Effect of scale on liquid recirculation in bubble columns

Chemical Engineering Science, Vol. 49, No. 248, pp. 5637-5652. 1994 Copyright 0 1995 Elscvicr Science Lfd

Printed in Great Britain. All rinbts mservcd m9-2509/9487.w3+0.00

0009-2509(94)00349-l

EFFECT OF SCALE ON LIQUID RECIRCULATION IN BUBBLE COLUMNS

SAILESH B. KUMAR, NARASIMHAN DEVANATHAN,+ DAVOOD MOSLEMIAN* and MILORAD P. DUDUKOVI6*

Chemical Reaction Engineering Laboratory, Dept. of Chemical Engineering, Washington University, St. Louis, MO-63 130, U.S.A.

(Received May 1994; accepted for publication October 1994)

Abstract-Successful scale-up of liquid hydrodynamics in bubble columns using a one dimensional model requires an adequate closure scheme for the Reynolds shear stress. The existing correlations for the prescription of the eddy viscosity or the mixing length scale are demonstrated to be applicable only for a limited range of conditions and consequently cannot be used for scale-up predictions. A method for estimating the mixing length scale has been explored and an attempt at unifying a wide range of data available in the literature within the purview of the method has been made. The futility of such an attempt is attributed to the nonreproducibility of the flow in different laboratories and the consequent lack of data obtained under identical conditions. It is demonstrated, however, that scale-up based on the mixing leneth distribution is nossible when it is obtained from a consistent set of data for liauid velocitv and pas voile fraction profiles..Such a data set was obtained in the authors’ laboratory using’CARPT (Computer Automated Radioactive Particle Tracking) and CT (Computed Tomogaphy). Using the present method for arescribinrz the mixing length scale, model predictions for scale-up compare satisfactorily for the data that were obtained in the au&ors’ laboratory.-

1. INTRODUCTION

The ultimate goal of the research efforts focussed on the bubble column reactor is to enable a rapid scale-up from the laboratory or pilot size to the industrial scale. The hydrodynamics is the primary phenomenon in the bubble column to which all other processes such as heat and mass transfer are intricately related. Since two phase flow (more generally multiphase flow) has escaped a successful fundamental theoretical descrip- tion as yet, methods for studying the hydrodynamics have been by and large experimental followed by a phenomenological assessment of the gathered data. In recent years, with the advent of computers with faster number crunching capabilities, there have also been several attempts at the numerical solution of the complete set of equations of the two fluid model (Ranade, 1992; Svendsen et al., 1992). Although it is well recognized that such numerical models are the ultimate solution to the problem, the information necessary for conclusive quantification of various closure schemes is not yet available. Hence, a simple one dimensional model provides a useful interim solution.

A one dimensional model for liquid recirculation in bubble columns is a simplified form of the two fluid model. In such a model the flow is assumed to be

+ To whom correspondence should be addressed. t Present address: Amoco Oil R t D, Naperville. IL, U.S.A. $ Present address: Florida Atlantic University, Boca

Raton, FL, U.S.A.

steady, fully developed, axisymmetric and the end effects are considered negligible. Experimental evidence for the existence of such steady one dimensional liquid velocity profiles exist (Menzel et al., 1990; Devanathan, 1991). Although, several one dimensional models have been proposed (Ueyama and Miyauchi, 1979; Clark et al., 1987, Luo and Svendsen, 1991; Geary and Rice, 1992) they are all essentially equivalent from the standpoint of the governing equations. The differences arise in the modeling of the turbulence parameters involved in the constitutive equation provided for the Reynolds shear stress, which is expressed either in terms of the eddy viscosity or in terms of the Prandti’s mixing length as:

The key to the success of the one dimensional model hinges on the prescription of the correct eddy viscosity, vt, or the Prandtl mixing length, 1. In earlier attempts these parameters were prescribed either by correlations based on experimental data obtained in bubble columns or by a simple extension of relations used in single phase flow. In the next section it is shown that these relations have no universal significance and consequently cannot be used for scale-up purposes. They appear to work well only for the data selected by the authors for comparison and even that is more often aided by an adjustable parameter. An alternative approach is to directly calculate the mixing length using eq. (1) in conjunction with experimentally

5638 S. B. KU~~AR er al.

measured Reynolds shear stress and velocity profiles. Although this method itself is not new, the focus is on the attempt to bring within its purview a wide body of experimental data found not only in the literature but also those obtained in the authors’ laboratory using the Computer Automated Radioactive Particle Tracking (CARPT) facility (for liquid phase velocities) and the scanner for Computed Tomography (for void fraction).

A one dimensionat model that is used as the basis for predictions is presented first. This is followed by comparison of the model predictions obtained using the various correlations in the literature for the mixing length against some bench mark experimental data. The method that is adopted for prescribing the mixing length is then outlined. This method is then used for predicting scale-up effects on the hydrodynamics using data available in the literature as well as the measurements from the authors’ laboratory.

2. ONE DIMENSIONAL MODEL FOR

LIQUID RECIRCULATION

Several models have been proposed for liquid circulation based on a one dimensional approximation of the flow. Typical of these are the ones proposed by Ueyama and Miyauchi (1979), and by Anderson and Rice (1989). These models are basically simplified forms of the governing equations of the two fluid model. The difference between them can be attributed to the specification of the boundary condition at the wall. In Ueyama and Miyauchi’s model, the velocity profile in the core is matched to the universal velocity profile of single phase turbulent pipe flow at the wall. The specification of the wall shear stress involves an empirical constant. However, in the Anderson and Rice model, the setting of the boundary condition is based on physical reasoning and involves no empiricism. It is proposed that there is a thin layer of liquid close to the wall which is in laminar flow and from which the bubbles are excluded. This layer is assumed to extend into the core up to the point of maximum downward velocity. The shear stress and the velocity profiles are matched at this point and a no slip condition is assumed at the wall. The model of Luo and Svendsen (1991) is similar to these two models. There appears to be no clear advantage in using one approach over the other. In the present work the one dimensional model makes use of the Anderson and Rice boundary condition at the wall.

For the steady, one dimensional, axisymmetric two phase flow in a bubble column the Reynolds equation of motion, neglecting end effects, is written as:

-i $ h,) = $f + ~(1 - E(rh (2)

where dP 2r, -- dZ

= 7 + m(l - 5))9

in which r, is the wall shear stress. An axisymmetric distribution of the void fraction

is assumed and is expressed by the following relation:

E(5) = E- m + * (1 - c<“) m

in which m is an arbitrary constant, c is a parameter that allows for the possibility of nonzero void fraction close to the wall (this precludes the thin liquid boundary layer at the column wall) and E’ is a parameter related to the cross-sectional mean voidage g by the relation:

E=E” m--c+2

m .

Equation (2) can be integrated following the procedure of Rice and Geary (1990) incorporating the power law void fraction distribution [eq. (4)] to obtain the shear stress distribution:

where J+ is the dimensionless radial position at which the downward liquid velocity is a maximum. It is also related to the dimensionless pressure gradient. The following expression was derived for the maximum downward liquid velocity (Rice and Geary, 1990):

u,(< = 1) = ua = -$$ (A’ - 1 - 2L2 In A). (7) Ill

To complete the model one needs to specify a constitutive equation for the shear stress. If the approach using Prandtl’s mixing length for providing the constitutive equation for the core region is adopted, then the expression for the shear stress is:

Combining these equations with eqs (5) and (6) and using the boundary conditions

and at e-1, uf=O

the equation for the liquid velocity profile in the two regions obtained by Rice and Geary is:

for t I 1 (10)

u:(r) = -$$(<2-l-2121n<); <rn. (11) m

Effect of scale on liquid recirculation in bubble columns 5639

j?(r) in eq. (10) is defined by:

,w=-$+-(;)m]. (12)

This equation for the liquid velocity profile is independent of the correlation that can be specified for the mixing length. In order to use the model presented above for predicting the liquid velocity profile, it is necessary to specify the radial void fraction distribution, and the mixing length or the turbulent kinematic viscosity. The specification for the void fraction is generally based on experimental data. For a given operating condition (i.e. for a given diameter of the column, and given gas and liquid superficial velocity) an iterative procedure is involved in obtaining the liquid velocity profile from the model. Once a correlation for either the turbulent kinematic viscosity or the mixing length has been chosen, the procedure for computing the liquid velocity profile is as follows:

1. A suitable value for the exponent m that best fits the void fraction data is chosen.

2. A value for 1 is assumed. The maximum downward liquid velocity Us is evaluated using eq. (7). The liquid velocity profile is evaluated using eqs (10) and (11). The computed velocity and assumed void fraction profiles are incorporated into eq. (13) for liquid continuity which allows iteration on 1 until convergence is achieved.

(1 - 4i=)MT)T d< + s I

u:(5)5 dr = v,. (13) 1

In eq. (13) U, is the superficial liquid velocity.

3. CORRELATIONS FOR MIXING LENGTH AND COMPARISON OF PREDICTIONS

For the successful scale-up of bubble column reactors one needs the ability to predict the flow structure in the actual system based on the data that are obtained in a laboratory or pilot scale plant. There are very few studies in the literature that directly address this issue. Geary and Rice (1992) advocate the use of two different length scales depending upon whether the generation of turbulence is dominated by the bubbles or by the wall.

For the bubble generated turbulence, they define the following scaling relationship for the mixing length in terms of the bubble diameter db as:

&5) = d&W. (14)

For the wall generated turbulence, they suggest using Nikuradse’s (Schlichting, 1979) single phase correlation for the mixing length in turbulent pipe flows.

r(c) = 0.14 - 0.085’ - 0.06c4. (15)

Depending on the mechanism that dominates the generation of turbulence, the mixing length scale (at any given radial position) that is chosen is the one

[eq. (14) or eq. (15)] whose magnitude is higher. According to Geary and Rice (1992), for small columns it is the bubble scale that dominates, and, for larger columns the single phase mixing length scale predominates. The authors provide comparison of model predictions with data obtained by Hills (1974) and claim good agreement. It has to be noted that in obtaining these predictions, the authors have used a value of the exponent m in the void fraction profile that is different from what would be obtained if the actual data obtained by Hills were to be fitted with eq. (4). For example, at superficial gas velocities of 0.019 m/s and 0.038 m/s the values of m (with c = 1) are 19 and 13, respectively. For these gas velocities the values of m used by Geary and Rice (1992) are 8 and 6. Figure 1 shows the power law fits to the experimental void fraction data of Hills using the m values adopted by Geary and Rice (1992) as well as the values of m which provide the fits with the minimum least square error. It perhaps can be argued that the differences in the two fits are not of much significance. However, the predictions of the liquid velocity profile are sensitive to the value of m used and can lead to different magnitudes of the liquid velocities. This is shown in Fig. 2 where the model predictions for the liquid velocity profile with the two different values of m have been compared with the experimental data of Hills (1974). For these predictions the single phase mixing length scale [eq. (15)] was used. The values of m adopted by Geary and Rice (1992) lead to a much better prediction for the liquid velocity in comparison to the one obtained using the values of m that best fit the experimental data. Also shown in Fig. 1 is the fit with c # 1 that is adopted for modeling in the present work. Clearly the fit of data with c # 1 is superior at the expense of having an apparent nonzero void fraction at the wall. However, physical evidence suggests that the wall layer devoid of bubbles is so thin that it is invisible on the scale presented in Fig. 1. Only upon expansion of the length scale close to the wall could such a layer be observed. The value of a that the model predicts is most often greater than 0.99. Hence, it can be argued that fitting the radial void profile with the power law expression eq. (4) with c = 1 exaggerates the magnitude of the layer with low voidage due to the lack of experimental measurements at close proximity to the wall.

Figure 3 shows the model predictions using the two mixing length scales defined in eqs (14) and (15) for two of the superficial gas velocities in Hills’ data. For these conditions, the bubble scale will be larger than the single phase mixing length scale. It is seen that when the values of m (with c # 1) that lead to minimum least square error of the fits to the experimental void fraction data are used, the predictions obtained by using the bubble scale are not as good. In fact the use of the single phase mixing length scale leads to better agreement with the liquid velocity data in this small diameter column.

The possibilities of scale-up by using the above two

5640 S. B. KUMAR et al.

r 7 B

UO!K=Jd P!OA

6

0 -

-.2

-

---

m=&

c=l

----

m=l

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l

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,

Data

of H

ills

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Cal.

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1 0

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of

Hills

(19

74)

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4 m

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.6

.8

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ompa

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of m

odel

pre

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VI w I 6 m

I ./ . . . - . - . - -’ - ’

m 0 In I’

SWJ ‘SPWA p!nb!i IWV

d X

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length scales were investigated by comparing the model predictions with experimental data obtained in a larger column. For this, the data obtained by Yao et al. (1991) is considered. Their data were chosen since, in addition to the liquid velocities and local void fraction, the authors have also measured the bubble sizes. In the bubble length scale proposed by Rice and Geary (1990) it is necessary that the bubble diameter be known under identical conditions, Figure 4 compares the model predictions with the experimental data for some of the superficial gas velocities. Also shown in these figures are the predictions from the model using the mixing length scale proposed by Kawase and Tokunaga (1991):

I=KD. (16)

The constant K was adjusted so as to minimize the difference between their model predictions and the experimental data for the liquid velocities. The proposed equation for the characteristic mixing length is:

1 = 0.045U-0.38D B c (17)

where U, is in m/s and DC is in m. It is clear from Fig. 4 that the mixing length scale proposed by Kawase and Tokunaga does not fare any better than the other two correlations discussed earlier. The agreement between the prediction and data for the liquid velocity seems the best when the single phase flow mixing length scale, eq. (15) is used. However, consistent agreement under a wide range of conditions was not obtained.

Menzel et al. (1990) have directly estimated the eddy viscosity from their experimental data The authors measured the Reynolds shear stress profiles along with the velocity and void fractions profiles in a bubble column. They obtained the turbulent kinematic viscosity profile by dividing the measured Reynolds shear stress by the gradient of liquid velocity for each radial position:

%x4 vt(r) = --.

dua,idr (18)

The computed eddy viscosity distribution was described using the following function derived by Reichardt (1951) for single phase pipe flow:

w1 = kNR(;‘pr) (1 + 2(r/R)3)(1 - (T/R)~) (19)

where k, is a constant, R is the column radius, ~~ is the wall shear stress and m is the liquid density.

The authors used the eddy viscosity profiles obtained along with the measured void fraction profiles as input to a one dimensional model that is essentially similar to the one discussed in Section 2. The model predictions for the liquid velocity compared well with the experimental data. The value of kN that was used for the predictions was 1.13. It is claimed that this value is optimal based on the results obtained for a smaller column (diameter = 0.15 m), although no comparisons for these were presented.

Luo and Svendsen (1991) also used Reichardt’s functional relationship for describing the turbulent viscosity profiles. They compare the predictions from their one dimensional model (based on the same equations as the model in Section 2) with a wide range of data available in the literature. For some of the cases they set k, equal to 0.42 and for others they set it to 1.13. The comparisons of the model predicted liquid velocities with the experimental data are claimed to be favorable. However, it has to be noted that they treat m, the exponent in the void fraction profile, as a parameter that can be. changed. For example, for predicting the data of Menzel et al. they set m to 4. The value of m that correctly fits the experimental data of Menzel et al. (1990) for the void fraction, is 2, except for the lower superficial gas velocity of 0.012 m/s. The parameter m is defined by the void fraction profile measured experimentally and cannot be treated as an adjustable parameter.

The predictive capabilities of Reichardt’s correlation for the turbulent kinematic viscosity were evaluated within the framework of the model in Section 2. For this, the constitutive relation for the shear stress was expressed in terms of the turbulent kinematic viscosity instead of the mixing length. The resulting expression for the liquid velocity profile is similar to those expressed by eqs (10) and (11) (Devanathan, 1991). The evaluation of the correlation was done by comparing the model predictions for the experimental data of Nottenkamper et al. (1983). Figure 5 shows the comparisons for three of the superficial gas velocities from the data of Nottenkamper et al. (1983). The model calculations were done for both kN = 0.42 and k, = 1.13 and no single value of k, was able to predict the measured liquid velocity profiles for all the superficial gas velocities used in the experiments. In Fig. 5 the predicted velocity profiles with kN = 0.42 are shown. Based on these results it appears that Reichardt’s correlation for the eddy viscosity cannot be used either, for c scale-up purposes.

4. ESTIMATION OF MIXING LENGTH SCALE FROM

THE DATA IN THE LITERATURE

In the last section it has been demonstrated that the correlations in the literature for characterizing the turbulence associated with the one dimensional approximation of the flow in bubble columns have limited range of applicability. Although some of these correlations have an adequate theoretical basis for single phase flows, their extension to two phase flow is not straightforward. The only method of estimating the turbulence length scale that appears to be valid, is to estimate it from the ratio of the Reynolds shear stress and the gradient of the liquid velocity. Although, this approach was followed by Menzel et al. (1990), the relation that they have proposed is not successful in predicting the liquid velocities unless the parameter k, is tuned. In this section we attempt to explore the reasons for the failure of such relations when they are used for scale-up purposes. To accomplish this objective, a wide range of experimental data that is

5644 S. B. KUMAR et al

7 0

s/u ‘Qaola~ pinbig lqxy

9 v! 0 Y 9 v- F

S/UJ ‘Awola~ pinby le!xy

Fig. 5. Comparison of ‘predictions using Reichardt’s correlation for eddy viscosity for data of Nottenkamper ‘eta1 (1983).

Effect of scale on liquid recirculation in bubble columns

0 Exp’tal data U, = 0.82 m/s 0 Exp’tal data U, = 0.32 m/s . Exp’tal data U, = 0.05 m/s

.----- Model prediction for U, = 0.82 m/s --- --- Model prediction for Ug = 0.32 m/s

Data of Nottenkamper et. al. (1983) *.g: z .

Y.

0 .2 .4 .6 .8 1.0

Nondimensional Radius, 5

available in the literature has been assembled and the mixing length scale for each of those data sets has been estimated using the one dimensional model. The experimental data base includes the data obtained by Hills (1974), Yu and Kim (1991), Yao et al. (1991), Nottenkamper et al. (1983) and Menzel et al. (1990). The procedure for computing the mixing length from the experimental data for liquid velocities and void fraction is as follows.

The experimentally obtained liquid velocity profile is fitted to a polynomial subject to the boundary condition of zero gradient at the column center. The order of the polynomial used was four or less. Although a higher order polynomial leads to a better closure of liquid continuity, the resulting profile of the derivative is not monotonic and has several inflexion points. The use of such a derivative profile for computing the mixing length leads to nonrealistic profile shapes. With a polynomial of degree four, the derivative of the liquid velocity was almost monotonic and negative. However, this leads to a slightly higher error (lo-15%) in closure of the liquid continuity.

Data on Reynolds shear stress in bubble columns are limited. For the data bank that was assembled, the Reynolds stress measurements were not made. Hence, we make use of eqs (5) and (6) from the one dimensional model to compute it. The experimentally determined local void fraction data are fitted to eq. (4) to evaluate 4 m and c. These parameters are then

5645

used in eqs (5) and (6) and the Reynolds stress profile is computed. The mixing length scale is then calculated using an equation similar to eq. (18). Figure 6 shows the computed mixing length scale for the experimental data of Hills (1974). As the center of the column is approached, the mixing length scale increases without bound. This can be attributed to the zero velocity gradient at the center as well as to the errors in the numerical differentiation procedure. For modeling purposes, this increase was ignored and hence the mixing length scale was simply extrapolated from a radial position close to the column center (typically r/R = 0.15P0.2). It was found that the results are not sensitive to the starting point for the extrapolation if it is selected in this range. The radial profile of the mixing length was fitted to an expression of the form:

41 - 0 &a = ~

(< + b) + d(l - 5)’ (20)

where the constants a, 6, c, d and e are obtained by a nonlinear regression process. Figure 6 also shows the fit to the calculated profile using this functional form. The effectiveness of the estimated mixing length scale is demonstrated by putting it back into the model along with the void fraction profiles to compute the liquid velocity profile. Figure 7 shows a comparison of the recomputed velocities with the experimental data of Nottenkamper et al. (1983).


.--- Profile used for modeling ‘------- Computed profile

Data of Hills (1974), U, = 0.038 m/s

0 .2 .4 .6 .8 1.0

Nondimensional Radius, E,

Fig. 6. The estimated and the assumed form for the mixing length profile.

An attempt was made to extract some trends in the variation of the mixing length scale with column diameter and gas superficial velocity. Figure 8 shows the mixing length scale obtained in columns of different diameters for approximately the same superficial gas velocity. It is clear from the figure that it is not possible to extract a monotonic or any simple functional dependence on the column diameter. However, it has to be noted that the liquid superficial velocities used in these studies are very different. Yu and Kim (1991) had a liquid superficial velocity of 0.04 m/s, while Yao et al. (1991) and Nottenkamper et al. (1983) had liquid superficial velocities of 0.01 m/s and 0.012 m/s, respectively. Hills (1974) was conducting the experiments in a batch bubble column. Figures 9(a) and 9(b) demonstrate the effect of superficial gas velocity on the mixing length profiles for the experimental data of Hills (1974) and Nottenkamper et al. (1983). While the magnitude of the mixing length increases with U, for the data of Hills (1974), it initially increases with V, and subsequently decreases for the data of Nottenkamper et al. (1983). Consistent trends could not be observed in the data reported by the other investigators either. One of the possible reasons for this is the differences in the liquid superficial velocities used in these studies. Consequently, based on currently available information it is not possible to develop a universal set of numbers for the constants

involved in eq. (20). Based on the above treatment of the experimental results in the literature, a scaling relation for the mixing length proved elusive.

It is not difficult to contemplate the reasons for the inability to unify available experimental data into a single framework. From a global perspective, the flow structure in bubble columns is well established and the gross features remain the same, irrespective of the column size, and to some extent the physical properties of the phases involved. However, on a microscopic scale there appear to be significant differences in the flow structure, especially in an air-water bubble column. The causes for these differences are also well known, the most important of these being the surfactant concentration. They can render the disper- sion to be either coalescing or noncoalescing in nature. Very different holdup values have been reported by Anderson and Quinn (1970) with the use of distilled, deionized and tap water as the liquid phase. At low superficial gas velocities the differences in the bubble generation characteristics of the distributor are important. Unless we are able to incorporate all the information corresponding to these factors into a model, efforts to unify all the bubble column data into a single framework will be futile. In fact, this would be true even for a more sophisticated multidimensional model solving the complete set of equations of the two fluid model. On the other hand, if it is possible


0 u, = 0.053 m/s a u, = 0.105 m/s .

L-Z. U, = 0.324 m/s

-* x U ~0.823 m/s .x l

‘\\ LJ; = 1.425 m/s

L ‘\ ----_ \

* -\ ‘\

‘Y ‘\

Col. i.d . = 0.45 m

Data of Nottenkamper et. al. (1983)

0 .2 .4 .6 .8 1.0


Fig. 7. Effectiveness of the fits to the mixing length scale.

to generate a consistent set of experimental data for bubble columns under a wide range of operating conditions, then perhaps it might be possible to use the information obtained in a smaller column to predict the behavior of an identical gas-liquid system in a larger column. To the best of the authors’ knowledge, such a data base is not available in the literature. Therefore, they used the existing Computer Assisted Radioactive Particle Tracking (CARPT) facility and the newly implemented scanner for Computed Tomography (CT) in their laboratory to obtain such a data base. While the CARPT facility enables the quantification of the liquid phase velocities (Moslemian et al., 1992; Yang et al., 1993) the CT scanner provides information on the void fraction distribution (Kumar er al., 1994).

5. MODELING OF THE DATA FROM CARPT AND CT

The experimental program included the characteri- zation of flow in five column sizes. The diameters of the columns were 0.10 m, 0.14 m, 0.19 m, 0.26 m and 0.3 m. Air and water were used as the gas and liquid phase, respectively. The superficial gas velocities used ranged from 0.05 m/s to 0.12 m/s. The static liquid height/diameter ratio in all columns exceeded five. For all the columns, a perforated plate distributor was used. The hole sizes for the five plates ranged from

0.4 to 0.74 mm. The open area of the plates ranged from 0.05 to 0.1% of the total plate area.

CARPT (Devanathan et al., 1990; Yang et al., 1993) provides a three dimensional velocity field for the liquid phase. For the purpose of keeping the data tractable, it is circumferentially averaged to provide a two dimensional axisymmetric distribution of the velocities. This is then further averaged along the column axis neglecting the regions corresponding to entrance and end effects. The region of the flow for this one dimensional averaging process is ascertained on the basis of small radial velocities of the liquid. The final result is a velocity profile that can be compared with model predictions. The void fraction profiles are similarly averaged from CT scans (Kumar, 1994) of the flow at three sections which are away from the end influences. This averaged profile is used as input to the one dimensional model.

To evaluate the scale-up capabilities of the one dimensional model, the experimental data for the liquid velocities and the void fraction obtained in the 0.19 m diameter column are analyzed using the procedure described in Section 4 to yield the profile of the mixing length scale for each superficial gas velocity studied. This is then used in conjunction with the void fraction profiles for each of the other columns as mode1 input and the predicted velocity profile is compared with the experimentaly determined

S. B. KVMAR et al.

1 .oo I I I

- x-c-- Menzel et. al., Col. i.d. = 0.6 m, U, = 0.096 m/S + -- --+ Nottenkmper et. al., Cal. i-d. = 0.45 m, U, = 0.102 m/s

.75 _ Q------9 Yao et. al., Cal. i-d. = 0.29 m, U, = 0.1 m/s . m----a Yu & Kim, Cal. i.d. = 0.25 m, U, = 0.1 I’ll/S - Hills, Col. i.d = 0.14 m, U, = 0.095 Ill/S

L

0 .2 .4 .6 .8 1.0


Fig. 8. Estimated mixing length in columns of different diameters.

Table 1. Parameters of the measured void fraction distribution and 1 values

Cal. diam. (m) U, (m/s) 2 m c L

0.10 0.05 0.159 6.65 0.974 0.992 0.12 0.183 3.50 0.981 0.993

0.14 0.05 0.128 3.50 0.976 0.992 0.12 0.173 3.34 0.966 0.992

0.26 0.05 0.145 2.45 0.678 0.995 0.10 0.187 2.11 0.814 0.995

0.30 0.05 0.145 2.43 0.892 0.995 0.10 0.170 1.71 0.889 0.996

velocity profiles. Table 1 lists the parameters of the void fraction profile used as model input. These parameters were obtained by fitting the power law distribution [eq. (4)] to the void fraction distribution obtained experimentally with the CT scanner in the laboratory. Also listed in the table are the values of A for which the continuity equation was satisfied. Figures 10(a)- 10(d) compare the model predicted velocities to experimentally determined liquid velocities in the columns of diameter 0.10 m, 0.14 m, 0.26 m and 0.3 m respectively. It has to be emphasized that the mixing length scale as obtained in the 0.19 m diameter column for a given U, was used without any adjustment of the parameters to predict the liquid velocity in the other columns for the same U,. Except for the 0.10 m and 0.14 m diameter columns, the predicted velocities match closely the experimentally determined ones. The lack of better agreement in the

small diameter columns can be attributed to wall effects. The column diameter has to be greater than 0.15 m for wall effects on the flow to be negligible (Shah et al., 1982; Wilkinson et aZ., 1992). During the experimental runs, a significant amount of slugging was observed in the 0.10 rn diameter column.

Although the experimental program included a superficial gas velocity of 0.08 m/s, the model predictions have been plotted for only the gas velocities corresponding to 0.05 m/s and 0.1 m/s for each column. For the flows obtained in their laboratory, it appears that these two velocities were in the transition between bubbly and churn turblent flow regimes. Consequently there is very little difference in the measured velocity profiles at U, = 0.08 m/s and U, = 0.1 m/s. This behavior was observed in all the columns. It supports their claim regarding the consistency of the flow in columns of different sizes. This analysis indicates that the mixing length scale, deduced from the data measured in smaller columns, can be used to predict the liquid velocities in larger diameter columns provided the radial void fraction distribution is known.

6. FINAL REMARKS

This study demonstrates the deficiency of the existing correlations for eddy viscosity and mixing length profiles, in satisfactorily predicting the one dimensional liquid velocity profiles under a wide range of operating conditions. A mixing length scale can be

Data of H&s (1974)

Cal. i.d. = 0.14 m

Effwt of scale on liquid retircuiation in bubble columns 5649

0 .2 .4 .6 .8 1.0

I

1 Da& of N#~n~~~ et. al. (1953) Cd. i.d. = 0.45 m

P

P 2

.2

Fig, 9. Dependen@ of mixing length an superficial gas velocity: data of (a) Rills (19%); (b) Nottenkamper EYE af. (1983).

deduced from the Reynolds stresses and the liquid However, a universal correlation expressing the velocity gradient when the appropriate data are mixing length as a function of cotumn diameter and available. The nature of this mixing length profile was superficial gas velocity could not be obtained. This is similar for all the experimental data Sets investigated, attributed to the inconsistency in the flows obtained

5650 S. B. KumR et al.

t “! 0 ? -7

S/LIJ ‘Aqoola/( p!nby le!xy

s/w ‘&CJOI~A p!nbg IS!XV


in different laboratories. Measurements of all the relevant physicochemical properties of the phases and hydrodynamic data in columns of different sizes in the same laboratory do not exist in the literature to enable a scaling relation to be deduced. In this study, liquid velocity and void fraction data were obtained using CARPT and CT, respectively, for five column sizes. The data from one column (D = 0.19 m) was used to deduce a mixing length scale which reasonably predicts the velocities in columns of larger diameter. Thus, for an identical gas-liquid system, if holdup and velocity measurements are made in a small diameter column (but larger than 0.15 m in diameter), the liquid velocity profiles can be predicted in a larger column for which the radial holdup distribution is available. For u priori prediction of both liquid velocities and void fraction, the complete solution of the two-fluid model equation with appropriate turbulence closure is inevitable.

c

4 D

9 kN I m

P’ P R

UI

% u:

NOTATION

parameter for non-zero void fraction at the wall bubble diameter column diameter acceleration due to gravity Reichardt’s constant mixing length scale power law void fraction exponent dimensionless pressure gradient pressure radius of test section local liquid velocity local gas velocity the liquid velocity close to the wall in Rice and Geary’s liquid circulation model maximum downward liquid velocity fluctuating components of velocity of the continuous phase in the r and z directions liquid superficial velocity gas superficial velocity

Greek letters & void fraction E cross-sectional average void fraction A dimensionless radial position where the

liquid velocity is maximum v* kinematic viscosity of the liquid “I turbulent eddy viscosity r dimensionless radial position h liquid density t shear stress Tw wall shear stress

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- .

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Effect of scale on liquid recirculation in bubble columns

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