Development of a hydrodynamic model for air-lift reactors

DEVELOPMENT OF A HYDRODYNAMIC MODEL FOR AIR-LIFT REACTORS

E.Carvalho1^*, E.Camarasa¹, L.A.C.Meleiro¹, R.Maciel Filho¹, A.Domingues2

Ch.Vial³, G.Wild³, S.Poncin³, N.Midoux³ and J.Bouillard⁴

¹School of Chemical Engineering, Campinas State University, Laboratory of Optimization, Design

and Advanced Control (LOPCA), CP 6066, CEP 13081-970, Campinas - SP, Brazil

Fone: + 55 (0)19 7883971, Fax: + 55(0)19 2894717, E-mail: [e-mail]

²Rhodia Brasil Ltda. Fazenda São Francisco S/N, CEP 13140-000, Paulínia - SP, Brazil

Fone: + 55(0)19 8748561, Fax: + 55(0)19 8748575, E-mail: [e-mail]

³LSGC/ENSIC/CNRS 1, Rue Grandville, BP 451 – 54001, Nancy Cedex – France, Fone:

+ 33(0)3 83175206, Fax: + 33(0)3 83322975, E-mail: [e-mail]

⁴Rhône-Poulenc Industrialisation/CRIT 24, Avenue Jean Jaurès, 69153 Decines

Charpieu Cedex - France, Fone: + 33(0)4 72935417, Fax: + 33(0)4 72935850,

E-mail: [e-mail]

(Received: November 29, 1999 ; Accepted: April 6, 2000)

Abstract - In this paper, a 1D hydrodynamic model has been developed for gas hold-up and liquid circulation velocity prediction in air-lift reactors. The model is based on momentum balance equations and has been adjusted to experimental data collected on a pilot plant reactor equipped with two types of gas distributors and using water and water/butanol as the liquid phase. Different techniques of signal analysis have also been applied to pressure fluctuations in order to extract information about flow regimes and regime transitions.

A good knowledge of the flow pattern is essential to establish adequate correlations for the hydrodynamic model.

Keywords: 1D hydrodynamic model , momentum balance equations, air-lift reactors.

INTRODUCTION

Air-lift loop reactors are commonly used in chemical and biotechnological industry to carry out slow reactions like oxidations and chlorinations (Joshi et al., 1990). For design, operation and control purposes, an accurate simulation of the reactor performance is essential. However, modeling of air-lift reactors represents a difficult task because the influence of operating conditions, reactor geometry and physico-chemical properties of the phases (particularly the non-coalescing behavior) on the hydrodynamics is not yet fully understood.

The present work proposes a mathematical model in order to predict hydrodynamic parameters: the gas hold-up and the liquid circulation velocity. The model has been fitted to experimental data collected on a pilot plant reactor.

Air-lift hydrodynamics is characterized by different flow patterns, depending on the gas flow rate. Two regimes are commonly distinguished: homogeneous and heterogeneous regimes. Hydrodynamic parameters depend strongly on the prevailing flow regime.

For proper modeling, the identification of the nature of the dispersion is therefore essential. Wall pressure signal analysis is applied to extract information about flow regimes: this technique has been shown to give a deeper insight into the complex hydrodynamics of the gas-liquid system (Drahoš and Cermák, 1989; Letzel et al., 1997). In this paper, regime transitions and hydrodynamic characteristics of the air-lift reactor have been investigated using statistical, fractal and spectral analysis of pressure fluctuations.

EXPERIMENTAL SET-UP AND MEASURING METHODS

Apparatus

The experiments are conducted in an external loop air-lift pilot reactor (Fig. 1). An air-lift reactor is a modified bubble column, characterized by three distinct parts, namely riser, gas-liquid separator and downcomer. The pilot reactor is composed of a glass riser with an inside diameter of 0.23 m and a height of 3.5 m, a separator made in steel (height: 0.9 m, diameter: 1.2 m ) and a glass downcomer with a diameter of 0.15 m. Three gas spargers are used: a single-orifice nozzle (10 mm in diameter) and two multiple-orifice spargers (the multiple-orifice nozzle 1 with 12 holes of 2 mm uniformly spaced and the 2 with 72 holes of 2 mm). The superficial gas velocity is varied from 5 mm s^-1 up to 11 cm s^-1 and is controlled by two rotameters. Water and an aqueous solution of butanol at 0.1 %vol. are used as the liquid phase. It is well-known that alcoholic aqueous solutions inhibit bubble coalescence (Keitel and Onken, 1982) and so they can serve to simulate the behavior of organic mixtures generally encountered in industry. Experiments are carried out at room temperature and atmospheric pressure.

Measuring Methods

The usual manometric method is applied to measure the average gas hold-up and the hold-up in the different regions of the riser, using the equation:

(1)

where DP is the difference of static pressure between two sensors placed at a distance Dh. Pressure measurements are performed with 4 piezo-resistive sensors imbedded in the wall of the riser (Hytronic H10, 0-500 mbar). For static hold-up measurements, signals are recorded at a frequency of 1 Hz during 5 min. Dynamic pressure measurements are performed at a frequency of 50 Hz with one probe located far enough from the distributor and the liquid level to avoid end effects. The total acquisition length is 10,000 for each experiment in order to minimize statistical error.

The circulation velocity of the liquid phase is measured with an ultrasonic transit-time flowmeter (Controlotron^â , System 1010P UNIFLOW) located in the downcomer.

THE HYDRODYNAMIC MODEL

When elaborating a mathematical model for air-lift reactors, the main difficulty is to describe gas hold-up as a function of superficial liquid and gas velocities which are not independent. Liquid circulation is induced by injecting gas at the bottom of the riser, thus creating a net density difference between riser and downcomer. This involves that the gas flow rate determines both gas hold-up and liquid flow rate. Hence the hydrodynamic system contains two variables, the gas hold-up and the liquid circulation velocity, which requires the use of two sets of equations.

Description of the Model

1D hydrodynamic modeling is generally based on a balance between driving and resisting forces. The driving force is generated by the net hydrostatic pressure difference resulting from hold-up difference between riser and downcomer. The resisting force is the total pressure drop around the flow circuit caused by friction, direction losses, contraction, expansion losses and etc. Two approaches, one based on a momentum balance, another on an energy balance can be used. Here, a slightly different approach has been used, which consists in equating driving and resisting forces.

Next to the balance equation, the model must also contain an equation relating the gas hold-up with the superficial gas and liquid velocities. This correlation depends on the hydrodynamic regime present in the column, on the gas operating regime and configuration of the distributor and on the physico-chemical properties of the gas-liquid system.

In the riser, the momentum balance equation (Wallis, 1969) and the correlation of the gas hold-up are applied to calculate the pressure along the axis. Knowing the liquid circulation velocity, these two equations can be solved simultaneously beginning from the gas-disengagement tank to the bottom of the riser in order to obtain the pressure and gas hold-up profiles in the riser. In the downcomer, the total pressure drop is estimated which leads to the value of the pressure at the bottom of the riser just below the distributor.An energy balance at the distributor yields a relation between pressures below and above the distributor. The parameters of the model are: the gas-liquid dispersion height, the loss coefficient in the downcomer (function of reactor geometry) and the constants of the gas hold-up correlation. The model consists in a set of non-linear equations. The equations are solved by an iterative procedure that leads to the profiles of pressure, gas velocity and hold-up in the riser and to the global liquid circulation velocity.

The complete equation system has been described in a previous paper (Carvalho et al., 1999). Here, it will be insisted on the way to choose the hold-up correlation.

Gas Hold-up Correlation

Numerous empirical correlations have been proposed in literature for the gas hold-up (Deckwer, 1992). Up to now, there are no correlations able to cover the whole range of physical properties and operating variables. These purely empirical equations have been established without distinction of flow regimes and have a limited capacity for extrapolation. In this work, only the gas hold-up models having some physical significance will be presented. Correlations for homogeneous regime and heterogeneous regimes will be treated separately.

Homogeneous Regime

The homogeneous regime is encountered at low gas velocity and is characterized by a narrow bubble size distribution and by a radially uniform gas hold-up. Coalescence and break-up phenomena can be neglected. As a consequence, variables do not change in the radial direction, so that gas hold-up can be obtained from the relative (or slip) velocity of the phases:

(2)

The actual slip velocity v_GL is less than the terminal rise velocity , this phenomenon is referred to as the hindrance effect:

(3)

The extent of hindrance increases with an increase in gas hold-up. Numerous equations have been proposed to describe the hindrance function f(e_G) (Joshi et al., 1990). It is well described by the expression of Richardson and Zaki (1954) originally proposed for particle sedimentation:

(4)

In the gas/liquid dispersion, n depends of bubble characteristics, which are function of the gas velocity, of the physico-chemical properties of the system and of the gas distributor.

Heterogeneous Regime

At higher gas velocities, the homogeneous regime cannot be maintained. Large bubbles with higher rise velocity than that of small bubbles are formed by coalescence. This flow pattern is referred to as heterogeneous regime and is characterized by a wide bubble size distribution and a marked radial gas hold-up profile. Consequently, averaging of properties is necessary. Zuber and Findlay (1965) have proposed a simple equation referred to as the drift-flux model that considers the actual slip velocity between the bubbles and the liquid phase:

(5)

C₀ is the distribution parameter that depends on the radial profiles of velocity and hold-up in the column. The flatter these profiles, the closer C₀ will be to unity. C₁ is a constant that represents the gas velocity relatively to the mixture. The knowledge of the prevailing flow pattern is therefore necessary in order to choose the appropriate equation for the gas hold-up.

EVOLUTION OF HYDRODYNAMIC PARAMETERS

Figure 2 summarizes the hydrodynamic parameters. The e_G vs. U_G curves were obtained with the different gas distributors in water and water/butanol media. In air-lift reactors, the influence of gas distributor is less pronounced than in classical bubble columns because of the stabilizing effect of the liquid circulation. In water/butanol, the e_G values are of course higher than in the pure coalescing liquid, because the bubbles are smaller and so they have a longer residence time in the reactor. We also note that the difference in e_G values between distributors is greater in non-coalescing medium. The evolution of liquid circulation velocity follows the same trend than e_G.

REGIME IDENTIFICATION

Drift-flux Analysis

A classical method to describe regime limits with e_G and U_L measurements is provided by the drift-flux analysis reported on Fig. 3. The plot of U_G/e_G against U_G+U_L reveals immediately which regime prevails in the column. A change in flow pattern is indicated by a change of slope in the curve. It can be seen from Fig. 3 that flow patterns can be correctly identified with this technique in water but not in water/butanol where limits between regimes are less clear (Camarasa et al., 1999). It should be noted that in the work mentioned just above, no regime transition was observed with the single-orifice distributor. This may be due to the differences in geometry between both plants. In addition, the drift-flux analysis does not give any information about the character of the flow in the respective regimes. This point will be examined in part 5 by analyzing pressure fluctuations.

 

Statistical Analysis

Statistical analysis of time series of the pressure signal involves calculation of the Probability Density Function (PDF). In bubble columns, since the shape of the PDF is always gaussian, the discrimination between regimes must be deduced from the moments of this distribution. The standard deviation s is a simple parameter giving interesting information about regime transitions. Figure 4 presents our experimental s vs. U_G curves with the multiple-orifice nozzle 2 and with the single-orifice nozzle. The results with the multiple-orifice nozzle 1 are similar to those obtained with the multiple-orifice nozzle 2. The minimum of the curves corresponds to the transition between regimes. In water/butanol medium, the values of s are smaller and the transition occurs at slightly higher gas velocity than in pure water. With the single-orifice nozzle, values of s are higher and the difference in values at low and high gas flow rates are smaller than with multiple-orifice nozzles.

Third (Skewness) and fourth moment (Kurtosis) of the PDF are rarely calculated because they are less robust than lower moments as they require large amount of precise data to have any signification. Skewness characterizes the degree of asymmetry of the distribution around its mean value while kurtosis measures the relative peakedness or flatness of the distribution relative to a normal distribution. Skewness does not reveal any significant dependence on flow pattern. But kurtosis exhibits a pronounced maximum at the passage from homogeneous to heterogeneous regime (Fig. 4). This means that the transition regime is characterized by a smaller range of possible values than in a gaussian distribution. This result is in agreement with the work of Vial et al. (1999).

Fractal Analysis

Fractal analysis has been applied using Hurst’s empirical analysis. The so-called Hurst exponent H and the fractal dimension of the time series (Feder, 1988) are determinated. H may be calculated from Hurst rescaled range (R/S)_t , which is a random function with the scaling property (R/S)_tµ t^H (t is a time lag). For self-affine curves, H can be related to the local fractal dimension d_FL by d_FL =2- H. For H>1/2, the processes are persistent or positively correlated (the processes exhibit long-term trends: the future of a time series tends to be similar to its past). For H<1/2 the processes are anti-persistent or negatively correlated (the processes have a short-term memory: the future of a time series tends to oppose to its past). For the singular case H=1/2, processes correspond to the uncorrelated gaussian white noise.

Hurst’s analysis has been applied to the time series of pressure signal in the riser of the air-lift with the different spargers and in water and water/butanol (Fig. 5). Calculations have been performed with a maximal length of time series of 10,000 points. It has been shown that using more points does affect significantly the results.

The transition point is characterized by a maximum in the curves. The position of these maxima is in agreement with the results of the statistical analysis. With the single-orifice nozzle, the maximum of the curve is less pronounced and it is difficult to determine regime transitions. Homogeneous flow presents a pronounced persistent character whereas heterogeneous flow yields white noise signal (H » 0.5). With multiple-orifice nozzles, H reaches values as high as 0.8 – 0.85 whereas with the single-orifice nozzle, H never passes over 0.7. It means that at small gas flow rates, the character of the flow with the single-orifice nozzle is less persistent than with the multiple-orifice spargers.

Spectral Analysis

It is well known that characteristic frequencies of gas-liquid flows can be extracted from the pressure signal by its Power Spectral Density Function (PSDF). Drahoš and Cermák (1989) have shown that the interesting frequencies range from 0 to 20 Hz for bubble columns. The PSDF is estimated dividing the time series of 10,000 points in segments of 512 points each with 10% overlapping. Typical spectra for the air-lift in water are reported on Fig. 6.

With the multiple-orifice spargers, the PSDF exhibits two main frequency bands, one below 1 Hz and another in the range 3-5 Hz. In the homogeneous regime, the spectrum is dominated by the low frequency band, which reflects the slow oscillations of the liquid level accordingly to Drahoš and Cermák (1989). In the heterogeneous regime, the PSDF exhibits a broader peak in the range 3-5 Hz. The sources of this peak are probably related to bubble behavior and to the macro-scale liquid circulation. The transition region is characterized by the simultaneous presence of both peaks.

With the single-orifice sparger, two main frequency bands are also observed, but at different frequencies: 3-5 Hz and 10-15 Hz. At low gas velocity, higher frequencies (10-15 Hz) dominate in the spectrum. The origin of this band is probably caused by the quite unsteady bubble formation process at the sparger which operates under the jet regime. With an increasing gas velocity, a development of the previously described lower frequency band (3-5 Hz) is observed. The intensity of this peak becomes greater than the 10-15 Hz band when U_G >3.5 cm s^-1. In the heterogeneous regime, the shape of the PSDF is similar for all distributors. But when uniform distribution is produced, the PSDF is smoother.

From the information provided by the drift-flux, statistical and fractal analysis, it appears that with the single-orifice nozzle, a transition point exists but the flow at low gas flow rates does not exhibit the same characteristics as with the multiple-orifice spargers. Actually, with this sparger, the flow is always heterogeneous at all gas velocities and the existing transition could correspond to a transition from a "low interaction bubbling regime" to a "high interaction bubbling regime."

It finally appears that the spectral analysis of pressure fluctuations does not provide a good determination of regime limits but is able to distinguish regimes and to give interesting information about characteristic frequencies of the phenomena in the respective regime.

In addition, application of spectral analysis to industrial reactors could be interesting as it is not necessary to cover all the range of gas flow rates to know the flow regime in the column.

COMPARISON MODEL / EXPERIMENTAL DATA

When analyzing the results of the regime identification techniques applied previously, it is possible to define a transition point: homogeneous/heterogeneous for multiple-orifice spargers and low interaction/high interaction for the single-orifice sparger. The position of this point has been determined visually by comparing the minimum of the curves of variance from statistical analysis with the maximum of fractal analysis. Then, the error is approximately 0.001 m/s. Of course, the obtained value is only one approximate estimation, wich is still better than the one obtained by drift-flux model. The position of the transition point for the different systems is listed in Table 1.

The model has been compared to experimental data obtained in water/butanol medium. The gas hold-up correlation used in the model (cf. &3.2) depends on the hydrodynamic regime and on the gas distributor. They are listed in Table 2.

The coefficients of the correlations have been determined by adjust of parameters experimental data in order to minimizing the least squares. The difference in the constants of the correlations is mainly due to the difference in bubble characteristics which are a function on distributor operating regime. With the single-orifice nozzle, only the correlation valid in the heterogeneous regime is applied.

A good agreement is obtained between model and experiment: about 5% mean error on liquid velocity values and 3% mean error on global gas hold-up values.

CONCLUSION

A hydrodynamic air-lift model based both on mathematical equations and on experimental data has been developed for the prediction of the circulation velocity, local values of pressure and gas hold-up in the riser. Comparison between model and pilot experimental data is quite satisfactorily.

Signal analysis of pressure fluctuations has been applied to discriminate regimes and to extract information about regime characteristics. It has been shown that this technique is a good tool to support for hydrodynamic modeling.

ACKNOWLEDGMENTS

The authors are grateful to FAPESP, CNPq, ANRT, Rhône-Poulenc Industrialisation and Rhodia Brasil Ltda for their financial support.

NOMENCLATURE

C₀, C₁ Parameters of the drift-flux model

(-)

g Gravitational constant

(m² s^-1)

h Height

(m)

H Hurst Coefficient

(-)

P Pressure

(Pa)

(R/S) Rescaled range

(-)

U_G Gas superficial velocity

(m s^-1)

U_L Liquid superficial velocity

(m s^-1)

Terminal rise velocity

(m s^-1)

v_GL Slip or relative velocity

(m s^-1)

e_G Gas hold-up

(-)

r_L Liquid density

(kg m^-3)

s Variance of the Probability Density Function

(volt)

t Time lag

(s)

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To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  16 Mar 2001
• Date of issue
  Dec 2000

History

• Accepted
  06 Apr 2000
• Received
  29 Nov 1999