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

Chemical Engineering and Processing 43 (2004) 823–830

Gas-liquid mass transfer coefficient in stirred tanks interpreted through bubble contamination kinetics S.S. Alves∗ , C.I. Maia, J.M.T. Vasconcelos Department of Chemical Engineering, Instituto Superior Técnico, Centro de Eng. Biológica e Qu´ımica, 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 of kL 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 gas–liquid contactors is most often expressed by means of the volumetric mass transfer coefficient (kL a). This may be correlated, for example, with power input per unit volume and gas superficial velocity, 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 phenomena. While separate determination of kL and a has been carried out by a number of authors in bubble columns [1–9] and air-lifts [10], problems related to reliable measurement 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 rigid be named kL in this case, is theoretically obtained by the equation proposed by Frössling [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).

0255-2701/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0255-2701(03)00100-4


rigid kL

=c·

vS · D2/3 · ν−1/6 d

(1)

where d is the bubble diameter, D is the diffusivity, vS is the bubble-liquid relative velocity (slip velocity), ν is the kinematic viscosity of the liquid and c is a constant value of ≈0.6. 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 kL mobile , is given by the penetration model solution [14]:
vS mobile kL = 1.13 (2) · D1/2 d A considerable amount of literature data on bubble absorption in water [15–24] 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. [1,4,7,25–27]). There is experimental evidence suggesting that bubbles may be free of surface-active impurities when they are formed, but that their behavior changes in

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time as contaminants accumulate at the interface. This would explain why kL depends on time, both under uncontrolled conditions [12,16,17,28–32] 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 observed asymmetry of interface circulation that is implicit in Savic’s stagnant cap model [35,36]. According to this model, the insoluble 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, Savic’s model describes a limiting 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 by surfactants [36]. It has also been demonstrated that the transition 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 Schlünder [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 suddenly followed by a much lower kL value, consistent with Eq. (1). The time span of the initial regime was so short in Schulze and Schlünder’s system that it could only be detected 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 orders 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 period, 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 successfully interpret experimental mass transfer data in airlift and bubble column contactors [42]. This paper is an attempt at extending the analysis to stirred tanks. kL a data from a double turbine stirred tank are combined with previous data on local bubble diameter and on local gas holdup obtained in the same apparatus [43,44] to determine experimental values of the film coefficient 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 n   k

Li ·dt

kL =

tRi

i=1

N  i=1

· di2 (3)

di2

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 of kL are considered, depending on rigid surface mobility: kmobile for fully mobile bubble and kL L for a rigid bubble of the given average diameter. Eq. (3) then becomes: rigid

kL =

kLmobile · t mobile + kL tR

· (tR − t mobile )

(4)

where tmobile stands for the time span where bubbles behave like mobile fluid particles. kmobile is given by the penetration L rigid model solution (Eq. (2)). kL is using Frössling’s equation Eq. (1). The average bubble residence time, tR , in Eq. (4) is calculated dividing the gas volume by the gas volumetric flowrate, Q: VL · ε tR = (5) Q · (1 − ε) where VL is the liquid volume and ε 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]: t mobile = k ·

d 1/2 · ln (d / htrans ) Csurf

(6)

where d is the bubble average diameter, Csurf is the surfactant 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 , which was found earlier [40] to have a value of 6×10−4 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

S.S. Alves et al. / Chemical Engineering and Processing 43 (2004) 823–830

825

Fig. 1. Tank dimensions and location of experimental sampling points (䊉). 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 solution of sodium sulphate with 20 ppm PEG (surface tension, 63 mN/m). Operation conditions are presented in Table 1. The overall volumetric mass-transfer coefficient, kL a, was measured at 25 ± 0.5 ◦ C, using the peroxide decomposition steady-state technique with manganese dioxide as the catalyst [45]. Measurements of the dissolved oxygen concentration CL were performed using two oxygen meters, WTW Oxi340, equipped with galvanic probes WTW Cell Ox 325. The kL a value was calculated from Qperoxide kL a = (7) 2VL log C Table 1 Experimental conditions Liquid phase

Ref.

N (s−1 )

Q (m3 s−1 )

Aqueous Na2 SO4 0.3 M

S-N1-Q1 S-N2-Q1 S-N3-Q1 S-N4-Q1 S-N5-Q1 S-N4-Q2

4.2 5.0 7.9 7.5 10.0 7.5

0.000167 0.000167 0.000167 0.000167 0.000167 0.000333

Aqueous Na2 SO4 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 liquid volume V and log C is the logarithmic mean between the oxygen concentration in the liquid bulk, CL and the one in equilibrium with the gas. The outlet oxygen concentration in the gas phase was calculated assuming a constant volumetric gas flow across the vessel, which is accurate within ± 5%. kL a was determined at least twice under the same experimental conditions with a reproducibility within ± 20%.

4. Results and discussion Data on local gas hold-up and local average bubble diameter [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 i Vi a = tank (8) V

Fig. 3. Local specific areas, a (m−1 ), 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.

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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

ε

d32 (m)

a (m−1 )

kL a (s−1 )

kL (ms−1 )

S-N1-Q1 S-N2-Q1 S-N3-Q1 S-N4-Q1 S-N5-Q1 S-N4-Q2 PEG-N4-Q1 W-N4-Q1

0.018 0.022 0.033 0.044 0.072 0.050 0.052 0.025

0.00230 0.00167 0.00152 0.00124 0.00090 0.00134 0.00121 0.00289

47 78 135 213 512 233 253 53

0.008 0.013 0.028 0.030 0.062 0.030 0.022 0.017

0.000169 0.000166 0.000207 0.000141 0.000121 0.000129 0.000087 0.000319

Conditions as explained in Table 1.

where ai are the local experimental specific areas, Vi the corresponding compartment volumes and V the 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 coefficient, kL a and film coefficient given by the ratio kL a/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 Higbie’s Eq. (2) for bubbles with mobile surface and from Frossling’s Eq. (1) for rigid bubbles are also presented in Fig. 4. These depend upon gas–liquid slip velocity, which, apart from the turbines discharge jet, may be assumed to be a rise velocity. For the relatively 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 estimated using correlations of experimental data, given in Clift et al. [21]. It is however known that turbulence considerably reduces bubble mean rise velocity, up to 50% [46,47]. A correction for turbulence was introduced by a factor assumed 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 concentration of contaminant, as observed in [42]. Their value of kL should therefore fall on the Frössling 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 6×10−4 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 Higbie’s 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 diameter. 䊏, Salt solution; 䊉, water; 䉱, PEG solution; - - -, simulated kL .

without PEG addition, behave in a manner that is intermediate between that of bubbles in tap water and in PEG solution. This is because they are of intermediate size, but also because the liquid medium has intermediate contaminant concentration. Preparation of the salt solution with technical 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 solution 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 bubbles take longer to rigidify. Thus, they tend to move away

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5. Conclusions

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

from Frössling’s line, while smaller bubbles spend a greater fraction of their residence time in the rigid regime, thus approaching 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 contaminant 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 previous work [43,44]. Results are presented in Table 3. What they indicate is rather surprising, namely, that the volumetric mass transfer coefficient for the salt solution is significantly 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].

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, kLmobile , to those rigid expected for rigid bubbles, kL ,which are much lower. Bubbles in PEG solution behave as rigid bubbles, while bubbles in tap water behave closer to having a mobile interface. Bubbles in salt solution have intermediate values of kL . For the same liquid medium (salt solution) smaller bubrigid bles result in lower values of kL , closer tokL . 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 gradients caused by surfactant accumulation balance out shear stress. Appendix A. Nomenclature a c C d, d32 D htrans k kL kL a n N Q Qperoxide t tR V

specific interfacial area based on the liquid volume (m−1 ) constant in Eq. (1) concentration (mol m−3 ) bubble diameter, Sauter mean diameter (m) gas diffusivity in the liquid (m2 s−1 ) height of the clean segment at the bubble front (m) constant in Eq. (6) (mole m−7/2 s) liquid-side mass transfer coefficient (m s−1 ) volumetric mass transfer coefficient referred to the liquid volume (s−1 ) number of bubbles agitation rate (s−1 ) gassing rate (m3 s−1 ) peroxide solution addition flow rate (mol s−1 ) time (s) residence time (s) volume (m3 )

Table 3 Experimental gas holdup and bubble size; calculated specific area; simulated fraction of clean bubbles, film coefficient and volumetric mass transfer coefficient for top and bottom halves of the tank for various tank conditions Conditions

Location

ε

d32 (m)

a (m−1 )

S-N2-Q1

Top Bottom

0.031 0.013

0.00167 0.00164

110 46

S-N4-Q1

Top Bottom

0.053 0.033

0.00122 0.00128

S-N4-Q2

Top Bottom

0.069 0.035

PEG-N4-Q1

Top Bottom

0.060 0.039

Conditions as explained in Table 1.

KL (ms−1 )

kL a (s−1 )

0 0.62

0.000078 0.000401

0.0085 0.0184

269 157

0 0.73

0.000085 0.000198

0.0229 0.0310

0.00144 0.00120

285 175

0 0.67

0.000082 0.000291

0.0233 0.0509

0.00125 0.00116

307 198

0.000084 0.000086

0.0259 0.0170

x

0 ∼0

S.S. Alves et al. / Chemical Engineering and Processing 43 (2004) 823–830

vS x

slip velocity (m s−1 ) fraction of bubbles with mobile interface

Greek symbol ε 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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