Axial Dispersion of Liquid in Gas-Liquid Cocurrent Downflow and Upflow Fixed-Bed Reactors with Porous Particles

0263±8762/98/$10.00+0.00 q Institution of Chemical Engineers Trans IChemE, Vol 76, Part A, January 1998

AXIAL DISPERSION OF LIQUID IN GAS-LIQUID COCURRENT DOWNFLOW AND UPFLOW FIXED-BED REACTORS WITH POROUS PARTICLES I. ILIUTA, F. C. THYRION and O. MUNTEAN* Chemical Engineering Institute, UniversiteÂCatholique de Louvain, Louvain-la-Neuve, Belgium *Department of Chemical Engineering, University Politehnica Bucharest, Bucharest, Romania

T

he authors present the experimental investigation of the residence time distribution of the liquid in a gas-liquid down¯ ow and up¯ ow ® xed-bed reactor with porous particles and air/Newtonian or non-Newtonian liquid systems. The in¯ uence of imposed gas phase distribution along the packing bed on the axial mixing of liquid was examined. A tracer injection technique was used and the dynamic responses were interpreted using the pistondispersion-exchange model with Danckwerts’ boundary conditions. The dynamic evolution of the tracer concentration in the particles was described in terms of diffusion phenomena. An imperfect pulse method was used to estimate the model parameters directly from the experimentally input nonideal response and output response. Keywords: three-phase reactors; ® xed bed; up¯ ow; down¯ ow; axial dispersion

staged regions) macromixing models. These models representing the structure of the liquid and including up to six parameters (the number of model parameters corresponds generally to the degree of model complexity which increases with its approach to physical reality) are summarized in the reviews of Shah et al.1 , Gianetto et al.2 , Shah3 , Calo4 , Zhukova et al.5 , Kastanek et al.6 . The axial dispersion model is the fundamental differential mixing model developed for the description of axial mixing, while other models of this group (multistage dispersion model, probabilistic time model7 , cross¯ ow or pistonexchange model8 , piston-dispersion-exchange model9 ) can be viewed as its modi® cations constructed for the approximation of more signi® cant deviations from the ideal case of plug ¯ ow. Lerou et al.1 0 developed a ¯ ow model which accounts for the major factors responsible for liquid phase dispersion in pulsed gas-liquid cocurrent down¯ ow in ® xed beds: the difference in velocity and rapid mixing between fast moving liquid slugs and slow moving liquid ® lms. This model views the liquid phase as two continuous, cocurently ¯ owing liquid regions with different velocities and continuous mass exchange between them. Series of perfectly mixed stages without back¯ ow between consecutive stages, sometimes called the mixing cell model, is the most elementary and the simplest stagewise model. This model can be extended in two directions: the introduction of back¯ ow between adjacent stages or the individual stages can exhibit deviations from this ideal limit state. If it is necessary, both these approaches can be combined to model perfection. Also, several models constructed as combinations of plug ¯ ow (or axially dispersed ¯ ow) and perfectly mixed zones have been developed to describe satisfactorily the

INTRODUCTION The conversion and /or selectivity of the reaction process in gas-liquid or gas-liquid-solid reactors may be affected by several factors, such as axial mixing, solid-liquid and gasliquid mass transfer, catalyst wetting (especially for twophase down¯ ow ® xed-bed reactors). Previous published studies have shown that axial mixing, a deviation from piston ¯ ow, may be a signi® cant design factor for multiphase packed-bed reactors and one of the decisive factors determining the reactor choice for a speci® c reaction process. The character of the mixing of individual phases in multiphase reactors depends primarily on the reactor type which determines whether the respective phase will be continuous or dispersed under working conditions. Also, axial mixing characteristics of phases depend on the geometry, working conditions, and on the reaction system properties. Because the exact mathematical description of the mixing in ® xed-beds based upon the knowledge of complete velocity ® eld distribution of individual phases in these reactors is not feasible, the hydrodynamics of two-phase ¯ ow in ® xed beds has been commonly described by various approximate models where the parameters can be evaluated from the residence time distribution which de® nes the residence time of ¯ uid elements within the reactor. This formulation is convenient because the residence time distribution can usually be determined experimentally by well-established stimulus-response techniques. The data for residence time distribution are evaluated by a variety of simple and complex differential (the mixing is described by a Fickian diffusion-like formulation) or stagewise (the mixing is described by perfectly mixed 64

AXIAL DISPERSION OF LIQUID IN G/L COCURRENT DOWNFLOW AND UPFLOW FIXED-BED REACTORS 65 hydrodynamics of two-phase ¯ ow in a trickle bed (Michell and Furzer1 1 and Rao and Varma 1 2 ). The majority of data of the axial mixing of liquid phase in gas-liquid cocurrent down¯ ow and up¯ ow were produced on beds in which the particles were nonporous, so that the response of the bed, in the case where a tracer input is introduced into a system, is affected only by the porosity, the length and diameter of the packed column and the wall and bulk ¯ ow conditions. Moreover, the porous particles exhibit a considerable internal porosity so that a signi® cantly different ¯ ow behaviour and consequently a different extent of axial mixing in the liquid phase must be expected in the case of multiphase reactors. Very little information is available on the axial mixing of the liquid phase in two-phase ¯ ow through porous beds (Charpentier et al.1 3 Ð two-phase down¯ ow and air-water systems; Yang et al.1 4 Ð twophase up¯ ow and air-water systems) and it seems that this kind of data is not present in the literature for air/nonNewtonian liquid systems, such as those encountered in biotechnological and waste treatment applications. Since the practical applications of multiphase reactors often concern porous catalytic particles and because in the biochemical process many ¯ uids do not exhibit Newtonian behaviour, it is considered important to study the axial mixing of the liquid for two-phase down¯ ow and up¯ ow using air/Newtonian or non-Newtonian liquid systems. Also, because the continuous production and accumulation of CO2 gas in the immobilized-cell packed-bed reactors can cause axial mixing, the hydrodynamic studies should include the effects of the additional feed of the gas phase at different levels in the bed on the hydrodynamic parameters. It seems that this kind of data is not present in the literature. The primary objective of this paper is to present new data on the axial dispersion of the liquid in a ® xed-bed reactor with cocurrent down¯ ow or up¯ ow of the gas and liquid with porous particles and, secondly, to evaluate the signi® cance of the results obtained in the case of the additional feed of the gas phase at different levels in the bed. A transient diffusion model of the tracer in the catalytic particle coupled with the PDE model was used to represent and analyse the experimental data. EXPERIMENTAL Figure 1 shows the experimental equipment. It mainly consists of a 51 mm i.d. packed column. The height of the bed was 0.92 m. The characteristics of the porous particles and the parameters of the ® xed-bed are given in Table 1. In the case of down¯ ow operation of the column, the gasliquid distributor was positioned at such a height above the bed that the liquid was evenly distributed without spraying it on the wall of the column. The device consists of 5 tubes for gas ¯ ow of 50 mm long and 8 mm i.d. and 16 holes for liquid ¯ ow of 2 mm i.d. For two-phase up¯ ow, the gas and liquid were introduced into a cone ® lled with Pall rings then passed through a perforated plate distributor; at the top of the bed another perforated plate maintained the packed bed in place. Air, the gas phase, was taken from the laboratory supply line and was fed at the top of the column (two-phase down¯ ow), or at the bottom of the column (two-phase up¯ ow). Also, the gas phase was fed at the top/bottom of the Trans IChemE, Vol 76, Part A, January 1998

Figure 1. Simpli® ed ¯ ow sheet of the experimental plant: (1) column, (2) gas-liquid distributor, (3) reservoir, (4) pump, (5) valve, (6) liquid rotameter, (7) gas rotameter, (8) conductivity cell, (9) conducticity apparatus, (10) pressure transducer, (11) power supply, (12) resistor, (13) voltmeter, (14) ampli® er, (15) data acquisition system, (16) computer.

column and at another two levels of the packed bed placed at 0.3 and 0.6 m. The diagram of the gas feed is given in Table 2. The liquid was water or pseudoplastic carboxymethyl cellulose (CMC) solutions at different concentrations. The CMC solutions were prepared by dissolving CMC powder (carboxymethyl-cellulose sodium salt, low viscosity) in water at 608 C. The rheological behaviour of pseudoplasticCMC solutions was characterized by the power law model of Ostwald-de Waale (C = kc n , l eff = kc n- 1 ). The steady shear stress± shear rate data of the non-Newtonian polymer solutions were measured using a Couette viscometer (Haake, model VT 180). The range of shear rates (63±1007 s- 1 ) corresponds to those reached in the ® xed bed experiments; the shear rate for ® xed bed of spheres was calculated by using the expression of Christopher and Middleman1 5 . The shear stress was measured with both increasing and decreasing shear rates and a mean value was used in evaluating the rheological model parameters. The consistency index k and the ¯ ow behaviour index n (Table 3) were evaluated by plotting the logarithm of the measured shear stress against the logarithm of shear rate; a linear regression then gave k and n from the power law model. All the experiments were conducted at room temperature and near atmospheric pressure. Hydrodynamics were investigated for super® cial liquid velocities from 0.0021 to 0.0135 m s- 1 (two-phase up¯ ow) and 0.0021 to 0.017 m s- 1 (two-phase down¯ ow), respectively super® cial gas velocities from 0.028 to 0.41 m s- 1 . The axial dispersion coef® cient values were determined from residence time distribution (RTD) measurements. In Table 1. Packing characteristics. Packing material Particle shape Average diameter, m Internal porosity, m3 / m3particle External porosity, m3 / m3 Structural solid holdup, m3 / m3 Particle density, kg/m3 Structural density, kg/m3

SiO2 / Al 2 O3 / Cr 2 O3 sphere 0.0033 0.464 0.356 0.345 1160 2120

66

ILIUTA et al. Table 2. Diagram of the gas feed. G1 (H

= 0.0 m), m

3

h- 1

G2 (H

= 0.3 m), m

0.6 0.6 1.0 1.0 1.6 1.6 1.6

3

h- 1

G3 (H

= 0.6 m), m /h3

0.0 0.2 0.0 0.3 0.0 0.3 0.65

order to compare the values of the axial dispersion coef® cient with literature correlations, the mean values of the repeated experiments, obtained for each pair of gas and liquid ¯ ow rates were considered. In the case of two-phase up¯ ow, the experimental residence time distribution measurements were carried out in bubble and pulsing ¯ ow regimes. For two-phase down¯ ow, the runs were performed in trickle, pulsing and dispersed bubble ¯ ow regimes. Also, the runs were performed in the transition between these regimes. A tracer technique was applied. The electrical conductivity of the potassium chloride tracer was simultaneously measured downstream or upstream of the injection point at the bottom and the top of the bed (the tracer mass balance for both the input and output responses was satis® ed within 0.32±0.8% when the total amount of tracer injected was compared to that determined from the zeroth moments and the measured value of the liquid volumetric ¯ ow rate1 6 ). The signals of all sensors were ampli® ed and transmitted to a computer by a data acquisition system. Since the presence of electrolyte and the increase of the temperature of the CMC solutions due to the heat of pumping can affect the rheological properties, a fresh liquid was taken for each experiment. However, the exploratory measurements suggested that the change of rheological properties was insigni® cant for each experiment. The transition from the high to low interaction regime in the case of two-phase down¯ ow was determined by cessation of pressure ¯ uctuations characteristic of the pulsing ¯ ow regime or by visual observations of the ¯ ow behaviour for dispersed bubble ¯ ow regimes. The pressure drop measurements were made with a pressure transducer connected to two pressure taps located at the bottom and the top of the bed. The stability of the column operation can be observed accurately by a continuous recording of the pressure drop signal.

0.0 0.2 0.0 0.3 0.0 0.3 0.65

particles, a global dispersive ¯ ux may be considered (the driving force being the concentration gradient in the ¯ uid phase) with simultaneous transfer at the surface of the particles and only the dispersion in the ¯ uid phase, together with interphase mass transfer, can be taken in consideration. In this work, the piston-dispersion-exchange model (PDE) with Danckwerts boundary conditions corresponding to the imperfect pulse concentration change at the reactor inlet was used to describe the liquid ¯ ow. The dynamic evolution of the tracer concentration in the particles was described in terms of diffusion phenomena. The external and internal wetting of the solid spherical particles was considered complete. Also, the reversible adsorption of the tracer was not considered as a mechanism to explain the observed residence time distribution curves distortion. If such a mechanism were acting, it would be coupled with the diffusion effects, and it would be very dif® cult to discriminate between the two effects without independent adsorption measurements. The dimensionless forms of the tracer mass balance equations are the following: e

n

CMC 0.1% CMC 0.5% CMC 1.0%

1000.30 1001.40 1004.67

4.96 17.78 55.99

0.936 0.900 0.849

(1) (2)

L,st

x = 0,

(3)

Cd | x=0-

=C |

d x=0 +

¶Cd 0 = ¶x C n = 0, ¶ p = 0 ¶n

1 ¶Cd - Pe x ¶

(4)

x = 1,

(5) (6)

n

= 1,

Bim Cp n =1 - Cd

t

= 0,

Cd

Table 3. Physical properties of non-Newtonian liquids (208 C). k ´ 103 kg m- 1 s2- n

¶Cd vSL ¶Cd N vSL (C C ) + H x + H d - st ¶t ¶ 1 vSL ¶2 Cd as ¶Cp = Pe H ¶x2 - Deff rp ¶n n =1

The boundary and initial conditions are the following:

In the case of porous particles, it is dif® cult to describe the mixing. However, there are several possibilities1 7 : separate dispersive ¯ uxes could be postulated in the ¯ uid and solid phases with interphase transfer at the surface of the

q L kg m- 3

L ,d

¶Cst N VSL (C C ) 0 + L st - d = ¶t ¶Cp Deff ¶ n 2 ¶Cp e L,int = 2 ¶t rp2 n ¶n ¶n e

THEORETICAL CONSIDERATIONS

Non-Newtonian liquids

1

=C =C =0 st

p

¶Cp

= ¶n

n

=1

(7) (8)

The interpretation of the RTD data requires the following parameters: the axial PeÂclet number, the number of transfer units (characterizing the transfer between the dynamic and static external liquid holdups), the actual values of the dynamic and static external liquid holdups, the effective Trans IChemE, Vol 76, Part A, January 1998

AXIAL DISPERSION OF LIQUID IN G/L COCURRENT DOWNFLOW AND UPFLOW FIXED-BED REACTORS 67 diffusion coef® cient of the tracer in the pores of the particle, and the liquid-solid mass transfer coef® cient. The effective diffusion coef® cient was obtained from the time-domain analysis of nonideal pulse tracer response data using a simpler RTD model with one single external liquid holdup and a mass transfer coef® cient between the bulk of the external liquid and the liquid-solid interface, but with the effective diffusion coef® cient of the tracer in the particles as parameter1 6 . In order to obtain the effective diffusion coef® cient of the tracer, the external liquid holdup was taken as the sum of the dynamic and static liquid holdups estimated by draining the bed and the PeÂclet number was taken from the experiments with non-porous particles1 6 . For air/water systems, the liquid-solid mass transfer coef® cientwas taken as theaverage of three correlationsproposed by by Specchia et al.1 8 , Goto and Smith1 9 and Satter® eld et al.2 0 (two-phase down¯ ow), respectively Specchia et al.2 1 , Delaunay et al.2 2 and Mochizuki2 3 (two-phase up¯ ow). The liquid-solid mass transfer coef® cient for air-CMC systems was obtained from the correlation of Kawase and Ulbrecht2 4 . The other model parameters: Pe, N, e L,d , and e L,st are simultaneously identi® ed by using an imperfect pulse method for time-domain analysis of nonideal pulse tracer response data2 5 . The method was based on the least square ® t of the normalized experimental output response with the model predicted response2 6 ,2 7 . The theoretical output response was calculated by solving the model equations, using a numerical solution procedure based on the method of orthogonal collocation. The partial differential equations of the tracer and the boundary conditions were discretized using the collocation method2 8 ,2 9 ,3 0 and the resulting set of coupled ordinary differential equations were integrated using a 4th order Runge-Kutta method. RESULTS AND DISCUSSION In general, the deviations in mixing pattern from ideal plug ¯ ow limit can be caused by nonuniform velocity pro® les, by the existence of differently de® ned by-pass streams, or inef® cient regions, completely or partly isolated from dynamic zone of the liquid phase, and by velocity ¯ uctuations caused by molecular of turbulent diffusion in the ¯ ow direction. These mechanisms, acting individually or in combinations, are known to contribute to the phenomena of the axial mixing of the liquid phase. When the solid phase is porous and contains considerable liquid in its pores, the tracer diffusion inside and outside the porous particle and reversible adsorption of the tracer can distort the shape of the residence time distribution curves and the apparent mixing characteristics of the dynamic zone of the liquid phase. Also, in the case of air-CMC systems, the bed permeability reduction (this phenomenon can by explained by the adsorption of the organic molecules on the particles surface in the bed and by the mechanical entrapment of these molecules) could cause the observed distortion of the residence time distribution curves. PARAMETRIC SENSITIVITY ANALYSIS The low average values of the difference area between the Trans IChemE, Vol 76, Part A, January 1998

Figure 2. Pro® les of the dimensionless concentration of the tracer (twophase down¯ ow, air-0.5 mass % CMC system: L = 5 kg m- 2 s- 1 , G1 = 0.6 m3 h- 1 , G2 = G3 = 0.2 m3 h- 1 , e L,d = 0.177, e L,st = 0.047, Pe = 31.2, N = 4.7, Deff = 0.563 ´ 10- 9 m2 s- 1): Cd (x = 1) and Cst (x = 1) vs. t (model: - Cd , - - Cst , experimental: s - Cd ).

theoretical and the experimental output response curves lead to the conclusion that the PDE model coupled with the transient diffusion equation of the tracer in the porous particles represent adequately the experimental concentration pro® les (Figure 2). The precision of the hydrodynamic parameters determinination for the case of the porous particles depends on the correctness of the evaluation of the effective tracer diffusivity and liquid-solid mass transfer coef® cients1 6 . The lower/higher values for the effective diffusion coef® cient causes high/low values of the theoretically tracer concentration at the tail of the distribution curves. This causes the dynamic liquid holdup to be unreliable. Thus, the model parameters were calculated using the values of the effective diffusion coef® cient evaluated from the time-domain analysis of nonideal pulse tracer response data1 6 . Because the effective intraparticle diffusivity of porous particles is about 15% of the molecular diffusivity1 6 , the dominant resistance is due to the intraparticle diffusion and, therefore, the sensitivity of the experimental response to the liquid-solid mass transfer coef® cient is too small for accurate estimates. The sensitivity of the error function F to variations in the values of the effective diffusion coef® cient was estimated. Also estimated, was the sensitivity of the error function to variations in the values of the other model parameters. The error is a strong function of dynamic liquid holdup and effective diffusion coef® cient and only a weak function of PeÂclet number and of the number of transfer units (Figure 3). The precision in the determination of the ® rst two parameters is expected to be much better than for the PeÂclet number and the number of transfer units.This is con® rmed by the good agreement between the dynamic liquid holdup obtained by draining the bed and ® tting the results of the model (two-phase up¯ ow)1 6 . Con® dence limits of the last two model parameters (the PeÂclet number and the number of transfer units) can be

68

ILIUTA et al.

Figure 3. Parametric sensitivity (two-phase up¯ ow, air-0.5 mass % CMC system: vSL = 0.005 m s- 1 , G1 = 1.0 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 , Pe = 12.25, e L,d = 0.212, e L,st = 0.06, N = 5.7, Deff = 0.563 ´ 10- 9 m2 s- 1 ): (1) dynamic liquid holdup, (2) effective diffusion coef® cient, (3) number of transfer units, (4) PeÂclet number.

Figure 4. Variation of the PeÂclet number with the gas and liquid velocities in the case of air/water system (two-phase down¯ ow): (s ) vSL = 0.0021 m s- 1 , (´) vSL = 0.005 m s- 1 , (*) vSL = 0.007 m s- 1 , (+ ) vSL = 0.0092 m s- 1 , (K) vSL = 0.00113 m s- 1 , (Å) vSL = 0.0013 m s- 1 , (8 ) vSL = 0.0017 m s- 1 .

assigned when data from repeated experiments are available for proper statistical error analysis. The scatter between the PeÂclet number values for repeated experiments and mean PeÂclet numbers was quanti® ed by calculation of the mean absolute relative error1 6 . The mean scatter for the whole data (two-phase down¯ ow and up¯ ow of air-water systems) is quite good: 5.6%.

The data analysis for air/water systems (Figure 4) shows that PeÂclet number is almost exclusively in¯ uenced by the liquid ¯ ow rate; the in¯ uence of the gas ¯ ow rate is practically negligible. As a consequence of the fact that the axial dispersion coef® cient is inversely proportional to the PeÂclet number, the axial dispersion coef® cient decreases by increasing the liquid ¯ ow rate. The PeÂclet number increases from about 0.07 in trickling ¯ ow to 1.2 in pulsing ¯ ow (in general for nonporous particles the PeÂclet number increases from about 0.4 in trickling ¯ ow to 1.7 in pulsing ¯ ow), the axial dispersion coef® cient being greater in trickling ¯ ow than in pulsing ¯ ow. The values of the PeÂclet number obtained using the transient diffusion model of the tracer in the porous particle coupled with the PDE model are larger than those obtained using the PDE model, Table 4. This shows that the

AXIAL DISPERSION FOR TWO-PHASE DOWNFLOW The analysis of the dynamic response of the column with the transient diffusion model of the tracer in the porous particle coupled with the piston-dispersion-exchange model led to the conclusion that, in trickling ¯ ow regime, the axial dispersion is important for all gas/liquid systems used.

Table 4. Values of the PeÂclet number for air-water system. Ped (two-phase up¯ ow) vSL m/s

vSG m/s

transient diffusion model coupled with PDE model

Ped (two-phase down¯ ow)

PDE model

transient diffusion model coupled with PDE model

PDE model

0.005

0.028 0.085 0.140 0.228 0.410

0.028 0.024 0.021 0.022 0.022

0.027 0.020 0.018 0.016 0.015

0.145 0.146 0.150 0.152 0.148

0.087 0.085 0.096 0.091 0.114

0.007

0.028 0.085 0.140 0.228 0.410

0.035 0.031 0.028 0.027 0.027

0.034 0.032 0.027 0.023 0.020

0.376 0.360 0.365 0.370 0.414

0.236 0.226 0.245 0.220 0.201

0.0092

0.028 0.085 0.140 0.228 0.410

0.041 0.034 0.032 0.030 0.031

0.038 0.034 0.031 0.026 0.026

0.460 0.455 0.450 0.466 0.502

0.326 0.283 0.275 0.314 0.281

Trans IChemE, Vol 76, Part A, January 1998

AXIAL DISPERSION OF LIQUID IN G/L COCURRENT DOWNFLOW AND UPFLOW FIXED-BED REACTORS 69

Figure 5. Variation of the axial dispersion coef® cient with the liquid velocity in the case of the additional feed of the gas phase at different levels in the bed (two-phase down¯ ow±air/water system): (s ) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (K) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.2 m3 h- 1 ; (+ ) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (Å) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 ; (´) G1 = 1.6 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (8 ) G1 = 1.6 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 (*) G1 = 1.6 m3 h- 1 , G2 = G3 = 0.65 m3 h- 1 .

occurrence of tracer diffusion inside and outside the porous particle is very important. For the trickling ¯ ow regime, the results agree very well with the results of Charpentier et al.1 3 , obtained using the PDE model. Figure 5 shows that the additional feed of the gas phase at different levels in the bed consequently amplify the axial dispersion in the trickling ¯ ow regime. The nonuniform velocity pro® le in the liquid ® lm, the variations in the ® lm path length and the additional turbulence, due to local velocity ® eld ¯ uctuations in both gas and liquid phases, seems to be a reasonable explanation of this behaviour. For the pulsing ¯ ow regime, the additional feed of the gas phase has no in¯ uence on the axial dispersion of the liquid phase (the liquid axial dispersion in this regime is essentially negligible). Also, Figure 5 shows that for the same gas ¯ ow rate at the bed entrance, the axial dispersion coef® cient increases by increasing the additional gas ¯ ow rate. Due to a larger pore accessibility when the contact time of the liquid with a given particle is longer, the increase of the axial dispersion coef® cient is more important in the case of lower gas ¯ ow rates at the bed entrance. Figure 6 shows the PeÂclet number as a function of the gas ¯ ow rate for different air/CMC systems at a given mass ¯ ow rate of the liquid (L < 2.1 kg m- 2 s- 1 ). The PeÂclet values for the air/0.1 mass % CMC system are higher than the values for the air/water system. This is due to the low increase of the characteristic time of diffusion and the decrease of the contact time of the liquid with the packing (the mean residence time is lower for air/0.1 mass % CMC system). The PeÂclet number values were found to be relatively independent of the ¯ ow consistency index of the liquid for air/0.5 and 1.0 mass % CMC systems. The lower values of the PeÂclet number for these gas/liquid systems compared with the values for the air/water system, are due to the fact that the accessibility of the pores is larger when the contact Trans IChemE, Vol 76, Part A, January 1998

Figure 6. Variation of the PeÂclet number with the gas velocity in the case of two-phase down¯ ow (L = 2.1 kg m- 2 s- 1 ). Air-water system: (s ) no CMC, (´) 0.1 mass % CMC, (*) 0.5 mass % CMC, ( + ) 1.0 mass % CMC.

time of the liquid with a given particle is longer. Also, the transitional regime observed in the column, characterized by local pulses, seems to be a possible explanation of this behaviour. For these gas/liquid systems, the transition between trickling and pulsing ¯ ow occurs at relatively low super® cial gas velocities. As the gas ¯ ow rate is increased, the incipient point of pulsing moves slowly to the upper part of the column, so the pulsing ¯ ow regime remains, in generally, at the bottom of the column. On the column scale, the two-phase ¯ ow is a combination of pulsing ¯ ow occurring at the bottom and trickling ¯ ow occurring at the top. In the case of air/0.5 and 1.0 mass % CMC systems, the additional feed of the gas phase at different levels in the bed has little effect on the liquid axial dispersion (Figure 7). This may be due to the occurrence of local pulses at the top of the column (in the regions of the gas feed). The contribution of the nonuniform velocity pro® le in the liquid ® lm and the variations in the ® lm path length on the liquid axial dispersion is reduced due to the reduction of the trickling ¯ ow zones. AXIAL DISPERSION FOR TWO-PHASE UPFLOW In the case of the two-phase up¯ ow, the axial dispersion is very important and cannot be ignored in comparison with the effect of stagnant zones. The effect of the liquid and gas velocities on the PeÂclet number for air-water systems is shown in Figure 8. As it can be seen, the gas ¯ ow rate affects the PeÂclet number especially in the bubble ¯ ow regime. This is due to the macrocirculation ¯ ow patterns induced in the bed by the nonuniform cross-sectional distribution of bubbles. In the fully-developed pulsing ¯ ow regime, the PeÂclet values are independent of the gas ¯ ow rate. Also, the PeÂclet number does not depend on liquid ¯ ow rate at high liquid velocities (vSL $ 0.0092 m s- 1 ). These results are in contradiction with the results obtained by Yang et al.1 4 who found that the PeÂclet

70

ILIUTA et al.

Figure 7. Variation of the axial dispersion coef® cient with the liquid velocity in the case of the additional feed of the gas phase at different levels in the bed (two-phase down¯ ow±air/0.5 mass % CMC system): (s ) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (K) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.2 m3 h- 1 ; (+ ) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (Å) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 .

number does not depend on the ¯ uid ¯ ow rates. The PeÂclet values obtained in the present work are lower than those of Yang et al.1 4 . The discrepancy between the values of the PeÂclet number may be attributed to the bed geometry (at low gas and high liquid ¯ ow rates the gas tends to ¯ ow in channels, especially in the case of packings with small particles). The values of the PeÂclet number obtained using the transient diffusion model of the tracer in the porous particle coupled with the PDE model are close to those obtained using the PDE model, Table 4. In these conditions the axial dispersion of liquid phase seems to be dominated by the

Figure 8. Variation of the PeÂclet number with the gas and liquid velocities in the case of air/water system (two-phase up¯ ow): (s ) vSL = 0.0021 m s- 1 , (´) vSL = 0.005 m s- 1 , (*) vSL = 0.007 m s- 1 , (+ ) vSL = 0.0092 m s- 1 , (K) vSL = 0.00113 m s- 1 , (Å) vSL = 0.0013 m s- 1 .

bulk phenomena (nonuniform velocity pro® les due to molecular and turbulent diffusion in the ¯ ow direction, mass transfer between dynamic and stagnant zones of the liquid, radial ¯ ow exchange, back¯ ow of liquid due to velocity differences between phases, and macrocirculation ¯ ow patterns induced in the bed by the nonuniform crosssectional distribution of bubbles). Figure 9 shows that the additional feed of the gas phase at different levels in the bed amplify the axial dispersion. The increase of the gas holdup due to the increase of the number of the bubbles along the packing bed, especially at two levels of the additional feed of the gas, seems to be a reasonable explanation of this behaviour. This increase of the number of the bubbles causes additional turbulence in liquid phase. The PeÂclet values for the air/CMC systems used are higher than the values for the air/water systems (Figure 10). Higher liquid viscosity causes aditional turbulence in the liquid phase which decreases bubble coalescence and results in smaller bubble size with a lower rising velocity. In contrast, as a result of the low increase of the characteristic time of diffusion, the bed permeability reduction, the decrease of the contact time of the liquid with the packing (the mean residence time is lower for air/0.1 and 0.5 mass % CMC systems) and the non-Newtonian behaviour of the liquid, the axial dispersion decreases. In the case of air/1.0 mass % CMC these effects are more or less balanced, but for air/0.1 and 0.5 mass % CMC systems the PeÂclet values are higher. The PeÂclet values decrease with increasing ¯ ow consistency index of the liquid. This is due to additional turbulence in the liquid phase and is due to the fact that the accessibility of the pores is larger when the contact time of the liquid with a given particle is longer. In the case of the air/CMC systems used, the additional feed of the gas phase at different levels in the bed amplify

Figure 9. Variation of the axial dispersion coef® cient with the liquid velocity in the case of the additional feed of the gas phase at different levels in the bed. (two-phase up¯ ow ± air/water system): (s ) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (K) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.2 m3 h- 1 ; (+ ) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (Å) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 ; (´) G1 = 1.6 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (8 ) G1 = 1.6 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 (*) G1 = 1.6 m3 h- 1 , G2 = G3 = 0.65 m3 h- 1 .

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AXIAL DISPERSION OF LIQUID IN G/L COCURRENT DOWNFLOW AND UPFLOW FIXED-BED REACTORS 71

Figure 10. Variation of the PeÂclet number with the liquid velocity in the case of two-phase up¯ ow (G1 = 0.6 m3 h- 1 , G2 = G3 = 0.2 m3 h- 1 ). Airwater system: (s ) no CMC, (´) 0.1 mass % CMC, (*) 0.5 mass % CMC.

the axial dispersion (Figure 11). The increase of the gas holdup due to the increase of the number of the bubbles, especially at the levels of the additional feed of the gas, and the aditional turbulence in the liquid phase at higher liquid viscosity may be a possible explanation of this behaviour. CONCLUSIONS The analysis of the dynamic response of the column with the transient diffusion model of the tracer in the porous particle coupled with the piston-dispersion-exchange model led to the conclusion that axial dispersion is very important in the case of the two-phase down¯ ow (trickling ¯ ow regime) and two-phase up¯ ow.

Figure 11. Variation of the axial dispersion coef® cient with the liquid velocity in the case of the additional feed of the gas phase at different levels in the bed. in the case of air/CMC 0.5% system (two-phase up¯ ow±air/0.5 mass % CMC system): (s ) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (K) G1 = 0.6 m3 h- 1 , G2 = G3 = 0.2 m3 h- 1 ; (+ ) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.0 m3 h- 1 ; (Å) G1 = 1.0 m3 h- 1 , G2 = G3 = 0.3 m3 h- 1 .

Trans IChemE, Vol 76, Part A, January 1998

In the case of two-phase down¯ ow, the larger values of the PeÂclet number obtained using the transient diffusion model of the tracer in the porous particle coupled with the PDE model with respect to those obtained using the PDE model show that the occurrence of tracer diffusion inside and outside the porous particle is very important. In the case of two-phase up¯ ow, the values of the PeÂclet number obtained using the transient diffusion model of the tracer in the porous particle coupled with the PDE model are close to those obtained using the PDE model. So, the axial dispersion of liquid phase seems to be dominated by the bulk phenomena. In the case of air/water systems, the additional feed of the gas phase at different levels in the bed consequently amplify the axial dispersion for two-phase down¯ ow (trickling ¯ ow regime) and two-phase up¯ ow. In the case of air/CMC systems used, the additional feed of the gas phase at different levels in the bed has little effect on the liquid axial dispersion for two-phase down¯ ow and amplify the axial dispersion in the case of two-phase up¯ ow. NOMENCLA TURE as Bim c c0 C Deff DL F

speci® c area of the packing (surface of the particles/volume of the bed), m2 m- 3 Biot number, Bim = ks rp / Deff tracer concentration, kmol m- 3 moles tracer injected/(liquid ¯ ow rate ´ mean residence time of input response), c0 = 0¥ cd z=H dh dimensionless tracer concentration, C = c/ c0 effective diffusion coef® cient, m2 s- 1 axial dispersion coef® cient, m2 s- 1 error function, F

G H k ka ks L m n N Pe Ped r rp t tm vSG vSL x z

=

n

i

2 exp C calc out (t) - C out (t) /m exp C out (t)

s- 1

gas ¯ ow rate, m bed height, m ¯ ow consistency index, kg m- 1 s- 2- n overall mass transfer coef® cient, 1 s- 1 liquid-solid mass transfer coef® cient, m s- 1 liquid super® cial mass ¯ ow rate, kg m- 2 s- 1 number of sampling points of each curve ¯ ow behaviour index number of transfer units, N = ka H / vSL PeÂclet number, Pe = H vSL / DL e L,d PeÂclet number, Ped = dp vSL / DL e L,d radial position within solid particle, m particle radius, m time, s mean residence time of the input response, s super® cial velocity of the gas, m s- 1 super® cial velocity of the liquid, m s- 1 dimensionless axial coordinate (x = z/ H) axial coordinate, m 3

Greek letters n dimensionless radial coordinate (n = r / rp ) eL liquid holdup c effective shear rate, 1 s- 1 shear stress, Pa C l eff effective viscosity of non-Newtonian liquids, kg m- 1 s- 1 h dimensionless time, h = t/ tin Subscripts calc calculated d dynamic zone of liquid exp experimental G gas

72 int L p st

ILIUTA et al. internal liquid solid particle stagnant zone of liquid

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ADDRESS Correspondence concerning this paper should be addressed to Dr F. C. Thyrion, Chemical Engineering Institute, Universite Catholique de Louvain, 1 Voie Minckelers, B-1348, Louvain-La-Neuve, Belgium. The manuscript was communicated via our International Editor for Continental Europe, Professor M. Roques. It was received for refereeing 6 March 1997 and accepted for publication after revision 28 August 1997.

Trans IChemE, Vol 76, Part A, January 1998