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Metallic iron used in steel industries is mostly obtained from a direct reduction process. The focus of this study is to simulate the furnace of the MIDREX technology. MIDREX technology which is the most important gas-based direct reduced iron (DRI) process in the world, includes reduction, transition and cooling zones. The reduction zone considered as a counter current gas-solid reactor produces sponge iron from iron ore pellets. The transition zone has sufficient height to isolate the reduction zone and cooling zone from each other and the cooling zone cools the solid product down to around 50°C. Each zone has a system of reactions. Simultaneous mass and energy balances along the reduction zone lead to a set of ordinary differential equations with two points of boundary conditions. The transitions and cooling zone are investigated at the equilibrium condition leading to a set of algebraic equations. By solving these systems of equations, we determined the materials concentration, temperature, and pressure along the furnace. Our results are in a good agreement with data reported by Parisi and Laborde (2004) for a real MID-REX plant. Using this model, the effect of reactor length and cooling gas flow on the metallization and the effect of cooling gas flow on the outlet temperature of the solid phase have been studied. These new findings can be used to minimize the consumed energy. NomenclatureA p pellet external area (m 2 ) C reactor gas concentration (mol/m 3 ) C p heat capacity (j/mol K) D diffusion constant h global heat transfer coefficient (pellets/gas) k kinetics constant of the surface reaction k g external mass transfer coefficient (m/s) L reduction zone length (m) M metalizing percent M w molecular weight n p number of pellets per unit volume P pressure (bar-g) Q m molar flow (mol/m 2 s) R reaction rate r 0 external radius of the pellet (m) r c radius of unreacted core (m) T temperature (°C) u velocity (m/s) X extent of reaction/extent of reactant conversation (mol/m 3 ) z space variable inside the reactor (m)

Liquid crystals (LCs) are self-organizing anisotropic viscoelastic soft materials that flow like viscous liquids and display anisotropies like crystals. When a nematic liquid crystal is confined to a capillary tube with strong anchoring conditions, disclination defects of higher (+1) and lower (+1/2) topological charges can coexist, connected through a defect branch point. The shape of the +1/2 disclination lines emanating from the branch point are functions of confinement and bulk elasticity. Previous work shows that nematic liquid crystals under cylindrical confinement display a radial (one +1 line)-to-planar polar (two +1/2 lines) defect texture transition through the nucleation and uniform motion of a disclination branch point. Here we present analysis, scaling and modeling based on a non-linear non-local nematic elastic equation that shows that a branch point also can be generated from disclinations in a liquid crystal confined to different conical geometries with homeotropic anchoring conditions. The cone aperture increases the bending stiffness but decreases the curvature of the disclination. These competing effects lead to a decrease in the total disclination curvature, increase in elastic energy and volume of the branching region. The results are summarized into power laws and integrated into a shape/energy diagram that reveals the effects of confinement and its gradient (cone angle) on disclination shape selection. These new findings are useful to assess the Frank elasticity of new nematic liquid crystals and to predict novel defect structures in complex confinement, including biological microfluidics and mesophase fiber spinning.

The combination of low elasticity modulus, anisotropy, and responsiveness to external fields drives the rich variety of experimentally observed pattern formation in nematic liquid crystals under capillary confinement. External fields of interest in technology and fundamental physics are flow fields, electromagnetic fields, and surface fields due to confinement. In this paper we present theoretical and simulation studies of the pattern formation of nematic liquid crystal disclination loops under capillary confinement including branching processes from a m=+1 disclination line to two m=+1/2 disclination curves that describe the postnucleation and growth regime of the textural transformation from radial to planar polar textures. The early postnucleation and growth of emerging disclination loops in cylindrical capillaries are characterized using analytical and computational methods based on the nematic elastica that takes into account line tension and line bending stiffness. Using subdiffusive growth and constant loop anisotropy, we found that the solution to the nematic elastica is a cusped elliptical geometry characterized by exponential curvature variations. The scaling laws that govern the loop growth reflect the tension to bending elasticity balance and reveal that the loop dilation rate depends on the curvature and normal velocity of the disclination. The line energy growth is accommodated by the decrease in branch-point curvature. These findings contribute to the evolving understanding of textural transformations in nematic liquid crystals under confinement using the nematic elastic methodology.

In this research, the volume of fluid (VOF) method is used to study the hydrodynamics of rotating packed beds (RPBs). The model is validated, and grid independence analyses are performed for cases with different operating conditions. The droplet size distribution is investigated to characterize the hydrodynamics of RPBs. Droplet size distributions are compared in two-dimensional and three-dimensional simulations, and it is demonstrated that two-dimensional simulations can provide an accurate prediction while significantly reducing the significant computational cost. Radial distributions of droplet diameter in the packing region are studied, and different trends are observed at different rotational speeds (fluctuating at ω = 250 rpm, increasing–constant at ω = 500 rpm, and decreasing at higher rotational speeds). These trends are explained using the breakup and coalescence of droplets during droplet–packing and droplet–droplet collisions. Breakup, coalescence, and deposition regimes of droplets depend on the Weber, Ohnesorge, and impact parameters. We observed that with increasing rotational speed, the average droplet diameter and its standard deviation decreased, while changing the liquid flow rate did not significantly affect the average droplet diameter. It is also observed that there is a critical rotational speed (depending on the bed configuration), beyond which the average droplet size does not decrease with increasing rotational speed.