Mass Transfer Models for Oxygen-Water Co-Current Flow in Vertical Bubble Columns. Quinta de Santa Apolónia, Bragança, Portugal and

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1 386 Single and Two-Phase Flows on Chemical and Biomedical Engineering, 212, CHAPTER 14 Mass Transfer Models for Oxygen-Water Co-Crrent Flow in Vertical Bbble Colmns Valdemar Garcia 1 and João Sobrinho Teixeira 2, 1 Institto Politécnico de Bragança, Escola Sperior de Tecnologia e de Gestão, Qinta de Santa Apolónia, Bragança, Portgal and 2 Institto Politécnico de Bragança, Escola Sperior Agrária, Qinta de Santa Apolónia, Bragança, Portgal Abstract: The present work reports a theoretical and experimental stdy of mass transfer for oxygen-water co-crrent flow in vertical bbble colmns. The axial dispersion of liqid phase was also stdied. Experiments were carried ot in a 32 mm internal diameter and 5.35 and 5.37 m height colmns. The sperficial liqid velocity ranged from.3 to.8 m/s and volmetric flow rate ratio of gas to liqid ranged from.15 to.25. Mathematical models were developed to predict concentration of gas dissolved in the liqid as fnction of different physical and dynamic variables for two-phase cocrrent downflow and pflow. We obtained for the ratio of the liqid side mass transfer coefficient to initial bbbles radis, k /r =.12 s -1. Keywords: Vertical bbble colmns, gas-liqid bbble flow, oxygen-water flow, oxygen-water mass transfer, mass transfer models, gas-liqid co-crrent downflow, gas-liqid co-crrent pflow, mass transfer coefficient, axial dispersion, liqid axial dispersion coefficient, U tbe. INTRODUCTION The simltaneos gas and liqid flow freqently occrs in several sitations and in many indstrial applications sch as distillation colmns, chemical and nclear reactors, pipelines for hydrocarbon mixtres transport, solar collectors, mass transfer eqipments like bbble colmns, packed colmns, air lifts pmps, among others. Bbble colmns, eqipment where the gas phase is dispersed in small bbbles in continos liqid phase, have been sed as gas-liqid contactors Address correspondence to João Sobrinho Teixeira: Institto Politécnico de Bragança, Escola Sperior Agrária, Qinta de Santa Apolónia, Bragança, Portgal. sobrinho@ipb.pt. Ricardo Dias, Antonio A. Martins, Ri ima and Teresa M. Mata (Eds) All rights reserved- 212 Bentham Science Pblishers

2 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 387 devices, like as absorbers and/or strippers, chemical reactors for hydrogenation, oxidation or chlorination of organic liqids sch as fermenters in wastewater biological treatment nits. Particlarly, the air/oxygen-water flow can be fond in areas where the water oxygenation is very important, like the aqacltre and the water treatment nits (drinking, waste and river). The simltaneos flow of oxygen and water in cocrrent bbbly regime, first in downflow and then in pflow, occrs in a U tbe, device with applications on referred areas [1-6]. The liqid mixing or axial dispersion is one of the parameters that can inflence the process of mass transfer of gas to liqid and therefore mst be known. The literatre on liqid axial dispersion for bbble colmns, especially for the airwater system, is extensive. Many empirical [7, 8] and theoretical [9-11] eqations have been developed in order to predict the axial dispersion coefficient. In the present work, the liqid phase mass transfer coefficients and the axial mixing (axial dispersion coefficients) of the liqid phase of vertical bbble colmns operating with co-crrent downflow and pflow were determined for the oxygen-water system. EXPERIMENTA The experiments were performed on two vertical bbble colmns, linked in the bottom in U tbe form, operating with co-crrent downflow and pflow of gas and liqid. The colmns are respectively, 5.37 and 5.35 m height and 32 mm inside diameter. Gas Absorption The absorption experiments were performed in acrylic glass tbes. The liqid, tap water, was circlated in the tbes by a centrifgal pmp and was introdced at the top of test section. The water flow rate was measred by a calibrated rotameter and was reglated by a ball valve. The gas sed for all experiments was pre oxygen spplied by a high pressre bottle. The oxygen was introdced into the test colmn throgh a poros gas distribtor installed at the top of the test section. The poros gas distribtor allowed the formation of small diameter bbbles (abot

3 388 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira 2-4 mm). The volmetric oxygen flow rate was determined with an electronic flow meter. The pressre at entrance of the colmn was measred by a pressre sensor. Oxygen concentration in the water (liqid phase) was measred by means of five calibrated electrodes at five levels. The temperatre was measred in 4 points by thermocoples. The operating temperatre was 17±2ºC. The recorded data was stored by an acqisition system connected to a PC nit. Experiments were performed at sperficial liqid velocities, U, ranging from.3 to.8 m/s and at gas and liqid volmetric flow rates ratio, χ=g /, ranging from.15 to.25. The experimental set-p is shown in Fig. 1. Figre 1: Schematic diagram of the experimental set-p. OD1-OD5-oxygen probes; PM-pressre meter. Axial Dispersion Determination of the axial dispersion parameters was based on the typical method of tracer injection. A small amont of tracer (1x1 5 g/m 3 NaCl aqeos soltion) was manally injected as fast as possible at the top of the downflow colmn and at the bottom of the pflow colmn.

4 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 389 The system response, consisting in the detection of the tracer concentration variation with the time, was obtained with two condctivity probes, made of a pair of platinm cells, located at distances of 4.26 m and 3.95 m from the tracer injection points. The condctivity probes, connected to a condctimeter and to a compter, were sed to follow this time variation. The experimental set-p is shown in Fig. 2. Figre 2: Experimental set-p for determination of the axial dispersion parameters. THEORY Axial Dispersion iqid mixing in a bbble colmn is sally described on the basis of the onedimensional axial dispersion model, particlarly if the ratio between length and diameter of the colmn is large. A mass balance to the tracer, in the infinitesimal "slice" sitated between the qotas x and x+dx of a colmn, with volme S x (see Fig. 3), leads to the eqation that represents the model of axial dispersion: C C U C t x 1 x 2 A A A D 2 (1)

5 39 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira where D represents the liqid axial dispersion coefficient, S is the colmn cross sectional area, is the gas holdp, U is the sperficial liqid velocity and C A is the concentration of tracer in the liqid at position x of colmn and at time t. G x x x+dx Tracer CA CA +d CA - Plse inpt l Response - Measrement of tracer concentration S Figre 3: Schematic representation for the mass balance of the tracer in a bbble colmn. Introdcing the following variables: x t l l Z ; ; with tm l t U m U 1 where l represents the axial distance between tracer injection point and tracer response point in the colmn, t m is the mean residence time that represents the ratio between the liqid volme in length l of colmn and the volmetric liqid flow rate what goes throgh it and U is the real liqid velocity. eqation (1) can be rewritten in the form: C 1 C C Pe Z Z 2 A A A 2 (2) where Pe represents the Peclet adimensional nmber for simltaneos gas and liqid flow:

6 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 391 U l U l Pe (3) 1 D D In physical terms, the Peclet nmber, Pe, represents the ratio between the rate of mass transport by convection and the rate of mass transport by dispersion. When the Pe vale is very low, tending towards zero, the dispersion is very elevated being the perfectly mixed liqid flow. In the opposite extreme, when Pe is very high tending towards infinity, the dispersion is negligible and the flow of the liqid is called a plg flow. The eqation (2) can be solved analytically in the aplace space. The soltion depends on bondary conditions. The determination of the axial dispersion coefficient is obtained from the Pe nmber. In order to calclate this parameter it is not necessary solve the eqation (2). In fact, its determination is done throgh two parameters of the residence time distribtion (RTD) crve of the tracer, obtained on basis of the concentrations in 2 fnction of the time: the variance of the distribtion,, and the mean time of residence, t m. The analytical calclation of these parameters with a plse inpt of tracer for several bondary conditions is perfectly established [12-15]. 2 2 The variance is obtained from the moments: 2 1, where 1 and 2 represent, respectively, the first and second moments of the residence time distribtion crve (RTD), E(t), of the tracer. The response of the system to the plse inpt of tracer, in form of concentration crve in fnction of time, C A (t), allows to obtain the RTD crve: Et () C () t A C () t dt A (4) The mean time of residence, t m (eqal to 1 ), is given by:

7 392 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira t m tca() t dt tc i Aiti i te() t dt C Aiti C () t dt A i (5) The variance of the E(t) crve is given by: 2 2 tca() tdt tc i Aiti i 2 tetdt () tm tm t m C Aiti CA() t dt i (6) Since injection of the tracer has been effectated in a point below entrance the flids in the colmn and the response, in terms of tracer concentration, has been obtained in a point before the end of the colmn, the system is considered "open" type [13, 15, 16], as shown in Fig. 3. Assming that the bbble colmn is open to the liqid dispersion, the Peclet nmber is calclated sing the following eqation [13-15]: (7) tm Pe Pe where tm and 2 are calclated for each experimental condition from the experimental vales of tracer concentrationc A vs. time, with eqations (5) and (6), respectively. Therefore, the liqid axial dispersion coefficient D can be determined by eqation (3). Mass Transfer Models The oxygen transfer from the bbbles to the water in a co-crrent gas-liqid flow leads to a continos increase of the dissolved oxygen concentration along the colmn, and can be applied a one-dimensional theoretical model to describe the mass transfer process. The concentration profile along the tbe can be obtained for a given operating condition (vales of gas and liqid flow rate) when knowing the vales of

8 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 393 variables and parameters at the tbe entrance. In the model that we develop, we se as operating variables, the sperficial liqid velocity, U, and the ratio of volmetric gas/liqid flow rate at entrance,. The vales of variables and parameters at entrance that is necessary to know are: the pressre, P, concentration, C, the temperatre, T, the Henry's constant, H, the density of water,, and the tbe height, l. We will present two models to describe the process of mass transfer from gas to liqid: one takes into accont the axial dispersion of liqid and the other not. Models withot dispersion Downflow For an infinitesimally length dx, as shown in Fig. 4, the material balance in steady state for the dissolved oxygen in the liqid gives ka C C S x C C C (8) ( ) d ( d ) where a is the interfacial area per nit of colmn volme, C is the local concentration of oxygen solte dissolved in liqid, k is the liqid side mass transfer coefficient, S is the colmn cross sectional area, is the liqid (water) volmetric flow rate and C is the solbility of gas solte in liqid (satration concentration). This eqation gives, after rearrangement, the basic eqation of the model S C k a C C x (9) d ( )d In this model it is admitted, for each flow rate of water and of oxygen, that the bbbles that are formed in the inlet of the gas in the colmn are all of the same size, spherical, with eqal velocity and the gas holdp is constant at entrance of the colmn. While the flids go down in the tbe, all the bbbles decrease in size eqally with the increase of the pressre, so that in a determined section x, the bbbles take the same size and the gas holdp and the interfacial area per nit of colmn volme are constant.

9 394 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira G x x=; P ; C x x+dx U b C C +dc S Figre 4: Schematic representation of a gas-liqid downflow, initial point and scheme for the mass balance. For a given volmetric gas flow rate, G, the total nmber of bbbles that crosses a section of colmn per nit of time, n t, is given by: n G V n S U (1) t / b b b where V b represents the volme of a bbble, n b is the nmber of bbbles per nit of colmn volme and U b is the bbbles velocity. Being a = n b A b, where A b is the sperficial area of a bbble, and combining with eqation (1) we obtain: 3G 1 a (11) rsu b where r is the radis of a bbble. The retention of gas or gas holdp,, at any point of the colmn and U b are obtained by the expressions of Nicklin [17]: G (12) SU U b b G U S S (13) where U represents the bbbles velocity in stagnant medim. This parameter, althogh it is dependent of operating variables, can be taken approximately

10 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 395 constant, being sed in this model the reference vale of U =.2 m/s, recommended by several athors as Deckwer [16] and Whalley [18]. In or experiments we obtained vales of U in the range.16 to.23 m/s with an average of.19 m/s. The eqilibrim concentration at the interface can be calclated by Henry's law. If the gas phase is only formed by the gas to dissolve, then: P H C (14) where P is the total operation pressre and H is the Henry s law constant. The pressre variation is given by: dp m g (15) dx where m is the mixtre density given by: (1 ) (16) m In the development of the model, we have into accont the pressre variation and the oxygen consmmation as reslt of its transfer for the liqid and, conseqently, the variation of G and r, variables that have vales of G and r, respectively, at tbe entrance. Therefore, the parameters a, and m are also variable. P increases along the tbe, C increases, V b decreases and conseqently r and G. Ths, while the flids go down in the tbe, a and will decrease. Gas consmption enhances the effect of the increase of P in these variables. Considering behavior of ideal gas for the oxygen, the bbbles volme in a determined point of the tbe it is calclated by: P 1 ( C C ) Vb V T (17) P P n b t where the sbscript () represents the vale of variables at the tbe inlet, i.e., x=, the negative term of right side of the eqation represents the redction of the bbbles volme de its transfer for the liqid (or the consmption of oxygen), is the gas

11 396 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira constant and T is the temperatre. Taking into accont the eqation (1), n t =G/V b =G /V b, so the volmetric gas flow rate at any point of the colmn, is given by: P ( C C) G G T (18) P P Similarly we obtain the bbbles volme and its radis: P ( C C) r r T 3 3 P G P and (19) P ( CC ) P P r r T 1/3 (2) where is the ratio between gas and liqid volmetric flow rates at the tbe entrance, =G /. Eqation (12) can be maniplated taking into accont the eqations (13) and (18), obtaining for gas holdp: P C C U U T P P P C C P P U 1 T U where U is the sperficial liqid velocity, U=/S. (21) The interfacial area is obtained by eqation (11) taking into accont the eqations (13), (2) and (21): a 3 P ( CC ) T r P P 1/3 (22) The pressre variation with x is obtained from the maniplation of eqation (15) taking into accont the eqation (21):

12 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 397 U U dp g (23) dx U U U C C P T P The basic eqation of the model, eqation (9), can be maniplated taking into accont the eqations (14), (21) and (22) that, whit dx f ( dp) obtained from eqation (23), gives: 2/3 1/3 1 U U C C dc 2/3 P k 3 dp H g P T C P r (24) The profile of pressre and concentration along the colmn is then given by solving simltaneos eqations system (23) and (24) [19]. The soltion can be obtained by nmerical integration, for example by the Rnge-Ktta method, with the bondary condition C C for P P and x. Upflow For the pflow of gas and liqid, the volmetric flow rate of liqid at the entrance (y = ) is and the volmetric flow rate of gas is G. For U-tbe, at this point the concentration is C and pressre is P, eqal vales at the end of the downflow colmn. The velocity of rising bbbles is now given by: U b G U S S (25) By a similar process to that described for downward flow of gas and liqid, we obtain the eqations of the model for pflow. The gas holdp and the specific area are, respectively: P C C U U T P P P C C U 1 T U P P (26) a 3 P ( C C ) T r P P 1/3 (27)

13 398 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira The differential eqation for the pressre variation is: dp g U U dy U U U C C P T P (28) G where. The basic eqation of the model, eqation (9), now for pflow, after rearrangement gives: 2/3 1/3 1 U U C C dc P k dp H 3 r 2/3 g P T C P (29) The profile of pressre and concentration along the colmn is given by solving the simltaneos set of eqations (28) and (29) [19]. The soltion can be obtained by nmerical integration, for example by the Rnge-Ktta method, with the bondary condition C C for P P and y. Models with Dispersion We will now develop a model for mass transfer which, in addition to the assmptions of previos model, incldes the axial dispersion. From a mass balance of dissolved oxygen in the liqid contained in a infinitesimal "slice" located between the coordinates x and x + dx, with volme Sdx, whit D and S constants, we obtain for the steady state, the following eqation: dc dc D S (1 ) C( x) k a S dx( C C) D S (1 ) C( xdx) dx x dx xdx (3) where C, C and a represent, respectively, the concentration of dissolved oxygen in liqid at level x, the satration concentration at same level and interfacial area

14 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 399 per nit volme of colmn. From eqation (3) we obtain the basic eqation of the model: d dc dc D (1 ) U ka( C C) dx dx dx (31) For the downflow colmn, the pressre variation is still given by eqation (23), the gas holdp by the eqation (21) and the interfacial area per nit volme by the eqation (22). In order to solve eqation (31) can be sed the following change of variable: v (1 ) dc (32) dx With this change of variable, making a' ra and taking into accont the Henry's law, eqation (31) can be rewritten in the following form: dv U k P D v a' C dx 1 r H (33) where a ' is calclated by the following eqation: a' 3 P C C T P P 1 3 (34) The profiles of concentration and pressre along the colmn and the fitting k parameter r eqations: (Eq. (33)) are obtained by solving the following set of differential dv dx k P U a' C v r H 1 (35) D dc v (36) dx 1 d

15 4 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira dp g dx U U P C C 1 P P U T U (37) The resoltion of this system of differential eqations was performed sing a nmerical integrator of problems with bondary conditions at two points ( x and x 4.94 m) sing the obatto-rnge-ktta formla [19]. In addition to the points sed as bondary conditions provided also the experimental vale of the concentration of dissolved oxygen in water in two other measring points located at 1.6 and 3.25 m from the entrance, ths determining the fitting parameter k /r. In the optimization of this parameter was sed the evenberg-marqardt sbrotine. For the pflow colmn the model involves a nmber of eqations analogos to the downflow colmn. The basic eqation is eqal to eqation (31), with the axial coordinate y, the gas holdp and the interfacial area per nit of volme a d (1 ) dc D U dc k a ( ) C C dy dy dy (38) The gas holdp,, is given by eqation (26) and the interfacial area, a, is obtained by eqation (27). Making a change of variable similar to the eqation dc ' (32), (1 ), also considering the change a r dy a and the Henry's law, eqation (38) becomes: d U k P D a C dy r H ' 1 (39) where ' a is calclated by the following eqation: a ' P C C 3 T P P 1 3 (4)

16 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 41 The pressre variation is given by eqation (28). The resoltion process is identical to that sed for the downward flow, and now to the bondary condition y the pressre is P and concentration isc. For the U-tbe, we consider the vales of variables at the pflow colmn entrance eqal to the vales at the end of downflow colmn, at the same operating conditions. RESUTS AND DISCUSSION Axial Dispersion The axial mixing in bbble colmns is sally nknown, so it is better to determine it from experiments. A typical method of tracer injection can be applied and the response interpreted assming the axial dispersion model. From the model, it is possible to estimate the dimensionless Peclet nmber and the coefficients of axial dispersion which can then be sed in the determination of the mass transfer coefficients. Some of the estimated vales of Peclet nmber, Pe, are plotted in Fig. 5 as a fnction of ratio between volmetric gas flow rate and volmetric liqid flow rate,, for downflow mode of operation. The vales of the axial dispersion coefficients, D, calclated by the above mentioned procedre, are presented in Table 1. Some vales of D are plotted in Fig. 6 vs. for both downflow and pflow mode. Figre 5: Peclet nmber as fnction of the gas and liqid volmetric flow rate ratio and sperficial liqid velocity for downflow mode of operation.

17 42 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira Figre 6: iqid axial dispersion coefficient as fnction of the gas and liqid volmetric flow rate ratio and sperficial liqid velocity for downflow and pflow mode of operation. Under the applied experimental conditions, the vales of D are in the range of 41.3 x1-4 and 87. x1-4 m 2 s -1. For the same operating conditions, these vales are consistent with those provided by two correlations often cited in literatre: the correlation of Joshi [9] and correlation of Field and Davidson [1]. It can be seen in Fig. 5, which in the range of vales of gas and liqid flow rates tested, with the colmn operating nder bbble regime, Pe increases with and U, which means we move towards plg flow. The increase of Pe with the liqid flow rate is consistent with stdies by several athors for different diameter colmns, different ways of introdcing the gas stream and varios vales of flow of liqid and gas, whether the system is homogeneos, transition or heterogeneos. The increase of Pe with U can be explained by the fact that increasing the velocity of flid elements, the residence time in the colmn is redced. The increase of Pe with can be explained similarly: more bbbles in the colmn increases the effective velocity of the elements of flid, redcing their time of residence. The variation of Pe with (or gas sperficial velocity, G/S, since U is fixed) is in agreement with reslts obtained by athors sch as

18 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 43 Zahradníck and Fialová [2], when the bbble regime was homogeneos, as was the case in which most part of or experiments took place. In this regime, the bbbles have approximately eqal sizes, which make them move with similar velocities. Several athors [2-23] obtained an opposite reslt, i.e., Pe decreases with G/S when operating conditions in colmn originated transition or heterogeneos bbble regime. The type of regime is one of the factors with significant inflence on the liqid mixtre. As can be seen in Fig. 6, the variation of D with the liqid sperficial velocity and with the volmetric flow rate of gas/liqid is similar for downflow and pflow bbble colmns. Gas Absorption In the presented model, the ratio of the mass transfer coefficient and the initial bbbles radis, k /r, from eqation (24) can be obtained from the experimental data, being a fitting parameter of model. The crves that best approximate the experimental data (presented as C=C-C vs. x) are shown in Fig. 7 for U =.38 m/s and in Fig. 8 for U =.64 m/s, to flow downward and then pward in the U tbe. Figre 7: Comparison between the predicted liqid phase oxygen concentration profiles and the experimental data at U=.38 m/s for downflow and pflow bbble colmns of U tbe.

19 44 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira Figre 8: Comparison between the predicted liqid phase oxygen concentration profiles and the experimental data at U=.64 m/s for downflow and pflow bbble colmns of U tbe. The vales of k /r can be estimated from the evenberg-marqardt non-linear optimization scheme [24-26]. The estimated k /r vales obtained from this procedre are close, ranging from.11 and.14 s -1, with a mean vale of.12 s -1 (see Table 1). Figs. 7 and 8 show a good agreement between the theoretical and experimental points for varios conditions tested. Table 1: Experimental conditions, liqid axial dispersion coefficient and fitting parameter k /r (s - 1 ) U (m/s) (%) C (g m -3 ) P x1-4 (Pa) U (m s -1 ) D 1 4 (m 2 /s) k /r (s -1 ) Model withot Dispersion k /r (s -1 ) Model with Dispersion

20 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 45 Table 1: cont Mass transfer coefficient can be calclated for each operating condition throgh one of several correlations existing in literatre. We sggest and se in this stdy the Hgmark s correlation [27] becase it was specifically obtained with gasliqid bbble colmns operating whit air-water and oxygen-water: /3 (2 rk ) (2 r) UR (2 rg ) /3 D D D.116 (41) where D is the diffsivity of solte in the liqid, is the viscosity of the liqid and U R is the relative velocity of two flid phases (modls of difference between the real velocities of gas and liqid). This eqation can be rearranged, allowing explicit k /r for downward flow: P C C U UT 1 P P U U U.233 k D D g 1/3.35 r P C C 2 P C C 1.15 T r T r P P P P.779 (42)

21 46 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira We obtain an identical eqation for pward flow simply by replacing the term U- U by U+U. This eqation can be sed with the differential eqation of the model, eqation (24), implying that the fitting parameter is the radis of the bbble at tbe entrance, r. If this vale is known then the model predicts the concentration at any point in the colmn known initial vales of all variables. Analysing eqation (41), can be conclded that for high vales of the bbbles radis, the second term on the right side is dominant, so the vale of the transfer coefficient is approximately constant for a given operating condition. Ths, given the vales of the bbbles radis in the bbble colmns, with vales greater than.1 mm, all r vales are high and k is approximately constant. This is in agreement with the vales of k /r presented in Table 1. C (mg/) U=.3 m/s; =.15 r =1.5 mm C 1 (mg/) 8 6 U=.3 m/s; =.1 r =1.5 mm C (mg/) Sé k /r =.12 s -1 Sé k /r variable x (m) U=.8 m/s; =.15 r =1.5 mm C (mg/) 4 2 k /r =.12 s -1 k /r variable x (m) U=.8 m/s; =.3 r =1.5 mm 4 2 Série1 k /r =.12 s -1 Série2 k /r variable x (m) Figre 9: Comparison between predicted concentration vales obtained with k /r =.12 s -1 and variable k /r for a downflow 2m colmn at for operating conditions. Simlations with the model show that the concentration profile does not vary mch whether yo se the constant vale of.12 s -1 for k /r, whether to se variable vales along the colmn obtained with eqation (42), as shown in Fig. 9, 4 2 k /r =.12 s -1 k /r variable x (m)

22 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 47 and therefore the model is not very sensitive to variations in the k /r vale. Variation of concentration as a fnction of axial distance in a 2 m downflow colmn is obtained from the model for varios operating conditions, according to Fig. 9. It was sed in all simlations the following vales for the variables: C =, P =11325 Pa, U =.2 m/s, T= K, H=2.765x1 6 J/kg, =999.1 kg/m 3, =1.149x1-3 Pa s and D=1.88x1-9 m 2 /s. Under the experimental conditions tested, Fig. 1 shows, for for operating conditions of Fig. 9, that the parameter k /r and therefore the mass transfer coefficient k, have a small change along the colmn, being obtained with eqation (42) k /r vales near.12s -1. This vale can be sed in most practical sitations, except in the operating conditions in which the gas volmetric flow rate is very small. k /r ( s -1 ) x (m) U=.3 m/s; =.15 U=.3 m/s; =.1 U=.8 m/s; =.15 r =1.5 mm U=.8 m/s; =.3 Figre 1: Variation of k /r parameter with axial coordinate in downflow 2 m colmn for several operating conditions. CONCUSIONS Models are presented for mass transfer in bbble colmns with simltaneos flow of gas and liqid in downflow and pflow co-crrent bbbly regime. The models developed for steady state take into accont: the variation of pressre along the tbe, the variation of bbbles size and the variation of gas volmetric flow rate, either by pressre variation or by consmption of solte absorbed by the liqid. Models that inclde the axial dispersion are also presented.

23 48 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira The reslts obtained with the models are in good agreement with the experimental reslts obtained with the oxygen-water system. The parameter that reslts from the ratio of the mass transfer coefficient and the initial bbbles radis, and therefore the mass transfer coefficient, k, in most practical sitations of interest, have a small change with the bbbles radis along the colmn. Therefore we can se a constant vale of k /r =.12 s -1, mean vale obtained with the fit of the experimental data to the model and that is close of the vale obtained with the correlation of Hgmark. Under tested conditions, axial dispersion had no inflence on mass transfer process. CONFICT OF INTEREST None declare. ACKNOWEDGEMENT None declare. NOTATION a -interfacial area per nit of colmn volme, m -1 A b -area of a bbble, m 2 C -concentration of gas solte dissolved in liqid (liqid-phase oxygen concentration), kg m -3 C -concentration of gas solte dissolved in liqid at tbe entrance (x=), kg m - 3 C -solbility of gas solte in liqid, kg m -3 C A -tracer (NaCl) concentration in the liqid, kg m -3 d b -bbble diameter, m

24 Mass Transfer Models for Oxygen-Water STP Flows on Chemical and Biomedical Engineering 49 D -moleclar diffsion coefficient of gas in liqid, m 2 s -1 D -liqid axial dispersion coefficient, m 2 s -1 G -acceleration de to gravity, m s -2 G -gas volmetric flow rate, m 3 s -1 G -gas volmetric flow rate at tbe entrance (x=), m 3 s -1 H -Henry s law constant, J kg -1 k -liqid phase mass transfer coefficient, m s -1 -liqid volmetric flow rate, m 3 s -1 P P -total pressre, Pa -pressre at x=, Pa Pe -Peclet nmber, see eqation 3 R r -bbble radis, m -bbble radis at colmn entrance (x=), m -gas constant, J kg -1 K -1 S -colmn cross sectional area, m 2 T -temperatre, K U b -bbbles velocity, m s -1 U -liqid sperficial velocity, m s -1 U -rise velocity of bbbles in stagnant medim, m s -1 U R -relative velocity between the moving phases = [(G/S)-(/(1-)S)], m s -1

25 41 STP Flows on Chemical and Biomedical Engineering Garcia and Teixeira V b -bbble volme, m 3 x y -vertical (axial) coordinate for downflow system, m -vertical (axial) coordinate for pflow system, m -ratio of gas flow rate and liqid flow rate (=G /) -liqid viscosity, Pa s -density of the liqid, kg m -3 m -density of gas-liqid mixtre, kg m -3 REFERENCES [1] Boyd CE, Watten B. Aeration systems in aqacltre. Rev Aqat Sci 1989; 1(3): [2] Petit J. Water spply, treatment, and recycling in aqacltre. In Barnabe G editor. Aqacltre, volme 1, Ellis Horwood Series in Aqacltre and fisheries spport, Ellis Horwood 199; pp [3] Speece RE. Oxygen spplementation by U-Tbe to the Tombigbee river. Water Sci Technol 1996; 34(12): [4] Rostan M, iné A, Wable O. Modeling of vertical downward gas-liqid flow for the design of a new contactor. Chem Eng Sci 1992; 47(13): [5] Teixeira JAS. Oxigenação em aqicltra: o sistema do Tbo em U. PhD in Chemical Engineering. Faclty of Engineering: Porto University; 1998 (in Portgese). [6] Garcia VRR, 26. Oxigenação em Borblhadores Verticais e Inclinados. PhD in Chemical Engineering. Faclty of Engineering: Porto University; 26 (in Portgese). [7] Deckwer WD, Brckhart R, Zoll G. Mixing and mass transfer in tall bbble colmns. Chem Eng Sci 1974; 29(11): [8] Hikita H, Kikkawa H. iqid-phase mixing in bbble colmns: Effect of liqid properties. The Chem Eng J 1974; 8(3): [9] Joshi JB. Axial mixing in mltiphase contactors-a nified correlation. Trans Inst Chem Eng 198; 58(3): [1] Field RW, Davidson JF. Axial dispersion in bbble colmns. Trans Inst Chem Eng 198; 58: [11] Kawase Y, Moo-Yong M. iqid phase mixing in bbble colmns with Newtonian and non-newtonian flids. Chem Eng Sci 1986; 41(8): [12] evenspiel O, Smith WK. Notes on the diffsion-type model for longitdinal mixing of flids in flow. Chem Eng Sci 1957; 6(4-5): [13] evenspiel O, Ed. Chemical Reaction Engineering. 2 nd ed. New York: John Wiley & Sons 1972.

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