Characterization of Transient CO 2 Transport in Two Convecting Aerosol Droplets in Tandem

A study of carbon dioxide absorbed by atmospheric aerosol droplets can account for the carbon cycle in the atmosphere in part. To figure out the microphysics of carbon dioxide transport in the atmosphere, a numerical method is developed to predict the transient absorption processes of CO2 by a pair of convecting droplets in tandem. Particular attention is paid to the mass transport influenced by the droplet-droplet interaction. The governing equations include the continuity and momentum equations in gas phase, and the continuity, momentum, and species equations in the liquid phase. Two important parameters of droplet spacing (0.5 and 3 droplet diameter) and Reynolds number (0.1, 1, and 10) are considered. The results show that radial diffusion dominates the H2CO3 transport in the droplets at Re = 0.1, whereas internal circulation plays an important role in moving the solute at Re = 10. As a result, the absorption rates of the two droplets are close to each other, regardless of what the droplet spacing is. For the case of Re = 1, the diffusion and the internal circulation are in a comparable state. The absorption rates of the droplets are affected by the droplet spacing to a small extent. The absorption rate of the downstream droplet is always lower than that of the upstream one, resulting from the disturbance of flow field around the former.


INTRODUCTION
Over the past several decades, burning fossil fuels, such as coal, oil, and natural gas, has become the most important way to gain heat and power.However, large amounts of greenhouse gases (GHGs) were produced and released to the atmosphere.The emissions of GHGs, especially for CO 2 , due to anthropogenic activities are the primary reason causing the global warming.The amount of CO 2 emission from electricity sectors in 2010 is about 65.7% of total GHG emissions (Liu et al., 2012).The total anthropogenic CO 2 emissions in 2011 were about 9.1 Gt/yr (Meyssignac and Cazenave, 2012) in which around 45% of CO 2 was retained in the atmosphere (Durand et al., 2011).With the growth of CO 2 concentration in the atmosphere, the global warming is getting worse (and this results in sea warming and land ice loss (Hansen et al., 2006;Vermeer and Rahmstorf, 2009).For example, the ice sheets of Antarctica and Greenland are losing rapidly and it is estimated that the sea level will rise about 56 cm by 2100 (Rignot et al., 2011).If the global mean temperature increases over 1.5-2.5°C,20-30% of animal and plant species are likely to be extinct (IPCC, 2007).For these reasons, the investigation of atmospheric carbon dioxide has received a great deal of attention in the last decade.
The carbon dioxide in the atmosphere plays an important role in the global carbon cycle and the carbon cycle relies on several sources and sink processes.These processes could be divided into short-and long-term time scales (Boucot and Gray, 2001).The short-term time processes include photosynthesis, respiration, air-sea exchange of carbon dioxide, and humus accumulation in soils (Berner, 2003).The exchange of carbon between active receptacles occurs at long-term time scales (Serrano-Ortiz et al., 2010).In the global carbon cycle, there are three active receptacles; they are the atmosphere, the ocean, and the terrestrial system including all sorts of stocks, such as forests and the organic carbon in soils (Schimel, 1995;Falkowski et al., 2000).In the three reservoirs, the ocean keeps the largest amount of carbon.In spite of the fact that the atmosphere holds the smallest amount of carbon, it plays the most important role as a conduit between the others in the carbon cycle (Post et al., 1990).Consequently, a number of investigators have focused their attention upon the interaction of carbon cycle between the atmosphere and the ocean or the terrestrial system (Franck et al., 1999;Schlesinger and Andrews, 2000).In addition, some researchers put their emphasis on the carbonaceous species in the atmosphere during the whole carbon cycle process.For example, Meinshausen et al. (2011) used an updated simple carbon cycle-climate model (MAGICC6) to predict the atmospheric concentrations of CO 2 in 2100, revealing that it will be in the range of 732 to 1025 ppm.Broström (2000) numerically explored the annual cycles for the airsea exchange of CO 2 and reported that the air-sea flux of CO 2 was about 0.4 Gt/yr for the area north of 30°N.
Carbon dioxide absorption by atmospheric aerosol droplets, such as cloud droplets, fog droplets, and raindrops, plays a notable role in carbon cycle in the atmosphere, so the absorption mechanisms of gas species by the aerosol droplets have attracted much attention by researchers.Chen and Lu (2003) explored the microphysics of atmospheric carbon dioxide uptake by a cloud droplet containing a solid nucleus at low Reynolds numbers.Their results indicated that the mass transfer through diffusion became significant with increasing the solid particle size, and the absorption rate of the aerosol droplet increased when the nucleus size rose.Ma et al. (2005) developed an equilibrium model and a non-equilibrium one at the gas-liquid interface to predict the absorption time for CO 2 uptake by a droplet.They concluded that the absorption time of the droplet under the equilibrium theory was shorter than that under the nonequilibrium model.Elperin et al. (2010) analyzed the nonisothermal absorption of CO 2 by a raindrop with internal circulation where the heat and mass transfer were simultaneously considered.Their predictions indicated that the non-uniform vertical temperature distribution in the atmosphere changed the rate of gas absorption by the raindrop during its fall.When one is concerned with solute movement in a convecting droplet, the mass transport is simultaneously contributed by mass diffusion and internal circulation (Chen et al., 2012).The mass diffusion is governed by concentration gradient (Fick's law), whereas the streamlines transport the solute when the internal circulation is involved.
From the above literature review, it is figured out that the topics of CO 2 and other gas species absorbed by single droplets have been studied extensively (Adewuyi and Carmichael, 1982;Ma et al., 2005;Chen, 2006;Elperin et al., 2010;Chen et al., 2011b).In fact, CO 2 uptake by liquid absorbents also plays an important role in carbon capture and storage (Yu et al., 2012).When one is concerned with CO 2 absorption by droplets in clouds or fogs, the atmospheric droplets are usually in a cluster.However, to the authors' knowledge, CO 2 absorption by multiple aerosol droplets in a convective flow has not been studied yet.Therefore, this study is intended to develop a numerical method to explore the CO 2 transport process in a pair of aerosol droplets in tandem.From the viewpoint of fluid dynamics, the flow field of downstream droplet will be disturbed by the upstream droplet, thereby affecting the mass transfer process of the trailing droplet (Rowe and Henwood, 1961;Tal et al., 1984, Baz-Rodríguez et al., 2012).The CO 2 transport processes in the two aerosol droplets will be discussed in detail.

Physical Description and Governing Equations
The schematic of CO 2 uptake by two convecting water droplets in tandem is shown in Fig. 1(a).The two droplets are stationary initially.At a certain moment, an air stream containing CO 2 flows through the droplets.The shear stress at the interfaces between the gas phase and liquid phase induces the internal circulations inside the droplets.Meanwhile, CO 2 physically dissolves in the droplets.The CO 2 absorption can be described by the following equation (Cents et al., 2005): To simplify the physical problem, the following assumptions are adopted: (1) the flow fields are twodimensional and symmetric along the droplet centerline; (2) the flows in the two phases are laminar and incompressible; (3) the geometry of the droplets remains spherical in the absorption process (Pruppacher et al., 1970); (4) the aqueous diffusion abides by Fick's law; (5) the CO 2 concentration at the gas-liquid interfaces is uniform because of its low mass diffusion number (Chen, 2002); (6) the physical properties of fluids are constants; and (7) the mass transfer processes are isothermal (at 298 K).

Governing Equations as well as Initial and Boundary Conditions
According to the assumption of uniform H 2 CO 3 concentration at the interfaces, the governing equations in the gas phase include the continuity and momentum equations.In the liquid phase, the species equation is also considered to account for the absorption and diffusion of H 2 CO 3 in the droplets.The equations are expressed as the following.

Gas Phase Governing Equations Continuity equation:
Liquid Phase Governing Equations Continuity equation: Species equation:

Initial Conditions
Considering the initial conditions, when the droplets expose to a uniform flow suddenly, the flow velocities, pressure, and H 2 CO 3 concentration in the liquid phase are given as follows.

Gas phase
Liquid phase

Boundary Conditions
The boundary conditions consist of the upstream inflow, downstream outflow, droplet surfaces, and axis of symmetry, as shown in Fig. 1

(b). They are written by
Upstream inflow: Downstream outflow: Droplet surface: Axis of symmetry:

Physical Scale
In the subsequent discussion, two physical scales of the dimensionless absorption amount  where r'(= r/r s ) is the dimensionless radial coordinate.The total drag coefficient is given by where F total means the total force acting on the body and A p is the cross-sectional area of the droplet.Meanwhile, the dimensionless droplet spacing L', is defined as where b (μm) is the half distance between the two droplets' surfaces.

Numerical Method
To solve the governing equations in association with the initial and boundary conditions, the commercial software COMSOL Multiphysics 4.0 was utilized.It is a software package for various physics or multiphysics in which the governing equations were solved through the finite element method and the solver PARDISO was utilized.PARDISO is a direct solver for solving sparse symmetric and unsymmetric linear systems of equationa (Schenk and Gärtner, 2004).The advantages of PARDISO solver are thread-safe, highperformance, robust, memory efficient, and easy to be used.A hybrid grid system combining the structured and unstructured grids was developed, as shown in Fig. 2. To reduce the numerical truncation errors, an structured orthogonal grid system was constructed in the two droplets.Alternatively, an unstructured triangular mesh was employed in gas phase to avoid the generation of singular points.The grid distributions were controlled to be finer near the droplet surface.The computational domain was extended to 20 times of the droplet radius from the origin of the coordinate (Fig. 1(b)).Three different grid systems, as listed in Table 1, were performed for the test of grid independence.Figs.
3(a) and 3(b) show the temporal profiles of dimensionless CO 2 uptake amounts in the upstream and downstream droplets where the droplet radius, droplet spacing (L'), and the Reynolds number are 10 μm, 2, and 1, respectively.It can be seen that the three curves in the upstream and downstream droplets almost overlap each other.This suggests that the second grid system (Grid system 2) satisfies the requirement of grid independence, and it is thus adopted for simulations.
The correlations of total drag coefficient (C DT ) of a single sphere in the studies of White (1974) and Seinfield (1986) are given as follows   spheres at Re = 30.Prahl et al. (2007) reported that the upstream sphere has a less influence on the downstream sphere when the distance between the two spheres is larger.Similar behavior is clearly observed in Fig. 4(c).In addition, the total drag coefficient of the leading sphere is  very close to that of a single sphere (Seinfield, 1986) as the value of L' is as large as 13.Consequently, the rigorous numerical validation is this work is accomplished.

RESULTS AND DISCUSSION
Two identical aerosol droplets in the atmosphere absorbing CO 2 at 298 K serve as the basis of the present study.The radii of the two droplets are 10 μm.At the present time the average CO 2 concentration in the atmosphere is around 390 ppm, so this concentration is regarded in the study.The droplet spacing and Reynolds number are two important parameters affecting the absorption process.Two dimensionless distances between the two droplets' surfaces (L') of 0.5 and 3 and three Reynolds numbers (Re) of 0.1, 1, and 10 are considered.Detailed fluid properties, aqueous mass diffusivity, Henry's law constant, and environmental conditions, such as pressure, temperature, gaseous CO 2 concentration, and gaseous velocities at various Reynolds numbers, are tabulated in Table 2.

Flow Structure
The streamline and velocity vector distributions of the gas phase at the absorption time of 50,000 μs under the conditions of Re = 0.1 and 10 as well as L' = 0.5 and 3 are presented in Fig. 5.It is of interest that a recirculation bubble develops between the two droplets when the droplet spacing is 0.5, whether the Reynolds number is 0.1 or 10.This arises mainly from the fact that the flow field behind the upstream droplet is affected by the downstream one, thereby resulting in the difficulty in pressure recovery at the aft region of the upstream droplet.It follows that the flow field between the droplets are disturbed notably and causes a droplet-droplet interaction in fluid dynamics.It should be pointed out that the flow direction of the recirculation bubble is opposite to those of the internal vortexes inside the two droplets.This will reduce the Table 2. Values of fluid properties, aqueous mass diffusivity, Henry's law constant and concentration of CO 2 in gas phase in terms of gas properties (Chen, 2001;Chen et al., 2011a).

Denotation
Value ρ g (kg/m 3 ) 1.184 ρ l (kg/m 3 ) 997 μ g (Pa s) 1.84 × 10 -5 μ l (Pa s) 8.91 × 10 -4 D l (m 2 /s) 1.72 × 10 -9 H (M/atm) 3.4 × 10 -2 P ∞ (atm) 1 bubble is triggered behind the upstream droplet.This implies that the flow interaction between the two droplets become insignificant.H 2 CO 3 is accumulated in the vicinity of the droplet surfaces (Fig. 6(a)).In contrast, the H 2 CO 3 concentration adjacent to the droplets' centers is extremely low, as a consequence of short exposure time.More CO 2 is absorbed and more H 2 CO 3 is transported into the droplets with increasing time; the H 2 CO 3 concentration in the droplets grows, (Figs.6(b)-6(c)).Physically, the transport of H 2 CO 3 in the droplets is carried out through two different mechanisms; one is the mass diffusion and the other is the internal circulation.The internal vortexes are able to deliver H 2 CO 3 from the droplet surfaces into the droplets' centers along the streamlines.At the same time, H 2 CO 3 is transported though the radial diffusion.With the condition of Re = 0.1 (i.e., u g,∞ = 0.0777 m/s), the radial diffusion is faster than internal circulation in that the lowest concentrations of H 2 CO 3 are almost located at the droplets' centers at 8000 μs (Fig. 6(d)).This elucidates the dominant mechanism of H 2 CO 3 transport by mass diffusion rather than convection.A comparison between the upstream droplet and the downstream one, the mass transport processes are similar to each other.It follows that the influence of droplet interaction or the recirculation bubble on the mass diffusion is equivalent.With the conditions of L' = 3 and Re = 0.1, no recirculation bubble is elicited and the fluid behaves as a creeping flow.Seeing that the influence of the leading droplet on the other one is slight, the mass transport in the two droplets is similar to that in a single droplet.This is the reason that the location of the vortex core in the upstream droplet is almost the same as that of the downstream one, and their H 2 CO 3 transport processes are similar to each other, as observed in Fig. 7.

CO 2 Transport Dynamics at Re = 10
Upon inspection of the H 2 CO 3 transport processes in the two droplets at L' = 0.5 and Re = 10 where the gas velocity is 7.77 m/s, Fig. 8 depicts that the mass transfer processes are different from those at Re = 0.1 (Fig. 6).By virtue of the stronger internal circulation at Re = 10, the H 2 CO 3 diffusion toward the droplet interior is suppressed.This is the reason why at the absorption time of 10 μs the concentration boundary layers of H 2 CO 3 in the droplets are thinner (Fig. 8(a)) when compared to those at Re = 0.1 (Fig. 6(a)).At the absorption time of 600 μs, while H 2 CO 3 diffuses toward the droplets' centers, the role played by internal circulation on H 2 CO 3 transport becomes notable (Fig. 8(b)), especially in the upstream droplet.H 2 CO 3 transported along the streamlines in the leading droplet is faster than in the other one, so the H 2 CO 3 concentration contours skewed in the former is more profound.When the absorption time reaches 1000 μs, the influence of internal circulation on H 2 CO 3 transport becomes more pronounced in that the concentration contours are skewed to a great extent (Fig. 8(c)).Once the time is as long as 2000 μs, in the upstream droplet the H 2 CO 3 concentration contours almost coincide with the streamlines (Fig. 8(d)), reflecting that the H 2 CO 3 transport is dominated by the internal vortex.At this moment, the minimum concentrations of H 2 CO 3 in the upstream droplet and the downstream one are 4.7345 × 10 -3 and 4.7613 × 10 -3 M, respectively (Fig. 8(d)), and they are 9.2187 × 10 -5 and 9.2029 × 10 -5 M at Re = 0.1 (Fig. 6(b)).The higher concentrations at Re = 10 suggest that an internal vortex along with higher strength is conducive to H 2 CO 3 transport.This can be explained by the shorter diffusion distance from the droplet surface to the vortex core than to the droplet center.Considering the case of L' = 3, Fig. 9 indicates that the H 2 CO 3 concentration contours in the two droplets are similar to each other at the uptake time no less than 600 μs, as shown in Figs.9(a) and 9(b).When the absorption time reaches 1000 μs, the influence of the leading droplet on the trailing one is exhibited in that the H 2 CO 3 transport in the downstream droplet is slower than in the upstream one to a small extent (Fig. 9(c)).At the uptake time of 2000 μs, the H 2 CO 3 transport in the two droplets is mainly governed by the internal vortexes, stemming from the consistency of H 2 CO 3 concentration contours and streamlines (Fig. 9(d)).The minimum H 2 CO 3 concentration in the upstream droplet at 2000 μs is 4.6962 × 10 -3 M, whereas it is 4.6045 × 10 -3 M in the downstream one.They are located at the vortex centers.The small difference in the minimum H 2 CO 3 concentration reveals that the absorption interaction between the two droplets at L' = 3 is slight.On the other hand, the minimum H 2 CO 3 concentrations in the upstream droplet and the downstream one at 2000 μs, L' = 3, and Re = 0.1 are 9.3126 × 10 -5 and 9.2754 × 10 -5 M, respectively (Fig. 7(b)).These two values are approximately smaller than those at Re = 10 by two orders of magnitude.It is thus summarized that the strength of internal vortex plays an important role in facilitating H 2 CO 3 transport over the entire uptake process.

Transient CO 2 Uptake
Fig. 10 displays the temporal distributions of CO 2 uptake amount in the droplets at three different orders of Reynolds number.When the Reynolds number is 0.1, the curves almost merge together, regardless of the case of L' = 0.5 or 3.This is attributed to the dominant mechanism of H 2 CO 3 diffusion along the radial direction in the droplets.Increasing Reynolds number intensifies the role played by internal vortex on H 2 CO 3 transport.The difference in the curves tends to be enlarged, especially at Re = 10 and L' = 0.5.Moreover, the CO 2 uptake accelerated at Re = 10 can be observed when comparing Fig. 10(c  At present, two time scales of half-saturated time (HST) and quasi-saturated time (QST) are defined to elucidate the CO 2 absorption behavior by the two droplets.The HST is identified at the moment when H 2 CO 3 accumulated in a droplet reached 50% of the saturated amount; the QST is identified when H 2 CO 3 accumulated in a droplet reached 99% of the saturated amount.In examining the profiles of shown in Fig. 11(a), the HST of the downstream droplet is longer than that of the upstream one at Re = 10.Additionally, the two droplets' HST at L' = 0.5 is longer than that at L' = 3 a bit, as a result of flow disturbance around the downstream droplet.As far as the QST of the two droplets is concerned, Fig. 11(b) depicts that whether L' = 0.5 or 3, they are almost equivalent at Re = 0.1 due to the radial diffusion governing the H 2 CO 3 transport in the two droplets (Figs. 6 and 7).The same results are obtained at Re = 10, as a consequence of H 2 CO 3 transport dominated by the internal circulation (Figs. 8 and 9).For the case of Re = 1, the QST of the droplets is sensitive to the location a bit.This may arise from the fact that the mass diffusion and the internal circulation are in a comparable state in H 2 CO 3 transport.In particular, Fig. 11(b) suggests that the QST of the two droplets almost linearly decreases with increasing Reynolds number, and all the absorption processes are implemented in 0.025 s.The absorption rates of the two droplets at various distances and Reynolds numbers are plotted in Fig. 12. Within the investigated ranges of droplet spacing and Reynolds number, as a whole, the influence of the latter on CO 2 uptake process prevails over the former.The saturated H 2 CO 3 amount in the droplet is determined by Henry's law and it independent of the droplet location and Reynolds number (Fig. 10), the absorption rates of the two droplets are thus governed by QST.Overall, in view of the shorter  Chen et al., Aerosol and Air Quality Research, 14: 207-219, 2014 217 QST at a higher Reynolds number, it leads to a higher droplet absorption rate.In contrast, the QST is relatively insensitive to the droplet spacing and location (Fig. 11(b)).Consequently, they merely affect the droplet absorption rate slightly, as observed (Fig. 12).

Fig. 2 .
Fig. 2. Grid distributions (a) in the vicinity of two droplets and (b) in the entire computational domain.(L' = 3).

Fig. 4 .
Fig. 4. Distributions of (a) the total drag coefficient of a sphere, (b) dimensionless SO 2 absorption amount in a droplet, and (c) the total drag coefficients of the upstream and the downstream spheres at Re = 30.

Table 1 .
A list of mesh numbers of three different grid systems.