Liquid phase mixing in bubble columns with Newtonian and non-Newtonian fluids

ChcmicaI Engineering Science, Vol. 41, No. 8, pp. 1969-1977, Printed in Great Britain.

LIQUID PHASE NEWTONIAN

1986.

ooo9-2509/86 Pemmmn

s3.oLl+0.00 Journals Ltd.

MIXING IN BUBBLE COLUMNS WITH AND NON-NEWTONIAN FLUIDS Y. KAWASE

and M. MOO-YOUNG

Department of Chemical Engineering, University of Waterloo, Waterloo,

Ontario, Canada, N2L 3Gl

(Received 31 May 1985) hydrodynamic model for the liquid phase in bubble columns is developed. The proposed model for fully developed turbulent Newtonian and non-Newtonian fluids is based on an energy balance and the mixing length theory. The predictions of the model are in reasonably good agreement with data on liquid velocity at the column axis and the axial dispersion coefficient. The liquid velocity data in an invertedconical bottom gas-liquid column contactorhave also been measured.They are correlated by the proposed model Abstract-A

reasonablywell.

For predicting axial dispersion coefficients, an iterative procedure is required. Riquarts (1981) proposed a model based on the assumption that the steady liquid circulation pattern has no effect on the axial dispersion and that mixing is due solely to the rising bubbles. It contains an adjustable parameter as well as a cell model developed by Zehner (1982). The objectives of the present work are two-fold: (a) to develop simple models for the liquid velocity at the column axis and the axial dispersion coefficient in the turbulent flow regime which is of practical interest in real commercial-scale bubble ~01~s; and (b) to obtain liquid velocity data in an inverted conical bottom column which was previously developed by Moo-Young et al. (1979) for an aerobic fermentation process. Despite the wide occurrence of non-Newtonian fluids in the chemical and biochemical industries, no theoretical analysis has been reported yet in the literature on non-Newtonian liquid mixing in a bubble column. Therefore, the analysis is developed to include the case of non-Newtonian fluids, as well, characterized by a power-law model.

INTRODUCTION

are widely used in industry as reactors, fermenters, absorbers and strippers because of their relatively low cost and simple construction. However, their design and scale-up are very difficult due to the complexity of their hydrodynamics. Therefore, extensive effort has been devoted to elucidating the mixing characteristics of bubble columns. Shah et al. (1978) and Joshi and Shah (1981) have recently given excellent reviews of this subject. Liquid circulation is developed in a bubble column because of the introduction of gas and it affects the performance of the bubble column. It is of importance to estimate the extent of the induced liquid phase mixing. While experimental study of the liquid circulation has been carried out by a number of investigators, theoretical analysis of this problem is rather limited. Recently, Ulbrecht and Baykara (1981) suggested that the liquid phase velocity at the column axis is one of the useful parameters to help interpret the liquid phase mixing. Some models based on the pressure balance or the energy balance have been proposed for the liquid phase flow patterns (Joshi and Shah, 1981). Although they agree reasonably well with the experimental data on liquid velocity at the column axis, they are rather complicated to handle. The axial dispersion model has also been widely used to characterize non-ideal mixing behaviour in the liquid phase and a large amount of information is available on a single parameter in the model, i.e. the axial dispersion coefficient. However, there is little understanding of the physical processes lumped in the axial diffusion coefficient and only a few theoretical studies of this have been attempted. Baird and Rice (1975) developed an isotropic turbulence model but the proportional constant in their model was determined from experimental data in the literature. Joshi (1980) extended Taylor’s equation for a single-phase turbulent pipe flow to the case of bubble columns but the calculation procedure is cumbersome. Bubble

columns

THEORY

Turbulent flow in the column The bubble-driven liquid flow has previously been discussed by a number of investigators. It should be noted that simplifying assumptions were required to allow analysis to proceed. Whalley and Davidson (1974) proposed a model for calculation of the turbulent liquid velocity field in bubble columns. The model is based on the folIowing energy balance: Ev (energy dissipation rate in the liquid motion) = E, (energy input rate) (energy dissipation rate at the gas-liquid -4 interface) (1) which was also used for laminar liquid circulation and

1969

Y.

1970

KAWASE

and M.

bubble street formation by Rietema and Ottengraf (1970). Later, Joshi and Sharma (1979) extended the approach of Whalley and Davidson (1974) and proposed a multiple cell circulation model. This model is in reasonably good agreement with the experimental data. From an engineering viewpoint, however, it is not so usable because the calculation procedure for the gas hold-up and the slip velocity of a bubble included in the resulting equation is complicated. The flow in a bubble column is actually a heterogeneous, gas-liquid two-phase flow. In the present analysis, it is assumed that it can be treated as a homogeneous single-phase flow, which also simplifies the deductions. By introducing this assumption, the term of the energy dissipation rate at the gas-liquid interface, Es, in eq. (1) is omitted. The hypothetical single-phase flow is discussed using a mixing length theory which is common for turbulent single-phase flow in a pipe (Schlichting, 1979). Of course, the difference between the hypothetical single-phase flow in a bubble column and the actual single-phase flow is taken into account. Scrizawa et al. (1975) measured the mixing length and the eddy diffusivity of momentum in two-phase flows and found that these values are much larger than

1.2 I 1 .o

0.6

MOO-YOUNG

those in single-phase flows. However, measurements for momentum transport properties in two-phase flows are very sparse and no reliable correlations are available. Therefore, we estimate the characteristic average mixing length and virtual kinematic viscosity based on the velocity profile in bubble columns by following von Karman’s similarity for single-phase flow (Schlichting, 1979). Despite the fluctuations in the local liquid velocity, relatively stable velocity profiles are observed in the column. If gas is injected in the centre of the column, in the core region (R -Z R,), the liquid rises with the bubbles and the velocity decreases with distance from the column centre. In the outer annular region (R > R,), liquid flows downward. Between these two sections there is a transition point (R = R,) at which the velocity is zero. In the case of high flow rate condition. the transition point occurs at around R, = 0.7R. As shown in Fig. 1, the liquid velocity distribution in a bubble column at high Reynolds numbers is satisfactorily approximated by the following simple correlation: U -=

-2

%

0

f

2+l.

In general, the liquid viscosity has little influence on liquid phase mixing in the turbulent flow regime. Furthermore, the liquid velocity profile data for nonNewtonian fluids (0.05 % and 0.75% Carbopol solutions) obtained by Walter and Blanch (1983) can be correlated by eq. (2). Considering these facts, eq. (2) may be applicable to non-Newtonian fluids as well as to Newtonian fluids. The average mixing 1engthTand the average virtual viscosity Z in a bubble column are assumed to be given as (3) and aty = R/2

x2 =yu$

respectively. The validity of these assumptions for Newtonian fluids was discussed by Kawase and MooYoung (1986). For power-law fluids represented by v Yamagoshi V Kojima

119691

q

Hills (1984)

.

NottenkBmper

0 Schiigerl Walter

et al. (1983)

x = 0.4n.

11984)

and Blanch

(19831

E,= aD2wpg 0

I 0.2

I 0.4

I 0.6

(6)

The liquid velocity at the column axis The energy input rate E, is given by the following equation:

-1.0 -1.21

(5)

we have an expression for x (Clapp, 1963):

11972)

0 Franz eta/. I

z=Kjr”

et al. (1980)

I 0.8

I 1.0

rlR

Fig. 1. Radial profile of axial velocity in the turbulentflow regime.

(7)

u,=

The energy dissipation rate in the liquid motion is (Rietema and Ottengraf, 1970):

R E,=

-

0

2arHT*dr dr

’

(8)

1971

Liquid phase mixing in bubble columns In the turbulent flow region, a shear stress is given as (Schlichting, 1979) (9) Substitution of eqs (2), (3) and (9) into eq. (8) yields E, = 0.512n”npHDu~.

(10)

By equating eqs (7) and (10) and rearranging, we have an expression for the liquid velocity at the column axis: a0

= 0 787n-2f3gif3

-

0.23 m

D1/3 fli/3

VI .

%

(11)

This equation may be rewritten as Ret = (0.787 n- 2/3)2-n Rezg FI-~;(~).

(12)

The axial dispersion coeficient The axial dispersion coefficient may be given as (Rodkey, 1966):

-

-

(Az,)~+ (AzJ2 2(At, + At.)

(13)

where AZ is the longitudinal displacement of a fluid element during the time period At. It may be reasonable that the contribution of the mixing in the outer annular region to the axial dispersion coefficient is assumed to be very small as compared with that in the core region. In fact, the error caused by this assumption is about 6%. This estimation is obtained by applying the following approach for the core region to the annular region. The characteristic average liquid velocity in the core region is assumed to be given by ii= = uatr= R,/2 -Since AZ, = uC Ato we have

=

0.755 u,.

E,+

(14)

(15)

where the average time of residence of the liquid

(16)

Substituting eqs (4), (14) and (16) into eq. (15) leads to E, = 0.436r~-~u,D.

(17)

Substitution of eq. (11) for u, in eq. (17) gives E, = 0.343 r~-~/“g~/~D~~~u;~~.

(18)

The dimensionless form of this equation is UD Pe = A E*

= 2.92n w3 Fr,‘,13.

Air

Fig. 2. Diagram of

elements in the core region At, is A+.

i0 orifices

(19)

EXPERIMENTAL

Measurements were carried out in a bubble column 0.23 m in diameter and 1.22 m in height (Fig. 2). The bottom of the column is an inverted conical section with a cone angle of 45”. Twenty orifices 1 mm in

inverted

conical

bottom bubble column.

diameter are equidistant and located on a circle of 0.215 m diameter centred on the bubble column axis. This so-called inverted conical bottom developed for et al.. an aerobic fermentation process (Moo-Young 1979) minimizes the occurrence of dead space at the column bottom where biomass and substrate particles could be trapped. The location of the orifices causes liquid circulation up in the annular region and down in the core region (Fig. 2). The direction of this circulation is opposite to that in a conventional bubble column where gas is fed in the centre of the column. The experimental technique for the measurement of the liquid velocity at the centre of the column is the same as that in the work of Ulbrecht and Baykara (1981) and Ulbrecht et al. (1985). A few solid tracer particles suspended freely in the liquid were used to measure the liquid velocity. The time required for a particle to travel between two markers on the wall of the column was measured. In this work, about 50 measurements were used for each calculation of a,,. The ratio of the clear liquid height to the column diameter was kept constant at 4 and column operation was batch with respect to liquid phase. The physical

Y.

1972

KAWASEand M. MOO-YOUNG

Table 1. Physical properties of liquids at actual test conditions K

n

(Pas”)

Water :&z-l

1.0 0.793 1.0

0.00089 0.0275 0.0045

992 990

CMC-2 Separan

0.625 0.902

0.165 0.0084

999 991

RESULTSAND

DISCUSSION

liquid velocity at the column axis Let us discuss the proposed models in the light of available experimental data. It can be seen from Fig. 3 that the present model, eq. (12), is in good agreement with the data for Newtonian fluids over a wide range of conditions (D = 0.10-5.5 m, ~1= 0.001-0.02 Pas). Zehner (1982) proposed a model which may be written as The

u, = 0.737#‘30”3 properties of the Newtonian and non-Newtonian fluids used in this work are given in Table 1. The rheological properties of viscous fluids [carboxymethyl cellulose (CMC), polyacrylamide (Separan), and methyl cellulose (MC)] were measured by a concentric cylinder viscometer (Fann, Model 35A) at shear rates of l-l--1,021 s-i. The viscosity of water was measured with a Cannon-Fenske viscometer. The departures from Newtonian flow behaviour were interpreted by the power-law model, eq. (5).

Re., 5

I(

us’,‘“_

This is very close to the proposed model, eq. (11). It should be noted, however, that the proportional constant in eq. (20) is an adjustable parameter. Figure 4 compares the data for non-Newtonian fluids obtained by Ulbrecht et al. (1985) with the present model. The agreement is seen to be very good. Wlbrecht et al. (1985) proposed the following empirical correlation for the liquid velocity at the column axis: Re: = 3.34 (Re:, Frs.0.25)0.92_ (21)

Fr,,-‘h

IO7

5

109

308

108

5 -Theory,

eq. (12)

/

IO’ 5

106 $ 5

.

Migauchi

and Shu (19701

A Kojima et al. (1980) 0 Yoshitome v

and Shirai (1970)

Pavlov I1 965)

* Hills 11994) A Nottenklmper V

et&.

-

IO5

11983)

Franz et a/. (19841

0 Ueyama and

Miyauchi

(1977)

Q Ulbrecht et al. (1986t .

Re,,

(20)

Preseot

result

Fr,,-”

Fig. 3. Comparison between the predictedand experimentalvalues of liquid velocity at the column axis (Newtonian fluids).

Liquid phase mixing in bubble columns

1973

lo5 5

Al

wt%PVP

a 3 wt% PVP I

0.5 wt% P#i

v

0.25 \M% PAA

Ulbrechtetal. D=0.10-0.15m

(1986)

lo4 d -3 * 6; E

5

lo3 Present work

5

n = 0.793

. CMC-2

n = 0.625

.

/

I

5

10

I 703

D = 0.23 m

0 CMC-1

Separann = 0.902

I

I

5

104

I

5

105

(ReLfL.4. Fig. 4. Comparison between the predictedand experimentalvalues of liquid velocity at the column axis (non-Newtonian fluids). This form is very similar to the present theoritical correlation. As shown in Fig. 7 in the paper of Ulbrecht et al. (19&Q eq. (21) can correlate the data of the wide range of conditions for Newtonian fluids. However, the agreement of the data for non-Newtonian liquids with eq. (21) is somewhat poor compared with that of water and the data points are scattered around the correlation, eq. (21). It is probable that in the empirical correlation proposed by Ulbrecht et al. (1985) the influence of non-Newtonian flow behaviour is somewhat inaccurately taken into account. On the whole, it may be concluded that the present model yields satisfactory predictions of liquid velocity at the column axis in Newtonian and non-Newtonian fluids. The axial dispersion coeficient The predictions of our model are shown in Fig. 5 along with the available correlations published so far. Baird and Rice (1975) derived the following expression using the isotropic turbulence theory of Kolmogoroff E, = 0.35g

l/3

p/3

U1/3 s9

(22)

where the constant 0.35 was determined from analysis of a wide range of published data. It can be seen from Fig. 5 that the present model is in excellent agreement with the semi-empirical correlation of Baird and Rice (1975). Riquarts (1981) developed a model based on a turbulent stochastic mixing process and the resulting correlation is E, = 0.068 g3/8 D312 ,fJ*,-

‘Is_

(23)

This correlation for water also agrees very well with the present model. Recently, Zehner (1982) proposed a correlation based on the cell model E, = 0.368 g”’ D413 u&”

(24)

where the proportionality constant 0.368 is an adjustable parameter. This correlation is also very close to the present model, eq. (18). As well as the model of Riquarts (1981), eq. (24) is based on the well-known relation for fixed beds: Pe = 2. (25) A number of measurements indicate that two important physical parameters, the superficial gas velocity and the column diameter, affect axial dispersion. It is surprising that all theoretical models can be expressed in the same form of E z cc D4/3

u1/3 SB -

(26)

From this equation it can be seen that E, varies as D4’” and ult3. These indices are in close agreement with 1.4, and 02 in the empirical correlation of Deckwer et al. ( 1974), respectively. The empirical correlations proposed by Towel1 and Ackerman (1972), Deckwer et al. (1974) and Hikita and Kikukawa (1974) are also in reasonable agreement with the present model. Joshi and Sharma (1979) obtained the following correlation using reported data for a wide range of conditions: E, = 0.31 D V,. Substituting

(27)

an expression for the average liquid

1974

Y.

KAWASE

iv

I

__

and M. MOD-YOUNG 1

I

Present

I

D = 0.5 m

work

I-

, -

I_

i-

- - - ___ _._--..

Baird and Rice (1975)

Deckwer et al. (1974)

-

Towell and Ackerman

-- -

---

*-

Hikita and Kikukewa 11984)

5

1

-2

Fig. 5. Comparison

(1972)

Riquarts (1981)

of the present model for the axial dispersion coefficient with various correlating equations (Newtonian fluids).

circulation velocity based on the cell model (Joshi and Sharma, 1979) into the above equation, we obtain E, = 0_435g”= DA/’ (u,~-

E,++,~)~‘~.

(28)

Using the data in Table 1 in the paper of Ueyama and Miyauchi (1979), we have sGuba, x 0.65 usg.

-

Present theory,

(2g)

Substitution of this approximation into eq. (28) yields E, = 0.307 g”= D4i3 I.$“.

(30)

This form is the same as that of eqs (1 S), (22~(24) but the constant 0.307 is somewhat smaller than those in the correlations mentioned above. A detailed comparison between the published data for axial dispersion coefficient in bubble columns with

eq. (19)

1

10-l

8

.

5

. v L1

A 1 o-2 1 o-5

5

104

5

10-s

Towell and Ackerman D = 0.406 and ‘1.07 m Deckwer era/.

(1972) water,

119741 water, D = 0.2 m

Hikita and Kikukawa (1984) water, D = 0.1 and 0.19 m Wendt

5

et al. (19S41) water. D = 0.196 and 1 m

1o-2

5

10-l

5

1

Fr,, Fig. 6. Comparison

between the predicted and experimental values of the axial dispersion coefficient (Newtonian fluids).

Liquid phase mixing in bubble columns

1975

5

1 5 :: 2 Q) P10-l 5

IJata by Dackwer eta/. .

1 o-2

” = 0.82

o 1.3 wt% CMC

n = 0.66

.

1.0 wt% CMC

” = 0.92

*

0.7 wt% CMC

” = 0.91

5

10-l

5

(1982l cl = 0.14 m

1.6 wt% CMC

.

1

We),,1 Fig. 7. Comparison between the predicted and experimental values of the axial dispersion coefficient (nonNewtonian fluids).

1.6 wt oA CMC despite this being the most viscous liquid in their measurement. However, it is not a surprising result. The present model, eq. (18), predicts that the axial dispersion coefficient increases with an increase of pseudo-plasticity. Similar results were obtained in aconcentric-tube airlift reactor by Fields et al. (1984). For dilute xanthan gum solutions, the effective dispersion coe5cients for a single passage around the loop were larger than those for tap water. This peculiarity may also be due to the pseudoplasticity of gum solutions. Unfortunately, they did

water and the proposed model, eq. (19X is given in Fig. 6. The proposed equation is seen to fit the data of axial dispersion coefficients in water reasonably well. Figure 7 compares the proposed model, eq. (19), with the axial dispersion coefficient data for nonNewtonian fluids obtained by Deckwer et al. (1982). They calculated the axial dispersion coe5cients from the profiles of the liquid-phase oxygen concentration measured under stationary conditions. The agreement is seen to be quite reasonable. Deckwer et al. (1982) found the highest axial dispersion coefficient for 5

I

Experimental ___ -----

Yoshitome -_

I

I

I

data

Miyauchi

and Shira! (1970) 0 = 0.15 m and Shyu (1970) 0 = 0.10 m

NottenkSmper

eta/.

(1963) Cl = 0.45 m

1 F .E

/

5

S

5 10”

5

1 o-2

5 4,

10-l

5

(m/s)

Fig. 8. Liquid velocity at the column axis in the inverted conical bottom column.

Y.

1976

KAWASE

and M. MOO-YOUNG

not measure the rheological properties of the gum solutions. Therefore, a quantitative discussion of the effects of non-Newtonian flow behaviour was impossible. It may be concluded that the present model, eq. (1 S), gives reasonably good predictions for axial dispersion coefficients in bubble columns with both Newtonian and non-Newtonian fluids. Liquid velocity at the column axis in the inverted conical bottom

column

As described previously, the direction of liquid circulation in the inverted conical bottom column is opposite to that in the conventionally operated column. The liquid velocities at the column axis in water and CMC-2 solution in the inverted conical bottom column are depicted in Fig. 8 as a function of the superficial gas velocity. For comparison are shown the data obtained for conventional bubble columns by Yoshitome and Shirai (1970), Miyauchi and Shyu (1970) and Nottenkiimper et al. (1983). The relationship between liquid velocity at the column axis and gas velocity in the inverted conical bottom column is found to be similar to that in the conventional column. The data for the CMC-2 solution lie somewhat above those for water. This trend coincides with the prediction of the present model. As shown in Figs 3 and 4, the data for Newtonian fluids (water and MC) and non-Newtonian fluids (CMC-1, CMC-2 and Separan) are in reasonably good agreement with the proposed model. It may be concluded therefore that the hydrodynamics in the inverted conical bottom column are similar to those in the conventional column despite the opposite directions of liquid circulations. In fact, Kawase and Moo-Young (1985) found that the gas hold-up and the volumetric mass-transfer coefficient in the inverted conical bottom column are only slightly higher than those for the conventional bubble columns in which the upflow of liquid occurs at the centre and downflow in the annular region near the wall. CONCLUSIONS

New models for deducing the liquid velocity at the column axis and the dispersion coefficient in bubble columns have been developed. They are in good agreement with the existing correlations and experimental data for Newtonian fluids over wide ranges of conditions. The theory predicts the increases of liquid velocity at the column axis and the axial dispersion coefficient with an increase of pseudo-plasticity and this agrees reasonably well with the experimental data. Measurements have also been carried out on the liquid velocity at the column axis in an inverted conical bottom bubble column. Although the direction of liquid circulation induced by the rise of bubbles is opposite to that in the conventional bubble column, the data in the inverted conical bottom column could be correlated by the correlation for the conventional column.

NOTATION

D EI Es

Ev EZ

Fr L K 1 Fe R RC Re Re* r t At u I(brn % ll ;“,”

Y z At Greek

letters

f

shear rate, s- ’ virtual kinematic viscosity, m2 s- ’ gas hold-up viscosity, Pas density, kg m- 3 shear stress, Pa constant

& EE

p P T X

-

column diameter, m energy input rate, W energy dissipation rate at the gas-liquid interface, W energy dissipation rate in the liquid motion, W axial dispersion coefficient, m s- 1 Froude number (= u’/Dg) gravitational acceleration, m s-’ clear liquid height, m consistency index in a power-law model, Pas” mixing length, m flow index in a power-law model Peclet number (= u,s D/E,) column radius, m locus of the liquid flow reversal, m Reynolds number ( = puD/p) modified Reynolds number (= p~‘-~ D”/K) radial coordinate, m time, s time period, s velocity, m s - 1 slip velocity of bubble, m s- ’ liquid velocity at column axis, m s- 1 superficial gas velocity, ms-’ liquid circulation velocity, m s- 1 distance (= R -r), m axial coordinate, m longitudinal displacement, m

Superscript

a C

cal exp 0

sg

and subscripts

average in the annular region in the core region calculated experimental based on u, based on I(_

REFERENCES

Baird,M. H. I. and Rice, R. G., 1975. Axial dispersionin large

unbatIled columns. Chem. Engng J. 9, 171-174. Brodkey, R. S., 1966, Mixing: Theory und Practice (Edited by Uhl, V. W. and Gray, J. B.), Vol. 1, pp. 7-10. Academic hess, New York. Clapp, R. M., 1963, Turbulent heat transfer in pseudoplastic non-Newtonian fluids. International Developments in 1961 Heat Transfer, International Heat Transfer Conference, D211, 652-661. Dcckwer, W. -D., Burckhart, R. and Zoll, G.. 1974, Mixing and mass transfer in tall bubble columns. C&m. Enana - ” Sci. 29, 2177-2188. Deckwer, W.-D., Nguyen-Tien, K., Schlumpe, A. and

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1977

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