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Hydrodynamics, stability and scale-up of slot-rectangular spouted beds Chen, Zhiwei 2008

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Hydrodynamics, Stability and Scale-up of Slot-Rectangular Spouted Beds by Chen, Zhiwei B.Sc., Tsinghua University, 1999 M.Sc., Chinese Academy of Sciences, 2002  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemical and Biological Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) March, 2008 c Chen, Zhiwei 2008  Abstract Slot-rectangular spouted beds, with rectangular cross-section and slotted gas inlets, have been proposed as a solution to overcoming scale-up difficulties with conventional axisymmetric spouted beds. They can be utilized in gas/particle processes such as drying of coarse particles and coating of tablets. However, application of this spouted bed was limited because of instability and insufficient hydrodynamic studies. The present work is therefore aimed at the study of hydrodynamics, stability and scale-up of slot-rectangular spouted beds. The hydrodynamic study was carried out in four slot-rectangular columns of various width-to-thickness ratios combined with various slot configurations, particles of different properties and a range of operating conditions. Hydrodynamics of slot-rectangular spouted beds showed major similarity with conventional spouted beds. However, equations and mechanistic models adopted from conventional axisymmetric spouted beds generally failed to provide good predictions for the three-dimensional slot-rectangular geometry. New empirical correlations were derived for the minimum spouting velocity and maximum pressure drop for different slot configurations. Slot-rectangular spouted beds also showed more flow regimes than conventional spouted beds. Nine flow regimes, as well as unstable conditions, were identified based on frequency and statistical analysis of pressure fluctuations. Slot geometrical configuration was found to be the main factor affecting the stability of slot-rectangular spouted beds. A comprehensive hydrodynamic study on the effect of slot configuration was therefore carried out. Slots of smaller length-to-width ratio, smaller length and greater depth were found to provide greater stability. Stable criteria for the  ii  Abstract slot configuration were found consistent with the conventional axisymmetric spouted beds with extra limitation on slot length-to-width ratio and slot depth. Local distributions of pressure, particle velocity and voidage, as well as spout shape and particle circulating flux, were compared for different slot configurations. Higher slot length-to-width ratios lead to slightly higher particle circulation rates. A previously proposed scale-up method involving multiple chambers was tested in the present work using multiple slots. Instability caused by the merging of multiple spouts and asymmetric flow was successfully prevented by suspending vertical partitions between the fountains. Some criteria and guidelines were also proposed for scale-up using multiple chambers.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Nomenclature  xx  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.2 Spouted bed technique . . . . . . . . . . . . . . . . . . 1.2.1 Conventional axisymmetric spouted beds . . . 1.2.2 Slot-rectangular spouted beds . . . . . . . . . . 1.2.3 Slot-rectangular spouted beds with draft plates 1.3 Flow regimes for slot-rectangular spouted bed . . . . . 1.4 Hydrodynamic studies . . . . . . . . . . . . . . . . . . 1.4.1 Global properties . . . . . . . . . . . . . . . . . 1.4.2 Local flow structure . . . . . . . . . . . . . . . 1.5 Scale-up for slot-rectangular spouted beds . . . . . . . 1.6 Stability of slot-rectangular spouted beds . . . . . . . 1.7 Mechanistic models . . . . . . . . . . . . . . . . . . . 1.8 Summary and research objectives . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  1 1 3 3 5 6 6 9 9 9 10 11 11 12  2 Experimental Set-Up . . . . . 2.1 Experimental units . . . . . 2.2 Slot-rectangular columns . 2.3 Gas supply . . . . . . . . . 2.4 Particulate materials . . . . 2.5 Pressure measurement . . . 2.6 Voidage and particle velocity  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  15 15 15 21 21 23 25  Velocity . . . . . . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  29 29 32 35 41  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . measurement  . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  3 Maximum Pressure Drop and Minimum Spouting 3.1 Pressure drop across the bed . . . . . . . . . . . . 3.2 Spouting pressure drop . . . . . . . . . . . . . . . 3.3 Minimum spouting velocity . . . . . . . . . . . . . 3.4 Correlation of minimum spouting velocity . . . . . iv  . . . . . . .  . . . . . . .  Table of Contents 3.5  Maximum pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . .  51  4 Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fluctuation of column pressure drop . . . . . . . . . 4.2 Visual characterization and frequency analysis . . . . 4.2.1 Fixed bed regime (F) . . . . . . . . . . . . . 4.2.3 Spouting regimes (S) . . . . . . . . . . . . . . 4.2.4 Multiple spouting regime (MS) . . . . . . . . 4.2.5 Merging of multiple spouting regime (MS(M)) 4.2.6 Incoherent spouting regime (IS) . . . . . . . . 4.2.7 Slugging regime (SL) . . . . . . . . . . . . . 4.2.8 Jet in fluidized bed regime (JF) . . . . . . . . 4.2.9 Unstable conditions (US) . . . . . . . . . . . 4.3 Flow regime map . . . . . . . . . . . . . . . . . . . . 4.3.1 Evolution of flow regime . . . . . . . . . . . . 4.3.2 Regime maps . . . . . . . . . . . . . . . . . . 4.4 Statistical analysis of flow regimes . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  59 59 61 61 63 66 69 70 71 72 76 78 78 79 79  5 Slot-Rectangular Spouted Bed Stability 5.1 Approaches to analyze stability . . . . . 5.2 Effect of slot configuration . . . . . . . . 5.2.1 Slot width . . . . . . . . . . . . 5.2.2 Slot length . . . . . . . . . . . . 5.2.3 Slot length-to-width ratio . . . . 5.2.4 Slot depth . . . . . . . . . . . . 5.2.5 Converging and diverging angle . 5.3 Slot velocity distributions . . . . . . . . 5.4 Termination of spouting . . . . . . . . . 5.4.1 Mechanisms . . . . . . . . . . . 5.4.2 Maximum spoutable bed height .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  84 84 85 86 88 91 96 99 102 104 104 107  6 Local Flow Structure . . . . . . . . . . 6.1 Measurement issues . . . . . . . . . 6.2 Shape of spout and dead zone . . . . 6.2.1 Measurement method . . . . 6.2.2 Spout shape and size . . . . 6.2.3 Prediction of spout shape . . 6.2.4 Dead zones . . . . . . . . . . 6.3 Distribution of pressure . . . . . . . 6.3.1 Axial pressure distribution . 6.3.2 Lateral pressure distribution 6.4 Distribution of voidage . . . . . . . 6.4.1 Axial distribution of voidage 6.4.2 Lateral distribution of voidage  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  108 108 110 110 110 116 118 121 121 122 123 123 124  v  . . . . . . . . . . . .  . . . . . . . . . . . . .  Table of Contents 6.5  Distribution of vertical component of particle velocity 6.5.1 Axial distribution of particle velocity . . . . . 6.5.2 Lateral distribution of particle velocity . . . . 6.6 Comparison of slots . . . . . . . . . . . . . . . . . . 6.6.1 Axial pressure distribution . . . . . . . . . . 6.6.2 Spout shape and dead zone boundaries . . . . 6.6.3 Axial distribution of particle velocity . . . . . 6.6.4 Axial distributions of voidage . . . . . . . . . 6.6.5 Lateral distributions of particle velocity . . . 6.6.6 Lateral distributions of voidage . . . . . . . . 6.6.7 Particle circulation flux . . . . . . . . . . . .  7 Spouting with Multiple Slots . . . . . . . . . . 7.1 Equipment . . . . . . . . . . . . . . . . . . . 7.2 Pressure evolution . . . . . . . . . . . . . . . 7.3 Minimum spouting velocity . . . . . . . . . . 7.4 Spouting pressure drop . . . . . . . . . . . . 7.5 Flow regimes . . . . . . . . . . . . . . . . . . 7.5.1 Onset of spouting . . . . . . . . . . . 7.5.2 Evolution of flow regimes . . . . . . . 7.5.3 Termination of spouting . . . . . . . . 7.6 Interaction between spouts . . . . . . . . . . 7.6.1 Effect of distance between slot centres 7.6.2 Effect of diverging base . . . . . . . . 7.6.3 Effect of vertical partition . . . . . . . 7.7 Scale-up of slot-rectangular spouted bed . . . 7.7.1 Single chamber . . . . . . . . . . . . . 7.7.2 Combined multiple chamber . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  125 125 135 136 137 137 138 138 139 139 139  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  147 147 149 150 153 154 154 155 158 159 159 160 162 166 167 172  8 Conclusions and Recommendation . . . . . . . . . . . . . . . . . . . . . 176 8.1 Conclusions from this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . 180 References  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181  Appendices A Summary of Previous Work  . . . . . . . . . . . . . . . . . . . . . . . . . 199  B Measurement Systems . . . . B.1 Particle velocity and voidage B.1.1 Measurement system B.1.2 Data analysis . . . .  . . . . . . . . measurement . . . . . . . . . . . . . . . .  vi  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  210 210 210 213  Table of Contents B.2 Gas velocity measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 216 C Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 C++ code for particle velocity analysis . . . . . . . . . C.2 MATLAB code for orifice discharge coeffcient calculation C.3 MATLAB code for spout shape prediction . . . . . . . . C.4 MATLAB code for axial particle velocity prediction . . .  vii  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  220 220 242 246 250  List of Tables 2.1 Dimensions of rectangular columns . . . . . . . . . . . . . . . . . . . . . 2.2 Properties of particles used in this project . . . . . . . . . . . . . . . . .  17 23  3.1 3.2  Comparison of correlations . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linearly fitted constant and exponents of equation 3.22 for all available data and for individual experimental studies . . . . . . . . . . . . . . . .  46  Slots with different length/width ratios in 300 × 100 mm column . . . . .  92  5.1  48  7.1  Dimensions and stability of slots for 1.33 mm glass beads in a plexiglass column of cross-section 300 mm × 100 mm, with air at 20◦ C and 1 atm pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.2 List of Archimedes numbers for the particles investigated in this work . . 172 A.1 A.2 A.3 A.4  Annotated list of previous work on slot-rectangular spouted beds . Correlation and criteria for slot-rectangular spouted beds . . . . . Experimental technique available for gas/solid flow measurement . Some literature models for spouted beds . . . . . . . . . . . . . .  viii  . . . .  . . . .  . . . .  . . . .  199 201 203 206  List of Figures 1.1 Diagram of slot-rectangular spouted beds (Dogan et al., 2000) . . . . . . 1.2 Flow pattern in a conventional conical-cylindrical spouted bed (He, 1995) 1.3 Schematic of slot-rectangular spouted bed column with draft plates (Kalwar et al., 1993). The dimension shown in the diagram as “BED LENGTH” is called “column thickness” in the current work. . . . . . . . . . . . . . . 1.4 Scale-up methods according to Kalwar et al. (1993). ”‘LENGTH”’ is named as ”‘thickness”’ in the current work. . . . . . . . . . . . . . . . . . 2.1 2.2  2 7  8 11  Flow chart of spouted bed . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic drawing of column C (a) Front view, (b) Side view, (c) Top view, (d) Detail of slot at bottom assembly. . . . . . . . . . . . . . . . . 2.3 Top views showing slot positions of all four experimental columns. . . . . 2.4 Diverging base allowing two parallel slots spouting. . . . . . . . . . . . . 2.5 Different type of slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Gas pressure and velocity in an empty column . . . . . . . . . . . . . . . 2.7 Sample calibration of pressure transducer against water manometer . . . 2.8 Sample fluctuation of pressure transducer signal when there is no gas flow through the bed: PX142-002D5V, for measuring the pressure drop across the bed. Column C: 300 × 100 mm, with diverging base and type b slot of 4 mm width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Comparison between known particle velocity and particle velocity determined with calibrated effective distance interval for 1.33 mm glass beads, with 90% confidence level. . . . . . . . . . . . . . . . . . . . . . . . . . .  16  3.1 3.2  31  3.3  3.4 3.5 3.6  Evolution of pressure drops with superficial gas velocity . . . . . . . . . . Difference in Ums caused by hysteresis. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type a, depth= 62.7 mm, see Figure 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference in ∆Pmax caused by hysteresis. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type a, depth= 62.7 mm, see Figure 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spouting pressure drop across the bed. Particles: Glass beads, dp =1.33 mm; Slot type a, depth 62.7 mm, see Figure 2.5; diverging base. . . . . . Spouting pressure drop across the bed. Particles: Glass beads; Slot type b, depth 12.7 mm, see Figure 2.5; diverging base. . . . . . . . . . . . . . . Effect of static bed height on Ums , Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  18 19 20 20 22 24  25  28  32  33 34 34  36  List of Figures 3.7  3.8  3.9  3.10  3.11  3.12  3.13  3.14 3.15 3.16 3.17  3.18 3.19  3.20  3.21  3.22  Effect of static bed height on Ums , Column: α=150 mm, β=200 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of static bed height on Ums . Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; flat bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of diverging base on Ums . Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of column aspect ratio on minimum spouting velocity. Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. All dimensions are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of column width to slot width ratio on minimum spouting velocity. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. All dimensions are in mm. . Effect of dimensionless slot width on minimum spouting velocity. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. All dimensions are in mm. . . . . . . . . Effect of dimensionless bed height on minimum spouting velocity. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. . . . . . . . . . . . . . . . . . . . . . . . Comparison between the correlation of all available experimental results, equation 3.23 and data by Dogan et al. (2000) and Freitas et al. (2000). . Comparison between the correlation of experimental results using type b slots, equation 3.24 and data by Dogan et al. (2000) and Freitas et al. (2000). Correlation predictions for type a slot, equation 3.25, compared with experimental data of Dogan et al. (2004). . . . . . . . . . . . . . . . . . . . Effect of static bed height on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base. . . . . . . . . . . . . . . . . . . Effect of static bed height on maximum pressure drop . . . . . . . . . . . Comparison between experimental maximum pressure drop and prediction by Equation 3.26. Columns C and D; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5. . . . . . . . . . . . . . . . . Effect of static bed height on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; flat base. . . . . . . . . . . . . . . . . . . . . . . Effect of diverging base on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of column aspect ratio on maximum pressure drop. Particles: Glass beads; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. All dimensions are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  36  37  37  39  40  40  41 49 50 51  52 53  53  54  54  56  List of Figures 3.23 Effect of width to slot width ratio on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. All dimensions are in mm. 3.24 Effect of dimensionless slot width on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. All dimensions are in mm. 3.25 Effect of dimensionless bed height on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. . . . . . . . . . . . . . . . 3.26 Comparison between experimental maximum pressure drop and prediction by Equation 3.26. Columns C and D; Particles: Glass beads, dp =1.33 mm; Slot: type b, depth 12.7 mm, see Figure 2.5; diverging base. . . . . . . . . 4.1 4.2  Natural fluctuation produced by pressure transducer . . . . . . . . . . . Fluctuations of overall bed pressure drop for Fixed Bed regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 550 mm; Ug = 0.256 m/s. . . . . . . . . . . . . 4.3 Fluctuations of pressure drop for Internal Jet regime . . . . . . . . . . . 4.4 Internal Jet and Steady State Spouting regimes observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. 4.5 Fluctuations of pressure drop for Steady State Spouting. Column C: 300 × 100 mm, with diverging base; Slot: type a, 5 mm width; 1.33 mm glass beads; Hs = 150 mm; Ug = 0.730 m/s. . . . . . . . . . . . . . . . . . . 4.6 Fluctuations of pressure drop for Spouting regime with fountain swaying parallel to slot. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 200 mm; Ug = 0.46 m/s. 4.7 Spouting with fountain swaying for two directions observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. 4.8 Top-view of spouting column with fountain swaying in two directions as observed in Column C: 300 mm x 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Fluctuations of pressure drop for spouting regime with fountain swaying normal to slot. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 1 50 mm; Ug = 0.566 m/s. 4.10 Fluctuations of pressure drop for swaying normal to slot with dilute fountain. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 50 mm; Ug = 0.374 m/s. . . . . . . 4.11 Spouting and merging of multiple spouts observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. . . . . . . . 4.12 Fluctuations of pressure drop for Multiple Spouting regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 180 mm; Ug = 0.826 m/s. . . . . . . . . . . . . . . .  xi  57  57  58  58 60  62 62 63  64  65 66  67  67  68 68  69  List of Figures 4.13 Fluctuations of pressure drop for Merging of Multiple Spout regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 180 mm; Ug = 0.769 m/s. . . . . . . . . . 4.14 Pressure evolution with Multiple Spouting flow regime. Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. All dimensions are in mm. Filled symbols show where 2 spouts co-exist. . . . 4.15 Pressure evolution with merging of multiple spouts. Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. All dimensions are in mm. Filled symbols show where 2 spouts co-exist . . . 4.16 Fluctuations of pressure drop in Incoherent Spouting regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 450 mm; Ug = 0.965 m/s. . . . . . . . . . . . . . . . 4.17 Frequency evolution of pressure fluctuations with Incoherent Spouting regime. Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Incoherent spouting and slugging observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. . . . . . . . . . . 4.19 Fluctuations of pressure drop for Slugging regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 5 mm width; Glass beads of 1.33 mm; Hs = 600 mm; Ug = 0.912 m/s. . . . . . . . . . . . . . . . . . . . 4.20 Fluctuations of pressure drop for Jet in Fluidized Bed regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 600 mm; Ug = 0.853 m/s. . . . . . . . . . . . . . . . 4.21 Jet and spouting fluidized bed observed in Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads. . . 4.22 Irrecoverable asymmetric conditions observed in Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads. 4.23 Irrecoverable asymmetric fluctuation in Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 450 mm; Ug = 0.828 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Irrecoverable asymmetric fluctuation in Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 450 mm; Ug = 0.789 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.25 Regime map for decreasing gas velocity. Column: 300 × 100 mm, with diverging base, type a slot, 4 mm width, 1.33 mm glass beads. . . . . . . 4.26 Regime map for increasing gas velocity. Column: 300 × 100 mm, with diverging base, type a slot, 4 mm width, 1.33 mm glass beads. . . . . . . 4.27 Comparison of the distribution of cycle frequency for different flow regimes. 4.28 Comparison of distribution of magnitude of fluctuations for different flow regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1  Drawing of a typical slot. . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  70  71  72  73  73 74  74  75 75 76  77  78 80 80 81 83 86  List of Figures 5.2  5.3  5.4  5.5  5.6  5.7  5.8  5.9  5.10  5.11  5.12  5.13  5.14  Comparison of relative magnitudes of pressure fluctuations for slots of different width. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot length: standard length of 100 mm, diverging base. . . . . . Comparison of pressure drops across slots of different width. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot length: standard length of 100 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . Comparison of minimum spouting velocities for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of maximum pressure drops for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of magnitudes of pressure fluctuation for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . Comparison of pressure drops across slot for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of minimum spouting velocities for slots of different length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. Horizontal lines are corresponding results for circular orifice. . . . . . . . . . . . . . . . . . . Comparison of maximum pressure drops for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. Horizontal line are corresponding results for circular orifice. . . . . . . . . . . . . . . . . . . . . . . . . Comparison of spouting pressure drops for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. Horizontal line are corresponding results for circular orifice. . . . . . . . . . . . . . . . . . . . . . . . . Comparison of magnitudes of pressure fluctuations for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base. . . Comparison of pressure drops across slot for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base. . . . . . . . . . Comparison of maximum pressure drops for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of minimum spouting velocities for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  87  87  89  89  90  90  93  94  95  95  96  97  98  List of Figures 5.15 Comparison of magnitudes of pressure fluctuation for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . 5.16 Comparison of pressure drops across slot for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 200 mm, slot width: 4 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Comparison of minimum spouting velocities for converging and diverging slots. Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. . 5.18 Comparison on magnitudes of pressure fluctuation for converging and diverging slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Comparison of pressure drops across slot for converging and diverging slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 CFD predicted distribution of gas velocity in the direction normal to the slot. Average outlet velocity: 0.66 m/s. Comparison of two straight slot of different depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 Distribution of gas velocity along the centreline of the slot. Average outlet velocity: 0.66 m/s. Comparison of two straight slot of different depth. . . 5.22 Pressure drop across the slot along the centreline of the slot. Average outlet velocity: 0.66 m/s. Comparison of two straight slots of different depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23 Distribution of gas velocity in the direction normal to the slot for three different geometries. Average outlet velocity: 0.66 m/s. . . . . . . . . . . 5.24 Distribution of gas velocity along the centreline of the slot for three different geometries. Average outlet velocity: 0.66 m/s. . . . . . . . . . . . 6.1 6.2  98  99 100  101  101  103 104  105 105 106  Top view of column showing where the probes were inserted. . . . . . . . 109 Sample voidage signal; X is the distance from the axis, Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 300 mm, Z = 50 mm, Ug = 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Power spectrum of sample voidage signal in Figure 6.2; X is the distance from the axis, Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 300 mm, Z = 50 mm, Ug = 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . 112 6.4 Flow structure. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 300 mm, Ug = 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . 112 6.5 Side view of spout shape for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Slot: 10 × 40 mm, flat base, Ug = 0.7 m/s.113 6.6 Front view of spout shape for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Slot: 10 × 40 mm, flat base, Ug = 0.7 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.7 Side view of spout shapes for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 xiv  List of Figures 6.8  6.9  6.10 6.11  6.12  6.13  6.14 6.15  6.16  6.17  6.18  6.19  6.20  6.21  6.22  Front view of spout shapes for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of spout shapes in two orthogonal directions. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot width: 10 mm, Slot length: 40 mm, flat base . . . . . . . . . . . . . . . . . . . . . . . . Structure of spout and annulus showing the spout expansion angle. . . . Comparison between the experimental spout shapes and predictions from two-dimensional Krzywanski et al. (1989) model. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundaries of dead zone for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. Dashed lines show 60◦ included angle. . . . . . . . . . . . . . . Boundaries of dead zone for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Slot: 10 × 40 mm, flat base, Ug : 0.7m/s. Dashed lines show 60◦ included angle. . . . . . . . . . . . . . . . . . . . . Axial pressure distributions at the wall. Column C: 300 × 100 mm, 1.33 mm glass beads, with flat base; Line is Equation 6.8. . . . . . . . . . . . Axial pressure distributions along column axis for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Ug : 0.7 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . Axial pressure distributions along column axis for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . Pressure profiles in direction normal to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Ug : 0.6 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure profiles in direction parallel to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Ug : 0.6 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure profiles in direction normal to slot for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Z: 100 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . Pressure profiles in direction parallel to slot for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Z: 100 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . Voidage distributions along column axis at different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Ug : 0.7 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . Voidage distributions along column axis at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . .  xv  114  115 117  119  120  120 122  123  124  125  126  126  127  127  128  List of Figures 6.23 Voidage profiles in direction parallel to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.24 Voidage profiles in direction normal to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.25 Voidage profiles in direction parallel to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . 6.26 Voidage profiles in direction normal to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . 6.27 Axial particle velocity profiles at different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Ug : 0.7 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.28 Axial particle velocity profiles at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.29 Comparison of prediction with and without drag force in deceleration zone from Thorley et al. (1955) model and experimental axial velocity profile. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base, B = 0.72. . . . . . . . . . . . . . . . . 6.30 Lateral profiles of vertical particle velocity in direction normal to slot at different levels. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . 6.31 Lateral profiles of vertical particle velocity in direction parallel to slot at different levels. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base. . . . . . . . . . . . . . . 6.32 Lateral profiles of vertical particle velocity in direction normal to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base. . . . 6.33 Lateral profiles of vertical particle velocity in direction parallel to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base. . . . 6.34 Axial pressure profiles along the column axis for three slots of equal area, but different length/width ratio. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . 6.35 Shapes of spout in direction parallel to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.36 Shapes of spout in direction normal to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xvi  128  129  129  130  130  131  134  136  137  138  139  140  141  141  List of Figures 6.37 Boundaries of dead zone for different slots for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. Dashed lines show 60◦ included angle. . . . . . . . . . . . 6.38 Axial velocity profiles along the column axis for different slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.39 Axial voidage profiles along the column axis for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.40 Velocity distributions in direction parallel to slot for different slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.41 Velocity distributions in direction normal to the slot for different slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . 6.42 Voidage profiles in direction parallel to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.43 Voidage profiles in direction normal to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.44 Particle circulation flux profiles in direction parallel to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . 6.45 Particle circulation flux profiles in direction normal to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base. . . . . . . . . . . . . . . . . . . 7.1 7.2  7.3  7.4 7.5 7.6  Top views showing slot positions of the experimental columns; all dimensions are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure evolution with increasing gas velocity, Column C: 300 × 100 mm, Two parallel: 4 × 30 mm slots as shown in Figure 7.1(a), diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure evolution with decreasing gas velocity, Column C: 300 × 100 mm, Two parallel: 4 × 30 mm slots as shown in Figure 7.1(a), diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of minimum spouting velocities for single and parallel 4 × 30 mm slots, 1.33 mm glass beads, diverging base. . . . . . . . . . . . . . . Comparison of minimum spouting velocity for columns with single and aligned 4 × 30 mm slots with 1.33 mm glass beads, diverging base. . . . Comparison of average minimum spouting velocities for columns with single and multiple 4 × 30 mm slots with 1.33 mm glass beads, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xvii  142  142  143  143  144  144  145  145  146 148  150  151 152 152  153  List of Figures 7.7  7.8  7.9  7.10  7.11  7.12  7.13 7.14  7.15 7.16 7.17  7.18 7.19  7.20 7.21  Comparison of spouting pressure drops for columns with single and multiple 4 × 30 mm slots, 1.33 mm glass beads, Hs = 200 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure fluctuations for swaying flow regime, Column C: 300 × 100 mm, two parallel 4 × 30 mm slots as shown in Figure 7.1 (a), 1.33 mm glass beads, Hs =300 mm, Ug =0.64 m/s, diverging base. . . . . . . . . . . . . . Pressure evolution showing where merging of fountain begins and ends, Column C: 300 × 100 mm, Two parallel 4 × 30 mm as shown in Figure 7.1(a), 1.33 mm glass beads, Hs =200 mm, diverging base. . . . . . . . . . Pressure fluctuations in Merging of fountain flow regime, Column C: 300 × 100 mm, Two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads, Hs =200 mm, diverging base. . . . . . . . . . . . . . . . Effect of distance between slots on Ums , Column C: 300 × 100 mm, two aligned 4 × 30 mm slots as shown in Figure 7.1(e), 1.33 mm glass beads, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of distance between slots on pressure fluctuation, Column C: 300 × 100 mm, two aligned 4 × 30 mm slots as shown in Figure 7.1(e), 1.33 mm glass beads, diverging base. . . . . . . . . . . . . . . . . . . . . . . . . . Effect of diverging base on Ums , Column C: 300 × 100 mm, two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads. . . . . Effect of diverging base on difference between values of Ums , Column C: 300 × 100 mm, Two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position of vertical partition in column C; all dimensions are in mm. . . Position of suspended vertical partition in column C; all dimensions are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of vertical partitions for parallel slots, Column C: 300 × 100 mm, Two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads, Hs =200 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . Position of suspended vertical partition in column B; all dimensions are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of vertical partition for parallel slots, Column B: 150 × 200 mm, Two aligned 4 × 30 mm slots as shown in Figure 7.1(c), 1.33 mm glass beads, Hs =300 mm, diverging base. . . . . . . . . . . . . . . . . . . . . . Diagram of column of combined chambers. . . . . . . . . . . . . . . . . . Diagram of column with multiple chambers for continuous processing of particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  B.1 Details of tip of the optical probe for velocity measurement. . . . . . . . B.2 dependence of calibrated effective distance of probe with distance from probe tip to surface of particles. Particle velocity: 0.24 m/s. . . . . . . . B.3 Dependence of calibrated effective distance of probe with particle velocities. Distance from probe tip to particles: 1 mm. . . . . . . . . . . . . . B.4 Typical two channel data series. . . . . . . . . . . . . . . . . . . . . . . . xviii  154  156  157  157  160  161 161  162 163 164  165 166  167 174 175 211 212 212 213  List of Figures B.5 Lag and corresponding correlation function of data in Figure B.4. . . . . 215 B.6 Diagram showing positions of orifice and the pressure measurement. . . . 216 B.7 Orifice coefficient as a function Reynolds number. . . . . . . . . . . . . . 219  xix  Nomenclature a, b, c, d, e  Constants (-)  A0 , B0 , b0 , n0  Intermediate parameters in calculating orifice discharge coefficient (-)  A2D  Ratio of energy needed to yield spout to that needed to sustain the spout, ReT Remf (dp /λ)/Ar (-)  Aa  Cross-sectional area of the annulus (mm2 )  Ac  Cross-sectional area of the column (mm2 )  Ai  Cross-sectional area of slot (mm2 )  Ar  Archimedes number, ρg (ρs − ρg )gd3p /µ2 (-)  As  Cross-sectional area of the spout (mm2 )  B  Intermediate parameter (-)  C  Constant (-)  C1  Constant (-)  C2  Constant (-)  CD  Drag coefficient (-)  CDs  Drag coefficient for single particle (-)  D0  Diameter of orifice (m)  D1  Diameter of pipe (m) xx  List of Figures Dc  Diameter of column (mm)  Dc,a  Equivalent area diameter of column (mm)  Dc,h  Hydraulic diameter of column (mm)  Di  Diameter of slot (mm)  di  Particle diameter of specific particle (mm)  Di,a  Equivalent area diameter of slot (mm)  Di,h  Hydraulic diameter of slot (mm)  dp  Average particle diameter (mm)  dp,no  Linear mean particle diameter (mm)  dp,wt  Weight mean particle diameter (mm)  dv  Diameter of sphere of same volume as particle (mm)  Fpeak  Peak frequency of pressure drop fluctuations (Hz)  Fr  Froude number (-)  g  Gravitational acceleration (m/s2 )  Hc  Column height, from the slot to the top (mm)  Hf  Fountain height (mm)  Hmax  Maximum spoutable bed height (mm)  Hs  Static bed height (mm)  K  Portion of momentum lost in the total momentum of the gas at the slot (-) xxi  List of Figures K0  Gas discharge coefficient of the orifice (-)  K1  Volume flow coefficient of the orifice (m3 /s)  Ke  Orifice discharge coefficient when Re = Ree (-)  M  Molar weight of gas (kg/mol)  m  Mass of gas (kg)  Mgain  Momentum got by the particles at the slot (kg·m/s)  Mlost  Momentum lost at the inlet by the fluid (kg·m/s)  N  Number of data points used in correlation (-)  n  Number of particles that can be spouted per unit time (-)  nz  Number of particles at vertical location Z (-)  ∆P  Pressure drop of column (kP a)  ∆Pmax  Maximum pressure drop of column (kP a)  ∆Pmax,corr  Measured minimum spouting velocity (kP a)  ∆Pmax,d  Maximum pressure drop of column in decreasing gas velocity process (kP a)  ∆Pmf,max  Fluidization pressure drop at Hmax (kP a)  ∆Ps  Spouting pressure drop (kP a)  ∆Ps,max  Maximum spouting pressure drop (kP a)  ∆Pz  Gauge pressure at vertical position Z (kP a)  xxii  List of Figures Q  Gas volume flow at 1 atm (m3 /s)  Q0  Gas volume flow through the orifice (m3 /s)  R  Universal gas constant (kg·m2 /s2 ·mol·K)  r  Radius of measurement position (mm)  R2  Goodness of fitting (-)  Re  Reynolds number (-)  Ree  Special Reynolds number,  Remf  Reynolds number of particle at Umf , dp Umf ρg /µ (-)  ReT  Reynolds number of particle at UT , dp UT ρg /µ (-)  ri  Radius of gas inlet nozzle (mm)  rs  Radius of spout (mm)  t  Time (s)  U0  Fluid velocity through orifice (m/s)  Ua  Superficial gas velocity in the annulus (m/s)  Ug  Superficial gas velocity in column (m/s)  Ui,ms  Inlet gas velocity at minimum spouting velocity (m/s)  Umf  Minimum fluidization velocity (m/s)  Ums  Minimum spouting velocity (m/s)  Ums,cor  Minimum spouting velocity by empirical correlations (m/s)  106 D0 15  xxiii  (-)  List of Figures Ums,exp  Measured minimum spouting velocity (m/s)  Ums,i  Minimum spouting velocity in increasing gas velocity process (m/s)  Ums,max  Highest value of Ums in multiple spouted bed column (m/s)  Ums,min  Smallest value of Ums in multiple spouted bed column (m/s)  Us  Superficial gas velocity in the spout (m/s)  us  Gas velocity in the spout (m/s)  UT  Terminal velocity of particle (m/s)  V  Volume of gas (m3 )  v  Vertical component of particle velocity in the spout (m/s)  v0  Vertical particle velocity at spout centre r (m/s)  vg  Gas velocity at the outlet of slot (m/s)  vi  Velocity of particles at the slot (m/s)  vr  Vertical particle velocity at radial position r (m/s)  vx  Vertical particle velocity at position X in the normal direction of slot (m/s)  vy  Vertical particle velocity at position Y in the direction along the slot (m/s)  Ws  Particle circulation rate (kg/s)  X  Distance from measurement point to the central line of column normal to the slot (mm) xxiv  List of Figures X0  X coordinate of spout centre (mm)  Y  Distance from measurement point to the central line of column parallel to the slot (mm)  Y0  Y coordinate of spout centre (mm)  z, Z  Vertical coordinate over the bottom (mm)  α  Column width (mm)  β  Column thickness (mm)  δa  Equi-area radius of spout (mm)  δh  Half-the hydraulic diameter of spout (mm)  δs  Half-width of spout (mm)  δx  Half-width of spout in direction normal to slot (mm)  δy  Half-width of spout along the slot direction (mm)  g  Voidage in the spout (-)  g,0  Voidage in static as-poured bed (-)  s,0  Solid fraction in static as-poured bed (-)  η  Length of slot (mm)  γe  Spout expansion angle (degrees)  κ  Depth of slot (mm)  λ  Slot width (mm)  xxv  List of Figures λe  Lagrangian multiplier (-)  µ  Dynamic viscosity of gas (kg/m·s)  ω  Column aspect ratio, α/β (-)  φi,no  Percentage of particles of diameter di (-)  ρg  Density of gas (kg/m3 )  ρs  Density of solid (kg/m3 )  θ  Included angle of the diverging base (◦)  ϕs  Particle sphericity (-)  xxvi  Acknowledgements I would like to express my sincere gratitude to Dr. C. J. Lim and Dr. J. R. Grace for their financial assistance, experienced and distinguished supervision and patience which allowed me to complete this work. I am also thankful to Dr. Norman Epstein for providing valuable literature, to Dr. Bushe for his important suggestions and to Dr. Zhiguo Wang and Dr. Zhong Zheng for help and suggestions in the experiment. Thanks are also due to the staff of the Department of Chemical Engineering Workshop and Stores for their invaluable assistance. Finally, I am grateful to my wife Xiumin and my parents, for their encouragement, support and understanding.  xxvii  Chapter 1 Introduction 1.1  Introduction  Since their development by Mathur and Gishler (1955), cylindrical-conical spouted beds have been used in a number of applications (Epstein and Grace, 1997; Mathur and Epstein, 1974). Conventional spouted beds of laboratory scale have proved to be quite effective for gas/particle contacting. However, the spouted bed technique has seldom been applied in large-scale industrial processes due to scale-up difficulties (Dogan et al., 2000), such as the inability to achieve good quality spouting in large vessels, and difficulties in predicting the performance of spouted beds larger than about 0.3 m in diameter. To address the scale-up issue, a modification of conventional axisymmetric spouted beds, denoted “Two-Dimensional Spouted Bed”, was proposed by Mujumdar (1984). This configuration was renamed “Slot-Rectangular Spouted Bed” (SRSB) by Dogan et al. (2000) and Freitas et al. (2000) because of significant three-dimensional effects as the thickness of the bed increases. A schematic of this geometry is shown in Figure 1.1, where the column is constructed with four vertical plane walls and a diverging base on two opposite sides. The spouting air enters through a horizontal rectangular slot at the bottom of two diverging planar surfaces. In this configuration, the column has a rectangular cross-section. The slot-rectangular spouted bed has been claimed (Mujumdar, 1984) to have greater flexibility, ease of design and simplicity of construction than conventional axisymmetric spouted beds. Another advantage attributed to this configuration is the possibility of achieving different flow regimes by changing only the inlet nozzle dimensions. Scaling-up  1  Chapter 1. Introduction  Figure 1.1: Diagram of slot-rectangular spouted beds (Dogan et al., 2000)  of the slot-rectangular columns was supposed to be achievable by simply moving the facing vertical walls further apart, or combining small units by sharing either the front and back faces or the sidewalls. Slot-rectangular spouted beds need to be characterized in terms of properties as for conventional spouted beds, key properties include minimum spouting velocity (Ums ); maximum pressure drop (∆Pmax ); maximum spoutable static bed height (Hmax ); operating pressure drop (∆Ps ); particle velocity (Vp ) distribution; solid volume fraction (s ) profiles; net solids mass flux (Ws ); height of fountain (Hf ); gas dispersion, gas2  Chapter 1. Introduction solid contact efficiency, etc. The size, shape and position of the spout differ from those of conventional spouted beds. Published papers on slot-rectangular spouted beds and two-dimensional spouted beds are listed and summarized in Table A.1, Appendix A.  1.2 1.2.1  Spouted bed technique Conventional axisymmetric spouted beds  Conventional axisymmetric spouted beds include the conical (San Jose et al., 1995; Wang, 2006), cylindrical and conical-cylindrical spouted beds (Mathur and Gishler, 1955). Conical-cylindrical spouted beds are contained in cylindrical vessels with a conical bottom. The particles are normally relatively coarse, e.g. 1-5 mm in mean diameter. Fluid flows into the column vertically upwards through a nozzle located at the centre of the bottom. Particles are entrained into the flow resulting in a jet of gas and solids. When the jet velocity is sufficient to penetrate the entire bed of particles, a fountain appears at the surface of the bed. The flow pattern at this stage is shown in Figure 1.2. The flow can be characterized into three zones: i) the “spout” along the axis of column, with a dilute phase upflow of fluid and solids at high velocity; ii) the “annulus” surrounding the spout, with particles moving downwards slowly as a moving packed bed and fluid percolating countercurrently upwards and outwards; iii) the “fountain” over the surface of the solids. Particles circulate in the bed by being carried rapidly upwards through the spout and fountain, falling back into the surface of the annulus, descending in moving packed bed flow and entering the spout again through the spout/annulus interface. Originally designed to overcome the difficulty in fluidizing wheat, the spouted beds were commonly used to contact coarse particles, dp ≥ 1 mm, with gas. These particles can normally be categorized as class D or B according to the fluidization classification of Geldart (1973). Fluidization of these particles suffers from the bypassing of gas in  3  Chapter 1. Introduction the form of big bubbles. The spouted bed technique “appears to achieve the same purpose for coarse particles as fluidization does for fine materials”, as noted by Mathur and Gishler (1955). For the same application, spouted beds can process solid particles of larger density and larger diameter. Conventional spouted beds are frequently applied to drying (Cowan et al., 1957; Passos et al., 1997), granulation (Uemaki and Mathur, 1976), coating (Oliveira et al., 2005; Publio and Oliveira, 2004), heterogeneous reactions (Kechagiopoulos et al., 2007; San Jose et al., 2006; Olazar et al., 2005), charcoal gasification (Abudul Salam and Bhattacharya, 2005), and heating or cooling of particles (Mujumdar, 1984). Scale-up of spouted beds is of great importance. However, most spouted bed equations are based on small-scale columns (Dc < 0.3 m), and they often do not work for larger vessels. The difficulty in scaling-up conventional spouted beds has been noted by He (1995) and Dogan et al. (2000). Because of the lack of data for conventional spouted beds in large vessels, no fully reliable models or equations have been developed for large columns. The difficulty of scale-up of conventional spouted beds also lies in the fact that inter-particle forces and particle-particle forces are omitted in the common scaling laws for fluidized beds (Glicksman, 1984, 1988; Glicksman et al., 1993). Because of the scale-up difficulty of conventional spouted beds, investigation and application of multiple spouted beds have been reported, e.g. by Murthy and Singh (1994); Saidutta and Murthy (2000); Perterson (1966); Mathur and Epstein (1974); Albina (2006); Gong et al. (2006); Xu et al. (2000); Foong et al. (1975). These multiple spouted beds normally consist of rectangular/square chambers and circular gas inlets. Multiple chamber stability issues have been noticed and the gas and solids exchange in both the upper and lower part is considered to play an important role in the stability of multiple spouted beds. Baffles (e.g. draft plates) (Perterson, 1966) and conical bases (Albina, 2006; Murthy and Singh, 1994; Saidutta and Murthy, 2000) have been utilized  4  Chapter 1. Introduction in the bed to improve the spouting stability. Multiple spouted beds are more suitable for processes with short gas-solid contact times, such as drying of heat-sensitive particles or coating of thin layers.  1.2.2  Slot-rectangular spouted beds  Slot-rectangular spouted beds were first investigated by Soviet Union researchers. Mujumdar (1984) noticed its potential ability to avoid the difficulty of scaling-up conventional spouted beds and suggested multiple column spouting. The slot-rectangular spouted bed was then investigated for hydrodynamics, stability and scale-up (Dogan et al., 2000, 2004; Freitas et al., 2000, 004a,b; Kalwar et al., 1989, 1992, 1993; Passos et al., 1991, 1993, 1994). However, reported work on this bed is still limited. Despite the different configurations for conventional and slot-rectangular spouted beds, the flow patterns of particles and gas are quite similar. Again, a spout, annulus and fountain can be identified. Slot-rectangular spouted beds can therefore be utilized in similar applications as conventional spouted beds. However, because of insufficient investigation on the hydrodynamics, stability and scale-up, reported work on the application of slot-rectangular spouted beds is limited. Applications can mainly be found for coating of particles (Donida and Rocha, 2002; Wiriyaumpaiwong et al., 2004, 2003; Taranto et al., 1997). Most tests have been in 2-dimensional columns, (i.e. with small thickness-to-width ratio < 1/6 ), with square gas inlets, with the slot width and length the same as the column thickness, and with draft plates (Wiriyaumpaiwong et al., 2004, 2003; Taranto et al., 1997). The studies of draft plates are covered in the next section. Application of columns with larger thickness-to-width level and with gas inlet of small width-to-column-thickness has not been reported. This may be due to instabilities in columns with increased thickness-to-width level, (e.g. up to 2/3) (Freitas et al., 2000). To overcome this instability, draft plates have been applied for slot-rectangular spouted  5  Chapter 1. Introduction beds corresponding to the draft tubes for conventional conical-cylindrical spouted beds.  1.2.3  Slot-rectangular spouted beds with draft plates  Draft plates have been used to improve the stability in slot-rectangular spouted beds, as shown in Figure 1.2.3. Kalwar et al. (1992, 1993) investigated the solid circulation with different separation distances, plate heights and angles to the vertical, and correlated circulation rates. Luo et al. (2004) performed some pressure measurements with draft plates. Their schematic configuration, shown in Figure 1.2.3, features two parallel vertical plates, one on either side of the air entry slot. The spout was constrained to prevent asymmetry and reduce fluctuations. Spouting with this geometry was found to be more stable than in un-baffled columns of otherwise similar geometry, and the pressure drop was greatly decreased. This geometry adds other geometric parameters such as the baffle separation distance, the length of the draft plates, and their distance above the air entry slot.  1.3  Flow regimes for slot-rectangular spouted bed  Dogan et al. (2000) determined flow regimes in a thin rectangular spouted bed. The flow patterns were characterized by 8 different regimes: Fixed Bed, Internal Jet, Jetin-Fluidized-bed, Spouting, Dilute-Phase Spouting, Incoherent Spouting, Transitional regime and Slugging. Freitas et al. (2000) reported flow regimes in slot-rectangular spouted beds of thickness from 30 mm to 100 mm. With increasing column thickness, some significant three-dimensional effects appeared, such as formation of multiple spouts, defined as a distinct flow regime: Multiple Spouts. Three spout termination mechanisms for axisymmetric beds were identified by Mathur and Epstein (1974): (i) fluidization of the annulus top, (ii) choking of the spout, (iii)  6  Chapter 1. Introduction  Figure 1.2: Flow pattern in a conventional conical-cylindrical spouted bed (He, 1995)  7  Chapter 1. Introduction  Figure 1.3: Schematic of slot-rectangular spouted bed column with draft plates (Kalwar et al., 1993). The dimension shown in the diagram as “BED LENGTH” is called “column thickness” in the current work.  8  Chapter 1. Introduction growth of instabilities. Passos et al. (1994) found that the first two of these also apply to slot-rectangular spouted beds. A dimensionless parameter A2D , defined as the ratio of the energy required to penetrate the particle layer to the energy required to sustain the spout relative to the minimum frictional energy loss across the spout at Hs /Hmax = 1, was used to correlate the spout termination mechanisms. Pressure fluctuations were adopted by Freitas et al. (004a) to identify the flow regimes of slot-rectangular spouted beds by statistical analysis in the time domain, spectral analysis in the frequency or Fourier domain, and chaos analysis in the state space of the system. The pressure fluctuations in each flow regime were identified and shown to differ.  1.4  Hydrodynamic studies  Most reported work on slot-rectangular spouted beds have been related to applications for drying or tablet coating. There is limited published information on the hydrodynamics of slot-rectangular spouted beds.  1.4.1  Global properties  The minimum spouting velocity, Ums , global pressure drop, ∆P , and maximum spoutable bed height, Hmax , of slot-rectangular spouted beds have been investigated by several researchers. Table A.2, Appendix A, lists empirical correlations of these properties.  1.4.2  Local flow structure  Voidage profiles in slot-rectangular spouted beds were reported by Freitas et al. (004b). The effects of air flow rate, Ug , slot width, λ, and column thickness, β, were investigated. The results showed significant three-dimensional behaviour with increasing column thick-  9  Chapter 1. Introduction ness. The local particle velocity distribution has not been previously reported.  1.5  Scale-up for slot-rectangular spouted beds  Beginning with the scaling laws for fluidized beds proposed by Glicksman (1984), Passos et al. (1993) provided some criteria to achieve stable flow and scale-up of slot-rectangular spouted beds. The maximum nozzle width and spoutable bed height were determined as functions of particle properties and column dimensions. Costa and Taranto (2003) used three columns of different widths and thicknesses to investigate scale-up, aided by the scale-up criteria of Glicksman (1984). Several non-dimensional parameters were combined to derive scale-up criteria. Correlations of Ums , ∆Pmax and Hmax were also provided. As already noted, the slot-rectangular spouted bed was originally proposed (Mujumdar, 1984) to overcome the difficulty in scaling-up conventional spouted beds by simple modification to the dimensions of the column. Multiple spouts can also increase the capacity of slot-rectangular spouted beds. Mujumdar (1984) suggested a bank of twodimensional chambers with common walls between adjacent beds. Kalwar et al. (1993) advocated scaling-up by adding units in any orthogonal direction such that the laboratory scale apparatus can be considered as a single unit in a complex of similar units. The shared wall between the adjacent units can be removed. In this case, the thickness of the column can be simply increased. A schematic of these concepts is shown in Figure 1.6. Huang and Chyang (1992) investigated the behaviour of multiple slots in a thin 2D column with three slots. The results showed that the stability of a bed with multiple spouts was affected by bed height, superficial gas velocity, particle size, and distributor design. The latter governed the interaction between adjacent spouts and whether or not all orifices were active. When Freitas et al. (2000) tested the effect of increasing the thickness of column, significant three-dimensional effects and multiple spouts appeared. 10  Chapter 1. Introduction  Figure 1.4: Scale-up methods according to Kalwar et al. (1993). ”‘LENGTH”’ is named as ”‘thickness”’ in the current work.  1.6  Stability of slot-rectangular spouted beds  The stability of slot-rectangular spouted beds has been investigated when seeking promising operating conditions. The relationships between the column dimensions, slot width and particle diameter to provide stable spouting have been explored (Passos et al., 1993; Dogan et al., 2000; Kalwar et al., 1989). These criteria are listed in Table A.2. They indicate that the configuration of the column and the slot play important roles in determining the stability of slot-rectangular spouted beds.  1.7  Mechanistic models  Few published papers have dealt with modeling of slot-rectangular spouted beds. Table A.4 lists some papers which focus on the modeling of spouted beds. Krzywanski et al. (1989) developed a fundamental model to predict the spout shape in both two11  Chapter 1. Introduction dimensional and cylindrical spouted beds based on the least-action principle. In this model, three-dimensional effects were not taken into account. In a spouted bed, the gas pressure at the inlet has to provide the vertical pressure drop and withstand the lateral pressures of the solid particles to make a spout. Kalwar et al. (1986) investigated the static particle pressure of grains on the slots. Their results showed that the static particle pressure force was influenced by the aspect ratio, solid density, diverging angle and static bed height. These four parameters were optimized. The ratio of the static lateral particle pressure force to static vertical particle pressure force was found to be independent on the depth of fill for all points on the horizontal cross-section.  1.8  Summary and research objectives  The research status on slot-rectangular spouted beds can be stated as follows: • Few published papers have dealt with the hydrodynamic properties and performance of slot-rectangular spouted beds. Most have instead focused on applications and global flow properties. Correlations for Ums , ∆Pmax and Hmax of the slotrectangular spouted beds have been provided by several researchers, but none of them has been widely accepted, mainly due to the differences in the slot design adopted by the researchers and the large number of variables. The configuration of the slot could have a major effect on the hydrodynamics of slot-rectangular spouted beds. • The local flow structure of slot-rectangular spouted beds has not received enough investigation. Profiles of particle velocity, voidage and pressure distribution are needed to understand the performance of slot-rectangular spouted beds. • Flow regimes in slot-rectangular columns have been investigated and identified. 12  Chapter 1. Introduction Three-dimensional effects have been found to be significant. • The basic configuration of slot-rectangular spouted beds has commonly been based on four vertical plates and a diverging base. In previous studies, the columns always have thickness < width and a slot length equal to the column thickness. An included angle of 60◦ has been normally adopted. Slot-rectangular spouted beds with a flat bottom or with width-to-thickness ratio less than 1 have not been investigated. Draft plates have been employed to modify slot-rectangular spouted beds. • While it was originally intended that scaling-up could be achieved by simply changing the thickness of column, there are three-dimensional effects which need to be taken into account. • Because of the complexity of gas/solid flow, there have been few attempts to model slot-rectangular spouted beds. None of the models apply to thick columns subject to three-dimensional effects. The main objectives of the present work are therefore to: • Study the hydrodynamics in columns of larger scale size than the prototype tested by Dogan et al. (2000), with increased width, thickness and slot dimensions. • Investigate the effect of diverging bases, slot configurations and particle properties. • Measure global flow properties as well as local flow structure of slot-rectangular spouted beds, including minimum spouting velocity, maximum pressure drop, spouting pressure drop, local profiles of particle velocity, voidage and pressure, and particle circulation rate. • Improve the stability of slot-rectangular spouted beds by optimizing the slot configuration. 13  Chapter 1. Introduction • Scale-up slot-rectangular spouted beds based on the methods proposed by Mujumdar (1984) and others, using multiple slots and multiple chambers.  14  Chapter 2 Experimental Set-Up 2.1  Experimental units  A generic schematic diagram of the basic slot-rectangular spouted bed experimental units used in most elements of this project is shown in Figure 2.1. Compressed air, the spouting gas, after being measured by an orifice flow meter, and adjusted by the main valve, entered the column at the bottom. A “windbox” chamber, with the same crosssection as the column and a height of 250 mm at the bottom of the column, provides a calming area for the air. Then the gas flows out through the slot into the bed of particles above. The pressure upstream and the pressure drop across the orifice were measured in order to determine the gas flow rate. The pressure in the plenum chamber was also measured to ensure it was below a safe limit of 40 kPa. An electromagnetic pressure relief valve was installed at the bypass to open if the pressure in the windbox should exceed this limit. Two identical pressure taps were connected just above the two ends of slot to measure the pressure drop cross the whole bed in all experiments.  2.2  Slot-rectangular columns  To investigate the performance of slot-rectangular spouted beds of different dimensions, four different plexiglass columns, labeled A to D, were used in this project. The column utilized by Dogan et al. (2000) and Freitas et al. (2000), column A, was also employed in this project as the prototype for the scale-up investigation, which is described in Chapter  15  Chapter 2. Experimental Set-Up  Figure 2.1: Flow chart of the spouted bed system 1: Rectangular column, 2: Diverging lower section, 3: Slot (Gas inlet), 4: Safety relief valve, 5: Windbox chamber, 6: Orifice flow meter, 7: Main valve, ∆P : Differential pressure transducer, P : Absolute pressure transducer. For dimensions, see Table 2.1.  16  Chapter 2. Experimental Set-Up 7. The other three columns (B, C and D) were manufactured with geometric similarity to the column A. The widths, thicknesses and heights of all four columns are summarized in Table 2.1. The similarity in configuration and differences in dimensions allows the investigation on the effects of column dimensions on the flow properties. Table 2.1: Dimensions of rectangular columns  Column  A  B  C  D  Width, α (mm)  150  150  300  300  Thickness, β (mm)  100  200  100  200  Height, Hc (mm)  800  1000  1000  1000  A schematic drawing of Column C, which is typical of all four columns except for the dimensions, is shown in Figure 2.2. This column basically consists of four parts: a rectangular column of 300 × 100 mm cross-section, a windbox chamber of height 250 mm and the same cross-section as the column, a removable diverging lower section, and a slot mounted into a flange right under the diverging lower section. Additional columns were designed to meet the needs of scale-up investigation with similar geometries and different dimensions. A schematic top view of the columns is shown in Figure 2.3, where all the slots have the “Standard” length, i.e. equal to the thickness of the column. With Column A served as the prototype, other columns were designed to test the hypothesis of Mujumdar (1984) and Kalwar et al. (1993). Of these columns, Column B, with the same width as column A and doubled thickness, was used to investigate effect of increased thickness. Effect of multiple slots was tested by Columns B and C with two slots mounted. In this case, Columns B and C can each be considered as the combination of two columns with the dimensions of Column A. The arrangement of multiple slots in these columns is presented in Chapter 7.  17  Figure 2.2: assembly.  Schematic drawing of column C (a) Front view, (b) Side view, (c) Top view, (d) Detail of slot at bottom  Chapter 2. Experimental Set-Up  18  Chapter 2. Experimental Set-Up  Figure 2.3: Top views showing slot positions of all four experimental columns.  To achieve basic stability and keep consistency with the geometry of the prototype, diverging bases were constructed for columns B, C and D with the included diverging angle (θ) of 60◦ in each case, this being an angle commonly used in spouted bed studies. A diverging base with a pair of parallel slots to provide two spouts was also fabricated for Columns C. The front view of the double-slotted diverging base is shown in Figure 2.4. Both the single and double slotted bases were equipped with a flange of standard size to fit the column and be removable. The bottom of the two plane plates constructing the diverging base do not extend to the edge of the slot. There is a flat bottom of 1/3 the width of column or chamber, which helps avoiding the choking of spout at the bottom. When the diverging plates are removed, each column can contain a flat bottom spouted bed. The columns were designed to allow different slot configurations. When slots were  19  Chapter 2. Experimental Set-Up  Figure 2.4: Diverging base allowing two parallel slots spouting.  mounted on the column, a silica rubber gasket sealed the gap between the slot and the diverging base, as shown in Figure 2.2 (d). A stainless steel screen of mesh size 0.5 mm was attached at the inlet of each slot to prevent particles from dropping into the windbox chamber below. As shown in Chapters 3 and 5, the slot configuration may have a major influence on spout stability for a given column. Slots of different shapes and sizes were tested in this project. Their configurations, in each case symmetrical, appear in Figure 2.5. These slots were made of 1/2” (12.7 mm) plexiglass sheets. Slot of type a consists of two separated identical opposite pieces, allowing the slot width to be adjusted. For slots of type a and b, slot widths from 2 to 8 mm were tested. Slots c and d have different inlet and outlet widths, which will be covered in Chapter 5. These slot configurations are called: extended rectangular slot, standard rectangular slot, short diverging slot and short converging slot respectively.  Figure 2.5: Different type of slots: (a) Extended normal slot, (b) Standard rectangular slot, (c) Short diverging slot, (d) Short converging slot  20  Chapter 2. Experimental Set-Up Slots of length equal to column thickness, i.e. extending from the front to the back wall, have normally been investigated in previously reported work. In the current study, slots of different lengths were also tested. The detailed dimensions and the effects of varying the slot dimensions and geometry are presented in Chapter 5.  2.3  Gas supply  Compressed air was used in this project. Air was provided by a building compressor at 50 psi (i.e. 345 kP a), and this was reduced to 200 kPa by an air regulator. The air flowrates were measured by an orifice meter of 1 inch (25.4 mm) inner diameter and 2 inch (50.8 mm) pipe diameter. The procedure to calculate the discharge coefficient of the orifice is provided in Appendix B.2 followed by the MATLAB code. This procedure was verified by using orifice with a known coefficient. The difference between the result of this procedure and the known coefficient was ≤ 1%. Because the air compressor works in an intermittent mode, the pressure of air measured before the orifice flow meter was not constant, but varied periodically when the blower started and shut down. The period depended on the flow rate, as shown in Figure 2.6(a). This caused the gas velocity to change periodically around the mean velocity, as shown in Figure 2.6(b). The pressure and velocity changed more frequently and more intensely for higher gas flow rates. This fluctuation had some effects on the stability of the spouting, as discussed in Chapter 5.  2.4  Particulate materials  The types of particles tested in this project included glass beads and polystyrene. Properties of these particles are listed in Table 2.2. The particle sphericities were close to 1.0 based on visual appearance. Average particle diameters of glass beads and polystyrene 21  Chapter 2. Experimental Set-Up  Gauge pressure before orifice (kPa)  190  Mean superficial gas velocity 0.527 m/s 0.337 m/s 0.202 m/s  180  170  160  150  140  130 0  100  200  300  400 Time (s)  500  600  700  800  (a) Gauge pressure before orifice 0.6  Superficial gas velocity (m/s)  0.5  0.4  0.3  0.2 Mean superficial gas velocity 0.527 m/s 0.373 m/s 0.202 m/s  0.1  0 0  100  200  300  400 500 Time (s)  600  700  800  (b) Superficial gas velocity in column  Figure 2.6: Gas pressure and velocity in an empty column. Column C: 300 × 100 mm, with diverging base; Slot: type b, 4 mm width.  22  Chapter 2. Experimental Set-Up particles were measured by determining the surface-to-volume diameters of 100 particles:  dp,no  P 3 (d φi,no ) = P 2i (di φi,no )  (2.1)  The minimum fluidization velocity of particles, Umf was measured in a two-dimensional fluidization column of 300 × 12.7 mm cross-section. The solid fraction in a relatively loose packed bed, s,0 , was measured by the water displacement method. A 1-litre beaker of known weight was filled with 500 ml of particles and weighed. The as-poured density of the particles could then be calculated. Water was next added until the beaker was filled with water to 1 litre, filling all the voids, and the beaker and its contents were weighed again. Since the water density is known, the as-poured voidage of the particles can be calculated. Table 2.2: Properties of particles used in this project screen dp range Material  dp (mm) (mm)  ρs (kg/m3 )  ϕs  s0  Umf (m/s)  Glass beads  0.66  0.6 − 0.7  2490  1.0  0.59  0.22  Glass beads  1.33  1.2 − 1.4  2490  1.0  0.58  0.74  Glass beads  2.45  2.2 − 2.8  2490  1.0  0.60  1.37  Polystyrene  3.79  3.2 − 4.2  930  1.0  0.65  1.2  2.5  Pressure measurement  Local pressures and pressure drops were measured by stainless steel tubes of 1/8” O.D. inserted into the column. A screen of 70 micron mesh was used to cover the tip of the tube. These tubes were connected to a set of pressure transducers of PX140 and PX160  23  Chapter 2. Experimental Set-Up series supplied by OMEGA. Voltage signals were collected and converted to digital values by a DAS08 A/D conversion board supplied by Measurement Computing Inc., capable of handling sampling frequencies up to 10 kHz. A program written in Visual Basic controlled the data sampling process. The sampling frequency ranged from 20 to 1000 Hz. 8 7  Pressure (kPa)  6 5 4  Model of Transducer PX164-010D5V PX142-001D5V PX142-002D5V PX142-005D5V  3 2 1 0 0.5  1  1.5  2  2.5  3 3.5 Voltage (v)  4  4.5  5  5.5  6  Figure 2.7: Sample calibration of pressure transducer against water manometer  Pressure transducers were calibrated with a water manometer. Sample calibration lines are shown in Figure 2.7. The time-average voltage signal was linearly related to the pressure reading from the manometer. The slope of this linear relationship did not vary significantly during each measurement interval, but the base point, with no pressurization, had to be checked from time to time. The measurement of pressure had a systematic error that varied with the effective range of the transducer and the sampling mode of the data acquisition board. This small systematic error results from the roundoff error of the A/D conversion process. It had some influence on the results when the measured magnitude of the pressure fluctuations was close to the magnitude of the systematic error. The signal produced by the data  24  ∆Pmax, (Pa)  Chapter 2. Experimental Set-Up 60 50 40 30 20 10 0 -10 -20 -30 0  1  2  3  4  5  40  50  PSD, (kPa/s2)  Time, t, (s) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0  10  20 30 Frequency, f, (Hz)  Figure 2.8: Sample fluctuation of pressure transducer signal when there is no gas flow through the bed: PX142-002D5V, for measuring the pressure drop across the bed. Column C: 300 × 100 mm, with diverging base and type b slot of 4 mm width.  acquisition board fluctuated ∼ 40 Hz with a harmonic frequency of ∼ 20 Hz, as shown in Figure 2.8. The fluctuation of the transducer was 0.015 kPa. This fluctuation affected the pressure fluctuation measurements, as discussed in Chapter 4.  2.6  Voidage and particle velocity measurement  The local particle velocity and local voidage in the spouted beds were measured by using a Particle Velocity Analyzer version 4 (PV4A) manufactured by the Institute of Process Engineering (IPE) of the Chinese Academy of Sciences (CAS). This analyzer uses crosscorrelation to find the lag between the moments when particles pass two measurement levels separated by a known distance. This instrument is also capable of measuring the voidage in the column at the same time by measuring the intensity of the reflected light. This feature made it possible to measure particle velocity and local voidage at the same time.  25  Chapter 2. Experimental Set-Up The velocity measurement system in this project consists of a three-fibre optical fibre probe, a box with a light source and a signal amplifier, a high-speed data acquisition board capable of dual-channel-data-sampling at up to 2 MHz, and a computer with companion software. A glass window of width 4 mm was attached to the tip of the optical probe, as shown in Figure B.1, to eliminate the influence of the blind zone of the probe and to achieve a constant effective length interval (Cui et al., 2001; Liu et al., 2003a,b). Software provided by the manufacturer, IPE, could be used for the data sampling and processing of voidage and particle velocity. However, to improve the data processing, a program was written in Visual C++. This program provided the same features but more convenience in the measurements, e.g. automatic sampling and saving when the operator was not available. The main part of the Visual C++ code is provided in Appendix C.1. The velocity measurement system was calibrated by the same method as was used by Wang (2006). The effective distance interval for the probe was determined using a rotating plate mounted with attached particles. It was found to be 1.057 mm. Comparison between known particle velocities and the particle velocities measured with the calibrated effective distance for particles of different mean diameter is shown in Figure 2.9. Our probe worked well with 1.33 mm particles, as used in most of the work described in this thesis. A more detailed description of the calibration process appears in Appendix B.1. Because of the difficulty in calibrating the probe with glass beads, the voidage in the column was assumed to be linearly proportional to the intensity of reflected light. A linear relationship over the whole range of voidage was confirmed by He (1995) for glass beads of mean diameter of 1.41 mm, close to the mean diameter (1.33 mm) of particles used in most of this work. Data were read in an empty column and in a packed bed after moving the probe slowly. The voidages at these two conditions were assumed to be 1.0 and (1-s0 ). A linear function between the voidage and intensity of reflected light was constructed and intermediate voidages were calculated by the calibration function.  26  Chapter 2. Experimental Set-Up Some factors, such as dirt and engine oil deposits from the compressor, and minor fluctuations in the power supply, may significantly affect measured voidages. So the tip of the probe was cleaned, and the calibration end points were redetermined each time before the probe was inserted into the column. Because the particle velocity in the column can be up to 20 m/s, collisions and friction between the particles and probe could damage the glass window of the probe. So the effective distance had to be checked from time to time.  27  Chapter 2. Experimental Set-Up 9 Glass beads of 1.33 mm diameter  Measured velocity (m/s)  8 7 6 5 4 3 2 1 0 0  1  2  3 4 5 Actual velocity (m/s)  6  7  8  Figure 2.9: Comparison between known particle velocity and particle velocity determined with calibrated effective distance interval for 1.33 mm glass beads, with 90% confidence level.  28  Chapter 3 Maximum Pressure Drop and Minimum Spouting Velocity The maximum pressure drop and the minimum spouting velocity have been widely investigated for spouted beds especially for axisymmetric ones, but also for slot-rectangular ones. In this research, these properties were studied in a factorial manner to understand the influence of column dimensions, slot type, and particle properties.  3.1  Pressure drop across the bed  As shown in Figure 2.1, the pressure drop from the slot outlet to the top of the column was measured by an absolute pressure transducer. To determine the maximum pressure drop and minimum spouting velocity, the pressure drop was measured according to a “Standard” procedure: • For each column with known particle properties and known static bed height, one test run was carried out before measurements began. The bed was spouted first and then shut down slowly to ensure that the particles were loose packed as the initial condition. • Pressures at different points were measured with no gas flow to calibrate the base point of the pressure transducers. • Measurements started with a very small superficial gas velocity. The gas velocity was then increased in very small increments.  29  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity • Increasing the gas velocity was stopped when the gas velocity reached the maximum capability of the blower, or when the fountain reached the top of column. • Gas velocity was then decreased, again in small steps, until the flow totally stopped. • Pressure and pressure drops were measured at each gas velocity 10 to 20 s after changing the gas velocity. • More points were taken around critical velocities where there was an abrupt change in flow behaviour. A typical pressure drop evolution is shown in Figure 3.1. At first, the pressure drop across the column increased with increasing gas velocity. Before the pressure drop reaches its maximum, the main part of the line follows a parabolic trend. Here, the particles were static, so the pressure drop was governed by the Ergun equation. Close to the maximum pressure drop, the parabolic trend ended, the surface of the particles started moving, a tiny inner spout formed at the bottom, and the pressure drop increased more slowly with increasing gas velocity. More points were sampled to find the point of maximum pressure drop, ∆Pmax . Beyond the maximum, the pressure drop decreased sharply. The inner spout, which could not be visualized directly, but can be inferred from the motion of the particles, kept expanding and approaching the bed surface. Then, abruptly, with a small further increase in gas velocity, the inner spout broke through the surface and a spouted bed formed. This breakthrough is called the “onset of spouting”. The pressure drop also dropped sharply at this point. This point was called Ums,i . After the onset of spouting, the pressure drop decreased slightly with increasing gas velocity. This decrease is notable, as the pressure drop tends to be constant once spouting has been achievedin conventional spouted beds. This decrease may be due to the change of spout shape when the gas velocity was increased.  30  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Pressure drop across bed, ∆P (kPa)  7 Increasing gas velocity Decreasing gas velocity  6  ∆Pmax  5 4 ∆Pmax,d 3 2 1 Ums  0 0  0.2  Ums,i  0.4 0.6 Superficial gas velocity, Ug (m/s)  0.8  1  Figure 3.1: Evolution of pressure drop with superficial gas velocity. Column C, diverging base, type a slot with width of 4 mm, Hs = 450 mm. When the gas velocity was reduced, the pressure drop showed the reverse trend until the spout collapsed. The gas velocity at the point of collapse was called Ums , the minimum spouting velocity, as is normal in spouted bed research. The pressure drop showed a sudden shift at Ums . It reached a maximum, ∆Pmax,d , and then started to decrease. An inner submerged spout appeared at this stage. At a low gas velocity, the inner spout disappeared, particles became static and the pressure drop curve rejoined the increasing line.  Comparison of the pressure drops for increasing and decreasing air flows shows  that there is a significant hysteresis, as is common for spouted beds. The hysteresis is mainly caused by packing effects and friction between the particles, with the inner spout needing to overcome this force to forge the spout. The extent of the hysteresis can be characterized by the percentage difference between the minimum spouting velocities, as well as the percentage differences in maximum pressure drops, for increasing and decreasing gas velocity. The percentage difference in the former, (Ums,i − Ums )/Ums , changes with static bed height and slot width, as shown in Figure 3.2. (Ums,i − Ums )/Ums is higher for a smaller slot width. For each slot width, except 8 mm, this difference 31  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity reached a maximum at an intermediate height. For a deep bed, Ums,i may even be less than Ums . It also happened for very shallow bed, when the bed surface was not flat. 0.2  Slot width,λ: 2 mm 4 mm 6 mm 8 mm  (Ums,i-Ums)/Ums  0.15  0.1  0.05  0  -0.05  -0.1 0  0.1  0.2  0.3 0.4 0.5 Static bed height, Hs (m)  0.6  0.7  0.8  Figure 3.2: Difference in Ums caused by hysteresis. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type a, depth= 62.7 mm, see Figure 2.5.  As shown in Figure 3.3, the percentage difference in maximum pressure drops, (∆Pmax − ∆Pmax,d )/∆Pmax , shows similar behaviour, but the maximum pressure drop with increasing Ug was always higher than that with decreasing Ug . This means that the hysteresis for the slot-rectangular column was smaller for the shallower and the deeper bed, and most significant at intermediate bed heights, i.e. for Hs ≈ 0.35 m in the present case. There was no significant change with varying slot width.  3.2  Spouting pressure drop  As shown in Figure 3.1, the pressure drop across the bed decreased with increasing superficial gas velocity for Ug > Ums . For consistency, the pressure drop at Ums (for the decreasing gas process) was taken as the reference spouting pressure drop to be compared. This pressure drop, ∆Ps , arises from two parallel resistances, the spout and the annulus 32  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity 0.6  Slot width,λ: 2 mm 4 mm 6 mm 8 mm  (∆Pmax-∆Pmax,d)/∆Pmax  0.5  0.4  0.3  0.2  0.1  0 0  0.1  0.2  0.3 0.4 0.5 Static bed height, Hs (m)  0.6  0.7  0.8  Figure 3.3: Difference in ∆Pmax caused by hysteresis. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type a, depth= 62.7 mm, see Figure 2.5.  resistances. The measured pressure drop increased with the increasing static bed height. It was also higher for wider slots, because the emerging gas spread more widely, with more gas passing through the annulus, as the slot width increased. Mamuro and Hattori (1968) predicted that the ratio of the maximum spouting pressure drop to the corresponding fluidization pressure drop is 0.75. Experimental results from this project plotted in Figures 3.4 and 3.5 agreed reasonably with this value. Most ratios of spouting pressure drop to fluidization pressure drop are less than 0.75, but there were some exceptions for a 150 mm × 200 mm column where the top part of the bed was fluidized, with spouting below. The spouting pressure drop increased with increasing bed height and increasing slot width. Particle diameter had only a small influence on ∆Ps , consistent with results of Dogan et al. (2000).  33  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  0.9 0.8  Mamuro and Hattori (1968)  ∆Ps/(ρ s(1-ε g,0)gH)  0.7 0.6 α,β,λ (mm) 300,100,2 300,100,4 300,100,6 300,100,8 150,200,1 150,200,2 150,200,3 150,200,4  0.5 0.4 0.3 0.2 0.1 0  0.2  0.4 0.6 Static bed height, Hs, (m)  0.8  1  Figure 3.4: Spouting pressure drop across the bed. Particles: Glass beads, dp =1.33 mm; Slot type a, depth 62.7 mm, see Figure 2.5; diverging base.  0.9  ∆Ps/(ρ s(1-ε g,0)gH)  0.8  Mamuro and Hattori (1968)  0.7 α,β,λ,dp (mm) 300,100,2,0.66 300,100,2,1.33 300,100,2,2.45 300,100,4,0.66 300,100,4,1.33 300,100,4,2.45 150,200,2,0.66 150,200,2,1.33 150,200,2,2.45 300,200,4,1.33  0.6 0.5 0.4 0.3 0.2 0  0.05  0.1  0.15 0.2 0.25 Static bed height, Hs, (m)  0.3  0.35  0.4  Figure 3.5: Spouting pressure drop across the bed. Particles: Glass beads; Slot type b, depth 12.7 mm, see Figure 2.5; diverging base.  34  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  3.3  Minimum spouting velocity  The minimum spouting velocity, Ums , is normally defined as the superficial gas velocity at the point where the spout collapses. Ums has been widely investigated for the conventional circular spouted beds. In slot-rectangular columns, additional factors may influence Ums , including all column and slot dimensions. In this section, “extended slots” (type a) with a slot depth of of 62.7 mm and “standard rectangular slots” (type b) with slot length the same as column thickness were investigated (see Figure 2.5. Both columns with and without the diverging bases, as shown in Figure 2.2, were also tested. Influences of slot configuration on the stability of spouting are considered in Chapter 5.  Column with “extended slot” and diverging base Figures 3.6 and 3.7 plot the minimum spouting velocity for a column with an “extended slot” (type a) and diverging base. Ums increased quickly at first with the static bed height. For deeper beds, Ums increased more slowly and approached the minimum fluidization velocity of the particles. For deep beds, Ums could exceed Umf , when the top part of the bed was already fluidized. Ums also increased with slot width. This result is different to that of Dogan et al. (2004), who stated that the slot width has negligible effect on the minimum spouting velocity in a half column of similar slot configuration. In this work, a wider slot also resulted in a lower maximum spoutable height, Hmax , although the latter was only roughly measured.  Column with “extended slot” and flat bottom Figure 3.8 presents the minimum spouting velocity for a column with an “extended slot” (type a) in a flat-bottomed column without any diverging base. The flat bottom results show the same trend as for the diverging-base column with respect to the effect of static 35  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Minimum spouting velocity, Ums, (m/s)  0.9 0.8  Umf=0.74m/s  0.7 0.6 0.5 0.4 Slot width, λ 2 mm 4 mm 6 mm 8 mm  0.3 0.2 0.1 0  0.1  0.2  0.3 0.4 0.5 Static bed height, Hs, (m)  0.6  0.7  0.8  Figure 3.6: Effect of static bed height on Ums , Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base.  Minimum spouting velocity, Ums, (m/s)  0.8 Umf=0.74m/s  0.75 0.7 0.65 0.6 0.55 0.5 0.45  Slot width, λ 1 mm 2 mm 3 mm 4 mm  0.4 0.35 0.3 0.25 0  0.1  0.2 0.3 Static bed height, Hs, (m)  0.4  0.5  Figure 3.7: Effect of static bed height on Ums , Column: α=150 mm, β=200 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base.  36  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Minimum spouting velocity, Ums, (m/s)  0.9 0.8  Umf=0.74m/s  0.7 0.6 0.5 0.4 Slot width, λ 2 mm 4 mm 6 mm 8 mm  0.3 0.2 0.1 0  0.2  0.4 0.6 Static bed height, Hs, (m)  0.8  1  Figure 3.8: Effect of static bed height on Ums . Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; flat bottom.  Minimum spouting velocity, Ums, (m/s)  0.8 0.75 0.7 0.65 0.6 0.55 0.5  Slot width, λ 6 mm, diverging base 8 mm, diverging base 6 mm, flat base 8 mm, flat base  0.45 0.4 0.35 0  0.1  0.2 0.3 0.4 0.5 Static bed height, Hs, (m)  0.6  0.7  Figure 3.9: Effect of diverging base on Ums . Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5.  37  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity bed height and slot width. Ums /Umf can be up to 1.1 for a deep bed. However, single spouting in flat bottom column normally terminated at much smaller Hmax than for diverging bases. For the same static bed height, the flat-bottomed column had higher minimum spouting velocities, as shown in Figure 3.9. This was caused by the flat bottom, which results in spreading of gas at the bottom, so that more gas is needed to support the particles.  Column with “standard rectangular slot” and diverging base “Standard rectangular slots” (type b) have been used in all previous reported works (Dogan et al., 2000, 2004; Freitas et al., 2000, 004a,b; Kalwar et al., 1989, 1992, 1993; Passos et al., 1991, 1993, 1994) on slot-rectangular spouted beds. For comparison purposes, a set of factorial experiments was carried out on columns with type b slots to determine the influence of each parameter. With the configuration of the slot and column fixed, the following factors were varied: column width, α; column thickness, β; slot width, λ; mean particle diameter, dp ; solid density, ρs and static bed height, Hs . The effects of these factors are represented by the effects of the dimensionless parameters α/β, α/λ, √ λ/dp , Hs /dp and the Froude number F rms = Ums / gHs . Glass beads and polystyrene particles with densities, ρs , of 2500 and 930 kg/m3 , respectively, were used to develop a correlation to predict the minimum spouting velocity. Ums was found to increase with increasing static bed height, as found in previous studies (Dogan et al., 2000, 2004). Consistent with the earlier results as well, Ums also increased with increasing particle diameter for type b slots. Larger particles have higher terminal settling velocities so that higher gas velocities are needed to initiate spouting. However, in this work, Ums increased with increasing slot width, which can also matches the finding of Dogan et al. (2000). As shown in Figure 3.10, the dimensionless minimum spouting velocity increased  38  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity as the column aspect ratio, α/β, increased. It is likely that with smaller aspect ratio, more gas entered the annulus, loosening the packed bed. Three-dimensional effects no doubt also played a role. With increasing column thickness, the slot length, equal to the column thickness, increased proportionally. For a column of smaller thickness at a given superficial gas velocity, it is easier for a long slot to support a single spout. The dimensionless minimum spouting velocity decreased monotonously with increasing α/λ, as shown in Figure 3.11. Spouting is related to the gas momentum at the slot outlet. A larger α/λ ratio causes a higher entry velocity and higher momentum of gas at the bottom for a given superficial gas velocity. As a result, it is easier to achieve spouting in a column with larger α/λ ratio. The dimensionless minimum spouting velocity was also found to decrease with increasing λ/dp ratio, as shown in Figure 3.12, and to increase with increasing Hs /dp , as shown in Figure 3.13.  Dimensionless minimum spouting velocity, Ums/(gHs)0.5  0.4 Hs/dp=75 0.35  112.5 150  0.3  0.25  0.2  0.15 0.5  1  α, β, Hs, λ, dp 300,100,200,4,1.33 300,200,200,4,1.33 150,200,100,2,0.66 300,100,150,4,1.33 300,200,150,4,1.33 150,200,75,2,0.66 300,100,100,4,1.33 300,200,100,4,1.33 150,200,50,2,0.66  1.5 2 2.5 Column aspect ratio, α/β  3  3.5  Figure 3.10: Effect of column aspect ratio on minimum spouting velocity. Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. All dimensions are in mm.  39  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Dimensionless minimum spouting velocity, Ums/(gHs)0.5  0.7  λ,  0.6 λ/dp=3  0.5  d p, H s  2, 0.66, 5 4, 1.33, 10 8, 2.42, 15  0.4 0.3 0.2 0.1 0 20  40  60 80 100 120 Column width / Slot width, α/λ  140  160  Figure 3.11: Effect of column width to slot width ratio on minimum spouting velocity. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. All dimensions are in mm.  Dimensionless minimum spouting velocity, Ums/(gHs)0.5  0.7  λ,  d p, H s  2, 0.66, 5 2, 1.33, 10 2, 2.42, 15 4, 0.66, 5 4, 1.33, 10 4, 2.42, 15  0.6 0.5 0.4 0.3 0.2 α/λ=150  0.1  75  0 0  1  2 3 4 5 Slot width / Particle diameter, λ/dp  6  7  Figure 3.12: Effect of dimensionless slot width on minimum spouting velocity. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5. All dimensions are in mm.  40  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Dimensionless minimum spouting velocity, Ums/(gHs)0.5  0.7  Glass beads dp 0.66 mm 1.33 mm 2.42 mm  0.6 0.5 0.4 0.3 0.2 0.1 0 0  50  100 150 200 250 300 Static bed height / Particle diameter, Hs/dp  350  Figure 3.13: Effect of dimensionless bed height on minimum spouting velocity. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5.  3.4  Correlation of minimum spouting velocity  A number of previous Ums correlations have been proposed for conventional and slotrectangular spouted beds. In this section, our data are compared to some of these correlations, as well as with the results of Dogan et al. (2000), Freitas et al. (2000) and Dogan et al. (2004), for which experiments had been carried out earlier in our laboratory on a half-column. Passos et al. (1991) proposed a correlation for slot-rectangular columns with A2D ≥ 0.044, which is outside the range covered in this project. Costa and Taranto (2003) provided an empirical correlation for their experimental results obtained in a column with a square inlet and column thickness, β, equal to slot width, λ. This creates ambiguity as √ to whether to use β or λ or λ × β. The following correlations, derived from the Costa and Taranto (2003) correlation, are therefore first compared with the experimental results  41  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity in Table 3.1.  α  α −0.75  H 0.16  ρ − ρ 0.23 Ums s s g √ = 3.0 λ dp φ α ρg 2gHs −0.75  0.16  0.23   Ums α Hs ρs − ρg α √ = 3.0 β dp φ α ρg 2gHs   −0.75  0.16  0.23 Ums α α Hs ρs − ρg √ = 3.0 √ dp φ α ρg 2gHs λβ  (3.1)  (3.2)  (3.3)  Mathur and Gishler (1955) proposed the first Ums correlation for axisymmetric spouted beds, and this correlation has been widely applied by subsequent researchers. A slotrectangular column requires at least one more dimension to characterize it than a column of circular cross-section. In Equations 3.1 to 3.3, the column diameter, Dc , and orifice diameter, Di , have been replaced by α, β and λ for slot-rectangular columns with the slot length equal to the column thickness. In order to compare the new results with correlations for the conventional spouted beds, the area-equivalent diameters and hydraulic diameters of the rectangular columns and slot in it are tested as replacements for the column diameter and orifice diameter. The original Mathur and Gishler (1955) correlation for the circular spouted beds is: U √ ms = 2gHs    dp Dc    Di Dc  0.33 s  ρs − ρg ρg  (3.4)  Equivalent area diameters are r Dc,a =  4αβ π  (3.5)  4λβ π  (3.6)  and r Di,a =  42  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity Hydraulic diameters are Dc,h =  2αβ (α + β)  (3.7)  Di,h =  2λβ (λ + β)  (3.8)  and  With different combinations of Dc and Di , the following correlations of minimum spouting velocity were derived and compared with the experimental results. With Dc replaced by Dc,a and Di by Di,a :  Ums  r  s 1/6 π dp λ 2gHs (ρs − ρg ) = 2/3 2 α β ρg  (3.9)  With Dc substituted by Dc,a and Di by Di,h :  Ums  4/3  s  1/6  1+ω dp λ1/6 β 1/6 2gHs (ρs − ρg ) 4 = 4/3 π 2 α ρg  (3.10)  where ω = α/β. With Dc replaced by Dc,h and Di by Di,a :  Ums =  π  2/3    2  1/3  dp λ α2/3 β 2/3  s  2gHs (ρs − ρg ) ρg  (3.11)  With Dc substituted by Dc,h and Di by Di,h :  Ums = 21/3    1+ω 2  4/3   dp λ1/3 α4/3  s  2gHs (ρs − ρg ) ρg  (3.12)  The form of equation 3.4 was based on dimensional analysis. A correlation of Ums of similar form was derived by Ghosh (1965) with the assumptions: 1. Particles at the bottom require a velocity of at least  43  √  2gHs to reach the surface of  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity the bed. 2. The momentum lost by fluid entering the column is equal to the momentum retained by the particles at the slot outlet, which is assumed to be proportional to the total momentum of the fluid. 3. In unit time, n particles are able to enter the spout at a velocity of  √  2gHs . Particles  need to move a length dp to vacate their original place. So the number of particles  √ that is spouted per unit time is n 2gHs /dp . A theoretical prediction based on these assumptions was derived for the slot-rectangular column. The momentum lost by the fluid is:  2 Mlost = K(λ × β)ρg Ui,ms  (3.13)  where K is the lost momentum divided by the total momentum. Ui,ms is the minimum spouting inlet gas velocity, Ui,ms =  λ Ums α  (3.14)  The momentum gained by the particles is:  Mgain =  where vi =  √   nvi πd3p (ρs − ρg )vi dp 6  (3.15)  2gHs is the velocity of particles needed to reach the top.  According to the assumption Mlost = Mgain  44  (3.16)  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity The minimum spouting velocity is then:   Ums  where C =  p  dp =C λ  1/2   dp β  1/2  s λ 2gHs (ρs − ρg ) α ρg  (3.17)  nπ/6K is an overall constant.  Table 3.1 compares the experimental results from this project and with the predictions from the above equations. The goodness of fit, R2 , was used in the comparison, defined as: P (Ums,exp − Ums,cor )2 R =1− P (Ums,exp − Ūms,exp )2 2  (3.18)  where Ums,exp is the experimental minimum spouting velocity, Ūms,exp is the mean of experimental value and Ums,cal is predicted value. The constant C in Equation 3.17 were calculated using linear regression in MATLAB. Not one of the above equations is able to predict the experimental results well. The equations derived from Costa and Taranto (2003) (Equation 3.1 to 3.3) produced ridiculous fitting results, due to the big difference in their slot configuration from this project. In a slot-rectangular spouted bed, the minimum spouting velocity was affected by the following controlling variables: α, β, λ, dp , Hs , ρg , ρs , g, µ, ϕs and particle size distribution. In the current work, µ, ϕs , ρg and particle size distribution were held constant. ρg was kept in the control variables for the purpose of non-dimensionlization and comparison to results from previous work. So the set of control variables in this work was: α, β, λ, dp , Hs , ρg , ρs and g. Ums was dependent variable and could be determined by these variables. According to the Buckingham π theorem, 6 dimensionless groups could be derived to account for the independent control variables the dependent variable: α β λ H s ρs U √ ms , , , , , gHs dp dp dp dp ρg where the first dimensionless group is the Froude number. 45  (3.19)  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity Table 3.1: Comparison of correlations with literature data and experimental data obtained in this project  R2 , goodness of fit Equation Dogan et al. (2000)  Freitas et al. (2000)  Dogan et al. (2004)  This work, type a slot  This work, type b slot  Overall  3.1  -615  -17600  -1420  -58400  -13600  -3220  3.2  -2.42  -3.67  -43.7  -2.26  -1.58  -2.93  3.3  -45.4  -461  -256  -584  -250  -89.2  3.9  -0.117  -0.445  0.424  -1.79  0.276  0.428  3.10  0.201  0.426  -5.95  -0.420  0.644  0.531  3.11  -0.404  -1.80  0.596  -4.42  -0.449  0.170  3.12  0.408  -0.688  -3.55  -2.83  0.062  0.508  3.17  0.249  0.385  0.107  0.054  0.565  0.315  C in 3.17  3.241  5.103  1.583  6.4  0.449  2.989  3.22  0.900  0.929  0.937  0.919  0.966  0.834  These dimensionless groups can be recast into the following set of more meaningful dimensionless groups: U α α λ Hs ρs − ρg √ ms , , , , , ρg gHs β λ dp dp  (3.20)  where α/β represents the effect of column shape, α/λ the relationship between column and slot width, λ/dp the relation between slot and particle sizes, and Hs /dp the bed height.  46  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity So the correlation for minimum spouting velocity can be written in the following form: U α α λ H ρ − ρg √ ms = f ( , , , s , s ) β λ dp dp ρg gHs  (3.21)  which is also consistent with the common form of all the previously derived correlations:  a    c  d  e Ums α α b λ Hs ρs − ρg √ =C β λ dp dp ρg gHs  (3.22)  Here C is a constant which accounts for the effect of the other independent controlling variables that have not been investigated in this work. Fitting based on this form of equation was carried out for each group of experimental results using non-linear regression and MATLAB. The resulting parameters are listed in Table 3.2, where N is the number of data points used to derive the correlation parameters. The normalized forms of correlation for different groups of data have different exponents, with the correlation depending strongly on slot shape.  Correlation for all available data Equation 3.23 is an empirical regression equation for Ums derived from all the available data including the result from this work and the experimental data from Dogan et al. (2000); Freitas et al. (2000); Dogan et al. (2004).  0.418    −0.996  0.192  0.847 Ums α α −1.18 λ Hs ρs − ρg √ = 0.0743 β λ dp dp ρg gHs  (3.23)  It was compared to the experimental data in Figure 3.14. With R2 = 0.880, the resulting correlation had a good fit with the experimental results. However, for type a slots in this work and Dogan et al. (2004), the correlation result deviated quickly from the experimental result with increasing bed height. The main difference between the set-ups  47  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity of these two groups and the others was the configuration of slot. For these two groups, gas went through a long channel before it entered the column, whereas for the other groups, the distance between the inlet and outlet of the slot is small. So these two types of configurations were treated separately.  Table 3.2: Non-linearly fitted constant and exponents of equation 3.22 for all available data and for individual experimental studies  Dogan et al. (2000)  Freitas et al. (2000)  Dogan et al. (2004)  This work, type a slot  This work, type b slot  All data  N  76  69  34  67  51  297  R2  0.900  0.929  0.937  0.919  0.967  0.880  C  0.388  0.552  1.27  1.08  0.173  0.0743  Exponents of Equation 3.22 on parameter in left column a  0.136  0.302  0.197  0.156  0.284  0.418  b  -0.973  -0.729  -1.03  -0.904  -1.131  -1.18  c  -0.783  -0.654  -1.11  -0.791  -1.139  -0.996  d  0.268  0.107  -0.063  -0.067  0.125  0.192  e  0.523  0.389  0.543  0.5  0.774  0.847  Column with “standard rectangular slot” and diverging base An empirical regression equation for Ums derived from all experimental data obtained in this project for the “standard rectangular slot” (type b slot), with all the independent  48  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Correlation result, Ums,cor, (m/s)  3.5 3 2.5 2 1.5 1  Dogan et. al (2000) Freitas et. al (2000) Dogan et. al (2004) Type b slot, this work Type a slot, this work  0.5 0 0  0.5  1 1.5 2 2.5 Experimental result, Ums, (m/s)  3  3.5  Figure 3.14: Comparison between the correlation of all available experimental results, equation 3.23 and data by Dogan et al. (2000) and Freitas et al. (2000).  dimensionless groups varied experimentally (see 6th column of Table 3.2), is  0.34    −0.898  0.276  0.598 α Hs ρs − ρg Ums α −1.09 λ √ = 0.256 β λ dp dp ρg gHs  (3.24)  With R2 = 0.946, this regression fits the experimental results well for type b slots. Since there was physical similarity between the type b slots and the slots used by Dogan et al. (2000) and Freitas et al. (2000), which were of 6.3 mm thickness, the results from these works are compared with Equation 3.24. As shown in Figure 3.15, the regression shows good agreement with the results of Freitas et al. (2000), but large deviations from the results of Dogan et al. (2000), obtained in a column of thickness of 30 mm where the wall may have had a major influence.  49  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Correlation results, Ums,cor, (m/s)  3.5 Type b slot, this work Dogan et. al (2000) Freitas et. al (2000)  3 2.5 2 1.5 1 0.5 0 0  0.5  1 1.5 2 2.5 Experimental result, Ums, (m/s)  3  3.5  Figure 3.15: Comparison between the correlation of experimental results using type b slots, equation 3.24 and data by Dogan et al. (2000) and Freitas et al. (2000).  Column with “extended slot” and diverging base The “extended slots” were only tested with glass beads. The experimental results for the type a slot are compared with the result of Dogan et al. (2004). The slot configurations for these two sets of results were similar. Gas entered the column through a long channel between two parallel plates. The resulting correlation for the experimental result in the current work is:  0.122    −0.689  −0.081  0.5 α α −0.721 λ Hs ρs − ρg Ums √ = 0.492 β λ dp dp ρg gHs  (3.25)  Calculated values of Ums based on this equation are compared with the experimental result of this work and literature data in Figure 3.16. The correlation worked well for the results of the current work at intermediate bed height, while for smaller and larger bed depths, the predictions tended to exceed the experimental values. The experimental results of Dogan et al. (2004) had the same trend as the results for type a slot in the current study. It deviated from the correlation as the static bed height increased. Note  50  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Correlation results, Ums,cor, (m/s)  1.2  1  0.8  0.6  0.4  0.2 Type a slot, this work Dogan et. al (2004) 0 0  0.2  0.4 0.6 0.8 Experimental result, Ums, (m/s)  1  1.2  Figure 3.16: Correlation predictions for type a slot, equation 3.25, compared with experimental data of Dogan et al. (2004).  that the correlations in this work are all based on the particles with sphericity equal to 1.  3.5  Maximum pressure drop  Column with “extended slot” (Figure 2.5(a)) and diverging base As shown in Figures 3.17 and 3.18, the maximum pressure drop increased almost linearly with static bed height. There was a negligible influence of slot width on ∆Pmax , unlike the results of Dogan et al. (2000) and Dogan et al. (2004). The maximum pressure drop was less than the minimum fluidization pressure drop (shown as dash-dot lines in Figures 3.17 and 3.18) for static bed heights less than 300 mm. For deeper beds, the maximum pressure drop exceeded the pressure drop at minimum fluidization, reaching as high as 1.2(ρs (1 − g,0 )gHs ). For a type a slot with diverging base, the maximum pressure drop can be correlated to  51  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Maximum pressure drop, ∆Pmax, (kPa)  12  10 ρ s(1-ε 0)gHs  8  6  4  Slot width, λ 2 mm 4 mm 6 mm 8 mm  2  0 0  0.1  0.2  0.3 0.4 0.5 Static bed height, Hs, (m)  0.6  0.7  0.8  Figure 3.17: Effect of static bed height on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base.  the column and slot dimensions in the following form, obtained by non-linear regression in MATLAB.  ∆Pmax ∝  Hs2 (α × β)  0.562  λ0.013  (3.26)  with R2 = 0.989 and proportionality constant = 2.154. The fit is excellent as shown in Figure 3.19. ∆Pmax seems to be related to Hs2 / (α × β), where α × β is the cross-section area of the column. The slot width, λ, only had a small effect on the maximum pressure drop.  Column with “extended slot” (Figure 2.5(a)) and flat bottom For columns with flat bottoms, ∆Pmax was directly proportional to the static bed height. Slot width had little influence on the maximum pressure drop, as indicated in Figure 3.20. A diverging base(Figure 2.2) had little effect on ∆Pmax , as shown in Figure 3.21. Compared to columns with diverging bases, ∆Pmax was slightly higher for flat-bottomed 52  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Maximum pressure drop, ∆Pmax, (kPa)  7 6 ρ s(1-ε 0)gHs  5 4 3  Slot width, λ (mm) 1 mm 2 mm 3 mm 4 mm  2 1 0 0  0.1  0.2 0.3 Static bed height, Hs, (m)  0.4  0.5  Figure 3.18: Effect of static bed height on maximum pressure drop. Column: α=150 mm, β=200 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; diverging base.  Correlation results, ∆Pmax,corr, (kPa)  12  10  8  6  4  2 Diverging base Flat bottom 0 0  2  4 6 8 Experimental result, ∆Pmax, (kPa)  10  12  Figure 3.19: Comparison between experimental maximum pressure drop and prediction by Equation 3.26. Columns C and D; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5.  53  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Maximum pressure drop, ∆Pmax, (kPa)  12  10 ρ s(1-ε 0)gHs  8  6  4  Slot width, λ 2 mm 4 mm 6 mm 8 mm  2  0 0  0.1  0.2  0.3 0.4 0.5 Static bed height, Hs, (m)  0.6  0.7  0.8  Figure 3.20: Effect of static bed height on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5; flat base.  Maximum pressure drop, ∆Pmax, (kPa)  12  10  8  6  4  Slot width, λ 2 mm, diverging base 6 mm, diverging base 2 mm, flat base 6 mm, flat base  2  0 0  0.1  0.2  0.3 0.4 0.5 Static bed height, Hs, (m)  0.6  0.7  0.8  Figure 3.21: Effect of diverging base on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot: type a, depth 62.7 mm, see Figure 2.5.  54  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity columns under corresponding conditions for shallow bed. Not surprisingly, the results also indicate that the effect of the diverging base mostly occurs in the lower part of bed. Therefore the results for a flat bottom can also be correlated by equation 3.26, as shown in Figure 3.19.  Column with “standard rectangular slot” (Figure 2.5(b)) and diverging base The maximum pressure drop was also measured in columns with diverging bases and “standard rectangular slots” (type b) and then normalized by the pressure drop of a fluidized bed of corresponding static bed height, i.e. by (ρs (1 − g,0 )gHs ) Experiments were also carried out to change the dimensionless groups. The dimensionless maximum pressure drop increased with increasing column aspect ratio, α/β, and dimensionless height, Hs /dp , but decreased with increasing column-width-to-slot-width ratio, α/λ and dimensionless slot width, λ/dp , as shown in Figures 3.22 to 3.25. The maximum pressure drop depends on the packing of the particles. More gas passing through the annulus caused the particles to spread more, i.e. to have higher voidage. As a result, the maximum pressure drop was lower. A smaller column aspect ratio caused the gas to be more widely distributed leading to a smaller pressure drop. The correlation 3.26 was found not applicable to the Hmax for “standard rectangular slot” (type b). With the independent controlling variables: α, β, λ, dp , Hs , ρg , ρs and g. The following dimensionless groups can be written: ∆Pmax α α λ H s ρs − ρg , , , , , g(ρs − ρg )Hs β λ dp dp ρg  (3.27)  So the maximum pressure drop can be represented as: ∆Pmax α α λ Hs ρs − ρg = f( , , , , ) g(ρs − ρg )Hs β λ dp dp ρg  55  (3.28)  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Dimensionless maximum pressure drop, ∆Pmax/(ρ s(1-ε g,0)gHs)  1.4  α,  β, Hs, λ, dp  300,100,200,4,1.33 300,200,200,4,1.33 150,200,100,2,0.66 300,100,150,4,1.33 300,200,150,4,1.33 150,200,75,2,0.66 300,100,100,4,1.33 300,200,100,4,1.33 150,200,50,2,0.66  1.3 1.2 1.1 1 0.9 0.8  Hs/dp = 75  112.5  0.7 0.6  150  0.5 0.5  1  1.5 2 2.5 Column aspect ratio, α/β  3  3.5  Figure 3.22: Effect of column aspect ratio on maximum pressure drop. Particles: Glass beads; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. All dimensions are in mm. The following correlation:   −0.174   −0.770  0.196  −0.207  α α −0.561 λ Hs ρs − ρg ∆Pmax = 2630 (3.29) g(ρs − ρg )Hs β λ dp dp ρg was found to best fit the data with R2 = 0.712 using non-linear regression in MATLAB. This regression equation indicates that the particle diameter has almost no effect on the pressure drop when the other parameters are fixed. The comparison between the experiment and correlation is shown in Figure 3.26.  56  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Dimensionless maximum pressure drop ∆Pmax/(ρ s(1-ε g,0)gHs)  1  λ,  d p, H s  2, 0.66, 5 4, 1.33, 10 8, 2.42, 15  0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 20  40  60 80 100 120 Column width / Slot width, α/λ  140  160  Figure 3.23: Effect of width to slot width ratio on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. All dimensions are in mm.  Dimensionless maximum pressure drop, ∆Pmax/(ρ s(1-ε g,0)gHs)  1.1  λ,  d p, H s  2, 0.66, 5 2, 1.33, 10 2, 2.42, 15 4, 0.66, 5 4, 1.33, 10 4, 2.42, 15  1 0.9 0.8 0.7 0.6 0.5 0.4 α/λ=150  0.3  75  0.2 0  1  2 3 4 5 Slot width / Particle diameter, λ/dp  6  7  Figure 3.24: Effect of dimensionless slot width on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base. All dimensions are in mm.  57  Chapter 3. Maximum Pressure Drop and Minimum Spouting Velocity  Dimensionless maximum pressure drop, ∆Pmax/(ρ s(1-ε g,0)gHs)  1 0.9 0.8 0.7 0.6 0.5 0.4  Particle diameter, dp 0.66 mm 1.33 mm 2.42 mm  0.3 0.2 0  50  100 150 200 250 300 Static bed height / Particle diameter, Hs/dp  350  Figure 3.25: Effect of dimensionless bed height on maximum pressure drop. Column: α=300 mm, β=100 mm; Particles: Glass beads, dp =1.33 mm; Slot type b, depth 12.7 mm, see Figure 2.5, diverging base.  Correlation results, ∆Pmax,corr, (kPa)  3  2.5  2  1.5  1  0.5  0 0  0.5  1 1.5 2 Experimental result, ∆Pmax, (kPa)  2.5  3  Figure 3.26: Comparison between experimental maximum pressure drop and prediction by Equation 3.26. Columns C and D; Particles: Glass beads, dp =1.33 mm; Slot: type b, depth 12.7 mm, see Figure 2.5; diverging base.  58  Chapter 4 Flow Regimes Several flow regimes have been reported for slot-rectangular spouted beds, (Dogan et al., 2000; Freitas et al., 2000, 004a), more than for conventional circular columns. The difference is attributed to 3-dimensional effects in rectangular columns. By analyzing the pressure fluctuations in a slot-rectangular column, column A in the present study, with 1.4 mm diameter glass beads, Freitas et al. (004a) identified the flow regimes found by Dogan et al. (2000); Freitas et al. (2000). In the present work, statistical analysis and frequency analysis are adopted to identify the flow regimes in larger slot-rectangular columns. Experiments were mostly carried out in column C (300 mm×100 mm) with glass beads of 1.33 mm diameter. Diverging base and “Extended normal slots” (Type a slots) with 62.7 mm slot depth and different slot widths were used. The pressure drop across the column was measured at a frequency of f = 100 Hz. For each measurement, 1024 (N = 210 ) data points were collected in 10.24 s, convenient for Fast Fourier Transforms. The precision of frequency analysis was 0.977 Hz, which can be calculated by f /N .  4.1  Fluctuation of column pressure drop  Spectral analysis of pressure fluctuations has been used to identify flow regimes of spouted beds (Taranto et al., 1997; Rocha et al., 1998; Xu et al., 2004; Piskova and Morl, 2007). This method was also recently employed by Dogan et al. (2004) to characterize the flow regimes in a rectangular column with the same dimensions as Column A. Pressure fluctuations were attributed to different factors, such as spout oscillation, particle flow  59  Chapter 4. Flow Regimes and bubble breaking etc. In this work, the following factors can be identified as causes of fluctuations. • Natural fluctuation caused by the pressure measurement system. A sample signal from the pressure transducer measuring the pressure drop across the empty column was analyzed. The amplitude was ±16 P a, less than the fluidization pressure drop for a 1 mm deep bed, an acceptable system error. As shown in Figure 4.1, there was a frequency of 40 Hz in the plot of PSD (Power Spectral Density). Since the fluctuation frequency of the spouted bed pressure drop was always < 20 Hz, natural  ∆P, (Pa)  fluctuations had negligible effect on the measurements. 60 50 40 30 20 10 0 -10 -20 -30 0  1  2  3  4  5  40  50  2  PSD, (kPa/s )  Time, t, (s) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0  10  20 30 Frequency, f, (Hz)  Figure 4.1: Nature fluctuation produced by pressure transducer: PX142-02D5V.  • Unstable gas supply: The gas supply pressure varies in a periodic manner due to its on/off control system. Typical fluctuations are plotted in Figure 2.6. The period of this fluctuation exceeds 1 minute, whereas the sampling period is normally 10.24 s. Hence, fluctuations of the gas supply system are not problematic. • Periodic entry of particles into the spout at the bottom for every spouting flow 60  Chapter 4. Flow Regimes regime with a frequency from 5 to 10 Hz. This frequency range is close to the range, i.e 5 ∼ 7.5 Hz, found by Freitas et al. (004a). The range is also similar to the dominant frequency of 4 ∼ 10 Hz reported by Piskova and Morl (2007) for two-dimensional units; • Fountain swaying: The fountain sways both parallel and normal to the slot; • Bubble expansion and breakage: Bubble phenomena can result in a frequency of 0 to 5 Hz for the Jet-in-Fluidized-bed (JF) regime, Incoherent spouting (IS) regime and some unstable conditions (US); • Slug flow: Slugging results in a frequency of ∼ 2 Hz in the column utilized in the experiments and a magnitude of 10% to 15% of the average pressure drop. Pressure drop fluctuations can be considered to arise from a combination of several of the above mechanisms. Frequency analysis can indicate the mechanism. Fast Fourier transformation was used to analyze the pressure drop signals in the frequency domain. The method of Fourier transformation has been discussed in detail by Press et al. (1992).  4.2 4.2.1  Visual characterization and frequency analysis Fixed bed regime (F)  For gas velocities too low to cause particles motion, the spouted bed is in the fixed bed flow regime and all particles are static. Only slight pressure fluctuation can be measured. The power spectrum distribution of the pressure drop across the column shows a peak at 40 Hz with this peak disappearing at higher Ug . The power spectral density then shows a peak at 0 Hz in Figure 4.2.  61  ∆P, (kPa)  Chapter 4. Flow Regimes  8.03 8.02 8.01 8 7.99 7.98 7.97 7.96 7.95 7.94  (a)  0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 14 12 10 8 6 4 2 0  (b)  0  2  4 6 Frequency, f, (Hz)  8  10  ∆P, (kPa)  Figure 4.2: Fluctuations of overall bed pressure drop for Fixed Bed regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 550 mm; Ug = 0.256 m/s.  7.96 7.94 7.92 7.9 7.88 7.86 7.84 7.82 0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 9 8 7 6 5 4 3 2 1 0 0  2  4 6 Frequency, f, (Hz)  8  10  Figure 4.3: Fluctuation of pressure drop for Internal Jet regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 500 mm; Ug = 0.548 m/s.  62  Chapter 4. Flow Regimes jet.  4.2.3  Spouting regimes (S)  Two different spout shapes were found, one dilute and one dense, as shown in Figure 4.4(b) and 4.4(c) respectively. A dilute fountain appeared in columns of low static bed height, with the fountain shape changing from dilute to dense as the static bed height increased and more particles entering the spout. For a dilute fountain, the spouting could either be steady or sway from side to side.  (a) Internal Jet  (b) Stable Dilute Spout- (c) Stable Dense Spouting ing  Figure 4.4: Internal Jet and Steady State Spouting regimes observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads.  63  Chapter 4. Flow Regimes Steady state spouting (SS) As in the conventional spouted beds, there was a steady state spouting regime in slotrectangular columns. In this case, the fountain was symmetrical, and the shape, position and height of the fountain did not vary with time. Gas and particles traveled in a very smooth manner. However, despite the apparent steady state condition, the pressure still fluctuated. The pressure drop and PSD of pressure fluctuation are shown in Figure 4.5. Note the significant energy content at frequencies of ∼ 5 to 10 Hz. This frequency is similar to the finding of Freitas et al. (004a). It may be attributed to the oscillations in spout diameter and discontinuous flow of particles from annulus to the spout. This  ∆P, (kPa)  frequency became less noticeable when the other fluctuations emerged. 0.855 0.85 0.845 0.84 0.835 0.83 0.825 0.82 0.815 0.81 0.805  (a)  0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0  6.934 (Hz)  (b)  0  5  10 Frequency, f, (Hz)  15  20  Figure 4.5: Fluctuations of pressure drop for Steady State Spouting. Column C: 300 × 100 mm, with diverging base; Slot: type a, 5 mm width; 1.33 mm glass beads; Hs = 150 mm; Ug = 0.730 m/s.  Spouting with fountain swaying parallel to slot (S(A)) This behaviour occurred after the onset and before the collapse of the spout. The spout swayed parallel to the direction of the slot in a cyclic manner, as shown in Figure 4.7(b) 64  Chapter 4. Flow Regimes and 4.8(b). There were two dominant frequencies. As shown in Figure 4.6, the main frequency was the one of the swaying of fountain, ∼ 0.6 Hz. This frequency decreased with increasing gas velocity. The fountain in this case was relatively dense. The other  ∆P, (kPa)  was the frequency of the discontinuous particle flow, between about 5 and 10 Hz. 1.4 1.39 1.38 1.37 1.36 1.35 1.34 1.33 1.32 1.31 1.3  (a)  0  2  4  6  8  10  PSD, (kPa/s2)  Time, t, (s) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0  0.586 (Hz) (b)  0  5  10 Frequency, f, (Hz)  15  20  Figure 4.6: Fluctuations of pressure drop for Spouting regime with fountain swaying parallel to slot. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 200 mm; Ug = 0.46 m/s.  Spouting with fountain swaying normal to slot (S(N)) This type of swaying occurred after the swaying parallel to the slot and before reaching steady state spouting. The character of the signal was similar to that found for back-andforth swaying. For swaying with a dense fountain, as shown in Figure 4.9, the swaying caused a small frequency of 0.2 Hz, while fluctuations also appeared from 5 to 10 Hz. For swaying with a dilute fountain, as observed for a bed of ≤ 200 mm height, the fluctuations were much simpler than for a dense fountain. As shown in Figure 4.10, two peaks could be identified. The smaller frequency of ∼ 1 Hz is the swaying frequency, which was higher than for a dense fountain. The larger one, ∼ 11 Hz, was due to particle 65  Chapter 4. Flow Regimes  (a) Swaying in direction (b) Swaying normal to slot parallel to slot  Figure 4.7: Spouting with fountain swaying for two directions observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads.  flow. Because there were few particles in the column, the particles could choke the gas flow periodically. The fluctuation energy was due to the particle flow at low gas velocity, and due to swaying at high Ug . The swaying may be due to minor asymmetry of the experimental set-up causing the fountain to tilt to one side. When more particles accumulated at this side, this caused a higher pressure drop than on the other side. The spout was then “propelled” by this pressure difference toward the other side.  4.2.4  Multiple spouting regime (MS)  Multiple spouts were observed for low bed heights and low gas velocities. In column B, more than 3 spouts were observed along the slot. In column C, normally two spout 66  Chapter 4. Flow Regimes  (a) Swaying in direction normal to slot  (b) Swaying parallel to slot  ∆P, (kPa)  Figure 4.8: Top-view of spouting column with fountain swaying in two directions as observed in Column C: 300 mm x 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads.  0.9 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.8  (a)  0  2  4  6  8  10  Time, t, (s)  PSD, (kPa/s2)  5 4  0.195 (Hz)  3 (b) 2 1 0 0  2  4 6 Frequency, f, (Hz)  8  10  Figure 4.9: Fluctuations of pressure drop for spouting regime with fountain swaying normal to slot. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 1 50 mm; Ug = 0.566 m/s.  67  ∆P, (kPa)  Chapter 4. Flow Regimes  0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0  2  4  6  8  10  PSD, (kPa/s2)  Time, t, (s) 35 30 25 20 15 10 5 0 0  5  10 Frequency, f, (Hz)  15  20  Figure 4.10: Fluctuations of pressure drop for swaying normal to slot with dilute fountain. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 50 mm; Ug = 0.374 m/s.  (a) Multiple (b) Merging of multi- (c) Merged multiple Spouting ple spouts spouted  Figure 4.11: Spouting and merging of multiple spouts observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads.  68  Chapter 4. Flow Regimes 1.85 ∆P, (kPa)  1.8 1.75  (a)  1.7 1.65 0  2  4  6  8  10  PSD, (kPa/s2)  Time, t, (s) 9 8 7 6 5 4 3 2 1 0  4.688 (Hz)  (b)  0  2  4 6 Frequency, f, (Hz)  8  10  Figure 4.12: Fluctuations of pressure drop for Multiple Spouting regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 180 mm; Ug = 0.826 m/s.  appeared and settled at the walls, one at the front and one at the back, as shown in Figure 4.11(a), both oscillating. Pressure signals were obtained at both the front and back wall of the column. The correlation coefficient of the two signals varied from 0.75 to 0.95, indicating that the two spouts oscillated mostly simultaneously. Sample signal and frequency is shown in Figure 4.12. The main frequency varied from 4 to 8 Hz.  4.2.5  Merging of multiple spouting regime (MS(M))  Multiple spouts tended to merge as the gas velocity or static bed height increased. Drawings of the merging and merged multiple spouts are provided in Figures 4.11(b) and 4.11(c). During the process of merging, there were significant interactions between the spouts. Their fountains spread in opposite directions. After they merged, the flow fluctuated significantly with a frequency from 6 to 10 Hz and an amplitude of 7 to 12% of the average. Multiple spouting and merging could also be identified from the evolution of pressure 69  ∆P, (kPa)  Chapter 4. Flow Regimes 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3  (a)  0  2  4  6  8  10  PSD, (kPa/s2)  Time, t, (s) 90 80 70 60 50 40 30 20 10 0  8.301 (Hz)  (b)  0  2  4 6 Frequency, f, (Hz)  8  10  Figure 4.13: Fluctuations of pressure drop for Merging of Multiple Spout regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 180 mm; Ug = 0.769 m/s.  drop across the column. In some cases, the two spouts formed one after the other. As shown in Figure 4.14, the pressure drop underwent two dips corresponding to the onsets of the two spouts. The pressure evolution for the merging of spouts is shown in Figure 4.15. With increasing gas velocity, when two spouts merged, a shift on the pressure drop appeared.  4.2.6  Incoherent spouting regime (IS)  In the incoherent spouting regime, the gas and particles in the spout appear to move incoherently. This flow regime was observed for Ug > 0.7 m/s in beds higher than 450 mm. In this condition, the fountain tended to settle at either the front or back wall of the column, as shown in Figures 4.18(a) and 4.18(b). The upper part of the spout can be observed visually. The diameter of the top of the spout oscillated periodically at a frequency of 6 to 8 Hz, and particles were entrained into the spout periodically in the upper part of the spout. 70  Chapter 4. Flow Regimes 3  Slot type, λ, Hs b, a,  Pressure Drop of Column, ∆P (kPa)  1st spout appears 2.5  4, 200 6, 100  2nd spout appears 2  1.5 2nd spout collapse 1  1st spout collapse  0.5  0 0  0.2  0.4 0.6 Gas Velocity, Ug (m/s)  0.8  1  Figure 4.14: Pressure evolution with Multiple Spouting flow regime. Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. All dimensions are in mm. Filled symbols show where 2 spouts co-exist.  As shown in Figure 4.16 the resulting pressure signal showed fluctuations concentrated at a specific frequency of 6.738 Hz, with a harmonic at twice the main frequency. Similar behaviour was noted previously by Freitas et al. (004b). The main frequency decreased with increasing gas velocity, as shown in Figure 4.17. Some hysteresis was also observed for this flow regime.  4.2.7  Slugging regime (SL)  Slugging was observed at gas velocities > 0.8 m/s. Large bubbles were initiated in the lower part of the column and rose to the bed surface one after another. Particles above the slugs were pushed upward. Then at the surface of the bed, slugs broke through, as shown in Figure 4.18(c). This flow regime is similar to slugging in fluidized beds. As shown in Figure 4.19, pressure signal variations were very simple (saw tooth) for slug flow, with the maximum corresponding to the breaking of the slugs. Harmonic frequencies could also be observed. 71  Chapter 4. Flow Regimes  Pressure Drop of Column, ∆P (kPa)  3  Slot type, λ, Hs a, a,  2.5  6, 150 6, 200  2  1.5  1 ∆P shift 0.5  0 0  0.2  0.4 0.6 Gas Velocity, Ug (m/s)  0.8  1  Figure 4.15: Pressure evolution with merging of multiple spouts. Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. All dimensions are in mm. Filled symbols show where 2 spouts co-exist  4.2.8  Jet in fluidized bed regime (JF)  The Jet-in-fluidized-bed regime was normally found for deep beds (Hs ≥ 500 mm) before the onset of spouting. In this regime, a jet formed at the bottom of the column. Particles at the bottom could be observed descending at the outer wall. Small bubbles detached from the surface, as shown in Figure 4.21(a). Bubbles were so small that no characteristic frequency could be identified. If the gas velocity was increased further, spouting in a fluidized bed could be reached, as illustrated in Figure 4.21(b). This condition was denoted as a Transition regime by Dogan et al. (2000). It has characteristics of both the Jet-in-fluidized-bed regime and incoherent spouting. The spout showed pulsations at a frequency of 2 Hz, corresponding to the bubbling, as shown in Figure 4.20. A frequency from 5 to 10 Hz could also be discerned.  72  Chapter 4. Flow Regimes  4.2 ∆P, (kPa)  4.1 4 3.9  (a)  3.8 3.7 3.6 0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 140 120 100 80 60 40 20 0  6.738 (Hz)  (b)  0  5  10 Frequency, f, (Hz)  15  20  Figure 4.16: Fluctuations of pressure drop in Incoherent Spouting regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 450 mm; Ug = 0.965 m/s.  7.4  Glass beads,dp=1.33 mm α=300 mm, β=100 mm Hs=400 mm, Slot width: λ  Peak Frequency, Fpeak, (Hz)  7.2  4 mm, increasing gas 4 mm, decreasing gas 6 mm, increasing gas 6 mm, decreasing gas  7  6.8  6.6  6.4  6.2 0.80  0.82  0.84  0.86 0.88 0.90 0.92 0.94 Superficial gas velocity, Ug, (m/s)  0.96  0.98  1.00  Figure 4.17: Frequency evolution of pressure fluctuations with Incoherent Spouting regime. Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads.  73  Chapter 4. Flow Regimes  (a) Front view of in- (b) Sidecoherent spout view of incoherent spout  (c) Slugging  ∆P, (kPa)  Figure 4.18: Incoherent spouting and slugging observed in Column C: 300 × 100 mm, with diverging base; Slot: type a; 1.33 mm glass beads. 6.4 6.3 6.2 6.1 6 5.9 5.8 5.7 5.6  (a)  0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 140 120 100 80 60 40 20 0  3.320 (Hz) (b)  0  5  10 Frequency, f, (Hz)  15  20  Figure 4.19: Fluctuations of pressure drop for Slugging regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 5 mm width; Glass beads of 1.33 mm; Hs = 600 mm; Ug = 0.912 m/s.  74  Chapter 4. Flow Regimes  8.85 ∆P, (kPa)  8.8 8.75 8.7  (a)  8.65 8.6 8.55 0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 35 30 25 20 15 10 5 0  1.953 (Hz)  (b)  0  2  4 6 Frequency, f, (Hz)  8  10  Figure 4.20: Fluctuations of pressure drop for Jet in Fluidized Bed regime. Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads; Hs = 600 mm; Ug = 0.853 m/s.  (a) Jet in fluidized (b) Spouting in flubed idized bed  Figure 4.21: Jet and spouting fluidized bed observed in Column C: 300 × 100 mm, with diverging base; Slot: type a, 4 mm width; 1.33 mm glass beads.  75  Chapter 4. Flow Regimes  4.2.9  Unstable conditions (US)  Irrecoverable asymmetric conditions In some cases, the spout and fountain settled in one corner of the column, as shown in Figure 4.22. The pressure fluctuated significantly at a frequency of 6 Hz, as shown in Figure 4.23. This regime normally appeared in a deep bed (Hs ≥ 400 mm), with a dense fountain swaying normal to the slot length. As in the swaying regimes, this condition may be triggered by minor asymmetry of the experimental set-up. However, in this condition, the gas/particle flow was so dense that the annulus was not able to push the spout back to the centre. The flow could only return to be symmetric at a much smaller gas velocity near Ums .  (a) Front view  (b) Side view  (c) Top view  Figure 4.22: Irrecoverable asymmetric conditions observed in Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads.  76  ∆P, (kPa)  Chapter 4. Flow Regimes 4.2 4.1 4 3.9 3.8 3.7 3.6 3.5 0  2  4  6  8  10  4 6 Frequency, f, (Hz)  8  10  PSD, (kPa/s2)  Time, t, (s) 35 30 25 20 15 10 5 0 0  2  Figure 4.23: Irrecoverable asymmetric fluctuation in Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 450 mm; Ug = 0.828 m/s.  Unstable conditions caused by multiple spouts As stated before, multiple spouts could merge at small static bed heights. For increased heights, a single stable spout could not be achieved. The flow appeared to fluctuate unpredictably. It is likely that internal jets were interacting. One special condition was observed in beds of 200 to 300 mm height. The flow kept switching intermittently between partial fluidization and brief spouting. The pressure fluctuations are shown in Figure 4.24. There seemed to be three dominant frequencies in this regime, one smaller and two greater. The smaller one could be the frequency at which the flow switched between the other two flow regimes, while the lager ones corresponded to the two individual regimes, in which the frequency around 4 to 5 Hz is the frequency of spouting and the frequency at 6 Hz is the frequency of bubbling. This regime normally appeared beyond the multiple spout regimes, and also existed for dimensions where no multi-spouts were observed. Beyond this regime, a swaying regime appeared. The spout could normally recover from the instabilities, but for large 77  ∆Pmax, (kPa)  Chapter 4. Flow Regimes 4.15 4.1 4.05 4 3.95 3.9 3.85 3.8 3.75 3.7 0  2  4  6  8  10  4 6 Frequency, f, (Hz)  8  10  Time, t, (s)  PSD, (kPa/s2)  30  Glass beads,dp=1.33 mm α=300 mm, β=100 mm λ=2 mm, Hs=450 mm Ug=0.789 m/s  25 20 15 10 5 0 0  2  Figure 4.24: Irrecoverable asymmetric fluctuation in Column C: 300 × 100 mm, with diverging base; Slot: type a, 2 mm width; 1.33 mm glass beads; Hs = 450 mm; Ug = 0.789 m/s.  slot (λ ≥ 6 mm), it underwent a transition to the slugging regime directly.  4.3 4.3.1  Flow regime map Evolution of flow regime  In “standard” spouting, measurements always started from a fixed bed. Near the point of maximum pressure drop, an internal jet was initiated. For deep beds (Hs ≥ 500 mm), there could be a Jet-in-Fluidized-bed regime before the onset of spouting. In the current study, for some wide slots (λ ≥ 8 mm) and deep beds (Hs ≥ 300 mm), the column could not be spouted. Instead, there was a direct transition to the unstable or slugging regime. For wide slots (λ ≥ 6 mm) and shallow beds (Hs ≤ 200mm), multiple spouts could appear. With increasing gas velocity, the multiple spouts started to merge. If the merging failed, there was a direct transition to slugging. Otherwise, spouting resulted with high 78  Chapter 4. Flow Regimes frequency fluctuations. If a single smooth spout formed, it normally started to sway parallel to the slot length, then normal to the slot length. The flow then calmed down to steady state. For bed depths less than 450 mm, steady state spouting was sustainable over a considerable range of gas velocities. For deeper beds, the spout became denser and denser, leading to incoherent spouting.  4.3.2  Regime maps  Regime maps provide information for the operation of spouted beds. Regime maps for slot-rectangular spouted beds of small scale were presented by Dogan et al. (2004) and Freitas et al. (004b). A regime map for column C with a 300 × 100 mm cross-section, glass beads of 1.33 mm diameter and a type a slot of 4 mm width is plotted in Figures 4.25 and 4.26. Because of the hysteresis, regime maps are plotted for both increasing and decreasing gas flow. While the maps are similar, there are differences in the boundaries of the two maps.  4.4  Statistical analysis of flow regimes  Cycle frequency is defined as the number of times a signal passes through its mean value. Because of the natural frequency of the pressure transducer, the cycle frequency for pressure was normally much higher than the fluctuation frequency of pressure drop. The distribution of cycle frequencies is plotted for each flow regime in Figure 4.27. The fixed bed showed the widest distribution. Internal jet and Jet-in-fluidized bed regimes showed similar distributions, as did the swaying regimes. The cycle frequency for steady state spouting mainly ranged from 7 to 25 Hz. The incoherent regime had cycle frequencies from 3 to 9 Hz. Multiple spouting and merging conditions showed similar distributions  79  Chapter 4. Flow Regimes  Figure 4.25: Regime map for decreasing gas velocity. Column: 300 × 100 mm, with diverging base, type a slot, 4 mm width, 1.33 mm glass beads.  Figure 4.26: Regime map for increasing gas velocity. Column: 300 × 100 mm, with diverging base, type a slot, 4 mm width, 1.33 mm glass beads.  80  Chapter 4. Flow Regimes  Fixed bed  Internal jet  Chance of appearance, %, (0 - 100%)  Jet in fluidized bed  Swaying normal to slot  Swaying along slot  Steady state spouting  Incoherent spouting  Merging of multiple spouts  Multiple spouting  Slugging  0  5  10  15 20 Cycle frequency (Hz)  25  30  Figure 4.27: Comparison of the distribution of cycle frequency for different flow regimes.  81  Chapter 4. Flow Regimes (6 -13 Hz). The cycle frequency for slugging had a relatively narrow distribution at lower frequencies. In Figure 4.28, the multiple spout merging, incoherent spout and the slugging regimes showed fluctuation of higher amplitude. The other regimes showed similar distributions between 0 and 0.05 %. This may imply that the magnitude of fluctuation was independent of the average pressure drop. Both frequency analysis and statistical analysis showed that the flow regimes had some unique characteristics. Nevertheless, it was not possible to determine the flow regime from a single factor. For example, some of the distributions overlapped. Determination of the flow regime without direct visualization requires that frequency and statistical analysis be accompanied by measurement of the evolution of pressure drop.  82  Chapter 4. Flow Regimes  Fixed bed  Internal jet  Chance of appearance, %, (0 - 100%)  Jet in fluidized bed  Swaying normal to slot  Swaying along slot  Steady state spouting  Incoherent spouting  Merging of multiple spouts  Multiple spouting  Slugging  0.00  0.05  0.10  0.15  0.20  0.25  0.30  0.35  -Magnitude of fluctuation, STD(∆P)/∆P Figure 4.28: Comparison of distribution of magnitude of fluctuations for different flow regimes.  83  Chapter 5 Slot-Rectangular Spouted Bed Stability 5.1  Approaches to analyze stability  Stability of the slot-rectangular spouted beds can be analyzed from several viewpoints. • Response to a small disturbance. If the system returns to its previous state, it is stable; if it moves away, it is unstable. This is the classic definition of stability. In this work, a disturbance was not applied. However, hysteresis between flow increasing and decreasing the gas indicates that the spouted bed is not a stable system according to this definition. Another example is that as soon as the system reaches the incoherent spouting flow regime, the gas velocity needs to be decreased much more to recover continuous spouting, as indicated in Figure 4.17. • Flow status over time. A system is stable if it can sustain constant operating conditions. According to this definition, the system is unstable if there are intermittently spouting and collapsing conditions. • Symmetry of flow. Because the spout and internal spout are not visually observable, the stability can only be judged from the fountain. In the swaying regimes, the flow at any instant is asymmetric and the system can be considered to be unstable. • The spouted bed is intended to feature a dilute up-flow region and a dense downflow region, in which different processes can be applied. To achieve this, there should  84  Chapter 5. Slot-Rectangular Spouted Bed Stability first be a permanent jet or spout. Second, smooth spouting with minimal fluctuations is preferred. However, the flow in the slot-rectangular spouted beds is always accompanied by fluctuations. High fluctuations can prevent the bed from being spouted. In the spouting flow regimes, high fluctuations can also lead to slugging. From this aspect, steady state spouting with minimal fluctuations is considered to be most stable. The slugging regime and multiple spouts merging regime have higher fluctuations, and are therefore considered to be most unstable. In the current work, stability is mainly based on the last three points. This means that smooth symmetric flow with minimal fluctuations is considered most stable, whereas flow with greater fluctuations is considered unstable.  5.2  Effect of slot configuration  The results with “extended normal slots” and “standard rectangular slots” in a column of 300 mm width and 100 mm thickness were presented in Chapter 3. The minimum spouting velocities and maximum pressure drops for these two kinds of slots were directly related to the slot configuration. A correlation derived for slots of one type is unlikely to apply to slots of another type. In this section, the effect of slot dimensions, including width, length and slot depth, as well as the length/width ratio and the slot expansion angle, are investigated to find the slot configuration able to provide the most stable flow. A typical drawing of a slot appears in Figure 5.1, slot width, length and depth are denoted as λ, η and κ respectively. The length of slot, η, is not necessarily the largest slot dimension, i.e. the width of slot under this definition could be larger than the length. Instead, the length is defined as the dimension parallel to the column thickness, and the width as the dimension parallel to the column width. All experiments were carried out in a column of 300 mm x 100 mm cross-section, so that the slot length is the dimension  85  Chapter 5. Slot-Rectangular Spouted Bed Stability  Figure 5.1: Drawing of a typical slot.  in the direction where the column dimension is 100 mm, and the slot width is in the direction in which the column dimension is 300 mm.  5.2.1  Slot width  The effect of slot width on the minimum spouting velocity and maximum pressure drop were investigated in Chapter 3. A wider slot provided a higher minimum spouting velocity and a higher maximum pressure drop. As shown in Figure 5.2, only the widest slot tested, λ = 8 mm, caused slightly higher fluctuations. The pressure drop across the slot was lower for a wider slot cross-section, as shown in Figure 5.3. The pressure drop across the slot was calculated by subtracting the pressure drop across the column from the gage pressure in the base chamber. For all the cases in this chapter, these two pressures were measured at the same positions for different slots and operating conditions.  86  __ Normalized amplitude of ∆P fluctuation, STD(∆P)/∆P  Chapter 5. Slot-Rectangular Spouted Bed Stability 0.3  Slot width (mm) 2 4 8  0.25  0.2  0.15  0.1  0.05  0 0  0.1  0.2 0.3 0.4 0.5 0.6 0.7 Superficial gas velocity, Ug (m/s)  0.8  0.9  Figure 5.2: Comparison of relative magnitudes of pressure fluctuations for slots of different width. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot length: standard length of 100 mm, diverging base.  Average pressure drop across slot (kPa)  25  Slot width (mm) 2 4 8  20  15  10  5  0 0  0.1  0.2  0.3 0.4 0.5 0.6 Superficial gas velocity (m/s)  0.7  0.8  0.9  Figure 5.3: Comparison of pressure drops across slots of different width. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot length: standard length of 100 mm, diverging base.  87  Chapter 5. Slot-Rectangular Spouted Bed Stability  5.2.2  Slot length  The slot lengths (see Figure 5.1) in Chapter 3 were all of “standard length”, i.e. equal to the thickness of the column. This length of slot has also been common in earlier research. As noted in Chapter 4, multiple spouts were found when there was a low static bed height. One cause of multiple spouting was suspected to be the length of the slot. With a shorter slot, the gas enters through a smaller area, and it is easier for the bed to spout. For a long slot, gas is more widely dispersed. As observed in Chapter 4, two separate spouts tend to stay attached to the column walls when the slot covers the whole thickness of the column. For shorter slots with gaps at both ends so that the spouts were not able to attach to the column wall, multiple spouts were forced to interact with each other. This caused unstable conditions. Four slots are examined in this section, all with the same width of 4 mm, but different lengths of 40, 60, 80 and 100 mm. The experiments were carried out in a column of 300 × 100 mm cross section with a diverging base. Slots with length η ≤ 60 mm were observed to provide steady state spouting. No steady state spouting was observed for slots of η ≥ 80 mm. Spouting was less unstable for a slot of 80 mm length than for a slot of 100 mm length. Incoherent spouting could be achieved for the 80 mm slot. However, for the 100 mm slot, there were always unstable conditions such as slugging or partial fluidization. The minimum spouting velocity and maximum pressure drop of these slots are plotted in Figures 5.4 and 5.5. The shortest slot (40 mm) caused the lowest Ums and the lowest ∆Pmax . This means that beds with shorter slots can be more easily spouted than those with longer slots. The crossing of curves is likely due to measurement errors and to the small Hs values, resulting in bed surfaces which were not exactly horizontal. Another factor could slight non-verticality of the column. Fluctuations caused by shorter slots were also smaller, as shown in Figure 5.6. Pressure  88  Chapter 5. Slot-Rectangular Spouted Bed Stability  Minimum spouting velocity (m/s)  0.7  Slot length (mm) 40 60 80 100  0.6  0.5  0.4  0.3  0.2  0.1 0  0.05  0.1 0.15 0.2 Static bed height (m)  0.25  0.3  Figure 5.4: Comparison of minimum spouting velocities for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base.  4  Slot Length (mm) 40 60 80 100  Maximum pressure drop (kPa)  3.5 3 2.5 2 1.5 1 0.5 0  0.05  0.1 0.15 0.2 Static bed height (m)  0.25  0.3  Figure 5.5: Comparison of maximum pressure drops for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base.  89  __ Normalized amplitude of ∆P fluctuation, STD(∆P)/∆P  Chapter 5. Slot-Rectangular Spouted Bed Stability  0.18  Slot length (mm) 40 60 80 100  0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0  0.1  0.2 0.3 0.4 0.5 0.6 0.7 Superficial gas velocity, Ug (m/s)  0.8  0.9  Figure 5.6: Comparison of magnitudes of pressure fluctuation for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot width: 4 mm, diverging base.  Average pressure drop across slot (kPa)  18  Slot length (mm) 40 60 80 100  16 14 12 10 8 6 4 2 0 0  0.1  0.2  0.3 0.4 0.5 0.6 Superficial gas velocity (m/s)  0.7  0.8  0.9  Figure 5.7: Comparison of pressure drops across slot for slots of different length. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot width: 4 mm, diverging base.  90  Chapter 5. Slot-Rectangular Spouted Bed Stability drops across the slots are compared in Figure 5.7. As expected, shorter slots generally show higher pressure drops than longer slots of the same width. However, slot of 100 mm length showed higher pressure drop than that of 80 mm length, which may be caused by the manufacture error. There could be several different explanations for the influence of slot length. • Mean gas velocity at slot: A shorter or narrower slot causes higher gas velocities at the slot outlet for a given superficial gas velocity in the column. • Gas distribution at the slot: Slots of greater length likely caused greater variations of velocity along the slot, causing the flow to be more unstable. • Proximity of the column wall. Spouts farther from the wall could be more stable. To avoid the effect of the mean gas velocity at slot, slots of equal cross-sectional area but different widths and lengths were investigated. For the same superficial gas velocity, these slots provided the same mean gas velocity through the slot outlets.  5.2.3  Slot length-to-width ratio  For a constant cross-sectional area, the slot length-to-width ratios were changed. Six rectangular slots were designed with length/width ratios of from 0.25 to 25 with a constant cross-sectional area of 400 mm2 and a constant slot depth of 12.7 mm. A circular orifice with the same cross-sectional area was also used for comparison. Dimensions of the slots are listed in Table 5.1. In each case, the slot was installed with its central (bisecting) axes collinear with the central (bisecting) axes of the column. It was observed that slots with aspect ratio smaller than or equal to 8 could spout stably. With higher length/width ratio, the column was unable to spout, with the flow being more like slugging. Considering that 4 × 60 mm slot was able to spout (see Section 5.2.2), there could be a limit on the slot’s length-to-width ratio ∼ 15. 91  Chapter 5. Slot-Rectangular Spouted Bed Stability Table 5.1: Slots with different length/width ratios in 300 × 100 mm column  Shape  Circular  Rectangular  Length/Width  N.A.  0.25  Length (mm)  22.5  10  Width (mm)  22.5  40  1  2  4  8  16  25  20 28.2  40  56.5  80  100  20 14.1  10  7.1  5  4  Area (mm2 )  400  Depth (mm)  12.7  Perimeter (mm)  71  100  80  85  Hydraulic Diameter (mm)  22.5  16  20 18.8  100 127.2 170 208 16  12.5  9.4  7.7  Swaying of the fountain for these slots was normally in the direction of the major dimension of the slot for small gas velocities near Ums . In only a few cases was the swaying in the direction normal to the major dimension of the slot. These cases were for deep beds and large slot length/width ratio. This differs from the results for slots with extended depth, where both modes of swaying occurred. One experiment was carried out without a diverging base. In this condition, swaying was significant in both directions. As noted in Chapter 4, swaying normal to slot occurred at higher Ug than swaying parallel to the slot. The diverging base appeared to play an important role in preventing the swaying normal to the major dimension of the slot. The minimum spouting velocities, maximum pressure drops and spouting pressure drops of these slots are compared in Figure 5.8. Ums reached a minimum when the slot length was equal to the slot width. Ums increased when the slot shape deviated further from a square, i.e. as the hydraulic diameter decreased. For slots further from square, gas probably spreads more widely, so that more gas is needed to penetrate the particle 92  Chapter 5. Slot-Rectangular Spouted Bed Stability layer.  Minimum spouting velocity (m/s)  1.2  1  Hs (mm) 100 200 300 400  0.8  0.6  0.4  0.2 0.125  0.25  0.5 1 2 4 Length-to-width ratio of slot  8  16  Figure 5.8: Comparison of minimum spouting velocities for slots of different length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. Horizontal lines are corresponding results for circular orifice.  As shown in Figures 5.9 and 5.10 respectively, ∆Pmax and ∆Ps , the pressure drop at Ums , showed similar dependence on slot length/width ratio to Ums . A slot closer to square provided a lower pressure drop, suggesting that less gas passes through the annulus. The voidage in the spout may also be affected. One explanation is that slots of length/width ratio further from 1 have larger perimeter, providing more gas/particle contact area so that more particles are entrained into the spout, causing an increase in pressure drop. This is also consistent with visual observations. Overall the results show that for the same cross-sectional area, a slot with a greater hydraulic diameter can cause spouting more readily than one with a smaller hydraulic diameter. The circular inlet usually provided lower Ums and lower pressure drops. However, rectangular slots with aspect ratio close to 1 provided slightly smaller minimum spouting velocities than the circular inlet. This could be due to the measurement errors in the experiments or to minor discrepancies in manufacturing the square and/or circular 93  Chapter 5. Slot-Rectangular Spouted Bed Stability  Maximum pressure drop (kPa)  10  Hs (mm)  8  100 200 300 400  6  4  2  0 0.125  0.25  0.5 1 2 4 Length-to-width ratio of slot  8  16  Figure 5.9: Comparison of maximum pressure drops for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. Horizontal line are corresponding results for circular orifice.  inlet. Another interesting comparison involves the results for slots of length/width ratio of 0.25 and 4. These have the same shape and dimensions, but were oriented in different directions relative to the column. From Figures 5.9 and 5.10, they have almost the same pressure drops. However, the slot with a length-to-width ratio of 0.25 had a higher Ums . Given that the column had a cross-section of 300 mm x 100 mm, we see that the behavior must also be affected by the distances between the end of the slot and the wall. As shown in Figure 5.11, slots with length/width ratio ≤ 8 had smaller fluctuations, consistent with the observation that ratios ≤ 8 led to greater spout stability. The comparison of “stable slots” showed that fluctuations were slightly smaller for slots with higher length/width ratio. Pressure drops across the slots of different length/width ratios are plotted in Figure 5.12. The circular inlet and square slot had the same pressure drop. The pressure drop decreased as the length/width ratio increased. The measured relationship between 94  Chapter 5. Slot-Rectangular Spouted Bed Stability  Spouting pressure drop (kPa)  6  Hs (mm) 100 200 300 400  5  4  3  2  1  0 0.125  0.25  0.5 1 2 4 Length-to-width ratio of slot  8  16  __ Normalized amplitude of ∆P fluctuation, STD(∆P)/∆P  Figure 5.10: Comparison of spouting pressure drops for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. Horizontal line are corresponding results for circular orifice.  0.14  Slot Length / Width 0.25 Circular 1 2 4 8 16 25  0.12 0.1 0.08 0.06 0.04 0.02 0 0.1  0.2  0.3 0.4 0.5 0.6 0.7 0.8 Superficial gas velocity, Ug (m/s)  0.9  1  Figure 5.11: Comparison of magnitudes of pressure fluctuations for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base.  95  Chapter 5. Slot-Rectangular Spouted Bed Stability  Average pressure drop across slot (kPa)  8  Slot Length / Width 0.25 Circular 1 2 4 8 16  7 6 5 4 3 2 1 0 0.1  0.2  0.3  0.4 0.5 0.6 0.7 0.8 Superficial gas velocity, Ug (m/s)  0.9  1  Figure 5.12: Comparison of pressure drops across slot for different slot length/width ratios but equal cross-sectional area of 400 mm2 . Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base.  pressure drop across slots and slot configurations is consistent with previous results in other configurations, as in Chapter 5.2.2, where higher pressure drop at the slot provided more stable flow.  5.2.4  Slot depth  Slot depth, κ, is defined as the height of the channel from the inlet to the outlet of the slot, as shown in Figure 2.5. For “standard rectangular slots” in this thesis, the depth of the slot was 1/2” (12.7 mm), equal to the thickness of the plexiglas used to make the base plate. Parallel plates were used to extend the channel to depths of 22, 33, 40, 50, and 62.7 mm. The slot width was 4 mm in all cases. For these tests, the slot length was equal to the thickness of the column, β, in all cases. These experiments were carried out in column C with glass beads of 1.33 mm diameter. Deeper slots were observed to provide greater stability. There was a large difference between slot depths of 22 and 33 mm. Steady spouting was never observed for slots 96  Chapter 5. Slot-Rectangular Spouted Bed Stability of depth 22 mm or less. Instead, the flow regime was mainly incoherent spouting and slugging. Spouting terminated at a static bed height of 300 mm. For slots of depth greater than 33 mm, a single steady state spout could always be observed at a specific gas velocity. An unstable condition, with the fountain forming and collapsing intermittently was also found for the slot of 33 mm depth. As shown in Figure 5.13, slots of 12.7 mm depth showed a lightly higher maximum pressure drop than others having depth ≥ 22 mm. This result suggests that slots of smaller depths cause slightly higher maximum pressure drops. Minimum spouting veloc-  Maximum pressure drop (kPa)  6  5  4  3 Slot depth (mm) 12.7 22 33 40 50  2  1  0 0  0.05  0.1  0.15 0.2 0.25 Static bed height, (m)  0.3  0.35  0.4  Figure 5.13: Comparison of maximum pressure drops for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base.  ities are plotted in Figure 5.14. Ums for different slot depths are the same for shallow beds (not extending beyond the diverging conical base). When the bed height exceeded 200 mm, Ums differed significantly between slots with depth ≤ 22 and those ≥ 33 mm. The magnitudes of ∆P fluctuations caused by these slots are compared in Figure 5.15. Slot depths ≤ 22 mm led to higher pressure drop fluctuations. Pressure fluctuations were smaller for slots of depth ≥ 40 mm. This result was consistent with visual observations. The pressure drops across the slots are compared in Figure 5.16. The pressure drop 97  Chapter 5. Slot-Rectangular Spouted Bed Stability  Minimum spouting velocity (m/s)  0.8 0.7 0.6 0.5 0.4  Slot depth (mm) 12.7 22 33 40 50 62.7  0.3 0.2 0.1 0  0.05  0.1  0.15 0.2 0.25 Static bed height, (m)  0.3  0.35  0.4  Figure 5.14: Comparison of minimum spouting velocities for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, slot width: 4 mm, diverging base.  __ Magnitude of fluctuation, STD(∆P)/∆P  0.18  Slot Depth (mm) 12.7 22 33 40 50 62.7  0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0  0.2  0.4 0.6 0.8 Superficial gas velocity, Ug (m/s)  1  Figure 5.15: Comparison of magnitudes of pressure fluctuation for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, slot width: 4 mm, diverging base.  98  Chapter 5. Slot-Rectangular Spouted Bed Stability  Average pressure drop across slot, (kPa)  9  Slot depth (mm) 12.7 22 33 40 50  8 7 6 5 4 3 2 1 0 0  0.2  0.4 0.6 Superficial gas velocity, Ug (m/s)  0.8  1  Figure 5.16: Comparison of pressure drops across slot for slots of different depth. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 200 mm, slot width: 4 mm, diverging base.  for a deeper slot is smaller, except for the slot of depth 62.7 mm, which consists of two separate parts and the slot width of which may change when there is gas flow. Considering the longer slots provided greater stability, this trend is reverse to those for slot lengthto-width ratio. There could be other factors affecting the stability, e.g. the gas velocity distribution, discussed in Section 5.3 below.  5.2.5  Converging and diverging angle  By changing the width of the slot inlet or outlet, it is possible to change the gas distribution at the bottom of the column. Converging orifices have been used in conventional conical spouted beds (Olazar et al., 1992) in accord with the suggestion by Mathur and Epstein (1974) that a converging orifice will be more suitable than an abrupt change in size from the gas supply pipe to the inlet. In the current work, the same technique was adopted not only for converging slot but also for a diverging slot to smooth the outlet. Four slots are compared: 99  Chapter 5. Slot-Rectangular Spouted Bed Stability 1. Converging slot, with outlet width of 4 mm and inlet width of 6 mm; 2. Straight slot, with constant width of 4 mm; 3. Two diverging slots, one with an outlet width (6 mm) bigger than the inlet width (4 mm), (simply the inverted converging slot), and the other with 2 mm as the inlet width and 4 mm as the outlet width. These slots all had one width of 4 mm. All of these slot had standard length (equal to the thickness of the column). The minimum spouting velocity varied somewhat for the different slots, as shown in Figure 5.17. For a low static bed height, Hs ≤ 200 mm, the diverging slot had the smallest Ums and the converging slot the highest Ums . The minimum spouting velocities were similar for deeper beds.  Minimum spouting velocity (m/s)  0.8  Slot width: Inlet / Outlet 6 mm / 4 mm 4 mm / 4 mm 4 mm / 6 mm  0.7  0.6  0.5  0.4  0.3  0.2 0  0.05  0.1 0.15 0.2 Static bed height (m)  0.25  0.3  Figure 5.17: Comparison of minimum spouting velocities for converging and diverging slots. Column C: 300 × 100 mm, 1.33 mm glass beads, diverging base. As indicated in Figure 5.18, before the bed was spouted, the diverging slot showed smaller fluctuations. After the onset of spouting, the fluctuations of diverging slots also appeared to be much smaller. 100  __ Normalized amplitude of ∆P fluctuation, STD(∆P)/∆P  Chapter 5. Slot-Rectangular Spouted Bed Stability  0.18  Slot width: Inlet / Outlet 6 mm / 4 mm 4 mm / 4 mm 4 mm / 6 mm 2 mm / 4 mm  0.16 0.14 0.12  Mean Ums 0.1 0.08 0.06 0.04 0.02 0 0  0.1  0.2  0.3 0.4 0.5 0.6 0.7 Superficial gas velocity (m/s)  0.8  0.9  1  Figure 5.18: Comparison on magnitudes of pressure fluctuation for converging and diverging slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base.  Average pressure drop across slot (kPa)  9  Slot width: Inlet / Outlet 6 mm / 4 mm 4 mm / 4 mm 4 mm / 6 mm 2 mm / 4 mm  8 7 6 5 4 3 2 1 0 0  0.1  0.2  0.3 0.4 0.5 0.6 0.7 Superficial gas velocity (m/s)  0.8  0.9  1  Figure 5.19: Comparison of pressure drops across slot for converging and diverging slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 250 mm, diverging base.  101  Chapter 5. Slot-Rectangular Spouted Bed Stability As shown in Figure 5.19, the pressure drop for the diverging slot with inlet and outlet widths of 4 and 6 mm was a little higher than for the converging slot. Diverging slot with 2 mm inlet and 4 mm outlet should have the highest pressure drop. However, straight slot with same inlet and outlet widths showed the highest pressure drop which may be caused by slight difference in the slots as fabricated. The results suggest that the diverging slot can stabilize the spout by providing higher pressure drop across the slot and slightly smaller spouting pressure fluctuations. It was more effective at lower bed depths.  5.3  Slot velocity distributions  The last section suggests that slot configurations with higher pressure drops can provide more stable flow. However, for slots of different depth and geometry, there are exceptions. In these cases, gas flow distribution must play an important role in initiating spouts. Because creation of a spout is related to gas momentum, more concentrated flow facilitates forging of a spout. For slots of different outlet size, e.g. constant width and various lengths, smaller outlet produces higher gas velocity. For slots of same outlet size and different lengthto-width ratio, the smaller ratio provides more concentrated gas velocity at the centre. However, for slots of same outlet width and length but different depth or geometry, calculating the gas velocity distribution can be helpful to understand the flow. Because of the complexity of modeling 3-dimensional gas/solid flow, the gas flow distributions for the slot configurations of interest were calculated with FLUENT 2.6 using the K −  turbulence model. The grid size was set to be 0.5 mm in all cases. The gas velocity at the slot inlet was supposed to be evenly distributed. In all the cases, gas entered the slot inlet with a pre-calculated value to cause 0.66 m/s in the column of 300 × 100 mm cross-section. The outlet of the slot was set to be at ambient pressure. The gas 102  Chapter 5. Slot-Rectangular Spouted Bed Stability distribution at the outlet of the slot was then calculated. First, straight slots of depth 62.7 mm and 12.7 mm are compared. Gas velocity profiles along two central lines are plotted in Figure 5.20, normal to the slot length, and in Figure 5.21 parallel to the slot length. 60  Gas velocity, vg (m/s)  50  40 Slot depth (mm) 62.7 12.7  30  20  10  0 -2  -1.5  -1  -0.5 0 0.5 Distance from center, (mm)  1  1.5  2  Figure 5.20: CFD predicted distribution of gas velocity in the direction normal to the slot. Average outlet velocity: 0.66 m/s. Comparison of two straight slot of different depth.  The maximum gas velocity at the centre is predicted to be 10% higher for the deeper slot than for the shallower one. This could help to create more stable spouting. The pressure drop across the slot is compared in Figure 5.22. The pressure drop for deeper slot is higher than for shallower slot. The pressure drop in this calculation is mainly caused by the friction of the channel wall. Deeper slot has longer wall, which will cause higher pressure drop. This trend is consistent with the experiment results, suggesting that the higher pressure drop across the slot provide greater stability. Calculations were also carried out to compare flow in converging, straight and diverging slots. The diverging slot had an inlet width of 2 mm and an outlet width of 4 mm, giving an included angle of 9◦ . The converging slot had the same included angle, but 103  Chapter 5. Slot-Rectangular Spouted Bed Stability 60  Gas velocity, vg (m/s)  50  40 Slot depth (mm) 62.7 12.7  30  20  10  0 -60  -40  -20 0 20 Distance from center, (mm)  40  60  Figure 5.21: Distribution of gas velocity along the centreline of the slot. Average outlet velocity: 0.66 m/s. Comparison of two straight slot of different depth.  an inlet of 6 mm width and an outlet of 4 mm width. The straight slot had a constant width of 4 mm. Hence all these had the same outlet of width 4mm and length 100 mm. Predicted velocity profiles are shown in Figures 5.23 and 5.24. The diverging slot was predicted to have the highest maximum gas velocity. Gas distribution was more concentrated. The calculations suggest that a more concentrated gas velocity distribution can provide a more stable flow. The maximum gas velocity caused by the slot seems to affect the flow stability.  5.4 5.4.1  Termination of spouting Mechanisms  Three mechanisms of spout termination were suggested by Mathur and Epstein (1974). 1. Fluidization of annular solids 2. Choking of the spout 104  Chapter 5. Slot-Rectangular Spouted Bed Stability  Pressure drop across the slot, (Pa)  350 300 250 200  Slot depth (mm) 62.7 12.7  150 100 50 0 -40  -20 0 20 Distance from center, (mm)  40  Figure 5.22: Pressure drop across the slot along the centreline of the slot. Average outlet velocity: 0.66 m/s. Comparison of two straight slots of different depth.  60  Gas velocity, vg (m/s)  50  40 Slot type Converging Straight Diverging  30  20  10  0 -2  -1.5  -1  -0.5 0 0.5 Distance from center, (mm)  1  1.5  2  Figure 5.23: Distribution of gas velocity in the direction normal to the slot for three different geometries. Average outlet velocity: 0.66 m/s.  105  Chapter 5. Slot-Rectangular Spouted Bed Stability 60  Gas velocity, vg (m/s)  50  40 Slot type Converging Straight Diverging  30  20  10  0 -60  -40  -20 0 20 Distance from center, (mm)  40  60  Figure 5.24: Distribution of gas velocity along the centreline of the slot for three different geometries. Average outlet velocity: 0.66 m/s. 3. Development of instabilities In this work, mechanism 1 occurred in columns of static bed depth > 600 mm for glass beads of 1.33 mm diameter. In a column equipped with a type a slot of small width, this condition did not lead to termination of spouting. Instead, the flow was more like an incoherent spout in a fluidized bed. For a wider slot and higher gas velocity, this could lead to slugging. This mechanism is more significant for 2.42 mm glass beads and 3.79 mm polystyrene particles, where the spouting terminated because of fluidization of the bed surface. Choking of the spout was also observed particularly with slots of intermediate length/width ratio. The flow passed directly from swaying to slugging. Before the spout collapsed, the voidage in the fountain was very small. Swaying appeared to be one way the spouted bed avoided overloading particles in the spout and fountain. Most spout termination in this work was caused by growth of instabilities, mainly associated with the interaction between multiple spouts. This instability was initiated at the bottom of the column, broadening the distribution of gas velocity. The flow 106  Chapter 5. Slot-Rectangular Spouted Bed Stability underwent a direct transition from internal spout to slugging or fluidization, limiting the maximum spoutable bed height.  5.4.2  Maximum spoutable bed height  The maximum spoutable bed height, Hmax , was not precisely measured in this work. However, it was clear that Hmax depended strongly on the slot configuration. Hmax appears to show a bimodal distribution. Those slots that could form a single steady spout could be spouted to static bed heights up to 800 mm. However, when multiple spouts formed, spouting typically terminated at a static bed height ∼ 300 mm.  107  Chapter 6 Local Flow Structure 6.1  Measurement issues  Local flow properties were measured at steady state in the α = 300 mm, β = 100 mm column, with slots of different length-to-width ratio to investigate the effect of slot configuration. Glass beads of 1.33 mm diameter and room temperature air were used in all these experiments. The measurements were only carried out for conditions where the flow was steady, i.e. the position and size of spout and the voidage in the spout were invariant with time. The local flow properties were measured one point at a time since it was not possible to handle more than one measurement point simultaneously. Local voidage distribution in a slot-rectangular spouted bed has been investigated by Freitas et al. (004a) and was proved to be three-dimensional. As a result, measurements should be obtained at hundreds, or even thousands of points to reveal the full status of the flow. However, in this work, the flow properties were only measured at two vertical surface, orthogonal to each other, intersecting at the vertical axis of the column. One direction is parallel the direction of the slot, measured by probes inserted along the dotted line shown in Figure 6.1. The other is normal to the direction of the slot, as shown by the dashed line in Figure 6.1. Y and X indicated the distance from the centre in these two directions, respectively. For points with X ≤ 0, probes were inserted from the west (left) in Figure 6.1, and east (right) for X ≥ 0, north for Y ≥ 0 and south for Y ≤ 0. The column was flat-bottomed for the convenience of measuring from different sides. It also provided the possibility of determining the position of the annulus/dead zone interface.  108  Chapter 6. Local Flow Structure  Figure 6.1: Top view of column showing where the probes were inserted.  Because of minor asymmetries in the experimental set-up, the flow in the column was not always exactly symmetric. Instead, the centre of the spout did not always coincide with the centre of the column. Axial profiles in this chapter are measured from the axis of the column. For each point in the measurement, the particle velocity and voidage were measured in 4 to 8 segments, each including 32768 (215 ) data points, with the sampling period depending on the sampling frequency which varied from 125 kHz to 2,000 kHz. The average value for the 4 to 8 measurements with 90% confidence level were adopted as the particle velocity and voidage at that point. For particle velocities, the 90% confidence interval are plotted in every figure. For voidage the 90% confidence interval based on successive measurement intervals of ≈ 1 s was found to be less than ±2% of the average. Given this small value, confidence intervals are not plotted on voidage plots. The probe inserted into the column influenced on the flow in the spout. It caused the particle velocity to decrease, voidage to increase and the spout to move away. The influence is more significant for the points near the spout centre and less significant for the points near the spout/annulus interface. In the current work, the measurements for each point were carried out successively without moving of the probes. The main cause  109  Chapter 6. Local Flow Structure of measurement error is the fluctuation of particle velocity in the spout. However, if the probes were moved in a random measurement, there could be larger effective errors on both the particle velocity and the voidage from the influence of probe on the flow.  6.2 6.2.1  Shape of spout and dead zone Measurement method  The spout/annulus interface was determined based on where the frequency of the voidage probe signal changed abruptly, the same method as was employed by He (1995). A voidage optical probe of 5 mm O.D. connected to a PC4 voidage analyzer was inserted from two directions, as shown in Figure 6.1, into the column. The signal was collected by a DAS-08 A/D data acquisition board. A C++ program read the signal and analyzed real time frequency. Sample series of the signal are shown in Figures 6.2. The signal changed significantly when the probe was moved from X = 13 (annulus) to 11 mm (spout). The signal at X = 12 mm was transitional. The power spectrum distribution of signal, as shown in Figure 6.3, in annulus showed appreciable signals at frequencies < 20 Hz. However, for the spout, no such frequency could be detected. The dead zone could also be delineated by the probe. When the signal is constant, with no particle movement, the probe is in a dead zone. The dead zone was only observed in the direction normal to the slot. The structure of the spout, annulus and dead zone for one set of conditions is shown in Figure 6.4.  6.2.2  Spout shape and size  Spout shapes were measured from two directions, in “side view”, which designated the spout shape in the direction of the slot length, and “front views”, showing the shape normal to the slot. 110  Signal, (v)  Chapter 6. Local Flow Structure  3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1  X=11 mm  X=12 mm  X=13 mm  0  0.5  1 Time, (s)  1.5  2  Figure 6.2: Sample voidage signal; X is the distance from the axis, Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 300 mm, Z = 50 mm, Ug = 0.8 m/s, Slot: 10 × 40 mm, flat base.  A variety of spout shapes have been summarized by Mathur and Epstein (1974). In the current work, two of these spout shapes were identified: (a), “continuously diverging”, normally observed in the direction normal to the slot; and (b), “expanding, necking and expanding”. The latter normally appeared in the direction along the slot. At lower gas velocities, the second shape could also be observed in the normal direction of slot. These two shapes shared the spout expansion at the bottom of the column, where gas left the slot with high pressure and distributed into the annulus. The gas pushed the spout/annulus interface outwards causing expansion. As shown in Figures 6.5 and 6.6, the spouts for different static bed heights had similar shapes and positions, i.e. the position of spout/annulus interface at specific level did not vary significantly with static bed height. Spout shapes at different gas velocities are shown in Figures 6.7 and 6.8. The spout size is seen to have increased with increasing gas velocity.  111  PSD  Chapter 6. Local Flow Structure  4000 3000 2000 1000 4000 0 3000 2000 1000 4000 0 3000 2000 1000 0  X=11 mm  X=12 mm  X=13 mm  0  20  40 60 Frequency, (Hz)  80  100  Figure 6.3: Power spectrum of sample voidage signal in Figure 6.2; X is the distance from the axis, Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 300 mm, Z = 50 mm, Ug = 0.8 m/s, Slot: 10 × 40 mm, flat base.  300  Height coordinate, Z (mm)  250  Interface of Spout/Annulus Annulus/Dead zone  200  150  Spout  Annulus  Annulus  100 Dead zone  Dead zone  50  0 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.4: Flow structure. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs = 300 mm, Ug = 0.8 m/s, Slot: 10 × 40 mm, flat base.  112  Chapter 6. Local Flow Structure  300  Hs = 200 mm Hs = 250 mm Hs = 300 mm  Height coordinate, Z (mm)  250  200  150  100  50  0 -40  -20 0 20 Distance from column axis, Y, (mm)  40  Figure 6.5: Side view of spout shape for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Slot: 10 × 40 mm, flat base, Ug = 0.7 m/s.  300  Height coordinate, Z (mm)  250  Hs = 200 mm Hs = 250 mm Hs = 300 mm  200  150  100  50  0 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.6: Front view of spout shape for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Slot: 10 × 40 mm, flat base, Ug = 0.7 m/s.  113  Chapter 6. Local Flow Structure  300  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  Height coordinate, Z (mm)  250  200  150  100  50  0 -40  -20 0 20 Distance from column axis, Y, (mm)  40  Figure 6.7: Side view of spout shapes for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base.  300  Height coordinate, Z (mm)  250  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  200  150  100  50  0 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.8: Front view of spout shapes for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base.  114  Chapter 6. Local Flow Structure  Longitudinal position, Z (mm)  300  Ug = 0.7 m/s Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.8 m/s Ug = 0.9 m/s Ug = 0.9 m/s  250  200  150  100  50  0 10  15  20  25  30 35 40 Size of spout, (mm)  45  50  55  Figure 6.9: Comparison of spout shapes in two orthogonal directions. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot width: 10 mm, Slot length: 40 mm, flat base, dotted line is in direction normal to slot, solid line is in direction parallel to slot.  Because of the minor asymmetry of the spout, the sizes of the spout in the two directions were calculated and they are compared in Figure 6.9, where the dotted line corresponds to the size in the direction parallel to the slot, and the solid line shows the size in the normal direction. It was found that the spout behaved as if there were “surface tension”, causing the spout to shrink in size in the wider direction and expand in the direction in which it was narrower. So particles in the wider direction travelled inwards toward the centre, whereas particles in the narrower direction spread outwards from the axis. Because of the inertia, the spout size in either direction shrank and expanded alternatively. As shown in Figure 6.9, the spout started to shrink in the direction parallel to the slot after the sizes in this directions reached a maximum, and the spout size started to expand in the normal direction.  115  Chapter 6. Local Flow Structure  6.2.3  Prediction of spout shape  Development of a mechanistic model predicting the variation in the spout diameter with bed level for a two-dimensional spouted bed was discussed in detail by Krzywanski et al. (1989). The model was recently adopted by Zanoelo and Rocha (2002) for normal axisymmetric columns. The model is based on the application of the least-action principle, which means that the work done to create the spout cavity, overcoming the external forces trying to prevent it from forming, is the minimum work needed to avert the collapse of the annular bed. After mathematical transformation, it turned out that the length of the spout/annulus interface should be a minimum. The spout/annulus in a two-dimensional spouted bed can be described by  rs = C2 −  1p 1 − (λe z + C1 ) λe  (6.1)  where C1 and C2 are constants derived from the boundary conditions (rs )z=0 = ri and (drs /dz)z=0 = tan(γe ), leading to  C1 = p  tan(γe ) 1 + (tan(γe ))2  p 1 − C12 C2 = ri + λe  (6.2)  (6.3)  γe is the spout expansion angle at the bottom, as shown in Figure 6.10. γe should be determined by minimizing the length of spout/annulus interface: Z L= 0  Hs   1/2 drs 2 1+( ) dz dz  (6.4)  Another unknown parameter in Equation 6.1, λe , is a Lagrange multiplier used in  116  Chapter 6. Local Flow Structure  Figure 6.10: Structure of spout and annulus showing the spout expansion angle.  solving the model. The experimental average radius, r¯s of the spout should be used to determine this parameter. For a given γe , λe can be determined by numerically solving Z  Hs  rs dz = r¯s Hs  (6.5)  0  The MATLAB code for the Golden-section method and the Newton-Rapson method, listed in Appendix C.3, were employed to solve the model In the current work, the spout shape was calculated in two orthogonal directions. In the direction along the slot, half the slot length was used as ri . In the other direction, half the slot width was adopted. The model predictions are compared to experimental data in Figures 6.11(a) and 6.11(b), where δx and δy are the lateral distance from the spout/annulus interface to the column centre. The predicted data do not agree well with the experimental data in the direction normal to the slot. However, in the direction along the slot, the model gives fair results at the top of the bed. Errors at the bottom are no doubt due to the 3-dimensional geometry of the column.  117  Chapter 6. Local Flow Structure The equi-area radius, δa , and the half hydraulic diameter, δh , of the spout were also compared to the prediction. δa and δh are defined as r δa =  4δx δy π  (6.6)  δh =  δx δy δx + δy  (6.7)  and  As shown in Figures 6.11(c) and 6.11(d), the prediction is still not very good. It is clear that three dimensional effects need to be considered in the modeling of the spout shape.  6.2.4  Dead zones  Because the experiments were carried out in a flat-bottomed column, there were significant dead zones, where particles did not move at all. Dead zones for different gas velocities and static bed heights are shown in Figures 6.12 and 6.13. A higher gas velocity caused a smaller dead zone but had little effect on the included angle of the annulus/dead zone interface. The included angle was affected by the static bed height, with deeper beds having smaller included angles. The diverging base introduced in Chapter 2.2 with 60◦ included angle was shown in Figures 6.12 and 6.13 as dashed lines. It was noticed that the included angles between dead zone boundaries in this tests were ≤ 60◦ , suggesting that a diverging base of included angle ≤ 60◦ is appropriate for the slot-rectangular spouted beds, at least for these particles. Smaller included angle is necessary for deeper bed. At the same time, these results also suggested that the width of the plane area at the bottom of the diverging base should be decreased to eliminate the dead zones.  118  300  300  250  250 Height coordinate, Z (mm)  Height coordinate, Z (mm)  Chapter 6. Local Flow Structure  200  150  100  200  150  100  50  50  δy Prediction  δx Prediction 0  0  0 10 20 30 40 50 Distance from column axis, Y, (mm)  0 10 20 30 40 50 Distance from column axis, X, (mm)  (b) Along the slot  300  300  250  250 Height coordinate, Z (mm)  Height coordinate, Z (mm)  (a) Normal to slot  200  150  100  50  200  150  100  50 δa Prediction  δh Prediction  0  0 0  10 20 30 40 50 Distance from column axis, (mm)  0  (c) Equi-Area radius  10 20 30 40 50 Distance from column axis, (mm)  (d) Half hydraulic diameter  Figure 6.11: Comparison between the experimental spout shapes and predictions from two-dimensional Krzywanski et al. (1989) model. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s Slot: 10 × 40 mm, flat base.  119  Chapter 6. Local Flow Structure  300  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  Height coordinate, Z (mm)  250  200  150  100  50  0 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.12: Boundaries of dead zone for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base. Dashed lines show 60◦ included angle.  300  Hs = 200 mm Hs = 250 mm Hs = 300 mm  Height coordinate, Z (mm)  250  200  150  100  50  0 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.13: Boundaries of dead zone for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Slot: 10 × 40 mm, flat base, Ug : 0.7m/s. Dashed lines show 60◦ included angle.  120  Chapter 6. Local Flow Structure  6.3  Distribution of pressure  The local pressure in the column was measured by a 1/8” (3.2 mm) tube inserted into the column. This probe measured the pressure drop from the measurement point to the freeboard.  6.3.1  Axial pressure distribution  The longitudinal pressure distribution in a flat-bottomed cylindrical spouted bed was investigated by Lefroy and Davidson (1969). They found that the longitudinal pressure distribution could be fitted by a quarter cosine function, ∆Pz = cos(πZ/(2Hs )) ∆Ps  (6.8)  where Z is the vertical coordinate measured from the bottom, Hs is the static bed height, ∆Pz is the gauge pressure at Z and ∆Ps is the spouting pressure drop. In the current work, the pressure was measured at the wall at one end of the slot for different static bed heights, gas velocities and slot configurations. Steady state results are shown in Figure 6.14. The measured pressure distribution is fitted well by the quarter cosine correlation, suggesting that Equation 6.8 can be applied to the slot-rectangular spouted beds. The pressure along the axis of the column for the same operating conditions as in Figure 6.14 was also measured. In this case the longitudinal profile was different, as shown in Figures 6.15 and 6.16. A pressure-ascending region appeared at the bottom. After reaching a maximum value, pressure descended with height. The pressure-ascending region is likely due to the suction at the outlet of the slot. Gas and particles are entrained into the spout area at the bottom, due to the high gas velocity and Venturi effect. As expected, the pressure was higher for a higher static 121  Chapter 6. Local Flow Structure 1  ∆Pz / ∆Ps  0.8  0.6 Slot Length / Width, Hs, Ug 0.4  40 mm / 10 mm, 200 mm, 0.7 m/s 40 mm / 10 mm, 250 mm, 0.7 m/s 40 mm / 10 mm, 300 mm, 0.7 m/s 40 mm / 10 mm, 300 mm, 0.8 m/s 40 mm / 10 mm, 300 mm, 0.9 m/s 20 mm / 20 mm, 300 mm, 0.8 m/s 28 mm / 14 mm, 300 mm, 0.8 m/s  0.2  0 0  0.2  0.4  0.6  0.8  1  z/Hs  Figure 6.14: Axial pressure distributions at the wall. Column C: 300 × 100 mm, 1.33 mm glass beads, with flat base; Line is Equation 6.8.  depth. However, it was lower for higher gas velocities and the effect of the gas velocity is more pronounced in the bottom region.  6.3.2  Lateral pressure distribution  The lateral pressure profiles from two orthogonal directions are shown in Figures 6.17 to 6.20 for different gas velocity and levels. The lateral distribution of pressure was similar for different operating conditions. In the annulus, the pressure increased as the measurement position moved from the wall to the spout/annulus interface, reaching a maximum close to the spout/annulus interface. In the spout, the pressure then decreased toward the centre of the spout, which did not coincide with the axis of the column when the flow was asymmetric. For both the annulus and spout, the pressure was normally higher closer to the bottom, as shown in Figures 6.17 and 6.18. One exception is at the centre of the spout, which is caused by the Venturi effect. Figures 6.19 and 6.20 indicates that lower superficial  122  Chapter 6. Local Flow Structure 2 1.8 Pressure drop, ∆Pz (kPa)  1.6 1.4 1.2 1 0.8 0.6 0.4  Hs = 200 mm Hs = 250 mm Hs = 300 mm  0.2 0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.15: Axial pressure distributions along column axis for different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Ug : 0.7 m/s, Slot: 10 × 40 mm, flat base.  gas velocity caused the local pressure drop to increase. This is consistent with the axial profile (see Figures 6.15 and 6.16). The lower height and higher gas velocity caused larger dip of pressure drop in the spout.  6.4 6.4.1  Distribution of voidage Axial distribution of voidage  The voidage along the axis of the column varied in a manner similar to the pressure along the centre line. The voidage was low at the bottom, then increased with height and finally decreased. This can be explained by the evolution of particle velocity. When particles are accelerated at the bottom, the gaps between particles increase, causing the voidage to increase. After a maximum is reached, the voidage decreases due to the velocity decreasing and the particles being drawn into the spout from the side. As shown in Figure 6.21, for the same gas velocity, a shallower bed caused higher 123  Chapter 6. Local Flow Structure 2 1.8 Pressure drop, ∆Pz (kPa)  1.6 1.4 1.2 1 0.8 0.6 0.4  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  0.2 0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.16: Axial pressure distributions along column axis for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base.  voidage. This probably occurred because that the spout was shorter for lower bed, giving a smaller spout/annulus interface area. The pressure is also smaller for a lower bed, so there is less force to push particles into the spout. The superficial gas velocity also affected the local voidage. As shown in Figure 6.22, a higher gas velocity led to higher voidages, attributed to the increase in particle velocity caused by increased gas velocity.  6.4.2  Lateral distribution of voidage  The voidage in the spout normally reaches a maximum near the centre of the spout, as shown in Figures 6.23 to 6.26. In some cases, the voidage in the spout had two peaks, with a dip inbetween (e.g. Figure 6.24). This finding differs from Freitas et al. (004b). As shown in Figures 6.23 to 6.26, voidage decreased with increasing height and increased with increasing gas velocity. Both of these trends are as expected.  124  Chapter 6. Local Flow Structure 1.2 Z = 50 mm Z = 100 mm Z = 150 mm Z = 200 mm  Pressure drop, ∆Pz (kPa)  1 0.8 0.6 0.4 0.2 0 -0.2 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.17: Pressure profiles in direction normal to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Ug : 0.6 m/s, Slot: 10 × 40 mm, flat base.  6.5  Distribution of vertical component of particle velocity  6.5.1  Axial distribution of particle velocity  The evolution of the vertical particle velocity with height along the axis of the column is shown in Figures 6.27 and 6.28. This velocity showed a pattern similar to previous findings (Mathur and Epstein, 1974), with particles rapidly accelerated at the bottom and then gradually decelerating until they reached the bed surface. The fountain was not investigated in this work. With the other operating conditions fixed, axial profiles were measured for three different static bed heights. As shown in Figure 6.27, particle velocities for different static bed heights were similar, especially near the bed surface. The velocity at a fixed height did not appear to be affected significantly by the static bed height. The velocity  125  Chapter 6. Local Flow Structure  1.2 Z = 50 mm Z = 100 mm Z = 150 mm Z = 200 mm  Pressure drop, ∆Pz (kpa)  1 0.8 0.6 0.4 0.2 0 -0.2 -40  -20 0 20 Distance from column axis, Y, (mm)  40  Figure 6.18: Pressure profiles in direction parallel to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Ug : 0.6 m/s, Slot: 10 × 40 mm, flat base.  1  Pressure drop, ∆Pz (kPa)  0.9  Ug = 0.5 m/s Ug = 0.6 m/s Ug = 0.7 m/s  0.8 0.7 0.6 0.5 0.4 0.3 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.19: Pressure profiles in direction normal to slot for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Z: 100 mm, Slot: 10 × 40 mm, flat base.  126  Chapter 6. Local Flow Structure  0.95  Ug = 0.5 m/s Ug = 0.6 m/s Ug = 0.7 m/s  Pressure drop, ∆Pz (kPa)  0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 -40  -20 0 20 Distance from column axis, Y, (mm)  40  Figure 6.20: Pressure profiles in direction parallel to slot for different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 200 mm, Z: 100 mm, Slot: 10 × 40 mm, flat base.  1.0  Time-mean local voidage  0.9  0.8  0.7  0.6 Hs = 200 mm Hs = 250 mm Hs = 300 mm  0.5  0.4 0.1  0.2  0.3  0.4 0.5 0.6 0.7 0.8 Normalized vertical position, Z/Hs  0.9  1.0  Figure 6.21: Voidage distributions along column axis at different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Ug : 0.7 m/s, Slot: 10 × 40 mm, flat base.  127  Chapter 6. Local Flow Structure  1.0  Time-mean local voidage  0.9  0.8  0.7  0.6 Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  0.5  0.4 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.22: Voidage distributions along column axis at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base.  1.0 Z = 100 mm Z = 200 mm Z = 300 mm  Time-mean local voidage  0.9  0.8  0.7  0.6  0.5  0.4 -30  -25  -20  -15 -10 -5 0 5 10 Distance from column axis, Y, (mm)  15  20  Figure 6.23: Voidage profiles in direction parallel to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base.  128  Chapter 6. Local Flow Structure  1.0 Z = 100 mm Z = 200 mm Z = 300 mm  Time-mean local voidage  0.9  0.8  0.7  0.6  0.5  0.4 -25  -20  -15  -10 -5 0 5 10 15 Distance from column axis, X, (mm)  20  25  Figure 6.24: Voidage profiles in direction normal to slot at different heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base.  0.8  Time-mean local voidage  0.75 0.7 0.65 0.6 0.55 0.5 Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  0.45 0.4 -25  -20  -15 -10 -5 0 5 10 Distance from column axis, Y, (mm)  15  20  Figure 6.25: Voidage profiles in direction parallel to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base.  129  Chapter 6. Local Flow Structure  1.0  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  Time-mean local voidage  0.9  0.8  0.7  0.6  0.5  0.4 -25  -20  -15  -10 -5 0 5 10 15 Distance from column axis, X, (mm)  20  25  Figure 6.26: Voidage profiles in direction normal to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base.  7  Verticle particle velocity, (m/s)  6 5 4 3 2 Hs = 200 mm Hs = 250 mm Hs = 300 mm  1 0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.27: Axial particle velocity profiles at different static bed heights. Column C: 300 × 100 mm, 1.33 mm glass beads, Ug : 0.7 m/s, Slot: 10 × 40 mm, flat base.  130  Chapter 6. Local Flow Structure 9  Vertical particle velocity, (m/s)  8 7 6 5 4 3 2 Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  1 0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.28: Axial particle velocity profiles at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Slot: 10 × 40 mm, flat base.  reached a maximum ∼ 50 to 100 mm above the bottom. A higher superficial gas velocity caused higher particle velocities, as shown in Figure 6.28. Higher gas velocity caused a longer acceleration zone. By modifying the previous model of Thorley et al. (1955), Mathur and Epstein (1974) provided a one-dimensional model for the prediction of axial particle velocity distribution in the spout. The spout is considered as three successive zones: an initial particle acceleration zone, a particle deceleration zone in the upper part of the bed and a further deceleration zone in the fountain. The axial particle velocity profile can then be predicted by a force balance model on the particles in the spout. The force balance of the particles in the spout can be written as 1 3 d(nz v) 1 1 1 πdv = (nz ρf (us − v)2 πd2p CD ) − (nz πd3v (ρs − ρf )g) 6 dt 2 4 6  (6.9)  where us is the superficial gas velocity in the spout, v is the particle velocity, dv is the diameter of the sphere of same volume as the particle. In this work, the particle is 131  Chapter 6. Local Flow Structure assumed to be spherical. dv can then be replaced by dp . The main accelerating force is the frictional drag of the ascending fluid and the main decelerating force is that of gravity. Equation 6.9 can be converted from a Langrangian to an Eulerian basis by means of the relationship dt = dz/v for the particle. This leads to v 2 dnz dv 3ρf (us − v)2 CD (ρs − ρf )g +v = − nz dz dz 4dp ρs ρs  (6.10)  The assumption after Thorley et al. (1955) that the number of particles, nz , is directly proportional to z, was adopted by Mathur and Epstein (1974). The governing equation for this zone is simplified to: 3ρf (us − v)2 CD (ρs − ρf )g v 2 1 d(v 2 ) + = − z 2 dz 4dp ρs ρs  (6.11)  The absolute gas velocity in the spout is us = Us /g , where g is the voidage in the spout. Experimental value is adopted in this work. Us is the superficial gas velocity in the spout, which can be predicted from the continuity.  Us As + Ua Aa = Ug Ac  (6.12)  Ua , the superficial gas velocity in the annulus has been empirically represented by Lefroy and Davidson (1969) as  Ua = BUmf sin(  132  πz ) 2Hs  (6.13)  Chapter 6. Local Flow Structure where B=  ∆Ps ∆Ps,max / Hs Hmax  ∆Ps,max =  2 ∆Pmf,max π  (6.14) (6.15)  As , Aa and Ac are the cross-section area of the spout, annulus and column, respectively, with As + Aa = Ac = αβ  (6.16)  If the cross-section of the spout is assumed to be elliptical, the area of spout can be calculated from the ellipse area equation:  As = πδx δy  (6.17)  where δx and δy are the distance from the spout/annulus interface to the column centre in the two orthogonal directions. Experimental spout size was adopted and linear interpolation was used for intermediate height. CD is the drag coefficient. CD = CDs g −4.7  (6.18)  CDs is the drag coefficient for single particle, which is related to Reynolds number. For Re≥1000 CDs = 0.44  (6.19)  24 (1 + 0.15Re0.687 ) Re  (6.20)  For Re < 1000 CDs =  Equation 6.11 can be integrated beginning with the boundary condition vz=0 = 0. 133  Chapter 6. Local Flow Structure 7  Vertical particle velocity, (m/s)  6 5 4 3 2 Acceleration Zone Deceleration Zone Experimental data  1 0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.29: Comparison of prediction with and without drag force in deceleration zone from Thorley et al. (1955) model and experimental axial velocity profile. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base, B = 0.72.  In the deceleration zone in the upper part of the bed, because the fountain height is unknown, the boundary condition to integrate Equation 6.11 is the experimental particle velocity at the surface of the bed. Because the velocity of particles in the spout follows the parabolic distribution, the average particle velocity in the spout, which was used to compare to the prediction, is half the velocity in the spout centre. MATLAB code of Runge-Kutta method was used to integrate these equations numerically as listed in Appendix C.4. The model prediction is compared to the experimental results in Figure 6.29. The predicted particle velocity is smaller than the experimental data in the acceleration zone. This may be due to under-estimation of the gas velocity in the spout, providing smaller acceleration to the particles in the model. The particle velocity in the deceleration is overpredicted. This also implies an under-estimation of gas velocity in the spout, which caused the particle to decelerate too soon. The under-estimation may come  134  Chapter 6. Local Flow Structure from the experimental voidage which is the voidage in the centre of spout and higher than the average voidage of the whole cross-section of spout.  6.5.2  Lateral distribution of particle velocity  The lateral profile of vertical particle velocity in the spout was measured from two directions at different levels and different gas velocities. Results are shown in Figures 6.30 to 6.33. The velocity gradient from the centre of the spout to the edge of the spout was normally greater at lower levels. Increasing the superficial gas velocity caused the particle velocity at the bed surface to be higher. Although, there is one exception in Figure 6.32, where the particle velocities for Ug = 0.9 m/s were measured to be smaller than those for Ug = 0.8 m/s. This anomaly may have been caused by asymmetry of the spout. The radial distribution of particle velocity in the spout of conventional axisymmetric spouted beds was summarized by Mathur and Epstein (1974). They noted that the velocity profile can be described by a parabolic equation,  vr /v0 = 1 − (r/rs )2  (6.21)  where vr is the particle velocity at radial position r, v0 is the particle velocity at the spout centre, and rs is the spout radius. In the current work, the flow was not symmetric. Hence the velocity of the particles was correlated to a best fit parabolic equation of the form  vx /v0 = 1 − ((X − X0 )/δx )2  (6.22)  vy /v0 = 1 − ((Y − Y0 )/δy )2  (6.23)  135  Chapter 6. Local Flow Structure for (X − X0 ) ≤ δx and (Y − Y0 ) ≤ δy , where δx and δy are the half width of the spout in the two orthogonal directions, X0 and Y0 are the position of spout centre in these directions, and vx and vy are the vertical component of particle velocities at (X, 0) and (0, Y ) and specific Z level, respectively. The lines plotted in Figures 6.30 to 6.33 are least-square fits based on this form of equations. As shown in Figures 6.30 to 6.33, the correlation between the experimental results and parabolic equations was favorable, indicating that the lateral particle velocity profile in the slot-rectangular spouted beds can also be represented by parabolic relationships. The data are also consistent with the observation that the centre of the spout did not coincide with the axis of the column. 9 Z = 100 mm Z = 200 mm Z = 300 mm  Vertical particle velocity, (m/s)  8 7 6 5 4 3 2 1 0 -20  -15  -10 -5 0 5 10 15 Distance from column axis, X, (mm)  20  25  Figure 6.30: Lateral profiles of vertical particle velocity in direction normal to slot at different levels. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base.  6.6  Comparison of slots  Three slots of equal area but different length-to-width ratio 1, 2 and 4 (see Table 5.1), were used to compare the effect of slot configuration for a constant slot cross-sectional  136  Chapter 6. Local Flow Structure 8 Z = 100 mm Z = 200 mm Z = 300 mm  Vertical particle velocity, (m/s)  7 6 5 4 3 2 1 0 -25  -20  -15  -10 -5 0 5 10 15 Distance from column axis, Y, (mm)  20  25  Figure 6.31: Lateral profiles of vertical particle velocity in direction parallel to slot at different levels. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, Slot: 10 × 40 mm, flat base.  area, i.e. equal mean inlet gas velocity or the same superficial gas velocity.  6.6.1  Axial pressure distribution  Axial pressure profiles are compared in Figure 6.34. Differences due to these slots mainly occurred near the bottom of the column. The slot with the smallest length-to-width ratio had the lowest pressure at the bottom.  6.6.2  Spout shape and dead zone boundaries  Spout shapes for slots of different length/width ratio appear in Figures 6.35 and 6.36. Note that for the same gas velocity, the spout for all three configurations reached essentially the same size above a certain level, despite the difference in initial slot shape. Below this level, the slot with a higher length/width ratio had a wider spout. However, there was negligible difference between the dead zones for different slots, as shown in  137  Chapter 6. Local Flow Structure  Vertical particle velocity, (m/s)  5  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  4  3  2  1  0 -20  -15  -10 -5 0 5 10 Distance from column axis, X, (mm)  15  20  Figure 6.32: Lateral profiles of vertical particle velocity in direction normal to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base.  Figure 6.37.  6.6.3  Axial distribution of particle velocity  Velocity distributions along the axis of the column are compared in Figure 6.38. Slots with smaller length/width ratio had higher particle velocities at the bottom. Particles accelerated and decelerated more rapidly for a smaller length/width ratio. Towards the bed surface, particle velocities were similar for all three slot configurations.  6.6.4  Axial distributions of voidage  The voidage profiles along the axis for the three slot configurations were similar, as shown in Figure 6.39. Voidage profiles at the centre show no big difference between slots of different length/width ratio.  138  Chapter 6. Local Flow Structure  Vertical particle velocity, (m/s)  5  Ug = 0.7 m/s Ug = 0.8 m/s Ug = 0.9 m/s  4  3  2  1  0 -30  -20  -10 0 10 Distance from column axis, Y, (mm)  20  Figure 6.33: Lateral profiles of vertical particle velocity in direction parallel to slot at different superficial gas velocities. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Slot: 10 × 40 mm, flat base.  6.6.5  Lateral distributions of particle velocity  Lateral profiles of particle velocities at the bed surface are compared in Figures 6.40 and 6.41. Consistent with the axial profiles, the velocities for different slot configurations were similar to each other.  6.6.6  Lateral distributions of voidage  Lateral profiles of voidage in two orthogonal directions are compared in Figures 6.42 and 6.43. The results suggest that the voidage tended to decrease with increasing slot length/width ratio. Note the asymmetry in the results.  6.6.7  Particle circulation flux  From the previous sections, the particle velocities for different slot configuration were similar, whereas the voidage was smaller for a slot of higher length/width ratio. Since  139  Chapter 6. Local Flow Structure 1.8  Pressure drop, ∆Pz (kPa)  1.6 1.4 1.2 1 0.8 0.6 0.4  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  0.2 0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.34: Axial pressure profiles along the column axis for three slots of equal area, but different length/width ratio. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  all results are for steady operating conditions, the particle circulation flux at the spout outlet can be estimated from ρp vs (1 − ). As shown in Figures 6.44 and 6.45, slots of higher length/width ratio provided greater local particle circulation. Overall, slots of higher length/width ratio provided slightly larger spouts, higher exit solid fractions and greater particle circulation fluxes. These results can be attributed to the greater slot perimeter, which encourages more particles to enter the spout.  140  Chapter 6. Local Flow Structure  300  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  Height coordinate, Z (mm)  250  200  150  100  50  0 0  10  20 30 Half-width of spout, (mm)  40  50  Figure 6.35: Shapes of spout in direction parallel to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  300  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  Height coordinate, Z (mm)  250  200  150  100  50  0 0  10  20 30 Half-width of spout, (mm)  40  50  Figure 6.36: Shapes of spout in direction normal to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  141  Chapter 6. Local Flow Structure  300  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  Height coordinate, Z (mm)  250  200  150  100  50  0 -150  -100  -50 0 50 Distance from column axis, X, (mm)  100  150  Figure 6.37: Boundaries of dead zone for different slots for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base. Dashed lines show 60◦ included angle.  Vertical particle velocity, (m/s)  12  10  8  6  4 Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  2  0 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.38: Axial velocity profiles along the column axis for different slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  142  Chapter 6. Local Flow Structure  1.0  Time-mean local voidage  0.9  0.8  0.7  0.6 Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  0.5  0.4 0  50  100 150 200 Height coordinate, Z (mm)  250  300  Figure 6.39: Axial voidage profiles along the column axis for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  8  Vertical particle velocity, (m/s)  7 6  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  5 4 3 2 1 0 -1 -20  -15  -10  -5 0 5 10 15 20 Distance from column axis, Y, (mm)  25  30  Figure 6.40: Velocity distributions in direction parallel to slot for different slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base.  143  Chapter 6. Local Flow Structure  6  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  Vertical particle velocity, (m/s)  5 4 3 2 1 0 -1 -30  -20  -10 0 10 Distance from column axis, X, (mm)  20  30  Figure 6.41: Velocity distributions in direction normal to the slot for different slots. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base.  1.0  Time-mean local voidage  0.9  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  0.8  0.7  0.6  0.5  0.4 -35  -30  -25  -20 -15 -10 -5 0 5 Distance from column axis, Y, (mm)  10  15  Figure 6.42: Voidage profiles in direction parallel to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  144  Chapter 6. Local Flow Structure  1.0  Slot length/width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  Time-mean local voidage  0.9  0.8  0.7  0.6  0.5  0.4 -20  -15  -10 -5 0 5 10 15 Distance from column axis, X, (mm)  20  25  Figure 6.43: Voidage profiles in direction normal to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Ug : 0.8 m/s, flat base.  4000  2  Particle circulation rate, kg/(m s)  3500 3000  Slot Length / Width 20 mm / 20 mm 28 mm/ 14 mm 40 mm / 10 mm  2500 2000 1500 1000 500 0 -40  -30  -20 -10 0 Distance from column axis, Y, (mm)  10  20  Figure 6.44: Particle circulation flux profiles in direction parallel to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base.  145  Chapter 6. Local Flow Structure 5000  Particle circulation rate, kg/(m2s)  4500 4000  Slot Length / Width 20 mm / 20 mm 28 mm / 14 mm 40 mm / 10 mm  3500 3000 2500 2000 1500 1000 500 0 -25  -20  -15  -10 -5 0 5 10 15 Distance from column axis, X, (mm)  20  25  Figure 6.45: Particle circulation flux profiles in direction normal to slot for three slots of equal area. Column C: 300 × 100 mm, 1.33 mm glass beads, Hs : 300 mm, Z: 300 mm, Ug : 0.8 m/s, flat base.  146  Chapter 7 Spouting with Multiple Slots It was shown in Chapter 4 that a slot of high length-to-width ratio caused multiple spouts in shallow beds. With increasing bed height and gas velocity, the multiple spouts interacted significantly with each other. This interaction was considered to be the main cause of the instability of slot-rectangular spouted beds. However, an exception was noticed during the experiment: When a particle of diameter close to the width of slot lodged at the middle of the slot, the two spouts became smooth and continuous, without obvious merging as described in Chapter 4. This implies that two separate slots may cause the flow to be more stable than a continuous slot. This possible way of improving the stability of slot-rectangular spouted beds is explored in this chapter. As a means of scale-up, Mujumdar (1984) suggested that it is possible to use a bank of parallel two-dimensional columns with common walls between adjacent beds. Kalwar et al. (1993) stated that in this method, the common walls can even be removed. This chapter investigates spouting in columns with twinned chambers and with multiple slots.  7.1  Equipment  As described in Chapter 2, Column C and Column B (see Figure 2.3) had twice the width and thickness of Column A respectively. In this chapter, they were equipped with multiple slots with a common base chamber windbox. The slot sizes were chosen according to the results of Chapter 5. These slots of lengthto-width ratio < 15 were found to provide stable spouting. So, in this chapter, slots of width 4 mm and lengths 30 and 40 mm were adopted to minimize the instability. Figure  147  Chapter 7. Spouting with Multiple Slots  Figure 7.1: Top views showing slot positions of the experimental columns; all dimensions are in mm.  148  Chapter 7. Spouting with Multiple Slots 7.1 shows alternative positions of the slots in top views of the column. Multiple slots were configured in two ways: First, slots were arranged with their major dimensions parallel to each other. This type of configuration with 2 or 3 slots was used in Column C, as shown in Figure 7.1 (a,b). Second, aligned slots were arranged in series, i.e. with their major dimensions in a straight line, as shown in Figure 7.1(c,d,e). As for the parallel arrangement, two and three slots were tested. For two parallel slots in Column C, as shown in Figure 7.1(a), the diverging base shown in Figure 2.4 was provided. For Column B (see Figure 7.1(c,d)), a diverging base was also always used. The pressure drop across the column was measured from just above each slot to the top of column. These experiments were carried out with glass beads of 1.33 mm diameter and compressed air. Different static bed heights were tested. Spouting in Column A with a single 4 × 30 mm slot was also tested for comparison, as shown in top view in Figure 7.1(f).  7.2  Pressure evolution  The typical pressure evolution for Column C with two parallel slots, as shown in Figure 7.1(a), and a diverging base is plotted in Figures 7.2 and 7.3. With increasing gas flow, the pressure drops from right above the slot outlet to the top of the bed feature two dips, corresponding to the onset of the two spouts and consistent with the results of Chapter 4. Two shifts of pressure drop could be observed with decreasing gas velocity, indicating the collapse of the two spouts. Note that, the right slot spouted and collapsed at a lower gas velocity in this case. With increasing gas velocity, after the onset of one spout, the pressure drop of the other slot was higher. But it was lower than the maximum pressure drop and never reached that maximum value. Apparently, gas from the actively spouting slot helped loosen the particles over the other slot, so that the other spout had a lower maximum pressure drop to overcome. 149  Chapter 7. Spouting with Multiple Slots It can also be noticed that, when one spout collapsed, the pressure drop across the column measured at the end of the two slots changed in a reversed way. If the pressure drop of the column for one slot increased, that for the other slot decreased, as shown in Figure 7.3.  Pressure drop across column, (kPa)  3.5 Right slot Left slot  3 2.5 2 1.5 1 0.5  Right spout  Left spout  0 0  0.2  0.4 0.6 0.8 Superficial gas velocity, Ug (m/s)  1  1.2  Figure 7.2: Pressure evolution with increasing gas velocity, Column C: 300 × 100 mm, Two parallel: 4 × 30 mm slots as shown in Figure 7.1(a), diverging base.  7.3  Minimum spouting velocity  As shown in Figure 7.3, there are two values of minimum spouting velocity corresponding to the collapse of the two spouts in a column with two slots. Ums normally increases with increasing static bed height. In our case, the difference between the two Ums values increased with increasing bed height. The two Ums values for Column C with two parallel slots of 30 mm length and 4 mm width are plotted in Figure 7.4. Both are higher than Ums for Column A, which has a single slot of 4 mm width and 30 mm length and the same particles at the same static bed height. Thus more gas is needed to achieve spouting when two slot-rectangular columns 150  Chapter 7. Spouting with Multiple Slots  Pressure drop across column, (kPa)  2.5 Right slot Left slot 2  1.5  1  0.5 Right spout collapse  Left spout collapse  0 0  0.2  0.4 0.6 0.8 Superficial gas velocity, Ug (m/s)  1  1.2  Figure 7.3: Pressure evolution with decreasing gas velocity, Column C: 300 × 100 mm, Two parallel: 4 × 30 mm slots as shown in Figure 7.1(a), diverging base.  were combined. Ums was measured with decreasing gas velocity, as recommended by Mathur and Epstein (1974). As has been noted, the two spouts collapsed one after the other. After the first spout collapsed, the gas from the second spout leaked into the upper part of the other slot. So more gas was needed to sustain the spout than for a single spout. The Ums values were also higher than that for Column C with a single slot of 60 mm length, which, indicated by inverted triangles in Figure 7.4, had the same slot width and total cross-sectional area as the multiple slots. This result means that more gas was needed to make an extra spout than to create a single spout. This can be explained by the positions of the slots. For multiple spouts caused by multiples slots, the gas could be more dispersed. Supposing that the resistance for each spout is the same, more gas is needed to overcome the resistance for two spouts. The pair of Ums values in Column B with two aligned slots of 30 mm length were also compared to that for the single slot in Column A. In this case, one of the two Ums values was higher than for a single slot column, whereas the other was smaller, as 151  Chapter 7. Spouting with Multiple Slots  Minimum spouting velocity, Ums (m/s)  0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 Right 4 x 30 mm slot in Column C Left 4 x 30 mm slot in Column C Single 4 x 30 mm slot in Column A Single 4 x 60 mm slot in Column C  0.5 0.45 0.4 140  160  180  200 220 240 260 Static bed height, Hs (mm)  280  300  Figure 7.4: Comparison of minimum spouting velocities for single and parallel 4 × 30 mm slots, 1.33 mm glass beads, diverging base.  Minimum spouting velocity, Ums (m/s)  0.9 0.85  Front 4 x 30 mm slot in Column B Back 4 x 30 mm Slot in Column B Single 4 x 30 mm slot in Column A  0.8 0.75 0.7 0.65 0.6 0.55 0.5 140  160  180  200 220 240 260 Static bed height, Hs (mm)  280  300  Figure 7.5: Comparison of minimum spouting velocity for columns with single and aligned 4 × 30 mm slots with 1.33 mm glass beads, diverging base.  152  Chapter 7. Spouting with Multiple Slots shown in Figure 7.5. This suggests that scaling up a slot-rectangular spouted bed by duplicating chambers in the direction of the column thickness is a better way to preserve the hydrodynamic properties of the prototype than by increasing the column width. The average values of Ums for the multiple spouts are compared in Figure 7.6. This again indicates that the average minimum spouting velocity for slots in series was closer to that of column A.  Minimum spouting velocity, Ums (m/s)  0.9 0.85 0.8 0.75 0.7 0.65 0.6 Parallel slots in Column C (300x100 mm) Aligned slots in Column B (150x200 mm) Single slot in Column A (150x100 mm)  0.55 0.5 140  160  180  200 220 240 260 Static bed height, Hs (mm)  280  300  Figure 7.6: Comparison of average minimum spouting velocities for columns with single and multiple 4 × 30 mm slots with 1.33 mm glass beads, diverging base.  7.4  Spouting pressure drop  The average pressure drops for different configurations, all at 200 mm static bed height, are compared in Figure 7.7. The pressure drop for columns with multiple slots, which was the average of the pressure drops measured for each of the two spouts, were higher than for a single-slot column, indicating more gas flows through the annulus. This is supported by the results in Figure 7.4. When more gas goes through the annulus, more gas is needed to support the spout, resulting in an increase in Ums . 153  Chapter 7. Spouting with Multiple Slots 1.3  Spouting pressure drop, (kPa)  1.25 1.2 1.15 1.1 1.05 1 Parallel slots in Column C Aligned slots in Column B Single slot in Column A  0.95 0.9 0.6  0.65  0.7  0.75 0.8 0.85 0.9 0.95 Superficial gas velocity, (m/s)  1  1.05  Figure 7.7: Comparison of spouting pressure drops for columns with single and multiple 4 × 30 mm slots, 1.33 mm glass beads, Hs = 200 mm, diverging base.  7.5 7.5.1  Flow regimes Onset of spouting  Although the slots were identical, multiple spouts exerted from these slots did not appear at the same gas velocity. Instead, they formed one by one as the superficial gas velocity was gradually increased. The order of their appearance was found to depend on the size and position of the slots, as well as the initial shape of the bed surface, i.e. the static bed height of each chamber. When two identical slots were arranged as shown in Figure 7.1(a), any minor difference could cause one slot to spout first. To investigate the relationship between the order and the size of slot, two slots, one of length 40 mm and the other 30 mm long, both of 4 mm width, were arranged as in Figure 7.1 (a). The longer slot spouted and collapsed at a lower gas velocity in all cases. When three slots were installed as shown in Figure 7.1 (b) and (d), the central slot spouted at a lower gas velocity, probably because the gas inlet of the base chamber was at the centre of the  154  Chapter 7. Spouting with Multiple Slots bottom, so the gas velocity at the centre was higher than for the side slots. When one spout formed, it spread particles to the other slot causing the height of particles above the other slot to be higher. The other slots then needed an even higher gas velocity to spout.  7.5.2  Evolution of flow regimes  The flow regimes in multiple slot spouting were similar to those for single spouting, but with some additional features. A typical evolution of regimes with increasing gas velocity was as follows: • Fixed bed regime; • Internal jet, before the bed spouted. The internal jet could be detected from the particle movement near the wall. As expected, there were two internal jets for two slots, and three for three slots. • After each of the slots made its own spout, each slot could exhibit swaying. Both swaying along the slot and normal to the slot were found. Unlike the regimes for single slot spouting, swaying in random directions was also found in Column B with two or three slots. However, there was a tendency for spouts to sway together or opposite to each other. Figure 7.8 shows typical pressure fluctuations where both spouts were swaying. There was a noticeable frequency around 2 Hz. The distribution of frequency was similar to that for swaying with a single slot. At this stage, the flow of multiple spout was similar to that noticed by Murthy and Singh (1994); Saidutta and Murthy (2000) in multiple spouted beds with square chambers and central circular gas inlet. In that case, the flow is stable and steady close to the minimum spouting velocity. • When the gas velocity increased further, the behaviour of the spout differed for 155  Chapter 7. Spouting with Multiple Slots  ∆P, (kPa)  2.2 2.18 2.16 2.14 2.12 2.1 2.08 2.06 2.04 2.02 0  1  2  3  4  5  Time, t, (s)  PSD, (kPa/s2)  6 5 4 3 2 1 0 0  5  10 Frequency, f, (Hz)  15  20  Figure 7.8: Pressure fluctuations for swaying flow regime, Column C: 300 × 100 mm, two parallel 4 × 30 mm slots as shown in Figure 7.1 (a), 1.33 mm glass beads, Hs =300 mm, Ug =0.64 m/s, diverging base.  parallel and aligned slots. For parallel slots, the spouts and fountains started to merge. For slots at the side, the outlet of the spout migrated towards the centre. The fountains also inclined towards each other. When the gas velocity was high enough, the two fountains finally merged into one. Because at this stage, the two spouts were still separate, we call this “Merging of fountain”. When the two spouts merged, there was a slight shift in the pressure drop of the column. In the reverse process, there was a slight dip, as shown in Figure 7.9. There was no noticeable fluctuation in the Merging of fountain regime. As portrayed in Figure 7.10, the pressure fluctuation and PSD lines are similar to those of the steady state spouting regime (see Figure 4.5). Merging of spouts also occurred in column C for three slots of length 30 mm and width 4 mm, as shown in Figure 7.1 (b). At a static bed height of 300 mm, the fountain had a low height, looking like a speed bump on the road. Significant fluctuations in the pressure drop of the column were found in this regime. 156  Chapter 7. Spouting with Multiple Slots  Pressure drop across column, (kPa)  3 Decreasing gas flow Increasing gas flow 2.5  2  1.5  Shift showing start of merging  1  Dip showing end of merging  0.5  0 0  0.2  0.4 0.6 0.8 Superficial gas velocity, Ug (m/s)  1  1.2  ∆P, (kPa)  Figure 7.9: Pressure evolution showing where merging of fountain begins and ends, Column C: 300 × 100 mm, Two parallel 4 × 30 mm as shown in Figure 7.1(a), 1.33 mm glass beads, Hs =200 mm, diverging base.  1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0  1  2  3  4  5  PSD, (kPa/s2)  Time, t, (s) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0  5  10 Frequency, f, (Hz)  15  20  Figure 7.10: Pressure fluctuations in Merging of fountain flow regime, Column C: 300 × 100 mm, Two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads, Hs =200 mm, diverging base.  157  Chapter 7. Spouting with Multiple Slots • For aligned slots, the spouts tended to avoid overlap with other spouts. They always moved to a random asymmetric position and settled there. The positions of these spouts did not change, even when the static bed height was varied. The incoherent spouting and swaying regimes were found after the spouts moved to a random asymmetric position. Each spout for multiple-slot spouting can differ from the others, depending on the gas and particle flow through it. For example, swaying could co-exist with an incoherent spouting, as found in Column B for three slots and a 350 mm static bed height. The flow of spouting with multiple slot-rectangular chambers is similar to that noted by Murthy and Singh (1994); Saidutta and Murthy (2000) in multiple spouted beds with square chambers and central circular gas inlet. In that case, the flow is stable and steady close to the minimum spouting velocity. When the gas velocity was increased, the spout oscillated and interference of fountains increased, which in turns increased the solids transfer between the cells through the fountains in the upper region of the chambers. When the superficial gas velocity was reduced, the flow sequence was the reverse of that for increasing velocity. However there was hysteresis for the fountain merging, as shown in Figure 7.9.  7.5.3  Termination of spouting  In multiple-slot spouting, the main mechanism for termination of spouting was the development of instabilities. In most cases, the instabilities were caused by the interaction between adjacent spouts. This normally happened when the distance between the slots was too small for the spouts to act independently.  158  Chapter 7. Spouting with Multiple Slots  7.6  Interaction between spouts  As noted in the last section, there were two main types of interaction between multiple spouts: (a) spouts merging with each other, and (b) spouts appearing to avoid overlapping with each other. Both types of interaction caused the spouts and fountains to be asymmetric. The interactions between spouts involved interchange of gas and particles in the annulus and fountain. To reveal the mechanism of the interactions and prevent interaction, the effects of distance between slot centres, diverging bases and vertical partitions were investigated.  7.6.1  Effect of distance between slot centres  As shown in Figure 7.1(e), two slots of 4 × 30 mm were arranged in series with their centres separated by 40 to 70 mm. It was observed that there was significant interaction between the two spouts in all cases. For a 40 mm slot separation distance, i.e. the distance from the centres of slots, merging of spouts appeared for Hs < 150 mm, similar to what was observed in Chapter 4 for a continuous long slot. With increasing Hs , the spouting appeared to be unstable and random. The merged spout alternated between spouting and collapsing. For greater separation distance, distinct spouts could persist at higher gas velocity and for deeper beds. Increasing the distance between the slots did help prevent the spouts from merging, but the interaction between the slots was still significant. When two spouts formed, the two fountains avoided overlapping with each other by stretching in opposite directions. Figure 7.11 shows the influence of distance between the centres of slots on the average Ums . A distance of 50 mm provided the smallest Ums . For this case, the centre of the slot was one-quarter of the thickness, i.e. equi-distance from the wall and the column centre. If the column was considered to be a combination of two identical chambers, these slots would be at the centre of each chamber. The pressure fluctuations for this separation 159  Chapter 7. Spouting with Multiple Slots distance was also smaller than for other separation distances, as shown in Figure 7.12. The greatly reduced fluctuations and smooth curve for a centre-to-centre spacing of 50 mm relative to the other curves may be due to the more sysmmetric geommetry in the former case.  Minimum spouting velocity, Ums (m/s)  0.9 0.8 0.7  Distance between slot centres 40 mm 50 mm 60 mm 70 mm  0.6 0.5 0.4 0.3 0.2 100  150  200 250 300 350 Static bed height, Hs (mm)  400  450  Figure 7.11: Effect of distance between slots on Ums , Column C: 300 × 100 mm, two aligned 4 × 30 mm slots as shown in Figure 7.1(e), 1.33 mm glass beads, diverging base.  7.6.2  Effect of diverging base  Flow was tested in Column C with and without diverging bases. As noted in Section 7.5, the flow regimes for these configurations were the same. The fountains tended to merge when the gas velocity increased. The diverging base prevented gas/particle exchange in the bottom part. Hence exchange of gas and particles at the bottom is not the main cause of interaction between spouts. As shown in Figure 7.13, the average minimum spouting velocity for the column with a diverging base was smaller than for the column without a diverging base. This is consistent with the result in Chapter 3.2.  160  Ums for the two spouts for the two cases  Magnitude of pressure fluctuation, STD(∆P) (kPa)  Chapter 7. Spouting with Multiple Slots  0.11 0.1 0.09 0.08  Distance between slot centres 40 mm 50 mm 60 mm 70 mm  0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.65  0.7  0.75 0.8 0.85 0.9 Superficial gas velocity, Ug (m/s)  0.95  1  Figure 7.12: Effect of distance between slots on pressure fluctuation, Column C: 300 × 100 mm, two aligned 4 × 30 mm slots as shown in Figure 7.1(e), 1.33 mm glass beads, diverging base.  Minimum spouting velocity, Ums (m/s)  1 0.95  With diverging base Without diverging base  0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 140  160  180  200 220 240 260 Static bed height, Hs (mm)  280  300  Figure 7.13: Effect of diverging base on Ums , Column C: 300 × 100 mm, two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads.  161  Chapter 7. Spouting with Multiple Slots are compared in Figure 7.14, where Ums,max and Ums,min denote the highest and smallest values of Ums . The difference in Ums for a pair of spouts without a diverging base was higher than when there was a diverging base. Because the slots in these two cases were identical, this implies that the diverging base helped to reduce the imbalance between two adjacent slots.  Difference in Minimum spouting velocity, Ums,max-Ums,min (m/s)  0.18 With diverging base Without diverging base  0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 140  160  180  200 220 240 260 Static bed height, Hs (mm)  280  300  Figure 7.14: Effect of diverging base on difference between values of Ums , Column C: 300 × 100 mm, Two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads.  7.6.3  Effect of vertical partition  A vertical partition was tested at different positions in the columns. As shown in Figure 7.15, a plexiglass plate of 150 mm height was mounted vertically in the centre of flatbottomed Column C parallel to the length of two 4 × 30 mm slots, arranged as shown in Figure 7.1(a). For a 100 mm static bed height, the particles piled up on one side of the plate when only one spout formed. The other spout could only be formed at very high gas velocity, but the bed height never recovered to be even on the two sides. When the bed surface 162  Chapter 7. Spouting with Multiple Slots  Figure 7.15: Position of vertical partition in column C; all dimensions are in mm.  was at (Hs = 150 mm) or above (Hs = 200 mm) the top of the vertical partition, the bed level was always even, but the fountains still tended to merge as the gas velocity increased. This confirms that the exchange of particles and gas in the bottom part of the bed is not the main cause of fountain merging. The result is different to that of Perterson (1966) (Mathur and Epstein, 1974), who used the merged vertical partition to separate the chambers in multiple spouted bed with rectangular chambers and circular gas inlet, where the merged vertical partitions significantly improved the stability of the multiple spouting by cutting off the lateral gas flow between chambers. However, in the current work, the gas and particle exchange inside the bed is either not significant or not an important factor that affect the stability of multiple chambers. This result is more similar to the result of Murthy and Singh (1994); Saidutta and Murthy (2000), who noted  163  Chapter 7. Spouting with Multiple Slots that the solids recirculation in the lower portion of the bed kept unchanged when there was a significant interference of fountains. As shown in Figure 7.16, a vertical partition of 300 mm height was next mounted in the upper part of Column C with a diverging base, with its bottom 250 mm above the bottom of the column. The behavior for a 200 mm static bed height was then compared to that of the same column without the vertical partition. It was observed that the two spouts were successfully separated by the vertical partition. In the range of gas velocity covered, up to 1 m/s, the spouts always achieved steady state and uniform spouting. It can also be seen from the pressure drop evolution, Figure 7.17, that the character of merging of spout disappeared when the vertical partition was present.  Figure 7.16: Position of suspended vertical partition in column C; all dimensions are in mm. This indicates that fountain merging was caused by exchange of particles and gases in the fountains themselves. Note that the drop in pressure decrease with increasing gas  164  Chapter 7. Spouting with Multiple Slots  Pressure drop across column, (kPa)  1.8 With draft plate Without draft plate  1.6 1.4 1.2 1 0.8 0.6  Dip showing end of merging 0.4 0.2 0.1  0.2  0.3  0.4 0.5 0.6 0.7 0.8 0.9 Superficial gas velocity, Ug (m/s)  1  1.1  Figure 7.17: Effect of vertical partitions for parallel slots, Column C: 300 × 100 mm, Two parallel 4 × 30 mm slots as shown in Figure 7.1(a), 1.33 mm glass beads, Hs =200 mm, diverging base.  velocity once spouting was reached is even sharper in this case than in Figure 3.1. Suspended vertical partitions were also applied for the case where two slots were aligned as in Figure 7.1(c). As shown in Figure 7.18, a vertical vertical partition of 300 mm height was mounted 250 mm above the bottom of column. The length of the plate was equal to the width of the column. The plate was mounted normal to the length of the slot at the centre of the column. Three static bed heights of 200, 250 and 300 mm were tested. In all three cases, the fountains were at the centre of each chamber and there was a steady state spouting. For a static bed height at or below the bottom of the vertical partition, the bed surface was flat. However, when the bed surface was above the bottom of the vertical partition, the bed surfaces on the two sides were uneven. As shown in Figure 7.19, the pressure drops of the two spouts were non-uniform after the second dip of the pressure drop when both spouts existed, indicating that particle and gas exchange at the upper part of the annulus is important to balance the two spouting chambers. 165  Chapter 7. Spouting with Multiple Slots  Figure 7.18: Position of suspended vertical partition in column B; all dimensions are in mm. Application of the suspended vertical partition allows columns with multiple slots to spout in a stable manner. If a single slot can be spouted stably, it appears that a column with multiple slots of the same size can be stably spouted with the addition of suspended vertical partitions. As observed, the main disadvantage of this type of scale-up is that a minor manufacturing difference in the slot or minor unevenness of the bed surface could cause a higher difference between the Ums of the spouts.  7.7  Scale-up of slot-rectangular spouted bed  It has been demonstrated above that stable spouting can be achieved by installing suspended vertical partitions in a column containing multiple chambers and slots. The scale-up method for slot-rectangular spouted beds proposed by Mujumdar (1984) can then be put into application with the assistance of vertical partitions as reported in the previous section.  166  Chapter 7. Spouting with Multiple Slots  Pressure drop across column, (kPa)  4.2 Front slot Back slot  4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 0.5  0.6  0.7 0.8 0.9 Superficial gas velocity, Ug (m/s)  1  1.1  Figure 7.19: Effect of vertical partition for parallel slots, Column B: 150 × 200 mm, Two aligned 4 × 30 mm slots as shown in Figure 7.1(c), 1.33 mm glass beads, Hs =300 mm, diverging base.  7.7.1  Single chamber  Slot designation To achieve stable spouting in a multiple slot chamber, the stability criteria for a single chamber should be satisfied first. In the current work, the stability of spouting was found to be mainly influenced by the slot configuration. Table 7.1 summarizes the stability of a number of slot width, length and depth combinations tested in Chapter 5. Here Ai is the cross-sectional area of slot. It can be concluded from the table that the slot-rectangular spouted bed column C can be stably spouted as long as the following criteria are satisfied. First, the slot cross-sectional area should be smaller than a maximum value, i.e.: √  Ai < 20 dp ∼  167  (7.1)  Chapter 7. Spouting with Multiple Slots This limitation is consistent with the criterion proposed by Passos et al. (1993): √  λβ ≤ 25.4 dp  (7.2)  where, λβ is the cross-sectional area of the slot. The cross-sectional area of slot in Equation 7.1 can be rewritten in the form of the equi-area diameter: Di,a < 23 dp ∼  (7.3)  which is also consistent with the stability criterion of Chandnani and Epstein (1986) for conventional spouted beds and for jet in fluidized beds (Grace and Lim, 1987): Di ≤ 25.4 dp  (7.4)  Limiting the slot cross-sectional area seems to be a necessary condition in all cases. According to Table 7.1, to achieve stable spouting, the dimensions of slots should satisfy either of the following two conditions: • Depth of slot: Slot Depth κ > 25 = dp dp ∼  (7.5)  Length η < 15 = W idth λ ∼  (7.6)  • Or slot Length/Width ratio:  These dimensionless groups could also be justified by dimensional analysis based on the Buckingham π theorem. With the two extra dimensions introduced in this work, the set of control variables in Chapter 3.4 becomes: α, β, λ, η, κ, dp , Hs , ρg , ρs and g. For  168  Chapter 7. Spouting with Multiple Slots Table 7.1: Dimensions and stability of slots for 1.33 mm glass beads in a plexiglass column of cross-section 300 mm × 100 mm, with air at 20◦ C and 1 atm pressure.  Slot dimension, mm Width Length  Depth  Dimensionless ratios √  Stable or not Ai /dp  Depth/dp  Length/W idth  2  100  62.7  10.6  47.1  50  Yes  4  100  62.7  15.0  47.1  25  Yes  6  100  62.7  18.4  47.1  16.7  Yes  8  100  62.7  21.3  47.1  12.5  No  4  100  22  15.0  16.5  25  No  4  100  33  15.0  24.8  25  Yes  4  100  40  15.0  30.1  25  Yes  4  100  50  15.0  37.6  25  Yes  2  100  12.7  10.6  9.5  50  No  4  100  12.7  15.0  9.5  25  No  6  100  12.7  18.4  9.5  16.7  No  8  100  12.7  21.3  9.5  12.5  No  4  40  12.7  9.5  9.5  10  Yes  4  60  12.7  11.6  9.5  15  Yes  4  80  12.7  13.5  9.5  20  No  40  10  12.7  15.0  9.5  0.25  Yes  20  20  12.7  15.0  9.5  1  Yes  14.1  28.2  12.7  15.0  9.5  2  Yes  10  40  12.7  15.0  9.5  4  Yes  7.1  56.5  12.7  15.0  9.5  8  Yes  5  80  12.7  15.0  9.5  16  No  169  Chapter 7. Spouting with Multiple Slots  determination of Ums , the following dimensionless groups can be identified: α β λ η κ H s ρs U √ ms , , , , , , , gHs dp dp dp dp dp dp ρg  (7.7)  These dimensionless groups can be combined to produce all of the dimensionless groups appearing in the criteria above, i.e.: √  λη η κ , , and dp λ dp  (7.8)  α α β β−η , , , and dp α 2dp λ  (7.9)  as well as:  Chamber size Dead zones in spouted beds could be harmful for processing some particles such as foodstuffs. To eliminate dead zones, a diverging base with an included angle of 60◦ should be utilized. Widths of the columns tested in this work were up to 225dp . As observed in the experiments, dead zones at the two side walls (shown in Chapter 6) can be eliminated by the diverging base. So a safe limit on the width of column can be estimated to be: α < 225 dp ∼  (7.10)  When the column thickness is increased, dead zones could appear at the front and back of the column. These dead zones were not observed in the current work. They should be related to the distance from the end of the slot to the column wall, (β − η)/2. The maximum value of this distance tested in the current work was 30dp . So the following  170  Chapter 7. Spouting with Multiple Slots criterion should be conservative: β−η < 30 2dp ∼  (7.11)  The stability of slot-rectangular spouted beds was mainly investigated in the column with β/α = 1/3 and 2/3. However, column B with β/α = 4/3 could still achieve stable spouting with type a slot of 62.7 mm depth. So a column thickness-to-width ratio: 4 β ≤ α 3  (7.12)  appears to be safe. In conventional spouted beds, Dc /Di is an important determinant of the stability of spouting. However, in the current work, spout stability appeared to be not affected by α/λ, which ranged from 35 ∼ 150. The ratios of the equi-area diameter or hydraulic diameter of the column and slot, in the range of 6.12 ∼ 13.6 and 7.5 ∼ 38.3 respectively, did not affect the spouting stability either. Fountain height and the maximum spoutable height were not investigated in the current work. They should be tested for the specific particles of interest in a single chamber before multiple chambers are designed.  Termination of spouting In conventional spouted beds, the Archimedes number based on particle diameter is related to the termination of spouting. For Ar < 223, 000, spouting terminated due to slugging for small particles or larger particles at elevated gas temperature; for Ar > 223, 000, the spouting terminated because of fluidization of the bed surface for large particles. In the current work, Ar ranged from 25,900 to 1,820,000, as listed in Table 7.2. As noted in Chapter 5, for 2.45 mm glass beads and 3.79 mm polystyrene particles, the spout terminated due to fluidization of bed surface. For 0.66 mm glass beads, the flow was  171  Chapter 7. Spouting with Multiple Slots Table 7.2: List of Archimedes numbers for the particles investigated in this work  Material  dp (mm)  ρs (kg/m3 )  Glass beads  0.66  2490  Glass beads  1.33  2490  ρg (kg/m3 )  Ar 25,800 221,000  1.225 Glass beads  2.45  2490  1,270,000  Polystyrene  3.79  930  1,820,000  unstable. For 1.33 mm glass beads, with Ar close to the limit for conventional spouting, fluidization could be observed with a properly designed slot. Otherwise spouting terminated because of instability. So it appears likely that a limitation similar to that for the conventional spouted beds exists for the slot-rectangular spouted beds, although dependent on the slot configuration.  7.7.2  Combined multiple chamber  When stable spouting is guaranteed in a single column, a column of large scale can be designed as multiples of the width and/or thickness of the single column. A diagram showing the geometry appears in Figure 7.20. Several guidelines are as follows: • To prevent the merging of fountains, vertical partitions should be applied to separate the large column into small chambers of the same size as the prototype. The bottom of the vertical partition should be positioned vertically just above the bed surface. It is suggested that the top of the vertical partition be above the summit of the fountain. • An independent gas supply is recommended for each chamber of columns with multiple chambers. This means separate base chambers for each section. If a 172  Chapter 7. Spouting with Multiple Slots common base chamber is used, a slight difference on the size of slots can cause great difference in hydrodynamic behaviour, especially in the onset of spouting, among the chambers. It is also suspected that, even with exactly identical slots, the flow could also be mal-distributed according to the analysis of Grace et al. (2007). In this case, the base chamber should be as small as possible to achieve the ’constant flow’ condition to minimize interaction among the slots. When a common gas supply is used for all chambers, diverging bases should be applied to minimize the lag between the onset of spouting above each slot. Diverging bases with an included angle of 60◦ are also recommended to eliminate dead zones. • It is also suggested that slots be put right at the centre of each chamber to minimize the interaction between them. • For drying of particles of high-temperature reactions, the perimeter of the scaled column should be minimized for a given number of chambers to minimize heat loss. • When the bed surface is lower than the bottom of suspended vertical partitions, exchange of particles between adjacent units is possible. This provides the possibility for continuous processing of particles, with particles added to one side of the overall large-scale column, and removed from the opposite side, as shown in Figure 7.21. Multiple stages of process can be achieved based on this design.  173  Figure 7.20: Diagram of column of combined chambers.  Chapter 7. Spouting with Multiple Slots  174  Chapter 7. Spouting with Multiple Slots  Figure 7.21: Diagram of column with multiple chambers for continuous processing of particles.  175  Chapter 8 Conclusions and Recommendation 8.1  Conclusions from this thesis  A comprehensive hydrodynamic study was carried out on four slot-rectangular spouted bed columns. The results were found to be similar to those for conventional axisymmetric spouted beds. Slot configuration was found to have significant influence on the stability of slot-rectangular spouted beds. Scale-up of slot-rectangular spouted beds was successfully achieved by applying multiple slots and suspended partitions. The following conclusions can be drawn from this study: • The pressure evolution of slot-rectangular spouted beds showed qualitative behaviour which was generally similar to that of conventional axisymmetric spouted beds. Hysteresis was found for the onset and collapse of the spout, with the hysteresis being more significant for intermediate static bed heights. • The spouting pressure drop, minimum spouting velocity and maximum pressure drop were compared to those of conventional spouted beds, and to earlier results reported for slot-rectangular spouted beds as well. Spouting pressure drop and maximum pressure drop have little relationship with the slot width and particle diameter. The ratio of spouting pressure drop to fluidization pressure drop was found to be less than 0.75, consistent with conventional axisymmetric spouted beds. However, previously proposed correlations for minimum spouting velocity for both conventional and slot-rectangular spouted beds did not give good agreement with results for the experimental units tested in the current work. The dependence of the minimum spouting velocity on the operating conditions is highly dependent on 176  Chapter 8. Conclusions and Recommendation the geometric configuration of the slot. Columns with similar slot configurations had similar trends in the minimum spouting velocity. Empirical correlations were developed based on the slot configurations for minimum spouting velocity and maximum pressure drop. • A number of distinct flow regimes were identified for slot-rectangular spouted beds. Fixed bed, internal jet, jet-in-fluidized bed, spouting, incoherent spouting, multiple spouting, merging of multiple spouts, slugging and spouting-in-fluidized-bed were observed. Spouting with swaying of fountain was found in addition to regimes identified in previous work. Flow regimes were identified based on the frequency, cycle frequency and the magnitude of fluctuations of the pressure drop of column. Each flow regime was found to have its own characteristic fluctuations. • All three mechanisms of spout termination: fluidization of the bed surface, choking of the spout and development of instability were found in slot-rectangular spouted beds, where the latter plays the most significant role. The instability was mainly caused by interaction and overlap among the multiple spouts. As long as a single steady state spout was formed, the bed could be spouted at static bed height > 600 mm. Otherwise, spouting terminated at heights < 300 mm. • Efforts were made to improve the stability of slot-rectangular spouted beds by eliminating multiple spouts along the slot length. To achieve this objective, slots of different configurations were tested with variables including slot depth, slot width and slot length. Slots of greater depth, smaller width, smaller length and smaller √ length/width provided more stable flow. With Ai /dp ≤21.3, stable spouting can be achieved if either of the following criteria is satisfied: slot length/width ≤ 15 or slot depth/dp ≥ 24.8. The distribution of gas velocity and pressure drop across the slot significantly affected the stability of slot-rectangular spouted beds. More  177  Chapter 8. Conclusions and Recommendation concentrated gas flow through the slot and higher pressure drop across the slot provided more stable spouting. This result is consistent with predictions from CFD simulations including the k −  model for gas velocity and pressure drop through slots of different configurations in single-phase flow. • Optical voidage probes successfully measured the location of the spout/annulus interface and the boundaries of dead zones. The dead zone was found to have an included angle close to 60◦ , suggesting that a diverging base with an included angle of 60◦ is an appropriate choice for slot-rectangular spouted beds. The spout shapes determined from two orthogonal directions differed, depending on the width and length of the slot. A wider slot produced a wider spout at the bottom. However, the spout tended towards a circular shape with increasing height. Three-dimensional effects for slot-rectangular apouted bed were then proved. The spout tended to “forget” the initial geometry of the slot. For different static bed heights, the spout shape was virtually the same at the same level. As a result, spouts from slots of equal cross-sectional area but different length/width ratios approached similarity in the upper part of the bed. A mechanism model proposed for two-dimensional column was used to predict the shape of the spout/annulus interface. Prediction deviated significantly from the experimental result because of the three-dimensional effect in the column. • Optical probes were able to measure the velocities and voidages of glass beads in the slot-rectangular spouted beds. The axial distributions of voidage and particle velocity showed that there is an acceleration zone of particles at the bottom. This zone is longer at higher superficial gas velocity. Particle velocity and voidage reached a maximum at the end of the acceleration zone. Higher superficial gas velocity caused higher particle velocities and higher voidages. Static bed height had no significant effect on the axial profile of particle velocity. However, higher static 178  Chapter 8. Conclusions and Recommendation bed heights led to smaller voidages in the spout. The lateral distribution of particle velocity in the spout followed a parabolic distribution. The voidage in the spout increased towards the centre of spout. In some cases, there were two peaks. A one dimensional model for conventional axisymmetric spouted bed based on the force balance of particle in the spout was used to predict the axial profile of the particle velocity in the current work. Predicted result was over-estimated but showed the same trend as the experimental result. • Particle velocities were higher for slots of smaller length/width ratios, but they appeared to approach the same value towards the top of the spout. The voidage for slots of higher length/width ratio was found to be higher. A slot of higher length/width ratio provided slightly higher particle circulation fluxes. • Scale-up by simply increasing the column thickness and slot length generally failed because of the instability caused by multiple spouts and the interaction between those spouts. However, spouting in columns with combined multiple chambers and multiple slots was proved to be a good way to scale up slot-rectangular spouted beds. Interaction among multiple fountains, including the merging of fountain and random positioning of fountains, was the main factor causing instability in the combined columns. Suspended vertical partitions with its bottom above the bed surface successfully separated the fountains, leading to stable and symmetric spouting. As long as the single column can yield stable spouting, the column with multiple chambers of the same configuration can spout stably. Some guidlines were proposed for this means of scale-up.  179  Chapter 8. Conclusions and Recommendation  8.2  Recommendations for future work  Flows in slot-rectangular spouted bed columns are much more complex than in conventional axisymmetric columns. Considerable work remains to reveal the behaviour of slot-rectangular spouted beds: • Gas/particle flows in the slot-rectangular spouted beds are three-dimensional. 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Chem. Eng. Sci. 58, 1641–1644. Zhu, J. X., G. Z. Li, S. Z. Qin, F. Y. Li, H. Zhang, and Y. L. Yang (2001). Direct measurements of particle velocities in gascsolids suspension flow using a novel five-fiber optical probe. Powder Technology 115, 184–192.  198  Appendix A Summary of Previous Work Table A.1: Annotated list of previous work on slot-rectangular spouted beds  Author  Comments  Mujumdar (1984)  Described the configuration of slot-rectangular spouted bed and the advantage of this configuration. Proposed combined columns for scaling-up.  Kalwar et al. (1989)  Studied vertical compressive stress of particle experimentally as a function of aspect ratio and diverging angle. An angle of 30◦ found to be preferred.  Krzywanski (1989)  et  Proposed a theoretical model to predict spout size and shape al. in thin two-dimensional bed, using least-action principle and minimizing the length of spout/annulus interface.  Kalwar et al. (1992)  Tested Hmax in two-dimensional columns with draft plates, with different separation distance, height and entrance angle.  Investigated flow regimes of multiple (3) spouts in a thin twoHuang and Chyang dimensional column with a perforated plate distributor. Pres(1992) sure fluctuations were the same in the spout and annulus.  Kalwar et al. (1993)  Investigated circulation of particles in two-dimensional spouted beds with draft plates.  Passos et al. (1993)  Proposed spoutability criteria, spout formation and termination conditions. Correlated maximum spoutable bed height in slotrectangular spouted beds.  Rocha et al. (1995)  Determined dynamics and heat transfer during coating of tablets in a two-dimensional spouted bed.  199  Appendix A. Summary of Previous Work Author  Comments  Dogan et al. (2000)  Report 8 flow regimes and spout termination conditions in a thin (30 mm) spouted bed.  Freitas et al. (2000)  Delineated flow regimes in columns of different thickness including a new regime Multiple Spout (MS). Hmax , Ums , ∆P , Hf were also investigated.  Madhiyanon et al. Devised two-region mathematical model for batch drying of (2001) grains in a two-dimensional spouted bed. Proposed criteria for scale-up of two-dimensional beds with geometrical similitude analysis. Some modification of nonCosta and Taranto dimensional parameters in conventional spouted beds. Ums , (2003) ∆Pmax , Hmax correlations were also provided.  Luo et al. (2004)  Performed experiments on a slot-rectangular column with draft plates. Measured pressure drop and compared it to conventional spouted beds.  Freitas et al. (004a)  Measured voidage profiles in slot-rectangular spouted beds of different thicknesses. The profiles showed significant threedimensional effects.  Freitas et al. (004b)  Identified different flow regimes in slot-rectangular spouted bed using statistical, spectral and chaotic analysis of pressure fluctuations.  Dogan et al. (2004)  Determined flow regimes, Ums and ∆Pmax in a half column with different slot width and divergence angle.  Singh et al. (2006)  One-dimensional force balance model to predict Ums and ∆Pmax slot-rectangular column, involving the inter-particle and particle/wall friction  200  Appendix A. Summary of Previous Work Table A.2: Correlation and criteria for slot-rectangular spouted beds  Author  Correlations α = 6 to 20 λ  Kalwar et al. (1989)  c0 c3 Hmax = + c2 + 2 α A2D A2D  2 β dp φ < 650 λ λ where ci is a constant, leading to √ βλ < 25.4 dp  (A.1)  (A.2) (A.3)  (A.4)  Passos et al. (1993)  γ ∆Pmax Dc dp − 1)−1.92 (tan( ))0.7 = 1 + 0.006( )5.04 ( Hgρp Di Dc φ 2 Qm s = 0.0000592( Rocha et al. (1995)  (A.5)  dp 0.05 Di 0.26 γ 2gHs (ρp − ρg ) ) ) ( ) (tan( ))0.7 ( Dc φ Dc 2 ρg (A.6) Ums ∝ Hs c  (A.7)  λ/dp < 25.4  (A.8)  Hs,max P c ∝ Di Di  (A.9)  where c = 0.57 to 1.07  Dogan et al. (2000)  Huang and Chyang (1992) where c is a constant  201  Appendix A. Summary of Previous Work Author  Correlations  α  α −0.75  H 0.16  ρ − ρ 0.23 Ums s s g √ = 3.0 (A.10) λ dp φ α ρg 2gHs  Costa and Taranto (2003)  ∆Pmax α α 0.18 H 1.15 = 1+32.6( )0.05 ( ) ( ) (Ar(φ))−0.41 (tan(γ/2))−2 Hs gρp λ dp φ α (A.11) Hmax α α 0.18 H −0.145 = 39.47( )0.075 ( ) ( ) (tan(γ/2))0.69 (A.12) α λ dp φ α  202  FB,SB  FB  CFB  voidage, particle velocity  Particle velocity  Particle velocity  Local Voidage, Particle Velocity  Voidage, solid flux  Optical probe  Optical grating  Non-intrusive Optical  Capacitance Probe  Electrical Capacitance Tomography  203 CFB  FB  System  Property  Technique  0.089 2.8  0.092  0.1-0.5  0.2  0.06-3  dp (mm)  19  <6  <4  Vp (m/s)  noninvasive  invasive  noninvasive  invasive  invasive  Invasive?  Able to determine the voidage in every local area of the whole crosssection.  Water-cooled probe designed to be used at temperatures up to 1000◦ C.  Limited to dilute flows.  A differential grating is used in the system.  Used by many researchers in cold units. Can measure particle velocity and voidage at the same time. May influence flow in the bed. Different numbers and arrangements of fibers have been used to determine the direction. For small particles, measures particle aggregation.  Comments  Table A.3: Experimental technique available for gas/solid flow measurement  al.  Pugsley and co-workers  Hage and Werther (1997)  Dyakowski and Williams (1993)  Fiedler et al. (1997)  He et (1994)  References  Appendix A. Summary of Previous Work  204  Particle Trajectory  Radioactive Particle Tracking (RPT)  Particle velocity  Particle Image Velocimeter (PIV)  Magnetic Flowmeter  Particle velocity  Laser Doppler Velocimeter/ Anemometer LDV/LDA  Particle Velocity, Circulating Time  Property  Technique  SB, FB  SB  SB,FB  FB  System  0.5-5.5  0.5-3  0.06-5  0.0620.1  dp (mm)  0.1-8  up to 6  0.03-0.6  2  Vp (m/s)  noninvasive  noninvasive  noninvasive  noninvasive  Invasive?  References  Waldie and Wilkinson (1986) Mann and Crosby (1975)  Larachi et al. (1995) Fangary et al. (2000)  Cycling time was measured with one coil, to calculate average flux. Particle velocity calculated using cross-correlation. Unlike other tracking methods, position of particle cannot be determined. Properties of the magnetic particles must be similar to those of the other particles. Path of the tracing particle is determined by detectors, which are calibrated first. Velocity and direction are calculated. Particle must also be similar to regular particles.  For slow velocity and measurements are limited to close to wall or dilute conditions.  Ibsen et al. (2003) Chen and Fan (1992) Fei et al. (2003)  Wang et al. (1993) Levy (1986) Limited to dilute phenomena or Ibsen et al. (2002) near the wall  Comments  Appendix A. Summary of Previous Work  205 SB  Particle Velocity and Direction  Bubble Position and Size  Stopwatch  X-ray Tomography and Xray Image FB  FB  Local Flux  Thermistor Anemometer  System  Particle/ Momentum FB Flux  Property  Momentum Probe and Particle Sampling Probe  Technique  0.02-1.5  0.9-9  0.33  dp (mm)  0.4  0.1  3  Vp (m/s)  noninvasive  noninvasive  invasive  invasive  Invasive?  Zhang et al. (1997)  References  Analysis of X-ray image, rather like PIV.  This simple method is only suitable for particles at the outer wall.  Kantzas et al. (2001)  Yang and Keairns (1983) Day et al. (1987)  This method and those using drag Marsheck and force are not direct and need to be Gomezplata calibrated using other methods. (1965)  Particle velocity is calculated from the signal of these two probes  Comments  Appendix A. Summary of Previous Work  206  and  Gas flow; Residence time;  Objective  Equations  1) Combination of verticalaxial and horizontal-radial force balance on different height of annulus 2) Darcy’s law for gas in annulus 3) Mass balance on gas in spout  1) Streamtube assumption 2) Plug flow and no mixing of gas in each stream tube Mass balance and pressure 3) Dispersion along the balance of gas phase in each streamtube streamtube  Assumption  Gas distriRovero et al. bution;Force No vertical shear stresses on (1983) balance model annulus boundaries  Lim Mathur (1974)  Reference  Table A.4: Some literature models for spouted beds  This type of model was already used and expanded by many researchers, in cylindrical, conical and cylinder-conical beds for gas dispersion, pressure distribution, residence time; It would be interesting to split a 3D column to streamtube;  Comments  Appendix A. Summary of Previous Work  207  Continuity equation in cylindrical coordinates; BC: the rate of entrainment into the spout and no flow across the bed walls  1) Voidage in annulus assumed constant; Two-region 2) Solid velocity negligibly model with small; Kursad and variable spout 3) Change of velocity in anKilkis (1983) diameter nulus also negligible  1) Particles move downward to fill empty spaces left by particles removed from below; Velocity dis- 2) Horizontal velocity is proBenkrid and tribution in portional to gradient of verCaram (1989) annulus tical velocities  The assumptions seem to be related; The rate of particle entrainment into the spout is needed  With another assumption of the spout shape, it will be suitable for 2dimensional column. This may be the first step;  1) Continuity equations and momentum conservation for both gas and solids in the spout; 2) Force balance for particles in both redial and axial direction in the annulus 3) Darcy’s law of pressure drop for gas phase in annulus BC: Cycling boundary conditions except near the spout  Assumption  Comments  Objective  Equations  Reference  Appendix A. Summary of Previous Work  Krzywanski et al. (1989)  208  Olazar et al. Gas distribu(1993) Olazar et al. tion;Expansion (1995) of streamtube Streamtube assumptions  Krzywanski et al. (1992)  Hamilton principle: least action law; derived that the shape of the spout was related to the expansion angle  Spout shape in 2D and cylindrical spouted beds  1) Steady state conditions 2) Constant voidage throughout the annulus 3) Material in annulus can be described as an isotropic, incompressible, rigid plastic, non-cohesive Coulomb powder  Assumption  Objective  Reference  Comments  Mass conservation  Eqns: 1) Modified vector Ergun equation and continuity equation of gas in annulus 2) Force balance and continuity equation for particles in annulus 3) Continuity and momentum equation for gas and solid flow in spout 4) Poisson pressure equation for whole bed  Bases spout shape on Krzywanski (1989).  Used empirical correlation for average diameter; Although the result is not so Eqns: Momentum equa- good, but the aptions, Minimize the length of plication of least acspout/annulus interface for tion law is interest2D beds and interfacial area ing, and may be usefor cylindrical beds ful  Equations  Appendix A. Summary of Previous Work  1) Force balance with interparticle and particle/wall friction stress 2) At Ums , stress at the roof of internal spout reaches zero  2D gas velocity; 3D particle velocity; Axisymmetric for gas, but spout shape not for solids  1) Friction is downward when increasing gas and vise versa 2) Two zones: cartesian and radius zones. Radius zone is the internal spout with round shape and extends to where gas flow equal to bed Ums and surface. Cartesian zone is Singh et al. ∆Pmax , force the rest of bed, has same gas (2006) balance velocity.  Kawanguchi et al. (2000)  The equations for solids are simple and interesting, but it’s important to find the parameters;  Gas phase: 1) Continuity equation; 2) Momentum equation Solid phase: 1) Newton’s equation in annulus 2) Mass and moment balance in spout  Assumption  Comments  Objective  Equations  Reference  Appendix A. Summary of Previous Work  209  Appendix B Measurement Systems B.1 B.1.1  Particle velocity and voidage measurement Measurement system  Many measurement techniques for particle velocity have been applied to fluidized and spouted beds, as listed in Table A.3. In the current work, particle velocity and voidage were measured by optical probes. A Particle Velocity Analyzer (PV4A) manufactured by the Institute of Process Engineering, Chinese Academy of Sciences, was used. The measurement system includes an optical probe, a light conversion box, a data conversion board and software to provide sampling and data analysis. This type of particle measurement has been widely used to determine particle velocity and voidage in fluidized beds. It has also been used in spouted beds by several researchers (He et al., 1994; Wang, 2006; San Jose et al., 1998). The measurement system in this project was the same unit as used and calibrated by Wang (2006), except for the probe. The optical probe in the current work is shown in Figure B.1. There are three bundles of fibres of diameter 15 microns. Each bundle has a diameter of ∼ 1 mm. The central bundle of fibres are light projectors, whereas the other two act as lighter receivers, corresponding to two sampling channels A and B. The basic mechanism to measure the velocity is to calculate the lag, τ , between the moments when the particles pass the two lighter receivers. The particle velocity, vp , can then be calculated by:  Le =  210  vp τ  (B.1)  Appendix B. Measurement Systems To prevent the blind zone reported by Liu et al. (2003a,b); Wang (2006); He et al. (1994), a glass window of 4 mm length was attached to the tip of the probe. The effective distance of the probe was then calibrated by the same method as was used by Wang (2006).  Figure B.1: Details of tip of the optical probe for velocity measurement.  In our case, the probe was calibrated employing the actual particles from this project. A rotating plate with a particle trap holding particles was driven by a motor. Particles rotated together with the rotating plate. Based on the known particle velocity, vp , the probe was used to measure the lag, τ , between the two channels. The effective distance was then calculated from Equation B.1. As shown in Figure B.2, the effective distance of the probe did not change when the probe tip moved from the surface of the particles to 3 mm from the surface. This confirms that the glass window effectively removes the blind zone of the probe. The effective distance when the tip of the probe was immersed below the surface of the particles was higher than the others. The difference was probably caused by the disturbance of the probe on the particles. The effective length between two channels was also calibrated for different particle velocities up to ≈ 8 m/s using rotating plate with attached particles. As shown in Figure B.3, the effective distance did not change for different particle velocities. The calibrated effective distance had an average of 1.057 ± 0.031 mm. The calibration error mainly 211  Appendix B. Measurement Systems  Effective distance of probe, (mm)  1.4  1.2  1  0.8  0.6 -2  -1 0 1 2 3 Distance from probe tip to particle surface, (mm)  4  Figure B.2: dependence of calibrated effective distance of probe with distance from probe tip to surface of particles. Particle velocity: 0.24 m/s.  2  Effective distance (mm)  Glass beads of 1.33 mm diameter Average 90% Confidence interval 1.5  1  0.5  0 0  1  2  3 4 5 Particle velocity (m/s)  6  7  8  Figure B.3: Dependence of calibrated effective distance of probe with particle velocities. Distance from probe tip to particles: 1 mm.  212  Appendix B. Measurement Systems comes from the calibration system where the angular velocity of the rotating plate had slight fluctuation.  B.1.2  Data analysis  In a typical measurement process, two series of data were read from the system. These data represent the intensity of the reflected light through the two channels. The measurement system was designed to be able to sample 2n points (with n= 9 - 15). In this work, 215 points were normally sampled for each measurement. These data series were divided into several groups in the data analysis process. Cross-correlation was carried out to find the lag between the corresponding groups in the two channels. The number of data points in each group was kept at 2n for convenience. A sample of one group of data is shown in Figure B.4. In this case the sample interval was 0.008 ms, and there were 2048 data points in each series.  Intensity, (v)  1  Channel A  0.8 0.6 0.4 0.2 0 0  20  40 60 Time, (ms)  Intensity, (v)  1  80  100  Channel B  0.8 0.6 0.4 0.2 0 0  20  40 60 Time, (ms)  80  100  Figure B.4: Typical two channel data series.  The lag between two channels was found by cross-correlating the signals from the two channels. The correlation coefficients C(X, Y )p between two discrete series of T data 213  Appendix B. Measurement Systems points with a lag, p, were defined as the covariance of the two series, R(X, Y )p divided by the product of the standard deviations of the two series: Pt=1 T −p X(t + p)Y (t) C(X, Y )p = ST D(X)ST D(Y )  (B.2)  The lag between two channels, τ is chosen to produce the maximum value of R(X, Y ). With this lag, the data for the two channels has the highest similarity. However, this method of finding the lag is too slow in terms of computation time. Another method using the Fourier transform and inverse Fourier transform has much higher efficiency. The relation between the correlation function and lag time was simply the inverse Fourier transform of the product of Fourier transforms of the two data series:  F F T (X)•(F F T (Y ))∗  InverseF F T  * )  FFT  R(X, Y )p  (B.3)  where R(X, Y )p =  t=1 X  X(t + p)Y (t)  (B.4)  T −p  The “∗” denotes the complex conjugate of the resulting Fourier transform of the corresponding data series. By finding the p that produces the maximum R, the lag was found. R0 is the correlation function without any lag. R1 is correlation function when the series X is one time interval later than Y. RT −1 is correlation function when Y is 1 time interval later than X. This method can find the maximum correlation when either series is leading. So that this method can be used to calculate the velocity in both the positive and negative directions. This method was applied to the software provided by the manufacturer. A sample calculation based on the data series in Figure B.4 is shown in Figure B.5. In this case data for channel A is 25 sampling intervals later than for channel B.  214  Appendix B. Measurement Systems 1  Correlation coefficient  0.8 0.6 0.4 0.2 0 -0.2 -0.4 -1000  25 -500  0 Lag  500  1000  Figure B.5: Lag and corresponding correlation function of data in Figure B.4.  After the lag of each group was calculated, the velocity is calculated, presumably by:  v¯p =  Le τ̄  (B.5)  To make this process clear and improve the convenience of the measurements, a program was written in C++ to process the data. The lag between the two channels is calculated by FFT and inverse FFT. The average velocity is calculated by: ¯ Le v¯p = τ The source code of this program is listed in Appendix C.1  215  (B.6)  Appendix B. Measurement Systems  B.2  Gas velocity measurement  In the current work, gas flow to the column was measured by an orifice flowmeter. The positions of the orifice and pressure measurement points are shown in Figure B.6. The gas flow direction is also shown. When this orifice was mounted in the pipeline, there was no valve, elbow or T connector within 50 cm of the upstream end of the orifice. The pressure drop across the orifice and the pressure just upstream of the orifice were measured. These were then used to calculated the gas flow from the usual orifice equation.  Figure B.6: Diagram showing positions of orifice and the pressure measurement.  According to Stearns et al. (1951), the orifice discharge coefficient can be calculated  216  Appendix B. Measurement Systems based on the orifice diameter, D0 and pipe diameter, D1 :  β0 =  D0 D1  (B.7)  530 B0 = √ D1  (B.8)  A0 = D0 (830 − 5000β0 + 9000β02 − 4200β03 + B0 )  (B.9)  0.007 D1  (B.10)  0.076 n0 = 0.364 + √ D1  (B.11)  Ke ≈ b0 + n0 β04  (B.12)  b0 = 0.5993 +  106 D0 15  (B.13)  ρg U0 D0 µg  (B.14)  1 + A0 /Re 1 + A0 /Ree  (B.15)  Ree = Re = K0 = Ke  Here Re is the Reynolds number at the orifice; K0 is the orifice discharge coefficient; Ke is the orifice discharge coefficient when Re = Ree ; U0 is the gas velocity through the orifice, µg is the gas viscosity, ρg is the gas density; the other parameters are intermediate parameters.  It should be noted that the equations and values of parameters are specific  to the type of orifice flowmeter, varying with the shape of the orifice and the position of gas pressure measurement. Formulas for other types of orifice were listed by Stearns et al. (1951). The gas velocity and gas volumetric flowrate, Q0 , through the orifice can then be calculated by: s U0 = K0  217  2∆P ρg  (B.16)  Appendix B. Measurement Systems s 2∆P Q0 = U0 (πD02 /4) = K0 (πD02 /4) ρg  (B.17)  By applying the ideal gas law, the gas density can be calculated.  ρg =  m PM = V RT  (B.18)  where ∆P and P are the pressure drop across the orifice and the absolute pressure before the orifice; V is the volume of gas; m is the mass of gas; R is the universal gas constant; M is the molar weight of gas; T is the absolute temperature with units of “K”. The volumetric flow rate of gas can then be represented by: r Q0 =  K0 (πD02 /4)  r r r 2∆P RT 2RT ∆P ∆P 2 = K0 (πD0 /4) = K1 PM M P P r K1 = K0 (πD02 /4)  2RT M  (B.19)  (B.20)  where K1 is the volumetric flow coefficient. Since Q0 is the flow through the orifice at P , it should be converted to the volumetric flow in the column. In this project, the flow was calculated at the pressure of the environment. Q = Q0  P 101325  (B.21)  where P is in Pascals. The orifice coefficients are functions of Reynolds number. As shown in Figure B.7, the coefficients become constant when the Reynolds number is high enough. In the experiments of the current work, D0 = 25.4 mm and D1 = 50.8 mm, the value of K1 = 0.1303 at Re = 106 and room temperature T = 293.15 K for gas with M = 0.0288 kg/mol was normally used. It should be noted that, because the coefficient is a function of Reynolds number, when the gas velocity was unknown, the coefficient for  218  Appendix B. Measurement Systems  Volumetric flow coefficient, (m3/s)  0.2  0.18  0.16  0.14  0.12  0.1 1000  10000 100000 Reynolds number  1e+006  Figure B.7: Orifice coefficient as a function Reynolds number.  small gas velocities should be calculated by iteration. The procedure is to first calculate the Reynolds number from the constant coefficient, then adjust the coefficient with the calculated Reynolds number until the result does not change. The MATLAB code written to calculate the orifice discharge coefficient is listed in Appendix C.2  219  Appendix C Programs C.1  C++ code for particle velocity analysis  // C D a t a f i l e : c a l s s t o read t h e d a t a and do c r o s s −c o r r e l a t i o n // D a t a f i l e . h #pragma once class CDatafile { public : C D a t a f i l e ( void ) ; ˜ C D a t a f i l e ( void ) ; % Some o b j e c t s fromt th e CEdit ∗ s t a t b o x ; // CListCtrl ∗ l i s t b o x ; // CFileFind ∗ m f i n d e r ; // BOOL m found ; bool bool void void void void void void void void void void void void void void  main GUI E d i t b o x t o show summary L i s t b o x t o show g r o u p s F i l e f i n d e r for batch process  ReadPva ( void ) ; ReadCsv ( void ) ; C o r r e l a t e ( void ) ; S t a t i c s ( int , int , int , double ∗ , double ∗) ; d f f t ( double ∗ , unsigned long , int ) ; C o r r e l a t e A ( void ) ; C o r r e l a t e B ( void ) ; C o r r e l a t e C ( void ) ; C o r r e l a t e D ( void ) ; Stop ( void ) ; R e f i l l ( void ) ; R e s t a t ( void ) ; Reminr ( void ) ; SaveCsv ( void ) ; SaveDat ( void ) ; D o f f t ( int ) ;  220  Appendix C. Programs int Working ; char m f i l e p a t h [MAX PATH ] ; char m path [MAX PATH ] ; char m name [MAX PATH ] ; t i t l e . extension BYTE datax [ 3 2 7 7 0 ] ; BYTE datay [ 3 2 7 7 0 ] ; BYTE d a t a f [ 3 2 7 7 0 ] ; int m fmin ; int m fmax ; int m u f f t ; // Parameters read double m f r e q ; // double m prbs ; // // int m pnts ; int m grpc ; // double double double double double double double double  m m m m m m m m  bedr ; bedd ; bedl ; bedw ; bedt ; bedh ; prba ; prbb ;  // D i r e c t o r y // D i r e c t o r y // F u l l pathname i n c l u d i n g p a t h \  // Data s e r i e f o r CH1 // Data s e r i e f o r CH2 // A l l o c a t e f o r FFT  Index of frequency | period Index of probe s i z e P o i n t s per group Number o f g r o u p s  // p a r t i c l e d e n s i t y // p a r t i c l e s i z e // s l o t w i d t h // bed w i d t h // bed t h i c k n e s s // s t a t i c bed h e i g h t // parameter a i n e p s=a∗ v o l t+b // parameter b i n e p s=a∗ v o l t+b  // Parameters t h a t // int m chno ; int m grps ; // int m mode ; // int m type ; // double m minr ; // double m rang ; //  s h o u l d be s a v e d Use which c h a n n e l s i g n a l f o r v o i d a g e Group number Direction Maximum t y p e Minimum c o e f f i c i e n t Search range  int m vcnt ; double m avgx ; double m avgy ; double m aepx ; double m aepy ; double m avgv ;  Average Average Average Average Average  // // // // //  of channel X of channel Y voidage voidage velocity  221  Appendix C. Programs double double double double  m m m m  avgm ; avgt ; avgp ; avgr ;  // // // //  Average Average Average Average  mass mass mass mass  flux flux flux flux  // Parameters t h a t s h o u l d be t h e r e s u l t f i l e // Number o f p e r i o d s int m optp [ 1 2 8 ] ; double m optt [ 1 2 8 ] ; // Delay tiem o f each group double m optr [ 1 2 8 ] ; // C o e f f i c i e n t on t h e peak double m optv [ 1 2 8 ] ; // V e l o c i t y o f e v e r y group double m optx [ 1 2 8 ] ; // V o l t a g e o f each group f o r double m opty [ 1 2 8 ] ; // V o l t a g e o f each group f o r double m epsx [ 1 2 8 ] ; // Voidage o f each group f o r double m epsy [ 1 2 8 ] ; // Voidage o f each group f o r // I f t h e group i s v a l i d bool m optu [ 1 2 8 ] ;  CH1 CH2 CH1 CH2  // Another f u n c t i o n s h o u l d be w r i t t e n t o t e r m i n a t e c o r r e l a t i o n bool m stop ; bool i s r u n n i n g ; };  // D a t a f i l e . cpp #include ” StdAfx . h” #include ”math . h” #include ” . \ d a t a f i l e . h” #include ” s h l w a p i . h” #include ”CommonProc . h” #include ” P r o c e s s . h” #define Swap ( a , b ) tempr = ( a ) ; ( a ) = ( b ) ; ( b ) = tempr C D a t a f i l e : : C D a t a f i l e ( void ) : m stop ( f a l s e ) , isrunning ( false ) { Working =0; } C D a t a f i l e : : ˜ C D a t a f i l e ( void ) { }  222  Appendix C. Programs // Read d a t a from f i l e m name bool C D a t a f i l e : : ReadPva ( void ) { int i , tmpint ; float tmpflt ; char tmpbuf [MAX PATH ] ; FILE∗ f i d ; f i d=f o p e n ( m name , ” r ” ) ; i f ( f i d !=NULL) { f s c a n f ( f i d , ”%d,% f ,% s ” ,& tmpint ,& t m p f l t , tmpbuf ) ; m prbs=t m p f l t ; s p r i n t f ( tmpbuf , ”%.3 f mm” , m prbs ) ; : : AfxGetApp ( )−>m pMainWnd−>GetDlgItem (IDC CPRBS)−> SetWindowText ( tmpbuf ) ; f s c a n f ( f i d , ”%d,% f ” ,&m pnts ,& t m p f l t ) ; m f r e q=t m p f l t ; s p r i n t f ( tmpbuf , ”%.3 f ms” , m f r e q ) ; : : AfxGetApp ( )−>m pMainWnd−>GetDlgItem (IDC CFREQ)−> SetWindowText ( tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; for ( i =0; i <m pnts ; i ++) f s c a n f ( f i d , ”%d” , &datax [ i ] ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; for ( i =0; i <m pnts ; i ++) f s c a n f ( f i d , ”%d” , &datay [ i ] ) ; fclose ( fid ) ; m pnts=pow ( 2 , ( int ) ( l o g ( ( double ) ( m pnts −1) ) / l o g ( 2 . ) ) +1) ; s p r i n t f ( tmpbuf , ”%d” , m pnts ) ; : : AfxGetApp ( )−>m pMainWnd−>GetDlgItem (IDC CPNTS)−> SetWindowText ( tmpbuf ) ; i f ( m pnts ==32768) { datax [ 3 2 7 6 6 ] = datax [ 3 2 7 6 7 ] = datax [ 3 2 7 6 5 ] ; datay [ 3 2 7 6 6 ] = datay [ 3 2 7 6 7 ] = datay [ 3 2 7 6 5 ] ;  223  Appendix C. Programs } return true ; } else return f a l s e ; } // R e t r i e v e p r o c e s s e d d a t a bool C D a t a f i l e : : ReadCsv ( void ) { int i , tmpint ; f l o a t t m p f l t , t m p f l t a , tmpfltb , t m p f l t c , tmpfltd , t m p f l t e , t m p f l t f , tmpfltg ; char tmpbuf [MAX PATH ] ; FILE∗ f i d ; char tmpname [MAX PATH ] ; s t r c p y ( tmpname , m name ) ; : : PathRemoveExtension ( tmpname ) ; s t r c a t ( tmpname , ” . c s v ” ) ; f i d=f o p e n ( tmpname , ” r ” ) ; i f ( f i d !=NULL) { f s c a n f ( f i d , ”%s %s ” ,&tmpbuf ,& tmpbuf ) ; f s c a n f ( f i d , ”%s %f %s %s ” ,&tmpbuf ,& t m p f l t ,&tmpbuf ,& tmpbuf ) ; m prbs=t m p f l t ; f s c a n f ( f i d , ”%s %d” ,&tmpbuf ,& tmpint ) ; m pnts=tmpint ; f s c a n f ( f i d , ”%s %f %s %s ” ,&tmpbuf ,& t m p f l t ,&tmpbuf ,& tmpbuf ) ; m f r e q=t m p f l t ; f s c a n f ( f i d , ”%s %d” ,&tmpbuf ,& tmpint ) ; m grps=tmpint ; f s c a n f ( f i d , ”%s %d %s %s ” ,&tmpbuf ,& tmpint ,&tmpbuf ,& tmpbuf ) ; m grpc=tmpint ; f s c a n f ( f i d , ”%s %f %s %s ” ,&tmpbuf ,& t m p f l t ,&tmpbuf ,& tmpbuf ) ; m rang=t m p f l t ; f s c a n f ( f i d , ”%s %f ” ,&tmpbuf ,& t m p f l t ) ; m minr=t m p f l t ; f s c a n f ( f i d , ”%s %d %s %s %s %s %s ” ,&tmpbuf ,& tmpint ,&tmpbuf ,& tmpbuf ,&tmpbuf ,&tmpbuf ,& tmpbuf ) ; m type=tmpint ; f s c a n f ( f i d , ”%s %d %s %s %s %s ” ,&tmpbuf ,& tmpint ,&tmpbuf ,& tmpbuf ,&tmpbuf ,& tmpbuf ) ;  224  Appendix C. Programs m mode=tmpint ; f s c a n f ( f i d , ”%s ” ,&tmpbuf ) ; for ( i =0; i <m grps ; i ++) { f s c a n f ( f i d , ”%d,%d,%d,% f ,% f ,% f ,% f ,% f ,% f ,% f ” , &tmpint ,& m optu [ i ] ,& m optp [ i ] ,& t m p f l t ,& t m p f l t a ,& tmpfltb ,& t m p f l t c ,& tmpfltd ,& t m p f l t e ,& t m p f l t f ) ; m optt [ i ]= t m p f l t ; m optr [ i ]= t m p f l t a ; m optv [ i ]= t m p f l t b ; m optx [ i ]= t m p f l t c ; m opty [ i ]= t m p f l t d ; m epsx [ i ]= t m p f l t e ; m epsy [ i ]= t m p f l t f ; } f s c a n f ( f i d , ”%s ,%d,% f ,% f ,% f ,% f ,% f ,% f ,% f ,% f ” , &tmpbuf ,& tmpint ,& t m p f l t ,& t m p f l t a ,& tmpfltb ,& t m p f l t c ,& tmpfltd ,& t m p f l t e ,& t m p f l t f ,& t m p f l t g ) ; m avgp=t m p f l t ; m avgt=t m p f l t a ; m avgr=t m p f l t b ; m avgv=t m p f l t c ; m avgx=t m p f l t d ; m avgy=t m p f l t e ; m aepx=t m p f l t f ; m aepy=t m p f l t g ; fclose ( fid ) ; R e f i l l () ; return true ; } else return f a l s e ; } // Do c o r r e l a t i o n a c c o r d i n g t o t h e s e t t i n g s void C D a t a f i l e : : C o r r e l a t e ( void ) { Working =1; i f ( m u f f t ) // Use FFT {  225  Appendix C. Programs switch ( m type ) { case 0 : C o r r e l a t e D ( ) ; break ; case 1 : C o r r e l a t e D ( ) ; break ; case 2 : C o r r e l a t e D ( ) ; break ; } } else { switch ( m type ) { case 0 : C o r r e l a t e A ( ) ; break ; case 1 : C o r r e l a t e B ( ) ; break ; case 2 : C o r r e l a t e C ( ) ; break ; } } Working =2; } void C D a t a f i l e : : Stop ( void ) { } // This f u n c t i o n r e t u r n s s t a n d a r t d e v i a t i o n as mstd , and mean v a l u e as mmean // Z h i w e i 04/14/04 void C D a t a f i l e : : S t a t i c s ( int mchno , int mmin , int mmax, double∗ mstd , double∗ mmean) { int i ; double tmp=0; i f ( mchno==0) { for ( i=mmin ; i<= mmax; i ++) tmp = tmp + datax [ i ] ; ∗mmean = tmp / (mmax−mmin+1) ; tmp=0; for ( i=mmin ; i<= mmax; i ++) tmp = tmp + ( datax [ i ]−∗mmean) ∗( datax [ i ]−∗mmean) ; ∗mstd = s q r t ( tmp / (mmax−mmin) ) ; } else  226  Appendix C. Programs { for ( i=mmin ; i<= mmax; i ++) tmp = tmp + datay [ i ] ; ∗mmean = tmp / (mmax−mmin+1) ; tmp=0; for ( i=mmin ; i<= mmax; i ++) tmp = tmp + ( datay [ i ]−∗mmean) ∗( datay [ i ]−∗mmean) ; ∗mstd = s q r t ( tmp / (mmax−mmin) ) ; } } // The a c t u a l FFT p r o c e d u r e m o d i f i e d from ”Math R e c i p e s ” void C D a t a f i l e : : d f f t ( double mdatat [ ] , unsigned long nn , int i s i g n ) { unsigned long n , mmax,m, j , i s t e p , i ; double wtemp , wr , wpr , wpi , wi , t h e t a ; double tempr , tempi ; double ∗ mdataf ; /∗ Use p o i n t e r i n d e x e d from 1 i n s t e a d o f 0 ∗/ mdataf = &mdatat [ − 1 ] ; m=( int ) ( l o g 1 0 ( ( double ) ( nn−1) ) / l o g 1 0 ( 2 . 0 ) ) ; nn=pow ( 2 ,m+1) ; n = nn << 1 ; j = 1; for ( i = 1 ; i < n ; i+=2 ) { if ( j > i ) { Swap ( mdataf [ j ] , mdataf [ i ] ) ; Swap ( mdataf [ j +1] , mdataf [ i +1]) ; } m = nn ; while ( m >= 2 && j > m ) { j −= m; m >>= 1 ; } j += m; } mmax = 2 ;  227  Appendix C. Programs while ( n > mmax) { i s t e p = mmax<<1; t h e t a = i s i g n ∗ 6 . 2 8 3 1 8 5 3 0 7 1 7 9 5 9 /mmax; wtemp = s i n ( 0 . 5 ∗ t h e t a ) ; wpr = −2.0∗wtemp∗wtemp ; wpi = sin ( theta ) ; wr = 1 . 0 ; wi = 0 . 0 ; for ( m = 1 ; m < mmax; m += 2 ) { for ( i = m; i <= n ; i += i s t e p ) { j = i + mmax ; tempr = wr ∗ mdataf [ j ] − wi ∗ mdataf [ j + 1 ] ; tempi = wr ∗ mdataf [ j +1] + wi ∗ mdataf [ j ] ; mdataf [ j ] = mdataf [ i ] − tempr ; mdataf [ j +1] = mdataf [ i +1] − tempi ; mdataf [ i ] += tempr ; mdataf [ i +1] += tempi ; } wr = ( wtemp=wr ) ∗wpr−wi ∗ wpi+wr ; wi = wi ∗wpr+wtemp∗ wpi+wi ; } mmax = i s t e p ; } } // This f u n c t i o n f i n d s t h e ” f i r s t peak ” v a l u e r e g a r d i n g t h e d i r e c t i o n mmode : // 0−d o u b l e , 1−u p f l o w , 2−downflow . // Z h i w e i 04/14/04 void C D a t a f i l e : : C o r r e l a t e A ( void ) { int i , j , k , n , g r p c n t ; int p f l a g , n f l a g ; double tmpv , tmpdbln , tmpdblp , n f c t , p f c t ; double miux , miuy , stdx , s t d y ; m grpc=g r p c n t =( int ) ( m pnts / ( m grps+m rang ) ) ; // f o r ( i=mgrps −1; i >=0; i −−) int tmpint=m grps ;  228  Appendix C. Programs for ( i =0; i <tmpint ; i ++) { n=g r p c n t ∗ i ; n f c t=p f c t =−1; tmpdbln=tmpdblp =1; n f l a g=p f l a g=f a l s e ; for ( j =0; j <g r p c n t ∗ m rang ; j ++) { i f ( m mode !=2) { tmpv=0; S t a t i c s ( 0 , n , n+grpcnt −1, &stdx , &miux ) ; S t a t i c s ( 1 , n+j , n+grpcnt −1+j , &stdy , &miuy ) ; for ( k=0; k<g r p c n t ; k++) tmpv=tmpv+(datax [ n+k]−miux ) ∗( datay [ n+k+j ]−miuy ) ; tmpv=tmpv / ( grpcnt −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpv=tmpv / ( s t d x ∗ s t d y ) ; i f ( tmpv>tmpdblp ) p f l a g=true ; else i f ( pflag ) { m optr [ i ]= tmpdblp ; m optp [ i ]= j −1; m optx [ i ]=miux ; m opty [ i ]=miuy ; break ; } tmpdblp=tmpv ; } i f ( m mode !=1) { tmpv=0; S t a t i c s ( 0 , n+j , n+grpcnt −1+j , &stdx , &miux ) ; S t a t i c s ( 1 , n , n+grpcnt −1, &stdy , &miuy ) ; for ( k=0; k<g r p c n t ; k++)  229  Appendix C. Programs tmpv=tmpv+(datax [ n+k+j ]−miux ) ∗( datay [ n+k]−miuy ) ; tmpv=tmpv / ( grpcnt −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpv=tmpv / ( s t d x ∗ s t d y ) ; i f ( tmpv>tmpdbln ) n f l a g=true ; else i f ( nflag ) { m optr [ i ]= tmpdbln ; m optp [ i ]=1− j ; m optx [ i ]=miux ; m opty [ i ]=miuy ; break ; } tmpdbln=tmpv ; } } m m m m  optt [ optv [ epsx [ epsy [  i ]= m optp [ i ] ∗ m f r e q ; i ]=( m optp [ i ]==0) ? 0 : m prbs / ( m optp [ i ] ∗ m f r e q ) ; i ]= m optx [ i ] ∗ m prba+m prbb ; i ]= m optx [ i ] ∗ m prba+m prbb ;  } R e f i l l () ; } // This f u n c t i o n f i n d s t h e ”maximum f i r s t ” v a l u e r e g a r d i n g t h e d i r e c t i o n mmode : // 0−d o u b l e , 1−u p f l o w , 2−downflow . // Z h i w e i 04/14/04 void C D a t a f i l e : : C o r r e l a t e B ( void ) { int i , j , k , n , g r p c n t ; int p f l a g , n f l a g , nlag , p l a g ; double tmpv , tmpdbl , n f c t , p f c t ; double miux , miuy , stdx , s t d y ; m grpc=g r p c n t =( int ) ( m pnts / ( m grps+m rang ) ) ;  230  Appendix C. Programs int tmpint=m grps ; for ( i =0; i <tmpint ; i ++) { n=g r p c n t ∗ i ; n f c t=p f c t =−1; tmpdbl =1; n f l a g=p f l a g=f a l s e ; for ( j =0; j <g r p c n t ∗ m rang ; j ++) { tmpv=0; S t a t i c s ( 0 , n , n+grpcnt −1, &stdx , &miux ) ; S t a t i c s ( 1 , n+j , n+grpcnt −1+j , &stdy , &miuy ) ; i f ( j ==0) { m optx [ i ]=miux ; m opty [ i ]=miuy ; } for ( k=0; k<g r p c n t ; k++) tmpv=tmpv+(datax [ n+k]−miux ) ∗( datay [ n+k+j ]−miuy ) ; tmpv=tmpv / ( grpcnt −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpv=tmpv / ( s t d x ∗ s t d y ) ; i f ( tmpv>tmpdbl ) p f l a g=true ; else i f ( pflag ) { p f c t=tmpdbl ; p l a g=j −1; break ; } tmpdbl=tmpv ; } for ( j =0; j <g r p c n t ∗ m rang ; j ++) { tmpv=0; S t a t i c s ( 0 , n+j , n+grpcnt −1+j , &stdx , &miux ) ; S t a t i c s ( 1 , n , n+grpcnt −1, &stdy , &miuy ) ; for ( k=0; k<g r p c n t ; k++)  231  Appendix C. Programs tmpv=tmpv+(datax [ n+k+j ]−miux ) ∗( datay [ n+k]−miuy ) ; tmpv=tmpv / ( grpcnt −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpv=tmpv / ( s t d x ∗ s t d y ) ; i f ( tmpv>tmpdbl ) n f l a g=true ; else i f ( nflag ) { n f c t=tmpdbl ; n l a g=1−j ; break ; } tmpdbl=tmpv ; } m m m m m m  optr [ optp [ optv [ optt [ epsx [ epsy [  i ]=( p f c t >n f c t ) ? p f c t : n f c t ; i ]=( p f c t >n f c t ) ? p l a g : n l a g ; i ]=( m optp [ i ]==0) ? 0 : m prbs / ( m optp [ i ] ∗ m f r e q ) ; i ]= m optp [ i ] ∗ m f r e q ; i ]= m optx [ i ] ∗ m prba+m prbb ; i ]= m optx [ i ] ∗ m prba+m prbb ;  } R e f i l l () ; } // This f u n c t i o n f i n d s t h e ”maximum peak ” v a l u e r e g a r d i n g t h e d i r e c t i o n mmode : // 0−d o u b l e , 1−u p f l o w , 2−downflow . // Z h i w e i 04/14/04 void C D a t a f i l e : : C o r r e l a t e C ( void ) { int i , j , k , n , g r p c n t ; int p f l a g , n f l a g , nlag , p l a g ; double tmpv , tmpdbl , n f c t , p f c t ; double miux , miuy , stdx , s t d y ; m grpc=g r p c n t =( int ) ( m pnts / ( m grps+m rang ) ) ; int tmpint=m grps ; for ( i =0; i <tmpint ; i ++)  232  Appendix C. Programs { n = i ∗ grpcnt ; nflag = pflag = false ; for ( j =0, tmpdbl =1, p f c t =−1;( j <g r p c n t ∗ m rang )&&(m mode!=2) ; j ++) { tmpv=0; S t a t i c s ( 0 , n , n+grpcnt −1, &stdx , &miux ) ; S t a t i c s ( 1 , n+j , n+grpcnt −1+j , &stdy , &miuy ) ; i f ( j ==0) { m optx [ i ]=miux ; m opty [ i ]=miuy ; } for ( k=0; k<g r p c n t ; k++) tmpv=tmpv+(datax [ n+k]−miux ) ∗( datay [ n+k+j ]−miuy ) ; tmpv=tmpv / ( grpcnt −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpv=tmpv / ( s t d x ∗ s t d y ) ; i f ( tmpv>tmpdbl ) p f l a g=true ; else { if ( pflag ) i f ( tmpdbl>p f c t ) { p f c t=tmpdbl ; p l a g=j −1; } p f l a g=f a l s e ; } tmpdbl=tmpv ; } for ( j =0, tmpdbl =1, n f c t =−1;( j <g r p c n t ∗ m rang )&&(m mode!=1) ; j ++) { tmpv=0;  233  Appendix C. Programs S t a t i c s ( 0 , n+j , n+grpcnt −1+j , &stdx , &miux ) ; S t a t i c s ( 1 , n , n+grpcnt −1, &stdy , &miuy ) ; i f ( j ==0) { m optx [ i ]=miux ; m opty [ i ]=miuy ; } for ( k=0; k<g r p c n t ; k++) tmpv=tmpv+(datax [ n+k+j ]−miux ) ∗( datay [ n+k]−miuy ) ; tmpv=tmpv / ( grpcnt −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpv=tmpv / ( s t d x ∗ s t d y ) ; i f ( tmpv>tmpdbl ) n f l a g=true ; else { if ( nflag ) i f ( tmpdbl>n f c t ) { n f c t=tmpdbl ; n l a g=1−j ; } n f l a g=f a l s e ; } tmpdbl=tmpv ; } m m m m m m  optr [ optp [ optv [ optt [ epsx [ epsy [  i ]=( p f c t >n f c t ) ? p f c t : n f c t ; i ]=( p f c t >n f c t ) ? p l a g : n l a g ; i ]=( m optp [ i ]==0) ? 0 : m prbs / ( m optp [ i ] ∗ m f r e q ) ; i ]= m optp [ i ] ∗ m f r e q ; i ]= m optx [ i ] ∗ m prba+m prbb ; i ]= m optx [ i ] ∗ m prba+m prbb ;  } R e f i l l () ; } // This f u n c t i o n f i n d s t h e ”maximum peak ” v a l u e f o r b o t h up and down f l o w u s i n g FFT.  234  Appendix C. Programs // Z h i w e i 04/14/04 void C D a t a f i l e : : C o r r e l a t e D ( void ) { int i , j , n , grpcnt , maxint , mrang ; double tmpdbl , tmpmin , miux , miuy , stdx , s t d y ; m grpc=g r p c n t=m pnts / m grps ; m rang=min ( 0 . 5 , m rang ) ; mrang=g r p c n t ∗ m rang ; for ( i =0; i <m grps ; i ++) // f o r ( i=m grps −1; i >=0; i −−) { double∗ f f t x =(double ∗) : : G l o b a l A l l o c (GPTR, g r p c n t ∗2∗ s i z e o f ( double ) ) ; double∗ f f t y =(double ∗) : : G l o b a l A l l o c (GPTR, g r p c n t ∗2∗ s i z e o f ( double ) ) ; n = i ∗ grpcnt ; S t a t i c s ( 0 , n , n+grpcnt −1, &stdx , &miux ) ; S t a t i c s ( 1 , n , n+grpcnt −1, &stdy , &miuy ) ; for ( j =0; j <g r p c n t ; j ++) { f f t x [ j ∗2]=( datax [ n+j ]−miux ) / 2 5 5 . ; f f t y [ j ∗2]=( datay [ n+j ]−miuy ) / 2 5 5 . ; f f t x [ j ∗2+1]= f f t x [ j ∗2+1]=0; } d f f t ( f f t x , grpcnt , 1 ) ; d f f t ( f f t y , grpcnt , 1 ) ; for ( j =0; j <g r p c n t ; j ++) { f f t x [ j ∗ 2 ] = ( ( tmpdbl=f f t x [ j ∗ 2 ] ) ∗ f f t y [ j ∗2]+ f f t x [ j ∗2+1]∗ f f t y [ j ∗2+1]) / g r p c n t / 2 ; f f t x [ j ∗2+1]=( tmpdbl ∗ f f t y [ j ∗2+1]− f f t x [ j ∗2+1]∗ f f t y [ j ∗ 2 ] ) / grpcnt /2; } d f f t ( f f t x , grpcnt , −1) ; maxint =0; tmpdbl=tmpmin=0; int m=g r p c n t / 2 ;  235  Appendix C. Programs for ( j =0; j <g r p c n t ; j ++) { f f t y [ j ]= f f t x [ 2 ∗ j ] ; i f ( tmpmin>f f t y [ j ] ) tmpmin=f f t y [ j ] ; i f ( tmpdbl<f f t y [ j ] ) // i f ( ( tmpdbl< f f t y [ j ] ) &&(( j >m−mrang )&&(j <m+mrang ) ) ) { tmpdbl=f f t y [ j ] ; i f ( ( j <mrang ) | | ( j >grpcnt−mrang ) ) { i f ( j <m) maxint=j ; else maxint=j−g r p c n t ; } } } for ( j =0; j <m; j ++) { f f t x [ j+m]= f f t y [ j ] ; f f t x [ j ]= f f t y [ j+m] ; } f f t x [ 2 ∗m]= f f t y [m] ; for ( j =0; j<=g r p c n t ; j ++) { d a t a f [ n+j ]=( f f t x [ j ]−tmpmin ) / ( tmpdbl−tmpmin ) ∗240+10; } tmpdbl =0; i f ( maxint >0) { S t a t i c s ( 0 , n , n+grpcnt −1−maxint , &stdx , &miux ) ; S t a t i c s ( 1 , n+maxint , n+grpcnt −1, &stdy , &miuy ) ; for ( j=grpcnt−maxint −1; j >=0; j −−) { tmpdbl=tmpdbl+(datax [ n+j ]−miux ) ∗( datay [ n+j+maxint ]−miuy ) ; } } else  236  Appendix C. Programs { S t a t i c s ( 0 , n−maxint , n+grpcnt −1, &stdx , &miux ) ; S t a t i c s ( 1 , n , n+grpcnt −1+maxint , &stdy , &miuy ) ; for ( j=g r p c n t+maxint −1; j >=0; j −−) { tmpdbl=tmpdbl+(datay [ n+j ]−miuy ) ∗( datax [ n+j−maxint ]−miux ) ; } } tmpdbl=tmpdbl / ( grpcnt−maxint −1) ; i f ( s t d x ∗ s t d y==0) continue ; else tmpdbl=tmpdbl / ( s t d x ∗ s t d y ) ; m m m m m m m m  optp [ optr [ optv [ optt [ optx [ opty [ epsx [ epsy [  i ]= maxint ; i ]= tmpdbl ; i ]=( m optp [ i ]==0) ? 0 : m prbs / ( m optp [ i ] ∗ m f r e q ) ; i ]= m optp [ i ] ∗ m f r e q ; i ]=miux ; i ]=miuy ; i ]= m optx [ i ] ∗ m prba+m prbb ; i ]= m opty [ i ] ∗ m prba+m prbb ;  } R e f i l l () ; } // R e c a l c u l a t e t h e mean v a l u e s f o r p r o c e s s e d d a t a g r o u p s void C D a t a f i l e : : R e s t a t ( void ) { int i ; m avgx=m avgy=m avgv=m avgt=m avgp=m avgr=m aepx=m aepy=m vcnt =0; for ( i =0; i <m grps ; i ++) { i f ( m optu [ i ] ) { m avgr+=m optr [ i ] ; m avgv+=m optv [ i ] ; m avgt+=m optt [ i ] ; m avgp+=m optp [ i ] ; m vcnt++; }  237  Appendix C. Programs m m m m  avgx+=m avgy+=m aepx+=m aepy+=m  optx [ opty [ epsx [ epsy [  i i i i  ]; ]; ]; ];  } i f ( m vcnt >0) { m avgr/=m vcnt ; m avgv/=m vcnt ; m avgt/=m vcnt ; m avgp/=m vcnt ; } m avgx/=m grps ; m avgy/=m grps ; m aepx/=m grps ; m aepy/=m grps ; char tmpbuf [MAX PATH ] ; s t a t b o x −>S e t S e l ( 0 , 0 ) ; s p r i n t f ( tmpbuf , ” V a l i d : %d Vp : %.3fm/ s Ep1 : %.2 f Ep2 : %.2 f \ r \n\ r \n” , m vcnt , m avgv , m aepx , m aepy ) ; CTime tmptime=CTime : : GetCurrentTime ( ) ; CS tring tmpstr=tmptime . Format ( ”%y−%m−%d %H:%M:%S” ) ; s t a t b o x −>S e t S e l ( 0 , 0 ) ; s t a t b o x −>R e p l a c e S e l ( tmpstr+” C o r r e l a t i o n r e s u l t s l i s t e d below : \ r \n\ r \n”+ T ( tmpbuf ) ) ; Working =2; } // Kick o u t t h e u n q u a l i f i e d g r o u p s w i t h r s m a l l e r than m minr void C D a t a f i l e : : Reminr ( void ) { int i ; int tmpint=Working ; Working =0; for ( i =0; i <m grps ; i ++) { // J u s t t o make s u r e m optu [ i ]= f a l s e ; // Check i f t h i s group i s v a l i d i f ( m optr [ i ]>=m minr ) { m optu [ i ]= true ;  238  Appendix C. Programs l i s t b o x −>SetCheck ( i ) ; } else { m optu [ i ]= f a l s e ; l i s t b o x −>SetCheck ( i , f a l s e ) ; } } Working =2; Restat ( ) ; } // Save t h e r e s u l t s i n a ” c s v ” f i l e r e a d a b l e t o M i c r o s o f t E x c e l void C D a t a f i l e : : SaveCsv ( void ) { char tmppath [MAX PATH ] ; // s t r c p y ( tmpname , m name ) ; LPSTR tmpname=(char ∗) ( : : PathFindFileName ( m name ) ) ; : : PathRemoveExtension ( tmpname ) ; s t r c p y ( tmppath , m f i l e p a t h ) ; s t r c a t ( tmppath , ” r e s u l t . c s v ” ) ; FILE∗ f i d ; f i d=f o p e n ( tmppath , ”a” ) ; i f ( f i d !=NULL) { f p r i n t f ( f i d , ” , | , %d ,% .3 f ,%. 3 f , %. 3 f , %.3 f , %.3 f ,%. 3 f ,%. 3 f , % . 3 f \ n” , m vcnt , m avgp , m avgt , m avgr , m avgv , m avgx , m avgy , m aepx , m aepy ) ; fclose ( fid ) ; } } // Convert t h e f i l e t o an E x c e l r e a d a b l e f i l e void C D a t a f i l e : : SaveDat ( void ) { char tmpname [MAX PATH ] ; s t r c p y ( tmpname , m name ) ; : : PathRemoveExtension ( tmpname ) ; s t r c a t ( tmpname , ” . dat . c s v ” ) ;  239  Appendix C. Programs FILE∗ f i d ; f i d=f o p e n ( tmpname , ”w” ) ; i f ( f i d !=NULL) { f p r i n t f ( f i d , ” Source , %s \n” , m name ) ; f p r i n t f ( f i d , ” ProbeSize , %4.3 f , mm\n” , m prbs ) ; f p r i n t f ( f i d , ” Points , %d\n” , m pnts ) ; f p r i n t f ( f i d , ” Frequency , %4.3 f , ms\n\n” , m f r e q ) ; f p r i n t f ( f i d , ” Groups , %d\n” , m grps ) ; f p r i n t f ( f i d , ”EachGrp , %d , Pnts \n” , m grpc ) ; f p r i n t f ( f i d , ” Search , %4.3 f , Grps\n” , m rang ) ; f p r i n t f ( f i d , ” MinCoef , %2.1 f \n” , m minr ) ; f p r i n t f ( f i d , ”MaxType , %d , 0− f i r s t ; 1− b i g g e r f i r s t ; 2−maximum \n” , m type ) ; f p r i n t f ( f i d , ” D i r e c t i o n , %d , 0−double ; 1−upflow ; 2−downflow \n \n” , m mode ) ; f p r i n t f ( f i d , ”No . , CH1, CH2\n” ) ; int i ; for ( i =0; i <m pnts ; i ++) { f p r i n t f ( f i d , ”%d,%d,%d\n” , i +1, datax [ i ] , datay [ i ] ) ; } fclose ( fid ) ; : : AfxMessageBox ( ”Data saved as ’ .DAT ’ f i l e . ” ,MB OK| MB ICONINFORMATION, 0 ) ; } else : : AfxMessageBox ( ”Data s a v i n g f a i l e d . ” ,MB OK|MB ICONSTOP, 0 ) ; } // F i l l t h e l i s t i n main d i s p l a y window void C D a t a f i l e : : R e f i l l ( void ) { int i ; char tmpbuf [MAX PATH ] ; l i s t b o x −>D e l e t e A l l I t e m s ( ) ; for ( i =0; i <m grps ; i ++) { s p r i n t f ( tmpbuf , ”%3d” , i ) ; l i s t b o x −>I n s e r t I t e m ( i , T ( tmpbuf ) ) ; m optu [ i ]= f a l s e ;  240  Appendix C. Programs i f ( ( m optr [ i ]>m minr )&&(abs ( m optp [ i ] ) >10) ) { m optu [ i ]= true ; l i s t b o x −>SetCheck ( i ) ; } s p r i n t f ( tmpbuf , ”%d” , m optp [ i ] ) ; l i s t b o x −>SetItemText ( i , 1 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.3 f ” , m optt [ i ] ) ; l i s t b o x −>SetItemText ( i , 2 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.3 f ” , m optr [ i ] ) ; l i s t b o x −>SetItemText ( i , 3 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.4 f ” , m optv [ i ] ) ; l i s t b o x −>SetItemText ( i , 4 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.3 f ” , m optx [ i ] ) ; l i s t b o x −>SetItemText ( i , 5 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.3 f ” , m opty [ i ] ) ; l i s t b o x −>SetItemText ( i , 6 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.3 f ” , m epsx [ i ] ) ; l i s t b o x −>SetItemText ( i , 7 , T ( tmpbuf ) ) ; s p r i n t f ( tmpbuf , ”%.3 f ” , m epsy [ i ] ) ; l i s t b o x −>SetItemText ( i , 8 , T ( tmpbuf ) ) ; } Restat ( ) ; Working =2; } // Some rearrangment b e f o r e FFT void C D a t a f i l e : : D o f f t ( int c f l a g ) { int i ; double∗ d a t a t=new double [ m pnts ∗ 2 ] ; i f ( c f l a g ==1) { for ( i =0; i <m pnts ; i ++) { d a t a t [ i ∗2]=( double ) ( datax [ i ] ) / 2 5 5 . 0 ; d a t a t [ i ∗2+1]=0; } } else { for ( i =0; i <m pnts ; i ++)  241  Appendix C. Programs { d a t a t [ i ∗2]=( double ) ( datay [ i ] ) / 2 5 5 . 0 ; d a t a t [ i ∗2+1]=0; } } d f f t ( datat , m pnts , 1 ) ; double tmpmax=0; int maxint , tmpint [ 3 2 7 7 0 ] ; m fmin=max( m pnts ∗ 0 . 0 0 1 ∗ m freq , 1 ) ; m fmax=10∗m fmin ; for ( i =0; i <m pnts ; i ++) { tmpint [ i ]=( int ) ( s q r t ( d a t a t [ i ∗ 2 ] ∗ d a t a t [ i ∗2]+ d a t a t [ i ∗2+1]∗ d a t a t [ i ∗2+1]) ) ; i f ( ( tmpint [ i ]>tmpmax)&&(i >m fmin )&&(i <m fmax ) ) tmpmax=tmpint [ i ] , maxint=i ; } for ( i =0; i <m pnts ; i ++) { d a t a f [ i ]=(BYTE) ( ( double ) ( tmpint [ i ] ) /tmpmax∗500) ; } m fmax=maxint ∗ 2 ; char tmpbuf [MAX PATH ] ; s t a t b o x −>S e t S e l ( 0 , 0 ) ; Peak Frequency : %.3 f ( Hz ) \ r \n\ r \n” , ( double ) s p r i n t f ( tmpbuf , ” ( maxint ) / ( double ) ( m fmin ) ) ; CTime tmptime=CTime : : GetCurrentTime ( ) ; CS tring tmpstr=tmptime . Format ( ”%y−%m−%d %H:%M:%S” ) ; s t a t b o x −>S e t S e l ( 0 , 0 ) ; s t a t b o x −>R e p l a c e S e l ( tmpstr+ T ( tmpbuf ) ) ; }  C.2  MATLAB code for orifice discharge coeffcient calculation  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % C a l c u l a t e c o e f f i c i e n t s f o r a s e r i e o f Re and draw a graph %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 242  Appendix C. Programs function O r i f i c e D1= 3 . 0 6 8 ;%2 . 0 ; D2= 0 . 7 0 2 ;%1 . 6 3 0 ; % 1 . 7 7 1 ; % 1 . 0 ; % 0 . 5 5 0 4 ; Re = [ 3 : 0 . 1 : 6 ] ’ ; Re=10.ˆRe ; %This K i s f o r g a s v e l o c i t y t h r o u g h t h e o r i f i c e , %d i m e n s i o n l e s s %Ug=K∗ s q r t (2∗ g ∗h ) %Ug=K∗ s q r t (2∗dP/ rou ) K=CalcK (D1 , D2 , Re ) semilogx ( Re ,K) ; %y l i m ( [ 0 . 5 8 , 1 . 1 ∗ max(K) ] ) ; ylim ( [ 0 . 5 , 1 . 1 ∗max(K) ] ) ; grid on ; %This Kv i s f o r g a s volumn f l u x t h r o u g h t h e o r i f i c e , (mˆ3/ s ) %Vg = Kv∗ s q r t (dP/P) ; %Kv = K∗A∗ s q r t (2∗R∗T/(M/1000) ) ; %A = p i ∗(D2/2) ˆ 2 ; R= 8 . 3 1 4 1 5 ; T=293.15+41.45; M= 2 8 . 8 / 1 0 0 0 ; A=pi ∗(D2∗ 0 . 0 2 5 4 / 2 ) ˆ2 Kv=K∗A∗ sqrt (2∗R/M) f i gu r e ; semilogx ( Re , Kv) ; ylim ( [ 0 , 1 . 1 ∗max(Kv) ] ) ; grid on ; %dP=5.125 %P=112.29 %Vg = Kv∗ s q r t (dP∗T/P) ∗P/101.325 %This Km i s f o r g a s mass f l u x t h r o u g h t h e o r i f i c e , (m∗ s ˆ2) %Mg = Km∗ s q r t (dP∗P) ; %Km = K∗A∗ s q r t ( 2 ∗ (M/1000) /(R∗T) ) ; %A = p i ∗(D2/2) ˆ 2 ; R= 8 . 3 1 4 1 5 ; T= 2 9 3 . 1 5 ; M= 2 8 . 8 ; A=pi ∗(D2/ 2 ) ˆ 2 ; Km=K∗A∗ sqrt ( 2 ∗ (M/1000) / (R∗T) ) f i gu r e ; semilogx ( Re ,Km) ; ylim ( [ 0 , 1 . 1 ∗max(Km) ] ) ;  243  Appendix C. Programs grid on ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Function t o c a l c u l a t e t h e d i s c h a r g e c o e f f i c i e n t %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function K=CalcK (D1 , D2 , Re ) %I n s i d e p i p e diameter , i n c h %D1=2.0 %O r i f i c e diameter , i n c h %D2=0.5504 % The d i a m e t e r r a t i o , d i m e n s i o n l e s s bt=D2/D1 ; %Reynolds number r e f e r r e d t o t h e d i a m e t e r o f t h e o r i f i c e , dimensionless %Re = [ 2 : 0 . 1 : 6 ] ’ ; %Re=10.ˆRe ; %R e c i p r o c a l o f Reynolds number , d i m e n s i o n l e s s l t =1./Re ; %Value o f l t when Reynolds number e q u a l s 1000000∗D2/15 , dimensionless l e =15/(1 e6 ∗D2) ; %Ke : The o r i f i c e d i s c h a r g e c o e f f i c i e n t when Reynolds number %e q u a l s 1000000∗D2/15 , d i m e n s i o n l e s s %Prepare f o r Ke bt4 =0.07+0.5/D1 ; E=0.4∗(1.6 −1/D1) ˆ 5 ; Z=0.009+0.034/D1 ; %S= Flange : 65/(D1ˆ2) +3; Throat : 4 0 / ( D1ˆ2) +9; Vena : 35/(D1ˆ2)+7 S=65/(D1ˆ 2 ) +3; r=r e a l ( S ∗( bt −0.7) ˆ ( 5 / 2 ) ) ; %Ke i s d e c i d e by t a p l o c a t i o n : f o r 1&2 Ke=b+n∗ b t ˆ4+ j+r j=r e a l (E∗( bt4−bt ) ˆ ( 5 / 2 )−Z∗(0.5 − bt ) ˆ ( 3 / 2 ) ) ; %b= Flange : 0.5993+0.007/D1 ; Throat : 0.5983+0.009/D1 ; Vena : 0 . 5 9 7 3 + 0 . 0 1 1 /D1 ; b =0.5993+0.007/D1 ; %n= Flange : 0.364+0.076/ s q r t (D1) ; Throat : 0.409+0.012/ s q r t (D1) ; Vena : 0.406+0.016/ s q r t (D1) ; n =0.364+0.076/ sqrt (D1) ; Ke=b+n∗ btˆ4+ j+r ; %Decide by t a p L o c a t i o n : 1  F l a g e and t h r o a t (530/ s q r t (D1) ) ;  244  Appendix C. Programs % 2 Vena c o n t r a c t a (530/ s q r t (D1) −100) ; % 3 Pipe (875/D1+75) B=530/ sqrt (D1) ; A=D2∗(830 −5000∗ bt +9000∗ bt ˆ2−4200∗ btˆ3+B) ; %K: The o f i f i c e d i s c h a r g e c o e f f i c i e n t , d i m e n s i o n l e s s K=Ke∗(1+A. ∗ l t ) /(1+A∗ l e ) ; %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Function t o c a l c u l a t e g a s v e l o c i t y by i t e r a t i o n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function CalcU D1= 2 . 0 ; D2= 0 . 5 5 0 4 ; P =101325∗3; %(Pa) %dP=228; %(Pa) dP = [ 7 . 5 2 9 , 1 1 . 7 7 6 , 1 0 . 9 3 8 , 2 2 . 1 3 6 , 2 9 . 9 8 5 , 3 0 . 2 5 9 , 5 6 . 0 6 8 , 75.622 , 100.836 , 113.741 , 145.106 , 155.048 , 204.856 , 228.341]; Pm=101325; R =8.31415; T =293.15; M =0.029; A =pi ∗(D2∗ 0 . 0 2 5 4 / 2 ) ˆ 2 ; %(mˆ2) Dg=P∗M/ (R∗T) ; %( kg /mˆ3) Mu=1.81E−5; %( Ps∗ s ) for i =1:14 nRe=1000; oRe =100; while abs ( nRe−oRe )>1 oRe=nRe ; %K i s f o r g a s v e l o c i t y t h r o u g h t h e o r i f i c e , %d i m e n s i o n l e s s %Ug=K∗ s q r t (2∗ g ∗h ) %Ug=K∗ s q r t (2∗dP/Dg) K=CalcK (D1 , D2 , nRe ) ; Ug=K∗ sqrt (2∗dP( i ) /Dg) ; %Kv i s f o r g a s volumn f l u x t h r o u g h t h e o r i f i c e , (mˆ3/ s ) %Vg = Kv∗ s q r t (dP/P) ;  245  Appendix C. Programs %Kv = K∗A∗ s q r t (2∗R∗T/(M/1000) ) ; %A = p i ∗(D2/2) ˆ 2 ; Kv=K∗A∗ sqrt (2∗R∗T/M) ; %Vg = Kv∗ s q r t (dP( i ) /P) ; Vg = Ug∗A; nRe=(D2 ∗ 0 . 0 2 5 4 ) ∗Ug∗(P∗M/ (R∗T) ) /Mu; end V( i )=Vg∗(P/Pm) ; end V’ %End o f f u n c t i o n CalcU  C.3  MATLAB code for spout shape prediction  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This program i s u s i n g do g o l d e n f a c t o r t o minimize % the spout length %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c le a r r i =5/1000; % h a l f w i d t h o f s l o t , f o r normal t o s l o t p r e d i c t i o n %r i =0.02 % h a l f length slot , for along the s l o t prediction r i =0.01 % e q u i −area r a d i u s , f o r area p r e d i c t i o n %r i =0.008 % h a l f hydraulic diameter of s l o t , for hydraulic diameter p r e d i c t i o n ha =0; % boundarys hb = . 3 ; % s t a t i c bed h e i g h t (mm) l t = −0.001; % i n i t i a l i z lambda i =0; eps = 0 . 0 0 0 0 0 1 ; % c o n v e r g e n c e l i m i t a t i o n dg=2∗eps ; d f =2∗eps ; c=(sqrt ( 5 ) −1) / 2 ; %t h e g o l d e n f a c t o r : 0 . 6 1 8 %I n i t i a l i z e t h e f i r s t 4 p o i n t s ga = 0 . 0 1 ;%0 . 1 0 ; gd = 0 . 5 ;%1 . 0 ; gc=ga+abs ( gd−ga ) ∗ c ; gb=gd−abs ( gd−ga ) ∗ c ; [ x y f b ]= p f u n c t i o n ( ha , hb , r i , gb , eps ) ; [ x y f c ]= p f u n c t i o n ( ha , hb , r i , gc , eps ) ; 246  Appendix C. Programs  dg=abs ( gb−gc ) ; d f=abs ( fb−f c ) ; disp ( [ fb , f c , dg , d f ] ) ; disp ( [ ga , gb , gc , gd ] ) ; while abs ( dg )>eps | abs ( d f )>eps %Update t h e p o i n t s i f f c >f b gd=gc ; gc=gb ; f c=f b ; gb=gd−abs ( gd−ga ) ∗ c ; [ x y f b ]= p f u n c t i o n ( ha , hb , r i , gb , eps ) ; else ga=gb ; gb=gc ; f b=f c ; gc=ga+abs ( gd−ga ) ∗ c ; [ x y f c ]= p f u n c t i o n ( ha , hb , r i , gc , eps ) ; end dg=abs ( gb−gc ) ; d f=abs ( fb−f c ) ; disp ( [ fb , f c , dg , d f ] ) ; disp ( [ ga , gb , gc , gd ] ) ; %pause ; end gamma=gb+gc /2 g=(gb+gc ) /2∗180/ pi f i gu r e ; x=x / . 3 0 0 ; plot ( y , x ) ; hold on ;  xx = [ 0 , 3 5 : 2 5 : 2 8 5 ] / 1 0 0 0 ; xx=xx / . 3 % four s e t of experimental data corresponding to f o r type of radius r e s p e c t i v e l y yy = [ 5 , 1 3 , 1 6 . 5 , 1 9 . 5 , 2 0 , 2 1 . 5 , 2 1 . 5 , 2 2 , 2 2 , 2 1 . 5 , 2 1 . 5 , 2 3 ] / 1 0 0 0 ; %yy = [ 2 0 , 2 4 , 2 3 , 2 2 , 2 1 , 2 0 , 1 9 . 5 , 1 9 . 5 , 2 0 . 3 , 2 0 . 5 , 2 1 , 2 0 . 5 ] / 1 0 0 0 ; yy  247  Appendix C. Programs =[10 ,17.66 ,19.48 ,20.71 ,20.49 ,20.73 ,20.47 ,20.71 ,21.23 ,20.99 ,21.24 ,21.71] %yy =[8 ,15.6 ,18.4 ,20.3 ,20.4 ,20.9 ,20.7 ,21.0 ,21.4 ,21.1 ,21.3 ,22.0]/1000 plot ( yy , xx , ’ r ˆ ’ ) ; xlim ( [ 0 , . 0 5 ] ) ; xlabel ( ’ r (mm) ’ , ’ F o n t S i z e ’ , 1 2 , ’ FontWeight ’ , ’ bold ’ ) ; ylabel ( ’ h/H ’ , ’ F o n t S i z e ’ , 1 2 , ’ FontWeight ’ , ’ b old ’ ) ; legend ( ’ Model ’ , ’ Experiment ’ ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Newton−Rapson t o s o l v e \ lambda % [ x , y , f ]= p f u n c t i o n ( xa , xb , r i , gm , e p s ) % xa , xb : I n t e g r a t e from ’ xa ’ t o ’ xb ’ % yy : y on xa % ri : i n l e t radius % gg : A d j u s t a b l e ’gm ’ % a r g s : Some c o n s t a n t s f o r d i f f % x,y : R e s u l t e d ’ z ’ and ’ r ’ % f : R e s u l t e d mean d i a m e t e r − e x p e r i m e n t a l mean d i a m e t e r %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [ x , y , f ]= p f u n c t i o n ( xa , xb , r i , gm, eps ) h=1; dv=2∗eps ; dd=0.1∗ eps ; % Four a v e r a g e s p o u t r a d i u s avgd =19/1000∗2; %avgd =21/1000∗2; avgd = 1 9 . 6 2 / 1 0 0 0 ∗ 2 ; %avgd =19.27/1000∗2; x=[ xa : 0 . 0 0 1 : xb ] ; l a = 0 . 0 0 1 ;%−0.01; c c=tan (gm) ; ca=c c / sqrt (1+ c c ˆ 2 ) ; cb=r i +1/( sqrt (1+tan (gm) ˆ 2) ∗ l a ) ; y=cb−sqrt (1 −( l a ∗x+ca ) . ˆ 2 ) / l a ; % Objective function f a=mean( y )−avgd / 2 ; i f fa >0  248  Appendix C. Programs hh= −0.001; else hh = 0 . 0 0 1 ; end f b=f a ;  while f a ∗ fb >0 l b=l a ; f b=f a ; l a=l a+hh ; i f l a==0 l a=l a +0.1∗hh ; end cb=r i +1/( sqrt (1+tan (gm) ˆ 2) ∗ l a ) ; y=cb−sqrt (1 −( l a ∗x+ca ) . ˆ 2 ) / l a ; f a=mean( y )−avgd / 2 ; %d i s p ( [ f a ∗ f b , l a , cb ] ) end i f hh<0 l c=l b ; f c=f b ; l b=l a ; f b=f a ; l a=l c ; f a=f c ; end disp ( [ l a , fa , lb , f b ] ) l c =( l a+l b ) / 2 ; f c=abs ( f a )+abs ( f b ) ; while abs ( f c )>eps l c =( l a+l b ) / 2 ; i f l c==0 l c =( l b / l c ) / 2 ; end cb=r i +1/( sqrt (1+tan (gm) ˆ 2) ∗ l c ) ; y=cb−sqrt (1 −( l c ∗x+ca ) . ˆ 2 ) / l c ; f c=mean( y )−avgd / 2 ; i f f c ∗ fa >0  249  Appendix C. Programs f a=f c l a=l c else f b=f c l b=l c end end  ; ; ; ;  disp ( [ l c , f c ] ) ; n=length ( y ) ; %Minimize t h e Length o f i n t e r f a c e f=sum( ( 1 + ( ( y ( 2 : n )−y ( 1 : ( n−1) ) ) / 0 . 0 0 1 ) . ˆ 2 ) . ˆ 0 . 5 ) ; %Change t o Chi−Square %yy = [ 1 1 . 5 , 1 3 . 5 , 1 5 , 1 6 , 1 7 , 1 7 , 1 6 , 1 5 ] ; %f=sum ( ( y ( 1 : 1 0 : n )−yy ) . ˆ 2 ) ; %Try o t h e r m i n i m i z a t i o n f u n c t i o n s %f=p i ∗sum ( ( y ( 2 : n )+y ( 1 : ( n−1) ) ) .∗(1+( y ( 2 : n )−y ( 1 : ( n−1) ) ) . ˆ 2 ) . ˆ 0 . 5 ) ; %f=p i ∗sum ( ( y ( 2 : n )−y ( 1 : ( n−1) ) ) .∗(1+( y ( 2 : n )−y ( 1 : ( n−1) ) ) . ˆ 2 ) . ˆ 0 . 5 ) ; %f=p i ∗sum ( ( y ( 2 : n )−y ( 1 : ( n−1) ) ) ) ;  C.4  MATLAB code for axial particle velocity prediction  function [ x , y ] = RKmain( xa , xb , yy ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Runge−Kutta ODE s o l u t i o n w i t h f i x e d or v a r i a b l e s t e p s i z e % xa , xb : I n t e g r a t i o n range % yy : I n i t i a l v a l u e on ’ t ’ % x , y : S i m u l a t i o n r e s u l t o f ’ z ’ and ’ v ’ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i =1; h=0.0005; h= −0.0005; dlta =0.01; xx=xa ; y ( 1 )=yy ; %w h i l e xb>xx 250  Appendix C. Programs while xb<xx %f l a g : 0 f i x e d s t e p s i z e , 1 v a r i a b l e o p t i m i z e d s t e p s i z e f l a g =0; i f f l a g==1 ya=RKstep ( xx , yy , h ) ; yb=RKstep ( xx , yy , h /2 ) ; yb=RKstep ( xx+h / 2 , yb , h /2 ) ; d l t b=max( abs ( yb−ya ) ) ; %I f t h e s t e p i s t o o s m a l l , e n l a r g e i t t i l l d l t >c r e t e r i a %Don ’ t worry ! I t w i l l come back . while d l t b <d l t a h=h ∗( d l t a / d l t b ) ˆ 0 . 2 ; ya=RKstep ( xx , yy , h ) ; yb=RKstep ( xx , yy , h /2 ) ; yb=RKstep ( xx+h / 2 , yb , h /2 ) ; d l t b=max( abs ( yb−ya ) ) ; end %I f t o o l a r g e , c o r r e c t i t while d l t b >d l t a h=h ∗( d l t a / d l t b ) ˆ 0 . 2 ; %P r e v e n t t h e h from e x c e e d i n g t h e range % a b s ( tm−t ) f o r t h e l a s t p o i n t h=min( h , abs ( xb−xx ) ) ; ya=RKstep ( xx , yy , h ) ; yb=RKstep ( xx , yy , h /2 ) ; yb=RKstep ( xx+h / 2 , yb , h /2 ) ; d l t b=max( abs ( yb−ya ) ) ; end yy=yb ; else yy=RKstep ( xx , yy , h ) ; end i=i +1; y ( i )=yy ; xx=xx+h ; x ( i )=xx ; end x sqrt ( y ) ’ y %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  251  Appendix C. Programs % S i n g l e s t e p f u n c t i o n o f Runge−Kutta method % x , y : c u r r e n t i n d e p e n d e n t and denpendent % h : step size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function f=RKstep ( x , y , h ) ka=h∗RKfuns ( x , y ) ; kb=h∗RKfuns ( x+h / 2 , y+ka /2) ; kc=h∗RKfuns ( x+h / 2 , y+kb /2) ; kd=h∗RKfuns ( x+h , y+kc ) ; f=y+ka/6+kb/3+kc/3+kd / 6 ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k=RKfuns ( x , y ) % x : I n d e p e n d e n t parameter % y : y ( i ) on x , i =1˜n % k : F i r s t o r d e r d i f f e r e n t i a l o f y on x , r e t u r n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function k=RKfuns ( x , y ) global r s a % S t a t i c bed h e i g h t Hs = 0 . 3 ; % Maximum bed h e i g h t Hm= 0 . 8 ; % Portion B= 0 . 7 2 ; Umf= 0 . 7 4 ; % Voidage Ua=B∗Umf∗ sin ( pi ∗x /(2∗ Hs ) ) ; %Ua=Umf∗(1−(1−x /Hm) ˆ3) ; i f x<= 0 . 0 3 5 ep = 0 . 8 5 9 ; As =0.0004+(0.000980177 −0.0004) ∗( x−0) /(0.035 −0) ; e l s e i f x<=0.06 ep =0.859+(0.86 −0.859) ∗( x −0.035) / ( 0 . 0 6 − 0 . 0 3 5 ) ; As =0.000980177+(0.001192234 −0.000980177) ∗( x −0.035) /(0.06 −0.035) ; e l s e i f x<=0.85 ep =0.86+(0.889 −0.86) ∗( x −0.06) / ( 0 . 0 8 5 − 0 . 0 6 ) ; As =0.001192234+(0.001347743 −0.001192234) ∗( x −0.06) /(0.085 −0.06) ; e l s e i f x<=0.11 ep =0.889+(0.842 −0.889) ∗( x −0.085) / ( 0 . 1 1 − 0 . 0 8 5 ) ; As =0.001347743+(0.001319469 −0.001347743) ∗( x −0.085)  252  Appendix C. Programs /(0.11 −0.085) ; e l s e i f x<=0.135 ep =0.842+(0.819 −0.842) ∗( x −0.11) / ( 0 . 1 3 5 − 0 . 1 1 ) ; As =0.001319469+(0.001350885 −0.001319469) ∗( x −0.11) /(0.135 −0.11) ; e l s e i f x<=0.16 ep =0.819+(0.806 −0.819) ∗( x −0.135) / ( 0 . 1 6 − 0 . 1 3 5 ) ; As =0.001350885+(0.001317113 −0.001350885) ∗( x −0.135) /(0.16 −0.135) ; e l s e i f x<=0.185 ep =0.806+(0.761 −0.806) ∗( x −0.16) / ( 0 . 1 8 5 − 0 . 1 6 ) ; As =0.001317113+(0.001347743 −0.001317113) ∗( x −0.16) /(0.185 −0.16) ; e l s e i f x<=0.21 ep =0.761+(0.727 −0.761) ∗( x −0.185) / ( 0 . 2 1 − 0 . 1 8 5 ) ; As =0.001347743+(0.001416858 −0.001347743) ∗( x −0.185) /(0.21 −0.185) ; e l s e i f x<=0.235 ep =0.727+(0.627 −0.727) ∗( x −0.21) / ( 0 . 2 3 5 − 0 . 2 1 ) ; As =0.001416858+(0.001384657 −0.001416858) ∗( x −0.21) /(0.235 −0.21) ; e l s e i f x<=0.260 ep =0.627+(0.662 −0.627) ∗( x −0.235) / ( 0 . 2 6 − 0 . 2 3 5 ) ; As =0.001384657+(0.001418429 −0.001384657) ∗( x −0.235) /(0.26 −0.235) ; e l s e i f x<=0.285 ep =0.662+(0.542 −0.662) ∗( x −0.26) / ( 0 . 2 8 5 − 0 . 2 6 ) ; As =0.001418429+(0.001481261 −0.001418429) ∗( x −0.26) /(0.285 −0.26) ; end % Minimum s p o u t i n g v e l o c i t y Umf= 0 . 7 4 ; % Annulus s u p e r f i c i a l g a s v e l o c i t y Ua=B∗Umf∗ sin ( pi ∗x /(2∗ Hs ) ) ; % I n l e t r a d i u s , e q u i −area r i =0.01; % lambda f o r s p o u t shape l d = −0.3197; % gamma f o r s p o u t shape gm= 0 . 0 9 6 1 ; c c=tan (gm) ; % Constants  253  Appendix C. Programs ca=c c / sqrt (1+ c c ˆ 2 ) ; cb=r i +1/( sqrt (1+tan (gm) ˆ 2) ∗ l d ) ; % Equi−area r a d i u s o f s p o u t r s=cb −(1/ l d ) ∗ sqrt (1 −( l d ∗x+ca ) ˆ2) ; % Using d a t a s e t o f s p o u t shape p r e d i c t i o n %As=p i ∗ r s a ( round ( x ∗1000+1) ) ˆ 2 ; % Using a n a l y t i c a l r e s u l t s w i t h s o l v e d lambda and gamma As=pi ∗ r s ˆ 2 ; % S u p e r f i c i a l gas v e l o c i t y Uc = 0 . 8 ; % Column c r o s s −s e c t i o n Ac = 0 . 3 ∗ 0 . 1 ; % Annulus c r o s s −s e c t i o n Aa=Ac−As ; % A c c e l e r a t i o n zone , r e a l g a s v e l o c i t y Us=(Uc∗Ac−Ua∗Aa) /As/ ep ; % D e c e l e r a t i o n zone , r e a l g a s v e l o c i t y %Us=(Uc∗Ac−Ua∗Aa) /As / 0 . 5 5 6 9 ; % Gas d e n s i t y r f =1.204; % Solid density r s =2500; % Gravity g =9.813; % P a r t i c l e diameter dp = 0 . 0 0 1 3 3 ; % Particle velocity v=sqrt ( y ) ; % Vicosity of air miu =1.81 e −5; % Reynolds number Re=dp ∗( Us−v ) ∗ r f /miu ; % Drag c o e f f i c i e n t i f Re >=1000 Cd= 0 . 4 4 ; else Cd=24/Re∗(1+Re ˆ 0 . 6 8 7 / 2 ) end Cd=Cd∗ ep ˆ( −4.7) ; %A c c e l e r a t i o n Zone  254  Appendix C. Programs %k =((3∗ r f ∗( Us−v ) ˆ2∗Cd) /(4∗ dp ∗ r s )−(rs−r f ) ∗ g / r s ) ∗ 2 ; %D e c e l e r a t i o n Zone , w i t h drag f o r c e k =((3∗ r f ∗( Us−v ) ˆ2∗Cd) /(4∗ dp∗ r s ) −( r s −r f ) ∗g/ r s −y/x ) ∗ 2 ; %D e c e l e r a t i o n Zone , w i t h o u t drag f o r c e %k=(−(rs−r f ) ∗ g / rs−y / x ) ∗ 2 ;  255  

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