Pergamon Chemical Engineering Science, Vol. 50. No 5, pp g27 8~6, 1995 Copyright V )995 Elsevier Science Lid Printed in Great Britain. All rights rese, rved 0009 2500'95 $050 ~ O(X) 0009-2509(94)00441-2 THREE-PHASE MASS TRANSFER: ONE-DIMENSIONAL HETEROGENEOUS MODEL ENDRE NAGY Research Institute of Chemical Engineering, Hungarian Academy of Sciences, H-8201 Veszprem, P.O. Box 125, Hungary (Received 8 April 1994; accepted in revised Jbrm 1 September 1994) Abstraet--A new one-dimensional heterogeneous model has been developed to describe the mass transfer rate for a three-phase system which contains larger particles, i.e. particles larger than or in the same order of magnitude as the gas-liquid boundary layer. The diffusionof the absorbed component into the bulk phase can occur through a composite medium--it can be a continuous + a dispersed phase (6~ + d > ~) or a continuous + a dispersed + a continuous phase (6t + d < 6}- or through the continuous phase, only. The mass transport through the heterogeneous medium has been solved using the film-penetration theory. For the sake of simplicity,the spherical particles are assumed to be cubic ones in the boundary layer. The model also takes into account the effects of the first- and zero-order reactions which can take place either in the continuous or in the dispersed phase. The effect of reaction and particle size on the enhancement of absorption rate has been discussed. In addition, the role of the boundary conditions and that of the bulk phase concentration (in brief) has been shown (or analysed). INTRODUCTION The presence of fine solid particles or a finely dis- persed second liquid phase in the continuous absorb- ent phase can have a very strong effect on the mass transfer rate between the gas and the continuous liquid phases. The mass transport into the fine par- ticles in the gas-liquid boundary layer can essentially alter the concentration gradient at the gas-liquid in- terface and, as a consequence, the absorption rate [see e.g. Alper and Deckwer (1980), Pal et al. (1982), Nagy et al. (1986), Janakiraman and Sharma (1985) Holstvoogd et al. (1986, 1988) Zarzycki and Chacuk (1993)]. The importance of the dispersed second liquid phase is regarded as more and more significant, in the last few years, especially for bioprocesses, where the oxygen uptake rate is very often the limiting rate. Using a second liquid phase with higher permeability and/or higher solubility for the gaseous component, the oxygen uptake rate can substantially be increased (Janakiraman and Sharma, 1985; Junker et al., 1990; Rols et al., 1990, 1991). Bruining et al. (1986), Mehra (1988) and, recently, Nagy and Moser (1995) gave theoretical equations for the absorption rate in the presence of a dispersed second liquid phase using the familiar pseudo-homogeneous model. The most im- portant assumption of this model is that, the particle size of the discontinuous phase must be much smaller than the so called "liquid film thickness" at the gas-liquid interface. This is the case when a micellar microphase is the dispersed phase (Mehra, 1988). This assumption can also be fulfilled when solid particles are the discontinuous phase (Mehra, 1988). However, the droplet size of a simple emulsified phase is mostly larger than 10 15pm, thus the assumption for the model is no longer fulfilled. Recently Junker et al. (1990) studied the oxygen transfer rate in aqueous/perfluorocarbon fermenta- tion solutions and suggested a semi-infinite composite model for the case when the droplet size is larger than the film thickness. The boundary layer at the gas-liquid interface was assumed to consist of both the continuous and discontinuous phases (Fig. 1). In regions where the droplets are located in the film layer, oxygen passes first from the gas phase to the continuous phase and thereafter through the discon- tinuous phase. The one-dimensional composite model using the penetration theory can be regarded as a first approach to mass transfer rate in systems, where dp> & A flat geometry was assumed for the gas-liquid boundary layer and for the discontinuous phase. A zero-order reaction was chosen to accom- pany the mass transfer either in the aqueous continu- ous or in the aqueous dispersed phase. In bio- processes, however, not only the zero-order reaction but also the first-order one is of great importance. Both reactions are limiting cases of the Michaelis- Menten equation. The aim of this work was to develop a general one-dimensional heterogeneous model, which can be used for the whole particle size region, where the pseudo-homogeneous model is not valid yet (i.e. when the particle size is somewhat less than or equal to ,5, or even when d~, > 6). The mass transfer may be accom- panied by a zero- or first-order chemical reaction in both the continuous and the dispersed liquid phase,,;. For the mathematical models presented, the most general unsteady film-penetration theory has been 827