(434g) Modeling Tube-Scale Fixed Bed Radial Heat Transfer From Sub-Particle-Scale Velocity Components

Heat
transfer in low tube-to-particle diameter ratio (N) fixed beds is important as
these devices are extensively used as reactors, in applications such as steam
reforming, partial oxidation and hydrogenation. Current approaches to radial
fixed bed heat transfer modeling all suffer from serious deficiencies, as they average
out the small-scale velocity components in the bed, replacing them with either plug
flow or grossly simplified velocity models. Transport processes are then based
on the classical k_r ? h_w effective
parameter modelor its extensions, in which transport mechanisms are
represented only on an averaged tube diameter scale. A CFD-based multi-scale approach
is proposed to developing a radial heat transfer model for fixed beds. In this
approach we do not use the effective thermal conductivity k_r for
thermal dispersion, a purely fluid mechanical phenomenon, or the apparent wall
heat transfer coefficient h_w. We have developed a new velocity-based
approach that depends on the use of computational fluid dynamics (CFD) to
obtain 3D flow fields on a sub-particle level; these are carefully
coarse-grained to retain the essential features and to provide the tube-scale
input to a new 2D model. CFD here also provided validated temperature profiles
to test the new model.

Computer-generated beds of spheres were
obtained through a modified soft-sphere collective rearrangement algorithm, for
N = 3.96, 5.96 and 7.99. Overall, the features and magnitude of experimental radial
void fraction profiles and bulk voidage were well-reproduced. CFD simulations
of flow and heat transfer were carried out using the commercial code Fluent^®
version 6.3.26, for the values of N and for four values of Re in the range 80 -
1900. The simulations for lower Re were run as
laminar (DNS) models, while the Re = 1900 simulations were run as turbulent k-ε
models. The spherical particles were nominally 0.0254 m in diameter,
reduced to 99% to avoid meshing problems. The mesh had boundary layer prism
cells on the tube wall, with tetrahedral cells in the main fluid volume. Mesh
refinement studies and validation against experimental data were done and have
been reported previously.

The conduction contribution to the heat transfer model was
obtained with a locally-varying stagnant thermal conductivity, k_e⁰(r)
from the Zehner-Schlünder cell model, as a function of true fluid thermal
conductivity, particle thermal conductivity and the radial bed voidage profile.
Our earlier work on conduction has validated this pointwise use of the
Zehner-Schlünder formula, which was originally developed to predict the
overall, or bed-scale thermal conductivity.

For
the convective contribution, we postulated that the dispersion of heat by small-scale
radial displacements of flow could be represented by a bed-scale two-dimensional
velocity field using the averaged components v_z(r) and v_r(r,z).
The new 2D model equation is (for the pseudohomogeneous case):

The
boundary conditions are T = T_in at z = 0, T
= T_w at r = R, and symmetry at the tube
centerline r = 0. COMSOL^® finite element software was used to
solve the pseudo-continuum 2D pseudohomogeneous heat transfer model. The pointwise
velocities were implemented by the incorporation of tables for void fraction
and axial velocity as functions of r, and radial velocity as a function of r
and z, which were then interpolated as input to the finite element method.

The comparisons between the averaged CFD 3D discrete particle
temperatures and the new 2D model temperatures showed excellent quantitative
agreement. The new model predicted the axial temperature distribution fairly
well; in addition the radial temperature distribution was predicted very well.
The temperatures of the fixed beds had rougher distributions in the axial
direction due to the particle heat transfer by conduction and the radial
velocities distribution between particles. Both models illustrated development
of the temperature into the center of the bed at the same locations.