New Approaches for Prediction of Gas Holdups and Validation of the Mixing Length Concept in Gas-Liquid and Slurry Bubble Columns

The successful prediction of gas holdups in (slurry) bubble columns is very important for both the design and scale-up of these reactors. In the literature hitherto there are only few reliable empirical gas holdup correlations (mainly for gas-liquid bubble columns). In this work, a new approach has been developed for predicting the gas holdups at ambient conditions in gas-liquid bubble columns (0.095 and 0.102 m in ID) operated with 21 pure organic liquids, 17 liquid mixtures and tap water. The same approach was also applied for prediction of gas holdups in a slurry bubble column (0.095 m in ID) operated with 7 three-phase systems under ambient conditions.
The new model for gas holdup prediction in (slurry) bubble columns is based on the theoretical calculation of the gas-liquid interfacial area: a=6ɛg/ds. This correlation is explicitly valid for rigid spherical bubbles. In the case of slurry bubble columns, an empirical correlation developed by Schumpe et al. (1987) for the interfacial area prediction is frequently used. When both correlations are set equal, then the theoretical gas holdup can be calculated provided that one knows how to estimate the Sauter-mean bubble diameter ds and the effective viscosity μeff. The same approach was also applied to gas-liquid bubble columns. However, the interfacial areas were estimated by the empirical correlation of Akita and Yoshida (1974).
In the above-mentioned approaches the estimation of the Sauter-mean bubble diameters ds was based on empirical correlations (Wilkinson et al. (1994) for bubble columns and Lemoine et al. (2008) for slurry bubble columns).
For given gas-liquid-solid system, gas distributor layout and column diameter, the ds value is a function of both the superficial gas velocity and gas holdup (1-ɛg)^1.56 (Lemoine et al., 2008). Following the above-described approach, the ɛg value was calculated (based on a trial and error method) from the ratio ɛg/(1-ɛg)^1.56. The obtained ɛg value in this way was multiplied by a correction factor (a function of Eӧtvӧs number Eo) since the formed bubbles under the tested experimental conditions were oblate ellipsoidal (i.e. non-spherical). In the case of slurry bubble columns, the Eo number was based on the slurry density ρSL.
Following the above-described approach in two-phase bubble columns, it was found that for given gas-liquid system, column diameter and Ug value the theoretical gas holdup could be estimated from the simplified correlation: ɛg^0.13=const. Then the obtained ɛg value was also multiplied by a correction factor (a function of Eo). So, the objective of this part of the research work was to find the best expressions for the correction factors in two-phase and three-phase bubble columns, which fit successfully the experimental gas holdups ɛg.
The determination of the scale of liquid mixing in the main hydrodynamic regimes of bubble column operation is also of essential importance for their design and scale-up. In this context, a new method (and correlation) has been proposed by Kawase and Tokunaga (1991) for the determination of the mixing length (L). The parameter L characterizes the degree and scale of mixing. It can be also associated with the distance over which a turbulent eddy retains its identity.
In this work, a new definition of entropy (E) has been developed on the basis of gas holdup time series data measured by a conductivity wire-mesh sensor in an air-water bubble column (0.15 m in ID). The new entropy has been estimated by means of multiple reconstructions of the signal. It was found that in the Ug range from 0.034 to 0.101 m/s (see Fig. 2), the entropy (E) decreased monotonously and it was correlated to the mixing length L (a function of both column diameter Dc and Ug^−0.38). Secondly, a newly defined information entropy (IE) has been also extracted from the gas holdup fluctuations and correlated to the mixing length in almost the same Ug range (0.022−0.101 m/s).
In a previous publication (Nedeltchev et al., 2014), it was shown that the Kolmogorov entropy (KE) and a new statistical parameter (called “maximum number of signal’s visits in a region” Nvmax) were capable of identifying the range of applicability (0.034≤ Ug≤0.112 m/s) of the mixing length concept. Another statistical parameter F (average/(3×average absolute deviation)) was also introduced by Nedeltchev and Schubert (2015) for validating the range of applicability of the mixing length concept. It was also found that the F index is a function of the mixing length L in the Ug range from 0.034 to 0.112 m/s.
In this work, a comparison of the results obtained by the five different parameters (E, IE, KE, Nvmax and F) is performed. Most of them (except for KE) are new and such a comparison has not been reported in the literature hitherto. It revealed that the determination of the boundaries of the transition flow regime and the range of applicability of the mixing length concept depends to some extent on the parameter used. Based on the above-mentioned parameters it was found that the mixing length concept was applicable only in the transition flow regime. Such a result has not been reported in the previous papers (for instance, in Kawase and Tokunaga, 1991).