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Radiative and total heat transfer in circulating fluidized beds Luan, Wenqi 1997

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RADIATIVE AND T O T A L H E A T TRANSFER IN C I R C U L A T I N G F L U I D I Z E D BEDS By  Wenqi Luan  B. A. Sc., Northwestern University, Xi'an, 1982 M . A. Sc., Northwestern University, Xi'an, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Chemical and Bio-Resource Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1997 ©Wenqi Luan, 1997  In  presenting  degree at the  this  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  scholarly purposes may be her  representatives.  permission.  Department of The University of British Columbia Vancouver, Canada  DE-6 (2/88)  for  an advanced  Library shall make it  agree that permission for extensive  It  publication of this thesis for financial gain shall not  Date  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  ABSTRACT Suspension-to-surface total and radiative heat transfer in circulating fluidized beds at elevated temperatures were studied using a dual tube assembly, a membrane wall and a multifunctional probe. All experiments were carried out in a pilot scale circulating fluidized bed combustor having a 0.152 m by 0.152 m square cross-section and a 7.3 m height. The effects of suspension density, suspension temperature and particle size on the total and radiative heat transfer coefficients were investigated. Results were obtained with silica sand particles of mean diameter 137, 286, 334 and 498 pm at suspension temperatures ranging from 794 to 913 °C for suspension densities up to 110 kg/m .  In one set of experiments, heat transfer rates were measured simultaneously for two tubes having different surface emissivities. The radiative component was estimated by comparing the total heat fluxes measured for each tube. It was found that the total suspension-to-tube heat transfer coefficients, which ranged from 150 to 250 W/m K, increased with increasing 2  suspension density and decreased with increasing mean particle size under the conditions of this study. The measured radiative suspension-to-surface heat transfer coefficients, which lay between 40 and 100 W/m K, also increased with increasing suspension density and decreased with increasing particle size. For the bulk suspension temperature range of 800 to 900 °C, it was found that radiative heat transfer comprises 25 to 45% of the total heat transferfromthe suspension to a tube located near the wall. An empirical correlation for the total suspension-to-surface heat transfer coefficient is proposed for broad ranges of suspension density, particle size and suspension temperature. This correlation represents within ±50% the results of this work, as well  ii  as the high-temperature results of previous workers.  Experiments were also carried out using a membrane wall containing embedded thermocouples as the heat transfer surface. Total suspension-to-pipe and suspension-to-fin heat transfer coefficients were estimated from the temperatures measured inside a pipe and a fin. The suspension-to-pipe heat transfer coefficient is higher than the suspension-to-fin coefficient when radiation is significant for the conditions investigated, indicating that the fin is not as efficient as the exposed pipe surface in extracting heat.  A multifunctional probe was designed and fabricated. This probe combines the advantages of the differential emissivity method and the window method for measuring radiative heat transfer fluxes. Cylinders of well-oxidized stainless steel 347 and polished stainless steel 316 with distinctly different surface emissivities were incorporated in the probe, while zinc selenide was chosen as the window material. The probe was calibrated using a cylindrical cavity heated by an electric furnace. Net radiation and ray tracing methods were employed to develop a model to calculate the radiative component through the window. Comparison of the results from these two methods showed that the radiative heat transfer coefficient determined by the window method was higher than that obtained by the differential emissivity method. The discrepancy between the two methods is attributed to the unreliable values of the surface emissivities used in the differential emissivity method. The heat transfer coefficients measured by the probe were higher than those obtained using either the dual tube or the membrane wall due primarily to the attenuating effect of the length of the heat transfer surface.  iii  A comprehensive radiative heat transfer model was developed based on the known coreannulus structure of the suspension in the circulating fluidized bed. In the core region, the temperature and solids concentration are assumed to be uniform, while the annulus region is composed of a gas layer and an emulsion layer in the vicinity of the wall, with heat being transferred through the parallel layers by conduction and convection, respectively. For radiative heat transfer, the gas layer is assumed to be transparent. The emulsion medium is considered to be non-gray, absorbing, emitting and scattering, while the scattering within the emulsion layer is taken as multiple, independent and anisotropic. The temperature profile as well as the solids concentration profile in the emulsion layer were considered. The predictions from the model are about 20% higher than the experimental data obtained by the probe using the differential emissivity method, while the predictions agree well with the results using the window method under the conditions in this study. The model predicts that radiation contributes about 35% of the total heat transfer coefficient for typical CFB combustion conditions.  iv  T A B L E OF CONTENTS Page Abstract  ii  Table of Contents  v  List of Tables  x  List of Figures  xii  Acknowledgment  xix  Dedication  xx  CHAPTER 1. INTRODUCTION  1  1.1. Circulating Fluidized Bed Technology  !  1  1.2. Previous Heat Transfer Investigations  4  1.2.1. Effect of Particle Diameter  8  1.2.2. Effect of Superficial Gas Velocity  14  1.2.4. Effect of Suspension Density  15  1.2.5. Effect of Length of Heat Transfer Surface  16  1.2.6. Effect of Bulk Temperature  16  1.2.7. Membrane Wall Studies  17  1.2.8. Experimental Studies on Radiation  18  1.3. Objectives of This Study  ;  CHAPTER 2. EXPERIMENTAL FACILITIES  24  26  2.1. CFBC System  26  2.2. Multifunctional Probe  29  2.2.1. Considerations for Probe Design  31  2.2.2. Probe Design  35  2.3. Dual Tube Heat Transfer Surface  38  2.4. Membrane Wall Heat Transfer Surface  41  2.4.1. Considerations for Membrane Wall Design  v  41  2.4.2. Design and Construction  41  CHAPTER 3. PROBE CALIBRATION  48  3.1. Cavity Design  48  3.2. Heat Exchange between Probe and Cylindrical Cavity  49  3.2.1. Local Apparent Emissivities  49  3.2.2. Radiative Heat Transfer through Opening of Cylindrical Cavity and Calculation of Integrated Cavity Emissivity  53  3.2.3. Radiative Heat Exchange between Cavity and Probe  55  3.2.4. Enhanced Exchange between Cavity and Probe by Free Convection  56  3.3. Heat Transfer EstimatedfromProbe Temperatures  58  3.3.1. Conduction Models for Cylindrical Probes  58  3.3.1.1. One-Dimensional Conduction  58  3.3.1.2. Determination of Probe Correction Coefficient  59  3.3.2. Indirect Parameter Estimation Model  66  3.3.3. Comparison of the Two Models  71  3.4. Radiative Heat Emission and Heat Exchange  73  3.4.1. Differential Emissivity Method (DEM)  73  3.4.2. Window Method (WM)  74  3.5. Conclusions  82  CHAPTER 4. SUSPENSION-TO-SURFACE HEAT TRANSFER: EXPERIMENTAL RESULTS  84  4.1. Heat Transfer to Dual Tube Surface  84  4.1.1. Total Suspension-to-Tube Heat Transfer Coefficients  86  4.1.1.1. Total Suspension-to-Surface Heat Transfer Coefficients  87  4.1.1.2. Effects of Suspension Density, Temperature and Particle Size  89  4.1.1.3. Correlation for Suspension-to-Tube Total Heat Transfer Coefficient  94  4.1.2. Suspension-to-Tube Radiative Heat Transfer Coefficients  vi  95  4.1.2.1. Calculation of Radiative Heat Transfer Coefficients  95  4.1.2.2. Results and Discussion  98  4.2. Heat Transfer to Membrane Wall Surface  104  4.2.1. Effect of Membrane Wall Length  104  4.2.2. Temperature Profiles  108  4.2.2.1. Lateral Temperature Profile in the Riser  108  4.2.2.2. Temperature Distribution in Pipe and Fin  111  4.2.3. Suspension-to-Surface Heat Flow Rates and Heat Transfer Coefficients 4.3. Conclusions  114 123  CHAPTER 5. SUSPENSION-TO-PROBE HEAT TRANSFER IN CFBC  .125  5.1. Total Heat Flux and Heat Transfer Coefficient  126  5.2. Radiative Heat Transfer  130  5.2.1. Effect of Suspension Density on Radiative Heat Transfer  133  5.2.2. Effect of Suspension Temperature on Radiative Heat Transfer  140  5.2.3. Effect of Surface Length on Radiative Heat Transfer  141  5.2.4. Ratio of Radiation to Total Heat Transfer  144  5.2.5. Emissivity of SS316 Probe  144  5.3. Convective Heat Transfer  144  5.4. Conclusions  148  CHAPTER 6. MODELING OF HEAT TRANSFER  150  6.1. Previous Radiative Heat Transfer Models  151  6.1.1. Packet Model of Basu and Nag  151  6.1.2. Uniform Emulsion Model of Chen et al  152  6.1.3. Grid Model of Werdermann and Werther  154  6.1.4. Two-Flux Model of Glatzer and Linzer  155  6.1.5. Network Model of Fang et al  156  6.1.6. Summary of Previous Radiative Heat Transfer Models  157  vii  6.2. Proposed Non-Uniform Emulsion Model  157  6.2.1. Framework  158  6.2.2. Assumptions  161  6.2.3. Heat Transfer Components  163  6.2.3.1. Conduction in Gas Gap  163  6.2.3.2. Conduction / Convection in Emulsion Layer  164  6.2.3.3. Radiation in Emulsion Layer  165  6.2.4. Solving for Radiative Heat Transfer Fluxes and Coefficients  168  6.2.5. Simulations of Heat Transfer Fluxes and Coefficients  170  6.2.5.1. Comparison of Model with Experimental Results from the Literature 6.2.5.2. Prediction of Heat Transfer under Isothermal Emulsion Layer Conditions 6.3. Conclusions  170 174 186  CHAPTER 7. OVERALL CONCLUSIONS AND RECOMMENDATIONS  187  7.1. Conclusions  187  7.2. Recommendations  190  NOMENCLATURE REFERENCES  1.91 ,  199  Appendix I. Determination of Total Surface Emissivity  211  Appendix II. Thermal Properties of Window Material-Zinc Selenide  221  Appendix III. Expressions for View Factors  224  III. 1. Definition of View Factor  224  III.2. View Factors in Cylindrical Cavity  226  III.2.1. Diffuse View Factor between Elemental Circular Band and Elemental Ring III.2.2. Diffuse View Factor between Two Elemental Coaxial Circular Bands  226 229  Appendix IV. Derivation of Equation (3-19)  232  Appendix V. Derivation of Equation (3-37) Using Net Radiation Method  234  viii  Appendix VI. Influence of Radiation from Refractory Walls to the Dual Tube Surfaces Appendix VII. Experimental Heat Transfer Data Appendix VIII. Program Listing.  237 ....241 253  ix  LIST OF TABLES Page Table 1.1. Parameters affecting heat transfer coefficients in CFBs  7  Table 1.2. Summary of experimental studies of total suspension-to-wall heat transfer in CFBs with short probe surfaces at low temperatures  9  Table 1.3. Summary of experimental studies of total suspension-to-wall heat transfer in CFBs with short probe surfaces at high temperatures  11  Table 1.4. Summary of experimental studies of total suspension-to-wall heat transfer in CFBs with long surfaces.  12  Table 1.5. Summary of experimental studies of total suspension-to-wall heat transfer in CFBs with membrane wall surfaces  13  Table 1.6. Summary of experimental studies of radiative suspension-to-wall heat transfer in circulating fluidized beds  22  Table 1.7. Summary of experimental studies of radiation in bubbling fluidized beds  23  Table 2.1. Properties of some window materials (Ivan'ko 1971, Touloukian and Dewitt 1972, Lide 1992) Table 2.2. Measured physical properties of the four probe cylinders (Touloukian and Dewitt 1970, 1972, Incropera and Dewitt 1990)  38  Table 3.1. Comparison of heat flux calculated by analysis of cavity emission and by temperature measurement  70  Table 4.1. Particle size analyses and fluidization properties for Ottawa silica sands used in dual tube heat transfer tests  85  Table 4.2. Symbols and operating parameters for heat transfer data plotted in Figures 4.3 to 4.6  91  34  Table 4.3. Particle size analyses and fluidization properties for Ottawa silica sand before and after membrane surface heat transfer test  105  Table 4.4. Symbols and operating conditions for heat transfer data plotted in Figure 4.16  106  Table 4.5. Symbols and parameters for temperature profiles plotted in Figure 4.17  110  Table 5.1. Experiments showing the effect of probe length on CFB heat transfer at high temperature. Particles were sand in all cases  127  x  Table 6.1. Major assumptions and parameters in previous CFB radiative heat transfer models  158  Table 6.2. Selected angles and weights used in the discrete ordinate method with 12 directions  169  Table 6.3. Predicted radiative heat transfer results compared with published experimental results  173  Table 6.4. SS347 and SS316 probe surface temperatures as a function of bulk suspension temperature  178  Table 1.1. Measured values of surface emissivities of stainless steel 316  218  Table 1.2. Measured values of surface emissivities of stainless steel 347  218  Table VI. 1. View factors calculated using Eqs. (VI-1) and (VI-2).  239  Table VII. 1. Suspension-to-membrane-wall heat transfer coefficients as functions of suspension density and suspension temperature Table VII.2. Suspension-to-dual-tube heat transfer coefficients as functions of suspension density and suspension temperature Table VII.3. Suspension-to-probe heat transfer coefficients as functions of suspension density and suspension temperature  xi  241 242 249  LIST OF FIGURES Page Figure 1.1. Simplified schematic diagram of an atmospheric circulating fluidized bed combustion system for power generation  2  Figure 2.1. Simplified schematic diagram of circulating fluidized bed combustion facility  27  Figure 2.2. View of principal refractory-lined riser showing location of heat transfer tubes (all dimensions in mm)  28  Figure 2.3. Cumulative percentage of blackbody thermal emission as function of wavelength (Touloukian and Dewitt 1972)  32  Figure 2.4. Spectral transmissivity of zinc selenide at different temperatures (Touloukian and Dewitt 1972)  35  Figure 2.5. Schematic of multifunctional probe.  36  Figure 2.6. Total hemispherical emissivities of stainless steel 316 and stainless steel 347 as functions of temperature (Touloukian and Dewitt 1972)  39  Figure 2.7. Schematic of dual tube heat transfer surface (all dimensions in mm)  40  Figure 2.8. Membrane wall (top view)  42  Figure 2.9. Membrane wall (side view)  :  43  Figure 2.10. Thermocouples embedded in the pipe wall and the fin  46  Figure 2.11. Refractory-lined section equipped with membrane wall composed of two pipe and fin materials (stainless steels 347 and 316) with different surface emissivities Figure 3.1. Cylindrical blackbody cavity showing geometric factors used to calculate heat transfer to the probe cylinder using the integral equation method  47 49  Figure 3.2. Temperatures measured by K type thermocouples embedded in the probe cylinders in calibration runs with several furnace temperatures  60  Figure 3.3. Schematic diagram showing quasi-symmetric probe cylinder assembly  61  Figure 3.4. Diagram of probe simplified to quasi-symmetric cylinder  62  Figure 3.5. Algorithm for calculation of temperature distribution in quasi-symmetric cylinder  65  xii  Figure 3.6. Simulated probe correction coefficients determined using two-dimensional and one-dimensional models  66  Figure 3.7. Comparison between probe correction coefficients in Eq. (3-25) determined using one-dimensional model and inverse model  72  Figure 3.8. Comparison between heat flux estimated using one-dimensional model and inverse method for window probe. Tf = 200 to 1000 °C  72  Figure 3.9. Schematic diagram of heat flow through ZnSe window probe  75  Figure 3.10. Procedure for computing window temperature distribution and heat transfer fluxes using the window method  80  Figure 3.11. Temperature distribution from the furnace to the window probe cylinder for different furnace temperatures and a water temperature of 5 °C.  81  Figure 4.1. Axial profile of bulk suspension temperature along CFBC riser. d =334 pm. group I: gas burning; group II: coal burning .'.  87  Figure 4.2. Suspension-to-tube heat transfer coefficient plotted against particle suspension density for different particle diameters at different suspension temperatures with oxidized tube. U =7.8 to 9.6 m/s, P/S=1.3 to 4.0. Solid lines are least squares linear fits  90  Figure 4.3. Comparison of total heat transfer coefficients as a function of suspension density with other reported data obtained at high temperature with surfaces 100 mm long or longer. Symbols and parameters are listed in Table 4.2  92  Figure 4.4. Effect of suspension temperature on total heat transfer coefficient at a suspension density of 20 kg/m . Symbols and operating parameters are listed in Table 4.2.  92  Figure 4.5. Effect of particle size on total heat transfer coefficient at high temperature. psusp 20 kg/m . Symbols and operating parameters are listed in Table 4.2  93  Figure 4.6. Comparison of experimental heat transfer coefficients with those predicted using Eq. (4-5). p =l to 100 kg/m , d =137 to 498 pm, T =680 to 1178 K. Symbols and operating parameters are listed in Table 4.2  95  Figure 4.7. Effect of suspension density on suspension-to-tube heat transfer coefficient for different suspension temperatures. d =498 pm, U =8.4 to 8.8 m/s, P/S=2.4-4.0. Solid symbols: oxidized tube; open symbols: unoxidized tube.  98  Figure 4.8. Effect of suspension density on suspension-to-tube heat transfer coefficient for different suspension temperatures. d =334 pm, U =8.4 to 8.8 m/s, P/S=2.4-4.0. Solid symbols: oxidized tube; open symbols: unoxidized tube  99  p  g  3  =  3  3  susp  p  p  b  g  p  g  xiii  Figure 4.9. Effect of suspension density on suspension-to-tube heat transfer coefficient for different suspension temperatures. d =137 um, U =7.8 to 8.9 m/s, P/S=1.6 to 2.4. Solid symbols: oxidized tube; open symbols: unoxidized tube  99  Figure 4.10. Effect of suspension density on suspension-to-tube radiative heat transfer coefficient for three types of sand at 905 ± 7 °C. Solid lines are least square fits  100  Figure 4.11. Effect of suspension density on suspension-to-tube radiative heat transfer coefficient for three types of sand at 851 ± 6 °C. Solid lines are least square fits  100  Figure 4.12. Effect of suspension density on suspension-to-tube radiative heat transfer coefficient for three types of sand at 803 ± 9 °C. Solid lines are least square fits  101  Figure 4.13. Relative importance of radiative heat transfer as a function of suspension density at 905 ± 7 °C  102  Figure 4.14. Relative importance of radiative heat transfer as a function of suspension density at 851 ± 6 °C  103  Figure 4.15. Relative importance of radiative heat transfer as a function of suspension density at 803 ± 9 °C.  103  Figure 4.16. Effect of heat transfer surface length on suspension-to-membrane-wall heat transfer coefficient at two bulk temperatures. d =286 pjn,U =8.0-8.5 m/s, P/S=2.0-2.4.  106  Figure 4.17. Lateral suspension temperature profiles near membrane walls, measured from pipe crest. The conditions are listed in Table 4.5  Ill  Figure 4.18. Temperature distribution in pipe and fin versus suspension density. T =804 + 5 °C, d =286 um, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m.  112  Figure 4.19. Pipe and fin temperature distributions measured by embedded thermocouples. d =286 um, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m. Open symbols: T =804 + 5 °C; Solid symbols: T =706 + 4 °C. Dotted lines: best linear fits for T =804 + 5 °C; Solid lines: best linear fits for T =706 ± 4 °C.  113  Figure 4.20. Schematic of exposed-pipe-radial, insulated-pipe-tangential, fm-one-dimensional heat transfer model of Bowen et al. (1991).  115  p  g  p  b  p  p  g  g  g  b  b  b  b  Figure 4.21. Effect of suspension density on suspension-to-membrane-wall heat transfer coefficient for different suspension temperatures. d =286 um, U =8.0 to 8.5 m/s, P/S=2.0 ...117 p  g  Figure 4.22. Suspension-to-pipe and suspension-to-fin heat transfer rates versus suspension density. d =286 um, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m. Q and Qf are evaluated using (T1-T2) and (T5-T6), respectively. The lines are the best linear fits p  g  p  xiv  119  Figure 4.23. Suspension-to-pipe, suspension-to-fin and total suspension-to-membranewall heat transfer coefficients versus suspension density. Solid symbols: Tb=804 ± 5 °C; open symbols: T =706 ± 4 °C. d =286 pm, P/S=2.0, U =8.0 to 8.5 m/s, z=5.44 m  120  Figure 4.24. Suspension-to-membrane-wall heat transfer rate versus suspension density. d =286 pm, P/S=2.0, U =8.0 to 8.5 m/s, z=5.44 m.  121  Figure 4.25. Temperatures of the insulated pipe at 0=0 versus suspension density. Solid symbol: measured temperature; open symbol: temperature predicted by model 2 of Bowenetal. (1991)  123  Figure 5.1. Suspension-to-probe heat transfer coefficient plotted against suspension density at different suspension temperatures for oxidized SS347 probe. d =334 um, U =7.8 to 9.6 m/s, P/S=1.3 to 4.0. Lines are least square linear fits with dashed line corresponding to Tb=851 °C. .  126  Figure 5.2. Suspension-to-probe total heat transfer coefficient as a function of vertical length of heat transfer surface. d =215 to 334 pm, p sp=50 to 60 kg/m  128  Figure 5.3. Heat transfer coefficients obtained using SS347 and SS316 probes as a function of the suspension temperatures. d =286 pm, ps =35-50 kg/m . The solid lines are the exponential-growth best fit results  130  Figure 5.4. Effect of suspension density on total heat transfer flux. d =334 um. Solid symbols: SS347; open symbols: SS316  131  Figure 5.5. Surface temperatures of the SS347 and SS316 probes at various suspension temperatures as a function of suspension density for particle size 334 pm  132  Figure 5.6. Radiative heat flux as a function of suspension density. Solid Symbols: window method; open symbols: differential emissivity method, (a): d =334 um; (b): d =286 um  134  Figure 5.7. Effect of suspension density on suspension-to-surface radiative heat transfer coefficient for different suspension temperatures determined using differential emissivity method, (a): d =334 pm, U =8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d =286 pm, U =7.8 to 8.9 m/s, P/S=1.5-2.3. The lines show the best linear fits  136  Figure 5.8. Comparison of radiative heat transfer fluxes estimated from window method using Eqs. (3-37) and (5-4) for different suspension densities, (a): d =334 pm, T =851 °C, U =-8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d =286 pm, T =850 °C, U =7.8 to 8.9 m/s, P/S=1.5-2.3. Solid lines are best linear fits  137  Figure 5.9. Effect of suspension density on suspension-to-probe radiative heat transfer coefficient for different suspension temperatures determined using the window method (Eq. 3-37). (a): d =334 pm, U =8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d =286 pm, U -=7.8 to 8.9 m/s, P/S=1.5-2.3. Solid lines are best linear fits  138  b  p  p  g  g  p  g  p  SU  p  Usp  p  p  p  p  g  p  g  p  b  g  p  p  g  b  g  p  g  xv  Figure 5.10. Effect of cooled surface temperature on radiative heat transfer coefficient. Solid symbols arefromthis work for particles of mean diameter 334 um for the five suspension densities given at the right and at a suspension temperature of 850 ± 6 °C. Lines are the calculated results passing through the points measured in this work using Eq. (5-5) and assuming that the probe surface temperatures varyfrom300 to 1000 K 140 Figure 5.11. Total heat transfer fluxes estimated using SS347 and SS316 probes at different suspension temperatures. d=286 um, U =7.8 to 8.9 m/s, P/S=l.5-2.3. Solid lines are the best linear fits for a second order polynomial p  g  Figure 5.12. Comparison of radiative heat transfer coefficients with literature results for different suspension temperatures. • , • : d=286 um; • : d=264 um; • : d=296 um p  p  141  p  142  Figure 5.13. Comparison of radiative heat fluxes estimated using Eq. (3-33) (differential emissivity method) with that for black body at the same suspension temperature. d=286 um. Solid line is obtained by regression using a second order polynomial  142  Figure 5.14. Comparison of radiative heat transfer coefficient obtained with the probe using differential emissivity method and window method and the tube surfaces (see Chapter 4). d=334 um. Tb=851 °C. Lines are least square linear fits  143  p  p  Figure 5.15. Relative importance of probe radiative heat transfer estimated using the window method and differential emissivity method as a function of suspension density, (a): d=334 um, (b): d=286 um  145  Figure 5.16. Apparent surface emissivity of SS316 probe estimated using radiative heat fluxes determinedfromthe window method (d=334 um). Runs were performed in order of increasing suspension temperature (Tt,)  146  p  p  p  Figure 5.17. Effect of suspension density on suspension-to-probe convective heat transfer coefficient, estimated by subtracting radiative componentfromtotal coefficient, for different suspension temperatures, (a): d=334 um, U =8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d=286 um, U=7.8 to 8.9 m/s, P/S=l.5-2.3 147 p  p  g  g  Figure 6.1. Schematic showing non-uniform emulsion heat transfer model: (a) physical model; (b) radiative heat transfer model  160  Figure 6.2. Procedures used to compute the heat transfer flux and coefficient  171  Figure 6.3. Effect of particle-surface contact time on the predicted emulsion convection heat transfer coefficients. T =851 °C 173 Figure 6.4. Extinction coefficient distribution in the emulsion layer. T =850 °C, T =490 °C, p =50 kg/m , p=2610 kg/m 175 b  b  3  susp  3  p  xvi  w  Figure 6.5. Radiative heat flux distribution in the emulsion layer. d =334 pm, p kg/m , e =0.9, e =0.9, 2X=0.152 m, L=0.006m. T = f(T ) (see Fig. 5.5) p  p 50 =  SUS  176  3  w  p  surf  b  Figure 6.6. Comparison of the predicted and experimental radiative heat transfer fluxes as a function of the bulk suspension temperature. p s p 5 0 kg/m , p =2610 kg/m , e =0.9. T f is listed in Table 6.4. Line shows predicted results. DEM=Differential Emissivity Method; WM=Window Method =  3  3  SU  w  p  s u r  177  Figure 6.7. Comparison of predicted radiative heat transfer coefficient with experimental results of the probe using the window method. Tb=850 °C, ep=0.9, ew 0.9, d =334 um  179  Figure 6.8. Effect of heat transfer surface emissivity on the predicted radiative heat transfer coefficient. T =850 °C, T -490 °C, psus=50 kg/m , pp=2610 kg/m  180  Figure 6.9. Effect of particle size on the predicted heat transfer coefficients (lines): T =850 °C, T =490 °C, p s 50 kg/m , e -=0.9, ^=0.9, d =334 um. Symbols are for measured radiative heat transfer coefficients (see Table 6.3 for details): • : Basu and Konuche (1988); • : Werdermann and Werther (1994); A : This study, (Differential Emissivity Method); A : This study, (Window Method)  181  Figure 6.10. Effect of particle size on the predicted relative importance of the radiative heat transfer component. T =851 °C, T rf=490 °C, p p=50 kg/m , e =0.9. DEM=Differential Emissivity Method; WM=Window Method  182  Figure 6.11. Comparison of predicted and experimental heat transfer coefficients as a function of suspension temperature. d =334 pm, p p 50 kg/m , e =0.9, T f is listed in Table 6.4. DEM=Differential Emissivity Method; WM=Window Method  183  Figure 6.12. Effect of suspension temperature on the relative importance of radiative heat transfer coefficients. d =334 pm, p s p 5 0 kg/m , ew=0.9. T rf is listed in Table 6.4. DEM=Differential Emissivity Method; WM=Window Method  183  Figure 6.13. Comparison of radiative heat transfer coefficient predicted by the nonuniform emulsion model with predictions from the packet model of Basu and Nag (1987) as well as the network model of Fang et al. (1995) at elevated temperatures. e =0.9, e =0.8, ed^O^, p s p 5 0 kg/m , d =334 pm. T f is shown in Figure 5.5 '.  184  Figure 6.14. Comparison of predicted radiative heat transfer coefficients with experimental results. • , B : This work, ep=0.9, ew=0.8, ps sp 18-90 kg/m . T f is shown in Figure 5.5 and Table 6.4; • : Werdermann and Werther (1994) (for details see Table 6.3); T : Basu and Konuche (1988) (for details see Table 6.3)  185  Figure L l . Schematic of apparatus for measurement of normal spectral emissivity (DeWitt and Nutter 1988)  212  =  p  3  b  surf  =  b  surf  SU  3  P  3  P  p  p  3  b  SU  SUS  w  =  p  3  SUS  w  SUI  =  p  SU  SU  =  p  w  SU  p  s u r  =  U  xvii  s u r  Figure 1.2. Typical arrangement of blackbody technique for normal spectral emissivity measurements (DeWitt and Nutter 1988)  213  Figure 1.3. Typical arrangement of calorimetric technique for hemispherical total emissivity measurements (DeWitt and Nutter 1988) 215 Figure 1.4. Schematic of spectral emissivity measurement using FT-IR (Markhamet al. 1996) 216 Figure I. 5. Measured and recommended total emissivities of stainless steel 347 and 316. Symbols are listed in Tables VII. 1 and2  219  Figure II. 1. Thermal conductivity of zinc selenide as a function of temperature  ,221  Figure II.2. Overall spectral absorptivity of zinc selenide at 373 K. Integrated average absorptivity for the wavelength range from 2.5 to 18.3 micrometers is 0.10 Figure II.3. Total transmissivity of zinc selenide as a function of temperature  221 222  Figure II.4. Analysis of the uncertainty of the transmissivity of zinc selenide window on the estimates of radiative heat flux, (a): d =334 pun; (b): d =286 um p  p  Figure III. 1. View factor and solid angle between two arbitrarily oriented surfaces  223 224  Figure III.2. Diffuse view factor between elemental circular band and elemental circular ring  226  Figure III.3. Diagram of cavity cylinder  228  Figure III.4. Diffuse view factor between two elemental coaxial circular bands  229  Figure IV. 1. Heat exchange between two coaxial disks  232  Figure V . l . Diagram illustrating heat balance for ZnSe window.  234  Figure V.2. Diagram illustrating heat balance for SS347 surface behind ZnSe window  234  Figure VI. 1. Schematic of dual tubes placed in the riser. Dimensions are in mm  237  Figure VI.2. View factor between cylinder and parallel rectangle  237  xviii  ACKNOWLEDGMENT  I would like to express my sincere gratitude and admiration to my supervisors, Dr. J. R. Grace, Dr. C. J. Lim, Dr. C. H. M . Brereton and Dr. B. D. Bowen for their guidance, responsible supervision, spiritual support and encouragement over the entire course of this work. My appreciation also goes to Dr. Z. H. Fang, Dr. K. S. Lim and Dr. J. X. Zhu for their excellent ideas, suggestions and assistance. I am thankful to all of the staff of the Workshop and Stores in the Department of Chemical and Bio-Resource Engineering and in the Pulp and Paper Center for their help with equipment set-up and maintenance. I am also indebted to Dr. J. Chen, Dr. A. Ergiidenler, Ms. I. Hwang, Dr. S. Julien, Mr. F. L. Lui, Dr. J. Muir and Mr. W. S. Tang for their efforts and participation. Also, I want to thank everybody in the Fluidization Group for helpful discussions and suggestions. The financial support of Energy, Mines and Resources Canada and the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.  Finally, I would like to express my thanks to my family, Mei Chen and Weijie Luan, for their thorough understanding, continuous encouragement, and wholehearted support.  xix  DEDICATED TO: my parents and my teachers  XX  Chapter 1: Introduction  Radiative and Total Heat Transfer in Circulating Fluidized Beds  CHAPTER 1 INTRODUCTION  1.1. Circulating Fluidized Bed Technology Circulating fluidized bed (CFB) technology dates back to the 1940's when it was first investigated for the cracking of high molecular weight hydrocarbons. It has evolved rapidly in the last two decades since the potential advantages of CFB combustion over conventional pulverized and bubbling fluidized bed boilers have been realized (e.g., see Grace and Bi 1997). There are now hundreds of circulating fluidized bed combustors (CFBCs) in operation or under construction worldwide (Lee 1997). CFB technology has also been used for various other gassolid reactions, e.g., in calcination of aluminum ores and phosphates (Reh 1986, 1995) and gasification (Dry and Beeby 1997) as well as for other catalytic gas-phase reactions (Matsen 1997).  A typical power station circulating fluidized bed facility is shown in Figure 1.1. The principal components include the riser (reactor) and particle return-to-riser system (cyclone and downcomer). Air enters the riser from its base through a distributor to suspend the particles and to provide the oxygen needed for reaction. The gas velocity in the riser is high enough that  1  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  particles are carried up and entrained out of the riser from its top. Most entrained particles are captured in one or more cyclones and returned to therisercontinuously through a loop seal, L valve or J-valve near the base of the riser. The gas entrained and fine particles leaving the top of the cyclone unit pass through a heat exchanger and a dust filter for further capture of fine particles in order to meet environmental requirements before it is vented through the stack. Fuel and sorbent are added into the bottom of the riser using feeders. Heat exchangers are installed in the riser and the external chamber. Steam generated from heat exchangers can be used to propel a steam turbine to supply electricity.  The typical advantages and disadvantages of the CFB have been reviewed by Grace and Bi (1997). The main advantages of the CFBC relative to other combustors include: 1. Low N O emissions. This is achieved by staging air injection and by operating with x  lower and uniform bed temperatures (typically 850 - 900 °C). 2. Low S 0 emissions. This is fulfilled by using sorbent particles, such as limestone, to 2  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. 4. High combustion efficiency. The combustion efficiency (i.e., carbon combustion efficiency, the ratio of fixed carbon combusted to that which is fed) commonly reaches 98-99% (Brereton et al. 1991).  3  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  5. Low fuel preparation cost. Fuels can be fed to CFBC with broad particle size distributions and with limited amounts of crushing. 6. Simple turndown and control. There are a number of independent variables that can be used to control the system under a fairly wide range of operating conditions.  In an industrial scale CFBC, the riser heights are generally in the range of 10 to 35 m and the width varies from about 5 to 10 m. The superficial gas velocity above the secondary air inlets is typically 4 to 8 m/s. Primary-to-secondary air ratios are generally in the range 1:3 to 3:1. Sand, commonly used as an inert medium, has average diameters from about 100 to 500 microns and densities between 2000 and 3000 kg/m . Solid fuel particles vary in size from about 20 to 3000 3  microns and typically comprise only about 5 % of the total bed inventory by weight. Suspension densities in theriserare usually of the order of 300 to 1000 kg/m . 3  1.2. Previous Heat Transfer Investigations The study of CFB heat transfer is very important because most CFB applications, including combustion, calcination, and hydrocarbon cracking, are operated at high temperature. With better knowledge of heat transfer mechanisms, design and operation can be improved and the energy created during the combustion process can be used with higher efficiency. Grace (1986), Glicksman (1988), Grace (1990a), Leckner (1990), Basu and Nag (1996), Glicksman (1997) have presented comprehensive reviews of heat transfer in CFBs. Most investigations of CFB heat transfer have been restricted to consideration of the suspension-to-wall heat transfer coefficients.  4  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  Total suspension-to-wall CFB heat transfer coefficients are strongly affected by the gas and particle motions. Simultaneous measurements of instantaneous heat transfer and particle concentrations in cold model units have shown that high local instantaneous particle concentrations adjacent to the transfer surface give rise to high heat transfer coefficients (Wu et al. 1989b, Dou et al. 1992). From previous studies (Grace 1986, Basu and Nag 1987, Wu et al. 1987, 1989a, Chen and Chen 1992, Leckner and Andersson 1992, Nag and Ali 1992, Hyre and Glicksman 1995), it is clear that the local suspension density is a dominant factor influencing CFB heat transfer. Several researchers have related their experimental heat transfer coefficient data to the cross-sectional average suspension density, estimated from the pressure drop between pressure taps above and below the heat transfer surface. Any factor that changes the contact mode or contact time between particles and a heat transfer surface affects the heat transfer coefficients (Grace 1990a, Glicksman et al. 1991, Leckner and Andersson 1992). Most experiments have been conducted using columns of circular cross-section operated at ambient temperature. Heat transfer data for high temperature CFBs have been presented by Kobro and Brereton (1986), Andersson et al. (1987), Wu et al. (1987, 1989a), Basu and Konuche (1988), Polpool (1991), Leckner and Andersson (1992), Nag and Ali (1992), Couturier et al. (1993), and Werdermann and Werther (1994). Less attention has been paid to heat transfer to membrane walls, the geometry used in commercial CFB boilers. Some preliminary investigations have been carried out at UBC and at Chalmers University of Technology to study heat transfer in CFBs with membrane walls (Wu et al. 1987,1989a, Andersson and Leckner 1992, Lockhart et al. 1995).  Another challenging area for heat transfer studies is to determine the importance of thermal radiation. Previous data obtained in conventional bubbling fluidized beds tend to be  5  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  inconsistent. Few experimental and theoretical studies of radiative heat transfer in CFBs have been carried out (Basu and Nag 1987, Basu and Konuche 1988, Chen et al. 1988, Basu 1990, Han et al. 1992, Werdermann and Werther 1994, Glatzer and Linzer 1995, Fang et al. 1995, Steward et al. 1995, Baskakov and Leckner 1997). Basu and Konuche (1988) reported that the radiative component is 70-90% of the total heat flux transferred from the bed to the wall, while the results of Han et al. (1992) showed that the radiative contribution to heat transfer in CFBs is about 40-50%. The high percentage of radiation to total heat transfer in CFBs compared with bubbling fluidized beds is attributed to the relatively low particle concentrations. Thermal radiation intensity increases with temperature according to the Stefan-Boltzmann law. Hence, the bulk suspension temperature, especially the temperature of particles close to the heat transfer surface, is a predominant factor affecting radiation. Fatah et al. (1992) measured radial temperature profiles of both particles and gas, while Leckner and Andersson (1992) and Golriz (1995) measured the temperature of a gas-particle mixture and the thermal boundary layer close to a membrane wall. Couturier et al. (1993) as well as Werdermann and Werther (1994) measured radial temperature profiles in commercial CFB power plants and showed that there existed significant radial temperature gradients near the membrane walls.  There are numerous variables that affect suspension-to-wall heat transfer coefficients. They can be classified into four groups as listed in Table 1.1. Most previous research has focused on several parameters which have been found to have a significant influence on heat transfer coefficients. Such parameters as particle diameter, superficial gas velocity, suspension density, solid circulation rate and bulk temperature have been extensively studied. The results and conclusions are generally consistent with each other. The experiments that have been carried out  6  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter I: Introduction  Table 1.1. Parameters affecting heat transfer coefficients in CFBs.  Physical properties:  Dependent parameters: k j u s p — suspension thermal conductivity  Gas: p ~ density  s  k g — thermal conductivity  P s u s p " density of suspension  p ~ viscosity  e  C p ~ specific heat  8— thermal boundary layer thickness  g  g  g  susp  ~ voidage of suspension  susp  ~ emissivity of suspension  e ~ emissivity g  Particles:  {Configuration:  p ~ density p  kp~ thermal conductivity  D~ riser diameter  dp— mean diameter  H ~ riser height  shape factor c ~ specific heat  D t u b e — tube diameter  e ~ emissivity  L ~ tube length  PSD— particle size distribution  Location of tubes  pp  p  Entrance and exit configuration  Heat transfer surface:  Distributor:  ksurf" thermal conductivity  e ~ emissivity  -- orifice diameter  C,— surface roughness  — orifice spacing  surf  -orifice open percentage — configuration  Operating conditions:  Membrane wall and fin: U ~ superficial gas velocity  — geometry and dimensions Internals  g  G ~ solids circulation rate s  P~ system pressure T  s u s p  ~ temperature of suspension  Other:  T s u n T " temperature of heat transfer surface g— acceleration due to gravity  7  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  to obtain these data have led to a better understanding of heat transfer in CFBs. The experimental conditions and heat transfer coefficients measured in studies using short probe surfaces at low temperatures are listed in Table 1.2; those for short probe surfaces at high temperatures are listed in Table 1.3; with long tube surfaces at high and low temperatures are listed in Table 1.4; and with membrane wall surfaces at high and ambient temperatures are listed in Table 1.5. The effect of the principal parameters on suspension-to-wall heat transfer coefficient is briefly reviewed in the following paragraphs.  1.2.1. Effect of Particle Diameter There is considerable evidence (Mickley and Trilling 1949, Kobro and Brereton 1986, Subbarao and Basu 1986, Wu et al. 1987, Furchi et al. 1988, Basu 1990) that heat transfer coefficients increase with decreasing mean particle diameter. At any suspension density, the heat transfer coefficient is typically proportional to d - ~ "°- (Mickley and Fairbanks 1955). Finer -0  3  6  p  particles give higher heat transfer rates because they result in shorter average distances for conduction between the wall and adjacent particles, i.e., the thermal conductive resistance due to the intervening gas is smaller. The influence of particle diameter is significant only for the results obtained with short heat transfer surfaces. If the heat transfer surface is sufficiently long in the vertical direction, the particles in contact with the surface reach thermal equilibrium as the particles fall down along the wall (Wu et al. 1990). 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This means that suspension densities drop with increasing superficial gas velocities at constant solids circulation rate. Hence, heat transfer coefficients decrease with increasing superficial gas velocities (Kiang et al. 1975, Basu et al. 1987). If the suspension density is maintained constant with increasing superficial gas velocity, the heat transfer coefficient does not change significantly (Wu et al. 1987) because the gas convective component is less important for typical particles than the particle convective component (Grace 1986). On the other hand, some results obtained with suspension densities in the ranges of 20 to 60 kg/m (Basu et al. 1987) and 240 to 368 kg/m (Fraley et al. 1983) show a small effect of gas 3  3  velocity, with heat transfer coefficients decreasing with increasing gas velocity at a given suspension density. However, the data of Mickley and Trilling (1949) indicate that heat transfer coefficients increase with superficial gas velocities for low gas velocities and dense suspensions.  If the superficial gas velocity is increased while adjusting the solids circulation rate to maintain constant suspension density, the voidage profile may change, thereby affecting heat transfer coefficients. Tung et al. (1988) found that the radial profile of voidage at a given average suspension density is solely a function of radial position. The effect of superficial gas velocity on the heat transfer coefficient mainly depends on its effect on solids motion near the wall. The particle renewal rate at the heat transfer surface is dependent on the exchange rate of particles  14  Chapter 1: Introduction  Radiative and Total Heat Transfer in Circulating Fluidized Beds  between the core and annulus regions in a CFB riser.  1.2.3. Effect of Solids Circulation Rate With increasing solids circulation rate, the suspension density increases for a constant superficial gas velocity. Heat transfer coefficients therefore increase with increasing solids circulation rate (Feugier et al. 1987). However, heat transfer coefficients increase with increasing solids recirculation rate for a given suspension density. This has been found for both high suspension density (group A particles) (Fraley et al. 1983) and low suspension density (group B particles) (Basu et al. 1987).  1.2.4. Effect of Suspension Density It is generally recognized that the suspension density is a dominant factor influencing CFB heat transfer. Heat transfer coefficients depend on the coverage of the heat transfer surface by particles and the renewal rate of particles at the surface. In a bubbling fluidized bed, the particle concentration is high and bubbles sweep the exchanger surface, causing renewal. This results in heat transfer coefficients which are higher than those in a CFB. In a CFB, as suspension density increases the particle concentration close to the wall also increases. Therefore, more particles contact the surface and the total heat transfer coefficient increases. Researchers have often related their experimental heat transfer data to the cross-sectional average suspension density. The suspension densities have usually been estimated by measuring the static pressure drop between pressure taps located below and above the heat transfer surface. This gives reasonable estimates of the suspension density when acceleration and frictional effects are negligible (Van Swaaij et al.  15  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter I: Introduction  al. 1970). Some advanced techniques have also been developed to measure local particle concentrations, e.g., capacitance probes (Brereton 1987, Wu 1989), optical fiber probes (Horio et al. 1988, Chen and Chen 1992, Zhou 1995) and y-rays (Wirth and Seiter 1991).  1.2.5. Effect of Length of Heat Transfer Surface Glicksman (1988) found that heat transfer coefficients obtained using short heat transfer probes are roughly proportional to the square root of the average suspension density, p s p (Kiang S U  et al. 1976, Subbarao and Basu 1986, Andersson et al. 1987, Basu et al. 1987). For long heat transfer surfaces, it has been consistently shown that the total heat transfer coefficient, hj, is proportional to ( p s p ) (Feugier et al. 1987, Wu et al. 1987, Furchi et al. 1988). Length-average n  S U  heat transfer coefficients for long surfaces are usually lower than those for short surfaces (Wu et al. 1989a, Bi et al. 1990). As particles move down along a longer heat transfer surface, their temperature tends to equilibrate with the surface temperature. Therefore, there is a decrease in the average temperature difference between particles and heat transfer surface, resulting in lower total heat transfer coefficients.  1.2.6. Effect of Bulk Temperature Heat transfer coefficients increase with increasing bulk temperature. This has been attributed to the increase in thermal conductivity of the fluidizing gas and to an increase in radiation at higher temperatures. The radiation contribution is negligible when the temperature is low. Above 400°C, heat transfer coefficients increase with temperature predominantly due to  16  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  radiation (Wu et al. 1989a, Grace 1990a). Few data are available for bulk temperatures of 800°C to 900°C and higher. However, all of the results (Kobro and Brereton 1986, Wu et al. 1987, Andersson et al. 1987, Basu and Konuche 1988, Nag and Ali 1992, Leckner and Andersson 1992) obtained at this higher temperature range for particles of diameter 200-300 p:m are in broad agreement. Radiation is an important mode of heat transfer in CFBCs as discussed in detail in section 1.2.8.  1.2.7. Membrane Wall Studies Information on heat transfer at high temperatures obtained using membrane walls as heat transfer surfaces is scarce. Membrane walls commonly used in industrial boilers are classified as two types. One is the "American type" where the fin is welded to the tube, while the other is the "Walther type" where the fin is a part of the tube. The heat transfer coefficients near the crest of the tubes and valley between adjacent tubes can be quite different. Lockhart et al. (1995) did experiments in a circularriserwith a membrane wall consisting of two groups of tubes with different diameters (19 mm and 32 mm), for three suspension densities at a superficial gas velocity of 7 m/s. The experiments were carried out using air at room temperature. The heat transfer coefficients were found to increase with particle concentration. Heat transfer coefficients were higher along the fin than on the crest. Wu et al. (1987) tested an "American type" membrane wall operating at high temperature. Their data show that the total heat transfer coefficients obtained with a membrane wall are lower than those obtained with a tube located close to a flat wall. Andersson and Leckner (1992) studied a "Walther type" membrane wall and embedded 10 thermocouples in one tube and its adjacent fin to examine the temperature differences between the crest and valley  17  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  regions. There was a variation of only 6 to 10 °C, a small difference compared to the temperature difference between the tube and the particles. They also measured the heat flux from the suspension to the fin with a carefully designed probe and found that heat transfer coefficients in the fin area, where particle concentration is higher, are lower than in the crest region. The explanation is that denser shielded particles along the fin equilibrate with the surface so that radiative and convective heat transfer decrease as particles near the wall are cooled (Leckner and Andersson 1992, Flament et al. 1992). Convection dominates at low temperature, while radiation is significant at high temperatures, consistent with the fourth power temperature dependence of the latter. The heat flux from a suspension to a membrane wall can be calculated using temperatures measured by thermocouples embedded in the pipes and fins (Jestin et al. 1990,1992, Bowen et al. 1991, Andersson and Leckner 1992, Taler 1992).  1.2.8. Experimental Studies on Radiation Few experimental data on radiation in CFB have been published. Basu and Konuche (1988) used a quartz window to distinguish between radiative and convective components. The radiation through the window onto a flux meter was determined. The total heat transfer was measured by another probe cooled by water. The radiative heat transfer coefficient was found to increase with suspension temperature and suspension density. The results indicated that the ratio of radiative to total heat transfer coefficient varied from 74 % to 91 % as the suspension density decreased from 20 to 4 kg/m . These experiments were carried out in a 3  200 mm x 200 mm riser operating at 923 K and 1158 K, with sand particles having a mean diameter of 296 \im. The investigation did not cover a broad range of suspension densities.  18  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter I: Introduction  Han et al. (1992) used a probe consisting of a heat flux transducer and a 2 mm thick zinc selenide window to investigate CFB radiation. The total heat transfer coefficient was obtained by measuring the cooling water temperature difference over a test section 2.2 m in length and 49 mm in diameter with water flowing in the jacket. The experimental measurements were obtained with two mean particle sizes, 137 and 264 p;m, and with suspension temperature in the range of 473 K to 873 K. Suspension densities varied from 2 to 35 kg/m . The probe was calibrated using 3  a blackbody source, and the emissivity of the zinc selenide was found not to change appreciably over the temperature range of 200 to 600 °C. Their results indicated that the ratio of radiative transfer to total heat flux was of the order of 40 % to 50 %. The lesser importance of radiation is probably due to the lower suspension temperature. The results reported by Han et al. (1992) and Basu and Konuche (1988) demonstrate that the ratio of radiative to total heat transfer is greater than in conventional bubbling fluidized beds where it seldom exceeds 30%. This is due to the lower convective component in fast fluidization compared with a bubbling bed. Because the emissivities of gas components are much lower than those of particles, the effective emissivity increases significantly with increasing suspension density, especially for dilute suspensions.  The probes used in these studies of radiation are fundamentally the same as those used in previous investigations of radiation in conventional fluidized beds. H'chenko et al. (1968) appear to have made the first attempt to measure total and radiative heat fluxes in a fluidized bed employing thermal probes similar to those used in studying flames. The radiometer consisted of a thermoelectric detector. A quartz window was used to protect the detector from the bed medium and to separate the conductive and convective heat transfer with respect to gas and particles from  19  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chavter 1: Introduction  the contribution due to radiation. Another probe, a total heat flux meter, was used to estimate the total heat transfer by using a differential thermocouple to measure the temperature difference between the two surfaces when the probe outer surface was exposed to the high temperature suspension while the inner surface was cooled by water. Both probes were calibrated using an ideal blackbody source. It was found that there was negligible influence of particle abrasion on the radiation measurement. The same principle with different configurations and different window materials has also been employed in other investigations (e.g., Basu 1978, Vadidel and Vadanurthy 1980, Ozkaynak et al. 1983, Mathur and Saxena 1987, Alavizadeh et al. 1990).  Jolley (1949) was the first investigator to apply an alternative measurement technique. He employed cylindrical blocks of copper and aluminum, with the copper surface blackened by sulfiding, while the aluminum was covered with a dull oxide film. Baskakov and Goldbin (1970) computed the radiative heat transfer coefficient by observing the change of temperature of two metal spheres of low Biot number immersed in a fluidized bed. Both spheres were made of stainless steel, one oxidized, the other plated with silver. Yoshida et al. (1974) used two pipes of different emissivities to measure the radiative component. The pipes, one polished with a low emissivity (0.17) and the other oxidized at 1273 K for three hours with effective emissivity 0.80, were located on the axis of a fluidized bed and had water flowing through them. The emissivity values were determined with air flowing through the empty bed, with radiative exchange occurring only between the heated walls of the column and tubes. Platinum (Panov et al. 1979) and gold (Botterill and Sealey 1970, Botterill et al. 1984) were chosen as bright surfaces, while the black surface was prepared by special heat treatment.  20  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter I: Introduction  With these two categories of probes, experimental radiation results have been obtained in conventional bubbling fluidized beds, and some conclusions can be drawn. The radiative component increases with increasing fluidized bed temperature. The heat flux, either total or due to radiation, was not influenced by increasing the fluidizing velocity (ITchenko et al. 1968, Basu 1978) or was not very sensitive to the superficial gas velocity (Ozkaynak et al. 1983, Alavizadeh et al. 1990). However, in some experiments (Vadidel and Vadanurthy 1980, Zhang and Xie 1985, Mathur and Saxena 1987), the radiative heat transfer coefficient increased with gas velocity. The ratio of radiation transfer to total transfer in the bubbling fluidized bed was from 5% to 50 % in the temperature range of 500°C to 1500°C. The effective emissivity of the fluidized bed is greater than the particle emissivity itself (Grace 1982) and is a function of bulk temperature as well as the heat transfer surface temperature (Baskakov et al. 1973, Baskakov 1985).  The conditions for and results of radiation studies in CFBs and FBs are summarized in Tables 1.6 and 1.7, respectively. A review of this work shows that: i.  Radiation has been studied in fluidized beds using a variety of experimental methods which are applicable to circulating fluidized beds.  ii. Radiation is more important in fast fluidization than in bubbling beds due to the lower particle concentrations and reduced convective transfer. iii. The total and radiation heat transfer coefficients (h and h ) are both significantly t  r  influenced by suspension density and temperature. The effect of superficial gas velocity tends to be insignificant. iv. The larger the particles, the higher the ratio of radiation to total heat transfer.  21  -s  CO  C  NO  1.1 p  3 O  -*  • 00  |T3  oo  ON  X  CM O  X  m CM o  o m  m  oq  NO CM  co  in NO ON CM  oo  00  o~ m  00  CM  3  o o m  o co  CO  o  T-H  ON  m  NO  O  CO  „  w  X  CM  |2 ^ in J  T-H  3  rH  CM CM  '1o  NO NO  m  ©  CO  X  CM  ON ON CM  NO  CM* CM CM  in  CO  o r» r-" oo m  OO I  co" co  NO  «n  00  NO  I  in  00  00  NO I  NO  CM CM  I  o  CM  NO  oo ca - . CU ON ON  1ST  CM  CM'  CO  ON  CM ©  X  O  o  o o  NO CM  '1  T-H  i>  CO  CO  NO I  o o  2  m co  <1  in co  o in • o  CM  CM  00  <x CO  T-H  CM  in o x  O O  1 m  2  CO  CO  ON  00  O  CM  CM  00  CO  ON  O  ON  oo  o  CM  CM  in  o  ON O  CO NO CO CO  m  CM  T-H  in  ca •<-> <u  in  i-s II co  CM CM  <x ON ON  ca  m  CO  NO ON  m  CO  o o  X  NO ON  '1  I>  © TH  00  CM  CO  CM  o o  CM  m  NO  o o  o m  CO  CO  P  CO  IIS  p  ca fa  p  •8  p  r ca 3  « -g X ts o  >  Xl  O bQ  CO  P  O <+-. CD 4-> O  ^  XI b? X  §3 S£  It OH »-H  ? CO  CD >  ca p  t s  p o  P  1t~ «» 53 U  co §  g  c  P 5 CD t> .rtP ca "T? p ° - , •E co p p  +-  M  '•§1 -B «4ca  rr « . ^ o c ca o 53 ^ - P a**—' c p  M  "3  •o S  co  a. CO  3  CO  '•a £  ca co  *  9  is  £  P  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 1: Introduction  Table 1.7. Summary of experimental studies of radiation in bubbling fluidized beds. Authors Jolly 1949 H'chenko et al. 1968 Szekely and Fisher 1969 Botterill et al. 1970,1984 Baskakov et al. 1970 Yoshida et al. 1974 Basu 1978 Panov et al. 1979 Vadivel and Vedamurthy 1980 Ozkaynak et al. 1983 Alavizadeh et al. 1990 Zhang and Xie 1985 Mathur and Saxena 1987  Method*  Probe position  II  center  I  50 mm from wall  T  T °C 790-980  1600  q/q % 33-43  100-200 water-jacket 40-100 air-cooled  430-1430  570-1750  18-50  315-377  100-150  insignificant  II  50 mm from wall center  80 water-cooled 125-775  500-1400 880 850  200-300 1150 350-1250  5-60 10-17 6-32  II  center  500-1500  180  1-6  I  center  800-900  280  5-10  II  —  30-150 water-cooled 20-30 water-cooled >500  800-1200  500-6000  <10  750  4000-6000  30-37  250-760  1030  7-35  770  2140  13  850-1050  1340-1580  20-30  600-900  560-750  10-12  III  I  w  °C 15-251  I  I  at r=D/4  I  —  30-200 air-cooled 80  I  —  —•  I  center  watercooled  * I:  b  d  P  um  t  Radiative flux is determined by isolating it from total flux by means of an optical window. II: Radiative flux is determined by two surfaces having different thermal emissivities. Ill: Radiative flux is determined using thermal radiation source which radiates to a cold fluidized bed.  23  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter I: Introduction  1.3. Objectives of This Study While a number of studies on CFB heat transfer have been carried out, leading to qualitative agreement on the influence of some significant parameters, most conclusions have been based on experimental results obtained with cold units. One must be very careful in making use of these results when designing a CFB boiler or other high-temperature equipment. Data must be obtained at the appropriate operating temperatures. Conclusions about radiation in CFBs cannot be drawn in a definitive manner because the results have been obtained with limited ranges of test conditions, such as bulk temperature, particle size and suspension density. Uncertainties exist regarding the mechanism of heat transfer, e.g., with respect to the effect of particle size on radiation and the effect of the primary-to-secondary gas flowrate ratio on total heat transfer. The dominant effect of the temperature gradient close to the heat transfer surface on radiation has been neglected in previous radiative studies. Experimental results based on two different measurement principles (Category I: transmissivity; Category II: differential emissivity) have not been compared with each other. Also, criteria for accounting for the radiative component are not clear. For example, the temperature at which radiation becomes significant has been estimated as 400°C (Wu et al. 1989a), 450°C (Han et al. 1992) and 880°C (Molerus 1994) in CFBs, and 500°C (Panov et al. 1979) and 1000°C (Szekely and Fisher 1969, Yoshida et al. 1974) in bubbling beds. The efficiency of the fins in membrane walls is unknown, and the relative importance of the tube crest area and the fin area to heat transfer has not been experimentally studied.  In view of these considerations the objectives of this project are:  24  Radiative and Total Heat Transfer in Circulating Fluidized Beds  i.  Chapter 1: Introduction  to investigate the extent and importance of thermal radiation in high-temperature circulating fluidized beds;  ii. to study heat transfer to membrane wall surfaces; iii. to achieve a better understanding of the mechanisms of radiative heat transfer in circulating fluidized beds; iv. to provide improved methods for predicting heat transfer in high temperature circulating beds.  25  Chapter 2: Experimental Facilities  Radiative and Total Heat Transfer in Circulating Fluidized Beds  CHAPTER 2 EXPERIMENTAL FACILITIES  2.1. CFBC System The experiments presented in this thesis were carried out in the UBC pilot scale CFBC facility located in the Pulp and Paper Center. A schematic of the major components is shown in Figure 2.1. The main components of the CFBC facility are the reactor (riser) and the solids return pipe (standpipe). The riser, shown in Figure 2.2, is 7.3 m in height and 0.152 m x 0.152 m in cross-sectional area. It is equipped with thermocouples at 0.61 m intervals and pressure taps. Primary air flows through a slot distributor at the bottom of the riser. Secondary air is introduced at one of two levels, either through an upper manifold (two opposed pairs of orifices of diameter 12.7 mm), located at 3.1 m above the base, or through a lower manifold, 1 m above the primary air distributor. Solid fuel stored in the hoppers is fed into the riser 0.6 m above the primary air distributor by a pneumatic feeder. Limestone can also be fed at the same level for in situ capture of sulfur dioxide. Gas and entrained solids leaving the top of the riser enter a primary cyclone of inside diameter of 0.305 m. Solids captured in the cyclone drop into a standpipe (0.1m in diameter) and are returned into the bottom of the riser, 0.2 m above the primary air distributor,  26  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  1. Reactor; 2. Windbox; 3. Primary cyclone; 4. Secondary cyclone; 5. Recycle hopper; 6. Standpipe; 7. Eductor; 8. Secondary air preheater; 9. Flue gas cooler; 10. Baghouse; 11. Induced draught fan; 12. Fuel hopper; 13. Sorbent hopper; 14. Rotary valves; 15. Secondary air ports; 16. Membrane wall or dual tube; 17. Pneumatic feed line; 18. External burner; 19. Ventilation; 20. Multifunctional probe.  Figure 2.1. Simplified schematic diagram of circulating fluidized bed combustion facility.  27  Chapter 2: Experimental Facilities  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Figure 2.2. View of principal refractory-lined riser showing location of heat transfer tubes (all dimensions in mm).  28  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  through an L-valve; the solids flow is controlled by aeration near the corner and along the horizontal lower leg. Fine particles which are not captured by the primary cyclone are separated from the flue gas by a secondary cyclone. The fine solids captured by the secondary cyclone are recycled back to the riser by the pneumatic feed system. The flue gas leaving the secondary cyclone is cooled by a three-stage heat exchanger and then vented. A baghouse is used to capture ash and fine solids to meet solids pollution control requirements. Full details regarding the CFBC facility are provided elsewhere (Brereton et al. 1993).  The local cross-sectional area average suspension density in the vicinity of the heat transfer surfaces is estimated from the corresponding measured axial pressure gradient by neglecting acceleration and wall friction (Van Swaaij et al. 1970). The pressure gradient is determined from the pressures measured by transducers located 4.57 and 5.79 m above the primary air distributor, where the heat transfer surfaces were placed. Bulk suspension temperatures along the riser are measured by K-type thermocouples inserted into the axis of the riser.  2.2. Multifunctional Probe Heat transfer between the fluidized material and the reactor walls is an essential aspect of circulating fluidized bed design. Most experimental studies of heat transfer in circulating fluidized beds have been carried out at ambient temperature in order to estimate heat transfer by conduction and convection due to the movement of gases and particles. Steam heated conductive heat flow meters (Basu and Nag 1987, Chen and Chen 1992, Nag and Ali 1992) and electrically heated probes (Wu et al. 1989b, 1990, Bi et al. 1990) have been successfully used to investigate  29  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  these conductive and convective components. However, circulating fluidized bed technology is applied at high temperatures. Applications such as combustion require such high temperatures that heat transfer by radiation cannot be neglected.  Various experimental methods have been introduced to measure heat transfer in CFBCs (Andersson and Leckner 1992). For example, a water-cooled conductive heat flow meter has been employed as a simple device for measuring heat fluxes at high bed temperatures (Andersson et al. 1987), but it only yields information about the total heat flux due to all three contributing components. Several methods have been employed in fluidized bed heat transfer studies to determine the radiative heat flux contribution. The measurement techniques for radiative heat transfer can be classified into two categories: (1) In Method I, termed the window method (WM), a single probe employs a transparent material as a window to block the conductive and convective heat fluxes and only permits the radiative heat flux to reach a flux meter behind the window. (2) In Method II, termed the differential-emissivity method (DEM), heat fluxes measured by two probes with different surface emissivities are compared and the difference in flux is used to estimate the radiative component.  Both methods have advantages and disadvantages. Radiative heat transfer is directly measured by the window method, while radiation is estimated indirectly by the differential emissivity method. The selection of the window material is of critical importance for the window method due to complex factors such as waveband, hardness, temperature resistance, etc.. Previous investigations of radiation in fluidized beds in which both methods have been utilized  30  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  have appeared to be inconsistent with each other. According to Saxena et al. (1989), the window method generally gave a higher ratio of radiation to the total heat transfer than the differential emissivity method. In this study, a probe is developed which uses both methods of determining the radiative heat transfer flux at the combustor wall. Hence, an additional objective of this present study is to compare these two measurement techniques.  2.2.1. Considerations for Probe Design The choice of material as the sensor for a heat flow meter depends on the following considerations: (1) The temperature gradient through the solid body of the meter should be high enough to give reliable calculation of heat flux over a reasonable range of operating conditions. In particular, the temperature difference between measurement positions should be much higher than the thermocouple sensitivity. (2) The temperature of the heated surface should be high enough to prevent unwanted occurrences such as moisture condensation or soot deposition in a CFBC. On the other hand, this temperature should not be too high because lateral heat flux increases with the increasing surface temperature. The lateral heat flux should be minimized to ensure that the temperature profile along the axis of the probe is close to linear.  According to the required range of thermal conductivities, stainless steel (k = 13-24 W/m K) is appropriate for relatively low heat fluxes, while brass (Cu 61%, Zn 38%, Pb 1%, k=l 10 W/m K) is better for high heat fluxes (Andersson et al. 1989).  31  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  Considerations for specifying the window material include: (1) Wave band: thermal emissions have wavelengths between 0.1 to 100 pm. For the combustion temperature range (750 - 950 °C), the thermal emission spectrum is shown in Figure 2.3, which indicates that more than 98.5% of the emissions have wavelengths shorter than 20 pm and less than 0.5% have wavelengths shorter than 1 pm (Touloukian and Dewitt 1972). (2) Transmissivity: the window should allow as great a fraction of the thermal emissions to pass as possible. (3) Melting and softening temperatures: the window should resist temperatures as high as the maximum temperature inside the reactor.  Figure 2.3. Cumulative percentage of blackbody thermal emission as function of wavelength (Touloukian and Dewitt 1972).  32  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  (4) Mechanical properties: the window should be sufficiently hard to withstand erosion by the particles.  Many potential window materials were considered. Their properties are listed in Table 2.1 (Ivan'ko 1971, Touloukian and Dewitt 1972, Lide 1992). The transmissivity of zinc selenide as a function of wavelength and the effect of temperature on the spectral transmissivity are shown in Figure 2.4. According to the definition of the total transmissivity, t, it is related to the spectral transmissivity, Tx, by co  *= \G,(X)dX  o where Gx is the spectral irradiation and X, is the wavelength. With the assumption that the spectral distribution of irradiation at the window can be approximated as the spectral emission from a blackbody at temperature T (T=l 173 K is a typical temperature in CFB boilers), we may rewrite Equation (2-1) using the Planck distribution as follows:  CO  (2-2)  '  jE„(X)d>.  a  T  where Ex,b (X) is the spectral distribution of blackbody emission, the first and second radiation constants are ci=27m'(c ) =3.742xl0 W um /m and C2=h c /k =1.439xl0 um K, h'=6.6256xl0- J s 2  8  4  2  0  ,  ,  4  34  o  and k-1.3 805x10" J/K are the universal Planck and Boltzmann constants, respectively, 23  c =2.998xl0 m/s is the speed of light in vacuum, and a=5.67xl0" W/m K is the Stefan-Boltzmann 8  8  o  33  2  4  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  Table 2.1. Properties of some window materials (Ivan'ko 1971, Touloukian and Dewitt 1972, Lide 1992).  2  0.17-5  >0.9  10.7  Melting Temperature (°c) . 1710  Fused SiC>2  0.15-5  >0.9  1.4  1470  461  Si  2.7-8.7  0.5  140 (40 °C)  1410  1150  KBr  0.2-40  0.9  4.8 (27 °C)  734  5.9  ZnSe  0.5-18  0.7  6.8-13(54 °C)  >1100  105-135  A1 0  0.15-7  0.85  25-30  2015  1370  CdTe  0.85-30  0.5-0.65  1-3.8 (27 °C)  1121  45  ZnS  0.4-15  0.5-0.7  15.5 (54 °C)  1700  355  CdS  0.5-15  0.75  N/A  1750  N/A  CdSe  0.7-25  0.65  N/A  >1350  90-130  NaF  0.18-15  0.95  N/A  993  N/A  2  0.13-10  0.95  8.0 (80 °C)  1423  158  2  0.15-11.5  0.93  11.7 (13 °C)  1355  82  KCi  0.2-20  0.9  5 (100 °C)  770  7  MgO  0.3-10  0.88  10 (723 °C)  2852  640  MgF  0.3-10  0.9  14.7 (56 °C)  1255  575  0.17-20  0.9  5(100°C)  801  15.2  Waveband (pm)  Window Material Crystal Si0  2  CaF BaF  3  2  NaCi  Conductivity (W/m K)  Transmissivity (Maximum at a specific wavelength)  Hardness (Knoop) 741  * The waveband is defined as the wavelength range in which there exists a dominant maximum transmissivity.  34  Chapter 2: Experimental Facilities  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1.0 0.9 0.8  — i — i — • — | — i — i — i — | — ' — i — i — i — • — i — i — i — r -  •  T=298 K (Total Transmissivity=0.664)  ©  T=523 K (0.653)  A  T=623 K (0.650)  n— —r 1  0.7 0.6  CO .S3  a 2  H  0.4 03 02  0.1 •  0.0  2  4  6  8  10  •  I  12  i  I  14  i  L  16  18  20  22  Wavelength, X (um)  Figure 2.4. Spectral transmissivity of zinc selenide at different temperatures (Touloukian and Dewitt 1972).  constant. Equation (2-2) is used to calculate the total transmissivities of the window at several temperatures, the spectral transmissivities of which are shown in Figure 2.4.  From the table and figures, a comparison of these materials indicates that zinc selenide is best able to meet all of our requirements since zinc selenide has a wide waveband with a consistently high transmissivity, an intermediate thermal conductivity and a sufficiently high hardness to withstand particle erosion.  2.2.2. Probe Design With the foregoing considerations in mind, a multifunctional probe was designed which combines the advantages of the differential emissivity method and the window method. Stainless  35  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  steel was chosen as the sensor material, and zinc selenide as the window. The probe, shown in Figure 2.5, consists of a stainless steel body in which four stainless steel cylinders are inserted. Each cylinder is surrounded by a concentric stainless steel sleeve which separates the cylinder and the probe body. The cylindrical sleeve does not contact the cylinder or the body; instead there is a thin air gap on both sides. That is, there is an air layer between the cylinder and the sleeve and also between the sleeve and the probe body. The cylindrical sleeve has two functions: first, it divides a thick air gap into two thin air gaps to suppress free convection in the gap, and second, it provides a parallel conduction path for heat absorbed at the hot surface in order to minimize radial heat conduction in the probe cylinder. One end of each probe cylinder is cooled by running water while the other faces the heat source. Three of the four cylinders were 6 mm in diameter and 30 mm in length; two were made of stainless steel 316 (C 0.08%, Cr 17%, Ni 12%, Mo 2.5%)  Figure 2.5. Schematic of multifunctional probe.  36  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  while the third was stainless steel 347 (C 0.08%, Cr 18%, Ni 11%, Nb 0.8%, Ta 0.2%). In order to obtain different surface emissivities, these probe cylinders were treated in different ways. The stainless steel 347 cylinder (head 1) was oxidized at 1050 °C for three hours; one of the stainless steel 316 cylinders (head 2) was polished using fine sandpaper, while the second one (head 3) was coated with a thin layer of silver. To minimize heat transfer from a cylinder to its sleeve or vice versa, the cylindrical sleeve surrounding each probe cylinder had the same exposed surface emissivity as the probe head itself. When the three heads are directly exposed to a heat source, axial temperature gradients develop along the cylinders. The axial temperature gradient in each probe cylinder is measured using three chromel-alumel (K type) thermocouples embedded along the axis of each cylinder at uniform intervals of 11 mm. The surface temperatures at both ends of each cylinder are obtained by linear extrapolation of these three temperature readings. The total heat flux from the hot solids and gases within the riser to the exposed probe surfaces by conduction, convection and radiation can then be determined from the temperature distributions in these cylinders. In addition, the radiative heat flux can be estimated by comparing the total heat fluxes calculated for the heads having different surface emissivities. The theory required to extract the radiative heat flux from the differential-emissivity measurements is developed in Section 3.4.1.  The fourth cylinder (head 4) is 6 mm in diameter and 24 mm in length and was made of stainless steel 347. It was oxidized at 1050 °C for three hours so that its surface emissivity was the same as that of head 1. A zinc selenide window, 12.7 mm in diameter and 2 mm thick, was placed between the head and heat source to virtually eliminate heat transfer by conduction and convection from the bed solids and gases. The air gap between the tip of head 4 and the window  37  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  was 1 mm. Hence, the heat flux estimated from its temperature distribution, measured by three, uniformly-positioned (9 mm interval), K-type thermocouples, can be attributed to radiative transfer from the heat source. The theory for the window probe is presented in Section 3.4.2.  The physical properties of the probe heads which were to be exposed to temperatures up to 850 °C in a CFBC riser are listed in Table 2.2. The emissivities of stainless steel 347 and stainless steel 316 from Touloukian and DeWitt (1970) are shown in Figure 2.6 as a function of temperature. The methods of emissivity measurement and a wider range of emissivities of stainless steel 316 and 347 from the literature are presented in Appendix I. The thermal conductivities and transmissivities of zinc selenide as functions of temperature and the reflectivity of zinc selenide as a function of wavelength are presented in Appendix II.  2.3. Dual Tube Heat Transfer Surface The tubular heat transfer surface is made of two separate stainless steel 316 tubes. One tube is polished, while the other is oxidized at 800-900 °C for more than three hours. The inside  Table 2.2. Measured physical properties of the four probe cylinders (Touloukian and Dewitt 1970,1972, Incropera and Dewitt 1990).  Transmissivity  Emissivity  Reflectivity  Conductivity (W/m K)  298-623 K  600-1200 K  600-1200 K  300-1000 K  SS347 head  0.00  0.87-0.90  0.13-0.10  14.2-24.7  SS316head  0.00  0.30-0.78  0.70-0.22  13.4-24.2  Silver coated head  0.00  0.03-0.08  0.97-0.92  13.4-24.2  0.66-0.65  0.10-0.11  0.18-0.24  6.8-13.0  (373 K)  (298 K)  Probe cylinders  ZnSe window  38  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1.0  Chapter 2: Experimental Facilities  1  1  1  1  p  0.9 h 0.8 0.7 .52  I* 3  0.6  _o  0.5  Oi .23  0.4  •§ •a  03  • A  02  •s  H  0.1 0.0 200  400  j 800  600  Stainless steel 316 Stainless steel 347  i_  1000  1200  1400  1600  Temperature, T (K)  Figure 2.6. Total hemispherical emissivities of stainless steel 316 and stainless steel 347 as functions of temperature (Touloukian and Dewitt 1972).  and outside diameters of the tubes are 9.5 mm and 12.7 mm, respectively, while the length of each tube is 1.524 m, bent into a U shape. The total exposed area of each tube is 0.061 m . The tubes are close to the wall but do not touch it, leaving a gap of about 1-2 mm between the wall and the tubes. The radius of curvature of the curved portion of the U tubes is 16 mm. The cooling water flow rate is kept constant at 3.13x10" m /s for each tube to ensure that the flow inside the 6  3  tubes is in the turbulent region. K-type thermocouples were used for both tubes to measure water inlet and outlet temperatures in order to calculate heat fluxes and radiative heat transfer coefficients, as described in Section 4.1.2. Figure 2.7 shows the structure and dimensions of the tubes.  39  Chapter 2: Experimental Facilities  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Figure 2.7. Schematic of dual tube heat transfer surface (all dimensions in mm).  40  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  2.4. Membrane Wall Heat Transfer Surface 2.4.1. Considerations for Membrane Wall Design The total area of the membrane wall depends on the length and diameter of the membrane pipes and the width of the fin. The ratio of the pipe diameter to fin width commonly used in industrial boilers is about 5. The determination of the pipe diameter is related to the energy generated by the CFBC unit and the requirement that a reasonable cooling water temperature difference be maintained between the inlet and outlet. The water flow rate should be large enough to keep the inside convection resistance sufficiently low that the pipe inside surface temperature is close to the water-side temperature in order to simulate conditions in industrial-scale operations.  According to the previous studies summarized in Section 1.2.5, a shorter surface length usually gives a higher heat transfer coefficient. To minimize the effect of the heat transfer surface length on heat transfer coefficients in a CFB, the length of the membrane wall should be greater than 1.0 m (Wu et al. 1989a).  2.4.2. Design and Construction The experimental membrane wall consists of pipes and connecting fins, divided into two parallel sections, side by side. A top view and a side view are shown in Figures 2.8 and 2.9, respectively. Stainless steels 316 (17% Cr, 12% Ni, 2.5% Mo) and 347 (18% Cr, 11% Ni, 0.8% Nb) were chosen as the construction materials for the two sets of fins and pipes. They have  41  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  152.4 mm THERMOCOUPLES EMBEDDED IN THE PIPE WALL AND THE FIN  203.2 mm  STAINLESS STEEL 347 AND 316 PIPES EXPOSED TO PARTICLES AND HOT GAS  254.0 mm  406.4 mm K-TYPE THERMOCOUPLES 254.0 mm FIBERFRAX INSULATION 203.2 mm 152.4 mm  Figure 2.9. Membrane wall (side view).  43  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  almost the same conductivities at the same temperature (16.2 W/m K for SS316 and 16.1 W/m K 2  2  for SS347 at 100 °C; 21.5 W/m K for SS 316 and 22.2 W/m K for SS 347 at 500 °C, 2  2  Touloukian and Dewitt 1972). However, SS347 has higher emissivity (about 0.90) when oxidized at a high temperature compared with unoxidized SS316 (about 0.30). The stainless steels are able to withstand sand erosion and they have low oxidation rates. Two stainless steel 347 pipes (21.3 mm O.D. and 14.1 mm I.D.) and three fins of the same material are welded together to form half of the membrane wall while the other half is made of stainless steel 316 with the same dimensions of pipes and fins (see Figure 2.8). The total length of the membrane section is 1.626 m and the total exposed area for each half of the membrane wall is 0.150 m . 2  When operating the CFBC facility with the membrane wall installed, the different emissivities of oxidized SS347 (e347=0.90) and unoxidized SS316 (e i6=0.30) (see Appendix I), allow 3  estimation of the radiative component of heat transferfromthe suspension to the membrane wall. The two half-membrane assemblies are separated by a 3.2 mm gap filled with fiberfrax insulation to prevent conduction between the two parts. The whole membrane wall was made symmetrical so that it could be turned upside down to determine if there are any systematic variationfromone side of the riser to the other.  The inlet and outlet cooling water temperatures for both SS347 pipes and one of the SS316 pipes were measured and eight K-type thermocouples were installed in the other SS316 pipe to measure water temperatures at different elevations, including the water inlet and water outlet (see Figure 2.9). These latter thermocouples were installed at the centerline of the pipe nearest to the corner of the combustor. To measure the temperature profile within the pipe wall and fin, six 0.508 mm diameter K-type thermocouples were embedded in one SS316 pipe  44  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  wall and adjacent fin (Figure 2.10). The installation of the membrane wall in the combustor is shown in Figure 2.11. The bottom of the membrane surface was 4.37 m above the primary air distributor when it was installed in the riser.  45  NO  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 2: Experimental Facilities  PROBE PORTS  REFRACTORY-LINED SECTION  Figure 2.11. Refractory-lined section equipped with membrane wall composed of two pipe and fin materials (stainless steels 347 and 316) with different surface emissivities.  47  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  CHAPTER 3 PROBE CALIBRATION  3.1. Cavity Design To calibrate the radiation probe, a thermal emission source of known emissivity is needed. An ideal radiant emitter is a blackbody with unit emissivity. However, few materials have emissivities greater than 0.9. Even graphite, which appears to be visually black, has an emissivity of 0.88 - 0.9, still significantly short of unity. Although there are no natural blackbody emitter materials, cavities can be constructed with emissivities close to unity. The use of such a cavity to calibrate the multifunctional probe is described in this chapter.  The cavity emissivity depends strongly on the ratio of the area of the opening to the total internal surface area of the cavity. The lower the ratio, the higher the cavity emissivity. The details of the internal shape are of secondary importance. The choice of the cavity shape is determined by the particular requirements of the application and the ease of construction. A cylindrical cavity was chosen in the current work due to the simple geometry and ease of manufacture.  48  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  The probe calibration was carried out using an electrically heated furnace. An oxidized stainless steel 316 cylindrical tube, 25.4 mm in internal diameter and 279.4 mm in length, with one end open, was put in the furnace. The probe assembly is aligned with the cylinder axis with the probe heads close to the opening. The probe does not touch the heated cylinder to minimize conduction to the probe. Both the probe and the ends of the cylinder are well insulated by fiberfrax so that heat losses to the environment and cold air flowing into the chamber of the cylinder can be minimized. A K-type thermocouple is inserted into the cylinder from its closed end to measure the temperatures along its centerline.  3.2. Heat Exchange between Probe and Cylindrical Cavity 3.2.1. Local Apparent Emissivities To illustrate the calculation of the integrated emissivity of a cylindrical cavity having diffusely reflecting walls, we consider the cylindrical cavity shown in Figure 3.1. probe cylinder  Figure 3.1. Cylindrical blackbody cavity showing geometric factors used to calculate heat transfer to the probe cylinder using the integral equation method.  49  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  The radiative heat flux emanating from any surface element in the cylindrical cavity includes not only that emitted by the element, but also that reflected by the elementfromother areas of the enclosure (Sparrow et al. 1962). The analysis is based on a radiative heat flux balance at a typical axial position, x,, along the cylindrical wall. The total radiant energy per unit time leaving per unit area due to emission and reflection is called the radiosity G, while the incident energy per unit time and per unit area from all other areas is the irradiation I. Thus, at location x , where the local temperature is T and the furnace wall emissivity is ef, the 0  radiosity is, by definition G(x ) = e aT (x ) + R I(x )  (3-1)  4  0  f  0  f  0  where Rf = 1 - ef denotes the gray furnace wall reflectivity.  The energy incident on x<, is from radiation leaving both the cylindrical surface and the closed end (i.e., base) of the cavity. First, consider the contributionfroma typical element of area dA =(7iD)dx located at x on the cylindrical surface. The energy leaving x in all directions is x  G(x)dA . The energyfromthe element at x reaching the element at x„ is G(x)dA dF -x ,ix-xj where x  dFx-xo.  ix-xj  x  x  0  is an infinitesimal view factor from x to x, separated by a distance I x-Xo I • The  reciprocity relation between view factors requires that dA dF x  x  - ,[x-xo| Xo  = dA^-xjx-xol  Hence, the energy incident at x„ per unit area dAx. is G(x)dF _ | _ | Xo  x x  Xo  The contribution per unit area at Xo of the basefroma typical annular ring of area 27rrdr is G(r)dF -r,(L-x Xo  o)  50  (-) 3  2  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  The total energy incident at x is obtained by integrating the energy contributions from all 0  elementary areas of the cylindrical surface and the closed base. Thus,  I(x ) =  G(x)dF _ , _ + fG(r)dF _ _  (3-3)  f  0  Xo  x |x  Xo|  Xo  x=0  r>|L  Xol  r=0  Substituting Eq. (3-3) into Eq. (3-1) and introducing dimensionless variables, we obtain the local apparent emissivity: x'=UD  e (x' ) = e a  0  f +  (l-e ) f  r'=l  j* e.(x')dF , _ , ,_ ,, + (1 - e ) J e ( r W , x  x j|x  x  f  x'=0  where e (x') = a  oT (x') 4  x' = —, D  a  x  ,  ,  (3-4)  r'=0  r' = —. R  In a similar manner, the radiosity of an annular ring located at radius r on the base is defined as: G(r)=e oT (r)+R I(r)  (3-5)  4  f  f  The total energy incident from the cylindrical area to the base is obtained by integration: x=L  I(r) = jG(x)dF _ _ r  x|L  (3-6)  xl  x=0  Combining Eqs. (3-5) and (3-6) with the dimensionless variables defined above leads to an expression for the local apparent emissivity of the base: x-=L/D  e (r') =e + (l-e ) J e (x')dF,_ ,, a  f  f  a  x  ,,  (3-7)  x'=0  Here e is termed the apparent emissivity because it is the ratio of the actual energy leaving a a  given position to the emission of a black surface at temperature T. Note that the apparent  51  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  emissivity e always exceeds the emissivity e of the cavity material. The apparent emissivity a  f  provides a more meaningful description of the radiation characteristics of a surface than the surface emissivity itself because it is the radiosity G, which includes the reflected energy, that is actually measured by a thermal radiation sensor.  To calculate the local apparent emissivities in the cylindrical cavity, it is essential to derive expressions for the view factors dFxVxMxVxi, dFxVr'.ixvn and dF -x,ir-xi. The derivations are ,  ,  ,  r  given in Appendix III. With these expressions, Eq. (3-4) and Eq. (3-7) can be rewritten respectively (Sparrow et al. 1962) as  x'=L/D  e (x' )=e (l-e ) a  0  f+  f  J e.odl-X '  n  -X'  x'=0  2(x'-x' ) +3  dx'  0  (x'-x'o) +1 4(^-x' ) +l-(rf r 2  r'=l  +4(  1  0  l_e )(L_ ' ) Je^f). f  x  •dr*  o  4(|-x!o) +l+(0 2  (3-8)  -4(r) t  2  and  X  e (r') = e +8(l-e ) a  f  f  ,  |4(^-x' ) l-(r') 2  L / D  T  0  ±-  fe.(x'X--x') D x'=0  2  +  -  J  J  rdx'  (3-9)  12 4 ( ^ - x ' ) + l (r') 2  0  2  +  •4(r')  where x' - x' is the absolute magnitude of the distance between two elements. 0  The apparent emissivities estimated from Eqs. (3-8) and (3-9) are isothermal emissivities, calculated assuming a uniform temperature throughout a cavity, i.e., the temperature of the  52  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  cylindrical and base surfaces is everywhere the same. In more practical cases, there may be an axial temperature gradient along a cylindrical cavity because the cavity becomes cooler near its opening since these parts communicate more with the outside temperature. The effect of a nonuniform axial temperature profiles on the apparent emissivity of a cavity has been studied by Bedford and Ma (1974) and Sparrow (1965). The apparent emissivity for the nonisothermal case can be obtained from the isothermal apparent emissivity (given by Eq. 3-8 above) using (DeWitt and Nutter 1988)  e:  nbo  (x ) = e , ( x ' ) ^ l  %  (3-10)  where T is the constant temperature of the cavity reached at distances sufficiently removed from f  its entrance. No similar modification is made for the cavity base since its temperature is usually uniform at T . f  3.2.2. Radiative Heat Transfer through Opening of Cylindrical Cavity an Calculation of Integrated Cavity Emissivity Heat transfer by radiation through the cavity opening can be estimated once the apparent emissivity distributions have been determined. The quantity essential to most applications is the integrated cavity emissivity, e , which can be estimated by the energy balance method (Sparrow c  et al. 1962).  The net local heat emission, q, from a unit area of surface in the cavity represents the difference between the emitted energy and the absorbed incident energy. Thus  53  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  q(x) = e (x)aT (x)-A I(x)  (3-11)  4  f  c  where A (= ef) is the absorptivity of the cavity surface. Eliminating I(x) using Eq. (3-1) or c  Eq. (3-5), and making the axial and radial length scales dimensionless, we obtain q(x') = - ^ - [ 1 - e (x')] CJTV) l-e  (3-12)  a  f  q ( r  .  _ X_ l_e ')]aT (r') l-e e  ) =  (3-13)  4  [  a(r  f  where T(r') = Tf . The total rate of radiative heat transfer outward from the cavity into the environment, denoted by (Q )totai»is calculated by integrating the local net heat transferfromall r  the cavity surfaces, i.e., x'=L/D  (Q )totai=47iR r  2 c  ^=1  jq(x )dx'+27 R Jq(f )f df ,  2  t  (3-14)  r'=0  x'=0  The cavity emissivity is defined such that (Q ),ota.=e 7rR^T r  c  (3-15)  4 f  Thus, combining Eqs. (3-14) and (3-15) yields fO c  A  \  X'=L/D  ry  r'=l  T = -=?- Jq(x*)dx'+—p J q ^ f d i * 7tR^aT oT; ,i~ aT; ^ f  (3-16)  x  Eq. (3-16) shows that the integrated cavity emissivity is the ratio of the total heat transfer from the cavity divided by the black body emission from an area equal to that of the cavity opening. From the definition of emissivity, this represents the apparent hemispherical emissivity of the whole cavity.  54  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  3.2.3. Radiative Heat Exchange between Cavity and Probe The heat emitted from a cylindrical cavity can be taken as equal to that emitted from a disc having the same area as the cavity mouth and apparent emissivity, ec. If a sensor, such as the probe to be calibrated, is aligned parallel to the cavity at a distance, s, from its mouth, the view factor, equivalent to that between the two concentric discs having unequal radii, can be calculated (Feingold 1978) by 2 , 2 , 2  ii +r +s  (3-17)  2  2f  Eq. (3-17) can be written in terms of dimensionless variables as 2" 2  •Y - Y -4  (3-18)  2  r, where r = —,  . r r =— 2  x  2  l + (r ) and Y = 1 +  2  2  The radiative heat exchange between any two flat gray surfaces denoted as 1 (probe head surface) and 2 (cavity) is obtained (Siegel and Howell 1992) as e oT [l-(l-e )(F _ , ) A / A ] - e e F 4  = q  '  Q  l  =  A,  1  2  1  2  fi  r2  s  1  2  i_ i_e )(l-e )(F _ (  1  2  ri  t  2  ) A /A  T O  <  2  r2iS  1  2  where ej is the constant emissivity of the probe surface and e (e ) is the integrated cavity 2  c  emissivity. F -r ,s is the view factor from the probe to the equivalent disc of the cavity. q is the ri  2  r  net heat transfer from the cavity to the probe by radiation. A positive value of q indicates net r  heat transfer from the probe to the cavity, while a negative value indicates net heat transfer  55  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  from the cavity to the probe. If the two surfaces are so close that they can be considered as infinite parallel flat plates, F -r = 1 and A1/A2 = 1. Eq. (3-19) then reduces to ri  2>s  (3-20) ' - A , - ±  +  _ L - I  ei  e  2  The derivation of Eq. (3-19) is given in Appendix IV.  3.2.4. Enhanced Exchange between Cavity and Probe by Free Convectio Heat is transferred between the cavity opening and the probe not only by radiation but also by free convection. A careful comparison of the experimental heat flux to the probe, based on its measured internal temperature distribution, and the flux calculated by the above theoretical analysis indicates that the reason for the observed deviation is the neglect of additional convective transfer due to circulation of air within the cavity close to its opening. This circulation is caused by the temperature difference between the cavity and the probe surface. Colder denser air near the probe surface moves towards the center of the cavity displacing the hotter lighter air in that region. This warmer air flows to the opening and is cooled in turn by the probe surface. The result is that air circulates, and this free convection enhances the overall rate of heat transfer between the cavity and the probe. Air circulation within an enclosure due to this mechanism was first reported by Hottel and Keller (1933). The free convective component in the heat transfer flux between the cavity and the probe can be written (3-21)  q = hfc(T -T ) fc  f  p  where hf is the average free convection heat transfer coefficient and Tf and T are the C  p  56  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calihrntinn  temperatures of the cavity and the probe surface, respectively. The average free convection heat transfer coefficient in a cylindrical cavity of length L and diameter D is assumed similar to that in a rectangular cavity with heated and cooled vertical surfaces and all other surfaces adiabatic. Based on this assumption, the free convective heat transfer coefficient in the cavity can be estimated (Catton 1978) by —  .hfcL k  i=  0 > 2 2 (  g  _Pr_ 0.2 + Pr  xo.28  R  D_  4  (3-22)  [LJ  for 2<D/L <10, and X  Nu  x =  -  k  ^ g  .18(—^Ra ) 0.2 + Pr 0  =  0  (3-23)  2 9  x  for 1<D/L <2, X  where Ra. = Gr Pr g|3L (T -T ) 3  G r  =  x  f  p  2  x  Pr = Gr is the Grashof number, Pr the Prandtl number, and Ra the Rayleigh number. L is the x  characteristic length of the cavity for free convection, assuming in this case to be equal to D. Thus, Eq. (3-23) is used to estimate the free convection coefficient. The influence of the cold surface on the nonuniformity of the temperature distribution is more significant at higher cavity temperatures than at lower cavity temperatures. The properties of air in the cavity are evaluated at the average of the cavity and probe surface temperatures, i.e., at (T +T^/2. f  57  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  If we assume that free convection does not influence radiation and vice versa, the total heat transfer from the cavity to the probe is the sum of heat transfer by radiation and by free convection, i.e. q = q +qfc t  r  (3-24)  3.3. Heat Transfer Estimatedfrom Probe Temperatures Measuring the temperatures at suitable locations along the centerline of each probe cylinder provides a relatively easy way to estimate the heat flux. The technique has the advantage of not disturbing the gas and solids flows in the CFB combustor. Two mathematical models were considered to calculate the heat flux from the measured temperature distributions. One is based on one-dimensional conduction along the probe cylinder, assuming that the cylinder is perfectly insulated over its entire circumferential surface so that there is no lateral heat transfer. In this model, the steady state heat flux to the heated probe surface is assumed to be identical to that at all internal cross-sections of the cylinder so that the heat flux can be calculated directly from the measured centerline temperature gradient. The second model requires solution of an inverse heat conduction problem and is introduced to relax the restriction of one-dimensional conduction in the cylinder and thereby to provide more accurate heat transfer coefficients.  3.3.1. Conduction Models for Cylindrical Probes 3.3.1.1. One-Dimensional Conduction Assuming that the two concentric layers of stagnant air incorporated into the probe design  58  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  (see Figure 2.5) restrict radial heat flow, heat transfer by conduction in the cylindrical probe occurs only in the axial direction and Fourier's law can be written: °- = q= —  k  A  T  t  A  p  rtocx (3-25)  CAx  where k is the average conductivity of the probe cylinder and C is a probe correction coefficient, which depends on the heat source temperature, as well as on the measured heat flux. The determination of C is discussed in Section 3.3.1.2.  The temperatures along the four probe cylinders (see Section 2.2 for details), measured when the ends of the cylinders were exposed to the heat source supplied by the cavity, are plotted in Figure 3.2. Eight temperatures of the cavity from 200 to 1000 °C were tested. Note that the temperature variation along the probe centerline is close to linear in each case, so that Eq. (3-25) yields direct values of the heat flux, and hence of the heat transfer coefficient, once the correction coefficient C is determined. The surface temperature for each probe is determined by extrapolating its inner temperature to the probe surface.  3.3.1.2. Determination of Probe Correction Coefficient A conduction analysis of the probe structure shown in Figure 2.5 was carried out. Each probe cylinder and its surroundings are considered to constitute a quasi-symmetric geometry, as shown schematically in Figure 3.3. The radius of each probe (R6) is determined using the quarter cross-sectional area of the entire probe as the cross sectional area of the quasi-cylinder (see Figure 3.4).  59  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  cylinder behind ZnSe window TfH200 °C  Distance from side exposed to cavity, x (mm) Figure 3.2. Temperatures measured by K type thermocouples embedded in the probe cylinders in calibration runs with several furnace temperatures.  60  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  Figure 3.3. Schematic diagram showing quasi-symmetric probe cylinder assemby.  61  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  (b) quasi-symmetrical cylinder  (a) front view of the probe assembly  Figure 3.4. Diagram of probe simplified to quasi-symmetric cylinder.  The governing partial differential equation for the axi-symmetric problem under steadystate conditions in cylindrical coordinates is:  LJL k r — rdr  dx  k— = o  (3-26)  The boundary conditions are: (i) For the surface cooled by running water  - k ^ = h (T-T ) dx c w  w  0<r<R6,x = X7  (ii) Along the probe cylinder axis, the radial temperature gradient is zero by symmetry, i.e.,  6T= 0 6V  r = 0, 0 < x < X7  (iii) For the heated surface, the surface temperature gradient is satisfied by  q, =-k  3T  0<r<R6,x = 0  dx  (iv) For the outer well-insulated boundary of the probe cylinder assembly, two alternative boundary conditions are considered:  62  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  (a) for probe calibration, since the outside surface of the probe assembly was well insulated, — =0  r = R6,0<x<X7  dr  (b) for the CFBC experiments, the probe was inserted into the port flush with the inside refractory wall facing the heated suspension. To account for possible heat flows from the outside surface of the probe assembly, the assumed boundary condition is: - k ^ = h (T-T ) cs  or  r = R6,0<x<X7  s  In the above boundary conditions, hc is the refractory-to-probe lateral heat transfer coefficient; T S  s  is the local refractory temperature, estimated as the arithmetic average temperature at the two ends of the probe; q is the heat flux from the heat source to the probe head surface; T is the t  w  cooling water temperature; and h<; is the cooling-water-to-probe-end impinging-jet heat transfer W  coefficient, computed (Incropera and DeWitt 1990) using: ^  h„ =2.74Re -°- Pr -°(  652)  (  513)  V  -0.774)  ^ probe V tube J D  where Re ( = Dtube number; D  p r 0  Ucw  4p V row  C  ^ 27)  cw 2 p,cw  7lD ™tube  Pew / Pew) is the Reynolds number; Pr ( = p  cw  C  P J C  W  / kc ) is the Prandtl W  b e and D t u b e are the probe inside diameter and cooling-water tube diameter,  respectively; p , V , cw  c w  C  P J C  W,  Pcw, kcw, and u<; are the cooling-water density, volumetric flow rate, W  heat capacity, viscosity, thermal conductivity and velocity, respectively.  The thermal conductivities of the probe cylinder, air, inner sleeve, cement and outer sleeve are all evaluated as functions of temperature,fittingthe data available in Incropera and DeWitt (1990):  63  Radiative and Total Heat Transfer in Circulating Fluidized Beds  k  k  347  =0.01503 T + 9.7811  (3-28-a)  k  316  = 0.01534 T + 8.9683  (3-28-b)  =2.9162x10-"T -7.8255xl0- T +0.1183xl0- T-2.4771xl03  air  Chapter 3: Probe Calibration  8  2  3  (3-28-c)  3  1^=1.13 ^innersleeve  =  k ersleeve  =  out  (3-28-d) ^347  Or k  (3-28-e)  316  where all temperatures must be in Kelvins and the units of thermal conductivity are W/m K in each case.  The procedure for computing the temperature distributions in the probe assembly is summarized in Figure 3.5. Eq. (3-26) is solved using a finite difference method with radial and axial grid sizes of 0.25 mm and 0.25 mm, respectively. Total heat fluxes (q -o) were specified 2  and used to compute two-dimensional temperature distributions, while corresponding onedimensional heat fluxes (qi„ ) were estimated by the one-dimensional model using the calculated D  temperatures at the two end positions (see Figure 2.5) where the thermocouples were embedded. The simulated probe correction coefficient, defined as C  =q2-D ' li-D» / (  s i m  w  a  s  then determined.  The cooling-water temperature and flow rate have been found in this study to have an insignificant influence on the probe correction coefficient. The convective heat transfer coefficient from the refractory, h , however, has an effect on the probe correction cs  coefficient. The correction coefficient increases with increasing convective heat transfer coefficient and with decreasing heat source temperature. These simulated probe correction coefficients for SS347 and SS316 probe heads are plotted in Figure 3.6.  64  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Start r Calculate incremental distances dr, dx  r Determine h o n cooling water side cw  r Initial temperature distribution, T a (r,x)  Evaluate thermal conductivity, k(T) \  Determine temperature distribution, T(r,x) \  No max  \f  *•*  Yes  End Figure 3.5. Algorithm for calculation of temperature distribution in quasi-symmetric cylinder.  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  T — > — i — • — i —  1  — i —  32  U  e  .Q  o « o O hi hi  2.0  o  M  •  1^=10 WArr' K, tvwxliirEnsional irocfel  •  h ^ ^ W A r i K,tvvo-dirrcrBioralrrDcfel  •  1^=40 WAr? K, UwHiirrensional rrodel  2  SS316PROEE 1^=0 WAr? K, two-dirrensiorHl model  1.8  1^=10 W/rr? K, hWKlirrEnsiorHl model h ^ ^ W A r ? K,tswxiimensionalrrrxfcl 1^=40 WAr? K,tvKKlirrensiorHl model  1.6  'A  <D  PH T3  — r ~  SS347FROBE • 1^=0 WAri* K, two-dirrensional model  2.4 h  o  1  1.4  12 1.0 100  200  300  400  500  600  700  800  I  I  900  1000  1100  Heat Source Temperature, T (°C) f  Figure 3.6. Simulated probe correction coefficients determined using two-dimensional and one-dimensional models.  The simulated correction coefficients fall in the narrow range 1.24 to 1.37 for the SS347 and SS316 probes with h varying from 10 to 40 W/m K for temperatures above 600 °C. 2  cs  For h equal to 0, the corresponding correction coefficients for the SS347 and SS316 are cs  1.15 and 1.22, respectively.  3.3.2. Indirect Parameter Estimation Model The above approach shows that the one-dimensional model gives higher heat flux estimates than the two-dimensional model; the latter is generally expected to be much closer to the real situation. Since suspension-to-probe heat flux is the parameter to be evaluated based on the temperatures measured, it is necessary to solve the so-called inverse problem.  66  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Many problems in science and engineering are described by ordinary and partial differential equations. For transient or steady state conduction problems, if appropriate initial and/or boundary conditions with all the temperature-dependent parameters, such as thermal conductivity, density and specific heat are known, the governing equations for the temperature distribution can be solved analytically or numerically. This is termed a direct problem. On the other hand, if any of the above mentioned conditions, such as the heat flux or temperature at the solid surfaces, physical properties of the solid or even geometric size or shape must be determined by temperature measurement at one or more interior locations, this becomes an inverse problem. The solution of an inverse problem is much more difficult than that of a direct problem. Mathematically, such a problem may also be classified as a parameter estimation problem.  In the present problem, the governing differential equation is known while some of the parameters are unknown. The purpose of the computation is to obtain an optimal estimation of these parameters using measured interior temperatures.  The first step of parameter estimation is to choose an objective function, S , to x  represent the deviation of the measured values from the model values. Then, the parameters are determined in such a manner that they minimize this objective function. The most commonly used function is the sum of squares of the deviations. The number of parameters to be estimated must be less than the numbers of values measured. In the current project, three local temperatures are measured in the probe; hence, at most two parameters can be estimated.  67  Chapter 3; Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Mathematically, the least square regression can be expressed as dS  T  dZ  ;  =0  (3-29)  n  (3-30)  s =|;lTc)i-(TM)i]2  where  T  i=l  and (T )i is the temperature at the i-th location computed by the heat conduction model with the c  parameters, Z i ,  Z 2 , Z n ,  to be estimated; (T )i is the temperature measured at the i-th location. M  There are many algorithms for multidimensional extremization. The Gauss-Newton method is the simplest and most effective for minimizing a function if good estimates of Z ; can be provided. In this method, the function's gradient, or the first partial derivative, is required. To avoid the calculation of derivatives, a derivative-free method, Simplex method (Press et al. 1989), was used here to find the value of the single parameter, q , which minimizes the objective t  function S for a given h^. T  The accuracy of the estimated heat flux depends on the initial values of the unknown parameters. This depends on how well the heat transfer mechanism is understood and some impossible results may be caused by the multiple extremes of this ill-posed problem using inadequate initial assigned values. Fortunately, if the parameters can be assigned close to the real heat flux, the divergence of the estimates can be controlled, leading to estimations which are better than those from the one-dimensional model. This is also valid when there are errors in the temperature measurement with the two-air gap design. Because the air gap serve as thermal insulation, the influence of the lateral surface boundary conditions were greatly suppressed when the probe was employed in the CFBC.  68  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  The heat flux was obtained for each probe head using the one-dimensional model, i.e., q  m>1  _ = - k(AT/Ax), using only the measured temperatures nearest the two probe surfaces. D  The corresponding thermal conductivity was evaluated at the average of these two temperatures. The results obtained using the one-dimensional model and the parameter estimation procedure are listed in Table 3.1, where T  f  = temperature in the cavity where the temperature is uniform;  T  0  = temperature at the cavity opening;  T  p  = temperature of probe surface close to cavity;  q,.  = radiative heat flux predicted from measured cavity temperature (Eq. 3-19);  q  fc  = free convective heat flux estimated using Eq. (3-23);  q  t  = total heat flux including radiation and free convection (Eq. 3-24);  qjn  tav  = measured heat flux obtained by fitting two-dimensional conductive model to measured probe axis temperatures;  qkn.i-D = measured heat flux obtained from the one-dimensional model using measured probe axis temperatures; A qnU-D = I C|m,l-D " Qt I  A Qm.inv  =  I Qm.inv " *lt I •  69  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  Table 3.1. Comparison of heat flux calculated by analysis of cavity emission and by temperature measurement. . Probe  T  f  q  qfc  r  qm,l-D  A qnu-t/qt  A qm,im/q  kW/m  kW/m  kW/m  %  %  t  188.8  66.69  1.57  68.26  65.70  77.01  12.8  3.8  900  0.83  165.3  57.01  1.70  58.71  57.39  67.31  14.7  2.3  800 700  0.88 0.93  134.0 108.4  46.98 37.40  1.89 2.81  48.87 40.31  46.77 37.13  54.79 43.51  12.1 8.0  4.3 7.9  600  0.97  88.4  28.89  35.78  12.9  3.5  1.00  71.0  16.50  31.70 19.61  30.60  500  2.91 3.11  23.68  42.6  400  1.00  3.97  13.28  18.37  1.00 0.78  9.31 2.00 57.35  20.8 38.4  200 1000 900  56.0 29.3 171.8  27.97 21.48  152.6  48.88  4.51 58.95 50.60  4.89 57.93  0.83  2.51 1.60 1.72  800  0.88 0.93  123.3  40.26 32.03  1.92  45.81  2.10  24.73  2.59  42.18 34.13 27.32  65.3 51.5  14.15  3.15 4.00  17.30  2.53 1.61  12.00 4.24 50.84  1000  700 600  0.97  99.9 81.3  2  kW/m  2  2  2  9.04 67.85 60.51  61.8 100.4  8.4 1.7 2.1  27.3  8.6  34.03 27.31  53.71 39.80 32.22  16.6  0.3 0.1  21.40 16.15  25.27 18.93  46.1  4.51  7.65 58.38  51.68  1.00  400 200  1.00 1.00  27.5  1000  0.78  162.8  1.71 49.23  900  0.83  147.7  42.77  1.73  44.50  50.63 42.67  800  0.88  121.0  34.50  1.93  36.43  700  0.93  97.6  27.44  2.60  600  0.97  79.3  21.16  500 400  1.00 1.00  63.7 49.2  12.11  200  1.00  1000  8.00  2  15.1 19.6  500  17.9 57.8 80.4  23.7 34.6 6.4 0.4  52.24  14.8 17.4  4.1  35.42  42.71  17.2  2.8  30.04  29.68  15.8  1.2  2.86  24.02  22.93  34.77 27.82  15.8  4.5  15.27 10.85  18.62  21.79  42.7  6.83  3.16 4.02  13.02  15.30  41.0  21.9 20.0  25.5  1.47  2.55  4.02  4.81  38.3  19.7  0.78  149.2  -  136.8  41.69  53.40  28.f  -  800  0.88  119.0  37.29  47.24  26.7*  -•  700  0.93  100.4  31.56  39.24  24.3*  600  0.97  81.8  24.05  29.73  23.6*  500  1.00  17.72  21.56  21.7*  -  400  1.00  63.1 49.4  12.92  15.49  19.9*  -  200  1.00  27.3  -  29.8*  0.83  -  44.64  900  -  5.56 57.92  5.45  6.55  20.2*  -  : A q i-D/q ,inv (%) m>  qm.inv  qt  0.78  SS347  ZnSe  P  kW/m  °C  Silver  T  °C  head  SS316  To/Tf  m  70  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  3.3.3. Comparison of the Two Models The probe correction coefficients for each of the cylinders are plotted in Figure 3.7, based on both the one-dimensional model and the parameter estimation model. The heat fluxes estimated by the one-dimensional model are higher than those from the parameter estimation model due to the radial diffusion of heat. Heat fluxes calculated using the onedimensional model are in error since the true multi-dimensional temperature distribution in the probe cylinder causes the temperature gradients to change along the probe cylinder axis. The accuracy of both models increases with increasing operating temperatures because lateral heat diffusion is more significant compares with axial heat conduction when the total heat transfer rates are low. When the total heat transfer rates are high, the heat flow through the probe cylinder is primarily axial. Although heat fluxes in the radial direction also increase, the relative importance of the radial fluxes is diminished. From the calculations and Figure 3.7, the measured correction coefficient, C , required for the one-dimensional m  model, i.e. Eq. (3-25), declined to an approximately constant value for temperatures above 600 °C. The measured correction coefficients were determined to be 1.13 for the SS347 probe and 1.17 for the SS316 probe under the conditions of this study.  The heat fluxes estimated for the window probe using the one-dimensional model and inverse method are compared in Figure 3.8. The one-dimensional model again shows higher fluxes than those obtained using the inverse method. The ratios of these fluxes are in the range of 1.20 to 1.30. The average value 1.25 was taken as the correction coefficient for the window probe.  71  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  2.6  1  -I  1  1  1  2.4  •  22  *  2.0  ^  G  'o  a o  V  Hi  r—  c  C  347  316 ^silver  316  1.6  C  A  <u  fc o  1  1  Inverse Model • C^j (inverse) o C (inverse)  1.8  .2  1  1  One-Dimensional Model  (inverse)  silver  1.4  12  O 1.0 0.8  I  200  •  '  I  400  600  .  I  i  L _  800  1000  1200  Calibration Furnace Temperature, T (°C) f  Figure 3.7. Comparison between probe correction coefficient in Eq. (3-25) determined using one-dimensional model and inverse model.  Heat Emission Estimated by Inverse Estimation, q " (kW/m ) 2  r  Figure 3.8. Comparison between heat flux estimated using one-dimensional model and inverse model for window probe. T = 200 to 1000 °C. f  72  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  3.4. Radiative Heat Emission and Heat Exchange 3.4.1. Differential Emissivity Method (DEM) Suppose there are two probes with different surface emissivities used to measure the rate of radiative heat transfer to the probe surface; one has an absolutely black surface (e = 1) while the other has a perfect white surface (e = 0). Radiant energy incident on the white surface is totally reflected, so that the white probe measures only heat transferred by convection. In contrast, the black surface totally absorbs all energy incident upon it, so that the black probe measures the sum of the radiative and convective heat transfer contributions. In this case, the difference between the heat transfer rates measured by these two probes is solely due to radiation.  For the real case where the two surfaces are not perfectly white and black, separate heat balances need to be written for each probe. For example, for probe head 1 (SS347) and probe head 2 (SS316) in the multifunctional probe described in Section 2.2.2, we may write q ,347 =q +q c.347-R 47q -q"p,347 ,,  m  ,,  q ,3i6 m  3  = q" +q r  (3-3D  ,,  r  M  r  (3-32)  c,3i6-R3i6q -q"pr.3 6 ,,  r  1  where q is the heat flux determined using the measured temperatures; q" is the radiant energy m  r  incident on the probe surface; q" is the radiant energy emitted from the probe surface, q" is the pr  c  heat flux due to convection and R is the reflectivity of the probe surface. Assuming that heat transferred by convection to the two probe heads is the same, i.e. q" i = q" ,347 . and that the Ci3  surface temperatures of the two heads are the same, i.e. T  p3 1 6  =T  p3 4 7  6  = T (see discussion in  Section 5.2), we subtract Eq. (3-32) from Eq. (3-31) and rearrange to give:  73  c  w  Radiative and Total Heat Transfer in Circulating Fluidized Beds  q  „ _ qm,347  qm,316  q pr,347  |  Chapter 3: Probe Calibration  q pr,316 _ q ,347  qm,316 ^ ^ 4  m  (3  33)  r  347  e  e  316  e  347  e  316  e  347  e  316  where a"  =e o T *  4 pr,347  a"  c  347 1 =  e  qm,347  and  q ,3i6 m  316  ul  =e o T  4 pr,316 e  347 p,347  =  c  —  4  316 p,316 u l  ^347  1 ~ ^316  are calculated by the methods described in Section 3.3, using either the one-  dimensional model or the inverse estimation model; T is the probe surface temperature. R and e p  are reflectivity and emissivity of the probe surface, respectively. Using this approach, the radiative heat emission may be estimated using any two surfaces of different emissivities. Clearly, however, the accuracy of the estimates improve as the difference between the two emissivities increases. The radiative heat flux between the probe and the heat source can be determined as: „  ~  „«  „  rp4  „  qm,347  qr,347 = 3 4 q r- 34 OTw = 3 4 7 e  e  7  qm,316 ~  e  7  e  347  e  ~ 316  _  (3-34)  3.4.2. Window Method (WM) In the window method, the window is used to separate convection and radiation heat transfer from the particles and gases of the circulating fluidized bed. A simplified diagram of the various transfers of heat associated with the window is shown in Figure 3.9, in which: A =  window absorptivity;  74  Radiative and Total Heat Transfer in Circulating Fluidized Beds  A  1  q'  c  Chapter 3: Probe Calibration  =  probe-cylinder absorptivity;  =  window-to-probe-cylinder-surface heat flux due to conduction;  q" = c  heat-source-to-window heat flux due to convection;  q^ =  heat flux through probe cylinder due to conduction;  q"  = radiative heat flux emitted from probe cylinder surface, = e 0(T ) ; 4  pr  q" = r  p  p  radiative heat flux emitted from heat source incident on the probe;  q ^ = radiative heat flux emitted from window surface facing probe, = e cr(T ) ; 4  w  wi  q"^ = radiative heat flux emitted from window surface facing furnace, = e 0(T ) ; 4  w  R =  window reflectivity;  R' =  probe-cylinder reflectivity;  t .=  window transmissivity.  wo  During calibration, heat emitted from the cavity transfers by radiation and convection to the window. Part of the radiation is transmitted through the window, while the rest is reflected and absorbed. The heat absorbed by the window, combined with that arriving via convection, raises the window temperature. The probe head receives the sum of the transmitted component, the window emission, the inwards reflection from the window and the heat conducted from this window to the probe surface. Energy is multiply reflected between the window and the probe head. At steady-state a heat balance for the window shows that the heat flux from the cavity due to radiation and convection is equal to the heat flux through the gas gap between the window and the head by conduction and/or convection plus radiation from the window itself and radiation transmitted through the window. Using the ray-tracing method, the heat balance for the window  76  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  yields: q' = q". +0 c  T-  R ) q " + ^ ^ ( T R ' q" ) + r  q' ) + J Z ^ < l \  r  -  m  (3-35)  Based on a heat balance for the probe head, q' can be expressed as: c  l  i  < ™-T^  q  q  m  = q  —  (xq  R'  r  1 R' —  H \  1 R —  .  ' -Y^ - \-^ )  q  +  q  II  -  -  „  "  (3  36)  is obtained as described in Section 3.3, either using the one-dimensional model or utilizing  the inverse model. Setting the right-hand sides of Eqs. (3-35) and (3-36) equal and rearranging, we obtain the following equation for the thermal emission from the heat source: „  qm-q"c+q"wr+4iq'wr+^2q"pr  q"r =  „  (3-37)  -  v  4  where:  3  TR' SI  1-RR* x  S2 =  1-RR T R 2  1-RR'  The same result can be derivedfromthe net-radiation method as shown in Appendix V. The heat flux between the heat source and the probe is then (3-38)  q ^ e ^ - e . G ^  With the total heat flux measured by either cylinder, SS316 or SS347, q and q" can be r  c  determined by solving Eqs. (3-37) and (3-38) together with qmj=q,j+q"^i  (3-39)  where i is SS316 or SS347 depending on the type of stainless steel used in the probe cylinder.  77  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  The thermal emission from the window to the probe head depends upon the window's inside temperature, while the convection depends upon the window outside temperature. The temperature profile across the window can be estimated by solving the following differential equation proposed by Kellett (1952): k - ^ i - 2 a o T ^ - a k T - a q x + a C =0 dx 2  w  2  w  w  r a  1  (3-40)  where 1^ is the thermal conductivity of the window, q is the measured heat flux, x is the axial m  distance measured from the heat source, and a is the absorption coefficient of zinc selenide estimated as a =  (3-41) x  The absorption coefficient for zinc selenide in the wavelength range of interest is 0.215 mm" , 1  consistent with the data reported by Neuberger (1963).  Analytical solution of Eq. (3-40) is not possible due to the 4th power of temperature term. The following approximation is introduced to make it possible to solve the equation:  Tt = Tt  where T  w  \  |  (T _-T») w  14  = 4TT w-3T v 3  4  (3-42)  is the arithmetic average temperature of the window. The error caused by this  approximation is within 10% so long as 0.89Tw<T <1.18T w  w  (Mathur and Saxena 1987). Substituting Eq. (3-42) into Eq. (3-40), we obtain  78  (3-43)  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  k  f- - (a k + 8aaTl)T = a q x - a C , - 6 a a f t 2  w  2  w  dx'  W  m  (3-44)  The boundary conditions are: T  =T  w|x=0  W 1  = T„,  TJ  Solving Eq. (3-44) subject to these boundary conditions yields: T  a q x-aC, -6aaT  =C e +C e- bx  w  m  ,,x  2  3  (3-45)  w  b k„ 2  where:  , 2 2 8aoTv b = a + —4  T - C ( l + e2b ^x„- ) - C w o  b k -6aaT  2  w  a T  w o  -T  w i  -C (l-e4  b x  -)- ^ b k„ a q  2  C =  1+e C —Ce 3  C = 4  bx„  , , 2bx„  2  2  +C  a q bk  -2e  4  r  bx„  b k„ 3  w  and X is the window thickness. The above equation for T is combined with the two heat w  w  balance equations, (3-35) and (3-36), to yield the radiative heat flux, q , the window inside r  temperature, T i, and outside temperature, T o, once the temperature gradient in the probe head W  W  and heat source temperature have been measured. The probe surface temperature is obtained by linear extrapolation of the temperatures measured in the probe head. Solving Eq. (3-44) with an iterative computation algorithm, shown in Figure 3.10, we obtain the inside and outside surface temperatures of the window. Typical values illustrated in Figure 3.11, show uniform temperature  79  Chapter 3: Probe Calibration  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Start with input of Tb, qm,347j C]m,316, C|m,win  I  Initial assignment of the inside and the outside window temperatures and accuracy criteria T*w4n y T'w.out y 6  I  Estimate the convective heat flux between the window inside surface and the probe head  I  Evaluation of the properties and the coefficients at average window temperatures kw, x, cc, b, Ci, C 2 , C 3 , C 4  I Estimatedcoefficients for calculating window temperature distribution, T». in  9  Tw.out  I no I ( T „ > -T . . » ) / T w .  t  a  | <  e  | ( T „ , out *" T w , out  and  Determine radiative heat emissionfromthe solids suspension in CFBC, q"r  I  Determine the convective heatfluxbetween the window inside surface and the probe head  I [(q'c) i-q'c]/(q'c)ini in  f  |<s  no  yes  Determine radiative heat emission and convective heat fluxfromthe solids suspension in CFBC to the probe head, q'V, q"c  End Figure 3.10. Procedure for computing window temperature distribution and heat transfer fluxes using the window method.  80  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  distribution in the window. The calculated window temperatures indicate that there were almost no difference between the inside and outside temperatures of the window and that the window temperatures were close to the temperatures of the heat source when heat transfer by convection is significant. This verifies the earlier statement that radiative heat transfer is "enhanced by the window, with the result that radiation may be overestimated by the window method if an oversimplified model is used. However, the overestimation is eliminated by the model employed, since Eq. (3-37) allows for the influences of convection and window temperature in estimating thermal emission from the heat source to the probe.  3.5. Conclusions 1. A multifunctional probe with a two air-gap structure and an assembly of four heat flux cylinders (see Figure 2.5) can be used to estimate the total heat transfer and radiative heat transfer and to examine the window and differential- emissivity techniques for radiation measurement. The probe has been calibrated using a cylindrical blackbody cavity, and the results are broadly consistent with theoretical analyses. 2. The heat flux for the heat flux cylinders can be calculated using either a parameter estimation (inverse problem) method or assuming one-dimensional conduction. The correction coefficient for the one-dimensional model is obtained as 1.25 for the window probe cylinder, and 1.13 and 1.17 for the SS347 and SS316 probe cylinders, respectively, for operating temperatures over 600 °C. 3. Two methods were employed to determine thermal radiation, a differential emissivity method based on Eq. (3-34), and a window method based on Eq. (3-38). The window  82  Radiative and Tola! Heat Transfer in Circulating Fluidized Beds  Chapter 3: Probe Calibration  temperature distribution and window emission are considered in the model to minimize their influence on the estimation. The window method is more reliable if the temperatures of two faces are determined, while the accuracy of the differential emissivity method is smaller due to variations in the surface emissivity of stainless steel 316.  83  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  CHAPTER 4 SUSPENSION-TO-SURFACE HEAT TRANSFER: EXPERIMENTAL RESULTS  In this chapter, the experimental results obtained with both the dual tube and the membrane wall heat transfer surfaces are presented and discussed. Total suspension-to-tube and suspension-to-membrane-wall heat transfer coefficients are estimated and compared with published data. The effects of suspension density, suspension temperature and particle size are investigated. Some radiative heat transfer data are derived from the dual tube results, but it proved to be impossible to obtain radiation data for the membrane wall surfaces.  4.1. Heat Transfer to Dual Tube Surface A preliminary test using the dual tubes described in Section 2.3 to demonstrate the possibility of using differences in emissivity to determine the magnitude of the radiative heat transfer is first presented. Total suspension-to-tube heat transfer fluxes can be calculated from the cooling water temperatures measured at the inlet and outlet of each tube. The radiative component is then estimated using the differential emissivity method described in Section 3.4.1.  84  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Three Ottawa silica sands of mean diameter 137, 334 and 498 um were used in the experiments. The size distributions and some key properties of these sands are listed in Table 4.1.  The cross-sectional area-averaged suspension density is estimated from the pressure gradient over the interval where the dual tube is located as measured by pressure transducers. Neglecting solids acceleration and the effects of gas-wall and solids-wall friction  Table 4.1. Particle size analyses and fluidization properties for Ottawa silica sands used in dual tube heat transfer tests. Size range (um)  Weight fraction (wt.%) #430 sand  #20/30 sand  1150  840  840  590  29  590  420  47  30  420  297  11  43  1  297  210  N/A  20  10  210  149  N/A  6  37  149  105  N/A  1  32  105  74  N/A  15  74  53  N/A  4  53  pan  8  1  +  #710 sand  5  Sauter mean ciameter (urn)  498  Particle density (kg/m ) *  334  137  2610  2610  2610  Sphericity *  0.8  0.8  0.8  Minimum fluidization velocity at T=20 °C and P=l atm (m/s)  0.163  0.078  0.014  3  .  provided by OCL Industrial Materials Ltd..  85  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  (Van Swaaij et al. 1970), we can write: j_AP P ™ P = — —  g Az  ( -o 4  This approximation is often used to estimate suspension densities in studies involving laboratoryscale CFBs as well as for commercial CFBCs. Previous workers have shown that pressure drop measurements yield results in favorable agreement with those obtained by a y-ray technique (Hartge et al. 1988), a capacitance technique (Brereton 1987) and an optical fiber technique (Zhou 1995). The technique is reasonably accurate, except near the entrance, for most CFB risers (Louge and Chang 1990, Weinstein and Li 1990).  4.1.1. Total Suspension-to-Tube Heat Transfer Coefficients The assumption that the bulk suspension temperature along the riser is uniform eliminates the complication caused by variations of the suspension temperature over the location where the heat transfer surfaces were installed. Figure 4.1 shows that, except near the bottom, the temperature along the riser is reasonably uniform at different bulk suspension temperatures. The temperature profile is not uniform at the bottom of the riser since the inflow of fresh particles, supplemented by recycled fines from the secondary cyclone, as well as cold air, disturbs the local temperature field. The influence of these inflows disappears witMn 1 to 2 m above the feed point and has little or no impact on the temperature profile over the heat transfer surfaces. Previous investigations (e.g. Grace 1990b) have confirmed that bulk temperature profiles in the upper parts of CFB reactors are uniform. Also, vertical profiles of heat transfer coefficient (Andersson et al. 1987) and axial suspension density (Glicksman et al. 1991) tend to be reasonably uniform,  86  Radiative and Total Heat Transfer in Circulating Fluidized Beds  N  o  hT  Chapter 4: Suspension-to-Surface Heat Transfer  groupI  Transfer Surface  u  -  6.9 7.6 7.9 9.9 6.9 7.3 7.6 125  .  Psusp  kg'rri  5  120 120 120 120 100 100 100 80  •  rg  o •  U  .a Q  o  •  T  Sec. Air  X  g m/s  " • .  PH  <U > 0  •s 1  X  Fuel Feed  is 200  400  600  800  1000  1200  1400  1600  Bulk Suspension Temperature, T (°C) b  Figure 4.1. Axial profile of bulk suspension temperature along CFBC riser. d = 3 3 4 um. group I points: gas burning; group II points: coal burning. p  except near the top and bottom. Hence, it is reasonable to assume that the bulk suspension temperature over the transfer surfaces is uniform. The average temperature at the center of the riser over the elevation range corresponding to the heat transfer surface is used as the bulk suspension temperature, Tb.  4.1.1.1. Total Suspension-to-Surface Heat Transfer Coefficients For steady state conditions, energy conservation requires that the total heat removed by the cooling water is equal to the heat transferred from the suspension to the cooling water. The overall riser-to-water heat transfer coefficient, U eraii, is then given by 0V  u  overall  _ Mcwep.cwCTouUet — T A t.out (Isusp  87  jnlet  f water )  )  (4-2)  Radiative and Total Heat Transfer in Circulating Fluidized Beds  where At^ut is the tube outside area, c  P)CW  Chapter 4: Susvension-to-Surface Heat Transfer  is the average cooling water specific heat at T  wate  r, M  c w  is the cooling water mass flow rate, Ti i t is the cooling water inlet temperature, T tiet is the n e  cooling water outlet temperature, T  ou  is the axial cross-sectional average suspension  s u s p  temperature over heat transfer surface in the riser (Tb) and T ter is the average cooling water wa  temperature ((T iet+T ti )/2). in  ou  et  The total thermal resistance from the suspension to the cooling water is the sum of the thermal resistance on the water side, on the suspension side and in the tube wall. The total suspension-to-tube heat transfer coefficient, including convective and radiative heat transfer, can therefore be calculated using:  r  U overall  <X o/ .)  ' i ^  r  r  K,  where r* and r are the inside and outside radii of the tubes, respectively, and kt is the thermal 0  conductivity of the tubes. The tube-to-water heat transfer coefficient, hc , is determined using the W  Dittus-Boelter equation (Incropera and DeWitt 1990): Nu =0.023(Re ) (Pr ) 08  cw  cw  (4-4)  04  cw  where Nucw is the Nusselt number (hcwdi/kcw), Rew is the Reynolds number (diUcwPcv/licw), Pr is cw  the Prandtl number (c  P)CW  p /kc ), dj (=2r;) is the inside diameter of the tubes, U cw  W  velocity of the cooling water and kc , p W  cw  c w  is average  and pc are the thermal conductivity, density and W  viscosity, respectively, of the cooling water. The properties of the cooling water are all evaluated at T ter. wa  88  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  4.1.1.2. Effects of Suspension Density, Temperature and Particle Size The suspension density is one of the important operating parameters influencing the heat transfer to the wall (Grace 1986). The length-averaged, time-averaged suspension-to-tube total heat transfer coefficients obtained with the oxidized tube are plotted in Figure 4.2 for silica sands of all three sizes at the three bulk suspension temperatures, 803 °C, 851 °C and 905 °C. The total suspension-to-tube heat transfer coefficient increases approximately linearly with increasing suspension density in the range of 10 to 100 kg/m . This is expected because the volumetric thermal capacity of solids is much higher than that of the gas, by a factor of about 800 to 1000 in the current case. The heat transfer coefficients lie between 120 and 250 W/m K, within the 2  usual range of values for long heat transfer surfaces in circulating fluidized bed combustors.  The total heat transfer coefficients obtained in the present study are compared with the results of previous investigations in Figures 4.3 to 4.6. These figures are keyed to the symbols listed in Table 4.2. The measured suspension-to-tube heat transfer coefficients obtained at elevated suspension temperatures and for larger surfaces are compared in Figure 4.3. As do the earlier measurements by others, our heat transfer coefficients increase with increasing suspension density with an exponent of about 0.44, obtained by the non-linear regression of our results and the results listed in Table 4.2, instead of 0.5, as might be expected based on the model of Mickley and Fairbanks (1955). Some studies suggested correlations with indices closer to the 0.5 power, e.g. Basu (1996) with 0.5, Divilio and Boyd (1994) with 0.55 and Werdermann and Werther (1994) with 0.562. However, these correlations are based on limited data measured by probes with short surfaces and chiefly account for the convective component. The effect of heat transfer  89  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  350 300  T =803±9°C A d=498um • dp=334om • dp=137um b  250  p  200 150 100 50  _i 20  10  40  30  50  60  70  I  i  L  80  90  100  110  80  90  100  110  350  350 300 250  T  1  r—j  1  T  1—i—r  1  r  1  1  1  1  1  '  1  r  T =905±7°C A dp=498iim b  •  200  dp=137um  150 100 50 J  10  i  I  20  i  L  30  _i  40  i  50  i_  J  i  60  Suspension Density, p  I  70  s u s p  i  L _l  80  L.  90  100  110  (kg/m)  Figure 4.2. Suspension-to-tube heat transfer coefficient plotted against particle suspension density for different particle diameters at different suspension temperatures with oxidized tube. U =7.8 to 9.6 m/s, P/S=1.3 to 4.0. Solid lines are least squares linear fits. g  90  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  Table 4.2. Symbols and operating parameters for heat transfer data plotted in Figures 4.3 to 4.6. Suspension temperature  Kobro and Brereton (1986)  Particle diameter um 250  850  Length of heat transfer surface m 0.10  Feugieretal. (1987) Basu and Konuche (1988) Basu (1990) Basu (1990)  215 296 296 296  400 650-885 600-895 530-850  0.95 0.10 0.10 0.81  Wuetal. (1989a) Wuetal. (1989a) Andersson and Leckner (1992) Andersson and Leckner (1994)  241 227 312 250  340-880 400-847 850 854  1.22 1.22  Nag and Ali (1991)  309 334  400, 600  Reference  Symbol  •  •  *  •  O  + T V  •  o  A B  This work  °c  804, 850, 905  1-10 1-10 0.30 1.52*  *: see Figure 2.7, the vertical length of the U tube is 0.75 m.  surface length on heat transfer coefficients is discussed below in Section 4.2.1.2. The results presented in Figure 4.3 illustrate that total heat transfer coefficients increase with particle concentration quickly at low suspension densities, for instance up to 20 kg/m , as in the model of Mickley and Fairbanks (1955). As suspension density increases further, the influence of solids concentration on the total heat transfer coefficient appears to decrease.  Figure 4.4 demonstrates the effect that suspension temperature has on the total heat transfer coefficient. The data plotted were all obtained at elevated suspension temperatures. The comparison is made with similar mean particle sizes (215 to 334 um) and at a common suspension density of 20 kg/m to minimize the influence of these variables. The influence of 3  suspension temperature on heat transfer coefficient is apparent despite the scatter. The total heat transfer coefficient is seen to increase with increasing suspension temperature. This can be  91  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  Suspension Density, p  (kg/nr)  susp  Figure 4.3. Comparison of total heat transfer coefficients as a function of suspension density with other reported data obtained at high temperature with surfaces 100 mm long or longer. Symbols and operating parameters are listed in Table 4.2. tf CN  s  250  1  1  1  I  1  I  1  I  1  I  1  I  > I  1  n  1  1  1  +  1  1  1  O  O  o  A  •  A  •  _  . T  4)  V  T  o "  T  cn  100  o  +  +  +  -  4)  "3  I  +  Q  §  1  O  50  „  -  o  _  •  o  V  •  *  1 •  300  •  "  350  i . i . i . i i I  400  450  500  550  600  I  .  650  I  .  1  I  700  750  Bulk Suspension Temperature, T  b  800  1 1 1 1 850  900  950  (°C)  Figure 4.4. Effect of suspension temperature on total heat transfer coefficient at a suspension density of 20 kg/m . Symbols and operating parameters are listed in Table 4.2. 3  92  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  attributed to the radiative component and, to a lesser extent, the increased gas thermal conductivity with increasing temperature.  Particle size appears to have some influence on the total heat transfer coefficients as shown in Figure 4.5, with the total heat transfer coefficient being higher for small particles than for larger ones. Some direct experimental measurements using y-rays (Wirth and Seiter 1991, Molerus 1994) or a protruding needle (Glicksman et al. 1991) and indirect results (Wu et al. 1990) indicate that there is a largely solids-free region near the wall of thickness 0.06 to 0.7 mm. The thermal resistance of the gas gap between the suspension and the heat transfer surface depends on the thickness of the gap, which in turn depends on the particle size. However, the differences in the heat transfer coefficients for the three particle sizes used in the present study  tf 1  1  •  1  1  1  1  1  r—i  1  170 *r c  160  •  o  .a  o O '«£  1  1  1  1  1  • •  150 140  •  Vi  u  - |  130  • •  120  s  110 100  H  100  -i  1  150  1  1 200  1  1 250  r——|  300  1  1 350  1  !  400  1  1 450  1  1 500  1  550  Particle Diameter, dp (nm) Figure 4.5. Effect of particle size on total heat transfer coefficient at high temperature. Psusp20 kg/m . Symbols and operating parameters are listed in Table 4.2. =  3  93  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  are not very significant since relatively long heat transfer surfaces are used here compared to the Kobro and Brereton (1986) study. Particles travelling parallel to the wall then have enough time to equilibrate with the surface temperature for long heat transfer surfaces (Wu et al. 1987, Glicksman 1988), and conduction through the gas gap then becomes less significant. Thus, the influence of particle size on heat transfer diminishes.  4.1.1.3. Correlation for Suspension-to-Tube Total Heat Transfer Coefficient The results obtained using one tube (ew=0.76) of the dual tube surface have been compared with other published data in the literature under similar operating conditions at atmospheric pressure. The total heat transfer coefficients can be correlated as a function of suspension density, particle size, suspension temperature and surface length. On the basis of nonlinear curve fitting of all the data plotted in Figures 4.3, 4.4 and 4.5, of an equation of form ht=A+B (p p) (d ) (T ) (L) , the coefficient b on dp was found to be close to 0. Hence the d a  SUS  b  c  p  d  b  p  was removed and the constants refitted. The following empirical dimensional correlation fits the total heat transfer coefficient for previous and the present experimental studies listed in Table 4.2: "_  „^-6  h =62 + 1.2x10 p  0.44 ^2.4 -0.042  t  susp  T  .„ „  T  L  b  x  (4-5)  for 1 < p p < 100 kg/m , 137 x 10 < d < 498 x 10 m, 680 < T < 1178 K, 0.1 < L < 10 3  SUS  -6  -6  p  b  m, 0.1 < D < 1.7 m. In this equation the units to be used for p p, L and T are kg/m , m and K, 3  SUS  b  respectively, giving h in W/m K. The expression includes the three major parameters which t  have a clear influence on the total heat transfer coefficient. Suspension-to-tube heat transfer coefficients evaluated using Eq. (4-5) are compared with the experimental results in Figure 4.6. The correlation predictions are seen to be in reasonable agreement with the measured results  94  Chapter 4: Susoension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  tf  Figure 4.6. Comparison of experimental heat transfer coefficients with those predicted using Eq. (4-5). p =l to 100 kg/m , d=137 to 498 um, T =680 to 1178 K. Symbols and operating parameters are listed in Table 4.2. 3  susp  p  b  obtained at elevated temperatures, with almost all deviations within ±50%.  4.1.2. Suspension-to-Tube Radiative Heat Transfer Coefficients 4.1.2.1. Calculation ofRadiative Heat Transfer Coefficients Suspension-to-tube radiative heat transfer coefficients can be estimated by comparing the heat removed by tubes of different surface emissivity. Assuming that the convective/conductive heat transfer from the particle suspension to the two tubes is the same and that the tube is surrounded by a homogeneous outside environment, and neglecting any difference in the outside surface temperatures of the two tubes, we can determine the radiative heat transfer  95  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  coefficient for the oxidized tube as: h = rl  q t l  T A  where q and  b  " ' q  2  _ T A  t,out  c  ^ — e —e sl  c  (4-6)  s2  are total heat fluxes removed by the oxidized tube and the unoxidized tube,  tl  respectively, T is the bulk bed temperature, T b  to u t  is the outside tube wall temperature, and e  sl  and e^ are the corresponding suspension-surface emissivities. The emissivity for each tube is estimated by (4-7)  where e is the tube surface emissivity and e t  susp  is the effective emissivity of the particle  suspension. In previous theoretical studies (e.g. Hottel and Sarofim 1967), considerable effort was devoted to accurately estimating the effective  suspension emissivity. As a first  approximation, an empirical relationship between the effective emissivity of a fluidized bed and the particle emissivity which does not include the particle voidage can be used (Grace 1982). Borodulya and Kovensky (1983) developed an isothermal model to calculate the effective emissivity, treating the bed as a stationary cubic lattice of solid spherical particles. Only two specific solids voidages, a packed (e=0.4) and an expanded lattice (e>0.95) were used in the calculation. An expression for the effective emissivity of an isothermal bed of dispersed solid particles was derived using the two-flux model of Brewster (1986). However, the scattering coefficients used in the derivation are only suitable for dilute particle streams. Also, the twoflux model can be inaccurate in an anisotropically scattering dispersion such as a circulating fluidized bed (Tien and Drolen 1987). Xia and Strieder (1994) used a random overlap model to estimate the effective emissivity of an isothermal semi-infinite slab. Their correlation  96  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  contains both the particle surface emissivity and the voidage. The predicted effective emissivities increase linearly with voidage, contrary to empirical findings (Han et al. 1992). The following approach due to Hottel and Sarofim (1967) is adopted here to estimate the effective emissivity, since it considers not only particle surface emissivity and voidage, but also particle size:  e =l-e- '  (4-8)  K 6  susp  where K is the extinction coefficient and 5 is the thickness of the downflowing annular layer of t  solids which can be estimated from the correlation of Bi et al. (1996):  V = l-Vl-34-1.30(l-esec) - + 0 2  (l-I J-  4  for 0.80 <  se  < 0.9985  (4-9)  where X is the half-width of the riser. To simplify the mathematics, we assume that the extinction coefficient is independent of temperature. K may then be estimated (Tien 1988) by: t  K =L5 ~ 1  (4.10)  e a n n  t  where dp is the diameter of the particles, and e  a n n  is the average voidage in the annulus region.  The voidage distribution in the annulus region is assumed to follow the correlation of Zhang and Tung (1991), i.e., e&) =lsJ *" * om+  with  8sec  +3  (4-11)  l>)  being the cross-sectional average voidage at the level of interest evaluated from the  pressure gradient and <5=x/X is the dimensionless distance from the riser center. Hence: 1.0  2J"[l-e(<D)]<Dd<D Bmn  1  2 (6/X)-(6/X) 6  97  2  ( 4  "  1 2 )  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  4.1.2.2. Results and Discussion Suspension-to-tube total heat transfer coefficients for both oxidized and unoxidized tubes are shown in Figures 4.7, 4.8 and 4.9 for all three particle sizes at three bulk suspension temperatures. It is seen that both suspension-to-tube heat transfer coefficients increase with increasing suspension density. Also there are clear differences between the total heat fluxes measured by the unoxidized (open symbols) and oxidized (solid symbols) stainless steel 316 tubes which have different surface emissivities. The suspension-to-tube radiative heat transfer coefficients calculated using Eq. (4-6) are plotted against the suspension density in Figures 4.10, 4.11 and 4.12. The radiative heat transfer coefficient is seen to lie in the range of 40 to 100 W/m K, to increase with increasing suspension density and to decrease with increasing 2  mean particle diameter. Consistent with Eqs. (4-8) and (4-10), suspension emissivity is enhanced by  <s  B  240 220  •  ,  A  T =802±8°C  •  ,  O  T =851±5°C  b  b  .100 80 I  1 30  1  1 40  1  1  1  50  1  1  60  Suspension Density, p  sus  1—:—i 70  1  1  80  1 90  (kg/m )  Figure 4.7. Effect of suspension density on suspension-to-tube heat transfer coefficient for different suspension temperatures. d=498 um, U =8.4 to 8.8 m/s, P/S=2.4-4.0. Solid symbols: oxidized tube; open symbols: unoxidized tube. p  98  g  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  220-|  A  T =803 ± 9°C  O  T=851 ± 6 °C  O  T =905± 5°C  b  b  • L  b  -  .1  i  200  "3 SB  •  I  g 1804  0  '  'a 160H  *  J  ° q > t £ of>rf  8  1  ^ g ^ C ^ A  100  I  10  I  I  Ho"  50 Suspension Density, p  20  A  •  120-1  o H  A^  A  o  -r70 (kg/m )  I  "eo"  40  susp  80  90  Figure 4.8. Effect of suspension density on suspension-to-tube heat transfer coefficient for different suspension temperatures. dp=334 um, U =8.4 to 8.8 m/s, P/S=2.4-4.0. Solid symbols: oxidized tube; open symbols: unoxidized tube. g  a 260  1  - i  240  1  1  1  T =800 ± 6 °C  O  T =849 ± 5 °C  220  i  1  A  1 — ' — r  b  b  T =900 ± 5 b  o 200 180 160 140  g 120 -a  oo • o dp  100 10  •  *5A  , A  B 20  30  40  A  _ %  50  60  Suspension Density, p  susp  70  80  90  100  (kg/m )  Figure 4.9. Effect of suspension density on suspension-to-tube heat transfer coefficient for different suspension temperatures. d =137 um, U =7.8 to 8.9 m/s, P/S=1.6 to 2.4. Solid symbols: oxidized tube; open symbols: unoxidized tube. p  99  g  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  tf  120 |  1  1  1  1  1  1  1  |  1  1  •—i  1  1  Suspension Density, p  r  1  (kg/m ) 3  Figure 4.10. Effect of suspension density on suspension-to-tube radiative heat transfer coefficient for three types of sand at 905 ± 7 °C. Solid lines are least square fits. tf  120 |  r  1  1  1  1  1  1  1  1  1  1  1  1  T  •  1  1  1  r  Figure 4.11. Effect of suspension density on suspension-to-tube radiative heat transfer coefficient for three types of sand at 851 ± 6 °C. Solid lines are least square fits.  100  Chavter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  tf  130 |  3  3  0  I  .  _•  30  1  1  1  1  1  ,  L 50  ,  40  1  1  1  1  60  Suspension Density, p  1  '  T  1  1  70  80  -.  1 90  (kg/m ) 3  Figure 4.12. Effect of suspension density on suspension-to-tube radiative heat transfer coefficient for three types of sand at 803 ± 9 °C. Solid lines are least square fits.  small particles and high particle concentrations. Previous experimental studies (e.g. Han et al. 1992) have also found that suspension emissivity increases with particle concentration. The emissivities of both the suspension and tube were set equal to 0.91 to estimate radiation at a suspension temperature of 701 °G (Wu et al. 1989a). They found that the radiative heat transfer coefficient is 68 W/m K regardless of suspension density. This value is comparable to the 2  radiative heat transfer coefficient shown in Figure 4.12. A key factor for measuring the radiative component using the differential-emissivity method is to have accurate surface emissivities for both tubes. The error bars of -7% to 15%, shown in Figures 4.10, 4.11 and 4.12 are based on ±10% uncertainty in the emissivities. That is, the errors of -7% to 15% of the calculated radiative heat transfer coefficients occur if the tube emissivities used in Eq. (4-6) were both in error  101  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  by -10% and 10%, respectively. The radiative heat transferfromthe refractory walls to the dual tube is estimated in Appendix VI.  The ratio of the radiative heat transfer coefficient for the oxidized tube to the total suspension-to-tube heat transfer coefficient is plotted against the solids suspension density in Figures 4.13, 4.14 and 4.15. The radiation component is in the range of 25% to 45% of the total heat transfer from the suspension to the tube. The relative importance of radiation appears to decrease slightly with increasing suspension density. The effect of particle size on the relative importance of radiation appears to be insignificant for the range of conditions studied. The method of calculating the radiative heat transfer coefficient by Eq. (4-6) assumes that streams close to the tube have a uniform emissivity and uniform temperature equal to the  0.60  §  is  0.55  8  A  d=498 um  •  d=334 um  X  d =137 um  i  i——r  1  p  p  p  0.50  X  0.45  *  •53 « H o  •4-*  x  X  x  t  0.40  X  >  0.35 O O  •  X X  X X  0.30  »X*& •A^Xx^t A A/A X* % A A A A A A A A  J  10  20  30  40  .  A  L  60  50  Suspension Density, p  70  80  (kg/m ) J  susp  Figure 4.13. Relative importance of radiative heat transfer as a function of suspension density at 905 ± 7 °C.  102  90  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  0.55  i  '  1  S  0.40  x  •a •s  0.35  H o  i  '  I  I  1  •  p  X  1  1  • X  •  X  -  X>  -  •  :  m  .xx /A  x  •><  X A  X  I . 20  I . 30  I  . 40  -  A&  x  > X c  i  .  -  xx x  •  50  X  V  X  •  10  -  AvX  x ^xSk x x xxtf* x^ x  X A<AA*  •  0.20  '  I  1  •  d =137 um  X  0.25  o Q  i '  •  p  •  «4-l  |  d =334 um  0.30  •*-»  '  p  •  0.45  |  d =498 um  A 0.50  1  1  T  i  1  1  I  60  70  Suspension Density, p  1  1  80  1  1  90  100  110  -3  susp  (kg/m )  Figure 4.14. Relative importance of radiative heat transfer as a function of suspension density at 851 ± 6 °C.  ^  0.60  x  1  1 <3  0.55  A  d =498 um  •  d =334 um  X  d =137 um  p  p  p  0.50 X  X  0.45 X  ts  H o  x  X  0.40  X  X A  4»  0.35 A  A  X  A  •  X •  •  0.30  X*  X  A  A  * X  X  X  X  o 0.25 30  40  X  X  X  X A  X  X  X _l_ 60  50  X  x^<x xx xx  Suspension Density, p  JL  70 susp  80  (kg/m )  Figure 4.15. Relative importance of radiative heat transfer as a function of suspension density at 803 ± 9 °C.  103  90  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  bulk suspension temperature. Therefore, the method is applicable to situations in which radial temperature profiles are uniform or close to isothermal.  4.2. Heat Transfer to Membrane Wall Surface Experiments were carried out using the membrane wall heat transfer surface described in Section 2.4. Temperature profiles close to the pipe crest and suspension-to-pipe and suspensionto-fin heat fluxes were measured using embedded thermocouples. The effects of suspension density, heat transfer surface length and bulk suspension temperature on suspension-tomembrane-wall total heat transfer coefficients were investigated. Unfortunately, the unoxidized portion of the membrane wall was twisted and the surfaces were oxidized at an early stage of experimentation because the cooling water for this section was accidentally shut off. Thus, the objective of investigating radiative heat transfer to membrane walls, based on differences in surface emissivities, could not be carried out. Ottawa silica sand having an initial mean diameter 286 pm was used in the test. The particles drained from the standpipe after the 20-hour run were collected and measured by screening to determine the change in particle size. The properties of the particles before and after the run are listed in Table 4.3. The mean particle diameter after the test was 264 pm, 22 pm smaller than the original value, presumably due to attrition. This change is small enough that the particle size is treated as constant during the heat transfer experiment.  4.2.1. Effect of Membrane Wall Length Earlier investigations have revealed the importance of the vertical length of the heat transfer surface on heat transfer coefficients. Wu et al. (1987) pointed out that long  104  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Tola! Heat Traitsfer in Circulating Fluidized Beds  Table 4.3. Particle size analyses andfluidizationproperties for Ottawa silica sand before and after membrane surface heat transfer test Sand weight fraction Sand size range (wt.%) (pm) #50 sand after run #50 sand before run + 1000  500  6.7  1.0  500  355  26.1  20.4  355  250  44.7  41.9  250  180  24.7  28.3  180  125  3.6  5.3  125  pan  0.2  3.1  Sauter mean diameter (pm)  286  264  Sand density (kg/m ) *  2596  2409  Minimum fluidization velocity  0.080  0.064  3  at T=20 °C and P=l arm (m/s)  -  *: measured by volume and weight using water displacement method.  heat transfer surfaces give lower heat transfer coefficients than small probes. To verify this conclusion, eight thermocouples were inserted into one of the SS316 pipes at different vertical positions as shown in Figure 2.9. The average heat transfer coefficients for different distances from the top, obtained using Eq. (4-3), are shown in Figure 4.16, along with data obtained by Wu et al. (1989a) using similar calculations. The more accurate estimation of suspension-tomembrane-wall heat transfer coefficient is discussed in Section 4.2.3. The symbols and operating conditions corresponding to each study are listed in Table 4.4. It is clear that heat transfer coefficients initially decrease quickly with distance, becoming essentially independent of distance beyond about 0.6 m.  In most circulating fluidized beds, solids are carried up in the core and fall downwards in an annular wall layer (e.g. see Rhodes 1992). Based on the study of thermal resistance of surfaces  105  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  in contact with fluidized bed particles (Gloski et al. 1984), Glicksman (1988) derived a time constant for particles to approach thermal equilibrium: P' P p ~D P P ^ tP C  (4-13)  d  36k Table 4.4. Symbols and operating conditions for heat transfer data plotted in Figure 4.16. Reference  Symbol  A  Wu et al. (1989a) (measured)  •  Wu et al. (1989a) (measured) This work (measured) This work (measured) This work (measured) This work (measured) Wu et al. (1989a) (Gaussian fit) This work (Gaussian fit) This work (Gaussian fit)  T  O  •  •  •  s  Particle diameter urn 241  Suspension temperature °C 860  Suspension density kg/m* 54  241 286  407 804 804 706 706 407,860 706,804 706,804  54 50-55 20-24 50-55 20-24 54 20-24 50-55  286 286 286 241 286 286  450 400 h  M o  U  M  350  •  This work: T =706 °C  O  This work: T =804 °C  b  b  Wu et aL (1989a): T =860 °C b  300  Wu et aL (1989a): T =407 °C b  250 200  ?=8  150  a o H  100  50 0.0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  1.8  Distance from Top of Membrane Assembry, 27 - Z (m) Figure 4.16. Effect of heat transfer surface length on suspension-to-membrane-wall heat transfer coefficient at two bulk temperatures. d=286 urn, U =8.0-8.5 m/s, P/S=2.0-2.4. p  106  g  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  For the particles used in this work (286 pm), the time constant evaluated at suspension temperatures of 700-800 °C is 0.067 to 0.072 s. The length traveled by particles before reaching thermal equilibrium is determined by their downward velocity. Solids in the annulus travel downwards at a velocity of 1 to 3 m/s (Zhou et al. 1995, Wang et al. 1993). Therefore, to reach 95% of thermal equilibrium, the particle travelling length is determined to be about 0.21 to 0.64 m, shorter than the distances required in Figure 4.16 to approach an asymptotic value. There is considerable lateral solids motion when solids travel downwards (Zhou 1995). When solids from the core region enter the annulus, there is an added thermal driving force because of the greater temperature difference. As solids fall along the wall surface, the solids temperature declines, resulting in a decreasing driving force. The lateral solids mixing slows down the decrease of solids temperature, prolonging the distance needed to approach steady state. As a consequence, the minimum heat transfer surface length beyond which there is no effect on heat transfer coefficient is longer than that calculated using the time constant and the solids velocity in the annulus.  Results obtained by Wu et al. (1989a) in the same column with a membrane surface of similar geometry are also plotted in Figure 4.16 for a suspension density of 54 kg/m , particle 3  diameter of 241 pm, and suspension temperatures of 880 and 407 °C. These results give similar trends, with the heat transfer surfaces being long enough in both studies to reach asymptotic values of the heat transfer coefficient. From this figure, it is seen that the heat transfer coefficient increases with increasing temperature and suspension density for a given heat transfer surface, especially for shorter surfaces.  107  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  4.2.2. Temperature Profiles Suspension axial temperature profiles in the riser were uniform over the interval where the membrane wall was located (see Figure 4.1). However, there tend to be significant gradients of temperature normal to the wall (i.e. lateral gradients) close to cooling surfaces in CFB furnaces (Leckner and Andersson 1992, Couturier et al. 1993, Golriz 1995). As indicated above, thermal boundary layers are developed along the transfer surface. To estimate and predict suspension-to-surface heat transfer coefficients under non-isothermal conditions, an expression is needed for the temperature profile close to the surface.  4.2.2.1. Lateral Temperature Profile in the Riser The local suspension temperature was measured with a shielded K-type thermocouple, 1 mm in diameter, inserted into the riser from the temperature measurement port opposite the membrane wall, 5.44 m above the air distributor (see Figure 2.11). The thermocouple was shielded by two concentric stainless steel pipes with the end of the pipe facing the combustor blocked to avoid radiation from the membrane wall. A conical hole with top and bottom diameters of 10 mm and 6 mm, respectively, was drilled at the blocked end with its axis vertical, and the thermocouple head was fixed at the center of the hole. The thermocouple was tested using an ice-water mixture and boiling water suggesting that the accuracy was better than ± 0.5 °C for the 0 to 100 °C range. The accuracy can be extended to 800 to 900 °C since the electromotive force (EMF) of nickel-cromium / nickel-aluminium thermocouple increases linearly with increasing temperature and its thermopower remains practically constant over a wide range of temperature, e.g. 40.95 pV/ °C in the range of 0 to 100 °C and 40.48 pV7 °C in the  108  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  range of 800 to 900 °C (Quinn 1990).  The experiments were performed at several bulk suspension temperatures. The results indicated that there were substantial temperature gradients close to the wall under the conditions studied. The lateral temperature profiles were uniform across the riser core cross-section, while there was a thermal boundary layer in the annular region. Temperature gradients have also been observed in commercial CFBC risers (Andersson and Leckner 1992, Jestin et al. 1992, Couturier et al. 1993, Werdermann and Werther 1994, Golriz 1995, Wirth 1995). Golriz (1995) correlated measured temperature profiles close to the membrane wall by: 0' =l-[-0.023Re +0.163(T /T ) + 0.294(z/H)]exp{-0.0054(x/d )} p  b  w  p  (4-14)  where 0' = (T-T )/(Tt,-T ) is a dimensionless temperature, Tb and T are the bulk suspension w  w  w  temperature and outside pipe wall temperature, respectively, H is the overall riser height, Re (= UgPgdp/pg) is the particle Reynolds number, d is the mean particle diameter, and x is the p  p  distance normal to the fin. Use of Eq. (4-14) is restricted to large risers and the operating conditions ranging from 5.6 to 13.5 in Reynolds number, 2.1 to 2.47 in Tb/T and 0.19 to 0.81 in z/H. w  According to their data, the ratio Tb/T is about 2.26 to 2.32 for bulk suspension temperatures of w  740 to 920 °C and silica sand particles of diameter 270 to 290 um. The thermal boundary thickness is in the range of 600 to 800 d (Golriz 1995) determined at (T-T )/(T -T ) = 0.99. Wirth (1995) p  w  b  w  recommended the following correlation for the lateral temperature profile based on the data of Werdermann and Werther (1994): T = 6155 + 154.6 x —arctan(35x) 180 V  109  V  (4-15) '  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  The measured temperature profiles in the vicinity of the pipe crest in this study are compared with the data reported by others in Figure 4.17, with the symbols and parameters listed in Table 4.5. There are steep temperature gradients close to the water-cooled surface. Temperature increased with increasing distance from the fin and pipe crest, and presumably there was a thermal boundary layer close to the water-cooled heat transfer surface. These results are qualitatively consistent with earlier observations. Figure 4.17 also shows the effect of riser scale on the temperature profile for similar operating conditions. In this study,*the riser is relatively small compared with other studies in which temperature profiles have been measured. The thermal boundary layer is about 15 to 20 mm thick at the location where the membrane wall was installed, determined as the thickness where T - T = 0.99 (Tb -T ). Equation (4-14) is the best x  w  w  Table 4.5. Symbols and parameters for temperature profiles plotted in Figure 4.17. Symbol  • O A V X  •  Reference Couturier et al. (1993) Golriz (1995) Jestin et al. (1992) Leckner and Andersson (1992) Werdermann and Werther (1994) This work  T °C 880  d pm N/A  Psusp kg/m N/A  P/S  u  N/A  m/s N/A  N/A  845  290  3-70  2.33  2.6-6.6  0.81  725  260  N/A  N/A  N/A  N/A  820  3570  N/A  3.35  4.0  0.86  850  215  5-7  N/A  N/A  0.74  808  286  45-72  2.0  8.0  0.73  b  p  3  g  z/H  •  This work  763  286  45-72  2.0  8.0  0.73  •  This work  711  286  45-72  2.0  8.0  0.73  T  This work  687  286  45-72  2.0  8.0  0.73  110  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  900  O  9-,  -X  850 800 750 700  H  c o  650 600 550  00  500 450 400  0  50  100  J i L 150j—i—i— 200 250 300  350  400  450  500  550  Distance from Membrane Wall, x (mm) Figure 4.17. Lateral suspension temperature profiles near membrane walls, measured fromfin.The conditions are listed in Table 4.5.  available equation for predicting the local lateral temperature in the annulus region for industrial-scale risers. The thickness of the thermal boundary layer in commercial CFB boilers is typically 100 to 200 mm (Golriz 1995, Wirth 1995).  4.2.2.2. Temperature Distribution in Pipe and Fin The heat fluxes from the suspension to the pipe and fin are important factors in determining the membrane wall efficiency. To estimate these heat fluxes requires knowledge of the temperature distributions in the pipe and fin. The experimental membrane wall with thermocouples embedded in the pipe and fin is shown in Figure 2.10. The temperatures in the pipe and the fin as well as the crest andfintemperature differences at two bulk bed temperatures are plotted in Figures 4.18 and 4.19, respectively.  Ill  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  From these figures, it can be observed that the temperature at the center of the fin (T5) is higher than that near the crest of the pipe wall (Ti and T2). The closeness of Ti and T6 indicates that the heat flux through the pipe wall is essentially one-dimensional in the radial direction over the exposed portion of the pipe. The fact that T3 and T4 are so similar shows that the heat flux through the insulation is small. The temperature difference (Ti - T2) is typically 10 to 18 °C, while (T5 - Te) is typically 21 to 26 °C for a suspension density of 40 kg/m at bulk suspension 3  temperatures of 700 to 800 °C. Note that the temperature differences in the pipe and the fin increase with increasing suspension density, indicating that more heat is transferred from the suspension to the membrane walls and there is less thermal resistance on the suspension side. Temperatures along the periphery of a pipe and fin measured by Andersson and Leckner (1992)  350  0  15  25  20  30  35  40  45  Suspension Density, p  50  55  60  65  70  , (kg/m )  Figure 4.18. Temperature distribution in pipe andfinversus suspension density. T =804±5 °C, d=286 um, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m. b  p  g  112  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  40 35 of 0  s  1 3 2  a s  .S  30  r  1  ~i  -i  •  AT =T -T  2  O  AT =T -T  6  1  1  2  5  1  1  1  r-  ....-er-  o,.°"c7« o a- "o G  25 20 15 10  HH "9  10  20  30  40  50  Suspension Density, p  60 susp  70  80  90  (kg/m )  Figure 4.19. Pipe andfintemperature distributions measured by embedded thermocouples. dp=286 um, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m. Open symbols: T =804±5 °C; Solid symbols: T =706±4 °C. Dotted lines: best linearfitsfor T =804±5 °C; Solid lines: best linearfitsfor T =706±4 °C. g  b  b  b  b  showed that the temperature difference between the pipe crest and the junction of the pipe and the fin was about 10 °C, while the difference between the junction of the fin and the center of the fin was about 5 °C at similar thermocouple locations for a pipe of diameter 60.3 mm and fin of width 8.8 mm. However, the exact locations of the thermocouples were not identified. The differences between the (Ti-T^) and (T5-T6) results of Andersson and Leckner (1992) and those of this study are no doubt caused by the larger pipe (diameter 60.3 mm compared with 21.3 mm) and shorter fin (8.8 mm compared with 12.7 mm). The measured temperatures are used to calculate heat fluxes and heat transfer coefficients in the next subsection.  113  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  4.2.3. Suspension-to-Surface Heat Flow Rates and Heat Transfer Coeffi Several models have been used to calculate heat fluxes and temperature profiles in pipe/fin assemblies (Raymond 1984, Bowen et al. 1991, Taler 1992, Andersson and Leckner 1992, Jestin et al. 1992, Montat et al. 1993). For practical cases, the temperature distributions in the pipe obtained using these models indicated that radial one-dimensional heat flow gives a good approximation for the exposed portion of the pipe wall. The exposed-pipe-radial, insulatedpipe-tangential, fin-one-dimensional model (Model 2) proposed by Bowen et al. (1991) has been used here to estimate the suspension-to-pipe-and-fin average heat transfer coefficient. The temperatures measured by the embedded thermocouples are also used to estimated the local suspension-to-pipe, suspension-to-fin and suspension-to-pipe-and-fin heat transfer coefficients.  The diagram of the exposed-pipe-radial, insulated-pipe-tangential, fin-one-dimensional model and the positions of the embedded thermocouples are shown in Figure 4.20. In the exposed-pipe-radial, insulated-pipe^tangential, fin-one-dimensional model (Model 2) (Bowen et al. 1991), heat is removed by the cooling water through two sections in parallel: the pipe section and the fin section. The thermal resistance of the pipe section consists of suspension convection to the outside surface of the pipe, radial conduction through the pipe to the inside surface and convection to the cooling water. The thermal resistance in the fin section includes one dimensional radial conduction through the fin and conduction through the pipe wall to the center of the pipe. From there, heat flows to the cooling water by two parallel paths: by conduction through the remaining half thickness of the pipe and by convection to the water; and by tangential conduction through the insulated portion of the pipe and convection to the water. An electric circuit analogue is applied to obtain the following expression:  114  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1  -in(g) cok L  tpo  tpl  p  -A/VVvV  coBi^kpL  -A/vVvV  A V W V  -lnr2g/(l c)]  1  + C  •ln[(l + a)/2]  1  *XBi  kL out  tfo  p  V V V v V - t ^ W v Y V tav  r  1  o  0 + o) 2m(l-a)tanh(mco)k L  ^A/WV-  p  VWVv  Figure 4.20. Schematic of exposed-pipe-radial, insulated-pipe-tangential, fin-one-dimensional heat transfer model of Bowen et al. (1991) and electrical circuit analogue.  4  Q.=2 k L | T p  T b  w.out  +T • ^ w,m  where Q is the total heat transfer rate (= M t  CO  1  C + 2  c w  c  p>cw  115  (T  C + 3  XBi out Wj0ut  Bi  - T i )) and: Wj  n  (4-16)  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  l-BLln]  C = 2 2  1 C  \  1  2a  (l + a)<)>  "  1 +a  1+a  -—In x  C,=-  2(1 - a)m tanh(mco)  Bi,  c,=-  2  -In a;  Bi , ;  1/2  B i j l + a) m= 2(1-a)  x =2 P B i Bi , ;  Y  =  1T  k  0 U t  ;  °  p  ^ ^  ;  Bi  tanh  =  ;  V2p^.  k  ut  R '  p  0  •• = ?sin- (P); 1  co =  R  0  7r — d>  The dimensional symbols used above are defined in Figure 4.20. h; is estimated using Sleicher and Rouse's correlation (Wu 1989). k and kf are evaluated, respectively, at the average pipe and p  average fin temperatures measured by the embedded thermocouples. Within the temperature range of the pipe andfin,the impact of kf / k on h can be neglected. Since Eq. (4-16) is implicit p  t  in h , the average outside heat transfer for each experimentally determined value of Q was t  t  determined numerically using the Newton-Raphson method.  As shown in Figure 4.21, the heat transfer coefficients obtained using Eq. (4-16) varied between 110 and 230 W/m K for the conditions investigated here, increasing with both 2  increasing suspension density and increasing bulk suspension temperature. The heat flow rates through the pair of SS316 pipes are estimated to be 5300 to 9300 W. The average heat transfer coefficients obtained by the model based on the exposed pipe and fin area as well as heat flow  116  Chavter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  rates obtained by measuring the water inlet and outlet temperatures will now be compared with the local values estimated using the temperatures measured by the embedded thermocouples.  From Figure 4.20, the total suspension-to-exposed-surface heat transfer rate is equal to the sum of die heat flow rates through the pipe and the fin. The heat transfer rate for the pipe, Q , p  is given by T,-T ln^/rJ/^Lkp)  (4-17)  2  where ri and t2 are the radial locations of thermocouples Ti and T2, respectively, and L is the length of the pipe. From the definition of the heat transfer coefficient, the suspension-to-pipe heat transfer coefficient based on the exposed pipe area can then be determined from  tf  260  CN  s  240 220  •  T =804 ± 5 °C  •  T =706 ± 4 °C  b  b  200  a  180  1"  160 U  140 120  o H  100  20  10  30  40  Suspension Density, p  50  60  susp  (kg/m )  70  80  Figure 4.21. Effect of suspension density on suspension-to-membrane-wall heat transfer coefficient for different suspension temperatures. dp=286 um, U =8.0 to 8.5 m/s, P/S=2.0. g  117  Radiative and Total Heat Transfer in Circulating Fluidized Beds  h =  Chapter 4: Suspension-to-Surface Heat Transfer  OP 2coR L(T -T 0  where T  Pj0U  b  P)OUt  (4-18)  )  t is the temperature of the outside surface of the pipe, determined by the logarithmic  extrapolation of Ti and T2 to the surface at r=Ro.  With the assumptions that (1) conduction through the fin is essentially one-dimensional; (2) the fin is adiabatic at the plane equidistant between the two pipes; and (3) the non-exposed side of the pipe and fin assembly is well insulated, the suspension-to-fin heat transfer rate can be determined from Q = 1^/28,k h (T - T )tanh(mco) f  f  f  b  6  (4-19)  and the suspension-to-fin heat transfer coefficient from: h =28 k m f  f  (4-20)  2  f  where 8f is the half thickness of the fin and m can be determined from the measured temperatures, Tb, T5, and T6 as: m = — cosh co  1  T -T  5  T -T  6  b  b  (4-21)  The total uniform suspension-to-fin-and-pipe heat transfer rate and coefficient based on the exposed area of the pipe and fin can now be obtained from Qt = Q + Q f  (4-22)  P  Qn+Qf  \=  and  -^V^  7  (A  p +  A XT -T f  b  x  s u r f  )  ( - ) 4  2 3  where Q and Qf are the heat transfer rates to the pipe and fin, respectively, calculated from p  Eqs. (4-17) and (4-19). A (= 2coR<,L) and Af (= 2wL) are the exposed pipe area andfinarea, p  118  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  respectively. T is the bulk suspension temperature. T b  the pipe and fin surfaces, i.e. T  surf  surf  is the weighted average temperature of  = [A (T + T ) / 2 + A T ] / [A + A ]. f  5  6  p  p out  p  f  The suspension-to-pipe and suspension-to-fin heat transfer rates versus suspension density are shown in Figure 4.22, and the corresponding coefficients are plotted in Figure 4.23. As can be seen from Figure 4.22, the heat transfer rates to the pipe and to the fin differ significantly. Suspension-to-pipe transfer is enhanced by increasing either the suspension density or the suspension temperature, while both operating parameters have less impact on the suspension-to-fin heat transfer rate. The suspension-to-fin heat flow rate accounts for about 3244 % of the total heat transfer rate at 706 °C and about 25-32% of the total heat transfer at 804 °C.  Visual observations (Wu et al. 1991, Andersson and Leckner 1992) of particle flow  8000 Q  Qf  P  7000  T =706°C b  1000 10  20  30  40  Suspension Density,  50  6 0 7 0  80  (kg/m) 3  Figure 4.22. Suspension-to-pipe and suspension-to-fin heat transfer rates versus suspension density. dp=286 urn, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m. Q and Q are evaluated using (TpT^ and (T -TJ, respectively. The lines are the best linear fits. g  p  f  5  119  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  patterns near membrane walls have shown that falling solids tend to gather in the troughs along the fins between adjacent pipes. When the suspension density is high enough that the emulsion boundary layer covers both the pipe and fin, this layer is thinner at the pipe crest than along the fin. The particle concentration is also lower at the crest of the pipe than in the trough (Leckner 1990, Lockhart et al. 1995). Hence, increasing the suspension density has a greater impact on heat transfer to the pipe than to the fin. In addition, the particle temperature close to the fin is low, decreasing the heat transfer driving force and resulting in decreased radiative transfer to the fin.  Figure 4.23 shows that h,, hp and h increase with increasing suspension density and hp > h f  f  for 804 °C while hp < h for 706 °C. This indicates that the fin is not as efficient as the crest of the f  pipe in extracting heat from the combustor for conditions where radiation is more important.  i  .a  Suspension Density,  (kg/m) 3  Figure 4.23. Suspension-to-pipe, suspension-to-fin and total suspension -to-membrane-wall heat transfer coefficients versus suspension density. Solid symbols: T =804±5 °C; open symbols: T =706±4 °C. <L=286 um, P/S=2.0, U =8.0-8.5 m/s, z=5.44 m. b  b  g  120  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  Thus, the width of fins should be as short as possible in order to increase the efficiency of membrane walls for those processes where thermal radiation is significant.  The total heat transfer coefficients determined using Eq. (4-23) lie between hf and h . The p  total suspension-to-membrane-wall heat transfer rates determined from the measured inlet and outlet water temperatures, (Eq. 4-16) are compared with the local values estimated from the embedded thermocouple readings (Eq. 4-22) in Figure 4.24. It is found that the length average heat transfer rate is about up to 26% higher than the local heat transfer rate at 804 °C and 14-41% higher at 706 °C. The inconsistency of the two measurements is probably due to the uncertainty of the local temperature values since it is difficult to precisely determine the thermocouple positions.  12000  i  •  i  •  •  i  1  '  1  1  1  1  • T =«04 CT ^706 C o  o  b  11000 10000  -  b  •  •  •  O  estimated using water temperature difference estimated using embedded thermocouples  9000  •_  f—I  u •  8000  I  •  6000  3!  • •  g  •  ••  •  5000  •a o  •  ••  7000  8  3000  :  °  * • •  o  o  •  o  o  o ° ° °  _  ° o i  10  •  o  4000  E-i  •  •  1  20  30  .  1  40  .  1  50  Suspension Density, p  .  1  1  60  70  80  (kg/m ) 3  susp  Figure 4.24. Suspension-to-membrane-wall heat transfer rate versus suspension density. d=286 um, P/S=2.0, U=8.0-8.5 m/s, z=5.44 m. p  g  121  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  The temperature distribution along the insulated portion of the pipe wall was derived by Bowen et al. (1991) as  T(e) = T  w  -(T.-T„,)i5!^§I  ,4.24)  cosh(mco) where T is the local cooling water temperature (approximated as ( T w  win  +T  wout  )/2), and T  t e v  is  temperature at the center of the pipe wall adjacent to the fin. T ^ can be calculated from T , the 6  temperature measured at the pipe-fin junction, and from Q , the fin heat transfer rate obtained f  from Eq. (4-19), using T. =T -Q V  6  y <J>kL + a  f  (4-25)  ;  p  where L is the pipe length and a, cj> and k have all been defined previously. The temperatures at p  0 = 0, estimated using Eq. (4-24), are compared with the measured temperatures T = (T3+T4V2 (see m  Figure 4-20) in Figure 4.25. The measured and calculated temperatures agree well with each other.  The ratio of the heat transfer rate through the insulated portion of the pipe (0 < 0 < co) to the heat transfer rate through the fin can be estimated from  1 §Bi  m  1 1 +ln[(l+a)/2a] (1+oc) <j) 2m(l-a)tanh(mco) (1+cc) 2m(1 - a) tanh(mco)  Under the conditions of this study,  (4-26)  accounts for about 64% of heatflowfrom the fin which the  remaining 36% is dissipated by the section of pipe immediately adjacent to the fin (co <0 < CO+(j>).  122  Chapter 4: Suspension-to-Surface Heat Transfer  Radiative and Total Heat Transfer in Circulating Fluidized Beds  9  75  H  70  -  65  -  1  1  • #  '  1  •  1  '  •  T =804 ± 5 ° C  o  T =706 ± 4 ° C  1  •  l  i  i  I  b  b  O  II  d?  a  1  •  "1 55 °  <U  50 -  • S  S 5  8  B  8  o • •  o •  • 10  •  •  n u  45  40  o  •  m  i  i  i  i  20  30  40  50  Suspension Density,  I . I . 60  70  80  (kg/m ) 3  Figure 4.25. Temperatures of the insulated pipe at 8 = 0 versus suspension density. Solid symbol: measured temperature; open symbol: temperature predicted by model 2 of Bowen et al. (1991).  4.3. Conclusions 1. The total suspension-to-wall heat transfer coefficients for tubes and membrane surfaces increase with suspension density and decrease with increasing mean particle size. The bulk suspension temperature has a major effect on the total heat transfer coefficient. An empirical correlation for total suspension-to-surface heat transfer is proposed for broad ranges of suspension density, particle size and suspension temperature. This expression has deviations generally within ±50% for the tube-results in this work and for the high-temperature results of previous workers.  123  Radiative and Total Heat Transfer in Circulatinz Fluidized Beds  Chapter 4: Suspension-to-Surface Heat Transfer  2. Radiative suspension-to-surface heat transfer coefficients decrease with increasing particle size. The effect of temperature on the radiative heat transfer coefficient is unclear given the restricted temperature range examined in this study (-800 - 900 °C) and the overlapping influence of other factors. For bulk suspension temperatures of 800 °C to 900 °C, radiative heat transfer comprises 25% to 45% of the total heat transfer from the circulating fluidized bed suspension to a pipe located near the wall for the operating conditions studied in this work. The suspension density had little influence on this ratio. 3. The thickness of the thermal boundary layer in the pilot CFB combustor is about 15 to 20 mm over the location where the heat transfer surfaces were placed, while it reaches 100 to 200 mm or 600 to 800 d in large boilers. p  4. Local suspension-to-pipe and suspension-to-fin heat transfer coefficients have been estimated from measured temperatures inside the membrane pipes and fins. The suspension-topipe heat transfer coefficient is higher than the suspension-to-fin heat transfer coefficient when radiative heat transfer is important, indicating that the fin is not as efficient as the exposed pipe surface in extracting heat from CFB combustors. About one third of the total heat transferred flows through the fin and approximately 64% of that is dissipated by the insulated portion of the pipe. Average suspension-to-membrane-wall heat transfer coefficients, obtained using inlet and outlet water temperature measurements, are up to 41% higher than local values, obtained using embedded thermocouple measurements. These results, while useful, did not yield accurate radiative heat transfer data. Therefore further work is needed, and this is addressed in the next chapter.  124  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Prohe Heat Transfer in CFBC  CHAPTER 5 SUSPENSION-TO-PROBE HEAT TRANSFER IN CFBC  All  experimental results discussed in this chapter were measured using the  multifunctional probe described in Section 2.2. The differential emissivity method and the window method demonstrated in Chapter 3 were employed to estimate suspension-to-probe radiative and convective heat transfer fluxes and coefficients. Unfortunately, the silver coated probe head was entirely oxidized during the tests in the CFBC so that no meaningful measurements were possible with this probe head.  The two methods are compared in this chapter in order to illustrate their relative differences and uncertainties. The effects of suspension density, suspension temperature and particle size on radiative and convective heat transfer coefficients are investigated. The ratios of radiative to total heat transfer in CFBCs operating at 800 to 900 °C are determined and compared with published results. Silica sands of mean diameters 334 and 286 um were used in this study. The size distributions and properties of these particles are listed in Tables 4.1 and 4.2, respectively.  125  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  5.1. Total Heat Flux and Heat Transfer Coefficient The local, time-averaged total suspension-to-probe heat transfer coefficients for the 334 um silica sand, estimated using the stainless steel 347 (SS347) probe, are plotted in Figure 5.1 as a function of the cross-sectional average suspension density for suspension temperatures of 407, 804, 851 and 905 °C. As shown in the figure, the total heat transfer coefficient increases with increasing suspension density and also with increasing suspension temperature. The suspensionto-probe total heat transfer coefficients are in the range 500 to 820 W/m K for 804 < T < 905 °C. 2  b  These coefficients are significantly higher than those obtained using the dual tube surface discussed in Section 4.1, where the total heat transfer coefficients were only about 150 to 250 W/m K under similar operating conditions. This difference occurs because the vertical length of 2  the probe is much less than that of the tubes.  1000  I  1  1  1  I  i  1  •  I  •  1  1  I  i  !  .  I  •  1  1  1  1  1  1  1  1 70  •  1 80  1  1  CM  H  200  1  0  10  20  30  40  1 • 50  Suspension Density, p  1 60  1  L  90  (kg/m ) 3  susp  Figure 5.1. Suspension-to-probe heat transfer coefficient plotted against suspension density at different suspension temperatures for oxidized SS347 probe. dp=334 um, U =7.8 to 9.6 m/s, P/S=1.3 to 4.0. Lines are least square linear fits with dashed line corresponding to T =851 °C. g  b  126  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  To clarify the influence of vertical surface length on the total heat transfer coefficient, Table 5.1 lists the operating conditions for several experimental investigations where different surface lengths were used. These experiments are classified into two groups according to operating temperature. In the first group, the operating temperature was in the range 800 to 900 °C, while in the second it was about 400 °C. The mean particle size and suspension density were approximately the same in all of these studies. Figure 5.2 plots total heat transfer coefficients obtained by Wu (1989) and by other authors. Our data are also included. For consistency with Wu's data (1989), the same procedure was used here to extract approximate heat transfer coefficients. Except for a few outlying points, the group 1 and group 2 total heat transfer coefficient data (in SI units) are reasonably well fitted by the respective expressions: h  t L  =166 + 469exp  L  800 °C<T <900 °C b  022  (5-1)  Table 5.1. Experiments showing the effect of probe length on CFB heat transfer at high temperature. Particles were sand in all cases. Authors  Riser Size (m)  Probe Radial Location r/R  Mean Particle Diameter (um)  Surface Length (m)  Suspension Temperature (°C)  Group 1: T =800 to 900 °C; d =227 to 334 um; p =50 to 60 kg/m . 3  b  Kobro and Brereton (1986) Wuetal. (1989a) Basu (1990) Andersson and Leckner (1992, 1994) This work  p  susp  0.20 dia.  1.00  250  0.100  850  0.152x 0.152 0.10 dia.  0.92, 1.00 -  227,241  850,880  1.00  296  1.220, 0.152-1.590 0.025  1.7x1.7  1.00  240,312  >1.000  850, 902  0.152x 0.152  0.92, 1.00  286, 334  0.152-1.626 0.753, 0.006  804, 851,905  885, 895  Group 2: Tb=400 °C; d =215 to 334 um; p =50 to 60 kg/m . 3  p  Feugieretal. (1987) 0.150 dia. Wuetal. (1989a) This work  0.152x 0.152 0.152x 0.152  susp  1.00  215  0.095  400  1.00  241,299  407  1.00  334  1.220, 0.152-1.590 0.006  127  407  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  700  CN  •  •  Group 1 Group 2  •  •  600  • • •  •tt 500  Kobro andBraeton(1986) Feugieretal.(1986) Wu(1989) Basu (1990) Andersson andLeckner(1992)  V  A  This work (Eq. 4-3) This work (Bowen's model 2) Eq.(5-1) Eq.(5-2)  +  400  o O  H  300  •B 200  k ±  -1±  33  • 0.0  0.5  •  •  1.0  #  1.5  Vertical Heat Transfer Surface Length, L (m) Figure 5.2. Suspension-to-probe total heat transfer coefficient as a function of vertical length of heat transfer surface. d =215 to 334 um, p =50 to 54 kg/m . 3  p  and  h  t L  susp  =103 + 340exp  L 0.23  Tb=400 °C  (5-2)  where L is the length of the heat exchange surface in m. As shown in Figure 5.2, heat transfer coefficients, calculated using the more precise method outlined in Section 4.2.3, are 7 to 17 % higher than those using Eq. (4-3).  Figure 5.1 shows that there is a significant increase (about 200-250 W/m K) in the total 2  heat transfer coefficients when the suspension temperature increases from 407 °C to 804 °C. This difference is primarily due to the radiation contribution, since the heat transfer enhancement by gas and solids convection related to the effect, for example, of increasing temperature on the thermal conductivity of air, is expected to be small. Subtraction of Eq. (5-2) from Eq. (5-1)  128  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Susoension-to-Probe Heat Transfer in CFBC  indicates that the radiative heat transfer coefficients may also depend on the length of the heat transfer surface. The radiative component and how it is influenced by heat transfer surface length are described and discussed in Section 5.2.  Heat transfer coefficients decrease quickly with increasing heat transfer vertical surface lengths for L < 0.5 m. Based on the results in Figure 5.2, heat transfer surfaces should be longer than about 0.7 m to establish accurate heat transfer coefficients for design purposes. From Eq. (5-1) the deviation of the heat transfer coefficient compared to that of an infinitely long heat transfer surface is approximately 10% for a 0.7 m long surface and 3% for a 1.0 m surface.  The influence of the heat transfer surface length on the heat transfer coefficient can be explained by the decrease in thermal driving force between the heated suspension and the cooling surface (Glicksman 1988, Wu et al. 1989a). At the top, the surface is exposed to the solids at (or near) the bulk suspension temperature. The temperature difference between the suspension and the cooling surface is then large. The temperature of the solids then drops as the particles travel downwards along the surface, ultimately approaching a fully developed temperature profile in emulsion layer. As a result, the heat transfer coefficient decreases because of the decreasing thermal driving force. Once the fully developed temperature profile is attained the surface length no longer influences the local heat transfer coefficient.  The total heat transfer coefficients for both the SS347 (oxidized) and SS316 (imoxidized) probes with dp = 286 pm and p  : sxisp  35-50 kg/m are shown in Figure 5.3 as a function of suspension 3  temperature. For the SS347 probe, the total heat transfer coefficient increasesfromabout 370 W/m^K  129  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  to 850 W/m K as the suspension temperature increases from 300 to 900 °C. Figure 5.3 also shows 2  that there is a significant difference between the results obtained using the SS316 and SS347 probes, presumably due to their different surface emissivities. This difference can be used to calculate the radiative heat transfer as discussed in the next section.  5.2. Radiative Heat Transfer The radiative heat transfer coefficient can be determined by either the differential emissivity method or the window method as discussed in Chapter 3. Figure 5.4 shows the total heat fluxes for the 334 um sand particles obtained from the measured axial temperature distributions of the SS316 and SS347 surfaces plotted against suspension density. The solid  300  400  500  600  700  800  900  1000  Suspension Temperature, T (°C) b  Figure 5.3. Heat transfer coefficients obtained using SS347 and SS316 probes as a function of the suspension temperatures. d=286 um, p =35-50 kg/m . The solid lines are the exponential-growth bestfitresults. 3  p  130  susp  Chavter 5: Susvension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  and open symbols represent the results obtained using the SS347 and the SS316 surfaces, respectively. The suspension-to-probe total heat fluxes are between 180 to 270 kW/m for 2  suspension temperatures between about 800 °C and 900 °C and for suspension densities in the range of 20 to 80 kg/m . The total heat flux increases with increasing temperature and suspension 3  density as expected. The difference in heat fluxes between the two stainless steel probes is a measure of the radiative component, assuming the SS347 and SS316 surface temperatures to be about the same. According to the analysis in Section 4.1.2, the radiative heat transfer coefficient determined by the SS347 probe can be calculated using an equation similar to Eq. (4-6): <lt,347 h  r,347  "8,347  ^t.316  (5-3)  = T b '  e  s,347  e  s,316  Here T is the surface temperature of the SS347 probe, while the other symbols are the same as w  in Section 4.1.2.1. The assumption that the surface temperatures are equal is supported by the  400  350  •  I.  300  •  T =905°C  O  T =851°C  A  T =804°C  B  B  B  h  250  200  g  1 5 0  73  "5  100 50  20  10  30  50  40  Suspension Density, p  60 susp  70  80  (kg/m )  Figure 5.4. Effect of suspension density on total heat transfer flux. d =334 um. Solid symbols: SS347; open symbols: SS316. p  131  90  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  experimental finding that the difference between these two surface temperatures, on average, is only about 20 °C (see Figure 5.5), much lower than the differences in temperature between the suspension and the probe surfaces.  The window method, in particular Eqs. (3-37) and (3-45), provides a second estimate of the suspension-to-probe radiative heat flux using the temperatures measured along the axis of the window probe. Calculations using Eq. (3-45) demonstrate that there was negligible temperature drop across the window for all conditions in this study. In fact, the window temperature approaches the suspension temperature since the convection component and the absorbed radiation contribute sufficient heat to raise the inner and outer window surface temperatures to very nearly Tb.  903  g  800  T  SS347 SS316 T=905 ±5°C • • • O T =851±6°C A A T=804 ±9°C T=407 ±4°C T V b  b  ture,  H  700  b  b  600  fc &<  500  <o  400  Sur  ca  300  o ll  200  B Ha o  PH  ee  100  10  20  8  30  40  50  Suspension Density, p  60  70  80  90  3x  s u s p  (kg/m )  Figure 5.5. Surface temperatures of the SS347 and SS316 probes at various suspension temperatures as a function of suspension density for particle size 334 um.  132  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  5.2.1. Effect of Suspension Density on Radiative Heat Transfer Figure 5.6 shows measured radiative heat fluxes for sand particles of average diameter 334 urn and 286 urn. The solid and open symbols represent the results from the window method and the differential emissivity method, respectively. Both methods show that the radiative heat flux from the suspension increases with increasing suspension density and with increasing suspension temperature. However, the results from the window method are consistently higher than thosefromthe differential emissivity method.  The radiative heat fluxes estimated by the window method range from about 55 to 110 kW/m , while the differential emissivity method gives q values between about 40 and 80 kW/m , in 2  2  r  both cases for suspension densities in the range 20 to 90 kg/m . The window transmissivity, T, used in 3  window method to estimate radiative heat flux is 0.65. The deviations of the radiative heat fluxes caused by the uncertainty in window transmissivity are within -5.6 to 6.8 % based on outer limiting values of T = 0.6 and 0.7 (see Appendix LT). The heat transfer fluxes obtained using the differential emissivity method are clearly somewhat lower than those using the window method. The former depends on the emissivities of both the SS347 and SS316 probes. The emissivity of the SS347 probe is more reliable since this surface was well oxidized at an elevated temperature prior to being exposed to the combustion conditions. The emissivity of the SS316 probe may have increased when it was inserted into the CFBCriserand exposed to the suspension under oxidizing conditions at elevated temperatures. From Eq. (5-3), the radiative heat transfer coefficient is inversely proportional to the difference in the emissivities between the SS347 and SS316 probes. The use of too low an emissivity for the S S316 probe would result in a too high emissivity difference causing the radiative heat transfer coefficients obtained assuming e3i6 = 0.3 to be underestimated. The SS316 probe was visually  133  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  140  1  1  120 "  1  1  i  1  .  i  1  I  1  I  1  •  •  T =905 ± 5 ° C  (a) •  •  O  T =851 ± 6 ° C  -  A  A  T =804±9°C  b  b  CN  I J  •  <D  •  •  80  •  60  * no  m  •  • •  ••  M_ Hi  m  o o  o  •  cfeoCfa  °rpA  «a g o * A  A  CD  .(5  •a  • •  ••  PH  c3  •  b  100  HAA ^ A  40  20  I  10  .  I  .  I  i  .  .  50  40  30  20  Suspension Density, p.  susp  140  i  i  60  70  .  i  80  (kg/m ) 3  (b)  o  T =913 ± 9 °C  O  T =850 ± 6 °C  A  T =794±ll°C  b  CN  120  b  b  100 PH «  80  S3 4)  (2  60  40  20  20  30  40  50  60  70  80  100  90  Suspension Density, p^p (kg/m ) Figure 5.6. Radiative heat flux as a function of suspension density. Solid Symbols: window method; Open symbols: differential emissivity method, (a): dp=334 um; (b): d =286 um. p  134  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  checked after each test and it was found to be somewhat blackened and in need of polishing for the next test. The corresponding radiative heat transfer coefficients using the differential emissivity method are in the range 110 to 220 W/m K, as shown in Figure 5.7. The radiative heat transfer coefficients obtained using the differential emissivity method with the 334 um and 286 um sand particles, for T =850 °C and p p=50 kg/m , are 170 W/m K and 150 W/m K, respectively. 3  b  2  2  SUS  Basu and Konuche (1988) estimated the radiative heat transfer flux using the window method assuming that the heat flux, q , determined by the sensor behind the window equals the m  true radiative heat transfer to the window, q , times the window's transmissivity, i.e., r  q =^ r  x  (5-4)  where x, the window transmissivity, is 0.65 for ZnSe (Touloukian and Dewitt 1972). The radiative heatfluxesestimated using Eqs. (5-4) and (3-37) are plotted against suspension density for a suspension temperature of 850 °C for both types of sand in Figure 5.8. As can be seen from the figure, the radiative heatfluxesestimated using Eq. (5-4) are significantly greater than those estimated using the more rigorous model developed in Section 3.4 which considers the influence of the window temperature. The difference in fluxes predicted by the two equations can be attributed to the heat emitted by the window and multiple reflections between the window inside surface and the probe head. The radiative heat transfer coefficients obtained using the more rigorous window method (i.e., Eq. 3-37) are shown in Figure 5.9 to demonstrate how they are influenced by suspension density, suspension temperature and particle size. The coefficient is in the range of 120 to 280 W/m K, increasing with increasing suspension density and increasing 2  particle size.  135  Chavter 5: Susvension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  tf  260  CN  B  1  240 t—  220  1  i  i  -  •  T =905 ° C  -  o  T =851°C  '  i  •  i  1  1  1  fl  b  •  b  A  1  1  OB  T =804 ° C b  200  M  •  o  180  s CD  -  160  VI  ^:  140 -  120  CD  —  -  A  **%  o  i  Q  l  >-^SA  •  ^  ..-Of'  •35 cu  •  o"  (a)  100 80  i 10  I  20  30  .  I  40  .  I  I  .  I  70  60  50  J  .  90  80  (kg/m ) Suspension Density, psusp . 3  tf  220  CN  s  " 200  •a  .8  180  1  160  O  V=850  A  T  -  H  1  1  1  1  1  B  b  =  B  =794 ° C  ...-••A'  o o  /  -  o ...  •  /  -  "o  -  "6  120  (b)  o  •cS  t3  T3  1  1  °C  B  CD  i  b  140  CD  •  T ==913 ° C  •  °  .  i  i  B  o  CD O  1  1  i  100 20  30  I  .  50  40  I  1  .  70  60  Suspension Density, p  susp  .  1  i  80  90  "  100  (kg/m )  Figure 5.7. Effect of suspension density on suspension-to-surface radiative heat transfer coefficient for different suspension temperatures determined using differential emissivity method, (a): d =334 um, U =8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d =286 um, U =7.8 to 8.9 m/s, P/S=1.5-2.3. The lines show the best linear fits. p  g  p  136  g  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  180  —I  160  _S  140  1  1  '  1  '  1  '  1  '  I  '  I  '  •  estimated using window model (Eq. 3-37)  •  estimated using Eq. (5-4) at T =851 °C  1  r  (a)  b  I  120  PH  100  « u  X 4>  80  .(2  60  40 '  >10  20  30  40  50  Suspension Density, p  180 i  160  _£  1  1  -  1  1  1  1  1—  70  60  80  90  (kg/m )  susp  1  1  1  1  •  estimated using window model (Eq. 3-37)  •  estimated using Eq. (5-4) at T =850 °C  1  1—  (b)  b  140  jr  1201  PH  100  ta « .6  80  60  40 I  i  I  20  30  i  I  i  40  50  i  Suspension Density, p  i  I 60  i  1 70  I  i  80  I 90  (kg/m ) 3  susp  Figure 5.8. Comparison of radiative heat transfer fluxes estimated from the window method using Eqs. (3-37) and (5-4) for different suspension densities, (a): dp=334 um, T =851 °C, U =8.4 to 8.8 m/s, P/S=2.4 - 4.0; (b): d =286 um, T =850 °C, U =7.8 to 8.9 m/s, P/S=1.5-2.3. Solid lines are best linear fits. b  g  p  137  b  g  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  320  r-i  1— —I— —I—'—r 1  300 -  •  T =905 °C  O  T =851 °C  A  T =804 °C  280 260  J3  1  b  b  b  240  "o  220  S  200  <4H  a  180 160 140  (a)  120  •a  100  (2  20  10  30  40  50  Suspension Density, p  280  s C  260 -  •  T =913 °C  240 -  O  T =850 °C  A  T =794 °C  220 -  a .a "o  1  '  60 susp  70  90  100  (kg/m )  1  r  80  '  1  «~  b  b  b  200 180 -  S O  cj  160 140  c3 4)  35 <u « PH  120  (b) J  100 80 I— 20  40  30  50  60  Suspension Density, p  70 susp  80  90  100  (kg/m )  Figure 5.9. Effect of suspension density on suspension-to-probe radiative heat transfer coefficient for different suspension temperatures determined using the window method (Eq. 3-37). (a): d =334 um, U =8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d =286 um, U =7.8 to 8.9 m/s, P/S=l.5-2.3. Solid lines are best linear fits. p  g  p  138  g  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Susvension-to-Probe Heat Transfer in CFBC  As demonstrated by Baskakov (1964), radiative heat transfer coefficients increase with increasing surface temperature. According to the definition of the radiative heat transfer coefficient,  K = ^r^r*b  ^surf  00  ^FT?^ *b  K (T  »  2 + T  ^ " ^ )(T  +  }  (5  "  5)  *surf  It is clear that h is not only a function of the suspension temperature, but also of the surface r  temperature. Figure 5.10 shows the influence of the heat transfer surface temperature on the radiative heat transfer coefficient. The solid symbols indicate experimental results obtained under the operating conditions of this study, while the dotted lines illustrate the results calculated using Eq. (5-5) with various surface temperatures. In general, radiative heat transfer coefficients published in the literature for CFB risers range from about 50 to 100 W/m K at 850 °C (Basu 2  and Konuche 1988, Andersson and Leckner 1992, Werdermann and Werther 1994, Steward et al. 1995). Experimental findings at 850 °C from these studies, as well as theoretical estimates of h  r  (Glatzer and Linzer 1993), are also represented in Figure 5.10 by open symbols. It is seen from Eq. (5-5) and Figure 5.10 that radiative heat transfer coefficients increase with increasing heat transfer surface temperature. Below 600 K the results of the present study are seen to be consistent with the earlier published values.  The literature results shown in Figure 5.10 were obtained with long heat transfer surfaces and/or with low heat transfer surface temperatures. The influence of heat transfer surface length on the radiative heat transfer coefficient was indicated by Eqs. (5-1) and (5-2). These two equations indicate that the radiative heat transfer coefficient may be a function of the heat transfer surface length. If the heat transfer surface length is long enough, the radiative heat  139  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  transfer coefficient can be estimated roughly by subtracting Eq. (5-2) from Eq. (5-1). For a suspension density of about 50 kg/m and a suspension temperature of 800 to 900 °C, the subtraction 3  gives hr equal to 63 W/m K. This is reasonably consistent with the radiative heat fluxes 2  obtained in this study, but only at relatively low surface temperatures.  5.2.2. Effect ofSuspension Temperature on Radiative Heat Transfer The influence of the suspension temperature on the total heat flux is demonstrated in Figure 5.11. The total heat fluxes measured by the SS347 and SS316 probes are significantly different when the suspension temperature exceeds about 600 °C. The estimated radiative heat  CN  800 700 600  •  Steward et al. (1995): T=850 °C, d=200 nm  O  Qatar & IinzHr (1995): T=850 °C, d=209 um  • susp '74kgm.3  A  Werdermann &Werther (1994): T=850 °Q d=209 um  -50 kg/m  V  Basu & Kounche (1988): T=885 °C, d=396 um  -38kgm lj  b  p  b  -  p  b  3  p  b  H  3  p  -28kgm  3  M o s c5 u CO  500  -21 kg/m -J 3  400 300  c5  200 100  O A  ^7%  •  •  •3 200  300  400  500  600  700  800  900  1000  1100  1200  Heat Transfer Surface Temperature, T ^ (K) Figure 5.10. Effect of cooled surface temperature on radiative heat transfer coefficient. Solid symbols are from this work for particles of mean diameter 334um for the five suspension densities given at the right and at a suspension temperature of 850± 6 °C. Lines are the calculated results passing through the points measured in this work using Eq. (5-5) and assuming that the probe surface temperatures vary from 300 to 1000 K.  140  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Prohe Heat Transfer in CFBC  transfer coefficients for sand and short heat transfer surfaces are plotted in Figure 5.12 while radiative heat fluxes obtained from the differential emissivity method are compared with those from a black body emitter having the same temperature as the suspension in Figure 5.13. The latter figure shows that the radiative heat flux is strongly affected by the suspension temperature, 2  2  increasing nonlinearly from approximately 1.6 kW/m to 65 kW/m as the suspension temperature is increased from 230 °C to 900 °C. As expected, the experimentally measured radiative heat flux is always lower than the black body limit.  5.2.3. Effect of Surface Length on Radiative Heat Transfer The radiative heat transfer coefficients obtained from the dual-tube surface (h  riiube  ) and the  probe surface Aerobe) using the differential emissivity method and the window method are plotted  300  i  • A  250  tNg  200|  •  1501  «  '  "I  1  '  1  ~i  '  r  probe SS 347 probe SS 316  100  50  200  300  400  500  600  700  800  900  1000  Suspension Temperature, T (°Q b  Figure 5.11. Total heat transfer fluxes estimated using SS347 and SS316 probes at different suspension temperatures. d =286 um, U =7.8 to 8.9 m/s, P/S=1.5-2.3. The solid lines are the best fit results for a second order polynomial. p  g  141  Chavter 5: Susvension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  tf  200  I  180  • •  160 140  M  120  •  V  "  A  - •  100 1  a  l  80  -  60  -  O  1  '  '  l  '  '  l  l  1  Han etaL (1992) Basu and Konuche (1988)  B  •  i  .  t  1  •  • o A V  u  A  •  20  •  This work: differential emissivity method This work: window method Steward et al. (1995) Werdermann and Werther (1994) •  40  X  l  -  !  0 i  200  i  300  .  t  400  .  i  .  500  i  600  ,  i  .  700  Suspension Temperature, T  b  i  i  800  900  1000  (°C)  Figure 5.12. Comparison of radiative heat transfer coefficients with literature results for different suspension temperatures. • , • : d =286 um •: d =264 um • : d =296 nm. p  p  p  120 h  Radiative fluxes estimated by Eq. (3-33) Radiative fluxes from a black body  100 80  c  .2  « .83  60 40  <u 20  200  300  400  500  600  700  800  900  1000  Suspension Temperature, T (°C) b  Figure 5.13. Comparison of radiative heat fluxes estimated using Eq. (3-33) (differential emissivity method) with that for black body at the same suspension temperature. dp=286 um. Solid line is obtained by regression using a second order polynomial.  142  Radiative and Total Heat Transfer in Circulating Fluidized Beds  in Figure 5.14. The figure shows that h p ri  ro  be  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  is greater than  h,tube for r  a suspension temperature of  850 °C with 334 um particles in the suspension density range 20 to 75 kg/m . The radiative heat 3  transfer coefficient not only increases with increasing suspension density, but also depends on the length of the heat transfer surface. The effect of suspension density on h was discussed in r  Section 5.2.1. The effect of heat transfer surface length on h is attributed to the r  suspension temperature close to the surface. The suspension temperature in the vicinity of the surface remains nearly as high as the suspension temperature in the core region for the probe surface, while a temperature profile is developed from the core region toward the surface for the much longer tube surface. Hence, a higher h is obtained for the short probe r  surface than for the tube surface even when the differential emissivity method is used in both cases.  400  a  • • •  350 h 300 h  10  probe surface (window method) probe surface (differential emissivity method) tube surface (differential emissivity method)  20  30  40  50  Suspension Density, p  60  70  80  90  (kg/m ) 3  Figure 5.14. Comparison of radiative heat transfer coefficient obtained with the probe using differential emissivity method and window method and the tube surfaces (see Chapter 4). dp=334 um. T=851 °C. Lines are least square linear fits. b  143  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  5.2.4. Ratio of Radiation to Total Heat Transfer The ratio of the radiative heat transfer flux to the total heat flux or the radiative heat transfer coefficient to the total heat transfer coefficient indicates the relative importance of the radiative component. The ratio q/q^ with the heat transferred from the suspension to the probe by radiation evaluated using both the differential emissivity method and the window method, is plotted as a function of suspension density in Figure 5.15. Changes in the suspension density and suspension temperature appear to have little impact on the relative importance of radiative component for the range of conditions covered in this study.  5.2.5. Emissivity ofSS316 Probe It was previously noted that the differential emissivity and window methods yield significantly different results for the radiative heat transfer component of the total heat transfer between the suspension and the probe, probably because the differential emissivity method uses too low a surface emissivity for the SS316 surface. If the window method is assumed to be correct, then radiative fluxes determined using the window method can be used to calculate the actual surface emissivity of the SS316 probe. The results, shown in Figure 5.16, indicate that the emissivity of the SS316 probe probably changed from the assumed value of 0.3 to about 0.5.  5.3. Convective Heat Transfer According to the common assumption that total heat flux is transferred independently by radiation and convection from the suspension to the heat transfer surfaces, suspension-to-surface  144  Chapter 5: Susvension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  0.60  r  0.55 h 0.50  (a) 1  905 °C 851 °C 804 °C • • A Estimated using window method • O A Estimated using differential-emissivity method  0.45 0.40  •v  0.35  A  0.30  °Q 0.25  8  OO 0.20 0.15 0.10  30  20  10  50  40  Suspension Density,  0.50  T  913 °C 850 °C 794 °C • • A • O A  0.40 H  0.35  2  0.30  Eo  0.25  .•s  0.20  r  i  (kg/m )  s u s p  i  80  1  1  1  Estimated using wndowmethod Estimated using differential-emissivity method  o  O  S  A *&0&?£  A  13  O  '  70  (b)  0.45  <4_  "i  p  60  A  •  ° •  0.15 0.10  j 20  i_  i  40  30  70  60  50  Suspension Density, p  susp  80  90  (kg/m )  Figure 5.15. Relative importance of probe radiative heat transfer estimated using the window method and differential emissivity method as a function of suspension density, (a): d =334 um, (b): d =286 um. p  p  145  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  0.8  1  1  & 0.7  1 £  VO  r  T  I  1  - i — i — i  •  T =905 °C  o  T =851 °C  •  T =804°C  i  i  i  i  ,  b  -  b  b  A  A  0.6  i—<  CO  % «  o  0.5  o °  0.4  0.3  1  10  1 20  i  i  30  i  •  i  1  1  40  50  60  Suspension Density,  .  1  70  .  1  80  90  (kg/m ) 3  Figure 5.16. Apparent surface emissivity of SS316 probe estimated using radiative heat fluxes determined from the window method (d =334 um). Runs were performed in order of increasing suspension temperature (Tb). p  convective heat transfer coefficients may be estimated by subtracting the radiative heat transfer coefficient obtained using the window method from thetotalheat transfer coefficient Figure 5.17 shows that convective heat transfer coefficients obtained in this manner are in the range of 400 to 600 W/m  K, and increase with increasing suspension density and suspension temperature. These  convective heat transfer coefficients are as much as twice those obtained using long heat transfer surfaces at high temperatures (e.g., Kobro and Brereton 1986, Wu et al. 1989a). The strong effect of heat transfer surface length has been discussed in Section 5.1. The convective heat transfer coefficients obtained in this work are in good agreement with other published results obtained with short surfaces at high temperature (Wu et al. 1989a, Nag and Ali 1992, Couturier et al. 1993). The influence of particle size on the convective heat transfer coefficient appears to be insignificant for  146  Chapter 5: Suspension-to-Probe Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  600 CM  T =905 °C  (a)  b  550  T =851 °C  ft*  b  S5  I  T =804°C b  500  o. '•95™  A  "oo  >  2  A ^ o A 450  CP  A© f ^ A  A A A B  400  350  I  I 8  CM  300  J 20  10  50  40  30  60  Suspension Density,  700  1  -1  650 600  1  1  1—  •  T =S13 °C  o  T =850 °C  A  T =794°C  T—>  1  •  70  80  90  100  (kg/m ) 3  r (b)  b  b  b  550  O 500  O  A  450  ^  H  a  o A  400  °  o  C ,5> A  A  A  350  8 > c 6  300  10  20  30  40  60  50  Suspension Density,  80  70  90  100  (kg/m ) 3  0 s t B p  Figure 5.17. Effect of suspension density on suspension-to-probe convective heat transfer coefficient, estimated by subtracting radiative component from total coefficient, for different suspension temperatures, (a): d =334 um, U =8.4 to 8.8 m/s, P/S=2.4-4.0; (b): d =286 pm, U =7.8 to 8.9 m/s, P/S=l .5-2.3. p  g  g  147  p  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Prabe Heat Transfer in CFBC  the restricted range investigated. The packet model of Mickley and Fairbanks (1955) may be used to explain the effect of the suspension density and some influence of suspension temperature on the convective heat transfer coefficient. From Eq. (6-34) in the next chapter, the convective heat transfer coefficient is proportional to the suspension concentration, the suspension conductivity and its specific heat. The suspension temperature affects the convective heat transfer coefficient because increasing temperature increases the suspension conductivity and specific heat.  5.4. Conclusions The following conclusions can be drawn from the results obtained in this chapter: 1. Heat transfer surface length has a significant influence on heat flux and on the corresponding heat transfer coefficients, both radiative and convective. 2. Radiative heat transfer coefficients and convective heat transfer coefficients increase with increasing suspension density and increasing suspension temperature. The effect of mean particle size on the radiative heat transfer coefficient appears to be not very significant for the very limited range (286 to 334 pm) studied. 3. Radiative heat transfer coefficients increase with increasing heat transfer surface temperature. 4. The radiative component is only about 6% of the total heat transfer coefficient when the suspension temperature reaches 400 °C. The ratio of the radiative heat transfer coefficient to the total heat transfer coefficient is about 22% to 38% determined using the window method and about 16% to 28% using the differential emissivity method for the 286 and 334 pm sand particles at temperatures in the range of 800 to 900 °C.  148  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 5: Suspension-to-Prohe Heat Transfer in CFBC  5. The results obtained by the window method differ significantly from those obtained using the differential emissivity method. This difference is probably caused by an increase in the emissivity of the SS316 probe surface due to oxidation. The window method is believed to give more accurate results. The emissivity of the SS316 probe surface is estimated fb have increased from 0.3 to about 0.5 during operation. 6. With a short surface, radiation contributes approximately 150 to 250 W/m K to the overall heat transfer coefficient for a bulk suspension temperature of 850 °C under the CFBC conditions explored.  149  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  CHAPTER 6 MODELING OF CFBC HEAT TRANSFER  The comprehensive design of a circulating fluidized bed combustor requires knowledge of hydrodynamics as well as heat and mass transfer. The scale-up of CFBs to large commercial applications is more reliable when based on mechanistic models than when based on experimental correlations. For example, heat transfer from the suspension to the wall depends on the particle size and concentration at the wall as well as the mode of contact with the wall. Even when the cross-sectional average particle concentrations in CFB units of different sizes are the same, the particle concentration profiles may differ significantly, resulting in different heat transfer coefficients. Therefore, there is a need for fundamental mechanism-based models of CFB heat transfer. In this chapter, previous models which account for radiation in addition to convection and conduction are briefly reviewed and a new non-uniform emulsion model is developed to predict radiative and total heat transfer coefficients. The predicted results are compared with those obtained using the probe (presented in Chapter 5) and with previous experimental data of other groups. The comparisons show reasonable agreement between model predictions and experimental results.  150  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  6.1. Previous Radiative Heat Transfer Models 6.1.1. Packet Model of Basu and Nag In "packet" heat transfer models heat transfer is represented as a transient process from "packets", i.e. groups of particles, to the wall, with the resulting heat flux related to the residence time of the packet at the surface. The radiative component may be taken into account in the packet model as the sum of the radiation from clusters andfromthe dispersed phase. Basu and Nag (1987), and Basu (1990) use the expression: h =fh r  + (l-f)h  r c l  (6-1)  rjd  where f is thefractionof the surface occupied by clusters, o-(T -T:) 4  b  -L+J--1 ( T L ci e  with  e  w  (6-2)  b  -T ) w  .  e =05(l'+e ) d  (6-3)  p  O-CT^-T:)  and  h  r,d  (6-4)  =  e,  (T -T ) b  e..,  w  The subscript cl stands for the cluster phase, while d denotes the dilute (i.e. non-cluster) phase.  Brewster (1986) showed that the dilute phase emissivity could be estimated as -10.5  •+2 (l-e )B [(l-e )B p  (6-5)  (l-e )B  p  D  for an isothermal boundary layer and: e (l-ep)B P  4! ^ n!(4-n)!  (Ti-l) £+ n  151  n  ^  ( 1 +  2B(l-e e  P  p ) )  I  + n  (6-6)  Radiative and Total Heat Transfer in Circulating Fluidized Beds  with  Chapter 6: Modeling of Heat Transfer in CFBC  r\ = ^ -  (6-7)  lb  3(1  and  ~  6j ) § e  re (2(l-e )B + e ? P  P  (6-8)  P  for a non-isothermal boundary layer. Here B is the back-scatter fraction, T is the temperature of w  the particles at the wall, and 8 is the non-isothermal layer thickness. e  6.1.2. Uniform Emulsion Model of Chen et al. A steady state energy balance for a control volume was employed in the model of Chen et al. (1988) with radiative and convective transport considered to occur simultaneously throughout the suspension. The radiative and convective components are considered together. The model can account for temperature variations in both the lateral and axial directions. It is based on the assumption that particles have a uniform lateral distribution throughout the entire riser. The twodimensional energy balance is written as  l ( - f ) - [ ( - a p ^ > ^ ^ " f - | ( r - r > - G ' (6-9) k  1  8  where x is the horizontal coordinate from the wall, y is the vertical coordinate from the bottom of the riser, G' is the heat generation rate per unit volume, oo  n/2  J = J 2K Jl^cosOsinOdOdX +  (6-10)  x=o e=o oo  and  3it/2  J " = J 2TT Jl cos0sin0d0dX, x  )i=o  (6-11)  e=jt  X" and J are the radiative heat fluxes in the positive and negative x directions, respectively. The 1  152  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  radiative heat transfer equations for J and J can be written as +  ^ - = - ( C + C ) j + C J- + C oT dx +  a  dJ dx  s  s  = (c + c ) r - c r - c o T a  s  (6-12a)  4  a  s  4  a  '  (6-i2b)  where C and C are the absorption cross-section and scattering cross-section, respectively. a  s  The boundary conditions are written: T(x = 0,y) = T  w  f^(x = 5 ,y) = 0 dx e  T(x,y = 0) = T  (6-13)  I  •J (x = O y) = e oT^+(l-e )J-(x = 0,y) +  t  w  w  J-(x = 5 ,y)=J (x = 6 ,y) +  e  e  where T; is the inlet temperature. The transverse radiative heat flux at any position is then: q = (J r  +  -r)  (6-14)  while radiative heat transfer coefficient at the wall is defined as: q j x = 0y) T„-T. Three transport parameters, k , C and C , are required to solve the partial differential e  a  s  equations. Other parameters, such as u , u , T and T , are directly influenced by the two-phase p  g  w  {  hydrodynamics and reaction kinetics, and are therefore affected by particle size and by the physical properties of the gas and particles. The absorption and scattering of radiative energy occur by photon interactions with the suspension particles. Tien and Drolen (1987) suggested that, for dilute suspensions of particles with diameters much greater than the radiation wavelength, independent scattering dominates. Tien and Drolen (1987) predicted the volumetric  153  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  absorption and scattering cross-sections to be: (6-16a) 3(l-s)(l-e ) p  and  (6-16b)  respectively, where C is a constant which needs to be determinedfromexperimental results. The e  effective conductivity of the particle emulsion, ke, is also difficult to evaluate.  6.1.3. Grid Model of Werdermann and Werther In CFBs, a thermal boundary layer exists near the wall. In the grid model of Werdermann and Werther (1994), the suspension is regarded as consisting of spherical particles of uniform size arranged in a regular cubic array. There are (i ax+l) particle layers, considered as surfaces m  exchanging radiation. The radiative heat transfer coefficient between layers i and j is calculated from:  qtf+TfXTj+Tj)  with  (6-17)  (6-18)  where  and  quitted,;  = e  i  c y T  i  4 +  ( - i)qr, 1  e  The radiation exchange coefficient F between layers of particles is determined from:  71A j  154  (6-19)  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Here i = 0 refers to the wall, i = i  m a x  Chapter 6: Modeling of Heat Transfer in CFBC  +1 refers to the hot gas, and s is the distance between  successive particle layers. The net heat flux to the wall is calculated by simultaneously solving the above equations for the radiative heat transfer for all layers. When the model is applied, the temperature profile of the particle layers is required. Particles are assumed to be arranged in a regular array, with an arbitrary number of layers.  6.1.4. Two-Flux Model of Glatzer and Linzer Glatzer and Linzer (1995) developed a two-flux model which considers the temperature distribution in a thermal boundary layer with a uniform particle concentration near heat transfer surfaces. The solids-gas mixture is assumed to constitute an absorbing, emitting and anisotropically scattering medium. The radiative transfer equations  ~T~ = - fc. + K.) r + K.r + K o T 2 dx  (6-20a)  4  a  and  = " ( a + K  K  s )  r  +  K  s  I  +  +  K  a°T  (6-20b)  4  2 dx are solved giving:  V (x) =  +K s  K e ^  +B  x  K  t  2  " ^fe^Le*^' + I (T) b  K +K +VKf-K t  s  (6-21)  2 s  where t'ix) is the intensity in the positive x direction; x is the lateral coordinate starting from the centerline of the riser; K = K + K , with K and K being the scattering and absorption t  s  a  s  a  coefficients, respectively; Bj and B are coefficients determined by the boundary conditions: 2  l+(x = 0) = i;(x = 0) l  + m  (x = X) = (1 - a  w  155  (6-22a)  (x = X) + e I (T ) w  b  w  (6-22b)  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  lL(x = X - 6 - 6 ) = i;(x = X - 6 - 6 )  (6-22c)  IL. (x = X - 6, - 5 ) = I" (x = X - 8, - 5 )  (6-22d)  g  e  g  e  e  e  Here X is the riser half-width; a,^ and e are the wall absorptivity and emissivity, respectively; 5 w  g  and 5 are the gas layer and wall layer thicknesses, respectively. The subscripts "c" and "ann" e  denote the core and annulus regions, respectively. The radiative heat flux is then predicted by q = ^ [ l l ( x = X ) - I ^ ( x = X)]  (6-23)  r  6.1.5. Network Model of Fang et al. Fang et al. (1995) considered that an exposed surface in a CFB furnace of 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, are represented using an electrical network analogy. Convection and radiation are not simply taken as additive. The final expression for the radiative heat transfer coefficient is e [e fe (l-e )]a(T -T:) 4  h  =  '  w  d+  CI  d  b  {e + (l-e )[fe + ( l - f ) e ] | ( T - T ) d  d  cl  w  b  w  where f is fraction of the wall covered by clusters; e and e are the emissivities of the dilute d  cl  suspension and clusters, respectively, with e being evaluated by d  L5(l-e )e  f  d  e = 1 - exp d  H  V  d  P  ^  p L  b  J  1^ (=0.95 D) is the mean beam length and e is the particle emissivity. p  156  (6-25)  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  6.1.6. Summary of Previous Radiative Heat Transfer Models The major assumptions and parameters in these previous radiative heat transfer models are listed in Table 6.1. Each of these models is claimed by the respective authors to give good agreement with experimental results. However, none of the models takes into account the lateral variation of solids concentration in streams or clusters. Most of the models assume that particle suspension density is uniform throughout the annulus region or even across the entire riser crosssection. A number of researchers (e.g., Zhang et al. 1994, Zhou et al. 1995) using various advanced experimental techniques have determined the annulus-core structure and lateral particle concentration distribution in CFB risers. There is an opportunity to develop a model accounting for radiative heat transfer which considers these recent experimental findings. A model is proposed below which considers both the temperature distribution and the solids concentration distribution in a fully developed hydrodynamic and thermal boundary layer close to a heat transfer surface. No parameter fitting is needed in the model, which is developed in the next section, beyond the fitting relationships, already determined by others, that must be inserted in the model.  6.2. Proposed Non-Uniform Emulsion Model Since more and more experimental studies have been carried out both on radiative heat transfer and on hydrodynamics, it should be possible to devise better models to represent the complicated phenomena involved in heat transfer, including radiative heat transfer, in CFB risers. The following approach extends previous models in the light of current understanding.  157  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  Table 6.1. Major assumptions and parameters in previous CFB radiative heat transfer models. Model Packet model  Uniform emulsion model  Grid model Two-flux model  Network model  Parameters  Authors Basu and Nag (1987), Basu (1990)  Assumptions  f> ci> d» T , • T e e • • • • •  spherical particles; radiation and conduction additive cluster is semi-infinite; T i =T =T ; isotropic scattering; uniform T and e in the cluster and dispersion. Chen et al. K g , K , kg, 6, • independent scattering; (1988) G\ U , U , e , • solids flow upwards; • axial radiation negligible; Tb. T • uniform solids concentration throughout the riser. Werdermann and e;, T , Fi.j, e(r) • spherical particles of uniform size; Werther (1994) • radial particle concentration distribution. Glatzer and K s , K , kg, T • ,spherical particles Linter (1995) T , ep, e , 8 , • independent scattering; • absorbing, emitting and anisotropically scattering medium; • one-dimensional radiation; • gray heat transfer surface; • transparent gas; • average temperature in the core and annulus regions; • uniform solids concentration; • temperature distribution in the emulsion layer. Fang et al. f» Tj,, T , e j, • spherical particles; (1995) • semi-infinite clusters; d> p> w •T =T . e  c  b  c  d  c l  b  5  a  p  g  w  w  ;  b  a  w  w  w  e  e  e  c  e  6.2.1. Framework The  core-annulus structure in CFB risers has been confirmed using various  measurement techniques (e.g., see Rhodes et al. 1988, Zhou et al. 1995). The cross-section of a riser can be divided into two regions, with particles in the core region transported upwards,  158  Chavter 6: Modeling of Heat Transfer in CFBC.  Radiative and Total Heat Transfer in Circulating Fluidized Beds  while solids in the annulus region travel down along the wall. At high solids suspension densities, the thickness of the solids layer along vertical heat transfer surfaces can be approximated as uniform. Senior (1992) showed that heat transfer surfaces were completely covered by particle layers when the emulsion layer thickness reached 8 mm. For industrial-scale boilers or reactors, operated under high suspension density conditions, heat transfer surfaces are likely to be fully covered by particle emulsions (Leckner and Andersson 1992). In such a layer, a temperature profile develops laterally from the water-cooled heat transfer surfaces. A particle concentration profile in the layer has also been identified (Zhang and Tung 1991).  The proposed model is shown schematically in Figure 6.1. As indicated in the figure, a heat transfer surface of constant temperature T is treated as a diffuse surface with emissivity e . w  w  A stationary thin layer of gas is assumed to be adjacent to the surface. The thickness 5 of this g  layer depends on the hydrodynamic conditions of the circulating fluidized bed and is estimated from an empirical correlation proposed by Lints and Glicksman (1994): 8  •0.581  (6-26)  The temperature in the gas gap is considered to vary linearly with distance x. Since gas and particles are in intimate contact and particles used in CFB processes have low Biot numbers, temperature gradients within the particles are neglected. The gas and particles are assumed to be locally at the same temperature. The emulsion formed near the surface is treated as a plane layer of finite thickness 8 , which can be estimated using the correlation of Bi et al. (1996) e  g  X  1-^134-130(1-^)"+o-s y.1.4 ilB  159  for  0.80 <  < 0.9985  (6-27)  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  x=0 x=5  x  g  Core region  Gas layer  Temperature distribution  i  Gas gap  Voidage distribution-  x=5 +8 g  e  Emulsion layer Core region  J  ; lTb(T ,ii-) 2  / / \  Heat transfer surface wall  u T  w  T  e  P  T(x)  T  T!=0  b  T  T =K 8 2  0  (b)  (a)  Figure 6.1. Schematic showing non-uniform emulsion heat transfer model: (a) physical model; (b) radiative heat transfer model.  160  t  e  Radiative and Total Heat Transfer in Circulating Fluidized Beds  where 8 sec (= l P s p / P p ) -  l s  m  e  SU  gradient along the riser, with p  Chapter 6: Modeling of Heat Transfer in CFBC  cross-sectional average voidage estimated from the pressure  susp  and p being the suspension density and particle density, p  respectively. The temperature distribution in the emulsion layer is estimated using the correlation of Golriz (1995), i.e., Eq. (4-14), obtained for 5.6 ^ Re ^ 13.5,2.1 ^ Tb/T <• 2.5 and 0.19 <• z/H p  w  ^0.81. The voidage distribution in the emulsion layer is described using the correlation of Zhang etal.(1994) „/VlS\  ~  (0.191+<D-+3<D )  S(<P) = 8 sec  2 5  11  (6-28) O  O  A  where O = 1 - x / X .  The correlation of Goedicke and Reh (1993) is used to estimate the local voidage at the wall: e =0.322+ 0.196(ssec)+15.09 (8sec)-0.4j  674  w  (6-29)  Equation (6-28) with O = 1 agrees well with Eq. (6-28) for the cross-sectional average voidage range 0.88 to 1.0.  6.2.2. Assumptions The following are the primary assumptions in formulating the radiative heat transfer equations: 1. The entire system is at steady state. 2. A core-annulus structure exists in the CFB. 3. The heat transfer surface is fully covered by an emulsion layer at all times. 4. There is a thin gas gap between the emulsion layer and the wall.  161  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  5. The solids concentration in the core region is homogeneous, while there is a lateral particle concentration profile in the annulus region. 6.  For long surfaces, the temperature in the solids is assumed to vary only in the emulsion layer and to be given by the Golriz (1995) equation. For short surfaces, temperature variation in the solids is ignored. Temperature gradients in the core are ignored in both cases.  7. The temperature of the heat transfer surface is uniform. 8. The heat transfer surface is a gray, diffuse surface, i.e., the absorption of thermal radiation is independent of wavelength and direction. 9. Variations of emulsion thickness in the z-direction are ignored. 10. The thermal conductivity of particles is large enough (Bi < 0.1) that temperature gradients within the particles can be neglected 11. The temperatures of the gas and particles are equal locally within both the emulsion and core regions. 12. Particles are spherical and uniform in size. 13. Both radiative and conductive/convective heat transfer are neglected in the axial direction compared with lateral heat transfer, i.e., one-dimensional (horizontal) radiative, conductive and convective heat transfer is assumed through the emulsion and gas layers. 14. The fluidizing gas between the wall and the emulsion layer is transparent, i.e. nonabsorbing and non-scattering. 15. The particle surfaces are non-gray and diffuse. The emulsion is a non-gray, absorbing, emitting and scattering medium. The scattering is multiple, independent and  162  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  anisotropic. 16. Any contribution to radiation from fines such as soot or ash particles is negligible. 17. Heat generation in both the annulus and core regions is ignored. 18. Convective and radiative heat transfer are additive.  6.2.3. Heat Transfer Components The total heat transfer coefficient can be approximated by the addition of the conduction and radiation components, accounting for the thermal resistances of the gas gap and the emulsion layer. Hence we write (6-30)  h  where he,, is the convective/conductive heat transfer coefficient; h and h are the heat transfer g  e  coefficients for conduction through the gas gap and for the emulsion, respectively, while h,, is the radiative heat transfer coefficient.  6.2.3.1. Conduction in Gas Gap The conductive heat flux through the gas gap perpendicular to the heat transfer surface is represented as one-dimensional, i.e. q.=-k,f  (6-31,  with the boundary conditions:  (6-32)  163  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  Allowing for the variation of the gas conductivity with temperature in the gas layer (McMordie 1962), we may write  h = * * T -T B  b  =— (T -T )5  (6-33)  l  g  w  b  w  8  The gas gap thickness, 8 , is estimated using Eq. (6-26). The thermal conductivity of the g  gas is a function of temperature; data are available in Incropera and Dewitt (1990).  6.2.3.2. Conduction / Convection in Emulsion Layer One of the most useful models for convective heat transfer from the suspension to the wall of CFB risers is derived from the packet model proposed by Mickley and Fairbanks (1955) for heat transfer in low-velocity fluidized beds. The heat transfer is represented as a transient process involving conduction from groups of particles arriving at the wall with the resulting heat flux related to the residence time of the packet at the surface. For a packet exposed to the wall at time 0 the local heat transfer coefficient after contact time t is  K^=,f^ V  Tt  (6-34) t  and the surface-length average heat transfer coefficient is given as (6-35)  where t is the contact time ( = L / U ), L is the surface length, U is the falling velocity of p  p  particles in the emulsion layer, and kg is the effective conductivity of the packet, estimated  164  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  from an expression proposed by Gelperin and Einstein (1971):  (l-e )(l--i) c  K  k e =kg 1 + ^- + 0.28  P  (6-36)  8  The effective density and heat capacity of the clusters are estimated, respectively, as Pe =Pp( -ee)  +Pg  1  N  n  and  V e  =  C  P . p (  -  1  8  E  P  (6-37a)  e  P  p  e ) — + p.g C  8  e  g  (6-37b)  —  The heat transfer coefficient, he, decreases with increasing contact time. In this study, the contact time, t, is estimated as the heat transfer surface length divided by the particle falling velocity.  6.2.3.3. Radiation in Emulsion Layer The radiative flux at any distance x from the wall in the emulsion layer can be obtained by integrating the local spectral radiation intensity, i. e., it can be written: 1  UJ  q (x)= r  J 2n J l  x  ( x , p ) p d p  dX  (6-38)  where p = cos 9, and 0 is the polar angle. The radiative intensity L.(x, p) must obey the radiative heat transfer equation (Siegel and Howell 1992):  =  dx  " W'ti  +  T W^X^p^dp' + (1 - n)I 2  _j J  XJ>  (T)  (6-39)  where y is the azimuthally averaged scattering phase function and Q is the spectral scattering albedo of the emulsion defined by Q = K / K , with PC. and K being the scattering and S  t  165  t  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  extinction coefficients, respectively, of the emulsion layer. In this equation, the x coordinate is replaced by the optical depth, defined by x = K, (x - 5 ). At the outside and inside edges of the g  emulsion layer, t = 0 and x = K 5 as shown in Figure 6.1, with 5 estimated using Eq. (6-27). x  z  t  e  e  The local spectral intensity of blackbody radiation in a vacuum is given in terms of the local temperature T(t) by Planck's law:  X [exp(c /XT(T;) )-l 5  2  j  where c ' = h c „ , c =h'c /k', c is the velocity of light in a vacuum (2.9979xl0 m/s), h' is ,  8  1  2  0  0  Planck's constant (6.6261 x 10~ J s ), X is the wavelength and k' is the Boltzmann constant 34  (1.3807xlO  -23  J / K ) . The extinction coefficient Kj is the sum of the scattering and absorption  coefficients, K = K + K . According to Tien and Drolen (1987), independent scattering occurs t  s  a  for inter-particle clearance c = Jt d / X > 5. This condition is satisfied for a CFB riser operating p  with T > 400 °C and group B particles. Hence, b  3R K (x) = —*-[l-e(x)]  (6-41a)  g  p  3(l-R ) i K . (x) = - — [ l - e(x)] 2d D  r  (6-41b)  p  K (x) = K, (x) + K (x) = t  [l - e(x)]  a  2d  (6-41c)  p  The albedo Q = K / K = R , where R , the reflectivity of the particles, is equal to 1 - e , while s  e  p  t  p  p  p  is the particle emissivity (Tien 1988). It is assumed that the anisotropic scattering is  independent of the angle of incidence. Hence, the phase function, y, for large diffuse particles is:  166  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1 8 •v(p,p') = — f — ( s i n 9 - © c o s © ) d i | ) 2jt 3jt  (6-42)  cosG = pp'+^/d- p ) ( l - p' ) cos(^ -i|>')  (6-43)  2 x  J  0  where  2  2  For a non-absorbing gas film, one of the boundary conditions associated with the radiative heat transfer equation, Eq. (6-39), is given by  l(0,p) = 2(l-e)jl(0,-p)pdp + e I .w w  where I  b w  w  0<p^l  b  (6-44)  is the spectral blackbody radiation intensity calculated at the surface temperature T . w  The other boundary condition at the optical thickness of the emulsion layer, x = K 5 , includes z  t  e  emission from the fluidizing gases and emission from the other walls:  I(t  2  ,-p) = e (fX^ d  0<p<l  +e JI , (t ) g  i=2  b  b  2  (6-45)  where Fj.j are the view factors from the other walls (i = 2, 3, 4) to the surface studied (j = 1), calculated (Siegel and Howell 1993) using:  Fl-2  ^2-1  —  —  ^1-4 — F4-I  2  —  ,  x  1 + — ,  , rx  1+  N  —  (6-46) F r  =F  l-3  =  +1  3-l  C  w_ 2Y  Here W is the distance between two parallel walls; X and Y are the geometric dimensions of the riser (i.e. half-widths of the riser cross-section). With a riser of square cross-section, W= 2X = 2Y giving F  LM2  e and e d  gas  =  F , . = 0.293 and F 4  w  = 0.414.  in Eq. (6-45) are respectively the emissivities of the suspension and the  167  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  fluidizing gas in the core region. They can be evaluated (Siegel and Howell 1992) using e =l-expfK |i.6(X-6 -6 )]}  (6-47)  e  (6-48)  d  and  t  gas  =e  C02  +e  e  H20  -Ae  C02  _  g  H20  where K is the extinction coefficient estimated using Eq. (6-41) with the voidage in the core t  region obtained from a solids mass balance, i.e. _ -  (1 -i ec)(2X) - (1 -iw)(46 (2X) -45 ) 2  2  S  e  e  :—n ;—;— ? (2X) -46 (2X)+48  1 - 6d =  2  f <  —  (6-49)  2  e  e  Once q^x) is calculated (see next sub-section), then the corresponding heat transfer coefficient is h. = T -T q  ( T  =  (6-50)  0 )  6.2.4. Solving for Radiative Heat Transfer Fluxes and Coefficients It is difficult to solve the coupled energy and radiative heat transfer equations. However, the temperature change across the emulsion layer can be estimated using the correlation of Golriz (1995), and only the radiative transfer equation then needs to be solved to obtain the radiative heat flux, and hence the radiative heat transfer coefficient. Equation (6-39) can be integrated using the discrete ordinate method (S-N method, Modest 1993). The terms in Eq. (6-39) are evaluated for a discrete number of ordinate directions with the integration being carried out by numerical quadrature as follows:  ^^^ _i ( ^ ) ^2w T(ii ^ )ix(t^ )+a-Q)ix*w =  dt  x  t  i  +  J  l  2  j  J  j = 1  i = 1, 2  168  2n  (6-51)  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  where w- is the weight coefficient in the jth direction. The directions are selected to ensure that 1  2n  2j"u du. = £ w u k  j  o  where k = 0, 1, 2,  (6-52)  k J  j=i  2n-l. Equal weight coefficients, i.e. w = w = ... =-w = 1/n, were x  2  2n  suggested by Fiveland (1987) and are used in this study with n=6; Wx to w are for the positive n  direction, while w  n+1  to w correspond to the negative direction. The selected direction angles, 2n  determined by solving Eq. (6-52), are listed in Table 6.2.  Table 6.2. Selected angles and weights used in the discrete ordinate method with 12 directions. Weighting coefficient (w) Selected Angles  fctu)  1/6  1/6  1/6  1/6  1/6  1/6  0.066877  0.288741  0.366682  0.633318  0.711259  0.933123  The temperature distribution in the emulsion layer is known so that the blackbody intensity at the local optical depth can be evaluated using Planck's law, Eq. (6-40). The corresponding boundary conditions in their discrete ordinate form are given by 2n  I(0,n,)=2(l-e )]^w |n |l(0,|i )+e I w  j  j  J  w  b>w  (6-53)  i = l,2,...,n  j=n+l  I(t - ( i , ) = e ( £ F _ e +e )I 2  d  j  i  j  gas  D>b  (T )  i = n + l,n + 2,...,2n  2  (6-54)  j=2  Evaluating Eq. (6-51) at the midpoint between two nodes gives:  [^-|]l (t,^ )+fE iT(l*..M,)Ix(V^' ^^^ ' w  l  ) +  1  ( t  ) + I  » ''« ) < t  >  ' B . I) +  ,AT  2)  k=l, 2 , N  169  (6-55)  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  The radiative heat transfer flux is evaluated with respect to both the wavelength and the cosine variables as M  2n  q (T = 0 ) = 2 j t £ j;w |p |i (T=o,p ) r  j  =i  j  x  j  (6-56)  P  where the entire wavelength range from 0 to  0 0  is replaced, without loss of accuracy, by a finite  sequence with M wavelength intervals A l . With bulk suspension temperatures lower than 950 °C p  as in the present study, more than 98.5% of the thermal radiation from a blackbody lies within the wavelength range of 1 to 21 pm. The stability of the numerical solution of Eq. (6-55) depends on the optical depth intervals. To ensure that the calculated intensity is positive, the following simple relationship can be written to maintain a stable solution:  Min  >2  i = 1, 2,.... 2n '  (6-57)  Fiveland (1987) suggested an incremental optical depth of one-fifth to obtain numerically converged solutions. This condition was used in this work, giving At<0.4Min|pi|  i = l,2,...,2n  (6-58)  Since the minimum angle listed in Table 6.1 is 0.066877, the optical depth is divided into more than 37.4(t -t ) equal intervals to meet the suggested requirement The algorithm used to 2  1  compute the radiative heat transfer flux and coefficient is shown in Figure 6.2.  6.2.5. Simulations of Heat Transfer Fluxes and Coefficients  6.2.5.1. Comparison of Model with Experimental Results from the Litera The proposed model predicts three components: conduction through the gas layer,  170  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Start  I  Data input dp, T b , T w , R p , C i , C2,psusp  Selected angle and weight  I  Wk, |ik  Boundary thickness 5e 8(5e),IC(5e),Ka(8e),K,(5e)  Scattering angle  I  Optical thickness T  I  Scattering phase function  Temperature and intensity profile  T(x), I«(T) J. Intensity profile K(x,p:,p,')  Radiative heat flux q (x = 0) r  Radiative heat transfer coefficient hr (T = 0)  End Figure 6.2. Procedures used to compute the heat transfer flux and coefficient.  171  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  convection in the emulsion layer and radiation within the emulsion layer. The gas layer heat transfer coefficient is determinedfromthe suspension temperature and the cold wall temperature as well as the temperature distribution in the emulsion layer. The convection heat transfer coefficient is obtained using Eq. (6-35) which depends on the exposure time, t, of particles at the edge of the gas layer, ke, p and c e  Pse  are estimated using Eqs. (6-36) and (6-37), while s is e  estimated using Eq. (6-29). Figure 6.3 shows the predicted influence of the contact time on the convective heat transfer coefficient for several suspension densities. The heat transfer coefficients are strongly dependent on the contact time, except for long times. The contact time is influenced by the length of the heat transfer surface and the particle velocity. Previous studies (Zhou et al. 1995) showed that streamers adjacent to a flat surface fall at a velocity of 1 to 3 m/s. In this study, the vertical probe length is only 6 mm. For a particle falling velocity of 2 m/s, the contact time is 0.003 s, much shorter than the time constant (t = 0.122 s estimated using p  Eq. (4-13)). The thermal driving force remains high enough that most of the thermal resistance to conduction/convection comesfromthe gas layer.  The model developed in Section 6.2.3.3 is employed to predict the radiative heat fluxes and coefficients for cases where the heat transfer surfaces are fully covered by particles. The predictions from the model are compared with previous experimental findings and with published theoretical data with short probe lengths in Table 6.3. From the table, it is seen that the model predictions are consistent with the experimentalfindings,except for the prediction of the data of Werdermann and Werther (1994) under isothermal emulsion layer conditions.  172  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Figure 6.3. Effect of particle-surface contact time on the predicted emulsion convection heat transfer coefficients. T=851 °C. b  Table 6.3. Predicted radiative heat transfer results compared with published experimental results.  2Xx2Y(m ) T CO T (°C) PsusoCkg/m) d (nm) U (m/s) 2  b  W  3  D  e  ew (-) e„(-) measured q predicted q (kW) measured h predicted h (W/m K)  r  r  r  r  Steward et al. (1995) non-isothermal 3.96x3.96 850 279 40 200 6.4 0.855 0.85 57 50 100 87  Werdermann & Werther (1994) isothermal non-isothermal 5.13x5.13 860 340 9 209 N/A 0.91 0.91 53 30 74 28 57 54  2  173  109 143  Basu & Konuche (1988) isothermal 0.2 x 0.2 885 60 20 296 8.0-11.0 0.97 0.9 91 103 110 126  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  The non-uniform emulsion model 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. With the conditions used by Glatzer & Linzer (1995), i.e., d = 209 pm, p = p  e  35 kg/m , p = 5.2 kg/m , e™ = 0.9, ep = 0.9, T = 857 °C, T = 340 °C and 2X = 2Y = 5.1 m, the 3  3  d  b  w  model proposed here gives reasonable agreement. Compared to the radiative heat transfer coefficient, h =61.7 W/m K , obtained using the two-flux model, the non-uniform emulsion 2  r  model yields h =54.3 W/m K . The difference between the two models is probably due to the 2  r  different radiation absorption mechanisms assumed. Since the non-uniform emulsion model assumes that there is a solids concentration profile in the emulsion layer, the solids concentration near the wall is greater than the average solids concentration, leading to a larger extinction coefficient than that estimatedfromthe average solids concentration (e.g., 1 - 8e = 0.016, Kt= 110 m"  1  for the model presented here, while 1 - s = 0.014, K = 101 m" for the Glatzer and Linzer two1  e  t  flux model). These results indicate that the radiative heat flux decreases with emulsion layer thickness more steeply than with a lower extinction coefficient. The attenuation of the thermal intensity by the emulsion layer is more than compensated for by increasing the emissivity of the layer by raising the extinction coefficient. As a result, the radiative heat transfer coefficient decreases with increasing extinction coefficient.  6.2.5.2. Prediction ofHeat Transfer under Isothermal Emulsion Layer Co The model can be applied for non-isothermal and isothermal conditions. In this section the model is used to predict heat transfer coefficients for comparison with the experimental results obtained using the probe given in Chapter 5. The probe surface is small enough that  174  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  thermal equilibrium between the probe and the emulsion layer is never reached. Hence, the thermal boundary layer is very thin near the probe surface. The emulsion layer temperature is considered to remain isothermal at the bulk suspension temperature, while a non-uniform voidage exists in the emulsion layer.  The variation of the voidage-dependent extinction coefficient with lateral position and particle size is shown in Figure 6.4. The extinction coefficient, which is proportional to the local particle concentration, decreases with optical depth (i.e. with x). Also, from the figure, it is seen that the extinction coefficient decreases with increasing particle size. For d = 334 pm and p sp 50 =  p  SU  kg/m , the thickness of the annular wall layer is 0.010 m from Eq. (6-27). Note that the 3  extinction coefficient K decreases nonlinearly from 350 to 130 m" over that layer. Since 1  t  the extinction coefficients very close to the surface are greater than the average extinction coefficient  600 h  50 0.000  0.002  0.004  0.006  0.008  /  0.010  Distance from Heat Transfer Surface, x (m) Figure 6.4. Extinction coefficient distribution in the emulsion layer. Tt,=850 °C, T =490 °C, p =50 kg/m , p=2610 kg/m . 3  w  susp  3  p  175  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  calculated from the average particle concentration and the attenuation happens in the first particle layers adjacent to the surface, the radiative heat fluxes are lower than those predicted by other models.  The predicted radiative flux distribution in the emulsion layer is shown in Figure 6.5 for a riser of 0.152 m square cross-section. The radiative heat fluxes between the wall and the emulsion layer depend on the radiation, reflection, scattering and emission of the gas-solids mixtures in the core and annulus regions. The rapid decrease of radiative heat flux with increasing x near the surface demonstrates that only layers of the emulsion phase close to the wall contribute significantly to the radiative surface flux. The subsequent slight increase of radiative flux with increasing distance x indicates the attenuation of radiation from the mixture of  80  T  '  1  '  1  r  T =905 °C  70  b  T =851 °C b  T =804 °C b  —'  50  T =407 °C b  i  40  E S3  30  <L> CD  .&  (3  20  io 0 0.000  / 0.002  0.004  0.006  0.008  0.010  Distance from Heat Transfer Surface, x (m) Figure 6.5. Radiative heat flux distribution in the emulsion layer. d =334 um, Psusp=50 kg/m , e^O.9, e =0.9,2X=0.152 m, L=0.006 m. T = f(T ) (see Fig. 5.5). p  3  p  surf  176  b  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  hot gas and dilute dispersion, as well as reflection from the other three vertical walls of the reactor. As expected, radiative heat transfer fluxes decrease with decreasing suspension temperatures. The radiative heat flux for T = 407 °C is very small. In fact, the radiative b  component can be neglected for bulk suspension temperatures lower than about 400 °C.  In Figure 6.6, the predicted radiative heat flux is plotted as a function of the suspension temperature and compared with the experimental results obtained with the probe described in Chapter 2 using both the differential emissivity method (DEM) and the window method (WM). The surface temperature, listed in Table 6.4 and shown in Figure 5.5, is determined by linear extrapolation of the measured temperatures along the probe heads. The predicted radiative fluxes range from 5 to 80 kW/m as the suspension temperature varies from 407 to 913 °C. The predicted radiative heat 2  fluxes are close to, but a little lower than, those obtained experimentally using window method.  100 90.  6  ~T—'  1  800  900  (^=334 nm d=286 nm p  • A  80  O A  Bcperimentalresults(DEM) Experimentalresults(WM)  70 60 50 40  X  30  M  20 10  o  0 200  ° 300  <& 400  500  600  700  1000  Bulk Suspension Temperature, T (°C) b  Figure 6.6. Comparison of the predicted and experimental radiative heat transfer fluxes as a function of the bulk suspension temperature. p =50 kg/m , p=2610 kg/m , e =0.9. T is listed in Table 6.4. Line shows predicted results. DEM=Differential Emissivity Method; WM=Window Method. 3  susp  3  p  177  w  surf  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chavter 6: Modeling of Heat Transfer in CFBC  Table 6.4. SS347 and SS316 probe surface temperatures as a function of bulk suspension temperature. dp = 286 pm, U =7.8 to 8.9 m/s, P/S=l.5-2.3, p p=30-90 kg/m . 0.152 m square column SS347 SS316 SS347 SS316 3  g  SUS  T (°C)  Tsurf (°C)  Tsurf (°C)  T (°C)  T urf (°C)  "T f (°C)  230 300 360 370 380 386 502 559 600 650 703 760 798 794 791 797 804 798 794 797 797 783 789  132.4 174.3 180.3 192.0 197.2 194.1 243.2 273.8 306.4 341.5 388.0 425.5 479.6 469.2 463.1 466.6 482.0 490.6 469.0 441.9 449.0 440.3 470.0  126.9 169.4 167.8 180.7 185.6 187.6 233.3 268.2 302.1 335.4 385.4 421.0 444.7 439.9 431.1 430.4 446.8 452.1 432.9 406.4 407.2 407.3 430.6  850 850 856 846 854 845 853 850 855 853 850 844 846 908 918 920 922 904 911 911 914 918 919  478.9 523.3 519.6 519.5 503.0 482.3 498.4 496.5 473.9 466.7 475.6 452.9 477.7 554.3 558.2 558.1 559.3 528.0 531.4 535.4 571.1 577.6 577.4  438.8 489.6 481.9 479.9 462.0 446.7 462.3 459.6 441.7 434.7 440.3 419.0 441.6 530.1 531.3 532.4 531.9 507.7 502.0 505.3 540.7 555.6 548.3  b  b  S  sur  The predicted radiative heat transfer coefficients using the non-uniform emulsion model, shown in Figure 6.7 as a function of suspension density at 850 °C, are compared with the experimental results obtained with the probe using the window method. It is expected that suspension density has insignificant influence on the predicted hr since the albedo (Q) and phase function (y) in Eq. (6-39) are independent of suspension density. Increasing suspension density results in an increasing extinction coefficient and optical depth for a given emulsion  178  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  i  1  1  1  1  1  1  1  1  1  1  i  i  1  1  i  i  1  1  i  i  1  1  i  i  f —  : Model predictions : Experimental results  350 [  • 300 [ £  250  «  CD O  200  CO  150  •  Id  H  •a 100 ffi 50  (2  j  "0  i  10  i  i  20  i  i  30  i  i  i  40  50  Suspension Density, p  60  70  80  i  90  (kg/m ) 3  Figure 6.7. Comparison of predicted radiative heat transfer coefficient with experimental results of the probe using the window method. Tb=850 °C, ep=0.9, ev=0.9, d=334 um. p  layer thickness. Hence, the predicted radiative heat transfer coefficients sUghtly decrease with increasing suspension density due mainly to the increase of the attenuation On the other hand, increasing suspension density results in increased emissivity of the emulsion layer. In fact, even without cooling surfaces in the riser, the radial temperature is not really uniform. Increasing suspension density enhances particle mixing inside the emulsion layer so that the temperature in the emulsion layer is close to the bulk suspension temperature. Hence, the measured radiative heat transfer coefficients increase with increasing suspension density and approach constant values.  Figure 6.8 shows the effect of the surface emissivity (varied between 0.3 and 0.9) on the radiative heat transfer coefficients predicted by the model. For an opaque surface, as the surface  179  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  <•£ 180 h  Q  .6 E  160h 140 h  •  d =286 um  O  dp=334 um  p  100  «S 80 H  60  ffi <o  40  .•a  20  03  04  0.6  05  0.7  05  0.8  Heat Transfer Surface Emissivity, e  (-)  Figure 6.8. Effect of heat transfer surface emissivity on the predicted radiative heat transfer coefficient. T =850 °C, 7,^90 °C, p p=50 kg/m , p=2610 kg/m . 3  b  SUS  3  p  emissivity decreases, the reflectivity increases and the surface reflects more heat so that the radiative heat transfer between the surface and the emulsion layer decreases. There is little difference between the predictions for the mean particle sizes used here, dp=334 and 286 um.  The predicted effect of particle size on the heat transfer coefficient is shown in Figure 6.9. The conductive/convective heat transfer coefficient, hcc is predicted to decrease with increasing particle size due to the thicker gas layer between the surface and the emulsion layer. Given the small probe surface used in this study, the thermal resistance in the gas layer dominates conductive/convective heat transfer so that hc is very sensitive to the thickness of the gas layer. C  According to Tien (1988), the extinction coefficient is inversely proportional to particle size. Hence, the radiative heat transfer coefficient increases with increasing particle size due to  180  Chavter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1800 j  0 I 100  1  i  1  1  I  i  200  1  1  I  i  300  1  1  I  i  400  1  •  I  i  500  1  •  1  1  600  1  •  1  1  700  1  '  1  1  800  1  •  1  •  900  1  r  1  • 1  1000  1100  Particle Size, dp (urn) Figure 6.9. Effect of particle size on the predicted heat transfer coefficients (lines): T =850 °C, T =490 °C, p =50 kg/m , ^=0.9, e^O.9, d=334 um. Symbols are for measured radiative heat transfer coefficients (see Table 6.3 for details): • : Basu and Konuche (1988); • : Werdermann and Werther (1994); • : This study, (Differential Emissivity Method); A : This study, (Window Method). 3  b  surf  susp  p  the decreased attenuation of the emitted and scattered radiation from different optical depths in the emulsion layer. However, the combination of the convective and radiative components results in the total heat transfer coefficient decreasing with increasing particle size, as reported in a number of experimental studies (e.g. Wu et al. 1987). Since ht decreases while h  r  increases with increasing particle size, the relative importance of the radiative component increases sharplyfrom20% to 50% as the mean particle size increasesfrom200 pm to 1 mm, as shown in Figure 6.10. The predictions are seen to be reasonably consistent with our experimental results.  181  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1g ||j  i  I  100  200  300  400  500  600  700  800  •  i  900  •  i 1000  •  I  1100  Particle Size, d (nm) p  Figure 6.10. Effect of particle size on the predicted relative importance of the radiative heat transfer component. T=851 °C, T =490 °C, p =50 kg/m , e =0.9. DEM=Differential Emissivity Method; WM=Window Method. 3  b  surf  susp  w  Figure 6.11 shows the effect of bulk suspension temperature (Tb) on the predicted total, conductive/convective and radiative heat transfer coefficients: h , hc and h . With bulk t  C  r  temperature increasing from 400 °C to 900 °C, the radiative heat transfer coefficient is predicted to increase from 40 to 200 W/m K. The convective heat transfer coefficient also increases with increasing temperature, but more slowly and mainly due to the increasing gas thermal conductivity. The predictions are in good agreement with the experimental results presented in Chapter 5. A significant difference between the model predictions and experimental results is that, at Tb = 400 °C, the predicted h is 40 W/m K, while no radiative component could be 2  r  detected experimentally when the temperature was that low. The corresponding relative importance of radiation (h /h ) is plotted in Figure 6.12. In the 400 - 900 °C temperature range, r  t  182  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulating Fluidized Beds  1000  1  Exp. 900 " (DEM)  1  1  Exp. (WM)  1  1  i  1  i  1  Model •  •  800 700  •  600  •  500  i  1  hoc  ------ h,  V  -  .—  •  „  V  - -  400 co  300 200  _----ar"  -  . A Y - - A  v  100  -  0 i  1  i  400  500  600  .  i  •  •  1  700  1  900  800  Bulk Suspension Temperature, T  (°C)  b  Figure 6.11. Comparison of predicted and experimental heat transfer coefficients as a function of suspension temperature. dp=334 um, p sp 50 kg/m , e =0.9, T is listed in Table 6.4. DEM=Differential Emissivity Method; WM=Window Method. =  3  SU  0.50  1  1  • 1  0.45  '  1  •  w  i  1  surf  '  i  Mode! predictions  co  0.40  H  M  Experimental results (DEM)  o  Experimental results (WM)  •  S3 0-35  g  030  g •  025  o  •  020  CD  I  0.15  SH  0.10  -  6 S>  0.05  i  400 £  0.00  .  i  500  .  i  .  600  i  i  700  800  Bulk Suspension Temperature, T  b  .  i  900  (°C)  Figure 6.12. Effect of suspension temperature on the relative importance of radiative heat transfer coefficients. dp=334 um, p =50 kg/m , e =0.9. T is listed in Table 6.4. DEM=Differential Emissivity Method; WM=Window Method. 3  susp  183  w  surf  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulatins Fluidized Beds  h /h is predicted to increase from 12% to 35%. The relative importance of radiation obtained by r  t  the window method is higher than that by the differential emissivity method, and the former is closer to the predicted values than the latter. Recall from Chapter 5 that the window method is also believed to give more accurate experimental values.  The predicted radiative heat transfer coefficients from the non-uniform emulsion model are compared in Figure 6.13 with those predicted from the packet model of Basu and Nag (1988) and the network model of Fang et al. (1995) (see Table 6.1) for p  SU  s  = 50 kg/m and dp = 334 um at 3  P  elevated temperatures, assuming e = 0.8, e = 0.9 and e i = 0.9. The results predicted from w  p  c  the packet model are in agreement with the network model, but both are lower than those from the non-uniform emulsion model. The fraction of the wall covered by clusters (f) assumed by  Figure 6.13. Comparison of radiative heat transfer coefficient predicted by the non-uniform emulsion model with predictions from the packet model of Basu and Nag (1987) as well as the network model of Fang et al. (1995) at elevated temperatures. e =0.9, e =0.8, ei=0.9, p =50 kg/m , d=334 um. T is shown in Figure 5.5. P  3  susp  p  SURF  184  w  c  Chapter 6: Modeling of Heat Transfer in CFBC  Radiative and Total Heat Transfer in Circulatins Fluidized Beds  the former two models appears to have little influence on the radiative heat transfer coefficients under the conditions assumed here.  Experimental radiative heat transfer coefficients are plotted against the predictions from the non-uniform emulsion model in Figure 6.14. The comparison shows that the measured radiative heat transfer coefficients are in the range of 0.75 to 1.50 times the predicted results. A major contributor to the deviations is likely to be the effect of suspension density on radiative heat transfer.  &  o  0  50  100  150  250  200  300  Predicted Radiative Heat Transfer Coefficient, hj. (W/m K) 2  Figure 6.14. Comparison of predicted radiative heat transfer coefficients with experimental results. • , • : This work, ep=0.9, e =0.8, p =18-90 kg/m . T is shown in Figure 5.5 and Table 6.4; • : Werdermann and Werther (1994) (for details see Table 6.3);T : Basu and Konuche (1988) (for details see Table 6.3). 3  w  susp  185  surf  Radiative and Total Heat Transfer in Circulatinz Fluidized Beds  Chapter 6: Modeling of Heat Transfer in CFBC  6.3. Conclusions A brief review and critique of the existing models of radiative heat transfer in circulating fluidized beds are first provided. A new non-uniform emulsion model is then proposed, with temperature and solids concentration varying in the emulsion layer, which is treated as a nongray, absorbing, emitting and anisotropically scattering medium. No newfittingconstants beyond those incorporated by other correlations used in the model are introduced. The predictions from this model are generally in reasonable agreement with experimental findings in the literature listed in Table 6.3 for the range of operating conditions investigated. The predictions are also consistent in most ways with those obtained using the probe in this work. The following conclusions can be drawnfromthe above comparisons: 1. A non-uniform emulsion model has been developed which predicts radiative heat transfer based on an improved representation of the radiative heat transfer mechanisms in a CFB. 2. The model predictions are in reasonable agreement with the limited published experimental data obtained in CFBs both for laboratory- and industrial-scale units. It can be used to indicate the effects of the main parameters such as suspension temperature and particle size on the total and radiative heat transfer coefficients, as well as the relative importance of the radiation component. The model can be simplified if the emulsion layer is assumed isothermal at the bulk temperature Tb. 3. The relative importance of the radiative component increases with increasing bulk suspension temperature and with increasing particle size. For mean particle sizes of about 300 pm and typical combustion temperatures, radiation contributes as much as 35% of the total heat transfer.  186  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 7: Conclusions and Recommendations  CHAPTER 7 OVERALL CONCLUSIONS AND RECOMMENDATIONS  Total heat transfer and radiative heat transfer in a pilot circulating fluidized bed combustor have been studied at elevated temperatures. Heat transfer coefficients and fluxes obtained using a dual tube, a membrane wall and a multifunctional probe under various operating conditions are presented, together with a comprehensive model. Conclusions can be drawn from the analyses of those experimental results and by comparing the findings with the predictions by the model. Further investigations are also suggested.  7.1. Conclusions Heat transfer in a pilot circulating fluidized bed combustor was studied using three different types of heat transfer surfaces: a dual tube, a membrane wall and a multifunctional probe, all of which are described in Chapter 2. Parameters that have a significant impact on total and radiative heat transfer fluxes and coefficients, namely, suspension density, suspension temperature and particle size, were investigated. The total suspension-to-tube heat transfer coefficients increase with suspension density and suspension temperature and  187  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 7: Conclusions and Recommendations  decrease with increasing mean particle size. Based on a regression of the experimental results obtained in this work, as well as others from the literature, an empirical correlation for the total suspension-to-surface heat transfer coefficient, h , is proposed for broad ranges t  of suspension density, particle size and suspension temperature. It was found that h is t  influenced by the suspension density to the power of 0.44 rather than 0.5 (Basu 1996), especially when the suspension density is greater than 20 kg/m . The correlation represented 3  all the experimental results within a +50% deviation. In this correlation, the total heat transfer coefficient is proportional to the absolute suspension temperature to the power of 2.4 showing the effect of temperature on radiation. Temperature had little influence on convective heat transfer. The radiative component is important at elevated suspension temperatures, especially for values exceeding 600 °C. For bulk suspension temperatures of 800 to 900 °C, radiative heat transfer accounted for 25 to 45% of the total heat transfer to the dual tube surface. The suspension density had little effect on this ratio.  The cross-sectional temperature profile was measured in the vicinity of the membrane wall. The thickness of the thermal boundary layer in the pilot CFB combustor was found to extend 15 to 20 mm from the fin. Suspension-to-pipe and suspension-to-fin heat transfer coefficients were estimated from temperatures measured inside the membrane pipes and fins. The suspension-to-pipe heat transfer coefficient is higher than the suspension-to-fin heat transfer coefficient for the conditions investigated, indicating that the fin is less efficient than the exposed pipe surface in extracting heat from CFB combustors.  188  Radiative and Total Heat Transfer in Circulating Fluidized Beds  Chapter 7: Conclusions and Recommendations  The effect of heat transfer surface length on heat flux and heat transfer coefficients was studied by comparing the heat transfer results obtained at different axial locations along the membrane wall assembly and the results from the short-surface probe and the much longer dual tube surface. Higher total and radiative heat transfer coefficients were obtained with shorter heat transfer surfaces. The radiative and convective heat transfer coefficients are found to increase with increasing suspension density and increasing suspension temperature. The effect of mean particle size on the radiative heat transfer coefficient was not found to be significant, but the range investigated was very small. As expected, radiative heat transfer coefficients increase with increasing heat transfer surface temperature. The ratio of radiative heat transfer coefficient to total heat transfer coefficient is about 22 to 38% from the window method, and about 16 to 28% using the differential emissivity method. The difference between the two methods is probably caused by the uncertainty in estimating the surface emissivities, especially that of the lower emissivity surface in the differential emissivity method.  A non-uniform emulsion model was developed to predict suspemion-to-surface total and radiative heat transfer coefficients in CFBs. The model is validated using published experimental heat transfer data obtained from both laboratory scale and industrial scale CFB units. The contribution of the radiative component increases with increasing bulk suspension temperature and with increasing particle size. For units using intermediate particle sizes and operating at typical combustion temperatures, radiation contributes as much as 35% of the total heat transfer. This model can be used for situations where the gas-solids medium is non-gray, absorbing, ernitting and scattering, and where scattering through the medium is multiple, independent and anisotropic.  189  Chapter 7: Conclusions and Recommendations  Radiative and Total Heat Transfer in Circulating Fluidized Beds  7.2. Recommendations Radiative heat transfer plays an important role in suspension-to-surface heat transfer at elevated temperatures in industrial processes. More experimental investigation and theoretical research are needed to understand the detailed mechanisms of thermal radiation in the presence of absorbing gases and scattering solids. The window method described in Chapter 2 is a useful technique for direct measurement of thermal radiation. Extending the fundamental principles of the window method to long heat transfer surfaces is a challenging task, but could be an invaluable approach. A key requirement for the reliable utilization of the differential emissivity method is to obtain accurate determination of the surface emissivities while the surface is exposed to a high temperature suspension, since emissivities, especially those of bright surfaces, are subject to change when exposed to the high temperature environments encountered during combustion and other high temperature experiments.  It is necessary to investigate in more detail the fluid and particle hydrodynamics in the vicinity of a membrane wall as well as the heat transferfromthe suspension to the pipes and  fins  of the membrane wall. The relative efficiencies of pipe surface and fin surfaces are important factors in optimizing the geometry of membrane walls for industrial scale CFB boilers.  The influence of soot and ash as well as different fuels on total and radiative heat transfer are other aspects that require investigation. A comprehensive and versatile model should take these factors into account. At the moment, data on this subject are very scanty.  190  NOMENCLATURE A  absorptivity of ZnSe window;  A'  absorptivity of probe heads;  A"  cross-sectional area of probe heads;  A  radiation absorption cross-section per unit volume;  a  Ac  absorptivity of cavity wall;  Af  cross-sectional area of fin, =L(|)Ro;  A  p  cross-sectional area of probe head;  A  s  cross-sectional area of riser; absorptivity of cylindrical cavity wall;  a a  w  absorptivity of heat transfer surface; back-scatter fraction;  B  Bi,B2 constants in Eq. (6-21);  Bi  Biot number, hR/k ;  b  parameter defined in Eq. (3-40);  C  correction coefficient in Eq. (3-25);  p  Ci, C2, C3, C4  parameters defined in Eq. (3-45'  C  a  absorption cross-section in Eq. (6-12);  C  e  constant in Eq. (6-16);  Cm  measured correction coefficient;  C  scattering cross-section in Eq. (6-12);  s  Csim  simulated correction coefficient;  C  dimensionless inter-particle clearance, =7cd M,; p  0  Co  speed of light m vacuum (2.998x10 );  Cl  first radiation constant (2Tch'(Co) =3.742x10 );  C2  second radiation constant (h'Co/k=1.439xl0 );  Cp  specific heat;  D  diameter of the cavity cylinder, or of riser;  d  tube diameter;  8  2  4  191  dp  particle diameter;  pm  Ey  mass flow rate of solids in riser;  kg/m s  Ex  emissive power at wavelength X;  W/m pm  Ea.,b  emissive power of blackbody at wavelength X;  W/m pm  e  emissivity;  -  e  apparent emissivity of cylindrical cavity;  -  b  surface emissivity of black body;  -  c  integrated cavity emissivity;  -  cl  cluster emissivity;  -  ef  emissivity of cylindrical cavity wall;  -  e  particle emissivity;  -  eps  probe surface emissivity;  -  total emissivity between suspension and probe surface;  -  emissivity of tube surface;  -  white probe emissivity;  -  e  e  e  a  p  e  .  2  2  F  view factor;  f  fraction of wall covered by solids or clusters;  -  G  heat source intensity per unit volume;  W/m  3  G'  heat generation rate per unit volume;  W/m  3  G  spectral radiosity at wavelength X;  W/m pm  Gr  Grashof number, gPATL /v ;  -  G  solids circulation rate;  kg/m s  g  gravitational acceleration;  m/s  H  height of riser;  m  h  heat transfer coefficient;  W/m K  h'  Planck's constant (6.6256x10" );  Js  cr  radiative heat transfer coefficient of clusters;  W/m K  cs u  refractory-to-probe lateral heat transfer coefficient;  W/m K  cooling-water-to-probe-end impinging jet heat transfer coefficient;  W/m K  x  s  h  h  cw  3  2  2  2  2  2  34  192  2  2  2  h  radiative heat transfer coefficient of dispersed phase;  W/m K  he  heat transfer coefficient of emulsion layer;  W/m K  hf  suspension-to-fin heat transfer coefficient;  W/m K  dr  2  2  2  hf  free convection heat transfer coefficient from furnace;  hp  suspension-to-pipe heat transfer coefficient;  W/m K  h,.  radiative heat transfer coefficient;  W/m K  h  t  total heat transfer coefficient;  W/m K  h  w  convection heat transfer coefficient from cooling water;  W/m K  C  . W/m K 2  2  2  2  2  I(t,u) local spectral diffuse radiation intensity at optical depth x in emulsion layer for a direction p.;  W/m umsr  Ix  monochromatic radiative flux;  W/m um  J"  total radiant energy of all wavelengths passing through unit  1  2  2  normal area in positive transverse direction per unit time; J  W/m  2  total radiant energy of all wavelengths passing through unit normal area in negative transverse direction per unit time;  W/m  K a  absorption coefficient;  1/m  K s  scattering coefficient;  1/m  K t  extinction coefficient;  1/m  k  thermal conductivity;  W/m K  k'  Boltzmann constant (1.3 805x10" );  J/K  k,.  effective thermal conductivity;  W/mK  L  cylindrical-cavity length, or membrane-wall length;  m  Lm  mean beam length;  m  L  cavity characteristic length in Eqs. (3-22) and (3-23);  m  23  ff  x  M M  number of measured temperatures; c w  mass flow rate of cooling water;  N  number of estimated parameters;  Nu  Nusselt number, h L / k ;  P  pressure;  fc  x  kg/s  g  N/m  193  2  2  P"  perimeter;  Pr  Prandtl number, p c / k ;  -  P/S  ratio of primary air flow to secondary air flow;  -  AP  pressure drop;  -N/m  Q  heat transfer rate;  q  heat flux;  W/m  2  conductive / convective heat flux;  W/m  2  q'c  conductive /convective heat fluxfromwindow to probe head;  W/m  2  q"c  conductive /convective heat flux from suspension to probe head;  W/m  2  qf  free convective heat flux;  W/m  2  q  suspension-to-membrane-tube heat flux;  W/m  2  q  m g  c  C  P  p g  g  2  " W  q" r  radiative heat flux emitted from probe cylinder surface, = epfj(Tp);  W/m  2  qr  heat flux transferred by radiation;  W/m  2  q"r  radiative heat flux emittedfromheat source incident on the probe;  W/m  2  q'wr  radiative heat flux from window surface facing probe, = ewa(T i) ;  W/m  2  q'V  radiative heat flux from window surface facing furnace, = ewa(T ) ;  W/m  2  R  reflectivity of ZnSe window;  -  or radius;  m  reflectivity of probe heads;  -'  P  R' Ri„ .Rs  4  4  w  4  W0  radius of probe heads in Figure 3.3;  m  Ra  Rayleigh number, GrPr;  Rc  radius of cylindrical cavity;  m  Re  Reynolds number, diU pcw/pcw;  -  Rf  reflectivity of cylindrical cavity wall;  -  Ri  inside radius of membrane pipe;  m  Ro  outside radius of membrane pipe;  m  r  coordinate in cylindrical coordinates, or radius;  m  S  back-scattering cross-section per unit volume;  1/m  s  distance between probe and cavity;  m  "-  cw  194  T  temperature;  Kor°C  Ti,...T6  temperatures of pipe and fin shown in Fig. 2.10;  K or ° C  TI...TN  measured temperatures;  Kor°C  T  b  time-averaged suspension temperature;  Kor°C  T  c  computed temperature in probe heads;  K or ° C  Tf  temperature of heated cavity;  Kor°C  Ti  bulk temperature of water inside membrane pipe;  Kor°C  Tiniet T  M  To  inlet cooling water temperature;  K or ° C  measured temperature in probe heads;  K or °C  temperature at cavity opening;  Kor°C  T o u t i e t outlet cooling water temperature;  Kor°C  T  p  probe surface temperature;  K or ° C  T  s  local refractory temperature;  Kor°C  Tsurf  probe surface temperature;  Kor°C  Tt  outside surface temperature of membrane pipe;  K or °C  T  window temperature, or wall temperature;  K or ° C  w  T w a t e r cooling water temperature; t t  p  U  c  U  g  w  K or °C  time;  s  time constant;  s  water velocity;  m/s  superficial gas velocity;  m/s  Uoveraii  overall heat transfer coefficient;  W/m K 2  V  volumetric flow rate;  m /s  w  half-width of fin;  m  W1...W12 X  x  Y  Fiveland's weight;  half-width of riser;  X1...X7  x  3  m  length of probe heads in Figure 3.3;  m  cavity longitudinal coordinate, or riser lateral coordinate; w  ZnSe window thickness;  m  half-width of riser;  m  195  Z  height from gas distributor to location of thermocouple in pipe;  m  Z'  heightfromgas distributor to top of membrane wall;  m  parameters used in parameter estimation method;  Zi...Zn  z  height above bottom of riser at which heat transfer surface is located;  G R E E K  m  LETTERS  a  absorption coefficient;  1/m  P  thermal expansion coefficient;  1/K  y(u.,u,')  azimuthally averaged scattering phase function;  8  thickness of thermal boundary, or annular layer;  m  8f  half thickness of fin;  m  8  thickness of gas layer;  m  g  s  time-averaged voidage of entire fluidized bed or suspension; or precision criterion;  \  coefficients defined in Eq. (3-37);  0  scattering angle;  rad  9  polar angle;  rad  9'  dimensionless temperature, = (T-T )/(Tb-T );  X  wavelength;  w  w  pm  p, p  cos 0;  p  gas viscosity;  Ns/m  v  kinematic viscosity;  m /s  p  density;  kg/m g/  a  Stefan-Boltzmann constant, 5.670x10~ ;  T  transmissivity, or optical thickness;  Ti, T 2  optical thickness of emulsion layer;  Tx  spectral transmissivity at wavelength X;  O  dimensionless lateral distance in the riser startingfromits axis;  1  g  2  2  3  8  196  W/m K ; 2  4  <])  angle subtended by base of fin at tube center in Figure 4.20;  rad  azimuthal angle;  rad  Q  spectral scattering albedo, = K /K ;  co  angle in Figure 4.20;  s  t  rad  SUBSCRIPTS  1- D  one-dimension;  2- D  two-dimension;  316  stainless steel 316;  347  stainless steel 347;  arm  annular region;  av  average;  b  fluidized bed;  CO2  carbon dioxide;  c  convection;  cc  conduction and convection;  cl  cluster;  cw  cooling water;  d  diffuse phase;  e  emulsion, or exposed outer surface of membrane pipe;  f  fin;  fr  furnace radiation;  g  gas;  H2O  water vapor;  i  inside;  ins  insulation;  inv  inverse;  m  measured;  o  outside;  197  p  particle, or pipe;  pr  radiation from probe heads;  r  radiation;  s  solids;  sec  riser cross-section;  susp  suspension;  t  total, or tube;  w  window or wall;  wi  window inside;  win  window method;  wo  window outside;  wr  radiation from window;  SUPERSCRIPTS  noniso  nonisothermal;  OVERBAR  average.  REFERENCES  Alavizadeh, N., R. 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Leckner, (1994) "Characteristics of the Lateral Particle Distribution in Circulating Fluidized Bed Boilers", in Circulating Fluidized Bed Technology IV, ed. A. A. Avidan, AIChE, New York, pp. 266-273. Zhang, W. and Y. Tung (1991), "Radial voidage Profiles in Fast Fluidized Beds of Different Diameters", Chem. Eng. Sci., 46, 3045-3052. Zhou, J., (1995) "Circulating Fluidized Bed Hydrodynamics in a Riser of Square Cross-Section", Ph.D. dissertation, University of British Columbia, Vancouver, Canada. Zhou, J., J. R. Grace, C. M . H. Brereton and C. J. Lim, (1995) "Particle Velocity Profile in a Circulating Fluidized Bed of Square Cross-Section", Chem. Eng. Sci., 50,237-244.  210  APPENDIX I: D E T E R M I N A T I O N O F T O T A L S U R F A C E EMISSIVITY  Determination of hemispherical total emissivity is crucial in the measurement of radiative heat transfer using the differential emissivity method. Two experimental techniques have been widely used to determine total surface emissivities (DeWitt and Nutter 1988). One involves radiometric emissivity measurement, made by measuring the radiance of a heated sample and of a blackbody radiator at the same temperature as that of the sample under the same spectral and geometric conditions. The emissivity is computed as the ratio of the two radiances. The other method involves calorimetric emissivity measurement in which the rate of radiative heat transfer to or from a sample is measured in terms of the heat lost or gained by the sample. The ratio of the measured rate of heat transfer to that of a blackbody radiator under identical conditions is the emissivity of the sample. The sample must be very well insulated thermallyfromits surroundings to prevent unwanted heat leaks to orfromthe sample. Pyroelectric thermal detectors, thermister bolometer thermal detectors and thermopile thermal detectors are commonly used in radiation thermometry (Richmond and DeWitt 1984, DeWitt and Nutter 1988).  Using the radiometric emittance technique, thermal emissivity may be measured directly by measuring the ratio of the radiance of a sample to that of a blackbody radiator of the same geometry with the same temperature and wavelength conditions. An example of the technique is shown in Figure I. 1. The sample, in the form of a flat strip, is heated by passing a current through it and is enclosed in a water-cooled shield to reduce thermal gradients due to convective cooling. Two identical blackbodies are used. One blackbody is always used as the source of a comparison beam for the spectrometer. Either the hot sample or the reference blackbody can  211  be used as the source for the sample beam of the spectrometer.  In operation, the comparison blackbody is brought to the desired temperature by adjusting the power input. The reference blackbody is placed in position to serve as the source for the sample beam of the spectrometer, and its temperature is controlled to be the same as that of the comparison blackbody by means of a differential thermocouple.  The integral blackbody approach is illustrated in Figure I. 2. A thin-wall tubular specimen is heated by passing a current through it. The small hole in the wall of the tube approximates a blackbody at the temperature of the wall if the tube walls are thin and of uniform thickness and the hole diameter is small compared to the diameter of the tube. Total normal emissivity can be measured by this technique if a detector is used that has essentially the same  212  THERMAL EXPANSION TAKEUP DEVICE  POWER LEAD COOLANT TUBE  THERMOCOUPLE TERMINAL (TYPICAL) Tc JUNCTION BACK SIDE (TYPICAL)  BLACK BODY HOLE  BOTTOM ENO HEATER  TUN6STEN HEATING  Tc LEAOS-TO FEED THRU (TYPICAI BOTTOM CLAMP  COIL  HIGH VACUUM TUBE FEED THRU (TYPICAL)  INSTRUMENTATION FLANGE  TO COOLANT MANIFOLD ELECTRICAL TERMINAL  8  Figure I. 2. Typical arrangement of blackbody technique for normal spectral emissivity measurements (DeWitt and Nutter 1988).  spectral response over the wavelength range of emitted radiant energy, or if the sample is a gray body emitter.  In the calorimetric technique the emissivity or absorptivity is measured from the heat lost or gained by the specimen through radiative heat transfer (DeWitt and Nutter 1988, Matsumoto and Ono 1993, Naws and Pouilloux 1994, Kola et al. 1995). The technique is suitable for total  213  emissivity and usually hemispherical total emissivity. One common factor in calorimetric methods is that the sample is thermally isolated from its surroundings so that essentially all heat transfer to or from it is by radiation. The rate of heat transfer is evaluated from the rate of temperature change of a sample of known heat capacity in dynamic measurements, or from the measured heat transferred to orfromthe sample to maintain it in thermal equilibrium in steadystate measurements. An example using calorimetric technique for measuring hemispherical total emissivity is shown in Figure I. 3. The sample, in the form of a long strip, is heated in vacuum by passing a current through it. If the sample is of uniform cross section and resistance, a section at the center quickly comes to a uniform equilibrium temperature. A small length of this central portion is instrumented with thermocouples, which are also used as potential leads. After the sample has come to equilibrium, its temperature, the current through it, the potential drop across its central section and the surrounding temperature are measured, allowing the total hemispherical emissivity to be determined. The overall accuracy of the emissivity measurement is dependent on the magnitude of the temperature difference between the sample and its surroundings and the relative magnitude of heat losses by conduction in the atmosphere of the chamber, through thermocouple leads andfromthe ends of the sample. With careful design and operation, accuracy on the order of ± 2% is easily attained (Touloukian and DeWitt 1970, DeWitt and Nutter 1988).  There are also two more recent methods for measuring spectral emissivity (Eckert and Goldstein 1976): (1) measurement of radiation emitted from a surface at a known temperature; (2) measurement of radiation reflected from a surface. Fourier Transform Infrared (FT-IR) Spectrometers have been used for spectral measurements of directional radiance  214  CONN. TO VACUUM PUMP  SEALED WIRES  TOP VIEW WITH SHELL REMOVED COOLING-COILS FOR BOTTOM PLATE  WATER-COOLED BUS BAR, PAINTED BLACK  WATER-COOLED BRASS SHELL. PAINTEO BLACK ON INSIOE  VACUUM TIGHT, ELECTRICALLY INSULATED SEAL  SYLPHON BELLOWS  WATER BEARING COPPER  WATER  OUTLETS  TUBE  WATER-COOLEO INPUT AND OUTPUT LEADS ROD FOR-1 WEIGHTS  CONN TO VACUUM PUMP  Figure I. 3. Typical arrangement of calorimetric technique for hemispherical total emissivity measurements (DeWitt and Nutter 1988).  and hemispherical-directional reflectivity to determine directional emissivity and surface temperature (e.g. Markham et al. 1993). Figure I. 4 shows a schematic of the FT-IR facility for measuring spectral emissivity employed by Markham et al. (1996). A sample is supported in a chamber facing a quartz window and cooled by water. A fiber/lens assembly (labeled "a") collects radiant energy from the sample. Assembly "b" provides  215  radiant energy from a high intensity lamp to the measurement spot at the specular reflection angle for "a" and "c' to "h" simultaneously provide radiant energy from a second lamp at progressively larger reflection angles. The timing circuit provides a means of synchronizing the position of the chopper blade with the FT-IR data collection system. Validation measurements indicate that the accuracy for spectral emissivity generally lies within ±2%. FT-IR has also been used to measure spectral emissivity with high temperatures samples heated with either an oxygen/hydrogen, propane, or oxygen/acetylene torch to 2000 K (Markham et al. 1990).  Experiments and tests have been carried but to determine the hemispherical total surface emissivities  of stainless steel 347 and stainless steel 316 using both the  calorimeter and radiometer methods (Wilkes 1954, Betz et al. 1957, Wade 1959, Tierney 1990). The specimens of stainless steel 347 were oxidized in air at 1367 K or gradually oxidized from 300 to 1367 K, while the specimens of stainless steel 316 were polished and cleaned before the tests. These experimental results have been adopted and recommended by Touloukian (1967), Touloukian and DeWitt (1970), Ozisik (1973), Thomas (1980), Incropera and DeWitt (1990), Siegel and Howell (1992) and Modest (1993). The values of the measured emissivities are listed in Tables I. 1 and 2.  Total emissivities of both stainless steel 347 and 316 increase with temperature, as shown in Figure I. 5. The emissivity of stainless steel 316 increases with increasing temperature rapidly because of oxidation on the surface while the emissivity of stainless steel 347 increases with temperature from 0.87 to 0.92 in a broad temperature range 600 K to 1300 K. Figure I. 5 indicates that the emissivity values of stainless 347 are consistent with each other, while the  217  Table I. 1. Measured values of surface emissivities of stainless steel 316. Symbol  Reference  O  Wilkes (1954)  V A  Betz et al. (1957) Touloukian(1967)  O  Touloukian and Dewitt (1970) Incropera and DeWitt (1990)  © X  * ffl  •  Siegel and Howell (1992) Modest (1993)  Temperature (K) 310-1282  Total Emissivity (SS316) 0.28-0.78  444-1222 300-1100 200-1405 423-1221  0.10-0.49 0.10-0.50 0.30-0.78 0.25-0.63  300-1000 300-1000 800-1000 800-1200 480-1310  0.17-0.30 0.22-0.35 0.33-0.40 0.67-0.76 0.24-0.31  302  0.17-0.28  Surface Preparation polished and cleaned rough surface rough surface 0.05 -0.38 ixm polished and cleaned polished cleaned lightly oxidized highly oxidized polished  Error (%)  + 10 ± 10  polished and cleaned  Table I. 2. Measured values of surface emissivities of stainless steel 347.  Wade (1959)  Temperature (K) 589-1367  Total Emissivity (SS347) 0.88-0.92  A  Toulpukian(1967)  589-1367  0.88-0.92  •  Touloukian and Dewitt (1970) Ozisik (1973)  589-1367  0.88-0.92  600-1200  0.88-0.92  *  Thomas (1980)  600-1200  0.88-0.92  T  Tierney (1990)  400-800  0.75-0.89  Incropera and DeWitt (1990)  600-1200  0.87-0.90  Symbol  Reference  B  •  •  218  Surface Preparation oxidized at 1367 K oxidized at 1367 K oxidized at 1367 K oxidized at 1367 K oxidized at 1367 K oxidized at 833-973 K stably oxidized  Error (%)  emissivities of stainless steel 316 show considerable scatter. The errors in the emissivity measurements have been reported to be less than ± 10% for a specific case (Touloukian and DeWitt 1970). The SS 347 values used in this work was 0.90, consistent with Figure I. 5, and it is believed to be accurate to ± 0.02. The emissivity values of unoxidized stainless steel 316 are seen to be in the range 0.1 to 0.3 for temperatures up to about 800 K. 0.3 was taken as the estimated radiation in this work because the SS316 surface had to face to a harsh environment in the combustor (involving impacts by burning particles, exposure to sulfur species, etc.). A lower emissivity value for the unoxidized surface would result in underestimating the radiative transfer by the differential emissivity method as discussed in Chapter 5. In fact, as outlined in Section 5.2.5, the emissivity of stainless steel 316 probably increased to 0.45 - 0.55. Hence there  1.0 0.9 h 0.8 0.7  O  &  o  0.6  O 8°  0.5  .53  a. w  o  °o  CP  0 4  © °  o E-t  O  ®d  o . o  o o  0.2 0.1 0.0  200  400  600  800  1000  1200  1400  1600  Temperature, T (K) Figure I. 5. Measured and recommended total emissivities of stainless steel 347 and 316. Symbols are listed in Tables I. 1 and 2.  219  is considerable uncertainty in this value, causing the differential emissivity method to be interior to the window method.  In summary, there are clear differential in emissivities between a polished and cleaned stainless steel 316 surface and well oxidized stainless steel 347. These result in measurable differences in total heat transfer that can be attributed to differences in radiative transfer. However, whereas the SS347 surface emissivity can be estimated with confidence, the SS316 surface emissivity is subject to considerable uncertainty and almost certainly changed significantly during the tests. As a result, the differential emissivity method is of limited utility for accurate determinations of radiative heat transfer.  220  APPENDIX  II:  THERMAL  PROPERTIES  OF  THE  WINDOW  MATERIAL-ZINC SELENIDE Key thermal properties of zinc selenide are plotted below based on data given by Touloukian and Dewitt (1972).  a  o  -§ e  3  •a  CO  400  500  600  Temperature (K)  Figure II. 1. Thermal conductivity of zinc selenide as a function of temperature. 1.0  ->  1  >  r  0.9 CO o  iS o •4-*  &  o  • •  0.8 0.7 0.6 0.5 0.4 0.3  13 D. CO  •  0.2  •  0.1 O  0.0  i  i_ 10  _i  i  15  i  i  20  i  i  _  25  Wavelength (um)  Figure II.2. Overall spectral absorptivity of zinc selenide at 373 K. The integrated average absorptivity for wavelengthsfrom2.5 to 18.3 micrometers is 0.10.  221  The transmissivity of zinc selenide as a function of temperature is shown in Figure II.3. The experimental values of the total transmissivity of zinc selenide are in the range of 0.61  to  0.66 at temperature from 25 to 840 °C (Touloukian and DeWitt 1972, Han et al. 1992). 0.65 was used (see Figure 2.4) to estimate radiative heat flux in Chapter 5 because the temperature of zinc selenide window for suspension temperatures of  800 - 900 °C were about 400-560 °C. The heat  transfer fluxes evaluated at the window transmissivities of 0.60 and 0.70 are within -5.6 to 6.8 % deviation from the heat fluxes evaluated at transmissivity 0.65.  The deviations of heat transfer  fluxes for all the window method radiative fluxes given in Chapter 5 with particles of mean diameter 334 um and 286 um are shown in Figure n.4.  In this case, given that the window  temperatures were always lower than 560 °C, it is reasonable to take 0.63 and 0.67 as outer limits on r. The deviations of radiative heat transfer fluxes then decrease to -2.4 to 2.6 %.  0.80  1  '  1  • •  0.75  '  1  '  1  '  1  1  1  1  1  1  1  1  1  >"  Touloukian and DeWitt (1972), (see Figure 2.4) Han et al. (1992)  0.70  0.65  |  co  0.60 O  H  0.55  0.50 0  100  200  300  400  500  600  700  800  900  Temperature, T (°C) Figure II.3. Total transmissivity of zinc selenide as a function of temperature.  222  -i  140  1  p  1  •  T =905 ± 5 °C  •  T =851±6°C  n  '  1  1  r  n  '  r~ (a)  b  U  120  T =804 ± 9 °C K  100 X  80  PH  t3 u .E5 t3 T3  (2  60 40 20 10  _i_  20  30  40  _i_  50  Suspension Density, p  60  70  80  (kg/m ) 3  susp  160 140 120  (b)  •  T =913 ± 9 °C  •  T =850 ± 6 °C  A  T =794±ll°C  b  1  b  I:  b  100 PH  to  a u >,  ••8 '-3  80 60 40 20 20  30  40  50  60  Suspension Density, p  susp  70  80  90  (kg/m )  Figure II.4. Analysis of the uncertainty of the transmissivity of zinc selenide window on the estimates of radiative heat flux, (a): d =334 um; (b): dp=286 um. p  223  APPENDIX III.  EXPRESSIONS F O R V I E W F A C T O R S  To compute radiation exchange between any two surfaces separated by a nonparticipating medium which does not absorb, emit or scatter radiation, we must consider the geometric orientation of the surfaces with respect to each other. The concept of view factor is introduced for convenience in the analysis of radiative heat exchange.  777.2. Definition of View Factor Consider two arbitrarily oriented diffusely-emitting, diffusely-reflecting surfaces Ai and • Aj maintained at uniform temperatures Ti and Tj , respectively. Let d A and dAj be two differential surfaces on Ai and Aj, respectively. The two differential surfaces are linked by a line of length S, which forms the polar angles 0i and 0j, respectively, with the surface normal unit vectors ni and nj. Figure III. 1 shows the geometry and coordinates considered.  Figure III. 1. View factor and solid angle between two arbitrarily oriented surfaces.  224  The view factor FAJ-AJ,S from a finite surface Ai to a finite surface Aj separated by distance S is defined as Radiative energy leaving surface A that strikes Aj directly Radiative energy leaving surface A in all directions in the hemispherical space f  Ai-Aj,s  s  The rate at which radiation leaves dAi and is received by dAj may be express as dqj_j = Ij cosOj dAj dcOj_j  (III-l)  where Ii is the intensity of the radiation leaving surface i and d j-i is the solid angle subtended by dAj when viewed from dAi. Substituting d j-i =cos0jdAj/S into Eq. (III-l), we obtain 2  cos9;cos0j , dA dAj  dq =I -T^2 i  H  1  ;  (01-2)  Since the surfaces emit and reflect diffusely, the radiosity Ji= Ii. Then, The total rate at which radiation leaves surface i and is received by surface j may be obtained by integrating over the two surfaces, it follows that f  COS9;COS9;  F  Assuming that radiosity Ji is uniform over the surface i, we obtain an expression for view factor FAJ-AJ,S  according to its definition: = J  qi_j =_ 1 r rcos9j cos9j dAi  T T T 11 A  i  J  i  A  iA,Aj  n  S  "  (III  4)  Similarly, the view factor FAJ-AJ,S may be written as q= i  l  f  cos9:cos9j  f  Comparing Eq. (III-4) and Eq. (III-5), we see that FAi-Aj.sAj  = F ._ . Aj  225  A  A  ; S  (III-6)  Eq. (III-6) is called the reciprocity relation and is useful in determining one view factor from knowledge of another.  III.2. View Factors In Cylindrical Cavity Radiative heat exchange inside a cylindrical cavity depends on two diffuse view factors including (1) an elemental circular band and an elemental circular ring, and (2) two elemental coaxial circular bands.  III. 2.1.. Diffuse View Factor between Elemental Circular Band and Ele Circular Ring Figure III.2 shows a cylindrical cavity of a radius R, an elemental circular band of width dx, and an elemental circular ring of width dr.  circular ring  circular band  dr  X  dx  Figure III.2. Diffuse view factor between an elemental circular band and an elemental circular ring.  226  The following notations are helpful in characterizing various diffuse view factors in the subsequent analysis: elemental diffuse view factor from a ring (r,dr) to a band (R, dx) separated by a  dFdr-dx,x:  distance x; Fdr-R,x:  diffuse view factorfroma ring (r,dr) to a disk of radius R at a distance x apart;  FR-dr,x:  diffuse view factorfroma disk of radius R to a ring (r,dr) at a distance x apart;  FR- ,x'.  diffuse view factorfroma disk of radius R to a disk of radius r, which are  r  parallel and concentric and a distance x apart;  By the law of conservation of energy, the radiative energy leaving ring (r,dr) that strikes band (R,dx), that is , F _ DR  DXX  , is equal to the radiative energy leaving the ring (r,dr) that strikes the  disk of radius R at x, that is, F _ , minus the radiative energy leaving the ring (r,dr) that strikes D R  R X  the disk of radius R at x+dx, that is, F _ DR  ^dr-dx,x  u  =  Fdr-R,x  R ( x + d x )  . Hence, we write  F^J^^+^J  —  = — —  (F^R , ^dx x  (IH 7) -  From the view factor reciprocity relation we have (27rrdr)F _ =(TcR )E _ 2  dr  R>x  R or  F _ dr  = R>x  R  pR-dr.x  where F _ ( R  r+dr)x  =  dr>x  (LII-8b)  F  dr x  (ffl-8a)  2  2 j " R-dr,x rf  By the law of conservation of energy F _  R  r  can be written as  FR-Cr+dO.x ~ ^R-r.x =  — (^R-r.x ^^T  (TH~9)  is the view factor from the disk of radius R to the disk of radius r+dr at a  227  distance x apart.  By substituting Eqs. (III-8b) and (III-9) into Eq. (III-7), we obtain  ir _a_  dFdr-dx.x where F _ R  r x  2r dx or  ( R-r.x) F  dx  (III-10)  is given by Jacob (1957) as R + r ' + x - , ^(R'+r'+x /(R +r +x ^ ) - 4 R.2_2 'r 2  L  z  z  2z  2  (III-l 1)  ^  R-r,x  2R Substituting Eq. (III-l 1) into Eq. (Ill-10) and performing the differentiation, we obtain 2  ^ 2 dFdr-ax,x =  2  2  R +x - r  2xR  (  2 R  +  x  2  -dx  2V  +r  (111-12)  2_2  -4r R  The diagram for the probe calibration is shown in Figure III.3 with y=L - x and dy= -dx. Hence  OPENING  CLOSED END  2R  L-y  X  Figure III.3. Geometry of cavity cylinder.  228  2  dF _ , _ =-2(L-y)R dr  dy  L  2  R +(L-y) - r  2  dy  y  (R +(L-y) +r ] -4r R 2  2  2  2  2  (III-13)  2  Additionally, by utilizing the reciprocity relation between view factors, it follows that dRdy-dr,L-y  27crdr dF _ 2rcRdy dr-dy,L-y  (111-14)  dr  Eq. (Ill-13) can be written in terms of dimensionless variables (Ozisik 1973) as 4(--y') +l-(r') 2  dF •  . • L  •  =4(  2  •dr  y) ' 4 ( ^ - y ' ) + l + (r') ] -4(r') 2  2  (III-15)  2  where r'=r/D and y'=y/D.  III.2.2. Diffuse View Factor between Two Elemental Coaxial Circular Bands Figure III.4 shows two elemental circular bands of widths dxi and dx2, respectively, separated by distance x.  circular band  Figure III.4. Diffuse view factor between two elemental coaxial circular bands.  229  The following notations help to characterize various diffuse view factors in the analysis below: elemental diffuse view factor from band (R,dx2) to band (R, dxi) at a distance x  dFdx2-dxi,x:  apart; diffuse view factorfroma band (R,dx2) to a disk of radius R at a distance x apart;  Fdx -R,x: 2  We first evaluate the elemental diffuse view factor  dFdx -dxi,x. 2  By the conservation of  energy, we write  (  ^ dx -dx ,x r7  2  Fdxj-R.diskatXi+dx,  =  1  —  F  =  dx -R,diskatx, 2  (III-16)  T" ^dx -R,x -  2  where F d x - R , x is diffuse view factorfromband (R, dx2) to a disk of radius R at a distance x apart. 2  From the reciprocity relation, we may write (27 Rdx )F. _ C  2  so that where  = (7iR )F _  (ffl-17a)  2  R>X  R  dx  F ^ ^ J L p ^  FR-<IX ,X 2  (III-17b)  is the diffuse view factor from a disk of radius R to a band of thickness dx2, a  distance x apart.  By the law of energy conservation F R . ^  ^R-dxj.x  =  f R-R,diskatx T  2  —  can be written as  pR-R.diskatCxj+dxj)  =  ~  (^R-R.x ) " 2 X  <3x  2  where F _ R  Rx  V  (III-18)  '  is the view factor from the disk of radius R at xi to the parallel disk of radius R at  X2, separated by a distance x.  230  Substituting Eqs. (Ill-17b) and (III-18) into Eq. (Ill-16), we obtain R  dF,dx -dx ,h 2  d  (III-19)  ^R-R,h ) ^ 1 X  1  2 6x^2  FR.R, is given by Jacob (1957) as x  2R + x - ^(2R + x J - 4R 2  L  2  2  R-R,x  2R  2  4  (111-20)  2  By substituting Eq. (111-20) into Eq. (Ill-19) and performing the differentiation, we obtain  ( X  X  2  X  2 -  X  + 6R  l )  l  dxj  (111-21)  4R + ( x - x , j 2  Here the absolute value x= | X2-X11 is used because the view factor depends on the absolute value of the distance of separation between the bands. Eq. (Ill-19) can be" written in terms of dimensionless variables as  ( (  dF,d(x )'-d(x,)',x  X  2 ) ' - (  X  +3  . )  1- (x ) -(x,)  d(x )  2  t  2  1 +( ( x ) ' - ( ) J 2  x , with D=2R. where (x.) = — and (x ) = — D D 0  2  231  X l  (in-22)  APPENDIX IV: D E R I V A T I O N O F E Q U A T I O N (3-19)  Consider the two surfaces shown in Figure IV-1.  /0\  qfa.1 = F 1-2.S  = e T,4+(i. o  1  p,)q««.2  Figure IV. 1. Heat exchange between two coaxial disks.  A complex radiative exchange occurs between the two surfaces as radiation leaves a surface, travels to the other surface, and is partially reflected. The quantities qin and q t are the ou  rates of incoming and outgoing radiant energy per unit area, respectively. A heat balance for the surface area provides the relation Q i = q i A i = ( q u U - q i n , i ) A i 0  i = l,2  (IV-  The energy flux leaving the surface is composed of emitted energy plus reflected energy. Thus q« =e o^+p q J  where p = l - a = l - e {  ;  ;  i  i  =e orl +(l-e )q 1  i l l f !  i  i  i  i l l i i  i.= l,2  (TV-2)  with i = 1, 2 for opaque gray surfaces. The incident flux qin,i is the  energy leaving surfaces 1 and 2 that arrives at surface i, i.e.,  232  qin,i = ZqoutjFi-j  i = l,2  (IV-3)  j=l  For flat surfaces, since the surface cannot view itself, Fi-i = 0. From the view factor reciprocity relation F _ A - F _iA , and Eqs. (IV-1) and (IV-3) can be rearranged to give 1  2  1  2  2  Y = L  A 7 A  T  ;  L  ^ ( ^ T  l-e  4 1  - q  o  u  t  i  i  i = l,2  )  (IV-4)  ;  i = l,2  = qou«>i-EF1_jqoutJ  i  (TV-5)  j=i  The solutions of Eqs. (IV-4) and (TV-5) give Q  e aT [l-(l-e )(F ) A, / A T - e F a T q. = ^ A, l-(l-e )(l-e )(F ) A /A 4  1  1  2  2  12  2  4 1  e i  2  2  1  233  2  1 2  1  2  1 2  4  (IV-6)  APPENDIX  V:  DERIVATION OF  EQUATION  (3-37)  USING  NET  RADIATION M E T H O D  The heat fluxes entering and leaving the ZnSe window and the SS347 surface behind the window are shown in Figure V . l and V.2, respectively.  q  c  q  qL  r  ZnSe window  qw,s  qout.w  qin.w  Figure V.l. Diagram illustrating heat balance for ZnSe window.  qw.s  qin.s  qout.s  sur.s  SS347 probe head behind ZnSe window  qm  Figure V.2. Diagram illustrating heat balance for SS347 surface behind ZnSe window.  234  The heat balance for the window (see Fig. V.l) is: qc+q +qin.w  =q ut,w+qL+qw.s  r  (v-i)  0  where = q ut. = e aT^_ + R'q  qin.w  qin.  q  0  s  s  s  (V-2)  out>w  = q ut.w = q + e aT^ + Rq^  -  T  s  r  0  w  =tqin.w+e o-T  L  w  w  4 0  utw  +Rq  w  (V-3) (V-4)  r  Solving Eqs. (V-2) and (V-3), we obtain: R'Tq,+e, Tl,+R'e g<„  "  q  4  4  xa + e oT v  1r  + e RaT  ^ w ^ ^m.w  s  sur.s  nr  IHS  - ~  q  ^  w  0  =  =  The convective heat transfer from the window, q  W)S  s\  ^  , is determined by substituting Eqs. (V-4),  (V-5) and (V-6) into Eq. (V-l), giving 1-R-  R't \ , (l-R-x)e aT q . =q +(i-R-. ^ ~ „ „„.)q +l-RR' l-RR'' l-RR' (l-R'+cR')e a< w , ss  2  n  |  %  c  s  r  w  w  4 s  j r i S  -  (V-7)  4  w  l-RR'  out.w  The heat balance for the probe head behind the zinc selenide window (see Fig. V.2) is (v-8)  qw.s+qm.s=qout.s+qm q in. = q o^.w = *q + w ^ , e  s  r  = qin.w = e,&rL,+ Solving Eqs. (V-9) and (V-10), we obtain: 0  T =  qr+ w^T e  -  Q  (v-9)  + Rq „*,  (v-io)  qin,  R,  q ut,s  n  w  in)W  s  +e RaT t s  s  iS  !_RR.  (  1 — 1 — RR Substituting Eqs. (V-l 1) and (V-12) into Eq. (V-8) leads to q ut,s - s s u r , s + e  al  0  _  a  q . w  s  n  -q ut, 0  +  +  s  _  n  q  m  n  _ -  4 P  f  y  e  s  a l  T  sur,  K  -  p.^qr+ew^+e.RaT^ U-R ; — — —+q  1 — RR  235  "  N  )  (V-l2)  ~  n s  V  m  (v-13)  Equating the right hand sides of Eqs. (V-7) and (V-13) gives 1 — R'  q r - 7 — — ^qr+q'wr+Rq r) + qm = q c + A q + B q - C q ' - q P  P  1 - RR  q  r  1 — R' R-B)q - q +(C-  +(1-  q = r  A+  p r  1 — R' )q' +q —  w  w r  (V-14)  _  (V-15)  T  1-RR' where T Qpr  4  sur,s  q wr  in,w  qwr  out.w  and A-l-R.  '- ' R  1-RR"  1-RR*  1-RR*  C=  I-R'+R'T  1-RR*  Then, rearranging Eq. (V-15), we obtain: _  T q m + q  =  l-RR'  q p r  "  R'T , q c + q w r  +  l-RR'  q w r  ^  1-RR' Simplifying the above equation results in: qm+^qpr-qc+qwr+^q'wr  q = r  -  (V-17)  ?  where: 1-RR' T  1-KK' 1-RR' R'T  2  1-RR'  236  A P P E N D I X VI:  INFLUENCE O F RADIATION F R O M  REFRACTORY  W A L L S T O T H E D U A L TUBE SURFACES  A top view of the dual tube surface placed in upper part of the riser is shown in Figure VI. 1.  p  RISER  2  LL. LU UL  it— '  UNOXIDIZED TUBE  OXIDIZED TUBE  • Z LU  -e  Q Q  Q <  28  1  32  1  32  1  32  1  7.8  28  Figure VI. 1. Schematic of dual tubes placed in the riser. Dimensions are in mm.  The tube length and the refractory length are much greater than the diameter of the tube so that the tube and the refractory are approximated as being of infinite length. Radiation from an infinitely long cylinder to a parallel, not symmetrically placed, infinite rectangle is shown in cross-section in Figure VI.2.  Figure VI.2. View factor between cylinder and parallel rectangle.  237  According to Siegel and Howell (1992), the view factors between the tube and the refractory wall behind the dual tube can be determined using - i . .„_-i tan" - +tan" b + av c cJ r  F^-tube  (>  b  a  1  (VI-1)  F ^ ^ ^ ^ - ^ t a n - ^ + tan- ^  (VI-2)  1  1  Since the oxidized and the unoxidized U-tubes were placed in the riser at equal distances from one of the bisecting lines of the cross-section, we need only consider one of the U-tubes. Each U-tube is comprised of a pair of separate cylindrical legs at the positions shown in Fig. VI. 1. The radiative heat flux between the tube and the refractory wall adjacent to the tube is calculated by 4  4  Qwaii.tube ~ A ^ F ^ ^ ^ e ^ a T ^ , — A ^ F ^ ^ ^ e ^ a T ^  (VI-3)  Using the reciprocity relation Siegel and Howell (1992), Eq. (VI-3) can be rewritten 4  4  Qwaii.tube A ^ F ^ ^ ^ o ^ e ^ T ^ , — e ^ T ^ ) =  (VI-4)  To estimate the maximum influence of the radiation from the refractory behind the U-tubes, we assume that (1) the emissivity of the refractory wall is 0.5; (2) the refractory temperature is the same as the suspension temperature; (3) there are no particles between the tube and the adjacent wall. The calculated view factors and radiative heat fluxes at a wall temperature of 850 °C and a tube temperature of 80 °C are listed in Table VI. 1. The radiative heat transfer coefficients based on the full tube surface for the conditions investigated in Chapter 4 are in the range of about 40 to 100 W/m K for the oxidized tube. The corresponding total radiative heat fluxes are about 940 to 2350 W.  238  Table VI. 1. View factors calculated using Eqs. ( V I - 1 ) and ( V I - 2 ) . oxidized tube left leg right leg  unoxidized tube right leg left leg e a b c  2 1 7  Fwall-tube Ftube-wall Qwall.tube  6  0.3 8 (mm) 2 4 (mm) . 8 (mm) 0 . 1 1 7 0 . 4 4 7 0 9 ( W )  6 9 7  6  0.3 0 (mm) 2 (mm) . 8 (mm) 0 . 1 2 2 0 . 4 6 6 3 5 ( W )  0.76 9 2 (mm) 6 0 (mm) 7 . 8 (mm) 0 . 1 2 2 0 . 4 6 6  6 2 9  ( W )  1 2 7  6  0.76 2 4 (mm) 8 (mm) . 8 (mm) 0 . 1 1 7 0 . 4 4 7 0 4 ( W )  Hence, the radiation from the refractory wall adjacent to the tubes could account for an appreciatefraction( 2 5 to 6 6 % based on this simple worst case scenario) of the radiation from the refractory walls and suspension.  Besides the influence of the refractory wall adjacent to the dual tube, the radiation from the other three refractory walls must also be considered. As the radiation passes through an absorbing, emitting and scattering medium, its intensity is reduced by absorption and scattering. The  intensity attenuated after the radiation passes through length S can be determined by  Bouguer's Law I(S) 1(0)  = exp  JK dS* t  (VI-5)  where K is the extinction coefficient of the media. It can be estimated (Tien and Drolen 1987) t  using 1-s K, = 1 . 5  The mean penetration distance of the radiation is defined Siegel and Howell (1992) as  2 3 9  (VI-6)  L  = ] s K e x p - J K d S ' dS  m  t  (VI-7)  t  L  o  o  For constant K , integration gives t  L  m  = K ]sexp(-K S) dS = — t  t  '  (VI-8)  t  o  The extinction coefficient K is calculated to be 34.6 m" from Eq. (VI-6) for an average solids 1  t  concentration of 0.0077 (corresponding to p =20 kg/m ) and a mean particle diameter 334 pm. susp  The mean penetration distance is then estimated using Eq. (VI-8) to be 28.9 mm. If the optical thickness or opacity, T=S/Lm »  1, the medium is optically thick. In the case studied, the riser is  152 by 152 mm square cross section. The optical thickness is 5.3 so that the medium can be considered optically thick. For higher suspension densities, the medium gets optically thicker, and fewer photons can penetrate through the medium. According to Eq. (VI-5), only 5% of the original intensity will penetrate through the path length of 0.087 m, while the intensity reduces to 1% of the original intensity for a path length of 0.133 m. Consequently, radiationfromthe other three refractory walls is attenuated and has little chance of reaching the tube surfaces, so their contribution can be neglected.  240  APPENDIX VII: E X P E R I M E N T A L H E A T T R A N S F E R D A T A The suspension-to-surface heat transfer coefficients obtained in this project using the window method and differential emissivity method are listed in Tables VII. 1 to VII.3 as functions of suspension density and suspension temperature for different sizes of particles.  Table VII. 1. Suspension-to-membrane-wall heat transfer coefficients for various surfaces used in this work as functions of suspension density and suspension temperature. T  b  (°C) 703 702 708 702 708 704 710 708 . 708 704 704 800 804 808 807 806 800 806 797 804 802 806 807 808 807 808 808  Psusp  h (W/m K) 131.4 132.7 117.8 124.5 183.1 111.5 75.1 84.9 80.8 102.8 123.7 157.5 133.0 125.8 142.7 133.3 171.5 179.7 187.9 184.1 200.8 192.1 171.0 155.1 135.6 137.7 136.9 t  (kg/m ) 50 43 59 65 71 54 26 18 24 40 57 46 40 32 21 19 47 49 54 55 64 51 38 41 28 23 28 3  2  241  h (W/m K) 127.2 123.1 103.7 111.8 183.2 91.2 61.9 69.7 70.1 95.0 98.1 161.7 130.0 123.4 152.7 144.5 174.9 192.7 198.2 191.7 208.3 201.1 181.9 159.6 144.0 141.7 145.8 p  2  h (W/m K) 141.8 156.2 152.0 155.0 184.2 159.8 105.4 120.4 105.6 121.2 185.8 148.8 140.8 132.0 121.0 108.8 164.8 151.4 165.6 168.0 185.0 172.8 147.0 145.8 117.4 129.6 117.6 f  2  Table V H 2 . Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 4 9 8 pm). p  T  h316  h347  h ,347  (°Q  (kg/m )  (W/m K )  (W/m K )  (W/m K )  8 0 2 8 0 3 8 0 0 8 0 5 7 9 5 8 0 0 8 0 1 7 9 8 8 0 0 8 0 2 8 0 5 8 0 4 8 0 3 7 9 6 8 0 6 8 1 0 8 0 9 8 1 0 8 0 2 7 9 4 8 5 0 . 8 5 3 8 5 6 8 5 3 8 4 6 8 5 3 8 5 3 8 5 5 8 5 1 8 5 0 8 5 0 8 4 9 8 5 2 8 4 8 8 5 1 8 5 1 8 4 8 8 5 2 8 5 4 8 4 8  4 7 6 2 4 9 5 4 5 9 6 0 6 4 7 6 5 7 81 7 3 5 4 61 6 2 5 5 4 6 3 5 4 4 3 8 3 5 6 0 7 0 5 6 7 8 71 5 8 5 6 5 6 6 2 7 5 6 8 5 9 5 6 5 2 5 0 3 8 4 0 4 1 3 6 3 2  109.3. 118.4 113.9 119.8 119.5 1 2 0 . 9 126.6 129.4 135.0 134.5 129.9 115.3 120.2 125.4 117.1 109.1 105.4 107.1 108.2 101.5 132.1 135.5 129.2 142.3 1 3 7 . 7 135.4 137.3 133.1 134.1 143.7 134.8 130.6 129.8 124.7 126.0 116.6 115.3 113.9 114.3 109.5  141.2 165.4 1 4 6 . 0 151.9 155.6 1 5 8 . 0 161.4 171.2 154.6 180.3 1 7 0 . 0 148.8 157.8 159.3 1 5 4 . 6 141.3 131.6 139.6 138.1 134.3 167.5 176.4 163.8 184.9 181.1 166.8 164.0 162.0 174.5 184.4 174.9 167.5 169.4 165.5 1 6 2 145.8 148.7 1 4 6 . 9 142.9 136.9  b  Psusp 3  2  2 4 2  2  r  2  5 2 . 7 7 7 . 7 5 3 . 0 5 3 . 0 5 9 . 6 61.3 57.5 69.1 5 0 . 6 7 5 . 7 66.3 "5 5 . 3 62.1 5 6 . 0 6 2 . 0 5 3 . 2 4 3 . 3 5 3 . 7 4 9 . 4 5 4 . 2 58.5 6 7 . 6 57.2 7 0 . 4 7 1 . 7 5 1 . 9 44.1 4 7 . 7 6 6 . 7 6 7 . 2 66.3 6 1 . 0 6 5 . 4 6 7 . 4 59.5 4 8 . 2 5 5 . 2 54.5 4 7 . 3 45.3  Table VII.2. Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 498 Lim). (continued) p  T  b  (°C) 900 908 912 904 902 902 901 902 899 900 903 901 902 903 899 902 900 900 . 898 904 898 899 901 902 901 899 901 902 900  Psusp  ll347  h316  (kg/m ) 72 66 59 63 72 59 73 69 77 81 76 73 70 62 48 54 56 57 33 52 38 41 43 43 45 41 51 45 48 3  (W/m K) 155.1 144.5 145.6 141.5 138.2 132.8 138.4 156.6 153.4 153.3 152.7 156.8 149.2 145.4 138.7 134.6 136.7 138.6 119.0 118.2 124.4 122.5 120.4 125.6 124.0 117.0 123.9 123.8 127.8  243  (W/m K) 196.2 189.7 186.9 181.5 188.7 181.9 189.0 197.0 201.9 200.8 195.5 202.9 191.1 184.0 175.7 175.1 175.5 175.5 151.0 161.0 160.0 159.8 157.2 161.5 161.7 148.6 164.0 154.8 168.7 2  2  r  h ,347 r  (W/m K) 65.8 72.3 66.1 64.0 80.8 78.6 81.0 64.6 77.6 76.0 68.5 73.8 67.0 61.8 59.2 64.8 62.1 59.0 51.2 68.5 57.0 59.7 58.9 57.4 60.3 50.6 64.2 49.6 65.4 2  Table VJJ.2. Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 334 pm). p  T  b  (°Q 794 792 797 794 803 805 807 809 806 801 804 812 812 812 812 800 807 797 797 794 798 847 850 857 859 854 851 850 846 850 854 849 849 854 855 852 854 845 853 850  Psusp  1*316  (kg/m ) 41 54 66 60 53 50 59 80 37 34 37 76 53 66 65 45 65 46 43 40 44 57 56 33 54 40 57 49 76 56 75 74 65 72 74 51 67 50 42 28 3  (W/m K) 134.2 134.6 143.5 134.2 139.9 149.4 149.0 141.2 141.7 142.7 139.7 150.4 155.3 155.3 152.9 142.9 146.5 133.6 136.0 134.1 131.0 146.0 154.8 151.1 148.4 140.1 151.9 157.1 159.7 145.4 151.7 155.0 152.6 156.3 160.7 154.4 160.9 155.7 142.5 143.1 2  244  h347  hr,347  (W/m K) 149.0 155.2 165.7 154.8 166.2 175.3 176.7 183.8 155.9 160.8 158.3 177.3 177.4 183.3 185.1 166.7 174.8 154.1 150.3 154.7 150.0 169.2 186.9 168.9 179.1 164.0 175.0 188.7 193.4 166.7 187.7 187.1 196.3 196.9 200.4 183.4 202.5 186.2 175.1 166.7  (W/m K) 46.8 51.1 55.0 51.1 65.2 64.2 68.6 80.6 41.9 44.9 46.1 66.7 54.8 69.4 79.8 59.0 70.1 50.8 44.2 51.1 47.1 53.7 74.2 41.2 71.0 55.3 53.4 73.1 77.9 68.2 83.3 74.2 74.1 78.4 81.0 67.1 83.4 70.5 75.4 54.6  2  2  Table VII.2. Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 334 um). (continued) p  T (°Q 852 843 840 851 855 850 853 851 855 903 906 901 896 903 902 907 906 900 903 907 909 907 906 907 903 906 904 901 906 909 900 902 909 906 904 902 903 903 902 b  psusp  (kg/m ) 56 39 28 21 34 42 32 18 43 63 57 51 58 53 41 53 57 53 59 69 64 70 65 61 65 68 73 51 64 73 46 34 40 42 37 44 40 36 31 3  h.316  h347  h ,347  (W/m K)  (W/m K)  (W/m K)  163.0 135.0 133.9 133.6 135.3 154.7 138.2 136.0 133.0 150.3 147.8 132.8 143.6 148.3 150.5 147.6 149.8 146.9 152.3 155.6 159.2 153.6 149.8 155.6 144.3 153.8 154.1 154.7 157.8 163.2 146.9 130.7 131.5 134.0 126.4 138.5 140.4 138.4 139.9  182.2 161.0 154.5 150.4 160.3 179.5 155.0 157.7 165.4 197.1 191.5 188.8 190.6 200.1 194.1 196.2 203.1 192.9 203.8 202.9 213.5 204.6 206.4 209.0 190.5 201.5 206.5 199.3 209.2 220.2 194.6 179.7 168.6 185.7 168.0 185.9 188.9 180.7 180.9  66.7 60.1 47.6 38.9 57.8 57.4 38.9 50.2 74.9 74.8 69.9 81.5 75.2 80.4 69.7 77.8 85.2 73.6 82.4 75.8 86.8 81.7 90.5 85.6 73.8 76.4 83.8 71.3 82.3 91.1 76.3 78.5 59.3 66.6 66.5 75.7 77.7 67.7 65.5  2  245  2  r  2  Table VII.2. Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 137 um). p  T  b  (°Q 806 805 805 798 798 798 797 796 797 798 800 797 800 801 798 801 800 796 797 799 802 801 803 795 806 804 796 796 796 799 804 801 798 801 802 798 795 795 804 802  Psusp  (kg/m ) 68 70 83 55 70 82 78 76 66 75 70 82 81 59 57 57 42 45 44 55 54 47 50 37 78 82 78 52 42 54 79 80 51 54 78 82 49 52 79 80 3  h316  h347  hr,347  (W/m K)  (W/m K)  (W/m K)  169.1 167.1 181.1 157.1 180.5 180.5 177.3 178.6 176.0 180.5 176.0 178.3 189.4 156.3 143.0 160.9 126.7 120.6 129.4 126.8 137.7 135.5 126.0 113.9 185.4 178.8 176.0 136.4 136.2 145.8 174.8 168.6 134.6 129.4 189.2 191.3 130.4 130.5 179.5 189.4  216.6 227.6 232.9 190.8 221.8 237.8 214.1 219.6 216.7 232.4 220.1 238.0 241.2 210.1 182.1 200.1 179.6 159.3 174.6 168.9 183.6 183.8 178.2 149.3 227.6 237.6 224.8 178,1 168.8 190.4 223.1 230.9 180.2 182.1 237.0 242.8 180.9 180.8 225.8 240.6  2  246  2  2  76.0 96.8 82.9 54.0 66.0 91.8 58.9 65.7 65.2 83.1 70.5 95.5 82.9 86.0 62.5 62.7 84.6 61.9 , 72.3 67.3 73.4 77.2 83.5 56.6 67.5 94.0 78.0 66.7 52.2 71.4 77.3 99.7 73.0 84.3 76.5 82.5 80.8 80.5 74.1 82.0  Table VII.2. Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 1 3 7 pm). (continued) p  T  ;  b  Psusp  h i6  h347  (W/m K)  (W/m K)  2 2 4 . 9 2 2 7 . 2 2 2 4 . 0 2 5 0 . 0 2 2 6 . 0 2 1 9 . 2 2 2 8 . 3 2 2 2 . 2 2 2 9 . 4 2 3 7 . 3 2 2 2 . 1 2 1 1 . 2 2 1 4 . 1 2 0 1 . 1 190.1 199.5 196.3 2 0 6 . 6 2 0 4 . 2 2 0 8 . 8 2 0 4 . 5 193.5 2 1 0 . 5 2 0 9 . 0 190.5 2 0 3 . 4 174.1 2 0 3 . 8 194.1 2 2 6 . 1 2 2 8 . 6 188.3 2 1 1 . 4 2 2 8 . 6 2 3 5 . 7 199.8 178.2 169.3 1 9 0 . 9 196.3  5 8 . 6 87.5 6 1 . 8 8 2 . 2 6 9 . 4 69.1 6 0 . 4 5 8 . 2 8 4 . 2 75.1 7 0 . 5 7 2 . 7 4 9 . 7 6 5 . 3 7 4 . 2 5 6 . 5 5 6 . 0 6 4 . 7 7 4 . 9 5 8 . 2 6 6 . 7 6 0 . 9 6 7 . 5 8 6 . 4 7 9 . 3 6 7 . 2 5 7 . 8 71.3 8 6 . 6 9 2 . 8 86.3 6 8 . 7 7 9 . 6 7 8 . 9 98.3 84.5 6 5 . 2 5 9 . 5 6 0 . 8 7 7 . 6  3  CO  (kg/m )  (W/m K)  8 4 9 8 5 0 8 5 2 8 5 1 8 5 4 8 5 2 8 5 2 8 5 0 8 5 1 8 5 0 8 5 2 8 5 1 8 5 0 8 5 3 8 4 4 8 4 9 8 5 2 8 5 0 8 5 0 8 5 1 8 4 9 8 4 9 8 5 0 8 5 1 8 4 6 8 5 2 8 5 0 8 5 2 8 5 1 8 5 2 8 4 7 8 4 5 8 5 1 8 5 2 8 5 3 8 4 8 8 4 1 8 4 7 8 5 4 8 5 2  7 6 83 83 9 9 7 4 7 2 8 5 7 8 7 8 8 8 7 0 6 6 7 9 5 8 4 7 5 4 6 3 6 9 6 6 6 7 61 4 4 5 5 5 8 4 4 4 9 3 3 6 0 5 2 8 0 8 0 5 2 5 3 8 3 8 5 53 3 5 3 4 5 2 4 3  188.3 172.5 185.4 198.6 182.6 176.0 190.5 185.9 176.8 190.4 178.0 165.8 183.1 160.3 143.7 164.2 161.3 166.1 157.4 172.4 162.8 155.4 168.3 155.0 1 4 0 . 9 161.4 138.0 159.2 140.0 168.1 174.7 145.4 161.6 179.3 174.2 147.0 137.4 132.1 152.9 147.8  3  2  2 4 7  2  hr,347 2  Table VII.2. Suspension-to-tube heat transfer coefficients as functions of suspension density and suspension temperature (d = 137 um). (continued) p  T (°C) 854 854 853 847 846 853 852 850 896 900 901 904 903 900 901 903 902 901 900 900 901 902 902 901 902 903 894 905 899 897 901 899 899 902 904 901 896 895 897 896 898 b  Psusp  (kg/m ) 78 78 80 52 49 51 53 81 52 59 56 61 60 54 58 50 51 50 45 44 46 38 40 38 39 39 26 26 21 23 48 42 51 38 48 31 18 13 12 29 41 3  h316  h347  h ,347  (W/m K)  (W/m K)  (W/m K)  178.5 174.1 183.2 164.7 151.7 150.3 148.4 188.3 172.4 177.5 170.1 165.3 169.7 145.7 166.0 145.2 145.4 168.1 139.6 147.7 149.6 139.3 137.3 135.4 133.3 143.2 116.4 118.9 123.7 123.9 149.6 133.7 154.6 139.9 151.7 132.0 116.7 114.9 113.0 124.7 138.6  228.0 240.5 238.3 210.1 183.1 198.5 198.7 244.4 210.3 224.3 214.5 206.7 221.1 203.1 216.9 200.1 200.3 214.5 193.8 193.8 202.6 188.7 191.1 189.0 186.5 186.3 169.9 165.4 164.4 157.9 192.0 183.3 199.4 178.0 195.9 173.7 145.3 145.5 145.1 156.6 176.6  79.2 106.3 88.2 72.6 50.2 77.1 80.5 89.8 60.6 74.9 71.0 66.2 82.2 91.8 81.5 87.9 87.9 74.2 86.7 73.8 84.8 79.0 86.2 85.8 85.1 69.0 85.6 74.4 65.2 54.4 67.9 79.4 71.6 61.0 70.7 66.7 45.7 48.9 51.3 51.0 60.8  2  248  2  r  2  Table VII.3. Suspension-to-probe heat transfer coefficients as functions of suspension density and suspension temperature (d = 334 im). j. p  T  b  CQ 407 407 407 407 407 794 792 797 794 803 805 807 809 806 801 804 812 812 812 812 800 807 797 797 794 798 847 850 857 859 854 851 850 846 850 854 849 849 854 855 852  Psusp  (kg/m ) 3  38 33 26 22 51 41 54 66 60 53 50 59 80 37 34 . 37 76 53 66 65 45 65 46 43 40 44 57 56 33 54 40 57 49 76 56 75 74 65 72 74 51  h-347  h ,347  r . ** Ur.win  (W/m K)  (W/m K)  (W/m K)  358.6 352.5 347.4 317.2 422.9 623.8 620.9 712.5 641.1 673.9 668.2 667.5 741.9 599.2 561.1 547.6 653.4 679.0 721.6 717.5 668.1 701.1 587.3 570.6 549.1 568.6 697.3 706.6 653.7 679.1 660.4 666.9 730.0 696.3 679.7 728.4 723.7 764.5 715.5 756.7 723.9  10.8 22.2 7.8 8.9 24.6 158.2 158.3 189.9 155.3 182.7 170.4 170.8 204.0 150.8 133.3 131.0 178.9 175.6 195.1 196.8 170.9 191.7 143.3 137.2 136.7 142.4 183.6 186.8 164.9 173.0 162.6 170.7 212.7 180.0 177.6 205.8 196.2 211.0 191.5 206.0 206.1  -  2  249  r  2  2  48.0 181.1 179.5 218.5 201.8 207.2 207.9 207.5 232.9 171.8 159.2 146.2 186.0 210.1 225.8 228.2 214.1 220.6 172.6 163.5 147.1 154.0 214.5 218.8 203.8 211.1 209.6 207.6 222.6 221.2 212.0 225.0 238.6 245.7 231.4 247.0 230.6  Table VII.3. Suspension-to-probe heat transfer coefficients as functions of suspension density T  b  (°Q 854 845 853 850 852 843 840 851 855 850 853 851 855 903 906 901 896 903 902 907 906 900 903 907 909 907 906 907 903 906 904 901 906 909 900 902 909 906 904 902 903  t i ** nr,win  h347  h,347  (kg/m )  (W/m K)  (W/m K)  (W/m K)  67 50 42 28 56 39 28 21 34 42 32 18 43 63 57 51 58 53 41 53 57 53 59 69 64 70 65 61 65 68 73 51 64 73 46 34 40 42 37 44 40  750.2 690.3 608.3 602.3 644.9 616.6 542.2 515.1 569.3 562.3 577.8 542.8 546.2 729.6 703.3 661.8 701.1 709.0 719.7 731.7 736.2 727.0 751.5 765.0 794.2 748.7 763.8 787.4 794.0 781.4 820.2 772.3 757.8 772.4 706.4 670.3 630.0 626.7 632.4 624.5 642.1  208.2 175.6 148.7 145.7 164.3 151.4 126.6 117.9 137.8 134.5 140.1 124.7 129.0 189.4 174.0 163.7 177.7 180.9 180.0 192.9 193.3 185.5 203.4 196.5 216.0 188.6 204.7 220.3 210.6 212.4 235.3 212.1 201.1 206.5 179.2 170.0 154.5 154.5 156.2 152.7 159.4  244.6 229.0 172.4 179.7 194.8 185.0 148.2 127.6 149.8 149.8 160.4 145.0 141.6 237.1 230.2 211.0 229.6 231.9 245.4 246.2 250.1 250.0 252.6 262.6 274.1 262.3 265.5 271.4 278.8 269.4 283.3 269.7 261.8 265.6 233.1 213.4 185.3 182.1 184.2 179.6 188.0  Psusp  2  3  -  250  r  2  2  Table VII.3. Suspension-to-probe heat transfer coefficients as functions of suspension density T (°C) 903 902 907 902 b  Psusp  (kg/m ) 36 31 28 34 3  1*347  h ,347  (W/m K)  (W/m K)  (W/m K)  633.7 621.0 360.7 568.2  155.9 152.6 140.0 155.6  188.0 178.9 162.9 152.4  2  r  2  Ur,win 2  Table VII.3. Suspension-to-probe heat transfer coefficients as functions of suspension density and suspension temperature (d = 286 xm). p  T CO 230 300 380 386 370 360 502 559 600 650 702 760 797 850 911 798 794 791 797 804 798 794 797 797 783 789 850 850 856 846 854 b  Psusp  (kg/m ) 35 35 35 35 38 39 44 42 43 40 43 50 38 44 46 62 62 54 63 67 64 65 38 34 33 51 45 75 78 67 53 3  I1347  h ,347  (W/m K)  (W/m K)  391.5 380.2 420.7 390.3 388.8 388.4 420.0 421.8 464.6 500.5 572.7 620.0 689.9 745.2 841.6 647.2 601.9 588.5 591.6 639.9 701.3 608.2 537.3 556.8 565.8 652.8 562.0 700.3 652.3 709.9 637.3  17.1 19.0 23.7 31.3 15.9 26.9 36.1 46.8 64.1 60.5 90.7 99.7 129.0 149.0 165.5 157.7 138.2 135.3 140.7 154.3 173.0 148.7 119.6 118.9 118.5 154.8 137.8 171.7 173.0 178.5 158.2  2  251  r  2  Ur.win  (W/m  2  K)  190.4 186.9 176.8 175.6 189.2 188.6 184.2 129.9 129.1 116.5 154.0 157.4 211.2 231.5 197.7 171.4  Table Vifl.3. Suspension-to-probe heat transfer coefficients as functions of suspension density T  b  (°C) 845 853 872 873 870 850 855 853 850 844 846 908 918 920 922 904 911 911 914 918 919  psusp  1*347  (kg/m ) 44 52 41 59 52 50 38 42 48 30 44 50 42 60 55 29 66 46 90 87 65 3  h ,347 r  , ** nr,win2  (W/m K)  (W/m K)  (W/m K)  576.6 623.1 672.7 662.0 634.1 617.0 553.7 513.9 547.2 511.0 566.3 724.4 744.3 689.5 688.9 632.3 599.7 647.0 760.6 765.7 774.6  134.9 147.7 156.8 161.8 157.0 148.3 118.5 114.6 130.9 111.9 133.3 167.0 169.9 166.3 169.7 132.7 144.7 151.5 210.0 199.0 193.3  158.4 166.4 177.3 180.0 168.6 171.2 129.0 146.3 159.9 122.3 154.3 198.9 172.4 219.3 222.3 167.3 205.2 178.8 250.2 260.7 234.4  2  obtained by differential emissivity method, obtained by window method.  252  2  APPENDIX VIII: P R O G R A M LISTING  PROGRAM 1 This program uses a red-black point-to-point relaxation method to determine the two dimensional steady-state temperature distribution in the probe head. ****************************************************** * alpha: over-relaxation factor * aw..ap,s: arrays of coefficients of finite differnce equation * ckk: array of conductivity values * delta: Stefan-Boltzmann constant (5.67* 10e-8 W/m K ) * dr: grid size in r-direction * dz: grid size in z-direction * eb: emissivity of bed * eps: convergence criterion * ebb: system emissivity of bed to probe body * ebh: system emissivity of bed to probe head * ebs: system emissivity of bed to probe sleeve * ewb: emissivity of probe body * ewh: emissivity of probe head * ews: emissivity of probe sleeve * hb: bed-probe convective heat transfer coefficient (W/m K) * hw: water-probe convective heat transfer coefficient (W/m K) * iter: iteration number * 1: number of bed temperatures investigated * m: node number in r direction * maxit: maximum iteration number * n: node number in z direction * niter: array for total iteration number * qa: average heat flux calculated by thermocouples 1 and 3 (W/m ) * qb: heat flux calculated by thermocouples 2 and 3 (k W/m ) * qf: heat flux calculated by thermocouples 1 and 2 (k W/m ) * qt: total heat flux between bed and probe (k W/m ) * r: array for radial distance * t: temperature (K) * taveh: average heated surface temperature (K) * tavec: average cooled surface temperature (K) * tb: array of bed bulk temperature (K) * tw: temperature of probe surface on the side of water (K) * z: array of vertical distance ******************************************************** 2  4  2  2  2  2  2  2  c c c  The program (fprobe.for) requires the following subprograms: fFin,fdrdz,fhw/im^,fcl<ck,fcoeff,fsolv,fout  253  parameter (m 1=19,n 1 =61,11=8 ,m2=6,n2=7) implicit real* 8 (a-h,o-z) corrjmon/ckJc7ckk(ml -1 ,nl -1) common/coeff7aw(m 1 ,n 1 ),ae(m l,nl),as(ml,nl ),an(m l,nl),ap(ml,nl), + s(ml,nl) conunon/contl/epSjalpha,delta,maxit,iter cormnon/cont2Mv,tw,denw,cpw,visw,c^ common/convec/hw,hb(l 1 ),eb(l 1 ),ewh,ews,ewb,ebh common/dimen 1 /r(in2),z(ii2),m(ni2),nz(n2) common/dimen2/iskip j skip common/drdz/dr,dz,rr(m 1) common/mri/l,m,n,mm,nn common/qflux/qa(ll),qb(ll),qf(ll),qt(ll) common/tdis/t(m 1 ,nl ),tt(m 1 ,nl ,11 ),taveh(l 1 ),tavec(l 1 ),niter(l 1) common/temp/tb(ll ),tm(l 1,3),tc(nl ,11) common/tk/tk(m 1-1, n 1-1) c c c  10 20 30 40 50 60 70 80 90 100 c c c  input starting datafromfile 'ffin.dat' open (unit=5, file-fpinqtl.dat',status-old') read (5,10) l,m,n,mm,nn format(4(il0/),il0) read (5,20) r format(dl0.4) read (5,30) z format(dl0.4) read (5,40) qt format(dl0.4) read (5,50) eps,alpha,delta,maxit format(3(dl0.4/),il0) read (5,60) vw,tw,denw,cpw,visw,condw format(5(dl0.5/),dl0.5) read (5,70) djet,dprobe format(dl0.5/dl0.5) read (5,80) hb format(dl0.4) read (5,90) tb format(dl0.4) read (5,100) ((tm(ij)j=l,8),i=l,3) format(3(dl0.4)) close (unit=5) calculate incremental distances in r and z directions  254  call fdrdz c c c  calculate convective coefficient of cooling water call fhw  c c c  do loop for different tb do200k=l,l  c c c  set initial temperature distribution call finitfk)  c c c  calculate conductivity matrix call fckkfk)  c c c  calculate aw,ae,as,an,ap,s matrices call fcoeff(k)  c c c  solve equation and determine converged temperature distribution  call fsolv(k) 200 continue c c write results c call fout c stop end c c subroutine fdrdz is used to calculate c nr(i) and nz(j) with uniform grid c subroutine fdrdz c parameter (ml=19,nl =61,11=8,m2=6,n2=7) implicit real* 8 (a-h,o-z) common/dimen 1/r(m2),z(n2),nr(m2),nz(n2) common/dimen2/iskip,jskip common/drdz/dr,dz,rr(m 1) common/mn/l,m,n,mm,nn  255  nr(i) values  c c  10 c c c  20 c  30  dr=r(rnm)/(m-l) do 10 i=T,mm nr(i)=r(i)/dr+l continue nz(i) values dz=z(nn)/(n-l) do 20 i=l,nn nz(i)=z(i)/dz+l continue do 30 i=l,m rr(i)=(i-l)*dr continue  c iskip=(m-l)/(r(mm)* 1 .d3) jskip=(n-l)/(z(nn)*l.d3) c return end c c c c  subroutine fhw is used to calculate heat transfer coefficient between running water and the probe surface subroutine fhw  c parameter (ml=l 9,nl=61 ,H=8,m2=6,n2=7) implicit real* 8 (a-h,o-z) common/cont2/vw,tw,denw,cp w, visw,condw,dj et,dprobe common/convec/hw,hb(l 1 ),eb(l 1 ),ewh,ews,ewb,ebh common/mn/l,m,n,mm,nn c pi=4.d0*datan(l.d0) uw=4.d0*vw/(pi*djet* *2) re=djet*uw*denw/visw pr=visw* cpw/condw st=2.74d0*re**(-0.652)*pr**(-0.513)*(dprobe/djet)**(-0.774) hw=st* denw* uw* cpw write (*,10) hw 10 format(lx,*hw=\ fl0.4)  256  c  return end  c c c c  subroutinefinitis used to set initial coefficient and temperature values subroutine finitfk)  c parameter (ml=l 9,nl=61,11 =8,m2=6,n2=7) implicit real* 8 (a-h,o-z) common/coefY/aw(ml,nl),ae(ml,nl),as(ml,nl),an(ml,nl),ap(ml,nl), + s(ml,nl) common/cont2/vw,tw,denw,dj et,dprobe,cp w, vis w,condw common/mn/l,m,n,mm,nn common/qflux/qa(l 1 ),qb(l 1 ),qf(l 1 ),qt(l 1) common/tdis/t(m 1 ,nl ),tt(m 1 ,nl ,11 ),taveh(l 1 ),tavec(l 1 ),niter(l 1) common/temp/tb(l 1 ),tm(l 1,3 ),tc(n 1,11) c c c  set initial temperature and coefficient for all nodes  do 20 j=l,n do 10 i=l,m t(i j)=tw+j *qt(k)*dz/l 7.d0 10 continue 20 continue c do30j=l,n  tcG,k)=tb(k)-(j-l)*(tb(k)-tw)/(n-l) 30 c  continue return end  c c c c  subroutine fckk is used to calculate the local conductivity values subroutine fckk(k)  c parameter (ml=19,nl =61,11 =8,m2=6,n2=7) implicit real* 8 (a-h,o-z) common/ckk/ckk(m 1 -1 ,n 1 -1) common/dimenl/r(m2),z(n2),nr(m2),nz(n2) common/mn/l,m,n,mm,nn common/tdis/t(ml ,nl ),tt(m 1 ,nl ,11 ),taveh(l 1 ),tavec(l 1 ),niter(l 1)  257  common/tk/tk(m l-l,nl-l) c c c  30 40 c c c c  calculate local average temperature for conductivity, do40j=l,n-l do30i=l,m-l tk(ij)=(t(ij+l)+t(i+lj)+t(ij)+t(i+lj+l))/4.d0 continue continue calculate local conductivity for ss347 head (note: temperature's unit is in (k)) do60j=l,n-l do50i=l,m-l if (i.le.nr(l)-l.and.j.le.nz(7)-l) then  c c c  ss347 conductivity ckk(ij)=9.7811d0+1.503d-2*tk(ij) ckk(ij)=17.d0 endif  c c  if (i.gt.nr(l)-l .and.i.le.nr(2)-l) then if (j-ge.l.and.j.le.nz(l)-l) then c c c  ss347 conductivity ckk(ij)=9.781 ld0+1.503d-2*tk(ij) ckk(ij)=17.d0 elseif (j.gt.nz(l)-l.and.j.le.nz(2)-l) then  c c c c  cement conductivity ckk(ij)=1.13d0 elseif (j.gt.nz(2)-l.and.j.le.nz(4)-l) then  c c c  air conductivity  + c c c  ckk(ij)=(2.9162d-8*tk(ij)**3-7.8255d-5*tk(ij)**2 +0.1183d0*tk(ij)-2.4771d0)/l.d3 ckk(ij)=0.03d0 elseif (j .gt.nz(4)-1 .and.j .le.nz(7)-1) then ss347 conductivity  258  ckk(ij)=9.7811d0+1.503d-2*tk(ij) endif endif if (i.gt.nr(2)-1 .and.i.le.nr(3)-1) then if (j.ge.l.and.j.le.nz(5)-l) then ss347 sleeve conductivity ckk(ij)=9.781 ld0+1.503d-2*tk(ij) elseif (j .gt.nz(5)-1 .and.j .le.nz(6)-1) then cement conductivity ckk(ij)=1.13d0 elseif (j.gt.nz(6)-l.and.j.le.nz(7)-l) then ss347 head conductivity ckk(ij)=9.781 ld0+1.503d-2*tk(ij) endif endif if (i.gt.nr(3)-l .and.i.le.nr(4)-l) then if (j.ge.l.and.j .le.nz(l)-l) then ss347 sleeve conductivity ckk(ij)=9.781 ld0+1.503d-2*tk(ij) elseif (j.gt.nz(l)-l.and.j.le.nz(2)-l) then cement conductivity ckk(ij)=1.13d0 elseif (j.gt.nz(2)-l.and.j.le.nz(3)-l) then air conductivity  +  ckk(ijH2.9162d-8*tk(ij)**3.d0-7.8255d-5*tk(ij)* +.1183d0*tk(iJ)-2.4771d0)/l.d3 ckk(ij)=0.03d0 elseif (j.gt.nz(3)-l.and.j.le.nz(5)-l) then ss347 sleeve conductivity  259  ckk(i j>9.7811 dO+1.503d-2*tk(i j) elseif (j.gt.nz(5)-l.and.j.le.nz(6)-l) then cement conductivity ckk(ij)=1.13d0 elseif (j.gt.nz(6)-l.and.j.le.nz(7)-l) then ss347 head conductivity ckk(i j>9.7811 dO+1.503d-2*tk(i j) endif endif if (i.gt.nr(4)-l .and.i.le.nr(5)-l) then if (j.ge.l.and.j.le.nz(5)-l) then ss347 sleeve conductivity ckk(i j)=9.781 IdO+l .503d-2*tk(i j) elseif (j.gt.nz(5)-l.and.j.le.nz(6)-l) then cement conductivity ckk(ij)=1.13d0 elseif (j.gt.nz(6)-l.and.j.le.nz(7)-l) then ss347 head conductivity ckk(i j>9.7811 dO+1.503d-2*tk(i j) endif endif if (i.gt.nr(5)-l.and.i.le.nr(6)-l) then ss347 body conductivity ckk(i J)=9.7811 dO+1.503d-2*tk(i j) endif continue continue ckmax=24.d0 do80j=l,n-l  260  70 80 c c c c c c  c c c  do70i=l,m-l if(ckk(i j).gt.ckmax) then ckk(ij)=ckmax endif continue continue return end subroutine fcoeff is used to calculate the local coefficient values subroutine fcoefffk) parameter (ml=19,nl=61,ll=8 ,m2=6,n2=7) implicit real* 8 (a-h,o-z) common/ckk/ckk(ml -1 ,n 1 -1) common/coeff/aw(ml ,nl),ae(ml ,nl),as(ml ,nl),an(ml ,nl),ap(ml ,nl), + s(ml,nl) cornmon/contl/eps,alpha,delta,maxit,iter common/cont2/vw,tw,denw,djet,dprobe,cpw,visw,condw common/convec/hw,hb(l 1 ),eb(l 1 ),ewh,ews,ewb,ebh common/dimenl /r(m2),z(n2),nr(m2),nz(n2) common/drdz/dr,dz,rr (m 1) common/mn/l,m,n,mm,nn common/qflux/qa(ll),qb(ll),qf(ll),qt(ll) common/tdis/t(ml ,nl ),tt(ml ,nl ,11 ),taveh(l 1 ),tavec(l 1 ),niter(l 1) common/temp/tb(l 1 ),tm(l 1,3),tc(nl ,11) for all interior nodes do 30 i=2,m-l  do20j=2,n-l  aw(i j)=(ckk(i-1 j - l)+ckk(i-1 j))*(rr(i)-dr/2.d0)*dz/dr ae(iJHckk(ij-l)+ckk(ij))*(rr(i)+dr/2.dO)*dz/dr as(ij)Kckk(i4j)*(rr(i)-025d0*dr)+ckk(ij)*(rr(i)+0.25d0 + . *dr))*dr/dz an(ij)=(ckk(i-lj-l)*(rr(i)-0.25d0*dr)+ckk(ij-l) + *(rr(i)+0.25d0*dr))*dr/dz ap(ij)=aw(ij)+ae(ij)+as(ij)+an(ij) s(ij)=0.d0  20 continue 30 continue c  261  c c  for side 1: the water cooled surface  do 40 i=2,m-l aw(i,n)=ckk(i-l,n-l)*(rr(i)-dr/2.d0)*dz/dr ae(i,n)=ckk(i,n-1 )*(rr(i)+dr/2.d0)*dz/dr as(i,n)=0.d0 an(i,n)=(ckk(i-1 ,n-1 )* (rr(i)-0.25d0*dr)+ckk(i,n-1 )* (rr(i) + +0.25d0*dr))*dr/dz ap(i,n)=aw(i,n)+ae(i,n)+as(i,n)+an(i,n)+2.d0*hw*rr(i)*dr s(i,n)=2.d0*hw*tw*rr(i)*dr 40 continue c c for side 2: the head centreline surface c do50j=2,n-l aw(lj)=0.d0 ae(lj)=4.d0*dz**2 as(lj)=dr**2 an(lj)=dr**2 ap(l j)=aw(l j)+ae(l j)+as(l j)+an(l j) s(lj)=0.d0 50 continue c c for side 3: the heated surface c do 60 i=2,m-l aw(i,l)=ckk(i-l, l)*(rr(i)-dr/2.d0)*dz/dr ae(i,l)=ckk(i,l)*(rr(i)+dr/2.d0)*dz/dr as(i, l)=(ckk(i-l, 1 )*(rr(i)-0.25d0*dr)+ckk(i, 1 )*(rr(i) + +0.25d0*dr))*dr/dz an(i,l)=0.d0 ap(i, 1 )=aw(i, 1 )+ae(i, 1 )+as(i, 1 )+an(i, 1) s(i,l)=2.d0*qt(k)*rr(i)*dr 60 continue c c for side 4: outside surface of the probe body c do70j=2,n-l aw(mj)=(cld<m-lj4)+c^^ ae(mj)=0.d0 as(m j)=ckk(m-l j)*(r(6)-0.25d0*dr)*dr/dz an(mj)=ckk(m-lj-l)*(r(6)-0.25d0*dr)*dr/dz apCmj^^aw^jj+aeCmj^+asCmj^+an^j) + +2.d0*hb(k)*r(6)*dz s(mj)=2.d0*hb(k)*tc(j,k)*r(6)*dz 262  70 continue c c for corner 1: cooling water + head centreline c aw(l,n)=0.d0 ae(l,n)=2.d0*ckk(l,n-l)*dz as(l,n)=0.d0 an(l ,n)=ckk( 1 ,n- l)*dr* *2/dz ap( 1 ,n)=aw( 1 ,n)+ae( 1 ,n)+as( 1 ,n)+an( 1 ,n)+hw* dr* * 2 s( 1 ,n)=hw* tw* dr * * 2 c c for corner 2: head centreline + heated surface c aw(l,l)=0.d0 ae(l,l)=2.d0*ckk(l,l)*dz as(l,l)=ckk(l,l)*dr**2/dz an(l,l)=0.d0 ap(l,l)=aw(l,l)+ae(l,l)+as(l,l)+an(l,l) s(l,l)=qt(k)*dr**2 c c for corner 3: heated surface + body surface c aw(mj)=ckk(m-l,l)*(r(6)-dr/2.d0)*dz/dr ae(m,l)=0.d0 as(m,l)=ckk(m-l,l)*(r(6)-0.25d0*dr)*dr/dz an(m,l)=0.d0 ap(m, 1 )=aw(m, 1 )+ae(m, 1 )+as(m, 1 )+an(m, 1) +. +hb(k)*r(6)*dz s(m,l)=qt(k)*dr*(r(6)-0.25d0*dr) + +hb(k)*tc(l,k)*r(6)*dz c c for corner 4: body surface + cooling surface c aw(m,n)=ckk(m-1 ,n- l)*(r(6)-dr/2.d0)*dz/dr ae(m,n)=0.d0 as(m,n)=0.d0 an(m,n)=ckk(m-1 ,n-1 )*(r(6)-0.25d0*dr)*dr/dz ap(m,n)=aw(m,n)+ae(m,n)+as(m,n)+an(m,n) + +hw*(r(6)-0.25d0*dr)*dr + +hb(k)*r(6)*dz s(m,n)=hw*tw*(r(6)-0.25d0*dr)*dr + +hb(k)*tc(n,k)*r(6)*dz c return end  263  c c c c  subroutine fsolv uses point over-relaxation method starting from left upward corner in a red-black pattern subroutine fsolv(k)  c parameter (m 1=19,n 1 =61,11=8 ,m2=6,n2=7) implicit real*8 (a-h,o-z) common/ckk/ckk(m 1 -1 ,n 1 -1) common/coeff/aw(m 1 ,n 1 ),ae(m 1 ,n 1 ),as(m 1 ,n 1 ),an(m 1 ,n 1 ),ap(m 1 ,n 1), + s(ml,nl) common/contl/eps,alpha,delta,maxit,iter common/convec/hw,hb(11 ),eb(l 1 ),ewh,ews,ewb,ebh common/dimenl/r(m2),z(n2),nr(m2),nz(n2) common/drdz/dr,dz,rr(ml) conimori/mri/l,m,n,mm,nn common/qflux/qa(ll),qb(ll),qf(ll),qt(ll) common/tdis/t(m 1 ,n 1 ),tt(m 1 ,n 1,11 ),taveh(l 1 ),tavec(l 1 ),niter(l 1) common/temp/tb(l 1 ),tm(l 1,3),tc(nl ,11) c alpham=l .dO-alpha c  do 60 iter=l,maxit c c c  update nodal temperature values using red-black point-point method  do 30 ipass=l,2 is=ipass do20j=l,n jm=j-l JP=J+1 do 10 i=is,m,2 im=i-l ip=i+l t(i J )=alpham* t(i j )+alpha* (s(i j )+aw(i J )* t(imj )+ + ae(ij)*t(ipj)+as(ij)*t(ijp)+an(ij)*t(ijm))/ap(ij) 10 continue is=3-is 20 continue 30 continue c c find maximum residual c resmax=0.d0 do 50 j=l,n  264  jm=j-l JP=J+1 is=l do 40 i=is,m,2 im=i-l ip=i+l res=s(ij)+aw(ij)*t(imj)+ae(ij)*t(ipj)+as(ij)*t(ijp) + +an(ij)*t(ijm)-ap(ij)*t(ij) resmax=dmax 1 (resmax, dabs(res)) 40 continue is=3-is 50 continue c c test for convergence c if (resmax.lt.eps) goto 70 c c reassign conductivity c callfckkfk) c c recalculate aw,ae,as,an,ap,s matrix c call fcoefffk) c 60 continue 70 niter(k)=iter do 75 i=l,m do 74j=l,n tt(ij,k)=t(ij) 74 continue 75 continue c c determine average temperatures on both head surfaces c sumhends=t(nr( 1), 1) *r( 1) sumcends=t(nr( 1 ),n) *r( 1) surnheven=0.dO sumceven=0.d0 do 80 i=2,nr(l)-l,2 sumheven=sumheven+t(i, 1 )*rr(i) sumceven=sumceven+t(i,n) * rr(i) 80 continue sumhodd=0.d0 sumcodd=0.d0  265  do85i=3,nr(l)-l,2 suirmodd=sumhodd+t(i, 1 )*rr(i) sumcodd=sumcodd+t(i,n)*rr(i) continue taveh(k)=2 .dO * (dr * (surnhends+4. dO * sumhe ven+2. dO * sumhodd) + /3.d0)/r(l)**2 tavec(k)=2. dO * (dr * (sumcends+4. dO * sumceven+2. dO * sumcodd) + /3.d0)/r(l)**2 determine average heat fluxes by measured temperatures nf=2.d-3/dz+l nc=13.d-3/dz+l nb=24.d-3/dz+l tavea=(t(l ,nf)+t(l ,nb))/2.d0 ckka=9.781 IdO+l .503d-2*tavea qa(k)=ckka*(t(l ,nf)-t(l ,nb))/((nb-nf)*dz* 1 .d3) tavef=(t(l ,nf)+t(l nc))/2.d0 ckkf=9.781 ld0+1.503d-2*tavef qf(k)=ckkf*(t(l ,nf)-t(l ,nc))/((nc-nf)*dz* 1 .d3) 5  taveb=(t(l ,nc)+t(l ,nb))/2.d0 ckkb=9.781 ld0+1.503d-2*taveb qb(k)=ckkb*(t(l ,nc)-t(l ,nb))/((nb-nc)*dz* 1 .d3) return end  subroutine fout prints out the information, such as the temperature distribution, in the file 'fout.dat' subroutine fout parameter (ml=19,nl =61,11=8,m2=6,n2=7) implicit real* 8 (a-h,o-z) common/ckk/ckk(m 1-1,n 1-1) common/contl/eps,alpha,delta,maxit,iter common/cont2/vw,tw,denw,cpw,visw,condw,djet,dprobe common/convec/hw,hb(l 1 ),eb(l 1 ),ewh,ews,ewb,ebh common/dimenl/r(m2),z(n2),nr(m2),nz(n2) common/dimen2/iskip j skip  266  common/drdz/dr,dz,rr(ml) cornmon/rrLn/l,m,n,mm,nn commonyqflux/qa(l 1 ),qb(l 1 ),qf(l 1 ),qt(l 1) comrnon/tdis/t(ml ,nl ),tt(ml ,nl ,11 ),taveh(l 1 ),tavec(ll ),niter(l 1) conrrnon/ternp/tb(l 1 ),tm(l 1,3),tc(nl,ll) c open (unit=8,file-fpoutqtl .dat',status old') write (8,10) 10 format( 18x,'conduction through a probe head') write (8,20) m,n,dr,dz 20 format (lx,'input data:'// +5x,'m =',i5,5x,'n =*,i5,5x,'dr =',fl0.4,5x,'dz =',fl0.4) write (8,30) eps,alpha,iskipjskip 30 format (lx,*eps ^flOASx/alpha^flO.^Sx/iskip =',i2, +5x,'jskip =',i2) write (8,35) hw,tw 35 format(lx,hw=*, flOAlx/tw^flO^) do90k=l,l write (8, 40) k,taveh(k),tavec(k),niter(k) 40 format (lx,i2,lx,'taveh -,fl0.2,lx,'tavec =', +fl0.2,lx,'niter=',i8) write (8, 50) qt(k),qf(k),qb(k),qa(k) 50 format (lx,'qt =',fl0.2,lx,'qf =',fl0.2,lx,'qb =',fl0.2,lx, +qa=',fl0.2,(kw/m**2)') write (8,60) tb(k),hb(k) 60 format (Ix;tb=',n0.2,5x,'hb=',n0.2) write (8,70) (rr(i),i=l,m,iskip) 70 format (lx,10f7.5) write (8,80) ((tt(ij,k),i=l,m,iskip)j=l,njskip) 80 format (lx, 10f7.2) 90 continue c close (unit=8) return end =l  ,  ,  ,  PROGRAM 2  This is a program for the calculation of conductive and radiative heat flux from a hot suspension to a probe by the window method and by the differential emissivity method. The window temperature distribution is also calculated. * * *  alfa: d: delta:  absorption coefficient of zinc selenide window thickness of window Stefan-Boltzmann constant  267  e316: e347: emit: eps: hwc: kg,kgas: kw: ql: q316: q347: qc: qcee: qmwin: qm316: qm347: qr: qr2: qr3: qrb: qree: qwri: qwro: ratio: roup: rouw: tavg: tb: tg: twin: t316: t347: tqi: tqo: xg:  emissivity of probe head surface ss316 emissivity of probe head surface ss347 emissivity of probe head surface behind window accuracy criteria heat transfer coefficient from bed to window conductivity of air conductivity of zinc selenide window heat transfer by conductionfromwindow to probe thermal emission by probe head-ss316 thermal emission by probe head-ss347 heat transfer by conductionfromsuspension to window conductive heat flux estimated by dual emissivities heat flux measured by probe head behind window heat flux measured by probe head-ss316 heat flux measured by probe head-ss347 thermal radiative emittance from solids suspension thermal radiative emittance evaluated by saxena 1987 thermal radiative emittance evaluated with qm=qr*tg thermal radiative emittance from black body radiative emittance estimated by dual emissivities heat emitted by window inside surface heat emitted by window outside surface radiative over total heat transfer to probe ss347 reflectivity of sensor behind window reflectivity of window average temperature of window heat source temperature (suspension temperature) transmittance of window surface temperature of window probe head surface temperature of probe head ss316 surface temperature of probe head ss347 surface temperature of window facing probe head surface temperature of window facing bed thickness of air between window and probe head  parameter (nm=37) implicit real* 8 (a-h,o-z) dimension tag(nm),tavg(nm),t347(nm),t316(nm),twin(nm),tb(nm) dimension kgas(nm), kw(nm) dimension tqi(nm),tqo(nm),ttqi(nm),ttqo(nm) dimension cl (nm),c2(nm),c3(nm),c4(nm),b(nm),bb(nm),cc(nm),dd(nm) dimension errl(nm),err2(nm),eeff(nm), ratio l(nm), ratio2(nm) dimension qwri(nm),qwro(nm),qpr(nm),qmwin(nm),qrb(nm),qr(nm) dimension q 1 (nm),qr2(nm),qc(nm),qr3 (nm),qcee 1 (nm),qcee2(nm) dimension qm347(nm),q347(nm),qm316(nm),q316(nm),qree(nm)  268  25  30 40  real kgas,kg,kw,kwi integer n dataemit,e316,e347,delta,eps/0.90,0.3d0,0.9d0,5.67d-8,l.Od-4/ datad,xg/2.0d-3,1.0d-3/ data kg,kwi/2.6d-2, l.Odl/ data n/37/ datatg,rouw,roup/0.65d0,0.25d0,0.1d0/ open (unit=20,file='run-l .dat',foim-formatted',status='old') open (unit=21 ,file='iun-2.dat',fonn-formatted',su^^ do 25 i=l, n read (20,*) tb(i),twin(i),t316(i),t347(i) read (21,*) qmwin(i),qm316(i),qm347(i) continue do40i=l,n write (*,30) tb(i),twin(i),t316(i),t347(i),qmwin(i),qm316(i), lqm347(i) format (lx,7fl 0.2) continue close (unit=20,status-keep') do50i=l,n tb(i)=273.15d0+tb(i) twin(i)=273.15d0+twin(i) t316(i)=273.15d0+t316(i) t347(i)=273.15d0+t347(i) continue  50 c c calculation of radiative heat flux by comparing ss316 and ss347 c do60i=l,n q316(i)=e316*delta*t316(i)* *4 q347(i)=e347*delta*t347(i)**4 qree(i)=(qm347(i)-qm316(i)+q347(i)-q316(i))/(e347-e316) qceel (i)=qm347(i)-e347*qree(i)+q347(i) qcee2(i)=qm316(i)-e316*qree(i)+q316(i) ratio 1 (i)=((l •d0-roup)*qree(i)-q347(i))/ 1 ((1 .d0-roup)*qree(i)-q347(i)+qceel(0) 60 continue c do 65 i=l,n tqo(i)=tb(i) qrb(i)=delta*tb(i)**4 65 continue c c calculation of heat flux and temperature distribution c  269  alfa=-dlog(tg)/d do 75 i=l,n ql(i)=0.4d0*qmwin(i) tqi(i)=twin(i)+q 1 (i) * xg/kg tqo(i)=tqi(i)+ql (i)*d/kwi 1000 if(tqi(i).gt.tb(i))then tqi(i)=tb(i) endif if (tqo(i).gttb(i)) then tqo(i)=tb(i) endif if(tqo(i).lt.tqi(i))then tqo(i)=tqi(i) endif tag(i)=(tqi(i)+twin(i))/2.d0 kgas(i)=(2.9162d-08*tag(i)**3-7.8255d-05 1 *tag(i)**2+0.1183d0*tag(i)-2.4771d0)* 1.0d-3 ql(i)=(tqi(i)-twin(i))*kgas(i)/xg tavg(i)=(tqo(i)+tqi(i))/2.d0 kw(i)=54.6648-0.229*tavg(i)+3.8294d-4*tavg(i)**2 1 -2.1989d-7*tavg(i)**3 b(i)=sqrt(alfa**2+8.dO*alfa*delta*tavg(i)**3/kw(i)) bb(i)=exp(-b(i)*d) cc(i)=b(i)*kw(i)*bb(i) dd(i)=b(i)**3*kw(i)*bb(i) c4(i)=(ql (i)/cc(i))-(alfa* *2*qmwin(i)/dd(i)) c2(i)=(tqo(i)-tqi(i)-c4(i)*(l .dO-bb(i))-alfa* *2*qmwin(i)*d 1 /(b(i)**2*kw(i)))/(l.d0+bb(i)**(-2)-2.d0*bb(i)**(-l)) c3(i)=c2(i)*bb(i)**(-2)+c4(i) cl (i)=((tqo(i)-c4(i)-c2(i)*(l .d0+bb(i)* *(-2)))*b(i)* *2*kw(i) 1 -6.d0*alfa*delta*tavg(i)**4)/alfa ttqi(i)=c2(i)*bb(i)**(-l)+c3(i)*bb(i)-(alfa**2*qmwin(i)*d 1 -alfa*cl(i)-6.d0*alfa*delm^ ttqo(i)=c2(i)+c3(i)+(cl(i)*alfa+6.d0*alfa*delm*tavg(i)**4)/ 1 (b(i)**2*kw(i)) errl(i)=dabs((ttqo(i)-tqo(i))/ttqo(i)) erT2(i)=dabs((ttqi(i)-tqi(i))/ttqi(i)) if (errl(i).gt.eps.or.err2(i).gt.eps) then if (ttqo(i).gt.tqo(i).or.ttqi(i).gt.tqi(i)) then tqo(i)=ttqo(i) tqi(i)=ttqi(i) else tqo(i)=tqo(i)* 1.001 dO tqi(i)=tqi(i)*1.001d0 endif  270  goto 1000 endif c c c  calculation of convective and radiative heat flux from bed to probe  aaa=l.dO-rouw*roup qwri(i)=delta*(l.d0-tg-rouw)*ttqi(i)**4 qwro(i)=delta* (1 .dO-tg-rouw)*ttqo(i)* *4 qpr(i)=delta* emit*twin(i)* *4 q347(i)=delta*e347*t347(i)**4 al=tg*roup/aaa a2=tg/aaa a3=l .dO-rouw-(roup*tg* *2)/aaa qr(i)=((qmwin(i)-ql(i))*aaa-(l .dO-roup)*qwri(i)+(l .dO-rouw) 1 *qpr(i))/((l.dO-roup)*tg) qc(i)=qmwin(i)-a3 *qr(i)+qwro(i)+al *qwri(i)+a2*qpr(i) eeff(i)=qr(i)/(delta*tb(i)* *4) ratio2(i)=((l.d0-roup)*qr(i)-q347(i))/ 1 ((1 .d0-roup)*qr(i)-q347(i)+qc(i)) qr2(i)=(qmwin(i)-ql (i))/tg qr3 (i)=qmwin(i)/tg 75 continue c c recording of results into file res-run.dat c open(l 1 ,file=Ves-run.dat\form-formatted',status- old ) 1  write (11,*)'* heat flux calculated by window probe *' write (11,*) ' trans, ref.w ref.p e316 e347' write (11,170) tg, rouw, roup, e316, e347 170 format (lx, 5f8.2//) write (11,*)' tb (k) qmwin qm316 qm347 qree qc347 1 qc316 ratio-ee qr/qm347' write (11,180) (tb(i),qmwin(i),qm316(i),qm347(i),qree(i), 1 qcee 1 (i),qcee2(i),ratio 1 (i),i=l ,n) 180 format(lx,f6.1,6fl0.1,f6.2) write(ll,*)'tb tqo tqi qrb qr ql qc qr2 qr3 1 ratio-w' write(ll,190)(tb(i),tqo(i),tqi(i),qrb(i),qr(i),ql(i),qc(i), 1 qr2(i), qr3(i),ratio2(i),i=l,n) 190 format(lx,3f6.1,6f9.1,f6.2) stop end  271  PROGRAM 3 This program is used to calculate the total suspension-to-membrane-wall heat transfer coefficient using Bowen's model-2 * alpha: ri/ro * belta: t/ro * biin: biot number inside the tube (Tu*ri/kt) * biout: biot number outside the tube (ho*ro/kt) * galma: w/ro * cpw: water capacity (J/kg K) * eps: converge criterion * hi: heat transfer coefficient on the cooling water side (W/m K) * ho: heat transfer coefficient on the suspension side (W/m K) * kf: thermal conductivity of the fin (W/m K) * kfkt: kf/kt=1.02 (constant) * kt: thermal conductivity of the pipe (W/m K) * 1: length of the fin (1.626 m) * nu: Nusselt number * omega: (pi-phi)/2 * phi: 2*arcsin(belta) * pr: Prandtl number * q: dimensionless heat flux * re: Reynolds number * ri: inside radius of the pipe (m) * ro: outside radius of the pipe (m) * t: half thichness of the fin (2.4 mm) * tave: (twi+two)/2 * tb: bulk suspension temperature (°C) * tfin: fin-base temperature (t6) * twi: inlet water temperature (°C) * two: outlet water temperature (°C) * vw: water volumn flow rate (m /s) * w: halfwidthofthefin(6.4mm) * wl: the length of the membrane wall (1.626 m) * wm: parameter m ******************************************** parameter (nn=37) implicit real* 8 (a-h,o-z) real* 8 kt, kfkt, nu common/Dlockl/belte,galma,pm,omega,kfkt,eps common/block2/c 1 (nn),c2(nn),c3 (nn),q(nn),qq(nn) dimension tb(nn), twi(nn), two(nn), tave(nn) dimension denw(nn), cpw(nn), visw(nn),condw(nn) dimension re(nn), nu(nn),pr(nn),kt(nn),wm(nn),biin(nn) 2  2  3  272  dimension hi(nn),ho(nn),al(nn),a2(nn) external f datari,ro,t,w,wl/7.05d-3,10.65d-3,2.4d-3,6.35d-3,1.270d0/ datakfkt/1.02d0/ data vw,tfin/30.d-6, 373 AO/ datan, eps/28, l.d-6/ C C C  70 C C C  input of cooling water temperatures open (unit=5, file-htmodel2',status='old') read (5,70) (tb(i),twi(i),two(i), i=l,n) format(3(fl0.4)) close (unit=5) calculation of constants and q, hi,cl,c2,c3 pi=4.d0*datan(l.d0) alpha=ri/ro belta=t/ro galma=w/ro phi=2.d0*dasin(belta) omega=(pi-phi)/2.d0 uw=vw/(pi*ri**2.d0) do 100 i=l,n tave(i)=273 .dO+(twi(i)+two(i))/2.dO condw(i)=0.08235d0+5.1091d-4*tave(i)+9.424d-6*tave(i)**2.d0 1 -1.7425d-8*tave(i)**3.d0 denw(i)=264.3353d0+6.4953*tave(i)-0.01799*tave(i)**2+ 1 1.4923d-5*tave(i)**3 cpw(i)=4.5359d4-4.9161d2*tave(i)+2.2005d0*tave(i)**2-4.3807d-3 1 *tave(i)**3+3.276d-6*tave(i)**4 visw(i)=(406389.01-4680.3*tave(i)+20.3281*tave(i)**2-0.03938 1 *tave(i)**3+2.86654d-5*tave(i)**4)* 1 .d-6 re(i)=2. dO *ri* uw* denw(i)/visw(i)  C pr(i)=visw(i) * cp w(i)/condw(i) al(i)=0.88d0-0.24d0/(4.d0+pr(i)) a2(i)=0.333d0+0.5d0*exp(-0.6d0*pr(i)) nu(i)=5.d0+0.015d0*re(i)**al(i)*pr(i)**a2(i) hi(i)=nu(i)*condw(i)/(2.d0*ri) kt(i)=8.9683d0+1.534d-2*((tave(i)+tfin)/2.d0) biin(i)=hi(i)*ri/kt(i) wm(i)=sqrt(biin(i)*(l .d0+alpha)/(2.d0*(l .dO-alpha))) q(i)=vw*denw(i)*cpw(i)*(two(i)-twi(i))/(2.d0*kt(i)*wl 1 *(tb(i)+273.d0-tave(i)))  273  qq(i)= vw* denw(i) * cp w(i) * (two(i)-twi(i)) cl (i)=biin(i)/(l .d0-biin(i)*dlog(2.d0*alpha/ 1 (1 .d0+alpha)))+2.d0*(l .dO-alpha)*wm(i)* 2 dtanh(wm(i)*omega)/((l .dO+alpha)*phi) c2(i)=l .dO/cl (i)-dlog((l .d0+alpha)/2.d0) c3(i)=l .dO/biin(i)-dlog(alpha) 100 continue C a=0.01d0 b=l.d0 do 200 i=l,n C C using newton-raphson method to find biout C 150 call rnewton (f,i,a,b,biouf) ho(i)=kt(i)*biout/ro 200 continue C open (unit=8,file=htmodel2.out',status='old') write (8,*) qq biin q m cl c2 c3' write (8,210) (qq(i),biin(i),q(i),wm(i),cl(i),c2(i),c3(i),i=l,n) 210 format (lx, 7fl0.4) write (8,*)' ho hi tb twi two write (8,220) (ho(i), hi(i), tb(i), twi(i), two(i), i=l,n) 220 format (lx, 5fl0.2) write (8,*)'re pr nu kt cpw denw condw visw' write (8,23 O)(re(i),pr(i),nu(i),kt(i),cpw(i),denw(i),condw(i), 1 visw(i), i=T,n) 230 format (lx, 7f8.2, fl0.6) C stop end C subroutine rnewton (f,i,xi,xf,x4) parameter (nn=37) implicit real* 8 (a-h,o-z) real*8 kfkt common /blockl/belta,galma,pW,omega,kfkt,eps external f C dx=l.d-4 xl=xi yi=f(xi,i) if(yl.eq.0.) Then write (*, 10) xl ,  1  1  274  10  20  30  40  format (lx, 'the root is', fl0.4) return endif x2=xl+dx if (x2.gt.xf) then write (*,30) format (lx, 'no root found in the interval selected ) return endif y2=f(x2,i) if (y2.eq.O.) Then write (*,10)x2 return endif if(yl*y2.gt.0.)Then xl=x2 yl=y2 goto 20 endif x3=(xl*y2-x2*yl)/(y2-yl) dfdx=(f(x3+dx,i)-f(x3-dx,i))/(2.d0*dx) x4=x3-f(x3,i)/dfdx if ((abs(x4-x3)/x3).lt.eps) then write (*, 10) x3 return endif x3=x4 goto 40 return end 1  C  double precision function f(x,i) parameter (nn=37) implicit real* 8 (a-h,o-z) real* 8 kfkt common /blockl /belta,galma,pM,omega,kfkt,eps common /block2/cl (nn),c2(nn),c3(nn),q(nn) cappa=sqrt(kfkt/(2.d0*belta*x))* 1 dtarm(sqrt(x/(2.dO*belta*kfkt))*galma) f=q(i)-(phi/(c2(i)+l .d0/(cappa*x))+omega/(c3(i)+l .d0/x)) C  return end  275  PROGRAM 4 This program uses the discrete ordinate method to calculate the emusion-to-wall radiative heat transfer coefficient. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  angle: cO: cl: c2: con: d: dp: dt: dwl: e: ed: eeff: eg: ep: ew: h: hO: hgas: hr: bint: inten: kO: ka: ken: kg: ks: kt: leng: n: na: nt: nwl: phase: re: rp: sden: susp: t: taul,2: tb:  selected angles (rad) i=l-6 for positive;i=7-12 negative light velocity in vacuum, 2.9998*10**6 (m/s) 2*h0*c0**2, 1.191*10**8 (W pm**4) h0*c0/k0, 1.439*10**4 (pmk) average particle concentration diameter or width of reactor (m) particle diameter (m) optical depth interval wavelength interval average voidage disperse phase emissivity (core region) effictive emissivity of emulsion layer gas emissivity particle emissivity heat transfer surface emissivity location of heat transfer surface (m) planck's constant (6.6261* 10* *(-34) J s) gas layer conductive heat transfer coefficient (W/m K) emulsion layer radiative heat transfer coefficient (W/m K) blackbody intensity w/(m2*micron*solid angle) radiative intensity w/(m2*micron*solid angle) Boltzmann constant (1.3807*10**(-23) J/K) absorption coefficient extinction coefficient for core region gas thermal conductivity (W/mK) scattering coefficient extinction coefficient for annulus region half thickness of core region (m) node number in gas layer number of cattering angles number of optical depth intervals number of wavelength intervals scattering phase function Reynolds number particle reflectivity (albedo) particle density (kg/m ) particle suspension (kg/m ) temperature (K) optical depth boundary bed temperature (K) 2  2  3  3  276  * * * * * * * * * * *  te: tw: ug: void: vis: weight: wll,wl2: xe: xg: xxg: z:  emulsion temperature at taul (K) surface temperature (K) gas velocity (m/s) voidage near wall gas viscosity (N*s/m ) phase function weight wavelength (pm) thickness of emulsion layer (m) gas layer thickness (m) integration interval (m) height of reactor (m) 2  c parameter (m 1 =41 ,m2=5 01 ,m3=12) implicit real* 8 (a-h,o-z) dimension phase(m2,m3),bint(ml ,m2,m3),iold(ml ,m2,m3) dimension t(m2),xxe(m2),wl(ml) dimension ratio(m2),void(m2),qr(m2) dimension ka(m2),ks(m2),kt(m2),tau(m2),inten(ml ,m2,m3) dimension w(m3),angle(m3) dimension suml6(m2),sum72(m2) real inten,ka,ks,kt external func data nwl,nt,na,nmax/40,500,12,10/ dataeps/l.d-1/ data tb,tw/850.d0,518.8/ data susp/51.d0/ dataew,ep,dp/0.9d0,0.85d0,334.d-6/ data d,h,z,sden/0.152d0,7.3d0,6.0d0,2.61d3/ datac0,h0,k0,rp/2.9979d8,6.6261d-34,1.3807d-23,0.1/ data cl,c2,rp/1.191d8,1.439d4,0.1d0/ data w/0.16667d0,0.16667d0,0.16667d0,0.16667d0,0.16667d0, + 0.16667d0,0.16667d0,0.16667d0,0.16667d0,0.16667d0,0.16667d0, + 0.16667d0/ datawll,wl2/l.dO,21.dO/ data angle/0.066877,0.288741,0.366682,0.633318,0.711259, + 0.933123,-0.066877,-0.288741,-0.366682,-0.633318, + -0.711259,-0.933123/ data vis,ug/146.65d-6,6.6d0/ data fl2,fl3,egas/0.293d0,0.414d0,0.305d0/ datataul/O.dO/ data delta/5.67d-8/ c c c  calculate boundary thickness and emulsion temperature  277  naa=na/2 pi=4.d0*atan(l.d0) a=0.dO  b=2.d0*pi re=ug*dp/vis dt=(tb-te)/nt dwl=(wl2-wl 1 )/nwl con=susp/sden e=l.dO-con xe=d/2.d0*(l .dO-sqrt(l .34d0-l .30d0*con**0.2d0+con** 1.4d0)) te=tw+(tb-tw)*(l .d0-(-0.023d0*re+0.163d0*tb/tw+0.294d0*z/h)) c voidave=0.322d0+0.196d0*e+15.09d0*(e-0.4d0)**6.74d0 xg=0.0287d0*dp*con**(-0.581d0) ec=(con*d**2.d0-(l.d0-voidave)*(4.d0*xe*d-4.d0*xe**2.d0))/ + (d**2.d0-4.d0*xe*d+4.d0*xe**2.d0) c c c  detemination of disperse phase emissivity ken= 1. 5d0*ep* (1 .dO-ec)/dp leng=d/2.d0-xe-xg ed=l .dO-exp(-ken*leng) ed=l.dO  c c c  determination of scattering phase function  do20i=l,na do 10 j=l,na angle l=angle(i) angle2=angle(j) call simp (func,a,b,anglel,angle2,s) phase(ij)=s 10 continue 20 continue c do 18j=nt+l,l,-l xxe(j -1 )* xe/nt ratioG)=l.dO-(0-l)*xe/nt)/(d/2.dO) voidG)=e**(0.191dO+ratio(j)**2.5dO+3.dO*ratioO)**H.dO) kaQ)=l .5d0*(l .d0-ep)*(l .d0-void(j))/dp ksG)=1.5dO*ep*(l.dO-void(j))/dp ktO)=kaG)+ksG) tauG )=xxeG) * ktQ) tG)=tw+(tb-tw)*(l.dO-(-0.023dO*re+0.163dO*tb/tw 1 +0.294dO*z/h)*exp(-0.0054dO*xxeG)/dp)) 278  18 c c c  continue initial assignment of i(optical depth, scattering angle) do 50 i=l,nwl+l wl(i)=wll+dwl*(i-l) do40j=l,nt+l do 30 k=l,na if(k.le.na/2) then inten(ij,k)=100.d0 iold(ij,k)=inten(ij,k) else inten(ij,k)=-100.d0 iold(ij,k)=inten(ij,k) endif continue continue continue  30 40 50 c c c  wavelength integration loop (i) do 100i=l,nwl+l  c c c  solving ordinate equation using discrete ordinate method by iteration do 90 iter=l,nmax resmax=0.0  c c c  optical depth loop (j) do80j=l,nt jj=nt+l-j  c c c  boundary conditions do 70 11=1,naa suml=0.d0 do 6012=naa+l,na suml =suml -w(12)*dabs(angle(12))*inten(i, 1,12)  c + 60  inten(i,nt+l,12)= -cl/(wl(i)**5.d0*(dexp(c2/(tb*wl(i)))-l.d0)) *ed*((fl3+2.d0*fl2)*ew+egas) bint(ij,12)=-cl/(wl(i)**5.dO*(dexp(c2/(tO)*wl(i)))-l.dO)) continue inten(i, 1,11 )=c 1 /(wl(i)* * 5 .d0*(dexp(c2/(tw* wl(i)))-1 .dO))  279  + 70 c c c c  +2.d0*(l.d0-ew)*suml bint(ij,ll)=cl/(wl(i)**5.dO*(dexp(c2/(tO')*wl(i)))-l.dO)) continue bint(ij)=cl/(wl(i)**5.0*exp(c2/(t(j)*wl(i))-1.0)) solving the ordinary differential equation do 76 k=na,l,-l suml6(j)=0.d0  1  sum720j)=0.d0 do751=l,na sum 16(j )=sum 16(j )+w(l) *phase(k,l) * dabs(inten(i j ,1) +inten(ij+l,l))/2.d0 sum72(jj) sum72(jj)+w(l)*phase(k,l)*dabs(inten(ijj,l) +inten(ijj+l,l))/2.d0 continue if(k.gt.6)then inten(ijj k)=-(((abs(angle(k))/(tau(jj+l)-tau(jj))-0.5d0) *abs(inten(ijj+l,k))+0.5d0*rp*sum72(jj)+0.5d0*(l.d0-rp) *abs(bint(i jj,k)+bint(i jj+1 ,k)))/(abs(angle(k)) /(tauOJ+l)-tauGJ))+0.5dO)) else inten(ij+l,k)=((abs(angle(k))/(tauG+l)-tauG))-0.5d0) *abs(inten(ij,k))+0.5dO*rp*suml6G)+0.5dO*(l.dO-rp) *abs(bint(ij,k)+bint(ij+l,k)))/(abs(angle(k)) /(tauG+l)-tauG))+0.5dO) ::::  75  1  + + + + + + c  5  endif  error=abs(iiiten(i j ,k)-iold(i j ,k)) resmax=amax 1 (resmax,error) ioldG J ,k)=inten(i j ,k) 76 continue 80 continue if (resmax.le.eps) then goto 100 endif 190 continue 100 continue c c calculation of radiative heatfluxand coefficient c do 130j=l,nt+l sum4=0.d0  do 120 i=l,nwl+l sum3=0.d0  280  do 110k=l,na sum3=sum3+w(k) * dabs(angle(k)) * inten(i j ,k) 110 continue sum4=sum4+dwl*sum3 120 continue qr(j)=2.d0*pi*sum4 130 continue hr=qr(l)/(tb-tw) eeffM .d0/(l .d0+(delta*(te**4.d0-tw**4.d0)/(-qr(l)))-l .dO/ew) c c record the results c open (unit=8,file=hradiat.out',status='old') write (8,200) 200 format(l 8x,'radiative heat transfer through emulsion layer') write (8,210) dp,susp,tb,te,tw,d,xe,ew 210 format(lx,dp=,f8.6,2x,*susp=',f9.4,2x,'tb=',f8.2,2x,'te=*, + f8.2,2x,tw==',f6.2,2x,*d=',f6.3,2x,'xe=',f8.4,2x,'ew=',f6.2) write (8,220) (angle(i),i=l,na) 220 format (lx,'angle=',fl 0.7) write (8,230) (k,inten(2,l,k),k=l,na) 230 format (lx,inten=(',i2,)=',fl 0.2) write (8,240) (j,t(j),tau(j),ktG),voidG)j=l,nt+l,10) 240 format (lx,'t(',i3,')=', fl0.2,2x,'optical depth*,fl0.4,lx,'kt=', + fl0.2,2x, Void=',fl0.6) write (8,250) (i,qr(i),i=l,nt+l,10) 250 format (Ix,'qr(',i3,*)=',f20.4) write (8,260) hr,eeff 260 format(2x,'hr=',fl 0.4,2x,'eeff=',fl 0.4) write (8,*) 'phase function' write (8,270) ((phase(ij),i=l,na)j=l,na) 270 format (12(lx,f5.3)) c stop end c subroutine simp(func,a,b,anglel ,angle2,area) implicit real* 8 (a-h,o-z) data eps,maxit/l.d-6,100/ c n=3 dx=(b-a)/2.d0 x=a+dx sumevn=func(x,angle 1 ,angle2) ends=func(a,angle 1 ,angle2)+func(b,angle 1 ,angle2) l  ,  ,  ,  ,  ,  :  281  areold=dx*(ends+4.d0*sumevn)/3 .dO sumodd=0.dO do 300 iter=l,maxit dx2=dx dx=dx/2.d0 n=2*n-l nm=n-l sumodd=surnodd+sumevn sumevn=0.dO x=a+dx do 250 i=2,nm,2 sumevn=sumevn+func(x,angle 1 ,angle2) x=x+dx2 250 continue area=dx*(ends+4.d0*surnevn+2.d0*surnodd)/3.d0 if(dabs((area-areold)/area).lt.eps) return areold=area 300 continue return end c double precision function func(x,anglel,angle2) implicit real*8(a-h,o-z) pi=4.d0*datan(l.d0) sca=dacos(anglel *angle2+sqrt((l .d0-anglel * *2.d0) + *(l.d0-angle2**2.d0))*dcos(x)) func=4.d0*(dsin(sca)-sca*dcos(sca))/(3 .d0*pi* *2.d0) return end  282  

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