Development of a Two-Dimensional Mathematical Model for Stress and Velocity Distribution in a Packed Bed
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INTRODUCTION
THE formation of cavity or raceway in the gas crossflow condition is common in various industrial processes such as blast furnace, COREX process, granular drying, etc. The size and shape of the raceway plays a very important role in the distribution of the gas flow within the bed. Hence, it affects the overall heat and mass transfer within the bed. Due to the importance of raceway, it has been an area of research interest since the middle of the last century. Many investigators studied the raceway phenomena and proposed a few correlations based on their experimental (cold[1–4] and hot[5,6] model) and theoretical studies.[3,7–10] One of the earliest mathematical models was developed by Szekeley et al.[3] based on simple force balance between the gas drag and the bed weight on the raceway roof, ignoring the frictional forces. MacDonald and Bridgewater[7] studied the formation of cavity in the stationary and moving bed under the gas cross-flow condition. Though they reported the importance of frictional force in the cavity formation, they did not consider it in their analysis due to the complexity. Apte et al.[8] studied the one-dimensional (1-D) stress distribution above the cavity formed by a vertical air jet introduced at the bottom of the bed. They had considered the force balance between the frictional forces, gas drag, and the bed weight along the tuyere axis. Takahashi et al.[9] studied the stress distribution and the S. SARKAR, formerly Graduate Student, Department of Materials Engineering, Indian Institute of Science, is Postdoctoral Fellow, Fundamentals of Chemical Reaction Engineering, Faculty of Science and Technology, University of Twente, 7522 LD, Enschede, The Netherlands. G.S. GUPTA, Associate Professor, is with the Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India. Contact e-mail: [email protected] Manuscript submitted October 25, 2006. Article published online November 29, 2007. METALLURGICAL AND MATERIALS TRANSACTIONS B
buoyancy force at the lower part of blast furnace assuming deadman as a conical body. They calculated vertical normal stress near the raceway boundary at the tuyere level. Rajneesh et al.[10] explained cavity formation considering the elemental force balance between the frictional forces, gravity force, and the gas pressure gradient, which is nothing but the manifestation of the gas drag force under gas cross-flow condition. They divided the entire system into two regions, Cartesian and Radial region. In their 1-D analysis they considered the frictional forces of two sidewalls, which were acting over the cross-sectional area. They pointed out that for better prediction of raceway size and shape a two-dimensional (2-D) model is essential. Solution of 2-D stress field in presence of gas flow has been a challenge since 1976.[11] Plasticity theory (Mohr–Coulomb failure criteria) is widely used to calculate the stresses for designing the storage vessels for granular material (packed bed) both in 1-D and 2-D cases.[12] A 2-D stres
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