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 model or 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.