M O D E L I N G OF H E A T TRANSFER IN CIRCULATING F L U I D I Z E D BEDS By Donglai Xie B. A . Sc., Shandong Institute of Architecture and Engineering, Jinan, 1992 M . A . Sc., Harbin University of Architecture and Engineering, Harbin, 1995 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR. OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Chemical and Biological Engineering) We accept this thesis as conforming to the r e a u r ie d standard THE UNIVERSITY OF BRITISH C O L U M B I A October 2001 © Donglai Xie, 2001 In presenting this degree at the thesis in University of freely available for reference copying of department this or partial fulfilment British Columbia, and study. of publication of this his or her requirements I agree that the I further agree thesis for scholarly purposes by the representatives. that may be It thesis for financial gain shall not Department of CkfiM\C&.\ OnA The University of British Columbia Vancouver, Canada D«e DE-6 (2/88) f)^ TUB I fciblryj by the that allowed without permission. '>rj advanced Library shall make it understood be an permission for extensive granted is for head of my copying or my written ABSTRACT Suspension-to-wall heat transfer in circulating fluidized beds is modeled considering both the reactor-side and wall-side heat transfer processes. The overall flow structure in fast fluidized beds is represented by a core-annulus flow pattern with a stagnant particlefree gas gap between the wall and wall layer. Descending particles are assumed to enter the heat transfer zone with the same temperature as the core suspension. As particles descend in the wall layer, they lose heat to the gas by convection and gain heat from fresh particles arriving from the bulk core region. Gas is dragged downwards in the heat transfer zone by the rapidly-descending annular particles. The gas receives heat from the immersed particles by particle-to-gas convection andfromthe core by conduction. Heat is then conducted to the wall through the stagnant gas gap, and then through the furnace wall to the coolant. The model is the first to include the coolant-side heat transfer in the overall process. Particles also participate in radiationfromthe core to the wall through the wall layer. They are assumed to constitute a gray continuous absorbing, emitting and scattering medium. The radiation heat transfer process is solved by the two-flux model in a twodimensional model for CFB units with smooth walls, while the moment method is employed for the three-dimensional case when membrane walls are present. Under highdensity CFB operating conditions with smooth walls, the model is extended by allowing the suspension in the vicinity of the wall to travel intermittently downwards and upwards as is observed experimentally. The two- and three-dimensional models are validated ii using experimental results from the literature and both yield satisfactory predictions of the suspension-to-wall heat transfer. The influences of key parameters on the heat flux are analyzed and are found to be consistent with experimental trends where these are known. The simulation results suggest that the particles participate in a significant way in determining the radiation flux through the wall layer. Therefore radiation cannot be uncoupled from particle and gas conduction and convection without introducing significant error for high temperature systems. Experiments were conducted in the 76 mm diameter jacketed riser of a dual-loop high-density CFB facility with FCC particles of 65 pm Sauter mean diameter as bed material. The superficial gas velocity varied from 4 to 9.5 m/s and the solids circulation flux was as high as 527 kg/m s. The suspension temperature and the average and local 2 suspension-to-wall heat transfer coefficients were measured. The suspension temperature distributions indicate that the particles in the vicinity of the wall do not move in one direction only, but oscillate downward and upward, leading to higher local heat transfer coefficients at the ends of the heated section. Experimental results also show that suspension-to-wall heat transfer coefficients are strongly influenced by suspension density. However, they are not significantly influenced by superficial gas velocity at a constant suspension density. By superimposing the heat transfer results when the suspension in the vicinity of wall is allowed to move downwards and upwards separately, the model predicts the experimental results well. ui TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES : x LIST OF FIGURES x ACKNOWLEDGEMENT xviii CHAPTER 1. INTRODUCTION 1 1.1. Circulating Fluidized Bed Technology 1 1.2. Circulating Fluidized Bed Hydrodynamics 3 1.3. Thermal Radiation in Fluidized Beds 4 1.3.1. Absorption and Scattering by a Single Particle 6 1.3.2. Absorption and Scattering of a Particulate System 8 1.4. Previous Models 9 1.4.1. Models for Reactor-Side Heat Transfer 9 1.4.1.1. Particle Convective Component 10 1.4.1.2. Gas Convective Component 12 1.4.1.3. Radiation Component 13 1.4.2. Models for Wall-Side Heat Transfer 18 1.5. Summary of Previous Models 19 1.6. Objectives of this Thesis Project 20 1.7. Structure of Thesis 20 iv CHAPTER 2. TWO-DIMENSIONAL MODEL FOR LOW-DENSITY CFB WITH SMOOTH WALLS 22 2.1. Introduction 22 2.2. Model Development 23 2.2.1. Background Assumptions 23 2.2.2. Governing Equations and Boundary Conditions 25 2.2.3. Parameter Determination 27 2.3. Numerical Method and Accuracy Analysis 35 2.4. Predictions for a Typical Case 38 2.4.1. Example Description 38 2.4.2. Heat Flux Distribution 41 2.4.3. Particle and Gas Temperature Distributions 42 2.4.4. Thermal Boundary Layer Thickness 43 2.4.5. Coolant and Wall Temperature Distribution 44 2.4.6. Radiation Flux 45 2.4.7. Heat Transfer Coefficient 47 2.5. Comparison of Model Predictions with Experimental Results 48 2.5.1. Comparison with Data of Pagliuso et al. (2000) 48 2.5.2. Comparison with Data of Furchi et al. (1988) 53 2.5.3. Comparison with Data of Han et al. (1996) 55 2.5.4. Comparison with Data of Luan et al. (1999) 60 2.5.5. Comparison with Data of Ahmad et al. (1997) 62 2.5.6. Comparison with Data of Tan et al. (1994) 64 2.6. Sensitivity Analysis 2.6.1. Influence of Gas Gap Thickness 65 66 2.6.2. Influence of Particle Diameter 67 2.6.3. Influence of Suspension Density 68 2.6.4. Influence of Wall Layer Thickness 69 2.6.5. Influence of Particle Velocity 70 2.6.6. Influence of Gas Velocity 71 2.6.7. Influence of Bulk Temperature 72 2.6.8. Influence of Particle Emissivity 73 2.6.9. Influence of Particle Volumetric Heat Capacity ip C ) 74 2.6.10. Influence of Particle Average Residence Length (L ) 75 p pp ar 2.6.11. Influence of Wall Thermal Resistance (LJk ) and Water-side Heat Transfer Coefficient 75 w 2.6.12. Summary of Influence of Various Parameters on Heat Flux 2.7. Summary 77 79 CHAPTER 3. HEAT TRANSFER IN A HIGH-DENSITY CIRCULATING FLUIDIZED BED 80 3.1. Introduction 80 3.2. Experimental Facilities 82 3.2.1. HDCFB System 82 3.2.2. Heat Transfer Measurement Equipment 84 3.2.3. Bed Material 86 3.2.4. Measurement Techniques 87 3.3. Experimental Results and Discussion 94 3.3.1. Suspension Temperature Distribution 94 3.3.2. Average Heat Transfer Coefficient 98 vi 3.3.3. Local Heat Transfer Coefficient 101 3.4. Modeling Heat Transfer in HDCFBs 107 3.4.1. Particle Motion in the Vicinity of the Wall 107 3.4.2. Extended Two-Dimensional Model for Smooth Wall Ill 3.4.3. Comparison of Model Prediction with Experimental Results 114 3.5. Influence of Particle Velocity on Heat Transfer Coefficient 118 3.6. Summary 120 CHAPTER 4. THREE-DIMENSIONAL MODEL FOR LOW-DENSITY CFB WITH MEMBRANE WALL 121 4.1. Introduction 121 4.2. Model Development 123 4.2.1. CFBC Hydrodynamics with Membrane Wall 123 4.2.2. Thermal Radiation in Wall Layer with Membrane Wall 125 4.2.3. Governing Equations 126 4.2.4. Parameter Determination 130 4.3. Numerical Method and Accuracy Analysis 132 4.4. Predictions for a Typical Case 136 4.4.1. Base Case 136 4.4.2. Heat Flux Distribution 138 4.4.3. Particle and Gas Temperature Distributions 141 4.4.4. Irradiance Distribution 145 4.4.5. Heat Transfer Coefficient Distribution 146 4.5. Comparison of Model Prediction with Experiments 4.5.1. Comparison with Data of Wuetal. (1987, 1989) vii 147 147 4.5.2. Comparison with Data of Luan (1997) and Luan et al. (2000) 150 4.5.3. Comparison with Data of Andersson and Leckner (1992) 156 4.5.4. Comparison with Data of Andersson (1996) 163 4.5.5. Comparison with Data of Andersson et al. (1996) 168 4.6. Influence of Fin Width on Heat Transfer Coefficient 171 4.7. Summary 176 CHAPTER 5. OVERALL CONCLUSIONS AND RECOMMENDATIONS 177 5.1. Overall Conclusions 177 5.2. Recommendations for Future Research 178 NOMENCLATURE 180 REFERENCES 184 APPENDICES .• 193 Appendix I: Application of Two-Flux Model for Radiation Process in Wall Layer 193 Appendix II: Application of Keller's Box Method 196 Appendix III: Application of Moment Method for Radiation in CFB Wall Layer 203 Appendix IV: Application of Finite Element Method and Finite Difference Method ... 207 Appendix V: Experimental Data 216 Appendix VI: Program Listing 221 vui LIST OF TABLES Table 2.1: Correlations for radial voidage distribution in CFB risers 28 Table 2.2: Measurement of downward particle velocity in wall layer in fast fluidization flow regime 30 Table 2.3: Base parameters used in sample calculations below 39 Table 2.4: Summary of predicted parameter influences on heat flux for base case conditions listed in Table 2.3 78 Table 3.1: Size distribution of FCC particles determined by screening 86 Table 3.2: Solids mass flux at rlR - 0.95 measured by Issangya (1998) 97 Table 3.3: Average particle velocity at r/R = 0.95 determined by Liu (2001) 113 Table 3.4: Operating conditions and fd for U = 6 m/s 114 g Table 3.5: Operating conditions and fd for U = 8 m/s 115 g Table 4.1: Base parameters used in example calculations below 136 Table 4.2: Main operating conditions of test runs of Andersson (1996) 163 Table V . l : Suspension temperature distribution for A: G = 282 kg/m s, p = 342 kg/m and U = 6 m/s 216 s sus g 2 3 Table V.2: Suspension temperature distribution for B: G = 213 kg/m s, p and U = 6 m/s s sus g 2 3 Table V.3: Suspension temperature distribution for C: G = 26 kg/m s, p and U = 6 m/s s sus g = 136 kg/m 217 = 19 kg/m 217 Table V.4: Average heat transfer coefficients 218 Table V.4: Average heat transfer coefficients (continued) 219 Table V.5: Local bed-to-wall heat transfer coefficient for U = 6 m/s. Operating conditions are listed in Table 3.4 220 Table V.6: Local bed-to-wall heat transfer coefficient for U = 8 m/s. Operating conditions listed in Table 3.5 220 g g ix LIST OF FIGURES Figure 2.1: Schematic of heat transfer process 24 Figure 2.2: Influence of vertical step on calculation of heat flux at the wall. 1. Azmax = 10 mm; 2. Azmax = 0.25 mm; 3. plot 1 after filtration with equation (2.33) 36 Figure 2.3: Influence of Ax on wall temperature calculation. Operating conditions are listed in Table 2.3 37 Figure 2.4: Particle concentration distribution in wall layer for base case 40 Figure 2.5: Variation of suspension absorption and scattering coefficients in wall layer for base case 40 Figure 2.6: Vertical variation of heat flux along the wall for base case conditions given in Table 2.3 41 Figure 2.7: Gas and particle axial temperature distributions for base case. Dashed lines: particle temperature; Solid lines: gas temperature. From left to right: x =x = 0.08, 0.24, 0.74, 1.13, 1.83, 2.94,10.48 mm, respectively 42 Figure 2.8: Growth of predicted particle and gas thermal boundary layer thicknesses for base case 43 Figure 2.9: Vertical variation of coolant, wall and bulk temperatures for base case 44 Figure 2.10: Lateral variation of radiation flux for base case 45 Figure 2.11: Lateral variations of net radiation flux and irradiance for base case 46 Figure 2.12: Vertical variation of local suspension-to-coolant heat transfer coefficient for base case 47 Figure 2.13: Local suspension density as a function of elevation and comparison of predicted local heat transfer coefficients with experimental results of Pagliuso et al. (2000) for d = 179 mm. Average suspension density from A to F: 11, 21, 22, 41, 48, and 53 kg/m . (+: suspension density; O: experimental heat transfer coefficient; Solid lines: predicted local heat transfer coefficients) ... 49 p Figure 2.14: Local suspension density as a function of elevation and comparison of predicted local heat transfer coefficient with experimental results of Pagliuso et al. (2000) for d = 230 um. Average suspension density, from A to F: 10, 20, 28, 38, 46 and 33 kg/m . (+: suspension density; O: experimental heat transfer coefficient; Solid lines: predicted local heat transfer coefficients) ... 50 p 3 Figure 2.15: Local suspension density as a function of elevation and comparison of predicted local heat transfer coefficients with experimental results of Pagliuso et al. (2000) for d = 460 pm. Average suspension density from A to F: 10, 18, 26,48, 56 and 62 kg/m . (+: suspension density; O: experimental heat transfer coefficient; Solid lines: predicted local heat transfer coefficients) 51 p 3 Figure 2.16: Predicted average heat transfer coefficients (open points) compared with experimental data (soild points) of Pagliuso et al. (2000). A: d = 179 um; B: d = 230 um; C: d = 460 um 52 p p p Figure 2.17: Suspension density as a function of elevation and comparison of predicted local heat transfer coefficients with experimental data of Furchi et al. (1988). A: d = 109 um, U = 5.8 m/s, GJGg = 11.3; B: d = 109 um, U = 8.9 m/s, Gs/G = 7.5; C: d = 196 um, U = 6.8 m/s, GJG = 13.4; D: d = 196 um, U = 8.9 m/s, GJGg = 3.9. +: suspension density; Solid circle: heat transfer p g g p p g g g p g coefficient; Solid line: predicted local heat transfer coefficient for L = 1.5 m; Dashed line: predicted local heat transfer coefficient for L = 0.3 m 54 ar ar Figure 2.18: Comparison of predicted average heat transfer coefficients (lines) with experimental data of Furchi et al. (1988) (points) 55 Figure 2.19: Comparison of predicted average heat transfer coefficients (open points) with experimental data of Han et al. (1996) for FCC particles (solid points) .56 Figure 2.20: Comparison of predicted average heat transfer coefficients (open points) with experimental data of Han et al. (1996) for sand particles (solid points). A: d = 136 um; B: d = 157 pm; C: d = 264 pm 57 p p p Figure 2.21: Comparison of predicted radiative heat transfer coefficient (open points) as a function of suspension density with experimental data of Han (1992) (solid points) for 137 pm sand 58 Figure 2.22: Comparison of predicted radiative heat transfer coefficient (open points) as a function of bed temperature with experimental data of Han (1992) (solid points) for 137 pm sand. (Suspension density: 10 kg/m ) 59 3 xi Figure 2.23: Schematic of showing integration intervals used to determine the average probe heat transfer coefficient 60 Figure 2.24: Comparison of predicted total and radiative heat transfer coefficients with experimental data of Luan et al. (1999). • : total; V: radiative by window method; A: radiative by differential emissivity method; *: predicted total; +: predicted radiative 61 Figure 2.25: Comparison of predicted heat transfer coefficients (open points) with experimental data of Ahmad et al. (1997) (solid points). A: p = 16 kg/m , T = 373 °C, U = 3.0 m/s; B: p 3 sus b T = 361 °C, U = 3.5 m/s; C: b D: p g g = 16 kg/m , 3 sus = 14 kg/m , T = 485 °C, U = 4.8 m/s; 3 P s u s b g = 12.5 kg/™ , T = 478 °C, U = 5.5 m/s " 3 sus b g 63 Figure 2.26: Comparison of predicted local heat transfer coefficients (solid lines) with experimental data of Tan et al. (1994). +: 1.06 m above distributor; x: 2.75 m above distributor. A: p^ = 20 kg/m ; B: p = 40 kg/m ; C: p = 80 kg/m 65 3 3 sus 3 ms Figure 2.27: Influence of gas gap thickness on vertical heat flux profile. (Base conditions, Table 2.3, except for gas gap thickness.) 66 Figure 2.28: Influence of particle diameter on vertical heat flux profile. (Base conditions, Table 2.3, except for particle diameter.) 67 Figure 2.29: Influence of suspension density on vertical heat flux profile. (Base conditions, Table 2.3, except for suspension density.) 69 Figure 2.30: Influence of particle velocity in wall layer on vertical heat flux profile. (Base conditions, Table 2.3, except for particle velocity.) 70 Figure 2.31: Influence of gas velocity in wall layer on vertical heat flux profile. (Base conditions, Table 2.3, except for gas velocity.) 71 Figure 2.32: Influence of bulk temperature on vertical heat flux profile. (Base conditions, Table 2.3, except for bulk temperature.) 72 Figure 2.33: Influence of particle emissivity on vertical heat flux profile. (Base conditions, Table 2.3, except for particle emissivity.) 73 Figure 2.34: Influence of particle volumetric heat capacity on heat flux profile. (Base conditions, Table 2.3, except for particle volumetric heat capacity.)... 74 xu Figure 2.35: Influence of particle residence length on vertical heat flux profile. (Base conditions, Table 2.3, except for average particle residence length.)... 75 Figure 2.36: Influence of water-side heat transfer coefficient on vertical heat flux profile. (Base conditions, Table 2.3, except for water-side heat transfer coefficient.) 76 Figure 3.1: Schematic diagram of UBC high-density CFB unit 83 Figure 3.2: Concentric-tube heat exchanger. All dimensions in millimeters 85 Figure 3.3: Wall, water and riser temperature and riser pressure measurement system... 88 Figure 3.4: Set-up for water temperature measurements. (All dimensions in mm.) 90 Figure 3.5: Schematic for calculation of average bulk temperature in the laminar flow through the annular space between two concentric tubes of circular crosssection 91 Figure 3.6: Temperature profile measured at 4 locations in water channel. A: Z = 3.00 m; B: Z = 2.95 m; C: Z = 2.79 m; D: Z = 2.62 m. Operating conditions: Run A listed in Table 3.4. Dashed lines: temperature distributions obtained by fitting equation (3.4); Solid lines: average values 93 Figure 3.7: Radial and vertical bed temperature distribution for U = 6 m/s. A: G = 282 kg/m s, p = 342 kg/m ; B: G = 213 kg/m s, p = 136 kg/m ; C: G = 26 kg/m s, p =19 kg/m 96 g 2 s 3 sus 2 s 2 s 3 sus 3 sus Figure 3.8: Suspension-to-water heat transfer coefficient vs. particle circulation flux.. 100 Figure 3.9: Suspension-to-wall heat transfer coefficient vs. particle circulation flux.... 100 Figure 3.10: Suspension-to-water heat transfer coefficient vs. suspension density 100 Figure 3.11: Suspension-to-wall heat transfer coefficient vs. suspension density 100 Figure 3.12: Local suspension-to-wall heat transfer coefficient as a function of height for U =6 m/s. Operating conditions are listed in Table 3.4 103 g Figure 3.13: Local suspension-to-wall heat transfer coefficient as a function of bed height for U = 8 m/s. Operating conditions are listed in Table 3.5 104 g Figure 3.14: Suspension temperatures at the wall and axis of the riser for U = 6 m/s. Operating conditions are listed in Table 3.4. O: rlR = 0; x : rlR = 1 105 g Figure 3.15: Suspension temperatures at the wall and axis ofthe riser for U = 8 m/s. Operating conditions are listed in Table 3.5. O: rlR = 0; x : rlR = 1 106 g xiii Figure 3.16: Schematic of particle motion in the vicinity of the wall and its influence on the suspension temperature 108 Figure 3.17: Fraction of particles moving downwards in the vicinity of the wall as a function of suspension density and superficial gas velocity 110 Figure 3.18: Comparison of predicted and experimental heat transfer coefficients for U = 6 m/s. Operating conditions are listed in Table 3.4 116 g Figure 3.19: Comparison of predicted and experimental heat transfer coefficients for U = 8 m/s. Operating conditions are listed in Table 3.5 117 g Figure 3.20: Influence of particle velocity on model predictions for U = 6 m/s. g Operating conditions are listed in Table 3.4 Figure 4.1: Configuration of membrane wall. Tube axes are normally vertical 119 121 Figure 4.2: Plan view of membrane wall and wall layer assembly 127 Figure 4.3: Three-dimensional view of membrane wall and wall layer showing Cartesian control volume 128 Figure 4.4: Finite element mesh and nodes for membrane wall and wall layer of UBC pilot CFBC 137 Figure 4.5: Lateral variation of heat flux along membrane wall surface at z = 1.5 m for base case conditions listed in Table 4.1 139 Figure 4.6: Vertical variation of integral heat fluxes along the wall for base case conditions given in Table 4.1 140 Figure 4.7: Gas and wall temperature distributions at z = 0 m for base case conditions listed in Table 4.1 142 Figure 4.8: Gas and wall temperature distributions at z = 1.5 m for base case condition listed in Table 4.1 143 Figure 4.9: Particle temperature distribution at z = 1.5 m for base case conditions listed in Table 4.1 144 Figure 4.10: Irradiance distribution at z = 1.5 m for base case conditions listed in Table 4.1. (Unit: kW/m ) 145 2 Figure 4.11: Vertical variation of average suspension-to-coolant heat transfer coefficient for base case conditions listed in Table 4.1 146 xiv Figure 4.12: Comparison of predicted average heat transfer coefficients (X ) with experimental data of Wu et al. (1987) for sand particles (O) Gas temperature: 150-400 °C. A: d = 356 urn; B: d = 188 um 148 p p Figure 4.13: Comparison of model predicted average heat transfer coefficients with experimental data (solid circles) of Wu et al. (1989) for p = 54 kg/m . solid line: e = 0.5; dashed line: e = 0.9; A: T = 860 °C; B: T = 407 °C. 149 3 sus w w b b Figure 4.14: Comparison of predicted average heat transfer coefficients ( X ) for e = 0.5 with experimental data of Wu et al. (1989) (O) A: T = 870 ± 14 °C; B: T =681±18°C;C: T = 410±15°C 150 w b b b Figure 4.15: Positions (T1-T6) of thermocouples embedded in pipe wall and fin by Luan et at. (1997) 151 Figure 4.16: Comparison of model-predicted average heat transfer coefficients (lines) with experimental data (points) of Luan (1997). A: T =8Q4°C,p =52 kg/m ; B: T = 706 °Q p = 52 kg/m ; C: T = 804 °C, p = 22 kg/m ; D: T = 706 °C, p = 22 kg/m 152 b 3 sus 3 b 3 sus b sus 3 b sus Figure 4.17: Comparison of predicted temperature differences in tube and fin (X ) with experimental measurements of Luan et al. (2000) (O). The positions of T i , T2, T5 and T6 are indicated in Figure 4.15. A: T = 804 ± 5 °C, B: T = 706 ± 4 °C 153 b b Figure 4.18: Comparison of predicted tube, fin and total heat transfer coefficients ( X ) at z = 0.559 m with experimental data (O) of Luan et al. (2000). Left panels: T = 804+5 °C; Right panels: T = 706 ± 4 °C 155 b b Figure 4.19: Suspension density profile in experimental study of Andersson and Leckner (1992) 157 Figure 4.20: FEM mesh generated for membrane wall and wall layer of CFB used by Andersson and Leckner (1992) 160 Figure 4.21: Comparison of predicted local heat transfer coefficients for L , = 1 m and L f = 2 m with experimental data of Andersson and Leckner (1992) O: method 2; +: method 3; X : method 4; solid diamond: method 2 with obstacle 0.5 m above meter. Dashed lines: prediction for e„, = 0.6; solid lines: prediction for e = 0.85. A: heat transfer coefficient at fin; B: total heat transfer coefficient for fin and tube. Lines 1 and 2: Predictions assuming heat transfer starts 0.5 m above heat flux meter 161 ar ar w xv Figure 4.22: Comparison of predicted local heat transfer coefficients for L = 0.5 m and L f= 1 m with experimental data of Andersson and Leckner (1992) O : method 2; +: method 3; X ; method 4; solid diamond: method 2 with obstacle 0.5 m above meter. Dashed lines: prediction for e = 0.6; solid lines: prediction for e = 0.85. A: heat transfer coefficient at fin; B: total heat transfer coefficient for fin and tube. Lines 1 and 2: Predictions assuming heat transfer starts 0.5 m above heat flux meter 162 art ar w w Figure 4.23: Measured local suspension densities and comparison of predicted (lines) and experimental (points) local heat transfer coefficients for runs A, B, C, D and E of Andersson (1996). Operating conditions are listed in Table 4.2 165 Figure 4.24: Measured local suspension densities and comparison of predicted (lines) and experimental (points) local heat transfer coefficients for runs F, G, H, I and J of Andersson (1996). Operating conditions are listed in Table 4.2 166 Figure 4.25: Comparison of predicted lateral variation of relative heat flux (solid lines) with experimental data of Andersson (1996) (dashed lines). Left panel: height is 10.5 m; Right panel: height is 3.4 m. Operating conditions are listed in Table 4.2 167 Figure 4.26: Geometry of the membrane wall and wall layer and finite element mesh generated for the 165 MW CFB boiler in Orebro, Sweden 169 th Figure 4.27: Comparison of predicted local heat transfer coefficients (solid line) with experimental average heat transfer coefficient (dashed line) for the 165 MW CFB boiler in Orebro, Sweden 170 th Figure 4.28: Influence offinwidth on vertical variation of heat flux based on total area and projected area. Dashed line: w = 12.8 mm: Solid line: w = 6.4 mm; Dashdotted line: w = 3.2 mm. A: T = 804 °C, T _ = 80 °C; B: T = 706 °C, b c r . , = 370 °C c out b .' o u 173 Figure 4.29: Influence offinwidth on lateral variation of radiative, conductive and total heat flux at z = 1.8 m for B: T = 804 °C and T = 80 °C. Dashed line: w = 12.8 mm: Solid line: w = 6.4 mm; Dash-dotted line: w = 3.2 mm 174 b cout Figure 4.30: Influence offinwidth on lateral variation of radiative, conductive and total heatfluxat z = 1.8 m for B: T = 760 °C and T = 370 °C. Dashed line: w = 12.8 mm: Solid line: w = 6.4 mm; Dash-dotted line: w = 3.2 mm 175 b Ci0Ul 193 Figure 1.1. Schematic of two-flux model Figure II. 1 Schematic of Keller's box method xvi 196 Figure III. 1: Schematic of moment method Figure IV. 1: A six-node triangular finite element xvii ACKNOWLEDGEMENT I wish to express my sincere gratitude and admiration to my supervisors, Dr. John R. Grace, Dr. Bruce Bowen and Dr. C. Jim Lim for their guidance, responsible supervision, spiritual support and encouragement over the entire course of this work without which this work would have not been possible. I would like to thank Mr. Jinzhong Liu for his assistance with the experiments. My appreciation also goes to Dr. Zhaohong Fang for his encouragement, suggestions and assistance. I am indebted to Mr. Peter Roberts and Mr. Robert Carrasco in the workshop for their professional work in the construction of the equipment and their critical inputs during the design phase. Thanks to Mr. Horace Lam and Ms. Qi Chen in the stores who helped me order the right items. The data acquisition system owes a great deal to Mr. Alex Thng. The friendship and helpful discussions of my fellow graduate students in the Fluidization Group will always be cherished. Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Finally I want to express my appreciation to my wife Yan for her encouragement and support. xviii CHAPTER 1 INTRODUCTION 1.1. Circulating Fluidized Bed Technology Gas/solid reactors are critical to numerous industrial processes in the chemical, petrochemical and metallurgical industries, in the manufacture of fine powders and ceramics, and in combustion and environmental remediation. The type of gas/solid reactor under consideration in this thesis, the Circulating Fluidized Bed (CFB), is finding significant applications industrially because of advantages such as efficiency and operational flexibility. The CFB consists of a riser in which a gas-solid suspension is transported upward. The two-phase mixture is separated at the top of the riser and solids are recycled to the bottom via a standpipe. The CFB is often accompanied by ancillary equipment such as strippers, regenerators and external heat exchangers. Most CFB applications, including combustion, calcination, and hydrocarbon cracking, are operated at high temperature with heat either added or removed. As in more conventional bubbling fluidized bed reactors, the design of the heat transfer surfaces is critical in ensuring efficient operation and control. Most CFBs require exchange of heat between the gas-solid suspension and heat transfer surfaces deployed at one or more levels. Possible locations for heat transfer surfaces in CFB combustion systems include: l . surfaces forming part of the wall ofthe riser; 1 2. surfaces inside the riser, e.g. suspended or panel walls; 3. an external low-velocity fluidized bed heat exchanger; 4. downstream surfaces, e.g. superheater or economizer tubes. Ideally, all in-reactor heat exchanger surfaces should form the outer wall since suspended heat transfer surfaces are subject to erosion and impede radial mixing of the gas and solids. They may also reduce hold-up of solids in the reactor and promote attrition. However, the amount of wall surface area per unit reactor volume decreases as the reactor is scaled up, so that a size is eventually reached where it becomes impossible to provide sufficient heat transfer via surfaces located on the wall. This problem can be avoided by employing an external heat exchanger. Two kinds of walls are employed in CFB systems: smooth walls (including flat walls and the wall of a cylindrical riser) and membrane waterwalls. The latter are commonly used primarily in industrial CFB boilers, while smooth walls are employed in other CFB applications and in most small-scale units. CFB reactors can be divided into two categories: gas-solids reactors and catalytic gasphase reactors. Gas-solids processes, such as coal combustion, alumina calcination and iron ore reduction, usually do not require very high gas velocities, nor high solids circulation fluxes. A typical Circulating Fluidized Bed Combustor (CFBC) is operated at a superficial gas velocity from 5 to 9 m/s and a net solids circulation flux of 10 to 100 kg/m s. Circulating fluidized beds operating at relatively low gas velocities and limited 2 2 solids circulation fluxes are classified as Low-Density Circulating Fluidized Beds (LDCFB). Catalytic gas-phase reaction processes usually require a relatively high gas velocity in the riser to promote plug flow and a higher solids circulation rate. For example, typical Fluid Catalytic Cracking (FCC) units are operated at gas velocities from 6 to 28 m/s and net solids circulation fluxes of 400 to 1000 kg/m s. CFBs operating under high solids flux and high solids concentration conditions are classified as High-Density Circulating Fluidized Beds (HDCFB) (Zhu and Bi, 1995). Grace et al. (1999) defined a HDCFB riser as one operating with G > 200 kg/m s and c > 0.10 throughout the entire riser. 2 s 1.2. Circulating Fluidized Bed Hydrodynamics The dependence of heat transfer on CFB hydrodynamics cannot be overemphasized. The suspension-to-wall heat transfer process is controlled by the hydrodynamics of the solids and gas mixture in the vicinity of the wall. A major feature of the overall flow structure in most LDCFB units is core-annulus segregation. In simple terms, there exists a relatively dilute core in which solid particles are entrained upward by the high velocity gas stream entering at the bottom of the riser, and a much denser annular layer near the column wall in which solid particles congregate and fall as dense structures such as strands or streamers. The particles are actively mixed in the core where the temperature is nearly uniform. The thickness of the annulus zone, whose boundary is commonly defined as the point where the time-average solids flux is 3 zero, tends to be a modest fraction of the bed diameter. While a net downflow of particles at the wall is a commonly observed feature of LDCFBs, it is absent in HDCFBs (Grace et al., 1999). Solids move upward on a timemean basis throughout the entire HDCFB riser cross-section. Yet there are still considerable radial gradients in suspension density, with higher particle concentrations near the wall than in the interior. While there do not appear to be clusters in HDCFB risers, there are certainly substantial fluctuations in local voidage. 1.3. Thermal Radiation in Fluidized Beds Gas-solid fluidized beds are multiphase systems consisting of solid particles and gases. In homogeneous media such as gases, absorption and emission are the major radiative mechanisms. If the media contains inhomogeneities, such as the particles in fluidized beds, the additional mechanism of scattering is introduced. Thermal radiation within the bed usually originates as emissions by the hot walls and the gas-particle mixture. This radiation undergoes complex interactions with the bed, primarily due to absorption and scattering processes (Tien, 1988). The three primary radiative properties that characterize the interactions of radiation with the particulate bed are the scattering coefficient, the extinction coefficient (i.e., the sum of the scattering and absorption coefficients), and the scattering phase function. Computation of the transport of thermal radiation in the particulate system requires an accurate knowledge of these primary radiative characteristics. This is made clear by considering the propagation of radiation within an absorbing, emitting and scattering medium, which is governed by the following 4 equation of transfer (Siegel and Howell, 1992; Ozisik, 1973): ^ = -a i,'(S)-aJ \S) dS x x + a i»'(S) x + ^f- f' ' V(S)<&(A,fl>,fl>,)<fc>,. An ' (1.1) io =0 The first two terms on the right-hand side represent the attenuation of intensity due to absorption and scattering, respectively. The third term represents energy gain due to emission. The last term is the gain due to scattering into the 5 direction from all other directions. The intensity ix' is defined as the energy per unit area per unit solid angle per unit wavelength. The scattering phase function O^,^,^.) represents the radiation intensity scattered from direction a>j into the direction under consideration, normalized by the isotropic scattered radiation intensity. Note that 0(2,co,co ) = \ for isotropic i scattering. The absorption and scattering coefficients are defined as the fraction of the corresponding energy loss from the propagating wave per unit length of travel. The units of these coefficients are inverse length, whereas the phase function is dimensionless. These absorption and scattering processes are governed by electromagnetic field equations and their associated boundary conditions at all interfaces. When particles or other materials are present or are injected into a gas to enhance its absorption or emission of radiation, the gas particle mixture may act as nearly gray (Siegel and Howell, 1992). For gray media, the absorption, emission and scattering coefficients are independent of radiation wavelength and hence equation (1.1) may be written as ~ = -ai' (S) + ai '(S) - cr/ (S) + ris: An ' h (Smco,co,)dco,. 4j =0 5 (1.2) Another important quantity, which is of greater interest than the intensity, is the heat flux. The radiative heat flux crossing an area dA due to intensities from all directions is (1.3) where <f> is the angle from the normal of dA to the direction of /"'. Integrating equation (1.3) over all angles yields the following equation for the divergence of the radiation flux (Ozisik, 1973; Siegel and Howell, 1992): (1.4) 1.3.1. Absorption and Scattering by a Single Particle The absorption and scattering characteristics of a single particle are described by the solution of the electromagnetic field equations. Physically they can be explained by the processes of reflection, refraction and diffraction and are governed by three factors: particle shape, particle size relative to the wavelength of the incident radiation, and the optical properties of the particle and the medium. General solutions are available for only a few common shapes such as spheres, cylinders, and spheroids, and these solutions are complicated even for these simple cases. The size factor is commonly expressed by the parameter a, defined as nd IX for spheres. Optical characteristics are represented by the complex refractive index, m, defined as n + IK, where n is the index of refraction and K is the index of absorption. The solution ofthe electromagnetic field equations yields the internal and scattered electromagnetic fields from which the corresponding absorption 6 and scattering cross-sections are obtained. The cross-sections are defined as the ratio of the energy loss to the incident energy flux and have units of area. Efficiencies are defined as the dimensionless ratios of cross-sections to the geometric cross-sectional area A, i.e., Q =CJA (1.5) Q.=CJA (1.6) a where A = xd„ /4 (1-7) 2 p for spheres of diameter d . p For large opaque spheres the efficiencies can be approximated (Siegel and Howell, 1992) as Q =e a (1-8) p The phase function is O{0 ) = ^-(sme -e cos0 ) p p p (1.10) p in if the sphere reflects diffusely. Since large particles in most circulating fluidized beds such as chemical reactors and coal combustors are diffuse 7 reflectors, the above expression permits a significant saving of computational resources. 1.3.2. Absorption and Scattering of a Particulate System Absorption and scattering characteristics of many particles in packed or fluidized beds can be obtained from the single particle characteristics. The procedure depends on the scattering regime to which the system of particles belongs (Tien, 1988). Brewster and Tien (1982b) found experimentally that the domain of packed and fluidized beds, for the most part, falls well within the independent scattering regime. The independent theory is based on the assumption that each particle assembly scatters and absorbs radiation unaffected by the presence of other particles. Thus the absorption and scattering of energy by the system is expressed by simple algebraic addition of the energy absorbed and energy scattered by each primary particle. The cross-section for the system of N particles is the sum of the cross-sections of each particle, and the individual particles are assumed to scatter and absorb radiation independently. For identical particles this leads (Ozisik, 1973; Siegel and Howell, 1992) to: ^-ZC,=/VC (1.12) i M 0 =O N (1.13) U 8 Here subscript M indicates the value for a single particle. The corresponding coefficients for the medium are C • NC a = =^- = - ^ C * NC ^ L = (1.16) £^SM_ = V (1.17) V where V is the volume containing N particles. For beds of monodisperse spherical particles of diameter d p 3 ce a = -—^ 2 d (1.18) p cr.=2f2z^. 2 (1.19) d 1.4. Previous p Models 1.4.1. Models for Reactor-Side Heat Transfer To explain heat transfer in CFBs and help scale up heat transfer coefficients, several models have been proposed. The heat transfer between the reactor wall and the suspension includes contributions from radiation, particle convection and gas convection. Though there are doubts about the additive nature of these components (e.g. Botterill, 9 1984; Fang et al., 1995a), many authors approximate the overall heat transfer coefficient as h = fJi,+Q-fJh +h . g (1.20) r 1.4.1.1. Particle Convective Component A number of mechanistic models have been proposed to describe the particle convective component and explain the nature of heat transfer at the walls of a CFB boiler. These models can be classified (Basu and Nag, 1995) broadly under three groups: single particle models, cluster renewal models and continuous film models. i. Single particle models The primary concern of this class of models (Ziegler et al., 1964; Sekthira et al., 1988) is the first layer of particles adjacent to the wall. Here particles are assumed to travel down the wall with an initial temperature equal to that of the bulk bed. Heat is convected from the particles closest to the wall, to the surrounding gas, which in turn transfers it to the wall. It is further assumed that die heat flux to the wall is controlled by the heat transfer from the particle to the gas film surrounding it. This film is assumed to be at the mean temperature of the inner suspension and wall. Assuming the gas-particle Nusselt number to be 2, Sekthira et al. (1988) derived an expression for the particle-towall heat transfer coefficient whose simplified form is 1.574(1-cU K=—. 0 5 • 0-2D 10 ii. Cluster renewal models Congregation of solid particles into clusters (or strands) is a major characteristic of most circulating fluidized beds, distinguishing CFB risers from entrained beds. The descent of these clusters takes place primarily in the gas-solid wall layer (annulus region) adjacent to the reactor wall. The clusters, after descending a certain distance, "dissolve" or detach themselves from the wall. Thus, there is a characteristic length over which clusters maintain their identity. When the clusters slide along the wall, unsteady-state heat conduction takes place. For modeling purposes, clusters are commonly considered to be semi-infinite. In addition, a gas film resistance is usually inserted at the wall limiting the rate of heat transfer from the first layer of particles to the wall. The existence of a thin and almost particle-free zone next to the wall has been observed by Wirth and Seiter (1991) and Glicksman (1988). This resistance is significant, especially for coarse particles and very short residence times. For a residence time, t, and heat conduction from a cluster and through a gas film of thickness d ln, p the time-average heat transfer coefficient due to particle convection is written (Basu, 1990; Wu et al., 1990; Glicksman, 1988) as: h = d, nk„ ^- d-22) tn ^ c,C P k pcl cl Fang et al. (1995a) improved the cluster renewal model by letting clusters reach the wall at different positions and travel different distances. Combined conductive and radiative 11 heat transfer between the surface and cluster was later modelled by including radiation across the gas gap (Fang et al., 1995b). iii. Continuous film models Models of this type assume that the walls of CFB boilers are always covered by a homogeneous film of gas and particles. Therisinggas in the core does not contact the wall. Thus the gas convection component, h , is not computed separately, but is built into g the particle convective term calculated through this model. The solids in the film descend while the gas is assumed to percolate upward. The particle convective heat transfer considering unsteady-state heat transfer to the falling film and a gas gap resistance equivalent to a layer of thickness dp/10, is (Mahalingam et al., 1991) 1 (1.23) -i 7C kJ 2 where h, : A + 2 ^ - y e x p'i 2 (1.24) ' PeCpe : 5 2 is the local heat transfer coefficient ofthe moving emulsion layer. 1.4.1.2. Gas Convective Component Since the particle convection coefficient, h , is much greater than the gas convection s coefficient, h , for the particle sizes used in most CFB applications, the dilute phase heat g transfer coefficient has received limited attention. 12 Johnsson et al. (1988) and Kudo et al. (1991) assumed that the entire wall receives heat through gas convection in addition to that received directly from the particles. The gas convective component, h , was calculated from an equation of Eckert and Drake g (1972): Nu g =0.009Pr A r 1/3 (1.25) ,/2 Here h is independent of the superficial gas velocity. A more realistic approach was g taken by Wu et al. (1990) who used a correlation for forced convection from particle free gas, and applied that only to the parts of the wall not exposed to clusters. Basu and Nag (1987) and Basu (1990) argued that the cluster-free parts of a CFB are not entirely solidfree. A small number of particles are dispersed in this up-flowing gas, with important consequences for forced convection. They used the following correlation for dust-laden gas to estimate the gas convection component: ^ _ g g PP /Pdis \0.3f d C p p ps r p 2 Y'' 2 ^— Pr. (1.26) W P ) 1.4.1.3. Radiation Component Radiation is a major component of heat transfer, especially for CFB boilers with high steam pressures, where the temperatures of both the tube wall and furnace are high, and for the low suspension densities encountered under turndown conditions. Several models have been proposed to calculate the radiation heat exchange between suspension and wall. These can be classified under two groups: surface interchange 13 models and continuous medium models. i. Surface interchange models Surface interchange models treat the radiation process between the wall and bed as classical surface interchange between two or more plates. Brewster (1986) represented the bed and wall as two very large parallel planes with effective bed absorptivity a ff, and e emissivity e //. The net heat flux to the wall is then given by e q = ^-,—r—• ( L 2 7 ) r —+—-1 eff a G » Basu and Nag (1987) and Basu (1990) accounted for radiation received by the wall from both clusters adjacent to the wall and the dispersed phase. They expressed the total radiative heat transfer coefficient as where h, = ^ d and (1.29) -L - L - i k - r J + ^ ** = T (1.30) t - j J ) ' l 1 ^ (T -T ) —+ 1 b w Fang et al. (1995b) considered that an exposed surface in a CFB furnace of 14 rectangular cross-section receives radiation not only from the dilute suspension, but also from clusters located along the opposite or side walls. The three components, including facing surfaces of alternating clusters, exposed wall and dilute suspension, were represented using an electrical network analogy, leading to e [e +Le (l-e MT:-T:) w d cl d + (i - O l M * + a - /„te.lto - T ) " (1.31) w In CFBs, a thermal boundary layer exists near the wall. In the model of Werdermann and Werther (1994), the suspension is regarded as consisting of spherical particles of uniform size arranged in a regular cubic array of (/ ax+l) particle layers, considered as m surfaces exchanging radiation. The radiative heat transfer coefficient between layers / and j is calculated from: o-iT^+T/W+Tj) (1.32) When this model is applied, the temperature profile within the particle layers is required. Particles are assumed to be arranged in a regular array, with an arbitrary number of layers. ii. Continuous medium models In this kind of model, the suspension covering the wall is treated as a continuous absorbing, emitting and scattering gray medium. Brewster and Tien (1982a) examined the scattering properties of suspensions in packed and fluidized beds and concluded that 15 most packeaVfluidized bed systems of engineering interest fall in the independent scattering regime. The traditional two-flux model (a special case of the discrete ordinate method), based on the assumption of semi-isotropic intensity distribution, is usually employed in the continuous medium models. As derived in Appendix I, it is obtained (e.g. Siegel and Howell, 1992) by integrating the complete transfer equation (1.2) for azimuthally symmetric radiation in a one dimensional plane-parallel slab over all directions (4n solid angle), giving 1 2 dV dx = -(a + cr B)I 1 dr 2 dx + s + al (S) + a BI' b s = -(a + o- B)r+aI (S) s b + cT BP. s (1.33) (1.34) Glatzer and Linzer (1995) adopted the two-flux model in a thermal boundary layer with a uniform particle concentration near heat transfer surfaces. The equations above were solved assuming • known absorption, emission and back-scattering coefficients • constant temperature T and homogeneous density within the solution range, Ax . Chen et al. (1988) developed a steady-state model for the case where radiative, conductive and convective transport occur simultaneously throughout the suspension. The model accounts for temperature variations in both the lateral and axial directions. It 16 is based on the assumption that particles are uniformly distributed laterally throughout the entire riser. The two-dimensional energy balance is written as il - i)A,P C„u ^-c C ^-l (r-n---Q (.35) k d P p M K ; where x is the horizontal coordinate from the wall, z is the vertical coordinate from the bottom of the riser, and Q is the heat generation rate per unit volume. Coupled with the two-flux model expression for f and I~ in equations (1.33) and (1.34), the problem can be solved numerically. The non-uniform suspension model of Luan et al. (1999) is similar to the two-flux model of Glatzer and Linzer (1995), except that the former considers a non-uniform solids concentration profile in the emulsion layer and a more general discrete ordinate method is used to solve transfer equation (1.2). Twelve fluxes were used, and the temperature distribution in the suspension layer was estimated using the correlation of Golriz (1995). Thus the radiative transfer equation could be solved separately, uncoupled from the temperature field. 1.4.2. Models for Wall-Side Heat Transfer Membrane waterwalls, consisting of parallel tubes connected longitudinally by fins or membrane bars, have long been a feature of many pulverized coal combustion boilers and recovery boilers used in the pulp and paper industry. They can also be used to contain and extract heat from large-scalefluidizedbeds, operated as conventional bubbling beds or as circulating beds. 17 Conductive transfer in the membrane wall assembly was considered by Bowen et al. (1991). Numerical solutions were obtained using a finite difference technique in conjunction with a boundary-fitted orthogonal coordinate transformation in the fin. Four simpler analytical or semi-analytical approximations were also derived for design purposes, and their accuracies were assessed by comparisons with the numerical solutions. It was demonstrated that heat transfer rate estimates within a few percent of the numerical solution values could be obtained from the more sophisticated simplified models for conditions of practical interest. Taler (1992) developed a numerical method that utilizes the tube temperature data from interior thermocouples to determine heat fluxes in boiler furnaces. This method requires that a set of nonlinear equations be solved. Andersson and Leckner (1992) presented four different methods to estimate bed-to-membrane wall heat transfer, one based on the fin-tube temperature difference. The relation between the bed-side heat flow and the fin-tube temperature difference at steady state is calculated using a finite element program. They later measured the local lateral distribution of heat flux (Andersson and Leckner, 1994; Andersson, 1996). Fang et al. (1997) considered conductive transfer not only in the membrane wall, but also in the insulation. The boundary condition on the furnace side was relaxed to "n" uniform flux rates. Based on the linear superposition theorem, a procedure was proposed for determining steady heat fluxes on boundaries, posing an inverse conduction problem. Error analysis indicates that the condition number of the resulting coefficient matrix is the decisive factor for successful application of this approach. Error transmission was 18 discussed further by Fang et al. (1998). 1.5. Summary of Previous Models 1. In previous models, the reactor side and wall side heat transfer processes have been treated separately. When reactor side heat transfer is considered, a constant wall temperature is usually assumed as a boundary condition. When wall side heat conduction is considered, uniform heat fluxes are usually applied on the furnace side surface as boundary conditions. 2. Except for the models of Fang et al. (1995a) and Chen (1988), most models consider the reactor side conduction/convection and radiation separately. When the radiative continuous media models are solved, suspension temperature distributions have to be provided. 3. None of reactor side heat transfer models have considered the geometry of the membrane walls. Even in the surface interchange radiation models, one relatively large surface is usually assumed, neglecting the fact that the tube and thefinhave different view factors to the bed. 4. Most models are based on experimental observations obtained in CFB combustors and other low-density CFBs. These models need to be extended to high-density operating conditions. 19 1.6. Objectives of this Thesis Project 1. Develop a model for flat smooth walls that can simultaneously account for both the reactor-side and wall-side heat transfer processes. The model should also couple the reactor side conduction, convection and radiation processes. 2. Conduct an experimental study of suspension-to-wall heat transfer in a highdensity circulating fluidized bed in order to extend the model to high-density flow regimes. 3. Extend the model to the geometry of membrane walls. Propose a model to deal with radiation heat transfer processes when membrane walls and coupled radiation, conduction and convection are all involved. 1.7. Structure of Thesis Chapter 2 develops a model for heat transfer to a low-density CFB with smooth walls. The governing equations and boundary conditions are discussed. The model is then validated using published experimental data. The sensitivity of the heat transfer process to variations in important operating parameters is also investigated. Chapter 3 describes experiments carried out in a high-density circulating fluidized bed. Suspension temperatures and bed-to-wall heat transfer coefficients (both average and local) are reported. The model developed in Chapter 2 is then extended to highdensity flow regimes. 20 Chapter 4 modifies the model to include membrane wall geometries. The moment method is employed to deal with radiation problems. The model is then validated using published experimental data from CFB combustors equipped with membrane wall assemblies. Overall conclusions and recommendations are given in Chapter 5. Key derivations and tabulations of the experimental results appear in the appendixes. 21 CHAPTER 2 TWO-DIMENSIONAL MODEL FOR LOW-DENSITY CFB WITH SMOOTH WALLS 2.1. Introduction In this thesis, the term smooth walls refer to flat walls and the walls of cylindrical risers with relatively thin annular layers of particles, i.e. non-membrane surfaces. For smooth walls, the system can be assumed to be two-dimensional, i.e. all transfer and gradients in the tangential direction are ignored. This leads to a heat transfer problem where only the vertical direction and direction normal to the wall surface are involved. A major feature of the overall flow structure in most low-density CFB units is a core-annulus flow pattern, confirmed using various experimental techniques. The crosssection of a CFB riser is commonly divided into two regions, with particles transported upwards in a dilute core, while denser layers of solids in an outer annulus are assumed to descend along the wall. Particles, after staying in the wall layer for an average residence length L , are re-entrained into the core and replaced by fresh particles with the bulk ar temperature. Typically the wall layer becomes denser due to an increased cross-sectional average suspension density with descending height, indicating that more fresh particles are transferred from the core to the wall layer than the other way around. Experiments in CFB combustors (Weimer et al., 1991) reveal that vertical waterwall surfaces experience very little wear. This suggests either that few particles actually touch the wall or that the 22 particle velocity adjacent to the wall is not very high. Lints and Glicksman (1994) determined that there is a particle-free gas layer along the wall whose thickness is of the order of Q.3d to p l.0d , p depending on the overall suspension density. It is common to assume that this gas gap is stagnant, and also that the temperature in the core region is uniform. With the latter assumption, suspension temperatures only change appreciably in the wall layer and it is reasonable to limit consideration of reactor-side heat transfer to the wall layer. 2.2. Model Development 2.2.1. Background Assumptions Based on considerations of hydrodynamics and thermal radiation in circulating fluidized beds, a coupled two-dimensional heat transfer process is assumed as illustrated schematically in Figure 2.1. Descending particles are assumed to enter the heat transfer zone at z = 0 with the same temperature as the core suspension. As particles descend in the wall layer, they lose heat to the gas by convection and gain heat fromfreshparticles arriving from the bulk core region. Temperature gradients within each particle are neglected since for typical particles used in CFB reactors, the Biot number is much smaller than 0.1. Particles also participate in radiation heat transfer from the core to the wall through the wall layer, enhancing the radiation flux by emission and attenuating the flux by absorption and scattering, while the gas is assumed to be optically transparent. To simplify the radiation analysis, the particles are assumed to constitute a gray continuous absorbing, emitting and scattering medium. For particle sizes and concentrations typical 23 Vertical heat transfer by conduction and radiation are much smaller than convective transfer in the z direction. Therefore the former two mechanisms are ignored, i.e. only convective transfer is considered in the vertical direction. 2.2.2. Governing Equations and Boundary Conditions Based on the above assumptions, the following governing equations and boundary conditions are written for steady conditions. Heat balance on gas in the wall layer (outside the gas gap) ignoring heat generation: ^-".f-^f)-^- 0 (21) where Q is the volumetric rate of heat convection from particles to gas given by = KA P -r ) = 6Nu Q T f Pg w (l-*)* (*; -T Vd> f t (2.2) with the particles assumed to be spherical and temperature gradients within the particles neglected. Heat balance on the particles: dT P C cu p pp p + E C (T x pp p -T ) b + Q p g + Vq = 0 r (2.3) where Vq , the local divergence of the radiative flux, based on the 2-flux model, is given r by (see Appendix I) 25 d r-n^_d_r dx dx dx ( ( With— = -2(a + cr B)I +2aa T dx + A s — = 2(a + a B)r dx t 0 +2a BI~ - 2aa T - 2a BI A 4 ) (1.33) s (1.34) + 0 2 s E is the particle exchange rate between the core and wall layer and is discussed later in x this chapter. Heat transfer through gas gap: Heat transfer from exposed wall surface to cooling water: q={T -T )l£+hw (2.6)' c Heat balance on cooling water: LPC u^ c c pc c + q = 0. (2.7) Boundary and interface conditions: Top of heat transfer zone: T =T = T at z = 0, 5 < x < 5 g Bulk side of wall layer: p b g T = T = T = /\z) at z > 0, x = 8 g p b 26 (2.8) (2.9) I' = e oT +(\-e )r atz>0,x 4 b b b Inner reactor side of wall: q = q -1* +1 at z > 0, x = 0 c Gas gap/wall layer interface: -q - —k c w dx at z (2.10) (2.11) (2.12) r=e aT:+CL-e )Iw =S > 0, x = 8.s (2.13) 2.2.3. Parameter Determination 1. Voidage distribution in wall layer Herb et al. (1989) used a capacitance probe with FCC particles in ariserof diameter 0.15 m and observed local solids concentration profdes at different elevations. They suggested that there may be a universal radial profde of bed voidage for time-averaged solids concentrations. Several researchers have proposed different correlations to predict the local voidage distribution as listed in Table 2.1. Equation (2.18) is correlated from data obtained in a riser of 76 mm diameter and 6 m height for FCC particles at superficial gas velocities between 4 and 8 m/s and solids circulation fluxes up to 425 kg/m s. This equation also works well for a wide range of 2 operating conditions and risers of large size and for other particles (Issangya et al., 2001). Hence this equation is employed in the present model. 27 Table 2.1: Correlations for radial voidage distribution in CFB risers. Correlation Researchers Zhang etal. (1991) , » Rhodes et al. (1992) (2.14) (0.191+^"+3*") (2.15) l-e(<P) = 2(l-£ )<f> sec 2 Patience and Chaouki (1995) Wang etal. (1996) (2.16) 1 - etf) = (l - £ )(0211 +1.92 sin (0.5*0)) 10 m Issangya et al. (2001) ^)=^ (^-^KJ- 1 + 5 + 2 1 ^ , + 5 V , ) (2.17) (2.18) 2. Thickness of gas gap Visual studies suggest that clusters are not in direct contact with smooth walls. Wirth et al. (1991) used gamma rays to measure the solid-to-wall spacing in a 0.168 m x 0.168 m x 11 m cold-operated CFB and reported values of about 0.5 mm. Using an impact probe that could be positioned at various distances from the wall, Lints and Glicksman (1994) measured the gas layer thickness between the wall and the nearest clusters for 182 pm sand and correlated the thickness as = 0.0287(1 - O " 0 ( - ) 581 2 This relationship is adopted in our model. 28 1 9 3. Particle downward velocity in wall layer Several measurements of particle velocity near the wall have been made (see Table 2.2). The falling velocity of solid particles close to the wall appears to be minimally influenced by particle size, particle density and superficial gas velocity. It should be noted that all these data were obtained for fast fluidization (LDCFB) operating conditions. Particle motion in the vicinity of wall, for high-density operating conditions is discussed in Chapter 3. Here a particle downward velocity of 1.2 m/s is assumed consistent with most experimental evidence in Table 2.2. The sensitivity of the predicted results to this assumption is explored below. 4. Gas downward velocity in wall layer Gas enters the heat transfer zone and is assumed to be dragged downwards by the rapidly-descending annular particles. There is no information in the literature regarding the gas velocity in the wall layer. We begin by assuming that u = 0.4 m/s. The sensitivity g to this assumption is shown below to be small. 5. Heat exchange between particles and gas: Zenz and Othmer (1960) predicted the particle-to-gas heat transfer coefficients in gas fluidized beds by the relation = 0.017( P s u s P d y.2, (Re < 80, Pr «1). 29 (2.20) o o "5 > wo CN oo CN* CN '•E © ca CO CN o O WO CN CN wo i o CN vo VO <N CN i 00 o CN © .P <u 3 cr O -o O o o <u p IU c 1) M~ 2 "tt> 0 •3 3 cd o p. o O 3 -4—» § oE o tl o '-4-> p< o p. o o O >-c \S g &, o p< '•*-» o o o i<D 6 u p CL) Xi o i-. P. a ca T3 <U <D &, ca 3 PH ca cn > .P '-*—» ca o o <u &, o l-l <u o Q o O (D CN WO wo o co vo 8 VO CN wo *—I S X o wo o wo O CN WO O CO 00 VO CN X CN in X o o VO as CN wo CO I VO I wo © as CO CO as CO CN vo © wo wo VO o VO CN o r- o <U ca £ .2 vo U V PH CN W0 ca O WO oo ca o •P cn ca U ffl O 00 PH PH u O O VO o ca~ oo c © CN o vo U CJ PH VO ca p o u 3 PH o wo oo* p N •a T3 00 S -o C 3 ca —< c ca oo OO wo ON ON ON 00 < ON oo ON OO oo oo ON oo ON ca (U IU .P o (H ca <u <u ca o VH ca CQ &, ca •a '6 oo oo o ON as CN ON ON ON o <u ca •4—* <*} ca .5 00 ON O w O <^ o •a o 30 00 (U ca 3 o -4—• ON ON ON ON ca ca •*-* '<u OO (U •a ca •P IU t ; 3 <U IU IU ON ON C ca 3 5 O N The following more general equation was given by Frantz (1962): Nu„„ =0.017 Re' Pr0.67 (Re < 80, Pr ~ 1). 21 (2.21) Ranz (1952) correlated experimental data in packed beds and suggested that: Nu = PS = 2 + 1.8xRe Pr1/3 05 kg (Re < 100, P r « l ) . (2.22) This equation has been recommended by Kunii and Levenspiel (1991) and is employed in our model. 6. Bulk emissivity Most researchers (e.g. Glicksman, 1988; Fang et al., 1995b) calculated the emissivity of the bed using the following equation, based on the mean beam length concept of Hottel and Sarofim (1967): <\-e )e L e = 1 -exp(-1.5 b b p b (2.23) •)• P d Equation (2.23), which is also adopted here, suggests that for very large and dense beds the bed interior may be assumed to be a black body. 7. Thermal radiation characteristics of particle suspension The particle and gas suspension in the wall layer of a CFB is assumed to constitute a continuous absorbing, emitting and scattering gray medium with monodisperse spherical 31 particles of diameter d . The absorption and scattering coefficients can be calculated from p equations (1.18) and (1.19) and the phase function for anisotropic scattering is represented by equation (1.10). 8. Wall layer thickness Three correlations have been previously proposed to predict the wall layer thickness. The correlation of Patience and Chaouki (1995) D 1 = 0.5 (2.26) .083 Fr l+ lAFv(G /p U f' s s g relates the wall layer thickness to the overall solids circulation rate, superficial gas velocity and particle density. The correlation of Werther (1994) D = 0.55 v P, , U ^ D j H-Z H (2.27) accounts for the axial variation of the wall layer thickness. Bi et al. (1996) correlated the wall layer thickness with the cross-sectional average voidage according to: = 0.5 _1 - V l . 3 4 - 1 . 3 0 ( 1 - 0 " + 0 - O ] m D (0.0015 <l-^ <0.2) sec (2.28) Since this correlation was obtained from regression of most of the data reported previously, it is selected for use in our model. To simplify the calculation when the bed cross-sectional average voidage changes with bed height, an average bed voidage over 32 the heat transfer zone is used in equation (2.28) so that the wall layer thickness is assumed to be constant over that section. 9. Average particle residence length L ar Wu et al. (1990) correlated the "characteristic" residence length of particles in the wall layer on a smooth-walled riser as Z = 0.0178,9J 5 9 6 (2.29) flr However, when applying this length to his heat transfer model, he found that L should ar be at least 1.7 m to fit his experimental data. Visual observations and data obtained in a 0.20 m diameter CFB by Lints and Glicksman (1992) indicated that some clusters remain near the wall for distances of at least 0.5 m, the limit of the field of view of their video system. Han (1992) obtained the following equation for L from non-linear regression of ar his own experimental data I =dx4961Fr'^ 46 or P = 61. lFr" 24 1 (Group A particles) (2.30) (Group B particles) Experiments by Golriz (1996) in a 12 MWu, circulating fluidized bed showed that L ar should be at least 2 m. The data available for L are clearly diverse; here 1.5 m is used. ar The sensitivity to this assumption is investigated below. 33 10. Particle exchange rate, E x There are two sources of fresh particles in the wall layer. One is the cross-flow exchange of particles between the wall layer and the core region. Assume a particle concentration of c{<p) at a certain height and consider a surface dxdy normal to the direction of particle velocity. In unit time, the mass of particles crossing this surface is p c(<p)u dxdy . p p Since the average particle residence length is L , after a distance of ar L, ar these particles are on average replaced by fresh particles. Hence the mass of particles p c((f>)u dxdy exchanged per unit volume per unit time is —- - L dxdy ar p„c(<p)u =— L . The other source ar is that the wall layer may become denser as particles descend along the wall due to an increase in the cross-sectional average suspension density lower in the riser. In this case, if at a certain position, the vertical particle concentration gradient is ^ m e exchange dz rate would be p u (^ Hence, summing the two terms gives dc p p dz c(<f>) E * = P p u P L„ | dc{<t>) (2.31) dz 34 2.3. Numerical Method and Accuracy Analysis Keller's box method (Keller, 1971) is employed to solve the model equations. The wall layer domain (8 < x <S, is divided into (m-1) horizontal and («-l) 0<z<H) g vertical grid intervals. Application of Keller's box method to the governing equations and boundary conditions described above leads to the following set of algebraic equations containing the (5m+7) unknowns (T , T w(J+l) , T g{iJ+l) , S , p{iJ+l) ij+l 7 , iand I~ i) for the + v+ iJ+ (J+l) vertical layer: \ «u«) " 2 g(\j i) +7(u+D -7"(V+D = i E T E T + E + ej&*wj« -7 + (y+1) +(l-eJI- J+l) =0 ^ o - W . ^ - ' W ^ ' '=U.../«-l 0 \i g(i,m \Jg(i+\,m ~BnTrt,m "^z^.+ij+i) B T ~n +B B M p(i.j \) ii n(M.m +Q,-7 /j+i +Q/VJ+1 +C ,J"/.; i T C +C T + + 5 + +CJ~MJ \ + =0 u paj+\) 2i p{i+u+i) +c i ij \ +c r \j^ +c,trij+i +Qy~, i,y i =0 c T +c T 5i + 6i i+ + A-C^/j+i) g[i i,j i)) 2i( p(ij r) (i ijH)) (I (ij*i) +T +D + g(m.j+\) 1 T +T + +D P + + -/"W+o -7"oj+D +7"<<+i.;+i))=A,.J + 3i b 1 =T ( b -OC^,) e T + i=\...fn—1 + 4 =e o-T b b (2.32) where B , C, D and E are coefficient matrices defined in Appendix II and S is the horizontal gas temperature gradient, 8T I dx. g The set of equations (2.32) is non-linear, not only because of the fourth-power radiation term, but also because some parameters are functions of temperature. The set can only be solved numerically and iteratively. Keller's box method is unconditionally 35 stable and has second-order accuracy even for a non-uniform grid in the x-direction. The accuracy is affected by both the horizontal and vertical grid dimensions, Ax and Az. If the vertical step is too large, the result will be inaccurate, even though the method is unconditionally stable (Keller, 1971). However, intervals which are too small cause a heavy computational load. I i 20 40 • T • 1 ' r i , i • i 60 80 100 , I 120 Heat flux (kW/m ) 2 Figure 2.2: Influence of vertical step on calculation of heat flux at the wall 1. Az = 10 mm; 2. Az = 2.5 mm; 3. plot 1 after filtration with equation (2.33) max max At the top of heat transfer surface, the gas temperature near the wall decreases sharply along the heat transfer surface because of the large temperature difference between the gas and the wall. Therefore, Az needs to be small near the top. With increasing distance below the top, the temperature difference decreases, so that zlz can be enlarged. To obtain smoothly increasing values of zlz, a Fibonacci series (i.e., zlz (/) = zlz (/-/) + zlz (i-2)) was 36 employed until a pre-set maximum step-size Az was reached. As shown in Figure 2.2, max over a certain range of Az^, the local flux due to gas conduction was found to oscillate. Decreasing the step-size reduced the amplitude of oscillation. To smooth the curve while retaining a reasonable value of Az^, a simple low-pass fdter was employed, i.e., ( / ) = q (i ~ 3) + q {i ~ 2) + q (i -1) + 2 x q (i) + q Q +1) + q (i + 2) + q {i + 3) c c e e c c c 8 Figure 2.3 shows the vertical wall temperature distribution for different values of Ax with uniform grids in the x-direction. It can be seen that the choice of Ax influences the calculated wall temperature. The largest deviation between the values occurs in the first z step. Then it remained almost constant along the remainder of the heat transfer surface. Hence it is essential for the values at z = Az{\) to be as accurate as possible. ' i 2.00 ji 1.98 1.96 r , r- 1 1 0.00 0.02 _ Ax- 0.05 mm Ax= 0.10 mm Ax= 0.20 mm 0.04-S- 1.94 0.06 1.92 0.08 1.90 ~ I 300 0.10 I t , I . 350 I 1 . . 400 I . 1 ' 450 Wall temperature (K) Figure 2.3: Influence o f Ax on wall temperature calculation. Operating conditions are listed in Table 2.3. 37 1 500 Near the top of heat transfer surface, the particle and gas temperatures only vary significantly very close to the wall. Hence the horizontal grid dimension near the wall needs to be very small to obtain accurate results. A Fibonacci series was also employed to produce grids of a smoothly increasing horizontal step-size. Simulation results show that the solution became essentially independent of Ax when its initial value is 1 * 10" m. In 5 the calculations below, Fibonacci series were employed for both the horizontal and vertical grid dimensions with Ax(\) = Az(\) while Ax max = 1 mm, Az max = lxlO" 4 mm, Ax(2) = Az{2) = 2x10~ 4 mm, = 1 0 mm. For iteration, the solutions for T and T from they'th p g level were used as a set of initial values to calculate the coefficients B, C , D and E in equation (2.32), allowing new T and T values for the (/'+l)th level to be obtained. The p g solution is then iterated in this manner until a preset convergence criterion is reached. In the calculations below, the convergence criteria are set to be (T max g,new T —T g,old g,new ^ < 10" and max 6 J p,new T p,new p,old £10 . (2.34) J 2.4. Predictions for a Typical Case 2.4.1. Example Description Consider a cylindricalriserof inside diameter 0.152 m cooled by water. To allow the solution to be marched from z = 0, the outlet water temperature rather than the inlet temperature is assumed to be fixed at a known value (80 °C). Physical properties of water and gas such as density, thermal conductivity, Prandtl number, heat capacity, etc. are functions of their temperatures, obtained by fitting standard property data. Other key 38 parameters are typical of those encountered in pilot-scale circulating fluidized bed combustors and are listed in Table 2.3. Table 2.3: Base parameters used in sample calculations below. Particle diameter 286 pm Particle emissivity 0.85 Particle heat capacity 840 kJ/kgK Particle density 2610 kg/m Particle thermal conductivity 1.9 W/mK Particle velocity in wall layer 1.2 m/s Suspension density 52.5 kg/m Wall layer thickness 10.5 mm Gas velocity in wall layer 0.4 m/s Gas gap thickness 77.1 um Bulk temperature 1076 K Bulk emissivity 0.99 Conductivity of wall 21 W/mK Wall surface emissivity 0.90 Riser inner diameter 0.152 m Thickness of wall 2.4 mm Particle average residence length 1.5 m Water-tube heat transfer coefficient 12270 W/m K 3 3 2 The particle concentration in the wall layer calculated from equation (2.18) is illustrated in Figure 2.4. The volumetric particle concentration is seen to be a maximum at the inner boundary of the gas gap and then to decrease continuously through the wall layer. The suspension absorption and scattering coefficients determined from equations (1.18) and (1.19) are shown in Figure 2.5. They also have maximum values at the gas gap boundary and decrease continuously through the wall layer since they are proportional to 39 the particle concentration. The thickness of the gas gap is too small to be seen in Figures 2.4 and 2.5. 2 4 6 8 Distance from wall, x (mm) Figure 2.4: Particle concentration distribution in wall layer for base case. c 400 absorption coefficent o co XI CO l_ 100 scattering^coefficient 0) >. CD 2 4 6 8 10 D i s t a n c e from wall, x (mm) Figure 2.5: Variation of suspension absorption and scattering coefficients in wall layer for base case. 40 2.4.2. Heat Flux Distribution The predicted heat flux distributions along the heat transfer surface are plotted in Figure 2.6. Both the conductive and radiative heat fluxes generally decrease with distance from the top of the surface due to particle cooling along the surface. At the top of the heat transfer surface, the conductive heat flux decreases quickly, because, when the particles and gas enter the wall layer, they are assumed to have the same temperatures as the CFB o -' ' V' / 11 3 Radiation E co CD X 2 L . i • i / Conduction 4 i ' i 1 ro o r- N 3| 1 • 1 0 . - 4 1 Ii 0i E < .! i i • i 1 2_ . i 50 . i . i . i . i , i . i .i 100 150 200 250 300 350 400 450 Heat flux (kW/m ) 2 Figure 2.6: Vertical variation of heat flux along the wall for base case conditions given in Table 2.3. core, a value substantially higher than the wall temperature. At z = 0, the wall temperature is fairly high (see Figure 2.9) but it decreases sharply with increasing z. This rapid decline in surface temperature causes a short-lived rise in radiative heat flux, which 41 is too small to be seen in Figure 2.6. 2.4.3. Particle and Gas Temperature Distributions Figure 2.7 shows the vertical profiles of particle and gas temperature at different distances from the wall up to the boundary between the wall layer and the core. The 1 - 1 / / ' / / / / - 1 / ' •* / / 1 1 / 1 1 l - ; .; [ ; / s / / / /' / ' / // // /' /' // I 1 J 1 i I F i // // /' /' Ii /i 2 1 N ! j - j - - 0 400 i ,i . , 500 1 . . 600 700 800 Temperature (K) II i 1 , 900 ill 1000 , 1100 Figure 2.7: Gas and particle axial temperature distributions for base case. Dashed lines: particle temperature; Solid lines: gas temperature. From left to right: x = 0.08, 0.24, 0.74, 1.13, 1.83, 2.94, 10.48 mm, respectively. particle and gas temperatures differ appreciably close to the wall, with the particles having a higher temperature because of their higher density and volumetric heat capacity. The predicted difference between the particle and gas temperatures are similar to those measured by Flamant et al. (1992). 42 2.4.4. Thermal Boundary Layer Thickness The thermal boundary layer thickness, defined as the distance from the wall to the point where the local temperature difference from the wall temperature is 99% of the temperature difference between the core and the wall, i.e. where T -T (2.35) ^ = 0.99 - 1 i, L 1 1 ' 1 i 1 i i i i 1 i > i i 1 _ VS. Gas thermal \\\ E ~ sz - boundary layer thickness • « 3 Ol <D - A E N Wall layer boundary— Particle thermal A boundary layer H " thickness \ • - • . 0 1 t . f . 2 i 3 . l 4 . I 5 l 6 i 7 i 8 r 9 i 10 11 12 Distance from wall, x (mm) Figure 2.8: Growth o f predicted particle and gas thermal boundary layer thicknesses for base case. is plotted as a function of elevation for particles and gas in Figure 2.8. As expected, the thermal boundary layer thickness grows continuously along the heat transfer surface. Note that the thicknesses of the thermal boundary layers are always significantly less than the total wall layer thickness. 43 2.4.5. Coolant and Wall Temperature Distribution The longitudinal distributions of coolant temperature, furnace-side wall surface temperature and bulk temperature are shown in Figure 2.9. The bulk suspension temperature is assumed to remain constant. When the moving gas and particles enter the heat transfer zone, they are at the temperature of the bulk, and because there is a thin stationary gas gap between them and the wall, the wall temperature is predicted to be -JTT TO — iCD 7 r •Furnace-side wall temperature Q_ E CD -.— ' E ro o o O N Bulk temperature- M 5 200 300 400 500 600 700 800 900 1000 1100 Temperature (K) Figure 2.9: Vertical variation of coolant, wall and bulk temperatures for base case. about 440 K at z = 0. This temperature decreases rapidly to about 390 K and then more slowly to about 360 K at z = 5 m. In practice, because of the steep axial temperature gradient near the lop, longitudinal conduction will likely be important in that region, reducing the maximum temperature. The coolant temperature does not change 44 significantly because of the high volumetric water flowrate assumed. 2.4.6. Radiation Flux Figure 2.10 shows the predicted core-to-wall and wall-to-core radiation flux distributions at 0, 0.3 and 5 m below the top of the heat transfer surface. The core-to-wall ('-' direction) radiation fluxes always exceed the wall-to-core ('+' direction) values at the same level with the differences being the net radiation fluxes through the wall layer, as shown in Figure 2.11. The net radiation flux distribution curves are. predicted to have *i 80 • r 70 <c-60 E §: 50 <§*= 40 .2 30 "-" direction at z = 0.0 m "-" direction at z = 0.3 m "-" direction at z = 5.0 m -4—* CO t "+" direction at z = 0.0 m 20 "+" direction at z = 0.3 m "+" direction at z = 5.0 m 10 o 2 4 6 8 Distance from wall, x (mm) 10 Figure 2.10: Lateral variation of radiation flux for base case, maxima for z > 0.3 m. Since the particle temperature is higher in the wall layer near the core region, more energy is emitted than attenuated by these particles so that the radiation flux increases towards the wall layer. However, the particles near the wall are cooled by 45 the wall, causing more heat to be attenuated than emitted by these particles, and the radiation flux therefore decreases. The lower the elevation, the thicker the thermal boundary layer, and the farther the peak from the wall. The net radiation fluxes in the core-side wall layer are almost zero because the radiation fluxes in both directions are nearly the same there. Figure 2.11 also shows the lateral irradiance distribution. The irradiances, which in this case are the summations of radiation fluxes in both directions, decrease along the heat transfer surface since the particles are losing heat. Both the net radiation flux and the irradiance are seen to change significantly in the wall layer. Hence the radiation cannot be decoupled from the conduction/convection process. 160 ,—i 1 1 1 1 1 1 1 1 1 1 1 Distance from wall, x (mm) Figure 2.11: Lateral variations of net radiation flux and irradiance for base case. 46 2.4.7. Heat Transfer Coefficient The variations of suspension-to-coolant heat transfer coefficients along the heat transfer surface are plotted in Figure 2.12. The profiles are similar to the heat flux profiles illustrated in Figure 2.6. The conduction contribution decreases sharply at the top and then becomes almost constant for z > 1.5 m. The radiation contribution decreases with distancefromthe top of the surface as well. • 5 1 i • I 1 1 • 0 / / / 4 1 - 3 H \ E 1 12 1 2 - 2 —3 ! - 3 1: Radiation 2: Conduction 1 3: Total i! 0 0 4 1 1 1 r i 100 200 300 400 500 5 600 Suspension-to-coolant heat transfer coefficient (W/m K) 2 Figure 2.12: Vertical variation o f local suspension-to-coolant heat transfer coefficient for base case. 47 2.5. Comparison of Model Predictions with Experimental Results 2.5.1. Comparison with Data of Pagliuso et al. (2000) Pagliuso et al. (2000) reported experimental local bed-to-wall heat transfer coefficients for temperatures at which radiation is unimportant. The riser was 72.5 mm in internal diameter, 6.0 m high, with six double pipe-annular, water-cooled heat exchangers, each 0.93 m high, located one above the other. Five narrow size fractions of quartz sand particles -d = 179, 230, 385, 460 and 545 um - were tested. The suspension p temperature was kept approximately constant at 423 K while the superficial gas velocity was 10.5 m/s. Water and gas-solid suspension temperatures were measured at the inlet and outlet of each jacketed section. Pressure drops were also recorded using U-tube manometers to determine the suspension density. The authors found a significant effect of particle size on heat transfer coefficient. The measured local suspension densities with a cubic spline interpolation are used as inputs for model simulations. Figures 2.13 to 2.15 show experimental and predicted local bed-to-wall heat transfer coefficients and corresponding suspension densities for different operating conditions. The model overestimates the heat transfer coefficients in the top meter or so especially for dilute conditions and small particles sizes, while it underpredicts them in some cases at the bottom. Elsewhere, the model does well. However, the model assumes that the particles in the wall layer are always descending. For the relatively high superficial gas velocity of 10.5 m/s, the particles may not be flowing directly downward in the wall layer, but fluctuating upward and downward along the wall, or even all rising when the 48 Suspension density (kg/m ) 3 6 0 20 40 60 80 100120140 * 1 r - + o 5 + o E~ CD '55 X 0 20 40 60 80 100120140 0 20 40 60 80 100120140 0 20 40 60 80 100120140 0 20 40 60 80 100120140 •— 4 r. o J 0 0 1 \ + o + ( 20 40 60 80 100120140 Heat transfer coefficient (W7m K) 2 Suspension density (kg/m ) 0 0 20 40 60 80 100120140 20 40 60 80 100120140 6 0 20 40 60 80 100120140 0 20 40 60 80 100120140 r 0 20 40 60 80 100120140 ,2 0 20 40 60 80 100120140 Heat transfer coefficient (W/m K) L Figure 2.13: Local suspension density as a function of elevation and comparison of predicted local heat transfer coefficient with experimental results of Pagliuso et al. (2000) for d = 179 um. Average suspension .density from A to F: 11,21, 22, 41, 48 and 53 kg/m . (+: suspension density; O: experimental heat transfer coefficient; Solid lines: predicted local heat transfer coefficients.) p 3 49 Suspension density (kg/m ) 3 6 0 20 40 60 80 100120140 +o 0 20 40 60 80 100120140 B + c A 0 20 40 60 80 100120140 5 If + o 4 r X { + c t|c 1 1 + c + o + < + ( + - 0 20 40 60 80 100120140 0 20 40 60 80 100120140 0 20 40 60 80 100120140 ,2. K) Heat transfer coefficient (W/m Suspension density (kg/m ) 3 0 0 20 40 60 80 100120140 20 40 60 80 100120140 0 20 40 60 80 100120140 6 0 20 40 60 80 100120140 0 20 40 60 80 100120140 r 0 20 40 60 80 100120140 Heat transfer coefficient (W/m K) Figure 2.14: Local suspension density as a function of elevation and comparison of predicted local heat transfer coefficients with experimental results of Pagliuso et al. (2000) for d = 230 um. Average suspension density from A to F: 10, 20, 28, 38, 46 and 33 kg/m . (+: suspension density; O: experimental heat transfer coefficient; Solid lines: predicted local heat transfer coefficients.) p 3 50 Suspension density (kg/m ) 3 0 20 40 60 80 100120140 0 20 40 60 80 100120140 +o ( +o 0 20 40 60 80 100120140 0 20 40 60 80 100120140 0 20 40 60 80 100120140 0 20 40 80 100120140 B 1 +0 1 1 +o + <! 0 20 40 60 80 100120140 0 20 40 60 80 100120140 Heat transfer coefficient (W/m K) 2 Suspension density (kg/m ) 3 0 20 40 60 HO 80 100120140 0 20 40 60 80 100120140 *~ E D fi lt •4—» '<D I / I 2 \ + 1 0 0 20 40 60 80 100120140 0 20 40 60 80 100120140 60 Heat transfer coefficient (W/m2 LK) Figure 2.15: Local suspension density as a function of elevation and comparison of predicted local heat transfer coefficients with experimental results of Pagliuso et al. (2000) for d = 460 u.m. Average suspension density from A to F: 10, 18, 26, 48, 56 and 62 kg/m . (+: suspension density; O: experimental heat transfer coefficient; Solid lines: predicted local heat transfer coefficients.) p 3 51 T 1 1 1 1 ' r S u s p e n s i o n density (kg/m ) 3 Figure 2.16: Predicted average heat transfer coefficients (open points) compared with experimental data (solid points) of Pagliuso et al. (2000). A: dp = 179 pm; B: dp = 230 um; C: dp = 460 um. bed is too dilute. The authors also plotted the heat transfer coefficients as a function of suspension density and particle size. Figure 2.16 plots the experimental and predicted heat transfer 52 coefficient as a function of suspension density for d = 179, 230 and 460 pm. Except for p some outlier points that are in poor agreement with the predictions near the top of the heat exchanger, the agreement is good. 2.5.2. Comparison with Data of Furchi et al. (1988) Furchi et al. (1988) reported experimental results from the same CFB facility as Pagliuso et al. (2000) for temperatures up to 250 °C. The particles were glass spheres of average diameter 109, 196 and 269 pm. The superficial gas velocity ranged from 5.8 to 12.8 m/s, and the particle circulation flux from 0 to 80 kg/m s. 2 Figures 2.17 shows the experimental and predicted local bed-to-wall heat transfer coefficients for different operating conditions. The model gives very good predictions for d = 196 pm, while for the 109 pm particles, the predictions tend to be low. If the particle p average residence length is changedfrom1.5 m to 0.3 m in the model for d = 109 pm, a p better match is obtained between the predictions and the experimental results. The authors also measured the average heat transfer coefficients for the first water jacket (vertical interval from 0 to 1 m) at different suspension densities. The model predictions and experimental results are compared in Figure 2.18. It can be seen that the measured heat transfer coefficient for d = 109 pm is much higher than for d = 196 pm, p p while the latter is only a little higher than for d = 296 pm. The model cannot predict the p high heat transfer for d = 109 pm unless the average particle residence length is reduced p to 0.3 m. The experimental work of Pagliuso et al. (2000) on the same column did not 53 Suspension density (kg/m ) 0 25 50 Suspension density (kg/m ) 0 25 50 150 200 Heat transfer coefficient (W/m K) Heat transfer coefficient (W/m K) 2 Suspension density (kg/m ) 0 25 50 Suspension density (kg/m ) 0 25 50 75 100 150 150 Heat transfer coefficient (W/m K) 2 Heat transfer coefficient (W/m K) 2 Figure 2.17: Suspension density as a function of elevation and comparison of predicted local heat transfer coefficients with experimental data of Furchi et al. (1988). A: d = 109 um, U = 5.8 m/s, GJG = 11.3; B: d = 109 um, U = 8.9 m/s, GJG = 7.5; C: d = 196 um, U = 6.8 m/s, GJG = 13.4; D: d„ = 196 um, U = 8.9 m/s, Gs/Gg = 3.9. +: suspension density; Solid circle: heat transfer coefficient. Solid line: predicted local heat transfer coefficient for L = 1.5 m; Dashed line: predicted local heat transfer coefficient for L = 0.3 m. p g g p g g p g g g ar ar 54 200 - i — • — i — • — i — • — i — 1 — i — 1 — i — • — i — — r 1 Experiments; L =1.5m; L =0.3 m a CM E c CD O CD O O 160 U cf =109um • —•— d = 196 urn o —•— d= v -*w— p p 296 urn p 8 —a— D • v 120 80 CO c CD CO Cl) 40 v o 10 15 20 25 30 Suspension density (kg/m ) 35 40 45 3 Figure 2.18: Comparison of predicted average heat transfer coefficients (lines) with experimental data of Furchi et al. (1988) (points). show this unexpected effect of particle size on the heat transfer coefficient. The effect may have arisen because for Group A particles the average residence length decreases with decreasing particle size as indicated in equation (2.30), while a constant particle average residence length is used in the model. 2.5.3. Comparison with Data of Han et al. (1996) and Han (1992) Han et al. (1996) determined the thermal performance of a CFB heat exchanger operating with vertical up-flow of a hot gas loaded with solid particles. Their facility consisted of a combustion chamber, a cylindrical heat transfer test section, and a solids recycle and feeding system. The test section was a 50-mm-ID tube inside a 75-mm-ID 55 shell. The gas-solid suspension flowed upward through the tube, while cooling water flowed downward through the shell side. The heat exchanger section was instrumented with thermocouples to measure suspension and water temperatures at the top and bottom of the test section. In the experiments, the suspension temperatures varied from 100 to 600 °C, while the inlet gas superficial velocity ranged from 1.5 to 13 m/s. Particulate materials were FCC (mean diameter 88 and 117 pm) and sand (mean diameter 136, 157, and 264 pm). 140 CM £ 120 |100 a) o • 80 it= <D O O Cfl C CD 60 40 O d = 88 nm • of = 117^m p CO CD 20 p X 10 20 30 40 S u s p e n s i o n d e n s i t y (kg/m ) 3 Figure 2.19: Comparison of predicted average heat transfer coefficients (open points) with experimental data of Han et al. (1996) for FCC particles (solid points). 56 100 o 80 A 0 •• 60 40 20 E c 0 "o <D O O t_ <D 0 1 ' 1 1 1 1 • 1 I 1 100 1 o 80 o 60 • 8 • 40 *+— V) c 03 20 0 i i 1 1 i 1 i | i i 1 100 * c 80 60 40 20 0 • 1 1 1 10 20 L. 30 40 S u s p e n s i o n density (kg/m ) 3 Figure 2.20: Comparison of predicted average heat transfer coefficients (open points) witli experimental data of Han et al. (1996) for sand particles (solid points). A: dp = 136 pm; B: dp= 157 um; C: dp = 264 pm. Figures 2.19 and 2.20 compare experimental average total heat transfer coefficients and model predictions. The model gives better predictions for low suspension densities than for higher ones. For higher suspension densities, the model underestimates the heat 57 transfer coefficients for FCC and overestimates it for sand particles. Note that the model always assumes a descending wall layer, while at suspension densities as low as 10 kg/m , the particles in the vicinity of wall may oscillate upward and downward, or even 3 travel upward on average, in which case the assumed core-annulus structure ceases to exist. E c cu 'o <L> O O cz (0 TO <d JZ c o •o ro Suspension density (kg/m ) Figure 2.21: Comparison o f predicted radiative heat transfer coefficient (open points) as a function o f suspension density with experimental data o f Han (1992) (solid points) for 137um sand. A radiometer probe was also employed in the experiments of Han (1992) to measure the radiant heat flux from the suspension to the wall. This probe consisted of a brass body, zinc selenide window and a heat flux transducer. As shown in Figure 2.21 the 58 predicted radiation heat transfer coefficients are lower than the experimental data. Moreover, the experimental coefficients tend to increase with increasing suspension density while the model predicts much less variation. The probable reason is that the model assumes a wall layer between the wall and the core, while in the experiments, the radiometer probe was not flush with the round riser wall causing the probe to be less shielded by the intervening particles. Figure 2.22 shows the influence of temperature on the radiation heat transfer coefficient. Again, the model underestimates the experimental results. E co o it cd O O co cz cd cd cd sz c o co T3 co or 300 600 400 B e d temperature (°C) Figure 2.22: Comparison o f predicted radiative heat transfer coefficient (open points) as a function o f bed temperature with experimental data o f Han (1992) (solid points) for 137 um sand. (Suspension density: 10 kg/m ) 3 59 2.5.4. Comparison with Data of Luan etal. (1999) Luan et al. (1999) utilized a multifunctional probe combining the differential emissivity and window methods to measure not only the radiation heat transfer coefficient, but also the total heat transfer coefficient. The probe consisted of a stainless steel body containing four stainless steel cylinders, each surrounded by a concentric stainless steel sleeve separating the cylinder and probe body by thin air gaps. The probe was located 2.13 m below the top of a 152 mm x 152 mm x 7.3 m tall CFB combustion riser. Silica particles having mean diameters of 286 and 334 pm were the bed materials. To determine the average probe heat transfer coefficient in the model simulation, a symmetric half of the probe cylindrical surface is divided into 5 vertical strips as shown in Figure 2.23. In each section, the wall layer is assumed to travel along the average distance indicated by the dashed lines. The average heat transfer coefficients for each section are then weighted by the area of each section to obtain the total average heat transfer coefficient for the whole surface. Figure 2.23: Schematic showing integration intervals used to determine the average heat transfer coefficient. 60 800 800 T„ = 407 + 4 °C 600 CM E 600 400 •• • • • 400 200 200 c CD o u— w— CD o o _—ii-jki—_ 0 10 20 30 40 * . . . 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 40 50 60 70 80 90 Suspension density (kg/m ) CD **— 0) c: 03 03 CD X 10 20 30 40 50 60 70 80 90 10 20 30 Suspension density (kg/m ) Figure 2.24: Comparison of predicted total and radiative heat transfer coefficients with experimental data of Luan et al. (1999). • : total; V: radiative by window method; A: radiative by differential emissivity method; *: predicted total; +: predicted radiative. Figure 2.24 shows the measured total heat transfer coefficients, radiative heat transfer coefficients by the window and differential emissivity methods, and the model-predicted total and radiative heat transfer coefficients. For this short heat transfer probe, the model substantially underpredicts the total heat transfer coefficients, while the predictions for the radiative coefficients are quite close. Note that in the experimental measurements, the probe was assumed to be perfectly insulated such that conduction took place in one 61 dimension only; in reality this assumption is doubtful. At least part of the discrepancy is likely due to this factor. Another possible cause is that the model does not consider particle movement in the tangential y-direction; while for heat transfer surfaces as small as this probe, such exchanges may be significant. 2.5.5. Comparison with Data of Ahmad et al. (1996) Ahmad et al. (1996) presented results of an experimental investigation into the effect of localized cooling on heat transfer in a circulating fluidized bed. The CFB facility consisted of a 6.2 m tall, 0.161 m ID riser with two cyclones, an external heat exchanger and an L-valve, with silica sand of density 2500 kg/m and a mean particle diameter of 3 100 pm. The suspension-to-wall heat transfer coefficient was measured with a 16 mm high water-jacketed probe (Wang, 2001), mounted 2 m above the air distributor plate and flush with the riser inner surface. Cooling was applied alternatively to two 0.3 m long sections of the riser, immediately above and below the heat transfer probe, by purging a mixture of compressed air and water vapor through a tube ring with thirty-two 4-mm diameter holes facing the riser wall. When the cooling was applied above the heat transfer unit, the heat transfer coefficient was found to decrease with decreasing temperature of the cooling section. This effect was more significant as the superficial gas velocity decreased. The effect of cooling was found to be negligible, however, when cooling was applied below the heat transfer probe. 62 240 E > o o • B: • C: 200 D: cz CD O CO A: • • -A-o-o- 160 cu in 120 • CD CU 80 _1 50 1 1_ 100. 150 200 250 300 350 400 450 500 T e m p e r a t u r e of c o o l i n g w a l l (°C) Figure 2.25: Comparison of predicted heat transfer coefficients (open points) with experimental data of Ahmad et al. (1997) (solid points). A: p =\6 kg/m , T = 373 °C, U = 3.0 m/s; B: p = 16 kg/m , T = 361 °C, U = 3.5 m/s; C: p us = 14 kg/m , T = 485 °C, U = 4.8 m/s; D: = 12.5 kg/m , T = 478 °C, U = 5.5 m/s. 3 3 sus b g sus 3 S b g 3 b g b g When the model is applied to this heat transfer process, the particles and gas are first assumed to travel downward along a 0.3 m long section of wall with a constant surface temperature. The particles and gas then meet the water-jacketed probe with an assumed constant water temperature. Figure 2.25 shows the effect of upper localized cooling on the measured heat transfer coefficients and model predictions for different operating conditions. The model predicts a much higher effect of upper cooling on the heat transfer coefficient than was measured. The author also reported the measured suspension temperature 5 mm from riser wall for operating condition A listed in Figure 2.25. At this position, the suspension temperature decreased with decreasing temperature of the cooling section above the heat flux probe. The wall layer thickness for this operating 63 condition is calculated to be 5.4 mm from equation (2.28). Thus in the model, the temperatures of the particles and gas are assumed to be at the bulk temperature at x = 5.4 mm, while in reality, they are much lower than the bulk temperature. This may explain the significant discrepancy between the measured and predicted results. 2.5.6. Comparison with Data of Tan et al. (1994) Tan et al. (1994) measured local heat transfer coefficients in a 4 m high, 175 mm x 175 mm cross-sectional area riser by means of a 320 mm high heat transfer probe. The probe consisted offive64 mm x 14 mm independent but adjacent heating and heat flux sensing units located 1.06 m, then 2.75 m, above the gas distributor. Tests were carried out on ferrosilica particles of density 6700 kg/m and mean diameter 110 um, at local 3 suspension densities of 0 to 100 kg/m . 3 Figure 2.26 compares the model predictions and experimental results. The model underestimates the rate of heat transfer over the lower part of the probe. This may be because the particles in the wall layer oscillate upward and downward, while the model assumes a unidirectional downflow. This point is discussed further in Chapter 3 below. 64 Heat transfer coefficient (W/m K) 2 Figure 2.26: Comparison of predicted local heat transfer coefficients (solid lines) with experimental data of Tan et al. (1994). +: 1.06 m above distributor; x : 2.75 m above distributor. A: p = 20 kg/m ; B: p = 40 kg/m ; C: p = 80 kg/m . 3 sus 2.6. Sensitivity The previous 3 sus 3 m Analysis section shows that the two-dimensional model gives reasonable predictions in most cases. This model will now be used to investigate the effect of various parameters on the heat transfer process. Numerous variables play roles in this process. In the sections below, several of the most important and uncertain ones are varied parametrically to observe their influence on heat transfer. In each study, the base case predictions of the model are indicated by solid lines in the respective diagram. Where experimental evidence on the influence of each variable is available, this is noted. 65 2.6.1. Influence of Gas Gap Thickness Figure 2.27 shows the influence of gas gap thickness on the conduction and radiation contributions to heat transfer. The gas gap thickness has a significant effect on the conduction term, but little influence on the radiation term. The thinner the gap thickness, the smaller the thermal resistance on the furnace side, and the more heat is conducted to the wall. Hence the wall temperature becomes higher, causing the radiation from the suspension to the wall to decrease. 5.0 0.0 0.5 E 1.0 1: S = 2.86 nm g 2: S = 28.6 nm s 3: S = 77 A nm B 4: 3.0 20 40 60 80 100 120 140 = 143 nm 160 180 1.5 2.0 200 H e a t flux ( k W / m ) 2 Figure 2.27: Influence o f gas gap thickness on vertical heat flux profile. (Base conditions, Tabic 2.3, except for gas gap thickness.) 66 ^ 2.6.2. Influence of Particle Diameter As shown in Figure 2.28, the conduction heat flux at the wall increases as the particle diameter decreases, while the radiation flux decreases. The influence of the particle diameter on conduction transfer is greatest near the top of the heat transfer surface. The particle diameter influences the heat transfer process in three ways: ~i—•—i— —i—•—r 1 5.0 Radiation 0.0 S r1f 4.5 .2> 0.5 4.0 1.0 <D 1:d„ = 80 nm 150 3:4.= 3.5 4 : d \im 286 nm 1.5 „ = 400 urn 3.0 0 20 40 60 80 100 120 140 160 180 200 2.0 220 Heat flux (kW/m ) 2 Figure 2.28: Influence of particle diameter on vertical heat flux profde. (Base conditions, Table 2.3, except for particle diameter.) 1. via its influence on the gas gap thickness. The smaller the particles, the thinner the gas gap (since S oc d), and hence the lower the gas gap conduction resistance; g 2. via its influence on heat convection between the gas and particles. Because finer particles have larger total surface areas (for the same voidage), more heat is convected 67 from finer particles to the gas, and hence by conduction from the gas to the wall; 3. via its influence on the suspension absorption and scattering coefficients. From equations (1.18) and (1.19), the suspension absorption and scattering coefficients are proportional to \ld . Finer particles (for the same voidage) function as a denser curtain p between the high temperature core and the wall, thus decreasing the net radiation flux. Hence, it can be expected that the total heat flux will increase with decreasing particle size at low bed temperatures, where radiation is not significant. At high suspension temperatures, the situation is more complex since the particle diameter affects the convection and radiation heat transfer processes in opposite directions. 2.6.3. Influence of Suspension Density It is commonly recognized that CFB heat transfer is strongly correlated with the overall suspension density. A higher suspension density results in a thicker wall layer and more particles accumulated in it. The additional particles also carry more energy from the core to the wall layer, causing more heat to be transferred from the bulk to the wall. On the other hand, a thicker wall layer and higher particle concentration augments the radiation resistance between the bulk and wall, thereby decreasing the radiation contribution. The conduction and radiation heat fluxes calculated for different suspension densities are plotted in Figure 2.29. The higher the suspension density, the smaller the relative contribution of radiation flux, not only because the radiation component is lower, but also because of the increased contribution of heat conduction. The predicted influence of 68 suspension density on conduction heat flux is consistent with experimental results (e.g., Grace, 1986; Glicksman, 1988; Divilio and Boyd, 1994; Werdermann and Werther, 1994). T 1 I 1 I ' I • I 1 I ' I ' I ' 1 ' 1 r H e a t flux ( k W / m ) 2 Figure 2.29: Influence of suspension density on vertical heat flux profde. (Base conditions, Table 2.3, except for suspension density.) 2.6.4. Influence of Wall Layer Thickness For the base case, the wall layer thickness calculated from Equation (2.28) is 10.5 mm. Simulation results for wall layer thicknesses of 5 and 15 mm show no observable difference from the base case so long as the wall layer is thicker than the particle and gas thermal boundary layers. 69 2.6.5. Influence of Particle Velocity Figure 2.30 shows the influence of the velocity of the descending particles in the wall layer on the heat flux. The higher the particle velocity, the higher the heat flux although the effect is limited. The particle velocity affects the convective term in equation (2.3) as well as the particle-to-gas transfer term (Q ) through equation (2.22). Since in equation pg (2.3) the particle velocity u acts together with particle concentration c, it is expected at p high-density operating conditions that the influence of u would be augmented by the p high particle concentration. This point is considered in Chapter 3. 0.0 A 0.5 N 1.0 1.5 80 100 120 140 2.0 Heat flux (kW/m ) 2 Figure 2.30: Influence o f particle velocity in wall layer on vertical heat flux profile. (Base conditions, Table 2.3, except for particle velocity.) 70 2.6.6. Influence of Gas Velocity The gas velocity affects the heat transfer through both the first convective term and the final term in equation (2.1). As shown in Figure 2.31, the conduction heat flux decreases as the gas velocity increases, while the radiation flux increases, although the influence is small. The density and heat capacity of the gas are not of the same order as those of the particles. Hence the gas velocity has less influence than the particle velocity. Figure 2.31: Influence of gas velocity in wall layer on vertical heat flux profile. (Base conditions, Table 2.3, except for gas velocity.) 71 2.6.7. Influence of Bulk Temperature Heat flux increases with increasing bulk temperature due to three factors: an increase in radiation, a larger driving force for conduction, and increased thermal conductivity of the gas. Figure 2.32 shows that the increase in the radiation term is higher than the increase in the conduction component for the conditions investigated, consistent with experimental results (e.g., Kobro and Brereton, 1986; Andersson et al., 1987; Wu et al. 1989; Andersson and Leckner, 1992; Luan et al. 1999). Heat flux (kW/m ) 2 Figure 2.32: Influence of bulk temperature on vertical heat flux profile. (Base conditions, Table 2.3, except for bulk temperature.) 72 2.6.8. Influence of Particle Emissivity Higher particle emissivity results in higher suspension absorption, emission and scattering coefficients. Figure 2.33 shows that higher particle emissivity enhances the radiation flux, whereas there is no significant influence on the conduction flux. 5.0 0.0 Radiation. I 4.5 0.5 4.0 1.0 E, N 1:e = 0.95 CD p X 2 : e = P 3: e = 0.45 3.5 3.0 0.85 A 1-5 I 20 2.0 140 *—i—L Heat flux (kW/m ) 2 Figure 2.33: Influence of particle emissivity on vertical profile of heat flux. (Base conditions, Table 2.3, except for particle emissivity.) 73 2.6.9. Influence of Particle Volumetric Heat Capacity (p C ) p pp Figure 2.34 shows the predicted influence of the particle volumetric heat capacity. Particles of higher volumetric heat capacity carry more heat, resulting in a higher conduction heat flux. Higher volumetric heat capacity also keeps the particles at higher temperatures for longer traveling distances, resulting in a higher radiation flux. 5.0 4.5 E N CD X Heat flux (kW/m ) Figure 2.34: Influence of particle volumetric heat capacity on vertical heat flux profde. (Base conditions, Table 2.3, except for particle volumetric heat capacity.) 74 2.6.10. Influence of Particle Average Residence Length (L ) ar A longer particle average residence length causes particles to be cooler in the wall layer, resulting in lower conduction and radiation fluxes. Simulation results for L = 0.5, a r 1.5, 3.0 m, as shown in Figure 2.35, verify these trends. 0.0 0.5 1.0 NI 1.5 120 2.0 Heat flux (kW/m ) Figure 2.35: Influence of particle residence length on vertical heat flux profde. (Base conditions, Table 2.3, except for average particle residence length.) 2.6.11. Influence of Wall Thermal Resistance (LJkw) and Water-side Heat Transfer Coefficient A novel feature of the model presented in this chapter is the coupling of reactor-side and wall-side heat transfer. This allows the influence of wall-side thermal resistance on 75 the total heat transfer process to be investigated. However, the wall-side resistance is so small compared to that of the reactor-side for the base conditions that the differences between simulation results are insignificant, even when the wall-side resistance is changed to half and two times its original value (1.2 x 10" m K/W). The corresponding 3 2 water-side heat transfer coefficient for the base case conditions is about 12,270 W/m K. As seen from the simulation, the total suspension-to-water heat transfer coefficient varies between 300 and 80 W/m K. The reactor side heat transfer process dominates the overall 2 process for the typical conditions considered in this chapter. Simulation results show that decreasing the water-side heat transfer coefficients does not change the total suspensionto-water heat transfer coefficients significantly until it is of order 1000 W/m K. Figure 2 2.36 shows this trend. 0.0 5.0 Radiation 4.5 0.5 4.0 1.0 N V.h = 12270 W/m K X 2 c 2:h = 1000 W/m K 2 c 3.5 - 3 : h = 500 W/m K 2 e 3.0 120 20 140 1.5 2.0 Heat flux (kW/m ) 2 Figure 2.36: Influence of water-side heat transfer coefficient on vertical heat flux profile. (Base conditions, Table 2.3, except for water-side heat transfer coefficient.) 76 2.6.12. Summary of Influence of Various Parameters on Heat Flux The predicted influences on the heat flux of the parameters investigated are summarized in Table 2.4. The upward and downward arrows show the direction of the influence, while the number of arrows roughly indicates the extent of the influence. A dash in the table means that the influence is negligible. It should be noted that all these trends are based on the standard operating conditions described in section 2.4.1. They might be different if the base operating conditions were changed significantly. 77 Table 2.4: Summary of predicted parameter influences on heat flux for base case conditions listed in Table 2.3. Parameter and change direction Radiative heat flux (q ) Conductive heat flux (q ) r Comment c Gas gap thickness t t u Particle diameter t ttt U Suspension density t u ttttt Particle velocity t t tt Gas velocity t t Bulk temperature t Particle emissivity t tt - Particle volumetric heat capacity t tt ttt Smaller effect at top Particle average residence length t ttt ttt Smaller effect at top Wall layer thickness t - - Wall-side thermal resistance t - - Water-side heat transfer coefficient t - - Higher effect on q at top c Higher effect at bottom tttt Not significant until below 1000 W/m K 2 78 2.7. Summary A new two-dimensional model that couples gas conduction, particle-to-gas convection, radiation through the particle layer, conduction through the wall, and convection on the coolant side is proposed for heat transfer in circulating fluidized beds. The two-flux model is adopted to represent the radiation transfer in the wall layer. Keller's box method is employed in obtaining numerical solutions of the set of non-linear, partial differential governing equations. The sensitivity of the heat transfer process to changes of various parameters is investigated. The model predictions are compared with experimental results from the literature and predictions of the suspension-to-wall heat transfer rate are generally satisfactory. The predicted influences of different parameters on the heat flux are also consistent with experimental trends where these are known. The model predicts that both the conduction heat flux and the radiation flux decrease as particles descend along the heat transfer surface for constant suspension density. The simulation results suggest that the particles participate in a significant way in determining the radiation flux through the wall layer. Therefore radiation cannot be uncoupled from particle and gas conduction/convection without introducing significant error for high temperature systems. 79 CHAPTER HEAT TRANSFER IN A H I G H - D E N S I T Y FLUIDIZED 3.1. 3 CIRCULATING B E D Introduction Circulating fluidized bed risers have been investigated extensively for the past two decades because of their practical applications, as well as their intrinsic interest. However, the overwhelming majority of such work has been conducted at net solids fluxes, G , less than 100 kg/m s, and at superficial gas velocities, U , between about 2 2 s g and 8 m/s. For these conditions, the overall volumetric solids concentrations, c, is less than about 0.1 (Zhu and Bi, 1995). While these conditions are relevant to CFB combustion, much higher solids fluxes and holdups are encountered in CFB risers used for solid catalyzed reactions like fluid catalytic cracking and production of maleic anhydride. In such cases, G is commonly 300 to 1200 kg/m s, with corresponding c 2 s ranging from 0.1 to 0.25. Grace et al. (1999) defined a flow regime of dense suspension upflow as an operation with G > 200 kg/m s, c > 0.10 and solids upflow on average 2 s throughout the entire riser. Their studies demonstrate that such operations differ in several important respects from low-density circulating fluidized bed systems. Some of the key observations concerning the behavior of gas-solid suspensions in risers under high-density conditions are as follows (Grace et al., 1999): 80 1. Net downflow of particles at the wall, a commonly observed feature of fast fluidized beds, is absent. Instead, solids move upward throughout the entire riser cross-section. 2. While there is no downflow at the wall, there are still considerable radial gradients in particle density, with higher particle concentrations near the wall than in the interior of the riser. While there also do not appear to be clusters, there are certainly substantial fluctuations in local voidage. 3. In view of (1), there is reduced segregation of particles by size, and a closer approach to plug flow of both gas and solids than in the fast fluidization flow regime. 4. Axial profdes of solids concentration become relatively flat, with solids volumetric concentrations, averaged over the cross-section, ranging from about 0.1 to 0.25. 5. Both statistical and chaotic properties of pressure and local voidage fluctuations show that the behavior of high-density beds differs markedly from that found when the same particles are fluidized under standard gassolids flow regimes in the same column, even when compared at locations where the time-mean voidages are equal. While numerous experiments have been carried out to investigate the heat transfer process in circulating fluidized beds, almost none of these apply to the high-density conditions defined above. CFB bed-to-wall heat transfer is strongly influenced by the 81 flow pattern in the riser, especially the particle motion in the vicinity of the wall. Experimental work is needed to elucidate the heat transfer behavior in the high-density flow regime and to modify the model developed in Chapter 2 for low-density operating conditions. In this chapter, experiments carried in a dual-loop high-density CFB facility are presented. The temperature distribution in the suspension is measured together with local and average bed-to-wall heat transfer coefficients. Based on the experimental information, the model developed in Chapter 2 is extended to cover both low-density and high-density operating conditions. 3.2. Experimental Facilities 3.2.1. HDCFB System The high-density circulating fluidized system is located at the University of British Columbia. A schematic of the major components is shown in Figure 3.1. The dual-loop CFB unit consists of two Plexiglas risers, two PVC downcomers, a curved plate impingment separator, cyclones and an air filter baghouse. The first riser (1) has a diameter of 76.2 mm and a height of 6.10 m. At the top of this riser the suspension is directed by a 33.1 mm diameter nozzle inclined at 30° from the vertical onto a curved plate (2) installed in a 0.91 m x 0.46 m x 0.61 m rectangular box. From the impingement separator solids fall into the first downcomer, a 304.8 mm diameter 4.24 m tall column. Additional recovery is obtained by a cyclone (8a), which returns solids via a flapper valve. Solids from the storage tank are then fed via a 76.5 mm gate valve into the 82 I. I riser; 2. Impingement separator; 3. Storage tank; 4. 2 riser; 5. Downcomer; 6. Butterfly valve; 7. Pinch valve; 8. Cyclone; 9. Baghouse 10. Orifice meter; II. Rotameter; 12. Root's blower; 13. Dual-tube heat exchanger; 14. Steam-water heat exchanger. s1 nd Figure 3.1: Schematic diagram of U B C high-density C F B unit. 83 bottom of the second riser (4) of diameter of 101.6 mm and height of 9.14 m. A smooth exit bend at the top exit guides the entrained suspension into a primary cyclone (8b), a conventional cyclone except that it lacks a conical base. The solids removed by this cyclone and a secondary cyclone fall into the second downcomer (5), a 304.8 mm diameter, 8.33 m tall column, completing the loop via a second 76.2 mm gate valve (7) and a ball valve. The air superficial velocity in this downcomer was maintained at U f. m The circulation rate in the system can be determined by closing a porous butterfly valve (6) in the downcomer. Air from both secondary cyclones passes through a fdter baghouse before being discharged to the atmosphere. More details of this system are provided by Issangya (1998) and Liu (2001). 3.2.2 Heat Transfer Measurement Equipment The heat transfer system added to the above set-up by the author consists mainly of a concentric-tube heat exchanger (which replaced one of the riser 1 sections), a steamwater heat exchanger, a steam trap, four needle valves and a rotameter. Figure 3.2 shows the structure and dimensions of the brass concentric-tube heat exchanger. The inner tube, which forms a section of the first circulating fluidized bed riser, is 76.2 mm in inner diameter (the same diameter as the first CFB riser in Figure 3.1) and 3.2 mm thick, while the outer tube is 92.1 mm in inner diameter and 1.6 mm thick. Four K-type thermocouples are mounted on the outside surface of the inner tube to measure the wall surface temperatures. Four ports are provided on the outer tube where temperature probes can be inserted into the water flowing through the annular jacket. Water enters the bottom of the exchanger through four inlets, uniformly distributed at 90° intervals. It leaves the 84 exchanger from the top, again through four evenly distributed ports. The water flow rate can be carefully adjusted by four needle valves installed upstream of the water inlets. The entireriser1 including the heat exchanger and water inlet and outlet tubes are wrapped in fiberglass insulation, with the heat exchanger being especially well insulated. 3.2.3. Bed Material The particles used in all experiments were FCC particles of mean diameter 65 pm and density 1600 kg/m . These particles have a minimum fluidization velocity, U f, of 0.0032 3 m m/s in air at atmospheric temperature and pressure and a loose packed bed voidage, £„,/, of 0.45. Their size distribution determined by screening is given in Table 3.1. Table 3.1: Size distribution of FCC particles determined by screening Mesh size (pm) Mass fraction (%) 125-150 11.4 90-125 22.0 75-90 20.5 63-75 20.2 53-63 15.3 43-54 1.4 0-45 9.3 86 3.2.4. Measurement Techniques 1. Riser superficial gas velocity Superficial gas velocities were measured in both risers using orifice meters, fabricated to ASME standards, with three interchangeable stainless steel orifices having bore diameters of 41.1, 47.0 and 52.3 mm. The pressure drops over these orifices were measured by pressure transducers and recorded by a data acquisition system. 2. Solids circulation flux The solids circulation flux was measured by the butterfly valve installed in the upper part of the downcomer. During measurements, the two halves are rapidly rotated upward to the horizontal position, thus trapping the downflowing solids. The solids circulation rate is calculated from the time to accumulate a known volume of solids on top of the valve. 3. Cross-sectional average suspension density The cross-sectional area-averaged suspension density is estimated from the pressure gradient over the interval where the dual-tube heat transfer section is located as measured by pressure transducers (Omega PX 162). If solids acceleration and the effects of gaswall and solid-wall friction are neglected, the average suspension density is given by P - ~ ^ 0.«> - g Az 87 The shell of each thermocouple is cut off at its tip so that the junction of the thermocouple wires is exposed to the suspension in order to determine its precise radial position in the bed. 5. Wall surface temperature Four Omega KMQSS-010U-36 thermocouples of 0.254 mm outer diameter are glued into small cavities drilled on the water-side surface of the inner tube. The wires then pass through the exit ports through special compression fittings. 6. Water flowrate The water flowrate is adjusted by four needle valves which control the flow to the four water inlets. The flowrate is monitored by a rotameter. It is further measured by collecting the water in a container and determining with a stopwatch the time taken to fill a container of known volume. 7. Water temperature The water is heated to about 85 °C to provide a sufficient driving force for the heat transfer to the bed suspension. To maintain an accurately measurable temperature difference (at least several degrees Celsius) between the water inlet and outlet temperatures, the water flowrate had to be kept very low, usually 30 ml/s. The corresponding water velocity in the annular channel between the two tubes was then only 0.02 m/s, yielding a Reynolds number of only 277 (for annuli, the diameter term in the Reynolds number is the equivalent diameter 2(r? - /*/))• Hence the water flow is laminar, 89 and significant radial temperature differences within the 4.8 mm wide annular gap are expected. Hence it was impossible to determine the bulk water temperature with a single thermocouple. Instead, a radial temperature distribution had to be measured to allow calculation of the average water bulk temperature. Figure 3.4: Set-up for water temperature measurements. (All dimensions in mm.) Four probes of the type shown in Figure 3.4 were specially constructed to measure the radial water temperature distribution in the annular channel. Each probe consists of an acrylic shell and 4 pairs of 0.38 mm O.D. K-type thermocouple wires (TFAL-015 and TFCY-015). Each pair of wires passing through the inside of the tube is joined together on the outer surface of the shell, with the distance from the tip to the shell end precisely determined. When installed, each probe is inserted until it makes contact with the inner tube of the heat exchanger and it is then locked in place. The vertical positions where the probes were inserted into the water annulus are indicated in Figure 3.2. The water inlet and outlet temperatures are also measured by standard Omega KQIN-18U 90 thermocouples. At sections sufficiently far from the entrance and exit, the velocity distribution for steady laminar, incompressible flow through an annular space between two concentric tubes of circular cross-section (Knudsen and Katz, 1958) is -> r Figure 3.5: Schematic for calculation of average bulk temperature in a laminar flow through the annular space between two concentric tubes of circular cross-section u - 2u, 2 , 2 -> (3.2) 2 where r is the distance from the axis of the concentric tubes and .2 r max = ..2 (3.3) 2 ln(r / r ) 2 x 91 The geometry is illustrated in Figure 3.5. Assume the fully developed temperature profile in the channel at any axial position is given by (3.4) T = Qr' + C r> + C^ + C r + C, 1 2 A where C/, C , C3, C4 and C5 are functions of z. Since the outside surface of the heat 2 exchanger is insulated, the boundary condition at r = r is 2 8T_ dr =0 (3.5) or 4r C, + 3r C + 2r C + C = 0 3 (3.6) 2 2 2 2 2 2 A When four temperatures 7), T , T3, and TV at positions yi, y , yi and r are measured, C;, 2 2 2 C , C3, C4 and C$ can be calculatedfromthe set of linear algebraic equations: 2 = 0 y; y' y\ y\ 4 y\ yi 3 ^3 y Arl 3r y 3 yi 2 3 2 2 (3.7) y 2r 3 r 1 The bulk average temperature is given by: 92 ^Tulnrdr ^ulnrdr r -r -2r /ln(r /r) 1 p (c> + C r 4 2 + C r + C r + C )• u 3 2 2 3 A 5 ma 2 , ( 2 2 2 0 rdr (3.8) r ~r -2r /ln(r /r) r-. — f{C\ > C , C , C , C , ^, 2 3 4 5 2 2 2 ma , 2 r r) i 2 +*\ 2 2 2 - ^max 2 2 Equation (3.8) can be integrated once C/, C2, C3, C4 and C5 have been determined for a given set of probe temperature measurements. The entrance transition length before the laminar flow is fully developed can be calculated (Langhaar, 1942) from T= 0.058Rex 2(r -#;). (3.9) 2 69 • 1 1 1 1 • 1 • y W • D " - / - / c - _ • 64 - B • _ -A- - " 62 61 - - * 63 60 1 • / 66 65 • . 68 67 - 1 1 -/ - / - / 1 0 . 1 . 1 1 1 3 2 . 1 . 1 4 5 Distance from out surface of inner tube (mm) Figure 3.6 Temperature profiles measured at 4 locations in water channel. A: Z = 3.00 m; B: Z = 2.95 m; C: Z = 2.79 m; D: Z = 2.62 m. Operating conditions: Run A listed in Table 3.4. Dashed lines: temperature distributions obtained by fitting equation (3.4); Solid lines: average values. 93 Equation (3.9) leads to an entrance length of 126 mm for our heat exchanger. Hence only the thermocouple at Z = 2.62 m falls into the developing flow region. However, in the entrance region, the water velocity distribution is flatter than in the fully developed region. Thus the weighted average temperature from equation (3.8) will not introduce a significant error. Hence, equation (3.8) was employed for all four positions. Figure 3.6 shows an example of the measured temperature profiles obtained by the four probes at the four different levels in the water channel. The solid lines are the water bulk average temperatures calculated from equation (3.8). 3.3. Experimental Results and Discussion 3.3.1. Suspension Temperature Distribution The suspension temperatures are recorded at the sections above and below the heat exchanger ( 3.85, 3.25, 2.35, 2.05, 1.45, 1.15 and 0.84 m above the gas distributor) while the bed is being heated by the heat exchanger. Initially all the bed thermocouples are pressed flush with the other side of the riser wall. After these data are obtained, the thermocouples are pulled back stepwise to other lateral positions until the center is reached, with the temperature being recorded at each position for each thermocouple. Figure 3.7 shows the vertical and radial suspension temperature distributions for three different operating conditions at U = 6 m/s. To simplify the description, we use T (x, Z) g s to denote the suspension temperature at a certain location, with x representing the radial dimensionless coordinate rlR and Z denoting the height above the distributor. It can be 94 seen that for condition A (G = s 282 kg/m s, p 2 = 342 kg/m ), the radial temperatures are 3 sus quite flat at all these vertical levels, except at Z = 3.25 m, which is immediately above the water jacket. The suspension temperature at the wall is significantly higher than at the center. The difference between 7X1, 2.35) and T (0, 2.35) (just below the heat exchanger s section) is much smaller than between 7X1, 3.25) and Ts(0, 3.25) . Also 7X1, 2.35) is greater than 7X1, 2.05), while 7X1, 3.25) is greater than 7X1, 3.85). Since the heat exchanger extends from Z = 2.50 m to Z = 3.10 m, these trends imply that most particles in the vicinity of wall are traveling upwards, yet there appears to be a smallfractionof particles falling near the wall, or perhaps oscillating back and forth close to the wall. For operating condition B, it can be seen that 7X1, 2.35) > 7X1, 3.25), and, contrary to run A, [7X1, 2.35) - 7X0, 2.35)] is greater than [7X1, 3.25) - 7X0, 3.25)]. This indicates that most of the particles in the vicinity of the wall are descending, causing the temperature below the water jacket to be higher than the one above. On the other hand, 7X1, 3.25) remains higher than 7X1, 3.85), suggesting that there may also be some rising particles in the wall region. In run C with G = 26 kg/m s, p 2 s sus = 19 kg/m , 7X1, 2.35) and 7X1, 3.25) are almost 3 the same, while [7X1, 2.35) - 7X0, 2.35)] is a little greater than [7X1, 3.25) - 7X0, 3.25)]. This indicates that while the 'majority of the particles in the vicinity of wall are descending, a significant fraction is ascending. 95 rlR H e i g h t Z (m) -•-3.85 - O - 3.25 - A - 2 . 3 5 -O-1.00 -O-0.78 -V-2.05 - O - 1.45 -V-0.26 9 — | — 1.15 - X - 0 . 8 5 -. - -A-0.52 -O-o.oo * O I_ <u 2 7 • TO CU 26 V em o % a. I- 25 x- 0.0 0.2 0.4 0.6 0.8 1.0 24 25 26 Dimensionless coordinate r/R 0.0 0.2 0.4 0.6 0.8 27 28 29 Temperature (°C) 1.0 24 26 Dimensionless coordinate r/R . 28 30 32 Temperature (°C) A oo. / o — sz 3 Interval occupied by heat e x c h a n g e r o> CD 0.0 0.2 0.4 0.6 0.8 1.0 22 24 Dimensionless coordinate r/R 26 28 Temperature (°C) Figure 3.7: Radial and vertical bed temperature distribution for U = 6 m/s. A: G, = 282 kg/m s, p = 342 kg/m ; B: G = 213 kg/m s, = 136 kg/m ; C: G = 26 kg/m s, p = 19 kg/m . s 2 3 2 sus s 2 s 3 sm 96 3 P a a 30 Issangya (1998) measured local solids mass fluxes in the same column using sampling probes. Table 3.2 summarizes his measurements at r/R = 0.95 and at a height of Z= 2.8 m, which is at the center of our heat exchanger. It can be seen that in all his tests, there are always particles in the vicinity of wall moving both upward and downward. In the relatively dilute cases, more particles are descending; when the net circulation flux increases, however, more particles are traveling upwards near the wall. Table 3.2: Solids mass flux at r/R = 0.95 measured by Issangya (1998). G (kg/m s) 38 194 210 250 253 325 U (m/s) 4.5 4.5 4.5 7.5 7.0 7.0 52 NA 325 345 NA 440 -100 NA -255 -330 NA -120 -48 -110 +70 +15 +60 +320 2 s g up (kg/m s) 2 G down (kg/m s) 2 w net (kg/m s) 2 Gsy V Liu (2001) measured local particle velocities using a multifunctional probe in the same column. He found that when G > 500 kg/m s, particles are moving upward on a 2 s time-mean basis across the whole riser cross-section. This result is consistent with the suspension temperature measurements made as part of the present study. As shown in Figure 3.14A, when G = 527 kg/m s and U = 8 m/s, 7X1, 2.35) is slightly lower than s g r,(0, 2.35) while 7X1, 3.25) is larger than 7/,(0, 3.25) and 7X1, 2.35), implying that the particles close to the wall are ascending for this operating condition. From these observations, one can construct a likely picture of particle motion in the 97 vicinity of wall. In the pneumatic conveying flow regime, when there are very few particles in the suspension, particles all move upwards in theriser.As G increases, more s particles accumulate in the column, and those close to the wall begin to descend (Senior and Grace, 1998). When the net solids flux in the vicinity of the wall is downwards, a wall layer forms and the fast fluidization flow regime is reached. If G continues to s increase, the suspension becomes denser, with more particles in the wall layer carried up until the net solids flux in the vicinity of the wall becomes upwards and the annular downflow wall layer is eliminated. The system is then operating as a dense suspension upflow. For G extremely high, > 500 kg/m s for our experimental set up, virtually no 2 s particles travel downward in the vicinity ofthe wall. 3.3.2. Average Heat Transfer Coefficient For steady state conditions with lossesfromthe outside of the jacket ignored, energy conservation requires that the total heat supplied by the hot water equals the heat transferred from the heating water to the suspension. The overall suspension-to-water heat transfer coefficient, is then given by, c m JJ (T. _ c pc\ ~~ A AT c —T cm ^ coul' JQ\ sus-to-water where A is the total inside area of the inner cylinder, and c IT _ r Vcitt .rp sus-to-waler nr< cjn i In -T )-(T sbc / T c,out V coul ) sic' /o nr V sbc -T stc 98 | j \ ' / is the logarithmic mean temperature difference between the water and core suspension. Since the water-side wall surface temperature is also measured, the suspension-towater-side wall surface heat transfer coefficient can be calculated from »-- - ;^- ^ m c , T c^^ (3 2) sus-to-wall A where Ann _ ( wjn svs-lo-wall she) ( rp w,oul "rp w,//t T • w oul t ^i(c) • \J.LJ) sbc -T sic The total thermal resistance from the suspension to the water-side wall surface is the sum of the thermal resistance on the suspension side and that due to conduction in the tube wall. The suspension-to-wall heat transfer coefficient can therefore be calculated using k = \ rMrJRY h' ( 3 - 1 4 ) k Figures 3.8 and 3.9 show the suspension-to-water and suspension-to-wall heat transfer coefficients as functions of particle circulation rate and superficial gas velocity. It has been widely reported that, at constant suspension density, the superficial gas velocity does not have a significant influence on the heat transfer coefficient (e.g., Wu el al., 1987; Furchi et al., 1988). Figures 3.10 and 3.11 plot the heat transfer coefficient against 99 "1 300 i V 1 1 | 1 1 I I I 1 I E «$><> „ o - • 400 It! O ° co cu X - 300 CO i_. • O A O A O 200 •— l l I f • i 1 I I t • co cu X i I I I —i 300 1 — r " T T T l r| U„ (m/s) E o • i 1 c A C0400 O 1—i • AGO "2 CO ro ro (0100 CO cu ' • • 1 1 1 1 Suspension density (kg/m ) i | i i / V - A 9.5 — - yi i - / / logarithmic fit — —equation (3.15) • J - - .1 1 I ! i i i - o o i 4.0 O 6.0 7.0 O 8.0 St _I • U„ (m/s) CU O300 o • Figure 3.9: Suspension-to-wall heat transfer coefficient vs. particle circulation flux. - O 4.0 6.0 V 7.0 O 8.0 A 9.5 i ° 2 T ~ O AO Particle circulation flux (kg/m s) Figure 3.8: Suspension-to-water heat transfer coefficient vs. particle circulation flux. 1 ° O 100 ~i °*A O 100 Particle circulation flux (kg/m s) tn o c ro AO ° 10 c 9.5 9 § 9- o o O o Itl c cu 'o £ cu o o 11 CO c ro E A V A A in * o 4.0 6.0 7.0 8.0 o o • ° I I I u m/s CM O 8.0 A 9.5 - CU o o 1 4.0 6.0 7.0 O _ »t 1 U„ (m/s) • c cu 'o 200 • • i -1 i i i i 1 1 1 1 I i i i i i Suspension density (kg/m ) Figure 3.10: Suspension-to-water heat transfer coefficient vs. suspension density. 100 Figure 3.11: Suspension-to-wall heat transfer coefficient vs. suspension density the suspension density. It can be seen that, for a given suspension density, the heat transfer coefficient is hardly influenced by the superficial gas velocity. Divillo and Boyd (1994) correlated the convective heat transfer coefficient as a function of suspension density based on datafromdifferent cold units and obtained: (3.15) h = 23.2xpj . 55 This relationship, plotted as dashed line in Figure 3.11, overestimates our heat transfer coefficient data for p sus > 200 kg/m . Our experimental data are better correlated by the 3 following logarithmic relationship, shown as a solid line in Figure 3.11: A = llllnp1BI-221 <400 kg/m ). 3 (\0<p sus (3.16) 3.3.3. Local Heat Transfer Coefficient As described in section 3.2.6, probes containing four thermocouples were inserted into the water flowing channel at four levels, while two regular thermocouples measured the average temperatures at the water inlet and outlet. The water jacket can therefore be divided into five sections in series bounded by these six measurement levels. For each section, the average heat transfer coefficient can be calculated from Equations (3.12) and (3.14) where the wall surface temperatures at the ends of each section can be calculated from cubic splines fitted to the four wall surface temperature measurements. The experimental results are illustrated in Figures 3.12 and 3.13 for superficial gas velocities of 6 and 8 m/s, respectively. The operating conditions are listed in Tables 3.4 101 and 3.5. For both gas velocities, the heat transfer coefficients are higher at the two ends of the water jacket, except for condition A, corresponding to the highest net particle circulation flux, where the heat transfer coefficient decreases with height along the entire length of the jacketed section. As discussed above, the suspension-to-wall heat transfer coefficient is closely associated with the particle motion in the vicinity of the wall, and the direction of this motion is indicated by the suspension temperature distribution. The suspension temperatures in the vicinity of the wall and at the axis of the column, recorded simultaneously with the heat transfer coefficient measurements, are shown in Figures 3.14 and 3.15. As can be seen for both runs designated as A, 7X1, 2.35) is almost the same as T (0, 2.35) while 7X1, 3.25) > 7X0, 3.25). This suggests that for these operating s conditions, particles in the vicinity of wall are all ascending, causing the heat transfer from the wall to the suspension to drop continuously as particles ascend from the bottom of the water jacket. However, for all the other operating conditions, it appears from the suspension temperature distributions that particles are moving both upwards and downwards in the vicinity of the wall, although the numbers ascending and descending probably differ. As a result, heat transfer is augmented at both ends of the heat exchanger. Quantification of the relative number of particles rising and falling is discussed in the next section. 102 ) sr. D) CD X 0 100 200 300 400 500 600 700 800 L o c a l heat transfer coefficient ( W / m K ) 2 Figure 3.12: Local suspension-to-wall heat transfer coefficient as a function of height for U = 6 m/s. Operating conditions are listed in Table 3.4. s 103 D) X 0 100 200 300 400 500 600 700 800 L o c a l heat transfer coefficient ( W / m K ) 2 Figure 3.13: Local suspension-to-wall heat transfer coefficient as a function of bed height for U = 8 m/s. Operating conditions are listed in Table 3.5. s 104 "i— —i—•—i—'—i—•—r 1 A I o Interval o c c u p i e d by heat exchanger 3 t JZ 2U X 1 -i—I—i 24 4 • 26 I i 28 "I ' I • i 30 I ' 1 1 32 < 1 34 1 36 24 4 r- 1 26 ~i 28 i 1 30 > i 32 34 36 •—i—'—r D o Interval o c c u p i e d by heat exchanger \ Interval o c c u p i e d by heat exchanger JZ <D 2 2L 1 24 28 30 32 1 34 36 24 '—•—•—-—•—•—• T —i—•—i—•—r 4 E 26 26 28 I 30 i I 32 • i • 34 36 Temperature (°C) 1 3 -i—I—i—I—i o Interval o c c u p i e d by heat exchanger O 1 I—i—1 24 i 26 I 28 • I 30 . I . 32 I . 34 36 Temperature (°C) Figure 3.14: Suspension temperatures at the wall and axis of the riser for U = 6 m/s. Operating conditions are listed in Table 3.4. O : r/R = 0; x:r/R=l. g 105 "I 1 I I 1 Interval o c c u p i e d 1 ' 1 " b y h e a t T rA e x c h a n g e r «—i— —i—•—i—•—r 1 Interval o c c u p i e d b y h e a t B e x c h a n g e r JZ T x 1 24 4 26 T 28 1 i 30 1 32 34 T —i— —r 1 1 36 24 4 J i L J i L 26 28 30 32 34 36 26 28 30 32 34 36 3 t Interval o c c u p i e d b y h e a t e x c h a n g e r JZ 2U X 1 24 4 E 26 28 30 32 34 1 36 24 .—.—.—.—.—.—.—.—.—.—.—.—- Temperature (°C) 3 b JZ "OJ X 2 L 1 24 26 28 30 32 34 36 Temperature (°C) Figure 3.15: Suspension temperatures at the wall and axis of the riser for U = 8 m/s. Operating conditions are listed in Table 3.5. O : r/R = 0; x : r/R = 1. g 106 3.4. Modeling Heat Transfer in HDCFBs 3.4.1. Particle Motion in the Vicinity of the Wall As discussed above, except for extremely dilute pneumatic conveying or extremely dense suspensions, where the particles and gas suspension all travel upwards across the entire cross-section of the riser, the motion of particles in the vicinity of the wall is not uni-directional. Instead, there is intermittent upward motion near the wall interspersed with periods of downward local particle motion. Depending on the superficial gas velocity and the net circulation flux or cross-sectional average suspension density, the net local flux at the wall may be downwards, forming a wall layer corresponding to the fast fluidization flow regime; or the net time-average local flow can be everywhere upward, corresponding to dense suspension upflow. When particles oscillate upwards and downwards in the vicinity of the wall, the suspension enters the heat transfer section from both below and above. To study the heat transfer process from wall to suspension, detailed hydrodynamic information, including the fraction of time the particles spend moving upwards and downwards, must be determined. There are several ways of estimating the latter parameter: 1. By measuring the local time-average upward solids mass flux and downward flux in the vicinity of the wall. If the upward flux is G , and the downward G j, the fraction su of particles moving downwards is simply 107 s 2. From the suspension temperature distribution. Tstc Tstw * Tsbc Tsbw Figure 3.16: Schematic of particle motion in the vicinity of the wall and its influence on the suspension temperature. As illustrated in Figure 3.16, if particles in the vicinity of wall are allrising,we would expect that Tsbw, the suspension temperature in the vicinity of wall below the heat exchanger, would equal to T b , the suspension temperature at the axis of the riser below s c the heat exchanger. Above the heat exchanger, T stw > T since the ascending particles slc near the wall have gained heat from the walls. Similarly, if the particles in the vicinity of the wall are all descending, we would expect T , = T and T shl slc sbw > T bc- If more particles s descend than ascend, we would expect [T b - T b ] > [T . - T \. In other words, \T bw s w s c shi slc s T bc] is an indicator of how many particles in the vicinity of the wall are descending while s [T - T ,c] indicates how many are rising. Therefore, the fraction of particles falling slw s 108 downward in the vicinity of wall can be approximated by ~( - d T V stw T 1 )+( - T s t c ) ^ V sbw T 1 )• ^ 6 ) sbc ) Since the suspension temperatures are measured simultaneously with the heat transfer coefficients, method 2 for determining f<t is employed here. The vertical thermocouple positions must be carefully considered. If the thermocouple is too close to the water jacket, radial gas conduction may dominate the heat transfer process so that the temperature may not reveal the vertical motion of the particles. If it is too far away, the particle temperature will decay too much due to particle exchange between the wall and bulk regions. In our experiments, the thermocouples were positioned 150 mm below and above the heat exchanger section. Figure 3.17 shows the estimated fraction of particles moving down in the vicinity of wall as a function of the cross-sectional average suspension density for superficial gas velocities of 6 and 8 m/s. The solid lines are quadratic fits to these data. It can be seen that a lower superficial gas velocity favors the downward movement of particles in the vicinity of wall. It should also be noted that fa never reached 1 for either gas velocity, though it approached it more closely for the lower U value. g Generally, for an idealized system where uniform spherical solid particles of diameter dp and density p are contacted with a steady upwards flow of gas of density p and p g viscosity p in the absence of inter-particle forces,famay be written g /, (3-19) =f(gAp,p ,d ,U ,ju ,e ). g p g g m 109 This leaves aside such secondary variables as column diameter, distributor plate geometry, static bed height, particle size distribution, particle shape and humidity. Since there are 7 quantities and three fundamental dimensions (length, mass and time), this equation can be written in terms of four independent dimensionless quantities, e.g., S u s p e n s i o n d e n s i t y (kgm / ) Figure 3.17: Fraction of particles moving downwards in the vicinity of the wall as a function of suspension density and superficial gas velocity. f =f(d \u',e ) d p (3.20) sec where d' =d [p gA /S] n p p s (3.21) P 110 '=U W<VgAp] U n g ^sec = Psus I Pp and Ap = p p (for P ( 3 p p » 2 2 ) Q p) g (3.24) g It can be expected that larger d * and lower if will favor the downward movement of p particles in the vicinity of the wall. In this study, Group A FCC particles having a mean diameter 65 pm were used, while in circulating fluidized bed combustion, Group B particles with mean diameter between about 100 and 300 um are generally employed. This may explain why, in our case, there are still many particles rising near the wall even when the suspension density is between 20 and 100 kg/m . 3 3.4.2. Extended Two-Dimensional Modelfor Smooth Wall The model developed in Chapter 2 assumes that all the particles in the wall layer are always descending, As we have seen above, this is not the case, even in the fast fluidization flow regime where the time average flux near the wall is downwards. The model needs to be extended to cover the true situation where particles oscillate upwards and downwards. As a first approximation, two independent layers can be assumed to exist in the vicinity of wall. One is falling from top to bottom; the other is traveling in the reverse direction. If the heat transfer coefficient due to the former motion is h and the coefficient d due to the latter motion is h , then the overall heat transfer coefficient can be obtained by u 111 superposition as h = ( l - / „ )K+ (3.25) fh d d where fd is the fraction of particles moving down in the vicinity of the wall. Note that this approach is a significant oversimplification of a complex, transient problem as it assumes steady fluid and particle motion in each direction. Before this simple model can be applied to the current heat transfer results, some parameters need to be clarified: 1. Thickness of the wall layer The correlation of Bi et al. (1996) (equation 2.28) was used to calculate the wall layer thickness in the fast fluidization flow regime. In dense suspension upflow, a distinct wall layer is absent. However, the sensitivity analysis carried out in Chapter 2 showed that the layer thickness has a negligible influence on the heat transfer coefficient so long as it is larger than the thermal boundary layer thickness. Hence equation (2.28) is still employed here, but only when determining the temperature distribution close to the wall. 2. Particle downward and upward velocities near the wall Liu (2001) measured the average particle velocity at several elevations in the same riser that was used for the present experiments. Some of his results at rlR = 0.95 and Z = 2.2 and 3.6 m, below and above our heat exchanger section respectively, are listed in Table 3.3. It can be seen that the particle average velocity in the vicinity ofthe wall is a function not only of such operating conditions as superficial gas velocity and net particle 112 circulation flux, but also of elevation. It is difficult enough to predict the average particle velocity in the vicinity of wall, without having to estimate the actual average upward and downward velocities. Hence for the current study as a first approximation, a velocity of 1.2 m/s is employed for both the upward and downward particle velocities. The influence of this assumption on the heat transfer coefficient is analyzed below. Table 3.3: Average upward particle velocity at r/R = 0.95 determined by Liu (2001). Height u G (m) (m/s) (kg/m s) 6 49 6 Average Height u G (m) (m/s) (kg/m s) -0.1 6 10 2.8 89 -0.5 6 47 -0.2 6 145 -0.2 6 79 -0.6 6 214 0.2 6 147 0.8 6 328 0.0 6 215 0.3 6 431 0.0 6 288 0.4 7 453 0.1 6 360 0.7 8 47 3.8 6 10 2.8 8 151 1.5 8 9 7.0 8 224 1.1 8 100 1.9 8 348 0.1 8 214 1.0 8 433 0.0 8 248 0.8 8 476 0.5 8 324 0.5 8 457 0.9 8 s 2 Up (m/s) 3.6 2.2 113 8 Average s 2 Up (m/s) 3. Particle concentration distribution in the vicinity ofthe wall Equation (2.18) is a correlation reported by Issangya et al. (2001) based on the data obtained from the same CFB facility as we used with very similar particles. It was also tested successfully against datafromother laboratories, both for the fast fluidization and dense suspension upflow regimes. Hence the equation is useful both for high-density and low-density conditions, and it is employed here. 3.4.3. Comparison of Model Predictions with Experimental Results Tables 3.4 and 3.5 show the operating conditions and fa values calculated from the suspension temperature measurements, for superficial gas velocities of 6 and 8 m/s, respectively. Table 3.4: Operating conditions and fa for U = 6 m/s. g A B C D E All 372 182 149 51 329 335 81 37 24 Tstw ~ Tstc 1.31 1.52 0.87 2.25 4.06 T bw 0.11 0.46 4.72 6.56 4.86 0.08 0.23 0.84 0.74 0.54 Run G (kg/m s) 2 s Psus (kg/m ) 3 s fd T bc s Local heat transfer coefficients predicted by the above model are compared with the experimental data in Figures 3.18 and 3.19 for U = 6 and 8 m/s, respectively. The dashg 114 dot lines are the heat transfer coefficients (hj) calculated assuming all particles in the vicinity of the wall to be descending; the dashed lines (h ) are obtained by assuming u those particles are traveling upward. The thick solid lines are the weighted average heat transfer coefficients based on equation (3.25). It can be seen that the model predictions fit the experimental data quite well, definitely better than the individual unidirectional (hj and h ) predictions. u Table 3.5: Operating conditions and f for U = 8 m/s. d Run g A B C D E G, (kg/m s) 527 373 225 135 79 Psus (kg/m ) 262 239 111 34 . 23 Tbtw ~ Tbtc 1.05 1.01 0.83 5.68 4.51 Tbbw ~ Tbbc 0 0.43 0.92 2.06 0.43 0 0.30 0.53 0.27 0.09 2 3 fd 115 0 3.2 200 1 400 1 1 3.1 600 1 1 800 'c 200 3.2 : 3.0 - 2.9 sz CD X - 3.0 ~• .• • 2.9 2.8 2.8 2.7 : 2.6 - \\ • i 3.1 2.7 I 1 | 400 i —, 600 • i 800 -i r - > - 2.6 2.5 2.5 i 2.4 0 200 . i 400 i 2.4 600 800 200 i i • 400 600 800 H e a tt r a n s f e rc o e f f i c i e n t( W / m K ) z • E x p e r i m e n t a l P r e d i c t e d h P r e d i c t e d h u P r e d i c t e d h_ 0 200 400 600 800 H e a tt r a n s f e rc o e f f i c i e n t (W/m K) 2 Figure 3.18: Comparison o f predicted and experimental heat transfer coefficients for U Operating conditions are listed in Table 3.4. g 116 : 6 m/s. Figure 3.19: Comparison of predicted and experimental heat transfer coefficients for U = 8 m/s. Operating conditions are listed in Table 3.5. g 117 I 3.5. Influence of Particle Velocity on Heat Transfer Coefficient As discussed in Chapter 2, higher particle velocities should increase the heat transfer coefficient. Since in equation (2.3) the particle velocity u acts in combination with p particle concentration c, it is expected that for high-density operating conditions, the influence of u will be greater than for low-density conditions. The simulation results p based on the experiments at U = 6 m/s are in agreement with this analysis. As shown in g Figure 3.20, if both particle upward and downward velocities are changed to 0.5 m/s, the heat transfer coefficients decrease compared to the base case where both velocities are assumed to be 1.2 m/s. If both velocities are increased to 2.0 m/s, the predicted heat transfer coefficients increase. At higher suspension densities, the differences between these three sets of simulation results are greater. 118 0 200 400 3.2 "1 3.1 600 1 1 800 <~ C j 3.0 2.9 sz cn <D X 2.8 2.7 2.6 2.5 2.4 j L 200 3.2 3.1 - • 400 i > 600 sz 2.8 2.7 »j 7i 0 200 400 - / ( i — Experimental u p i = 1.2 m/s u = p - 0 • I 200 - l i 400 600 l 800 2 i 2.6 2.5 2.4 600 H e a t transfer coefficient ( W / m K ) 3.0 2.9 800 H e a t transfer coefficient 2.0 m/s u = 0.5 m/s 800 (W7m K) 2 Figure 3.20: Influence o f particle velocity on model predictions for U = 6 m/s. Operating conditions are listed in Table 3.4. s 119 3.6. Summary Experiments carried out with FCC particles in the 76 mm diameter riser of a dualloop high-density CFB facility show that particles move both upwards and downwards in the vicinity of the wall. The direction of this motion is indicated by the suspension temperature distribution when heat is being transferred to or from the wall. The average suspension-to-wall heat transfer coefficients are strongly influenced by suspension density. However, they are almost independent of the superficial gas velocity at constant suspension density. The local heat transfer coefficient profiles are strongly associated with the direction of particle motion. This alternation of direction leads to higher heat transfer coefficients at both ends of the heat exchanger. The heat transfer model developed in Chapter 2 is extended to cover both highdensity and low-density operating conditions considering the actual particle motion in the vicinity of wall by introducing the factorfa,defined as the fraction of particles traveling downwards. This fraction is estimated from the suspension temperature distribution. The resulting model predictions compare well with the experimental data. 120 CHAPTER 4 THREE-DIMENSIONAL MODEL FOR LOW-DENSITY C F B WITH M E M B R A N E W A L L 4.1. Introduction Heat extraction in circulatingfluidizedbed combustors is usually accomplished by using membrane walls. Membrane walls, composed of parallel tubes connected by fins, as indicated in Figure 4.1, typically form the inside surface of C F B combustors. The correct sizing of these membrane wall surfaces is important to ensure proper operation, load turndown, and optimization of C F B boiler systems. It is essential to thoroughly understand the mechanisms of heat transfer between the gas-solid suspension and the membrane wall surface, to consider the heat conduction in membrane walls, and to develop an appropriate model to predict the rate of heat transfer. Tube Insulation Figure 4.1: Configuration of membrane wall. Tube axes are normally vertical. 121 In an industrial scale CFBC, the riser heights are generally 20 to 35 m and the width varies from about 5 to 10 m. The superficial gas velocities above the secondary air inlets are typically 4 to 8 m/s. Primary-to-secondary air ratios are generally in the range of 1:3 to 3:1. Sand, commonly used as an inert medium, and/or limestone sorbent particles have average diameters from about 150 to 400 pm and densities between 2000 and 3000 kg/m . Solid fuel particles vary in size from about 20 to 3000 pm and typically comprise 3 less than about 5% of the total bed inventory by weight. Suspension densities in the riser are usually in the range of 10 to 100 kg/m . 3 The main advantages of the CFBC relative to other combustors include (Grace and Bi, 1997): 1. Low NO emissions. This is achieved by staging air injection and by x operating with lower and more uniform bed temperatures (typically 850900 °C). 2. Low SO2 emissions. This is fulfilled by using sorbent particles such as limestone to capture sulfur dioxide in-situ. 3. Fuel flexibility. The CFBC can be operated with a broad array of fuels, including low-grade fuels (high ash and/or high moisture content) and high sulfur fuels. Many industrial wastes and by-products (peat, wood chips, petroleum coke, pitch, sewage) can also be utilized to generate steam or electrical power. 122 4. High combustion efficiency. The combustion efficiency (i.e., ratio of fixed carbon combusted to that which is fed) commonly reaches 98-99% (Brereton etal., 1991). 5. Low fuel preparation cost. Fuels can be fed to CFBCs with broad particle size distributions and with limited crushing. 6. Simple turn-down and control. There are a number of independent variables that can be used to control the system for a wide range of operating conditions. 4.2. Model Development 4.2.1. CFBC Hydrodynamics with Membrane Wall As in the case for low-density CFB risers with flat walls, CFB risers with membrane walls also have a core-annulus structure. However, the geometry of the membrane may have a profound influence on the dynamics of gas and particle flows in CFB risers, especially in the wall layer. Visual observations of a transparent section having the shape of a membrane wall assembly (Wu et al., 1991) showed that particles traveled downwards predominately in the sheltered flat web or fin section, leaving the curved tubes exposed to the dilute core. This suggests that heat transfer results or models developed for experimental columns having smooth surfaces may fail to represent the behavior in industrial units with membrane walls or rough heat transfer surfaces (Grace, 1990). 123 Lockhart et al. (1995) showed that the time-averaged solids volumetric concentration at the junction between thefinand the tube is much higher than that at the crest or at the center of the fin of a cold model circulating fluidized bed. For higher overall suspension densities, the local solids concentration can be as high as 40% near the junction. The solids holdup in this region tended to be higher for smaller rather than for larger diameter tubes. Zhou et al. (1996) measured the local voidage distribution near a membrane tube in a circulating fluidized bed riser having a square cross-section. They found that the valley or trough formed by the fin and two adjacent membrane tubes protects particles from the upflowing gas. However, there was little influence of the membrane walls on the voidage and particle velocity in the core of the riser. Particle streamers tend to move downward along the fin in the protected valleys between adjacent membrane tubes. The particle concentration is highest in the fin area and lowest near the crest of the tube. The membrane wall also influences the average particle residence length in the fin and crest regions. Visual observations made in the region near the walls of a 12 MWth boiler (Golriz, 1994) show that particles concentrate over the fins and stay there longer than those traveling near the crest of the tubes. Generally, except for the particle concentration distribution and average particle residence length, the mechanism of heat transfer from the suspension to a membrane wall is the same as in a CFB with flat walls. However, in order to accommodate the geometry of a membrane wall, the problem must be transformed from two dimensions to three 124 dimensions. 4.2.2. Thermal Radiation in Wall Layer with Membrane Wall As discussed in Chapter 1, the particulate suspension in the wall layer formed by a membrane wall also constitutes an emitting, absorbing and scattering gray medium. However, because of its geometry, the cross-section of the wall layer becomes twodimensional for a membrane wall and so does the radiation transfer. Radiation from the fin and tube surfaces as well as from the core must be considered together. For a twodimensional system, equation (1.2) becomes: cos <p— + cos 6 sin <p — = -ai' (x, y) + ai ' (x, y) - cr/ (x, y) + — (* i' (x, y)<P(a), co )dco dx dy b i An (4.1) where <p is the polar angle measured from the positive x axis and 6 is the azimuthal angle. Several approximate methods are available to solve this integrodifferential equation including the discrete ordinate, spherical harmonic and moment methods (Ozisik, 1973). It has been shown by Krook (1955) that these methods are closely related and that their solutions are completely equivalent. Appendix III describes the application of the lowest-order moment method (equivalent to the PI approximation of the spherical harmonic method) to the equation of transfer. First, the intensity distribution at a certain position in the medium is assumed to obey 125 i (4.2) i'= i (x,y) + b(x, y) cos <p + c{x, y) sin <p cos 6 0 where i , b and c are unknown functions of position only. Upon substituting equation 0 (4.2), and after performing a few mathematical manipulations, equation (4.1) becomes 1 3a + (l + /)cr dG 1 dx s + • dx 1 1 dG dy 3a + (l + f)a s = dy a(G-4a T'). 0 (4.3) By integrating equation (4.3), the divergence of the radiative flux can now be written as V »q = Aae - a £ J\co,S)dco = 4ae -aG = 4aa T* -aG . r h b 0 (4.4) 4.2.3. Governing Equations Consider the plan view of the membrane wall and wall layer shown in Figure 4.2. Because of symmetry, the domain of the wall and wall layer need only correspond to a half-tube and a half-fin. Similar to those described in Chapter 2, the governing equations for steady-state heat transfer in the wall layer of a CFB riser with a membrane wall can be written as follows. Heat balance on gas in wall layer Consider a Cartesian control volume in the wall layer as shown in Figure 4.3. The energy balance for the control volume ignoring heat generation (e.g. due to reaction) can be written 126 (4.5) where Q is the volumetric heat convection rate from the particles to the gas and given pg by equation (2.2). Figure 4.2: Plan view of membrane wall and wall layer assembly 127 Heat conduction through gas gap (Q2): Heat conduction through membrane wall (Q.3): Heat balance on cooling water (Q.4): i P c R C p c u c ~ + o = 0. (4.9) Boundary and interface conditions: Top of heat transfer zone, at z = 0, (x,y) czQ.1 (i.e., belonging to domain QI in Figure 4.2.): T =T =T , (4.10) r (4.11) g p b =T c c,out Bulk side of wall layer, at z > 0, (x, .y) c Tl: (4.12) T =T =T =f(z), s p b dG _ 3(a + (\ + f)a )dn s e b 2(2 -e„) (G -Aa T ). 0 b 129 (4.13) Symmetry boundaries of wall layer, at z> 0, (x, y) e T2, T8: ox ox Membrane wall-wall layer interface, at z>0, (x, y) <z T9: z. 8 T . i t dT Insulated or symmetry boundaries of membrane wall, at z> 0, (JC, .y) <= T3, T4, T5, T7: (4.17) dn Cooling water - membrane wall interface, at z > 0, (x, y) c T6: • ^ ^ = *c(^-3;)- (4.18) where /z can be calculated from a standard correlation. c 4.2.4. Parameter Determination Unless otherwise indicated, the parameters in equations (4.6) to (4.18) are the same as those described in Chapter 2. 130 1. Voidage distribution in wall layer It is commonly agreed (e.g., Wu et al., 1991; Lockhart et al., 1995; Zhou et al., 1996) that the particle concentration in the fin region is higher than that in the crest region. However, no quantitative correlations are available to describe the particle concentration profile in the wall layer near a membrane wall. Hence, the correlation of Issangya et al. (2001) for flat walls (i.e., equation (2.18)), is again employed here to predict the particle concentration in the wall layer. In equation (2.18), the particle concentration is only a function of the distance from the center of theriser.The particle concentration it predicts for the fin region is therefore higher than in the crest region, in agreement with experimental observations. 2. Wall layer geometry For membrane wall heat transfer surfaces, the thickness of the annular layer on the fin surface will be a little thicker than at the tube crest. For a relatively thin wall layer, we may begin by approximating the geometry of the layer boundary as part of a circle concentric with the tube, with the correlation of Bi et al. (1996), equation (2.28), selected to predict the mean wall layer thickness relative to the membrane surface. The boundaries were then adjusted manually to provide the shape shown in Figure 4.4. For a relatively thick wall layer, we may just assume that the geometry of the boundary layer is a straight line as shown in Figure 4.20. The sensitivity analysis in Chapter 2 showed that the wall layer thickness does not influence the heat transfer coefficient so long as it is larger than the thermal boundary layer. 131 3. Average particle residence length L ar For flat walls, the average particle residence length was assumed to be 1.5 m. Apparently the particle residence length in the fin region is larger than this value since the membrane tubes help prevent particles from exchanging with the core. On the other hand, the residence length in the crest region is shorter because the tube extends further into the reactor and the particle motion there is more strongly influenced by the upward moving particles and gas in the core region. For the present study, we assume 2.0 m in the fin region 1.0 m in the crest region (4.19) The corresponding particle exchange rate between the wall layer and the core is calculated from equation (2.31) based on this new set of L values. ar 4.3. Numerical Method and Accuracy Analysis Equations (4.6) to (4.18) couple reactor side radiation, gas conduction, gas and particle convection, particle-to-gas convection and wall side conduction. There are fourth-power radiation terms, as well as parameters such as gas conductivity, heat capacity and density which are functions of temperature. Hence, this set of equations is non-linear and highly coupled, and therefore can only be solved numerically. Considering the complex irregular geometry of the solution domain, the Galerkin finite element method is employed to solve the conduction problem in the membrane wall and annular wall layer as well as the radiation problem in the wall layer. To reduce the number of 132 finite elements and increase the calculation accuracy, six-noded triangular elements and quadratic shape functions are employed, i.e. the temperature or irradiance distribution in each element is assumed to be T(x,y) or G(x,y) = C,x + C y 2 2 2 + C xy + C x + C y + C 3 4 s 6 (4.20) After calculating local approximations of the solution over each element, the equations describing the approximations are assembled for the entire mesh by adding up the contributions to these equations furnished by each element. Then the gas convection term 8T I dz in the resulting equation is approximated by a Galerkin finite difference method g that has an error of 0(zfz ) and is unconditionally stable. Since there is no axial conduction involved, the particle heat balance equation (4.7) is also approximated by a Galerkin finite difference method. After these numerical approximations have been made (see Appendix IV), the governing equations reduce to the set of algebraic equations: SRxK = F (4.21) where 5?, K and F are defined in Appendix IV. 91 contains capacitance matrices and conductance matrices; some of its terms are functions of gas temperature, while some are functions of T . F consists of vectors related to the boundary conditions and vectors p which contain the information from the last iteration at z — dz. Hence equation (4.21) can only be solved iteratively. Usually we use the solution K from the last iteration as set of initial values to calculate 9?, then a new K can be obtained from equation (4.21). The solution is then iterated in this manner until a preset converge criteria is reached. In the 133 calculations presented below, the convergence criterion is set to be ( -6 new old <10 V new J (4.22) max In the application of the finite element method, the solution domain is divided into six-noded triangular elements in the x and y directions. Due to the high thermal conductivity of the membrane wall, the temperature variations there are small and hence the elements can be coarse. A much finer mesh is needed in the particle layer nearest to the wall, where the temperature and irradiance gradients are expected to be large. In the wall layer closer to the bulk, where the temperature and irradiance distributions are expected to be flat, relatively coarse elements can once again be employed. Thefinerthe mesh, the more accurate are the expected results. However, finer meshes result in a heavier computational load. To test the accuracy of the finite element solution, the mesh used to generate all of the calculations presented in Section 4.4 was further refined by a factor of four (i.e., a factor of two in each of the x and y directions). The differences between the temperature and irradiance profiles obtained with the original and refined meshes were negligible. In the z-direction, the wall layer is divided into finite difference steps for vertical space discretization. A two-level algorithm is used, where the solution at a particular height z is calculated from the temperatures and irradiances at the previous height z - Az. At the top of the heat transfer surface (i.e., at z = 0), the gas temperature near the wall decreases sharply along the heat transfer surface because of the large temperature 134 difference between the gas and the wall. Therefore, Az needs to be very small near z = 0. With increasing distance below the top, the temperature differences decrease, so that Az can be enlarged. To obtain smoothly increasing values of Az, a Fibonacci series (i.e., Az(j) = Az(i-1) + Az(i-2)) was employed until a pre-set maximum step-size Az max was reached. To ascertain the accuracy, tests with shorter vertical steps were tried to see what effect that had on the results. 135 4.4. Predictions for a Typical Case 4.4.1 Base Case Consider the pilot scale CFBC facility previously located in the Pulp and Paper Center at UBC. The riser was 7.3 m in height and its cross-section was 0.152 m x 0.152 m. The experimental membrane wall consisted of pipes and connecting fins, divided into two parallel sections, side by side. Two stainless steel 347 pipes (21.3 mm O.D. and 14.1 mm LD.) and three fins (6.4 mm half-width) of the same material were welded together to form half ofthe membrane wall while the other half was made of stainless steel 316 pipes and fins of the same dimensions. The total length of the membrane section Table 4.1: Base parameters used in example calculations below Particle diameter 286 pm Particle emissivity 0.85 Particle heat capacity 840 kJ/kgK Particle density 2610 kg/m Particle thermal conductivity 1.9 W/mK Particle velocity in wall layer 1.2 m/s Suspension density 52.5 kg/m Wall layer thickness 10.5 mm Gas velocity in wall layer 0.4 m/s Gas gap thickness 77.1 pm Bulk temperature 1077 K Bulk emissivity 0.99 Conductivity of wall 21 W/mK Wall surface Emissivity 0.90 Water outlet temperature 80 °C Particle average residence length Fin: 2 m; Crest: 1 m Water velocity in tube 2 m/s 3 136 3 was 1.626 m and the total exposed area for each half of the membrane wall was 0.150 m . 2 The membrane wall was cooled by water flowing upward through the tubes. To simplify 0 0.005 0.01 0.015 x(m) Figure 4.4: Finite element mesh and nodes for membrane wall and wall layer o f U B C pilot C F B C . the calculations, the outlet water temperature rather than the inlet temperature is assumed to be fixed at a known value (80 °C). Other key parameters are typical of those encountered in pilot-scale circulating fluidized bed combustors and are listed in Table 137 4.1. The finite element mesh generated in the membrane wall and wall layer and the corresponding nodes are shown in Figure 4.4. The grid contains a total of 120 elements and 281 nodes, zlz(l) is set to 1 x 10~ m and Az 8 max to 0.1 m. 4.4.2. Heat Flux Distribution The predicted lateral heat flux distribution at different parts of the exposed pipe and fin surfaces at z = 1.5 m is illustrated in Figure 4.5. From the crest of the pipe to the junction of the tube and fin, the conductive heat flux increases due to the effect of particle concentration. The conductive heat flux is minimal on thefinside of the junction, and then increases slightly toward the center of the fin. The radiative heat flux has its lowest value at the junction since the particles have their lowest temperature there, i.e., the junction is shielded by the cool particles around it. The particles near the crest ofthe tube are lower in concentration and higher in temperature; hence the radiative heat flux is highest there. 138 100 90 L . • 80 CJ ^ E ^—' • Total 70 • Radiaton Conduction 60 X Z3 *4— -*—' 50 He CD 40 J t_ 30 Tube Fin 20 5 10 15 20 Distance from crest (mm) Figure 4.5: Lateral variation of heat flux along membrane wall surface at z = 1.5 m for base case conditions listed in Table 4.1. The predicted conductive, radiative and total heat fluxes, obtained by integrating the local values tangentially over the pipe and fin surfaces, are plotted as functions of vertical distance in Figure 4.6. In general, both the conductive and radiative heat fluxes decrease from z = 0 due to particle cooling along the surface. The conductive heat flux decreases very rapidly near z = 0 because when the particles and gas enter the wall layer, they are assumed to have the same temperature as the CFB core, a value substantially higher than the wall temperature. The radiative heat flux actually increases for a short distance near z = 0 because the wall temperature initially decreases sharply with distance from the top of the membrane wall. 139 Figure 4.6: Vertical variation of integral heat fluxes along the wall for base case conditions given in Table 4.1. 140 4.4.3 Particle and Gas Temperature Distributions The particles and gas are assumed to have the same temperature as the bulk when they enter the heat transfer zone. Figure 4.7 shows the gas and wall temperatures at z = 0. Most of the temperature drop takes place in the gas gap (which is too thin to be seen in the figure) while the predicted wall temperature is close to that of the water. The temperatures at the fin tip are higher than those at the crest. As the wall layer descends along the membrane wall, both the particle and gas temperatures decrease due to heat conduction and radiation to the membrane wall. Figure 4.8 shows the gas and wall temperatures at z = 1.5 m, while Figure 4.9 shows the particle temperature distribution at the same level. It can be seen that both the gas and particle temperature profiles are quite flat close to the core, but drop sharply near the membrane wall. 141 4.4.5 Heat transfer coefficient distribution The variation of the average suspension-to-coolant heat transfer coefficient along the heat transfer surface is similar to the heat flux profile and is plotted in Figure 4.11. The conduction term decreases sharply at z = 0 and then becomes almost constant for distances of 1.5 m or more below the,top of the membrane wall. The local lateral heat transfer coefficient distribution is similar to the local heat flux distribution as well. Figure 4.11: Vertical variation of average suspension-to-coolant heat transfer coefficient for base case conditions listed in Table 4.1. 146 4.5. Comparison of Model Prediction with Experiments 4.5.1. Comparison with Data ofWu et aL (1987,1989) Wu et al. (1987) reported experimental suspension-to-membrane waterwall heat transfer coefficients obtained from the CFBC facility described in Section 4.4. Heat transfer data were obtained from two nearly identical heat transfer surfaces, each 1.53 m long and 148 mm wide located on one wall of the column and beginning at 1.22 and 4.27 m, respectively, above the distributor plate. Each surface consisted of four identical, halfembedded, schedule 80 vertical stainless steel water-cooled tubes connected longitudinally by flat fins to form a membrane wall. One of the central tubes in the upper surface was instrumented with twelve thermocouples, approximately 150 mm apart, which allowed the vertical temperature distribution of the cooling water to be measured. The cooling water flow rate was also recorded. This information enabled the calculation ofthe average heat transfer coefficient between thermocouples 1 and 12 or between any pair of thermocouples. The suspension-to-wall heat transfer coefficient was determined from h= 1 R (4.23) R ln(/? /i?,) a a 0 where U = c pc( c,nu, m C - c.in) T T (4.24) A (T -T ) 0 b c 147 is the suspension-to-water heat transfer coefficient. The experimental gas temperatures were between 150 and 400 °C. Figure 4.12 compares the model predictions and experimental results for the column average heat transfer coefficients. For both particle sizes, the model overestimates the heat transfer for low suspension density conditions and underestimates for higher suspension densities. Figure 4.12: Comparison of predicted average heat transfer coefficients (x ) with experimental data of Wu et al. (1987) for sand particles (O) Gas temperature: 150-400 °C. A: d = 356 pm; B: d = 188 pm. p p Wu et al. (1989) further reported experimental average and local heat transfer coefficients obtained with the same CFB facility and membrane surface under higher suspension temperature conditions. Figure 4.13 compares the model predictions and 148 experimental heat transfer coefficients averaged over the interval between z = 0 and z = z for p sus = 54 kg/m . Since the stainless steel wall emissivity was unknown, the two values 3 0.5 and 0.9 were tried. The actual value likely lies between these two extremes. The model predicts these averaged heat transfer coefficients very well when e = 0.5. w 100 200 300 400 100 200 300 400 Heat transfer coefficient (W/m K) 2 Figure 4.13: Comparison of model predicted average heat transfer coefficients with experimental data (solid circles) of Wu et al. (1989) for p = 54 kg/m . solid line: e = 0.5; dashed line: e = 0.9; A: T = 860 °C; B: T = 407 °C. 3 sus w w b b Figure 4.14 compares the predicted column-average heat transfer coefficients with experimental results. A wall emissivity of 0.5 is used in the simulations. The model 149 slightly underpredicts the experimental results. 180 150 120 "I f A o o X 90 s 8® 8 ' T ~~I 8 9. ® 8 60 30 0 150 120 U B • c 8<8 « «X 8 90 60 30 0 150 120 n x x° « 90 60 30 0 10 20 40 30 50 60 70 80 Suspension density (kg/m ) Figure 4.14: Comparison of predicted average heat transfer coefficients (x ) for e„. = 0.5 with experimental data of Wu et al. (1989) (O) A: 7 = 8 7 0 ± 14 °C; B: 7i = 6 8 1 ± 18 °C; C: r = 4 1 0 ± 15 °C , i A 4.5.2. Comparison with Data of Luan (1997) and Luan et al. (2000) Luan (1997) carried out experiments in the same U B C C F B C facility as Wu et al. (1987, 1989). The membrane wall in his experiments is described in Section 4.4.1. Eight 150 K-type thermocouples were installed at the centerline of one of the pipes to measure water temperature at different elevations, including the water inlet and water outlet. To measure the temperature profde within the wall of that pipe, six 0.508 mm diameter Ktype thermocouples were embedded in the tube and adjacent fin 5.44 m above the primary air distributor (i.e., at z - 0.56 m) as indicated in Figure 4.15. Figure 4.15: Positions (T T ) of thermocouples embedded in pipe wall and fin by Luan (1997). r 6 Like Wu et al. (1989), Luan (1997) reported local heat transfer coefficients based on the cooling water temperature change measured between adjacent thermocouples. To maintain consistency, the experimental average heat transfer coefficients (i.e., integrated over z) were calculated from equation (4.23) as well. Figure 4.16 compares the modelpredicted average coefficients with the experimental results. In these simulations, a wall emissivity of 0.9 was used. 151 _l 1 100 I I I 200 I 300 I I 400 I I 1 100 200 . I . 300 I 400 Heat transfer coefficient ( W / m K ) 2 _l i 100 I I i 200 300 i l I 400 l 100 . I I i 200 300 i_ i 400 Heat transfer coefficient ( W / m K ) 2 Figure 4.16: Comparison of model-predicted average heat transfer coefficients (lines) with experimental data (points) of Luan (1997). A: T = 804 °C, = 52 kg/m ; B: T = 706 °C, = 52 kg/m ; C: T = 804 °C, p = 22 kg/m ; D: T = 706 °C, p = 22 kg/m . 3 b P a 3 a b P a a 3 b 3 sus b 152 sus 35 i 1 i i > i 1 • i 35 1 30 1 25 25 °o Oo o 15 20 ° 1 o -X X 10 I 0 • I • I • ' • ' • I 10 20 30 40 50 60 70 80 Suspension density (kg/m ) 1 i i 1 i 1 i 1 A 30 1 i 1 i 1 . 0 T" o 20 o -x- 15 . 1 o>t5" — 8— I I . I . . I - o o t-—x . ~ I . I . 10 20 30 40 50 60 70 80 Suspension density (kg/m ) - • 24 - B 1 1 ' 1 1 o OD 1 1 1 0 1 1 1 o ° o o X 1 ' o • . O .—-x—— x 16 10 1 O i -X -x- I 1 32 < '9 oo 1 3 o. 25 1 o o o 3 35 1 1 15 0 • I i 1 - 5 0 i 1 B 10 5 t- I i I i i i 30 h A 20 1 i x 8 I 0 • I . I . I . I . I . I . . 10 20 30 40 50 60 70 80 0 Suspension density (kg/m ) . 1. I . I . I . 10 20 30 40 50 60 70 80 I . I . I Suspension density (kg/m ) 3 3 Figure 4.17: Comparison of predicted temperature differences in tube and fin ( x ) with experimental measurements of Luan et al. (2000) (O). The positions of T T , T and T are indicated in Figure 4.15. A: T = 804 ± 5 °C, B: T = 706 ± 4 °C. h 2 s 6 b b Figure 4.17 compares the predicted temperature differences between T\ and Tj, and between Ts and T^ (see Figure 4.15) with experimental measurements. Except for (T\-Tj) at the bed temperature of 706 ± 4 °C, the predictions are always low. One possible reason 153 is that the embedded 0.508 mm diameter thermocouples introduced contact resistances in the membrane wall. Bowen et al. (1991) proposed an exposed-pipe-radial, insulated-pipe-tangential, finone-dimensional model (model 2) that was used by Luan et al. (2000) to estimate the local suspension-to-pipe, suspension-to-fin and suspension-to-pipe-and-fin heat transfer coefficients. Based on the model: 0 where (4.25) R HR,IR )(T -T ) p T Pi0ut 0 b pfiul is the temperature of the outside surface of the pipe, determined by the logarithmic extrapolation of 7/ and T to the surface at r = R , 2 h, = 28 k f w — cosh T ~T b (4.26) 5 co 6j l (Ti -T )cok 2 h. = -1 0 w HRJR ) + J0.58 k h (T -T )tanh f w f b 6 0 ( w + R )(T -T^) o0) cosh" ^T —T ^ (4.27) b T +T o p put 2 w h e r e T mif = (4.28) w + coR„ is the weighted average temperature of the pipe andfinsurface. Most of the additional symbols are defined in Figure 4.15. 154 250 1 ' I 1 ' 1 1 1 1 200 CN E o°<* o o o _ox— . 1 1 o o 150 o°o o x-—~~ 100 (50 1 , 20 1 x X er . 1 30 . 1 . 40 50 1 60 70 S u s p e n s i o n density (kg/m ) S u s p e n s i o n d e n s i t y (kg/m ) 3 250 250 200 200 150 150 L 100 L 100 20 30 40 50 60 50 70 250 "i— —I— —I— —I—•—i—•—r 1 1 200 i i i_ i | i | . 30 i 40 I i 50 I i -I 20 ' 1 1 1 |—i—r- I 60 U 70 i 80 ' 1 • 1 ' 1 ' 1 ' -• O CN o °<S> oo 150 L E 150 o I 100 L o I—L 20 1 30 40 50 60 70 50 S u s p e n s i o n density (kg/m ) 10 20 . t 30 o , 1 40 O ° o o J3 C O , -CM00 50 i . 200 CN E i S u s p e n s i o n d e n s i t y (kg/m ) 3 1 _i 10 S u s p e n s i o n density (kg/m ) 250 -i -X x . 1 50 . 1 . 60 1 . 70 80 S u s p e n s i o n density (kg/m ) 3 Figure 4.18: Comparison o f predicted tube, fin and total heat transfer coefficients ( x ) at z = 0.56 m with experimental data ( O ) o f Luan et al. (2000). Left panels: T = 804 + 5 °C; Right panels: T = 706 ± 4 °C. b b 155 Figure 4.18 compares the predicted tube,finand total heat transfer coefficients at z 0.56 m with the experimental results of Luan et al. (2000). The model underestimates the majority of heat transfer coefficients, although it predicts very well the local tube and total heat transfer coefficients as shown in Figure 4.16. These results are not surprising considering that the heat transfer coefficients are calculated from the differences in wall temperatures, which, according to Figure 4.17, are underestimated by the model. 4.5.3. Comparison with Data of Andersson and Leckner (1992) Andersson and Leckner (1992) reported the results of experiments carried out in a 12 MWth circulating fluidized bed boiler built at Chalmers University of Technology in Sweden. The combustion chamber, consisting of membrane tube walls, has a crosssection of 1.7 m by 1.7 m and is 13.5 m tall. The outer tube diameter is 60.3 mm, tube wall thickness 5.6 mm, fin length 8.8 mm, fin thickness 6.0 mm and the radius of the fintube junction 4.0 mm. The bed material was silica sand with a density of 2600 kg/m , and 3 the fuel was 0-20 mm bituminous coal. The suspension densities in the case considered here varied from 45 kg/m at 2 m to 10 kg/m at 11 m, as shown in Figure 4.19. Four methods were employed by the authors to determine the heat transfer coefficients: Method 1: Heat balance An average heat transfer coefficient between the bed and the cold membrane wall can be calculated from: 156 where T be is the temperature of the furnace side crest of the tube (position tu T\ in Figure 4.20) which must be averaged over the height of the combustion chamber since the tube temperature increases along the heat of the cooling water. The average heat transfer coefficient calculated by this method was 127 W/m'K. i—•—i—•—i—'—i— —i—•—i—•—i—•—i—•—i—•—i— 1 i . i . 0 5 i . i . i . i . i 10 15 20 25 30 r . i . i • i • i i i i i • i i i 35 40 45 50 55 60 65 70 S u s p e n s i o n density (kg/m ) 3 Figure 4.19: Suspension density profile in experimental study of Andersson and Leckner (1992). 157 Method 2. Local heat flow meters A water-cooled, conductivity-type heat flow meter was inserted through holes between two adjacent tubes. The meter was provided with wings to cover the hole and to avoid disturbing the flow of particles along the wall. The heat transfer coefficient is calculated from the measured heat flow and the temperature difference between the core of the riser and the surface of the meter (position I2 in Figure 4.20). The latter temperature was obtained by extrapolation assuming a linear temperature profile in the brass cylinder of the meter. The heat transfer coefficients calculated using this method are shown as open circles in Figures 4.21 and 4.22. Method 3. Local water temperature In method 3 the local temperature of the cooling water inside the tube is determined as the measured wall temperature on the insulated side of the tube (position I4 in Figure 4.20). Local values of absorbed heat are then estimated from the increase in water temperature along the tube. Also, in this case, the bed-to-wall heat transfer coefficient is based on the temperature difference between the core and the furnace-side crest of the tube (position T\ in Figure 4.20). The heat transfer coefficients calculated using this method are shown as + symbols in Figures 4.21 and 4.22. Method 4: Fin-tube temperature difference In method 4, the temperature field in the membrane wall and tube was analyzed by the finite element program Ace, and a linear relationship between the furnace-side heat 158 flux and the temperature difference between the fin and insulation-side crest was established. Hence by measuring the temperature at those two positions, an estimate of heat flux was obtained. The calculated heat transfer coefficients are shown as x symbols in Figures 4.21 and 4.22. To further analyze the heat transfer in thefinregion, a heat flow meter was positioned at a height of 3.8 m where the suspension density was about 25 kg/m . An obstacle in the 3 form of a tube with a 48 mm diameter was inserted into the combustion chamber 0.1m from the fin surface, 0.5 m above the meter. This arrangement increased the heat transfer coefficients measured by the heat flow meter by 50% as shown by the solid diamonds in Figures 4.21 and 4.22. Figure 4.20 shows the FEM mesh used to solve the membrane wall and wall layer heat transfer model for this case. Since the wall layer is quite thick, a line parallel to the fin surface is assumed to be the bulk side boundary of the wall layer. Figures 4.21 and 4.22 show the predicted heat transfer coefficients obtained for fin and crest particle residence lengths of L art = 1 m, L f = ar 2 m and L , ar - 0.5 m, L /= 1 m, respectively. Since ar the wall emissivity is unknown, values of 0.6 and 0.85 are tried. 159 0.12 Figure 4.20: FEM mesh generated for membrane wall and wall layer of CFB used by Andersson and Leckner (1992) Method 2 actually measured the heat transfer coefficient at the fin surface. Hence when comparing the predictions to the experimental results obtained from method 2, the heat transfer coefficient at the fin region is used and the temperature predicted at position T is used as the wall temperature. The predicted fin heat transfer coefficients are 2 illustrated in panel A of both Figures 4.21 and 4.22, while the predicted total heat transfer coefficients are plotted in panel B. 160 I 0 I I 50 I I I 100 I 150 I I 200 I I 250 I I 300 0 I I I I 50 I 100 I I 150 I I 200 I I 250 I I 300 Heat transfer coefficient ( W / m K ) 2 Figure 4.21: Comparison of predicted local heat transfer coefficients for L = 1 m and L f= 2 m with experimental data of Andersson and Leckner (1992). O : method 2; +: method 3; x : method 4; solid diamond: method 2 with obstacle 0.5 m above meter. Dashed lines: prediction for e = 0.6; solid lines: prediction for e = 0.85. A: heat transfer coefficient at fin; B: total heat transfer coefficient for fin and tube. Lines 1 and 2: Predictions assuming heat transfer starts 0.5 m above heat flux meter. ari w ar w 161 0 50 100 150 200 250 300 0 50 100 150 200 250 300 Heat transfer coefficient ( W / m K ) 2 Figure 4.22: Comparison of predicted local heat transfer coefficients for L = 0.5 m and L f= 1 m with experimental data of Andersson and Leckner (1992). O : method 2; +: method 3; x : method 4; solid diamond: method 2 with obstacle 0.5 m above meter. Dashed lines: prediction for e = 0.6; solid lines: prediction for e = 0.85. A: heat transfer coefficient at fin; B: total heat transfer coefficient for fin and tube. Lines 1 and 2: Predictions assuming heat transfer starts 0.5 m above heat flux meter. arl ar w w When the obstacle was inserted 0.5 m above the heat flux meter, it destroyed the wall layer within the fin region and fresh particles were introduced from the core to the wall layer. Hence a new heat transfer process starts from the obstacle. Lines 1 and 2 in both 162 figures show the predicted heat transfer coefficients based on this assumption. Experimental results from the same boiler obtained by Golriz (1996) showed that if the obstacle was placed 2 m above the heat flux meter, the influence of the obstacle was negligible. Our simulations also demonstrate that, after 2 m, the heat transfer coefficients in the presence and absence of the obstacle are very similar. 4.5.4 Comparison with Data of Andersson (1996) Table 4.2: Main operating conditions of Andersson (1996) Case A B C D E F G H I J U (m/s) 1.76 1.76 1.83 3.58 4.53 2.66 2.68 2.65 3.68 6.39 u, (m/s) 3.47 2.38 1.25 2.26 3.33 3.43 2.39 1.32 1.90 3.45 883 873 830 857 845 837 845 826 865 881 637 659 777 777 826 711 747 799 842 871 dp (pm) 440 335 220 323 425 435 335 227 288 438 0.38 0.42 0.46 0.39 0.36 0.38 0.41 0.44 0.39 0.18 g Ts, CQ #*(m) Andersson (1996) measured local suspension densities and suspension-to-membranewall heat transfer coefficients in 10 sets of experiments carried out in the same CFB boiler at Chalmers University. Three different narrow-sized Tractions of the same silica sand were used as the bed materials. The Sauter mean particle diameter of the active bottom bed material, including sand, fuel and ash, varied between 0.22 and 0.4 mm, and the particle density of the sand was 2600 kg/m . The bed temperature was measured near 3 163 the bottom of the bed, just above the primary air distributor, and at the gas exit. For tests in which the top and bottom temperatures were different, a linear profile was assumed between Z = 2ra and the gas exit, which yielded the core temperature for evaluation of local heat transfer. The important operating conditions for the ten runs are listed in Table 4.2. The cross-sectional average suspension densities were evaluated from pressure drop measurements, and are shown in the upper panels of Figures 4.22 and 4.23. The measured local suspension densities with linear interpolation between each pair of adjacent points are used as inputs for the model simulations. In these simulations, the bulk temperatures are also assumed to vary linearly from Z = 2mtoZ=Has described above. The lower panels of Figures 4.22 and 4.23 compare the predicted local heat transfer coefficient with the experimental results. In these simulations, a wall emissivity of 0.8 is employed. The author also evaluated the lateral variation of heat transfer around the tube and fin of the membrane wall. The surface was divided into three parts, the tube crest, the tube side and the fin. Average values of heat flux were estimated for each part. In order to compare results from different locations obtained under different operating conditions, the local values (q , q and qj) are normalized by the average total heat flux, i.e. they are c divided by {{q l c c s +qj +Q lf)l{l s f c where +h +1/)) l, l c s and // are the lateral distance along the crest, side and fin surfaces, respectively. Figure 4.25 compares the predicted lateral variation of relative heat flux with experimental data at the heights of 10.5 m and 3.4 m. For most cases, the model predicts well in the crest region but underestimates the relative experimental values in thefinregion. 164 eg o i- 00 T- ~\ CD I I I 1 1 1 1 eg CM T- I T 1 o I T- I 1 oo co •<* I I I I 1 1 1 1 o CM ID CM o O CN O ID I • 11 T— o o o • • •• I I1 ' I ' 1• 1I 1' I • 1 I • I o o CM o ID O O Q -o I i TV*" 1 - M l I • 1• 1' I i I ' 1 I c c CM O ID O O CO o CD •o ,• •••.* o c 0 1 • I • I • I • I O o CO o CL CO ID I ' I • I ' I • I •I O ZZi CN O ID O CO o o f »l ' I A ID I 1 I 1 I 1 I 1 I 1 1 o o CM O ID O O O ID i . mm, • CN O CO 7 CD (LU) 1U.6J8H c CD 4— M— CD o o CD *+— GO c CO o CO 1 ' 1*' E ID i . i *i t CN (LU) 1L|6!9H 165 CO CD X <- I T- I 1 00 I 1 CO I 1 I I 1 1 CM CN O T- 1- 00 O O CN O lO v— o o • ••• I I I I I I o I • 1 II I • I ' I ' I ' 1 ' o o CN O in o o -o ••••••• I i ! ' 1 I ' I • I • I ' I ' I ' 1' I 1 I o o to c CN in CD T3 o o c o o in CO c I 1 I 1 1 1 I 1 I E in 1 I CD Q. CO I i I • I i I ' 1 CN O CD CD in o o • I I I 1I H 0 o o — i CD CO c ' I o o (0 — CO CD 0 X o in I ' I ' 1 II o o CN O in o o o m CN O (oi) CO CO ^1- 3 CN (OI) m6|8H 166 1.5 1 1 ' I < I I 1 _ I 1 1 1 i . 1 | 1 1 . 1 1 1 1 1.0 X ZD 0.5 - CO CD 0.0 x; <D > 1.5 CD 1.0 F l , 0.5 . l 10 i l 1 1 i l 30 20 , 1 i l 40 1 1 50 1 0 10 _ 1 - G i 0.0 F . 10 i . 20 i . 30 i . t 40 50 0 50 0 i . 20 i • i . i . 30 i| • i 1 i i 40 50 i • i i . G 10 i , 20 i . 30 40 i 50 X 3 co cu j= CD > cu 1.5 1 1 ' 1 I 1 I 1 ' 1 i ! i | i | i | i | 1 . 1 , 1 . 1 . 1 1.0 0.5 X 3 CO CD CD > TO CD " "' 1 0.0 1 1 1.5 1 1 1 20 10 1 1 30 1 1 40 1 " 1 50 1 1 1 1 1 I , . , 1 1 1 . 1 . 1 1 1 0 10 1 I 20 1 30 1 1 1 . 40 1 1 1 . 1 50 I 1 1.0 0.5 " 0.0 J " 10 20 30 40 50 Distance from crest (mm) J 1 0 10 20 30 1 40 1 50 Distance from crest (mm) Figure 4.25: Comparison o f predicted lateral variation o f relative heat flux (solid lines) with experimental data o f Andersson (1996) (dashed lines). Left panels: height is 10.5 m; Right panels: height is 3.4 m. Operating conditions are listed in Table 4.2. 167 4.5.5. Comparison with Data of Andersson etal (1996) Andersson et al. (1996) reported experimental results carried out on a 165 MWu, circulating fluidized bed boiler located in Orebro, Sweden. The combustion chamber, consisting of membrane tube walls, has a cross-section of 4.7 m by 12 m and is 33.5 m tall. The membrane tubes are 63 mm in outer diameter with a fin half-length of 10 mm. The bed material was sand with a density of 2600 kg/m and particle diameter 250 pm, and the fuel was coal (Golriz, 1994). The bed temperature in the case presented here varied from 780 °C at the top to 850 °C at the bottom, while the suspension density varied from 4.5 kg/m at the top to 5.8 kg/m at the bottom (Golriz, 2000). Details of the 3 3 membrane wall geometry were also provided by Golriz (2000). Figure 4.26 shows the geometry of the membrane wall and wall layer as well as finite element mesh generated for this case. For the operating conditions described above, Andersson et al. (1996) reported that the average suspension-to-wall heat transfer coefficient determined by method 4 described in Section 4.4.3 was about 96 W/m K. Figure 4.27 shows the predicted local suspension-to-wall heat transfer coefficients assuming linear variations of suspension density and bulk temperature from top to the bottom, and a wall emissivity of 0.6. 168 -0.02 -0.04 0 0.02 0.04 Figure 4.26: Geometry o f the membrane wall and wall layer and finite element mesh generated for the 165 MW„, C F B boiler in Orebro, Sweden. 169 —I 200 . I 250 , I 300 H e a t transfer coefficient ( W / m K ) 2 Figure 4.27: Comparison of predicted local heat transfer coefficients (solid line) with experimental average heat transfer coefficient (dashed line) for the 165 MWd, C F B boiler in Orebro, Sweden. 170 4.6. Influence of Fin Width on Heat Transfer Coefficient Two novel features of the model presented in this chapter are the consideration of the membrane wall geometry and coupling of the reactor-side and wall-side heat flows. Hence, it allows one to investigate the influence of system parameters on the total rate of heat transfer. Here thefinhalf-width is changed from 6.4 mm (corresponding to the base case)firstto 3.2 mm and then to 12.8 mm. Two operating conditions were tested: A: T = b 804 °C, T , = 80 °C; B: T = 706 °C, T , = 370 °C. c oul b c oul Figure 4.28 shows the predicted vertical heat flux profile based on the total exposed area and also on the projected area of the membrane wall. It can be seen for operating conditions A, that heat fluxes based on total area decrease for the narrowerfin,while those based on the projected area increase. For operating conditions B, the heat fluxes based on the total area are almost the same forfinhalf-widths of 3.2 mm and 6.4 mm, and both are higher than for afinhalf-width of 12.8 mm, while for heat fluxes based on projected area, narrowerfinsagain yield higher values. Figure 4.29 compares the lateral distributions of radiative, conductive and total heat flux along the membrane surface at z = 1.8 m for operating conditions A. Both the predicted radiative and conductive heat fluxes increase when thefinis wider. The reason for this is that when thefinis wider, the pipes are separated farther; hence both the pipe and fin see more of the bulk (i.e., have increased view factors). Wider fins further increase the particle temperature close to the wall and consequently increase the rate of 171 conductive heat transfer. However, if the fin is too wide, the fin conductive resistance becomes significant and reduces the total heat flux delivered to the tube and hence to the cooling water. As shown in Figure 4.29, for a fin half-width of 12.8 mm, the conductive heat flux at the far end of the fin is smaller than in the rest of the fin. Figure 4.30 compares the lateral distributions of radiative, conductive and total heat fluxes along the membrane surface at z = 1.8 m for operating conditions B. Under these conditions, the conductive and total heat fluxes in the fin region are smaller for a fin halfwidth of 12.8 mm than for 6.4 mm, while in the pipe region, the conductive heat fluxes at 12.8 mm are still higher than those for half widths at 6.4 mm and 3.2 mm. In practice, a number of practical considerations are likely to determine the width of the fins, or at least to limit the range of choice. First, the fin width should be at least 30 50 particle diameters to prevent particles from becoming wedged in the gap. Second, manufacturing the membrane walls, in particular welding the fins and tubes, is facilitated by having a wider fin. The model presented in this chapter is useful in considering the heat transfer consequences of different fin widths. 172 Figure 4.28: Influence of fin width on vertical variation of heat flux based on total area and projected area. Dashed line: vc = 12.8 mm: Solid line: w = 6.4 mm; Dash-dotted line: w = 3.2 mm. A: T = 804 °C, T . = 80 °C; B: T„ = 706 °C, T , = 370 °C. b c ml c out 173 60 1 r 1 r 50 *> 40 _ _ _ _ j 30 U - I 20 cT£ W H 1 r- t ^ - i — h h i—i—i—i—i—i- 50 I 40 30 20 90 H n 1 h—| h Fin side 80 ""1 70 Cr Pipe side 60 50 i 10 15 20 25 Distance from crest (mm) Figure 4.29: Influence o f fin width on lateral variation o f radiative, conductive and total heat fluxes at z = 1.8 m for A : T = 804 °C, T _ = 80 °C. Dashed line: w = 12.8 mm: Solid line: w = 6.4 mm; Dash-dotted line: w = 3.2 mm. b 174 c oul 30 35 T 30 "£ *S A; 1 r 1 " "1 1 1 r- 1 H L 2 5 20 -17=-= ' v. 15 10 5 H cC~- 25 E %, 1 1 1 H i 1 h f—i—I—i—I—H 1 1 1 \^J^' l._?J 1 20 Cr 15 10 H 1 1 60 ^ 50 § 40 f- Cr Pipe side Fin side 20 \ 10 10 _. Li _L 15 20 25 30 D i s t a n c e from crest (mm) Figure 4.30: Influence o f fin width on lateral variation o f radiative, conductive and total heat fluxes at z = l .8 m for B : T = 760 °C and T _ , = 370 °C. Dashed line: vc = 12.8 mm: Solid line: vc = 6.4 mm; Dash-dotted line: vc = 3.2 mm. b 175 c ou 4.7. Summary The model developed in Chapter 2 is extended to circulating fluidized beds with membrane walls, a more complex geometry. The governing equations are solved by finite element and finite difference methods using the moment method for radiation transfer. The solution is first demonstrated for a typical example. The model is then tested against experimental results from the literature and gives mostly satisfactory predictions of the suspension-to-wall heat transfer rate. The influence of fin width on the heat transfer is also analyzed. 176 CHAPTER 5 O V E R A L L CONCLUSIONS AND RECOMMENDATIONS 5.1. Overall Conclusions A two-dimensional model that couples the reactor side particle and gas convection, particle-to-gas convective exchange, gas conduction, particle radiation and wall side conduction is developed for heat transfer in low-density circulating fluidized beds with smooth walls. The model is then extended to high-density CFB operation with smooth walls by allowing the suspension in the vicinity of the wall to travel intermittently downwards and upwards as observed experimentally. A parameter fd, defined as the fraction of particles moving downward in the vicinity of the wall over total particles moving in both directions, is introduced. The model is also extended for fast fluidization to the three-dimensional geometry of membrane walls. The two-flux model is employed to solve radiation transfer in the two-dimensional model, while the moment method is employed for the three-dimensional case. The two- and three-dimensional models are validated using experimental results from the literature and both yield satisfactory predictions of the suspension-to-wall heat transfer. The predicted influences of different parameters on the heat flux are consistent with experimental trends where these are known. The low-density model predicts that both the conduction heat flux and radiation flux decrease while particles descend along the heat transfer surface if the suspension density 177 is constant. The simulation results suggest that the particles participate in a significant way in determining the radiation flux through the wall layer. Therefore radiation cannot be uncoupled from particle and gas conduction and convection without introducing significant error for high temperature systems. Experiments in a dual-loop high-density CFB facility show that the direction of particle motion in the vicinity of the wall of a circulating fluidized bed can be deduced from the suspension temperature distribution. Due to the heat transfer from the heat exchanger to the suspension and particle upward and downward motion in the vicinity of the wall, the suspension temperatures near the wall below and above the heat exchanger are higher than those at the axis of the riser. The average suspension-to-wall heat transfer coefficients are strongly influenced by suspension density. However, they are not significantly influenced by superficial gas velocity at a constant suspension density. The local heat transfer coefficients are strongly associated with particle motion in the vicinity of the wall. In most cases, the wall layer motion is not unidirectional, but oscillates downward and upward, leading to higher local heat transfer coefficients at the ends of a heated surface. By superimposing the heat transfer results when the suspension in the vicinity of wall is allowed to move downwards and upwards separately, the high-density model predicts the experimental results well. 5.2. Recommendations for Future Research The current model for heat transfer in dense suspension upflow assumes that the steady-state results obtained for two independent layers moving-upwards and downwards 178 can simply be superimposed. In reality, the particles in the vicinity of the wall oscillate upward and downward, yielding an unsteady problem for which the current approach may be insufficient. The model should be further extended to provide a more realistic representation of particle motion near the wall. Suspension-to-wall heat transfer is strongly associated with the system hydrodynamics, especially the particle and gas motion in the vicinity of the wall. However, hydrodynamic studies on high-density circulating fluidized beds are very limited. Both experimental and modeling studies are needed to extend understanding of particle and gas motion in high-density flow regimes. Most high-temperature studies have focused on circulating fluidized bed boilers, i.e. in the low-density flow regime. In high-density flow regimes where the particle concentration is high, independent scattering theory may not apply and optical properties such as the absorption and scattering coefficients and the scattering phase functions of the suspension may differ from those under low-density conditions. Additional theoretical and experimental studies are needed to resolve these issues. Solids exchange between the core and the wall region is important in the suspensionto-wall heat transfer process. Studies are needed to confirm the particle average residence length in the vicinity of wall in both low-density and high-density flow regimes. 179 NOMENCLATURE a a ff A A Ao a e c w ax B c C a pc Qpel •pe c,Pg PP C D s Up eb ei ed e e e E c eff p w x h fw G G Gd G h h' h hd s S su c hf h hp PS h suspension absorption coefficient for gray medium effective suspension absorptivity geometric cross-sectional area total inside area of the inner cylinder total outside area of the membrane surface effective wall absorptivity suspension absorption coefficient at wavelength X back-scattering fraction particle volumetric concentration absorption cross-section cooling water heat capacity cluster heat capacity effective suspension heat capacity gas heat capacity particle heat capacity scattering cross-section diameter or hydraulic diameter of riser particle diameter dimensionless particle diameter bulk emissivity effective cluster emissivity effective dispersed phase emissivity effective suspension emissivity particle emissivity wall emissivity particle exchange rate between core and wall layer time-averaged fraction of particles moving downward in the vicinity of wall time-averaged fractional area of the wall covered by particles irradiance solid circulation flux solid circulation flux moving downward in the vicinity of wall solid circulation flux moving upward in the vicinity of wall bed to wall heat transfer coefficient bed to water-side wall heat transfer coefficient heat transfer coefficient from tube to coolant bed to wall heat transfer coefficient due to downward particle motion in the vicinity of wall heat transfer coefficient in the fin region of membrane wall heat transfer coefficient due to gas convection heat transfer coefficient in the pipe region of membrane wall particle-to-gas heat transfer coefficient 180 m m m m 2 2 m m J-kg'-K" J-kg'-K" J-kg-'-K" J-kg'-K" J-kg'-K" m m m 1 1 1 1 1 2 kg-nf -s"' 3 W-m • kg-m"-s"' kg-m"-s"' 2 2 2 1 kg-m" -s" W-m- -K-' W-m- -K"' W-m^-K" W-m" -K-' W-m" -K"' W-m" -K"' W-m" -K"' W-m" -K"' 2 2 1 2 2 2 2 2 h r h i rx hd h h, r s h u H H V I f lb i' ixb' ki k kg k k L L' L Lan x x c e p w a r Lf ar L L m N q q q Q Q Q Q r ri r R Ri R s S T c w c c r a pg s 2 0 heat transfer coefficient due to radiation heat transfer coefficient due to cluster radiation heat transfer coefficient due to dispersed phase radiation heat transfer coefficient due to particle convection heat transfer coefficient at both the pipe and fin regions of membrane wall bed to wall heat transfer coefficient due to upward particle motion in the vicinity of wall length of heat transfer surface height of bottom bed radiation intensity radiative heat flux in"-" direction radiative heat flux in"+" direction emissive power of black body spectral radiation intensity spectral radiation emission intensity by black body cluster conductivity suspension effective conductivity gas conductivity particle conductivity wall conductivity length of cross-section where coolant is heated entrance transition length particle average residence length in wall layer particle average residence length in wall layer (tube region) particle average residence length in wall layer (fin region) hydraulic diameter of cross-section occupied by coolant wall thickness cooling water flowrate particle number heat flux through wall heat flux through stagnant gas layer radiative heat flux heat generation rate per unit volume absorption efficiency volumetric heat convection ratefromparticles to gas scattering efficiency radius outer radius ofthe inner tube ofthe concentric heat exchanger inner radius of the outer tube of the concentric heat exchanger inner radius of the inner tube of the concentric heat exchanger inner radius of the membrane tube outer radius of membrane tube particle surface area per unit volume horizontal gas temperature gradient temperature 181 W-m" -K"' W-m" -K"' W-m" -K"' W-m" -K" W-m" -K"' 2 2 2 2 ! 2 W-m'^K" 1 m m W-m"-sr W-m" W-m" W-m" W-m"-pm"'-sr" W-m"-pm"-sr' W-nf'-K" W-nf'-K" W-nf'-K" W-nf'-K" W-nf'-K" m m m m m m m kg-m" 2 2 2 2 2 2 1 1 1 1 1 1 3 W-m" W-m" W-m" W-m" 2 2 2 3 W-m" 3 m m m m m m nf' Km" K 1 bulk temperature T cross-sectional average coolant temperature gas temperature T particle temperature T T suspension temperature suspension temperature in the bottom part of the furnace Tb Tsbc suspension temperature at the center of the riser below the heat exchanger Tsbw suspension temperature in the vicinity of the wall below the heat exchanger suspension temperature in the top part of the furnace T suspension temperature at the center ofthe riser above the heat T tc exchanger Tstw suspension temperature in the vicinity of the wall above the heat exchanger wall temperature T overall bed to cooling water heat transfer coefficient U U* dimensionless superficial gas velocity u average axial coolant velocity (upwards) u axial gas velocity (downwards) u axial particle velocity (downwards) u particle terminal velocity (downwards) ug superficial gas velocity umf particle minimum fluidization velocity V volume w half-width of fin X horizontal distance from inner furnace wall z vertical coordinates, directed vertically downward Z height above air distributor grid step length in horizontal direction Ax grid step length in vertical direction Az T b c g P s S sl s w c g p t K K K K K K K K K K K K W-m" -K" 2 - m-s" m-s" m-s" m-s" m-s" m-s" m m m m m m M 1 1 1 1 1 1 3 Greek Letters o 5 8 S £ £b £mf £sec scattering phase function wall layer thickness gas gap thickness suspension voidage suspension voidage in bulk region loosely packed bed voidage cross-section average suspension voidage dimensionless lateral distance in theriser(=l-x/D) Polar angle gas viscosity 182 - m m - kg-nf'-s" 1 e Pel Pdis Pe Pg Pp Psus a o- s CO azimuthal angle cluster density effective density of dispersed phase suspension effective density gas density particle density suspension density Stefan-Boltzmann constant scattering coefficient for gray medium scattering coefficient at wavelength X solid angle kg-m" kg-m" kg-m" kg-m kg-m" kgm W-nf -K m" m" sr 3 3 3 -3 3 -3 2 1 1 Dimensionless N u m b e r s Ar Archimedes number Fr Froude number Fr' Modified Froude number gd p Apl p U„ g g (p -p )gd p Nu Nusselt number g hL g Pr Prandtl number Re Reynolds number p g g pg uL v 183 p REFERENCES Ahmad, M., X. 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Schematic of two-flux model ~ dS = -ai'(S) + ai \S) - cr i\S) b s +^ t' HSWa^da), (1.2) An *'.- =0 Assume that, for one-dimensional energy transfer, the intensity in the positive direction is isotropic and that in the negative direction is also isotropic but has a different 193 value (see Figure 1.1). The equation of transfer for the intensity in each hemisphere is written as cos <p — dx = -af (S) + ai (S) - a f(S) cos <p ~ dx = -ai~ (S) + ai (S) - a f(S) b s b s + An f* •. /' (S)O(o), a>, )da>, (1.1) b =0 + An • . (S)0(a>, co )dco t i (1.2) }iJ =0 These equations are integrated over their respective hemispheres to give 1 df = -(a + a B)f s 2 dx 1 di 2 dx + ai (S) + a Bi~ = -(a + a B)r where B = ^— s £ b (1.3) s + ai (S) + a Bf b (1.4) s (1.5) ^(co^Adoy^rnqxiq) is the back scatter coefficient. For gray media, large particles (nd IX > 5), and independent scattering (Siegel and p Howell, 1992), the phase function <> t can be described by equation (1.10). When O is substituted into equation (1.5), it can be calculated that B = 0.667 (1.6) Based on equation (1.3), the total radiative fluxes in the + and - directions are related to the radiative intensity by f = n f (1.7) r=nf (1.8) 194 Hence, 1 dl + 2 dx = -(a + asB)r +aI (S) + o-sBr (1.33) b i— - -(a + a B)r 2 dx s + al (S) + cr BI (1.34) + b s From equation (1.4), the local divergence of the radiative flux Vq can then be r derived as: = ^~ Kdr__dr_ d Vq r r dx dx dx =A a h _ 2a(I+ 195 +r ) (I9) Appendix II: Application L Application of Keller's Box Method of Keller's box method to the gas phase heat transfer equation For spherical particles, equations (2.1), (2.2) and (2.22) lead to dT dT, ai. i) where P 1 = 01.2) Pfpz z £ u 6(2 +1.8 x ( P ( U p u ' ) d ')" Pr )(1 1/3 e)k (II.3) b=P D PZ PZ Z £ C U Note that p, b and k are functions of x and z. g Figure II. 1 Schematic o f Keller's box method Let s- dT s (II.4) g dx 196 Then equation (II. 1) becomes dT dk dS g (II.5) g The finite difference form of Equation (II.4) is written for point (i+\/2,j+l), i.e. (II.6) Similarly equation (II.5) is replaced at the mid-point, (/+l/2,y'+l/2), by T -T T -T (<+i/2,y+i/2) SlJ Ax, Ax, Az, Az, S<J+l (II.7) +l + (7+l/2,y*l/2) + b( • • l / 2 . y t i / 2 ) Equation (II.7) may be rewritten as U gVJ+l) B T + \l g(MJ*\) B T ~ 2l KU*i) B T ~ 2i p(MJ l) B T + + 3i i B S J + l ~ Ai MJ*l B S ~ 5i B ( ) IL8 where ii = 2l ^(,„/2.y,./2) >-\ (II.9) j 2 (11.10) ^ J 197 dk„ (M )(/,./2,,>,/2) 5,41 Az (11.11) zlz, (11.12) ; AX: g , ( / W2.y+l/2) ? /lx.. = a - - B » ) ^ +0-5,)r + 2 ^ , - 5 , ^ . +BJS,i+lj (11.13) For heat conduction through the gas layer (11.14) 2. Application of Keller's Box method to the particle heat transfer equation Heat transfer in the particles is described by equations (2.2), (2.3) (2.4) and (2.31). Hence 9T dl — = f~ dz dx + n J -1 - where / = P Cp u dl~ f — + gT„+lT+r dx * " (11.15) g ,g= 5c a c p p p C D Z PP PP P C a ,1 = PP PP P CU C CU Equation (11.15) is replaced at the mid-point (z+1/2,/'+1/2), i.e. 198 , 5c r=— T b C D Z (11.16) I TPi'+hj+i)—T p(i+Uj) T - T p(JJ+i) p(i,j) 1 2 1 1 Az,. Az.. I (/+!, y+l) ~ I (i.j+l) ^ I~(i+\.j) - 1 (ij) (11.17) 7+1/2) Ax, 4-T T + Ax 4-T %(i+\ll,j+\l2) Ax, 4-T +1 (/+1/2J+1/2) Ax, 4-T 4-T 4-T g('J) ^^sii+lj) " -'g(/.;>l) ~*~ g(/+l,y+l) T r 1 + r, (/+1/2./+1/2) or ( gOJ+i) T + + ) 2i ( p(ij \) +D g(M,j l) T + + p(MJ \) T ) T + + • 3 / ( ^ ( ' . 7 + i ) - / (/+i,y+i) - /"(/.y+i) + /"(1+i.y+i)) = D D + + 4 (11.18) , where D = _W ^ ) ^ 2 ^4* = -Du(T + g >, , (J = 2 7 + 7;, , ) (2 - D )(T +I s ( i J ) y) + 2i (-I*OJ)+I*{Mj)+I D /2 +1/2) 7 +r , p0J) Ax. 3 1 + 1 1 ,) aj)-I (u\j)) + 2r.((+i/2.y+i/2)^2 3i (11.19) (11.20) Given equations (1.33) and (1.34) for the radiation flux in the medium, we can write for point (/,/'+!) dl — + =-2(a +cr , /i)/ , + (0 i( ) ( y+I) +2a cr T ; (i) 0 p(i + 2a BI s{i) (n (/,y + i) dx = 2(a (/) +cr , /i)/- ,, i( ) ( +1) -2a <Tj w A m Similarlyforpoint (/+1, y'+l) 199 -2^,)BI + (i) (11.21) (11.22) dl* ~77 =- 2 0(, +I) + ^S{M)S)I\MJ*\) + 2a aJ (i+l) +2a 4 p(M) S ( M ) BI ( M ) (11.23) ' (*X (»+l,y+l) dl dx „ . . ..... = 2 ( a c > D +o- B)r s(M) (,+i.y+i) -2a cr T {Mj+l) (M) 0 -2a BI A p(M) s0+l) (11.24) + (M) Thus, for the mid-point (z'+l/2,y'+l), _ 7 ,+l,y+l — I iJ+\ dl + + dx ( i/2j+i) Ax (+ dP 1 + 2 i dl +• + dx (ij+i) dx o+ij+i) (11.25) + ^ ( , i ) £ y V w i ) +a o- T -( (/ D a + + + 0+l) _ I /+i,y+i — / c?/ (I+I/2J+1) 0 _ 1 Ax s{M) (M) <fr d7 _ dx (ij+i) 2 t +o- BI ~ 4 p(M) + •dx (i+ij+i) 4 -a o- T ( (i i)+°- « i) )I~(MJ+i) + B a + s (11.26) (i+l) + 0 - 4 p(i+1) a BI s(i+1) + (i+1) Hence lFp(ij+\) C C\Fp(i,j \) + + Qi^(i+l.y+l) 4 +CT 2i A + G 3 i ri \ + C ,/ / l,y+i + + J + 4 + + + 5/ 6i + /,y+l + 5j +C + C / / , y + l + C I Mj i p0+lj+]) Cl I Vl Q,7 ,,y+i + C g / , i,y i = + + 0 / ,+i.y+i = 0 (11.27) (11.28) where \, = a^Ax,, c C , = -1 4 C = a <j Ax,, C . = 1 - ( a 2i {M) 0 + a B)Ax sli+i) i 3l ( 0 + cT 5)ztx,. j(l) C = cr BAx , C , = cy , /3zk,. Si s(i) i 200 6 j( +1) (11.29) (11.30) C , = -1 - (a + cr B)Ax ,C , = 1 - (a 7 v) s(l) t g 0+l) + o- B)Ax s(M) i For radiation through the transparent gas layer / («j+D + = oT e w +(l-e )I~ 4 w w J+]) For radiation from the bulk: ^j*\ = b b I e aT ) +( - b) *j+\) l e oT\ + (1 - e j / ^ b e y+1) I - I- mj+l) o r =0 For heat transfer through the wall ? 1 ( ~ g(0,M) K , ^(Q.y+O >, r o ^g(Q.y+0 + r I + or where 201 0.7+1) J C.7+I) f * . „ - n r 1 0 r » - The (5/n+l) unknowns ( ^ , 7 ^ , , , T , p{iJ+i) from the set of equations given in equation (2.32). 202 S, , / V i a n d / V O are obtained J+l Appendix III: Application of Moment Method for Radiation in CFB Wall Layer Consider radiation of intensity i'(S) within a continuous gray absorbing, emitting and scattering medium. Attention is directed to the change of intensity as the radiation passes through the distance dS. Considering energy loss by media absorption and scattering and gain by media emission and the scattering into 5 direction, the equation of transfer can be written: S y Figure III. 1: Schematic of moment method. ^ dS = -oi'(S) + ai \S)-<j i'(S) b s +^ (\i'(S)0{a,co )dco i AK • . {J =0 l (1.2) In two-dimensional rectangular coordinates, the directional derivative dldS is related to the partial derivatives with respect to x and y by 203 d d d •—= coses— + .sines cost?— dS dx (WEI) /TIT dy I N where cp is the polar angle measured from the positive x axis and 6 is the azimuthal angle. Thus, in rectangular coordinates, equation (1.14) becomes di' di' <J r*f cos <p (- cos 6 sin <p — = -ai' (x, y) + ai ' (x, y) - aj' (x, y) + —— b dx dy /' (x, y)0(a>, co )dco i An .-= A, 0 (III.2) The intensity can be represented by an equation having the form /'= /' (x,>') + 6(x,>')cos(/3 + c(x,>')sint^cos(9 0 (4.2) where io, b and c are functions of position only and are to be determined. Equation (4.2) is substituted into (III.2). The resulting expression is multiplied successively by cos<r>, s'm<pcos0, and, in each case, the resulting expression is integrated over the solid angle 4 n. This procedure yields b=- ^ a + (l + f)a s c= - (III.3) dx ^ l a + (l + f)a s (III.4) dy where/depends on the scattering phase function O and, for the phase function described in equation (l.lO), 204 j / = 23.397100x3 An x 4 x = 0 4 4 5 (III.5) 8 Equation (4.2) is now substituted into equation (III.2) and the resulting expression is integrated over the solid angle 4 n to obtain db dc _ . „ . 1 = -3ai + 3ai dx dy Q (III.6) b Substituting equations (III.3) and (III.4) into equation (III.6) yields: 1 dx « + (l + / di K & + df 1 b\ — dy a + (l + f)a dy (111.7) = 3ai - 3ai 0 b s Based on its definition, the irradiance (incident radiation) G is given by G= ^Vdoo- ^ (i (x,y) + b(x,y)cos(p + c(x,y)s'm<pcos8)da> = 4ni Q 0 (III.8) 0 Replacing io with G in equation (III. 7) then gives, 1 dG d_ 1 dx 3a + (l + / ) c T dx s 1 1 dG = a(G-Ao-X ) dy 3a + (l + f)a dy +- P (4-3) s From equation (1.4), the divergence of radiative flux is V • q - Aae - a \ ?(a),S)da) = Aae -aG = Aaa T - aG r b b 0 Marshak's boundary conditions: 205 p (4.4) For a diffuse gray surface, the outgoing intensity /' from a boundary by emission and reflection is i'(0,u) = e + (1 -e )i'(0,-u), w (III.9) u> 0 w where u is the direction cosine of f relative to the surface normal. The intensity is related to G by cos <p di 1 dn An 0 a + (l + f)a s cos cp a + (1 + f)a G- dG s (III. 10) dn Substituting equation (III. 10) into (III.9) yields 1 L cos(p dG a + (l + f)a Any dn s An K G+ cos (p a + (l + f)o- dG s dn j 0<<p < — 2 (III.ll) Integrating equation (III. 11) over the hemisphere of incident solid angles then gives 2(2-O 3 1 dG a + (1 + f)cr dn s = (e G-Ae \ W aT) A W n O n ' / (III. 12) or dG Qr = 3(af + (l + /)o-Ja« (4.16) 2(2 - O 206 Appendix IV: Difference Application of Finite Element Method and Finite Method Finite Element Mesh Generation In the x and y directions, the solution domain is divided into six-noded triangular finite elements with 3 nodes on the corners of the triangle and three at the center of each side. The domain of each triangle is comprised of only one material, i.e., the membrane wall, the gas gap or the gas wall layer. The nodes are numbered counter-clockwise as shown in Figure IV. 1. Where there is a boundary, it always falls to the side consisting of nodes 2, 3 and 5. Figure IV. 1: A six-noded triangular finite element. Shape Function The temperature or irradiance profiles in the elements are assumed to be quadratic, i.e., 207 T(x,y) or G(x,y) = C,x +C y 2 +C y + C x + C y + C 2 2 iX 4 5 6 (4.20) The shape function can therefore be defined as tf," N 2 *6_ 'A A A M -r V y\ l x yi A A x yi A A A A A A X X 2 1 i x y* i x y$ i 6 y i_ 3 4 xy X (IV. 1) 5 We X 6 1 Hence the temperature or irradiance at the position (x, y) in the element is (IV..2) T(x,y) = f N (x,y)T j i i i=i G(x^) = 2 ^ ( x ^ ) G (IV.3) ( i=i Element Analysis Element analysis for heat conduction through membrane wall Heat conduction through the membrane wall is governed by equation (4.9). The Galerkin representation for this heat conduction problem is (IV.4) Using integration by parts on the left-side terms in equation (IV.4) and applying a 208 boundary condition of the second kind on T9 and a boundary condition of the third kind on T6 yields (the boundaries are defined in Figure 4.3) , dT dN. , dT, 8N, dxdy - \N —dx —dx + k„ dy dy (IV.5) dT, - \N,h(T -T )dT =0 iqr c 6 Note that the line integration on boundary T9 or boundary T6 only applies to the elements who have such boundaries. Inserting the temperature approximation of equation (IV.2) yields dN dN. , dN. dN. K—^—+K —-— dxdy + jhN,Njdr dx dx dy dy 6 \{T } = \hN T dT - JN,q dT WJ t c 6 r 9 (IV.6) Element analysis for heat conduction through gas gap Similarly to heat conduction through the membrane wall, the finite element representation for heat conduction through the gas gap (equation (4.8)) can be written , dN dN , dN, dN k„—dx dx- + k„ dy dy i g i g dxdy^{T } = 0. s) (IV.7) Element analysis for radiation transfer through wall layer Radiation transfer through the wall layer is governed by equation (4.3). The Galerkin representation for the radiation transfer problem is 209 1 dG dG +dx a + {\ + f)a dx) dy a + (\+f)a dy d_ s Splitting -3aG + l2aa T 0 p dxdy = 0 (IV.8) s into T T , inserting the temperature and irradiance approximations of T p p p equations (IV.2) and (IV.3) and applying Marshak's boundary condition on T9 yields f J l dN^dN^ dNj dN i - dx dx 0+ + / K a+a + \lacrJl \N N, dxdy\T \= t •3aN Nj t dy dy \ ,dr pj N 9 (IV.9) Element analysis for energy balance on gas wall layer The Galerkin representation for the gas energy balance equation (4.6) is 9T„ d dz 1 dT dx * dx V d dy dT g dy 2 g 2 " dxdy = 0 (IV. 10) where (IV.ll) A =sp C u i g _ A 2 pg 6 N g u c P S k (IV. 12) s o; Using integration by parts on the conductive terms in equation (IV. 10) yields 210 dT dT dN. 8T dN. A N —^- + k -^—^ + k^^+ A NT l l dz g n dx dx 8 dy 1 dy i g (IV. 13) -A N,T dxdy = 0 2 p Inserting the temperature approximations for gas and particles then gives dN^dN^ * dx •\\A N,N dxdy\T } 1 J ri dNj dN, dx ~dy~oy^^ "J +kg i dxdy\{T } s (IV. 14) =Q Global Assembly Having calculated the matrices and equations describing our approximation over each finite element, the next step in the analysis is to assemble the equations describing the approximation on the entire mesh by adding up the contributions to these equations furnished by each element. When this is done, the following set of equations is obtained: (IV. 15) (TV. 16) (IV. 17) 3T (IV. 18) M — - + K , T + K,„ T = F, r Q LO n % LP p L where K is the conductance matrix and M is the capacitance matrix. Equation (IV. 15) mainly describes heat conduction in the membrane wall. It has a T term because its g neighbor is the gas gap who contributes to the heat transfer in the membrane wall. Heat 211 conduction through the gas gap is represented by equation (IV. 16) where a contribution from the wall also appears. Equation (IV. 17) represents the radiation transfer through the particle wall layer, which is coupled to the particle temperature by the T term. The p complex heat transfer process in the wall layer involving gas conduction, gas convection and particle-to-gas convection is described by equation (IV. 18). The treatment of gas ST convection term M —- is discussed below. dz Finite Difference Treatment for the Convection Term M — L dz The following finite difference approximation is applied to the convection term in equation (IV. 18): (dT\ (dT + (l-<x) \ dz , Az (T -T ) (IV. 19) When cr = 0, the approximation is called the forward difference (Euler, explicit) method. It has a truncation error of O(Az) and is unstable. When cr = 1, it is called the backward difference (implicit) mediod. It has a truncation error of O(Az) and is unconditionally stable. The Crank-Nicolson difference (implicit) method applies when a = 1/2. It has a truncation error of 0(Az ) and is unconditionally stable. Finally, 2 a -2/3 yields the Galerkin difference (implicit) method which has a truncation error of 0(Az ) and also unconditionally stable. Here the Galerkin difference method is employed 2 and hence the convection term is replaced by 212 < dT. ^ -M 3 + - dz 2 M2 - A z ( dT. ^ ^ V 3 Z ( M / T ^ - M ^ J (IV.20) . ) z-tz Equation (IV. 18) can be rewritten at z and z - dz as f dT, ^ 3 (IV.21) : z-4z + — v T T 4-—K" (IV.22) ——F y & Combining equations (IV.20), (IV.21) and (IV.22) yields ( ^ + ! - K ) T - + ! K " T - = R (IV.23) S where R „ = - F + - F . „ + M*I*T 3 ' 3 zlz - - K „T z-Az 1 tz-Az ^ P. „.x„ z-Az p (IV.24) Galerkin difference application to particle heat balance The heat balance for the particles in the wall layer is governed by equation (4.7). Substituting equations (2.2) and (4.4) into equation (4.7) yields dT. ^ + B T + B T + B G + B,=0 dz l p 2 g (IV.25) 3 where 213 Sc P C -u +AacTj; p pp p 6 p„C „„cu. N u _ j 6 N v « k k g (IV.26) p C u d PP B _ PS p P z (IV.27) Pp PP P p C U d 2 a 5, (IV.28) p C cu„ n r p pp p Sc cSz B 4 (IV.29) Applying the Galerkin finite difference method to the convection term gives 3V| dz + = -UT i_f^A 3 Az - T *' K (TV.30) ) *-* P J Hence, at z and z-dz, equation (IV.25) becomes dz j dz 3 + 3 f ! \ B i > - * T P : - * 3 J i (IV.31) 3 + ^ 2 , - ^ ^ . + ^ 3 ^ 0 ^ + ±B ,_ 4 A =0 (IV.32) Combining equations (IV.30), (IV.31) and (IV.32) finally yields 2 B J , +(~ + 2B VT,,, + 2B .G. = R , f Az i: 3 ; 214 (IV.33) where R, = -*2,-«1 ,g +(_ /b + B L L _ )T A | > R _ A -B 3 l ^G_ A -2B 4 l -B,_ . A (IV.34) Finally, the entire set of nodal equation can be written as <K x K = F . (4.21) where O O ^WG ^GW GG K o Az + I K o 2B o O N = [T„ T ? 2 o o 2 —K,n o 3 3 2B, — Az + 2B. 1 Gf T, (IV.35) (IV.36) and F =K O R g R (IV.37) F,] . r 215 Appendix V: Experimental Data 1. Suspension Temperature Distribution Table V . l : Suspension temperature distribution for A: G = 282 kg/m s, p and U = 6m/s. 2 s = 342 kg/m 3 sus e ^ \ rlR Z <mj\ 1 0.78 0.52 0.26 0 3.85 27.2 27.2 27.1 27.1 27.1 3.25 27.9 27.7 27.0 27.1 26.7 Heat exchanger 2.35 26.7 26.3 26.4 26.4 26.6 2.05 25.8 25.9 26.0 26.0 26.1 1.45 24.9 25.1 25.2 25.5 25.5 1.15 24.9 25.0 25.2 25.3 25.4 216 Table V.2: Suspension temperature distribution for B: G and U = 6m/s. s = 213 kg/m s, p 2 = 136 kg/m 3 sus e r/R ^ \ 1 0.78 0.52 0.26 0 3.85 26.8 26.9 26.8 26.7 26.4 3.25 28.0 27.5 27.0 26.7 26.7 Z(m)>\ Heat exchanger 2.35 31.7 28.1 26.7 26.4 26.3 2.05 29.7 27.2 26.2 25.7 25.8 1.45 25.2 24.6 24.5 24.3 24.1 1.15 24.4 24.2 24.3 24.2 24.0 Table V.3: Suspension temperature distribution for C: G = 26 kg/m s, p^ = 19 kg/m and < X = 6m/s. 2 3 s \ r/R 1 0.78 0.52 0.26 0 3.85 27.2 26.7 25.6 25.12 25.0 3.25 29.3 26.1 24.2 24.4 24.8 Z(m)\. Heat exchanger 2.35 29.5 24.8 23.0 22.7 22.8 2.05 27.5 24.5 22.5 22.6 22.6 1.45 23.5 22.9 22.1 22.1 22.2 1.15 22.8 22.6 22.0 21.8 21.8 217 2. Average Heat Transfer Coefficient Table V.4: Average heat transfer coefficients U (mJs) g 4 6 G (kg/m s) 2 s Psus (kg/m ) 3 £/(W/m K) 2 h (W/m K) 2 24 50 123 144 63 117 220 288 270 166 301 437 356 193 303 437 19 26 88 99 137 214 272 400 342 283 300 436 15 10 50 53 36 73 130 146 88 126 204 250 198 228 278 366 305 299 291 394 334 463 296 406 11 13 60 63 13 67 89 97 40 123 128 150 225 205 273 374 289 306 291 403 349 455 305 423 218 Table V.4: Average heat transfer coefficients (continued) U g (m/s) 7 8 9 G,(kg/m s) 2 (kg/m ) 3 p sus tV(W/m K) 2 h (W/m K) 2 267 463 297 395 16 45 79 84 29 103 141 159 41 144 173 202 107 197 236 292 231 302 269 353 258 449 304 405 16 38 77 84 30 132 123 142 68 236 195 243 90 283 219 280 219 3. Local Heat Transfer Coefficient Table V.5: Local bed-to-wall heat transfer coefficient for U = 6 m/s. Operating conditions are listed in Table 3.4. g Height (m) Run A Run B RunC RunD RunE 3.03 121 194 238 171 55 2.98 159 109 117 68 47 2.87 217 138 73 49 14 2.70 332 279 287 140 96 2.58 457 587 316 147 92 Table V.6: Local bed-to-wall heat transfer coefficient for U = 8 m/s. Operating conditions listed in Table 3.5. g Height (m) Run A RunB RunC RunD RunE 3.03 154 304 291 126 89 2.98 200 286 190 34 63 2.87 163 131 60 40 23 2.70 256 206 172 157 73 2.58 555 372 360 194 123 220 Appendix VI: Program Listing 1. Example for two-dimensional model main.m g l o b a l dp ep roup cpp up k l k2 L c Tbulk e b u l k D L p i p e u L w a l l k w a l l ewall; g l o b a l EPS Nv SIGMA Qbulk a dx N kp Dr r e l a x s ss L a r dcdz_p dcdz_midp d e t a c d z _ p detacdz_midp; g l o b a l ca aa c c detag ug a i r T a i r R o u a i r C p a i r K airMu a i r P r dxs voidmf; parameter21; [XI,Qr(1),Qc(l)]=horitrans(Tbulk-130,Twat(1)); Tw(l)=Xl(1); Tg(l,:)=X1(2:N+1); II(1,:)=X1(N+2:2*N+1); 11(1,:)=X1(3*N+2:4*N+1); 12(1,:)=X1(4*N+2:5*N+1); for i=2:Nv i dcdz_p=0; dcdz_midp=0; detacdz_p=dcdz_p+cc/Lar ; detacdz_midp=dcdz_p+c/Lar; [X2,Qr(i),Qc(i),Twat(i)]=KellerBox(XI,Twat(i-1) ,Qr(i-1)+Qc(i1),dz(i-l)); Tw(i)=X2(1); Tg(i,:)=X2(2:N+1); Tp(i,:)=X2(N+2:2*N+1); II(i,:)=X2(3*N+2:4*N+1); 12(i,:)=X2(4*N+2:5*N+1); X1=X2; end qcf=(Qc(l:Nv-6)+Qc(2:Nv-5)+Qc(3:Nv-4)+2*Qc(4:Nv-3)+Qc(5:Nv-2)+Qc(6:Nvl)+Qc(7:Nv))/8; qrf=(Qr(1:Nv-6)+Qr(2:Nv-5)+Qr(3:Nv-4)+2 *Qr(4:Nv-3)+Qr(5:Nv-2)+Qr(6:Nvl)+Qr(7:Nv))/8; Twf=(Tw(1:Nv-6)+Tw(2:Nv-5)+Tw(3:Nv-4)+2*Tw(4:Nv-3)+Tw(5:Nv-2)+Tw(6:Nvl)+Tw(7:Nv))/8; Tgf=(Tg(1:Nv-6,:)+Tg(2:Nv-5,:)+Tg(3:Nv-4,:)+2*Tg(4:Nv-3,:)+Tg(5:Nv2, : )+Tg(6:Nv-l, : )+Tg(7:Nv, : ) ) /8,Tpf = (Tp (1: Nv-6, :)+Tp(2:Nv-5,:)+Tp(3:Nv-4,:)+2*Tp(4:Nv-3,:)+Tp(5:Nv2,:)+Tp(6:Nv-l,:)+Tp(7:Nv,:))/8; I l f = ( I l ( 1 : N v - 6 , :)+11(2:Nv-5, :)+11(3:Nv-4, :)+2 * 11(4:Nv-3, :)+11(5:Nv2, :)+11(6:Nv-l, :)+11(7:Nv, :) )/8; 12f=(12(l:Nv-6,:)+12(2:Nv-5,:)+12(3:Nv-4,:)+2*12(4:Nv-3,:)+12(5:Nv2, :)+12(6:Nv-l, :)+12(7:Nv, : ) ) / 8 ; Twatf=Twat(4:Nv-3); 221 lps=lp(4:Nv-3); hcbwat=qcf./(Tbulk-Twatf) ; hrbwat=qrf./(Tbulk-Twatf) ; htbwat=hcbwat+hrbwat; save main21 parameter.m g l o b a l dp ep roup cpp up k l k2 L c Tbulk e b u l k D L p i p e u L w a l l k w a l l ewall; g l o b a l EPS Nv SIGMA Qbulk a dx N kp Dr r e l a x s ss L a r ; g l o b a l ca aa c c detag ug a i r T a i r R o u a i r C p a i r K a i r M u a i r P r dxs voidmf %%%%%%Thermal boundary l a y e r c a l c u l a t i o n voidmf=0.071; % minimum f l u i d i z a t i o n v e l o c i t y %PARAMETERS FOR PARTICLES o o % dp -- p a r t i c l e diameter (m) dp=0.000286; % ep -- e m i s s i v i t y / a b s o r p t i v i t y o f p a r t i c l e s ep=0.85; % roup -- d e n s i t y (kg/m3) roup=2610; % cpp — heat c a p a c i t y kj/kgK cpp=84 0; % up -- downflowing v e l o c i t y (m/s) up=1.2; % kp -- t h e r m a l c o n d u c t i v i t y w/mK kp=l.9; % R i s e r Dimension m Dr=0.152; %Suspension D e n s i t y kg/m3 rous=55.25; %Calculate the cross-section ca=rous/roup; average p a r t i c l e c o n c e n t r a t i o n % PARAMETERS FOR GAS o O %Thermal c o n d u c t i v i t y as a f u n t i o n of temperature k=k2+kl*T a i r T = [ 0 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 600 700 800 900 1000 1100 1200]+273; airRou=[1.293 1.247 1.205 1.165 1.128 1.093 1.06 1.029 1 0.972 0.946 0.898 0.854 0.815 0.779 0.746 0.674 0.615 0.566 0.524 0.456 0.404 0.362 0.329 0.301 0.277 0.257 0.239 ] ; airCp=1000*[1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.009 1.009 1.009 1.009 1.013 1.017 1.022 1.026 1.038 1.047 1.059 1.068 1.093 1.114 1.135 1.156 1.172 1.185 1.197 1.21 ] ; airK=[0.0244 0.0251 0.0259 0.0267 0.0276 0.0283 0.029 0.0296 0.0305 222 0.0313 0.0321 0.0334 0.0349 0.0364 0.0378 0.0393 0.0427 0.046 0.0491 0.0521 0.0574 0.0622 0.0671 0.0718 0.0763 0.0807 0.085 0.0915 ] ; airMu=[13.28 14.16 15.06 16 16.96 17.95 18.97 20.02 21.09 22.1 23.13 25.45 27.8 30.09 32.49 34.85 40.61 48.33 55.46 63.09 79.38 96.89 115.4 134.8 155.1 177.1 199.3 233.7. ]/1000000; a i r P r = [ 0 . 7 0 7 0.705 0.703 0.701 0.699 0.698 0.696 0.694 0.692 0.69 0.688 0.686 0.684 0.682 0.681 0.68 0.677 0.674 0.676 0.678 0.687 0.699 0.706 0.713 0.717 0.719 0.722 0.724 ] ; %ug--gas v e l o c i t y ug=0.4; %PARAMETERS FOR WALL LAYER % % L -- w a l l l a y e r t h i c k n e s s L=Dr*(1-sqrt(1.34-1.3*ca~0.2+ca"l.4))12; xtmp=(Dr-2*L)/Dr; sumc=quad8('cdis',xtmp,1) ; cbulk=(ca-sumc)/xtmp; %cbulk=0.016165233241814; % c -- p a r t i c l e v o l u m t r i c c o n c e n t r a t i o n %c=sumc/(1-xtmp); %PARAMETERS FOR BULK Q. O %Tbulk — Temperature (K) Tbulk=803+273; %ebulk — e m i s s i v i t y Lbulk=0.95*(Dr-2*L); ebulk=l-exp(-1.5*cbulk*ep*Lbulk/dp) ; %PARAMETER FOR THE COOLANT % %D -- Diameter o f p i p e (m) D=0.0141; % L p i p e -- v e r t i c a l l e n g t h o f t h e p i p e Lpipe=2; %u -- water v e l o c i t y i n t h e p i p e (m/s) u=2 ; %PARAMETER FOR WALL %Gas l a y e r t h i c k n e s s detag detag=0.02 87*dp*ca (-0.581) ; A % L w a l l - - T h i c k n e s s o f w a l l (m) Lwall=0.0024; % k w a l l - - c o n d u c t i v i t y of t h e w a l l ( i n v a r i a n t w i t h temperature) kwall=21; % e w a l l - - e m i s s i v i t y of the w a l l surface ewall=0.9; % CALCULATION PARAMETRE % 223 %EPS -- a c c u r a c y EPS=0.000001; Lh(l)=detag; dxs(l)=le-7; dxs(2)=2e-7; Lh(2)=Lh(l)+dxs(l); Lh(3)=Lh(2)+dxs(2); i=2; while(Lh(i+1)<L) i=i+l; dxs(i)=dxs(i-2)+dxs(i-1); i f dxs(i)>0.001 dxs(i)=0.001; end Lh(i+1)=Lh(i)+dxs(i); end if Lh(i+1)>L Lh(i+1)=L; dxs(i)=Lh(i +l)-Lh(i) ; end %N -- Number o f elements i n h o r i z o n t a l - d i r e c t i o n N=i+1; r=(Dr-Lh(l:N-l)-Lh(2:N))/Dr; c=cdis(r); rr=(Dr-2*Lh)/Dr; cc=cdis(rr); dz ( l ) = i e - 7 ; dz(2)=2e-7; lp(l)=0; lp(2)=le-7; i=2; while(lp(i)<Lpipe) i=i+l; dz(i)=dz(i-2)+dz(i-l); % dz(i)=1.01*dz(i-l); if dz(i)>0.01 dz(i)=0.01; end lp(i)=lp(i-l)+dz(i-l); end Nv=i; %CONSTANTS D. "5 ISIGMA — S t e f a n - B o l t z m a n c o n s t a n t SIGMA=5.67e-8; % INITIAL CALCULATIONS %heat t r a n s f e r c o e f f i c i e n t between c o o l a n t and t h e w a l l Twat(1)=80+273; % a -- a b s o b t i v i t y / e m i s s i v i t y o f the l a y e r based on p a r t i c l e 224 concentration %and e m i s s i v i t y [ H o t t e l and Sarotim] a=l.5.*c*ep/dp; aa=l.5.*cc*ep/dp; s=1.5.*c*(1-ep)/dp; ss=l.5.*cc*(1-ep)/dp; relax=l; l a v e r a g e p a r t i c l e s r e s i d e n c e l e n g t h meter Lar=l.5; function [XI,QI, Q2]=horitrans(Tguess,Twater) g l o b a l dp ep roup cpp up k l k2 L c Tbulk ebulk D Lpipe u L w a l l kwall ewall; g l o b a l EPS Nv SIGMA Qbulk a dx N kp Dr r e l a x s ss L a r ; g l o b a l ca aa c c detag ug a i r T a i r R o u a i r C p a i r K airMu a i r P r dxs; [kwat,Prwat,viswat,rouwat,cpwat]=watpropt(Twater) ; %Use D i t t u s - B o e l t e r t o c o r r e l a t i o n t o c a l c u l a t e Nu Nu=0.023*(u*D/viswat) 0.8*Prwat"0.4; hwat=Nu*kwat/D; A %Guess temperature o f furnace s i d e Tl=Tguess; temper=(1/hwat+Lwall/kwall); Q=(Tl-Twater)/temper; Qmax=(Tbulk-Twater)/temper; f=l; while f>0 [II,I2,Ql]=radiation(Tl); TgO=(Tl+Tbulk)/2; k g O = s p l i n e ( a i r T , a i r K , TgO); Q2=(Tbulk-Tl)*kg0/detag; Qt==Ql+Q2; fl=Qt-Q; error=abs(fl/Q) ; if wall error<EPS return end i f Qt>Qmax Qt=Qmax; end T2=Tl+sign(fl)*100; Qn=(T2-Twater)/temper; [II,I2,Ql]=radiation(T2); Tg0=(T2+Tbulk)12; k g O = s p l i n e ( a i r T , a i r K , TgO); Q2=(Tbulk-T2)*kg0/detag; f2=Ql+Q2-Qn; 225 error=abs(f2/Qn); i f error<EPS return end end f=fl*f2; i f f>0 T1=T2; Q=Qn ; end w h i l e error>EPS/1000 T3=(T1+T2)/2; Q=(T3-Twater)/temper; [II,12,QI]=radiation(T3); TgO=(T3+Tbulk)/2; k g O = s p l i n e ( a i r T , a i r K , TgO); Q2=(Tbulk-T3)*kgO/detag; f3=Ql+Q2-Q; error=abs((T3-T1)/T3); i f fl*f3<0 T2=T3; f2=f3; else T1=T3; fl=f3; end Twall=T3; end X1=[T3 ones(1,2*N)*Tbulk (Tbulk-T3)/detag z e r o s ( l , N - l ) function I I ' 12']; [X2,Qr,Qc,Twat2]=KellerBox(XI,Twatl,Qt,dz); g l o b a l dp ep roup cpp up k l k2 L c Tbulk ebulk D L p i p e u L w a l l k w a l l ewall; g l o b a l EPS Nv SIGMA Qbulk a dx N kp Dr r e l a x s ss ddcdz_p dcdz_midp d e t a c d z _ p detacdz_midp; g l o b a l ca aa c c d e t a g ug a i r T a i r R o u a i r C p a i r K airMu a i r P r d x s ; [kwat,Prwat,viswat,rouwat,cpwat]=watpropt(Twatl) ; Twat2=Twatl-Qt*dz*4/rouwat/cpwat/u/D; Nu=0.023*(u*D/viswat)^0.8*Prwat 0 . 4 ; hwat=Nu* kwat/D; A X2=X1; Error=l; w h i l e (Error>EPS) Tg_point=(Xl(2:N+1)+X2(2:N+1))/2; kg_point=spline(airT,airK,Tg_point); 226 pkg_px_box=(kg_point(2:N)-kg_point(1:N-1))./dxs; Tg_box=(Xl(3:N+1)+X1(2:N)+X2(3:N+1)+X2(2:N))/4; Tp_box=(XI(N+3:2*N+1)+X1(N+2:2*N)+X2(N+3:2*N+1)+X2(N+2:2*N))/4; k v i s c o s i t y _ g a s _ b o x = s p l i n e ( a i r T , a i r M u , Tg_box); Rep_box=(up-ug)*dp./kviscosity_gas_box; Prg_box=spline(airT, kg_box=spline(airT, airPr,Tg_box); airK, Tg_box); Nu_box=2+l.8*sqrt(Rep_box).*Prg_box. (1/3); r o u g _ b o x = s p l i n e ( a i r T , a i r R o u , Tg_box); cpg_box=spline(airT, airCp,Tg_box); A o o o o ' o o o ' O ' o o ' o o ' o o o a_box=Nu_box.*kg_box/dp/dp*6.*c; ooo'ooooo'ooo'o'ooo p_box=(l-c).*roug_box.*cpg_box*ug; b_box=6*Nu_box.*c.*kg_box/dp/dp./p_box; B2=b_box/2*dz; Bl=l+B2; B3=l./p_box.*(kg_box./dxs-pkg_px_box/2)*dz; B4=l./p_box.*(kg_box./dxs+pkg_px_box/2)*dz; B5=(l-B2).*(XI(3:N+1)+X1(2:N))+B2.*(XI(N+3:2*N+1)+X1(N+2:2*N))B3.*X1(2*N+2:3*N)+B4.*X1(2*N+3:3*N+1); O O ' O O O O ' O O ' O O O O O O O ' O f_box=-l./(c.*roup*cpp*up); g_box=-detacdz_midp./c+a_box.*f_box; l_box=-a_box.* f_box; r_box=detacdz__midp./c*Tbulk; Dl=-l_box/2*dz; D2=l-g_box/2*dz; "o "o 15 ti "o "6 o "6 "5 o 15 "6 %D3=f_box./dxs*dz; D3=0; D4=-D1.*(XI(2:N)+X1(3:N+1)) + (2D2).*(X1(N+2:2*N)+X1(N+3:2*N+1))+D3.*(XI(3*N+2: 4*N)+X1(3*N+3:4*N+1)+X1(4*N+2:5*N)X1(4*N+3:5*N+1))+r_box*2*dz; ' O ' O O ' O O O O O O ' O O ' O O O O O ' O O ' O ' O O O O O O ' O O O O O O ' O O Cl=aa(1:N-1)*SIGMA.*dxs; C2=aa(2:N)*SIGMA.*dxs; C3=l-(aa(1:N-1)+ss(1:N-1)).*dxs; C4=-l-(aa(2:N)+ss(2:N)).*dxs; C5=ss(1:N-1).*dxs; C6=ss(2:N).*dxs; C7=-l-(aa(l:N-l)+ss(l:N-l)).*dxs; 227 C8=l-(aa(2:N)+ss(2:N)).*dxs; AA(1,1,1:N-1)=1; %%%%%%%%%%%%%%% AA(2,2,:)=C1.*X2(N+2:2*N). 3; AA(2,4,:)=C3; AA(2,5,:)=C5; AA(3,2, :)=AA(2,2, :) ; AA(3,4,:)=C5; AA(3,5,:)=C7; A %%%%%%%%%%%%%%%%%%%%% AA(4,1,:)=B1; AA(4,2,:)=-B2; AA(4,3,:)=B3; AA(5,1,:)=D1; AA(5,2,:)=D2; AA(5,4,:)=D3; AA(5,5,:)=-D3; BB(1,1,1:N-1)=-1; • • BB(1, 3,1:N-1)=dxs; "5 O'O'O"© 0'O'O'5'0'D"O'O'D'0'0'O"D BB(2,2,:)=C2.*X2(N+3:2*N+1). 3; BB(2,4,:)=C4; BB(2,5,:)=C6; BB(3,2, :)=BB(2,2, :) ; BB(3,4,:)=C6; BB(3,5,:)=C8; BB(4,1,:)=B1; BB(4,2,:)=-B2; BB(4,3,:)=-B4; BB(5,1,:)=D1; BB(5,2,:)=D2; BB(5,4,:)=-D3; BB(5,5,:)=D3; A DD(1:3,:)=zeros(3,N-1); DD(4,:)=B5; DD(5,:)=D4; TgO=(X2(1)+X2(2))12; kgO=spline(airT,airK,TgO); tmp=l/(Lwall/kwall+l/hwat)+kgO/detag; Right=[0 Twat2/tmp/(Lwall/kwall+l/hwat) 0 ebulk*SIGMA*Tbulk 4]' L e f t = [ l -1 0 detag 0 0 1 -kgO/detag/tmp 0 0 1/tmp -1/tmp ewall*SIGMA*X2(1) 3 0 0 0 - 1 1 - e w a l l ] ; A Left2=-l+ebulk; 228 A for end i=l:(N-l) LC(l+5*(i-1):5*i,1+5*(i-1):5*i)=AA(:,:,i); L C ( l + 5 * ( i - l ) : 5 * i , l + 5*i:5*(i+1))=BB(:, : , i ) ; LR(l+5*(i-1):5*i)=DD(:,i); A(l:(N-l)*5,2:N*5+1)=LC; A(((N-l)*5+l):((N-l)*5+3),1:6)=Left; A(N*5-1,N*5-3)=1; A(N*5,N*5-2)=1; A(N*5+1,N*5)=Left2; A(N*5+1,N*5+l)=1; B=[LR' Right(1:3) Tbulk Tbulk Right(4)] ; X3=A\B; X(1)=X3(1) ; f o r i=0:N-l X(i+2)=X3(i*5+2-) ; X(i+N+2)=X3(i*5+3); X(i+2*N+2)=X3(i*5+4); X(i+3*N+2)=X3(i*5+5); X(i+4*N+2)=X3(i*5+6); end X3=X; % save x 3 . t x t X3 - a s c i i Error=max(abs((X3(1:2*N+1)-X2(1:2*N+1)) ./X3(1:2*N+1))) ; X2=(l-relax)*X2+relax*X3; end Qr=X2(4 *N+2)-X2(3*N+2); Qc=kg0*(X2(2)-X2(1))/detag; function c=cdis(re) g l o b a l ca voidmf void=voidmf+(1-ca-voidmf)*(1-ca). (-1.5+2.l*re. 3.1+5*re. 8.8); c=l-void; A function A A [kwat,Prwat,viswat,rouwat,cpwat]=watpropt(Twat) %This program c a l c u l a t e the p r o p e r t i e s o f water a t t e m p e r a t u r e Twat %Convert from K t o C twat=Twat-273.18; k=[55.1 57.4 59.9 61.8 63.5 64.8 65.9 66.8 67.4 68.0 68.3 68.5 68.6 68.6 68.5 68.4 68.3 67.9 67.4 67.0 66.3 65.5 64.5 63.7 62.8 61.8 60.5 59.0 57.4 55.8 54 52.3 50.6 48.4 45.7 43.0 39.5 33.7]/100; Pr=[13.67 9.52 7.02 5.42 4.31 3.54 2.99 2.55 2.21 1.95 1.75 1.60 1.47 1.36 1.26 1.17 1.10 1.05 1.00 0.96 0.93 0.91 0.89 0.88 0.87 0.86 0.87 0.88 0.9 0.93 0.97 1.03 1.11 1.22 1.39 1.6 2.35 6.79]; vis=[1.789 1.306 1.006 0.805 0.659 0.556 0.478 0.415 0.365 229 0.326 0.295 0.272 0.252 0.233 0.217 0.203 0.191 0.181 0.173 0.165 0.158 0.153 0.148 0.145 0.141 0.137 0.135 0.133 0.131 0.129 0.128 0.128 0.128 0.127 0.127 0.126 0.126 0.126]/1000000; rou=[999.9 999.7 998.2 995.7 992.2 988.1 983.1 977.8 971.8 965.3 958.4 951 943.1 934.8 926.1 917 907 897.3 886.9 876 863 852.3 840.3 827.3 813.6 799 784 767.9 750.7 732.3 712.5 691.1 667.1 640.2 610.1 574.4 528.0 450.5]; cp=[4.212 4.191 4.183 4.174 4.174 4.174 4.179 4.187 4.195 4.208 4.220 4.233 4.250 4.266 4.287 4.313 4.346 4.380 4.417 4.459 4.505 4.555 4.614 4.681 4.756 4.844 4.949 5.07 5.23 5.485 5.736 6.071 6.574 7.244 8.165 9.504 13.984 40.231]*1000; m=ceil(twat/10); kwat=k(m) + (k(m+1)-k(m))*(twat-(m-1)* 10)/10; Prwat=Pr(m)+(Pr(m+1)-Pr(m))*(twat-(m-1)*10)/10; viswat=vis(m)+(vis(m+1)-vis(m))*(twat-(m-1)*10)/10; rouwat=rou(m)+(rou(m+1)-rou(m))*(twat-(m-1)*10)/10; cpwat=cp(m)+(cp(m+1)-cp(m))*(twat-(m-1)*10)/10; 230 2. Example for three-dimensional model main.m clear a global global global global global global global global global global global global global ll eq p t e r eh s r e r r l e r r 2 NEM NEG NEL NER NPM NPW NPG NPK KLc I L h IRq IRh IL1 I I 12 13 I r w l Irw2 I r b l I r b 2 Tbulk Twato EPS SIGMA xx k w a l l e w a l l ebulk Twato u D voiclmf ug up dp cone conp dcdzpe dcdzpp dcdzgp dcdzge absorp absore krp kre Lavp Lave ep re rp Qt mold f o l d B o l d t o l d roup cpp r e l a x a i r T airMu a i r K airRou airCp a i r P r FluxR FluxTM F l u x T c T parameterA; initialint; [cone, c o n p ] = p c o n ( l p ( 1 ) ) ; [TMN, FluxR, FluxT]=InitialMembrane(Twato,dz(1)); TT(:,1)=T; FR(:,l)=FluxR; FT(:,1)=FluxT; lps(l)=lp(l); TTwat(1)=Twato; QQts(l)=Qt; TTbulk(l)=Tbulk j=l ttl=clock for i = l : l e n g t h ( l p ) - 1 i [cone c o n p ] = p c o n ( l p ( i + 1 ) ) ; [TN, FluxR, FluxT, Twat(i+1), Q t ] = f e m ( T w a t ( i ) , d z ( i ) , d z ( i + 1 ) ) ; i f rem(i,2)==0 j=j+i; TT(:,j)=TN; FR(:,j)=FluxR; FT (:,j)=FluxT; lps(j)=lp(i); TTwat(j)=Twat(i); QQts(j)=Qt; TTbulk(j)=Tbulk; if end end rem(j,3)==0 save AnderssonA end j=j+i; TT(:,j)=TN; 231 FR(:,j)=FluxR; FT(:,j)=FluxT; lps(j)=lp(i); TTwat(j)=Twat(i) ; QQt s ( j ) = Q t ; TTbulk(j)=Tbulk; % % % P o i n t a t t h e c r e s t 132 s=sum(sr(1:NER)); %heat t r a n s f e r c o e f f i c i e n t based on p o i n t %Chalmers Method 3 T4=TT(132,:); ht4=QQts' ./s./(TTbulk'-T4') ; h3=[13.5-lps' ht4 ] ; 132 c r e s t tt2=clock time=tt2-ttl save AnderssonA parameterA.m global global global global global global global global global global global load load load load wall NEM NEG NEL NER NPM NPW NPG NPK KLc I L h IRqIRh I L 1 I I 12 13 I r w l Irw2 I r b l Irb2 Tbulk Twato EPS SIGMA r e l a x xx k w a l l e w a l l ebulk Twato u D voidmf ug up dp roup cpp ep cone conp dcdzpe dcdzpp dcdzgp dcdzge absorp absore k r p k r e Lavp Lave ep r e r p cz d e n s i t y CoefT a i r T airMu a i r K airRou airCp a i r P r Qt mold f o l d B o l d t o l d eq pt er eh % % % % matrix t o record matrix t o record Elements who has Elements who has f i n i t e elements NUMBER OF ELEMENTS *6 nodes NUMBER o f N0DES*2 i n mm r a d i a t i o n b o u n d a r i e s i n t h e mebrane w a l l c o n v e c t i o n b o u n d a r i e s i n t h e membrane pt=pt/1000; % c o n v e r t mm t o m NEM=60; % Number o f elements i n Membrane w a l l NEG=28; % Number o f elements i n gas gap NEL=153; % Number o f elements i n w a l l L a y e r NER=length(er); % Number o f elements who has r a d i a t i o n b o u n d a r i e s i n membrane w a l l NPM=17 9; % Number o f nodes i n membrane w a l l NPW=34 4; % Number o f nodes i n w a l l l a y e r NPG=29; % Number o f nodes i n gas gap (excepy i n w a l l l a y e r and membrane w a l l ) NPK=7; % Number o f elements on t h e b u l k s i d e who has f i r s t boundary conditions errl=NEM+NEG+[15:28]; Elements i n w a l l l a y e r who has % 232 t h i r d r a d i a t i o n boundaries i n w a l l side err2=NEM+NEG+NEL-2:NEM+NEG+NEL; % Elements i n w a l l l a y e r who has t h i r d r a d i a t i o n boundaries i n bulk side xx=[2 5 3 ] ; % Constant EPS=0.000001; % a c c u r a c y SIGMA=5.67e-8; %SIGMA — S t e f a n - B o l t z m a n c o n s t a n t relax=2/3; CoefT=[22.2636 581.09+273]; %Tbulk — Temperature (K) Twato=190 + 273; % Water o u t l e t temperature (K) 0.0.5.^0.00.000,0.000.0,0.0,0. "B"5'5'o'6i5'o^'o'5"5'o'o'D'o 0*5 o kwall=50; %kwall--conductivity of the w a l l ( i n v a r i a n t with ewall=0.8; % e w a l l — e m i s s i v i t y o f the w a l l surface ebulk=0.99; % b u l k e m i s s i v i t y D=(60.3-5.6*2)/1000; %D — Diameter o f p i p e (m) Lpipe=12; % L p i p e -- v e r t i c a l l e n g t h o f the p i p e Dr=1.65; % R i s e r Dimension m temperature) %42kg/s f o r 88 p i p e s whose d i a m t e r i s 60.3 mm u=0.252;- %u -- water v e l o c i t y i n t h e p i p e (m/s) OO"©"© O "O O "5 "5 O O OOO "O O "O O 0 " D ^ ^ " 0 ' 0 " 0 t > ' 5 ' 0 ' 0 ^ ' 0 ' 0 O'O'D'O'O O "D "O dp=0.00044; % dp — p a r t i c l e d i a m e t e r (m) roup=2600; % roup -- d e n s i t y (kg/m3) cpp=840; % cpp — heat c a p a c i t y k j / k g K up=1.8; % up -- downflowing v e l o c i t y (m/s) kp=1.9; % kp — t h e r m a l c o n d u c t i v i t y w/mK ep=0.85 % p a r t i c l e e m i s s i v i t y voidmf=0.43; % minimum f l u i d i z a t i o n v e l o c i t y %PARAMETERS FOR GAS a i r T = [ 0 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 600 700 800 900 1000 1100 1200]+273; airRou=[1.293 1.247 1.205 1.165 1.128 1.093 1.06 1.029 1 0.972 0.946 0.898 0.854 0.815 0.779 0.746 0.674 0.615 0.566 0.524 0.456 0.404 0.362 0.329 0.301 0.277 0.257 0.239 ] ; airCp=1000*[1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.009 1.009 1.009 1.009 1.013 1.017 1.022 1.026 1.038 1.047 1.059 1.068 1.093 1.114 1.135 1.156 1.172 1.185 1.197 1.21 ] ; airK=[0.0244 0.0251 0.0259 0.0267 0.0276 0.0283 0.029 0.0296 0.0305 0.0313 0.0321 0.0334 0.0349 0.0364 0.0378 0.0393 0.0427 0.046 0.0491 0.0521 0.0574 0.0622 0.0671 0.0718 0.0763 0.0807 0.085 0.0915 ] ; airMu=[13.28 14.16 15.06 16 16.96 17.95 18.97 20.02 21.09 22.1 23.13 25.45 27.8 30.09 32.49 34.85 40.61 48.33 55.46 63.09 79.38 96.89 115.4 134.8 155.1 177.1 199.3 233.7 ]/1000000; a i r P r = [ 0 . 7 0 7 0.705 0.703 0.701 0.699 0.698 0.696 0.694 0.692 0.69 0.688 0.686 0.684 0.682 0.681 0.68 0.677 0.674 0.676 0.678 0.687 0.699 0.706 0.713 0.717 0.719 0.722 0.724 ] ; ug=0.6; %ug--gas v e l o c i t y 233 % CALCULATION PARAMETRE % lp(l)=0; lp(2)=lp(l)+le-8; dz(1)=lp(2)-lp(1); i=l; while(lp(i)<Lpipe) i=i+l; dz(i)=1.5*dz(i-l); if dz(i)>0.2 dz(i)=0.2; end lp(i)=lp(i-l)+dz(i-l) ; end Nv=i; % Number o f v e r t i c a l steps %INITIAL CALCULATIONS %heat t r a n s f e r c o e f f i c i e n t between c o o l a n t and the w a l l Twat(1)=Twato; Lavf=2; Lavt=l; % m, average r e s i d e n c e % m, average r e s i d e n c e length i n f i n region l e n g t h i n tube r e g i o n Lavp=Lavt*ones(NPW,1); Ro=0.03 %m, o u t e r r a d i u s o f t h e tube f o r i=l:NPW i f pt(NPM+NPG+i,1)>Ro Lavp(i)=Lavf; end end y=pt(NPM+NPG+l:NPM+NPG+NPW,2); rp=(Dr/2-y)/Dr*2; Lave=Lavt*ones(NEL,1); f o r i=NEM+NEG+1:NEM+NEG+NEL y=(pt(eq(i,1),2)+pt(eq(i,2),2)+pt(eq(i,3),2)+pt(eq(i,4),2)+pt(eq(i,5),2 )+pt(eq(i,6),2))/6; re(i-NEM-NEG)=(Dr/2-y)/Dr*2; x=(pt(eq(i,1),1)+pt(eq(i,2),l)+pt(eq(i,3),1)+pt(eq(i,4),l)+pt(eq(i,5),1 )+pt(eq(i,6),l))/6; i f x>Ro Lave(i-NEM-NEG)=Lavf; end end density=[1.8567 1.6567 2.5887 4.4525 234 0.3106 3. 417 7.0412 ] ; cz=13.5-[10.1105 9. 547 7.7569 6.7956 3.2155 2.5193 2.0221 ] ; i n i t i a l i n t .m global global global global global global global global global global global NEM NEG NEL NER NPM NPW NPG NPK KLc I L h IRq IRh IL1 I I 12 13 I r w l Irw2 I r b l Irb2 Tbulk Twato EPS SIGMA xx k w a l l e w a l l e b u l k Twato u D ug up dp cone conp dcdzpe dcdzpp dcdzgp dcdzge absorp absore krp kre Lavp Lave ep r e r p Qt mold f o l d B o l d t o l d roup cpp r e l a x FluxR FluxTM F l u x T c KLc=zeros(NPM+NPG*2); ILh=zeros(3,3,length(eh)) ; IRh=zeros(3,length(eh)); IRq=zeros(3,length(eq) ) ; I L l = z e r o s ( 6 , 6, NEG) ; Il=zeros(6,6,NEL) ; I2=zeros(6,6,NEL); I3=zeros(6,NEL); Irwl=zeros(3,length(errl) ) ; Irw2=zeros(3,3,length(errl) ) ; Irbl=zeros(3,length(err2) ) ; Irb2=zeros(3,3,length(err2)); FluxTM=zeros(6,6,length(er)); for i=l:NEM temp=[pt(eq(i,1),1)"2 p t ( e q ( i , 1 ) , 2 ) 2 pt(eq(i,1),1) pt(eq(i,1),2) 1 pt(eq(i,2),l) 2 pt(eq(i,2),2) 2 pt(eq(i,2),l) pt(eq(i,2),2) 1 pt(eq(i,3),1) 2 pt(eq(i,3),2) 2 pt(eq(i,3),1) pt(eq(i,3),2) 1 pt(eq(i,4),1)"2 pt(eq(i,4),2) 2 pt(eq(i,4),1) pt(eq(i,4),2) 1 pt(eq(i,5),l) 2 pt(eq(i,5),2) 2 pt(eq(i,5),1) pt(eq(i,5),2) 1 pt(eq(i,6),1) 2 pt(eq(i,6),2)"2 pt(eq(i,6),1) pt(eq(i,6),2) 1]; Shape=(inv(temp))'; xy=[pt(eq(i,1),1) pt(eq(i,1),2) A pt(eq(i,1),1)*pt(eq(i,1),2) A A pt(eq(i,2),1)*pt(eq(i,2),2) A A pt(eq(i,3),1)*pt(eq(i,3),2) A pt(eq(i,4),1)*pt(eq(i,4),2) A pt(eq(i,5),1)*pt(eq(i,5),2) A A 235 pt(eq(i,6),1)*pt(eq(i,6),2) pt(eq(i,2),1) pt(eq(i,2),2) pt(eq(i,3),1) pt(eq(i,3),2)]; for ii=l:6 for jj=l:6 C=[4*Shape(ii,1)*Shape(jj,1)+ Shape(ii,3)*Shape(jj,3) Shape(ii,3)*Shape(jj,3)+4*Shape(ii,2)*Shape(jj,2) 2*(Shape(ii,1)*Shape(jj,3)+Shape(jj,1)*Shape(ii,3)+Shape(ii,2)*Shape(jj ,3)+Shape(jj,2)*Shape(ii,3)) 2 * ( S h a p e ( i i , 1 ) * S h a p e ( j j , 4 ) + S h a p e ( j j , 1 ) * S h a p e ( i i , 4 ) ) + S h a p e ( i i , 3)*Shape(j j,5)+Shape(jj,3)*Shape(ii,5) Shape(ii,3)*Shape(jj,4)+Shape(jj,3)*Shape(ii, 4)+2*(Shape(ii, 2)*Shape(jj ,5)+Shape(jj,2)*Shape(ii,5)) Shape(ii,4)*Shape(jj,4)+Shape(ii,5)*Shape(jj,5)] ' ; % KK(ii,jj)=InteT2(C,xy); KLc(eq(i,ii),eq(i,jj))=KLc(eq(i,ii),eq(i,jj))+kwall*InteT2(C,xy); end end end KLc(NPM+NPG+1:NPM+NPG*2,NPM+NPG+1:NPM+NPG*2)=eye(NPG); M=length(eh); for i=l:M xl=pt(eq(eh(i),2),1) ; x 2 = p t ( e q ( e h ( i ) , 3 ) , 1) ; y l = p t ( e q ( e h ( i ) ,2) ,2) ; y 2 = p t ( e q ( e h ( i ) ,3) ,2) ; s=sqrt((x2-xl)"2+(y2-yl) 2); temp=[0 0 1 s 2/4 s/2 1 s 2 s 1]; C=(inv(temp))'; A A A for end end ii=l:3 f o r j j =1:3 CC=£C(ii,l)*C(jj,l) C ( i i , l ) * C ( j j , 2 ) + C ( i i , 2 ) * C ( j j , 1) C(ii,l)*C(jj,3)+C(jj,l)*C(ii,3)+C(ii,2)*C{jj,2) C ( i i , 2 ) * C ( j j , 3 ) + C ( j j , 2 ) * C ( i i , 3) C(ii,3)*C(jj,3)]'; I L h ( i i , j j,i)=Integration(CC,0,s); end IRh(ii,i)=Integration(C(ii,:),0,s); M=length(er); for i=l:M s r ( i ) = s q r t ( ( p t ( e q ( e r ( i ) , 3) , 1) - 236 pt(eq(er(i),2),1))~2+(pt(eq(er(i),3),2)-pt(eq(er(i),2),2))"2); end for i=l:M temp=[0 0 1 sr(i) 2/4 sr(i)/2 1 s r ( i ) 2 s r ( i ) 1] ; C=(inv(temp))'; for ii=l:3 IRq(ii,i)=Integration(C(ii,:),0,sr(i)); end A A end f o r i=NEM+l:NEM+NEG temp=[pt(eq(i,1),1) 2 p t ( e q ( i , 1 ) , 2 ) 2 p t ( e q ( i , 1) ,1) p t ( e q ( i , l ) , 2 ) 1 p t ( e q ( i , 2 ) , l ) 2 pt(eq(i,2),2)"2 p t ( e q ( i , 2 ) ,1) p t ( e q ( i , 2 ) , 2 ) 1 pt(eq(i,3),1) 2 pt(eq(i,3),2) 2 p t ( e q ( i , 3) , 1) p t ( e q ( i , 3 ) , 2 ) 1 p t ( e q ( i , 4 ) , i r 2 pt(eq(i,4),2) 2 pt(eq(i,4),1) pt(eq(i,4),2) 1 pt(eq(i,5),1) 2 pt(eq(i,5),2) 2 pt(eq(i,5),1) pt(eq(i,5),2) 1 pt(eq(i,6),l) 2 pt(eq(i,6),2) 2 p t ( e q ( i , 6) , 1) p t ( e q ( i , 6 ) , 2 ) 1] ; Shape=(inv(temp)) ; xy=[pt(eq(i,1),1) pt(eq(i,1),2) pt(eq(i,2),1) pt(eq(i,2),2) pt(eq(i,3),1) pt(eq(i,3),2)]; A A A A A p t ( e q ( i , 1) , 1 ) * p t ( e q ( i , 1 ) , 2 ) pt(eq(i,2),1)*pt(eq(i,2),2) pt(eq(i,3),1)*pt(eq{i,3),2) A p t (eq ( i , 4 ) , 1) *pt (eq ( i , 4 ) , 2) A A p t ( e q ( i , 5 ) , 1 ) * p t ( e q ( i , 5) , 2) A A p t ( e q ( i , 6 ) , 1 ) * p t ( e q ( i , 6) , 2) 1 for ii=l:6 f o r j j =1:6 C=[4*Shape(ii,1)*Shape(jj,1)+ Shape(ii,3)*Shape(jj,3) Shape(ii,3)*Shape(jj,3)+4*Shape(ii,2)*Shape(jj,2) 2*(Shape(ii,1)*Shape(jj,3)+Shape(jj,1)*Shape(ii,3)+Shape(ii,2)*Shape(jj , 3)+Shape(jj,2)*Shape(ii,3)) 2*(Shape(ii,1)*Shape(jj,4)+Shape(jj,1)*Shape(ii,4))+Shape(ii,3)*Shape(j j , 5)+Shape(jj,3)*Shape(ii,5) Shape(ii,3)*Shape(jj,4)+Shape(jj,3)*Shape(ii,4)+2*(Shape(ii,2)*Shape(jj ,5)+Shape(jj,2)*Shape(ii,5)) S h a p e ( i i , 4 ) * S h a p e ( j j , 4 ) + S h a p e ( i i , 5 ) * S h a p e ( j j , 5)] IL1(ii,jj,i-NEM)=InteT2(C,xy); end end end f o r i=NEM+NEG+l:NEM+NEG+NEL temp=[pt(eq(i,1),1) 2 p t ( e q ( i , 1 ) , 2 ) 2 pt(eq(i,1),1)*pt(eq(i,1),2) pt(eq(i,1),1) pt(eq(i,l),2) 1 pt(eq(i,2),l) 2 pt(eq(i,2),2) 2 pt(eq(i,2),1)*pt(eq(i,2),2) A A A A 237 pt(eq(i,2),1) pt(eq(i,2),2) 1 pt(eq(i,3),l) 2 pt(eq(i,3),2) pt(eq(i,3),l) pt(eq(i,3),2) 1 pt(eq(i,4),1) 2 pt(eq(i,4),2) pt(eq(i,4),l) pt(eq(i,4),2) 1 pt(eq(i,5),1) 2 pt(eq(i,5),2) pt(eq(i,5),1) pt(eq(i,5),2) 1 pt(eq(i,6),l) 2 pt(eq(i,6),2) pt(eq(i,6),l) pt(eq(i,6),2) 1]; Shape=(inv(temp)) ; xy=[pt(eq(i,1),1) pt(eq(i,1),2) pt(eq(i,2),1) pt(eq(i,2),2) pt(eq(i,3),1) pt(eq(i,3),2)]; A A A A A A A A 2 p t (eq ( i , 3) , 1) *pt (eq ( i , 3) , 2) 2 pt(eq(i,4),1)*pt(eq(i,4),2) 2 p t ( e q ( i , 5 ) , 1 ) * p t ( e q ( i , 5),2) 2 pt(eq(i,6),1)*pt(eq(i,6),2) 1 for ii=l:6 f o r j j =1:6 C=[4*Shape(ii,1)*Shape(jj,1)+ Shape(ii,3)*Shape(jj,3) Shape(ii,3)*Shape(jj,3)+4*Shape(ii,2)*Shape(jj,2) 2*(Shape(ii,1)*Shape(jj,3)+Shape(jj,1)*Shape(ii,3)+Shape(ii,2)*Shape(jj ,3)+Shape(jj,2)*Shape(ii,3)) 2*(Shape(ii,1)*Shape(jj,4)+Shape(jj,1)*Shape(ii,4))+Shape(ii,3)*Shape(j j , 5)+Shape(jj,3)*Shape(ii,5) Shape(ii,3)*Shape(jj,4)+Shape(jj,3)*Shape(ii,4)+2*(Shape(ii,2)*Shape(jj ,5)+Shape(jj,2)*Shape(ii,5)) Shape(ii,4)*Shape(jj,4)+Shape(ii,5)*Shape(jj,5)]'; 12(ii,jj,i-NEM-NEG)=InteT2(C,xy); CC=[Shape(ii,1)*Shape(jj,1) S h a p e ( i i , 3 ) * S h a p e ( j j , 1 ) + S h a p e ( i i , 1 ) * S h a p e ( j j,3) Shape(ii,1)*Shape(jj,2)+Shape(ii,2)*Shape(jj,1)+Shape(ii,3)*Shape(jj,3) Shape(ii,2)*Shape(jj,3)+Shape(ii,3)*Shape(jj,2) Shape(ii,2)*Shape(jj,2) Shape(ii,4)*Shape(jj,1)+Shape(ii,1)*Shape(jj,4) Shape(ii,1)*Shape(j j,5)+Shape(ii,3)*Shape(jj,4)+Shape(ii,4)*Shape(jj,3) +Shape(ii,5)*Shape(jj,1) Shape(ii,3)*Shape(j j,5)+Shape(ii,4)*Shape(jj,2)+Shape(ii,5)*Shape(jj,3) +Shape(ii,2)*Shape(jj,4) Shape(ii,5)*Shape(jj,2)+Shape(ii,2)*Shape(jj,5) Shape(ii, 4)*Shape(jj,4)+Shape(ii,6)*Shape(jj,1)+Shape(ii,1)*Shape(jj,6) Shape(ii,4)*Shape(jj,5)+Shape(ii,5)*Shape(jj,4)+Shape(ii,3)*Shape(jj,6) +Shape(ii,6)*Shape(jj,3) Shape(ii, 5)*Shape(jj,5)+Shape(ii,6)*Shape(jj,2)+Shape(ii,2)*Shape(jj,6) S h a p e ( i i , 6 ) * S h a p e (jj,4)+Shape ( i i , 4 ) * S h a p e ( j j,6) Shape(ii,5)*Shape(jj,6)+Shape(ii,6)*Shape(jj,5) Shape(ii,6)*Shape(jj,6)] II(ii,jj,i-NEM-NEG)=InteT4(CC,xy); 238 end 13(ii,i-NEM-NEG)=InteT2(Shape(ii,:),xy); end end M=length(errl) ; for i=l:M x l = p t ( e q ( e r r l ( i ) ,2) ,1) ; x2=pt(eq(errl(i),3),1) ; yl=pt(eqferrl(i),2),2); y2=pt(eq(errl(i),3),2); s=sqrt((x2-xl) 2+(y2-yl) 2) ; temp=[0 0 1 s 2 / 4 s/2 1 s 2 s 1]; C=(inv(temp)) ' ; A A A A for ii=l:3 for jj=l:3 C C = [ C ( i i , 1 ) * C ( j j , 1) C(ii,l)*C(jj,2)+C(ii,2)*C(jj,l) C(ii,1)*C(jj,3)+C(jj,1)*C(ii,3)+C(ii,2)*C(jj,2) C(ii,2)*C(jj,3)+C(jj,2)*C(ii,3) C(ii,3)*C(jj,3)]'; Irw2 ( i i , j j , i ) i n t e g r a t i o n (CC, 0, s) ; end I r w l ( i i , i ) i n t e g r a t i o n (C ( i i , : ) , 0, s) ; end end M=length(err2); for i=l:M x l = p t ( e q ( e r r 2 ( i ) , 2) , 1) ; x2=pt(eq(err2(i),3),1) ; yl=pt(eq(err2(i),2),2); y2=pt(eq(err2(i),3),2); s=sqrt((x2-xl) 2+(y2-yl) 2); temp=[0 0 1 s 2 / 4 s/2 1 s 2 s 1] ; C=(inv(temp)) ' ; A A A A for ii=l:3 for jj=l:3 CC=[C(ii,1)*C(jj,1) C ( i i , 1)*C(jj,2)+C(ii,2)*C( jj,1) C(ii,1)*C(jj,3)+C(jj,1)*C(ii,3)+C(ii,2)*C(jj,2) C(ii,2)*C(jj,3)+C(jj,2)*C(ii,3) C(ii,3)*C( jj,3)] ' ; Irb2 ( i i , j j , i ) i n t e g r a t i o n (CC, 0, s) ; end I r b l ( i i , i ) = I n t e g r a t i o n ( C ( i i , :),0,s); end end for j=l:length(er) 239 i=er(j); c=sqrt((pt(eq(i,2),l)-pt(eq(i,3),1)) 2+(pt(eq(i,2),2)pt(eq(i,3),2)) 2); a=sqrt((pt(eq(i,1),1)-pt(eq(i,3),1)) 2 +(pt(eq(i,1),2)pt(eq(i,3),2)) 2); b=sqrt((pt(eq(i,l),l)-pt(eq(i,2),1)) 2+(pt(eq(i,1),2)pt(eq(i,2),2)) 2); alpha=acos((b*b+c*c-a*a)/2/b/c); temp=[ 0 0 0 0 0 1 c*c 0 0 c 0 1 (b*cos(alpha)) 2 (b*sin(alpha)) 2 b*cos(alpha)*b*sin(alpha) b*cos(alpha) b*sin(alpha) 1 c*c/4 0 0 c/2 0 1 (c+b*cos(alpha)) 2/4 (b*sin(alpha)) 2/4 ( c + b * c o s ( a l p h a ) ) * b * s i n ( a l p h a ) / 4 (c+b*cos(alpha))/2 b * s i n ( a l p h a ) 1 2 1 ( b * c o s ( a l p h a ) ) 2/4 ( b * s i n ( a l p h a ) } 2/4 b * c o s ( a l p h a ) * b * s i n ( a l p h a ) / 4 b*cos(alpha)/2 b*sin(alpha)12 1]; FluxTM(:,:,j)=inv(temp); FluxTc(j)=c; end A A A A A A A A A A A A function y=integration(C,xl,x2) M=size(C,2); for i=l:M CC(i)=C(i)/(M-i+1); end CC=[CC,0]; y=polyval(CC,x2)-polyval(CC,xl); function global global global global global global global global global global global global global [TN, FluxR, FluxT, TwatN, Qt]=fem(Twat,dz,dz2) eq p t e r eh s r e r r l e r r 2 NEM NEG NEL NER NPM NPW NPG NPK KLc I L h IRq IRh I L 1 I I 12 13 I r w l Irw2 I r b l Irb2 Tbulk Twato EPS SIGMA xx k w a l l e w a l l ebulk Twato u D ug up dp cone conp dcdzpe dcdzpp dcdzgp dcdzge absorp absore k r p k r e Lavp Lave ep r e r p Qt mold f o l d B o l d t o l d roup cpp r e l a x FluxR FluxTM FluxTc Qt T a i r T airMu a i r K airRou airCp a i r P r % C a l c u l a t i n g New water temperature [kwat,Prwat,viswat,rouwat,cpwat]=watpropt(Twat); TwatN=Twat-2*Qt*dz*4/pi/rouwat/cpwat/u/D/D; [kwat,Prwat,viswat,rouwat,cpwat]=watpropt(TwatN); Nu=0.023*(u*D/viswat) 0.8*Prwat 0.4; hwat=Nu*kwat/D; A A 240 Ki=zeros(NPM+NPG+3*NPW); Fi=zeros(NPM+NPG+3*NPW,1); Ki(1:NPM,1:NPM)=KLc(1:NPM,1:NPM); for i=l:length(eh) for i i = l : 3 for jj=l:3 K i ( e q ( e h ( i ) , x x ( i i ) ) , e q ( e h ( i ) , x x ( j j ) ) ) = K i ( e q ( e h ( i ) , x x ( i i ) ) , eq(eh(i) , x x ( j j)))+hwat*ILh(ii,jj,i); end Fi(eq(eh(i),xx(ii)))=Fi(eq(eh(i) , xx(ii)))+hwat*TwatN*IRh(ii, i) ; end end %%%Matrix f o r G f o r i=NEM+NEG+l:NEM+NEG+NEL for i i = l : 6 for j j = l:6 Ki(eq(i,ii)+2*NPW,eq(i,jj)+2*NPW)=Ki(eq(i,ii)+2*NPW,eq(i,jj)+2*NPW)+kre (i-NEM-NEG)*I2 ( i i , j j , i - N E M - N E G ) + a b s o r e ( i - N E M - N E G ) * I 1 ( i i , j j , i-NEM-NEG); end end end for i=l:length(errl) for i i = l : 3 for jj=l:3 Ki(eq(errl(i),xx(ii))+2*NPW,eq(errl(i),xx(jj))+2*NPW)=Ki(eq(errl(i),xx( ii))+2*NPW,eq(errl(i),xx(jj))+2*NPW)+ewall/2/(2-ewall)*Irw2(ii,jj,i); end end end for i=l:length(err2) for ii=l:3 f o r j j =1:3 Ki(eq(err2(i),xx(ii))+2*NPW,eq(err2(i),xx(jj))+2*NPW)=Ki(eq(err2 (i),xx ( ii))+2*NPW,eq(err2(i),xx(jj))+2*NPW)+ebulk/2/(2-ebulk)*Irb2(ii,jj,i); end end end for i=l:length(err2) for i i = l : 3 Fi(eq(err2(i),xx(ii))+2*NPW)=Fi(eq(err2(i),xx(ii))+2*NPW)+2*ebulk*SIGMA *Tbulk 4/(2-ebulk)*Irbl(ii, i) ; end A 241 end Error=10; w h i l e Error>EPS K=Ki ; F=Fi; f o r i = l .-length (er) for i i = l : 3 F(eq(er(i) ,xx(ii)))=F(eq(er(i),xx(ii)))+FluxR(i)*IRq(ii,i); end end m=zeros(NPW); k=zeros(NPW,NPM+NPG+NPW); p=m; f=zeros(NPW,1); for i=NEM+l:NEM+NEG Tmean=(T(eq(i,1))+T(eq(i,2))+T(eq(i,3))+T(eq(i,4))+T(eq(i,5))+T(eq(i,6) ) ) /6; k g = s p l i n e ( a i r T , a i r K , Tmean); for i i = l : 6 i f eq(i,ii)<=NPM+NPG f o r j j =1:6 K(eq(i,ii),eq(i,jj))=K(eq(i,ii),eq(i,jj))+kg*ILl(ii,jj,i-NEM); end end i f eq(i,ii)>NPM+NPG f o r j j =1:6 k(eq(i,ii)-NPM-NPG,eq(i,jj))=k(eq(i,ii)-NPMNPG,eq(i,jj))+kg*ILl(ii,jj,i-NEM); end end end end for i=NEM+NEG+l:NEM+NEG+NEL Tmean=(T(eq(i,1))+T(eq(i,2))+T(eq(i,3))+T(eq(i,4))+T(eq(i,5))+T(eq(i,6) ) ) /6; Tpmean=(T(eq(i,1)+NPW)+T(eq(i,2)+NPW)+T(eq(i,3)+NPW)+T(eq(i,4)+NPW)+T(e q(i,5)+NPW)+T(eq(i,6)+NPW))/6; p r o p e r t i e s = s p l i n e ( a i r T , [airMu; a i r C p ; a i r R o u ; a i r P r ] , Tmean); k v i s c o s i t y = p r o p e r t i e s (1); Re=(up-ug)*dp./kviscosity; Prg=properties(4); Nu=2+1.8*sqrt(Re).*Prg. (1/3); roug=properties(3); cpg=properties(2); A 242 Al=roug*cpg*(1-cone(i-NEM-NEG))*ug; A3=6*Nu*kg*cone(i-NEM-NEG)/dp/dp; A4=-roug*cpg*ug*dcdzge(i-NEM-NEG)*Tbulk; A2=-A3+A4/Tbulk; for i i = l : 6 for jj=l:6 m ( e q ( i , ii)-NPM-NPG,eq(i,jj)-NPM-NPG)=m(eq(i,ii)-NPMNPG,eq(i,jj)-NPM-NPG)+A1*I1(ii,jj,i-NEM-NEG) ; k(eq(i,ii)-NPM-NPG,eq(i,jj))=k(eq(i,ii)-NPMNPG, e q ( i , j j ) ) + k g * I 2 ( i i , j j , i - N E M - N E G ) - A 2 * I 1 ( i i , j j , i - N E M - N E G ) ; p(eq(i,ii)-NPM-NPG,eq(i,jj)-NPM-NPG)=p(eq(i,ii)-NPMNPG, e q ( i , j j ) - N P M - N P G ) - A 3 * I I ( i i , j j , i - N E M - N E G ) ; end f ( e q ( i , ii)-NPM-NPG)=f(eq(i,ii)-NPM-NPG)-A4*I3(ii,i-NEM-NEG); end end %%%%%%%%% K(NPM+NPG+1:NPM+NPG+NPW,1:NPM+NPG)=relax*k(:,1:NPM+NPG) ; K(NPM+NPG+1:NPM+NPG+NPW,NPM+NPG+1:NPM+NPG+NPW) = (m+mold)/dz/2 + r e l a x * k ( : , NPM+NPG+1:NPM+NPG+NPW); K(NPM+NPG+1:NPM+NPG+NPW,NPM+NPG+NPW+1:NPM+NPG+2*NPW)=p*relax; F(NPM+NPG+1:NPM+NPG+NPW)=relax*f+fold+m*told(NPM+NPG+1:NPM+NPG+NPW)/2/d z; K(NPM+NPG+NPW-NPK+1:NPM+NPG+NPW,:)=0; K(NPM+NPG+NPW-NPK+1:NPM+NPG+NPW,NPM+NPG+NPWNPK+1:NPM+NPG+NPW)=eye(NPK) ; F(NPM+NPG+NPW-NPK+1:NPM+NPG+NPW)=Tbulk; k g = s p l i n e ( a i r T , a i r K , T(NPM+NPG+1:NPM+NPG+NPW)); k v i s c o s i t y = s p l i n e ( a i r T , airMu, (T(NPM+NPG+1:NPM+NPG+NPW)+T(NPM+NPG+NPW+1:NPM+NPG+2*NPW)) /2 ) ; Re=(up-ug)*dp./kviscosity; Prg=spline(airT, airPr, (T(NPM+NPG+1:NPM+NPG+NPW)+T(NPM+NPG+NPW+1:NPM+NPG+2*NPW))/2); Nu=2+1.8*Re. (1/2).*Prg. (1/3); r o u g = s p l i n e ( a i r T , a i r R o u , T(NPM+NPG+1:NPM+NPG+NPW)); B2=-6*Nu.* kg/dp/dp/roup/cpp/up; B3=-absorp(1:NPW)/roup/cpp./conp(1:NPW)/up; B4=-dcdzpp(1:NPW)./conp(1:NPW)*Tbulk; Bl=dcdzpp(1:NPW)./conp(1:NPW)-B2B3.*4*SIGMA.*T(NPM+NPG+NPW+1:NPM+NPG+2*NPW). 3; f o r i=l:NPW-NPK K(NPM+NPG+NPW+i,NPM+NPG+i)=B2(i)*relax; K(NPM+NPG+NPW+i,NPM+NPG+NPW+i)=B1(i)*relax+l/dz; K(NPM+NPG+NPW+i,NPM+NPG+2*NPW+i)=B3(i)*relax; F(NPM+NPG+NPW+i)=-B4(i)*relax+Bold(i ) ; end A A A K(NPM+NPG+2*NPW-NPK+1:NPM+NPG+2*NPW,NPM+NPG+2*NPWNPK+1:NPM+NPG+2*NPW)=eye(NPK); 243 F(NPM+NPG+2*NPW-NPK+1:NPM+NPG+2*NPW)=Tbulk; f o r i=NEM+NEG+l:NEM+NEG+NEL Tp3mean=(T(eq(i,1)+NPW) 3+T(eq(i,2)+NPW) 3+T(eq(i,3)+NPW) 3+T(eq(i,4)+N PW) 3+T(eq(i,5)+NPW) 3+T(eq(i,6)+NPW) 3)/6; for ii=l:6 for jj=l:6 A A A A A A K(eq(i,ii)+2*NPW,eq(i,jj)+NPW)=K(eq(i,ii)+2*NPW,eq(i,jj)+NPW)4*absore(i-NEM-NEG)*SIGMA*Tp3mean*Il(ii,jj,i-NEM-NEG); end end end for i=l:length(errl) Tw4mean=((T(eq(er(i),2))) 4+(T(eq(er(i),5))) 4+(T(eq(er(i),3))) 4)/3; for i i = l : 3 A A A F(eq(errl(i),xx(ii))+2*NPW)=F(eq(errl(i),xx(ii))+2*NPW)+2*ewall*SIGMA*T w4mean/(2-ewall)*Irwl(ii,i); end end TN=inv(K)*F; Error=mean(abs((TN(NPM+1:NPM+NPG+2*NPW)T(NPM+1:NPM+NPG+2*NPW))./TN(NPM+1:NPM+NPG+2*NPW))) T=TN*0.5+T*0.5; for i=l:length(errl) qr=0; for i i = l : 3 qr=qr+ewall/2/(2-ewall)*(T(eqferrl(i),xx(4-ii))+2*NPW)4*SIGMA*T(eq(er(i),xx(ii))) 4); end FluxR(i,1)=qr/3; end A %TN(NPM+NPG+1:NPM+NPG+NPW) end for j=l:length(er) i=er(j); Ta=[T(eq(i,2)) T(eq(i,3)) T ( e q ( i , l ) ) T(eq(i,5)) T(eq(i,4))] ' ; at=FluxTM(:,:,j)*Ta; F l u x T ( j , l ) = - k w a l l * ( a t ( 3 ) * F l u x T c ( j ) / 2 + at(5)) ; end 244 T(eq(i,6)) Qt=sr(1:NER)*FluxT(1:NER) ; told=T; mold=m; fold=(1-relax)*f+m*T(NPM+NPG+1:NPM+NPG+NPW)/dz2/2-(1relax)*k*T(1:NPM+NPG+NPW)-(1-relax)*p*T(NPM+NPG+NPW+1:NPM+NPG+2*NPW) Bold=-(1-relax)*B2.*T(NPM+NPG+NPW+1:NPM+NPG+2 *NPW) + ( l / d z 2 - ( 1 relax)*B1).*T(NPM+NPG+NPW+1:NPM+NPG+2*NPW)-(1relax)*B3.*T(NPM+NPG+2*NPW+1:NPM+NPG+3*NPW)-(1-relax)*B4; 245
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Modeling of heat transfer in circulating fluidized beds Xie, Donglai 2001
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Title | Modeling of heat transfer in circulating fluidized beds |
Creator |
Xie, Donglai |
Date Issued | 2001 |
Description | Suspension-to-wall heat transfer in circulating fluidized beds is modeled considering both the reactor-side and wall-side heat transfer processes. The overall flow structure in fast fluidized beds is represented by a core-annulus flow pattern with a stagnant particlefree gas gap between the wall and wall layer. Descending particles are assumed to enter the heat transfer zone with the same temperature as the core suspension. As particles descend in the wall layer, they lose heat to the gas by convection and gain heat from fresh particles arriving from the bulk core region. Gas is dragged downwards in the heat transfer zone by the rapidly-descending annular particles. The gas receives heat from the immersed particles by particle-to-gas convection and from the core by conduction. Heat is then conducted to the wall through the stagnant gas gap, and then through the furnace wall to the coolant. The model is the first to include the coolant-side heat transfer in the overall process. Particles also participate in radiation from the core to the wall through the wall layer. They are assumed to constitute a gray continuous absorbing, emitting and scattering medium. The radiation heat transfer process is solved by the two-flux model in a twodimensional model for CFB units with smooth walls, while the moment method is employed for the three-dimensional case when membrane walls are present. Under highdensity CFB operating conditions with smooth walls, the model is extended by allowing the suspension in the vicinity of the wall to travel intermittently downwards and upwards as is observed experimentally. The two- and three-dimensional models are validated using experimental results from the literature and both yield satisfactory predictions of the suspension-to-wall heat transfer. The influences of key parameters on the heat flux are analyzed and are found to be consistent with experimental trends where these are known. The simulation results suggest that the particles participate in a significant way in determining the radiation flux through the wall layer. Therefore radiation cannot be uncoupled from particle and gas conduction and convection without introducing significant error for high temperature systems. Experiments were conducted in the 76 mm diameter jacketed riser of a dual-loop high-density CFB facility with FCC particles of 65 pm Sauter mean diameter as bed material. The superficial gas velocity varied from 4 to 9.5 m/s and the solids circulation flux was as high as 527 kg/m²s. The suspension temperature and the average and local suspension-to-wall heat transfer coefficients were measured. The suspension temperature distributions indicate that the particles in the vicinity of the wall do not move in one direction only, but oscillate downward and upward, leading to higher local heat transfer coefficients at the ends of the heated section. Experimental results also show that suspension-to-wall heat transfer coefficients are strongly influenced by suspension density. However, they are not significantly influenced by superficial gas velocity at a constant suspension density. By superimposing the heat transfer results when the suspension in the vicinity of wall is allowed to move downwards and upwards separately, the model predicts the experimental results well. |
Extent | 8312909 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-10-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0058636 |
URI | http://hdl.handle.net/2429/13826 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2001-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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