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Dissolution rate of spherical carbon dioxide bubbles in strong alkaline solutions
Chemical Engineering Science 55 (2000) 3907}3917
Dissolution rate of spherical carbon dioxide bubbles in strong alkaline solutions Fumio Takemura *, Yoichiro Matsumoto Institute of Environmental Studies, Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan Department of Mechanical Engineering, The University of Tokyo 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan Received 3 August 1999; received in revised form 4 November 1999; accepted 11 January 2000
Abstract The dissolution rate of single carbon dioxide (CO ) bubbles in a strong alkaline solution was investigated experimentally and numerically. We developed a system that uses a charged-coupled device (CCD) camera coupled with a microscope to track the rising bubble and we photographed the rising CO bubbles in 0.01}1 M sodium hydroxide (NaOH) solutions. From these photographs we measured the bubble size and the rising speed, and from this data we estimated the drag coe$cient, C , and the Sherwood number, Sh, " for CO bubbles dissolving in NaOH solutions with simultaneous chemical reactions. Assuming chemical equilibrium at the bubble gas}liquid interface, we also estimated the dissolution rate of bubbles in alkaline solutions using numerically estimated dissolution rates in water. Comparing the numerical and experimental results indicates that chemical equilibrium is not achieved at the bubble surface because the values of the calculated Sh were larger than the measured Sh. Next, we numerically estimated C and Sh " corresponding to the `stagnant cap modela by directly solving the Navier}Stokes and the convection}di!usion equations for a CO bubble dissolving in a strong alkaline solution with simultaneous chemical reactions. We assumed that chemical reactions near the bubble were nonequilibrium. We included the species source}sink terms for the chemical reactions in the convection}di!usion equation. We compared these results with the measured rising speed and dissolution rate. This comparison shows that the experimental and numerical results are in good agreement and that the dissolution rate with chemical reactions can be estimated within about 10% of measured values, even for nonequilibrium chemical reactions near the bubble. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Multiphase #ow; Mass transfer; Dissolution; Liquid-phase chemical reaction; Sodium hydroxide; Numerical analysis
1. Introduction The gas dissolution process of bubbles rising in a liquid is important in liquid-phase reactions, which occur in chemical and bioengineering applications. Absorption of carbon dioxide (CO ) gas into strong alkaline solutions is a typical example of dissolution combined with chemical reactions. In this process, absorbed CO changes "rst into bicarbonate ions and then into carbonate ions. Because the dissolution of CO is promoted by these chemical reactions, this process has been used for gas absorption technology and much research has been performed on absorption towers that use packed bed reactors (Hirose, Toda & Sato, 1974) and bubble
* Corresponding author. Tel.: #81-3-5841-8586; fax: #81-3-38180835. E-mail address: [email protected] (F. Takemura).
columns (Fleischer, Becker & Eigenberger, 1995; Rocha & Guedes De Carvalho, 1987). Pinsent, Pearson and Roughton (1956) estimated the rate constant of reactions here CO changes into bicarbonate ions in a 0.005}0.05 M sodium hydroxide (NaOH) solution. Fukunaka et al. (1989) used a nozzle to generate CO bubbles and measured the subsequent absorption rates of CO . They also modeled the chemical reactions during absorption and numerically estimated the absorption rate. Most researchers, however, only measured the total mass transfer coe$cient in multi-bubble systems, and few results on the dissolution process with chemical reactions of a single bubble have been made. With increasing water contamination, a bubble in water changes from behaving like a #uid sphere to behaving like a solid particle. Water contamination is therefore important in the gas dissolution process, because it in#uences both the rising speed and the dissolution rate. When rising in clean water, a bubble behaves like a #uid
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 2 2 - 1
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sphere (Duineveld, 1995). Conversely, when rising in contaminated water, a bubble behaves like a solid particle if the bubble is su$ciently a!ected by the contamination (Clift, Grace & Weber, 1978). Much research in this transient regime has been performed and is reviewed by Clift et al. (1978) and by Cuenot, Magnaudet and Spennato (1997). The `stagnant cap model,a shown in Fig. 1, has been proposed for explaining the behavior in the transition regime. In this model, contamination accumulates from the rear of the bubble, increasing the immobile area of the surface. This immobile area eventually covers the entire surface, and the bubble then behaves like a solid particle. Many researchers have used this model to analyze the e!ect of contamination on the rising speed of a bubble (Sadhal & Johnson, 1983; Fdhila & Duineveld, 1996). In particular, Cuenot et al. (1997) analyzed the transient behavior of bubble drag by considering the di!usion of surfactants in the water and adsorption and accumulation of surfactants at the bubble surface. They obtained the following results about the e!ect of the adsorbed surfactants on the bubble motion. (1) When the surfactants are adsorbed onto the surface a distribution of surface tension is generated, creating Marangoni convection, which alters the #ow "eld around the bubble. (2) Independent of the kind of surfactants used, the concentration of the surfactant on the surface reaches equilibrium when the tangential velocity at the surface becomes zero. (3) The steady solution of the simple Stagnant Cap Model shown in Fig. 1 is a reasonable representation of the unsteady solution including adsorption of surfactants. Takemura and Yabe (1999) developed an experimental system that uses a charged-coupled device (CCD) camera coupled with a microscope to track rising bubbles. By precisely measuring the bubble size and rising speed, they were
able to accurately estimate the drag coe$cient and the Sherwood number, Sh, for the dissolution rate of gas bubbles. By directly solving the coupled Navier}Stokes and convection}di!usion equations, they also numerically estimated the drag coe$cient, C , and Sh corre" sponding to the `stagnant cap modela. They showed that the experimental and numerical results are in good agreement and that the Stagnant Cap Model can explain the mechanism of the transient dissolution process when the bubble behavior changes from that of a #uid sphere to a solid particle. In this study, we measured the rising speed and dissolution rate of a single CO bubble in a strong alkaline solution. We photographed rising CO bubbles in 0.01}1 M NaOH solutions. From sequences of photographs, we measured the bubble size and rising speed. From the rising speed we estimated C , and from cha" nges in the bubble size we estimated Sh. Assuming that chemical equilibrium was attained at the gas}liquid interface, we estimated the dissolution rate of the bubbles in an alkaline solution by using the dissolution rate in water calculated by Takemura and Yabe (1999). A comparison of numerical and experimental results shows that chemical equilibrium is not achieved at the surface because the calculated Sh is larger than the measured Sh. For a CO bubble dissolving in a strong alkaline solu tion with chemical reactions, we estimated C and Sh " corresponding to the `stagnant cap modela by directly solving the Navier}Stokes and convection}di!usion equations. We included the chemical reaction term in the convection-di!usion equation and we assumed that reactions near the bubble surface are not in equilibrium. The measured and simulated drag coe$cients and dissolution rates are in good agreement, and the dissolution rate including chemical reactions can be estimated within an error of about 10%, even for nonequilibrium chemical reactions near the bubble.
2. Experimental apparatus and procedures
Fig. 1. Schematic of the stagnant cap model.
Fig. 2 shows a diagram of the apparatus we used to measure changes in bubble radius, R, and rising speed, ;. It consists of a test section, a bubble generator, an optical microscope, a CCD camera, a Z-axis stage, a stage controller, a video capture board, and a personal computer (PC) for controlling the entire system. The test section was a pipe 40 mm on each side and 500 mm long. The bubble generator consisted of a 0.1 mm thick stainlesssteel plate with a 40-lm diameter hole, a solenoid coil, a rod with a rubber stopper, and a spring to keep the hole closed under normal conditions. When an electric current is supplied to the coil, the rod is pulled down, and when the electric current is stopped, it rises. The bubble generator produced single bubbles of CO because the pressure inside the bubble generator at the bottom of the
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We made measurements under atmospheric pressure (101.0 kPa) and room temperature (253C). Under these conditions, the kinematic viscosity of water is 0.89 mm/s (Sengers & Watson, 1986), the di!usivity of CO in water is 1.85;10U mm/s (Himmelblau, 1956), the solubility (Henry's constant based on density) of CO in water is 65 MPa (Wilhelm, Battino & Wilcock, 1977), and the Schmidt number (Sc) is about 500. When an aqueous solution includes ions, the viscosity of the solution, k, and the di!usivity of a gas, D, in the solution diverges from that of pure water and can be expressed as (Funazukuri & Nishio, 1995) D "1.013;10\k\ . ¹
Fig. 2. Schematic of the experimental apparatus.
test section was held about 0.5 kPa higher than the liquid pressure at the bottom of the test section. This produced bubbles with 0.1(R(0.5 mm. The CCD camera that we used had a resolution of 640;480 pixels and was connected to an optical microscope. We adjusted the CCD camera to about 1.8 lm per pixel and took photographs at 120 frames per second. The camera was moved in the direction of bubble motion with a Z-axis stage, and we took photographs of the bubbles while moving the camera at a constant speed. The depth of the "eld of view of the microscope was about $50 lm and light was supplied from behind the bubble column. We used 0.01, 0.1, and 1 M aqueous NaOH solutions (WAKO Chemical Co.). The water used for the solutions had a speci"c resistance of 6.7 M) cm and the total concentration of ions was less than 0.05 ppm. When we introduced the solution into the test section, the impurity level of the solution increased due to exposure to the atmosphere. After the measurements, we measured the number of 1}45 lm diameter particles by using an automatic particle size analysis system (HIAC, PA-720). The solution introduced into the test section included about 400 particles/ml and no particles greater than 15 lm were observed. The quality of the solution was the same as that of the water used by Takemura and Yabe (1999) to measure the dissolution rate of CO bubbles. They showed that the contaminants attached to the interface do not restrict the dissolution of CO when the bubble rises in the water. Therefore, because our experimental conditions were similar to theirs, we conclude that contamination at the bubble surface did not prevent the dissolution of CO gas into the solution.
(1)
We measured k by using a rotating viscometer with double cylinders. For CO in 0.01 and 0.1 M NaOH solutions, we used the values of k and D for pure water because they di!ered from the true values by less than 2%. For k"1.20;10\ Pa s and for a 1 M solution, we estimated D"1.5;10\ mm/s from Eq. (1). From the measured liquid density of 1034 kg/m, we estimated l"1.16 mm/s and Sc"800. Henry's constant, H, is also a!ected by the existence of ions and can be estimated from the ionic strength (I) as H H" U , I"([Na>]#[OH\]), 10\IQ '
(2)
where the subscript w denotes the value of pure water and k "0.135 l/mol (see, Fukunaka et al., 1989). Q We used the following experimental procedure. We "rst cleaned the test section with pure water, degassed an NaOH solution, and then introduced it into the test section. We injected CO gas bubbles into the test sec tion and took photographs of the bubbles at 120 frames per second while moving the camera at a constant speed. We measured both R and ; using the observed bubble position and known camera speed. We measured R in every frame, and we calculated ; by measuring the change in position of the center of the bubble from frame to frame, calculating the relative speed from the movement of the center in two consecutive frames, and adding the camera speed to this value. Regarding the reproducibility of our measurements, because the NaOH solutions we used included unknown surfactants and particles, for similar size bubbles the dissolution rate and rising speed varied by about several percent from run to run. Therefore, it is di$cult to accurately conclude anything about the unsteady behavior of the #ow and concentration "elds. However, for a "xed stagnant cap angle we can assume that the #ow "eld around the bubble is steady (Cuenot et al., 1997). For comparisons of the measured steady #ow and concentration "elds, we do not need to consider the kind and/or concentration of the surfactants.
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mass, buoyancy, drag forces, and history force (Mei, Klausner & Lawrence, 1994) and is written as
1 d 4 4 1 poR; " noRg!C nR o;#F , (3) " &0 2 dt 3 3 2
Fig. 3. Photographs showing a rising bubble in (a) 0.01, (b) 0.1, and (c) 1 M sodium hydroxide solutions.
Fig. 3 shows typical photographs taken with the CCD camera, which show the dissolution process of the rising CO bubbles in the NaOH solution. The initial radius was 0.3 mm. In the 0.01 M solution, the initial ; and the Reynolds number, Re, were 89.7 mm/s and 60.2, respectively, in the 0.1 M solution, the initial ; and Re were 83.0 mm/s and 55.5, respectively, and in the 1 M solution, the initial ; and Re were 71.5 mm/s and 37.0, respectively. The di!erences in the rising speed were caused by di!erences in l and the amount of contaminants attached to the bubble surface. Fig. 3 also shows that as the concentration of NaOH in the solution increases, the dissolution rate increases due to chemical reactions. The pictures also demonstrate that the bubble rose vertically, because every photograph is in focus within a depth of "eld of 50 lm. Therefore, the #ow around the bubble retained axial symmetry under our experimental conditions. We also estimated the nonsphericity of the bubbles by the quantity 1!R /R ,
where R and R are the measured minimum and
maximum radii of the bubble, respectively. For R"0.3 mm, the non-sphericity attained a maximum value of 0.01 and could be regarded as negligibly small. Therefore we were able to assume that the bubbles were spherical under our experimental conditions.
3. Estimation of C and Sh " Generally the unsteady motion of a rising bubble can be calculated from the force balance including the added
where F denotes the history force. We calculated &0 C normalized by the Hadamard}Rebzcynski solution " (Clift et al., 1978) as C Re/16. " We considered the e!ect of the walls on the bubble rising speed. Clift et al. (1978) showed that for R(0.06 ¸ (¸ is the distance between the center of the bubble and the wall) the e!ect of the wall could be neglected. We calculated the position of the bubble from the bottom of the column and excluded the data that did not satisfy the condition of R(0.06 ¸. Takagi and Matsumoto (1996) numerically analyzed the motion of bubbles released into a quiescent liquid and initially at rest, and showed that for Re"50, the time, t , for the drag coe$cient to come within 5% of the drag coe$cient at steady state can be expressed as t "R/(5l). Their analysis was done for constant}radius bubbles. In our experiments, the bubble radius changed due to dissolution of CO into the liquid. Magnaudet and Legendre (1998) investigated the viscous drag force on a spherical bubble with a time-dependent radius and showed the existence of a history force caused by the radius change. However, because ratios of the rate of change in bubble radius to the rising velocity under our experimental conditions were much smaller (i.e. (0.01) than those considered by Magnaudet and Legendre, we did not consider the history force caused by the radius change. For our experimental conditions, t +20 ms. Therefore, we can neglect the history force if we exclude the data within 30 ms after bubble generation. Finally, we considered the order of the added mass term. For our experimental conditions, the added mass term was negative because the bubble radius and the rising speed decreased monotonically with time. Therefore, the measured rising speed was always greater than the terminal velocity. In this work we neglected the history force, so that from Eq. (3) we can express C normalized by the measured Re as " C Re R 2g 1 d; 3 dR " " ! ! . (4) 16 6l ; ; dt R dt
Note that the measured Re, used in Eq. (4), is greater than Re estimated from the terminal velocity. We found from our experimental results that the ratio of the added mass to the buoyancy terms was a maximum of 6% for dissolution, even in a 1 M NaOH solution. Because this value is relatively small, we assumed the velocity "eld to be at steady state. Therefore the normalized C esti" mated from Eq. (4) must be close to the steady-state C normalized by the measured Re. We used the value of " C calculated from Eq. (4) in our analysis. Using Eq. (4) "
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
and the error analysis of Benedict et al. (1985) for the measurements of R, ;, and k, we estimated the uncertainty of the normalized C to be about 5%. " Clift et al. (1978) also showed that when the concentration "eld near a bubble is changed in a stepwise manner, the time for Sh to approach 10% of the steady-state value of Sh, t , can be expressed as t "1.8R/;. Because t was about 10 ms for our experimental conditions, we also assumed the concentration "eld to be in steady state. Sh for the dissolution rate of a CO bubble dissolving in water can be estimated as follows (Takemura & Yabe, 1999): 2RRQ H Sh"! . D oR¹
(5)
In Eq. (5) we equate the partial pressure of CO inside the bubble with the total pressure, without accounting for the vapor pressure, because the di!usion between the vapor and the gas inside the bubble is not the limiting resistance for mass transfer inside the bubble. As shown in Eq. (5), to calculate Sh we must estimate R and the rate of change of radius, RQ . Although we can directly measure R, to determine RQ we must "t a function of time to the time histories of R and calculate its time di!erential. The time histories of R can be "tted by a second-order polynomial function. The correlation is better than 0.999 when the maximum deviation of the "tted values from the experimental value is less than 1.5%. We estimated the rate of radius change from the time di!erential of these "tted functions. Finally, we estimated Sh from Eq. (5). For 0.01 and 0.1 M solutions, the maximum uncertainty of Sh was 8% and 10 for 1 M solutions. The error analysis included uncertainties in R, RQ , the solubility, D, and other experimental factors. Table 1 shows typical ratios of RQ for the solution to RQ of pure water, measured by Takemura and Yabe (1999). The subscript w denotes pure water. We selected the data for the solutions with Re and C the same as for " pure water, because the bubble behavior in our experiments included the transient region from a #uid sphere to a solid particle. For the 0.01 and 0.1 M solutions, the rates of radii changes in the NaOH solutions were 110}120% and 170}180%, respectively, of the rate of
Table 1 Transport properties of rising bubbles [NaOH] (M)
Re
C Re/16 "
RQ /RQ U
0.01 0.01 0.1 0.1 1.0 1.0
10.5 13.8 10.3 15.0 10.1 20.2
2.6 2.7 2.6 2.9 2.5 2.8
1.0 1.16 1.73 1.77 9.9 5.8
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radius change in pure water. Therefore chemical reactions enhanced the dissolution rates in 0.01 and 0.1 M NaOH solutions by 10}20% and 70}80%, respectively, of the dissolution rate in pure water. For the 1 M solution, we cannot directly compare the value of the dissolution rate in the solution with that of the dissolution rate in pure water because both the di!usivity and the kinematic viscosity in the solution and in the pure water were di!erent. But by comparing with the dissolution rates in 0.01 and 0.1 M solutions, with and without chemical reactions, we showed that chemical reactions signi"cantly enhanced the dissolution rate, because the rates of radii change in the NaOH solutions were from 600 to 1000% of the rate of radius change in pure water.
4. Comparison with equilibrium model The simplest model of the dissolution of CO bubbles in NaOH solutions includes chemical equilibrium at the gas}liquid interface. Carbon dioxide dissolved into the NaOH solution partly ionizes to bicarbonate and carbonate ions. The chemical reaction and the ion equilibrium constants can be expressed as [HCO\][H>] CO #H O B HCO\#H>, K " , (6) [CO ] [CO\][H>] HCO\ B CO\#H>, K " , (7) [HCO\] H>#OH\ B H O, K "[H>][OH\], (8) U where the temperature dependence of K , K , and K can be expressed as (see Plummer & Busenberg, 1982) U log(K )"!356.31!0.061¹#21834.4/¹ #126.8log(¹)!1684915/¹, log(K )"!107.89!0.03253¹#5151.8/¹ #38.93log(¹)!563713.9/¹, K "10\ (at 253C). U The condition of electric neutrality in the solution is as follows: [H>]#[Na>]"[OH\]#[HCO\]#2[CO\]. (9) The saturated concentration of CO in the solution is calculated from Eq. (2), and the equilibrium concentration of each ion can then be calculated from Eqs. (6)}(9). On the other hand, steady-state mass conservation of CO , bicarbonate ions, and carbonate ions is expressed as ( uo z )c "D c !k , ( uo z )c "D c #k !k , ( uo z )c "D c #k ,
(10a) (10b) (10c)
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where subscripts 1}3 correspond to CO , bicarbonate ions, and carbonate ions, respectively, and k and k denote the source terms for CO and bicarbonate ions due to chemical reactions, respectively. The di!usivity of bicarbonate and carbonate ions may be di!erent from that of CO in the solution. However, Funazukuri and Nishio (1995) showed that when the water has the same viscosity as the solution, the di!usivity of CO in water is the same as that of bicarbonate and carbonate ions in NaOH solutions. Therefore we assumed that the diffusivities of bicarbonate and carbonate ions are the same as that of CO in the solution. By summing Eqs. (10a)}(10c), we can write the mass conservation for all of species (c "c #c #c ) without the source term due to chemical reactions as ( uo z )c "D c . (11) To solve Eq. (11), we must know c at the bubble surface. We can use the equilibrium concentrations of c at the bubble surface because we assume it is in equilibrium there. We numerically estimated C and Sh that correspond " to the stagnant cap model in the chemical equilibrium model. To accurately compare with experimental results, we "rst determined h , because the calculated drag coe$cients coincided with the drag coe$cients obtained from the experimental results. We then calculated Sh from this value for h . However, because this method requires large computational e!ort, for Re"10, 20, 30, 40, and 50, in the stagnant cap model we calculated the drag coe$cients and Sh every 153 from 0 to 1803. We then selected experimental data that was within 3% of Re"10, 20, 30, 40, or 50. For h '1203, Sh varied signi"cantly, although the drag coe$cient was nearly invariant. Therefore, we excluded the data that did not satisfy the condition of h '1203. When the measured and calculated drag coe$cients coincided, we used the calculated Sh. When the measured and calculated drag coe$cients did not coincide, we estimated Sh by interpolating the measured values. For 0.01 and 0.1 M solutions, we used Sh for pure water calculated by Takemura and Yabe (1999), because Sc in solution is the same as Sc in water. For a 1 M solution, because Sc"800, we calculated C and Sh in the same way as Takemura and " Yabe (1999). Because the mass conservation equation of c does not include the source term, the calculated Sh corresponding
Fig. 4. Normalized Sh as a function of Re in 0.01, 0.1, and 1 M sodium hydroxide solutions: (䢇) 0.01 M; (䊏) 0.1 M; (䉱) 1 M.
to c are on the basis of the equilibrium concentration of c at the bubble surface. On the other hand, when we use Eq. (5) and the concentration of CO at the bubble surface to calculate Sh for the experimental results, Sh corresponds to the mass transfer of CO . Therefore, to calculate Sh corresponding to [CO ] from Sh corre sponding to c , the calculated Sh must be multiplied by the ratio of the total equilibrium concentration of carbon to CO at the bubble surface. For a temperature of 253C and a CO pressure of 100.0 kPa, Table 2 shows the equilibrium concentrations of CO , bicarbonate ions, carbonate ions, pH, and the ratio between the c in equilibrium and the concentration of CO at the bubble surface. Fig. 4 shows Sh calculated assuming equilibrium concentrations, Sh , normalized by measured values of Sh, Sh . For 0.01 M solutions, the normalized value is close to 1, indicating that Sh can be estimated assuming an equilibrium concentration. However, the di!erence between the calculated and measured values of Sh increases with increasing NaOH concentration. This indicates that ion equilibrium is not maintained at higher solution concentrations. Table 2 shows that the pH value of the 1 M solution is 7.8 when CO is saturated at a pressure of 100.0 kPa. The assumption of the equilibrium condition at the surface implies that for a 1 M solution, which has a pH of 14, the pH must decrease to 7.8 on the inside
Table 2 Species concentration and pH at equilibrium [NaOH] (mol/l)
[CO ] (mol/l)
[HCO\] (mol/l)
[CO\] (mol/l)
pH
c /[CO ]
0.01 0.1 1.0
3.33;10\ 3.24;10\ 2.43;10\
0.01 0.1 0.994
0 3.15;10\ 3.12;10\
5.8 6.8 7.8
1.3 4.1 42.0
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
of the bubble surface. The data in Fig. 4 indicates that this does not occur, and that nonequilibrium chemical reactions must be considered. By comparing the equilibrium constant of Reactions (6) and (7), Pinsent et al. (1956) noted that the chemical reaction described by Reaction (6) governs the overall chemical reaction rate. They estimated the reaction-rate constant of Reaction (6) in NaOH solutions from 0.005 to 0.05 M, assuming the chemical equilibrium represented by Reaction (7). Therefore, to numerically estimate the dissolution rate, we cannot assume equilibrium conditions at the bubble surface, but we must account for nonequilibrium in Reaction (6).
5. Simulation of dissolution of NaOH solutions 5.1. Assumptions In this section we describe the assumptions used to formulate the governing equations. Our experimental results demonstrate that the #ow around the bubbles was axially symmetric and that the bubbles remained spherical. Therefore, we used twodimensional, axisymmetric coordinates to model the bubble motion. Because we excluded both the data corresponding to the initial, unsteady motion and the data a!ected by the wall, we assumed that the velocity and concentration "elds reached steady state. Our measured C indicates intermediate behavior, between values for " a #uid sphere and a solid particle. Takemura and Yabe (1999) estimated C and Sh corresponding to the stag" nant cap model and showed that the stagnant cap model can explain the transient behavior when the bubble behavior changes from that of a #uid sphere to that of a solid particle. Therefore, we used the stagnant cap model in our numerical analysis. Because our comparison between the experimental results and the equilibrium model indicate that we must consider nonequilibrium gas concentrations in the chemical reactions of Reaction (6), we used the reaction-rate constant that Pinsent et al. (1956) estimated in 0.005}0.05 M NaOH solutions by assuming the chemical equilibrium of Reaction (7). In our analysis, we did not consider the back reactions of Reaction (6) because bicarbonate ions change to carbonate ions instantaneously. 5.2. Governing equations All physical quantities were nondimensionalized through the following normalization: c u";uH, r"RrH, cH" , p"o;pH, c Q
(12)
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where c is the saturated concentration of CO at a Q pressure of 100.0 kPa. The continuity, Navier} Stokes, and the convection}di!usion equations can be written as div uo H"0, 2 ( uo H z ) uo H"! pH# uo H, Re 2 kR[OH\] ( uo H z ) cH" cH! cH, Pe ;
(13)
2 ( uo H z )(cH#cH#cH)" (cH#cH#cH), Pe where k is a reaction-rate constant. Pinsent et al. (1956) estimated the reaction-rate constant of Reaction (6) in 0.005}0.05 M NaOH solutions by assuming chemical equilibrium of Reaction (7). Using the data of Pinsent et al. (1956), Fukunaka et al. (1989) derived the following function of the ion strength CO #OH\PHCO\, k"13.635!2895/¹#0.2I!0.0182I.
(14)
We used the following boundary conditions in the stagnant cap model of Sadhal et al. (1983) and Fdhila et al. (1996) (see Fig. 1) uH"0 (rH"1), P uH"0 (rH"1, 0(h)h ), F 1 *uH * uH P #rH F "0 (rH"1, h )h)p), rH *h *rH rH
(15)
uH"cos h, uH"!sin h (rHPR). P F We used the following boundary conditions for the concentration: cH"1 (rH"1), cH"0 (rHPR), *(cH#cH#cH) *cH " (rH"1), *rH *rH
(16)
cH#cH#cH"0 (rHPR). Here, we assumed that among the species in solution, only CO could pass through the bubble interface, because other ions can only exist in the solution (see Fukunaka et al., 1989). Therefore, *(cH#cH)/*rH"0. We also assumed that the concentra tion of CO at the surface attains the equilibrium value calculated from Henry's constant (Eq. (2)). We used Reactions (8)}(10) to determine the concentration of each species.
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The drag coe$cient and Sh can be expressed as follows (Clift et al., 1978): C "!4 "
F
6.1. Comparison with measured concentration xelds around a bubble
pH cos h sin h dh Q
F 2 *uH F sin h dh Re *rH Q
#
#4
p
F
(17)
4 *uH P cos h sin h dh, !pH# Re *rH Q
Sh"!
p *cH sin h dh. *rH Q
For our measurements the thickness of the concentration boundary layer was much thinner than that of the momentum boundary layer because 500(Sc( 800. Therefore, for increasing Pe, to capture the progressively thinner boundary layers, we used a "ner grid and the number of grid points around the bubble was increased. We therefore transformed the Navier}Stokes and the convection}di!usion equations as follows: 1 rH" , (1!y)?
(18)
where
a"
6. Results and discussion
1 (Pe*100), a"1 (Pe(100, Re(50). Pe
We di!erentiated the Navier}Stokes and the convection}di!usion equations by using the K}K scheme (Kawamura & Kuwahara, 1984), which is a thirdorder upwind scheme, and solved it by using the Successive Over-Relaxation (SOR) method (Young, 1954; Frankel, 1950). We divided the calculation domain into 100 grid points in the radial direction and into 60 grid points in the tangential direction. Takemura and Yabe (1999) estimated the uncertainty of calculated values of C by comparing their values with " those of Cuenot et al. (1997), and con"rmed that Takemura's values are su$ciently accurate. Takemura and Yabe (1998a, b) also con"rmed that their model has acceptable accuracy for estimating the dissolution rate at Pe"10. The temperature at the surface may increase due to chemical reactions. We therefore solved the energy conservation equation accounting for the heat source due to chemical reactions and calculated the temperature "eld. The maximum temperature increase was about 0.13C. Therefore, we neglected the e!ect of the change in temperature.
Fig. 5 shows typical photographs (a) of a rising CO bubble in a 1 M NaOH solution. The direction of gravity is from the left to the right side of the image. The operating conditions were R"0.23 mm, ;"51.0 mm/s, Sc"800, Re"20, C Re/16"3.05, and Sh"760. " In Fig. 5, we observe a line and a shadow near the angle of 963. This shadow results from the density di!erence between the solution with and without CO and ions, because the dissolution rate of CO is relatively high in the 1 M NaOH solution. Generally, Sh for a slip condition is larger than Sh for a nonslip condition. This indicates that the thickness of the boundary layer at a slip surface is thinner than at a nonslip surface. Because Fig. 5 shows that the thickness of the boundary layer becomes thicker after an angle of 963, the condition of the surface may change at this point from a slip to a nonslip surface. We assumed that the starting point of this line indicates the position of the stagnant cap angle. Fig. 5b shows the calculated concentration "eld of c around the bubble when the stagnant cap angle is 963. The black part represents the high concentration region, and indicates that the concentration boundary layer becomes very thick near 963. Figs. 5c and d show the concentration "elds of c for a solid particle and for a #uid sphere at the same Re, respectively. Both images show that the changes of the concentration boundary layer are smooth. It is di$cult to explain this phenomenon as the e!ect of separation, because under these conditions no separation occurs around the bubble. Even when Re"50, where the separation occurs on the solid particles (Clift et al., 1978),
Fig. 5. (a) Photograph of a rising bubble in 1 M sodium hydroxide solution and for the same Re, calculated concentration "elds of c around the bubble for a (b) stagnant cap angle of 963, (c) a solid particle, and (d) a #uid sphere.
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
the change of the concentration boundary layer is smooth. Therefore, it is reasonable to explain this phenomenon as the change of the surface condition near 963. Moreover, the calculated C "2.94 and the calculated " Sh"780, and these values agree well with the measured values. Therefore it is reasonable to explain this phenomenon by the stagnant cap model and we regard the starting point of the line as the position of the stagnant cap angle. When we observe Fig. 5 in detail, we can understand that the wake region of the experimental result is wider than that of the numerical result. Cuenot et al. (1997) showed that the calculated wake region including the Marangoni convection becomes wider than the calculated wake region from the stagnant cap model. Our results support their conclusions. 6.2. Comparison with experimental measurements of dissolution rate Figs. 6}8 show Sh as a function of C Re/16 for 0.01, " 0.1, and 1 M solutions, respectively. When we solved the convection}di!usion equation, we used the same values of Sc as the measured values: Sc"500 for 0.01 and 0.1 M solutions and Sc"800 for 1 M solutions. In the calculation, we assumed nonequilibrium chemical reactions. We obtained the steady-state solutions of C and Sh " every 153 from 0 to 1803 in the stagnant cap and for "xed Re. Figs. 6}8 show the results for "xed values of Re (solid lines) and for h "60, 90, and 1203 (dotted lines). The symbols show the experimental data that was within 3% of Re"10, 20, 30, 40, or 50. We consider that the numerical results agree with the experimental results when Re, C Re/16, and Sh of the experimental and " numerical results are equal. Fig. 6 shows that the calculated Sh agrees with the experimental value for 0.01 M solutions. However,
Fig. 6. Sh as a function of normalized drag coe$cients for 0.01 M solutions.
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Fig. 7. Sh as a function of normalized drag coe$cients for 0.1 M solutions.
Fig. 8. Sh as a function of normalized drag coe$cients for 1 M solutions.
the results in Fig. 4 that Sh /Sh +1 indicate that the equilibrium model produces nearly the same Sh. Therefore, the nonequilibrium model is required for estimating Sh when Sh /Sh '1, which occurs under conditions of nonequilibrium chemical reactions. Fig. 7 shows that the numerical and experimental results agree well for 0.1 M solutions. Pinsent et al. (1956) estimated the chemical reaction-rate constant in the range of 0.005}0.05 M, however, his values can also be applied for 0.1 M solutions. These results show that our numerical model can explain the dissolution process when we consider nonequilibrium in the chemical reactions. Fig. 8 shows that the di!erence of the numerical and experimental results increases for 1 M solutions. This may be because the maximum uncertainties of C and Sh "
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F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
for 1 M solutions are larger than those for 0.01 and 0.1 M solutions. The di!erence may also originate from the inapplicability of the chemical reaction constant obtained by Pinsent et al. (1956) for the relatively high 1 M solutions, or the e!ect of the ion strength on the solubility may not be described precisely. However, when we measure the bubble radius and the rising speed, we can estimate Re and C with our numerical model, and we " can then estimate Sh within about 10%, even though there is little data for Re'40.
(3) Comparing the equilibrium model and experimental results indicates that chemical equilibrium is not achieved at the bubble surface because the calculated Sh were larger than the measured Sh. (4) The comparison between the results of the nonequilibrium chemical reaction model and the experimental results shows that the dissolution rate with chemical reactions can be estimated within about 10% of the measured values, even using the nonequilibrium chemical reaction model near the bubble surface.
7. Conclusions
Notation
The dissolution rates of single carbon dioxide (CO ) bubbles rising in a strong alkaline solution and undergoing simultaneous chemical reactions was investigated experimentally and numerically. We developed a system that uses a charged-coupled device (CCD) camera coupled with a microscope to track the rising bubble and we photographed the rising CO bubble dis solving in 0.01 to 1 M sodium hydroxide (NaOH) solutions. By using a computer algorithm to analyze and determine the bubble size and rising speed from the pictures taken with the CCD camera, we accurately estimated the drag coe$cient, C , and the Sherwood num" ber, Sh, for the dissolution rate of the CO bubbles. We also numerically estimated C and Sh corresponding to " the `stagnant cap modela by directly solving the Navier}Stokes and the convection}di!usion equations for a CO bubble dissolving in a strong alkaline solution with simultaneous chemical reactions. From a comparison of the numerical and experimental results we conclude that:
c D g H I p Pe R R Re r Sc ¹ t ; u y
(1) As the concentration of NaOH in the solution increases, the dissolution rate increases due to the chemical reactions ionizing carbon dioxide to either bicarbonate or carbonate ions. In particular, chemical reactions signi"cantly enhanced the dissolution rate for 1 M solutions, because the rates of radii change in 1 M solutions were from 600 to 1000% of the rate of radius change in pure water. (2) Because the dissolution rate of CO is relatively high in 1 M NaOH solutions, we observed that the boundary-layer thickness at certain points on the surface of the bubble was larger for bubbles with CO and ions, because of the larger density than for solutions without CO and ions. Assuming that this indicates the position of the stagnant cap angle, we numerically calculated the velocity and concentration "elds near the bubble. Because the calculated concentration "eld, drag coe$cient, and Sh coincide well with the measured values, this point can be regarded as the position of the stagnant cap angle.
k l h o
concentration, mol/l di!usivity of a gas in a liquid, mm/s acceleration due to gravity, mm/s Henry's constant based on density, Pa ionic strength, mol/l pressure, Pa Peclet number, 2RU/D bubble radius, mm gas constant of CO , J/kg/K Reynolds number, 2RU/l coordinate of radial direction, mm Schmidt number, l/D Temperature, K time, s rising speed, mm/s velocity, mm/s transformation coordinate used in the numerical calculation
Greek letters viscosity of liquid, Pa s kinematic viscosity of liquid, mm/s coordinate of tangential direction density of liquid, kg/m
Superscripts and subscripts s *
gas}liquid interface nondimensional
References Benedict, R. P. et al. (1985). Measurement uncertainty, ANSI/ASME PTC 19-1. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, drops and particles. New York: Academic Press. Cuenot, B., Magnaudet, J., & Spennato, B. (1997). The e!ects of slightly soluble surfactants on the #ow around a spherical bubble. Journal of Fluid Mechanics, 339, 25}53. Duineveld, P. C. (1995). The rise velocity and shape of bubbles in pure water at high Reynolds number. Journal of Fluid Mechanics, 292, 325}332.
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917 Fdhila, R. B., & Duineveld (1996). The e!ect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet number. The Physics of Fluids, 8, 310}321. Fleischer, C., Becker, S., & Eigenberger, G. (1995). Transient hydrodynamics, mass transfer, and reaction in bubble columns: CO absorp tion into NaOH solutions. Transactions of the Institute of Chemical Engineers, 73, 649}653. Frankel, S. P. (1950). Convergence rates of iterative treatments of partial di!erential equations. Mathematical Tables Aids Computers, 4, 65}75. Fukunaka, Y., Jiang, M.-F., Yamamoto, T., Asaki, Z., & Kondo, Y. (1989). Nonuniformity of NaOH concentration and e!ective bubble diameter in CO injection into aqueous NaOH solution. Metallurgi cal Transactions, 20B, 5}12. Funazukuri, T., & Nishio, M. (1995). Prediction of di!usion coe$cients of carbon dioxide in water and in aqueous electrolytic solutions. Kagaku Kogaku Ronbunshu, 21, 824}828. Himmelblau, D. M. (1956). Di!usion of dissolved gases in liquids. Chemical Reviews, 64, 527}550. Hirose, T., Toda, M., & Sato, Y. (1974). Liquid phase mass transfer in packed bed reactor with cocurrent gas}liquid down#ow. Journal of Chemical Engineering of Japan, 7, 187}192. Kawamura, T., & Kuwahara, K. (1984). Computation of high Reynolds number yow around a circular cylinder with surface roughness. AIAA paper 84-0340, January, 9}12. Magnaudet, J., & Legendre, D. (1998). The viscous drag force on a spherical bubble with a time-dependent radius. The Physics of Fluids, 10, 550}554. Mei, R., Klausner, J. F., & Lawrence, C. J. (1994). A note on the history force on a spherical bubble at "nite Reynolds number. The Physics of Fluids, 6, 418}420. Pinsent, B. R. W., Pearson, L., & Roughton, F. J. W. (1956). The kinetics of combination of carbon dioxide with hydroxide ions. Transactions of the Faraday Society., 52, 1512}1520.
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Plummer, L. N., & Busenberg, E. (1982). The solubilities of Calcite, Aragonite, and Vaterite in CO }H O solutions between 0 and 903C, and evaluation of the aqueous model for the system CaCO }CO } H O. Geochimica et Cosmochimica Acta, 46, 1011}1040. Rocha, F. A. N., & Guedes De Carvalho, J. R. F. (1987). Absorption during gas injection through a submerged nozzle Part 2. Interfacial areas. Chemical Engineering and Research Design, 65, 279}284. Sadhal, S. S., & Johnson, R. E. (1983). Stokes #ow past bubbles and drops partially coated with thin "lms. Journal of Fluid Mechanics, 126, 237}250. Sengers, J. V., & Watson, J. T. R. (1986). Improved international formulations for the viscosity and thermal conductivity of water substance. Journal of Physical and Chemical Reference Data, 15, 1291}1314. Takagi, S., & Matsumoto, Y. (1996). Force acting on a rising bubble in a quiescent liquid. In Crow et al., Proceedings of the ASME Summer Meeting on Numerical Methods for Multiphase Flow, vol. 1 (pp. 575}580). Takemura, F., & Yabe, A. (1998a). Gas dissolution process of spherical rising gas bubbles. Chemical Engineering Science, 53, 2691}2699. Takemura, F., & Yabe, A. (1998b). Erratum to &&Gas dissolution process of spherical rising gas bubbles''. Chemical Engineering Science, 54, 1159. Takemura, F., & Yabe, A. (1999). Rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water. Journal of Fluid Mechanics, 378, 319}334. Wilhelm, E., Battino, R., & Wilcock, R. J. (1977). Low-pressure solubility of gases in liquid water. Chemical Reviews, 77, 219}262. Young, D. (1954). Iterative methods for solving partial di!erential equations of elliptic type. Transactions of the American Mathematical Society, 76, 92}111.
Dissolution rate of spherical carbon dioxide bubbles in strong alkaline solutions Fumio Takemura *, Yoichiro Matsumoto Institute of Environmental Studies, Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan Department of Mechanical Engineering, The University of Tokyo 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan Received 3 August 1999; received in revised form 4 November 1999; accepted 11 January 2000
Abstract The dissolution rate of single carbon dioxide (CO ) bubbles in a strong alkaline solution was investigated experimentally and numerically. We developed a system that uses a charged-coupled device (CCD) camera coupled with a microscope to track the rising bubble and we photographed the rising CO bubbles in 0.01}1 M sodium hydroxide (NaOH) solutions. From these photographs we measured the bubble size and the rising speed, and from this data we estimated the drag coe$cient, C , and the Sherwood number, Sh, " for CO bubbles dissolving in NaOH solutions with simultaneous chemical reactions. Assuming chemical equilibrium at the bubble gas}liquid interface, we also estimated the dissolution rate of bubbles in alkaline solutions using numerically estimated dissolution rates in water. Comparing the numerical and experimental results indicates that chemical equilibrium is not achieved at the bubble surface because the values of the calculated Sh were larger than the measured Sh. Next, we numerically estimated C and Sh " corresponding to the `stagnant cap modela by directly solving the Navier}Stokes and the convection}di!usion equations for a CO bubble dissolving in a strong alkaline solution with simultaneous chemical reactions. We assumed that chemical reactions near the bubble were nonequilibrium. We included the species source}sink terms for the chemical reactions in the convection}di!usion equation. We compared these results with the measured rising speed and dissolution rate. This comparison shows that the experimental and numerical results are in good agreement and that the dissolution rate with chemical reactions can be estimated within about 10% of measured values, even for nonequilibrium chemical reactions near the bubble. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Multiphase #ow; Mass transfer; Dissolution; Liquid-phase chemical reaction; Sodium hydroxide; Numerical analysis
1. Introduction The gas dissolution process of bubbles rising in a liquid is important in liquid-phase reactions, which occur in chemical and bioengineering applications. Absorption of carbon dioxide (CO ) gas into strong alkaline solutions is a typical example of dissolution combined with chemical reactions. In this process, absorbed CO changes "rst into bicarbonate ions and then into carbonate ions. Because the dissolution of CO is promoted by these chemical reactions, this process has been used for gas absorption technology and much research has been performed on absorption towers that use packed bed reactors (Hirose, Toda & Sato, 1974) and bubble
* Corresponding author. Tel.: #81-3-5841-8586; fax: #81-3-38180835. E-mail address: [email protected] (F. Takemura).
columns (Fleischer, Becker & Eigenberger, 1995; Rocha & Guedes De Carvalho, 1987). Pinsent, Pearson and Roughton (1956) estimated the rate constant of reactions here CO changes into bicarbonate ions in a 0.005}0.05 M sodium hydroxide (NaOH) solution. Fukunaka et al. (1989) used a nozzle to generate CO bubbles and measured the subsequent absorption rates of CO . They also modeled the chemical reactions during absorption and numerically estimated the absorption rate. Most researchers, however, only measured the total mass transfer coe$cient in multi-bubble systems, and few results on the dissolution process with chemical reactions of a single bubble have been made. With increasing water contamination, a bubble in water changes from behaving like a #uid sphere to behaving like a solid particle. Water contamination is therefore important in the gas dissolution process, because it in#uences both the rising speed and the dissolution rate. When rising in clean water, a bubble behaves like a #uid
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 2 2 - 1
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sphere (Duineveld, 1995). Conversely, when rising in contaminated water, a bubble behaves like a solid particle if the bubble is su$ciently a!ected by the contamination (Clift, Grace & Weber, 1978). Much research in this transient regime has been performed and is reviewed by Clift et al. (1978) and by Cuenot, Magnaudet and Spennato (1997). The `stagnant cap model,a shown in Fig. 1, has been proposed for explaining the behavior in the transition regime. In this model, contamination accumulates from the rear of the bubble, increasing the immobile area of the surface. This immobile area eventually covers the entire surface, and the bubble then behaves like a solid particle. Many researchers have used this model to analyze the e!ect of contamination on the rising speed of a bubble (Sadhal & Johnson, 1983; Fdhila & Duineveld, 1996). In particular, Cuenot et al. (1997) analyzed the transient behavior of bubble drag by considering the di!usion of surfactants in the water and adsorption and accumulation of surfactants at the bubble surface. They obtained the following results about the e!ect of the adsorbed surfactants on the bubble motion. (1) When the surfactants are adsorbed onto the surface a distribution of surface tension is generated, creating Marangoni convection, which alters the #ow "eld around the bubble. (2) Independent of the kind of surfactants used, the concentration of the surfactant on the surface reaches equilibrium when the tangential velocity at the surface becomes zero. (3) The steady solution of the simple Stagnant Cap Model shown in Fig. 1 is a reasonable representation of the unsteady solution including adsorption of surfactants. Takemura and Yabe (1999) developed an experimental system that uses a charged-coupled device (CCD) camera coupled with a microscope to track rising bubbles. By precisely measuring the bubble size and rising speed, they were
able to accurately estimate the drag coe$cient and the Sherwood number, Sh, for the dissolution rate of gas bubbles. By directly solving the coupled Navier}Stokes and convection}di!usion equations, they also numerically estimated the drag coe$cient, C , and Sh corre" sponding to the `stagnant cap modela. They showed that the experimental and numerical results are in good agreement and that the Stagnant Cap Model can explain the mechanism of the transient dissolution process when the bubble behavior changes from that of a #uid sphere to a solid particle. In this study, we measured the rising speed and dissolution rate of a single CO bubble in a strong alkaline solution. We photographed rising CO bubbles in 0.01}1 M NaOH solutions. From sequences of photographs, we measured the bubble size and rising speed. From the rising speed we estimated C , and from cha" nges in the bubble size we estimated Sh. Assuming that chemical equilibrium was attained at the gas}liquid interface, we estimated the dissolution rate of the bubbles in an alkaline solution by using the dissolution rate in water calculated by Takemura and Yabe (1999). A comparison of numerical and experimental results shows that chemical equilibrium is not achieved at the surface because the calculated Sh is larger than the measured Sh. For a CO bubble dissolving in a strong alkaline solu tion with chemical reactions, we estimated C and Sh " corresponding to the `stagnant cap modela by directly solving the Navier}Stokes and convection}di!usion equations. We included the chemical reaction term in the convection-di!usion equation and we assumed that reactions near the bubble surface are not in equilibrium. The measured and simulated drag coe$cients and dissolution rates are in good agreement, and the dissolution rate including chemical reactions can be estimated within an error of about 10%, even for nonequilibrium chemical reactions near the bubble.
2. Experimental apparatus and procedures
Fig. 1. Schematic of the stagnant cap model.
Fig. 2 shows a diagram of the apparatus we used to measure changes in bubble radius, R, and rising speed, ;. It consists of a test section, a bubble generator, an optical microscope, a CCD camera, a Z-axis stage, a stage controller, a video capture board, and a personal computer (PC) for controlling the entire system. The test section was a pipe 40 mm on each side and 500 mm long. The bubble generator consisted of a 0.1 mm thick stainlesssteel plate with a 40-lm diameter hole, a solenoid coil, a rod with a rubber stopper, and a spring to keep the hole closed under normal conditions. When an electric current is supplied to the coil, the rod is pulled down, and when the electric current is stopped, it rises. The bubble generator produced single bubbles of CO because the pressure inside the bubble generator at the bottom of the
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
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We made measurements under atmospheric pressure (101.0 kPa) and room temperature (253C). Under these conditions, the kinematic viscosity of water is 0.89 mm/s (Sengers & Watson, 1986), the di!usivity of CO in water is 1.85;10U mm/s (Himmelblau, 1956), the solubility (Henry's constant based on density) of CO in water is 65 MPa (Wilhelm, Battino & Wilcock, 1977), and the Schmidt number (Sc) is about 500. When an aqueous solution includes ions, the viscosity of the solution, k, and the di!usivity of a gas, D, in the solution diverges from that of pure water and can be expressed as (Funazukuri & Nishio, 1995) D "1.013;10\k\ . ¹
Fig. 2. Schematic of the experimental apparatus.
test section was held about 0.5 kPa higher than the liquid pressure at the bottom of the test section. This produced bubbles with 0.1(R(0.5 mm. The CCD camera that we used had a resolution of 640;480 pixels and was connected to an optical microscope. We adjusted the CCD camera to about 1.8 lm per pixel and took photographs at 120 frames per second. The camera was moved in the direction of bubble motion with a Z-axis stage, and we took photographs of the bubbles while moving the camera at a constant speed. The depth of the "eld of view of the microscope was about $50 lm and light was supplied from behind the bubble column. We used 0.01, 0.1, and 1 M aqueous NaOH solutions (WAKO Chemical Co.). The water used for the solutions had a speci"c resistance of 6.7 M) cm and the total concentration of ions was less than 0.05 ppm. When we introduced the solution into the test section, the impurity level of the solution increased due to exposure to the atmosphere. After the measurements, we measured the number of 1}45 lm diameter particles by using an automatic particle size analysis system (HIAC, PA-720). The solution introduced into the test section included about 400 particles/ml and no particles greater than 15 lm were observed. The quality of the solution was the same as that of the water used by Takemura and Yabe (1999) to measure the dissolution rate of CO bubbles. They showed that the contaminants attached to the interface do not restrict the dissolution of CO when the bubble rises in the water. Therefore, because our experimental conditions were similar to theirs, we conclude that contamination at the bubble surface did not prevent the dissolution of CO gas into the solution.
(1)
We measured k by using a rotating viscometer with double cylinders. For CO in 0.01 and 0.1 M NaOH solutions, we used the values of k and D for pure water because they di!ered from the true values by less than 2%. For k"1.20;10\ Pa s and for a 1 M solution, we estimated D"1.5;10\ mm/s from Eq. (1). From the measured liquid density of 1034 kg/m, we estimated l"1.16 mm/s and Sc"800. Henry's constant, H, is also a!ected by the existence of ions and can be estimated from the ionic strength (I) as H H" U , I"([Na>]#[OH\]), 10\IQ '
(2)
where the subscript w denotes the value of pure water and k "0.135 l/mol (see, Fukunaka et al., 1989). Q We used the following experimental procedure. We "rst cleaned the test section with pure water, degassed an NaOH solution, and then introduced it into the test section. We injected CO gas bubbles into the test sec tion and took photographs of the bubbles at 120 frames per second while moving the camera at a constant speed. We measured both R and ; using the observed bubble position and known camera speed. We measured R in every frame, and we calculated ; by measuring the change in position of the center of the bubble from frame to frame, calculating the relative speed from the movement of the center in two consecutive frames, and adding the camera speed to this value. Regarding the reproducibility of our measurements, because the NaOH solutions we used included unknown surfactants and particles, for similar size bubbles the dissolution rate and rising speed varied by about several percent from run to run. Therefore, it is di$cult to accurately conclude anything about the unsteady behavior of the #ow and concentration "elds. However, for a "xed stagnant cap angle we can assume that the #ow "eld around the bubble is steady (Cuenot et al., 1997). For comparisons of the measured steady #ow and concentration "elds, we do not need to consider the kind and/or concentration of the surfactants.
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mass, buoyancy, drag forces, and history force (Mei, Klausner & Lawrence, 1994) and is written as
1 d 4 4 1 poR; " noRg!C nR o;#F , (3) " &0 2 dt 3 3 2
Fig. 3. Photographs showing a rising bubble in (a) 0.01, (b) 0.1, and (c) 1 M sodium hydroxide solutions.
Fig. 3 shows typical photographs taken with the CCD camera, which show the dissolution process of the rising CO bubbles in the NaOH solution. The initial radius was 0.3 mm. In the 0.01 M solution, the initial ; and the Reynolds number, Re, were 89.7 mm/s and 60.2, respectively, in the 0.1 M solution, the initial ; and Re were 83.0 mm/s and 55.5, respectively, and in the 1 M solution, the initial ; and Re were 71.5 mm/s and 37.0, respectively. The di!erences in the rising speed were caused by di!erences in l and the amount of contaminants attached to the bubble surface. Fig. 3 also shows that as the concentration of NaOH in the solution increases, the dissolution rate increases due to chemical reactions. The pictures also demonstrate that the bubble rose vertically, because every photograph is in focus within a depth of "eld of 50 lm. Therefore, the #ow around the bubble retained axial symmetry under our experimental conditions. We also estimated the nonsphericity of the bubbles by the quantity 1!R /R ,
where R and R are the measured minimum and
maximum radii of the bubble, respectively. For R"0.3 mm, the non-sphericity attained a maximum value of 0.01 and could be regarded as negligibly small. Therefore we were able to assume that the bubbles were spherical under our experimental conditions.
3. Estimation of C and Sh " Generally the unsteady motion of a rising bubble can be calculated from the force balance including the added
where F denotes the history force. We calculated &0 C normalized by the Hadamard}Rebzcynski solution " (Clift et al., 1978) as C Re/16. " We considered the e!ect of the walls on the bubble rising speed. Clift et al. (1978) showed that for R(0.06 ¸ (¸ is the distance between the center of the bubble and the wall) the e!ect of the wall could be neglected. We calculated the position of the bubble from the bottom of the column and excluded the data that did not satisfy the condition of R(0.06 ¸. Takagi and Matsumoto (1996) numerically analyzed the motion of bubbles released into a quiescent liquid and initially at rest, and showed that for Re"50, the time, t , for the drag coe$cient to come within 5% of the drag coe$cient at steady state can be expressed as t "R/(5l). Their analysis was done for constant}radius bubbles. In our experiments, the bubble radius changed due to dissolution of CO into the liquid. Magnaudet and Legendre (1998) investigated the viscous drag force on a spherical bubble with a time-dependent radius and showed the existence of a history force caused by the radius change. However, because ratios of the rate of change in bubble radius to the rising velocity under our experimental conditions were much smaller (i.e. (0.01) than those considered by Magnaudet and Legendre, we did not consider the history force caused by the radius change. For our experimental conditions, t +20 ms. Therefore, we can neglect the history force if we exclude the data within 30 ms after bubble generation. Finally, we considered the order of the added mass term. For our experimental conditions, the added mass term was negative because the bubble radius and the rising speed decreased monotonically with time. Therefore, the measured rising speed was always greater than the terminal velocity. In this work we neglected the history force, so that from Eq. (3) we can express C normalized by the measured Re as " C Re R 2g 1 d; 3 dR " " ! ! . (4) 16 6l ; ; dt R dt
Note that the measured Re, used in Eq. (4), is greater than Re estimated from the terminal velocity. We found from our experimental results that the ratio of the added mass to the buoyancy terms was a maximum of 6% for dissolution, even in a 1 M NaOH solution. Because this value is relatively small, we assumed the velocity "eld to be at steady state. Therefore the normalized C esti" mated from Eq. (4) must be close to the steady-state C normalized by the measured Re. We used the value of " C calculated from Eq. (4) in our analysis. Using Eq. (4) "
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
and the error analysis of Benedict et al. (1985) for the measurements of R, ;, and k, we estimated the uncertainty of the normalized C to be about 5%. " Clift et al. (1978) also showed that when the concentration "eld near a bubble is changed in a stepwise manner, the time for Sh to approach 10% of the steady-state value of Sh, t , can be expressed as t "1.8R/;. Because t was about 10 ms for our experimental conditions, we also assumed the concentration "eld to be in steady state. Sh for the dissolution rate of a CO bubble dissolving in water can be estimated as follows (Takemura & Yabe, 1999): 2RRQ H Sh"! . D oR¹
(5)
In Eq. (5) we equate the partial pressure of CO inside the bubble with the total pressure, without accounting for the vapor pressure, because the di!usion between the vapor and the gas inside the bubble is not the limiting resistance for mass transfer inside the bubble. As shown in Eq. (5), to calculate Sh we must estimate R and the rate of change of radius, RQ . Although we can directly measure R, to determine RQ we must "t a function of time to the time histories of R and calculate its time di!erential. The time histories of R can be "tted by a second-order polynomial function. The correlation is better than 0.999 when the maximum deviation of the "tted values from the experimental value is less than 1.5%. We estimated the rate of radius change from the time di!erential of these "tted functions. Finally, we estimated Sh from Eq. (5). For 0.01 and 0.1 M solutions, the maximum uncertainty of Sh was 8% and 10 for 1 M solutions. The error analysis included uncertainties in R, RQ , the solubility, D, and other experimental factors. Table 1 shows typical ratios of RQ for the solution to RQ of pure water, measured by Takemura and Yabe (1999). The subscript w denotes pure water. We selected the data for the solutions with Re and C the same as for " pure water, because the bubble behavior in our experiments included the transient region from a #uid sphere to a solid particle. For the 0.01 and 0.1 M solutions, the rates of radii changes in the NaOH solutions were 110}120% and 170}180%, respectively, of the rate of
Table 1 Transport properties of rising bubbles [NaOH] (M)
Re
C Re/16 "
RQ /RQ U
0.01 0.01 0.1 0.1 1.0 1.0
10.5 13.8 10.3 15.0 10.1 20.2
2.6 2.7 2.6 2.9 2.5 2.8
1.0 1.16 1.73 1.77 9.9 5.8
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radius change in pure water. Therefore chemical reactions enhanced the dissolution rates in 0.01 and 0.1 M NaOH solutions by 10}20% and 70}80%, respectively, of the dissolution rate in pure water. For the 1 M solution, we cannot directly compare the value of the dissolution rate in the solution with that of the dissolution rate in pure water because both the di!usivity and the kinematic viscosity in the solution and in the pure water were di!erent. But by comparing with the dissolution rates in 0.01 and 0.1 M solutions, with and without chemical reactions, we showed that chemical reactions signi"cantly enhanced the dissolution rate, because the rates of radii change in the NaOH solutions were from 600 to 1000% of the rate of radius change in pure water.
4. Comparison with equilibrium model The simplest model of the dissolution of CO bubbles in NaOH solutions includes chemical equilibrium at the gas}liquid interface. Carbon dioxide dissolved into the NaOH solution partly ionizes to bicarbonate and carbonate ions. The chemical reaction and the ion equilibrium constants can be expressed as [HCO\][H>] CO #H O B HCO\#H>, K " , (6) [CO ] [CO\][H>] HCO\ B CO\#H>, K " , (7) [HCO\] H>#OH\ B H O, K "[H>][OH\], (8) U where the temperature dependence of K , K , and K can be expressed as (see Plummer & Busenberg, 1982) U log(K )"!356.31!0.061¹#21834.4/¹ #126.8log(¹)!1684915/¹, log(K )"!107.89!0.03253¹#5151.8/¹ #38.93log(¹)!563713.9/¹, K "10\ (at 253C). U The condition of electric neutrality in the solution is as follows: [H>]#[Na>]"[OH\]#[HCO\]#2[CO\]. (9) The saturated concentration of CO in the solution is calculated from Eq. (2), and the equilibrium concentration of each ion can then be calculated from Eqs. (6)}(9). On the other hand, steady-state mass conservation of CO , bicarbonate ions, and carbonate ions is expressed as ( uo z )c "D c !k , ( uo z )c "D c #k !k , ( uo z )c "D c #k ,
(10a) (10b) (10c)
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F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
where subscripts 1}3 correspond to CO , bicarbonate ions, and carbonate ions, respectively, and k and k denote the source terms for CO and bicarbonate ions due to chemical reactions, respectively. The di!usivity of bicarbonate and carbonate ions may be di!erent from that of CO in the solution. However, Funazukuri and Nishio (1995) showed that when the water has the same viscosity as the solution, the di!usivity of CO in water is the same as that of bicarbonate and carbonate ions in NaOH solutions. Therefore we assumed that the diffusivities of bicarbonate and carbonate ions are the same as that of CO in the solution. By summing Eqs. (10a)}(10c), we can write the mass conservation for all of species (c "c #c #c ) without the source term due to chemical reactions as ( uo z )c "D c . (11) To solve Eq. (11), we must know c at the bubble surface. We can use the equilibrium concentrations of c at the bubble surface because we assume it is in equilibrium there. We numerically estimated C and Sh that correspond " to the stagnant cap model in the chemical equilibrium model. To accurately compare with experimental results, we "rst determined h , because the calculated drag coe$cients coincided with the drag coe$cients obtained from the experimental results. We then calculated Sh from this value for h . However, because this method requires large computational e!ort, for Re"10, 20, 30, 40, and 50, in the stagnant cap model we calculated the drag coe$cients and Sh every 153 from 0 to 1803. We then selected experimental data that was within 3% of Re"10, 20, 30, 40, or 50. For h '1203, Sh varied signi"cantly, although the drag coe$cient was nearly invariant. Therefore, we excluded the data that did not satisfy the condition of h '1203. When the measured and calculated drag coe$cients coincided, we used the calculated Sh. When the measured and calculated drag coe$cients did not coincide, we estimated Sh by interpolating the measured values. For 0.01 and 0.1 M solutions, we used Sh for pure water calculated by Takemura and Yabe (1999), because Sc in solution is the same as Sc in water. For a 1 M solution, because Sc"800, we calculated C and Sh in the same way as Takemura and " Yabe (1999). Because the mass conservation equation of c does not include the source term, the calculated Sh corresponding
Fig. 4. Normalized Sh as a function of Re in 0.01, 0.1, and 1 M sodium hydroxide solutions: (䢇) 0.01 M; (䊏) 0.1 M; (䉱) 1 M.
to c are on the basis of the equilibrium concentration of c at the bubble surface. On the other hand, when we use Eq. (5) and the concentration of CO at the bubble surface to calculate Sh for the experimental results, Sh corresponds to the mass transfer of CO . Therefore, to calculate Sh corresponding to [CO ] from Sh corre sponding to c , the calculated Sh must be multiplied by the ratio of the total equilibrium concentration of carbon to CO at the bubble surface. For a temperature of 253C and a CO pressure of 100.0 kPa, Table 2 shows the equilibrium concentrations of CO , bicarbonate ions, carbonate ions, pH, and the ratio between the c in equilibrium and the concentration of CO at the bubble surface. Fig. 4 shows Sh calculated assuming equilibrium concentrations, Sh , normalized by measured values of Sh, Sh . For 0.01 M solutions, the normalized value is close to 1, indicating that Sh can be estimated assuming an equilibrium concentration. However, the di!erence between the calculated and measured values of Sh increases with increasing NaOH concentration. This indicates that ion equilibrium is not maintained at higher solution concentrations. Table 2 shows that the pH value of the 1 M solution is 7.8 when CO is saturated at a pressure of 100.0 kPa. The assumption of the equilibrium condition at the surface implies that for a 1 M solution, which has a pH of 14, the pH must decrease to 7.8 on the inside
Table 2 Species concentration and pH at equilibrium [NaOH] (mol/l)
[CO ] (mol/l)
[HCO\] (mol/l)
[CO\] (mol/l)
pH
c /[CO ]
0.01 0.1 1.0
3.33;10\ 3.24;10\ 2.43;10\
0.01 0.1 0.994
0 3.15;10\ 3.12;10\
5.8 6.8 7.8
1.3 4.1 42.0
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
of the bubble surface. The data in Fig. 4 indicates that this does not occur, and that nonequilibrium chemical reactions must be considered. By comparing the equilibrium constant of Reactions (6) and (7), Pinsent et al. (1956) noted that the chemical reaction described by Reaction (6) governs the overall chemical reaction rate. They estimated the reaction-rate constant of Reaction (6) in NaOH solutions from 0.005 to 0.05 M, assuming the chemical equilibrium represented by Reaction (7). Therefore, to numerically estimate the dissolution rate, we cannot assume equilibrium conditions at the bubble surface, but we must account for nonequilibrium in Reaction (6).
5. Simulation of dissolution of NaOH solutions 5.1. Assumptions In this section we describe the assumptions used to formulate the governing equations. Our experimental results demonstrate that the #ow around the bubbles was axially symmetric and that the bubbles remained spherical. Therefore, we used twodimensional, axisymmetric coordinates to model the bubble motion. Because we excluded both the data corresponding to the initial, unsteady motion and the data a!ected by the wall, we assumed that the velocity and concentration "elds reached steady state. Our measured C indicates intermediate behavior, between values for " a #uid sphere and a solid particle. Takemura and Yabe (1999) estimated C and Sh corresponding to the stag" nant cap model and showed that the stagnant cap model can explain the transient behavior when the bubble behavior changes from that of a #uid sphere to that of a solid particle. Therefore, we used the stagnant cap model in our numerical analysis. Because our comparison between the experimental results and the equilibrium model indicate that we must consider nonequilibrium gas concentrations in the chemical reactions of Reaction (6), we used the reaction-rate constant that Pinsent et al. (1956) estimated in 0.005}0.05 M NaOH solutions by assuming the chemical equilibrium of Reaction (7). In our analysis, we did not consider the back reactions of Reaction (6) because bicarbonate ions change to carbonate ions instantaneously. 5.2. Governing equations All physical quantities were nondimensionalized through the following normalization: c u";uH, r"RrH, cH" , p"o;pH, c Q
(12)
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where c is the saturated concentration of CO at a Q pressure of 100.0 kPa. The continuity, Navier} Stokes, and the convection}di!usion equations can be written as div uo H"0, 2 ( uo H z ) uo H"! pH# uo H, Re 2 kR[OH\] ( uo H z ) cH" cH! cH, Pe ;
(13)
2 ( uo H z )(cH#cH#cH)" (cH#cH#cH), Pe where k is a reaction-rate constant. Pinsent et al. (1956) estimated the reaction-rate constant of Reaction (6) in 0.005}0.05 M NaOH solutions by assuming chemical equilibrium of Reaction (7). Using the data of Pinsent et al. (1956), Fukunaka et al. (1989) derived the following function of the ion strength CO #OH\PHCO\, k"13.635!2895/¹#0.2I!0.0182I.
(14)
We used the following boundary conditions in the stagnant cap model of Sadhal et al. (1983) and Fdhila et al. (1996) (see Fig. 1) uH"0 (rH"1), P uH"0 (rH"1, 0(h)h ), F 1 *uH * uH P #rH F "0 (rH"1, h )h)p), rH *h *rH rH
(15)
uH"cos h, uH"!sin h (rHPR). P F We used the following boundary conditions for the concentration: cH"1 (rH"1), cH"0 (rHPR), *(cH#cH#cH) *cH " (rH"1), *rH *rH
(16)
cH#cH#cH"0 (rHPR). Here, we assumed that among the species in solution, only CO could pass through the bubble interface, because other ions can only exist in the solution (see Fukunaka et al., 1989). Therefore, *(cH#cH)/*rH"0. We also assumed that the concentra tion of CO at the surface attains the equilibrium value calculated from Henry's constant (Eq. (2)). We used Reactions (8)}(10) to determine the concentration of each species.
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F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
The drag coe$cient and Sh can be expressed as follows (Clift et al., 1978): C "!4 "
F
6.1. Comparison with measured concentration xelds around a bubble
pH cos h sin h dh Q
F 2 *uH F sin h dh Re *rH Q
#
#4
p
F
(17)
4 *uH P cos h sin h dh, !pH# Re *rH Q
Sh"!
p *cH sin h dh. *rH Q
For our measurements the thickness of the concentration boundary layer was much thinner than that of the momentum boundary layer because 500(Sc( 800. Therefore, for increasing Pe, to capture the progressively thinner boundary layers, we used a "ner grid and the number of grid points around the bubble was increased. We therefore transformed the Navier}Stokes and the convection}di!usion equations as follows: 1 rH" , (1!y)?
(18)
where
a"
6. Results and discussion
1 (Pe*100), a"1 (Pe(100, Re(50). Pe
We di!erentiated the Navier}Stokes and the convection}di!usion equations by using the K}K scheme (Kawamura & Kuwahara, 1984), which is a thirdorder upwind scheme, and solved it by using the Successive Over-Relaxation (SOR) method (Young, 1954; Frankel, 1950). We divided the calculation domain into 100 grid points in the radial direction and into 60 grid points in the tangential direction. Takemura and Yabe (1999) estimated the uncertainty of calculated values of C by comparing their values with " those of Cuenot et al. (1997), and con"rmed that Takemura's values are su$ciently accurate. Takemura and Yabe (1998a, b) also con"rmed that their model has acceptable accuracy for estimating the dissolution rate at Pe"10. The temperature at the surface may increase due to chemical reactions. We therefore solved the energy conservation equation accounting for the heat source due to chemical reactions and calculated the temperature "eld. The maximum temperature increase was about 0.13C. Therefore, we neglected the e!ect of the change in temperature.
Fig. 5 shows typical photographs (a) of a rising CO bubble in a 1 M NaOH solution. The direction of gravity is from the left to the right side of the image. The operating conditions were R"0.23 mm, ;"51.0 mm/s, Sc"800, Re"20, C Re/16"3.05, and Sh"760. " In Fig. 5, we observe a line and a shadow near the angle of 963. This shadow results from the density di!erence between the solution with and without CO and ions, because the dissolution rate of CO is relatively high in the 1 M NaOH solution. Generally, Sh for a slip condition is larger than Sh for a nonslip condition. This indicates that the thickness of the boundary layer at a slip surface is thinner than at a nonslip surface. Because Fig. 5 shows that the thickness of the boundary layer becomes thicker after an angle of 963, the condition of the surface may change at this point from a slip to a nonslip surface. We assumed that the starting point of this line indicates the position of the stagnant cap angle. Fig. 5b shows the calculated concentration "eld of c around the bubble when the stagnant cap angle is 963. The black part represents the high concentration region, and indicates that the concentration boundary layer becomes very thick near 963. Figs. 5c and d show the concentration "elds of c for a solid particle and for a #uid sphere at the same Re, respectively. Both images show that the changes of the concentration boundary layer are smooth. It is di$cult to explain this phenomenon as the e!ect of separation, because under these conditions no separation occurs around the bubble. Even when Re"50, where the separation occurs on the solid particles (Clift et al., 1978),
Fig. 5. (a) Photograph of a rising bubble in 1 M sodium hydroxide solution and for the same Re, calculated concentration "elds of c around the bubble for a (b) stagnant cap angle of 963, (c) a solid particle, and (d) a #uid sphere.
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
the change of the concentration boundary layer is smooth. Therefore, it is reasonable to explain this phenomenon as the change of the surface condition near 963. Moreover, the calculated C "2.94 and the calculated " Sh"780, and these values agree well with the measured values. Therefore it is reasonable to explain this phenomenon by the stagnant cap model and we regard the starting point of the line as the position of the stagnant cap angle. When we observe Fig. 5 in detail, we can understand that the wake region of the experimental result is wider than that of the numerical result. Cuenot et al. (1997) showed that the calculated wake region including the Marangoni convection becomes wider than the calculated wake region from the stagnant cap model. Our results support their conclusions. 6.2. Comparison with experimental measurements of dissolution rate Figs. 6}8 show Sh as a function of C Re/16 for 0.01, " 0.1, and 1 M solutions, respectively. When we solved the convection}di!usion equation, we used the same values of Sc as the measured values: Sc"500 for 0.01 and 0.1 M solutions and Sc"800 for 1 M solutions. In the calculation, we assumed nonequilibrium chemical reactions. We obtained the steady-state solutions of C and Sh " every 153 from 0 to 1803 in the stagnant cap and for "xed Re. Figs. 6}8 show the results for "xed values of Re (solid lines) and for h "60, 90, and 1203 (dotted lines). The symbols show the experimental data that was within 3% of Re"10, 20, 30, 40, or 50. We consider that the numerical results agree with the experimental results when Re, C Re/16, and Sh of the experimental and " numerical results are equal. Fig. 6 shows that the calculated Sh agrees with the experimental value for 0.01 M solutions. However,
Fig. 6. Sh as a function of normalized drag coe$cients for 0.01 M solutions.
3915
Fig. 7. Sh as a function of normalized drag coe$cients for 0.1 M solutions.
Fig. 8. Sh as a function of normalized drag coe$cients for 1 M solutions.
the results in Fig. 4 that Sh /Sh +1 indicate that the equilibrium model produces nearly the same Sh. Therefore, the nonequilibrium model is required for estimating Sh when Sh /Sh '1, which occurs under conditions of nonequilibrium chemical reactions. Fig. 7 shows that the numerical and experimental results agree well for 0.1 M solutions. Pinsent et al. (1956) estimated the chemical reaction-rate constant in the range of 0.005}0.05 M, however, his values can also be applied for 0.1 M solutions. These results show that our numerical model can explain the dissolution process when we consider nonequilibrium in the chemical reactions. Fig. 8 shows that the di!erence of the numerical and experimental results increases for 1 M solutions. This may be because the maximum uncertainties of C and Sh "
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F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917
for 1 M solutions are larger than those for 0.01 and 0.1 M solutions. The di!erence may also originate from the inapplicability of the chemical reaction constant obtained by Pinsent et al. (1956) for the relatively high 1 M solutions, or the e!ect of the ion strength on the solubility may not be described precisely. However, when we measure the bubble radius and the rising speed, we can estimate Re and C with our numerical model, and we " can then estimate Sh within about 10%, even though there is little data for Re'40.
(3) Comparing the equilibrium model and experimental results indicates that chemical equilibrium is not achieved at the bubble surface because the calculated Sh were larger than the measured Sh. (4) The comparison between the results of the nonequilibrium chemical reaction model and the experimental results shows that the dissolution rate with chemical reactions can be estimated within about 10% of the measured values, even using the nonequilibrium chemical reaction model near the bubble surface.
7. Conclusions
Notation
The dissolution rates of single carbon dioxide (CO ) bubbles rising in a strong alkaline solution and undergoing simultaneous chemical reactions was investigated experimentally and numerically. We developed a system that uses a charged-coupled device (CCD) camera coupled with a microscope to track the rising bubble and we photographed the rising CO bubble dis solving in 0.01 to 1 M sodium hydroxide (NaOH) solutions. By using a computer algorithm to analyze and determine the bubble size and rising speed from the pictures taken with the CCD camera, we accurately estimated the drag coe$cient, C , and the Sherwood num" ber, Sh, for the dissolution rate of the CO bubbles. We also numerically estimated C and Sh corresponding to " the `stagnant cap modela by directly solving the Navier}Stokes and the convection}di!usion equations for a CO bubble dissolving in a strong alkaline solution with simultaneous chemical reactions. From a comparison of the numerical and experimental results we conclude that:
c D g H I p Pe R R Re r Sc ¹ t ; u y
(1) As the concentration of NaOH in the solution increases, the dissolution rate increases due to the chemical reactions ionizing carbon dioxide to either bicarbonate or carbonate ions. In particular, chemical reactions signi"cantly enhanced the dissolution rate for 1 M solutions, because the rates of radii change in 1 M solutions were from 600 to 1000% of the rate of radius change in pure water. (2) Because the dissolution rate of CO is relatively high in 1 M NaOH solutions, we observed that the boundary-layer thickness at certain points on the surface of the bubble was larger for bubbles with CO and ions, because of the larger density than for solutions without CO and ions. Assuming that this indicates the position of the stagnant cap angle, we numerically calculated the velocity and concentration "elds near the bubble. Because the calculated concentration "eld, drag coe$cient, and Sh coincide well with the measured values, this point can be regarded as the position of the stagnant cap angle.
k l h o
concentration, mol/l di!usivity of a gas in a liquid, mm/s acceleration due to gravity, mm/s Henry's constant based on density, Pa ionic strength, mol/l pressure, Pa Peclet number, 2RU/D bubble radius, mm gas constant of CO , J/kg/K Reynolds number, 2RU/l coordinate of radial direction, mm Schmidt number, l/D Temperature, K time, s rising speed, mm/s velocity, mm/s transformation coordinate used in the numerical calculation
Greek letters viscosity of liquid, Pa s kinematic viscosity of liquid, mm/s coordinate of tangential direction density of liquid, kg/m
Superscripts and subscripts s *
gas}liquid interface nondimensional
References Benedict, R. P. et al. (1985). Measurement uncertainty, ANSI/ASME PTC 19-1. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, drops and particles. New York: Academic Press. Cuenot, B., Magnaudet, J., & Spennato, B. (1997). The e!ects of slightly soluble surfactants on the #ow around a spherical bubble. Journal of Fluid Mechanics, 339, 25}53. Duineveld, P. C. (1995). The rise velocity and shape of bubbles in pure water at high Reynolds number. Journal of Fluid Mechanics, 292, 325}332.
F. Takemura, Y. Matsumoto / Chemical Engineering Science 55 (2000) 3907}3917 Fdhila, R. B., & Duineveld (1996). The e!ect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet number. The Physics of Fluids, 8, 310}321. Fleischer, C., Becker, S., & Eigenberger, G. (1995). Transient hydrodynamics, mass transfer, and reaction in bubble columns: CO absorp tion into NaOH solutions. Transactions of the Institute of Chemical Engineers, 73, 649}653. Frankel, S. P. (1950). Convergence rates of iterative treatments of partial di!erential equations. Mathematical Tables Aids Computers, 4, 65}75. Fukunaka, Y., Jiang, M.-F., Yamamoto, T., Asaki, Z., & Kondo, Y. (1989). Nonuniformity of NaOH concentration and e!ective bubble diameter in CO injection into aqueous NaOH solution. Metallurgi cal Transactions, 20B, 5}12. Funazukuri, T., & Nishio, M. (1995). Prediction of di!usion coe$cients of carbon dioxide in water and in aqueous electrolytic solutions. Kagaku Kogaku Ronbunshu, 21, 824}828. Himmelblau, D. M. (1956). Di!usion of dissolved gases in liquids. Chemical Reviews, 64, 527}550. Hirose, T., Toda, M., & Sato, Y. (1974). Liquid phase mass transfer in packed bed reactor with cocurrent gas}liquid down#ow. Journal of Chemical Engineering of Japan, 7, 187}192. Kawamura, T., & Kuwahara, K. (1984). Computation of high Reynolds number yow around a circular cylinder with surface roughness. AIAA paper 84-0340, January, 9}12. Magnaudet, J., & Legendre, D. (1998). The viscous drag force on a spherical bubble with a time-dependent radius. The Physics of Fluids, 10, 550}554. Mei, R., Klausner, J. F., & Lawrence, C. J. (1994). A note on the history force on a spherical bubble at "nite Reynolds number. The Physics of Fluids, 6, 418}420. Pinsent, B. R. W., Pearson, L., & Roughton, F. J. W. (1956). The kinetics of combination of carbon dioxide with hydroxide ions. Transactions of the Faraday Society., 52, 1512}1520.
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Plummer, L. N., & Busenberg, E. (1982). The solubilities of Calcite, Aragonite, and Vaterite in CO }H O solutions between 0 and 903C, and evaluation of the aqueous model for the system CaCO }CO } H O. Geochimica et Cosmochimica Acta, 46, 1011}1040. Rocha, F. A. N., & Guedes De Carvalho, J. R. F. (1987). Absorption during gas injection through a submerged nozzle Part 2. Interfacial areas. Chemical Engineering and Research Design, 65, 279}284. Sadhal, S. S., & Johnson, R. E. (1983). Stokes #ow past bubbles and drops partially coated with thin "lms. Journal of Fluid Mechanics, 126, 237}250. Sengers, J. V., & Watson, J. T. R. (1986). Improved international formulations for the viscosity and thermal conductivity of water substance. Journal of Physical and Chemical Reference Data, 15, 1291}1314. Takagi, S., & Matsumoto, Y. (1996). Force acting on a rising bubble in a quiescent liquid. In Crow et al., Proceedings of the ASME Summer Meeting on Numerical Methods for Multiphase Flow, vol. 1 (pp. 575}580). Takemura, F., & Yabe, A. (1998a). Gas dissolution process of spherical rising gas bubbles. Chemical Engineering Science, 53, 2691}2699. Takemura, F., & Yabe, A. (1998b). Erratum to &&Gas dissolution process of spherical rising gas bubbles''. Chemical Engineering Science, 54, 1159. Takemura, F., & Yabe, A. (1999). Rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water. Journal of Fluid Mechanics, 378, 319}334. Wilhelm, E., Battino, R., & Wilcock, R. J. (1977). Low-pressure solubility of gases in liquid water. Chemical Reviews, 77, 219}262. Young, D. (1954). Iterative methods for solving partial di!erential equations of elliptic type. Transactions of the American Mathematical Society, 76, 92}111.