Abstract
This communication deals with the Abductive Network modeling approach to investigate the phase holdup distributions of a liquid–solid circulating fluidized bed (LSCFB) system. The Abductive Network model is developed/trained using experimental data collected from a pilot scale LSCFB reactor involving 500-μm size glass beads and water as solid and liquid phases, respectively. The trained Abductive Network model successfully predicted experimental phase holdups of the LSCFB riser under different operating parameters. It is observed that the model predicted cross-sectional average of solids holdups in the axial directions and radial flow structure are well agreement with the experimental values. The statistical performance indicators including the mean absolute error (~4.67%) and the correlation coefficient (0.992) also show favorable indications of the suitability of Abductive Network modeling approach in predicting the solids holdup of the LSCFB system.
Acknowledgements
The author would like to acknowledge the support provided by King Abdulaziz City for Science and Technology (KACST) through the Science & Technology Unit at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through Project No. NSTIP # 11-ENV1644-04, as part of the National Science, Technology and Innovation Plan. The author also acknowledges Dr. Mohammad Mozahar Hossain and Dr. Syed Mashiur Rahman of KFUPM, for their comments and advices.
Notation list
| Ar | Archimedes number |
 | Acceleration, m/s2 |
 | Drag coefficient |
| CPM | Complexity penalty multiplier |
 | Particle diameter, m |
 | Dimensionless particle diameter |
| K | Total number of coefficients used in the model |
| N | Total number of training data |
| PSE | Predicted squared error |
| FSE | Fitting squared error |
 | Weights obtained through training the network |
 | Radial position, m |
 | Radius of the riser, m |
 | Reynolds number at terminal settling velocity |
 | Time, s |
 | Superficial liquid velocity, cm/s |
 | Superficial solids velocity, cm/s |
 | Auxiliary liquid velocity, cm/s |
 | Terminal settling velocity, cm/s |
 | Normalizing liquid velocity |
 | Net superficial liquid velocity, cm/s |
 | Dimensionless terminal settling velocity, cm/s |
 | Velocity, m/s |
 | Riser height, m |
Greek letters
 | Density, kg/m3 |
 | Holdup |
 | Spehricity |
 | Estimated true unknown model error variance |
 | Viscosity, cP |
 | Time lag, s |
Subscripts
| g | Gas phase |
| l | Liquid phase |
| s | Solid phase |
 | Slip |
| bed | Fluidized bed |
| b | Bulk |
| d | Downer |
| r | Riser |
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