Chemical Engineering Science 60 (2005) 5962 – 5970 www.elsevier.com/locate/ces Exploiting the Bjerknes force in bubble column reactors Juerg Ellenberger, Jasper M. van Baten, Rajamani Krishna ∗ Van’t Hoff Institute for Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands Received 23 October 2004; received in revised form 18 December 2004; accepted 4 March 2005 Available online 17 May 2005 Abstract A bubble column, subjected to low-frequency vibrations, displays maxima in the gas holdup when operated at certain frequencies. These maxima represent various harmonics created by standing waves. The axial distribution of gas holdup was measured for these harmonics to demonstrate that the gas holdup at the anti-nodes is higher than at the nodes; this phenomena is a manifestation of the primary Bjerknes force acting on the bubbles. The Bjerknes force can be exploited to obtain the optimum increase in the gas holdup for a given set of operating conditions.  2005 Elsevier Ltd. All rights reserved. Keywords: Harmonics; Sound waves; Bubble columns; Anti-nodes; Nodes; Gas holdup 1. Introduction A bubble column reactor is commonly used in the process industries for carrying out a variety of liquid phase reactions (Deckwer, 1992). In many applications, especially for mass transfer limited reactions, it is necessary to have a precise control on the bubble sizes for improved conversion and se- lectivity. Our earlier work has shown that the application of low-frequency vibrations in the 20–100 Hz range can signif- icantly improve the gas holdup  and volumetric mass trans- fer coefficient k L a (Ellenberger and Krishna, 2002, 2003; Krishna and Ellenberger, 2002). For a specified set of op- erating conditions (dispersion height H and superficial gas velocity U), the improvement in the gas holdup  was found to be a non-monotonous function of the vibration frequency f (Krishna and Ellenberger, 2002), displaying maxima at a set of frequencies suggesting that the bubble column was operating at different harmonics. However, the harmonic op- eration of the bubble column was not investigated in detail and neither was the theoretical background explored to any extent. ∗ Corresponding author. Tel.: +31 20 525 7007; fax: +31 20 525 5604. E-mail address: R.Krishna@uva.nl (R. Krishna). 0009-2509/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.03.036 The theoretical background on the influence of sound waves on single gas bubbles is well documented in the literature (Brennen, 1995; Leighton, 1994; Leighton et al., 1990). When a gas bubble in liquid is subjected to an acoustic pressure field, it can undergo volume pulsa- tions. If the acoustic pressure gradient is non-zero, then it can couple with the bubble oscillations to produce a translation force on the bubble. This is the primary Bjerk- nes force (Bjerknes, 1909). Consider a bubble of volume V = 4 3 r 3 subject to an oscillating pressure field given by P(z,t) = A p sin(2ft) sin(2z/H ) where f is the vibra- tion frequency and  is the wave length of the sine wave. The pressure profiles are illustrated in Fig. 1a for five differ- ent harmonics, HM-1, 2, 3, 4 and 5 corresponding to wave lengths  = 4H, 4H/3, 4H/5, 4H/7 and 4H/9. The Bjerk- nes force acting on the bubble due to volume oscillations is given by the time average of - V(t) dP(z,t) dz , i.e. F vib =-  4 3 r(t) 3 A p 2 H sin(2ft) cos  2 z H   , where the braces represent time-averaging. .