CFD-Aided Modeling of Convective Radial Transport in Fixed Beds of Low Tube-to-Particle Diameter Ratio
Anthony G. Dixon and Nicholas Medeiros
Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA
Convective heat and mass transfer in low tube-to-particle diameter ratio (N) fixed beds is important as they are extensively used as reactors, in applications such as steam reforming, partial oxidation and hydrogenation.1 Current approaches to radial fixed bed convective transport modeling are usually based on the effective medium approach, in which radial dispersion of heat or mass is superimposed on axial plug flow, based on a constant effective radial transport parameter (either kr or Dr).2,3 For packed beds of small N the experimentally observed decrease in this parameter near the tube wall is accounted for by a lumped resistance located at the tube wall, the wall heat (or mass) transfer coefficient hw (or kw). Variations on this approach exist in which either the axial velocity or the radial transport parameter (or both) is allowed to vary radially. Experimental discrimination between the various modeling approaches is problematic due to the difficulty in obtaining enough measurements sufficiently close to the tube wall. In addition there is little agreement on the form of the velocity profile inside the tube, and most experimental data have to be obtained at the exit of the tube, or risk disturbing the packing with intrusive sampling tubes. We have developed a new approach that depends on the use of validated computational fluid dynamics (CFD) simulations to obtain detailed radial velocity and concentration profiles for packed tubes of spheres in the range 5.04 ≤ N ≤ 9.3, at various lengths of packing and over a range of flow rates 87 ≤ Re ≤ 870 where Re is based on superficial velocity and the particle diameter dp. Different effective medium models are then fitted to the simulated developing radial profiles to evaluate whether the models can account for the wall effects on radial convection.
Computer-generated beds of spheres were obtained through a modified soft-sphere collective rearrangement algorithm, for eight tubes of different N. The various tubes were comprised of 1000 to 1250 spheres, with L/dp from 17.0 to 50.13. Overall, the features and magnitude of literature radial void fraction profiles4 and velocity profiles5 were well reproduced. CFD simulations of flow and mass transfer were carried out using the commercial code ANSYS Fluent® version 14.5 for four values of Re in the range 87 - 870. The simulations were run as laminar (DNS) models. Contact points between the spheres were handled by the “caps” method, in which the spheres are locally flattened.6 Boundary layer prism cells were implemented on the tube wall and particle surfaces. Mesh refinement studies ensured grid-independence. The simulations were run first with pure air to obtain axial velocity profiles vz(r) averaged over the length of the packing and also local profiles at different bed depths. Following this, simulations were run with the tube wall coated with a diffusing species, methane, which yielded developing radial concentration profiles.
In the second stage, two-dimensional effective medium models of the fixed beds were solved using the finite element analysis software COMSOL Multiphysics®. In these models, axial velocity profiles and radial methane concentration profiles taken from the 3-D CFD simulations were supplied, and a fitting procedure by use of the Levenberg-Marquardt Least-Squares optimization algorithm was completed to fit the radial mass transfer parameters. It was quickly established that a model with only a single radial transport parameter, the radial dispersion coefficient Dr, was incapable of describing the concentration variation near the tube wall. The standard two-parameter model was then evaluated, with a radial dispersion coefficient and a wall mass transfer coefficient, which were fitted to the CFD data in dimensionless form, as radial Peclet number Per = v0dp/Dr and a wall Biot number Bim = kwR/Dr. Typical results are shown in Figs. 1 and 2 below.
The standard two-parameter model was clearly unable to reproduce the sharp decreases in concentration at the tube wall. A length dependency of both parameters was also noted, particularly in the developing sections of the bed. An explanation for these results is that there is very low (molecular) radial diffusion next to the wall, then a region of higher voidage and enhanced flow near the wall through which the transverse dispersion is increasing until far enough away from the wall the “bed center” dispersion is reached. The two-parameter model cannot properly account for the development of the radial concentration profiles using constant DT over the whole tube radius, followed by a concentration jump at the wall. Two sub-studies were conducted in which a constant velocity profile and a local velocity profile were supplied to the 2-D model, with no significant qualitative change in the results.
Our current work evaluates a model based upon the classical two-layer mixing length theory, which implements a wall-function which accounts for the decrease in transverse radial convective transport in the near-wall region.7,8 Initial studies using this model show improved ability to reproduce the near-wall concentration profiles. Further results will be reported in our presentation at the meeting.
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