Theoretical prediction of gas–liquid mass transfer coefficient, specific area and hold-up in sparged stirred tanks
Introduction
Mass transfer from the gas to the liquid phase has a decisive importance for the description of systems involving absorption, chemical reactions and fermentations. Usually the mass transfer rate is described as proportional to the concentration gradient, where the proportionality is given by the volumetric mass transfer coefficient, kLa. This coefficient must be known in order to carry out the design and scale-up of contactors, chemical reactors and bioreactors. This is a very complex task, especially in systems with chemical reactions and viscous media, such as production of antibiotics, polysaccharides and waste treatments.
A stirred tank is a very often used contactor, mainly as a reactor, in which a gas or mixture of gases is distributed in the liquid, in the form of bubbles, by an appropriate distributor and an agitation system which cause an intense mixing of the liquid phase. The fractional hold-up gas in the stirred tank reactor is a basic measurement of the efficiency of gas–liquid contacting. This, together with the knowledge of the mean bubble diameter, determines the gas–liquid interfacial area, which strongly depends on the physico-chemical properties of the system and geometrical parameters of the contactor.
Therefore, the most important characteristics affecting the mass transfer between the gas–liquid phases are the energy dissipated by turbulence (ε), the gas hold-up (φ), the size of the bubbles (db), or their distribution within the volume being mixed. Those variables are a function of operational conditions (power input or stirrer speed and gas flow), physical properties of the solution and gas phase (viscosity, surface tension and density) and the geometry of the vessel, mainly the stirrer and the gas distributor. At the same time, the influences of almost all parameters are interrelated, which in turn makes accurate design and scale-up of these contactors–reactors a difficult multi-parameter problem.
The rheological behaviour of the liquid can be described in terms of the Ostwald–de Waele model:as well the Casson model:Usually, the effects of those variables on kLa have been taken into account by means of simple empirical correlation, using dimension or dimensionless groups raised to several powers. A high number of correlations are available for the volumetric mass transfer coefficient, but often the results from different equations do not agree with others (Garcia-Ochoa and Gomez, 1998; Gogate et al., 2000). In recent years, another type of models based on artificial intelligence, such as neural networks, have been developed (Garcia-Ochoa and Gomez, 2001), but also with an empirical base. Until now, these empirical models have been the most frequently employed because they are useful for the scale-up. Nevertheless, a considerable experimental effort is necessary; therefore, they are being displaced by theoretical or predictive models, based on more fundamental principles. Thus, in recent years, several authors have developed theoretical models capable of describing the mass transfer rate in reactors. Most of these studies have been proposed for bubble column and airlift contactors (Kawase et al., 1987; Garcia-Calvo 1989, Garcia-Calvo 1992; Tobajas et al., 1999).
In stirred tank reactors, due to the complexity of two-phase fluid dynamics, the application of theoretical equations is very limited. With the exception of the semi-theoretical model proposed by Kawase and Moo-Young (1988), there is no theoretical model in the literature able to predict kLa. Probably this is due to the difficulty in obtaining a fluid-dynamic model that, encompassing all the factors affecting the system, is applicable to the different equipments and can predict the different parameters (db,φ,a,kL) under a wide range of operational conditions, and for different tank volumes.
The aim of this work is the proposal of a fundamental approach for the estimation of the volumetric mass transfer coefficient in stirred tank reactors. In a first step, it is necessary to separate kLa into the two parameters, kL and a. The coefficient mass transfer, kL, is estimated according to Higbie's penetration theory, for the description of the rate of mass transfer process in the continuous phase around the bubbles. Then, the specific interfacial area, a, is determined from gas hold-up and mean bubble size. The capability of prediction of equations proposed is discussed using the kLa experimental data and empirical correlations obtained in stirred tank reactors of different volumes and stirrer geometries, for both water and viscous fluids.
Section snippets
Mass transfer coefficient
The mass transfer coefficient, kL, in stirred tank reactors can be estimated by a large number of equations. Most of these are empirical (Johnson and Huang, 1956; Calderbank and Moo-Young, 1961; Perez and Sandall, 1974) and others have a theoretical base (Lamont and Scott, 1970; Prasher and Wills, 1973; Kawase 1987, Kawase 1992; Zhang and Thomas, 1996). The theoretical models for the prediction of the mass transfer coefficient are divided according to different approaches. Some of them are
Parametric sensitivity
Several simulations for different parameter values to examine the parameter sensitivities of the previous model have been performed. Using the equations proposed, the model is able to predict the influence of operational conditions, the properties of the liquid and the geometrical parameters of the vessel on the coefficient mass transfer, gas hold-up and specific interfacial area and, therefore, for combinations of these magnitudes, on the volumetric mass transfer coefficient.
Predicitions of kLa values under different operational conditions
Using the equations proposed, it is possible to predict the influence of the operational conditions, the properties of the liquid and the geometric parameters of the vessels on the volumetric mass transfer coefficient, kLa. The accuracy of those equations has been checked by making a comparison of kLa experimental values and the data obtained for correlations published for stirred tank reactors of different geometries, for Newtonian and non-Newtonian fluids. The simulations have been carried
Conclusions
A method based on theoretical principles for determination of the volumetric mass transfer coefficient, kLa, in stirrer tank reactors with Newtonian and non-Newtonian fluids has been derived. This model is based on Higbie's penetration theory, which establishes a relationship between the mass transfer coefficient, kL, and the contact time between two different phase elements. This exposure time can be estimated from turbulence isotropic of Kolmogoroff theory as the eddy length to fluctuation
Notation
specific interfacial area, exponents in Eq. (22) constant in , bubble diameter, m vessel diameter, m diffusivity on the liquid, constant gravitational, blade height of stirrer, m consistency index in a power-law model, volumetric oxygen mass transfer coefficient, diameter of the bubble formed in the turbulent stream, m length defined by Eq. (15) flow index in a power-law model, dimensionless stirrer speed, rps power dimensionless number, dimensionless power
Acknowledgements
This work has been supported by Plan Nacional I+D, Programa de Procesos y Productos Quı́micos, under contract no. PPQ2001-1361-C02-01.
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