Wall Effects In Packed Beds

Abstract

Packed beds find extensive application in a wide variety of industries. The objective of the present work is to analyze and evaluate the effects of the wall on structural characteristics, hydrodynamics and heat transfer in packed beds of spheres. As a first attempt, spheres of uniform size are considered.

The cylindrical wall of the bed confines the location of the particles thus leading to significant radial variations in void fraction and specific lateral surface area. The two characteristics at any given radial position r* are estimated by defining a concentric cylindrical channel (CCC) of an arbitrary thickness such that its boundaries are equidistant from the cylindrical surface passing through r* and accounting for the solid volumes or lateral surface areas of the segments of spheres (cap, slice, rod and annular ring) contained in the CCC and with centers lying within a distance of a particle radius from r*.The curved boundaries of the sphere segments are rigorously accounted for. The low aspect ratio beds (aspect ratio less than or equal to 2) show three distinct types of behavior. In beds of aspect ratio 2, the void fraction starts from a value of unity at the wall and decreases to a minimum and then increases to unity at the center of the bed. In beds with aspect ratio between l\/¯3/2, there is a continuous decrease in void fraction from unity at the wall to a fairly low value towards the axis and then a slight increase followed by another decrease. The profiles for aspect ratio less than l\/¯3/2 show a continuous decrease from a value of unity at the wall to zero towards the axis. In contrast, beds of high aspect ratio show heavily damped oscillations in the void fraction up to about five particle diameters from the wall and then a constant value. The lateral surface area variations in low aspect ratio beds show a steep fall from a very high value near the wall, and in high aspect ratio beds an oscillatory nature, though not as strong as in the corresponding void fraction profiles.

The distribution of flow in packed beds for steady flow of an incompressible Newtonian fluid under isothermal conditions is modeled by using Ergun equation with Brinkman-type correction to account for the viscous effects in the region close to the wall. The confining effect of the wall is incorporated through the radial variations in void fraction and specific lateral surface area. The hydraulic radius in the region next to the wall is modified to take into account the resistance of the wall surface to flow. The resulting model equations with appropriate boundary conditions are solved numerically by collocation technique. The influence of aspect ratio in the range 1.25 to 20.3 and Reynolds number from 0.1 to 1000, the two most important factors affecting the flow behavior, is evaluated. The velocity profiles show a peak in the region close to the wall thus indicating severe channeling effect in this region. The magnitude and location of the peak depend on aspect ratio and Reynolds number. The model predictions agree remarkably with reported experimental data on velocity profiles in a bed of aspect ratio 10.7, and on the effect of Reynolds number on friction factors in beds of low aspect ratio.

The radial variations in void fraction, velocity and effective thermal conductivity are incorporated in the two-dimensional pseudo-homogeneous steady-state model to analyze the wall effects on heat transfer in packed beds. Both constant wall temperature and constant wall flux boundary conditions are adopted. The equations are solved numerically using finite difference technique. The radial temperature profiles are seen to be fairly uniform in beds of low aspect ratio thus showing that the often made assumption of complete radial thermal mixing in low aspect ratio beds is valid. Beds of high aspect ratio show strong radial gradients. For constant heat flux condition the slope of the temperature profile remains constant after a small distance from the Inlet thus leading to thermally fully-developed flow. For this condition the heat transfer equations are solved analytically to obtain expressions for Nusselt number and the radial temperature profiles. There is a significant difference in the temperature profiles evaluated in the presence and absence of wall effects. Good agreement is found between the Nusselt numbers obtained from the model and reported experimental data.