Total Liquid Saturation in GasLiquid Cocurrent Downflow and Upflow

In this work, the effects of the gas and liquid flow rates, as well as of the size and shape of particles, on the total liquid saturation in gas-liqui...
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Ind. Eng. Chem. Res. 2004, 43, 1096-1102

Total Liquid Saturation in Gas-Liquid Cocurrent Downflow and Upflow through Packed Beds and Analysis of Correlations for Predicting the Total Liquid Saturation Marcos F. P. Moreira, Maria C. Ferreira, and Jose´ T. Freire* Department of Chemical Engineering, Federal University of Sa˜ o Carlos, Rodovia Washington Luiz km 235, 13565-905 Sa˜ o Carlos SP, Brazil

In this work, the effects of the gas and liquid flow rates, as well as of the size and shape of particles, on the total liquid saturation in gas-liquid cocurrent downflow and upflow through packed beds are investigated. Experimental values of the liquid saturation are obtained using air and water as the percolating fluids. The results indicate that the air and water flow rates affect the liquid saturation under downflow and upflow conditions. Concerning the particle characteristics, it was observed that the particle shape affects the values of βt for downflow and upflow conditions, whereas the influence of particle size is more significant in the downflow configuration. The experimental values are compared to the predictions of some correlations presented in the literature, and new correlations are proposed for predicting βt in downflow and upflow configurations. Introduction Packed beds with cocurrent gas-liquid flow are applied in many industrial operations of the chemical, petrochemical, and food industries. So-called packedbed reactors can be operated in either the downflow or upflow configuration. The liquid saturation in these beds is an important parameter for characterization of twophase flow, because it affects the effective thermal conductivity1,2 and also the mass-transfer rates. Downflow reactors are the most widely employed in industrial applications, so the number of works on liquid saturation reported in the literature1,3-13 is greater for the downflow configuration than for the upflow configuration.14-18 Although downflow reactors are the most common, cocurrent upflow reactors are also applied, such as in oxidation and hydrogenation reactors.18 In both downflow and upflow configurations for beds of nonporous particles, the total liquid saturation is the sum of the static and dynamic saturations.7,13,17 The dynamic liquid saturation is the volume of liquid that flows continuously through the bed, and the static liquid saturation corresponds to the volume of liquid that remains stagnant, adhered to the bed inner walls and to the particle surfaces. Different techniques can be applied for experimental measurements of the total liquid saturation. Ellman et al.8 mentioned the following: (i) weighing the packedbed, (ii) using tracing techniques, (iii) exposing the liquid to a beam of electromagnetic radiation, (iv) measuring the apparent electrical conductivity in the bed, and (v) measuring the volume of liquid outside the bed when a known quantity of liquid circulates in a closed system. Reinecke and Mewes19 and Moreira and Freire20 also reported that the measurement of the system capacitance is an effective technique for determining the total liquid saturation. A classical technique for measuring the dynamic liquid saturation consists * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: 55162608264. Fax: 55162608266.

of interrupting the gas-liquid flow and subsequently draining the bed and measuring the volume of liquid collected. This procedure, known as the drainage method, is widely applied7,12,13,17 because it is simple to perform, but another technique for measuring the static saturation, such as the drying method,20 must be used to estimate the total liquid saturation. Different types of particles have been tested in studies on the total liquid saturation of packed beds, such as spheres, cylinders, and Rashig rings (mainly in downflow reactors). However, despite the great number of correlations reported in the literature, few equations take into account the influence of particle shape on the bed liquid saturation, and significant disagreements between the experimental values and the predictions of the empirical equations are often reported. The purpose of this work is to contribute to a better understanding of fluid dynamics in two-phase flow through packed-bed reactors by obtaining experimental data over an extensive range of experimental conditions, including both downflow and upflow configurations, with different types of particles. The effects of the size and shape of the particles and the flow direction on the total liquid saturation are investigated. Correlations from the literature for predicting the total liquid saturation are evaluated by comparing their predictions with experimental data. The curvature measurements of Bates and Watts21 and the parameter biases of Box22 are calculated to evaluate the nonlinearity of the model and the reliability of the adjusted parameters. Materials and Methods The particles employed in this study were glass spheres with diameters equal to 1.9, 3.1, and 4.4 mm; glass cylinders with a diameter of 5.0 mm and a height of 3.4 mm; and glass parallelepipeds with dimensions equal to 2.9 mm × 5.3 mm × 5.3 mm. The characteristics of these particles and their packings are presented in Table 1. It must be noted that the 4.4-mm spheres, cylinders, and parallelepipeds have the same product of sphericity

10.1021/ie030469d CCC: $27.50 © 2004 American Chemical Society Published on Web 01/15/2004

Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 1097 Table 1. Particle Characteristics particles

dpa (mm)

φb

c

spheres cylinders parallelepipeds

1.9, 3.1, 4.4 5.0 5.4

1 0.86 0.77

0.37 0.32 0.31

a

Equivalent diameter. b Sphericity. c Average bed porosity.

(φ) and equivalent particle diameter (dp), approximately equal to 4.4 × 10-3 m. The product φdp can be considered as a characteristic dimension of the particle, thus making the comparison among the results physically consistent. The fluids percolating the bed were water and air. The experimental apparatus was composed of a column made of acrylic with a height of 80 cm and an inner diameter of 5 cm. Details of the column and the experimental apparatus can be found in Moreira and Freire.23 The packing of particles in the bed was carried out with both air and water flowing. The bed homogeneity was checked by comparing the pressure drops measured in two different sections of the column with water flowing through it. The total liquid saturation was determined by combining the drainage and drying methods. Initially, only water was supplied to the column, at a given water flow rate. In the sequence, the valve at the entrance was opened, and the air was fed into the column. Because of hysteresis phenomena (observed by Ravindra et al.24 and Moreira and Freire23), the air flow rate was increased from the initial value to the required value in small steps, while the water flow rate was kept constant. To allow equilibrium conditions to be reached, the measurements were made after a time interval of about 2 min. Then, the valve located at the bed entrance was closed, and the flow was interrupted. The water was drained for 15 min (dynamic liquid saturation), and its volume was determined by weighing. The volume of residual water on the surface of particles (static liquid saturation) was determined by the drying method.20 The wetted particles were initially weighed and dried in a furnace at 105 °C and then weighed again at time intervals of 1 h until their mass reached a constant value. By subtracting the final constant mass from the mass of the wetted particles, the liquid volume could be obtained, as the water density was known. The sum of water volumes obtained by the drainage and drying methods was divided by the bed void volume to give the total liquid saturation. The identification of flow regimes under each set of experimental conditions was carried out on the basis of visual observations. (Details of this procedure can be found in Moreira and Freire.23) The air flow rate was varied from 0.04 to 0.6 kg‚m-2‚s-1 under downflow conditions and from 0.04 to 1.27 kg‚m-2‚s-1 under upflow conditions. The water flow rates used were 2, 9, and 20 kg‚m-2‚s-1 The experiments were conducted at a constant temperature of 25 °C. To determine the total liquid saturation under downflow conditions, the column position was inverted, and the tests were repeated using the same measuring systems as those used in upflow tests. In both downflow and upflow experiments (dynamic and static saturations), the tests were replicated twice, and the results presented are the averaged values. Curvature Measures. In general, correlations are evaluated only by the deviations between the predicted and measured values, but the use of statistical analysis

is proving to be an important tool for improving analysis, as it allows for the consistency of the adjusted parameters to be checked. The curvature measurement of nonlinear models, a technique developed by Bates and Watts,21,25 was successfully applied by Barrozo et al.26 to select the best correlation for predicting soybean equilibrium isotherms. According to Ratkowsky,27 nonlinear models with good curvature measures require fewer iterations to converge in the regression procedures and can be treated using the same statistical methodologies as employed for linear models. In this work, the intrinsic (IN) and the parametereffects (PE) curvature measures of Bates and Watts21,25 were obtained. These authors developed a technique for measuring the curvature of a nonlinear model to assess the distance from the real case to the asymptotic one. This curvature has a component intrinsic to the model (IN) representing the behavior of the sample space in the region where the estimate is being realized (locus of the solution) and another attributed to the parameter effects (PE) that evaluates the nonparallelism of the parametric lines in the expectation space. Increasing the sample data set and the curvature PE by reparametrization of the model can decrease the curvature IN. The first and second partial derivative matrices are given by

V4 )

V 2 )

[( ) ] [( ) ]

1 ∂βt sxp ∂θj

(1)

θp

∂2βt 1 sxp ∂θr∂θj

(2)

θp

where θ represents the correlation parameters, estimated with our measured values of βt through the minimization of the function F (least squares). The arrays Q and R are obtained from the QR decomposition of eq 1. An array U is obtained from

2 R-1 U ) (R-1)TV

(3)

The combined acceleration array is given by

A ) APE|AIN ) QTU

(4)

where APE is the parameter-effects acceleration array, consisting of the first p faces of A, and AIN is the intrinsic acceleration array, consisting of the last N p faces of A. The PE and IN curvatures are the largest absolute values of the acceleration arrays APE and AIN, respectively. The statistical significance of IN and PE can be obtained by comparing these values with the value of the radius of curvature, given by 1/(2F′1/2), where the Fisher distribution F′ ) F(p, N - p;R) is obtained from statistical tables. In this work, a significance level, R, of 5% was used. The curvature measures PE and IN are considered excessive when their values surpass the radius of curvature. Nonexcessive curvature measures indicate that a nonlinear equation behaves similarly to a linear model in terms of the equation’s adjusted parameters. The significance of the discrepancy between the estimates of the parameters and their true parameter values are determined by Box22 percentage biases.

1098 Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004

Figure 1. Total liquid saturation for spheres with diameters of 1.9, 3.1, and 4.4 mm in two-phase cocurrent downflow.

According to Ratkowsky,27 percentage biases smaller than 1% indicate that the discrepancy is not significant. The Box26 biases are given by (Ratkowsky25)

bias(θ) θ

% bias (θ) ) 100

(5)

with

bias(θ) ) -

s2 2

N

(

∑ u)1

FuFuT)-1

N

∑ t)1

N

Fttr[(

FuFuT)-1Ht] ∑ u)1 (6)

where Ft () Fu) is the p × 1 vector of first derivatives of βt and Ht is the p × p matrix of second derivatives of βt with respect to each element of θ (parameters of the correlation), evaluated at each experimental condition. In eq 6, tr is the array trace. Results and Discussion Total Liquid Saturation under Gas-Liquid Downflow Conditions. The effects of the air and water flow rates on the total liquid saturation for twophase cocurrent downflow are shown in Figure 1. Increasing the water flow rate causes a significant increase in the total liquid saturation. At a constant liquid flow rate, the value of βt tends to decrease as the air flow rate is raised. The dependence of βt on the air flow rate is stronger for the higher water flow rates (L ) 9 and 20 kg.m-2‚s-1). These results show good agreement with those reported by Larachi et al.,9 Rode et al.,10 Fu and Tan,11 and Iliuta et al.17 It can also be noted in Figure 1 that, for the lowest water flow rate (2 kg‚m-2‚s-1), the dependence of βt on the air flow rate is not significant for the largest particle diameters (3.1 and 4.4 mm). This is probably because, for the largest particles, the water flows preferentially on the particle surface, resulting in poor interaction between air and water. For the 1.9-mm spheres, βt tends to increase as the air flow rate is decreased. In this case, the pores in the bed are expected to be smaller than those in the beds formed with the two other particle sizes, making the interaction between air and water more effective, thus

Figure 2. Total liquid saturation for spheres with diameters of 4.4 mm, cylinders, and parallelepipeds in downflow.

accentuating the dependence of βt on the air flow rate, a behavior that was also observed by Rao et al.5 The total liquid saturation is also affected by the particle size. By comparing the results obtained for the 4.4- and 1.9-mm spheres, it can be observed that βt increases as the particle size decreases. Larachi et al.9 and Fu and Tan11 reported similar behavior for the dependence of the total liquid saturation on the particle diameter. This increase observed in the total liquid saturation as the particle diameter is reduced is caused partially by an increase in the static saturation. The static saturation changed from 0.11 for the 4.4-mm spheres to 0.14 for the 1.9-mm ones. This is expected because the specific contact area between the particles and the fluid is larger for the smallest particle diameters. Moreover, the dynamic saturation also increases as the particle diameter is reduced, contributing to the increase observed in the value of the total liquid saturation. A reduction in the size of the pores within the bed is expected as the particle diameter is decreased, favoring the retention of water in the bed. Figure 2 shows the saturation values for 4.4-mmdiameter spheres, cylinders, and parallelepipeds. Identical values of static saturation were obtained for these three types of particles. A slight increase in βt can be observed as the particle sphericity is reduced from φ ) 1.0 to φ ) 0.86 for the cylinders and to φ ) 0.77 for the parallelepipeds. Note that the bed porosity also changes as the particle shape is altered, from  ) 0.37 in the bed of spheres to  ) 0.32 for the cylinders and  ) 0.31 for the parallelepipeds. The decrease in the bed average porosity might be related to a reduction in the size of the pores, which might favor the dynamic retention of liquid in the bed. For cylinders and parallelepipeds, the total liquid saturation presents the same dependence on the air and water flow rates as observed for the packing of spheres (see Figure 2). Total Liquid Saturation under Gas-Liquid Upflow Conditions. The dependence of the total liquid saturation on the air and the water flow rates for spheres with diameters of 1.9, 3.1, and 4.4 mm in twophase upflow is shown in Figure 3.

Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 1099

Figure 3. Total liquid saturation for spheres with diameters of 1.9, 3.1, and 4.4 mm in cocurrent two-phase upflow.

Figure 4. Total liquid saturation for spheres with diameters of 4.4 mm, cylinders, and parallelepipeds in cocurrent two-phase upflow.

For similar conditions, the values of the total liquid saturation in two-phase upflow are higher than those observed in downflow, a behavior that was also reported by Larachi et al.28 Comparing the data obtained in the downflow (Figure 1) and upflow (Figure 3) configurations, it can be noted that the differences between the values of βt are larger at the lowest water flow rate (L ) 2 kg/m-2‚s-1). This occurs because of the changes in the flow regimes observed as the flow direction is altered. Visual observation of the flow under these conditions (L ) 2 kg/m-2‚s-1) indicates that, under downflow conditions, the liquid flows preferentially on the particle surfaces. In the range of air flow rates investigated here (for L ) 2 kg/m-2‚s-1) regimes designated as trickle flow, transition II flow, and spray flow were identified. (Details of flow characteristics and regime maps can be found in Moreira and Freire.23) Under upflow conditions, the liquid flows through the entire volumes of the bed voids, and the bubble flow, transition I flow, and pulsed flow regimes were identified. At the highest water flow rate, βt is scarcely dependent on the flow direction, because, for similar ranges of L and G, the flow regimes are similar in both the downflow and upflow configurations (bubble flow, transition I flow, and pulsed flow regimes). It is worth noting that, for the smallest particles (spheres with diameters of 1.9 mm in this work), βt was not affected by the flow direction. As in the downflow configuration, βt in the upflow configuration increases as the water flow rate is increased and decreases as the air flow rate is increased. In contrast to the downflow configuration, however, the decrease in βt observed as the air flow rate is increased occurs at both high and low water flow rates. This is because the flow regimes differ depending on the flow direction. In cocurrent downflow, the trickle flow and the transition II regimes were observed, whereas in cocurrent upflow, the regimes detected were the bubbling and pulsating ones (Moreira and Freire23). Similarly to downflow, the dependence of the saturation on the air flow rate is stronger at the lower air flow rates. Similar dependences of βt on the gas and liquid flow rates in cocurrent upflow have also been reported by Larachi et al.,28 Molga and Westerterp,29 and ColliSerrano and Midoux.2

Concerning the influence of the particle size on the total liquid saturation under upflow conditions, no significant differences among the measured values of βt for the 1.9-, 3.1-, and 4.4-mm spheres were detected. The increase in the static saturation observed as the particle diameter was decreased (from 0.11 for the 3.1and 4.4-mm spheres to 0.14 for the 1.9-mm spheres) was probably balanced by the decrease in the dynamic saturation. Lamine et al.,16 who evaluated the effect of particle size on the liquid saturation through a packed bed by testing spheres of 1- and 4-mm diameters, reported higher values of the liquid saturation for the largest particle diameter. A separate flow regime was detected for the smaller particles by Lamine et al.,16 but for the particle diameters investigated in this work, such a regime was not observed. The influence of the particle shape on the liquid saturation was also investigated for cocurrent upflow. Figure 4 depicts the effects of particle shape on βt. An increase in the liquid saturation is observed when both the particle sphericity and the bed average porosity decrease, a behavior similar to that observed in cocurrent downflow. Despite the reduction in the bed average porosity, which might be related to a reduction in the size of pores, the major influence on the total liquid saturation seems to be the type of pores obtained depending on the particle sphericity. This is assumed because, for the range of conditions in the upflow configuration investigated in this work, the size of the pores did not alter the flow regimes significantly (Moreira and Freire23). It is worth noting that the static saturation was not affected by the particle shape. The differences in the saturation values are due only to the changes in the bed dynamic saturation. Figure 4 also shows that, for cylinders and parallelepipeds, the total liquid saturation presents the same dependence on the fluid flow rate as observed for the packing of spheres. Estimates of βt Using Empirical Correlations. The correlations tested in this work to predict the total liquid saturation are shown in Table 2. A comparison between the values of liquid saturation for air-water downflow predicted by the correlation from Larachi et al.,9 given in Table 2, and the experimental values is presented in Figure 5.

1100 Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 Table 2. Correlations for Predicting the Total Liquid Saturation in Gas-Liquid Flow through Packed Beds authors Stepanek15

Achwal and Stiegel and Shah30 Larachi et al.9 Lamine et al.16 Colli-Serrano and Midoux2 a

flow

correlation

upflow upflow downflow upflowa upflow

βt ) 1 - (1 + 0.59ul0.13ug-0.563)-1 βt ) 1.47Reg-0.14Rel0.11(apdp)-0.41 βt ) 1 - 10-(1.22Wel0.15/Xg0.15Rel0.2) βt ) (0.6ug + ul)/(ug + ul) βt ) 1 - (1.28 + 1.7ul0.508 ug-0.264)-1

(7) (8) (9) (10) (11)

Bubble flow.

Figure 5. Predicted βt as a function of measured βt under downflow conditions.

Figure 6. Predicted βt as a function of measured βt under upflow conditions.

An average deviation of 15.4% was obtained. This result is in good agreement with that reported by Iliuta et al.,17 who obtained an average deviation of 13.3% using this same correlation to predict experimental values of liquid saturation in air-water downflow. A comparison between the values of liquid saturation for air-water upflow predicted by the correlations listed in Table 2 and the experimental values is depicted in Figure 6. The equation from Lamine et al.16 provided a small average deviation (D % ) 16.7%), but as can be seen in Figure 6, it did not provide a good correlation between the predicted and experimental values. The correlation proposed by Achwal and Stepanek15 for upflow yielded an average deviation of 25%. The equation proposed by Colli-Serrano and Midoux2 for the upflow configuration presented the highest deviations for the conditions investigated here, about 48%.

The best predictions for upflow were obtained using the correlation proposed by Stiegel and Shah,30 which considers the effect of particle shape, with an average deviation of 11.2%. A similar result was reported by Iliuta et al.,17 with an average deviation of 9.5% between the experimental and predicted values. For most of the conditions investigated in this work, however, the values of βt were underestimated by almost all correlations tested, both in the downflow configuration and in the upflow configuration. Evaluation of the Curvature Measures of the Correlations. The parameters in the equations presented in Table 2 were re-estimated using the experimental data of this work, for upflow and downflow configurations, aiming at the evaluation of the structure of these equations concerning the PE and IN curvatures. It was observed that the curvatures were similar for both flow directions. In both cases (PE and IN curvatures), the occurrence of nonexcessive curvatures was observed only for the modified correlations from Lamine et al.16 In this case, the nonexcessive curvatures were already expected because βt is linearly dependent on a single parameter. Thus, eq 2 is always equal to 0, and so are the PE and IN curvatures. Nevertheless, this equation provided poor predictions for the data of this work, so the presence of nonexcessive curvatures is not the most important shortcoming. Both the Stiegel and Shah30 and the Larachi et al.9 modified correlations presented excessive curvatures (PE and IN), indicating a strong nonlinearity of βt with respect to the parameters used in this equations. These behavior probably appears because of the equations’ functional relationships and the parameters employed. In the Achwal and Stepanek15 and Colli-Serrano and Midoux2 modified correlations, the IN curvatures were nonexcessive, and the PE curvatures were excessive. Such similar behavior was expected, because the equations have similar functional structures. The results suggest that a new parametrization of these equations should be tried to reduce the PE curvature, for instance, by expressing the liquid and gas velocities in different units. Correlations Proposed in this Work. The results indicate that different equations must be used to predict the liquid saturation depending on the flow direction. Although some of the correlations tested here yielded small average deviations, in general, these correlations yielded poor residual distributions between the predicted and experimental values. This indicates that, statistically, the adjusted equations cannot be recommended for the experimental conditions of this work. In addition, the curvature measures were not good, thus preventing the application of simpler statistical procedures to the mathematical methodology developed. New equations to predict the liquid saturation were fitted for the upflow and downflow configurations. The quasi-Newton method was applied to the experimental

Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 1101

Figure 7. Predicted versus measured values of βt under cocurrent downflow and upflow conditions.

data obtained in this work, and the resulting equations were as follows

βt ) 0.45Reg-0.10Rel0.31(dp′)-0.51φ-0.64

Acknowledgment

(12)

The authors thank the Brazilian funding agencies Fundac¸ a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo (FAPESP), Programa Nacional de Exceleˆncia (PRONEX/ FINEP), and Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) for their financial assistance.

(13)

Nomenclature

(valid for downflow) (r2 ) 0.97 and D % ) 4.4%)

βt ) 0.88Reg-0.20Rel0.10(dp′)0.05φ-0.48

operating conditions), particularly at low water flow rates because of the change in the flow regimes caused by the inversion of the flow direction. The influence of the flow direction on βt is not significant for small particles at high water flow rates. As for the downflow configuration, the total liquid saturation is also affected by the gas and liquid flow rates and the particle shape. In cocurrent upflow, the influence of the particle size on the total liquid saturation is not significant for the range of particle diameters investigated. The correlations proposed here provided the best predictions of the total liquid saturation through packed beds in both air-water cocurrent downflow and airwater cocurrent upflow for the conditions of this work. Of all of the equations investigated, they provided the lowest average deviations. The proposed correlations also presented good residual distributions and nonexcessive curvature measures.

(valid for upflow, except under conditions of separated flow regimes) (r2 ) 0.96 and D % ) 3.8%). Equation 12 is valid for values of G from 0.04 to 0.6 kg‚m-2‚s-1 and L from 2 to 20 kg‚m-2‚s-1 for all particles investigated. Equation 13 is valid for G ) 0.04-0.6 kg‚m-2‚s-1 and L ) 2-20 kg‚m-2‚s-1 for the spheres, whereas for cylinders and parallelepipeds, it applies for G varying from 0.06 to 0.9 kg‚m-2‚s-1 and for L in the range from 2 to 20 kg‚m-2‚s-1. Equations 12 and 13 were obtained using a first group of experimental values of total liquid saturations to fit the parameters. To validate the results, these equations were then used to predict the values of a second group of data, measured under different experimental conditions from those used for the fitting. The comparison between the predicted and experimental data is shown in Figure 7. The average deviations obtained using eqs 12 and 13 are 6.8 and 4.3%, respectively, for these conditions. These deviations are smaller than those obtained using the correlations from Table 2. For the parameters estimated in our correlations, it was verified that the percentage bias was always less than 1%, indicating an insignificant discrepancy in the estimates of the parameters in eqs 12 and 13. Also, the correlations fitted in this work presented nonexcessive curvatures (PE and IN). Conclusions The results of this study show that the values of total liquid saturation in cocurrent downflow are affected by the gas and liquid flow rates and also by the particle shape and size. At low liquid flow rates and large particle diameters, the total liquid saturation is not affected by the gas flow rate. In general, the total liquid saturation is higher for upflow configuration than for downflow (under similar

ap ) specific area of a particle, Ap/Vp, m2/m3 Ap ) area of a particle, m2 dp ) equivalent diameter of a particle, (6Vp/π)1/3, m dp′ ) equivalent diameter of particle, mm F′ ) F distribution N (βt,measuredi - βt,predictedi)2 F ) residual function, ∑i)1 G ) gas flow rate, kg‚m-2‚s-1 L ) liquid flow rate, kg‚m-2‚s-1 N ) number of experiments p ) number of parameters Q ) array obtained from the QR decomposition of V4 R ) array obtained from the QR decomposition of V4 s ) standard deviation, [F/(N - p)]1/2 ug ) superficial velocity of gas, m‚s-1 ul ) superficial velocity of liquid, m‚s-1 U ) defined in eq 3 Vc ) total inner volume of the column, m3 Vl ) volume of liquid in the column, m3 VP ) volume of a particle, m3 V4 ) first partial derivative matrix V 2 ) second partial derivative matrix Dimensionless Numbers D % ) percentage average absolute relative deviation, (100/ N N)∑i)1 |βt,measuredi - βt,predictedi|/βt,measuredi 2 r ) explained variance Reg ) Reynolds number of the gas, Gdp/µg Rel ) Reynolds number of the liquid, Ldp/µl Wel ) Weber number of the liquid, L2dp/(Flσ) Xg ) modified parameter of Lockhart-Martinelli, G/L xFl/Fg Greek Letters R ) significance level βt ) total liquid saturation, Vl/(Vc)  ) average bed porosity φ ) sphericity, [6Vp/(Apdp)] µg ) viscosity of the gas, kg‚m-1‚s-1 µl ) viscosity of the liquid, kg‚m-1‚s-1

1102 Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 Fg ) density of the gas, kg‚m-3 Fl ) density of the liquid, kg‚m-3 σ ) surface tension of the liquid, N‚m-1

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Received for review June 2, 2003 Revised manuscript received December 1, 2003 Accepted December 3, 2003 IE030469D