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Spouted bed and spout-fluid bed hydrodynamics in a 0.91 m diameter vessel He, Yan-Long 1990

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SPOUTED BED AND SPOUT-FLUID BED HYDRc, 0.91 M DIAMETER VESSEL By Mr. Yan-Long He B. A. Sc., East China Institute of Technology, 1982 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S C H E M I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A February 1990 © Mr. Yan-Long He, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Experiments were conducted in a 0.91 m diameter half-cylindrical spouted bed/spout-fiuid bed column equipped with a 60° conical base and semi-circular inlet orifice diameters of 76 to 140 mm. Three particulate solid materials were studied: 3.25 mm polystyrene, 4.72 mm brown beans and 6.71 mm green peas. Beds with static depths of 0.55 to 2.60 m were contacted with air, both in the standard spouted bed and the spout-fluid bed mode. The dependent hydrodynamic parameters studied included minimum spouting ve-locity, maximum spoutable bed height, spout shape and diameter, fountain height, dead zone dimensions, overall bed pressure drop, fluid distribution in the annulus, longitudinal and radial pressure profiles in the annulus, and regime maps for the spout-fluid bed. Correlations for minimum spouting velocity developed on smaller vessels generally gave poor predictions for the large diameter vessel employed in this work and failed to predict the observed dependence of U m s on the static bed height. The empirical corre-lation due to McNab (1972) was found to predict the average spout diameter very well for standard spouted beds, while the correlation due to Hadzisdmajlovic et al. (1983) gave a reasonable prediction for spout-fluid beds. Substantial dead zone regions where particles were stagnant were observed in the lower portion of the vessel. The Littman et al. (1977) equation overestimated the maximum spoutable bed height, while the McNab and Bridgwater (1977) equation gave a value which appeared to be far too high. The observed fountains were extremely dilute, and their heights always exceeded the corre-sponding static bed heights for the conditions studied. The Epstein and Levine (1978) equation gave good estimates of overall bed pressure drop. n The longitudinal fluid velocity in the annulus was well predicted by the modified Lefroy-Davidson (1969) equation due to Epstein et al. (1978) and was reasonably pre-dicted by the Mamuro-Hattori (1968) model in the cylindrical portion. However, both equations gave poor predictions in the conical base portion. In the conical base section, the Rovero et al. (1983) equation predicted the correct trend, but consistently overesti-mated U Q by a considerable margin. Both the Epstein and Levine (1978) equation and the Lefroy and Davidson (1969) equation were found to be in good agreement with the experimental longitudinal pressure profiles. The radial distribution of pressure in the annulus for any bed level was observed to be nearly constant when there was auxiliary flow. A computer model based on the Ergun equation gave useful qualitative predictions of the fluid flow distribution in the annulus. Four fairly distinct flow regimes were delineated in this work for cases where there were auxiliary air flow: (1) spouting-with-aeration; (2) spout-fluidization; (3) submerged jets, slugs and bubbles in fluidized bed, and ( 4 ) packed bed. The minimum total fluid flowrate for spouting-with-aeration always exceeded the minimum spouting flowrate, but was smaller than the minimum fluidization flowrate. The minimum total fluid flowrate for spout-fluidization was found to be equal to the minimum fluidization flowrate. in Table of Contents Abstract ii List of Tables ix List of Figures - xii Acknowledgment xx 1 Introduction 1 2 Literature Review 6 2.1 Minimum Spouting Velocity 6 2.1.1 Mathur and Gishler equation 7 2.1.2 Fane-Mitchell modification of Equation (2.1) 7 2.1.3 Correlation of Grbavcic et al 8 2.1.4 Littman-Morgan equation 9 2.1.5 Wu et al. modification of Equation (2.1) 10 2.2 Maximum Spoutable Height 10 2.3 Average Spout Diameter 12 2.4 Fountain Height 13 2.5 Dead Zone Boundary 13 2.6 Fluid Distribution in the Annulus 13 2.6.1 Longitudinal annulus fluid velocity 13 2.6.2 Radial profiles of annular superficial gas velocity 15 i v 2.7 Pressure Profiles in the Annulus and Overall Bed Pressure Drop 16 2.7.1 Longitudinal pressure profiles in the annulus 16 2.7.2 Radial pressure profiles 17 2.7.3 Overall bed pressure drop 18 2.8 Regime Maps and Minimum Fluid Flowrate for Spout-Fluid Bed . . . . 18 2.8.1 Regime maps 18 2.8.2 Minimum fluid flowrate for spouting with aeration 22 2.8.3 Minimum total fluid flow for spout-fluidization 23 3 Experimental Equipment and Particulate Materials 24 3.1 Experimental Equipment 24 3.2 Particulate Materials 33 4 Minimum Spouting Velocity, Spout Shape and Diameter, Dead Zone Boundary, Maximum Spoutable Bed Height, Fountain Height and Over-all Pressure Drop 35 4.1 Measurement Technique 35 4.1.1 Minimum spouting velocity 35 4.1.2 Spout shape and diameter 35 4.1.3 Dead zone boundary 36 4.1.4 Maximum spoutable bed height 36 4.1.5 Fountain height 36 4.1.6 Overall bed pressure drop 36 4.1.7 Points of entry for auxiliary air flow 37 4.2 Results and Discussion 37 4.2.1 Minimum spouting velocity 37 4.2.2 Spout shape and spout diameter . 43 v 4.2.3 Dead zone boundary 50 4.2.4 Maximum spoutable bed height 60 4.2.5 Fountain height 61 4.2.6 Overall bed pressure drop 61 5 Gas Flow Distribution and Pressure Profiles in the Annulus 67 5.1 Measurement Technique 67 5.1.1 Vertical component of gas velocity in the annulus 67 5.1.2 Pressure profiles in the annulus 67 5.2 Results and Discussion 68 5.2.1 Vertical component of gas velocity 68 5.2.2 Pressure profiles in the annulus 83 5.3 A Mathematical Model for Predicting the Gas Flow Distribution and Pres-sure Profile in the Annulus 95 5.3.1 Introduction . 95 5.3.2 The mathematical model 95 5.3.3 Method of solution 101 5.3.4 Computing method 109 5.3.5 Results and discussion 110 6 Regime Maps and Minimum Fluid Flowrate for Spouting-With-Aeration and Spout-Fluidization 116 6.1 Preliminary Measurement of Minimum Fluidization Velocity (Umf) . . . 116 6.2 Experimental Techniques 116 6.2.1 Transition from spouting-with-aeration to fixed packed bed . . . 117 6.2.2 Transition from spouting-with-aeration to spout-fluidization . . . 117 vi 6.2.3 Transition from spout-fluidization to jets, slugs, and bubbles in fluidized bed 119 6.3 Results and Discussion 119 6.3.1 Regime maps 119 6.3.2 Minimum fluid flowrate 131 7 Conclusions and Recommendations for Further Work 135 7.1 Conclusions 135 7.2 Recommendations for Further Work 138 Notation 139 Bibliography 144 Appendices 149 A Pressure Drop vs Superficial Velocity for Polystyrene Particles 149 B Experimental Velocity Profile Data 151 C Experimental Pressure Profile Data 157 D Derivation of the Velocity Field Equation from the Ergun Equation 163 E Derivation of Equation (5.10) From Equation (5.6) 164 F Derivation of Equations (5.20) and (5.23) 166 G Derivation of the Pressure Field Equation from the Ergun Equation 168 H The Computer Program 170 vii Predictions from the Mathematical Model 191 Experimental Data for Minimum Spouting Velocity, Spout Shape, Spout Diameter, Dead Zone Boundary, Maximum Spoutable Bed Height, Fountain Height and Overall Pressure Drop 204 T i l l List of Tables 3.1 Properties of particulate materials 4.1 Experimental spout diameters and predicted values from two correlations. See Table 3.1 for particle properties. 4.2 Experimental overall bed pressure drops and predicted values from Equa-tion (2.32). See Table 3.1 for particle properties 4.3 Effect of increasing gas velocity on the bed pressure drop for polystyrene particles, H = 2.00 m, dp — 3.25 mm, Di = 88.9 mm 4.4 Effect of auxiliary flow on the bed pressure drop, Di = 88.9 mm. See Table 3.1 for particle properties 5.1 Experimental values of annulus gas velocities and extrapolated UaHm-(Particles: polystyrene, dp = 3.25 mm, Di = 88.9 mm, H — 2.0 m, U/Um. = 1.1) 6.1 Values of empirical constant A and slopes (K) of the hne A - H . . . . 6.2 Values of minimum total flow for spout-fluidization B . l Longitudinal annulus velocity profile for the polystyrene particles. (U/U, 1.1, dp = 3.25 mm, A = 88.9 mm, H = 2.0 m, ga = 0.0 m3jh) . . . . B.2 Longitudinal annulus velocity profile for the polystyrene particles. (U/U, 1.3, dp = 3.25 mm, A = 88.9 mm, H = 2.0 m, qa = 0.0 rn3/h) . . . . B.3 Longitudinal annulus velocity profile for the polystyrene particles. (U/U, 1.5, dp = 3.25 mm, A = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) . . . . B.4 Longitudinal annulus velocity profile for the polystyrene particles. (U/Ume = 1.7, dp = 3.25 mm, D,- = 88.9 mm, H = 2.0 m, qa = 453 m3/h) 155 B. 5 Longitudinal annulus velocity profile for the polystyrene particles. {U/Ums = 2.05, dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, qa = 721 mzjh) 156 C. l Annulus pressure profile for the polystyrene particles. (U/Ums = 1.1, dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, qa = 0.0 m 3 / / i ) 158 C.2 Annulus pressure profile for the polystyrene particles. (U/Ums — 1.3, dp = 3.25 mm, £>,- = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) 159 C.3 Annulus pressure profile for the polystyrene particles. (U/Umt = 1.5, dp = 3.25 mm, £>,- = 88.9 mm, i f = 2.0 m, qa = 0.0 m 3 / / i ) 160 C.4 Annulus pressure profile for the polystyrene particles. (U/Ume = 1.7, dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, qa = 453 m 3 / / i ) 161 C.5 Annulus pressure profile for the polystyrene particles. (U/Ums = 2.05, dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, qa = 721 m3/h) 162 J . l Experimental and predicted minimum spouting velocities for polystyrene particles, dp = 3.25 mm, D{ = 88.9 mm 205 J.2 Experimental and predicted minimum spouting velocities for brown beans, dp = 4.7 mm, D{ = 88.9 mm 206 J.3 Experimental and predicted minimum spouting velocities for green peas, dp = 6.7 mm, D{ = 88.9 mm 207 J.4 Spout diameters for polystyrene particles, dp = 3.25 mm, Di — 88.9 mm, U/Ume = 1.1 208 J.5 Spout diameters for brown beans, dp — 4.7 mm, Di — 88.9 mm, U/Ums = 1.1209 J.6 Spout diameters for green peas, dp = 6.7 mm', D; = 88.9 mm, U/Um, = 1.1 210 J.7 Spout diameters for polystyrene particles, dp = 3.25 mm, Di = 88.9 mm, H = 2.00 m 211 J.8 Spout diameters for spout-fluid beds with = 88.9 mm for polystyrene (dp = 3.25 mm) and brown beans (dp = 4.7 mm) 212 J.9 Dead zone boundaries as viewed from front plate for polystyrene particles, dp = 3.25 mm, Dt = 88.9 mm, U/Um. = 1-1 213 .1.10 Dead zone boundaries as viewed from front plate for brown beans, dp = 4.7 mm, Di = 88.9 mm, U/Umt = 1.1 214 J.11 Dead zone boundaries as viewed from front plate for green peas, dp = 6.7 mm, Di = 88.9 mm, U/Ume = 1.1 215 J.12 Dead zone boundaries as viewed from front plate for polystyrene particles, dp = 3.25 mm, Dt = 88.9 mm 216 J.13 Dead zone boundaries as viewed from rear column wall for polystyrene particles, dp = 3.25 mm, Di = 88.9 mm, U/Uma = 1.1 217 J.14 Dead zone boundaries as viewed from rear column wall for brown beans, dp = 4.7 mm, D{ = 88.9 mm, U/Um, = 1.1 218 J.15 Dead zone boundaries as viewed from rear column wall for green peas, dp = 6.7 mm, D{ = 88.9 mm, U/Um. = 1.1 219 J.16 Dead zone boundaries as viewed from front plate for spout-fluid beds, polystyrene particles, dp — 3.25 mm, Di — 88.9 mm, H=1.51 m 220 J.17 Dead zone boundaries as viewed from rear column wall for spout-fluid beds, polystyrene particles, dp = 3.25 mm, Di = 88.9 mm, H=1.51 m. . . 221 J.18 Dead zone boundaries for polystyrene particles, dp = 3.25 mm, Di — 88.9 mm, H=1.51 m 222 xi List of Figures 1.1 Schematic diagram of a spouted bed. Arrows indicate direction of solids movement 2 1.2 Schematic diagram of a spout-fluid bed. Arrows indicate direction of solids movement 4 2.1 Geometric parameters used in the cone-modified Mamuro-Hattori model 15 2.2 Radial profiles of annular superficial gas velocity for polystyrene beans with Di = 92 mm, U/Ums = 1.1, due to Lim and Grace (1987) 16 2.3 Regime map due to Nagarkatti and Chatterjee (1974). (Particles: glass beads,-p p = 2410 kg/m3, H = 0.185 m, dp = 0.60 mm, D{ = 5.0 mm, Dc = 90 mm, fluid used: air) 19 2.4 Regime map due to Vukovic et al. (1982). (Particles: calcium carbonate, pp — 2600 kg/m3, dp = 1.8 mm, Dc — 70 mm, Di = 15 mm, H = 1 m, fluid used: air) 20 2.5 Regime map due to Sutanto et al. (1985). (Particles: polystyrene, Di — 19.1 mm, dp = 2.9 mm, pp = 1040 kg/m3, Dc = 0.15 m, H = 0.60 m, fluid: air) 21 3.1 Details of the conical base cylindrical half-column. (All dimensions in mm) 25 3.2 Details of the semi-circular orifice plates 27 3.3 Central air inlet orifice configuration.(All dimensions in mm) 28 3.4 Conical base section. (All dimensions in mm) 29 xii 3.5 Expanded cone showing hole pattern for auxiliary flow. Each row of orifices corresponds to a separate subdistributor. (All dimensions in mm) . . . . 30 3.6 Schematic diagram showing overall equipment layout 31 3.7 Photograph of the overall equipment layout 32 3.8 Photograph of particulate materials 34 4.1 Experimental values of minimum spouting velocity for polystyrene par-ticles, dp — 3.25 mm, Di = 88.9 mm, Dc — 0.91 m, compared with predictions from some correlations 38 4.2 Experimental values of minimum spouting velocity for brown beans, dp — 4.7 mm, Di = 88.9 mm, Dc — 0.91 m, compared with predictions from some correlations 39 4.3 Experimental values of minimum spouting velocity for green peas, dp = 6.7 mm, Di = 88.9 mm, Dc = 0.91 m, compared with predictions from some correlations 40 4.4 Natural log of C/ m g versus natural log of H 42 4.5 Spout shapes: (a) polystyrene, dp = 3.25 mm, Di = 88.9 mm, U/Ums — 1.1; (b) brown beans, dp — 4.7 mm, Di — 88.9 mm, U/Ume - 1.1; (c) green peas, dp — 6.7 mm, Di — 88.9 mm, U/UmB = 1.1 44 4.6 Influence of flowrate on spout shape for polystyrene particles, H = 2.00 m, dp = 3.25 mm, D{ = 88.9 mm 45 4.7 Influence of auxiliary flow on spout shape in spouting with aeration regime for polystyrene particles, H = 1.51 m, dp = 3.25 mm, Di — 88.9 mm, Q = 772 m3/h; (a) configuration #1; (b) configuration #2 46 xm 4.8 Influence of auxiliary flow on spout shape in spout-fluidization regime for polystyrene particles, H =•- 1.51 m, dp — 3.25 mm, D,- = 88.9 mm, Q = 772 m3/h; (a) configuration #1; (b) configuration #2 47 4.9 Influence on spout shape of auxiliary flow with configuration #1 for brown beans, H = 1.28 m, dp = 4.7 mm, Di = 88.9 mm, Q = 798 m3/h; (a) spouting-with-aeration regime; (b) spout-fluidization regime 48 4.10 Photograph of spout shape for polystyrene particles with auxiliary air via configuration #3, H — 1.51 m, dp = 3.25 mm, Di — 88.9 mm, U/Ume = 1.96, Q = 460 m3/h, qa = 605 m3/h 49 4.11 Dead zone boundaries viewed from front plate: (a) polystyrene, dp = 3.25 mm, Di = 88.9 mm, U/Ume = 1.1; (b) brown beans, dp =. 4.7 mm, Di = 88.9 mm, U/Ums — 1.1; (c) green peas, dp = 6.7 mm, Di = 88.9 mm, U/Um. = 1.1 54 4.12 Influence of superficial gas velocity on dead zone boundary for polystyrene particles viewed from front plate, H — 2.00 m, dp = 3.25 mm, Di = 88.9 mm 55 4.13 Dead zone boundaries viewed from rear column wall: (a) polystyrene, dp = 3.25 mm, Di = 88.9 mm, U/Umt = 1.1; (b) brown beans, dv = 4.7 mm, Di = 88.9 mm, U/UmB = 1.1; (C) green peas, dp — 6.7 mm, Di — 88.9 mm, U/Ums — 1.1 56 4.14 Influence of superficial gas velocity on dead zone boundary for polystyrene particles viewed fron rear column wall, H = 2.00 m, dp — 3.25 mm, Di = 88.9 mm 57 xiv 4.15 Influence of auxiliary flow on dead zone boundary for polystyrene particles viewed from front plate, H = 1.51 m, dp = 3.25 mm, Di = 88.9 mm, Q — 772 m3/h; (a) spouting with aeration; (b) spout-fluidization; (c) jet, slugging and bubbling fluidized bed 58 4.16 Influence of auxiliary flow on dead zone boundary for polystyrene particles viewed from rear column wall, H = 1.51 m, dp = 3.25 mm, Di = 88.9 mm, Q — 772 m3/h; (a) spouting with aeration; (b) spout-fluidization; (c) jet, slugging and bubbling fluidized bed 59 4.17 Fountain height versus static bed depth for polystyrene, brown beans and green peas, U/Ums — 1.1, D{ = 88.9 mm, qa = 0 m3/h 62 4.18 Bed pressure drop versus static bed depth for polystyrene, brown beans and green peas, U/Ums = 1.1, Di — 88.9 mm, qa = 0 m3/h . 63 5.1 Longitudinal annulus gas velocity distribution compared with equations in the literature where extrapolated values of H m and U a # m have been employed. (Particles: polystyrene, dp — 3.25 mm, Di — 88.9 mm, H = 2.0 m, U/Umt = 1.1, qa = 0.0 m3/h) 69 5.2 Longitudinal annulus gas velocitj' distribution compared with equations from literature. (Particles: polystyrene, dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, i7/i7m. = 1.3, qa = 0.0 m3/h) 71 5.3 Longitudinal annulus gas velocity distribution compared with equations from the literature. (Particles: polystyrene, dp — 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Um. = 1.5, qa = 0.0 m3/h) 72 5.4 Influence of superficial gas velocity on average annulus superficial velocity. (Particles: polystyrene, dp — 3.25 mm, Di = 88.9 mm, H — 2.0 m, qa = 0.0 m3/h) 73 xv 5.5 Longitudinal annulus gas velocity distribution compared with Equation (2.29) proposed by Littman et al. (1976), assuming H m a y = H m . (Particles: polystyrene, dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Umi = 1.7, qa = 435 m3/h) 75 5.6 Longitudinal annulus gas velocity distribution compared with Equation (2.29) proposed by Littman et al. (1976), assuming H m a y = H m . (Particles: polystyrene, dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Ume = 2.05, qa = 721 m3/h) 76 5.7 Influence of auxiliary gas flow on average annulus superficial velocity. (Par-ticles: polystyrene, dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, Q/Qm, = 1.1) 77 5.8 Radial profile of annulus superficial gas velocity for polystyrene particles. {dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Ums = 1.1, qa = 0.0 m3/h) 78 5.9 Radial profile of annulus superficial gas velocity for polystyrene particles. {dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Ums = 1.3, qa = 0.0 m3/h) 79 5.10 Radial profile of annulus superficial gas velocity for polystyrene particles. {dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Um. = 1.5, qa = 0.0 m3/h) 80 5.11 Radial profile of annulus superficial gas velocity for polystyrene particles. {dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Ume = 1.7, qa = 435 m3/h) 81 5.12 Radial profile of annulus superficial gas velocity for polystyrene particles. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Um. = 2.1, qa = 721 m3/h) 82 5.13 Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Uma = 1.1, qa = 0.0 m3/h) 84 5.14 Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, Di — 88.9 mm, H = 2.0 m, U/Um„ = 1.3, qa = 0.0 m3jh) 86 x v i 5.15 Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Ums = 1-5, qa = 0.0 rn3/h) 5.16 Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Ums — 1-7, qa = 435 m3/h) 5.17 Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Ums = 2.05, qa = 721 m3/h) 5.18 Radial pressure profile for polystyrene particles. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Ums = 1.1, qa = 0.0 m3/h) 5.19 Radial pressure profile for polystyrene particles. (dp = 3.25 mm, Di — 88.9 mm, H = 2.0 m, U/Um} = 1.3, qa = 0.0 m 3 ) 5.20 Radial pressure profile for polystyrene particles. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Ums = 1.5, qa = 0.0 m3/h) 5.21 Radial pressure profile for polystyrene particles. (dp = 3.25, mm, Di = 88.9 mm, H = 2.0 m, U/Um. = 1.7, cja = 435 m3//i) 5.22 Radial pressure profile for polystyrene particles. (dp = 3.25, mm, Di = 88.9 mm, H = 2.0 m, U/Ums = 2.05, 9 a = 721 m3/h) 5.23 Geometrical parameters used in the mathematical model 5.24 The finite difference grid 5.25 The classification of grid points 5.26 Predictions of vector Ergun equation model for qa = 0.0 m3js. Solid lines give gas streamlines; corresponding numbers indicate fraction of total flow between streamline and wall. Dashed lines show isobars, with numbers indicating the fraction of total pressure drop xvii 5.27 Predictions of vector Ergun equation model for qa = 0.12 m3/s 112 5.28 Predictions of vector Ergun equation model for qa = 0.20 m3/s 113 5.29 Comparision of pressure profile predicted from various models with exper-imental values. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Umt = 1-7, qa = 435 m3/h) 115 6.1 Typical pressure drop vs. central air flowrate for the polystyrene particles, (<fp=3.25 mm, D c=0.91 m, A=88.9 mm, 9 a=384 m 3 /hr) 118 6.2 Typical pressure drop vs. auxiliary air flowrate for the polystyrene par-ticles, (cip=3.25 mm, D c=0.91 m, A=88.9 mm, H - 1.5 m, Q = 492 m 3 /h) 120 6.3 Regime map of the polystyrene particles. (Di = 88.9 mm, H = 1.5 mm, dp = 3.25 mm, G—Fluidization at minimum condition, A—Spouting at minimum condition, C—Spout-fluidization at minimum condition.) . . . 121 6.4 Transition from spout—fluidization (SF) to jets, slugs, and bubbles in fluidized bed (JSBF) for polystyrene particles: a —slugs in fluidized bed, Q=561 m 3 /h ; b —jets in fluidized bed, Q=715 m 3 /h ; c—spout-fluidization, Q=773 m 3 /h . (dp=3.25 mm, £,-=88.9 mm, H = 1.5 m, qa=1080 m 3 /h). . 123 6.5 Appearance of a slug in polystyrene bed. (dp = 3.25 mm, Di = 88.9 mm, H = 1.5 m, qa = 1012m3//i, Q — 825m3/h). Spacing between adjacent grid lines: 50 mm 126 6.6 Transition from slugs to bubbles in fluidized bed: a —slugs in fluidized bed Q=595 m 3 /h ; b —large bubbles in fluidized bed Q=315 m 3 /h ; c — bubbles in fluidized bed Q=255 m 3 /h . (Particles: polystyrene, dp = 3.25 mm, Di = 88.9 mm, H = 1.5 m, qa = 10i2m3/h) 127 xviii 6.7 Regime map for brown beans: G —fluidization at minimum condition; A —spouting at minimum condition; C —co-ordinates for spout-fluidization at minimum condition. (Di = 88.9 mm, dp = 4.7 mm, H = 1.25 m). . . . 128 6.8 Transition from spouting with aeration to spout-fluidization: a—spouting with aeration, qa=756 m 3 /h ; b —spout-fluidization, qa=940 m 3 / h . (Par-ticles: polystyrene, dP = 3.25 mm, Di = 88.9 mm, H — 1.5 m, 0=621 m 3 /h) 130 A . l Calibration curve of pressure gradient vs. superficial velocity in a packed bed column. (Particles: polystyrene, dp = 3.25, Dc = 0.152, fluid: air) . . 150 F . l Coordinate transformation 167 xix Acknowledgement I would like to express my sincere gratitude to Professors J . R. Grace and C. J . Lim for their guidance and suggestions which played a very important role in the completion of this study. Thanks are also due to the staff of the Department of Chemical Engineering Workshop and Stores for their invaluable assistance. Finally, to my wife, Li Zhang, for her continued encouragement, patience and under-standing from thousands of miles away. xx Chapter 1 Introduction The spouted bed technique has been widely accepted as an alternative to fluidization for handling particulate solids which are too coarse (dp > 1 mm) and uniform in size for good fluidization. Spouting involves contacting of solids with fluid by injecting a steady axial jet of the fluid into a bed of solids. Figure 1.1 illustrates a spouted bed schematically. The bed is made up of two distinct regions: a dilute core called the spout, and a surrounding annular dense region called the annulus. Solid particles are carried up rapidly with the fluid, usually a gas, in the spout to the top of a fountain above the bed surface, and fall down onto the surface of the annulus by gravity. In the annulus, the particles move slowly downward and, to some extent, inward as a loosely packed bed. Fluid from the spout leaks into the annulus and percolates through the moving packed solids there. These solids are re-entrained into the spout over the entire bed height. The overall system thereby becomes a composite of a centrally located dilute-phase cocurrent-upward transport region surrounded by a dense-phase moving packed bed with countercurrent percolation of fluid. The systematic cyclic movement of the fluid and solids leads to effective contact between them. Spouted beds have been used in a number of industrial applications. Mathur and Epstein (1974) summarized the various applications in the early decades. Examples of such applications include: drying of granular materials, coohng of fertilizers, solids blending, food processing, and charcoal activation. Some other applications that have not been used on an industrial scale, but have seemed to be feasible based on bench scale 1 Chapter 1. Introduction 2 Figure 1.1: Schematic diagram of a spouted bed. Arrows indicate direction of solids movement Chapter 1. Introduction 3 operation, include shale pyrolysis and thermal cracking of petroleum. One disadvantage of a spouted bed is that local solids mixing is not as high as in a fluidized bed because most of the particles move slowly as a loosely packed bed in the an-nulus and only about 15% are in the spout at any instant (Thorley et al. 1959). In a large vessel, this slow-moving zone with little fluid flow can be a serious liability, especially if the solids are sticky and tend to sinter. The spout-fluid bed aims at overcoming the limi-tations of spouted and fluidized beds by suitably superimposing one system on the other, to achieve higher mixing of the solids, and better annulus solid-fluid contact, thereby improving the heat and mass transfer characteristics. In a spout-fluid bed, auxiliary fluid is supplied through a porous or perforated distributor, in addition to supplying spout-ing fluid through a centrally located opening as shown in Figure 1.2. The spout-fluid bed gives reduced stratification with mixed particles of different diameters and densities. This is particularly useful in systems such as roasting, calcining or gasification, where the particles are continuously deformed or shrink and change their chemical and physical properties. Flow of fluid through the sloping sides also reduces the tendency for particles to sit for extended periods in contact with each other or with the wall of the vessel, giving spout-fluid beds an advantage when treating sticky solids such as agglomerating coals. Spout-fluid beds provide an important Hnk between spouting and fluidization and permit beds to be designed with varying portions of the characteristics of each (Littman et al. 1974). Lim and Grace (1987) stated: "The vast majority of experimental spouted bed studies have been carried out in experimental columns of diameter 0.3 m or less. Studies on columns of the order of 1 m or larger in diameter are rare, but are crucial for the rational scale-up of spouted bed equipment. Otherwise, correlations and models are based solely on small-scale vessels, and the results may not scale well to beds of commercial size. For example, there are strong indications (Mathur and Epstein, 1974; Kmiec, 1983; Green Chapter 1, Introduction 4 FLUID INLET Figure 1.2: Schematic diagram of a spout-fluid bed. Arrows indicate direction of solids movement Chapter 1. Introduction 5 and Bridgwater, 1983; Fane and Mitchell,1984) that the well-known minimum spouting velocity correlation of Mathur and Gishler (1955) gives poor predictions when applied to beds of diameter greater than 0.5 m. Obtaining data on large-diameter vessels was identified as 'the most pressing research need' for spouted beds at the 2nd International Symposium on Spouted Beds (Epstein, 1983)". The primary objectives of the present research are: (a) to collect hydrodynamic data on a "half-column" of diameter 0.91 m with beds operated both as a standard spouted bed and as a spout-fluid bed with a conical base; (b) to compare the results with the findings of Sutanto (1983) on a similar spout-fluid bed of diameter 0.152 m; (c) to compare data with the results obtained by Lim and Grace (1987) on the same 0.91 m diameter column with a flat base without auxiliary fluid; and (d) to compare the data with predictions from existing equations and correlations. The dependent hydrodynamic parameters of interest are: minimum spouting velocity, maximum spoutable bed height, spout shape and diameter, fountain height, dead zone dimensions, overall bed pressure drop, fluid distribution in the annulus, longitudinal and radial pressure profiles in the annulus, and regime maps for the spout-fluid bed. Chapter 2 Literature Review Although spouted beds have been actively studied for over two decades, there is still considerable uncertainly about suitable scale-up criteria. Beds greater than 0.3 m in diameter are rarely discussed in the open literature, even through units of almost 3 m diameter" are used commercially (Epstein, 1982). Many equations based on small-scale vessels are available for predicting hydrodynamic parameters. Mathur and Epstein (1974) and Epstein and Grace (1984) have reviewed some of these equations. The information available on spout-fluid beds is very limited. All of the experimental spout-fluid bed studies have been carried out in columns of diameter 0.15 m or smaller. 2.1 Minimum Spouting Velocity The minimum superficial fluid velocity at which a bed will remain in the spouted state is called the minimum spouting velocity (Umi). It is determined experimentally by reducing the fluid flow rate to a point at which a further decrease of flowrate will- cause the spout to callapse and the bed pressure to increase suddenly. This point is taken as the minimum spouting velocity. It is known that Ums depends on solid and fluid properties, bed geometry and bed depth. For given material and fluid properties, U m j l increases with increasing bed depth, increasing fluid inlet diameter, and with decreasing column diameter. However, results plotted by Mathur and Epstein (1974) and confirmed by Fane and Mitchell (1984) show that Ums first decreases with increasing Dc, but then goes through a minimum and begins to increase as Dc increases beyond about 0.4 to 0.6 m 6 Chapter 2. Literature .Review 7 for the systems studied. A number of empirical correlations have been developed for predicting Uma (Mathur and Epstein, 1974). 2.1.1 Mathur and Gishler equation This equation was developed by Mathur and Gishler by dimensional analysis in 1955. It is considered to be the most reliable equation for predicting Ume for cylindrical vessels up to 0.6 m in diameter (Epstein and Grace, 1984). The equation has the form: As noted by Mathur and Epstein (1974), Equation (2.1) underestimated the minimum spouting velocity by a factor of nearly 2 for a single measurment (on wheat) in a 0.91 m diameter vessel. This was confirmed by Lim and Grace (1987) for spouting polystyrene particles, brown beans and green peas in a 0.91 m diameter flat-based half-cylindrical column. Mathur and Epstein also pointed out that Equation (2.1) gives the wrong limit as Dc increases for geometrically similar beds (i.e. fixed Di/Dc and H/Dc) since the orifice velocity would eventually fall below the terminal settling velocity of the particles. 2.1.2 Fane-Mitchell modification of Equation (2.1) Fane and Mitchell (1984) proposed an empirical dimensional correction to Equation (2.1) based on experimental data in a 1.1 m diameter column for Dc > 0.4 m, (2.1) (2.2) where n = 1 - exp{-7£> 2 } (2.3) and all units follow the SI system. Chapter 2. Literature Review 8 Both Equations (2.1) and (2.2) predict Ums to be directly proportional to Hu where u> is equal to 0.5. Lim and Grace (1987) found that u> is of order 1.0 ~ 1.4. Green and Bridgwater (1983) also indicated that the exponent on H is greater in larger diameter vessels. 2.1.3 Correlation of Grbavcic et al. Using the model of Mamuro and Hattori (1968), Grbavcic et al. (1976) proposed the following correlation for predicting Ums for spherical particles: where a s is denned as the ratio of the area of the spout to that of the column. Since ae is much smaller than 1 in most cases, Equation (2.4) can be further simplified to (2.4) (2.5) where Umf is given by the Ergun (1952) equation: with fj and f2 given by h = 1-75 U = 150 fi(l - e m / ) 2 ,pf(l - e m / ) ' — z r ~ 3 — ) Hm is given by (2.7) which was based on data for water-spouted beds of spherical particles. Chapter 2. Literature Review 9 2.1.4 Littman—Morgan equation Littman and Morgan (1983) derived a general correlation for predicting the minimum spouting velocity based on the annulus pressure gradient at the top of the bed at minimum spouting: fi If*, fl + 1 + fl (2.8) Umf 2f2Umf ''\\2f2UmfJ ' \~ • f2Umii The parameter Cp represents the dimensionless pressure gradient at the top of the annulus and determines the flow there. It is defined by H w here Cp = l - Y D, •X' [2Y + (X - 2) + (Ar - 0.2) - (3.24/fl)] [2Y + 2(X - 0.2) - 1.8 + (3.24/6)] (2.9) X = + (H/De)] Y = l-(APm./APm/) with APmf, Ao and g((j)) given by <t>dp(pp - p^ge^f APmf = 1.75/9/ Hf2 A0 < 0.07 A0 = T r 9{<P) (Pp-Pf) 9 g(<j>) = [5(f>3 - 7.57(f)2 + 4.09<£ - 0.516] •{[85.7(1 - emf)/<j>}2 + [IM&rfAr]1" + 85.7(1 - e ^ / f l ; {[85.7(1 - em/)P + [2,29e3n/Ar]i/2 + 8 5. 7( 1 _ em/)} (2.9a) (2.96) (2.9c) (2.9d) (2.9c) (2-9/) Chapter 2. Literature Review 10 2.1.5 Wu et al. modification of Equation (2.1) A modified version of the Mathur and Gishler (1955) equation, with best fit values of the coefficient and of the exponents on the dimensionless groups for conditions including elevated temperature, was given by Wu et al. (1987): 0.256 U„ V2gH = 10.6 1.05 [Al 0.266 -0.095 Pp-Pf' I A J L A J . Pi . (2.10) The most significant difference from the original Mathur and Gishler equation is that the exponent on (pp — pf)jpj is 0.256 instead of 0.5. 2.2 Maximum Spoutable Height The maximum spoutable bed height, Hm, is the maximum height at which steady or stable spouting can be obtained. A common equation for predicting Hm is, A 2 \ / A \ 2 / 3 /5686 2 A / \ AJ 1 + 35.9 x 10- 6 Ar - 1 (2.11) where b= U m e = 1.0 ~ 1.5 Ar dp(Pr ~ Pf)pf9 P2 This equation was tested by McNab and Bridgwater (1977) for b = 1.11, and it was found to give a good fit for most existing experimental data. Differentiating Equation (2.11) with respect to Ar, substituting dp from Ar = dp(pp — p f)gp j jfi2, and setting dHmjd(Ar) equal to zero yields the critical value, Ar — 223,000, or P2 CTlt 60.6 Tl / 3 9(Pv~ Pi)Pi (2.12) below which Hm increases with dp and above which Hm decreases as dp increases. Chapter 2. Literature Review 11 Littman et al. (1977) proposed a correlation for predicting i / m for spherical particles -0.384 HmDs (D. W^T) = 0 3 4 5 U (2.13) where D„ is calculated from McNab's correlation, presented below. Combining Equation (2.13) with McNab's correlation and assuming that 1 >^ (D„/Dc)2, they obtained for spouting with gases 0.842 nl-45„0.569 Uc Pblk p mf (1 Pr 0.677 ' Ri •mf < 10 Hm = 0.0741 nl-45 ,.0.569 Uc Pblk Remf > 1000 (2.14) (2.15) [ d°p-33&emf {PvP]r™\ ' Grbavcic et al. (1976) developed a correlation based on water spouted beds of spherical particles. This correlation, Equation (2.7), was discussed in Section 2.1.3. Littman et al. (1976) reported that the maximum spoutable height in a spout-fluid bed (Hmsf) is the same as for a spouted bed (Hm) for a given system. However, realizing that the maximum spoutable heights for a spout-fluid bed and a spouted bed are not the same, Hadzisdmajlovic et al. (1983) redeveloped the equation for predicting minimum spout-fluid flowrate presented earlier by Littman et al. (1976). Since Hmef is a param-eter in those equations and its knowledge essential for predicting minimum spout-fluid flowrate, Hadzisdmajlovic et al. (1983) developed a semi-theoretical correlation for Hmsf-HmefDef + n=0 qJAa mf mf 0.345 ( | -0.384 0.596 (2.16) where A0 = 0.5962, A1 = -0.4316, A2 = 0.05617, A3 = -0.2972, AA = 0.5675, A5 = — 0.425, and DBj is given by n31 1/2 \D,f] = 1 -i Ds J l 1 msf Hn (2.17) Chapter 2. Literature Review 12 with Hm calculated from Equation (2.13). Equation (2.16) is observed to involve certain discrepanies (Rao et al., 1985). A new correlation was proposed by Rao et al. (1985) for predicting the maximum spoutable height in a spout-fluid bed, H™-H™S = 0.998 ( ^ ] - 1.087 ( ^ ) + 1.01 ( ^ ) (2.18) Hm \ Qmf ) V Qmf J \ Qmf J 2.3 Average Spout Diameter The longitudinal average spout diameter, D„, is an important parameter for determining the flow distribution between spout and annulus (Epstein and Grace, 1984). There are a number of equations available for estimating the average spout diameter. The correlation of McNab (1972) is most widely used. It has the form: ( S I u M t s ) ( 2 ' 1 9 ) Equation (2.19) was tested in 30° large sector beds by Green and Bridgwater (1983). It was found that the McNab's correlation underestimated Ds by about 15%, 30% and 40% in the 0.57 m, 1.17 m and 1.57 m diameter columns respectively. However, Lim and Grace (1987) established that measured data in a 0.91 m diameter column were in excellent agreement with Equation (2.19), with the average absolute deviation less than 10%. For spout-fluid beds, Hadzisdmajlovic et al. (1983) proposed the following equation for predicting the average spout diameter, D.f if Q 1/2 Ds [\Qmsf, where h = H/Hm. l - ( l - / 0 3 (2-20) Chapter 2. Literature Review 13 2.4 Fountain Height The shape of the fountain is roughly parabolic. Grace and Mathur (1978) formulated a theoretical model, based on a force-balance analysis, for the prediction of maximum height of a fountain. They also proposed a simpler approximate equation for estimating fountain height, U — C 1 A 6 lg{pv - Pi) 2.5 Dead Zone Boundary The interface between an inner "active annulus" region where particles are in moving packed bed flow and an outer "dead zone" where there is no particle motion in the annulus is called the dead zone boundary (Lim and Grace, 1987). In large-scale spouted bed applications, the dead zone may occupy a considerable fraction of the bed volume. Green and Bridgwater (1983) pointed out that the dead zones were inclined at about 65° to the horizontal near the column base in beds of 30° sector angle and radius up to 0.78 m. Lim and Grace (1987) noted the same behaviour in a 0.91 m diameter vessel. 2.6 Fluid Distribution in the Annulus 2.6.1 Longitudinal annulus fluid velocity The first and still most successful derivation of the longitudinal fluid velocity profile in the annulus of a spouted bed was made by Mamuro and Hattori (1968). It was based on a force balance on a differential height dz of the annulus, assuming that fluid flow in the annulus is governed by Darcy's law, with boundary conditions: at z = 0, Ua = 0 at Z — Hm, 11 a = UaHm = Umf Chapter 2. Literature Review 14 They obtained: fe^-O-i)' ( 2 - 2 2 ) Lefroy and Davidson (1969) proposed an empirical equation for predicting longitudi-nal fluid distribution in the annulus, Epstein et al. (1978) recommended that Equation (2.23) should be modified to U a ~ ' ( T Z ) ( 2 2 4 ) UaHm \2HmJ in which the flow regime index n for the loose-packed moving bed annulus varies from unity (Darcy's law) at the bottom of the annulus to maximum possible value of 2 (for inviscid flow) at z = Hm. Based on the Mamuro-Hattori model, Rovero et al. (1983) proposed the following equation for the conical-base spouted bed: Un i BHl L z l 3 UaHm 18AlcUaHm where l8UaHmAl B = ^ p m For the geometry shown in Figure 2.1, it can be shown by geometry that Rx = z tan 7 + Rb (2.26) in the cone (z < Hc) so that A a = ir(R2 - R2) = TT{(2 tan 7 + Rbf - R2} (2.27) In the cylinder (z > HC), the annulus area is independent of height, i. e. A a = A a c = rr(R2c - Rl) = TV[(HC tan 7 + Rb)2 - R2e] (2.28) Chapter 2. Literature Review 15 Axis of column A Wall of column Figure 2.1: Geometric parameters used in the cone-modified Mamuro-Hattori model In Equations (2.27) and (2.28), it is assumed that Rs (= D3/2) is invariant with z, and its value is determined either experimentally or from the empirical equation of McNab (1972), Equation (2.19). In an attempt to account for the auxiliary flow, Littman et al. (1976) modified Equa-tion (2.22) and obtained: where h^f = HjHm,j. 2.6.2 Radial profiles of annular superficial gas velocity The superficial gas velocity in the annulus generally shows little radial variation for small columns. For a larger column (Dc = 0.91 m), Lim and Grace (1987) found that the local velocity decreased with increasing radial distance as shown in Figure 2.2. (2.29) Chapter 2. Literature Review 16 10 •8 \ £ 3 -4 •2 Z , m H,ro • 0 7 6 0 -91 • 1 . 0 7 1 - 2 2 • 1 - 3 7 1 - 5 2 o °-1 in _A I C : E 0 0-1 02 0-3 0-4 0-5 r, m Figure 2.2: Radial profiles of annular superficial gas velocity for polystyrene beans with Di = 92 mm, U/Ums = 1.1, due to Lim and Grace (1987) 2.7 Pressure Profiles in the Annulus and Overall Bed Pressure Drop 2.7.1 Longitudinal pressure profiles in the annulus The longitudinal pressure gradient in the annulus of a fully spouted bed at any level is given by the Ergun (1952) equation, Equation (2.6). Epstein and Levine (1979) reworked the theory of Mamuro and Hattori using Equation (2.6) instead of Darcy's law and proposed: Chapter 2. Literature Review 17 where = whT)w-2){l-i(h'-*2) -(h3 - x3) + 0.25(h4 - x*)} +3{3(ft3 - x3) - 4.5(fe4 - x4) +3(h* - x5) - [h6 - x6) +0.U3{h7 - x7)}} (2.30) h = H/Hm 8 = 2 x = z/Hm 3 / i 2f2Umf A simpler empirical relation for vertical pressure distribution was proposed by Lefroy and Davidson (1969), P - PH ( \ , 2.7.2 Radial pressure profiles The radial pressure gradient for small columns has been found to be negligible, except in the conical section (Mathur and Epstein,1974; Lim,1975). For large columns, however, the pressure was found to decrease towards the column wall (Green and Bridgwater, 1983; Lim and Grace, 1987). More information on radial pressure profiles in the annulus is provided in Chapter 5. Chapter 2. Literature Review 18 2.7.3 Overall bed pressure drop The overall pressure drop, —AP„ = P0 — P#, is obtained by putting x = 0 in Equation (2.30), i . e. -AP, 2-(4/0), - A P / 2-(1/0) (1.5/i - r + 0.25/T) + 2 - ^ ( 3 ^ - 4 . 5 ^ + 3 ^ -h5 + 0.143/i6) (2.32) 2.8 Regime Maps and Minimum Fluid Flowrate for Spout-Fluid Bed The concept of spout-fluidization was first introduced by Chatterjee (1970). Since then several publications have appeared on this subject. Sutanto et al. (1985) observed that the term spout-fluid bed means different things to different authors. They noted that Littman et al. (1976) and Vukovic et al. (1982) refer it to a condition in which normal spouting is accompanied by additional aeration, but not fluidization of the annulus; hence it is more appropriate to call it "spouting with aeration". A l l of the previous experimental studies have been carried out in columns of diameter 0.15 m or smaller. 2.8.1 Regime maps Regime maps determined experimentally by Nagarkatti and Chatterjee (1974), Vukovic et al. (1982), Sutanto et al. (1985) are shown in Figures 2.3, 2.4 and 2.5 respectively. These regime maps, though they appear to be different from each other, share many common features: • a packed bed region, • a fluidization region, Chapter 2. Literature Review 19 120 160 Spouting (Central) flow (Ndm /min air) Figure 2.3: Regime map due to Nagarkatti and Chatterjee (1974). (Particles: glass beads, pp = 2410 kg/m3, H = 0.185 m, dP = 0.60 mm, D{ = 5.0 mm, Dc = 90 mm, fluid used: air) Chapter 2. Literature Review 20 4 Figure 2.4: Regime map due to Vukovic et al. (1982). (Particles: calcium carbonate, pp = 2600 kg/m3, dp = 1.8 mm, Dc — 70 mm, D{ 15 mm, H = 1 m, fluid used: air) Chapter 2. Literature Review 21 1.6 1.4 1.2-Legend O Observed spout with oerot ion O Observed spout - f lu id i za t ion Jet in fluidized bed (slugging) JF(I) Spout - f lu id izat ion (SF) 0.4 0.2-\ JF(H) (bubbling) Packed bed (P) \ . \ A \ Spouting with aerat ion (SA) 0.2 0.4 0.6 0.8 Q / Q 1.2 1.4 1.6 mf Figure 2.5: Regime map due to Sutanto et al. (1985). (Particles: polystyrene, Di — 19.1 mm. dP = 2.9 mm, pp = 1040 kg/m3, Dc = 0.15 m, H = 0.60 m, fluid: air) Chapter 2. Literature Review 22 • a spouting with aeration region, • a jet in fluidized bed region, and • a spout-fluidization region. 2.8.2 Minimum fluid flowrate for spouting with aeration Nagarkatti and Chatterjee (1974) found that the minimum total fluid flow remains con-stant along the hne L - M in Figure 2.3 and almost equal to the minimum spouting flow, i.e. QT,msa ^ Qms (2.33) which means that the sum of the auxiliary and central flow at the minimum spouting with aeration condition is equal to QmB. On the other hand, Littman et al. (1974) stated that Qmsa — Qmf ~p\ Qmsa (2.34) . ms A correlation obtained by Sutanto et al. (1985) based on their experimental data gives 1 _ | 9inl] (9^1 ] (2.35) Q mf Qmsa i Qms J \ Qmf ) _ The empirical constant K varied between 1.2 and 1.71 for different cases investigated. In all cases K > 1, and the sum of the auxiliary and central flow required for spouting with aeration was always greater than Qms-, even when Qms > Qmf-The annular velocity in a spout-fluid bed, Equation (2.29), was derived separately by Littman (1976). It was assumed that for spouting with aeration at the minimum condition, • U,H — Umf , qa = qmsa • QT = QT,msa With H < Hmsf Chapter 2. Literature JReview 23 2.8.3 Minimum total fluid flow for spout-fluidization The only correlation available for estimating the minimum total flow required for spout-fluidization, corresponding to point C in Figures 2.3, 2.4 and 2.5, was proposed by Na-garkatti and Chatterjee (1974). It combines the fluid needed to spout a static bed at the minimum condition with the fluid needed for minimum fluidization of the annulus: UT,m.f = Ume + Umf(l - $) (2.36) The function $ was determined empirically to be $ = o .20ay°- 3 2 0 z>?- 2 3 5 # 0 1 6 0 (2.37) The value of [7 m 8 is obtained from the Mathur and Gishler correlation, Equation (2.1). Chapter 3 Experimental Equipment and Particulate Materials 3.1 Experimental Equipment Experiments were carried out in a conical base cylindrical half-column constructed of plexiglass with an internal diameter of 0.91 m and a cylindrical height of 3 m. This column is the same one that was used in a previous study (Lim and Grace, 1987), except that it has been moved from outdoors to the inside of the Pulp and Paper Center and, instead of having a fiat base, a new 60° conical base section has been installed. The height of the column was also extended by a further 2 m by installing a wooden frame and a fine mesh screen, to permit operation of deep beds and high gas velocities. The front plate was made of abrasion-resistant plastic. Supports were installed at 1 m intervals to prevent distortion of the column. A solids feeding door and four solids feeding fines were installed on the back of the wooden frame and along the back of the column, respectively. Figure 3.1 shows a drawing of the column. The column was supported by a steel frame. Bed pressure drop measurements were provided by two pressure taps. One was in the approach section, 63.5 mm upstream from the central air inlet orifice plate. The other was installed at the same vertical level as the inlet orifice plate surface. Thirty pressure taps were installed at 100 mm intervals along the curved surface back, extending the full length of the column, and six pressure taps were installed at 114.3 mm intervals along the conical base back. Six semi-circular orifice plates of different sizes were made with orifice diameters 24 Chapter 3. Experimental Equipment and Particulate Materials 25 *914 Mesh screen Solids feeding door Solids feeding lines  Pressure taps Semi-cylindrical plexiglass column J ~ Abrasion-resistant plastic flat plate Auxiliary flow lines Five layers of calming chambers •Steel cone Pressure taps Discharge lines Central inlet SIDE VIEW REAR VIEW Figure 3.1: Details of the conical base cylindrical half-column. (All dimensions in mm) Chapter 3. Experimental Equipment and Particulate Materials 26 from 76 to 140 mm. For each plate the upper surface was machined smoothly. Details of the orifice plates are provided in Figure 3.2. The air entry configuration for the central (spouting) flow is shown in Figure 3.3. A piece of fine mesh screen was welded on the inlet orifice holder to prevent particles from falling into the air inlet pipe during shut-off, as shown in Figure 3.3. The conical base section is shown in Figure 3.4. There were five layers of auxiliary flow subdistributors: the upper three subdistributor chambers each had two auxiliary flow inlet lines, while the two lowest ones each had one auxiliary flow inlet line. Each subdistributor was supplied by a separate auxiliary gas hne with its own flowmeter and control valve. Gas flow through each of the five layers could therefore be controlled independently. The distributor design provided great flexibility, enabling the auxiliary flow to be introduced in a variety of different ways, similar to the system employed in a much smaller column by Sutanto (1984). On the upper curved surface of the conical base there were 35 openings, each 18 mm in diameter, evenly spaced over the entire conical surface, as shown in Figure 3.5. A piece of 16 mesh steel screen was installed over each opening to prevent particles from finding their way into the auxiliary air subdistributor chambers. Air flowrates were determined using six orifice meters, one for the central spouting flow hne and one for each of the five auxiliary flow subdistributors. A further improvement in the current work was the provision of a much better com-pressor to allow operation at higher flow rates and for deeper beds. The compressor was a diesel-driven unit with a rated capacity of 1870 N m 3 / h (1100 SCFM) at a pressure drop of 101.3 kPa (14.7 psi). A schematic diagram and a photograph of the overall equipment layout appear in Figure 3.6 and Figure 3.7, respectively. Chapter 3. Experimental Equipment and Particulate Materials 27 No. Dimensions (mm) D M N H E R 1 76.2 152.4 152.4 63.6 0.0 19.1 2 88.9 152.4 152.4 63.6 0.0 15.9 3 101.6 152.4 165.1 57.2 6.4 12.7 4 114.3 152.4 165.1 57.2 6.4 9.5 5 127.0 152.4 165.1 57.2 6.4 6.4 6 139.7 152.4 165.1 57.2 6.4 3.2 Figure 3.2: Details of the semi-circular orifice plates Fi gure 3.3: Central air inlet orifice configuration.(All dimensions in mm) Chapter 3. Experimental Equipment and Particulate Materials 29 914.4 381.1 Front view AUXILIARY FLOW LINE 2"SCH40X3"IONG Rear view Figure 3.4: Conical base section. (All dimensions in mm) Chapter 3. Experimental Equipment and Particulate Materials 30 Fountain Half-column Viewing^ 36 pressure tap openings Orifice plate flowmeters Control valves Auxiliary flow line#1 # 3 #4 # 5 V d v e Q Rl. Valve Central flow line .Orifice plate flowmeter for central (spouting) air Manometer Blower Figure 3.6: Schematic diagram showing overall equipment layout Figure 3.7: Photograph of the overall equipment layout Chapter 3. Experimental Equipment and Particulate Materials 33 3.2 Particulate Materials Three different particulate materials were employed in the experiments, all with relatively narrow size distributions. They were polystyrene, brown beans, green peas, the same materials as used in the previous work (Lim and Grace, 1987). The green peas were nearly spherical, while the polystyrene particles and brown beans were nearly elliptical cylinders and oblate ellipsoids, respectively. The sphericities of the particles were taken as in Lim and Grace (1987). Equivalent particle diameter (volume diameter) is defined as the diameter of a sphere having the same volume as the particle (Geldart, 1986). Particle dimensions for each particle were obtained by measuring (using calipers) the sizes of 20 particles chosen at random. The particle density of the polystyrene particles was determined first by weighing a large quantity of particles and then measuring their volume by water displacement in a 1000 ml graduated cylinder under loosely packed conditions. The loosely packed condi-tions were achieved by inverting the cylinder and then quickly returning it to its original upright position. In the cases of brown beans and green peas, which are permeable to water, the particle densities were obtained by weighing 20 particles chosen at random for each material and measuring their volumes. The voidage of polystyrene particles in the loosely packed bed condition was calculated from particle and bulk densities a,s follows: , Pblk eB = 1 PP The voidages of brown beans and green peas were taken as those used by Lim and Grace (1987). The key properties of the materials are listed in Table 3.1. Chapter 3. Experimental Equipment and Particulate Materials 34 Table 3 . 1 : Properties of particulate materials Material (mm) PP (kg/m3) 4 Polystyrene 3.25 1018 0.44 0.87 Brown beans 4.72 1296 0.36 0.93 Green peas 6.71 1250 0.42 0.98 1,8 1,9 21° f i f j j i I n n IIIII Figure 3 .8 : Photograph of particulate materials C h a p t e r 4 M i n i m u m S p o u t i n g Veloc i ty , S p o u t Shape a n d D i a m e t e r , D e a d Z o n e B o u n d a r y , M a x i m u m Spoutable B e d H e i g h t , F o u n t a i n H e i g h t and O v e r a l l Pressure D r o p 4.1 M e a s u r e m e n t Techn ique 4.1.1 M i n i m u m spout ing velocity The minimum spouting velocity was measured at the collapse of the spout and at the point where the bed pressure drop increased suddenly while reducing the gas flowrate. The pressure drop across an orifice meter was recorded and then converted into volu-metric flowrate using the orifice meter calibration curve. The superficial gas velocity was subsequently determined by dividing the volumetric flowrate by the cross-sectional area of the column (TTD2/8), with the absolute bed pressure taken as: Ps = Patm + (-APe)/2 (4.1) 4.1.2 Spout shape a n d diameter Spout diameter was measured directly at intervals of 100 mm starting from the orifice up to the bed surface. The area-averaged spout diameter was calculated as: where D„(z) was the measured spout diameter at height z. A computer program was writ-ten to do the numerical integration using the Adaptive Newton-Cotes Method (Bowen, 35 Chapter 4. Um3, Ds> Dead Zone, Hm, HF and AP, 36 1988). 4.1.3 D e a d zone b o u n d a r y The dead zone boundary was measured directly at intervals of 100 mm starting from the orifice surface to the height above which the entire annulus was in moving packed bed condition. Data were recorded both from the front fiat plate and from the back wall of the column. 4.1.4 M a x i m u m spoutable b e d height The maximum spoutable height has been characterized in the literature (Mathur and Epstein, 1974) as corresponding to choking of the spout, slugging developing from the top of the spout, or fluidization of the annulus. In this work, the static bed height was progressively increased until choking of the spout was observed. The static bed height was then lowered slowly until spouting was just achieved, and that height was taken as the maximum spoutable height. 4.1.5 F o u n t a i n height Fountain height was measured directly as the distance from the upper bed surface to the top of the fountain. 4.1.6 O v e r a l l bed pressure d r o p The overall pressure drop across the bed was determined by the equation proposed by Mathur and Epstein (1974): (4.3) Chapter 4. Ums, D„, Dead Zone, Hm, Hp and A P , 37 where P# is the measured absolute upstream pressure for the bed and PE is the corre-sponding value at the same flowrate for the empty column. 4.1.7 Points of entry for auxiliary air flow Three configurations were used in this work: • Configuration #1—Auxiliary air supplied at all levels (i.e. at each of the five sub-distributors shown in Figures 3.4, 3.5 and 3.6). • Configuration #2—Auxiliary air supplied through the upper two subdistributors only. • Configuration #3—Auxiliary air supplied through the two lowest subdistributors only. For each configuration, the air flow to each subdistributor was adjusted in proportion to the number of openings. In other words, the air velocity through each opening was the same for all subdistributors which were in operation. 4.2 Results and Discussion 4.2.1 Minimum spouting velocity The minimum spouting velocity was determined for polystyrene particles, brown beans and green peas using an inlet orifice of diameter 88.9 mm. The results are shown in Figures 4.1, 4.2 and 4.3, respectively. The minimum spouting velocity was found to increase rapidly with increasing static bed height in shallow beds and then increase slightly with bed height in deeper beds for the three different kinds of particles. Particularly for brown beans, Umis almost leveled Chapter 4. Ums, D,, Dead Zone, Hm, HF and A P , 38 1 o.H ( H — i — i — i — i — i — i — i — i — ' — i — ' — I 0 0.5 1 1.5 2 2.5 3 H (m) Figure 4.1: Experimental values of minimum spouting velocity for polystyrene particles, dp — 3.25 mm, D, = 88.9 mm, Dc = 0.91 m, compared with predictions from some correlations Chapter 4. Um,, D,, Dead Zone, Hm> HF and AP, 39 (0 1.3 1.2 1.H 1 0.9-0.8-0.7 | 0.6 0.5 0.4 0.3 0.2 0.1 Littman & Morgan (1983) Fane & Mitchell (1984) Wu et al.(l987) Mathur & Gishler (1955) 0 i — — i — i — r - 1 — n — i — 1 — i — 1 — i — 1 — i — 1 — i — 1 — i — r 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 H (m) Figure 4.2: Experimental values of minimum spouting velocity for brown beans, dp — 4.7 mm, Di — 88.9 mm, Dc = 0.91 m, compared with predictions from some correlations. Chapter 4. Ums, Ds, Dead Zone, Hm, HF and APS 40 1.5 1.4 1.3 1.2-1.1-1-0.9 '(0 £ 0.8-| | 0.7 H 0.6-0.5-0.4-0.3-0.2 0.1 Littman & Morgan (1983) Grbavcic et al.(1968) Fane & Mitchell _ (1984) 4 ^ Wu et al.(l987) Mathur & Gishler (1955) - i | i | i | i | i | -0 0.2 0.4 0.6 0.8 1 l 1 l 1 I 1 I 1.2 1.4 1.6 1.8 H (m) Figure 4.3: Experimental values of minimum spouting velocity for green peas, dv = 6.7 mm, Di = 88.9 mm, Dc = 0.91 m, compared with predictions from some correlations. Chapter 4. Ums, De, Dead Zone, Hm, HF and AP, 41 off to a constant value at H > 1.0 m. At constant bed height, U m s was found to increase with increasing particle size. A plot of the natural log of Uma versus the natural log of H is shown in Figure 4.4. It can be seen from the slopes that Ume is proportional to H1S for polystyrene particles at H < 1.35 m, H15 for brown beans at H < 1.0 m and i f 1 - 2 5 for green peas at H < 1.0 m. This is in a very good agreement with the results reported by Lim and Grace (1987). However, the dependence is of the order of H03 for polystyrene particles at H > 1.35 m, H02 for brown beans at H > 1.0 m and H0A for green peas at H > 1.0 m. Some comparisons are also given in Figures 4.1, 4.2 and 4.3 between the experimen-tal data and Ums values predicted by some correlations. As in previous studies cited in Chapter 2 for large diameter vessels, predictions from the Mathur-Gishler (1955) cor-relation, Equation (2.11), consistently he below the experimental values, with errors up to a factor of nearly 2 for the three different materials; the errors are seen to increase with the bed height. The correlation of Littman and Morgan (1983), Equation (2.8), overestimates Um, values by a considerable margin for all these types of particle. The correlation of Grbavcic et al. (1976) is only valid for spherical particles, and so it has only been compared with the data for green peas, which were very nearly spherical. The agreement with the experimental data must again be described as disappointing. The Fane and Mitchell (1984) empirical modification, Equation (2.2), results in a considerable reduction in deviations from the Mathur and Gishler (1955) equation, but the trend with increasing H appears to be somewhat in error. The predictions from the Wu et al. (1987) correlation, Equation (2.10), have the same tread as the predictions from the Fane and Mitchell (1984) correlation for polystyrene particles and brown beans. For green peas the agreement at H > 0.95 m is excellent, but the trend with decreasing H appears to be in error. The experimental data and predicted Ume values for the three materials are listed in Chapter 4. Ums, Ds, Dead Zone, Hm, HF and A P S Chapter 4. Ume, D,, Dead Zone, Hm, Hp and AP, 43 Tables JA, J.2 and J.3 in Appendix J . 4.2.2 Spout shape and spout diameter Spout shapes are shown in Figures 4.5 and 4.6 for pure spouting (q a=0). The spout diameter for each of the three particulate materials diverged in the conical region and then narrowed further up the column for shallow beds. For deeper beds the spout diameter diverged slightly near the bed surface. The influence of flowrate on the spout shape is shown in Figure 4.6. It is seen that the spout shape remained the same, but the spout diameter increased with increasing flowrate. Some comparisons between the spout shapes without auxiliary flow and the spout shapes with auxiliary flow are shown in Figures 4.7, 4.8 and 4.9. It is apparent for polystyrene particles in the regime of spouting with aeration with configuration #1 that the spout diameter diverged in the conical region for both cases with and without auxil-iary flow and then narrowed further up the column without auxiliary flow, but diverged slightly with auxiliary flow (Figure 4.7a). With configuration #2 the spout diameter decreased with auxiliary flow in the conical region and increased with auxiliary flow in the cylindrical section (Figure 4.7b). In the regime of spout-fluidization both configura-tions #1 and #2 had the same trend as described above. However, a marked change of spout shape in the upper section of the bed was noted, with the spout diameter appear-ing to shrink and expand in a periodic manner. The recordings were taken both in the shrinking and expanding period and then the average diameter was taken as the spout diameter (see Figures 4.8a and 4.8b). With configuration #3 stable spouting could not be achieved, and the spout-annulus interface created a significantly "distorted" spout shape, as shown in Figure 4.10. For brown beans only configuration #1 was used, and the result is shown in Figure 4.9. It is found that the spout diameter only increased slightly with auxiliary flow in the spouting-with-aeration regime. The spout diameter Chapter 4. Uma, D,, Dead Zone, Hm, HF and AP, 44 ( U i ) 2 ri «=> II II (X s 6 as bo co ,—, oo o II . J O (tu)Z LO " O •rt a S3 oo o S II 2 S 11 3 O e <U GO ("1 y V i C D U O  r& OO L O r ,, (ui)Z Chapter 4. Uma, D„ Dead Zone, Hm, HF and A P , 45 Legend O U / U m s = 1.1 O U / U m s = 1 . 3 V U / U m s = 1.5 0.2 r (m) Figure 4.6: Influence of flowrate on spout shape for polystyrene particles, H — 2.00 m, dp = 3.25 mm, Di = 88.9 mm. Figure 4.7: Influence of auxiliary flow on spout shape in spouting with aeration regime for polystyrene particles, H = 1.51 m, dp = 3.25 mm, Dx = 88.9 mm, O = 772 m3/h] (a) configuration #1; (b) configuration #2. Chapter 4. C 7 m s , Ds, Dead Zone, Hm, HF and A P S 47 Figure 4.8: Influence of auxiliary flow on spout shape in spout-fluidization regime for polystyrene particles, H = 1.51 m, dp = 3.25 mm, Dt = 88.9 mm, Q = 772 m3/h; (a) configuration #1; (b) configuration #2. Figure 4.9: Influence on spout shape of auxiliary flow with configuration #1 for brown beans, H = 1.28 m, dp = 4.7 mm, D{ = 88.9 mm, O = 798 m3/h; (a) spout-ing-with-aeration regime; (b) spout-fluidization regime. Figure 4.10: Photograph of spout shape for polystyrene particles with auxiliary air via configuration #3, H = 1.51 m, dp = 3.25 mm, D{ = 88.9 mm, U/Umi = 1.96, Q = 460 m3/h, qa = 605 m 3//i. Chapter 4. Ume, D„, Dead Zone, Hm, HF and A P , 50 appeared to shrink and expand in the upper section of the bed in the spout-fluidization regime. A l l experimental data are given in Tables J.4, J.5, J.6, J.7 and J.8 in Appendix J. The average spout diameter was determined using Equation (4.2), and the corre-sponding results are presented in Table 4.1. The McNab (1972) correlation, Equation (2.19). was used to predict D„ values for both pure spouted beds and spout-fluid beds and the Hadzisdmajlovic et al. (1983) correlation, Equation (2.20), was used to predict Ds values only for spout-fluid beds. The measured values are seen to be in excellent agree-ment with the McNab (1972) correlation for pure spouted beds, with most deviations less than 15%; somewhat larger error (23%) tends to be recorded for the shallowest bed of polystyrene particles. Both the McNab (1972) correlation and the Hadzisdmajlovic et al. (1983) correlation give poor predictions for spout-fluid beds. 4.2.3 Dead zone boundary For a standard spouted bed the dead zone boundaries are shown in Figure 4.11 for polystyrene particles, brown beans and green peas for different static bed depths. It is seen that the dead zones increased with increasing static bed depth. The dead zones were inclined at about 60° to the horizontal near the inlet orifice, and this angle decreased towards the top of the bed. These results are similar to those described by Green and Bridgwater (1983) and Lim and Grace (1987). The superficial gas velocity had little influence on the dead zone boundary for 1.1 < U/Ums < 1-5 as shown in Figure 4.12. This again is consistent with findings reported by Lim and Grace (1987) Chapter 4. Ums, Ds, Dead Zone, Hm, HF and APS 51 Table 4.1: Experimental spout diameters and predicted values from two correlations. See Table 3.1 for particle properties. D. D. H Expt'l Pred." Deviation Particles (m) u/un. {m3/h) (m) (m) (%) PS+ 0.55 1.1 0.0 0.090 0.069 -23 PS 0.83 1.1 0.0 0.105 0.095 -10 PS 1.20 1.1 0.0 0.113 0.113 0 PS 1.51 1.1 0.0 0.119 0.125 5 PS 1.76 1.1 0.0 0.116 0.127 10 PS 2.00 1.1 0.0 0.115 0.129 12 PS 2.30 1.1 0.0 0.120 0.130 8 PS 2.60 1.1 0.0 0.116 0.134 15 B B + 0.55 1.1 0.0 0.081 0.070 -13 BB 0.78 1.1 0.0 0.103 0.094 -9 BB 0.96 1.1 0.0 0.112 0.106 -5 BB 1.28 1.1 0.0 0.109 0.110 1 BB 1.60 1.1 0.0 0.113 0.111 -2 GP+ 0.55 1.1 0.0 0.105 0.095 -10 GP 0.78 1.1 0.0 0.120 0.121 1 GP 0.96 1.1 q.o 0.128 0.132 3 GP 1.18 1.1 0.0 0.130 0.139 7 GP 1.44 1.1 0.0 0.136 0.144 6 Continued Chapter 4. Ums, Ds, Dead Zone, Hm, HF and APS 52 Table 4.1 (continued) Dt D. H Expt'l Pred." Deviation Particles (m) u/um, (m3/h) (m) (m) (%) GP 1.64 1.1 0.0 0.137 0.148 8 PS+ 2.00 1.1 0.0 0.115 0.129 12 PS 2.00 1.3 0.0 0.127 0.140 10 PS 2.00 1.5 0.0 0.142 0.150 6 PS 1.51 2.3 865 0.145 0.181 25 PS 1.51 2.3 865 0.145 0.177* 22 PS 1.51 2.2 815 0.131 0.177 35 PS 1.51 2.2 815 0.131 0.171* 30 PS 1.51 2.5 991 0.162 0.188 16 PS 1.51 2.5 991 0.162 0.191* 18 PS 1.51 2.45 972 0.152 0.186 22 PS 1.51 2.45 972 0.152 0.189* 24 BB+ 1.28 2.0 667 0.117 0.148 26 BB 1.28 2.0 667 0.117 0.141* 20 BB 1.28 2.1 750 0.128 0.151 18 BB 1.28 2.1 750 0.128 0.149* 16 + PS = Polystyrene; BB = brown beans; GP = green peas. "Predicted using McNab (1972) correlation, Equation (2.19). "Predicted using Hadzisdmajlovic et al (1983) correlation, Equation (2.20), with H m predicted from Equation (2.13) and D 5 from McNab (1972) correlation. Figure 4.11: Dead zone boundaries viewed from front plate: (a) polystyrene, dp = 3.25 mm, Di = 88.9 mm, U/Umt = 1.1; (b) brown beans, dp = 4.7 mm, Di = 88.9 mm, U/Um. = 11; (c) green peas, dp = 6.7 mm, D{ = 88.9 mm, U/Um, = 1.1. Chapter 4. Ums, D„, Dead Zone, Hm, tip and AP„ 54 Figure 4.12: Influence of superficial gas velocity on dead zone boundary for polystyrene particles viewed from front plate, H = 2.00 m, dp = 3.25 mm, D{ = 88.9 mm. Chapter 4. Ums, Ds, Dead Zone, HM, HF and APS 55 Because the corner between the front flat plate and rear column wall had some influ-ence on the dead zone boundary, the measurements were also taken from the rear column wall. The results corresponding to Figures 4.11 and 4.12 are presented in Figures 4.13 and 4.14, respectively. The dead zone is seen to increase with static bed depth for brown beans and green peas (see Figures 4.13b and 4.13c), consistent with the measurements from front plate, and to decrease with static bed depth near the axis of symmetry along the rear column wall for polystyrene particles (see Figure 4.13a), contrary to the mea-surements from the front plate. Near the corner between the front flat plate and the rear column wall the dead zone boundary increased with static bed depth because the polystyrene particles were more angular and the "corner effect" had a strong influence on the dead zone. In other words, the measurements from the front flat plate for the polystyrene particles did not represent the real dead zone behaviour of a full column spouted bed. It is also noted that the superficial gas velocity had little influence on the dead zone boundary, in agreement with the results taken from the front plate (see Figure 4.14). For the spout-fluid bed tests, only polystyrene particles and brown beans were used to examine the dead zone boundary. Some comparisons are given in Figure (4.15) between the data from a standard spouted bed (qa — 0 m3/h) and from a spout-fluid bed (qa > 0 m3/h) for polystyrene particles. It is found that the volume of the dead zone decreased with auxiliary flow when the bed was operating in the spouting-with-aeration regime (Figure 4.15a), spout-fluidization regime (Figure 4.15b) and jet,slugging and bubbling fluidized bed regimes (Figure 4.15c). In addition, the dead zone was smaller for config-uration #1 (auxiliary flow supplied at all levels) than with configuration #2 (auxiliary air supplied through the upper two subdistributors only). The measured data from the rear column wall corresponding to Figure 4.15 (except that data could not be taken in N 0.0 0.2 0.4 0.6 0.8 r flattened (m) a 1.8-1.8 1.4-1.2 3 G a > O N 0.8 - x r ^ o --O' 0.6-0.4 0.2-Dead zone Conical base Legend O H=0.96 m O H=1.28 m V H=1.60 m T 0.0 0.2 0.4 0.6 0.8 r flattened (m) b 1.8 0.8 0.6 0.4 0.2 Dead zone Conical base Legend O H=0.96 i O H=t.tfi rr V H£l.64_n i 1 1 1 1-0.0 0.2 0.4 0.6 0.8 r flattened (m) c 8 &> r ti ( X ^3 Figure 4.13: Dead zone boundaries viewed from rear column wall: (a) polystyrene, dp = 3.25 mm, Dx = 88.9 mm, U/Umt = 1.1; (b) brown beans, dp = 4.7 mm, Di = 88.9 mm, U/Um, = 1-1; (C) green peas, dp — 6.7 mm, Di = 88.9 mm, U/Umt — 1.1. Chapter 4. Uma, Ds, Dead Zone, Hm, HF and A P , 57 Figure 4.14: Influence of superficial gas velocity on dead zone boundary for polystyrene particles viewed fron rear column wall, H = 2.00 m, dp — 3.25 mm, D{ = 88.9 mm. Figure 4.15: Influence of auxiliary flow on dead zone boundary for polystyrene particles viewed from front plate', H = 1.51 m, dp = 3.25 mm, Di = 88.9 mm, Q = 772 m3/h; (a) spouting with aeration; (b) spout-fluidization; (c) jet, slugging and bubbling fluidized bed. 1.6 0.4 0.2 q =0.0 m'/h U/Ums=2.3 qa=865 m 3/h Configuration #1 U/Um9=2.2 q„=815 m3/h Configuration #2 -1—'—r T T 0.0 0.2 0.4 0.6 0.8 r f lattened (m) a 0 . 6 -0 . 4 -0.2 o o UAJm.=i.i q.=0.0m3/h q11=991 m3/h Configuration #1 U/Un,,=2.45 qa=972 m3/h Configuration #2 1.6 J " 1 — 1 — I — 1 — I — 1 — I — r 0.0 0.2 0.4 0.6 0.8 r f lattened (m) b 0 . 6 -0.4 0 . 2 -Conical base O q.-O.O m J/h U/Uma=2.6 qa=1065 m 3/h Configuration #2 T" 0.0 0.2 0.4 0.6 0.8 r f lattened (m) C 1 I to Bi a . S3 e CL Figure 4.16: Influence of auxiliary flow on dead zone boundary for polystyrene particles viewed from rear column wall, H = 1.51 m, dp —- 3.25 mm, Di = 88.9 mm, Q = 772 m3/h; (a) spouting with aeration; (b) spout-fluidization; (c) jet, slugging and bubbling g fluidized bed. Chapter 4. Ums, Da, Dead Zone, Hm, Hp and AP, 60 the conical base section where the five layer subdistributors were made of steel) are pre-sented in Figure 4.16. It is seen that the dead zone reduction with auxiliary flow had the same trend as the results taken from front plate. In the case of the bed operating in jet, slugging and bubbling fluidized bed regimes with configuration #1, the dead zone disappeared altogether when viewed from the rear column wall. For brown beans with only a little auxiliary flow, the dead zone disappeared both when viewed from the front plate and when viewed from the rear column wall, as expected. Al l the experimental dead zone boundary data are listed in Tables J.9 through J. 18 in Appendix J. 4.2.4 Maximum spoutable bed height Due to the hmited quantities of the various particulate materials and finite height of the column, the maximum spoutable bed height was reached for only one material, brown beans. The measured H m value for this material was 1.94 m. Beyond this height, choking of the spout and slugging of the bed were observed. The values for H m predicted by Equations (2.11) and (2.15) are 18.3 m and 4.2 m, respectively, i.e. both equations overestimated Hm by a considerable margin. Following the same derivation procedure of Equation (2.11) and using Fane and Mitchell (1984) correlation, Equation (2.2), instead of Mathur and Gishler (1955) equation, Equation (2.1), give the following expression: where n = 1 - e x p { - 7 £ > 2 } (4.5) The value of H m predicted by Equation (4.4) is 5.52 m and it is also too high. An attempt was made to determine the H m value for the polystyrene particles, but the particles were Chapter 4. Um,, Da, Dead Zone, Hm, HF and APa 61 only enough to fill the bed to a depth of 2.60 rn and the maximum spoutable bed height was not reached. 4.2.5 Fountain height Few data on fountain heights could be obtained because the column height limitations restricted the range of measurements for deeper beds, higher gas velocities and spout-fluid beds. All observed fountain shapes were nearly parabolic and extremely dilute. The experimental results are presented in Figure 4.17 and Table J.19 in Appendix J . Fountain heights increased with increasing gas velocity. The data reported by Grace and Mathur (1978) and Lim and Grace (1987) indicate the same trend. Fountain heights increased with increasing bed height at a given U/Ums as shown in Figure 4.17. It was also observed that the fountain height was always greater than the static bed height for all the conditions studied. 4.2.6 Overall bed pressure drop The experimental overall bed pressure drops and the predicted values from Equation (2.32) are presented in Table 4.2 and Figure 4.18 for polystyrene, brown beans and green peas at U/Uma — 1.1. The pressure drop corresponding to fluidization in Equation (2.32) was estimated from: -APf = (pp-pf)(l-ea)gH (4.6) The predicted values of —AP„ are generally in good agreement with experimental values. The ratio of the pressure drop across the spouted bed to that across a comparable fluidized bed at minimum fluidization was always much less than unity. The effect of increasing gas velocity on the bed pressure drop is shown in Table 4.3. It is clear that the bed pressure drop decreased with increasing gas velocity for the range Chapter 4. Ums, Ds, Dead Zone, Hm, HF and A P , 62 3.5 • I 1 1 ' 1 1 1 1 1 1 1 1 1 0 0.5 1 1.5 2 2.5 3 H(m) Figure 4.17: Fountain height versus static bed depth for polystyrene, brown beans and green peas, U/Ums = 1.1, Di = 88.9 mm, qa = 0 m3/h Figure 4.18: Bed pressure drop versus static bed depth for polystyrene, brown beans and green peas, U/Umil = 1.1, D{ = 88.9 mm, qa — 0 m3/h Chapter 4. Um3, Ds, Dead Zone, Hm, HF and A P , 64 Table 4.2: Experimental overall bed pressure drops and predicted values from Equation (2.32). See Table 3.1 for particle properties. Particles H (m) H/Hm - A P . Expt'l (kN/m2) - A P . Pred." (kN/m2) -AP. -AP, Deviation (%) PS+ 0.55 0.16 0.80 0.33 0.11 -59 PS 0.83 0.24 1.05 0.82 0.18 -22 PS 1.20 0.35 1.63 1.83 0.27 12 PS 1.51 0.44 2.33 2.97 0.35 28 PS 1.76 0.51 3.21 4.04 0.41 26 PS 2.00 0.58 3.81 5.18 0.46 36 PS 2.30 0.67 5.13 6.71 0.52 31 PS 2.60 0.76 5.92 8.32 0.57 41 BB+ 0.55 0.28 1.59 0.90 0.20 -43 BB 0.78 0.40 1.87 1.94 0.31 4 BB 0.96 0.49 2.41 3.00 0.39 25 BB 1.28 0.66 4.16 5.26 0.51 26 BB 1.60 0.83 6.65 7.77 0.60 17 GP+ 0.55 0.18 1.42 0.37 0.10 -74 GP 0.78 0.26 1.89 0.87 0.16 -54 GP 0.96 0.32 2.46 1.42 0.21 -42 GP 1.18 0.39 3.66 2.30 0.27 -37 GP 1.44 0.48 4.83 3.56 0.35 -26 GP 1.64 0.54 5.97 4.69 0.40 -21 + PS — Polystyrene; BB = brown beans; GP = green pejus. "Predicted from Equation (2.32). Chapter 4. Ums, Ds, Dead Zone, Hm, Hp and APS 65 Table 4.3: Effect of increasing gas velocity on the bed pressure drop for polystyrene particles, H = 2.00 m, dp = 3.25 mm, Dl = 88.9 mm U/Um. - A P , (kN/m2) 1.1 3.81 1.3 3.66 1.5 3.34 1.1 < U/Ums < 1.5. The decreased pressure drop may be due to the increase in D 5 , to a decrease in gas flow in the annulus caused by increasing solids circulation rate (Epstein et al., 1978), and/or to increase in gas jet suction. The bed pressure drops for spouting with aeration are presented in Table 4.4. The data indicate that the bed pressure drop increased with increasing auxiliary flow, except for H = 2.00 m and qa = 721 m3/h with the polystyrene particles. This latter case was near the boundary between the spouting-with-aeration regime and the spout-fluidization regime. Configuration #2 (see section 4.1.7) was found to yield lower bed pressure drops than configuration #1. In the spout-fluidization regime the bed pressure drop was found to fluctuate. In the jet, slugging and bubbling fluidized bed regimes an increase in the magnitude of pressure fluctuations was noted. Further details are given in Chapter 6. Chapter 4. Ums, Ds, Dead Zone, Hm, Hp and A P , 66 Table 4.4: Effect of auxiliary flow on the bed pressure drop, Di = 88.9 mm. See Table 3.1 for particle properties. H 9a - A P , Particles (m) U/Ums (m3/h) Configuration (kN/m 2 ) P S + 1.51 1.1 0.0 - 2.33 PS 1.51 2.3 865 #1 4.73 PS 1.51 2.2 815 #2 3.81 PS 2.00 1.1 0.0 - 3.81 PS 2.00 1.7 453 #1 5.38 PS 2.00 2.1 721 #1 2.86 BB+ 1.28 1.1 0.0 - 4.16 BB 1.28 2.0 667 #1 6.37 + P S = Pohystyrene; BB = brown beans. Chapter 5 Gas Flow Distribution and Pressure Profiles in the Annulus 5.1 Measurement Technique 5.1.1 Vertical component of gas velocity in the annulus In order to determine the air flow through the annulus, a pressure drop versus superficial fluid velocity curve for a loosely packed bed of the polystyrene particles was first obtained experimentally using a 0.15 m I.D. glass column. The calibration curve is given in Appendix A. With the assumption that the annulus in a spouted bed behaves as a loosely packed bed of the same material, the static pressure gradient along the annulus in the vertical direction was measured and the readings were converted to the vertical component of annulus gas velocities using the calibration curve. The pressure gradient measurements in the annulus were taken using the probes installed along the back of the column. The longitudinal positions of the probes extended from the upper surface of the inlet orifice to 0.10 m below the bed surface. Measurements were taken at 0.3 m intervals in the cylindrical section and at 0.114 m intervals in the conical section. 5.1.2 Pressure profiles in the annulus The overall pressure drop across the bed was measured using a pressure tap located just below the gas inlet to the bed. The longitudinal pressure profiles in the annulus were measured by the same method, as described in the previous section. Radial pressure 67 Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 68 profiles were measured using the same probes by moving the taps radially through equal intervals (of 50 mm) from the interface between the spout and annulus to the column wall. 5.2 Results and Discussion 5.2.1 Vertical component of gas velocity Figure 5.1 plots the experimental data for the superficial gas velocity variation with height z in the annulus for the polystyrene particles. It is seen that in the cylindrical portion of the column the gas velocity increased with bed level, and the increase was more rapid in the lower section of the cylindrical portion. In order to compare the experimental data for Ua with existing equations from the hterature, it is important to determine UaHm, a quantity which could not be measured experimentally. However, Lim (1975) has shown that under any fixed conditions of Dc, Di, U/Ums, bed material, and spouting fluid, Ua at any given height above the orifice (z) is independent of bed height (H); also Ua rises rapidly with z and then levels off to a constant value. These two features make it possible to extrapolate the Ua data obtained at different heights below Hm to UaHm-The experimental results of Ua versus z together with the value of UaHm obtained by extrapolation are given in Table 5.1. Hm is another key parameter in the existing equations from the hterature. Unfor-tunately, the maximum spoutable height for polystyrene particles was not reached due to the hmited amount of this material, only enough to cover 2.60 m. Equation (2.11) (Littman et al. 1977), Equation (2.13) (Littman et al. 1977) , and Equation (2.7) (Gr-bavic et al. 1976) were employed to estimate Hm\ corresponding predicted values were equal to 22.4 m, 3.43 m, and 4.7 m, respectively. Because Hm = 22.4 m appears to be too high and Equation (2.7) was based on data for water-spouted beds of spherical Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 6 9 0.5 Mamuro-Hattori Model Height above orifice (m) Figure 5.1: Longitudinal annulus gas velocity distribution compared with equations in the literature where extrapolated values of H m and U c W m have been employed. (Particles: polystyrene, dp = 3.25 mm, D, = 88.9 mm, H = 2.0 m, U/Ums = 1.1, qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 70 Table 5.1: Experimental values of annulus gas velocities and extrapolated Uajj-cles: polystyrene, dp = 3.25 mm, Dt = 88.9 mm, H = 2.0 m, U/Ums = 1.1) z (m) 0.00 0.13 0.23 0.34 0.45 0.54 0.71 1.01 1.32 1.63 1.89 Ua (m/s) 0.14 0.15 0.16 0.18 0.19 0.20 0.26 0.31 0.35 0.37 0.39 Extrapolated: UaHm = 0.52 m/s particles, Hm = 3.43 m (from Equation 2.13) is used in present work. In Figure 5.1 the experimental data are compared to Equation (2.22) (Mamuro and Hattori, 1968) with UaHm replaced by Umj, Equation (2.23) (Lefroy and Davidson,1969), and Equation (2.24) (Modified L-D due to Epstein et al. 1978) with n = 2. The correlation of Lefroy and Davidson (1969) underpredicts Ua by a considerable margin, whereas the modified L-D Equation (2.24), shows excellent agreement in the cylindrical portion. Equation (2.22) (the Mamuro-Hattori model) does not give a good representation with increasing z. The data showing annulus superficial gas velocity variation with height for U/Ums — 1.3 and U/Ums = 1.5 are plotted in Figures 5.2 and 5.3, respectively. In the conical base portion the annulus superficial gas velocity increased with height, reaching a maximum in the cone region, and then falling to a minimum. The minimum corresponds to the position of the cone-cylinder junction. In the conical base section, Equation (2.25) (Rovero et al. 1983) predicts the same trend as the experimental result, but it consistently overestimates Ua by a considerable margin. In the cylindrical portion both Equation (2.25) and Equation (2.24) (modified L-D equation) give reasonable predictions. The influence of superficial gas velocity on the average annulus superficial velocity is shown in Figure 5.4. It is seen that in the cylindrical portion, Ua was not strongly Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 71 0.5 Rovero et al.Eq. Height above orifice (m) Figure 5.2: Longitudinal annulus gas velocity distribution compared with equations from literature. (Particles: polystyrene, dp = 3.25 mm, A = 88.9 mm, H = 2.0 m, U/Ums = 1.3, ga = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 72 0.5 Rovero et al.Eq. Height above orifice (m) Figure 5.3: Longitudinal annulus gas velocity distribution compared with equations from the literature. (Particles: polystyrene, dp = 3.25 mm, Di — 88.9 mm, H — 2.0 m, U/Ums = 1.5, qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 73 0.6 Legend 0.5 0.4 H • r=1.625 m • I=.1.320 m_ O 1=1.010 m A 1=0.710 m X £ = q ; 5 4 p _ m V JfP .-fil .T. -f- r=0.33S in O r=0 .232 m O 1=0.125 m r = 0 . 0 0 0 m CO 0.3 H co i : 0.2 H - • o.H — x 0.0 1.1 1.2 1.3 U / U 1.4 ms 1.5 1.6 Figure 5.4: Influence of superficial gas velocity on average annulus superficial velocity. (Particles: polystyrene, dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 74 influenced by the overall superficial velocity (at least over the range 1.1 < U/Uma < 1.5), although the general tend appears to be a decrease in annulus flow as U / U m a increases, consistent with earlier evidence of Epstein et al. (1978). In the conical base portion Ua first increased and then decreased with increasing U/Um,. Comparison of Figures 5.5 and 5.6 indicates that annulus superficial gas velocity increased with increasing auxiliary gas flow. Experimental data are not well fitted by Equation (2.29), the only existing equation derived for a spout-fluid bed, proposed by Littman et al. (1976) for aflat-base column. Sutanto (1983) has shown that the maximum bed height for spout-fluidization is always smaller or equal to the maximum spoutable bed depth, i.e. Hmsf < Hm. If i J m s / is replaced by Hm, Equation (2.29) is found to correlate the data for polystyrene particles quite well. So in Equation (2.29), Hm8f is replaced by Hm in the present work. Equation (2.29) then gives a reasonable prediction for UjUm, = 1.7 and qa = 435 m3/h, but overpredicts Ua for V'/Uma = 2.05 and qa = 721 m3/h. The influence of auxiliary gas flow on average annulus superficial velocity is shown in Figure 5.7. Ua was found to increase with increasing auxiliary flow. The increase was more substantial in the upper section of the cylindrical portion (z > 1.625 m) and in the entire conical base section (z < 0.54 m) than in the lower cylindrical portion. The results indicate that additional external fluid supplied directly into the annulus of a standard spouted bed can be accommodated without upsetting the basic behaviour of spouting. However, there is a hmit to how much auxiliary flow can be introduced before spout instability sets in, resulting in spout-fluidization. Radial profiles of annulus air velocities (expressed as local superficial velocity) are plotted in Figure 5.8 through Figure 5.12, all for polystyrene particles, with U/Ums = 1.1, 1.3, 1.5, 1.7, and 2.05, respectively. It is seen that the local superficial gas velocity decreased with increasing radial distance and increased, as expected, with height in the cylindrical portion as in earlier work (Lim and Grace, 1987) in the same scale flat-based Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 75 0.5 0.4 0.3 0 .2 -0.1 Fitted curve for r=0.10 m Legend • r=0.10 m A r=0.15 m + r=0.20 m O r=0.25 m o r=0.30 m r=0.35 m X r=0.40 m o r=0.45 m 0.0-r 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 H e i g h t a b o v e o r i f i c e (m) 1.8 2 Figure 5.5: Longitudinal annulus gas velocity distribution compared with Equation (2.29) proposed by Littman et al. (1976), assuming H m s y = H m . (Particles: polystyrene, dp = 3.25 mm, Dt = 88.9 mm, H = 2.0 m, U/Ums = 1.7, qa = 435 rn3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 76 0.5 _o CD > GO o CD "6 CJ CD > 0.4 0.6 0.8 1 1.2 1.4 1.6 He igh t a b o v e o r i f i c e (m) Figure 5.6: Longitudinal annulus gas velocity distribution compared with Equation (2.29) proposed by Littman et al. (1976), assuming H m s y=H m . (Particles: polystyrene, dp = 3.25 mm, D% = 88.9 mm, H = 2.0 m, U/Ums = 2.05, qa = 721 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 0.6 0.1 • z=1.625 m • z=1-320m O z=1.010 m A z=0.710 m X 2=0._5_40_m V z=0.445 m + z=0.335 m O z=0.232 m O z=0.125 m 0.0 0.0 0.1 0.2 0.3 q a / Q m f 0.4 — I — 0.5 0.6 Figure 5.7: Influence of auxiliary gas flow on average annulus superficial velocity. (P tides: polystyrene, dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, Q/Qms = 1-1) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 78 0.5-1 >~ "o _o co > V) D O) ~D O 2 0.0 0.1 0.2 0.3 0.4 0.5 Radia l d i s tance f r o m spou t ax is (m) Figure 5.8: Radial profile of annulus superficial gas velocity for polystyrene particles. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Um. = 1.1, q* = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 79 0.5 to 0.4 0.3-"o _o CD > If) D O 0.2 o 0.1 0.0 o CL CO D — • • -• - a — • -o c £ O O O-A" V -x-- o — o ...A A A - - - A • A —+- —+• — v V . . . X X' x \ ,.--x -: ~>?—x Legena • 1=1.890 m • I=1.62S m • 1-1.320 m_ O 1=1.010 m A 2=0.710 m X r=0.S40 m V 1=0.445 m + 7=0.3S5 m o r=0 232 m o r=0.125 m 1=0.000 m 0.0 0.1 0.2 0.3 0.4 0.5 Radial distance f rom spout axis (m) 0.6 Figure 5.9: Radial profile of annulus superficial gas velocity for polystyrene particles. (dp = 3.25 mm, Dt = 88.9 mm, H = 2.0 m, U/Ums = 1-3, qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 80 co D O 2 0.5 0.4 0.3 ' o _o > CO O cn 0.2 0.1-0.0 3 O CL CO A ' cr A A - - - - A A — A — - A " '1 - v t. X . X X X »x:-;-x-x! Legena • 1=1.890 m • 1=1.625 m • z=.1.320 m_ O z=1.010 m A z=0.710 m X z=0.540 m V z=0.445 m + 7=0-335 m o z=0.232 m o z=0.125 m B z = 0 . 0 0 0 m 0.0 0.1 0.2 0.3 0.4 0.5 Radia l d i s tance f r o m s p o u t ax i s (m) 0.6 Figure 5.10: Radial profile of annulus superficial gas velocity for polystyrene particles. ( i p = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Um, = 1.5, qa = 0.0 m3jh) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 81 0.45-1 0.40-0 .35 -"o _o CD > CO D fj 0.30 O 2 0 .25-0.20 =3 O CL CO • — -• z=1.890 m • z=1.625 m • z=.1.320 m O 7=1.010 m A z=0.710 m X z=0.540_m V J f 0^ .445 H + z=0.335 m O z=0.232 m O z=0.125 m 0.0 0.1 0.2 0.3 0.4 0.5 Radia l d i s tance f r o m spou t ax is (m) 0.6 Figure 5.11: Radial profile of annulus superficial gas velocity for polystyrene particles. (dp = 3.25 mm, Dt = 88.9 mm, H = 2.0 m, U/Umi = 1.7, qa = 435 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 82 0.55 0.1 0.2 0.3 0.4 0.5 Radia l d i s tance f r o m s p o u t ax is (m) Figure 5.12: Radial profile of annulus superficial gas velocity for polystyrene particles. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Umi = 2.1, qa = 721 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 83 spouted bed. In the conical base portion (z < 0.60 m) the local superficial gas velocity first increased slightly and then decreased with increasing radial distance for the cases without auxiliary flow (see Figures 5.8, 5.9, and 5.10). These three figures also indicate that the overall radial profiles of annulus air velocities were not strongly influenced by the overall superficial velocity, at least for 1.1 < U/Uma < 1.5. When auxiliary flow was added (see Figures 5.11 and 5.12), the radial profiles of annulus air velocities in the conical base portion did not change very much with increasing radial distance, but average annulus air velocities increased with increasing auxiliary flow. All experimental data are listed in Appendix B. 5.2.2 Pressure profiles in the annulus Longitudinal Pressure Profiles Some comparisons are given in Figure 5.13 between experimental data for the longitudi-nal pressure profile in the annulus for polystyrene particles and Equation (2.30) due to Epstein and Levine (1978) and Equation (2.31) due to Lefroy and Davidson (1969). The experimental data are more scattered in the lower section than in the upper section. In the cylindrical portion the longitudinal pressure profile was nearly independent of radial distance from spout axis, while in the conical base portion it was more strongly depen-dent on radial position. From Equation (2.30), the term (P — PH)/( —AP„) is derived by combining Equations (2.30) and (2.32) to give P-PE H = 1 -(23-4)f(x) + 3g(x) (23 - A)f(h) + 3g(h) (5.1) where f(y) = 1 V - y3 + 0.257/4 (5.2) g(y) = 3y3 - 4 V + 3y5 - y6 + 0.143y7 (5.3) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 84 Figure 5.13: Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Ums = 1.1, qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 85 Both equations are seen to be in good agreement with the experimental results. Com-parison of Figures 5.13, 5.14 and 5.15 shows that the longitudinal pressure profile in the annulus is only shghtly influenced by the superficial gas velocity for 1.1 < U/Umg < 1.5. The influence of auxiliary flow on the longitudinal pressure profile is shown in Figures 5.16 and 5.17. It is seen that the pressure profiles were almost independent of radial position when there was auxiliary flow. Note that the highest value of pressure drop was not at the orifice level (z = 0.0 m) in these two figures. Equations (2.30) and (2.33) both give good predictions in the cylindrical portion, while overpredicting in the conical base section of the column. Radial Pressure Profiles The radial pressure profiles for polystyrene particles at three different U/Uma values are shown in Figures 5.18, 5.19, and 5.20. Comparing these three figures, we find that the superficial velocity only shghtly influences the radial pressure profiles. The pressure in the annulus was almost uniform radially in the upper cylindrical portion (z > 1.625 m), but decreased with radial distance in the lower part of the cylindrical portion (0.6 m < z < 1.32 m), contrary to previously reported results (Mathur and Epstein, 1974) that radial pressure in the annulus is constant for any given bed levels above the conical region for columns of much smaller cross-section. The pressure also decreased in the overall conical portion. The effect of auxiliary gas flow on radial pressure profiles in the annulus is shown in Figures 5.21 and 5.22. The radial pressure in the annulus for any bed level is seen to be nearly constant, even in the conical base section. However, the overall pressure drop increased with increasing auxiliary air flowrate, consistent with the higher interstitial flow through the annulus already discussed. The experimental data are presented in Appendix C. Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 86 Figure 5.14: Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm: D{ = 88.9 mm, H = 2.0 m, U/Uma = 1.3, qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 87 CO Q_ O Q L I Q_ Figure 5.15: Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Um, - 1-5, qa - 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 88 CO Q_ <] I Q_ I Figure 5.16: Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm; D t = 88.9 mm, H = 2.0 m, U/Um. = 1.7, qa = 435 m3/^) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 89 C L <] Q_ I Q_ 0 . 8 0 .6 0 . 4 -0 . 2 -Figure 5.17: Longitudinal pressure profile for polystyrene particles at different radial positions. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Umi = 2.05, qa = 721 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 90 2-1-3 O CL if) v.. +- . A - . -o V — ' " X .V -x- -v-- A - - A -x" - A D C O o • — • a — - - a — • -L e g e n d • 2 = 1.890 m • 2 - 1 . 6 2 5 m • 2=1.320 m. O 2 = 1.010 m A 2=0.710 m X 2=0 .540 m V z=0.-445 m + 2= 0 . 3 3 5 m o 2=0.232 m o 2=0.125 m 2= 0 . 0 0 0 m 0.0 0.1 0.2 0.3 0.4 0.5 Radia l d i s tance f r o m s p o u t ax is (m) 0.6 Figure 5.18: Radial pressure profile for polystyrene particles. (dp = 3.25 mm, Di = 88.9 mm, H = 2.0 m, U/Um. = 1-1, Qa = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 91 CO CO a> Q_ O 3 -2 -0 3 o CL 8-o-+-- V X A -• — - a — • - a — • -Legena • 7=1.890 m • 7=1.625 m • 7=J.3_20 m_ O 7=1.010 m A 7=0.710 m X 2=0 .540 m V 7=0.445 m + 2= 0 . 3 3 5 m o 2=0 .232 m o 2=0.125 m IE) 2 = 0 . 0 0 0 m 0.0 0.1 0.2 0.3 0.4 0.5 Radial d is tance f r o m s p o u t ax is (m) 0.6 Figure 5.19: Radial pressure profile for polystyrene particles. (dp = 3.25 mm, D{ = 88.9 mm, H = 2.0 m, U/Umf = 1.3, qa = 0.0 m3) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 92 CD i_ Z5 CO CO CD i_ Q_ O ~o 00 3 -"A- - A d. 1 CO • - a — • . o-D 3 1 O CJ Legend • 2=1.890 m • 2=1.625 m • z=l.320 m O 2=1.010 m A 2=0.710 m X 2=0.540 m V 2=0.445 m + 2=0.335 m o 2=0.232 m o 2=0.125 m 2=0.000 m 0.0 0.1 0.2 0.3 0.4 0.5 Radia l d i s tance f r o m spou t axis (m) 0.6 Figure 5.20: Radial pressure profile for polystyrene particles. (dp = 3.25 mm, £),• = 88.9 mm, H = 2.0 m, U/Um. = 1.5, ga = 0.0 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 93 CD i_ Zi CO CO CD i_ Q_ O 15 00 2 -1-o Q_ cn i o — o o — o — +—+- -v V — X X - - -A A-•o •+-•V-• X -• A •V-•X--+ •V--• X - - • X -—  A - - A A A - •1 I D C E D 1 U --O-^-O- •-o • • L e g e n d • 2 = 1.890 m • 2 = 1.62S m • 2=.1.320 m O 2 = 1.010 m A 2 = 0.710 m X 2=0 .540 m V 2=0.445 m 2= 0 . 3 3 5 m o 2=0 .232 m o 2 = 0.125 m 3 2 = 0 . 0 0 0 m 0.0 0.1 0.2 0.3 0.4 0.5 Radia l d i s t ance f r o m spou t ax is (m) 0.6 Figure 5.21: Radial pressure profile for polystyrene particles. (dp = 3.25, mm, D{ = 88.9 mm, H = 2.0 m, U/Um. = 1.7, qa = 435 ms/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 94 to to <D C L (J ]o 4 -2 -O CL O-o- -o -o--+ v V x—-X A A O +— V — X--A - -—V---X---v v -•X—-x--• A — A -1 I D ••o—•- • - o — • • Legend 2=1.890 m • i=l.62S m • 12.'-3.2P "I Q 2=1.010 m A 2=0.710 m X £=9.540.1' V 1=0.445 rn 2=0.335 m O 2=0.232 m O 2=0.125 m 13 2=0.000 m 0.0 0.1 0.2 0.3 0.4 0.5 Radial d i s tance f r o m s p o u t ax is (m) 0.6 Figure 5.22: Radial pressure profile for polystyrene particles. (dp = 3.25, mm, Di = 88.9 mm, H = 2.0 m, U/Uma = 2.05, qa = 721 m3/h) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 95 5.3 A M a t h e m a t i c a l M o d e l for P r e d i c t i n g the G a s F l o w D i s t r i b u t i o n a n d Pressure Profi le in the A n n u l u s 5.3.1 In troduct ion A number of attempts have been made to model the hydrodynamics of spouted beds. One of the more promising techniques for modelling the annulus region, the region where intimate gas-solid contacting is achieved, involves finite difference solutions of a vector form of the well-known packed bed pressure drop equation due to Ergun (1952). This model was originally applied to a flat-bottomed spouted bed by Littman et al. (1977) and extended to the more commonly involved conical base geometry by Rovero et al. (1983). This model has also been extended by Brereton et al. (1984) to treat the case of a promising modified spouted bed geometry introduced in Japan (Hattori and Takeda, 1978), involving a side-outlet column and an impermeable inner draft tube. In the present work this model is extended to the conical base spout-fluid bed. 5.3.2 T h e mathemat ica l m o d e l The model extends earlier work due to Littman et al. (1977), who used the Ergun (1952) equation in its vector form to describe the flow field in the annulus and to predict the maximum spoutable bed depth of spouted beds. The annulus was treated as a loosely packed bed. Brereton (1982) used this approach to predict velocity and pressure distri-butions in a spouted bed. The model developed here is very similar to that employed by Brereton (1982), but with the following modifications: • It has been adapted for a spout-fluid bed (rather than a pure spouted bed) by changing the boundary condition along the conical base. Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 96 • The finite difference two-way line-by-line method with over-relaxation is used, which ensures the most rapid transfer of all the boundary information to the interior of the domain and proves to converge much faster than the Gauss-Siedel successive over-relaxation method. • The adaptive Newton-Coates integration method is used to calculate the annulus flow rates at different bed levels. 5.3.2.1 The velocity equation The continuity equation for the fluid, assumed to be incompressible, is V-{7 = 0 (5.4) The vector form of the Ergun equation, - Vp = U(f1 + f2\{j\) (5.5) is assumed to govern the velocity field. Since the curl of a gradient is equal to zero, taking the curl of Equation (5.5) produces an equation purely in velocity, V x ( - V p ) = 0 = V x L 7 ( / 1 + / 2 | L r | ) (5.6) Algebraic manipulation of this equation (see Appendix D) gives V x l 7 = C / x V l n ( / 1 + /2|^|) (5.7) Equation (5.7) must be solved simultaneously with the continuity equation to obtain the velocity solution. This is achieved most simply by introducing the streamfunction tp. For axisymmetic motion the streamfunction is defined by Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 97 r dr (5.9) where z is the height and r the radial coordinate. Since the geometry is assumed to be perfectly axially symmetric, the streamfunction (tp) automatically satisfies the continuity equation. Equations (5.8) and (5.9) are now substituted into Equation (5.7). After some manipulation, an equation for the stream-function is obtained: 1/2 fir h + dz I I dr , I?* [dz fir + d2j> dr2 +2 _2 r / a l dijj dip d2ip 1/2 + d2j> dz2 dz dr drdz h 1/2 dr * = 0 Or Details are provided in Appendix E. The boundary conditions are as follows: (5.10) At the bed surface, the pressure is uniformly equal to atmospheric pressure and the bed surface is horizontal. Hence / 9 A =0 (5.11) dr Equation (5.5) can be rewritten in cylindrical coordinates as (5.12) Equating the radial components of Equation (5.12), and substituting Equation (5.11),we obtain - ( f ) = 0 = c7 r(/ 1 +/ 2 |c7|) at z = H (5.13) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 98 From Equation (5.8), the assumption of uniform radial pressure at the upper bed surface [z = H) implies that (5.14) At the spout—annulus interface, the pressure is assumed to be governed by the Epstein-Levine (1978) 'hybrid' pressure distribution, that is — A P P-PH = -ypzrqW - 2){i-^ 2 - *2) -(h3 - x3) + 0.25(/i4 - z4)} + 3{3(/i3 - x3) -4.5(/i4 - a;4) + 3(/i5 - z 5) - [h6 - x6) +0.143(/i7 - x7)}} (5.15) where ,5 = 2+ (3/ 1/2/ 2L r m^), h = H/Hm, x = z/Hm and —AP/ is the pressure drop through a fluidized bed of height H and voidage ea of the same material. The packed bed flow equation must also apply at this interface. Thus from Equation (5.5), equating the z component of the vectors and recognizing that the z component of the pressure gradient is equal to the derivative of the right-hand side Equation (5.15) with respect to z, uA\fi + f2\U\) = ~^^W-2) x (3x - Zx2 + x3) + 3(9z2 - 18a;3 + 15z4 -6x5 + 1.001a;6)] (5.16) Finally, writting the left hand side in terms of the streamfunction produces the spout-annulus boundary condition in the desired form, Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 99 Figure 5.23: Geometrical parameters used.in the mathematical model 1 dxp r dr \ dz I \ or l/2> -AP, H(2f3-1) x[2(/3 - 2)(3a: - 3x2 + a:3) + 3(9x2 - 18 +15x4 - 6x5 + 1.001a:6)] (5.1' The boundary at the wall of the bed, is physically characterised by there being no flow across it. In terms of the streamfunction this is written as, ip-wali — constant For simplicity and to give the streamfunction a physical meaning, we set IP-wall = 0 (5.18) (5.19) Along the conical base (see Figure 5.23) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 100 uz = Un sin 7 (5-21) -c7 ncos 7 (5.22) where 7 is the half included cone angle and qa is the auxiliary flowrate. By solving ur Equation (5.9) with boundary condition ip = 0 at r = rc, we obtain i> = \Un[Tl-r* + 2rs{r-rc)} on the conical base surface (5.23) The details of developing Equations (5.20) and (5.23) are presented in Appendix F. 5.3.2.2 The pressure equation Taking the divergence of Equation (5.5) and imposing the continuity condition, Equation (5.4), gives the pressure field equation. V 2 p = -/ 2(c7.V|c7|) (5.24) Substitution of the appropriate pressure derivatives into Equation (5.24) now yields the pressure field equation in a form suitable for solution (see Appendix G). r dr I dr J dz2 2duT u.— h uTuz duz duT 2duz (5.25) r dr T z \ dr dz J ' 2 dz It should be noted that the right hand side of the pressure distribution equation is a function of the point velocity; it is hence necessary to solve for the velocity field before the pressure field can be computed. The boundary conditions are as follows: At the bed surface, the gauge pressure is set uniformly to zero, i.e. P(z=H) = 0 (5.26) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 101 At the spout—annulus interface, the pressure is given by the Epstein-Levine distri-At the wall, the directional derivative of the pressure perpendicular to the wall is where s is a coordinate normal to the wall. 5.3.3 Method of solution 5.3.3.1 Formulation of dimensionless equations The differential equations and boundary conditions governing both velocity and pressure distributions have been derived. For simplification these are written in terms of dimen-sionless variables describing geometry and flow: Dimensionless radius, bution, equation (5.15). equal to zero, i.e. (5.27) (5.28) Dimensionless height, (5.29) Dimensionless streamfunction, (5.30) Dimensionless pressure, P = V (5.31) Dimensionless radial velocity, (5.32) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 102 Dimensionless vertical velocity, Uz = ~ (5.33) 5.3.3.2 Summary of dimensionless partial differential equations governing the pressure and velocity fields The dimensionless strearnfunction equation and boundary conditions are now summarised as follows: h(28 - l)Hmf2Umf x [2(8 - 2)(3x - 3x2 + x3) + 3(9x2 - 18x3 + 15z4 - 6a;5 + l.OOlx6] for rs/rc < R < 1, 0<Z<1 (5.38) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 103 Solution of above equation yields the streamfunction and thence using Equations (5.8) and (5.9), radial and vertical velocity components. The corresponding dimensionless pressure equation and its boundary conditions are: RdR R d P  RdR T j 2 dUR TV\ DUR HJ dZ2 dUz + U » U z ( ^ + { i l ) " 8 i ) + U h H J dZ dU2 (U2R + U2) 2^ /2 PRA = 0 rjrc <R<1 (5.39) (5.40) P(T,/T,,Z) --APF -(h3 - x3) + 0.25(/J4 - x4)} + 3{3(/i3 - a;3) -4.5(/i4 - x4) + Z{h6 - x6) +0.143(/i7 - x7)}} DP ds = 0 The annulus flow rate at different bed heights is calculated from Q* = 2irUmfr2 [' Uzrdr (5.41) (5.42) (5.43) 5.3.3.3 F i n i t e d i f ference g r id The finite difference line-by-line method with over-relaxation relies upon expressing the value of the scalar field to be solved at a point explicitly in terms of surrounding points such that the governing partial differential equation is satisfied. A grid must first be constructed to cover the geometry under consideration. For the conical base spout-fluid bed: • The radial grid spacing is chosen as Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 104 Figure 5.24: The finite difference grid rs/rc N (5.44) • Set AR tan 7 (5.45) AZ is thus chosen such that the cone intersects grid points along its length (see Figure 5.24). However, by establishing the grid in this manner the upper bed surface will generally not lie on a grid line. This is simply catered as: 1 M = (mod AZ Then set Z(MP) = 1 where MP = M + 1 and M is the number of vertical grid points. (5.46) (5.47) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 105 (i+i.j+l) 0+l.j) (i+l.j-l) 0.i+D (i.j) O.j-l) (i-l.j+l) (i-U) (i-U-1) Figure 5.25: The classification of grid points 5.3.3.4 D e r i v a t i o n o f t h e f in i te d i f fe rence s t r e a m f u n c t i o n Figure (5.25) shows the grid nomenclature on which the finite difference approximations are based. The central difference partial derivatives are expressed for a scalar field F as - + 0[(Ai?)2] (5.48) dR 2AR " ' + 0[(A2f] (5.49) + 0[(Ai?)2] (5.50) dF F -^ - F i dZ 2AZ d2F Fl+1j — 2FtiJ + Fi-ltj dR2 (AR)2 d2F _ Fj,j+i — 2Fj}3 4- Fjj-i dZ2 ~ (AZ)2 d2F _ Fj-ij+i — Fi+ij+i — -f Fi+ij+i 4- 0[(AZ)2} (5.51) + 0[(Ai? 4- AZ)2] (5.52) dRdZ AARAZ When the preceeding expressions for the finite difference expressions are substituted into the continuous differential equation for the dimensionless streamfunction, Equation Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 106 (5.34), one obtains + Fxtti+x,.,- - 2(Fa + F2)%tj -f FiVij-! + F2ViJ+1 + F3 = 0 (5.53) where fi Umff2 ,+2 a* 1 i 1/2 ai? y V # / \dZ J (AR) (5.54) / i Umff: Rr a* az + -,1/2 + 2 { 8 R a#J j (AZ)2 F 3 = ^2 #J va^J vazj \a/?azj / i TV a* fry \az + a* ai? 1/2 a#\ ai?J a * y / a * ir ; \ai? j Ma i? a#\ 1 j_ ai? j i Ri (5.55) (5.56) All of the partial derivatives included in Fi ,F 2 , and F 3 are functions only of SI/i+1]j, and not of At the wall, equation (5.35) equates the streamfunction to zero, hence (5.57) Along the conical base equation (5.37) gives, 1 Un 2U„ [ l - i ? 2 + 2 - ^ ( i ? t - l ) j (5.58) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 107 At the bed surface, one can write + Q[AZ] (5.59) dZ AZ Equating this to zero yields the finite difference boundary condition, %, m = *;,m-i (5.60) Where m is the number of vertical grid points. At the spout annulus interface, equation (5.38) is a first order non-linear partial differential equation which must be solved each time this boundary is met in the relax-ation. Writing dR ~ AR— ( 5 - 6 1 ) and approximating the vertical derivative by the central difference equation (5.38), this equation may be solved. The governing equation is first manipulated to yield a fourth order polynomial in d*/dR. - * " ) ( © ' + 2 * ^ - R}F; = 0 (5.62) where Umff2, +3(9x2 - 18x3 + 15a;4 - 6a;5 + 1.001a;6)] (5.63) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 108 This polynomial was solved by the root finding subroutine "UBC Root" which solves for all roots, both real and imaginary. The single real negative root is now selected and designated "rootl". Substituting the forward difference definition of the radial derivative yields, *(nj) = *(n-ij) - AR(rootl) (5.64) where n is the number of radial grid points. The Mamuro-Hattori relationship was used to initialize the streamfunction matrix. This relationship in streamfunction form is, ( 1 - ^ ) 1 1 - \Hm). (5.65) 5.3.3.5 Finite difference pressure field equation In the bulk annulus, where 1 1 + AR2 - 2 2RAR) Pi~hj + \AR2 AR2 \H AZ' 2RAR} TV HAZ HAZ Pij+i - F(UR,UZ) = 0 F(UR,UZ) = -dUz fr, dR AP, dUR OR HJ dZ + U Z [ , H ) dZ 1 (5.66) (5.67) Boundary conditions: At the bed surface and spout-annulus boundaries respectively, equations (5.40) and (5.41) may be applied directly. At the column wall, setting the normal pressure derivative to zero gives, 2,i (5.68) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 109 At the conical base wall set ( Pi + i,j C O S T I Pi,i + i sin 7  rcAR HAZ  cos 7 i sin 7 rcAR ' HAZ As with the velocity field the pressure field required an initial estimate of the pressure values at any point. Except for the spout-annulus interface where the Epstein and Levine relationship, Equation (5.41), is used and the bed surface where P is set equal to 0, P was initially set equal to 0.5 at all points. 5.3.4 C o m p u t i n g m e t h o d The finite difference line-by-line method with over-relaxation on a uniform grid is used. In this method, the finite difference equations at each point are written in the forms: FlVi-u + F2Vij + F3Vi+lij = FA (5.70) in a row C l t f i j-i + C 2 * i j - + C 3 * i | j + 1 = C4 (5.71) in a column DlPi-ltj + D2Pid + D3Pi+1J = D4 (5.72) in a row S l P i j - i + E2Piti + EZPiJ+l = E4 (5.73) in a column where F l , F2, F3, F4, CI , C2, C3, C4, D l , D2, D3, D4, E l , E2, E3 and E4 are all constants. These sets of equations are easily solved because they are tridiagonal. The two-way line-by-line method involves solution first of rows from bottom to top, and secondly of (5.69) Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 110 columns from the outside to the inside radius in order to ensure the most rapid transfer of all the boundary information to the interior of the domain. The procedure is repeated until convergence is obtained. After the radial and vertical velocity components are obtained, the adaptive Newton-Coates method is used to calculate the annulus flow rates at different bed heights. The computing program is presented in Appendix H. 5.3.5 R e s u l t s a n d d i scuss ion The values of the input variables were • bed radius rc = 0.457 m • bed height H = 2.0 m • mean particle diameter dp = 3.25 mm (polystyrene) • annulus porosity ea = 0.44 • particle density pp = 1040 kg/m3 • auxiliary flow rate qa = 0.0 m3/s, 0.12 m3/s and 0.20 m3js • maximum spoutable bed height Hm — 3.43 m (predicted by Equation 2.13) • spout radius r„ = 0.064 m (calculated from the equation of McNab (1972)) • minimum fluidization velocity Umf = 0.53 m/s. Typical results of this model are shown in Figures 5.26, 5.27 and 5.28 for particular sets of conditions. Streamlines in the annulus are shown by solid curves and pressure profiles by dashed fines. Predictions for a spout-fluid bed (qa > 0) are shown in Figure 5.27 and Figure 5.28, with the corresponding results for a "pure" spouted bed (qa = 0.0) in Figure 5.26. The total flow and distribution of flow leaving through the top of the Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 111 RADIAL COORDINATE i R Figure 5.26: Predictions of vector Ergun equation model for qa = 0.0 m3/s. Solid lines give gas streamlines; corresponding numbers indicate fraction of total flow between streamline and wall. Dashed lines show isobars, with numbers indicating the fraction of total pressure drop. Chapter 5. Gas Flow Distribution and Pressure Profiles m the Annulus 112 RADIAL COORDINATE, R Figure 5.27: Predictions of vector Ergun equation model for qa = 0.12 m3/s. Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 113 0 0 0.2 0.4 0.6 0.8 1 .0 RADIAL COORDINATE, R Figure 5.28: Predictions of vector Ergun equation model for qa = 0.20 m3/s. Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 114 annulus is virtually the same for the three cases. However, the distribution of flow entering the annulus from the spout is quite different in the two cases, with more gas entering the annulus from the spout near the central inlet for the spouted bed (see Figure 5.26). As auxiliary flow rate increases, some gas even enters the spout from the annulus (see Figure 5.28, 9 = 0.400 line). This is consistent with observations in the experiments when the auxiliary flowrate was high enough, and it was observed that some particles were blown from the annulus into the spout by auxiliary gas flow in the conical base portion. In Figure 5.26 the varying separation of the 9 = 0.085 streamline from the outer wall indicates that the annulus gas velocity clearly passes through a maximum in the cone region and a minimum at the cone-cylinder junction, in qualitative agreement with the experimental results (see Figures 5.1, 5.2 and 5.3). The pressure distribution along the the spout-annulus boundary is the same for each case, since the same boundary condition, Equation (5.15), was applied in both cases. With or without the auxiliary flow, the predicted shape of the isobars in the cylindrical portion of the column is quite similar, consistent with the experimental results in Figures 5.18, 5.21 and 5.22. As the auxiliary flow increases, the radial pressure profiles in the conical base section only changed very little, whereas the experimental results indicate, with auxiliary flow, that the radial pressure in the annulus was nearly uniform for any given bed level. For the case of qa = 0.12 m3/s (435 m3/h), the predicted longitudinal pressure profile at r=150 mm is plotted in Figure 5.29. It is seen that the model based on the Ergun equation gives accurate predictions in the conical base portion and reasonable predictions in the cylindrical portion of the column. All of computed results are listed in Appendix I. Chapter 5. Gas Flow Distribution and Pressure Profiles in the Annulus 115 Figure 5.29: Compansion of pressure profile predicted from various models with exper-imental values. (dp = 3.25 mm, D{ = 88.9 mm. H = 2.0 m, U/Umt = 1.7. qa = 435 m3/h) Chapter 6 Regime Maps and Minimum Fluid Flowrate for Spouting-With-Aeration and Spout-Fluidization 6.1 Preliminary Measurement of Minimum Fluidization Velocity (Umf) The minimum fluidization velocities of the polystyrene particles and the the brown bean particles were determined experimentally using a 0.15 m I.D. fluidized bed glass column. For a suitable amount of particles (H=0.65 m) the solid bed was fluidized and shut down first. Then the air flowrate was increased stepwise and the bed pressure drop and flowrate were recorded. The minimum fluidization velocity was determined by the standard method (intersection of two linear portions) from a plot of pressure drop versus air flowrate. The minimum fluidization flowrates (Qmf) of the polystyrene particles and the brown bean particles in the 0.91 m I.D. half-cylindrical column were calculated by multiplying the minimum fluidization velocity (Umf) by the bed cross-sectional area (Ac). 6.2 Experimental Techniques Two methods were used to determine the hydrodynamic regime: • Method #1—The central air flowrate was increased/decreased continuously while maintaining the auxiliary air flowrate constant. In this case, the jet enters either a packed or a fluidized bed, depending on whether the auxiliary flowrate is above or below that required for minimum fluidization. 116 Chapter 6. Regime Maps and Minimum Fluid Flowrate 117 • Method #2—The auxiliary air flowrate was increased/decreased stepwise while maintaining the central air flowrate constant. From all the runs which were made, it was found that the flow regimes were quite distinct and could be identified visually. Usually the transition from one regime to another was accompanied by either a sudden increase or decrease in the magnitude of bed pressure drop fluctuation. These observations thus formed the basis for distinguishing the different hydrodynamic regimes. 6.2.1 Transition from spouting-with-aeration to fixed packed bed For a given bed height1, bed material and inlet orifice, spouting was initiated by supplying sufficient central spouting air. Auxiliary flow was then supplied and maintained constant while the central air flowrate was gradually decreased (Method #1), until a point was reached where there was a sudden increase in bed pressure drop as shown in Figure 6.1 and the spout was observed to collapse. Point (C) was taken as the condition for minimum spouting-with-aeration. The auxiliary flow was supplied in the range from no flow to a value where fluidization of the upper bed annular solids was attained. Thus, a set of points on the boundary between a fixed packed bed and spouting-with-aeration was estabhshed. 6.2.2 Transition from spouting-with-aeration to spout-fluidization In this case, Method $-2 was employed. Spouting was estabhshed initially and the central air flow was then kept constant while the auxiliary flow was increased stepwise until a point was reached where spout instability was observed. This marked the onset of spout-fluidization. At this condition the annular solids at the bed surface surrounding JThe bed height was measured from the top surface of the orifice plate to the upper surface of the solid bed which was in a loose packed bed condition. Chapter 6. Regime Maps and Minimum Fluid Flowrate 118 CQ » n i 1 1 1 1 1 1 1 200 300 400 500 600 Central air flowrate (m 3 / h ) Figure 6.1: Typical pressure drop vs. central air flowrate for the polystyrene particles, (dp=3.25 mm, Dc=0.91 m, A=88.9 mm, c;a=384 m3/hr) Chapter 6. Regime Maps and Minimum Fluid Flowrate 119 the spout were fluidized, and the behaviour of the bed resembled that of a spouted bed at rl—Hm where spout instability just begins to develop. The transition was accompanied by fluctuation of the bed pressure drop, and it could also be determined by the curve of bed pressure drop versus auxiliary flowrate as shown in Figure 6.2 (where point C was taken as the transition point). Several initial spouting flowrates (central air flow) in the range of 1.0 Qmt to 2.0 Q m s were used to obtained different experimental points. 6.2.3 Transition from spout-fluidization to jets, slugs, and bubbles in flu-idized bed Auxiliary air was introduced to a spouted bed until spout-fluidization was obtained as described in the previous section. Once this state was attained, the central air flow was decreased slowly and the bed was observed to gradually undergo a transition state where the condition of jet in fluidized bed and spout-fluidization alternated. A point was reached, upon further reducing the central air flow, where the bed was in a completely bubbling or slugging2 mode. This transition was accompanied by an increase in the magnitude of pressure fluctuations due to the rise and collapse of the bed surface as bubbles and slugs broke through. 6.3 Results and Discussion 6.3.1 Regime maps A typical phase diagram, plotted in the same manner as by Sutanto et al. (1985), is shown in Figure 6.3. The ordinate and abscissa denote the auxiliary flowrate and central flowrate normalized with respect to minimum fluidization flowrate, Qmf. The transition 2If the bubble size, (deq), is greater than 0.6D, where D is the column diameter, it is called a slug. Please refer to Clift (1986) for greater detail. Chapter 6. Regime Maps and Minimum Fluid Flowrate 120 60 Figure 6.2: Typical pressure drop vs. auxiliary air flowrate for the polystyrene particles, (rfp=3.25 mm, JDC=0.91 m, £>,=88.9 mm, ff = 1.5m,0 = 492 m3/h) Chapter 6. Regime Maps and Minimum Fluid Flowrate 121 A Observed spouting with aeration O Observed spout-fluidization 0 Observed transition between SF and JSBF • Observed transition between JSBF and LF • Observed transition between SA and F Figure 6.3: Regime map of the polystyrene particles. (D{ = 88.9 mm, H = 1.5 mm, dp = 3.25 mm, G—Fluidization at minimum condition, A—Spouting at minimum con-dition, C—Spout-fluidization at minimum condition.) Chapter 6. Regime Maps and Minimum Fluid Flowrate 122 points, determined as described in Section 6.2, have been used to define the boundaries of the various flow regimes. Four fairly distinct flow regimes have been delineated: • Spouting with aeration (SA), • Spout-fluidization (SF), • Submerged jets, slugs, and bubbles in fluidized bed (JSBF), • Packed bed (P). 6.3.1.1 Spout-fluidization In spout-fluidization, stable spouting was not achieved. The spout diameter at the upper section of the bed appeared to shrink and expand in a periodic manner. The spout was observed to be continuous. This differs from the findings of Sutanto (1983) in a half-column of diameter 152 mm who observed that the spout was not continuous. Instead, it broke into large bubbles near the bed surface. The front of the half-column was examined by means of a Strobotac (stroboscopic light generator) and super-8 mm movie film taken at 64 frames per second. In the larger spout-fluid bed column explored in the present work the discontinuous spout appeared only in the transition zone between spout-fluidization (SF) and submerged jets, slugs, and bubbles in fluidized bed (JSBF). As the central air flow was decreased while maintaining auxiliary air flow constant, the unstable spout first broke into fast jets and then into slugs (big bubbles), as shown in Figures 6.4a and 6.4fc, respectively. In spout—fluidization, the annular solids at the upper section of the bed were being fluidized, but overall fluidization could not be observed. This condition was determined by the appearance of unstable spouting or by examining the curve of bed pressure drop Chapter 6. Regime Maps and Minimum Fluid Flowrate 123 Figure 6.4: Transition from spout—fluidization (SF) to jets, slugs, and bubbles in flu-idized bed (JSBF) for polystyrene particles: a —slugs in fluidized bed, Q=561 m 3 /h : b —jets in fluidized bed, Q=715 m 3 /h : c —spout-fluidization, Q=773 m 3 / h . (dp=3.25 mm, £>,-=88.9 mm, H = 1.5 m, qQ = 1080 m 3 /h). Chapter 6. Regime Maps and Minimum Fluid Flowrate 124 versus auxiliary flowrate as shown in Figure 6.2. The appearance of this regime is shown in Figure 6.4c. The shape and structure of the fountain were entirely different from those observed for stable spouting. The fountain was formed by solid material periodically ejected by the spout shrinking and expanding in a periodic manner. A maximum height was achieved with the expansion of the spout and a minimum immediately thereafter ( Figure 6.4c). The movement of annular solids near the bed surface exhibited slip—stick type be-haviour, and the solids movement gradually became more continuous with decreasing height. The behaviour of spout-fluidization was not similar to that of a standard spouted bed whose height equals the maximum spoutable bed depth, unlike behaviour in a 152 mm bed examined by Sutanto (1983). 6.3.1.2 Jets, slugs, and bubbles in fluidized bed For the jets, slugs, and bubbles in the fluidized bed regime there were four states of fluidization. Near the boundary between spout-fluidization (SF) and jets, slugs, and bubbles in fluidized bed (at higher central flow), jets were noted. At an intermediate central flow, slugging of the sohd material in the upper section of the bed was observed. At a lower central flow, bubbles were observed. Finally, with only a little central flow, the total upper section of the bed resembled the behaviour of a bubbling bed. The transition from spout-fluidization (SF) to submerged jets, slugs, and bubbles in fluidized bed (JSBF) was gradual. The appearance of the transition is shown in Figure 6.4. When the central air flow was decreased gradually, the unstable spout first gave way to fast jets and then to slugs (big bubbles) as shown in Figure 6.5. Also the frequency of slug formation decreased. This allowed slugs to grow larger in size before reaching the bed surface. The slugs moved along the axis vertically; they were non-spherical voids with a particle core inside. As the central air flow was decreased, bubbles which were Chapter 6. Regime Maps and Minimum Fluid Flowrate 125 large but smaller in size than slugs appeared and moved along the axis vertically like the slugs (see Figure 6.66). The bubble shape was almost spherical on the bubble bottom and flat at the top. A further decrease of central air flow resulted in smaller bubbles which did not move along the axis but migrated away from the axis as shown in Figure 6.6c. Finally, when only a httle central air flow was maintained, the upper section of the bed resembled the behaviour of a bubbhng fluidized bed. In all cases described above, the bubbles formed in the middle section of the bed, in contrast to Sutanto's work (1983) where the formation of the bubbles occurred quite early, generally a few centimeters above the inlet orifice plate. Inside the slugs (see Figure 6.5) it was observed that there were many particles which moved in the same direction as the slug. For the bubbles in the fluidized bed, particles were carried upwards in the wake of the bubble (defined as the sohds occupying the bottom of the completed sphere) and in the drift (in the region behind the completed sphere). Particles appeared to be pulled into the wake or drift, carried up the bed a distance, and then shed. Thus, particles travelled upwards where there were bubbles and downwards elsewhere. For the polystyrene particles there is a local fluidization regime in the jets, slugs, and bubbles in fluidized bed (see Figure 6.3). In this case, the central area of the bed surface was fluidized. The regime map for the brown bean particles is shown in Figure 6.7. In all cases hne C-E has a positive slope, indicating that penetration of the spout through the bed surface in a fluidized state requires greater central spouting flow; other-wise, slugging and bubbhng are achieved. Polystyrene particles, which have an elliptical shape, showed a greater tendency to interlock and bridge. They are more likely to show slugging behaviour. This is reflected in the smaller slope of hne C-E compared with that for brown bean particles (see Figures 6.3 and 6.7). Chapter 6. Regime Maps and Minimum Fluid Flowrate 126 Figure 6.5: Appearance of a slug in polystyrene bed. (dp = 3.25 mm, Di = 88.9 mm, H = 1.5 m, qa = 1012m3/h, 0 - $25m3/h). Spacing between adjacent grid lines: 50 mm. Chapter 6. Regime Maps and Minimum Fluid Flowrate 127 a b c Figure 6.6: Transition from slugs to bubbles in fluidized bed: a —slugs in fluidized bed Q=595 m 3 /h ; b —large bubbles in fluidized bed Q=315 m 3 /h ; c —bubbles in fluidized bed Q=255 m 3 /h . (Particles: polystyrene, dp = 3.25 mm, D{ = 88.9 mm, H = 1.5 m. qa = 1012m3//i). Chapter 6. Regime Maps and Minimum Fluid Flowrate 128 Figure 6.7: Regime map for brown beans: G —fluidization at minimum condition: A —spouting at minimum condition; C —co-ordinates for spout-fluidization at minimum condition. (D; = 88.9 mm, dp = 4.7 mm, H = 1.25 m). Chapter 6. Regime Maps and Minimum Fluid Flowrate 129 6.3.1.3 Spouting-with-aeration Spouting with aeration was observed to be similar in appearence to spouting in a standard spouted bed. This regime was stable and its extent depended on several factors. The regime area for polystyrene particles is greater than for brown bean particles. Brown bean particles have a very low maximum spoutable bed height (Hm) of 1.94 m for an orifice diameter of 88.9 mm, whereas the corresponding Hm for polystyrene particles is greater than 2.60 m (see section 4.2.4). The transition from spouting with aeration to spout-fluidization is shown in Figure 6.8. The transition points separating spout-fluidization and spouting with aeration were obtained at the highest possible auxiliary air flow that does not cause fluidization of the annular solids at the bed surface. For both the polystyrene particles and the brown beans, the boundary shown by hne C-D has a negative slope which became smaller in magnitude with increasing central air flow but was always negative. This may occur because there is an increase of annulus gas velocity for an increasing central air flow. However, Lim (1975) pointed out that there was a decrease of annulus gas velocity for an increasing central air flow. That suggests that if spouting with aeration is to be maintained for a greater central flow the auxiliary flow has to be decreased. The transition with decreasing flowrate from spouting with aeration to fixed bed was direct (see Figures 6.3 and 6.7), but in Sutanto's work (1983) the transition was first from spouting with aeration to bubbhng in fluidized bed (JF(II)) and then to the fixed bed condition (see Figure 2.5 in Chapter 2). Regime maps of other workers discussed in Chapter 2 were determined from ex-periments in full columns except in the case of those of Sutanto (1983). The visible characteristics of the spout were found to be distinct in each flow regime in the present half-column work. The inability of previous workers to observe changes in spout shape Chapter 6. Regime Maps and Minimum Fluid Flowrate 130 Figure 6.8: Transition from spouting with aeration to spout-fluidization: a —spout-ing with aeration, qa=756 m 3 /h ; b —spout-fluidization, qa=940 m 3 /h . (Particles: polystyrene, dP = 3.25 mm, D{ = 88.9 mm, H = 1.5 m, Q=621 m 3/h) Chapter 6. Regime Maps and Minimum Fluid Flowrate 131 between the different regimes no doubt contributed to difficulties in interpreting flow be-haviour and distinguishing regimes. Visual observations and pressure drop measurements were of equal importance in the present work. The half-column geometry thus provided an important means of observation which would have been impossible if a full column had been used. 6.3.2 Minimum fluid flowrate 6.3.2.1 Minimum fluid flowrate for spouting with aeration The hnes marked, A - H in Figures 6.3 and 6.7 give the condition of 'minimum spouting with aeration'. The corresponding central and auxiliary flow rates, Qmsa and qmsa, were correlated by Littman et al. (1974) as Qmsa = Qmf ~ Qmsa (61) The unmodified Equation (6.1) failed to adequately represent the experimental data; it was necessary to introduce a constant A ' such that qmsa = (Qmf + A') - ( ^ ± ^ ) Qmsa (6.2) Dividing both sides of Equation (6.2) by Qmf and rearranging, we obtain \Qmf where qmsa\ = ( 1 + A ) * j Qmsa " \Qm~s~ (6.3) A = JL (6.4) Wmf The linear relationship suggested by lines A - H in Figures 6.3 and 6.7 can also be repre-sented by K / Q m . _Qmsa\ ^ Qmf \Qmf Qmf Chapter 6. Regime Maps and Minimum Fluid Flowrate 132 where K is a constant equal to the slope of line H-A. The two linear equations, (6.3) and (6.5), terminate at point H due to the formation of another flow regime. Equations (6.3) and (6.5) are therefore valid for \QmfJ \Qmf/max ( Qmsa \ I Qmsa \ f n ~\ Values for Qms and Q m f were determined experimentally. Since <?T,m»a = Qmsa + Qmsa (6.8) the total minimum fluid flow at a given bed height under this condition is obtained by combining Equations (6.8), (6.3) and (6.5) to give ' Q m , 0 ^ = { l + A ) Wmf Qr + (6.9) Qmf or Qr,msa Tr Qms M .IQ msa \ / - *t r\\ — = K-— + K)\—-\ (6.10) Wmf Wmf \ Wmf ) The values of A and the slopes (K) of line A-H for polystyrene and brown bean particles are summarized in Table 6.1. Since the slope is greater than unity in both cases, the sum of the auxiliary flow and central flow required for spouting with aeration is always equal to or greater than the minimum spouting flowrate (Qms), but always less than the minimum fluidization flowrate (Qmf)- Equation (6.3), together with Equations (6.4) and (6.5), provides an easy means of estimating the minimum fluid flowrate for spouting with aeration. er 6. Regime Maps and Minimum Fluid Flowrate Table 6.1: Values of empirical constant A and slopes (K) of the line A-H Material H Q Di A K (m) (mm) Polystyrene 1.50 88.9 0.27 2.83 Brown beans 1.25 88.9 2.35 6.97 Table 6.2: Values of minimum total flow for spout-fluidization H Di Expt. Eq.(6.11) Material (m) (mm) UT,msf UT,msf Error (m/s) (m/s) (%) Polystyrene 1.50 88.9 1.15 0.651 43.6 Brown beans 1.25 88.9 1.25 0.876 30.1 Chapter 6. Regime Maps and Minimum Fluid Flowrate 134 6.3.2.2 Minimum fluid flowrate for spout-fluidization The rninimurn total superficial gas velocity required for spout-fluidization, given by Equa-tion (2.36) due to Nagarkatti and Chatterjee, is given by UT,msf = U m s + Umf(l - $) (6.11) where the function $ was determined empirically to be $ = 0.20d P-°- 3 2 0A 0 - 2 3 5 F 0 - 1 6 0 (6.12) The values of £7ms and Umf were determined experimentally in the present work. The values of ITr,™*/ c a n be obtained from the experimental results or estimated from Equa-tions (6.9) and (6.10). Values from the two approaches are compared in Table 6.2. The values of UT,msf predicted by Equations (6.11) and (6.12) are less than the experimental values of t7 T m^ . The error is more than 30% for the brown beans and over 40% for the polystyrene particles. If a straight line between points G and J in Figure 6.3 or Figure 6.7 is drawn, repre-senting the total flowrate (qa+Q) = Qm/) it will be found that point C (minimum flowrate for spout-fluidization) lies on the straight line. That means that the minimum total flowrate for spout-fluidization equals the minimum fluidization flowrate for the same particles. This is also true for the results of Vukovic et al. (1982) but not for the results of Sutanto (1984) (see Figures 2.4 and 2.5). Chapter 7 Conclusions and Recommendations for Further Work 7.1 Conclusions Experiments were carried out successfully in a 0.91 m diameter half-column equipped with a 60° conical base for three types of particles, with and without auxiliary air. Experimental values of U m s were compared to predictions of correlations developed for smaller vessels. These correlations gave disappointing results. Um„ was found to be proportional to fT" where u> was of order 1.0 ~ 1.5 for shallow beds and 0.2 ~ 0.3 for deeper beds. Spout shape remained the same with increasing flowrate or auxiliary flow except for the case when the bed operated in the spout-fluidization regime. In that case the spout diameter appeared to shrink and expand in a periodic manner in the upper section of the bed. Spout diameter increased with increasing flowrate or auxiliary flow. The empirical correlation due to McNab (1972) was found to predict the average spout diameter very well for standard spouted beds, while the correlation due to Hadzisdmajlovic et al. (1983) gave a reasonable prediction of D a for spout-fluid beds. The large-diameter spouted bed and spout-fluid bed gave rise to large dead regions. The dead zones were found to increase in extent with increasing static bed depth. The measurements from the front flat plate were not representative of the dead zone behaviour of a full column spouted bed for the more angular polystyrene particles, but was more accurate for the more smooth brown beans. The dead zone was found to decrease in size 135 Chapter 7. Conclusions and Recommendations for Further Work 136 when the auxiliary flow increased, especially for the brown beans. The superficial gas velocity had only a small influence on the dead zone boundary. Experimental values of H m for brown beans were compared to the McNab and Bridg-water (1977) and Littman et al. (1977) equations. The latter overestimated H m , while the McNab and Bridgwater equation gave a value which appeared to be far too high. Fountain height was found to increase with increasing gas velocity and with increasing bed height. It was also observed that the fountain was extremely dilute, and its height always exceeded the static bed height for the conditions studied; these were the major differences from the observations of a smaller spouted bed. In the spout-fluidization regime the fountain shape was entirely different from that observed for stable spouting. In this regime a maximum fountain height was achieved with the expansion of the spout and a minimum immediately thereafter. The Epstein and Levine (1978) equation gave good estimates of overall bed pressure drop. The bed pressure drop decreased with increasing gas velocity for spouted beds and increased with increasing auxiliary flow in the spouting-with-aeration regime. In the spout-fluidization regime the bed pressure drop was found to fluctuate, while in the jetting, slugging and bubbling fluidized bed regimes an increase in the magnitude of pressure fluctuations was noted. The longitudinal fluid velocity in the annulus was well predicted by the modified Lefroy-Davidson (1969) equation, Equation (2.24), and was reasonably predicted by the Mamuro-Hattori (1968) model in the cylindrical portion. However, both equations gave poor predictions in the conical base portion. In the conical base section, the Rovero et al. (1983) equation predicted the correct trend, but consistently overestimated U a by a considerable margin. The average annulus superficial gas velocity was not strongly influenced by the overall superficial gas velocity. In the conical base portion Ua first increased and then decreased with increasing U / U m 4 . The annulus superficial gas velocity Chapter 7. Conclusions and Recommendations for Further Work 137 was found to increase with increasing auxiliary flow, and the experimental data were not well fitted by the Littman et al. (1976) equation. The local superficial gas velocity decreased with increasing radial distance and increased with height in the cylindrical portion. In the conical base portion the local superficial gas velocity first increased shghtly and then decreased with increasing radial distance. Both the Epstein and Levine (1978) equation and the Lefroy and Davidson (1969) equation were found to be in good agreement with the experimental longitudinal pressure profiles. The longitudinal and radial pressure profiles in the annulus were only shghtly influenced by the superficial gas velocity. On the other hand, the radial distribution of pressure in the annulus for any bed level was seen to be nearly constant when there was auxiliary flow. A computer model based on the Ergun equation gave a good qualitative prediction of fluid distribution in the annulus. This model predicted a maximum velocity in the cone region and a minimum at the cone-cylinder junction, in agreement with experimental measurements. The model also gave accurate predictions of the longitudinal pressure profile in the conical base portion and reasonable predictions in the cylindrical portion of the column. Four fairly distinct flow regimes were delineated in this work: (1) spouting-with-aeration; (2) spout-fluidization; (3) submerged jets, slugs and bubbles in fluidized bed, and (4) packed bed. Flow regimes maps shared many similarities to the regime maps of previous workers cited in Chapter 2. The transition from a packed bed to spouting-with-aeration was direct, which is different from observations in a much smaller column due to Sutanto et al. (1985) where there was a bubbhng regime between the packed bed and spouting-with-aeration. Another difference was that in the jets, slugs and bubbles in fluidized bed regime there were four states of fluidization in the present work. Near the boundary between spout-fluidization and jets, slugs, and bubbles in fluidized bed Chapter 7. Conclusions and Recommendations for Further Work 138 (at higher central flow), jets were noted. At an intermediate central flow, slugging of the solid material in the upper section of the bed was observed. At a lower central flow, bubbles were observed. Finally, with only a little central flow, the total upper section of the bed resembled the behaviour of a bubbhng bed. The minimum total fluid flowrate for spouting-with-aeration always exceeded the minimum spouting flowrate, but was smaller than the minimum fluidization flowrate. The minimum total fluid flowrate for spout-fluidization was found to be equal to the minimum fluidization flowrate. 7.2 Recommendations for Further Work Particle velocities both in the annulus and in the spout should be investigated. The solid mass circulation rate should then be calculated. The stability of spouting is another key point which should be examined in further work. The influence of orifice diameter on minimum spouting velocity, spout shape and diameter, dead zone boundary, maximum spoutable bed height, fountain height and overall pressure drop should receive further attention. A cyclone should be installed on the top of the column to collect dust and fine particles to keep the surroundings clean. The top screen arrangement should also be replaced by a more steady assembly. If appropriate particles can be found, scaling procedure similar to those being devel-oped for fluidized beds (eg. Fitzgerald et al., 1984; Glicksman, 1984) should be investi-gated using the two geometrically similar columns, i.e. the 0.15 m diameter column used previously by Sutanto et al. (1985) and the 0.91 m diameter column employed in the present work. Notation A E m p i r i c a l constant in Equation (6.3), A = A ' / Q m y A a Cross-sectional area of annulus A a c Annulus area throughout the cylindrical part of the column A c Cross-sectional area of the column A c 6 Conical base surface area A n Constants in Equation (2.16) A r Archimedes number, d>*~pJ)pf3 A 0 A ' Empir ica l constant in Equation (6.2) a s Ratio of spout area to column area B b Ums/Um/ c P -, V H V2\2Y + (X-2)+(X-0.2)-(3.24/'8)] X 1 Dc [2y + 2(A'-0 .2 ) - l .S+(3 .24 /«) ] D c Inside diameter of column D i Diameter of inlet orifice D s Mean spout diameter M e a n spout diameter of a spout—fluid bed dp Particle mean diameter Volumic equivalent bubble diameter F A n y scalar field 139 Notation 140 fi 150[(1 - eaflel\[pldl] (kg/m3s) f2 1.75[(1 - eaflel][Pildp} (kg/m4) g Acceleration due to gravity (m/s2) g(tfS) [5c*5- 7.57<£2 + 4.09^-0.516] I -{[85.7(l-em /)/^] 2 + [ 2 . 2 9 e ^ / M r ] 1 ^ + 8 5 . 7 ( l - c m / )/4>}l _ X [ {\&.7'\-snt)]* + [2.2*ln}ATyl*+%&.7'\-^i)} \ \ > H Static bed height (m) H c Height of conical base section (m) Hp Fountain height, measured from static bed upper surface (m) H m Maximum spoutable bed depth (m) H m s / Maximum spoutable bed depth of spout-fluid bed (m) h H / H m ( - ) h m s / H / H m s / ( - ) i Radial grid point index ( - ) j Vertical grid point index ( - ) K Empirical constant in Equation (2.35) ( - ) M Number of vertical grid points ( - ) N Number of radial grid points ( - ) P Dimensionless pressure, p / A p s ( - ) Patm Atmospheric pressure (Pa) P# Absolute pressure measured just below inlet orifice with solids in the bed (Pa) P# Absolute pressure measured just below inlet orifice without solids in the bed (Pa) VH Annulus fluid pressure at bed surface (Pa) P« Absolute pressure in the bed (Pa) Notation 141 Po Annulus fluid pressure at z=0 (Pa) p Annulus fluid pressure (Pa) Q Flowrate through central (spouting) orifice (m 3/s) Q Q Auxiliary flowrate at any bed level (m 3/s) Q m / Minimum fluidization flowrate (m 3/s) Q m e Minimum spouting flowrate (m 3/s) Qm«a Flowrate through centra] orifice at the condition of minimum spouting with aeration (m 3/s) Q m , / Flowrate through central orifice at the condotion of minimum spout-fluidization (m 3/s) QT Sum of central spouting flowrate and auxiliary flowrate (m 3/s) QT,msa Sum of central spouting flowrate and auxiliary flowrate at the condition of minimum spouting with aeration, =Q m j ,a+q m «a (m 3/s) q a Auxiliary flowrate (m 3/s) ^msa Auxiliary flowrate at the condition of minimum spouting with aeration (m 3/s) R Dimensionless radius, r / R c ( - ) R x Column radius in the cone region (m) R b Basal radius (m) R c Radius of cylindrical part of the column (m) R„ Spout radius (m) U Superficial gas velocity ( m / s ) U Velocity vector ( m / s ) U a Superficial gas velocity in the annulus ( m / s ) Notation 142 Ua Radial average superficial gas velocity in the annulus ( m / s ) UaHm Superficial gas velocity in the annulus at z=H T O ( m / s ) JJmf Minimum fluidization gas velocity ( m / s ) U m , Minimum spouting velocity ( m / s ) U n Superficial gas velocity perpendicular to the conical base in the annulus ( m / s ) Ufl Ur/Umf ( - ) LV Terminal setthng velocity of the particles ( m / s ) U^'ms/ Minimum fluid superficial velocity for spout-fluidization ( m / s ) Uz u 2 / U m / ( - ) u r Radial component of gas velocity in the annulus ( m / s ) u 2 Vertical component of gas velocity in the annulus ( m / s ) v„ Local upward particle velocity in spout or fountain core at any level ( m / s ) X 1/[1+ (H/De)] (-) x z / H m ( - ) Y l - ( A P m f / A P m / ) ( - ) y Dummy variable ( - ) Z Dimensionless height, z / H ( - ) z Vertical distance from inlet orifice (m) ft Flow regime parameter, 2 + 3 / i / (2 / 2 L7 m y) ( - ) 7 Cone included half-angle ( °) A P / Pressure drop across bed of particles Notation 143 at minimum fluidization (Pa) A P m J Overall pressure drop at minimum spouting condition (Pa) A P S Overall pressure drop (Pa) ea Voidage of the annulus region ( - ) emf Voidage at minimum fluidization ( - ) 9 7.18 1.07 ( - ) 0 Angular coordinate ( - ) fi Fluid viscosity (kg/ms) v Fluid kinematic viscosity, p/Pp (m2/s) puk Bulk density of particles (kg/m3) pj Fluid density (kg/m3) pp Particle density (kg/m3) $ Empirical constant in Equations (2.36) and (6.11) ( - ) cb Particle sphericity ( - ) $ Dimensionless streamfunction ( - ) tp Streamfunction (m3/s) Bibliography [1] Baeyens, J . and D.Geldart, "Chapter 5. Solids Mixing", Gas Fluidization Technol-ogy, 97-122, Ed. by D.Geldart, John Wiley & Sons Ltd. (1986). [2] Bowen, B. , "Mathematical Operations in Chemical Engineering", lecture notes, Uni-versity of British Columbia, Vancouver (1988). [3] Brereton, C .M.H. , "The/Ergun Model—A Model for Prediction of Velocity and Pressure Profiles in Spouted Bed Systems", B.A.Sc. thesis, University of British Columbia, Vancouver (1982). [4] Brereton, C .M.H. , N.Epstein and J.R.Grace, "Gas Flow Distribution in the Annulus of Spouted Beds", 33rd Can. Chem. Eng. Conference, Toronto, 785-790 (1984). [5] Clift, R., "Chapter 5. Hydrodynamics of Bubbhng Fluidized Beds", in "Gas Flu-idization Technology", Ed. D.Geldart, John Wiley &; Sons Ltd., Chichester, (1986). [6] Epstein, N . , Plenary Address, 2nd Intern. Symp. on Spouted Beds, Vancouver (1982). [7] Epstein, N . and J.R. Grace, "Spouting of Particulate Solids", Chapter 11 in "Hand-book of Power Science and Technology", Ed. Fayed, M.E . and Otten, L. , Van Nos-trand Reinhold Co., New York (1984). [8] Epstein, N . and S. Levine, "Non-Darcy Flow and Pressure Distribution in a Spouted Bed", in "Fluidization", Ed. J.F. Davidson and D.L. Keairns, Cambridge Univ. Press., England, 98-103 (1979). 144 Notation 145 [9] Epstein, N . , C.J . Lim and K . B . Mathur, "Data and Models for Flow Distribution and Pressure Drop in Spouted Bed", Can. J . Chem. Eng., 56, 436-447 (1978). [10] Ergun, S., "Fluid Flow Through Packed Columns", Chem. Eng. Progr., 48, 89-94 (1952). [11] Fane, A . G . and R.A. Mitchell, "Minimum Spouting Velocity of Scaled-up Beds", Can. J . Chem. Eng., 62, 437-439 (1984). [12] Fitzgerald, T., D. BushneU, S. Crane and Y - C . Shieh, "Testing of Cold Scaled Bed Modeling for Fluidized Bed Combustors", Powder Technol., 38, 107-120 (1984). [13] Geldart, D., "Chapter 2. Single Particles, Fixed and Quiescent Beds", in "Gas Flu-idization Technology", Ed. D. Geldart, John Wiley & Sons Ltd., Chichester (1986). [14] Glicksman, L.R., "Scaling Relationships for Fluidized Beds", Chem. Eng. Science, 39, No. 9, 1373-1379 (1984). [15] Grace, J.R. and K . B . Mathur, "Height and Structure of the Fountain Region above Spouted Beds", Can. Chem. Eng., 56, 533-537 (1978). [16] Grbavcic, Z.B., D.V.Vukovic, F .K. Zdanski and H.Littman, "Fluid Flow Pattern, Minimum Spouting Velocity and Pressure Drop in Spouted Beds", Can. J . Chem. Eng., 54, 33-42 (1976). [17] Green, M.C. and J.Bridgwater, "An Experimental Study of Spouting in Large Sector Beds", Can. J . Chem. Eng., 61, 281-288 (1983). [18] Hadzisdmajlovic, Dz.E., Z.B. Grbavcic, D.V.Vukovic and H.Littman, "The Mechan-ics of Spout-Fluid Beds at the Minimum Spout-Fluid Flowrate", Can. J . Chem. Eng., 61, 343-347 (1983). NotatioD 146 [19] Krniec, A., "The Minimum Spouting Velocity in Conical Beds", Can. J. Chem. Eng., 61, 274-280 (1983). [20] Lefroy, G.A. and J.F.Davidson, "The Mechanics of Spouted Beds", Trans. Instn. Chem. Engrs., 47, T120-128 (1969). [21] Lim, C.J., "Gas Residence Time Distribution and Related Flow Patterns in Spouted Beds", Ph.D. Thesis, University of British Columbia, Vancouver (1975). [22] Lim, C.J. and J.R. Grace, "Spouted Bed Hydrodynamics in a 0.91 m Diameter Vessel", Can. J. Chem. Eng., 65, 366-372 (1987). [23] Littman, H. and M.H. Morgan, "A General Correlation for the Minimum Spouting Velocity", Can. J. Chem. Eng., 61, 269-273 (1983). [24] Littman, H.,M.H. Morgan, III, D.V.Vukovic, F.K.Zdanski, and Z.B. Grbavcic, "The-ory for Predicting the Maximum Spoutable Height in a Spouted Bed", Can, J. Chem. Eng., 55, 497-501 (1977). [25] Littman, H., D.V.Vukovic, F.K. Zdanski and Z. B. Grbavcic, "Pressure Drop and Flowrate Characteristics of a Liquid Phase Spout-Fluid Bed at the Minimum Spout-Fluid Flowrate", Can. J. Chem. Eng., 52, 174-179 (1974). [26] Littman, H., D.V. Vukovic, F.K.Zdanski and Z. B. Grbavcic, "Basic Relations for the Liquid Phase Spout-Fluid Flowrate", Fluidization Technology, Ed. D.L. Keairns, Hemisphere, 1, 373-386 (1976). [27] Mamuro, T. and H.Hattori, "Flow Pattern of Fluid in Spouted Beds", J. Chem. Eng. Japan, I, 1-5 (1968) / correction, J. Chem. Eng. Japan, 3, 119 (1970). Notation 147 [28] Mathur, K . B . and N.Epstein, "Developments in Spouted Bed Technology", Can. J . Chem. Eng., 52, 129-144 (1974). [29] Mathur, K . B . and P.E.Gishler, "A Technique for Contacting Gases with Coarse Solid Particles", A.I.Ch.E.J. , 1, 157-164 (1955). [30] Mathur, K . B . and N.Epstein, "Spouted Beds", Academic Press, New York (1974). [31] McNab, G.S. and J.Bridgwater, "Spouted Beds Estimation of Spouted Pressure Drop and the Particle Size for Deepest Bed", Proc. European Congress on Particle Technology, Nuremberg (1977). [32] Nagarkatti, A. and A.Chatterjee, "Pressure and Flow Characteristics of a Gas Phase Spout-Fluid Bed and the Minimum Spout-Fluid Condition", Can. J . Chem. Eng., 52, 185-195 (1974). [33] Rao, K . B . , A.Husain and Ch.D.Rao, "Prediction of the Maximum Spoutable Height in Spout-Fluid Beds", Can. J . Chem. Eng., 63, 690-692 (1985). [34] Rovero, G., C.M.H.Brereton, N.Epstein, J.R.Grace, L.Casalegno, and N.Piccinini, "Gas Flow Distribution in Conical-Base Spouted Beds", Can. J . Chem. Eng., 61. 289-296 (1983). [35] Sutanto, W., "Hydrodynamics of Spout-Fluid Beds", M.A.Sc. Thesis, University of British Columbia, Vancouver, B.C. (1983). [36] Sutanto, W., N.Epstein and J.R.Grace, "Hydrodynamics of Spout-Fluid Beds", Powder Technology, 44, 205-212 (1985). [37] Thorley, B. , J.B.Saunby, K.B.Mathur, and G.L.Osberg, "An Analysis of Air and Solid Flow in a Spouted Wheat Bed", Can. J . Chem. Eng., 37, 184-192 (1959). Notation 148 [38] Vukovic, D.V. , Dz.E.Hadzismajlovic, Z.B.Grbavcic, R.V.Garic and H.Littman, "Regime Maps for Two-Phase Fluid-Solids Mobile Beds in a Vertical Column with Nozzle and Annular Flow", 2nd Int. Symp. on Spouted Bed, C.S.Ch.E., Vancouver, 93-102 (1982). [39] Wu, S.W.M., C.J.Lim and N.Epstein, "Hydrodynamics of Spouted Beds at Elevated Temperatures", Chem. Eng. Comm., 62, 251-268 (1987). Appendix A Pressure Drop vs Superficial Velocity for Polystyrene Particles 149 Appendix A. Pressure Drop vs Superficial Velocity for Polystyrene Particles 150 0.6 o.o-O—; 1 ; 1 ; 1 ; 1 • 1 ; 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 P r e s s u r e g r a d i e n t ( c m H 2 0 / c m ) Figure A . l : Calibration curve of pressure gradient vs. superficial velocity in a packed bed column. (Particles: polystyrene, dp = 3.25, Dc = 0.152, fluid: air) Appendix B Experimental Velocity Profile Data 151 Appendix B. Experimental Velocity Profile Data 152 Table B.l: Longitudinal annulus velocity profile for the polystyrene particles. (U/Ums = 1.1, dp = 3.25 mm, £>,- = 88.9 mm, H = 2.0 m, qa = 0.0 rn3/h) Vertical Component of Velocity (m/s) Tap z Radial Position, r (m) No. (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 2 1.890 0.386 0.379 0.374 0.370 0.368 0.366 0.365 0.364 3 1.625 0.371 0.366 0.364 0.362 0.360 0.359 0.358 0.357 4 1.320 0.346 0.343 0.340 0.337 0.335 0.334 0.333 0.332 5 1.010 0.306 0.303 0.300 0.298 0.296 0.295 0.294 0.293 6 0.710 0.264 0.258 0.253 0.248 0.243 0.239 0.230 • 0.227 7 0.540 0.195 0.186 0.179 0.173 0.167 0.162 0.157 0.155 8 0.445 0.187 0.193 0.193 0.193 0.180 0.163 0.159 9 0.335 0.176 0.200 0.200 0.200 0.181 10 0.232 0.156 0.188 0.188 11 0.125 0.146 0.198 12 0.000 0.138 Appendix B. Experimental Velocity Profile Data 153 Table. B.2: Longitudinal annulus velocity profile for the polystyrene particles. (U/Ums = 1.3, dp = 3.25 mm, Dx = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) Vertical Component of Velocity (m/s) Tap z Radial Position, r (m) No. . (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 2 1.890 0.407 0.404 0.402 0.401 0.400 0.399 0.399 0.398 3 1.625 0.377 0.374 0.372 0.370 0.368 0.367 0.365 0.364 4 1.320 0.344 0.341 0.339 0.336 0.334 0.333 0.331 0.330 5 1.010 0.301 0.298 0.296 0.292 0.290 0.288 0.287 0.286 6 0.710 0.215 0.219 0.227 0.226 0.222 0.219 0.217 0.217 7 0.540 0.153 0.159 0.159 0.166 0.174 0.153 0.153 0.148 8 0.445 0.180 0.180 0.180 0.187 0.187 0.155 0.155 9 0.335 0.195 0.195 0.195 0.200 0.181 10 0.232 0.244 0.247 0.247 11 0.125 0.198 0.213 12 0.000 0.121 Appendix B. Experimental Velocity Profile Data 154 Table B.3: Longitudinal annulus velocity profile for the polystyrene particles. (U/Ums = 1.5, dv = 3.25 mm, Dt = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) Vertical Component of Velocity (m/s) Tap z Radial Position, r (m) No. (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 2 1.890 0.393 0.385 0.379 0.373 0.368 0.364 0.360 0.357 3 1.625 0.372 0.369 0.366 0.364 0.363 0.361 0.355 0.344 4 1.320 0.330 0.327 0.324 0.321 0.319 0.318 0.316 0.315 5 1.010 0.296 0.293 0.291 0.289 0.287 0.286 0.285 0.284 6 0.710 0.199 0.201 0.206 0.206 0.206 0.206 0.206 0.206 7 0.540 0.133 0.133 0.133 0.153 0.153 0.153 0.153 0.110 8 0.445 0.174 0.167 0.167 0.163 0.155 0.142 0.129 9 0.335 0.176 0.176 0.176 0.176 0.176 10 0.232 0.202 0.202 0.202 11 0.125 0.163 0.175 12 0.000 0.022 Appendix B. Experimental Velocity Profile Data 1 Table B.4: Longitudinal annulus velocity profile for the polystyrene particl (U/Ums = 1.7, dp = 3.25 mm, Dt = 88.9 mm, H = 2.0 m, qa = 453 m3/h) Vertical Component of Velocity (m/s) Tap z Radial Position, r (m) No. (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 2 1.890 0.419 0.417 0.415 0.414 0.414 0.413 0.413 0.412 3 1.625 0.378 0.376 0.375 0.375 0.375 0.374 0.373 0.373 4 1.320 0.367 0.366 0.364 0.363 0.363 0.363 0.362 0.362 5 1.010 0.336 0.336 0.333 0.333 0.334 0.334 0.334 0.329 6 0.710 0.302 0.306 0.308 0.308 0.308 0.308 0.308 0.307 7 0.540 0.268 0.270 0.284 0.282 0.281 0.281 0.276 0.242 8 0.445 0.317 0.319 0.322 0.311 0.310 0.300 0.283 9 0.335 0.312 0.313 0.313 0.313 0.304 10 0.232 0.317 0.318 0.326 11 0.125 0.278 0.280 12 0.000 -0.374 Appendix B. Experimental Velocity Profile Data 156 Table B.5: Longitudinal annulus velocity profile for the polystyrene particles. (U/Ums = 2.05, dv = 3.25 mm, Dx = 88.9 mm, H = 2.0 m, qa = 721 m3/h) Vertical Component of Velocity (m/s) Tap z Radial Position, r (m) No. (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 2 1.890 0.502 0.497 0.494 0.492 0.490 0.489 0.488 0.487 3 1.625 0.464 0.459 0.457 0.455 0.453 0.452 0.452 0.451 4 1.320 0.422 0.418 0.415 0.412 0.409 0.407 0.406 0.405 5 1.010 0.351 0.349 0.347 0.349 0.343 0.342 0.342 0.342 6 0.710 0.321 0.321 0.321 0.325 0.327 0.325 0.325 0.325 7 0.540 0.291 0.300 0.306 0.306 0.303 0.303 0.291 0.291 8 0.445 0.384 0.371 0.356 0.357 0.358 0.360 0.343 9 0.335 0.351 0.344 0.348 0.348 0.336 10 0.232 0.338 0.343 0.364 11 0.125 0.307 0.299 12 0.000 -0.402 Appendix C Experimental Pressure Profile Data 157 Appendix C. Experimental Pressure Profile Data 158 Table C l : Annulus pressure profile for the polystyrene particles. (U/Ums = 1.1, dp = 3.25 m m , Dt = 88.9 m m , H = 2.0 m, qa = 0.0 m3/h) Pressure (kPa) Tap z Radial Position, r (m) No. ( m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2 000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 1 890 0.386 0.372 0.361 0.353 0.348 0.345 0.342 0.341 3 1 625 1.242 1.203 1.180 1.162 1.149 1.137 1.129 1.123 4 1 320 2.081 2.025 1.986 1.956 1.933 1.914 1.900 1.890 5 1 010 2.728 2.652 2.596 2.551 2.515 2.487 2.465 2.450 6 0 710 3.151 3.061 2.993 2.937 2.891 2.854 2.815 2.794 7 0 540 3.305 3.203 3.125 3.060 3.007 2.963 2.920 2.896 8 0 445 3.385 3.288 3.210 3.145 3.081 3.025 2.979 9 0 335 3.467 3.393 3.314 3.249 3.168 10 0 232 3.529 3.480 3.401 11 0 125 3.588 3.579 12 0 .000 3.650 Appendix C. Experimental Pressure Profile Data. 159 Table C.2: Annulus pressure profile for the polystyrene particles. (U/Ums = 1.3, dp = 3.25 mm, D, = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) Pressure (kPa) Tap z Radial Position, r (m) No. (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 1.890 0.426 0.421 0.417 0.414 0.412 0.411 0.409 0.409 3 1.625 1.309 1.292 1.277 1.264 1.254 1.244 1.237 1.231 4 1.320 2.138 2.106 2.078 2.055 2.034 2.016 2.001 1.989 5 1.010 2.753 2.701 2.656 2.607 2.574 2.546 2.524 2.507 6 0.710 3.071 3.027 2.999 2.948 2.907 2.872 2.847 2.830 7 0.540 3.171 3.134 3.106 3.063 3.032 2.972 2.947 2.925 8 0.445 3.246 3.208 3.181 3.142 3.112 3.029 3.004 9 0.335 3.345 3.308 3.281 3.247 3.199 10 0.232 3.475 3.440 3.412 11 0.125 3.574 3.552 12 0.000 3.624 Appendix C. Experimental Pressure Profile Data 160 Table C.3: Annulus pressure profile for the polystyrene particles. (U/Ums = 1.5, dp = 3.25 mm, D, = 88.9 mm, H = 2.0 m, qa = 0.0 m3/h) Pressure (kPa) Tap z Radial Position, r (m) No. (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 1.890 0.398 0.383 0.371 0.358 0.348 0.340 0.332 0.326 3 1.625 1.257 1.227 1.202 1.180 1.161 1.145 1.104 1.048 4 1.320 2.016 1.970 1.930 1.895 1.867 1.843 1.796 1.735 5 1.010 2.596 2.533 2.478 2.431 2.392 2.361 2.307 2.241 6 0.710 2.877 2.819 2.776 2.729 2.691 2.659 2.606 2.540 7 0.540 2.957 2.898 2.856 2.829 2.790 2.759 2.706 2.597 8 0.445 3.027 2.963 2.921 2.891 2.847 2.809 2.748 9 0.335 3.109 3.045 .3.003 2.973 2.930 10 0.232 3.208 3.145 3.103 11 0.125 3.278 3.224 12 0.000 3.283 Appendix C: Experimental Pressure Profile Data 161 Table C.4: Annulus pressure profile for the polystyrene particles. {U/Ums = 1.7, dp = 3.25 mm, D, = 88.9 mm, H = 2.0 m, qa = 453 m3/h) Pressure (kPa) Tap No. z (m) Radial Position, r (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 1.890 0.662 0.617 0.612 0.609 0.605 0.602. 0.600 0.597 3 1.625 1.511 1.498 1.488 1.482 1.478 1.473 1.466 1.461 4 1.320 2.474 2.452 2.434 2.423 2.417 2.409 2.399 2.392 5 1.010 3.276 3.253 3.218 3.207 3.206 3.201 3.191 3.159 6 0.710 3.875 3.880 3.853 3.842 3.841 3.836 3.826 3.791 7 0.540 4.122 4.129 4.132 4.116 4.112 4.107 4.087 4.002 8 0.445 4.338 4.348 4.356 4.322 4.316 4.294 4.241 9 0.335 4.580 4.592 4.600 4.566 4.540 10 0.232 4.814 4.829 4.849 11 0.125 4.980 4.998 12 0.000 4.570 Appendix C. Experimental Pressure Profile Data 162 Table C.5: Annulus pressure profile for the polystyrene particles. (U/Ums = 2.05, dj, = 3.25 mm, A = 88.9 mm, H = 2.0 m, qa = 721 m3/h) Pressure (kPa) Tap No. z (m) Radial Position, r (m) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 1.890 0.631 0.619 0.611 0.605 0.601 0.597 0.595 0.592 3 1.625 1.938 1.900 1.880 1.864 1.853 1.844 1.841 1.837 4 1.320 3.207 3.145 3.105 3.071 3.045 3.024 3.013 3.007 5 1.010 4.091 4.014 3.961 3.943 3.884 3.858 3.844 3.838 6 0.710 4.795 4.718 4.666 4.664 4.616 4.580 4.566 4.560 7 0.540 5.094 5.051 5.019 5.018 4.959 4.923 4.865 4.859 8 0.445 5.422 5.358 5.298 5.299 5.243 5.209 5.121 9 0.335 5.736 5.656 5.604 5.605 5.527 10 0.232 6.005 5.935 5.923 11 0.125 6.229 6.142 12 0.000 5.756 Appendix D Derivation of the Velocity Field Equation from the Ergun Equation The vector form of the Ergun equation is, -VP = U(f1 + f2\U\) (D.l) Taking the curl of Equation (D.l) produces an equation in velocity, V x ( -VP) = V x [U(h + f2\U\)} = 0 (D.2) The term (/j -f j^l^l) in equations (D.l) and (D.2) is a scalar term, hence, 0 - V ( / x + f2\U\) xU + (h+ f2\U\)(V x U) (D.3) Rearranging equation (D.Z) gives, ^ X 0 = ( f ."rVm^1 + /2^ ! ) X & (D-4) Since ^VS = V l n S and (lxl) = -(B x A) equation (DA) becomes, V x U = U x Vln(/ a +/2|17|) (D.5) 163 Appendix E Derivation of Equation (5.10) From Equation (5.6) From Equation (5.6) V x U = U x Vln(A + f2\fj\) (E.l) In cylindrical polar coordinates U = uTr + ue6 + uzz (E.2) The vector operators for cylindrical coordinates are = (E.4, [ V « « l = ^ - ^ (E.5) r or r oo Axial symmetry requires that u§ and all angular derivatives be zero, so that V x JJ = [V x L7H = " ^ ) « (E.6) Consider the right hand side of equation (E.l) RHS = V x Vln(/ a + f2\U\) (E.7) Rewriting Equation (E.7) RHS = (uTf + uzz) x ( _ L _ J V C A + /2|L7|) (E.8) 164 Appendix E. Derivation of Equation (5.10) From Equation (5.6) 165 Since and in cylindrical coordinates, for a scalar S - dS „ ldS- dS . VS = — r + - — 0 + — z ar r 88 dz Evaluating the gradient term of Equation (E.8) gives RHS = (urf + u,z) x — — — \u\(fi + f3\v\) duT duz z + dur du, uT— h uz —— or dr dz dz Evaluating the cross product and equating to the left hand side yields h | t f | ( / l + /2 |tf|) du u, duT duz dr dr —ur + uz du. duT duz dz dr In cylindrical coordinates, the streamfunction is denned by 1 Ur = r dz i dip r dr (E.9) (E.10) (E.ll) (E.12) (E.13) IE.14) Substitution of (E.13) and (E.14) into (.E.12) and algebraic manipulation yield Equation (5.10). Appendix F Derivation of Equations (5.20) and (5.23) In Figure F. 1 Un is perpendicular to the conical base wall, hence Un = f - (F.l) where Acb is the conical base surface area given by Ach = ir(rt + rc)rsr'c = rr(rs + r c)(r c - rs)/ sin7 (F.2) Substituting Equation (F.2) into Equation (F.l) gives U. = - S ^ (P.3) In z'o'r' coordinates On the conical base surface jjr> equals zero, which means that ^ is independent of z ' . By integrating Equation (F.5) with the boundary condition ip = 0 at r' = r^ , we obtain V» = ^ n ( r f - r ' 2 ) (F.6) The transformation between coordinates z'o'r' and zor is r = r'cos(90°-7) + r, (F.7) and z = z'sin(90°-7) (F.8) 166 Appendix F. Derivation of Equations (5.20) and (5.23) 167 Z Column spout <& 0 rs rc r 0' Figure F.l: Coordinate transformation Substituting Equations (F.l) and (F.8) into (F.6) yields V> = \un[r2c - r 2 4- 2rs(r - rc)] (F.9) Appendix G Derivation of the Pressure Field Equation from the Ergun Equation The vector form of the Ergun equation is -VP = U(h+f2\U\) (G. l ) Taking the divergence of both sides gives V - V P = - V . c 7 ( / 1 + / 2 | r J | ) (G.2) which, applying the basic principles of vector algebra, expands to - V 2 P = V ( / i + h\U\) •U + (f1 + f2\U\)(V • U) (G.3) The continuity equation for an imcompressible fluid gives V - L 7 = 0 (G.4) Combining Equation (G.3) and (G.4) yields -V2P = V(f1+f2\U\)-U (G.5) However, since fi is constant, i . e. the bed properties remain constant, V / A = 0 (G.6) The pressure field is thus obtained as - V 2 P = V / 2 | L 7 | • U (G.T) 168 Appendix G. Derivation of the Pressure Field Equation 169 For a cylindrical coordinate system with axial symmetry, r dr \ dr I dz2 (G.8) which can be equated to the left hand side of Equation (G.7). The right hand side of equation (G.7) can be expanded according to the definition of the gradient and by writing |tf| = K 2 + t # 1 / 2 (G.9) to give RHS = duT duz\ „ ur — h u2 - r - I r (u2 + u2)1!2 \ T dr dr • \urr 4- uzz duT du + [ U r - d z - + U z T z (G.10) Multiplying out the scalar product and equating the RHS to the LHS gives Appendix H The Computer Program 170 Appendix H. The Computer Program 171 £ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c C THE LINE-BY-LINE TECHNIQUE WITH OVER-RELAXATION IS USED IN C THIS PROGRAM TO SOLVE THE STREAMFUNCTION,VELOCITY DISTRIBUTIONS, C AND PRESSURE PROFILE IN THE ANNULUS OF A SPOUT-FLUID BED BASED C UPON THE ERGUN EQUATION IN ITS VECTOR FORM. THE ADAPTIVE C NEWTON-COTES METHOD IS USED TO CALCULATE THE ANNULUS FLOW RATES C AT DIFFERENT BED HEIGHTS. SUBROUTINES BSOLVE AND PINIT AND C FUNCTIONS FUNC1, FUNC2, FUNC3, PWALLC, AND ELPD WERE COPIED C FROM BRERETON'S (1982) WORK. C C VARIABLES: C C MAX MAXIMUM NUMBER OF ITERATIONS C N NUMBER OF RADIUS POINTS IN BED MATRIX C M NUMBER OF VERTICAL GRID POINTS IN MATRIX C RC COLUMN RADIUS (M) C RS SPOUT RADIUS (M) C H BED DEPTH (M) C HM MAXIMUM SPOUTABLE BED DEPTH (M) C UMF MINIMUM FLUIDIZATION VELOCITY OF SOLIDS (M/S) C RHO DENSITY OF SPOUTING GAS (KG/M**3) C DP PARTICLE DIAMETER (M) C VS VISCOSITY OF SPOUTING GAS (KG/MS) C ALPHA ACCELERATING FACTOR C RHOP PARTICLE DENSITY (KG/M**3) C E VOIDAGE IN THE BED ANNULUS C GAMA INCLUDED CONE ANGLE DIVIDED BY 1/2 (DEGREES) C C R ARRAY OF R VALUES C SR ARRAY OF SINGLE PRECISION R VALUES SCALED C Z ARRAY OF Z VALUES C SZ ARRAY OF SINGLE PRECISION Z VALUES SCALED C FI ARRAY OF STREAMFUNCTION C SFI ARRAY OF SINGLE PRECISION STREAMFUNCTION SCALED C P ARRAY OF PRESSURE C SP ARRAY OF SINGLE PRECISION STREAMFUNCTION SCALED C UR ARRAY OF RADIUS VELOCITY COMPONENTS C UZ ARRAY OF VERTICAL VELOCITY COMPONENTS C UZ1 ARRAY OF ONE DIMENSIONAL VERTICAL VELOCITY COMPONENTS C QAUX ARRAY OF AUXILIRAY FLOW RATES C QA ANNULUS FLOW RATE AT BED SURFACE C EPS ACCURACY USED IN LINE-BY-LINE TECHNIQUE C EPS2 ACCURACY USED IN SUBROUTINE NC4AD C c C MAIN PROGRAM C IMPLICIT REAL*8(A-H,0-Z) REAL*4 SR,SZ,SFI,SP,CONT1,C0NT2,CNTOUR,DELZS,ZN DIMENSION UR(91,91),UZ(91,91),?(91,91),FINEW(91),A(92),B(92), C(92),D(92),PNEW(91),UZ1(91),RR(91),QAUX(3),SR(91), SZ(91) ,SFI(91,91) ,SP(91,91) Appendix II. The Computer Program 172 COMMON Z(91),FI(91,91) ,CK,DELZ,OLDFI,DELR,PI,R(91)/VI COMMON/BLKA/RR,UZ1,N EXTERNAL FUNC DATA MAX1,MAX2/200,200/ DATA EPS,RC,RS/H,HM/UMF/l.D-4/.457D0,0.64D-1,2.DO,3.43D0,.53D0/ DATA RHO,DP,VS,ALPHA,RHOP,E/1.18D0,.33D-2,.18D-4,1.7D0,1020.DO, .44D0/ DATA QAUX,EPS2,GAMA/0.0D0,.12D0,.20D0,1.D-4,30.DO/ C C Function definitions for later use C N=14 ALPHAM=1.DO-ALPHA PI=4.D0*DATAN(1.D0) GAMA=GAMA* PI /18 0 . DO DELR=(RC-RS)/(N-l)/RC DO 10 1=1,N R(I)=(RS/RC)+DELR*(N-I) 10 CONTINUE DELZ=DELR*RC/DTAN(GAMA)/H DELZS=DELZ MMI=INT(1./DELZS) REM=AMOD(1.,DELZS) M=MMI+1 MP=M+1 DO 20 J=1,M Z(J)=(J-l)*DELZ 20 CONTINUE Z(MP)=Z(M)+REM M=MP M1=M-1 N1=N-1 Vl=(RC/H)**2 F1 = 150.D0*((1.D0-E) **2/E**3) *VS/DP**2 F2 = 1.75D0*((1.D0-E) /E**3)*RHO/DP CK=F1/UMF/F2 BETA=3.D0*F1/2.D0/F2/UMF+2.D0 DELPF= (RHOP-RHO) ME-1.D0) *H*9.81D0 PM=DELPF/(2.D0*BETA-1.D0) /H/F2/UMF/UMF C C Initialization of streamfunction according to Mamuro Hattori C Model C C Initialisation of wall points C DO 30 J=1,M FI(1/J)=0.D0 30 CONTINUE C C Initialisation of interior points C DO 50 J=1,M DO 40 1=2,N Appendix H. The Computer Program FI(I,J)=.44D0*(1.D0-R(I)**2)*(1-D0-(1.D0-Z(J)*H/HM)**3) 40 CONTINUE 50 CONTINUE C C DO 480 IQAUX=1,3 C UN=DSIN(GAMA)*QAUX(IQAUX) /PI/(RC**2-RS**2) WRITE (6,70) IQAUX,QAUX(IQAUX) 70 FORMAT(//3X,'(',11,'): THE AUXILIARY FLOW RATA = ',F7.4) C C C USE THE LINE-BY-LINE TECHNIQUE TO CALCULATE STREAMFUNCTION C C Iterations begin C DO 170 ITER=1,MAX1 DIFMAX=0.D0 C C Use the boundary conditions C DO 80 1=2,N FINEWW=FI(I,M1> DIFMAX=DMAX1(DIFMAX,DABS(FINEWW-FI(I,M))) FI(I,M)=FINEWW 80 CONTINUE DO 90 J=2,M1 CALL BSOLVE(CORECT,PM, J, I,H,HM,BETA,TOTER,N) 90 CONTINUE C C Rows C DO 120 J=2,M1 C C BOUNDARY CONDITION ON THE CONICAL BASE C IF(J.GT.N) GO TO 99 DO 95 L=1,N FI(N+1-L,L)=2.D0*UN/UMF*(1.-R(N+1-L)**2+2.*RS/RC *(R(N+1-L)-1.)) 95 CONTINUE C DO 79 11=1,Nl IN=N-II DO 78 JJ=1,IN FI(II,JJ)=FI(II,N-II+1) 78 CONTINUE 79 CONTINUE 99 DO 100 1=2,Nl DR= (FI (1-1, J) -FKI + l , J) ) /2.D0/DELR DZ=(FI(I,J+l) -FI(I, J-l) ) /2.D0/DELZ DR2=DR*DR DZ2=DZ*DZ Appendix H. The Computer Program 174 DRZ1=FI ( I - l , J+l) +FKI + 1, J-l) DRZ2=FI(1+1,J+l)+FI(I-l,J-l) DRZ=(DRZ1-DRZ2)/4.DO/DELR/DELZ TC=(V1*DZ2+DR2)**.5D0 FF1=FUNC1(V1,DR2,DZ2,TC,R,CK/DELR,I) FF2=FUNC2(DZ2,DR2,VI,TC,R,CK,DELZ,I) FF3=FUNC3(TC,DZ,DR,DRZ,R,CK,V1,I) A(I)=FF1 B(I)=-2.D0*(FFl+FF2) C(I)=FF1 D(I)=-FF2*FI(I,J-l)-FF2*FI(I,J+l)-FF3 100 CONTINUE D(2) =D(2) -A(2) *FI(1,J) D(N1) =D(N1)-C(N1) *FI(N, J) CALL TDMAR(A,B,C,D,FINEW,N) DO 110 1=2,Nl FINEW(I)=ALPHAM*FI(I,J) +ALPHA*FINEW(I) DIFMAX=DMAX1(DIFMAX,DABS(FINEW(I)-FI(I,J))) DIFMAX=DIFMAX FI(I,J)=FINEW(I) 110 CONTINUE 120 CONTINUE DO 121 1 = 2,N FI(I,M)=FI(I,Ml) 121 CONTINUE C C C Columns C DO 150 I=2,N1 C DO 122 L=1,N FI(N+1-L,L)=2.D0*UN/UMF*(l.-R(N+l-L)**2+2.*RS/RC * (R(N+1-L)-1.) ) 122 CONTINUE C DO 124 11=1,Nl IN=N-II DO 123 JJ=1,IN FI(II,JJ)=FI(II/N-II+1) 123 CONTINUE 124 CONTINUE 125 DO 130 J=2,M1 DR=(FI(I-l,J)-FI(1 + 1,J) ) /2.D0/DELR DZ=(FI(I,J+1)-FI(I,J-1) )/2.D0/DELZ DR2=DR*DR DZ2=DZ*DZ DRZ1=FI(I-1, J+l) +FKI + 1, J-l) DRZ2 = FI (1 + 1, J+l) +FKI-1, J-l) DRZ=(DRZ1-DRZ2)/4.DO/DELR/DELZ TC=(V1*DZ2+DR2)**.5D0 FF1=FUNC1(VI,DR2,DZ2,TC,R,CK,DELR,I) FF2 = FUNC2(DZ2,DR2,VI,TC,R,CK,DELZ,I) Appendix H. The Computer Program 175 FF3=FUNC3(TC,DZ,DR/DRZ,R,CK,V1,I) A(J)=FF2 B(J)=-2.DO*(FF1+FF2) C(J)=FF2 D(J) =-FFl*FI(1-1/J)-FF1*FI(1 + 1,J)-FF3 130 CONTINUE D(2) =D(2)-A(2) *FI(I,1) D(M1) =D(M1) -C(M1) *FI(I,M) CALL TDMAR(A,B,C,D,FINEW,M) DO 140 J=2,M1 FINEW(J)=ALPHAM*FI(I,J)+ALPHA*FINEW(J) DIFMAX=DMAX1(DIFMAX,DABS(FINEW(J)-FI(I, J) ) ) FI(I,J)=FINEW(J) 140 CONTINUE 150 CONTINUE 160 IF(DIFMAX.LT.EPS) GO TO 190 170 CONTINUE WRITE(6,180) DIFMAX 180 FORMAT(//IX,'WARNING',F7.4) C C Print the results of streamfunction C 190 WRITE(6,200) EPS 200 FORMAT(//5X,'ACCURACY= ',D8.D WRITE(6,210) ITER 210 FORMAT(5X,'ITERATIONS= ',14) WRITE(6,220) 220 FORMAT(/24X,'STREAMFUNCTION MATRIX'/) CALL PRINT(FI,M,N,Z,R) C C Calculate the velocity distribution and print the results C WRITE(6,230) 230 FORMAT(//24X,'DIMENSIONLESS VELOCITIES;UPPER-HORIZONTAL CMPT.') WRITE(6,240) 240 FORMAT(50X,'LOWER-VERTICAL CMPT.') CALL VELSOL(FI,N,R,DELR,DELZ,Nl,M,RC,H,Ml,UR,UZ,PI,RS, UN,UMF,Z,GAMA) DO 248 J=1,N1 IN=N-J DO 245 1=1,IN UZ(I,J)=UZ(IN+1,N-I+1) UR(I,J)=UR(IN+1,N-I+1) 245 CONTINUE 248 CONTINUE C C USE THE LINE-BY-LINE TECHNIQUE TO CALCULATE PRESSURE PROFILE C C Iterations begin C IF(IQAUX.GT.l) GO TO 250 CALL PINIT(N,P,Z,H,HM,BETA,M,Nl,DELPS,DELPF) Appendix H. The Computer Program 176 250 DENOM=2.D0*(l.D0/(DELR*DELR)+Vl/(DELZ*DELZ)) TNUM3=V1/(DELZ*DELZ) DO 340 ITER=1,MAX2 DIFMAX=0.D0 C C Use the boundary c o n d i t i o n s C DO 270 J=2,N I=N+1-J P(I,J)=PWALLC(P,GAMA,I,J,DELR,DELZ,H,RC) 270 CONTINUE DO 275 J=2,M P ( l , J) =P(2,J) 275 CONTINUE DO 279 J=1,N1 IN=N-J DO 278 1=1,IN P(I,J)=P(IN+1 /N-I+1) 278 CONTINUE 279 CONTINUE C C C C Rows C DO 300 J=2,M1 DO 280 I=2,N1 IREV=N+1-I TNUM1=1.D0/(DELR*DELR)+1.DO/(2.D0*R(IREV)*DELR) TNUM2=1.DO/(DELR*DELR)-1.DO/(2.DO*R(IREV)*DELR) FU=RHSP(DELR,DELZ,UR,UZ,IREV,J,PI,H,F2,RC,DELPS,UMF) A(IREV)=TNUM1 B(IREV)=-DENOM C(IREV)=TNUM2 D(IREV)=-TNUM3*(P(IREV,J-l)+P(IREV,J+l))+FU 280 CONTINUE D(2) =D(2) -A(2) *P(1, J) D(N1) =D(N1) -C(N1) *P(N, J) CALL TDMAR(A,B,C,D,PNEW,N) DO 290 1=2,Nl PNEW(I)=ALPHAM*P(I,J)+ALPHA*PNEW(I) DIFMAX=DMAX1(DIFMAX,DABS(PNEW(I)-P(I,J) ) ) DIFMAX=DIFMAX P(I,J)=PNEW(I) 290 CONTINUE 300 CONTINUE C C C Columns C DO 330 1=2,Nl IREV=N+1-I DO 305 JJ=2,N Appendix H. The Computer Program 177 II=N+1-JJ P (11,JJ)=PWALLC(P,GAMA, 11, J J,DELR,DELZ,H,RC) 305 CONTINUE DO 310 J=2,M1 TNUM1 = 1.DO/(DELR*DELR) +1.D0/(2.D0*R(IREV)*DELR) TNUM2 = 1.DO/(DELR*DELR) -1.DO/(2.D0*R(IREV)*DELR) FU=RHSP(DELR,DELZ,UR,UZ, IREV, J, PI,H,F2,RC,DELPS,UMF) A(J)=TNUM3 B(J)=-DENOM C(J)=TNUM3 D(J)=-TNUM1*P(IREV-1,J) -TNUM2*P(IREV+1,J)+FU 310 CONTINUE D(2) =D(2) -A(2) *P(IREV,1) D(M1) =D(M1) -C(M1) *P(IREV,M) CALL TDMAR(A,B,C,D,PNEW,M) DO 320 J=2,M1 PNEW(J)=ALPHAM*P(IREV,J)+ALPHA*PNEW(J) DIFMAX=DMAX1(DIFMAX,DABS(PNEW(J)-P(IREV,J))) P (IREV,J)=PNEW(J) 320 CONTINUE 330 CONTINUE 335 IF(DIFMAX.LT.EPS) GO TO 360 340 CONTINUE WRITE(6,350) DIFMAX 350 FORMAT(//lX,'WARNING',F7.4) C C Print the results of pressure distribution C 360 WRITE(6,370) EPS 370 FORMAT(//5X,'ACCURACY= ',08.1) WRITE(6,380) ITER 380 FORMAT(5X,' ITERATIONS= ',14) WRITE(6,390) 390 FORMAT(/24X,'PRESSURE PROFILE'/) CALL PRINT(P,M,N,Z,R) GO TO 2000 C C Calculate the annulus flow rate C DO 420 J=M,M DO 400 1=1,N IREV=N+1-I UZ1(I)=UZ(IREV,J) RR(I) =R(IREV) 400 CONTINUE CALL SPLINE(RR,UZ1,N,3) CALL NC4AD(FUNC,RR(1),RR(N),EPS2,QA) QA=2.D0*PI*UMF*RC*RC*QA WRITE(6,410) QA 410 FORMAT(//5X,'THE ANULUS FLOW RATE AT Z=H:',F10.6) 420 CONTINUE C C Appendix H. The Computer Program 178 C C Scaling and plotting C 2000 DO 440 1=1,N DO 430 J=1,M SFI(I,J) =FI(I,J) SP (I, J)=P(I,J) 430 CONTINUE 440 CONTINUE DO 450 I=1,N SR(I)=R(I)/.2+3. 450 CONTINUE DO 460 J=1,M SZ(J)=Z(J)/.1+2. 460 CONTINUE C C SET NEGATIVE VALUE SO AS TO AVOID CONTOURING C DO 467 J=1,N1 IN=N-J DO 465 I=1,IN SP(I,J)=-10.0E30 465 CONTINUE 467 CONTINUE DO 469 J=1,N1 IN=N-J DO 468 1=1,IN SFI(I,J)=-10.0E30 468 CONTINUE 469 CONTINUE C CALL AXIS(3.,2.,'RADIUS (R)',-10,5.,0.,0.,.2) CALL AXIS(3.,2.,'HEIGHT (Z) ',10,10.,90 .,0 ., . 1) CALL PLOT(3.,12., 3) CALL PLOT(8.,12.,2) CALL PLOT(8.,SZ(N),2) CALL PLOT(SR(N),2., 2) CALL PLOT(SR(N) ,12.,2) CALL SYMBOL(1.8,12.25,.25,'STREAMFUNCTION AND PRESSURE PROFILE' ,0.,35) CALL SYMBOL(3.5,1., .15,'AUXILIARY FLOW RATE = ',0.,22) CALL NUMBER(6.5,1., .15,QAUX(IQAUX),0.,3) DO 470 ICONT=1,20 CONT1=ICONT*0.035+0.05 CONT2=ICONT*0.05 CALL CNTOUR(SR,N,SZ,M,SFI,91,CONT1,-10.,CONTl) CALL CNTOUR(SR,N,SZ,M,SP,91,CONT2,-10.,CONT2) 470 CONTINUE CALL PLOT(11.,14.,-3) 4 80 CONTINUE CALL PLOTND STOP END Appendix H. The Computer Program 179 C c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE BSOLVE(CORECT,PM,J,I,H,HM,BETA,TOTER,N) C C SOLVE THE STREAMFUNCTION AT THE SPOUT ANNULUS INTERFACE. C**************************************************************** C IMPLICIT REAL*8(A-H,0-Z) DIMENSION POSSRT(4),DEV(4),ROOTR(4) ,ROOTI(4),C(5) COMMON Z(91) ,FI(91,91) ^ K/DELZ/OLDFI^ELR/PI^OD ,V1 I=N Y=Z(J)*H/HM RA=2.D0*(BETA-2.D0)*Y*(3.D0+Y*(Y-3.D0)) RB=3.D0*Y*Y*(9.D0+Y*(-18.D0+Y*(15.D0+Y*("6.D0+Y*1.001D0) ) ) ) FZ=PM*(RA+RB) DZ=(FI(I/J+l)-FI(I,J-l))12.DO/DELZ NR=4 OLDRT=(FI(1-1/J)-FI(I,J))/DELR C(l)=1.D0 C(2)=0.D0 C(3)=V1*DZ*DZ-R(I) *R(I) *CK*CK C(4)=2.D0*R(I)**3*CK*FZ C(5)=-l.DO*FZ*FZ*R(I)**4 CALL RPOLY1(C,NR,ROOTR,ROOTI,96) IF(NR.EQ.O) STOP NUNRT=0 DO 10 L=1,NR CROOT=DABS(ROOTI(L) ) IF(CROOT.GT.l.D-4) GO TO 10 IF(ROOTR(L).GT.0.D0) GO TO 10 NUNRT=NUNRT +1 POSSRT(NUNRT)=ROOTR(L) 10 CONTINUE IF(NUNRT.EQ.l) GO TO 40 DO 20 L=l,NUNRT DEV(L)=DABS(POSSRT(L)-OLDRT) 20 CONTINUE IBSF=1 DO 30 L=2,NUNRT IF(DEV(1) .LT.DEV(L) ) GO TO 30 IBSF=L DEV(l) =DEV(L) 30 CONTINUE CORECT=POSSRT(IBSF) GO TO 50 40 CORECT=POSSRT(l) 50 OLDFI=FI(1/J) FI (I, J) =FI (1-1, J) -CORECT * DELR RETURN END C Appendix H. The Computer Program ISO Q************************************************************** SUBROUTINE PRINT(FI,M,N,Z,R) C C THIS IS THE STREAMFUNCTION AND PRESSURE DISTRIBUTION PRINT C SUBROUTINE Q************************************************************** c IMPLICIT REAL*8(A-H,0-Z) DIMENSION FI(91,91),H0LD(91),Z(91) ,R(91) WRITE(6,10) 10 F0RMAT(/1X,' HEIGHT',18X,'DIMENSIONLESS RADIUS') DO 20 1=1,N HOLD(I)=R(N+1-I) 20 CONTINUE WRITE(6,30) (HOLD(I),I=1,N) 30 FORMAT(/9X,20F7.3, /) DO 60 J=1,M JREV=M+1-J DO 40 IN=1,N HOLD(IN)=FI(N+1-IN,JREV) 40 CONTINUE WRITE(6,50) Z(JREV),(HOLD(K),K=1,N) 50 FORMAT(IX,F7.3,1X,20F7.3) 60 CONTINUE RETURN END C Q*************************************************************** DOUBLE PRECISION FUNCTION FUNCl(VI,DR2,DZ2,TC,R,CK,DELR,I) C C THIS IS A FUNCTION SUBROUTINE CALLED BY LINE-BY-LINE PROCEDURE. Q*************************************************************** c IMPLICIT REAL*8(A-H,0-Z) DIMENSION R(91) T1=2.D0*DR2+V1*DZ2 FUNC1=(R(I)*CK*TC+T1)/(DELR*DELR) RETURN END C Q*************************************************************** DOUBLE PRECISION FUNCTION FUNC2(DZ2,DR2,VI,TC,R,CK,DELZ,I) C C THIS IS A FUNCTION SUBROUTINE CALLED BY LINE-BY-LINE PROCEDURE. Q* ************************************************************* * c IMPLICIT REAL*8(A-H,0-Z) DIMENSION R(91) T2=2.D0*V1*DZ2+DR2 FUNC2=(R(I)*CK*TC+T2)*V1/(DELZ*DELZ) RETURN END C Appendix H. The Computer Program 181 Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DOUBLE PRECISION FUNCTION FUNC3(TC,DZ,DR,DRZ,R,CK,VI,I) C C THIS IS A FUNCTION SUBROUTINE CALLED BY LINE-BY-LINE PROCEDURE. Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C IMPLICIT REAL*8(A-H,0-Z) DIMENSION ROD T3=2.D0*V1*DZ*DR*DRZ FUNC3=T3-CK*TC*DR-2.D0*TC*TC*DR/R(I) RETURN END C C************************************************************* SUBROUTINE TDMAR(A,B,C,D,X,N) C C THOMAS ALGORITHM £************************************************************* C IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(N) ,B(N) ,C(N) ,D(N) ,X(N) ,P(91) ,Q(91) NM=N-1 NM2=N-2 P(2) =-C(2) /B(2) Q(2) =D(2)/B(2) DO 10 1=3,NM2 IM=I-1 DEN=A(I) *P(IM) +B(I) P(I)=-C(I)/DEN Q(I)=(D(I)-A(I)*Q(IM)) /DEN 10 CONTINUE X(NM)=(D(NM)-A(NM)*Q(NM2))/(A(NM)*P(NM2)+B(NM)) DO 20 11=2,NM2 I=N-II X(I) =P(I) *X(I + 1)+Q(I) 20 CONTINUE RETURN END C C* ************************************************************** SUBROUTINE VELSOL(FI,N,R,DELR,DELZ,Nl,M,RC,H,Ml,UR,UZ,PI,RS, UN,UMF,Z,GAMA) C C SUBROUTINE VELSOL TAKES THE STREAMFUNCTION MATRIX FI AND C DIFFERENTIATES THE MATRIX TO CREATE VELOCITY COMPONENTS BOTH C HORIZONTAL AND VERTICAL. THE VELOCITY COMPONENTS ARE RENDERED C DIMENSIONLESS FOR SIMPLICITY OF USAGE IN THE PRESSURE SOLUTION C*************************************************************** C IMPLICIT REAL*8(A-H,0-Z) DIMENSION FI(91,91),R(91),UR(91,91),UZ(91,91),Z(91),HOLD(91) C Appendix H. The Computer Program 182 C Solution at the bed surface C J=M DO 10 I=l/N UR(I,J)=0.D0 10 CONTINUE UZ (1, J) =-l.D0* (FI (1, J) -FK2, J) ) /DELR/R(1) DO 20 1=2,Nl UZ(I,J)=-1.DO*(FI(1-1,J)-FI(I + 1,J) )/DELR/R(I)/2.D0 20 CONTINUE UZ(N,J)=-1.DO*(FI(Nl,J)-FI(N,J) ) /DELR/R(N) WRITE(6,30) 30 FORMAT(/lX,' HEIGHT',18X,'DIMENSIONLESS RADIUS') DO 40 1=1,N HOLD(I) =R(N+1-I) 40 CONTINUE WRITE(6,50) (HOLD(I),I=1,N) 50 FORMAT(/9X,20F7.3) CALL VPRINT(N,UZ,UR,J,Z) C C Solution in the bulk of the bed C MFINAL=M-2 DO 80 JR=1,MFINAL J=M-JR UZ (1, J) =-l.D0* (FI (1, J) -FK2, J) ) /DELR/R (1) UR(1, J) =RC/H* (FI (1, J+l) -FI (1, J-l) ) /DELZ/RU) /2.D0 DO 60 1=2,Nl UZ(I,J)=-l.D0*(FI(I-l,J)-FI(I+1,J))/2.D0/DELR/R(I) UR (I, J) =RC/H* (FI (I, J+l) -FKI, J-l) ) /DELZ/Rd) /2.D0 IF(J.GE.N) GO TO 60 UZ(N+l-J,J)=UN*DSIN(GAMA)/UMF UR(N+l-J,J)=-UN*DCOS(GAMA)/UMF 60 CONTINUE 70 UZ(N,J)=-l.D0*(FI(Nl,J)-FI(N, J) ) /DELR/R(N) UR(N,J)=RC/H*(FI(N,J+1)-FI(N,J-1) ) /2.D0/DELZ/R(N) CALL VPRINT(N,UZ,UR,J,Z) 80 CONTINUE C C Solution at the baseplate C DO 105 1=1,N UZ(I,1)=0.1D-10 UR(I,1)=0.1D-10 105 CONTINUE J=l CALL VPRINT(N,UZ,UR,J,Z) RETURN END C C c c Appendix H. The Computer Program 183 C*************************************************************** SUBROUTINE VPRINT(N,UZ,UR,J,Z) C C THIS IS THE VELOCITY PRINT SUBROUTINE. Q*************************************************************** c IMPLICIT REAL*8(A-H,0-Z) DIMENSION UZ(91,91) ,UR(91,91)/RKEEP(91),ZKEEP(91),Z(91),HOLD(91) DO 10 1 = 1,N RKEEP(I)=UR(N+1-I, J) ZKEEP(I)=UZ(N+l-I,J) 10 CONTINUE WRITE(6,20) Z(J),(RKEEP(I),I=1,N) 20 FORMAT(/IX,F7.3,1X,20F7.3) WRITE(6,30) (ZKEEP(I) ,I = 1,N) 30 FORMAT(9X,20F7.3) RETURN END C C**************************************************************** DOUBLE PRECISION FUNCTION PWALLC(P,GAMA,I,J,DELR,DELZ,H,RC) C C THE FUNCTION PWALLC CALCULATES THE PRESSURE VALUES ON THE C CONICAL BASE. £ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c IMPLICIT REAL*8(A-H,0-Z) DIMENSION P(91,91) DR=DELR*RC DZ=DELZ*H CNUM=P(1 + 1,J)*DCOS(GAMA) /DR+P(I,J+l) *DSIN(GAMA)/DZ DENOM=DCOS(GAMA)/DR+DSIN(GAMA)/DZ PWALLC=CNUM/DENOM RETURN END C Q**************************************************************** DOUBLE PRECISION FUNCTION RHSP(DELR,DELZ,UR,UZ,I, J,PI,H,F2,RC, DELPS,UMF) C C THE FUNCTION RHSP CALCULATES THE FUNCTION FU FOR SOLVING C PRESSURE DISTRIBUTIONS. C ****** * ********************************************************* c IMPLICIT REAL*8(A-H,0-Z) DIMENSION UR(91,91),UZ(91,91) C C Derivative definitions C DURDR=(UR(I-l,J)-UR(1+1,J))12.DO/DELR DUZDR=(UZ(I-l,J)-UZ(1+1,J))/2.D0/DELR DURDZ=(UR(I,J+l)-UR(I, J-l)) /2.DO/DELZ DUZDZ=(UZ(I,J+l)-UZ(I,J-l))/2.D0/DELZ Appendix H. The Computer Program 184 SPEED=DSQRT(UR(I/J)*UR(I,J) +UZ(I,J)*UZ(1/J)) PRE=F2*RC*UMF*UMF/DELPS C C C C T1,T2,T3 are 3 terms of the function and PRE is a C premultiplier term. C T1=UR(I/J)*UR(I,J)*DURDR T2=UR(I,J)*UZ(I,J)*(DUZDR+RC*DURDZ/H) T3=RC*UZ(I,J)*UZ(I,J)*DUZDZ/H RHSP=-1.D0*PRE*(T1+T2+T3)/SPEED RETURN END C Q*************************************************************** SUBROUTINE PINIT(N,P,Z,H,HM,BETA,M,Nl,DELPS,DELPF) C C THIS IS THE SUBROUTINE FOR INITIALISATION OF PRESSURE MATRIX. C THE UPPER BED SURFACE IS INITIALISED TO ATMOSPHERIC AND C THE SPOUT ANNULUS INTERFACE INITIALISED ACCORDING TO THE C EPSTEIN LEVINE PRESSURE DISTRIBUTION. Q*************************************************************** c IMPLICIT REAL*8(A-H,0-Z) DIMENSION Z(91),P(91,91) M1=M-1 C C I n i t i a l i s e bed surface C DO 10 1=1,N P(I,M)=0.D0 10 CONTINUE C C I n i t i a l i s e the spout annulus interface C DO 20 J=1,M P(N,J)=ELPD(Z,H,HM,BETA,J,DELPS,DELPF) 20 CONTINUE C C I n i t i a l i s e remainder of matrix C DO 40 J=1,M1 DO 30 1=1,Nl P(I,J)=.1D0 30 CONTINUE 40 CONTINUE RETURN END C C C C Appendix H. The Computer Program 185 Q********************************** ************** DOUBLE PRECISION FUNCTION ELPD(Z,H/HM,BETA,J,DELPS,DELPF) C C THIS IS THE EPSTEIN LEVINE PRESSURE DISTRIBUTION EQUATION WHICH C IS NORMALISED WITH RESPECT TO THE SPOUT PRESSURE DISTRIBUTION. C THE FUNCTION IS CALLED BY. THE SUBROUTINE PINIT IN THE COURSE C OF THE PRESSURE MATRIX INITIALISATION. Q* ************************************************************ * c IMPLICIT REAL*8(A-H,0-Z) DIMENSION Z(91) UL=H/HM CLL=Z(J)*H/HM T1=UL*UL*(1.5D0+UL*(-1.DO+UL*.25D0)) T2=CLL*CLL*(1.5D0+CLL*(-1-DO+CLL*.25D0) ) T3=UL*UL*(3.DO+UL*(-4.5D0+UL*(3.DO+UL*(-1.DO+UL*.143D0)))) T4=CLL*CLL*(3.DO+CLL*(-4.5D0+CLL*(3.DO+CLL*(-1.DO+CLL*.143D0) ))) T5=2.D0*(BETA-2.D0)*(T1-T2) T6=3.D0*(T3-T4) DELPS = -1.D0*DELPF*(2.DO*(BETA-2.D0)*T1+3.D0*T3) /(2.D0*BETA-1 .DO) ELPD=(T5+T6)/(2.DO*(BETA-2.D0)*T1+3.D0*T3) RETURN END C Q* *********************************************************** * SUBROUTINE NC4AD(F,A,B,EPS,AREA) C C INTEGRATION USING THE ADAPTIVE NEWTON-COTES METHOD Q************************************************************* C IMPLICIT REAL*8(A-H,0-Z) DIMENSION H(20),TOL(20),SR(20)rXR(20) DIMENSION FK20) ,F2(20) ,F3(20) ,F4(20) ,F5(20) , F6(20) ,F7(20) ,F8(20) ,F9(20) IMAX=20 N=5 AREA=0.D0 XI = A C C SET STRIP WIDTH AND TOLERANCE FOR EACH LEVEL C H(l)=(B-A)/4.D0 TOL(l)=50.D0*EPS DO 10 I=2,IMAX IM=I-1 H(I) =H(IM) /2.D0 TOL(I)=TOL(IM)/2.D0 10 CONTINUE C C CALCULATE INITIAL AREA S USING FOUR STRIPS C XR(1)=A+4.D0*H(1) Fl(1)=F(A)*A Appendix H. The Computer Program F3(l) =F(A+H(1) ) * (A+H(l) ) F5(l)=F(A+2-D0*H(l))*(A+2.D0*H(1) ) F7(1)=F(A+3.D0*H(1))*(A+3.D0*H(1) ) F9(l) =F(B) *B S=2.D0*H(1)*(7.D0*F1(1)+32.D0*F3(1)+12.D0*F5(1) +32.D0*F7(1) +7-D0*F9(l))/45.D0 1 = 1 C C RECALCULATE SL AND SR UNTIL SL+SR~S<TOL C 20 N=N+4 F2(I)=F(X1+H(I)/2.D0)*(X1+H(I)/2.D0) F4(I)=F(X1 + 3.D0*H(I)/2.D0)*(X1+3.D0*H(I) /2.D0) F6(I)=F(X1+5.D0*H(I)/2.D0)*(Xl+5.D0*H(I)/2.D0) F8(I)=F(X1+7.D0*H(I)/2.D0)*(Xl+7.D0*H(I) /2.D0) SL=H(I)*(7.D0*F1(I)+32.D0*F2(I)+12.D0*F3(I)+32.D0*F4(I)+ 7.D0*F5(I))/45.D0 SR(I)=H(I)*(7.D0*F5(I)+32.D0*F6(I)+12.D0*F7(I) +32.D0*F8(I) +7.D0*F9(I))/45.D0 IF(DABS(SL+SR(I)-S) .LT.TOL(I) ) GO TO 30 C C C IF SL+SR-S>TOL,INCREASE LEVEL AND SUBDIVIDE LEFT STRIP C IM=I 1 = 1 + 1 IF(I.GT.IMAX) GO TO 60 S = SL Fl(I)=F1(IM) F3(I)=F2(IM) F5(I)=F3(IM) F7(I)=F4(IM) F9(I)=F5(IM) XR(I)=X1+4.D0*H(I) GO TO 20 C C IF SL+SR-S<TOL, ADD SL+SR ONTO S AND LOCATE CORRECT LEVEL C 30 AREA=AREA+SL+SR(I) X1=X1+4.D0*H(I) DO 40 J=1,I IF(DABS(Xl-XR(J)).LT.H(IMAX)12.DO) GO TO 50 40 CONTINUE 50 I = J IF(I.EQ.l) RETURN IM=I-1 S=SR(IM) Fl(I)=F5(IM) F3(I)=F6(IM) F5(I)=F7(IM) F7(I)=F8(IM) F9(I)=F9(IM) GO TO 20 Appendix H. The Computer Program 187 60 WRITE(6,70) XI 70 FORMAT(IX,'WARNING - INTEGRATION FAILS BEYOND X =',F18.10) RETURN END C C************************************************************* SUBROUTINE SPLINE(X,Z,N,L) C C Interpolation using cubic splines. Three different kinds C of boundary conditions are availabe. C C Input: X Array of independent x-values C Z Array of dependent z-values C N Number of data points C C Output: Q,R,S Coefficients of cubic spline equations C************************************************************* C IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKB/Q(90),R(91),S(90) DIMENSION H(90),A(91),B(91),C(91),D(91),TAB(1001,101), X(91),Z(91) C C Cofficient matrices for cubic polynomials passing through C end points C DATA M/4/ NM=N-1 C C Calculate H(I) C DO 10 1=1,NM 10 H(I) =X(I + 1)-X(I) GO TO(20,30,70) ,L C C Use the natueal cubic splines C 20 A(1)=0. B(l) =1. C(l) =0. D(l) =0. A(N)=0. B(N) =1. C(N) =0. D(N) =0. GO TO 80 C C Use the Clamped cubic splines C 30 A(1)=0. B(l) =2.*H(1) Cd) =H(1) D(l) =3.* ( (Z(2)-Z(l) ) /H(l)-0.) Appendix H. The Computer Program 188 A (N) =H (NM) B(N) =2.*H(NM) C(N)=0. GO TO(40,50)/LL 40 GB=0.D0 GO TO 60 50 GB=0.D0 60 D(N)=-3.*((2(N)-Z(NM))/H(NM)-GB) GO TO 80 C C Use the cubic splines with f i t t e d end points C 70 CALL DIVDIF(X,Z,N,M,TAB) A4=TAB(1,M) K=N-M+1 B4=TAB(K,M) C C Coefficients of tridiagonal equations C A(l) =0. B(l) =-H(l) C(1)=H(1) D(l) =3.*H(1) *H(1) *A4 A (N) =H (NM) B(N)=-H(NM) C(N) =0. D(N)=-3.*H(NM)*H(NM)*B4 80 DO 90 I=2,NM IP=I+1 IM=I-1 A(I)=H(IM) B(I) =2.* (H(IM) +H(I) ) C(I) =H(I) 90 D(I) =3.* ((Z(IP) -Z(I) ) /H(I) -(Z(I) -Z(IM) ) /H(IM) ) C C Call Thomas algorithm to solve tridiagonal set C CALL TDMA(A,B,C/D/R/N) C C Determine Q(I) and S(I) C DO 100 1=1,NM IP=I+1 Q(I) = (Z(IP) -Z(I) ) /H(I) -H(I) * (2.*R(I) +R(IP) ) /3. 100 S(I) = (R(IP)-R(I))/(3.*H(D) RETURN END C Q************************************ ************************* SUBROUTINE TDMA(A,B,C,D,X,N) C C Thomas algorithm C************************************************************* Appendix H. The Computer Program 189 IMPLICIT REAL*8(A-H,0-Z) DIMENSION A (N) ,B(N) ,C(N) ,D(N) ,X(N) ,P(91) ,Q(91) NM=N-1 P( l ) =-C(l) /B(l) Q(l) =D(1) /B(l) DO 10 I=2,N IM=I-1 DEN=A(I)*P(IM)+B(I) P(I)=-C(I)/DEN 10 Q(I) = (D(I)-A(I) *Q (IM) ) /DEN ' X(N)=Q(N) DO 20 11=1,NM I=N-II 20 X(I) =P(I) *X(I + 1)+Q(I) RETURN END C Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE DIVDIF(X,Z,N,M,TAB) C C CALCULATE DIVIDED DIFFERENCES c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(N),Z(N),TAB(1001,100) IF(M.GT.N) M=N DO 10 1 = 1,N-TAB(I,1)=Z(I) 10 CONTINUE DO 30 J=2,M JM=J-1 NM=N-JM DO 20 1=1,NM TAB(I,J) = (TAB(I + 1,JM) -TAB(I,JM))/(X(I+JM)-X(I)) 20 CONTINUE 30 CONTINUE RETURN END C c c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DOUBLE PRECISION FUNCTION FUNC(Z) C C Sp l i n e i n t e r p o l a t i o n f u n c t i o n . I n t e r p o l a t i o n i n t e r v a l found C by b i s e c t i o n . Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKA/X(91),Y(91),N COMMON/BLKB/Q(90),R(91),S(90) 1 = 1 NM=N-1 IF(Z.LT.Xd) ) GO TO 30 Appendix H. The Computer Program 190 IF(Z.GE.X(NM) ) GO TO 20 J=NM 10 K=(I + J)/2 IF(Z.LT.X(K)) J=K IF(Z.GE.X(K)) I=K IF(J.EQ.I+1) GO TO 30 GO TO 10 20 I=NM 30 DX=Z-X(I) FUNC=Y(I)+DX*(Q(I)+DX*(R(I)+DX*S(I))) RETURN END Appendix I Predictions from the Mathematical Model 191 (1) : AUXILIARY FLOWRATE • 0.00 M**3/S STREAMFUNCTION MATRIX HEIGHT DIMENSIONLESS RADIUS 0 . 140 0 . 206 0 . 2 7 2 0 . 338 0 . 4 0 5 0 . 471 0 . 5 3 7 0 . 6 0 3 0 . 669 0 , , 7 3 5 0 . 8 0 2 0 . 8 6 8 0 . 9 3 4 1 . 0 0 0 1 . 0 0 0 0 . 7 0 2 0 . 6 8 5 0 . 6 6 1 0 . 6 3 2 0 . 596 0 . 554 0 . 506 0 . 4 5 2 0 . 3 9 2 0 . , 326 0 , , 254 0 . 175 0 . 0 9 1 0 . 0 0 0 0 . 9 9 5 0 . 7 0 2 0 . 6 8 5 0 . 6 6 1 0 . 6 3 2 0 . 596 0 . 554 0 . 5 0 6 0 . 4 5 2 0 . 3 9 2 0 , , 326 0 , . 254 0 . 175 0 . 0 9 1 0 . 0 0 0 0 . 9 6 9 0 . 7 0 0 0 . 6 8 3 0 . 6 6 0 0 . 6 3 0 0 . 5 9 5 0 . 5 5 3 0 . 505 0 . 451 0 . 391 0 , , 3 2 5 0 , , 2 5 3 0 . 175 0 , 0 9 1 0 . 0 0 0 0 . 9 4 3 0 . 6 9 7 0 . 6 8 0 0 . 6 5 7 0 . 6 2 7 0 . 5 9 2 0 . 551 0 . 5 0 3 0 . 449 0 . 3 9 0 0 . , 3 2 4 0 , . 2 5 2 0 . 174 0 . 0 9 0 0 . 0 0 0 0 .9 16 0 . 6 9 2 0 . 6 7 6 0 . 6 5 3 0 , 6 2 4 0 . 5 8 8 0 . 547 0 . 5 0 0 0 . 447 0 . 387 0 . , 3 2 2 0 , , 251 0 . 173 0 . 0 9 0 0 . 0 0 0 0 . 8 9 0 0 . 6 8 6 0 . 6 7 0 0 . 6 4 7 0 . 6 1 9 0 . 5 8 4 0 . 543 0 . 4 9 6 0 . 443 0 . 3 8 5 0 , . 3 2 0 0 , . 249 0 . 172 0 . 0 8 9 0 . 0 0 0 0 . 8 6 4 0 . 6 8 0 0 . 6 6 3 0 . 6 4 1 0 , . 6 1 3 0 . 5 7 8 0 . 5 3 8 0 . 4 9 2 0 . 4 3 9 0 . 381 0 , , 3 1 7 0 , , 2 4 7 0 . 170 0 . 0 8 8 0 . 0 0 0 0 . 8 3 8 0 . 6 7 2 0 . 6 5 6 0 . 6 3 4 0 , , 6 0 6 0 . 5 7 2 0 . 5 3 2 0 . 4 8 6 0 . 4 3 5 0 . 377 0 , , 3 1 3 0 , , 2 4 4 0 . 169 0 . 0 8 7 0 . 0 0 0 0 . 8 1 2 0 . 6 6 3 0 . 6 4 8 0 . 6 2 6 0 , . 598 0 . 5 6 5 0 . 5 2 5 0 . 4 8 0 0 . 4 2 9 0 . 3 7 2 0 , , 3 1 0 0 , , 241 0 . 166 0 . 0 8 6 0 . 0 0 0 0 . 7 8 5 0 . 6 5 4 0 . 6 3 8 0 . 6 1 7 0 , , 5 9 0 0 . 5 5 7 0 . 5 1 8 0 . 4 7 4 0 . 4 2 3 0 . 367 0 . 3 0 5 0 , . 238 0 . 164 0 . 0 8 5 0 . 0 0 0 0 . 7 5 9 0 . 6 4 3 0 . 6 2 8 0 . 6 0 7 0 , . 581 0 . 5 4 8 0 . 5 1 0 0 . 4 6 6 0 . 4 1 7 0 . 361 0 . , 3 0 0 0 , , 2 3 4 0 . 162 0 . 0 8 4 0 . 0 0 0 0 . 7 3 3 0 . 6 3 2 0 . 6 1 7 0 . 5 9 7 0 , , 5 7 0 0 , . 5 3 9 0 . 501 0 . 4 5 8 0 . 409 0 . 355 0 , , 2 9 5 0 , , 2 3 0 0 . 159 0 . 0 8 2 0 . 0 0 0 0 . 707 0 . 6 1 9 0 . 6 0 5 0 . 5 8 5 0 , , 5 5 9 0 , . 5 2 8 0 . 4 9 2 0 . 4 4 9 0 . 4 0 2 0 . 348 0 , , 2 9 0 0 , , 2 2 5 0 . 156 0 . 0 8 1 0 . 0 0 0 0 . 6 8 1 0 . 6 0 6 0 . 5 9 2 0 . 5 7 3 0 , , 5 4 8 0 , . 5 1 7 0 . 481 0 . 4 4 0 0 . 393 0 . 341 0 . 284 0 , , 221 0 . 153 0 . 0 7 9 0 . 0 0 0 0 . 6 5 5 0 . 5 9 2 0 . 5 7 8 0 . 5 5 9 0 , , 5 3 5 0 , . 5 0 5 0 . 4 7 0 0 . 4 3 0 0 . 384 0 . 333 0 . , 2 7 7 0 , , 2 1 6 0 . 149 0 . 0 7 7 0 . 0 0 0 0 . 6 2 8 0 . 5 7 6 0 . 5 6 3 0 . 5 4 5 0 , , 521 0 , , 4 9 2 0 . 4 5 8 0 . 4 1 9 0 . 374 0 . 325 0 . 2 7 0 0 , , 2 1 0 0 . 145 0 . 0 7 5 0 . . 0 0 0 0 . 6 0 2 0 . 5 6 0 0 . 547 0 . 5 3 0 0 , , 507 0 , , 479 0 . 4 4 5 0 . 4 0 7 0 . 364 0 . 316 0 . 2 6 3 0 , , 2 0 4 0 . 141 0 . 0 7 3 0 . 0 0 0 0 . 576 0 . 542 0 . 5 3 0 0 . 5 1 3 0 , , 491 0 , , 4 6 4 0 . 4 3 2 0 . 395 0 . 353 0 . 306 0 . 2 5 5 0 , , 198 0 . 137 0 . 07 1 0 , 0 0 0 0 . 5 5 0 0 . 5 2 3 0 . 5 1 2 0 , . 4 9 6 0 , , 4 7 4 0 , , 4 4 8 0 . 4 1 7 0 . 381 0 . 341 0 . 296 0 . 246 0 , . 192 0 . 132 0 , 0 6 9 0 . 0 0 0 0 . 524 0 . 5 0 3 0 , 4 9 2 0 , , 4 7 7 0 , , 4 5 6 0 , , 4 3 1 0 . 4 0 2 0 . 367 0 . 328 0 . 2 8 5 0 . 237 0 , , 184 0 . 127 0 . 0 6 6 0 , , 0 0 0 0 , 407 0 . 4 8 1 0 ,47 1 0 , 4 5 6 0 , 437 0 , , 4 1 3 0 . 385 0 . 352 0 . 3 1 5 0 . 2 7 3 0 , 227 0 , 177 0 . 122 0 , 0 6 3 0 . 0 0 0 0 , 4 7 1 0 , 4 5 8 0 , 449 0 , , 4 3 5 0 , 417 0 , , 3 9 4 0 . 367 0 , , 336 0 . 300 0 . 260 0 . 2 1 7 0 , , 169 0 . 117 0 , 0 6 0 0 , 0 0 0 0 . , 4 4 5 0 , , 4 3 3 0 , 424 0 , 4 1 2 0 , , 394 0 , , 373 0 . 348 0 , 3 1 8 0. 284 0 . 247 0 . 2 0 5 0 , , 160 0 . 1 10 0 , 0 5 7 0 , 0 0 0 0 4 10 0 , 4 0 0 0 , , 300 0 , , 3 0 7 0, 371 0 , 361 0 . 327 0 . 2 0 0 0. 207 0 . 2 3 2 0 . 193 0 , 160 0 , 104 0 , 0 5 4 0 . 0 0 0 0 . 393 0 , 378 0 , , 371 0 , , 3 6 0 0 . 346 0 , , 3 2 7 0 . 305 0 , , 2 7 9 0. 249 0 . 2 1 6 0 . 179 0 , , 1 3 9 0 . 0 9 6 0 . 0 5 0 0 . 0 0 0 0 , 367 0 . 349 0 , 342 0 , , 332 0 . 3 1 9 0 , 301 0 . 281 0 , , 256 0 . 228 0 . 198 0 . 164 0 , , 127 0 . 0 8 7 0 . 0 4 5 0 , 0 0 0 0 . 340 0 . 3 1 8 0 , , 3 1 2 0 , 303 0 . 291 0 , 274 0 . 2 5 5 0 , , 2 3 2 0 . 206 0 . 177 0 . 145 0 , , 1 1 1 0 . 0 7 4 0 . 0 3 6 0 . 0 0 0 0 . 3 14 0 . 286 0 . 281 0 . 2 7 3 0 . 261 0 . 246 0 . 227 0 , , 2 0 5 0 . 180 0 . 152 0 . 122 0 , , 0 9 0 0 . 0 5 6 0 . 0 0 0 0 . 288 0 . 254 0 , 250 0 , 242 0 . 231 0 . 2 1 6 0 . 198 0 , , 176 0 . 152 0 . 125 0 . 0 9 6 0 , , 0 6 5 0 . 0 0 0 0 . 2 6 2 0 . 2 2 2 0 . 218 0 , 2 1 1 0 . 2 0 0 0 . 185 0 . 167 0 , , 146 0 . 122 0 . 0 9 5 0 . 0 6 7 0 , , 0 0 0 0 . 236 0 . 190 0 , 186 0 , 179 0 . 169 0 . 155 0 . 137 0 , , 1 1 6 0 . 091 0 . 0 6 6 0 . 0 0 0 0 . 209 0 . 159 0 . 155 0 , 149 0 . 138 0 . 124 0 . 107 0 , , 0 8 5 0 . 0 6 2 0 . 0 0 0 0 . 183 0 . 129 0 . 126 0 . 120 0 . 109 0 . , 0 9 5 0 . 077 0 , , 0 5 7 0. 0 0 0 0 . 157 0 . 101 0 . 098 0 . 0 9 2 0 . 0 8 2 0 . 0 6 8 0 . 051 0 , , 0 0 0 0 . 131 0 . 0 7 5 0 . 0 7 3 0 . 0 6 7 0 . 0 5 7 0 . 0 4 4 0 . 0 0 0 0 . 105 0 . 0 5 2 0 . 0 5 0 0 . 0 4 5 0 . 0 3 6 0 . 0 0 0 0 . 0 7 9 0 . 0 3 3 0 . 032 0 . 0 2 7 0 . 0 0 0 0 . 0 5 2 0 . 0 1 9 0 . 0 1 8 0 . 0 0 0 0 . 0 2 6 0 . 001 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 D I M E N S I O N L E S S V E L O C I T I E S ; U P P E R - H O R I Z O N T A L CMPT. *c L O W E R - V E R T I C A L CMPT. H E I G H T D I M E N S I O N L E S S R A D I U S 5_ *' J—I 0) D_ ft' 5' a CO O ti 0 . 140 0 . 2 0 6 0 . 2 7 2 0 . 3 3 8 0 . 4 0 5 0 . 4 7 1 0 . 5 3 7 0 . 6 0 3 0 . 6 6 9 0 . 7 3 5 0 . 8 0 2 0 . 8 6 8 0 . 9 3 4 1 . 0 0 0 1 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 1 . 8 6 2 1 . 4 9 3 1 . 4 7 3 1 . 4 5 9 1 . 4 4 9 1 . 441 1 . 4 3 5 1 . 4 3 0 1 . 4 2 6 1 . 4 2 3 1 . 4 2 1 1 . 4 1 9 1 . 4 1 9 1 . 3 7 1 0 . 9 9 5 0 . 0 5 9 0 . 0 3 6 0 . 0 2 5 0 . 0 1 9 0 . 0 1 4 0 . 0 1 1 0 . 0 0 9 0 . 0 0 7 0 . 0 0 5 0 . 0 0 4 0 . 0 0 3 0 . 0 0 2 0 . 0 0 1 0 . 0 0 0 1 . 8 6 2 1 . 4 9 3 1 . 4 7 3 1 . 4 5 9 1 . 4 4 9 1 . 4 4 1 1 . 4 3 5 1 . 4 3 0 1 . 4 2 6 1 . 4 2 3 1 . 4 2 1 1 . 4 19 1 .4 19 1 . 371 0 . 9 6 9 0 . 163 0 . 103 0 . 0 7 2 0 . 0 5 4 0 . 041 0 . 0 3 2 0 . 0 2 5 0 . 0 2 0 0 . 0 1 5 0 . 0 1 1 0 . 0 0 8 0 . 0 0 5 0 . 0 0 2 0 . 0 0 0 1 . 8 4 4 1 . 481 1 . 4 6 5 1 . 4 5 3 1 . 4 4 4 1 . 4 3 7 1 . 4 3 1 1 . 4 2 6 1 . 4 2 3 1 . 4 2 0 1 . 4 1 8 1 . 4 1 7 1 . 4 1 6 1 . 369 0 . 943 0 . 247 0 . 159 0 . 1 1 3 0 . 0 8 5 0 . 0 6 6 0 . 0 5 2 0 . 0 4 1 0 . 0 3 2 0 , 0 2 5 0 . 0 1 9 0 . 0 1 3 0 . 0 0 0 0 . 0 0 4 0 . 0 0 0 1 . 8 2 3 1 . 4 6 6 1 . 4 5 4 1 . 4 4 4 1 . 4 3 6 1 . 4 2 9 1 . 4 2 4 1 . 4 2 0 1 . 4 1 7 1 . 4 1 4 1 . 4 1 2 1 . 4 1 1 1 . 4 1 0 1 . 363 0 . 9 1 6 0 . 321 0 . 209 0 . 150 0 . 1 1 3 0 , , 0 8 8 0 , 0 7 0 0 . 0 5 5 0 . 044 0 . 0 3 4 0 . 0 2 5 0 . 0 1 8 0 . 0 1 1 0 , 0 0 6 0 . 0 0 0 1 . 8 0 0 1 . 4 5 0 1 . 440 1 . 4 3 2 1, , 4 2 5 1 , , 4 1 9 1 . 4 1 5 1 . 4 1 1 1 . 408 1 . 4 0 6 1 . 4 0 4 1 . 4 0 3 1 , 4 0 2 1 . 356 0 . 8 9 0 0 . 389 0 . 2 5 4 0 . 183 0 . 139 0 , , 109 0 . 0 8 6 0 , , 0 6 9 0 . 0 5 4 0 . 0 4 2 0 . 0 3 2 0 . 0 2 3 0 . 0 1 4 0 , 0 0 7 0 . 0 0 0 1 . 7 7 4 1 . 4 3 1 1 . 424 1 . 4 1 8 1, , 4 1 2 1 , . 4 0 7 1, , 4 0 3 1 . 4 0 0 1 . 397 1 . 395 1 . 3 9 3 1 . 3 9 2 1 . 392 1 , 346 0 . 0 6 4 0 . 4 5 3 0 . 2 9 6 0 . 2 1 5 0 . 163 0 , , 128 0 , . 102 0 , , 001 0 . 0 6 4 0 . 0 5 0 0 . 0 3 8 0 . 0 2 7 0 . 0 1 7 0 , , 0 0 8 0 , 0 0 0 1 . 747 1 , 4 1 1 1. , 4 0 7 1 , , 4 0 2 1. 397 1 . 393 1, 389 1, , 386 1, , 384 1 . 3 8 2 1 . 381 1 . 3 8 0 1 . 379 1, , 333 0 . 8 3 8 0 . 5 1 3 0 , 336 0 , , 244 0 , , 186 0 . 146 0 , , 1 16 0 , , 0 9 3 0 , , 0 7 3 0 , , 0 5 7 0 . 0 4 3 0 . 0 3 1 0 . 0 2 0 0 , 0 0 9 0 , , 0 0 0 1 . 7 1 8 1 . 3 9 0 1, , 3 8 8 1 , , 384 1. 380 1. , 376 1. 373 1, , 371 1, , 369 1 . 3 6 7 1 . 3 6 6 1 . 365 1 . 364 1, , 3 1 9 0 . 8 1 2 0 . 571 0 , , 3 7 5 0 , , 2 7 2 0 , 208 0 . 164 0 . 130 0 . 104 0 , , 0 8 2 0 , , 0 6 4 0 , . 0 4 9 0 . 0 3 5 0 , . 0 2 2 0 , 0 1 1 0 . 0 0 0 1 . 6 8 6 1 , , 366 1, 367 1. . 364 1. 361 1. 358 1. 356 1, . 3 5 3 1, , 351 1, , 350 1 . 3 4 9 1, . 348 1. 348 1. 303 0 . 785 0 . 627 0 . 4 1 3 0 . 300 0 . 229 0 . 180 0 . 144 0 . 115 0 . 091 0 . 071 0 , , 0 5 4 0 , , 0 3 8 0 , , 0 2 4 0 . 0 1 2 0 . 0 0 0 1, , 6 5 3 1. 341 1. 344 1. 343 1. 341 1. 338 1. 336 1. 334 1. 332 1. , 3 3 1 1, , 3 3 0 1, , 329 1. 329 1. 2 8 5 0 . 759 0 . , 6 8 4 0 . 4 5 0 0 . 327 0 . 251 0 . 197 0 . 157 0 . 126 0 . 100 0 . 0 7 8 0 , , 0 5 9 0 , , 0 4 2 0 , , 0 2 7 0 . 0 1 3 0 . 0 0 0 1. 617 1. 314 1. 320 1. 3 2 0 1. 319 1. 317 1. 3 1 5 1. 3 1 3 1. 311 1, , 3 1 0 1, , 309 1, , 309 1. 308 1. 265 0 . 733 0 . 742 0 . 488 0 . 355 0 . 2 7 2 0 . 2 1 4 0 . 171 0 . 137 0 . 108 0 . 0 8 5 0 , , 0 6 4 0 , , 0 4 6 0 , , 0 2 9 0 . 0 1 4 0 . 0 0 0 1. 578 1. 285 1. 294 1. 2 9 5 1. 2 9 5 1. 293 1. 291 1. 2 9 0 1. 289 1. 287 1. 287 1. 286 1. 286 1. 243 0 . 707 0 . 801 0 . 527 0 . 384 0 . 294 0 . 231 0 . 185 0 . 148 0 . 117 0 . 0 9 2 0 , 0 6 9 0 , , 0 4 9 0 . , 0 3 2 0 . 0 1 5 0 . 0 0 0 1. 537 1. 254 1. 266 1. 268 1. 269 1. 268 1. 266 1. 2 6 5 1. 264 1. 2 6 3 1, , 2 6 2 1, , 2 6 2 1. 261 1. 2 1 9 0 . 681 0 : 8 6 2 0 . 567 0 . 4 1 3 0 . 316 0 . 249 0 . 199 0 . 159 0 . 126 0 . 0 9 9 0 . 0 7 5 0 , , 0 5 3 0 . 0 3 4 0 . 0 1 6 0 . 0 0 0 1. 4 9 2 1. 221 1. 2 3 5 1. 240 1. 241 1. 240 1. 239 1. 2 3 8 1. 237 1. 236 1, , 2 3 6 1. 2 3 5 1. 2 3 5 1. 194 0 . 6 5 5 0 . 9 2 6 0 . 609 0 . 443 0 . 340 0 . 267 0 . 2 1 3 0 . 171 0 . 136 0 . 106 0 . 0 8 0 0 . 0 5 7 0 . 0 3 6 0 . 0 1 8 0 . 0 0 0 1. 444 1. 185 1. 2 0 3 1. 209 1. 211 1. 211 1. 2 1 0 1. 2 1 0 1. 209 1. 208 1. 207 1. 207 1. 2 0 7 1. 167 0 . 6 2 8 0 . 9 9 3 0 . 654 0 . 4 7 5 0 . 364 0 . 287 • 0 . 229 0 . 183 0 . 145 0 . 113 0 . 0 8 6 0 . 061 0 . 0 3 9 0 . 0 1 9 0 . 0 0 0 1. 392 1. 146 1. 167 1, 176 1, 178 1. 179 1. 179 1. 179 1, 178 1 . 177 1. 177 1. 176 1. 176 1. 137 0 . 6 0 2 1. 0 6 5 0 . 701 0 . 5 1 0 0 . 390 0 . 307 0 . 245 0 . 196 0 . 156 0 . 121 0 . 0 9 2 0 . 0 6 6 0 . 0 4 2 0 . 0 2 0 0 . 0 0 0 1. 336 1. 104 1. 129 1. 140 1. 144 1. 145 1. 146 1. 146 1. 145 1. 145 1. 144 1. 144 1. 144 1. 106 0) o o a. CO CO I 0 . 576 1 . 142 0 . 751 . 0 .546 0 .418 0 . 329 0 .262 0 .210 0. 166 0 . 130 0 .098 0 .070 0 .045 0 .022 0 .000 1 .275 1 .058 1 .086 1 . 101 1 . 107 1 . 109 1 . 1 10 1. 110 1 . 1 10 1 .110 1 . 109 1 . 109 1 . 109 1 .072 0 . 550 1 . 224 0 .805 0 .585 0 .447 0 . 352 0 . 280 0 . 224 0. 178 0 . 139 0 . 105 0 .075 0 .048 0 .023 0 .000 1 .210 1 .009 1 .044 1 .059 1 .067 1 .070 1 .072 1 .072 1 .072 1 .072 1 .072 1 .072 1 .072 1 .036 3_ 0 .524 1 .312 0 .863 0 .627 0 .479 0 .377 0 . 300 0 .240 0.191 0 . 149 0 . 112 0 .080 0 .051 0 .025 0 .000 1 . 140 0 .956 0 .996 1 .014 1 .023 1 .028 1 .031 1 .032 1 .032 1 .032 1 .032 1 .032 1 .032 0 .998 0 .497 1 .405 0 .925 0 .672 0 .513 0 .404 0 . 322 0 .257 0.204 0 . 159 0 . 121 0 .086 0 .055 0 .026 0 .000 1 .066 0 .900 0 .944 0 .966 0 .977 0 .983 0 .986 0.988 0 .989 0 .989 0 .989 0 .989 0 .989 0 .957 0 .471 1 . 503 0 .990 0 .719 0 .550 0 .432 0 .345 0 .276 0.219 0 . 171 0 . 130 0 .093 0 .059 0 .028 0 ,000 0 .988 0 .840 0 .889 0 .914 0 .928 0 .935 0 .939 0.942 0 .943 0 .943 0 .944 0 .944 0 .944 0 .912 8 0 .445 1 .602 1 .057 0 .769 0 . 5B9 0 .464 0 .370 0 .297 0. 236 0 , 185 0 . 140 0 . 101 0 .064 0 .031 0 .000 05 0 .908 0 . 777 0 .830 0 .859 0 .875 0 .884 0 .889 0.892 0 .894 0 .894 0 .894 0 .894 0 .894 0 .864 0 .419 1 .700 1 . 124 0 .820 0 .629 0 .497 0 . 399 0 .321 0.257 0, ,202 0 .154 0 .111 0, ,072 0, ,035 0, 000 5 0 .829 0 .714 0 .770 0 .802 0 .821 0. .831 0 .837 0.840 0, ,841 0, ,841 0, ,841 0. ,840 0. 840 0. 812 si 0 . 393 1 .793 1 . 188 0 .870 0 .671 0 .533 0 . 430 0 , 349 0.282 0. 225 0, , 174 0. , 128 0. 084 0. 042 0. 000 0 .751 0. .653 0 .711 0 .746 0 .766 0 .778 0, .784 0.786 0. 787 0, ,785 0, ,783 0. 781 0. 779 0. 752 0 . 367 1 .875 1 .246 0 .916 0. .710 0 . 569 • 0. 464 0, , 381 0.314 0. 256 0, ,204 0. 157 0. 111 0. 065 0. 000 0 .678 0. .595 0 .656 0 .693 0 .715 0 .727 0, ,733 0.734 0. 732 0, ,728 0. 722 0. 714 0. 705 0. 677 2 0 . 340 1 .942 1 ,293 0 .955 0 .746 0 .602 0 ,497 0, 416 0.350 0. 294 0. 245 0. 201 0. 155 0. 209 0. 000 to 0 .610 0, 542 0 ,808 0 ,847 0 ,672 0, ,684 0. 690 0. 690 0. 686 0. 678 0. 668 0. 651 0. 599 0. 542 0 .314 1 .988 1. ,326 0 .983 0 .773 0 ,631 0, , 528 0. 449 0.3B7 0. 336 0. 293 0. 248 0. 372 0. 000 £. 0 .547 0. 495 0, ,565 0, ,610 0, ,638 0. ,652 0. 659 0.658 0. 653 0. 644 0. 622 0. 782 0. 000 0. . 288 2 .010 1. 344 1 .000 0. 791 0, ,650 0. 550 0. 476 0.418 0. 373 0. 328 0. 489 0. 000 a . 0 .490 0. 453 0. 531 0. 583 0, ,615 0. 632 0. 641 0.642 0. 639 0. 617 0. 901 0. 000 • D 0. 262 2 .005 1. 342 1. 002 0. 796 0. 659 0. 563 0. 492 0.439 0. 389 0. 567 0. 000 0. .438 0. 418 0. 505 0. 564 0. 601 0. 623 0. 636 0.642 0. 624 0. 979 0. 000 0. 236 1 . 971 1. 321 0. 988 0. 788 0. 656 0. 564 0. 497 0.437 0. 621 0. 000 0. 391 0. 387 0. 485 0. 552 0. 595 0. 623 0. 641 0.631 1. 033 0. 000 0. 209 1. 906 1. 278 0. 958 0. 767 0. 640 0. 553 0. 478 0.662 0. 000 o. 348 0. 360 0. 470 0. 545 0. 595 0. 629 0. 629 1.069 0. 000 0. 183 1. 809 1. 213 0. 911 0. 730 0. 612 0. 516 0. 693 0.000 0. 308 0. 337 0. 459 0. 542 0. 599 0. 613 1. 088 0.000 0. 157 1, 676 1. 124 0. 844 0. 678 0. 555 0. 716 0. 000 0. 270 0. 316 0. 450 0. 542 0. 578 1 .087 0.000 0. 131 1. 506 1. 009 0. 759 0. 596 0. 7 30 0. 000 0. 235 0. 297 0. 444 0. 515 1. 060 .0- 000 I—> 0. 105 1. 294 0. 866 0. 640 0. 732 0. 000 0. 201 0. 280 0. 409 0. 997 0. 000 0. 079 1. 033 0. 683 0. 715 0. 000 o, mn 0. 23(1 0, 81)3 0, 000 0.052 1.020 0.673 0.000 0.107 0.702 0.000 0.026 0.596 0.000 0.070 0.000 0) 0.000 0.000 0.000 PRESSURE PROFILE HEIGHT DIMENSIONLESS RADIUS 0. , 140 0. 206 0, ,272 0, ,338 0. ,405 0. ,471 0. 537 0. 603 0, 669 0. 735 0. 802 0, ,868 0, ,934 1 . 000 1. .000 0. .000 0. .000 0, ,000 0, .000 0, ,000 0, .000 0, ,000 0. 000 0, ,000 0. 000 0. 000 0. ,000 0, ,000 0. ,000 0. 995 0. .032 0. 032 0. ,032 0. ,032 0. ,032 0. 032 0. 032 0. 032 0, ,032 0. 032 0. 032 0. 032 0. 032 0. 032 0. .969 0. .065 0. ,065 0, ,065 0, ,065 0. ,065 0. ,064 0. 064 0. 064 0, ,064 0. 064 0. 064 0. 064 0. ,064 0. 064 0, .943 0 .098 0, ,097 0. ,097 0, ,097 0. ,097 0. ,097 0. ,097 0. 096 0, ,096 0. 096 0. 096 0, .096 0, .096 0, ,096 0 .916 0 . 131 0, , 130 0, , 129 0, , 129 0. , 129 0, , 129 0, , 128 0. 128 0, , 128 0. , 128 0. 128 0, , 128 0, , 128 0. , 128 0 .890 0 . 162 0 . 161 0 . 161 0, .161 0, , 160 0, . 160 0, , 160 0. 160 0 . 160 0. , 160 0. 160 0, , 160 0, , 160 0, . 160 0 . 864 0. . 194 0. , 193 0, , 192 0, , 192 0. 192 0. 192 0. 192 0. 191 0, , 191 0. 191 0. 191 0. , 191 0. 191 0, 191 0 .838 0 .226 0. .224 0, ,224 0, .223 0, ,223 0. ,223 0, ,223 0. 223 0, . 222 0. 222 0. 222 0, .222 0, ,222 0, .222 0 .812 0 .256 0 .255 0, ,254 0, ,254 0, ,254 0, ,254 0. 253 0. 253 0 ,253 0, ,253 0. 253 0. ,253 0, ,253 0, ,253 0 . 785 0 .287 0 .285 0. , 285 0 .284 0, , 284 0, , 284 0, , 284 0. 284 0 ,284 0. . 284 0. 284 0, . 284 0 , 284 0, , 284 0 . 7 b u 0 . 3 1 / 0 , 3 1 6 0 , 3 IB 0 ,314 n , 3 14 0 , 3 14 0,311 0. 314 n ,314 n 314 0 . 314 n ,314 0 ,314 0 ,314 0 . 733 0 . 347 0 . 344 0 .344 0 .344 0 , 344 0 , 344 0, , 343 0. 343 0 . 343 0 , 343 0. 343 0 . 343 0 . 343 0 , 343 0. . 707 0 . 369 0. ,371 0, ,372 0, ,373 0, ,373 0, ,373 0, 373 0. 373 0, , 372 0. 372 0. 372 0, ,372 0, , 372 0, 372 0. .681 0 . 403 0. .402 0, ,402 0, ,402 0, ,402 0, ,402 0, ,402 0. 401 0 .401 0. 401 0. 401 0, .401 0 .401 0, , 401 0, 655 0 .436 0, ,434 0, ,433 0, ,432 0, ,431 0, ,431 0, ,430 0. 430 0 .429 0, ,429 0. 429 0. .429 0 . 429 0 .429 0. 628 0 .469 0, ,465 0, ,463 0, .461 0, , 460 0, ,459 0, ,458 0. 458 0 .457 0, ,457 0. 456 0 .456 0 .456 0 .456 0. .602 0 . 502 0, ,496 0, ,493 0 ,490 0, .488 0, .487 0, ,486 0. 485 0 .484 0, ,484 0. 483 0 .483 0 .483 0 .483 0. 576 0, 535 0. 527 0, ,522 0, ,519 0, ,516 0, ,514 0. 512 0. 511 0, .510 0, , 509 0. 509 0 .509 0, . 508 0 . 508 0. 550 0, ,567 0. 557 0, ,551 0, ,546 0, ,543 0, ,540 0, ,538 0. 537 0. . 535 0, ,534 0. 534 0 . 533 0 .533 0 .533 0. 524 0. .599 0, ,586 0, ,578 0, ,573 0, ,569 0, ,566 0, ,563 0. 561 0 . 559 0, ,558 0. 557 0. .557 0. .556 0, .556 0, .497 0, ,631 0. ,615 0, ,605 0, ,599 0, .594 0, ,590 0, ,587 0. 584 0 .582 0, ,581 0. 580 0 .579 0 .579 0 .579 0. .471 0. .661 0, ,642 0 .631 0 .623 0, ,617 0, .612 0, ,609 0. 606 0 .603 0, ,602 0. 600 0 .599 0 . 599 0 . 599 0 . 445 0 .692 0. .669 0 .656 0 .646 0 .639 0 .634 0 .629 0. 626 0 .623 0 .621 0. 619 0 .618 0 .618 0 .618 0. 419 0. ,721 0, .695 0, .679 0, ,668 0, ,660 0, .653 0. ,648 0. 644 0 .641 0, ,638 0. 636 0 .635 0 .635 0 .635 0. 393 0, . 750 0, ,720 0, , 702 0, ,689 0, .679 0, .672 0, ,666 0. 661 0 .657 0 ,654 0. 651 0 .650 0 .649 0 .649 0. 367 0, ,777 0, .743 0, .723 0, .708 0, ,697 0, ,688 0, ,681 0. 675 0 .671 0 ,667 0. 664 0 .662 0 .661 0 .661 0. 340 0, .804 0, ,766 0 ,743 0, .726 0 .714 0 .704 0, ,695 0. 688 0 .682 0 .678 0. 674 0 .671 0 .670 0 .670 0. 314 0, , 629 0, . 788 0 . 762 0 .744 0 .730 0 ,718 0 .708 0. 700 0 .693 0 .687 0. 681 0 .677 0 .672 0. 288 0. 854 0, ,809 0, , 781 0, , 761 0, , 745 0, , 732 0, , 721 0. 711 0 . 702 0 .695 0. 688 0 .680 0. 262 0, ,876 0, ,829 0, ,799 0, ,777 0, .760 0, ,745 0, .733 0. 722 0 .712 0 .703 0. 693 0. 236 0, ,898 0, ,848 0, ,817 0 ,793 0, ,775 0, ,759 0, .746 0. 733 0 .722 0 .710 0 . 209 0, 918 0, ,867 0, ,834 0,810 0 , 790 0 , 773 0, , 759 0. 746 0 .731 0. 183 0 ,936 0, ,885 0 ,851 0.826 0 ,806 0 ,709 0 .773 0. 756 0. 157 0, 952 0, ,902 0, .869 0, . 844 0 .823 0 .805 0 ,785 0. 131 0. ,966 0, ,918 0, ,886 0, ,862 0, ,841 • o . ,818 0. 105 0. ,978 0, ,934 0 ,904 0 .880 0 ,856 0. 079 0 , 987 0 .949 0 ,922 0 ,895 0. .052 0 .994 0 ,963 0 ,936 0. .026 0 .999 0 .974 0. 000 1 , 000 2-5' 3 > Cn (2) : AUXILIARY FLOWRATE = 0.12 M**3/S STREAMFUNCTION MATRIX HEIGHT 0, 140 0 , 20Q 1. 000 0. 703 0, .686 0. 995 0. 703 0, ,686 0. 969 0. 701 0, ,684 0. 943 0. .698 0, ,681 0. 9 16 0 .694 0 ,677 0. 690 0. 688 0, 672 0. 864 0. 682 0.666 0. 838 0, ,674 0, ,659 0. 012 0 ,666 0 ,651 0. 785 0, ,657 0 ,642 0. 759 0 ,647 0 ,632 0. 733 0, ,637 0, .822 0. 707 0 .626 0, ,611 0, ,681 0. .614 0 .600 0, ,655 0. .601 0 ,588 0 .628 0 . 588 0 .575 0 , G02 0 . 574 0. .50 1 0 . . 576 0. . 559 0 .547 0. 550 0. , 544 0. ,532 0. 524 0. . 528 0, ,517 0. 497 0, ,511 0, 501 0. 47 1 0, 494 0, ,484 0. 445 0, 477 0, 467 0. 419 0. 458 0, ,449 0. 393 0. 440 0, .431 0. 367 0. 421 0. 413 0. 340 0. 402 0, ,394 0. 314 0. 383 0, ,376 0. 288 0. 364 0, ,357 0. 262 0. 346 0, 340 0. 236 0. 328 0, 323 0. 209 0. 312 0. 307 0. 183 0. 298 0, 293 0 . 1G7 0 . 205 0, 20 1 0. 131 0. 274 0, ,271 0. 105 0. 266 0, ,283 0. 079 0. 260 0, ,258 0 . 052 0. 257 0 . 255 0. 026 0. 260 0, ,259 0 . 000 0. 260 DIMENSIONLESS 0, ,272 0. 330 0. 405 0, 663 0. 633 0. 597 0, ,663 0. 633 0. 597 0, ,661 0. 632 0. 596 0, ,658 0. 629 0. 593 0, ,854 0.625 0. 590 0, 649 0. 620 0. 586 0, 643 0.615 0. 580 0, ,636 0.608 0. 574 0, ,629 0. 601 0. 568 0, ,621 0. 593 0. 560 0.611 0. 585 0. 552 0. 602 0. 575 0. 543 0, 591 0. 565 0. 534 0. ,560 0. 555 0. 524 0, ,568 0. 544 0. 513 0 ,556 0. 532 0. 502 0, , 543 0. 519 0. 491 0 , 529 0. 506 0. 479 0. ,515 0. 493 0. 466 0, ,500 0. 479 0. 452 0, ,405 0. 464 0. 439 0, ,469 0. 449 0. 424 0, ,452 0. 433 0. 410 0, ,435 0. 417 0. 395 0, ,418 0. 401 0. 379 0. 400 0. 384 0. 363 0. ,383 0. 367 0. 347 0, ,365 0. 350 0. 331 0. ,347 0. 333 0. 315 0, , 330 0. 316 0. 299 0, ,314 0. 301 0. 204 0. 299 0. 286 0. 271 0, 285 0. 274 0. 259 0, ,273 0 . 263 0. 240 0, ,264 0. 254 0. 241 0, ,257 0. 248 0. 236 0, .252 0. 246 0 .254 0. 47 1 0. 537 0. 603 0. 555 0. 507 0. 453 0. 555 0. 507 0. 453 0. 554 0. 506 0. 452 0. 552 0. 504 0. 451 0, 549 0. 501 0. 448 0. 545 0. 498 0. 445 0. 540 0. 493 0. 441 0. 534 0. 488 0. 436 0. 520 0. 483 0. 431 0. 521 0. 476 0. 426 0. 514 0. 470 0. 420 0. 506 0. 462 0. 413 0. 497 0. 454 0. 406 0. 488 0. 446 0. 399 0. 478 0. 437 0. 391 0. 468 0. 426 0. 382 0. 457 0. 418 0. 373 0. 445 0 . 407 0. 364 0. 4 34 0. 397 0. 355 0. 421 0. 385 0. 345 0. 409 0. 374 0. 334 0. 395 0. 362 0. 323 0. 382 0. 349 0. 312 0. 368 0. 336 0. 301 0. 353 0. 323 0. 289 0. 338 0. 310 0. 277 0. 323 0. 296 0. 264 0. 308 0. 281 0. 251 0. 293 0. 267 0. 238 0. 278 0. 253 0. 225 0 . 264 0 . 240 0 . 213 0. 251 0. 228 0. 202 0. 240 0. 218 0. 185 0. 231 0. 205 0. 222 0. GC9 0. 735 0. 002 0. 393 0. 327 0. 254 0. 393 0. 327 0. 254 0. 392 0. , 326 0. 254 0. 391 0. ,325 0. 253 0. 389 0, , 323 0. 251 0. 386 0. 321 0. 250 0. 382 0. 318 0. 247 0. 379 0, 315 0. 245 0. 374 0, 311 0. 242 0. 369 0, , 307 0. 239 0. 364 0, ,303 0. 236 0. 358 0, ,298 0. 232 0. 352 0, 293 0. 228 0. 346 0, .288 0. 224 0. 339 0, ,282 0. 219 0. 332 0, ,276 0. 215 0. 324 0 , 269 0. 210 0. 316 0 , 203 0. 205 0. 308 0, .256 0. 199 0. 299 0, . 249 0. 194 0. 290 0, .241 0. 188 0. 281 0, .234 0. 182 0. 271 0, . 226 0. 176 0. 261 0 .217 0. 169 0. 251 0 .209 0. 162 0. 240 0, .200 0. 155 0. 229 0, . 190 0. 147 0. 217 0, . 180 0. 138 0. 205 0, . 169 0. 129 0. 193 0 . 158 0. 106 0. 102 0 . 136 0. 162 0. 868 0. 934 1 , ,000 0. 176 0, ,091 0. ,000 0. 176 0. ,091 0 ,000 0. 175 0, ,091 0 .000 0. 175 0 ,090 0 .000 0. 174 0 .090 0 .000 0. 172 0. ,089 0 ,000 0. 171 0, ,089 0, ,000 0. 169 0. ,088 0 .000 0, 167 0 ,087 0 .000 0. 165 0 ,086 0 .000 0. 163 0.084 0 .000 0. 160 0 .083 0 .000 0. 158 0, ,082 0, ,000 0. 155 0, .080 0 .000 0. 152 0 .078 0 .000 0. 148 0 .077 0 .000 0. 145 0 .075 0 .000 0 . 14 1 0 .073 0 .000 0. 138 0 .071 0 .000 0. 134 0 .069 0 .000 0. 130 0 .067 0 .000 0. 126 0 .065 0 .000 0. 121 0 .063 0 .000 0. 117 0 .061 0 .000 0. 112 0 ,058 0 .000 0. 107 0, .055 0 .000 0. 101 0 .052 0 .000 0. 094 0 .039 0. 074 DIMENSIONLESS VELOCITIES; UPPER-HORIZONTAL CMPT. LOWER-VERTICAL CMPT. X; 2 HEIGHT DIMENSIONLESS RADIUS S_ ' X *0 0> a. o' o S l-J & o a. : 0> 0 . 140 0 .206 0 .272 0 .338 0 .405 0 .471 0 .537 0 .603 0 .669 0 .735 0 .802 0 .868 0 .934 1 .000 1 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 0 .000 1 .863 1 .493 1 .474 1 .461 1 .451 1 .443 1 .437 1 .432 1 .429 1 .426 1 . 424 1 .422 1 .422 1 . 374 0 .995 0 .057 0 .035 0 .024 0 .018 0 .014 0 .011 0 .008 0 .006 0 .005 0 .004 0 .003 0 .002 0 .001 0 .000 1 .063 1 . 493 1 .474 1 . 46 1 1 .451 1 .443 1 .437 1 .432 1 .429 1 .426 1 .424 1 .422 1 .422 1 . 374 0 .969 0 . 158 0 . 100 0 .070 0 .052 0 .040 0 .031 0 .024 0 .019 0 .015 0 .011 0 .008 0 .005 0 .002 0 .000 1 . 844 1 .482 1 .467 1 .455 1 .446 1 .439 1 .434 1 .429 1 .426 1 .423 1 .421 1 .420 1 .4 19 1 . 372 0 .943 0 .238 0 . 153 0 . 109 0 .082 0 .063 0 .050 0 .039 0 .031 0 .024 0 .018 0 .013 0 .008 0 .004 0 .000 1 .824 1 .467 1 .455 1 .446 1 .438 1 .432 1 .427 1 .423 1 .420 1 .417 1 .416 1 .414 1 .414 1 . 367 0 .916 0 .309 0 .201 0 . 144 0 . 109 0 .085 0, .067 0 .053 0 .042 0 .032 0 .024 0 .017 0 .011 0 .005 0 .000 1 .801 1 .451 1 .442 1, .434 1 .428 1 .422 1 .418 1 .414 1 .412 1 .409 1 . 408 1 .407 1 .406 1 .359 0 .890 0 .373 0 .243 0, ,175 0, .133 0 . 104 0, ,082 0, .065 0, .052 0 .040 0 .030 0 .021 0 .014 0 .007 0 .000 1 .776 1 .433 1, .427 1, .420 1 ,415 1, .410 1, .407 1, .404 1 .401 1 .399 1 . 398 1 .397 1 . 396 1 . 350 0 .864 0 .431 0, , 282 0, 204 0, , 155 0, , 122 0. 096 0, ,077 0, ,061 0 .047 0 .036 0 .025 0 ,016 0 ,008 0 ,000 1 .750 1 , ,414 1, ,410 1, ,405 1, ,400 1, ,397 1, ,393 1, ,391 1 . 389 1 . 387 1 . 386 1 . 385 1 . 384 1, , 338 0 .838 0 . 485 0, ,318 0. 231 0. , 176 0, , 138 0. 1 10 0. ,087 0. ,069 0 .054 0 .041 0, .029 0. ,018 0. ,009 0, ,000 1 .721 1 , , 393 1, ,391 1. ,388 1, , 384 1. 381 1, ,378 1. , 376 1 .374 1 . 373 1, .371 1, , 371 1. . 370 1, , 325 0, .812 0 .536 0, ,352 0. 256 0. . 195 0, , 153 0. 122 0, ,097 0, ,077 0 .060 0 .045 0, ,032 0 .021 0. .010 0, ,000 1 .691 1 . 370 1. 371 1. 369 1. 366 1. 364 1. 362 1. 360 1 ,358 1, . 357 1, , 356 1, ,355 1. . 354 1. 310 0. 785 0. . 585 0. 384 0. 279 0. 213 0. 168 0. 134 0. 107 0, ,085 " 0, .066 0 .050 0, ,035 0, .023 0. .011 0. 000 1. 660 1. 347 1. 350 1. 349 1. 347 1. 345 1. 343 1. 341 1, , 340 1, , 339 1. , 338 1. .337 1. 337 1. 293 0. 759 0, ,632 0. 415 0. 302 0. 231 0. 182 0. 145 0. 116 0. 092 0, .072 0 .054 0. ,039 0, .025 0. 012 0, 000 1. 626 1. 321 1. 327 1. 327 1. 326 1. 325 1. 323 1. 322 1, , 321 1 , 320 1. ,319 1, .318 1. 318 1. 274 0. 733 0. 677 0. 446 0. 324 0. 248 0. 195 0. 155 0. 124 0. 099 0, .077 0, ,058 0. 041 0, .026 0. 013 0. 000 1. 590 1. 294 1. 302 1. 304 1. 304 1. 303 1. 302 1. 301 1, , 300 1, .299 1. ,298 1, ,298 1. 298 1. 255 0. 707 0. 722 0. 475 0. 345 0. 264 0. 208 0. 166 0. 133 0. 105 0, ,082 0, ,062 0. 044 0, 028 0. 014 0. 000 1, 552 1. 266 1. 276 1. 279 1. 280 1. 280 1. 279 1. 278 1, ,277 1, .277 1, ,276 1, ,276 1. 276 1. 233 0 . 681 0. 765 0. 504 0. 366 0. 281 0. 221 0. 176 0. 14 1 0. 112 0. 087 0, 066 0. 047 0. 030 0. 014 0. 000 1. 512 1. 236 1. 249 1. 253 1. 255 1. 255 1. 255 1. 254 1. 254 1. 253 1. 253 1. 252 1. 252 1. 211 0. 655 0. 808 0. 532 0. 387 0. 296 0. 233 0. 186 0. 149 0. 1 18 0. 092 0. 070 0. 050 0. 032 0. 015 0. 000 1. 470 1. 204 1. 220 1. 226 1. 228 1. 229 1. 229 1. 229 1. 228 1. 228 1. 228 1. 227 1. 227 1. 187 0. 628 0. 851 0. 560 0. 407 0. 312 0. 245 •0. 196 0. 156 0. 124 0. 097 0. 073 0. 052 0. 033 0. 016 0. 000 1. 426 1. 170 1. 189 1. 197 1. 200 1. 201 1. 202 1. 202 1. 202 . 1. 202 1. 201 1. 201 1. 201 1. 161 0. 602 0. 892 0. 587 0. 427 0. 327 0. 257 0. 205 0. 164 0. 130 0. 102 0. 077 0. 055 0. 035 0. 017 0. 000 1. 379 1. 135 1. 157 1. 166 1. 170 1. 172 1. 173 1. 174 1. 174 1. 174 1. 174 1. 174 1. 174 1. 135 - • 7 0 . 576 0 .934 0 .614 0 .447 0 . 342 0 .269 0 .214 0 .171 0, , 136 0 . 106 0 .080 0 .057 0 .037 0 .018 0 .000 1 . 330 1 .098 1 . 123 1 . 134 1 . 139 1 . 142 1 . 144 1. 144 1 . 145 1 . 145 1 . 145 1 .145 1 . 145 1 . 107 0 .550 0 .974 0 .641 0 . 466 0 . 357 0 .281 0 .224 0 . 179 0, 142 0 .111 0 .084 0 .060 0 .038 0 .018 0 .000 1 . 278 1 .059 1 .088 1 . 100 1 . 107 1 .111 1 .113 1. 114 1 .114 1 .115 1 .115 1 . 115 1 . 115 1 .078 0 . 524 1 .013 0 .667 0 .485 0 .371 0 .292 0 .232 0 . 186 0. 147 0 . 115 0 .087 0 .062 0 .040 0 .019 0 .000 1 . 225 1 .019 1 .051 1 .065 1 .073 1 .078 1 .080 1. 082 1 .083 1 .083 1 .083 1 .084 1 .084 1 .048 0 . 497 1 .051 0 .691 0 .503 0 . 385 0 . 302 0 .241 0 . 193 0. 153 0 . 1 19 0 .090 0 .064 0 .041 0 .020 0 .000 1 . 169 0 .976 1 .012 1 .029 1 .038 1 .043 1 .046 1. 048 1 .050 1 .050 1 .051 1 .051 1 .051 1 .016 0 .471 1 .086 0 .715 0 . 520 0 . 398 0 .313 0. ,249 0 . 199 0. 158 .0 . 124 0 .093 0 .067 0 .043 0 .020 0 .000 1 .111 0 .932 0 .972 0 .991 1 .002 1, .008 1 .012 1. 014 11 .015 1.016 1 .017 1 .017 1 .018 0 .984 0, ,445 1 . 119 0 .736 0 .535 0 .410 0 . 322 0, ,257 0 .206 0. 163 0 . 128 0 .097 0 .069 0, .044 0 .021 0 .000 1 .051 0 .887 0 .930 0 .952 0 .964 0, ,971 0 .976 0. 979 0 .980 0 .981 0 ,982 0 .983 0 .983 0 ,950 0. 419 1 . 148 0 .755 0 .549 0 .421 0 . 331 0, ,264 0 .212 0. 169 0, .132 0, , 100 0 .072 0 .046 0, ,022 0. .000 0 .990 0 .841 0 .888 0 .912 0 .926 0, ,934 0 .939 0. 942 0, .944 0, .946 0 ,946 0, .946 0 ,947 0, ,915 0. . 393 1 . 170 0 .771 0 .561 0, .430 0 .339 0. ,271 0, ,218 0. 174 0. , 137 0, , 105 0. ,076 0, ,049 0, ,024 0. ,000 0 .929 0 .794 0 .845 0, .872 0 .888 0. ,897 0, .902 0. 906 0, .908 0, .909 0, .909 0, .909 0, .909 0, ,878 0. . 367 1 . 185 0. , 781 0. , 569 0, ,437 0. , 345 0. 277 0, .224 0. 180 0. , 143 0, ,111 0. ,082 0. ,056 0. ,031 0. ,000 0 .867 0. ,747 0, .803 0, ,833 0. .850 0. 860 0, .866 0. 870 0. ,872 0, .872 0. .872 0. ,870 0. .868 0, ,837 0. , 340 1 . 188 0. . 784 0, ,572 0. 440 0. , 349 0. 281 0. 228 0. 186 0. 150 0. 119 0. ,092 0. 067 0. 079 0. 000 0 . 807 0, , 702 0, , 762 0, 794 0 ,813 0. 825 0, .832 0. 836 0. 838 0. 839 0, ,838 0. 835 0. 819 0. 779 0. 314 1. . 173 0. 778 0, 568 0. 4 38 0. 348 0. 282 0. 231 0. 189 0. 155 0. 127 0. 101 0. 137 -0. 152 o. . 748 0, ,657 0, ,722 0, 758 0, ,779 0. 792 0. 801 0. 806 0. 809 0. 810 0. 808 0. 870 0. 088 0. 288 1. . 152 0, ,761 0, ,556 0. 429 0. ,342 0. 278 0. .228 0. 189 0. 158 0. 130 0, , 175 -0 . 152 0. 692 0. 615 0. 684 0. 723 0. ,747 0. 762 0. 772 0. 779 0. 784 0. 784 0. 893 0. 088 0 . 262 1. 107 0. 731 0, ,535 0. 412 0, . 328 0. 267 0. 220 0. 183 0. 152 0. 196 -0. 152 0. 639 0. 575 0. 648 0. 690 0. 717 0. 734 0. 747 0. 756 0. 761 0. 892 0. 088 0. 236 1. 042 0. 688 0. ,502 0. 387 0, 307 0. 248 0. 203 0. 167 0. 204 -0 . 152 0. 588 0. 536 0. 613 0. 658 0. 687 0. 707 0. 723 0. 732 0. 872 0. 088 0. 209 0. 957 0. 631 0. 459 0. 351 0. 276 0. 221 0. 177 0. 202 -0 . 152 0. .540 0. 499 0. 578 0. 625 0. 657 0. 680 0. 695 0. 835 0. 088 0. 183 0. 852 0. 560 0. 405 0. 306 0. 237 0. 184 0. 191 -0 . 152 0. 494 0. 462 0. 541 0. 590 0. 623 0. 646 0. 780 0. 088 0. 157 0. 728 0. 477 0. 340 0. 251 0. 187 0. 171 -0 . 152 0. 449 0. 424 0. 502 0. 551 0. 583 0. 704 0. 088 0. 131 0. 589 0. 381 0. 265 0. 188 0. 141 -0 . 152 0. 403 0. 383 0. 457 0. 504 0. 604 0. 088 0. 105 0. 437 0. 276 0. 183 0. 099 •0. 152 0. 354 0. 337 0. 405 0. 473 0. 088 0. 079 0. 283 0. 169 0. 043 •0 . 152 0. 297 0. 287 0. 308 0. 088 - a fci ' -a. a a. o" 5' o 3 IS r t ) £3 tt> o a. CO 00 0.052 0.012 -0.026 -0.152 0.235 0.104 0.088 0.026 -0.104 -0.152 0.123 0.088 0.000 0.000 0.000 PRESSURE PROFILE HEIGHT DIMENSIONLESS RAOIUS 0 , 140 0. 206 0. , 272 0 ,338 0, ,405 0. ,471 0. 537 0 .603 0. .669 0, 735 0. 802 0. 868 0, ,934 1 . ,000 1 .000 0 .000 0. 000 0. ,000 0 .000 0, ,000 0 .000 0, 000 0 .000 0, ,000 0, ,000 0. 000 0. 000 0, ,000 0, ,000 0, 995 0 .033 0. 032 0, 032 0 .032 0. 032 0 ,032 0. 032 0 .032 0. .032 0, .032 0. 032 0. 032 0, ,032 0, .032 0 .969 0 .065 0. 065 0. 065 0, ,065 0, 065 0 .065 0, 065 0 .065 0, ,064 0, ,064 0. 064 0. 064 0, ,064 0, ,064 0. 943 0 ,098 0. 097 0. 097 0, ,097 0. 097 0, .097 0. 097 0, .097 0. ,097 0, .097 0. 097 0. 097 0, ,097 0, 097 0. 916 0 ,131 0. 130 0. 129 0, 129 0. 129 0. , 129 0. 129 0, , 129 0. 129 0. 129 0. 129 0. 128 0. 128 0. 128 0. , 890 0 , 163 0. 162 0. 161 0, , 161 0. 181 0, , 161 0. 101 0, , 100 0. 160 0. 100 0, 160 0. 100 0, 160 0. 160 0 , 864 0 . 195 0. 193 0. 193 0, , 193 0. 192 0, , 192 0. 192 0, , 192 0. 192 0, 192 0. 192 0. 192 0. 192 0. 192 0 .838 0 . 226 0. 225 0. 224 0, ,224 0. 224 0, ,223 0. 223 0, ,223 0. 223 0, 223 0. 223 0. 223 0. 223 0. 223 0 .812 0 .257 0. 256 0. 255 0, ,255 0. 255 0, , 254 0. 254 0, ,254 0. 254 0, ,254 0. 254 0. 254 0. 254 0, 254 0 . 785 0 . 280 0. 206 0. 286 0, ,285 0. 285 0, ,285 0. 285 0, ,285 0, ,284 0, ,284 0. 284 0. 284 0. 284 0, ,284 0 . 759 0 . 318 0. 316 0. 316 0, ,315 0, 315 0 ,315 0. 315 0, ,315 0, ,315 0, ,315 0. 315 0. 314 0. ,314 0, ,314 0 .733 0 . 348 0. 345 0. 345 0, , 345 0, 345 0 , 345 0. 345 0 . 344 0. 344 0, , 344 0. 344 0. 344 0. . 344 0, , 344 0 . 707 0 .369 0. 372 0, 373 0, ,374 0, 374 0 .374 0. 374 0 ,374 0, ,374 0, ,374 0. 374 0. ,374 0. . 374 0, ,374 0 .681 0 .403 0. 403 0. ,403 0 .404 0, .403 0 .403 0. ,403 0 .403 0, .403 0, .403 0. 403 0. ,402 0. .402 0, ,402 0. .655 0 .436 0. 434 0. 434 0, ,433 0. 433 0, ,432 0. 432 0, ,432 0, 431 0, ,431 0. 431 0. 431 0. ,431 0. 431 0. . 628 0 .469 0. 466 0. 464 0, ,463 0. 462 0, ,461 0. 460 0, ,460 0, ,459 0, ,459 0. 459 0. 459 0, ,458 0. ,458 0. 602 0, , 502 0. 497 0. 495 0, ,492 0. ,491 0, .489 0. 488 0 .488 0. ,487 0, .486 0. 486 0. ,486 0, ,485 0, ,485 0. 576 0, ,535 0. 528 0. 524 0, ,521 0. ,519 0, ,517 0. 516 0 .514 0, ,513 0, ,513 0. 512 0. ,512 0, .512 0. .512 0. 550 0. ,567 0. 559 0. 553 0, ,550 0. 546 0, ,544 0. 542 0 .540 0. ,539 0, ,538 0. 538 0. ,537 0. .537 0. ,537 0. 524 0, , 599 0. 589 0. 582 0, ,577 0. 573 0, , 570 0. ,568 0 .566 0, ,564 0 .563 0. 562 0. ,561 0 .561 0 .561 0. 497 0. ,631 0. 618 0. 610 0, .604 0. 599 0 ,595 0. 592 0 .590 0. ,588 0 ,586 0. 585 0. ,584 0 .584 0 .584 0. 471 0, ,661 0. 646 0. 636 0, .629 0. 623 0 .619 0. ,615 0 .612 0, ,610 0 .608 0. 607 0, ,606 0 .606 0 .606 0. 445 0, ,692 0. 674 0. 662 0, .654 0, 647 0 ,641 0, 637 0 .633 0, ,631 0 ,628 0. 627 0 .626 0 .625 0 .625 0 . 4 19 0 ,721 0. 700 0. 687 0, .677 0. 669 0 ,662 0, 657 0 .653 0 ,650 0 .647 0. 645 0 ,644 0 .643 0 .643 0. 393 0. 750 0. 726 0. 710 0, .699 0, 689 0 .682 0, 676 0 .67 1 0 ,667 0 ,664 0. 66 1 0 .660 0 .659 0 .659 0 . 367 0. 777 0. 751 0. 733 0. ,719 0. 709 0, , 700 0. 693 0 .687 0, 682 0, .678 0. 675 0, ,673 0 .672 0 .672 0. 340 0. 804 0. 774 0. 754 0, ,739 0. 726 0, ,716 0. 708 0 .701 0, ,695 0. ,690 0. 686 0, .684 0 .682 0 .682 0 . 314 0. 829 0. 796 0. 774 0, , 757 0. 743 0, ,732 0. 722 0 .714 0, , 706 0, , 700 0. 695 0 ,690 0 .684 0 . 288 0. 854 0. 818 0. 793 0, ,774 0. 759 0, ,746 0, 735 0 .725 0. .716 0 , 709 0. 702 0 .693 0. 262 0. 876 0. 838 0. 811 0, ,791 0. 774 0, .759 0. 747 0 .736 0, ,726 0, ,717 0. 706 0. 236 0. 898 0. 857 0. 828 0, ,806 0. 788 0, .773 0. 759 0 .747 0, ,736 0 .723 0 . 209 0. 918 0. 875 0. 845 0, ,822 0. 803 0 . 787 0. ,772 0 . 759 0 ,743 0. 183 0. ,936 0. 892 0. ,862 0, .838' 0. 818 0 .801 0. ,785 0 .767 0. 157 0. ,952 0. 908 0, ,878 0 .853 0. .833 0 .816 0, , 796 0, 131 0. ,966 0. 924 0. ,893 0 .870 0, .849 . 0 .827 0. 105 0. 978 0. 938 0. 909 0, ,887 0. 862 0. 079 0, 987 0. 952 0, ,926 0, ,900 0. 052 0. ,994 0. 965 0. ,939 0. ,026 0 .999 0. 975 0 .000 1 .000 (3) AUXILIARY FLOWRATE « 0.20 M**3/S STREAMFUNCTION MATRIX HEIGHT DIMENSIONLESS RADIUS 0, , 140 0, ,206 0 .272 0. 338 0, ,405 0, 471 0, ,537 0. 603 0. 669 0. 735 0. 802 0. 868 0, ,934 1 , ,000 1 .000 0, 704 0, ,687 0, .663 0. 633 0, ,598 0, 556 0, ,508 0. 454 0. 393 0, 327 0. 254 0. 176 0, ,091 0, 000 0, ,995 0. 704 0, ,687 0, .663 0. 633 0, ,598 0, ,556 0. ,508 0. 454 0. 393 0. 327 0. 254 0. 176 0, ,091 0, ,000 0 , 969 0. 702 0, 685 0, .662 0. 632 0, ,596 0, 555 0, ,507 0. 453 0. 393 0. 326 0. 254 0. 175 0, ,091 0. ,000 0. 943 0. 699 0, 682 0, .659 0. 6 30 0, ,594 0. 552 0, ,505 0. 451 0. 391 0. 325 0. 253 0. 175 0, ,091 0, .000 0. ,916 0. 695 0, ,678 0. .655 0. 626 0, ,591 0. 549 0, ,502 0. 449 0. 389 0. 323 0. 252 0. 174 0, .090 0, ,000 0 , 890 0. 689 0, ,673 0, .650 0. 621 0, ,586 0. 545 0, ,498 0. 445 0. 386 0. 321 0. 250 0. 173 0, ,089 0, ,000 0. ,864 0. ,683 0, ,667 0, ,644 0. 616 0, ,581 0, 541 0. ,494 0. 442 0. 383 0. 318 0. 248 0. 171 0, .089 0, ,000 0, .838 0. ,676 0, ,660 0, ,638 0. 609 0, ,575 0. ,535 0. ,489 0. 437 0. 379 0. 315 0. 245 0. 170 0, ,088 0, ,000 0 .812 0. ,668 0, ,652 0 ,630 0. 602 0, ,569 0, ,529 0, ,484 0. 432 0. 375 0. ,312 0. 243 0. 168 0, .087 0, ,000 0 . 785 0, .659 0, ,644 0, ,622 0. 595 0, ,562 0, ,523 0, ,478 0. 427 0, 370 0, ,308 0. 240 0. 166 0, ,086 0, ,000 0 . 759 0, ,650 0, ,634 0, .613 0. 587 0, ,554 0. ,515 0, ,471 0. 421 0. 365 0, ,304 0. 236 0. 163 0, .085 0, ,000 0 .733 0 .639 0. ,625 0, ,604 0. 578 0, ,545 0, ,508 0, ,464 0. 415 0. 360 0, ,299 0. ,233 0. 161 0, .083 0, ,000 o . 707 0, ,629 0, .614 0, . 594 0. 568 0, .537 0, ,499 0, ,457 0. 408 0. 354 0, ,294 0. 229 0. 158 0 .082 0, .000 0 .681 0, ,617 0.603 0 ,584 0. 558 0, ,527 0, ,491 0, ,449 0. 401 0. 348 0, ,289 0, ,225 0. 156 0 .081 0 .000 0 . 655 0, ,606 0, ,592 0, ,573 0. 548 0, ,517 0, ,482 0, ,440 0. 394 0. 342 0, ,284 0. ,221 0. 153 0 .079 0 .000 o . 628 0, , 593 0, ,580 0, ,561 0. 537 0, ,507 0, ,472 0, ,432 0. 386 0. 335 0, ,279 0. ,217 0. 150 0 .078 0 .000 0 . 602 0, , 581 0, ,568 0 ,549 0. 525 0, .496 0, ,462 0, .423 0. 378 0. 328 0, ,273 0, ,212 0. 147 0 .076 0 .000 o , 576 0, , 567 0, ,555 0, ,537 0. 514 0, ,485 0, ,452 0, ,413 0. 370 0. 321 0, , 267 0, .208 0. 144 0 .074 0 .000 o , 550 0, , 554 0, ,542 0, , 524 0. 502 0, ,474 0, ,442 0, .404 0. 361 0. 313 0, ,261 0, ,203 0. 140 0 .073 0 .000 0 . 524 0 . 540 0. , 528 0, ,511 0. 490 0, ,463 0, ,431 0, , 394 0. 353 0, 306 0, , 255 0, . 198 0. 137 0 .07 1 0 .000 0 . 497 0 , 526 0, ,515 0 ,498 0. 477 0, , 451 0, ,420 0, . 384 0. 344 0. 298 0, , 248 0, , 193 0. 134 0 .069 0 .000 o ,47 1 0. ,512 0. ,501 0, ,485 0. 465 • 0, ,439 0, ,409 0, , 375 0. 335 0, 291 0, , 242 0, , 189 0. 130 0 .067 0 .000 0 , 445 0 . 498 0, , 487 0, ,472 0. 452 0, ,428 0, , 399 0, , 365 0. 326 0. 283 0, , 236 0, , 184 0. 127 0 .066 0 .000 0 . ,419 0. 484 0, 474 0, ,459 0. 440 0, ,416 0, , 388 0, ,355 0. 318 0. 276 0, , 230 0, , 179 0. 124 0 .064 0 .000 0 , 393 0. 470 0. 461 0, ,447 0. 428 0, ,405 0, , 378 0, , 346 0. 310 0. 269 0, ,224 0, , 175 0. 121 0 .063 0 .000 0, 367 0. 458 0. 448 0, ,435 0. 417 0, , 395 0, , 368 0, ,338 0. 302 0. 263 0, ,219 0, ,171 0. 118 0 .061 0 .000 0, 340 0. 445 0. 437 0, ,424 0. 407 0, , 385 0, ,360 0, , 330 0. 296 0. 257 0, ,215 0, , 168 0. 116 0 .061 0 .000 0, ,314 0. 4 3'4 0. ,426 0, ,414 0. 397 0, ,377 0, ,352 0, ,323 0. 290 0. 253 0, ,212 0, , 166 0. 116 0 .064 0, , 288 0. 425 0, ,417 0, ,405 0. 389 0. , 370 0, , 346 0, ,318 0. 287 0. ,251 0, ,211 0, , 166 0. 123 0. . 262 0. 416 0. 409 0, . 398 0. 383 0, , 364 0, 342 0, .315 0. 285. 0. 250 0 .212 0, , 177 0. , 236 0. 410 0, 403 0, ,392 0. 378 0. , 361 0, . 339 0, ,314 0. 286 0. 253 0 .226 0, , 209 0. 405 0. 398 0, .389 0. 376 0, ,359 0, ,339 0, .316 0. 289 0. ,270 0, , 183 0. 402 0, 396 0, ,387 0. 376 0, ,361 0, . 342 0 .321 0. 308 0, , 157 0, 402 0, 396 0, . 388 0. 378 0, , 364 0, , 348 0 .341 0, 131 0. 403 0, 399 0 , 392 0. 383 0, , 372 0 , 370 0. 105 0. 406 0. 403 0, , 398 0. 391 0, , 393 0. 079 0. 411 0. 408 0, .405 0. 411 0. 052 0. 416 0. 414 0, ,424 0. 026 0. 432 0. 431 0. 000 0. 434 Appendix 1. Predictions from the Mathematical Model 201 o o o O CO O O m O CO O c» O c n O r : O i» O s~. O CO O CO O c n o — O CO O c n O O J O i n C c n o o o -=r o m O I -O CVJ O f n O f n o O c n O CO O OT O C J O cn O r » O CVJ o o O CO O CVJ o o o •» O CVJ O CO o — O CVJ O c o O CD o O c n O t -o — O CD o o o — o o — o — o — o •- o *~ o — O — o — o — o •- o — O — o — o — o •-2 I-c_> Q . co cn O rn o cu O -<r *- m O C O O -c-cu c O C -O *r T m o <-o in co O O O -o-C O C O O cn O m C O C O O CO O cn co rn O r--O rn O CO *- in O rn O — — O rn m - - cu O rn cu m — o O rn n cu — C O O cu m o — C O O O J *- cn O cu tn m O cu in co C O o O O — O — o — o — o — o «- o *- O — O <- O — O — o *- o o *- O — O — O -C O C O C O O "3 -O cu O T O J T T O cu O in — O e g O *r C O C O o *-O * - C O *- o O T n cn * - C D O m co r*-» - C O O cn co cn r-s. O cn O C O cu m O cn cu oj O rn cn m cu cu O cn in cn cu o O rn r*. O J O J co O CU C O O O J C D O cu cn O J cn O cu O rn m O cu CU C O m co o --o o *- o *- o — o «- O *~ O — o O «- O o *- o o *- o o ^ o *- o O — CO o C O o m o cu o T m m o cu O C O Cg O C - ' O ir CU r*. O r-. Cn — o O — O cu o O T in co cu C O O cn co -«T cu r-O cn * - C D cn in O rn cu m O m m m cu O rn C D "3-m o O rn oj cn C D O cu «— C D O O J C D 1"^ tr m O O J co m «-O C M C D C O T T C O O — o o — o — o — O *- o o — O O — o o <- O *- o *- o — o O — o *- o ^~ in m r^-O O O J O " T r-» O cu O *— ^ — Cy O T r-. C D O CU * -O C D — cu o O « T L O cn m co O rn C D in m r*» O cn o T to O rn co cn O rn O J - «T tn O J O rn I O L O O O rn C D cn in co O cu CU y-C D C D O CU co m O O J r- cn co •— O oj C D C O to C O o O o »- o *- o — o *- o o — o O O o O — o »- O — O — o O — O — C D C O C O o o O ro O to o O ro o ^ •>x «— f g O T co •»— cu cu O T cu ro m «— O cn m m O O -<T co *-- 3 - C D o m cu r-m f-O cn co *— in co O m m C D O m co tn C D C M O rn m m o o m C O r>«. C O o cu CU C O C O O CU tn co co m o O J o m C O — o cu CU C O C D C O o — o O •— o *- o — o «- o *- O — O *- O — O — O ' - O — O o — o O — o *- o z <c O c_> r s i #-i >-i (— cr cr o UJ x > cr cr U J lu Q . S CL O 1 3 _I cn O « T C D ro c O C D o m C D cn CO in cu r- m O O cn O m n cn cu in o m cn C D C O co C O o O ^ O O T o O O cn O cn O m O rn o o *- O — O — o o o O "~ O O o CO C D CO C U O cn C D O O rn C D "*T C D CO O cu in «-o co *- cu C D CO O rn cu * - ' - C O •»- CU T-O c o CO CO <a- in CO CO c u cn CO in in in c n t o CO 1*-. m in c u c u CO CO c n c u m O r n O r n c u n m c u in «- t o o r-- C D CO CO CO C O O T— c u o C U CO m t o c n c n x^ in O T O O T O O O O m o m O c n r~ m — c n — n — c u — c u c u c u O O — O — o — o o *- o — o ^~ o O O i - o o O o — O o — •«- O T O o o C D m in a O J CO CO c n CO CD cn c o CD CO c o r*. tn tn c u c n r-. o c o in O cn T c n CO c u C D C D o CO •r- CO c u c n c u •^r o m CO CD CD c n CO CO CO o O o •«=- O O T O o m m T - c n *— c n m *— m O J * - c u c u c u o o — O o — o o *- o *~ o *- o O — o *- O — o — o o — O o O i—• i—i in O O J cn cu C D rs- C U C D m C D cu C O C O cu C D C O C D cu o •^ r O C O C O C J O o o m in m co m C O cu o T— o cn co ^ C D co in f * - C O C O O o < o - o - O T O o O "<x •^r cn »— cn cn cn T - cn _ 1 C C L U o o o — o - o *- o — o »— o o »- o ^~ o *- o «~ o , — > CO CO CO L U C O O cu C O cu C O O f - r^. m o »— r - C O C O C D -r- co cu o r-. C O tn 1 cn O C O • - C D in - i C0 O rn cn cu in o r-» co co r^. O L O cu ro cn o UJ ~z cn O o O T O «~ r~ cn »— m O J m oj cn cu m i o • 2 t—1 o o »- o o — o o o •t— O O ^ o *- O — O — O O CO < CO L U cu o m T in C D rs* r- I D cn cu r- C D * ~ < T cn co m O cu O O ro to z o i-s- cu C O C 2 O tn cu C D C U C D r-. to C D cn o o LU O J o o O *~ cu m cu cn cu cn cu cn cn cn Oil o o O o — o o O *- o ^~ O O — o O *- O C O o cn to m C O C U C O ca • T tn in O T *~ cu *— ro cn T C O C O o O C D cn C D C D C O tn co C D m cn cn " ~ C D C D cu C U C D cu O O o *— cu O J - * T cn m cn cn cn cn cn cn T C U o o — o o — o o — O o O ' - O »- O — O — O o O rn m co cn »- to y- r- cu m O -o- in cn O L O T O C D m co to m cu O O C D cu in r- O J O J C D C D C O O cn C D * " O C O O C O o C U C O cn co cn in co tn co C O C O co in o o «- o — o — O — o — O o o — o — O — O *- o r- in CD CO *- cu O co cu cu C D «— CO cu cu C D O C O C U oj m co O J O J O J «— o o — m co cu cu C D C O r » c n c u c u r- co r*. co co O Cn r - -cu cu O J T-O — O — co »-O J CO cn cu in in •<r to m c u o «-T o j T cu O C O O J C O O J C O O J ro •^r o j O — O — to cn ca to co in O — cu cu r - CO m in co O — o — o — o in co co ro r- co co co co — n T— O ^ o *-co — co co on L O — m «— O — O *-cn o — t-» co *— cn r-~ -cr co m O - O — X ca o o ao o Appendix I. Predictions from the Mathematical Model 202 O M O co O CVJ O (O O O O-^T O cn O CD o CO O r-O Od O CO O r - O ^ t OOJ OCT) O CO O M O oj o — O — O O O O O O O O O CO OO) O CD O O ) OCT O * - O — O — o — o — O O O O O O O O O O COCM tOCO C O C D C O O J C O LO C O CO Lf) C M CO C O OT r- M C U — to — m — O '-co — in *— O J *— o r- o co — M in M o — O — O * - O O O O O O O O O C D O O ) O O ) C U — O — O * - O — O — O — o — o — O O O O O O O O C M O J m co M O T cu m M n o *- *— cor- cu M — m ^ ^ M r-n e o m m m o n e o m m m C M m o cum — m neo in M O * ~ O — O * - O O O O O O O O O C D O C D O O ) O C O C M — o * - o *— o — o — o •— o — o * - O O O O O O O O O O — O J cum n co m «— C M M — r- cn O m m co — coco C O M o in -sr r-u"> co m n m o mco m m i n C M O M r - m m oj C M o o cow in M O — O *— o — o o O O o o O O o cn o cn O C D o cn o co C M — o — o — o * - o — o — o — o — o o o o O O O O O O O O — C M m m M C O M O M C M com coco M C M I O N cn C M M C O - C O O M r-r- co r - m r - o r- co r- in r- C M co cn tor- m M M cu C M cn o co cn i n mM O — O — O — O O O O O O O C D O O OO) OO) o CO o co o r - cu *— o " ~ o * - o * - o * - o — o * - o o o o o o o o o o o o o o o o M — co M cor- co O ) co*— t o n m m coco r - r u mco M C O C O C O m M M M M r -cn co cn cn c n o o r - cn m co C M cn cn coco r - M co* - M C O *— in *—cu cur- mM o * — o * — o — O O O O O O OO) OO) O O ) OO) O C O O C O O C O — co C U — o — o * - o * — o — o * - o * - O O O O O O O O O O O O O O O O o o *— O M n m m tor- coco M O cn C M n M C M C O r- OT too) o c o co C M O C M cn cn M r -OJ co C M m C M o cur-. C U M C M C M — cn -—to o n c o o cor- M M O * - n r - coco m M "~~ "~~ — — -— — o * — o *— o *— o — o *—cn o o ) o c o o c o o c o o r - *—m cu — o * ~ o * - o * - o * ~ o — o * - o o o o o o o o o o o o o o o o o o o o m o co*- c o n cn M o m coco cur- M C O C M cn m o M C D C O C O m o coco C M O C O * — M r-m m m m m o m r - m M m*— mco M m mcu — o o c o c o n n o o m tn — o c o m M *~" T — — — — *~~ *—O *- O *— O *—O — C D * — C D *— O OCO O c o OCO O r - O r - C U M cu — o *— o * - o * ~ o * - O — O * - O O O O O O O O O O O O O O O O O O O O O O *—r- m o c o o O*— OT — r-*— *— — cu— OT*— — o r-co r - n cum — m com *—co M O T M r-o m cn C M C D O o r - C D M C O — o c o com co C M m o cum cn C M toco C U M C M cn com mco m M * ~ * ~ *"~ • — * — C M o * — o *—O * - o * - o — o —co *-co o c o o r - O r - O c o o co cum cu — o * - o * - o * - o * - o — o * - O O O O O O O O O O O O O O O O O O O O O O O O . O -=r m m cn i n — m * - M com cu cu * - — t o o m r- cn M coco c o o mco cur- M cu o M C O C O <<r r-<tf in M cu M O mco m m M O *a- r- cn M — o o r - C D M n o o c o mcu o r - mcu cum o - t f m M cu*- C M — C M o w o w o O J o c u o cuo) C M co —co —co *-co o r - o r - o c o o c o —in m C M cu — o * ~ o * - o — o * - o — o * - o o o o o o o o o o o o o o o o o o o o o o o o o o in co cu c o r - r - o c o o M to*- o o r - t o o * r m o m r-co n o cum m to r - * - M r - r- cn o to M r-O ^ r *~" , — *T 0 0 O J in O J C M * - o o m cn C M r-cn m co C M C M co co M M O O m m c u o o n com t o o m M m *~ m -— n o n o n o m o m o c u o C M co C M co C M co * - r - *-r- . o r - Oco O co o m — M n * - cu — ° * ~ 0 ^ ~ o * - o * - o — o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c o coco m m o o o c o m C M toco O M O O o m m cn co*- mcu M C O r-.*- M C O * - M m o M M com o n o o * - r - * - * - o * - r - o n OT o com cn cn o o mco o cu M r - r- cn o r - r - — M m * - n — to f n * — " t f * — *t o -*r o ^ r o - T O M O n o mco n e o n r - cur- cur- to o co o m o m -— M C M m M O ° * ~ ° " ~ o * - o — o — o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O o m c u o *— in r - o r-cu *-co OTOT o n *—r- o j * — M C O C O O co*- * - * - mco M i n cntn r - M t o o -tfr--**— mco r - M r - o r- i*- r - n m o neo *—cu r - o cum co*- o c o C U M m cn M M M C O cn -~ o n M — in*— m o m o m o men m o mco mco mco M r - Mr- n r - C M C O C M C O *—m o m O M — M cum C M C M ° *~ 0 *"~ o*- o — o o o o o o o o o o o o o o o o o o o o o o o o o o o b o o o o ^ °2 ^ S? co* - to-tf r-co o o o cu * - m c o o — m o cu M C O m M n r- *-to n m com o o c o n r - m 2^ *£? £ O com r- *- r - m to «- m r- cu cu r-co cu n m o m M m o m m n o O M cur- m o M O C O O c o n com C O C M co C M co— co— c o o c o o r-o) r-o) coco mco M C O m r - cur- —to o m — M cum C M C M ° * ~ ° * ~ ° ^ ~ o * ~ o — o — o — o * - o o o o o o o o o o o o O O O O O O o o o o o o 0.052 -0.649 -0.484 -0.254 0.142 -0.288 0.147 0.026 -0.565 -0.254 0.072 0.147 0.000 0.000 0.000 PRESSURE PROFILE HEIGHT DIMENSIONLESS RADIUS 0. . 140 0. . 206 0, . 272 0. 338 0. ,405 0, 471 0. 537 0. 603 0. .669 0. 735 0. 802 0. 868 0. 934 1 . 000 1 .000 0. ,000 0. .000 0, .000 0. 000 0. .000 0. ,000 0. 000 0. 000 0, ,000 0, ,000 0, .000 0. 000 0. 000 0. .000 0 . ,995 0. .033 0, ,032 0, .032 0. 032 0, ,032 0. 032 0. 032 0. 032 0. .032 0, ,032 0. 032 0. 032 0. 032 0. .032 0 . 969 0. ,065 0 ,065 0, .065 0. 065 0, .065 0, ,065 0. ,065 0. 064 0 .064 0, ,064 0, ,064 0, 064 0. 064 0 .064 0 .943 0. .098 0, ,097 0, ,097 0. 097 0, ,097 0. ,097 0. 097 0. 097 0 .097 0, ,096 0, ,096 0. 096 0. 096 0. .096 0 .916 0 . 131 0. . 130 0 . 129 0. 129 0 . 129 0, , 129 0, 129 0. 129 0 . 128 0 . 128 0, , 128 0, , 128 0, , 128 0 . 128 0 . 890 0 , 163 0 , 162 0, , 161 0. 161 0, , 161 0, , 161 0. 160 0. 160 0 . 160 0, . 160 0, , 160 0, 160 0. 160 0, , 160 o . 864 0 . 194 0. . 193 0. . 193 0. 192 0, . 192 0. 192 0. 192 0. 192 0 . 192 0. , 192 0, 192 0. 192 0. 192 0. . 192 0 . 01(1 0 . 2 2 6 0 . 225 0 , 224 0. 224 0 , 223 0. 223 0, 223 0. 223 0 . 223 0 , 223 0. 223 0. 223 0. 223 0 . 223 0 .812 0 . 257 0. . 256 0, , 255 0. 255 0, , 254 0, .254 0. 254 0. 254 0 . 254 0. ,254 0, 254 0. 254 0. 254 0 . 254 0 .7 85 0 . 288 0 .206 0 ,285 0. 285 0 ,205 0 .205 0, 204 0. 284 0 .284 0 ,204 0 ,204 0. ,284 0. 204 0 .204 o . 759 0 .318 0 .316 0, ,315 0. 315 0. ,315 0, 315 0, 315 0. 314 0 .314 0. ,314 0, ,314 0. 314 0. 314 0 .314 o , 733 0 . 340 0, . 345 0, 345 0. 345 0, ,345 0, 344 0. 344 0. 344 0. . 344 0, , 344 0. 344 0. 344 0. 344 0 . 344 0 . 707 0 . 369 0 . 372 0, , 373 0. 374 0 , 374 0. 374 0, 374 0. 374 0 . 374 0, , 373 0, , 373 0, 373 0, 373 0 , 373 o , 681 0. . 403 0. .403 0, ,403 0. 403 0, ,403 0. 403 0, 403 0. 403 0 .403 0. .402 0, ,402 0, 402 0. ,402 0 .402 o ,655 0 .436 0 ,435 0, .434 0. 433 0 ,433 0. .432 0, 432 0. 432 0 .431 0 .431 0, ,431 0, ,431 0, ,431 0 .431 0 628 0, 469 0 . 466 0, 464 0. 463 0 , 462 0, 461 0. 460 0. 460 0 .459 0, ,459 0, ,459 0, 458 0, 458 0 ,458 0 , 002 0 . 502 0 . 490 0, 495 0. 493 0, ,491 0. 490 0 . 400 0. 400 0 .407 0. ,400 0, 400 0 , 480 0. 405 0 .405 0 . . 576 0, . 535 0. . 529 0, , 525 0. 522 0 ,519 0, .517 0. 516 0. 515 0, .514 0 .513 0. .512 0, 512 0, ,512 0 .512 0 . 550 0, 567 0. .559 0, 554 0. 550 0, , 547 0. 544 0. 542 0. 541 0 ,539 0, , 530 0, ,538 0, ,537 0, 537 0 .537 o , 524 0. 599 0 ,589 0, ,583 0. 578 0 ,574 0, .571 0, 560 0. 566 0 .564 0 .563 0 ,562 0 ,562 0, ,561 0 .561 0 , , 497 0 , 631 0, .618 0, ,610 0. 604 0 ,600 0, 596 0, 593 0. 590 0 .500 0 , 587 0, , 586 0, , 585 0, , 504 0 . 584 0 ,47 1 0 , . 66 1 0 . 647 0. ,637 0. 6 30 0, ,624 0, 6 20 0. 616 0. 6 13 0 ,611 0. ,609 0 .607 0. ,607 0, ,606 0 .606 0 , 445 0, . 692 0. .675 0, ,664 0. 655 0 .648 0, ,643 0, ,638 0. 635 0 .632 0 ,629 0 ,628 0, ,627 0, ,626 0 .626 0. 419 0, 721 0, 702 0. 689 0. 679 0, ,671 0. 664 0, 659 0. 655 0 ,651 0, ,648 0, ,646 0, ,645 0, ,645 0 .645 0 , 393 0 . . 7 50 0 .728 0, ,712 0. 701 0 .692 0, ,684 0, 678 0. 673 0 .669 0 .665 0 ,663 0 .662 0. ,661 0 .661 0 . 367 0, 777 0 , .752 0, 735 0. 722 0 ,711 0, , 702 0. 695 0. 669 0 .684 0, ,680 0, ,677 0, ,675 0, ,674 0 ,674 0. , 340 0 . .804 0, .776 0, .757 0. 741 0. .729 0. ,719 0, 711 0. 704 0 .698 0, ,693 0. ,689 0. .686 0. ,685 0 .685 0 . 314 0. 829 0, , 798 0, .777 0. 760 0 .746 0, ,735 0, 725 0. 717 0 .709 0 .703 0 .698 0, ,693 0 .687 0. 288 0. 854 0, ,820 0, .796 0. 777 0. , 762 0, , 749 0. 738 0. 728 0 .720 0. ,712 0, ,705 0. .695 0 . 262 0. 876 0, , 840 0, .814 0. 794 0 .777 0, , 763 0, ,750 0. 739 0 .729 0 .720 0 . 708 0. 236 0 . 898 0, ,859 0. .831 0. 809 0 .791 0, , 776 0, , 762 0. 7 50 0 .739 0 .725 0 . 209 0 . 918 0, ,877 0, .847 0. 824 0 .805 0, , 789 0, ,775 0. 761 0 .745 0. 183 0. 936 0. .893 0. .863 0. 840 0 .820 0, ,803 0, ,787 0. 769 0 . 157 0 . 952 0, 909 0. 879 0. 855 0 .835 0. .817 0 , 797 0 . 131 0 . 966 0 .924 0. .894 0. 870 0 .850 . o , 828 0. 105 0 . 978 0 ,938 0 910 0. 887 0 ,863 0. ,079 0. .967 0. .952 0 .928 0. 900 0. ,052 0 .994 0 .966 0 .939 0 ,026 0 9 9 9 0 .975 0 .000 1 ,000 Appendix J Experimental Data for Minimum Spouting Velocity, Spout Shape, Spout Diameter, Dead Zone Boundary, Maximum Spoutable Bed Height, Fountain Height and Overall Pressure Drop 204 Appendix J. Experimental Data: Ums, Ds, Dead Zone, Jfm, HF and APS 205 Table J.l: Experimental and predicted minimum spouting velocities for polystyrene par-ticles, dp = 3.25 mm, Dt = 88.9 mm. Expt'l Predicted values of Umt (m /*) H u m i MG" FM" LM" WU" (m) (m/s) (Eq.(2.1)) (Eq.(2.2)) (Eq.(2.8)) (Eq.(2.10)) 0.55 0.180 0.16 0.29 0.61 0.30 0.71 0.250 0.18 0.32 0.67 0.33 0.83 0.340 0.19 0.35 0.71 0.35 1.03 0.400 0.21 0.39 0.76 0.38 1.20 0.490 0.23 0.42 0.79 0.41 1.36 0.570 0.25 0.45 0.82 0.43 1.51 0.600 0.26 0.47 0.85 0.45 1.62 0.610 0.27 0.49 0.86 0.46 1.76 0.620 0.28 0.51 0.88 0.48 1.86 0.625 0.29 0.52 0.89 0.49 2.00 0.630 0.30 0.54 0.90 0.50 2.16 0.634 0.31 0.56 0.92 0.51 2.30 0.640 0.32 0.58 0.93 0.53 2.48 0.660 0.33 0.60 0.94 0.54 2.60 0.675 0.34 0.61 0.95 0.55 "MG=Mathur-Gishler (1955) equation; FM=Fane-Mitchell (1984) equation; LM=Littman-Morgan (1983) method; WU=Wu et al. (1987) equation. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, Hp and A P S 206 Table J.2: Experimental and predicted minimum spouting velocities for brown beans, dp = 4.7 mm, D7 = 88.9 mm. Expt'l Predicted values of Ume (m A) H M C FM" LM" wu-(m) (m/s) (Eq.(2.1)) (Eq.(2.2)) (Eq.(2.8)) (Eq.(2.10)) 0.55 0.250 0.25 0.46 0.86 0.46 0.67 0.350 0.28 0.51 0.92 0.50 0.78 0.460 0.30 0.55 0.97 0.53 0.87 0.520 0.32 0.58 1.00 0.56 0.96 0.580 0.33 0.61 1.03 0.58 1.05 0.600 0.35 0.64 1.06 0.60 1.14 0.610 0.36 0.66 1.09 0.62 1.28 0.620 0.38 0.70 1.12 0.65 •1.45 0.625 0.41 0.74 1.16 0.68 1.60 0.633 0.43 0.78 1.19 0.71 1.75 0.642 0.45 0.81 1.21 0.74 "MG=Mathur-Gishler (1955) equation: FM=Fane-Mitchell (1984) equation: LM=Littman-Morgan (1983) method; WU=Wu et al. (1987) equation. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, Hp and APS 207 Table J.3: Experimental and predicted minimum spouting velocities for green peas, dp 6.7 mm, D, = 88.9 mm. Expt'l Predicted values of Umt (m/s) H MG" FM" GVZL" LM- WU" (m) (m/s) (Eq.(2.1)) (Eq.(2.2)) (Eq.(2.5)) (Eq.(2.8)) (Eq.(2.10)) 0.55 0.42 0.36 0.65 0.61 1.04 0.67 0.69 0.54 0.40 0.73 0.73 1.12 0.73 0.78 0.68 0.42 0.77 0.80 1.17 0.77 0.87 0.76 0.45 0.81 0.87 1.21 0.80 0.96 0.82 0.47 0.85 0.94 1.25 0.84 1.06 0.85 0.49 0.90 1.01 1.29 0.87 1.18 0.90 0.52 0.95 1.08 1.33 0.91 1.44 0.96 0.57 1.04 1.23 1.40 0.98 1.64 1.01 0.61 1.11 1.31 1.45 1.03 " MG=Mathur-Gishler. (1955) equation; FM=Fane-Mitchell (1984) equation; GVZL=Grbavic et al. (1976) method for spherical particles; LM=Littman-Morgan (1983) equation; WU=Wu et al. (1987) equation. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, Hp and APS 208 Table J.4: Spout diameters for polystyrene particles, dp = 3.25 mm, Dx = 88.9 mm, U/Ums = 1.1. Spout Diameter (m) z (m) Bed height, H (m) 0.55 0.83 1.20 1.51 1.76 2.00 2.30 2.60 0.1 0.095 0.115 0.115 0.120 0.110 0.120 0.120 0.115 0.2 0.100 0.120 0.125 0.130 0.130 0.130 0.135 0.130 0.3 0.095 0.115 0.130 0.135 0.135 0.135 0.140 0.135 0.4 0.075 0.115 0.130 0.140 0.135 0.135 0.145 0.140 0.5 0.075 0.105 0.125 0.140 0.135 0.140 0.145 0.140 0.6 0.095 0.120 0.140 0.135 0.135 0.145 0.145 0.7 0.085 0.115 0.125 0.125 0.135 0.135 0.145 0.8 0.080 0.100 0.110 0.115 0.117 0.120 0.115 0.9 0.095 0.110 0.110 0.105 0.115 0.110 1.0 0.095 0.105 0.105 0.105 0.115 0.110 1.1 0.095 0.100 0.100 0.100 0.115 0.110 1.2 0.095 0.105 0.100 0.100 0.110 0.105 1.3 0.100 0.100 0.100 0.110 0.105 1.4 0.100 0.100 0.105 0.100 0.105 1.5 0.110 0.100 0.110 0.100 0.115 1.6 0.100 0.100 0.100 0.115 1.7 0.105 0.100 0.115 0.110 1.9 0.105 0.115 0.100 2.0 0.100 0.115 0.100 2.2 0.115 0.100 2.3 0.130 0.105 2.5 0.120 2.6 0.130 Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 209 Table J.5: Spout diameters for brown beans, dT = 4.7 mm, Di = 88.9 mm, U/Ums = 1.1 Spout Diameter (m) z (m) H=0.55 m H=0.78 m H=0.96 m H=1.28 m H=1.60 m 0.1 0.085 0.100 0.105 0.105 0.105 0.2 0.090 0.110 0.120 0.120 0.125 0.3 0.080 0.110 0.125 0.125 0.125 0.4 0.070 0.110 0.120 0.120 0.130 0.5 0.065 0.105 0.120 0.120 0.125 0.6 0.095 0.115 0.120 0.125 0.7 0.090 0.110 0.110 0.120 0.8 0.095 0.100 0.115 0.9 0.095 0.095 0.105 1.0 0.095 0.100 1.1 0.095 0.100 1.2 0.095 0.100 1.3 0.100 1.4 0.100 1.5 0.100 1.6 0.115 Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 210 J.6: Spout diameters for green peas, dp = 6.7 mm, Dx = 88.9 mm, U/Ums = 1.1 Spout Diameter(m) z (m) H=0.55 m H=0.78 m H=0.96 m H=1.18 m H=1.44 m H=1.64 m 0.1 0.110 0.115 0.125 0.125 0.130 0.120 0.2 0.120 0.130 0.140 0.140 0.140 0.135 0.3 0.105 0.135 0.150 0.145 0.150 0.150 0.4 0.095 0.130 0.145 0.145 0.150 0.150 0.5 0.090 0.120 0.135 0.140 0.150 0.150 0.6 0.110 0.125 0.140 0.150 0.150 0.7 0.095 0.110 0.130 0.145 0.150 0.8 0.105 0.115 0.130 0.135 0.9 0.105 0.115 0.125 0.135 1.0 0.115 0.125 0.135 1.1 0.130 0.125 0.135 1.2 0.125 0.130 1.3 0.125 0.130 1.4 0.150 0.130 1.5 0.135 1.6 0.150 Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 211 Table J.7: Spout diameters for polystyrene particles, dp — 3.25 mm, Dx — 88.9 mm, H = 2.00 m Spout Diameter (m) z H=2.0 m (m) U / U m . = l . l U/Ura.=1.3 U/Um J=1.5 0.1 0.120 0.125 0.130 0.2 0.130 0.135 0.135 0.3 0.135 0.140 0.140 0.4 0.135 0.150 0.150 0.5 0.140 0.150 0.155 0.6 0.135 0.150 0.155 0.7 0.135 0.145 0.165 0.8 0.117 0.125 0.155 0.9 0.105 0.115 0.140 1.0 0.105 0.115 0.140 1.1 0.100 0.115 0.140 1.2 0.100 0.115 0.140 1.3 0.100 0.115 0.140 1.4 0.105 0.110 0.140 1.5 0.110 0.115 0.135 1.6 0.100 0.120 0.135 1.7 0.100 0.125 0.135 1.8 0.110 0.125 0.145 1.9 0.105- 0.125 0.145 2.0 0.100 0.115 j 0.145 Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and A P 5 212 Table J.8: Spout, diameters for spout-fluid beds with D : = 88.9 mm for polystyrene (dp = 3.25 mm) and brown beans (dp = 4.7 mm). Spout Diameter (m) Polystyrene particles Brown beans H=1.51 m H=1.28 m SA" SF + SA" SF+ Con ® #1 Con.® #2 Con.® #1 Con.€ #2 Con.® #1 Con.® #1 U/Um,=2.3 U/Um,=2.2 U/Ura.=2.5 U/Um,=2.45 U/Um,=2.0 U/Ura.=2.1 z gQ=865 ga=815 <7a=991 <?a=972 ga=667 <7a=750 (m) m3/h m3/h m3/h m3/h m3/h m3/h 0.1 0.115 0.110 0.115 0.105 0.115 0.120 0.2 0.135 0.120 0.130 0.120 0.125 0.130 0.3 0.145 0.125 0.145 0.125 0.135 0.140 0.4 0.160 0.130 0.165 0.125 0.130 0.145 0.5 0.160 0.130 0.165 0.130 0.125 0.150 0.6 0.160 0.130 0.165 0.135 0.125 0.145 0.7 0.155 0.130 0.160 0.140 0.115 0.125 0.8 0.140 0.130 0.140 0.130 0.105 0.100 0.9 0.130 0.125 0.153 0.130 0.115 0.125 ' 1.0 0.135 0.130 0.171 0.155 0.105 0.120 1.1 0.135 0.135 0.178 0.170 0.105 0.110 1.2 0.140 0.137 0.178 0.185 0.105 0.150 1.3 0.155 0.145 0.193 0.195 1.4 0.155 0.155 0.193 0.215 1.5 0.160 0.155 0.200 0.220 "SA—Spout with aeration + SF—spout-fluidization ® Con.—Configuration Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and A P 5 213 Table J.9: Dead zone boundaries as viewed from front plate for polystyrene particles, dp = 3.25 mm, D{ = 88.9 mm, U/Ums = 1.1. Dead zone boundary as viewed from front plate (m) z (m) Bed height, H (m) 0.55 0.83 1.20 1.51 1.76 2.00 2.30 2.60 0.1 0.155 0.190 0.215 0.225 0.230 0.220 0.225 0.225 0.2 0.210 0.245 0.270 0.285 0.285 0.290 0.290 0.290 0.3 0.265 0.290 0.330 0.340 0.340 0.335 0.335 0.330 0.4 0.360 0.350 0.375 0.390 0.400 0.390 0.390 0.390 0.5 0.580 0.400 0.425 0.430 0.435 0.430 0.430 0.430 0.6 0.830* 0.460 0.450 0.456 0.485 0.470 0.460 0.450 0.7 0.520 0.470 0.480 0.490 0.490 0.470 0.470 0.8 0.800 0.520 0.520 0.560 0.525 0.510 0.510 0.9 0.910° 0.620 0.590 0.620 0.575 0.570 0.550 1.0 0.680 0.640 0.660 0.640 0.610 0.590 1.1 0.820 0.680 0.710 0.690 0.630 0.610 1.2 0.910^  0.730 0.750 0.710 0.670 0.650 1.3 0.800 0.770 0.740 0.715 0.675 1.4 0.910* 0.820 0.790 0.755 0.720 1.5 0.910 0.830 0.820 0.750 1.6 0.860 0.840 0.810 1.7 0.910 0.910A 0.860 1.8 0.910 *At z = 0.53 m; °at z = 0.83 m; 55at z — 1.15 m: *at z = 1.35 m; Aat z — 1.75 m. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and A P , Table J.10: Dead zone boundaries as viewed from front plate for brown bean mm, Di = 88.9 mm, U/Uma = 1.1. Dead zone boundary as viewed from front plate (m) z (m) H=0.55 m H=0.78 m H=0.96 m H=1.28 m H=1.60 m 0.1 0.205 0.225 0.205 0.200 0.185 0.2 0.300 0.320 0.285 0.250 0.250 0.3 0.380 0.390 0.355 0.305 0.305 0.4 0.480 0.475 0.420 0.355 0.350 0.5 0.670 0.535 0.480 0.390 0.390 0.6 0.830* 0.580 0.530 0.415 0.420 0.7 0.740 0.580 0.465 0.470 0.8 0.910° 0.710 0.525 0.530 0.9 0.910^  0.590 0.570 1.0 0.645 0.640 1.1 0.770 0.700 1.2 0.910 0.755 1.3 0.820 1.4 - 0.910 *At z = 0.55 m; °at z - 0.75 m; cat z = 0.92 m. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 215 Table J .H: Dead zone boundaries as V i e w e d from front plate for green peas d = 6 7 mm, Di = 88.9 mm, U/Ums = 1.1. ' Dead zone boundar}7 as viewed from front plate (m) z (m) H=0.55 m H=0.78 m H=0.96 m H=1.18 m H=1.44 m H=1.64 m 0.1 0.230 0.250 0.265 0.280 0.265 0.240 0.2 0.310 0.339 0.340 0.350 0.335 0.305 0.3 0.375 0.400 0.410 0.415 0.400 0.355 0.4 0.450 0.470 0.470 0.460 0.450 0.400 0.5 0.680 0.530 0.510 0.505 0.485 0.440 0.6 0.830* 0.580 0.580 0.540 0.520 0.465 0.7 0.760 0.650 0.600 0.575 0.510 0.8 0.910° 0.770 0.670 0.635 0.565 0.9 0.910 0.730 0.710 0.615 1.0 0.830 0.760 0.670 1.1 0.910c 0.810 0.770 1.2 0.850 0.835 1.3 0.910* 0.855 1.4 0.910A *At z = 0.55 m; °at z = 0.74 m; ^ 'at z = 1.05 m; *at z = 1.25 m; Aat z = 1.35 m. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 216 Table J.12: Dead zone boundaries as viewed from front plate for polystyrene particles, dp = 3.25 mm, D, = 88.9 mm. Dead zone boundary as viewed from front plate (m) z H=2.0 m (m) U/U m ,= l . l / l-:m« — 1-3 U/Um 4=1.5 0.1 0.220 0.230 0.230 0.2 0.290 0.295 0.295 0.3 0.335 0.345 0.345 0.4 0.390 0.390 0.410 0.5 0.430 0.425 0.435 0.6 0.470 0.460 0.450 0.7 0.490 0.470 0.460 0.8 0.525 0.545 0.540 0.9 0.575 0.610 0.620 1.0 0.640 0.665 0.690 1.1 0.690 0.700 0.720 1.2 0.710 0.750 0.760 1.3 0.740 0.810 0.800 1.4 0.790 0.840 0.840 1.5 0.830 0.910 0.910 + 1.6 0.860 1.7 0.910 At z=1.45 m Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and AP , 217 Table J.13: Dead zone boundaries as viewed from rear column wall for polystyrene par-ticles, dp = 3.25 mm, D{ = 88.9 mm, U/Ums = 1.1. Dead zone boundary as viewed from rear column wall U/Umt = 1.1 U/Umt = 1.3 U/Um. = 1.5 H=1.20 m H=2.00 m H=2.60 m H=2.00 m H=2.00 m z DB+ z D B + z DB+ z DB+ z DB+ (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) 0.78 0.000 0.73 0.000 0.72 0.000 0.68 0.000 0.85 0.000 0.8 0.100 0.8 0.340 0.8 0.260 0.8 0.440 0.9 0.240 0.9 0.600 0.9 0.640 0.9 0.720 0.9 0.840 1.0 0.680 1.0 0.840 1.0 0.840 1.0 0.880 1.0 1.000 1.1 1.140 1.1 0.960 1.1 1.020 1.1 1.060 1.1 1.140 1.15 1.420 1.2 1.080 1.2 1.160 1.2 1.200 1.2 1.240 1.3 1.160 1.3 1.240 1.3 1.260 1.3 1.300 1.4 1.220 1.4 1.280 1.4 1.340 1.4 1.360 1.5 1.300 1.5 1.300 1.5 1.420 1.45 1.420 1.6 1.360 1.6 1.360 1.7 1.420 1.7 1.380 1.8 1.420 + DB=Dead zone boundary Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 218 Table J.14: Dead zone boundaries as viewed from rear column wall for brown beans dv = 4.7 mm, £>,- = 88.9 mm, U/Ums = 1.1. Dead zone boundary as viewed from rear column wall (m) H=0.96 m H=1.28 m H=1.60 m z DB+ z DB- z D B + (m) (m) (m) (m) (m) (m) 0.85 0.000 1.05 0.000 1.1 0.000 0.9 1.200 1.1 0.840 1.2 1.000 0.92 1.420 1.15 1.200 1.3 1.220 1.2 1.420 1.4 1.420 +DB=Dead zone boundary Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 219 Table J.15: Dead zone boundaries as viewed from rear column wall for green peas, dp = 6.7 mm, Dt = 88.9 mm, U/Ums = 1.1. Dead zone boundary as viewed from rear column wall (m) H=0.96 m H=1.18 m H=1.44 m H=1.64 m z DB+ z D B + z DB+ z D B + (m) (m) (m) (m) (m) (m) (m) (m) 0.82 0.000 0.92 0.000 0.95 0.000 0.98 0.000 0.85 1.200 0.95 0.700 1.0 0.820 1.0 0.440 0.88 1.420 1.00 1.240 1.1 1.160 1.1 0.760 1.05 1.420 1.2 1.260 1.2 1.240 1.25 1.420 1.3 1.380 1.35 1.420 + DB=Dead zone boundary Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and A P , 220 Table J.16: Dead zone boundaries as viewed from front plate for spout-fluid beds, polystyrene particles, dp = 3.25 mm, Dx = 88.9 mm, H=1.51 m. Dead zone boundary as viewed from front plate (m) SA- SF+ 1 JSBF* Con ® #1 Con.® #2 Con ® #1 Con.® #2 Con.® #1 Con.® #2 j U/Uma=2.3 U/Um,=2.2 U/Um.=2.5 U/Um.=2.45 U/Uraf=2.6 U/Ums=2.6 ! z ga=865 <Za =815 qa=991 ga=972 9o=1080 <7a=1060 i (m) m 3 / h m 3 /h m 3 / h m 3 / h m 3 / h m 3 / h 0.1 0.210 0.220 0.225 0.225 0.220 0.230 0.2 0.270 0.290 0.280 0.295 0.270 0.310 0.3 0.330 0.350 0.315 0.350 0.310 0.340 0.4 0.410 0.405 0.375 0.420 0.350 0.410 0.5 0.475 0.465 0.420 0.450 0.435 0.470 0.6 0.530 0.500 0.450 0.480 0.485 0.490 0.7 0.545 0.535 0.480 0.500 0.650 0.510 0.8 0.670 0.580 0.540 0.560 -0.910* 0.600 0.9 0.740 0.640 0.690 0.650 0.690 1.0 0.855 0.690 0.825 0.750 0.780 • 1.1 0.910 0.770 0.910 0.910 0.910 1.2 0.805 1.3 0.910 "SA—Spout with aeration; ~SF—spout-fluidization; ^•JSBF—Jet,slug and bubble in fluidized bed; ® Con.—Configuration; *At z=0.75 m. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 221 Table J.17: Dead zone boundaries as viewed from rear column wall for spout-fluid beds, polystyrene particles, dp = 3.25 mm, Dx = 88.9 mm, H=1.51 m. Dead zone boundar}' as viewed from rear column wall (m) SA" SF+ JSBFi C o n . e #1 Con.° #2 Con. G #1 Con . e #2 Con.® #2 U/U m .=2 .3 U/U m .=2.2 U/U m ,=2.5 U / U m,=2.45 U/U T n < =2.6 ga=865 m3/h <7a=815 Tn3/h gQ=991 m3/h ga=972 m3/h ga=1065 m3/h z DB§ z DB§ z DB§ z DB§ z DB§ (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) 0.7 1.000 0.7 0.700 0.7 0.800 0.7 0.760 0.7 0.760 0.8 1.160 0.8 0.800 0.8 0.960 0.8 0.840 0.8 0.900 0.9 1.200 0.9 0.960 0.9 1.070 0.9 1.000 0.9 1.000 1.0 1.280 1.0 1.040 1.0 1.200 1.0 1.240 1.0 1.240 1.1 1.420 1.1 1.200 1.1 1.420 1.1 1.420 1.1 1.420 1.2 1.360 1.3 1.420 "SA—Spouting witli aeration; +SF—spout-fluidization; ^"JSBF—Jet,slug and bubble in fluidized bed; ®Con.—Configuration; §DB—Dead zone boundary. Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and A P S 222 Table J.18: Dead zone boundaries for polystyrene particles, dp = 3.25 mm, Dt = 88.9 mm, H=1.51 m. Dead zone boundary for configuration #3 viewed from viewed from front plate rear wall z DB+ z DB+ (m) (m) (m) (m) 0.1 0.240 0.2 0.340 0.3 0.420 0.4 0.420 0.5 0.480 0.6 0.530 0.7 0.570 0.8 0.590 0.9 0.620 0.95 0.000 1.0 0.715 1.0 0.360 1.1 0.755 1.1 1.420 1.2 0.805 1.3 0.910 + DB=Dead zone boundary Appendix J. Experimental Data: Ums, Ds, Dead Zone, Hm, HF and APS 223 Table J.19: Fountain heights, U/Ums = 1.1, D, = 88.9 mm, see Table 3.1 for particle properties. H H F H H F Particles (m) (m) Particles (m) (m) PS+ 0.55 1.25 BB+ 0.55 0.65 PS 0.83 1.55 BB 0.78 1.62 PS 1.20 2.40 BB 0.96 2.75 PS 1.51 2.80 BB 1.28 2.95 PS 1.76 2.95 BB 1.60 3.10 PS 2.00 2.60 GP- 0.55 2.00 PS 2.30 2.80 GP 0.78 2.15 PS 2.60 3.00 GP 0.96 1.95 PS 2.00 3.7" GP 1.18 2.22 PS 2.00 >3.7' GP 1.44 2.56 GP 1.64 3.26 + PS = Polystyrene; BB = brown beans; GP = green peas; "At L7/<ym5=1.3; 'At U/Um. = 1.5. 

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