• Acid Red 27;
  • azo dyes;
  • ozonation;
  • kinetic rate constants;
  • mass-transfer coefficient;
  • gas–liquid reactions;
  • reactor design;
  • reactor scale-up


  1. Top of page
  2. Abstract
  9. Acknowledgements

The design of a semibatch bubble column reactor with its mathematical description is proposed for the study of ozonation reactions. The mathematical model used to describe the gas–liquid mass transfer rate in the reactor is based on the unstationary film theory and the resulting model is theoretically analysed to identify its relevant parameters. After its structural identifiability analysis, the parameters are reduced to five, that is, the gas hold-up, the ratio of diffusivities of the reacting species, the volumetric mass transfer coefficient and two time constants related with the kinetic rate constant. From the sensitivity analysis of this reduced model, we conclude that it is not sensible to the gas hold-up and the diffusivity ratio of the reacting species for optimization purposes in moderate and slow kinetic regimes. The model is tested with the reaction between the ozone and the azo-compound Acid Red 27. The experimental data match quite well the model allowing the estimation of the volumetric mass transfer coefficient together with the kinetic constant. The kinetic rate constant for the direct reaction between the ozone and the Acid Red 27 is estimated in k2 = 3723 ± 127 M−1 s−1 at 21.2 ± 0.5°C. The self-coherence of the model, the absence of hypothesis about the state of the film together with the proposed optimization procedure, allows to consider the proposed methodology as a viable alternative for the study of gas–liquid systems in semi-batch bubble columns reactors in comparison with classical approaches.

La conception d'un réacteur semi-continu de type colonne à bulles avec sa description mathématique est proposée pour l'étude des réactions d'ozonation. Le modèle mathématique utilisé pour décrire le taux de transfert de masse gaz–iquide dans le réacteur est fondé sur la théorie des pellicules en régime instationnaire et une analyse théorique du modèle en résultant est faite pour identifier ses paramètres pertinents. Après l'analyse d'identifiabilité structurale, les paramètres sont réduits à cinq, c.-à-d., la retenue de gaz, le rapport de diffusivité des espèces en réaction, le coefficient volumétrique de transfert de masse et deux constantes de temps liées à la constante du taux cinétique. À partir de l'analyse de sensibilité de ce modèle réduit, nous concluons qu'il n'est pas sensible à la retenue de gaz et au rapport de diffusivité des espèces de réaction pour les besoins d'optimisation. Le modèle est examiné par rapport à la réaction entre l'ozone et l'azocomposé acide rouge 27. Les données expérimentales s'accordent tout à fait au modèle permettant l'estimation du coefficient volumétrique de transfert de masse ainsi que de la constante cinétique. La constante de taux cinétique pour la réaction directe entre l'ozone et l'acide rouge 27 est estimé à k2 = 3723 ± 127 1/(M.s) à 21,2 ± 0,5°C. L'auto-cohérence du modèle, l'absence d'hypothèse relative à l'état de la pellicule, ainsi que le processus d'optimisation proposé, nous permettent de considérer la méthodologie proposée comme une alternative viable pour l'étude des systèmes gaz-liquides dans des réacteurs semi-continus de type colonne à bulles comparée aux approches classiques.


  1. Top of page
  2. Abstract
  9. Acknowledgements

Bubble column reactors (BCR) are widely used in chemical, petrochemical, biochemical, and metallurgical industries. The absence of moving parts, their low operating and maintenance costs and the excellent mass and heat transfer rates explain the large number of applications developed with this kind of reactor against the others (Deckwer, 1992; Kantarci et al., 2005). However, the design and scale-up of bubble columns are difficult because of the complexity of the gas and liquid flow patterns coupled with mass transfer and chemical reactions. Key factors such as gas hold-up, ε, volumetric mass transfer coefficient, kLa, specific interfacial area, a, bubble size, r32, and kinetic rate constants, kn, are fundamental for the proper design and the operational control of gas–liquid reactors. There is an extensive list of works, both theoretically or experimentally, oriented to the determination of the physical parameters ε, kLa, a, and r32 as a function of operational conditions and physico-chemical properties of the gas–liquid system (see Kantarci et al., 2005 and references therein). Although the experimental conditions for each author differ significantly to each other, there is a relatively good agreement among them about which are the best experimental and numerical procedures used to determine these magnitudes. This agreement allows the definition of dimensionless expressions useful for the estimation of these parameters.

Otherwise, no such kind of agreement can be found when kinetic data are analysed in gas–liquid systems. Let us consider for instance the CO2–alkanolamine system which is of great technological importance. Recently Vaidya and Kenig (2007) have reviewed the kinetic data present in the literature concerning this system observing significant discrepancies between the kinetic rate constants. Neither an agreement between different authors is found for example in the study of the oxidation of textile azo dyes with the ozone, a system with evident applications in environmental engineering (Liakou et al., 1997; Wu and Wang, 2001; Sevimli and Kinaci, 2002; Sevimli and Sarikaya, 2002; Choi and Wiesmann, 2004; Gokcen and Ozbelge, 2005). In a significant number of papers related with this problem, the kinetic rate constants are reduced to apparent pseudo-first order rate constants which are not useful for reactor design, scale-up or control purposes. Two reasons are mainly behind these discrepancies: (i) the differences in the conception of the experimental set-up; (ii) the difficulties to relate the data with the theoretical models. The aim of this work is to contribute to the modelling of unstationary semi-batch bubble column reactors for understanding the experimental observation of gas–liquid reacting systems and for kinetic constants determination.

Different kinds of devices have been proposed to do kinetic studies in gas–liquid systems. Such devices have in common that their specific interfacial area is known, the flow pattern is nearly well defined and that mostly of them operates under stationary state conditions. Let describe in the following the conventional devices used for the characterization of kinetic gas–liquid systems with a brief description of their experimental advantages and disadvantages. The Stirred-Cell Reactors consist of two CSTR, one for the gas phase and the other for the liquid one, where the gas–liquid interface has a fixed geometrical value (Kucka et al., 2003). Measuring the gas pressure depletion in the upper chamber it is possible to determine the enhancement factors when the same depletion experiment is measured in absence of chemical reaction. The principal advantage of this device is that no chemical analysis of the liquid phase is necessary for the kinetic constant determination, but only apparent kinetic rate constants could be determined. However, the inherent difficulty of this method is to relate the enhancement factor, E, with the absolute kinetic rate constant. While the E definition is done in steady state conditions and thus, is a constant, the experimental enhancement factor determined in the stirred-cell reactors change with the reaction extent. An additional problem related with the experimental determination of E is how it is related with kinetic constants when the gas–liquid mass-transfer process is coupled with a complex chemical kinetic mechanism. The Laminar Jet Absorber consists in a rodlike liquid jet flowing enclosed in an absorption chamber where the absorption rate is determined in steady state conditions measuring the gas flow absorbed by the liquid jet (Rinker et al., 1995; Aboudheir et al., 2004). Assuming the penetration theory, the gas–liquid contact time could be modified changing the jet length and the liquid flow rate and, consequently, the absorption rate. In absence of chemical reaction this device allows the determination of gas-to-liquid diffusion coefficients which is a key parameter for kinetic studies (Alghawas et al., 1989). Despite of the complexity of the instrument and that only it is suitable for fast kinetic regimes, the most significant advantage of the jet absorber is the possibility to work close to true steady state conditions. Finally, the Wetted-Wall Absorber is another apparatus where a thin liquid film flows over a cylindrical rod and the absorption rate is determined measuring the differences between the inlet and the outlet gas concentrations at the chamber. Again it is assumed stationary conditions which allow the calculation of the enhancement factors from a physical absorption experiment (Cullinane and Rochelle, 2006). As it was noted for stirred-cell reactor, in the wetted-wall absorber we have the same problems about the interpretation of the experimental enhancement factors in terms of kinetic rate constants or chemical kinetic mechanisms. Finally, the Bubble Column Reactors have been widely used to study the gas–liquid reactions although the kinetic rate constants derived from these works must be considered as apparent kinetic constants because of a lack of complete description of the physical parameters of these devices (ε, a, kLa, r32, etc.). Considering for instance the reactions between the ozone and the azo dyes, these reactions are almost exclusively analysed using BCR's but only a relatively small number of studies provide absolute kinetic rate constants (Benbelkacem and Debellefontaine, 2003; Benbelkacem et al., 2003, 2004a; Lopez et al., 2004). This aspect is especially important when the ozonation is compared with other wastewater treatments such as advanced oxidation processes, that is, Fenton process, UV, UV/H2O2, UV/O3, O3/H2O2, etc.

The second source of discrepancies between different experiments is related with the theoretical framework linking the experiments with the physicochemical model describing the gas–liquid system. The vast majority of the references given above, together with the references therein, base the determination of the kinetic rate constants on the experimental determination of the enhancement factor which in turn is related with the absolute rate constant using some gas–liquid transfer model, for example, two-film, penetration or surface renewal models. Additionally to these models, a kinetic scheme is coupled with the transport equations leading to different expressions for the enhancement factors in accordance to the particular boundary conditions assumed for each author. An important aspect in common with all these models for the calculation of the enhancement factors is that the boundary conditions of the partial differential equations defining the problem are assumed to be in steady state regime, in other words, they are constants and bounded in a semi-infinite or in a finite space (Danckwerts, 1970; Froment and Bischoff, 1990). These stationary conditions at the boundaries ensure the analytical solution of the resulting stationary diffusion equations which allow the calculation of mass fluxes at the boundaries, and hence, the enhancement factors. However, these stationary conditions at the boundaries cannot be guaranteed in the experiments, especially when these experiments are carried out in semibatch reactors which are intrinsically unstationary in nature.

The purpose of this work is to describe a BCR operating in semibatch conditions based on the unstationary film model, allowing the direct estimation of the absolute kinetic rate constants without the intermediate calculation of the enhancement factor. Experiments of ozonation of azo dye Acid Red 27 were carried out as a reference of gas–liquid reactive system in order to check the model. The paper has been structured as follows. In the first part, the mathematical model of the unstationary interface linked to the reactor model is described in some detail. Because of the structure of the model, it is necessary to carry out a sensitivity analysis of the parameters sorting them by significance. After that, the description of the bubble column reactor specifically designed to meet the restrictions of the model follows. Then, finally, the experimental results are shown followed with their discussion.


  1. Top of page
  2. Abstract
  9. Acknowledgements

Model Description

For modelling purposes of the gas–liquid mass transfer process, we have considered the Lewis and Whitman (1924) quiescent two-film model. This theory has the advantage, against the surface renewal or eddy-diffusion theories, that the time scale is the same for the microscopic description of the gas–liquid interface and for the macroscopic reactor description. The model we propose assumes the following assumptions: (i) the hydrodynamic flow of the gas and liquid bulk in the reactor has been considered as CSTR for both phases because the height-to-diameter ratio of our reactor is very close to one; (ii) the gas film resistance has been considered negligible under the operating conditions and, consequently, the molecular diffusion in the liquid film is the only resistance to mass transport across the interface; (iii) ideal gas and Henry laws are valid under the operating conditions; (iv) there is only two reacting species, A and B, where A is the gas transferred to the liquid and B is a non-volatile substance dissolved in the liquid; (v) a global second order irreversible chemical reaction is considered together with unitary stoichiometric coefficients:

  • equation image(1)

For describing the mass-transfer phenomenon through the gas–liquid interface, a quiescent liquid film is considered separating the gas and liquid bulk. Inside this liquid film of length δ, the substances are spatially distributed as a result of the diffusion-reaction process. Assuming an unstationary diffusion-reaction equation to model the mass transport across the film with constant diffusion coefficients, the mass balances of “A” and “B” in the gas, film and bulk phases lead to the following coupled system of ordinary and partial differential equations:

  • equation image(2)
  • equation image(3)
  • equation image(4)
  • equation image(5)
  • equation image(6)

where y(t) stands for the molar fraction of A in the gas phase, CA and CB are the concentrations at the liquid film and, equation image and equation image are the concentrations at the liquid bulk. Because the gas–liquid interface is not accessible to measurement, the only known state variables of the system of Equations (2)–(6) are y(t), equation image and equation image.

Practically, in all bubble columns the existence of a gas chamber on the top of the column cannot be avoided, that is, the head space, which should be considered in the model to account for the dilution effect on gas molar fraction (Figure 1). This chamber has been modelled as a CSTR with its volume as a function of the gas hold-up. Assuming that VR is the geometrical volume of the reactor and V the volume of liquid, the equation accounting the head space is given by:

  • equation image(7)

where yout(t) is the molar fraction of A at the exit of the head space, y(t) is the solution of the Equation (2), yin(t) is the input molar fraction of A at the bottom of the BCR and Q is the gas flow rate at the same point. Notice that yin has been considered as a function of time. Because of the particular design of our bubble column reactor, it is convenient to express this concentration in this way. In fact, in the experimental section, it will be detailed the structure of the reactor but we mention here that at the bottom of our column there is a gas-mixing chamber leading to the following equation for yin(t):

  • equation image(8)

where y0 is the molar fraction of A entering the mixing chamber and VB is its volume. For the derivation of this equation, no mass accumulation has been considered in the mixing chamber.

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Figure 1. Diagram of the bubble column reactor for ozonation studies. The diameter of the reactor is 19 cm with a total volume of 10.32 L. The volume of the gas-mixing chamber is 2.55 L. The capillaries are 14 cm long with an inner diameter of 0.4 mm. They are disposed uniformly distributed at the bottom of the reactor with 2.1 cm of separation.

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All the initial conditions of the model described by Equations (2)–(8) are zero except for the variables CB and equation image which are equal to the dye initial concentration CB0. For the Equations (2)–(6) the initial and boundary conditions are:

  • equation image(9a)
  • equation image(9b)
  • equation image(10a)
  • equation image(10b)

The conditions (10a) ensure, for the component A, the equilibrium between the gas and the liquid phase at the interface and the continuity between the liquid film and the liquid bulk. In addition, the conditions (10b) express the non-volatility of “B” and the continuity between the liquid film and the liquid bulk.

No rigorous analytical solution of the system (2)–(8) can be envisaged by conventional mathematical techniques because of the coupling of the non-linear ordinary and partial differential equations with the mixed boundary conditions (Equation 10b). The system has been simplified using the method of lines and a finite differences scheme of Equations (3) and (4) along the spatial coordinate, z, to compute an approximated numerical solution of the concentrations inside the liquid film. The simplification is based on a second-order forward, backward, and central differences scheme for the flux terms in Equations (2)–(6). Finally, considering the input gas molar fraction, y0, the saturation concentration of the transferred gas, C* = y0P/H and the initial dye concentration, CB0, the system (2)–(10) in dimensionless form can be written as:

  • equation image(11)
  • equation image(12)
  • equation image(13)
  • equation image(14)
  • equation image(15)
  • equation image(16)
  • equation image(17)
  • equation image(18)

where the dot over the symbols stands for time derivative. The Equations (11) and (12) are the dimensionless ones equivalent to Equations (8) and (2) respectively. The Equation (13) is the discrete version of Equation (3) while the Equations (15) and (16) are the discrete version of Equation (4). The dimensionless bulk concentrations are given by Equations (14) and (17) and the Equation (18) is the dimensionless concentration of A at the reactor head. With this transformation, all the initial conditions of the above system of ODEs are equal to zero. The constants of the model are:

  • equation image(19)

and the parameters are given by:

  • equation image(20)

Model Analysis

In a precedent work, the structural identifiability analysis was applied to identify the accessible parameters of the unsteady-state gas–liquid interface model (Navarro-Laboulais et al., 2006, 2008). This analysis is pertinent in such models where the number of state variables (2N + 2, in our case) is greater than the number of observable variables (xN+1, x2N+1, and x2N+2). The analysis of the model described in those references, which is similar to Equations (2)–(6), had shown seven identifiable parameters:

  • equation image(21)

This group of parameters is the minimum number of parameters needed to characterise the reacting gas–liquid systems which is higher than the parameters considered in classical theories. We remark here that the mass transfer coefficient, kL, is defined as the ratio between the diffusion coefficient of A and the size of the liquid film, that is, equation image. This definition remains unchanged even when there are chemical reactions in the film. Following this idea, it has also been defined a new mass transfer coefficient based on the diffusion of the dissolved reactant B, that is, equation image. Additionally, the structural analysis of the model also results in the definition of two time constants, equation image and equation image, which are typical in mass transport phenomena related to bounded diffusion problems (Crank, 1999). Another significant result of this model is that the Hatta modulus is inherent to the model given by Navarro-Laboulais et al. (2006):

  • equation image(22)

This initial model could be reduced to the five parameters model (Equations 11–18) if (i) the diffusion coefficient of one of the species A or B is known and, (ii) the size of the bubbles in the BCR is known. Under these circumstances the parameter definition (20) holds and the Hatta modulus can be written as:

  • equation image(23)

Notice that the definition of Ha considered here is stationary. It is not defined from the enhancement factor, E, which classically is defined as a flux ratio (Danckwerts, 1970). If we consider an unstationary experiment, we expect that E should change with time, and thus, Ha too. However, in our model, Ha is defined implicitly and does not change with time. In our case the enhancement factor is calculated always a posteriori to the kinetic rate constant determination and never is used as a derived experimental magnitude.

Finally, even though the model could be reduced to the five parameters given by Equation (20), not all of them have the same effect on the observable state variables. The original set of partial and ordinary differential Equations (2)–(8) has been transformed into a set of ODE. This system is used further in the numerical algorithm for parameter determination. The same algorithm allows the sensitivity analysis of the model, that is, to analyse the effect of an infinitesimal change in the parameters on the observable state variables. This kind of mathematical analysis is local in the sense that depends on the values of the parameters around which the sensitivity of the model is analysed. Additionally, this analysis allows a better experimental design reducing or eliminating the unimportant data and determining the effect of the parameter variation on the system behaviour (Ionescu-Bujor and Cacuci, 2004). The first order relative sensitivity coefficients are defined as the partial derivative of the state variables with respect to each parameter of the model as Englezos and Kalogerakis (2001):

  • equation image(24)

This function is evaluated solving its differential equation simultaneously with the ODE system (11)–(18). Because the sensitivities of the model are evaluated locally near some particular point p0 located in the parameters space, the results are different depending on the kinetic regime under analysis. In Figure 2 the sensitivities of gas-phase and the dye concentrations against the parameters (20) are plotted for slow and fast kinetic regimes (Beltran, 2007). In Table 1 the principal figures of the sensitivities are collected. The higher the value in the table, the better the observable variable for parameter determination. From Table 1 we conclude that in the slow and fast kinetic regimes the gas hold-up, ε, and the diffusion coefficients ratio, ξ, will never be accessible by dynamic measurements whichever the observable variables are selected to fit the model because their sensitivities are much lower than the other parameters. Comparing the values of the second and third columns of Table 1 for slow kinetic regimes, the differences between the relative sensitivity of bulk concentrations, equation image and equation image, are not too. Also, it seems slightly more appropriate to consider equation image (the ozone in the solution) for the determination of kinetic rate constants instead of equation image. However, in the fast kinetic regime the relative sensitivities point to equation image as the best candidate to be used for the kinetic rate constant determination. In particular, the volumetric mass transfer coefficient and the kinetic parameters p4 and p5 could be evaluated with increasing accuracy using the states variables x2N+2 (the ozone gas concentration, yA), xN+1 (the dissolved ozone concentration, equation image) and x2N+1 (the dye concentration, equation image) in this order.

thumbnail image

Figure 2. Sensitivity analysis of the mathematical model under (a and b) slow kinetic conditions and (c and d) fast kinetic conditions. The simulation values are close to the experimental operating conditions. For all simulations Patm = 960 mbar, t = 21°C, [O3]g = 57.6 g m−3, ΔP3 = 22 mbar, ΔP2 = 142 mbar, kLa = 0.005 s−1, ξ = 1. CB0 = 0.03 mM, Q = 2 NL min−1. (a and b) k2 = 1 M−1 s−1, resulting in Ha2 = 0.0042. (c and d) k2 = 106 M−1 s−1, resulting in Ha2 = 4.25. (a and c) relative sensitivity equation image of the dimensionless ozone gas concentration, x2N+2 (see Equation 18), for the five parameters (20); for ε and ξ, the parameters are not sensitive to this observable state variable. (b and d) Relative sensitivity equation image of the dimensionless dye concentration, x2N+1 (see Equation 17); again ε and ξ are not sensitive. Notice the change of time scale for fast chemical regime.

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Table 1. Maximum absolute values of the relative sensitivity curves plotted on Figure 2
 yAequation imageequation image
  1. In bold has been signalled the maximum sensitivity of the parameters. For slow kinetic regime the parameter will be better determined following the concentration of the gas transferred to the solution, equation image, while in the fast kinetic regime this is better done with the measurement of the dye concentration equation image.

Slow kinetic regime
Fast kinetic regime

In conclusion, the unstationary gas–liquid interface model (2)–(6) together with the gas chambers Equations (7) and (8) are reduced to a ODE system (11)–(18) characterised by the five parameters (20). After the sensitivity analysis of the model we conclude also that only the volumetric mass transfer coefficient, kLa, and the kinetic parameters p4 and p5 would be determined by fitting the experimental data to this model directly. The gas hold-up, ε, and the diffusion coefficients ratio, ξ, should be determined by independent experiments.


  1. Top of page
  2. Abstract
  9. Acknowledgements

The reactor designed for the study of the ozonation processes is shown on Figure 1. It consists in three parts: (i) the gas mixing chamber; (ii) the gas distributor; (iii) the reactor body comprising the upper gas chamber. The gas mixing chamber has a volume of 2.55 L and its function is to equalise the gas composition and the pressure at the entrance of each capillary. This chamber avoids the nonuniformities in flow and composition during the ozone injection. The experiments consist in a sudden injection of ozone in the reacting media without changing the total gas flow rate in the reactor. The gas distributor is formed by 61 capillaries uniformly distributed on the reactor surface (see inset in Figure 1) separated 2.1 cm between them. The capillaries are 14 cm length with an inner diameter of 0.4 mm. With this configuration and the operating conditions applied, no bubble coalescence has been observed. The study of bubble formation when some gas flows through a submerged rigid orifice has been carried out both experimental (Jamialahmadi et al., 2001) and theoretically (Gerlach et al., 2007; Das and Das, 2009). The generalised expression given in those works for the bubble volume calculation has been used here to estimate the Sauter's bubble radius needed for the mathematical unstationary model. Following the authors aforementioned:

  • equation image(25)

where vB is the bubble volume, R0 the orifice radius, and Bo, Fr, and Ga, are the Bond, Froude, and Galileo dimensionless numbers respectively. These numbers are function of water properties as density, viscosity, and surface tension (IAPWS, 1994, 2008; Tanaka et al., 2001).

The total geometrical volume of the reactor, VR, is 10.32 L. The working solution volume, V, has been fixed to 9.0 L for each experiment in order to prevent the reactor overflow by water expansion.

A scheme of the instrumentation fitted to the ozonation reactor is shown in Figure 3. The ozone generator used in the experiments is an Anseros COM-AD-04 where the ozone production is changed modifying the pulse frequency in the discharge lamps. The ozonator is feed with pure oxygen (Carburos Metálicos) at constant pressure of 0.9 bar. Under these circumstances, the maximum ozone concentration measured was around 80 g Nm−3 for an oxygen flow rate of 2.0 L min−1.

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Figure 3. Scheme of the experimental set-up. The digital mass-flow controllers and meter (DMFC and DMFM) are connected to a computer using a RS485 protocol. The analogue mass-flow controller and meter (AMFC and AMFM) are connected to the analogue inputs or outputs of the data acquisition card. The solenoid valves (V1–V6) are controlled by the digital input/output of the acquisition card. The pressures are measured at the gas mixing chamber (P1), the reactor head (P2), and at the ozone-meter line (P3).

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An analogue mass-flow meter (M+W Instrumentation GmbH, model D6210; see AMFM N2 in Figure 3) was used to measure the nitrogen flow rate used to degas the reactor. This flow was fixed by hand with a manual regulation valve. The gas mixture used in the reactions was set using three digital mass-flow controllers (Bronkhorst, Mod. EL-Flow F201CV; see DMFC O3+O2, O2, Air in Figure 3) and the mixture was measured with a digital mass-flow meter (Bronkhorst, Mod. EL-Flow F111B; see DMFM Gas Mixture in Figure 3). These digital instruments were connected through a RS485 bus, and the communication with them is done using a Dynamic Data Exchange (DDE) server. Additionally, the RS485 is connected to the computer via RS485-to-USB converter (National Instruments, NI USB-485) giving the maximum connectivity of the instruments with the minimum computer resources.

The gas mixture is then sent to the reactor gas-mixing chamber or to the venting point of the system using several two-port solenoid valves (SMC, VDW Series; see V1–V6 in Figure 3). These valves are controlled with the digital port of the data acquisition card using a self-made interface which converts the state of each digital gate to a 0–24 V signal which commands the solenoid valve. With the combination of the valves shown in Figure 3 it is possible to measure the ozone gas not only at the output of the reactor, but also at the input.

The ozone has been measured in the gas and in the liquid phases. For the liquid phase an electrochemical electrode sensor (ATI, Q45H/64) has been used. Instead to measure the dissolved ozone directly inside the reactor, this magnitude has been measured using a low-volume flow cell set in a closed loop where the solution is recycled with a peristaltic pump. The analyser allows the simultaneous measure of the dissolved ozone and the solution temperature. On the other hand, the ozone in the gas phase has been measured using an UV-absorption O3-meter (Anseros, Ozomat GMRTI) which is able to measure up to 200 g m−3. It is important to consider here the flow rate and the pressure at which the measure is done. The gas flow through the measuring cell in the O3 meter must be constant because if this value changes, different delay times should be applied in the kinetic curves. Then, an AMFC (Aalborg, Mod. GFC17) has been linked to the Ozomat meter, fixing the flow rate to 0.3 L min−1. The ozone-gas concentration measure depends on the pressure and the constancy of this property cannot be ensured in all the installation. In order to avoid this problem, instead of the measurement of the ozone in the units given by the instrument, that is, g m−3, it is better to change the units to molar fraction. This unit conversion has the advantage that the ozone molar fraction is insensitive to pressure changes along the reactor. Additionally, the gas pressure is measured in other two critical points in the system, the pressure at the reactor head (see P2 in Figure 3) and in the gas mixing chamber (P1 in Figure 3). All the pressure transmitters are from Druck Limited (Mod PTX 1400) and give a 4–20 mA output proportional to pressure gauge.

The ozonation reactor was fitted to a UV–Vis spectrophotometer to measure the abatement of the reacting substances in the reactor. An Unicam Helios-Gamma spectrophotometer fitted with a Hellma Ultra-Mini Immersion Probe (Mod 661.622) has been used in the experiments. The spectrophotometer data is acquired with the serial RS232 port of the computer.

Finally, all the analogue inputs and outputs of the instruments are linked to a computer using a data acquisition card (Advantech, PCI1710-HG). The card has 8 analogue inputs configured in differential mode and 16 digital inputs/outputs which are used to control the solenoid valves states. The software controlling all the process was developed using LabView 8.20 (National Instruments) which manages properly all the devices linked to the computer.

The dye used in the kinetic experiments was the Acid Red 27 (Amaranth, CAS 915-67-3, Mr = 604.74 g mol−1) provided by Sigma–Aldrich (ref A1016). The absorbance measured at 520 nm has been used to follow this substance. All the solutions were set to pH 2 with HClO4 (Panreac, PA, ref. 132175) and the ozone radical reactions were blocked adding 0.01 M of tert-Butanol (2-Methyl-2-Propanol, Panreac PS, ref. 161903).


  1. Top of page
  2. Abstract
  9. Acknowledgements

In the precedent sections it has been demonstrated, through the sensitivity analysis, that the gas hold-up cannot be evaluated from kinetic data although there is no theoretical limitation to do this. The model is not sensitive to this quantity and then, it must be measured with another complementary technique because ε is still needed for kinetic calculations. The gas hold-up has been estimated independently by a manometric method. The results are plotted on Figure 4 showing a good agreement between the manometric data and the equation proposed by Wilkinson et al. (1992). The gas hold-up is linear with the gas flow rate and thus, with the superficial gas velocity too. This behaviour is characteristic of the bubbly homogeneous regime (Shaikh and Al Dahhan, 2007). The usual flow rate for kinetic experiments was fixed to 2 L min−1 giving a mean bubble diameter of 3.5 mm after Equation (25). Under these conditions we ensure the no coalescence of the bubbles that can change the predicted interfacial area in the reactor given by:

  • equation image(26)

where r32 is the Sauter's mean radius of the bubble calculated after Equation (25). Considering a pressure of 1100 mbar at the reactor, the interfacial area at 2 L min−1 is near 8.0 m−1.

thumbnail image

Figure 4. Gas hold-up for the reactor determined by the manometric method. The linear relation confirms the homogeneous flow regime in the reactor. The line is the prediction of the gas hold-up in the operating conditions, according to the equation proposed by Wilkinson et al. (1992).

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Only the data corresponding to the dye in the solution and the ozone in the gas outlet have been fitted to the model for the determination of the second order kinetic rate constant, k2. Because an analytical solution of this system is not available, a numerical method based on the Gauss Newton algorithm has been used instead. This is a gradient based algorithm which can be implemented in optimization problems involving systems of ordinary differential equations (Englezos and Kalogerakis, 2001) and it has been previously applied for gas–liquid models (Navarro-Laboulais et al., 2008). The method consists in the minimisation of the objective function defined as the sum of the square of residuals assuming that the model could be linearised around some parameter vector. The details of this algorithm is beyond the scope of this paper and we refer the reader to the book of Englezos and Kalogerakis for a complete description of the mathematics and the programming of the method.

In Unstationary Gas–Liquid Interface Model Section, it has been demonstrated that the unstationary film model is reduced to five parameters with only three of them sensitive to the observable state variables, that is, p2, p4, and p5 (see Equation 20). Because p4 and p5 are related with the kinetic rate constant, two series of experiments were designed in order to check the viability of the model. The first one consisted in maintaining the initial dye concentration constant while the initial ozone gas concentration entering the reactor was changed along the series. In this situation, p4 = k2 × CB0 remains constant and p5 = k2 × C* changes in each experiment. In the second series, the inlet ozone concentration remains constant while the initial dye concentration was changed. The results of the first and the second series of experiments are shown on Figures 5 and 6 respectively.

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Figure 5. Results for Series #1. Evolution of Acid Red 27 (a) and ozone gas concentration (b) for different ozone concentration at the input. The values are shown in the legend. CB0 = 3 x 10−5 M. The lines are the best fit to the ODE system (11)–(18).

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thumbnail image

Figure 6. Results for Series #2. Evolution of Acid Red 27 (a) and ozone gas concentration (b) for different initial dye concentration. The values are shown in the legend. [O3]g ∼ 14 g m−3. The lines are the best fit to the ODE system (11)–(18).

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In addition, the parameters p2, p4, and p5 could be evaluated simultaneously but the shape of the sensitivities (see Figure 2) together with the obvious proportionality between p4 and p5 makes the parameters strongly correlated. In order to uncouple these correlations an additional restriction to the minimisation of the objective function has been considered. Because the ratio p4/p5 only depends on the experimental conditions as a proportionality constant between the ratio of dye and ozone initial concentrations, the minimum of the objective function must be sought along the line p4/p5 = CB0/C*.

Figure 5a shows the fit of the dimensionless dye concentration and Figure 5b the dimensionless ozone gas molar fraction in the first series of experiments. From the graphs we confirm that the gas phase measurements do not show any feature (singular or characteristics points) which could be used on kinetic rate constant determination. This experimental evidence seems to contradict previous analytical results which suggested that the measurement of the gas phase could be considered alone for kinetic rate constants determination (Navarro-Laboulais et al., 2006, 2008). The dilution and integration effect of the reactor head chamber can be in the origin of this information loss. In fact, when exclusively the gas phase data are considered for fitting the model, the algorithm does not converge to any physical plausible solution. The use of dye concentration alone or in combination with ozone gas data, leads to a convergence of the algorithm to a minimum which is faster in this second case than in the first one. Figure 6a and b shows respectively the response of the system in the second series of experiments, where the input ozone concentration remains constant and the initial dye concentration is varied. As in the previous series, the ozone gas evolution does not give relevant kinetic information but gives stability to the optimization process.

All the results for the fits shown on Figures 5 and 6 are collected on Table 2. The volumetric mass transfer coefficient is almost constant for all the experiments with independence of the initial conditions even considering the different conditions of the chemical reaction. This is because the mass transfer coefficient defined in this work as equation image does not depend on flux measurements which can introduce errors in its determination due to the lack of knowledge about the instantaneous enhancement factor. Under these circumstances, considering a confidence level of 95%, the mean volumetric mass transfer coefficient measured with the kinetic experiments is:

  • equation image
Table 2. Experimental conditions for series 1 and 2
t (°C)PR (mbar)[O3]g (g/m3)C* (mg/L)C* (µM)CB0 (µM)CB0/C*kLa (s−1)p5 = k2C* (s−1)
Series #1
 211094.9257.613.34277.9300.10790.00298 ± 0.000070.76 ± 0.04
 21.71087.5940.18.99187.2300.16030.00263 ± 0.000070.59 ± 0.05
 21.81096.5932.27.18149.6300.20050.00263 ± 0.000060.65 ± 0.06
 20.21094.2525.66.16128.3300.23380.00300 ± 0.000060.21 ± 0.01
 20.91093.2512.82.9862.1300.48340.00282 ± 0.000070.16 ± 0.02
t (°C)PR (mbar)[O3]g (g/m3)C* (mg/L)C* (µM)CB0 (µM)CB0/C*kLa (s−1)p4 = k2CB0 (s−1)
  1. PR is the absolute pressure in the reactor. [O3]g is the ozone gas concentration measured without the pressure correction in the detector. C* is the saturation ozone concentration evaluated from gas measurements with the Henry constants values recommended by the International Ozone Association (Bin, 2006). The values of kLa, p4 and p5 are the results of the fit.

Series #2
 20.91093.2512.82.9862.1300.48340.00282 ± 0.000070.077 ± 0.008
 21.91106.9213.83.0663.8250.39180.00291 ± 0.000130.090 ± 0.016
 22.31101.9213.42.9260.8200.32890.00266 ± 0.000110.075 ± 0.011
 19.71107.5914.43.5573.9150.20300.00236 ± 0.000110.064 ± 0.008
 20.91105.5914.23.3068.8100.14530.00251 ± 0.000140.033 ± 0.003
 21.31102.5914.23.2467.650.07400.00296 ± 0.000070.018 ± 0.001

In order to check this value and because of the problems related with the dissolved ozone electrode calibration and its membrane characterization (Monzó et al., 2008), a series of oxygen absorption experiments were performed at different gas flow rates. The experiment consisted in degassing the solution with N2 and following the O2 concentration with time in an air absorption experiment. The equation image was calculated considering the model (11)–(18) without chemical reaction and fitting only the liquid phase. The results are shown on Figure 7. The volumetric mass transfer coefficient was evaluated at different gas flow rates showing a linear relation. For the kinetic experiments the gas flow rate is fixed to 2 L min−1 and considering an error confidence level of 95%, the oxygen mass transfer coefficient is:

  • equation image
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Figure 7. Volumetric mass transfer coefficient equation image measured from oxygen absorption experiments at different air flow rates. The line is the best fit to data. The discontinuous lines are the confidence bands for a 95% confidence level.

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Using this value to check the previously obtained with the ozone kinetic experiments, we need to know the diffusion coefficients of the oxygen and ozone in water together with some hypothesis about the structure of the liquid film, that is, the surface renewal theory or the liquid film theory:

  • equation image(27)

where d is 1/2 for the surface renewal and 1 for the liquid film model. Considering the diffusion coefficient for the oxygen (Ferrell and Himmelblau, 1967) and for the ozone (Winkelmann, 2007), and considering a mean temperature in all the experiments of 21.2 ± 0.5°C, the ratio (27) is 0.7978 for the surface renewal theory and 0.6367 for the liquid film. Consequently, the resulting volumetric mass transfer coefficient for the ozone is 0.00335 s−1 for the surface renewal model and 0.00267 s−1 for the film model. Considering that all the calculations are based on the microscopic unstationary film model, this last result is more coherent with the definition of the mass transfer coefficient in the model.

Once we have the mean mass transfer coefficient of the system, the data are again fitted to the model (11)–(18) but in this run considering this mean value for kLa, 0.00267 s−1. Then, the problem is again simplified because the value of kLa is fixed and considering that p4 and p5 are bonded by p4/p5 = CB0/C* only one parameter is fitted at one time. With the first experiment series the new value of p5 is recalculated while the second series is used for recalculating p4. Plotting p5 and p4 against the ozone saturation concentration and the initial dye concentration respectively, by definition we must have a line which slope is the kinetic rate constant. These plots are shown on Figure 8a and b respectively. The kinetic constant for the direct reaction between the ozone and the Acid Red 27 is 2933 ± 351 M−1 s−1 from the first experimental series and 3723 ± 127 M−1 s−1 from the second series.

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Figure 8. Kinetic constant determination from parameter p5 (a) calculated from data shown on Figure 5, and from parameter p4 (b) from data of Figure 6. The dotted and continuous lines are the confidence bands of the fit for 95% and 99% of confidence level.

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The origin of the discrepancy between these two values can be understood if we know how the systematic errors are propagated through the experiments and the model. In the first series, the inlet ozone gas concentration is modified resulting in a change in the saturation ozone concentration in the solution. The numerical value of this quantity depends on the accuracy of the ozone gas concentration measurement or on its direct measurement in the solution. Both alternatives are not free of instrumental problems resulting in a higher inaccuracy of this quantity. In fact, considering the Figure 8b, because there is not a significant variation in the ozone for these series, the data dispersion is lower than the shown in Figure 8a.

Finally, the kinetic constant determined by the methodology proposed in this paper is several orders of magnitude under the values reported in the bibliography. Most papers devoted to ozonation of organic dyes give just the apparent first order kinetic rate constants which are only valid for comparison purposes between different experiments when the geometry and operational conditions of the chemical reactor do not change very much (Sevimli and Sarikaya, 2002; Gokcen and Ozbelge, 2005; Wu et al., 2005). In a recent work, Tokumura et al. (2009) give a value of the second order kinetic rate constant for the reaction of the ozone with the Orange II (an azo compound) which is between 183 and 216 L/(g s), but the units used to express this second order constant do not make easy the comparison with the value we have obtained here because the authors express the concentration of the dye in TOC units (Tokumura et al., 2009). A special mention should be done to the work of Lopez et al. (2004) where they estimate the second order kinetic rate constant for the reaction of the ozone with the Acid Red 27 between 2 × 107 and 7 × 107 M−1 s−1. The mathematical model developed by the authors in a series of independent papers (Benbelkacem and Debellefontaine, 2003; Benbelkacem et al., 2003, 2004a,b) is similar to the unstationary film model shown in our work but there are significant differences in the data analysis and parameter estimation procedure. These authors base their calculation on the estimation of the enhancement factor and Hatta modulus along the reaction course. As it was stated in the introduction, the use of these magnitudes implies the existence of a steady state regime at the interfacial level which cannot be always guaranteed theoretically nor experimentally. An evidence of this last statement is that the instantaneous slope of the Hatta modulus evaluated experimentally at the different reaction stages is not null which contradicts its definition done under steady state conditions. Additionally, both models differ on the reactor definition. While in the present work the corresponding hydrodynamic equations of the reactor are coupled with the unstationary film model being able to distinguish both contributions to the observable variables, the model considered by Lopez et al. (2004) considers that the macroscopic description of the reactor does not contribute significantly to the system response.


  1. Top of page
  2. Abstract
  9. Acknowledgements

The experimental response of the bubble column reactor designed in this work for gas–liquid studies shows a good agreement with the unstationary film model. The model incorporates a macroscopic description of the reactor together with the microscopic description of the gas–liquid mass transfer coupled with chemical reactions. The theoretical analysis of the model gives that there are five parameters characterising the response of the system, that is, the gas hold-up, the ratio of diffusivities of the reacting species, the volumetric mass transfer coefficient and two time constants related with the kinetic rate constant. However, the sensitivity analysis of the model considering a semi-batch experiment reduces the number of accessible parameters to three because the model is not sensitive to gas hold-up and the diffusivity ratio.

The model has been tested with the reaction between the ozone and the azo-compound Acid Red 27. The experimental data fitted quite well the model to calculate the mass transfer coefficient and the kinetic rate constant simultaneously. The mass transfer coefficient derived from kinetic experiments matches the one derived from oxygen absorption experiments considering the quiescent film theory for the interface description. The rate constant for the direct reaction of the ozone with the Acid Red 27 is estimated in k2 = 3723 ± 127 M−1 s−1 at 21.2 ± 0.5°C. Additional experiments with a deeper analysis of the model and data will be needed to explain the discrepancy between the value of this kinetic rate constant and those reported in the bibliography. The self-coherence of the model, the absence of hypothesis about the state of the film together with the applied optimization procedure, allow to consider the method proposed in this paper as a viable alternative for the study of gas–liquid systems in semi-batch bubble column reactors.


  1. Top of page
  2. Abstract
  9. Acknowledgements

(1 − ε)S/V = interfacial specific area (m−1)


equation image = Bond number


molar concentration in the liquid phase (mol m−3)


y0P/H = saturation concentration in the liquid bulk (mol m−3)


diffusion coefficient in the liquid (m2 s−1)


enhancement factor


equation image = Froude number


equation image = Galileo number


gravity constant


Henry constant (atm L mol−1)


Hatta modulus


ith constant of the model; see Equations (19) for definition


volumetric mass transfer coefficient in the liquid side (s−1)


DA/δ = mass transfer coefficient in the liquid side for species A (m s−1)


DB/δ = mass transfer coefficient in the liquid side for species B (m s−1)


second order kinetic rate constant (M−1 s−1)


number of equally spaced grid points dividing the interval [0, δ]


absolute pressure (bar)


ith parameter of the model; see Equations (20) for definition

equation image

ith parameter of the model defined in Navarro-Laboulais et al. (2006, 2008); (see Equation 21 for definition)


total gas flow rate at the inlet of the mixing chamber (N L s−1)


gas flow rate through a capillary in gas asperger (N L s−1)


universal gas constant (8.314 J K−1 mol−1)


gas asperger capillary radius (m)


Sauter's mean radius (m)


geometrical gas–liquid exchange surface (m2)


sensitivity coefficient of the ith state variable respect the jth parameter; see Equation (24)


temperature (K)


volume of the liquid phase in the reactor (m3)


geometrical reactor volume (m3)


bubble volume, see Equation (25) (m3)


mixing-gas chamber volume at the bottom of the reactor (m3)


ith dimensionless state variable of the ODEs system (11)–(18)


mole fraction of A at the gas phase in the reactor


mole fraction of A in the gas phase at the reactor outlet


mole fraction of A in the gas phase at the reactor mixing chamber outlet


mole fraction of A in the gas phase at the reactor mixing chamber inlet


spatial coordinate in Equations (2)–(6)

Greek Letters


quiescent liquid film thickness (m)


reactor gas hold-up


DB/DA = diffusion coefficients ratio


liquid density (kg m−3)


liquid surface tension (kg s−2)


liquid dynamic viscosity (kg m−1 s−1)

Subscripts and Superscripts


bulk of the liquid phase


reactor inlet (gas phase) and/or initial concentration (liquid phase)


substance A transferred from gas to liquid phase; associated to O3


non-volatile substance B dissolved in the liquid phase; associated to the dye


  1. Top of page
  2. Abstract
  9. Acknowledgements

The authors wish to acknowledge the Ministerio de Educación y Ciencia for the financial support for this work under contract CTQ2006-10783/PPQ (Project DATO3) and to Generalitat Valenciana for the project ACOMP/2009/006.


  1. Top of page
  2. Abstract
  9. Acknowledgements
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