Theoretical prediction of gas–liquid mass transfer coefficient, specific area and hold-up in sparged stirred tanks

https://doi.org/10.1016/j.ces.2004.02.009Get rights and content

Abstract

A prediction method for calculating the volumetric mass transfer coefficient, kLa, in gas–liquid sparged stirred tanks is proposed. A theoretical equation based on Hibie's penetration theory and the isotropic turbulence theory of Kolmogoroff is used for kL determination. The values of the interfacial area have been calculated from a hold-up theoretical equation and the mean size of the gas bubble. Both Ostwald–De Waele and Casson models are used to describe the rheological properties of the fluid. The model predicts the mass transfer coefficient and the interfacial area values in stirred tank reactors, analysing the influence of different variables. The values of the volumetric mass transfer coefficient can be calculated for different geometries of the reactor, different physicochemical properties of the liquid and under different operational conditions. The capability of prediction has been examined using experimental data available in the literature for Newtonian and non-Newtonian fluids, for very different vessel sizes, different numbers and types of stirrers and a wide range of operational conditions, with very good results.

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:τ=kγnas well the Casson model:τ1/201/2c1/2γ1/2Usually, 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

aspecific interfacial area, m−1
a,b,cexponents in Eq. (22)
Cconstant in ,
dbbubble diameter, m
Dvessel diameter, m
DLdiffusivity on the liquid, m2s−1
gconstant gravitational, ms−2
hblade height of stirrer, m
kconsistency index in a power-law model, Pasn
kLavolumetric oxygen mass transfer coefficient, s−1
ldiameter of the bubble formed in the turbulent stream, m
Llength defined by Eq. (15)
nflow index in a power-law model, dimensionless
Nstirrer speed, rps
NPpower dimensionless number, dimensionless
Ppower

Acknowledgements

This work has been supported by Plan Nacional I+D, Programa de Procesos y Productos Quı́micos, under contract no. PPQ2001-1361-C02-01.

References (51)

  • V. Linek et al.

    Gas–liquid mass transfer in vessels stirred with multiple impellers-I. Gas liquid mass transfer characteristics in individual stages

    Chemical Engineering Science

    (1996)
  • A.G. Pedersen et al.

    A novel technique based on Kr-85 for quantification of gas-liquid mass transfer in bioreactors

    Chemical Engineering Science

    (1994)
  • M. Tobajas et al.

    Hydrodynamics and mass transfer prediction in a three-phase airlift reactor for marine sediment biotreatment

    Chemical Engineering Science

    (1999)
  • V. Abrardi et al.

    Sparged vessels agitated by multiple turbines

    Proceedings of the European Conference Mixtures

    (1988)
  • S.J. Arjunwadkar et al.

    Gas liquid mass transfer in dual impeller bioreactor

    Biochemical Engineering Journal

    (1998)
  • M. Barigou et al.

    Gas holup and interfacial area distributions in a mechanically agitated gas-liquid contactor

    Transactions of the Institution of Chemical Engineers

    (1996)
  • S.M. Bhavaraju et al.

    The design of gas sparged devices for viscous liquid systems

    A.I.Ch.E Journal

    (1978)
  • R. Billet et al.

    Predicting mass transfer in packed columns

    Chemical Engineering Technology

    (1993)
  • P.H. Calderbank

    The interfacial area in gas–liquid contacting with mechanical agitation

    Transactions of the Institution of Chemical Engineers

    (1958)
  • P.H. Calderbank et al.

    The power characteristic of agitators for the mixing of Newtonian and non-Newtonian fluids

    Transactions of the Institution of Chemical Engineers

    (1961)
  • K. Chandrasekharan et al.

    Further observations on the scale-up of aerated mixing vessels

    Chemical Engineering Science

    (1981)
  • P.V. Danckwers

    Significance of liquid-film coefficients in gas absortion

    Industrial Engineering Chemistry

    (1951)
  • S.M. Davies et al.

    The application of two novel techniques for mass transfer coefficient determination to the scale up of gas sparged agitated vessels

    Proceedings of European Conference on Mixtures

    (1985)
  • L.M. Figueiredo et al.

    The scale-up of aereated mixing vessels for specified oxygen dissolution rates

    Chemical Engineering Science

    (1979)
  • H. Gagnon et al.

    Power consumption and mass transfer in agitated gas–liquid columnsa comparative study

    Canadian Journal of Chemical Engineering

    (1998)
  • Cited by (154)

    View all citing articles on Scopus
    View full text