Gas-Liquid Mass Transfer Coefficient in Stirred Tanks Interpreted

7/28/2019 Gas-Liquid Mass Transfer Coefficient in Stirred Tanks Interpreted

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Chemical Engineering and Processing 43 (2004) 823830

Gas-liquid mass transfer coefficient in stirred tanks interpretedthrough bubble contamination kinetics

S.S. Alves, C.I. Maia, J.M.T. Vasconcelos

Department of Chemical Engineering, Instituto Superior Tcnico, Centro de Eng. Biolgica e Qumica, 1049-001 Lisboa, Portugal

Received 23 December 2002; received in revised form 18 May 2003; accepted 19 May 2003

Abstract

Experimental data on the average mass transfer liquid film coefficient (kL) in an aerated stirred tank are presented. Liquid media used were

tap water, electrolyte solutions and water with controlled addition of tensioactive material. Values ofkL range from those expected for bubbles

with a mobile surface to those expected for rigid bubbles. These data are quantitatively interpreted in terms of bubble contamination kinetics,

using a stagnant cap model, according to which bubbles suddenly change from a mobile interface to a rigid condition when surface tension

gradients, caused by surfactant accumulation, balance out shear stress.

2003 Elsevier B.V. All rights reserved.

Keywords: Tank; Bubble; Kinetics

1. Introduction

Mass transfer effectiveness in gasliquid contactors is

most often expressed by means of the volumetric mass trans-

fer coefficient (kLa). This may be correlated, for example,

with power input per unit volume and gas superficial veloc-

ity, but the resulting correlations do not achieve any degree

of generality. Too many phenomena contribute to the values

of the film coefficient, kL and of the specific area a and their

combined effect cannot easily be predicted. Separation of

kL and a in the volumetric mass transfer coefficient is thus

a first step for a better understanding of the underlying phe-

nomena. While separate determination ofkL and a has been

carried out by a number of authors in bubble columns [19]

and air-lifts [10], problems related to reliable measure-

ment of kL make determination in stirred tanks particularly

difficult.

The mass transfer film coefficient kL is a major function

of bubble rigidity. If a bubble is rigid, then kL, which will

be named krigidL in this case, is theoretically obtained by the

equation proposed by Frssling [11] from laminar boundary

layer theory:

Corresponding author. Tel.: +351-1-8417-188;

fax: +351-1-8499-242.

E-mail address: [email protected] (S.S. Alves).

krigid

L= c vS

dD2/3 1/6 (1)

where d is the bubble diameter, D is the diffusivity, vS is

the bubble-liquid relative velocity (slip velocity), is the

kinematic viscosity of the liquid and c is a constant value

of0.6. Experimental values of c have been found to vary

from 0.42 to 0.95 [12,13].

If the bubble has a mobile interface, then kL, which will be

named kLmobile, is given by the penetration model solution

[14]:

kmobileL = 1.13

vS

dD1/2 (2)

A considerable amount of literature data on bubble ab-

sorption in water [1524] shows that experimental kL falls

between the limits defined by Eqs. (1) and (2), which may

differ by a factor of >5 for small bubbles. The scatter of

data is attributed to different methods of bubble release, to

different measurement techniques and to different system

purities.

The dependence of bubble rigidity on bubble size and

on surface contamination has been widely recognized (e.g.

Refs. [1,4,7,2527]). There is experimental evidence sug-

gesting that bubbles may be free of surface-active impurities

when they are formed, but that their behavior changes in

0255-2701/$ see front matter 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0255-2701(03)00100-4


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824 S.S. Alves et al. / Chemical Engineering and Processing 43 (2004) 823830

time as contaminants accumulate at the interface. This would

explain why kL depends on time, both under uncontrol-

led conditions [12,16,17,2832] or controlled conditions

[33,34].

Contaminants with the greatest effect on the interface

mobility, thus on mass transfer, are insoluble [21]. Surface

contamination is particularly consistent with the observedasymmetry of interface circulation that is implicit in Savics

stagnant cap model [35,36]. According to this model, the in-

soluble monolayer of adsorbed surfactant is dragged by the

adjacent liquid towards the bottom of the bubble, where a

stagnant cap region builds up. When the amount and type of

contaminant are unknown, Savics model describes a lim-

iting case where the surface tension varies from its value

for a pure system, at the bubble front, to a minimum at

the rear [21]. The gradient of surface tension so generated

opposes the surface flow and increases the drag, up to the

point where it balances the viscous stress at the surface,

leading to immobilization. This hypothesis [37] is generally

accepted to explain the retardation of terminal velocity bysurfactants [36]. It has also been demonstrated that the tran-

sition bubble behaviour from that of a fluid sphere to that

of a rigid particle is sharper for increased Reynolds number

[38].

A particularly clear picture of the phenomenon started

to emerge from the experiments of Schulze and Schlnder

[39]. Mass transfer coefficient (kL) was determined from the

dissolution rate of free-floating bubbles held stationary in a

downward water flow. A period of large initial kL was sud-

denly followed by a much lower kL value, consistent with

Eq. (1). The time span of the initial regime was so short in

Schulze and Schlnders system that it could only be de-tected with highly soluble gases. More recently, the use of

a water cleaning system in a similar apparatus [40] allowed

the duration of the regime of large kL to be expanded by or-

ders of magnitude, so that it could easily be observed with

air and other slowly dissolving gases. Moreover, the initial

large kL value was consistent with Eq. (2), thus showing that

the abrupt change to the slower mass transfer regime was

connected with surface immobilization. A simple model was

developed [40] based on the stagnant cap concept [37,41] to

theoretically interpret contamination times for various sizes

of bubbles. Larger bubbles remain mobile for a longer pe-

riod, as they are slower to accumulate enough contaminant

for transition to rigidity, explaining the common knowledge

that larger bubbles tend to be mobile while smaller bubbles

tend to be rigid [21]. The model was also used to success-

fully interpret experimental mass transfer data in airlift and

bubble column contactors [42].

This paper is an attempt at extending the analysis to

stirred tanks. kLa data from a double turbine stirred tank

are combined with previous data on local bubble diame-

ter and on local gas holdup obtained in the same apparatus

[43,44] to determine experimental values of the film coeffi-

cient kL and try to interpret these in terms of the proposed

theory.

2. Model

The model employs average values of bubble size, gas

holdup, specific interfacial area and bubble residence time

in the tank to calculate an average gas liquid film coefficient,

kL.

For a population of n bubbles, kL is defined as

kL =

ni=1

kLidt

tRi d2i

Ni=1

d2i

(3)

where tR, is the bubble residence time. This equation may

be simplified if all bubbles are assumed of average size and

only two possible values ofkL are considered, depending on

surface mobility: kmobileL for fully mobile bubble and krigidL

for a rigid bubble of the given average diameter. Eq. (3) then

becomes:

kL =kmobileL t

mobile + krigidL (tR t

mobile)

tR(4)

where tmobile stands for the time span where bubbles behave

like mobile fluid particles. kmobileL is given by the penetration

model solution (Eq. (2)). krigidL is using Frsslings equation

Eq. (1). The average bubble residence time, tR, in Eq. (4)

is calculated dividing the gas volume by the gas volumetric

flowrate, Q:

tR =VL

Q (1 )(5)

where VL is the liquid volume and is the overall gas

holdup. An expression for the calculation of tmobile has been

deduced [40] assuming the stagnant cap model of bubble

surface contamination [36]:

tmobile = k d1/2 ln (d/htrans)

Csurf(6)

where d is the bubble average diameter, Csurf is the surfac-

tant concentration, k is a constant of unknown value related

with surfactant properties and htrans is the bubble clean

segment height at the transition point from mobile to rigid.

Eqs. (1)(6) constitute a model for predicting kL. Besides

bubble diameter, two parameters are required: htrans, whichwas found earlier [40] to have a value of 6104 m and the

ratio k/Csurf, which depends on experimental conditions.

3. Experimental

The experimental set-up consisted of a 0.292 m diameter,

flat-bottomed, fully baffled Perspex vessel. Agitation was

provided by two 0.096-m standard Rushton turbines set at

clearances of 0.146 and 0.438 m, respectively above the tank

base. The tank dimensions are shown in Fig. 1, together with

the location of sampling points for local gas hold-up and


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S.S. Alves et al. / Chemical Engineering and Processing 43 (2004) 823830 825

Fig. 1. Tank dimensions and location of experimental sampling points

(). Distances in mm.

bubble size, data which were determined in previous work

[43,44]. The liquid media used were tap water 0.3 M aqueous

solution of sodium sulphate and 0.3 M aqueous solutionof sodium sulphate with 20 ppm PEG (surface tension, 63

mN/m). Operation conditions are presented in Table 1.

The overall volumetric mass-transfer coefficient, kLa, was

measured at 25 0.5 C, using the peroxide decomposition

steady-state technique with manganese dioxide as the cat-

alyst [45]. Measurements of the dissolved oxygen concen-

tration CL were performed using two oxygen meters, WTW

Oxi340, equipped with galvanic probes WTW Cell Ox 325.

The kLa value was calculated from

kLa =Qperoxide

2VLlogC(7)

Table 1

Experimental conditions

Liquid phase Ref. N (s1) Q (m3 s1)

Aqueous Na2SO4 0.3 M S-N1-Q1 4.2 0.000167

S-N2-Q1 5.0 0.000167

S-N3-Q1 7.9 0.000167

S-N4-Q1 7.5 0.000167

S-N5-Q1 10.0 0.000167

S-N4-Q2 7.5 0.000333

Aqueous Na2SO4 0.3 M

with 20 ppm PEG

PEG-N4-Q1 7.5 0.000167

Water W-N4-Q1 7.5 0.000167

Fig. 2. Local bubble size distributions for operating conditions S-N4-Q2

(see Table 1).


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where Qperoxide is the peroxide molar addition to the liq-

uid volume V and logC is the logarithmic mean between

the oxygen concentration in the liquid bulk, CL and the one

in equilibrium with the gas. The outlet oxygen concentra-

tion in the gas phase was calculated assuming a constant

volumetric gas flow across the vessel, which is accurate

within 5%. kLa was determined at least twice under thesame experimental conditions with a reproducibility within

20%.

Fig. 3. Local specific areas, a (m1), for various operating conditions (see Table 1): (a) S-N2-Q1; (b, c) two runs of S-N4-Q1; (d) S-N4-Q2; (e)

PEG-N4-Q1; (f) W-N4-Q1.

4. Results and discussion

Data on local gas hold-up and local average bubble di-

ameter [43,44], obtained from bubble size distributions, as

shown in Fig. 2, allow local specific areas to be determined

for the tank. From local data, the average interfacial specific

area may easily be calculated using

a =

tank aiVi

V(8)


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Table 2

Experimental gas holdup, bubble size and volumetric mass transfer coefficient; calculated specific area and film coefficient for various tank conditions

Conditions d32 (m) a (m1) kLa (s

1) kL (ms1)

S-N1-Q1 0.018 0.00230 47 0.008 0.000169

S-N2-Q1 0.022 0.00167 78 0.013 0.000166

S-N3-Q1 0.033 0.00152 135 0.028 0.000207

S-N4-Q1 0.044 0.00124 213 0.030 0.000141S-N5-Q1 0.072 0.00090 512 0.062 0.000121

S-N4-Q2 0.050 0.00134 233 0.030 0.000129

PEG-N4-Q1 0.052 0.00121 253 0.022 0.000087

W-N4-Q1 0.025 0.00289 53 0.017 0.000319

Conditions as explained in Table 1.

where ai are the local experimental specific areas, Vi the

corresponding compartment volumes and Vthe tank volume.

Results are shown in Fig. 3. While large specific areas near

the turbines discharge are due to lower bubble diameter (see

Fig. 2), in other points of the tank they result from high local

gas hold-up.Table 2 brings together average tank data on gas hold-up

and bubble diameter [43,44], specific area (as calculated

through Eq. (8)), experimental volumetric mass transfer co-

efficient, kLa and film coefficient given by the ratio kLa/a.

The resulting kL values are plotted against bubble diameter

in Fig. 4. These results have an estimated random error of

30%.

Theoretical values obtained from Higbies Eq. (2) for bub-

bles with mobile surface and from Frosslings Eq. (1) for

rigid bubbles are also presented in Fig. 4. These depend

upon gasliquid slip velocity, which, apart from the turbines

discharge jet, may be assumed to be a rise velocity. For therelatively low gas holdups at play, rise velocities are close

to single bubble terminal velocities, which, both for rigid

and for mobile bubbles rising in still water, may be esti-

mated using correlations of experimental data, given in Clift

et al. [21]. It is however known that turbulence consider-

ably reduces bubble mean rise velocity, up to 50% [46,47].

A correction for turbulence was introduced by a factor as-

sumed equal for bubbles with both rigid and mobile surface.

This factor was adjusted by noticing that bubbles in 20 ppm

PEG solution are mostly rigid due to the relatively high con-

centration of contaminant, as observed in [42]. Their value

of kL should therefore fall on the Frssling line. This was

achieved by a 35% reduction on rise velocities, as calculated

from terminal velocities.

Simulated kL, calculated by applying the simple model

described in Section 2, is also superimposed on Fig. 4. While

the previously determined value of 6104 m was used for

parameter htrans [40], parameter k/Csurf is expected to vary

with the liquid medium, since both the contaminant and its

concentration are probably different, thus affecting k and

Csurf.

Bubbles in tap water are the closest to Higbies line, since

the water is relatively clean and the bubbles are relatively

large. Bubbles in salt solution (non-coalescing medium), but

Fig. 4. Liquid film mass transfer coefficient versus average bubble dia-

meter. , Salt solution; , water; , PEG solution; - - -, simulated kL.

without PEG addition, behave in a manner that is interme-

diate between that of bubbles in tap water and in PEG solu-

tion. This is because they are of intermediate size, but also

because the liquid medium has intermediate contaminant

concentration. Preparation of the salt solution with techni-

cal sodium sulphate is likely to have introduced a level of

contaminant higher than what existed in the tap water. This

explains why parameter k/Csurf that fits the data for salt so-

lution is approximately half of that which fits tap water. If

the contaminant were similar, this would mean contaminant

concentration in the salt solution about double of that in

tap water.

PEG (20 ppm), on the other hand, is certainly a higher

concentration than the trace levels of surfactant present either

in tap water or in salt solution. It causes bubbles to rigidify

very quickly after formation. In previous work carried out

both in airlifts and in a bubble column, with considerably

larger bubbles and lower overall residence times, antifoam

concentrations >10 ppm invariably led to rigid bubble values

of kL [42].

The effect of bubble size on kL which is apparent in

the simulation curves is also clear from the salt solution

experimental points. It is due to the fact that larger bub-

bles take longer to rigidify. Thus, they tend to move away


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Fig. 5. Fraction of bubbles with mobile interface versus average bubble

diameter.

from Frsslings line, while smaller bubbles spend a greater

fraction of their residence time in the rigid regime, thusapproaching it. The estimated fraction of bubbles in the

tank that are mobile, x = tmobile/tR, is presented in Fig. 5.

The above theory may be used to estimate average values

of kL for each of the two halves of the tank, assuming that

there is no bubble recirculation from the top to the bottom

half of the tank and that the air/liquid interface lose no con-

taminant in the top turbine. kL calculation for each half of

the tank follows the same steps as for the whole tank, only

using local average values of bubble size and gas holdup

for the two halves of the tank, obtained from data in pre-

vious work [43,44]. Results are presented in Table 3. What

they indicate is rather surprising, namely, that the volumet-

ric mass transfer coefficient for the salt solution is signifi-cantly higher in the bottom half of the tank, in spite of the

lower specific transfer area in that region. This is because

the film coefficient is much higher in the bottom half, since

the fraction of clean bubbles is much larger there. There are

very few experimental data to assess these simulated results.

While they appear to agree with the experimental results by

Moucha et al. [48], they disagree with results by Alves and

Vasconcelos [49] and Linek et al. [50].

Table 3

Experimental gas holdup and bubble size; calculated specific area; simulated fraction of clean bubbles, film coefficient and volumetric mass transfercoefficient for top and bottom halves of the tank for various tank conditions

Conditions Location d32 (m) a (m1) x KL (ms

1) kLa (s1)

S-N2-Q1 Top 0.031 0.00167 110 0 0.000078 0.0085

Bottom 0.013 0.00164 46 0.62 0.000401 0.0184

S-N4-Q1 Top 0.053 0.00122 269 0 0.000085 0.0229

Bottom 0.033 0.00128 157 0.73 0.000198 0.0310

S-N4-Q2 Top 0.069 0.00144 285 0 0.000082 0.0233

Bottom 0.035 0.00120 175 0.67 0.000291 0.0509

PEG-N4-Q1 Top 0.060 0.00125 307 0 0.000084 0.0259

Bottom 0.039 0.00116 198 0 0.000086 0.0170

Conditions as explained in Table 1.

5. Conclusions

Experimental data on the average mass transfer liquid film

coefficient, kL, in an aerated stirred tank range from those

expected for bubbles with a mobile surface, kmobileL , to those

expected for rigid bubbles, krigidL ,which are much lower.

Bubbles in PEG solution behave as rigid bubbles, whilebubbles in tap water behave closer to having a mobile inter-

face. Bubbles in salt solution have intermediate values ofkL.

For the same liquid medium (salt solution) smaller bub-

bles result in lower values of kL, closer tokrigidL .

These data can be quantitatively interpreted in terms of

bubble contamination kinetics, using a stagnant cap model,

according to which bubbles suddenly change from a mobile

interface to a rigid condition when surface tension gradi-

ents caused by surfactant accumulation balance out shear

stress.

Appendix A. Nomenclature

a specific interfacial area based on the liquid

volume (m1)

c constant in Eq. (1)

C concentration (mol m3)

d, d32 bubble diameter, Sauter mean diameter (m)

D gas diffusivity in the liquid (m2 s1)

htrans height of the clean segment at the bubble

front (m)

k constant in Eq. (6) (mole m7/2 s)

kL liquid-side mass transfer coefficient (m s1)

kLa volumetric mass transfer coefficient referredto the liquid volume (s1)

n number of bubbles

N agitation rate (s1)

Q gassing rate (m3 s1)

Qperoxide peroxide solution addition flow rate (mol s1)

t time (s)

tR residence time (s)

V volume (m3)


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vS slip velocity (m s1)

x fraction of bubbles with mobile interface

Greek symbol

overall fractional gas holdup

Superscripts and subscripts

i refers to zone i in the tank or to an

individual bubble

L liquid

mobile mobile interface

rigid rigid interface

surf surfactant

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Gas-Liquid Mass Transfer Coefficient in Stirred Tanks Interpreted