Method for inferring contact angle and for correlating static liquid hold-up in packed beds

Abstract

Static liquid hold-up (SLH), which is an upper bound measure of the passive liquid volume fraction in packed beds, comes into play in several reaction–separation–transport design models and in correlations of fundamental hydrodynamic variables such as total liquid hold-up, wetting efficiency and pressure gradient. Early reported experimental SLHs in packed beds seemed to level off at a plateau of ca. 5% when capillary force dominates over gravity. Hence, most of the published correlations were constructed taking this asymptotic feature into account, despite the fact that reliable SLH values well beyond 5% were also reported particularly for Bond numbers tending towards zero. Thermodynamics and dimensional analysis of the interfacial momentum balance (Young–Laplace) equation suggests that SLH depends, besides the Bond number ( Bo) and some packing and bed characteristic lengths, on an effective contact angle at the junction where the gas–liquid interface and the packing–gas boundary meet. The contact angle, being difficult to measure, was generally disregarded in most analyses, and in practice, seldom incorporated as a computable quantity in the majority of published SLH correlations. Therefore in this work, a method was proposed for inferring theoretically the contact angle from an energy analysis of the meniscus interfaces’ area obtained from solving the 2-D Young–Laplace equation for two vertically aligned touching equivalent spheres. The approach showed that the intuited contact angles matched the measured ones, when available, with an average error of 8.6%. Following this determination, a new SLH correlation based on the contact angle ( θ c ), the Bond number ( Bo), the solid-to-void volume ratio (1− ε)/ ε, the packing sphericity ( φ), and the bed-to-packing volume ratio, BPR, was proposed and analysed in the light of most published SLH data, i.e., 239 measurements over the past five decades. The SLH correlation yielded an average error of 23%. Sensitivity analysis of its weights showed an important impact of the contact angle as an input variable. The correlation was also found to capture SLH values exceeding 5% in the limit of Bo→0.