A study of influence of acoustic excitation on carbon dioxide capture by a droplet

Energy 37 (2012) 311e321

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A study of influence of acoustic excitation on carbon dioxide capture by a droplet Wei-Hsin Chen a, *, Yu-Lin Hou b, Chen-I Hung b a b

Department of Greenergy, National University of Tainan, Tainan 700, Taiwan, ROC Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 June 2011 Received in revised form 10 October 2011 Accepted 11 November 2011 Available online 10 December 2011

Carbon dioxide capture by a droplet plays a fundamental role for reducing atmospheric greenhouse effect stemming from anthropogenic activities. To recognize the CO2 capture dynamics by a quiescent water droplet, a theoretical analysis on the mass transfer of CO2 from the gas phase to the liquid phase is performed in this work, with emphasis on the effect of acoustic excitation upon the absorption process. The dimensionless fluctuated amplitude and frequency of the acoustic wave are in the ranges of 0.1e0.99 and 0.1e1000, respectively. The analyses suggest that an acoustic wave with smaller amplitude and higher frequency leads to a more significant energy decay in the droplet due to the role of damping played by the droplet. In contrast, pressure excitation along with larger amplitude and lower frequency has a pronounced effect on the mass transfer process, as a result of higher efficiency of energy penetrating into the liquid phase. From the perspective of achieving carbon capture and storage (CCS), the acoustic excitation with the dimensionless frequency of unity is recommended to capture CO2 by a droplet and the exposure time of the droplet should be controlled at the dimensionless aqueous diffusion time (sl ) between 0.31 and 0.36. The present study has provided a useful insight into the design and application of scrubbers for enhancing CO2 capture by sprays. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Carbon capture and storage (CCS) Greenhouse effect Water droplet Acoustic excitation Damping Scrubber and spray

1. Introduction The report of IPCC [1] indicated that the atmospheric concentration of carbon dioxide (CO2) before the pre-industrial times was between 255 and 280 parts of million (ppm). However, it has been increased to around 390 ppm at the present time, stemming from the anthropogenic activities. It is well know that CO2 is the most important greenhouse gas in the atmosphere and it has led to a serious problem of global warming [2e4]. In the end of 21st century, the CO2 concentration in the atmosphere may reach 700 ppm or more and the Earth’s surface temperature is expected to increase by 1.5e6.4
C [5]. Therefore, the reduction of CO2 emissions, especially from industrial activities, has become the prime concern of people for environment sustainability. In order to lessen the greenhouse effect, carbon (or carbon dioxide) capture and storage (CCS) is an effective countermeasure to reduce anthropogenic CO2 emissions into the atmosphere [6]. From fuel burning point of view, three different methods can be employed for the mitigation of CO2 emissions; they include pre-combustion, post-combustion and oxy-combustion [6]. Furthermore, four main routes have been developed for the purpose of capturing CO2,

* Corresponding author. Tel.: þ886 6 2605031; fax: þ886 6 2602205. E-mail address: [email protected] (W.-H. Chen). 0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.11.025

consisting of absorption, adsorption, membrane diffusion and cryogenic separation [7e10]. When one is concerned with the techniques of absorption, a variety of solutions have been developed. The commonly adopted mediums for CO2 absorption comprise Selexol for pre-combustion capture [10], the solutions of sodium hydroxide (NaOH) [11], potassium hydroxide (KOH) [12], potassium carbonate (K2CO3) [13], methyldiethanolamine (MDEA) [14], potassium glycinate (PG) [15], monoethanolamin (MEA) [16] and their mixtures [17]. Furthermore, to practice the capture process, scrubbers are the most frequently used facility in which absorbents are injected in chambers in the form of sprays [18]. When the absorption process is examined in detail, it can be found that the fundamental mechanism is the mass transfer from the gas phase into a liquid droplet. For this reason, attention of this study is focused on CO2 capture by a water droplet. As far as the control of flow fields is concerned, it is an important subject in fluid mechanics and the control can be classified into active control and passive control. Generally speaking, dynamic control is required in the active control method, and vice versa for the passive control method [19]. Passive control has been employed in some combustion processes by varying the geometries of injector, combustor or flameholder for achieving specific performance [20]. In contrast, active control is adopted for some physical processes in which oscillations can occur, especially for acoustic excitation. For instance, Fujisawa and Takeda [21] introduced acoustic excitation to reduce the drag and the variation of the flow

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W.-H. Chen et al. / Energy 37 (2012) 311e321

Nomenclature a0 c D Dm f H H0 m p p0 r rs t T u x

Y M

Nondimensional amplitude (e) Molar concentration (M) Diffusion coefficient (m2 s1) Mass diffusion number (e) Nondimensional frequency (e) Nondimensional Henry’s law constant (e) Henry’s law constant (M atm1) Nondimensional solute absorption amount (e) Pressure (atm) Pressure fluctuation (atm) Radial coordinate (m) Aerosol radius (m) Time (s) Temperature (K) Nondimensional concentration (e) Nondimensional radial coordinate (e)

Greek letters s Nondimensional time (e) Subscript c g i l s t N

cg ðr; tÞ ¼ cgN ;

r/N

(3)

t>0

(4)

cg ðrs ; tÞ ¼ cgs ðtÞ;

Meanwhile, the liquid-phase diffusion equation as well as its initial and boundary conditions are expressed as

vcl v2 cl 2 vcl þ ¼ Dl vt r vr vr 2

! (5)

cl ðr; 0Þ ¼ 0;

0  r  rs

(6)

 vcl  ¼ 0; vr r¼0

t>0

(7)

cl ðrs ; tÞ ¼ cls ðtÞ;

t>0

(8)

At the gaseliquid interface without acoustic excitation, it obeys the mass flux conservation and Henry’s law, that is

Dg

    vcg vcl ¼ Dl vr r¼rs vr r¼rs

cgs ¼ Hcls

2. Mathematical formulation

(9)

(10)

To analyze the above equations theoretically, a set of dimensionless parameters are defined as the following

2.1. Physical description Considering CO2 uptake by a liquid droplet, the governing equations are made up of the gas-phase and liquid-phase mass diffusion equations [28e32]. In the diffusion equation, it is assumed that the mass diffusion processes abide by Fick’s law [28], that is, the mass flux is determined by concentration gradient. By employing an unsteady one-dimensional spherical coordinate system, the gas-phase diffusion equation as well as its initial and boundary conditions are given as follows

cg ðr; 0Þ ¼ cgN ; r>rs

Diffusive characteristic time Gas phase or continuous phase Gaseous species i Liquid phase or discrete phase Droplet surface or interface Total pressure Infinity

Superscript k Iterative number

field around a circular cylinder. Cottet et al. [22] demonstrated an acoustic absorption technique to characterize the level of gas mixing at the molecular level. Nilsson et al. [23] successfully developed particle separation techniques in silicon microchips by means of acoustic forces. Doinikov and Bouakaz [24] conducted an improved theory to investigate acoustic microstreaming behavior around a gas bubble in an ultrasound field. It follows that acoustic excitation has been extensively applied for controlling flow fields. As describe earlier, CO2 uptake by a droplet plays a fundamental role in determining the performance of CCS [25e27]. However, relatively little research has been performed on the aforementioned topic, especially for the study concerning mass transfer in an environment with acoustic excitation. Inspired from the reviewed literature, attention of this study is paid to the influence of acoustic excitation on the transport phenomena. To the authors’ knowledge, theoretical solutions of CO2 absorption from the gas phase into a droplet with acoustic fluctuation have yet been developed. For these reasons, the CO2 absorption process affected by acoustic waves will be analyzed theoretically in the present study via introducing pressure fluctuation into the absorption system. Furthermore, feasible fluctuated amplitudes and frequencies for controlling CO2 capture by a liquid droplet will be suggested from the analyses.

vcg v2 cg 2 vcg þ ¼ Dg r vr vt vr 2

Molar fraction (e) Molecular weight (g mol1)

!

r t t t t ; sg ¼ 2 ¼ ;s ¼ 2 ¼ ; rs tgc l tlc rs =Dg rs =Dl cg ðr; tÞ c ðr; tÞ and ul ¼ l H ug ¼ cgN cgN x ¼

(11)

When these parameters are substituted into Eqs. (1)e(10), the obtained nondimensionalized equations are listed in Table 1. By further defining two parameters

(1)

  vg ¼ x 1  ug

(2)

and substituting them into the equations shown in Table 1, the transformed diffusion equations as well as the initial and boundary conditions of the two phases are tabulated in Table 2.

and

vl ¼ xul

(12)

W.-H. Chen et al. / Energy 37 (2012) 311e321 Table 1 Two-phase mass diffusion equations as well as boundary, initial and interfacial conditions.

Zh vg ¼ A þ B

313

ex dx 2

(17)

0

Gas phase Governing equation

vug v2 ug 2 vug ¼ þ x vx vsg vx2 ug ðx; 0Þ ¼ 1; x>1   ug x; sg ¼ 1; x/N

Initial condition Boundary conditions

  cgs ug 1; sg ¼ ; sg >0 cgN

Liquid phase Governing equation Initial condition Boundary conditions

Interface Mass flux conservation Henry’s law

(T1)

where

 2  B ¼ pffiffiffiffi ugs  1

(T2) (T3)

A ¼ 1  ugs ;

(T4)

Therefore, the dimensionless gas-phase concentration ug is obtained as

vul v2 ul 2 vul ¼ þ x vx vsl vx2 ul ðx; 0Þ ¼ 0; 0  x  1  vul  ¼ 0; sl >0 vx x¼0

(T6)

c ul ð1; sl Þ ¼ ls H; sl >0 cgN

(T8)

  vug vul ¼ Dm = vx vx x¼1 ugs ¼ uls

(T9)

(T5)

(T7)

(T10)

ug ¼ 1 

    1  ugs x1 ,erfc pffiffiffiffiffi 2 sg x

N X

2 2 ð1Þn ðnpÞen p sl sinðnpxÞ 

Hence, the expressed by

(13)

d2 vg dvg þ 2h ¼ 0 dh dh2

(14)

vg ð0Þ ¼ 1  ugs

(15)

vg ðhÞ ¼ 0; h/N

(16)

The solution of Eq. (14) in association with the conditions of Eq. (15) and Eq. (16) is

dimensionless

Table 2 Nondimensional two-phase mass diffusion equations as well as boundary, initial and interfacial conditions.

Initial condition Boundary conditions

liquid-phase

(20)

concentration

Z N 2 2 2 2 2X ð1Þn ðnpÞen p sl sinðnpxÞ  en p l uls ðlÞdl ul ¼  x n¼1

is

vvg v2 vg ¼ vsg vx2 vg ðx; 0Þ ¼ 1; x>1   vg x; sg ¼ 0; x/N

(T11) (T12) (T13)

    cgs vg 1; sg ¼ 1  ¼ 1  ugs sg ; cgN vvl v2 vl ¼ vsl vx2 vl ðx; 0Þ ¼ 0; 0  x  1 vl ð0; sl Þ ¼ 0; sl >0 vl ð1; sl Þ ¼

cls H ¼ uls ðsl Þ; cgN

sg >0

(T14)

(T15) (T16) (T17)

sl >0

(T18)

(21)

0

2.2.3. Interface In the above-obtained solutions, clearly, corresponding to the gas phase and the liquid phase an exact solution and an analytical one are obtained, respectively. So far, the dimensionless concentrations of the gas phase and the liquid phase are established in terms of the interfacial concentrations ugs and uls , respectively, as shown in Eq. (19) and Eq. (21). When the two-phase solutions are simultaneously solved, the mass flux conservation and Henry’s law are required to link ug and ul at the interface. The mass flux conservation of the dimensionless concentrations (T9) is expressed by



Liquid phase Governing equation

2 2 en p l uls ðlÞdl

0

Eq. (T11) thus becomes an ordinary differential equation and the counterparts of the initial and boundary conditions are given by

Initial condition Boundary conditions

Zsl

sl

x1

h ¼ pffiffiffiffiffi 2 sg

Gas phase Governing equation

(19)

2.2.2. Liquid phase For the liquid phase, the technique of separation of variable is PN and utilized. Specifically, defining vl ¼ n ¼ 1 vn ðsl ÞsinðnpxÞ substituting it into Eq. (T15), the solution of Eq. (T15) in conjunction with the conditions of Eqs. (T16)e(T18) is given by [34]

n¼1

2.2.1. Gas phase To solve the gas-phase equation shown in Eq. (T11), a similarity parameter [29] is defined as

(18)

p

In the above equation, erfc is the complimentary error function [33].

vl ¼ 2

2.2. Analytical analysis

and

v ug vx

  v ul ¼ Dm v x x¼1

where Dm ð ¼ Dl =Dg HÞ represents the mass diffusion number [30,31]. Meanwhile, the solute concentration at the droplet surface is expressed as

cls ¼ H 0 pi ¼ H0 pt Yi ¼

pt Y HRT

(22)

where H 0 ð ¼ 1=HRTÞ is the Henry’s law constant. To account for the influence of acoustic excitation on the interfacial concentrations, the pressure fluctuation p0 is expressed in terms of a sinusoidal function of pressure, that is, p0 ¼ a0 sinð2putÞpt . Then Eq. (22) becomes

cls ¼ H 0 Yi ðpt þ p0 Þ ¼

cgs ð1 þ a0 sinð2putÞÞ H

(23)

where a0 , u and t are the nondimensional amplitude, physical frequency and physical time, respectively. Accordingly, the nondimensional form of Eq. (23) is expressed by

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W.-H. Chen et al. / Energy 37 (2012) 311e321

ugs ð1 þ a0 sinð2pf sl ÞÞ ¼ uls

(24)

where f ð ¼ utlc Þ is dimensionless frequency which is the product of physical frequency and characteristic time of liquid phase. By virtue of the piecewise function encountered at the droplet surface, viz., Eq. (21), one is unable to obtain a fully analytical solution for describing the mass transfer process. Instead, a semianalytical method is developed. At the interface, Eq. (T9) can be discretized as the following

ugx  ugs x1  xs uls  ulx ¼ xs  x2

ugx  ugs Dxg uls  ulx Dxl

¼ Dm

(25)

where ugx is the gaseous concentration at x1 ð ¼ xs þ Dxg Þ and ulx is the aqueous concentration at x2 ð ¼ xs  Dxl Þ ; x1 and x2 designate the grid locations adjacent to the interface in the gas phase and the liquid phase, respectively. With the condition of Eq. (24), the interfacial aqueous concentration becomes

uls ¼

Dxl ugx þ ulx Dm Dxg Dxl þ Dm Dxg 1 þ a0 sinð2pf sl Þ

(26)

2.3. Numerical method Seeing that the interfacial concentrations of the two phases are the function of time and the semi-analytical method is developed, a numerical method in association with the finite difference technique is carried out to predict the absorption phenomena. In examining Eq. (26), it is apparent that an iterative procedure is required to find uls and ugs . As long as uls and ugs are obtained, ug and ul can be solved subsequently. The numerical procedures for solving the gaseous and aqueous concentrations are given as follows. Step 1. At a given time, perform an initial guess of interfacial concentrations of the gas phase ugs and the liquid phase uls ð ¼ ugs ð1 þ a0 sinð2pf sl ÞÞÞ. Step 2. Solve ug and ul via Eq. (19) and Eq. (21), respectively. Step 3. Renew ugs and uls followed by rerunning Step. 2. The procedures are reiterated until the set criterion is satisfied. At present, the criterion for convergence between two different iterative steps (i.e. kth and kþ1th iterative steps) is less than 108. That is,

 kþ1  us  uks    < 108   ukþ1 s

(27)

Once the interfacial concentration at a specified time is obtained, the calculation of the next time step proceeds until the steady-state absorption is attained. In this work, Dxg ¼ Dxl ¼ 103 are set and the time step multiplied by nondimensional frequency is less than or equal to 0.1 for time marching. To validate the developed analytical method, the predicted concentration distributions of CO2 in the gas phase and the liquid phase at various diffusion times have been compared to the numerical results [30]. The analytical results are in good agreement with the numerical predictions, verifying the validity of the analytical method. 2.4. Physical scale In the following discussion, the dimensionless absorption amount m is defined as

Z2p Zp Zrs m ¼

0

0

0

Z2p Zp Zrs

ul r sin qdrdqdj 2

4prs3 =3

¼

H

0

0

0

cgN

cl r 2 sin qdrdqdj 4prs3 =3

Z1 ul x2 dx

¼ 3

(28)

0

where rs is the radial of a droplet. When the environmental temperature varies, the values of gaseous and aqueous diffusivities as well as Henry’s law constant will be affected. Nevertheless, it should be illustrated that the following observed phenomena are presented in terms of dimensionless scales, and the effect of altered environmental temperature on the transport phenomena has inherently considered [35]. 3. Results and discussion The interest of the present study is on the CO2 absorption process by a water droplet in an environment with acoustic fluctuation. The temperature and pressure of the environment are 25
C and 1 atm, respectively. Therefore, the important properties of the fluids, such as gas and liquid diffusivities, Henry’s law constant and the mass diffusion number for CO2 absorbed by a water droplet at the aforementioned conditions are tabulated in Table 3 [30]. Because all physical phenomena are observed in terms of dimensionless scales, the droplet size and the ambient CO2 concentration are not assigned. To evaluate the effect of pressure fluctuation upon the uptake process, the dimensionless fluctuated amplitude (a0 ) and frequency (f) are in the ranges of 0.1e1000 and 0.1e0.99, respectively. 3.1. The effect of acoustic frequency on CO2 absorption Fig. 1a first shows the threeedimensional profiles of temporal and spatial concentrations in the liquid phase with the conditions of a0 ¼ 0:1 and f ¼ 0.1. Physically, f ¼ 0.1 means that it takes 10sl to implement an acoustic fluctuation cycle. Therefore, when the absorption time is less than unity (i.e. sl  1), it can be seen that the aqueous concentration monotonically increases with increasing absorption time. To provide a more detailed observation in the droplet, the aqueous concentration distributions at a number of selected absorption times are displayed in Fig. 1b. With the condition of sl  0:4, the transient behavior of CO2 penetrating into the droplet is clearly observed. Once the absorption time sl is as long as 0.5, the variation of CO2 concentration inside the droplet is slight and it turns to be influenced by acoustic waves in that the aqueous concentration is higher than unity. When the dimensionless frequency is changed from 0.1 to 1 (i.e. f ¼ 1), this implies, in turn, that the acoustic wave accomplishes a cycle at sl ¼ 1, it is of interest that the concentration fluctuation is also exhibited at the droplet center while the droplet surface undergoes a cycle of acoustic fluctuation at sl ¼ 1, as shown in Table 3 A list of gas and liquid diffusivities, Henry’s law constant and the mass diffusion number for CO2 absorbed by a water droplet at 25
C and 1 atm. Species

CO2

Dg (m2 s-1) Dl (m2 s-1) H Dm

1.66  105 1.96  109 1.2 9.839  105

W.-H. Chen et al. / Energy 37 (2012) 311e321

Fig. 1. (a) Three-dimensional profile of aqueous concentration and (b) Spatial distributions of aqueous concentration at various absorption time (a0 ¼ 0:1 and f ¼ 0:1).

Fig. 2a. Nevertheless, the amplitude of the fluctuation decays a bit at the droplet core because of the role of damping played by the droplet. For the situations of sl ¼ 0.4 and 0.5, the acoustic fluctuation facilitates the absorption process in that the concentration inside the droplet is always larger than 1, whereas it is smaller than unity at sl ¼ 0.8e1.0. This reflects that CO2 uptake by the droplet is intensified within the half-period of acoustic fluctuation, and vice versa after the half-period. Once the frequency is enlarged to 10, more acoustic waves are excited within sl ¼ 1. Fig. 3a shows that the excited waves are clearly observed at the droplet surface for the absorption time between 0.1 and 0.3. However, the concentration at the droplet center is featured by a monotonic increase. For the absorption time in the range of 0.6e0.8, the effect of acoustic fluctuation is exhibited at the droplet center; however, the amplitude is tiny, implying that it is harder for the acoustic waves penetrating into the droplet’s

315

Fig. 2. (a) Three-dimensional profile of aqueous concentration and (b) Spatial distributions of aqueous concentration at various absorption time (a0 ¼ 0:1 and f ¼ 1).

interior. Fig. 4a and b further demonstrate the distributions of aqueous concentration at the conditions of f ¼ 100 and 1000, respectively. A comparison between the two figures suggests that the wave propagation into the droplet declines significantly with increasing frequency. In particular, for the frequency of 1000, the depth of wave fluctuation in the droplet is fairly slight. From the above observations, it is summarized that the higher the frequency, the more notable the role of damping played by the droplet. As a consequence, the influence of acoustic fluctuation in the droplet is weakened. 3.2. The effect of acoustic frequency in the gas phase Subsequently, the effect of acoustic excitation on the gaseous concentration (i.e. ug ) at various frequencies and amplitudes are examined in Figs. 5 and 6. For the case of f ¼ 0.1 along with a0 ¼ 0.1, seeing that the period of acoustic fluctuation is much longer than

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W.-H. Chen et al. / Energy 37 (2012) 311e321

Fig. 3. Three-dimensional profiles of aqueous concentration at sl ¼ (a) 0.1e0.3 and (b) 0.6e0.8 (a0 ¼ 0:1 and f ¼ 10).

that of droplet absorption, Fig. 5a depicts that the gaseous concentration approaches unity as the absorption time is close to 1. With the conditions of f ¼ 1 (Fig. 5b) and 1000 (Fig. 5c), the oscillation of concentration in the gas phase is clearly exhibited. However, it should be addressed that the amplitude of fluctuation is slight compared to that in the liquid phase. This arises from the fact that the fluctuation in the gaseous concentration is induced by the droplet surface rather than by the pressure directly. Basically, for the condition of f ¼ 1, when the amplitude of fluctuation is amplified to 0.5 (Fig. 6a) and 1 (Fig. 6b), the oscillation of the gaseous concentration in the vicinity of the droplet surface grows as well. As a whole, the fluctuated amplitude in gaseous concentration is smaller than that in aqueous concentration approximately by three to four orders of magnitude. 3.3. Concentration distributions at the droplet surface and center as well as phase lag With attention focused on the droplet surface, the distributions of aqueous concentration at various frequencies and

Fig. 4. Three-dimensional profiles of aqueous concentration with the conditions of f ¼ (a) 100 and (b) 1000 (a0 ¼ 0:1).

amplitudes are plotted in Fig. 7, whereas the profiles of gaseous concentration are displayed in Fig. 8. The amplitude of fluctuation in the two figures is 0.1, namely, a0 ¼ 0.1. As expressed in Eq. (23), the aqueous concentration at the droplet surface is directly affected by the acoustic fluctuation. Consequently, the distribution of the interfacial aqueous concentration completely coincides with the acoustic fluctuation (Fig. 7a and b), regardless of what the frequency and amplitude are. With regard to the gaseous concentration at the interface, it is worthy of note that a higher frequency results in a higher amplitude in concentration fluctuation, as shown in Fig. 8, stemming from a higher energy contained in the acoustic wave. The distributions of aqueous concentration at the droplet center under three different amplitudes are presented in Fig. 9. By virtue of the role of damping played by the droplet, as can be

W.-H. Chen et al. / Energy 37 (2012) 311e321

317

Fig. 6. Spatial distributions of gaseous concentrations with the conditions of a0 ¼ (a) 0.5 and (b) 0.99 (f ¼ 1).

Fig. 5. Spatial distributions of gaseous concentrations with the conditions of f ¼ (a) 0.1, (b) 1 and (c) 1000 (a0 ¼ 0:1).

seen in the figures, increasing frequency tends to lessen the amplitude at the droplet center. In particular, once the frequency is as high as 500, the oscillation at the droplet core can be found no more, even though the fluctuated amplitude at the droplet surface is as high as 0.99 (Fig. 9c). Alternatively, it can be seen that a higher excitation state at the droplet surface also results in a smaller descent in the central fluctuation. Specifically, corresponding to a0 ¼ 0.1, 0.5 and 0.99 with the condition of f ¼ 1, the maximum values of the fluctuation in the first cycle are 0.058, 0.388 and 0.802, respectively. In other words, 58 (¼0.058/ 0.1  100), 78 and 81% of fluctuated energy can be delivered to the core. It is thus recognized that a higher fluctuated amplitude at the droplet surface can transfer more energy into the droplet center.

318

W.-H. Chen et al. / Energy 37 (2012) 311e321

Fig. 8. Temporal distributions of gaseous concentration at the interface with the conditions of f ¼ (a) 0.1e100 and (b) 100e1000 (a0 ¼ 0:1). Fig. 7. Temporal distributions of aqueous concentration at the interface with the conditions of f ¼ (a) 0.1e100 and (b) 100e1000 (a0 ¼ 0:1).

3.4. CO2 absorption process Furthermore, the phase lag between the interface and the droplet center at a0 ¼ 0.1 is explored in Fig. 10 where the phase lag is defined as

Phase lag ð
Þ ¼

jsl;s  sl;0 j

sl;s

 360

(29)

Upon inspection of the aqueous concentrations at the two different locations, they indicate that the values of phase lag at f ¼ 1 and 2 are 304
(Fig. 10a) and 623
(Fig. 10b), respectively. The above results suggest that more time is required for an acoustic wave reaching the droplet center when the acoustic frequency goes up. Considering the case of f ¼ 10, the damping effect of the droplet has more influence on the acoustic fluctuation. As a consequence, it is impossible to identify a complete cycle at the droplet center (Fig. 10c) and thereby the phase lag cannot be pointed out.

Finally, the CO2 absorption processes by a droplet at various frequencies along with a0 ¼ 0.1 are demonstrated in Fig. 11. When emphasis is placed on the case of f ¼ 1, because one cycle of acoustic fluctuation is implemented at sl ¼ 1.0 and the acoustic wave is able to reach the droplet center (Fig. 9a), it is not surprising that the CO2 uptake by the droplet also undergoes one cycle (Fig. 11a). Meanwhile, it is noted that the maximum uptake amount of CO2 takes place at sl ¼ 0.356. From the viewpoint of practical operation for CO2 capture, if the traveling time of the droplet in a scrubber is controlled at sl ¼ 0.356, not only more CO2 can be captured by the droplet, the absorption time can also be shortened drastically when compared to the absorption without acoustic excitation. With attention shifted to the case of f ¼ 2, the damping effect of the droplet results in the maximum absorption amount in the second cycle being higher than in the first cycle and the optimal absorption

W.-H. Chen et al. / Energy 37 (2012) 311e321

Fig. 9. Temporal distributions of aqueous concentration at the droplet center with the conditions of a0 ¼ (a) 0.1, (b) 0.5 and (c) 0.99 (f ¼ 0.1e500).

319

Fig. 10. Temporal distributions of aqueous concentration at the droplet’s surface and center with the conditions of f ¼ (a) 1, (b) 2 and (c) 10 (a0 ¼ 0.1).

320

W.-H. Chen et al. / Energy 37 (2012) 311e321

Fig. 11. Temporal distributions of solute absorption amount at the conditions of f ¼ (a) 0.1e100 and (b) 100e500 (a0 ¼ 0.1). Fig. 12. Temporal distributions of solute absorption amount at the conditions of a0 ¼ (a) 0.5 and (b) 0.99 (f ¼ 0.1e100).

time occurs at sl ¼ 0.667. When the frequency is as large as 10, the influence of the acoustic excitation on the absorption process withers markedly, and the aforementioned behavior is especially pronounced at higher frequencies such as f  100 (Fig. 11b). When the amplitude of fluctuation is amplified to 0.5 and 0.99, basically, similar phenomena of absorption process to Fig. 11a are also observed in Fig. 12a and b. That is to say, the frequency of 1 is able to provide a higher absorption amount of CO2 at a shorter time, thereby giving higher absorption rate and efficiency. For instance, corresponding to a0 ¼ 0.5 and 0.99, the maximum absorption amounts of CO2 are 1.416 and 1.851, respectively, and they occur at sl ¼ 0.318 and 0.311, respectively. In summary, the condition of f ¼ 1 with a higher amplitude of acoustic excitation is recommended for enhancing CO2 absorption by a droplet if the exposure time of the droplet in the CO2 environment is controlled at around 0.31e0.36sl .

4. Conclusions A theoretical analysis to approach CO2 uptake by a liquid droplet in an acoustic-excited environment has been performed in this study. By treating acoustic waves as a sinusoidal function and varying the dimensionless fluctuated amplitude (0.1e0.99) and frequency (0.1e1000), the influence of the two parameters on the absorption process has been examined. With the pressure fluctuation on the droplet surface, the predictions suggest that the amplitude of the fluctuated gaseous concentration is smaller than that of the aqueous one approximately by three to four orders of magnitude. An increase in frequency results in the growth of the damping role played by the droplet; the penetration of the fluctuated pressure into the droplet is thus abated. Meanwhile, it is found that a longer

W.-H. Chen et al. / Energy 37 (2012) 311e321

time is required for the pressure wave reaching the droplet center when the acoustic frequency is lifted. In contrast, a higher pressure excitation state at the droplet surface is relatively able to deliver more energy into the droplet center. Accordingly, from the perspective of practical operation, acoustic waves with lower frequencies and higher fluctuated amplitudes are conducive to the enhancement of CO2 transfer from the gas phase into the liquid phase. From the current analyses, the results reflect that the acoustic excitation along with the condition of f ¼ 1 and a higher amplitude, say, a0 ¼ 0.99, is recommended for enhancing CO2 absorption by a droplet if the exposure time of the droplet in the CO2 environment is controlled at around 0.31e0.36sl . Rather than water, if MEA is used as a solvent to absorb CO2, the process of chemical absorption is encountered. Under such a situation, the diffusion equations of all dissociated species should be taken into account and the chemical reactions should be included in the source terms of the equations. This makes the physical problem become much complicated but it deserves further investigation in the future. Acknowledgments The authors gratefully acknowledge the financial support of the National Science Council, Taiwan, ROC, on this study. References [1] Climate change 2007: synthesis report. IPCC, http://www.ipcc.ch/ publications_and_data/publications_ipcc_fourth_assessment_report_synthesis_ report.htm; 2007.  Jobson M, Smith R. Estimation and reduction of CO2 [2] Gadalla M, Oluji c Z, emissions from crude oil distillation units. Energy 2006;31:2398e408. [3] Zevenhoven R, Beyene A. The relative contribution of waste heat from power plants to global warming. Energy 2011;36:3754e62. [4] Darko Matovic D. Biochar as a viable carbon sequestration option: global and Canadian perspective. Energy 2011;36:2011e6. [5] Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt KB, et al. Climate change 2007: the physical science basis. Contribution of working group I to the fourth assessment report of the Intergovernmental Panel on climate change. New York: Cambridge University Press, http://www.ipcc.ch/publications_and_ data/publications_ipcc_fourth_assessment_report_wg1_report_the_physical_ science_basis.htm; 2007. [6] Davison J. Performance and costs of power plants with capture and storage of CO2. Energy 2007;32:1163e76. [7] Hammond GP, Ondo Akwe SS, Williams S. Techno-economic appraisal of fossil-fuelled power generation systems with carbon dioxide capture and storage. Energy 2011;36:975e84. [8] Li H, Ditaranto M, Berstad D. Technologies for increasing CO2 concentration in exhaust gas from natural gas-fired power production with post-combustion, amine-based CO2 capture. Energy 2011;36:1124e33. [9] Bounaceur R, Lape N, Roizard D, Vallieres C, Favre E. Membrane processes for post-combustion carbon dioxide capture: a parametric study. Energy 2006; 31:2556e70. [10] Olajire AA. CO2 capture and separation technologies for end-of-pipe applications e A review. Energy 2010;35:2610e28. [11] Stolaroff JK, Keith DW, Lowry GV. Carbon dioxide capture from atmospheric air using sodium hydroxide spray. Environmental Science and Technology 2008;42:2728e35.

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