.0 oo Main of Br Hm, HB' Not to Scle. A '' A m1 i/ V9 D, ~ 00 91

92 at the point in the bed where the gas velocity is such as to exert terminal drag on the particles of a lower slug surface. This is possible if the free cross section of the lower surface is low enough to yield the high interstitial velocity required for terminal drag0 Existing drag correlations predict precisely where in the bed this event will occuro This configuration is not particular to any one solid or bed supporto Nozzles, porous plugs and wire cloth were all used as distributors without any noticeable change in the level curveso One therefore concludes that the bed support is relatively unimportant in determining bed configurationo Neither size nor density of solid would appear to be important, since the level curves for monazite and molecular sieve cylinders did not deviate in shape from the curves for the 100-150 mesh sando Thus the type of slugs existing in a system is known once the level curves for the system are knowno No correlative techniques could be found which allowed a good prediction of these curveso The variables of bed diameter, bed height, gas properties and solid properties all affect these curves0 Some comments can be made however0 The HT curve slope is a strong function of bed diameter, as is shown in Figure 34o The height of the bed for determining these slopes is the fractional increase in static bed heighto The effect of increasing bed diameter is to decrease this slope0 Since the HT curve should pass thrugh te point of incapiens fluidiztion, the n is slope correlation allows a fair prediction of the HT curveo

U - I4 l -I~i l ~r/ t0 o - 8 0.0 0 w Z wo/o3S' 3Ao 8H 3H.L IO 3dolS 93

94 The level HB, the top of the bubbling zone of the bed, in all cases drops as velocity increases. It very quickly approaches a height corresponding to HB/Dt"2o This, for must slugging beds, is an insignificant part of the bedo The HM curves presents the greatest difficulty. Figure 35 shows a plot of the ratio of HM to static bed height at the minimum of the HM curve versus the velocity at that minimum, Considerable scatter is apparento It is impossible to generalize at all from these datao

95 0.8 - 1-, l. - -l - ----- 0.7 -~ - 0.6 - --..- --- 0.6 I.. 0.3 is.... A 0.3 A CO - SAND A 0 AIR - SAND A 9 HELIUM-SAND A AIR-MONAZITE i 02 0 10 20 Vg,GAS VELOCITY, CM/SEC AT MINIMUM IN NM CURVE Figure 35. Location of Minimums in HM Curves.

SLUG GENERATION The mechanism of slug generation is a statistical process in that slugs of varying lengths are generatedo The height at which slugs form has been treated in the preceding sectiono The two remaining questions are, then^- what: is the length of a slug when it forms and by what distance is it separated from the bubbling portion of the bed? Let us treat the separation problem firsto Suppose a slug of length yo has just been formedo This means that the bottom bubbling section of the bed at some prior instant was HB plus yoo These additional solids, of length yo, must have built up due to solids downflowo This rate is known to be 9 cm/seco Therefore the time required to obtain length yo is given by yo/0o In this length of time gas will have been creating void spaces in the form of bubbles. Since the rate of gas flow in excess of the minimum fluidization velocity has been shown to proceed through the slugging portion of the bed as void spaces, it is reasonable to assume that this excess gas proceeds through the bubbling zone as bubbleso It is also reasonable that there is a relatively constant bubble holdup in that zoneo If this is the case, then the excess gas which entered the bed during the interval yo/Q must be located beneath the newly formed slugo The separation of this slug from the bubbling zone will therefore be Vxsso/0 The second point, the size of the generated slugs, strongly suggests that some minimum length of bed above the level HB will be 96

97 stable in that a slug can be formed which does not immediately disintegrate. There has been some work done in this areao Zenz and Othmer(l2) state that the length-to-diameter ratio of slugs in slugging fluid beds may be used as a measure of the tangent of the angle of internal friction,, for the fluidized solid0 A reasonable value foro for the sands used in this study appears to be 64 (12) This value seems to be characteristic of sands in general. The tangent of this angle is 2.0450 This then means that the bubbling section of bed should terminate at an elevation of HBg-2.ODto This was verified visually. A generated slug in a 5008 cm. bed was approximately 10 cm. in lengtho There is some variation in Yo observed. An arbitrary method for accounting for such variation is to assume that the generated slug lengths are normally distributed about a mean of Yo=2Dt with some arbitrary standard deviation,O.o Experience dictates a value of OC= 0.2Dt. The slug generation mechanism suggested is therefore the random selection of consecutive values from such a distribution.

USE AND VERIFICATION OF PROPOSED MODEL I. Prediction of Bed Configurationo Procedure and Results The model proposed, together with the information already obtained, make possible the prediction of total bed configuration as a function of time. The first step in the procedure is to determine all measured variables in the system. A numerical example will be carried out to illustrate the techniqueo Bed diameter: 2o00t Bed support: wire cloth SolidS 100-150 mesh sand Bulk solid density. lo44 gm/em3 Mean particle size: 120 microns Gas velocity: 7o00 cm/sec Fluidizing velocity: 3ol4 cm/sec Excess gas velocity: 3086 cm/sec SolidS downflow rates 31o5 gm/cm2 sec Buildup velocity, S: 21o9 cm/sec Bubbling zone heighte HB: 25 cm Stabilization level, HM: 48 cm Mexirmun rise, HT: 86 cm Let us begin with an initial bed configuration chosen arbitrarilyo The bed behaviour will be worked out for a sufficient period of time to allow the solution to reach a steady stateo Refer to Figure 33o 98

99 Numbering slugs from the top down, this initial configuration is assumed to be: Slug lo x1 80 cm l- 12 cm Case 2 l = 68 cm Slug 2. x2- 66 cm 2 = 20 cm Case Z2 = 46 cm Slug 3 X3 =- 44 cm 3l = 10 cm Case 4 Z3 = 34 cm Bubbling XB = 32 cm zone. B = 32 cm ZB = 0 cm The slug generation height, HB + yo, is chosen from a random set of such numbers with mean 10 cmo and standard deviation 1 cmo Such a number for the first generation is 11 cmo At this point, let us recall the equations governing slug movemento These are: Case 1 slugo No material inflow or outflows Y+ ( Xo + vxs-to-) =.o + Vxs (t td,)

100 Case 2 slugo Material outflow onlyo Case 3 slugo Material inflow only. y y,-6e(t-ta) x xo + (vxs+ e)(t-t s- + yxs(e-to Case 4 slugo Material inflow and outflow - ot + (i Vse)(b-,t) These equations will govern the movement of the assumed slugs until (1) one of the slugs changes from one case to another9 (2) a slug disappears. or (3) a slug is generatedo When one of these events occurs, one must review the system and reassign a new case9 possibly the same as before, to every slugo At this point9 the values of x 'y. Yo zo and to are also changed appropriatelyo For the example at handy we have. +VKS = 5o76 Since only three slopes apply to all x, y and z curves, a graphical solution is the least tedious routeo Figure 36 indicates this graphical solutiono The heights x1 and '1 change according to the prescribed slopes tuntil t = 0~09 secondso At this time the lower surface of the se:cond slug has reached HM 48 cmo Hence the slug

101 Lii~... u,,,a 11111 0 w < lip *d CO CLCo. %.., 0 0 0 0 0 0 0,0.,. 0 In CM O- (O IO to CY 5 0, UL <- ' ~... -, " '.....( 00"" "2 ':"............ '4..............................'~':"]' '~";:;':: ':;"'' o~~, o ( o o 8oo (L ~ ~ ~ o IN3( c Z:

102 types change to Slug 1- Case 2 Slug 2 - Case 3 Slug 3 - Case 2 The slugs change according to the prescriptions for these cases until t = Oo20 seconds0 At that time a new slug is generated and the slug types are then, Slug 1 - Case 2 Slug 2 - Case 3 Slug 3 - Case 2 Slug 4, - Case 4 At the instant before generation, the bubbling zone had attained a height of 37 cm. After generation, the height reverted to 25 cm. with a new 11 cm. slug in the system0 The bottom surface of this slug is moved upward an appropriate distance to account for the excess gas which must have accumulated beneath the slug during its generation. This distance is the separation of two lines of slopes V`QG and( 9 drawn from the point of height Hg = 25 cm. at the instant of the previous generations at the time of this generation0 The system obeys this new set of rules until t = 0.52 seconds0 At that time the third slug disappears and the first or top slug disappears0 This is a coincidence, not the usual occurrence0 The system now changes to: Slug 1 - Case 2 Slug 2 - Case 3

103 Figure 36 shows this general procedure carried out for five full cycles of the bedo Due to a fortunate choice of initial configurations, this is sufficient to obtain the steady state variation in bed configurationo This example demonstrates the use of the model cand auxiliary information for the prediction of bed configuration as a function of timeo Two results are immediately apparento The maximum height to which the bed should rise is 86 cmo in Figure 36, comparing very well with the observed HT of 86 cmo Second, the period of one fluctuation, measured peak to peak, is predicted to be lo35 secondo This compares well with the observed period of lo44 secondso Conclusions o Sufficient visual observations had been made at this point to qualitatively verify assumptions one and two of the proposed modelo There existed, in the systems observed no radial variations outside of the bubbling zoneo Slugs and void spaces are clearly distinguishable The model has been shown to be a fairly simple one to use to predict a fairly complex phenomenon-bed configurationo It remains to be seen whether or not it can accurately predict bed behaviouro IIo Period of fluctuations. Procedure and Resultso There is a second method 6f predicting the period of bed fluctuations in the zone between HM and HT which is much less tedious

104 than the above procedureo One requires an estimate of the maxi;M m length of the top slug in the systemo The best experimental method for obtaining this estimate is to measure HM and HTo The fraction (HT4-HM)/HT is a measure of the fraction of the expanded bed height which lies in the zone of the two uppermost slugso It is therefore an estimate of the expanded length of the top slug in the system at the moment of its formationo To. translate this to the true maximum length of the top slug, one need only multiply by the static bed height (82) Y(,Tmox - ~ - M L Once this estimate of YV)ma has been obtained, the period of upper bed fluctuation is easily obtainedo The only way that this top slug can disappear is via solids downfcloW. Thi solids dowflow ate is known; it is Qo Therefore the time for disappearance which is the period, is given by (83) t = yom, /e In the example being considered, one finds 8e- 38 Y~~w^x = 86.7$ 34.7 cm hence =r;39/oA 9 = 9 ec which again agrees quite well with the observed period of lo.44 secondso This procedure will be referred to as the shortcut procedureo Figures 37 through 44 show the cu.rves predicted by this

--- ___-_ —__- 0....... -... _............ 5 Q t f.____.X__.___ _ _.._.....__ __..,__ _ |i_ ( -no - - 8 I Z J \ 3S 0 I,,'3 auo,. 105

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113 shortcut procedure for period versus velocityO The observed data are also shown on these plotso This period applies to the upper section of the beds that region bounded by HT and HMo This constitutes from 40 to 95% of the bed, a good average figure being 75%. The remainder of the bed is a region of bubbling and of small slugs. No attempt has been made to correlate these period data since all components of the period, HT1 HM, and Q- have been correlated in previous sectionso The measurement of period was done in two wayso The first was visual observation coupled with a high speed strip chart and markero Each time a maximum was observed in the upper bed surface, the strip chart was marked, An average over 30 or more readings was recordedo The second method was the measurement of peak to peak distances on a pressure-time recording of the upper section of the bed. This type of measurement will be discussed in more detail in the succeeding sectiono Conclusions The good agreement between the predicted and observed periods supports the model used to arrive at the predictiono All the assumptions which go into this estimate of period are thereby verifiedo III Presaure Profileso Prediction from Modelo An acid test of the model is the use of it to attempt a prediction of bed pressure profileso The prediction of configuration

114.erves as the baSis for determining pressureso Let us recall as'sumption 10 of the proposed model This concept is that gas flows thr.h a slug at the velocity for minimum fluidizationo If this is true, then the pressure drop per udiit length of such a slug muqt be the pressure drop per unit length of bed at the point of incipient fluidizationo This fact, coupled with the supported hypothesis (assumption 9) of no pressure drop in a void space, is enough to enable one to calculate pressures from the bed configurationO Let us return to the example begun in the discussion of bed configurationo One more piece of information is necessary; the pressure drop per unit length at incipient fluidization~ p PI 0,o03/7 psi/cm Suppose now that we wish to compute the pressure difference between two taps located 60 cm and 7507 cm above the bed support at a time of 200 seecnds.o Referring to Figure 36, we find that at that instant the entire region between taps is filled with solids in the form of a slugo The pressure differential will therefore be: AP O oa1i7 x 5rs7 o. 343 psi Proceeding in this manner, one can establish the pressure differential between these taps as a function of timeo Suppose that such a pressure versus time recording is availableo A much easier check on the accuracy of the model than working out the bed configuration, and then pressures, is availableo The first check is on the observed frequency of sluggingo This has already been checked and verifiedo The second check is on the slopes of the pressure-time

115 curveso These slopes must fall into three categories for the slugging zoneo They must either arise from an interfacial increase slope of Vxs0 ~, or of s + 9, the only three cases allowed in the.modelo Therefore measurement of the slopes of the pressure-time curve must be one of the following three: i, (VXS AP/L) 20 C SV5+G)(iAP/L) 3. G (LPIL) Measurement of slopes then, in conjunction with period9 provides a check on the modelo In practice the third one occurs only at the extreme bottom of the bed and is not measuredo Experimental Procedureo The apparatus for measuring pressure-time curves had been shown in Figure 15o The only modification used for this work was the addition of more pressure taps, The gases used in these experiments were humidified by bubbling through approximately 10 inches of water in order to minimize electrostatic effectso This technique proved completely effective, The experimental procedure is discussed in the section on the pressure technique for measuring solids downflow rateO Results0 The first result to be discussed is the agreement between the predicted psressuetime curves and the experimentally observed curves for two different gases in a 2000 inch diameter tubeo Figures 45

116 and 46 show predicted and observed pressure-time curves for airsand A in a 2o00 inch tube at a gas velocity of 7000 cm/seco Figure 47 shows the "noise" produced at the bottom of the bed due to the bubbling actiono At this point it would be in order to disgress for a moment and explain a feature of all oscillographic pressure-time records produced in this studyi None of these records. it will be observed, have any specified units for the ordinateo The units are mullivoltSo The characteristic curve for the particular pressure transducer used9 together with the calibratioa of the Sanborn recorder permitted all values taken from these curves to be translated into pressure-time datao These in turn can be converted via the quantity AP/L to velocity data. - 9 vXsg etco This will be done in all cases preceding and succeeding without further comment The transducer characteristics are shown in Appendix Co The Sanborn recorder was alWays calibrated for a sensitivity of 5~00 millivolts per ordinate centimeter Figure 48 shows the results of pressure-time measurements at two other air velocities for the 2o00" bed of 100-150 mesh sando Figure 49 shows pressure records at three elevations in the system of 100-150 mesh sand in a 2o00" diameter bed fluidized with helium at a velocity of 30o2 cm/seco It is seen that these curves are extremely regular0 Figures 50 and 51 show typical profiles of this same system at different gas velocitieso

I ~-I SEC TIME, SECONDS " T F I I I I I I ' ' " I SE TIME, SECONDS --- Figure 45. Experimental and Predicted Pressure-Time Curves Between Taps Located at T5.8 and 120 cm Above Bed Support. Upper Curve is Experimental, Lower is Predicted. System is 100-150 Mesh Sand in a 2.00" Bed at an Air Velocity of 7.00 cm/sec. The Indicated Slope Should Be (vx + e)(^) (See Table XV). Static Bed Height is?4.cm. 117

118 go- - — I SEC TIME.SECONDSdrlISEC TIME, SECONDS -—,* Figure 46.. Experimental and Predicted Pressure-Time Curves Between Taps Located at 60.0 and 75.8 cm Above Bed Support. Upper Curve is Experimental; Lower is Predicted. System is 100-150 Mesh Sand in a 2.00" Bed at an Air Velocity of 7.00 cm/sec. Static Bed Height is 74 cm.

W! 0 I1| 7,ISEC TIME, SECONDS - * Figure 47. Experimental Pressure-Time Curve Between Taps Located at 15.0 cm Above Bed Support and Just Beneath Bed Support. System is 100-150 Mesh Sand in a 2.00" Bed at an Air Velocity of 7.00 cm/sec. Static Bed Height is 74 cm. 119

I.. I — I — - IM, I I I —I -.... -- I TIME, SECONDS — Figure 48. Experimental Pressure-Time Curves Between Taps Located at 75.8 and 120 cm Above Bed Support. System for Upper Curve is 100-150 Mesh Sand in a a 2.00" Bed at an Air Velocity of 10.0 cm/sec. Slope 1 Should be vxs()i Slope 2 Should Be (.v, + e)(Ap/L). Static Bed Height is 74 cm. 120

121 I SEC U) TIME, SECONDS - ~ 0/ T 0' tI) a. 0L; — I -/ —I SEC,1 I, /, I I, i i I TIME, SECONDS -- Figure 49. Experimental Pressure-Time Curves for the System of 100-150 Mesh Sand in a 2.00" Bed at a Helium Velocity of 50.2 cm/sec. Upper Curve is Between Taps 60.0 and 75.8 cm Above Bed Support. Middle Curve is Between Taps 45.0 and 60.0 cm Above Bed Support. The Slope Indicated Should Be (G + Uks)(Ap/L). (See Table XV). Lower Curve is Between a Tap Below the Bed Support and a Tap at 15.0 cm Above the Bed Support. Static Bed Height is 72 cm. &1 1 7 1 1 I&mW G~ ' Bed Height is 72 cm.

122 I —... II '' r... i, I ' i 1 I "I SEC TIME., SECONDS -- - I —1 SEC TIME, SECONDS -- Figure 50. Experimental Pressure-Time Curves for the System of 100-150 Mesh Sand in a 2.00" Bed Between Taps Located 60.0 and 75.8 cm Above Bed Support. The Upper Curve is for a Helium Velocity of 12.7 cm/sec. Slope Should be (9 + vxs)(Ap/L). (See Table XV). Static Bed Height is 72 cm.

123 I SEC OI ^ -- E C..,...,. _ _,.., l,,......., TIME, SECONDS - of 100-150 Mesh Sand in a 2.00" Bed Between hps Located 60.0 and 75.8 cm Above Bed Support. The Upper Curve is for a Helium Velicity of 18.5 cm/sec. The Slpe Should be (i + v a)(Ap/L). (See Table XV). Static Bed Height is 72 cm.

124 Figures 529 539 and 54 show pressurd-time profiles for the system of 100-150 mesh sand in a lo00" diameter bed fluidized with air at three velocitieso These curves are presented for two reasonso First9 they demonstrate the regularity of fluctuations in pressures in these systems and to show how these fluctuations change with elevation in the tubeo Second9 they demonstrate the raw data from which the model-confirming slopes were taken, and how this data varies with gas velocity for different gases and different tube diameterso A summary of predicted and observed values of + Vxs and Vxs fo~{$ll systems studied is given in Table XVo The two values agree quite 'Wll for the quantity Q + vxo for 20O0 inch diameter bedso The few data available for ixs do not check as wello This is at least in part due to the large errors likely to exist in the measurements underlying both values of tse Good agreement is also found for data for the 21o2 cm bed in the loOO inch tube fluidized with air9 and for the 42 cm bed in the 1o00 inch tube fluidized with carbon dioxideo Data for the remaining systems is out of lineo This is due to the frictional effects noticed in the some of these small (lo00 inch) diameter systemso The assumption of no slug friction with the bed wall apparently is not valid for beds with a static L/Dt greater than about 15o This is not the case in the larger diameter (2~00 inch) beds9 where an L/Dt of 15 causes no trouble whatsoevero

.1..1 1I I ISEC TIME, SECONDS --- Figure 52. Experimental Pressure-Time Curves for the System of 100-150 Mesh Sand in a 1.00" Bed at an Air Velocity of 8.7 cm/sec. Upper Curve is Between Taps Located 45.0 and 60.0 cm Above Bed Support. Lower Curve is Between Taps Located 60.0 and 75.0 cm Above Bed Support. Slope 1 ShoUld be v (Ap/L). Slope 2 Should be (vxs)(jp/L). (See Table XV). Static Bed Height is 21.2 cm. 125

....... I I I ~ I I- 1 I. I SEC TIME, SECONDS — Figure 53. Experimental Pressure-Time Curves for the System of 100-150 Mesh Sand in a 1.00" Bed at an Air Velocity of 24.0 cm/sec. Upper Curve is Between Taps Located 60.0 and 75.0 cm Above Bed Support. Slope 1 Should be v (Ap/L). Slope 2 Should be (9 + v )(tIp/L). Lower Curve is Between Taps Lwatea 45.0 and 60.0 cm Above Bed Support. Slope Should be (e + Vxs)(Ap/L). (See Table XV). Static Bed Height is 21.2 cm. 126

00 C) I Id \O EA Js o m 'p \.H'0 ~. Qo rj r4 127 \^~~~~~~~~~~c rt g ^ r~~~~~~~~~~~~~~~~~~~~~~~~~~~ \'' r i0 127

128 TABLE XV Summary of Results of Pressure Profile Measurements Bed Dia. Bed Qas Solid Gas Vgmf vx + Vxs + Vxs cm Height Velocity cmec cmsec cm/sec Pred. Obs. Os. cm cm/sec cm/sec cm/sec cm/sec 5.08 74.0 Air Sand A 7.00 3.14 3.9 22.8 26.7 26.6 7.3 5.08 74.0 Air Sand A 10.00 3.14 6.9 22.8 29.7 28.6 9.6 5.08 74.0 Air Sand A 13.00 3.14 10.9 22.8 33.7 33.2 5.08 72.0 Helium Sand A 12.7 3.60 9.1 23.7 32.8 31.5 - 5.08 72.0 Helium Sand A 16.4 3.60 12.8 23.7 36.5 367 - 5.08 72.0 Helium Sand A 18.3 3.60 14.7 23.7 38.4 38.0 - 5.08 72.0 Helium Sand A 21.4 3.60 17.8 23.7 41.5 42.5 -- 5.08 72.0 Helium Sand A 30.2 3.60 26.6 23.7 50.3 43.3 -- 2.54 21.2 Air Sand A 8.7 1.10 7.6 17.7 24.3 23.0 9.2 2.54 21.2 Air Sand A 24.0 1.10 22.9 17.7 40.6 39.4 24.0 2.54 21.2 Air Sand A 36.4 1.10 35.3 17.7 53.0 52.0 -- 2.54 58.0 Air Sand A 11.8 1.82 10.0 17.7 27.7 >100 40.8 2.54 42.0 002 Sand A 7.0 1.52 5.5 13.8 19.3 22.3 - 2.54 42.0 C02 Sand A 17.5 1.52 16.0 13.8 29.8 32.4 - 2.54 42.0 Helium Sand A 26.7 6.60 20.1 16.1 26.2 58.3 9.8 2.54 42.0 Helium Sand A 43.7 6.60 37.1 16.1 53.2 >300 >100 2.54 33.0 Air Sand B 25.2 2.95 22.2 17.9 40.1 20 * Results of this run were erratic.

129 The effect of wall-slug friction is to create pressure buildups in the bedo This pressure buildup becomes very noticeable in attempts to predict pressures, but has no great effect on the period of fluctuations nor the assumption of linear pressure and level changes in the system. Figure 55 demonstrates this linearity in the worst case encounteredo An attempt to include this pressure buildup in the proposed model is certain to encounter great difficultyo The problem created is the one of unsteady compressible fluid flow through a porous medium of varying length and varying motion, An extremely simplified version of this problem is unsteady flow of a compressible fluid through a porous bed of constant size and possessing no motiono The equation for pressure variation in such a solids plug is Xa - 'at where c is a constant, P is pressure, t is time, and x is lengtho No analytical solution for this simplified problem is available, indicating the hopelessness of engaging with the parent problemo Conclusions The model meets all requirements for the prediction of the shapes and slopes of pressure curves in the bed in the majority of caseso There is an upper limit of static bed L/Dt, at least for small diameter beds9 above which pressure buildup invalidates the use of

EMPTY CHANNEL I IiI SEC TIME, SECONDS -- Figure 5 Experimental Pressure-Time Curve tor the System of 100-150 Mesh Sand in a 1.00" Bed at a Helium Velocity of 4357 cm/sec The Curve is Taken Between Taps Located 90.0 and 105.0 cm Above the Bed Support. Static Bed Height is 42 cm. 130

131 the model for predicting pressure variations. This limit does not greatly affect the prediction of bed fluctuation period. The lower limit of L/Dt appears to be merely the limit for the slugging regimec of fluid bed operation.

EXTENSION OF THE MODEL TO BUBBLING SYSTEMSo A pertinent question isO what value has this model in fluidized systems which are not in the slugging regime? A thorough investigation of this question is beyong the scope of this work,, but a quick check on the applicability of the model to the bubbling regime will (3) be made hereo Baumgarten and Pigford(3) have obtained data on the rate of bubble rise versus bubble diameter in systems which contained much the same solids as the sands used in this studyo The fluidizing gas used was airo Suppose one considers that a bubble is just a slug proceeding up through a column the walls of which are composed of solidso If this is true9 then the velocity of the "void space" should be given by the modelo There is, however9 one important difference between this sort of "slugging" system and the ones studied in this worko It is that the friction between solids and a "column" wall of solids will be very much greater than between solids and a smooth wallo Let us assume that the bubble9 or "slug"9 cannot push solids up the 'column" at allo Then the only way that the bubble can proceed upward is by the transfer of solids vertically downward through the bubbleo This solids downflow rate has been investigate'd'"in this studyo The resultant bubble velocity must be: (85) V8 -= w^ 132

133 Since the effect of tube diameter on 0 has been established, it is no great task to translate the words "tube diameter" to "bubble" diameter" and thus havea-predictioncf bubble velocity versus bubble diameter~ The information of Figure 23 is applicable, Figure 56 shows the curve predicted from this work together with Baumgarten and Pigfordvs datao The scatter in their data is large, but the predicted curve goes through the midst of their data, Thus the scope of this study is not necessarily limited,

134 Q Aqd w o 0;b U) _i O 0 )+)0 i0 kH Q\~ sD - *-0 m ~ x O w w \ dqLJ LJ I.ffi PL, P. z z -. 0. \ 0oc~~~ 0, 0 0 O 0 0 O 0 O )3S/w 3 * A110O13A 318anl ' ^

SUMMARY Let us now re-examine the results of this study, A microscopic approach was tried and abandonedo The useful result is the acquistion of an equation which should predict collision frequencies in gas-particle systemso An unsteady-state macroscopic approach was tried and abandoned. The useful result is the solution of the mass and momentum balances for slugs of changing size and masso A steady-state macroscopic slug model was then proposed. The procedure was then to inve'tigate the necessary areas, verifying assumptions in route, and then to verify the results predicted by the model, The first topic considered was the total bed pressure drop as a function of velocity. The normal "break point" at incipient fluidization was noted in each caseo Pressure drops at this point were observed to be the weight of the bed per unit areaO The pressure drop per unit length of bed was tabulated for future use. Solids downflow rates were checked nexto Bin-flow techniques yielded a solids downflow rate which varied with gas velocity, exhibiting a maximum at the velocity for incipient fluidization. This maximum was subsequently shown to be a good estimate of the single solids downflow rate existing in the operating system0 Pressure measurement techniques showed that a single solids downflow rate exists in each system at all velocitieso This rate was found to be linear in solids density and moderately dependent on 135

136 bed diameter. Gas properties do not appreciably affect this rate, This work verified the assumptions that (1) there is a unique solids downflow rate for a given system for all velocities, (2) gas flows through a slug at the velocity of minimum fluidization, (3) There is negligible:gas pressure drop in void spaces, (4) slugs do not accelerate. Auxiliary measurements verify the assumption that particles fall through void spaces at relatively large velocitieso The system-to-system differences point out the validity of the assumption that only one system at a time may be dealt witho The problem of determining what type of slugs exist at various points in the bed was treated nexta The answer was found in three characteristic elevations in the bed which yielded the necessary informationo The effect of operating variables on these levels was discussedo This work verified, qualitatively via visual observation9 the assumptions of no variation in radial solids concentration and well defined slugso The final preliminary9 the matter of slug generation9 was treated in a manner suggested by the experience of others0 The process is a reasonable representation of actual eventso None of this work guaranteed the accuracy of the model. This was tested in three wayso Fluctuation periods were observed to agree well with the predictions for all systems studiedo Internal bed pressure changes were predicted well for the majority of caseso

137 In a few extremes of low diameter and large static bed height, the assumption of no wall friction was observed to be invalido A third, and very important, verification was found in the fact that bubble velocities could be predicted from bubble diameter within the accuracy of the available data. Thus the proposed model appears reasonable in most respectso

APPENDIX A Details of the Development of Average Collision Frequency~ The assumed forms of the distribution functions for the random cartesian components of velocity are. - vx /a dx (85) P(V)dv e v (86) P(Vy) dVy l/yar ev / Y Vdy (87) P(v )dv l /o% \0 e -dv/ \. The distribution function for particle speed c is then obtained as follows: (88) P(vxvv), v)dv vyd v - P ) P(v ) P v) dvx dvy dv, Introducing spherical coordinates: (89) V\ C COSe Sin ) (90) Vy = C s 6 SinS (91) V = C os ( (92) dVdv dv d c~ si;nr d /dedc If one also assumes that (93) = 0( 02 138

139 _ c si' 4,+ (ccosd-.) then - + d (94) P(C, e ) ) Te One then integrates over the entire ranges of Q and ~ to remove angular dependence and obtain speed distribution: (sa)Pc s + CCc cos:,);r P^cjV, ^ ye/Ha dC J r (95) 3 s ); ode (96) 6 (C Cy Thus one angular dependence is removedo _ [- C S" si + (C co - (97) P(C) ') (98) p C - Ce a slPnh( / ) The remaining angular dependence has been removed and the speed distribution obtained. Now consider a partitle moving with constant speed c through a collection of identical and stationary particleso The particle will execute a series of zig-zag motions, moving in straight lines between encounterso If one imagines this trajectory as the axis of a cylinder of radius equal to the particle diameters, then any particle with its center inside this cylnder will collide with the moving particleo The "collision volume" swept out per unit time will then be equal tor D.o If the number density of particles is Np, the

140 collision f requency will be given by: (99) -C -= T Dp c Np The assumption underlying Equation (99) is that the particles are spheres of identical diameter. A more realistic case may readily be handled; namely, when the particles are spheres with a normal distribution of diameters. One assumes a distribution function of the form: -(Dp- Op)/D a (100) P(Dp)= I/o H e One can now consider a moving particle of diameter Dpo-The "collision volume" swept out will be 7(Dp+Dp)Co However, since half the encounters will be, on the average, with particles smaller than t, the collision frequency will be (101) 7c(Dp) F TT(DIp+p) NpC If this collision frequency is averaged over the entire distribution of particle diameters, one obtains cns,.. (qo-D~p! (103) 2N, c p r or For simplicity, let (104) C) + ] ~ ~',

141 Then (19) may be written (105) RC Np c T d P One must now average this collision frequency over the speeds of particles colliding~ First, consider a particle moving with speed cao Now, instead of the remaining particles being stationary, let them move with some speed c2, It can be shown (4) for this case that: (106) ic NpiTr Vr (107) Vr-= CQ + Ca/ C, C4 C2 (108) Vr C; - C /3C; CI< C Consequently, if we permit the considered particle to move at c1 and allow a distribution of speeds c2 according to Equation (98), the collision frequency will be given by: i ( C (C +W/) C A-r"a - -L r'tC C3 C This is the expression for the average collision frequency of a particle moving at speed c1 with particles moving at distributed speeds C2. The integration in Equation (109) is long and involved. However the result may be obtained analytically, and has the following form:

(o10) me e^(f37^f ( ( at ) it 3(y)]gerf(a+ e LPfU)3 e - )+i where Cl One more step remains before the problem of interest is solved, The speed of the considered particle is not constant but is also distributed according to Equation (98). The desired collision frequency is obtained by averaging the frequency given by Equation (110) over all speeds cl, or corresponding values of o (n11) = S P(ci)dc| c0 The procedure is to change variable from cl to E and substitute expression (110) for iC o The hyperbolic sine in the distribution function is expanded into the sum of two exponential termso This results in the production of 16 complex integrals (eight terms in C times two terms in P(C ))which can be evaluated analytically. It would perhaps be of interest to demonstrate one such integration: (112) I4 = e d~ 4 ea

143 change variable to Q( = + ~t (113) 4 e Z c o change variable to (3 = \a o 00 23f (114) 4 \ fd e rf = 3LJ erf (115) 14 ( a ( This is the simplest of the integrations. The result for average collision frequency is of the form: (116) 4CTr Vp,*d A check on the accuracy of this result is provided by the fact that the limit of 2 as 0 vanishes, or in other words, as the non-random component of vertical velocity approaches zero, is precisely the result demonstrated for pure random motion by the kinetic theory of gases(4)

APPENDIX Bo Solution of the mass and momentum balances for a slugo Case lo No solids inflow or outflow. The solution of the mass balance is trivial: (117) y = (118) y o) = yo (119) Y yo The solution of the momentum balance is also quite easyc (120) X = C Vs v - V- mf (121) K(o) Xo (122) X(o) = o bK where C In this case the solution is trivial: (123) X X/ + ct + t5 V -Vf The second form of the momentum balance is (124) i 4 AX Ad vg Vs V 5Vf (125) (0) = X (126) X ( ) = Yo where A '- Laplace transforms are usedo 144 flLU,

145 The transformed equation is: (127) SaX() - s-S A SX(S)-x,] -= (Av^ - ) Solving for x(s): (2 ( ) OiSX,(S(AVX-9)( S (S+A ) Taking the inverse transform, c ~t A ( -At (129) = X e At)(7L A(et Vs > V-j Vwvf Equations (119), (123), and (129) are the solutions to the mass and momentum balances for a Case 1 slugo Case 2< Solids outflow only. The solution of the mass balance is obvious: (130) y= - (131) \/(O = YO (132) y = yo-<^ The momentum balance in the first instance is: (133) >X g c4 vs v- F (134) X () = (135) Xy ) = Xb Substituting y from mass balance: (136) X) C + ( ecfa C(Y,,- et

146 Using direct integration once: (137) X-X Ct- C y Yo Integrating again: (138) X0 X,6^^tS Ctl. tt -+ It Vs ^N Vwhich is the desired result. The momentum balance in the second instance is: (139) X!-fAA +^ Vs -f (140) X(o)= = O (i4) X(o) = X The solution can proceed as follows# Define (142) P Then the equation is reduced to (143) +Ap Av3-94 y (144) p(O)= P = The solution of this first order linear equation is: At J" I 3 At / (145) t! Av 4;I t + cu5) pe = \JA^"9-dt + c or At pit /Ava \ At C ^h (146) pt ^ A 3

147 Which upon evaluating the integral becomes:, AB //~y"- At 4 A( (1/+7) A C 4 5 eI/ n n "+CL Using the boundary condition (144) to evaluate C.. one obtains: At At ('148) + 9 Ao II MT / I Now one may solve for x by noting (149) X y = p Performing this integration, one obtains: -At ~] (Iro)0 e, A K ) V. -C ye~~en. Which upon integration of the last integral yields r-AO -A( O)At(151), X Y L['Y s] t 3t[ A][- tj Vs V M, f L4.2 eh) Euqations (132), (8) an (151) are the solutions to the mass and momentum balances for a case 2 slug, Case 3. olids inflow only4 The solution of the mass balance is elementary: (152) Y

148 (153) y() = y (154) y= y e- &t The momentum balance in the first instance is: (155) V+ C- - LV- V m (156) K(o) X (157) X(o) = / The solution proceeds as followso Define (158) p X Substituting, the momentum balance becomes: (159) p p C + ^ 'SG (160) p(o) = pO - Xo, Solving the first order linear differential equation, one obtains: (161) yp = - e(6t-+5,e)t + Evaluating C, from boundary condition (44) and substituting: (162) p 1 +~ cLY d r 4 7 e,+c, et, Y Noting thiat (163) x = x + p dt 0 one obtains (164) + = (a +[ -,.,oSp ]g - -S — V-o) y 4 e ^ \-^

149 which is the desired result. The momentum balance in the second instance is: (165) X+ - A $)X A)- ( -g + \/ (166) \/ XV - V 9A{ (167) X(o)= Xo Again, define (168) p X Whereupon the momentum balance is reduced to (169) A)p A(V)- ( (170) p(o)= pc = Xo To solve this first order linear differential equation, make a change of variable from to to y: (171) y= Yo + et (172) dy= edt (173) p = = ex = p (174) = = 0 /X e p/ Then Equation (169) becomes: (175) (A- ) p - D+ - where dx = P dy and D = A(VgtO)- 9

150 and E C Il + 05' Solving (175) one obtains: (176) pa ee M eX c/ Using boundary condition (170), with variable changed: (177) p (yo) _ = X/ One obtains C and substituting into (176) the result is A (y-yo) (178) P -Xf ( = a )-F e.... + P w- %6 A D One final integration yields x: (179) X =d The result is: (180) x- XX eY - 0 _ 4 - 0 LeA ye )KJL Yo Vs > VVgVmf Case 4 o Solids inflow and solids outflow The mass balance may be solved by inspections (l81) y (182) =(o)= y

151 (183) 7 - ~ o The momentum balance in the first instance is, after substituting the solution for the mass balance: (184) Yo X + g X Cy, ) v Vcif& (185) (0) Ro (186) X/() = Xo Rearranging and taking the Laplace transform: (187) = ( +) = x C (88) S X(SJ- - S Xo S Xsx(s) SX Solving for x(s): (189) X(S) yo + S o s ' -+v S Taking the inverse transform: (190) X X + t Ye vs - M T where c0 = Ct + ) Thus the first instance is solvedo The momentum balance for the second alternative, after substitution of the mass balance, is: (1) X +(A^ ) - A - T +y + Vs - f > va r

152 (192) X(0) 0 xo (193) X( ) ( X= This equation, (191), is seen to be precisely similar to Equation ^9 (187). Therefore, if one substitutes for in Equation (190) the quantity (Aan ), d for o, the result will be the desired solution: (194) X / ) X, + zo/m IL C YeI - (A+Q)~ e) c~n, 7e.) L + e /Yd O e J where (mom ^Q g+^ F Vs^-lgf Equations (183), (190), and (194) represent the solutions of the mass and momentum balances for Case 4o In all four cases, the solutions of all equations have been checked bjy differentiation,

APPENDIX C 10 Calibration of Pressure Gage C2-174 -Figure 57 shows the calibration curve. This pressure gage was used to measure gas rotameter pressure, II Pressure Pickup Calibration,. Six pressure pickups were used to measure pressures in the fluid bed. Figures 58 through 63 show their calibration, These calibrations all involved a simultaneous calibration of pickup and loscillographic recorder. III Rotameter Calibrations,. The Manostat- company publishes a set of curves and data which are supposed to yield an accurate a priori calibration of their instruments, "Predictability" flowmeters0 It was deemed wise to check their data with one water displacement calibration, Figure 64 shows their predicted calibration and my data. It was judged from these results that the prediction could be trusted0 153

154 "\\~~~~ (Uw~~~(r 0 0 \ Q W 0 \ 0 \I~d '38nSS311d 3ma d~-0 VU U 915d ' 3urt$S3~ld 3N~IJ. 'd ~I~d '3~IflSS3~Id 3A8.L Qci

155 0 o rd....._..........I........ "\'~~~~~c ' \f, <<Hd wa,.,._...._.............................\ _. - 0 V -,~ d,..... _....... o N CM N N - - 6 6 6 NO ' ONIQV3I 3aQd0U3do ' crj ~ ~ ~ ~................. ~.................103ll

156 0 E X\o'0 d 0 \i ~U | ~ a. I\I I I I I T\C l \O 'WaV3N 8a 8a- ^;____N I____ o. a o N N -N - 0 o flD'9NIav38 83G80038' ^o'O~~~~~~~~iovad dsayooad^~~~~~~~~~~~~~~~~~r

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158 0 + A d T X ~ x o r 0 o - \ --- —3'~~~~~~~~~~ ' 0 c5 ) IN::)' 9NIQV3\ Q3Q~O::D3a OD 0 _ _ _ _ _ _ _ _ _G^ _ _ _ _ _ _ 5 I ' cJ~~ ~ ~ ~ ~ ~~~~~~~ *Hi ------ -A —N1~~~ 30)03t

159 Go ____ __________________ 0 - - -- - -l -_ _ —__ _ 6 -— ~ — - — _ -- 6^ P4 _ _ _ _ _ _ _-\~~~_ _ ____ 0 \~~~~~~~~c A ci -H P4 a. I ---------- V --- —-— d ie ^ $4 w C 0 0 w _r. 0 cik __ a 0. ~ C\~~ 0 L --- --- --- --- -- -- ---- --- --- 0 — 0 — -- -- -- -- no ~ DNIGV3\1 ~i3QIi O \ ^ o. ^~~~ __ __ __ __ __ __ __ __ __ __ -^ __ __ __ __ __ ^ ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r ____ ^ 0~~~~~~~~~~~~~~~~~~~~~~~~~C - - - _ - _ _ _ _ _ _ ^ - _ _ _ _~~~~~~~~~~~~~~~~~( ___ ___ ___ ___ ___ __ ___ ___ ___ ___ __ \ ___ ___ __ ___ ~ ^ -----------— ^ ---- ^ ^d ^ ^~~ --- -- -- -- -- -- -- --- -- -- -- --.^ -- -- - -- --,c3 c3 d d V _ _ _ _ _ _ _ __ _ _ _\ - _ _ ' ^0P3 J3 t0 31 '~

o,d '~ ~ a.~. \... W o i<< t<m m - 0 \ I.. y..... ' N": - 0 0.: i~ 9NIOV3T' E3:1< 0O ':,160 160

161 o P k~~~ 0~1*0 Wa: a. w -, 0 I I I I I I < TL o 0 r l 0.TII<X AL LIIO I- - 0 X I' 0 -- _ __ _ _ _ _4 - - I ()., 1^ od ~ - 0 8 N OH 8 8llOd 8N 0 9H WI09. O ]NV -Io02. IV NIIN/;I) ' ~O

APPENDIX Do Response of pressure measuring system, The times involved in the pressure variations measured were.small enough to cause worry about the dynamic response of the measuring systems Therefore several checks were made on the response of the pressure measuring system* I, Response of measuring system to step pressure inputs. Equipments The apparatus used for applying a step input to t1e measuring system is shown in Figure 65. An air line was led through a valve into a 28-liter tank. The tank pressure was read from ea manometer. An outlet line led from the tank to a two-way solenoid valve. One outlet of this solenoid valve was vented through a valve to the atmosphere. The other outlet was tied directly to the measuring system used for pressure determination.. The solenoid valve was actuated by a switch which, when closed, also caused a mark to be made on the margin of the oscillographic record. Procedure. In order to put a step impulse into the system, the tank was pressurized to the desired level. The vent of the solenoid was closed for this purpose. The solenoid was switched on and a record of the pressure change was made. 162

163 0 I I — E I 811 w U) 4, i. E E Q_ PI

164 Result S Figure 66 shows a typical response curve. The usual exponential curve is observed. The lag observed is about 0,05 seconds. Conclusions. The system responds satisfactorily to step pressure impulses. II Response of measuring system to ramp pressure inputs. Equipment0 The apparatus of Figure 65 was used, Procedure. The vent of the solenoid was openbd and a reasonable air flow rate established through the tank and solenoid. The solenoid was then switched to direct this flow into the measuring system by closing the vent and opening to the measuring system. A record of the response was made. Re sults, Figure 67 shows a typical response curve. A lag of about 0,03 seconds is found in the response. Conclusionj, The system responds satisfactorily to ramp pressure input changes,

P --- TT -' I ' I TIME SECONDS --- Figure 66. Response of Redord-ig System to a Step Pressure Increase, 165

EMPTY CHANNEL 1 SEC I I, I, I,1 I... 4* --- TIME, SECONDS Figure 67. Response of the Measuring System to a Ramp Pressure Input. 166

167 IIIo Response of measuring system to sinusoidal pressure inputso Equipmento A large diameter flexible tube was filled with mercury and one end attached to the measuring system~ The other end was attached to an eccentric shaft on an electric motoro When the eccentric was at the high position, it tripped a switch which marked the margin of the record, The resulting pressure input was very nearly sinusoidalo The apparatus is shown in Figure 68. Procedureo The motor was turned on and the variations in pressure recordedo Figure 69 shows an example of the results of these measurements. The mark at input maximum corresponds well to the output maximumn A lag of less than Oo02 seconds is observedo Conclusions ~ The system responds satisfactorily to sinusoidal pressure input s

168 -P) Ir ZW Cd a O,4 CQ Jo r- --- -I~ --. C,)u ~~~~~~~~~~~~~~~~~~~~~~~~~~Qa a) o^ Vi a F^l

I I II I [ EMPTY CHANNEL I I,.,,I I I SEC I TIME, SECONDS D Figure 69. Response of the Measuring Syststem to a Siuusoidal Input. 169

170 ^^ Ealibration ertiticate FOR PRESSURE PICKU.P TYPE _ -3/~_ SERIAL NUMBER - /.. PRESSURE RANGE / PSI D DATE //- EXCITATION K 0 VOLTS D-C OR A-C RMS TEST TEMPERATURE SENSITIVITY: MILLIVOLTS FULL RANGE,,lo 5 ~. -/1'. ~.Y76 LINEARITY: % FULL RANGE /7! i HYSTERESIS: % FULL RANGE ZERO SHIFT WITH TEMPERATURE —, JL % FULL RANGE/~F INPUT IMPEDANCE J__ __ OHMS r- - g - - -i -- r- - - v -i.. I OR I r. r,o ] I 11 OUTPUT IMPEDANCE -3 '-' OHMS f{, ^> —L^^sl~l 1N~UI g | piIu i ' - -i Sr I) K NPUT | -PICKUP - L-I a+ OR-D SIGNED. 2-OR OR lUAITY CONTROL ENGIEGNEER BLUE YELLOW OUTPUT Consolidated Electrodynamics CORPORATION 2083 Transducer Dlvision MONROVIA, CALIFORNIA

BIBLIOGRAPHY lo Bailie, Ro E,, Liang-Tseng Fan and J, J. Stewart, Industrial and Engineering Chemistry, 53, 567-69 (1961)o 2, Baron, Thomas, and Ro A. Mugele, "Quality of Fluidization," Shell Development Company, Emeryville, Californiao 30 Baumgarten, PO Ko, and Ro Lo Pigford, AIChE Journal 6, 115 (1960)o 4o Benson, S, WO, "The Foundations of Chemical Kinetics," McGrawHill, New York, 1960. 5o Dotson, Jo Mo, AIChE Journal 5, 169-74 (1959)o 60 Duncan, Ao Jo, "Quality Control and Industrial Statistics," Richard Do Irwin, Inco, Homewood, Ill. 1955. 70 Gregory, So Ao, Journal of Applied Chemistry 2, supplo issue 1, si (1952). 80 Kelley, Ao Eo, Petroleum Engineer, 16, 136 (1945)o 9o Leva, Max, "Fluidization," McGraw-Hill, New York, 1959. lOo Morse, Ro Eo and Co Oo Ballou, Chemo Engr. Progro, 55, 49 (1959)o llo Shuster, No Wo and PO Kisliak, Chemo Engro Progro, 48, 455-8 (1952)o 120 Zenz, Fo Ao and Do Fo Othmer, "Fluidization and Fluid Particle Systems," Reinhold, New York, 1960o 171