(410f) CFD-Aided Modeling of Convective Radial Transport in Fixed Beds of Low Tube-to-Particle Diameter Ratio

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 k_r
or D_r).^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 h_w (or k_w). 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 d_p. 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/d_p 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
v_z(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 D_r,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 Pe_r = v₀d_p/D_r
and a wall Biot number Bi_m = k_wR/D_r. 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 D_T 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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