Influence of droplet mutual interaction on carbon dioxide capture process in sprays

Applied Energy 92 (2012) 185–193

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Influence of droplet mutual interaction on carbon dioxide capture process in sprays 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

Article history: Received 22 August 2011 Received in revised form 17 October 2011 Accepted 21 October 2011 Available online 29 November 2011 Keywords: Carbon capture and storage (CCS) Greenhouse gas Scrubber and spray Droplet mutual interaction Number density Mass diffusion number

a b s t r a c t Sprays are an important tool for carbon dioxide capture through absorption. To figure out CO2 capture processes in sprays, the gas absorbed by a single droplet under droplet mutual interaction is investigated. In the study, the number density of droplet is in the range of 103–106 cm3. By conceiving a bubble as the influence distance of the droplet–droplet interaction, the predictions indicate that the mutual interaction plays an important role on the absorption process and uptake amount of CO2 when the number density is as high as 106 cm3 with droplet radius of 30 lm. Specifically, the absorption period and CO2 uptake amount of a droplet are reduced by 7% and 10%, respectively, so that the absorption rate is decreased compared to the droplet without interaction. Though the droplet mutual interaction abates the CO2 uptake amount of a single droplet, a higher number density is conducive to the total uptake amount of CO2 from the gas phase to the liquid phase. With the number density of 106 cm3 and increasing the droplet radius from 10 to 50 lm, CO2 capture from the gas phase to the liquid phase is intensified from 0.35% to 47.8%, even though the droplet–droplet interaction lessens the CO2 uptake amount of a single droplet by a factor of 48%. In conclusion, a dense spray with larger droplet radii enhances the droplet– droplet interaction and thereby reduces CO2 capture capacity of single droplets; but more solute can be removed from the gas phase. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Currently, the atmospheric greenhouse effect has become the prime concern of the public in that it has induced the serious problem of global climate change. In the atmosphere, the noticeable greenhouse gases due to anthropogenic emissions include carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O) and chlorofluorocarbons (CFCs) [1,2]. Among these greenhouse gases, CO2 is the most important one because it accounts for over 50% of the total radiative forcing of the greenhouse gases. For this reason, there has been increasing demand for a pronounced mitigation of CO2 emissions from anthropogenic activities, especially from industrial sources [3,4]. Combustion of fossil fuels for the purpose of getting heat and power is the main source of CO2 emissions in which thermal power plants play the most important role in liberating CO2. To lessen CO2 emissions, carbon capture and storage (CCS) has been thought of as a promising route for carbon management [5–7]. As a matter of fact, it has been reported that the capture of CO2 contributes 75% to the overall CCS cost and CCS increases the electricity production cost by 50% [8]. To date, three different capture techniques of postcombustion, pre-combustion and oxyfuel combustion have been ⇑ Corresponding author. Tel.: +886 6 2605031; fax: +886 6 2602205. E-mail address: [email protected] (W.-H. Chen). 0306-2619/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2011.10.035

developed for abating CO2 emissions from power plants [9–11]. Furthermore, there have been four means developed for CO2 separation; they are absorption, adsorption [12], membrane diffusion and cryogenic separation [11]. As far as the absorption is concerned, it can be cataloged into chemical absorption and physical absorption. In chemical absorption, the commonly used absorbents include aqueous ammonia (NH3) [13,14], amine solution such as monoethanolamine (MEA) [15,16], potassium carbonate (K2CO3) [17], sodium carbonate (Na2CO3) [18] and sodium hydroxide (NaOH) [19]. With regard to physical absorption, Selexol (dimethylether of polyethylene glycol) solvent [20], chilled methanol in Rectisol process [21] and propylene carbonate in Fluor process [18] are commercially available for CO2 removal. In contrast to chemical absorption, physical absorption can be utilized in an environment of high CO2 partial pressure such as the product gas or syngas from a shift converter in IGCC. Another benefit of physical absorption is that it consumes less energy for solvent regeneration [22]. However, it is necessary to cool syngas before CO2 is captured in that the performance of physical absorption is better at lower operating temperatures. To practice gas absorption processes, scrubbers have been extensively employed to capture CO2 [3,19,23,24]. In scrubbers, aqueous absorbents are usually injected into chambers in the form of sprays by means of fragmenting liquid mediums into droplets. Fthenakis [25] has reported that gaseous solutes absorbed by

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Nomenclature c D Dm H m n R t u x D

molar concentration (M) diffusion coefficient (m2 s1) mass diffusion number (–) nondimensional Henry’s law constant (M gas/M aqueous) nondimensional solute absorption amount number density (cm3) radial coordinate (m) time (s) nondimensional concentration (–) nondimensional radial coordinate (–) difference

Greek letters a total mass of CO2 (–) s nondimensional time (–)

a c d g j l p qss ss s

influence radius diffusive characteristic time droplet gas phase or continuous phase grid point liquid phase or discrete phase time step quasi-saturated state saturated state droplet surface or interface

Superscript i time step k iterative number l iterative number

Subscript 0 initial

water sprays are extremely effective because of extended contact surface between the solutes and the droplets. When one is concerned with CO2 uptake by water, Taniguchi et al. [26] carried out experimental studies with emphasis on the mass transfer of CO2 by water droplets in sprays. They pointed out that the amount of absorbed CO2 increased with the increase of mass flow rate of liquid. Dimiccoli et al. [27] also conducted spray-tower-loop absorbers to explore the behavior of CO2 uptake by droplets. The relationship between the droplet absorption time and the design of spray absorbers was addressed. Seeing that sprays are composed of droplets and the basic masstransfer unit in sprays is single droplet absorption [28], the recognition of CO2 uptake by a droplet plays an important role in the design of scrubbers and modeling of spray towers. In previous studies, CO2 absorption by a stationary water droplet has been analyzed theoretically [2]; the mass transfer phenomena for CO2 absorbed by a convecting droplet with a solid nucleus have also been investigated numerically [29]. Unfortunately, these studies merely considered single droplet absorption and disregarded the impact of the droplet–droplet interaction upon the absorption process. When CO2 is captured by droplets in a spray, in fact, the droplet mutual interaction may become remarkable if a dense spray is performed. Under such a situation, the influence of droplet–droplet interaction upon the solute absorption process in a spray should be taken into consideration. In reviewing past studies of droplet combustion in sprays, Sangiovanni and Kesten [30] developed experimental and theoretical method to examine the effect of droplet mutual interaction on the ignition of a single stream of monosized fuel droplets. Their study indicated that, as droplet spacing was smaller, the interaction effect became increasingly more important because the droplet tended to exert the maximum possible increase in ignition delay time. Labowsky [31] used suitable variable transformations and the method of images to investigate the evaporation rates of rapidly evaporating interacting spherical particles. They found that the interaction substantially affected particle evaporation rates even at large particle spacing, and the effect of interaction decreased when the particle spacing increased. However, these studies were performed with treating droplets as mass source rather than mass sink. Although there has been much research conducted concerning gaseous solute absorption in sprays, detailed information for CO2 uptake by single droplets in sprays under the influence of

droplet–droplet interaction is still absent so far. As illustrated earlier, the study of CO2 uptake by single droplets is of the upmost importance in thoroughly understanding greenhouse gas capture in sprays and modeling mass transfer processes. For this reason, the objectives of this study are to develop theoretical and numerical methods for approaching CO2 capture by droplets in sprays and to explore the mass transfer processes by altering number density and radius of droplet in a spray, particular emphasis is on the effect of droplet–droplet interaction upon the capture consequence. From the results, a practical suggestion from the perspectives of spray operation and scrubber design will be outlined. 2. Mathematical formulation 2.1. Physical description and assumptions This study focuses on CO2 absorption by single droplets in sprays. To simplify the physical problem, the following assumptions are included: (1) droplets are uniformly distributed in space and all droplets’ size is identical; (2) only physical absorption is considered; (3) the droplets and their surrounding are stationary and isothermal; (4) the absorption process at the droplet surface obeys Henry’s law; (5) the mass diffusion abides by Fick’s law; and (6) the effect of the mutual interaction upon CO2 capture by a droplet is restricted inside a bubble. The last assumption is adopted in accordance with the hypothesis of Bellan and Cuffel [32]. In other words, it is assumed that all of the droplets in a spray are individually surrounded by bubbles, namely, the spheres of influence so that the bubble surface is the interactive boundary. Detailed physical configuration is shown in Fig. 1a in which the space (i.e. the control volume) in a spray is partitioned into three regions: the droplets, the spheres of influence (i.e. the bubbles) and the space outside the bubbles. 2.2. Formulations The mass diffusion equations in the two phases are regarded to account for CO2 capture phenomena. Accordingly, the governing equations in association with the initial, boundary and interfacial conditions are tabulated in Table 1. It is desirable to describe the capture process in terms of non-dimensional scales, a set of dimensionless parameters are thus defined as follows:

(a)

W.-H. Chen et al. / Applied Energy 92 (2012) 185–193

187

ug ðx; 0Þ ¼ 1; x > 1

ð3Þ

A control volume

 ug



Ra ; Rd

sg ¼

ug ð1; sg Þ ¼

(b)

Rd

ð6Þ

ul ðx; 0Þ ¼ 0; 0 6 x 6 1

ð7Þ

 @ul  ¼ 0; @x x¼0

ð8Þ

Fig. 1. Schematics of (a) hypothesis accounting for droplet–droplet interaction in a control volume and (b) physical sizes of droplet and bubble in the study.

@cg @t

Initial condition Boundary condition

@cl @t

Initial condition Boundary condition

¼ Dg



@ 2 cg @r2

þ 2r

¼ Dl



@ 2 cl @r2

@cg @r

l þ 2r @c @r





ð10Þ

ugs ¼ uls

ð11Þ

" Z   # Z Ra Rd d 4 3 2 2 4pr cl dr þ n 4pr cg ðrÞdr þ 1  n pRa cg ¼ 0 n dt 3 0 Rd

n

cl ðr; 0Þ ¼ 0; 0 6 r 6 Rd  @cl  ¼ 0; t > 0 @r  cl ðRd ; tÞ ¼ cls ðtÞ;

Interface Max flux conversation

Dg

Hanry’s law

cgs ¼ Hcls

t

¼

t ; tgc

sl ¼

t R2d =Dl

¼



@cg @r

r¼Rd

¼ Dl

Z

Rd

4pr2 cl dr þ n

Z

0

Ra

Rd

  4 4pr 2 cg ðrÞdr þ 1  n pR3a cg 3

¼ constant

t>0

ð13Þ

Furthermore, the nondimensional form of above equation is obtained as:



@cl @r r¼R d

n H

t cg ðr; tÞ cl ðr; tÞ ; ug ¼ ; ul ¼ H t lc cg0 cg0 ð1Þ

Substituting the above parameters into the equations shown in Table 1, the diffusion equation as well as the initial and boundary conditions of the gas phase are transformed to:

@ug @ 2 ug 2 @ug ¼ þ x @x @ sg @x2

ð9Þ

   @ug @ul ¼ Dm @x @x x¼1

r¼0

R2d =Dg

sl > 0

where n is the number density, standing for the total number of droplets in a cubic centimeter (cm3). Accordingly, the gas-phase and liquid-phase diffusion equations are solved numerically to account for the mass transfer process for CO2 capture from the bubbles into the droplets. With regard to CO2 outside the bubbles, it is determined in accordance with the conservation of CO2 in the control volume. The above equation means that the total mass of CO2 in a control volume remains unchanged with time, that is:

cg ðr; 0Þ ¼ cg0 ; r > Rd cg ðr; tÞ ¼ cg ðtÞ; r ¼ Ra cg ðRd ; tÞ ¼ cgs ðtÞ; t > 0

Liquid phase Governing equation

cls H; cg1

ð12Þ

Table 1 Two-phase mass diffusion equations as well as boundary, initial and interfacial conditions. Gas phase Governing equation

sl > 0

where Dm( = Dl/DgH) is the mass diffusion number [33,34], representing the driving force ratio of mass transfer between the gas phase and the liquid phase. When attention is paid to single droplets in association with their bubbles and the space outside the bubbles, the conservation of the total mass of CO2 is taken into account [32], this implies:

Droplet

sg ¼

ð5Þ

At the droplet surface, the mass flux conservation and Henry’s law become:

Sphere of influence Ra

r ; Rd

sg > 0

@ul @ 2 ul 2 @ul ¼ 2 þ x @x @ sl @x

ul ð1; sl Þ ¼

x¼

cgs ; cg0

ð4Þ

Meanwhile, it gives the aqueous diffusion equation as well as the initial and boundary conditions as the following:

Rd

Ra

cg ðsg Þ cg0

ð2Þ

Z

1

4px2 ul dx þ n

Z 1

0

Ra Rd

4px2 ug dx þ

  4 3 ug ¼ a 1  n p R a 3 R3d 1

ð14Þ

In Eq. (12), Ra is the radius of bubble [32], as shown in Fig. 1b, and it is defined by:

 1=3 3 Ra ¼ 0:74 4pn

ð15Þ

The constant a shown in Eq. (14) is chosen by the initial state:

a¼

1 R3d

n

4p 3

ð16Þ

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W.-H. Chen et al. / Applied Energy 92 (2012) 185–193

When the capture process reaches the saturated state, Eq. (14) becomes:

n 4p 4p 1 ul;ss  n ug;ss þ 3 ug;ss ¼ a H 3 3 Rd

1.2

1

ð17Þ

0.8

Accordingly, the dimensionless saturated concentrations in the gas phase and the liquid phase are obtained as:

ul;ss ¼ ug;ss ¼

n 4p H 3

 n 43p



n 43p

þ

ul

1 R3d

1 R3d

CO 2 (Present) CO 2 [28]

τl=0.6

0.2

0.6

ð18Þ 0.1

0.4

2.3. Numerical method

0.2 0.01

Seeing that the interfacial concentrations of the two phases are the function of time, a finite difference method is developed to predict the physical phenomena. When discretizing the diffusion equations, i.e. Eqs. (2) and (6), the central difference scheme is adopted. At the same time, the fully implicit scheme is utilized in treating time marching procedure for the sake of numerical stability. The discretized equations are cast into the following general form: pþ1 p Aupþ1 þ Cupþ1 j1 þ Buj jþ1 ¼ Duj

ð19Þ

A¼

2 2  Dxj ðDxj þ Dxjþ1 Þ xj ðDxj þ Dxjþ1 Þ

ð20Þ

B¼

2 2 1   Dxjþ1 ðDxj þ Dxjþ1 Þ Dxj ðDxj þ Dxjþ1 Þ Ds

ð21Þ

2 2 þ Dxjþ1 ðDxj þ Dxjþ1 Þ xj ðDxj þ Dxjþ1 Þ

ð22Þ

C¼

D¼

1 Ds

ð23Þ

where the superscript p refers to the time increment Ds and the subscript j refers to the space increment Dx. Regarding Eqs. (10), (11), and (14), it is realized that the two-phase diffusion equations, interfacial concentrations and mass conservation equation are coupled; hence an iteration procedure is employed. During solving the equations, the Newton–Raphson scheme is applied at the interface to accelerate the numerical convergence. As far as the grid system is concerned, 100 grids with uniform distribution in the droplet is employed; in the gas phase, the grid distribution is non-uniform and the grid number is adjusted in accordance with the investigated number density. In the current study, corresponding to the number densities of 103, 104, 105 and 106 cm3, the grid numbers in the gas phase are 1871, 869, 403 and 188, respectively. 2.4. Numerical procedure When the governing equations are discretized, two matrixes with one in the gas phase and the other in the liquid phase are encountered. As mentioned earlier, an iterative procedure is required to solve the equations which have to simultaneously satisfy the boundary conditions and mass conservation equation. The numerical procedures for solving the gaseous and aqueous concentrations are stated as follows: Step 1. At a given time, perform the initial guesses of the gasphase concentrations at the bubble surface ulbs and the droplet surface us(=ugs = uls). Step 2. Solve ug and ul from Eq. (19), followed by renewing us(=ugs = uls) through Eqs. (10) and (11).

0.001

0

0

0.2

0.4

0.6

0.8

1

x Fig. 2. Comparisons of spatial concentration in the liquid phase at various nondimensional times.

Step 3. Repeat Step 2 and reiterate the procedure until the set criterion is satisfied. At present, the criterion for convergence between two different iterative steps (i.e. k  1th and kth iterative steps) must be less than 108, i.e.:

 k  us  uk1  s   < 108   uk

ð24Þ

s

l a Step 4. Then, renew ubs (ulþ1 bs ¼ ubs  al ) to satisfy Eq. (14) and repeat Step 2 and 3 until the set criterion is reached. Currently, the criterion of convergence of the total mass of CO2 at a specific iterative step (i.e. lth iterative step) must be less than 1016, i.e.: 

 a  al  16    a  < 10

ð25Þ

Step 5. Once the converged concentrations at the droplet surface and the bubble surface at a specific time are obtained, the calculation of the next time step (i.e. ith time step) proceeds until the saturated state of absorption is attained. The saturated state is defined by:

 i m  mi1    < 108   mi

ð26Þ

In this work, the increment of time step 103sl is chosen for time marching. In examining past literature, it was found that no experimental studies concerning CO2 capture by single droplets have been conducted and thereby no available experimental data can be used for comparison. To validate the accuracy of the developed numerical method, the distributions of aqueous concentration at various diffusion times in the absence of droplet mutual interaction are predicted and compared to the analytical solutions [33]. As can be seen in Fig. 2, the good agreement between the predictions and the analytical solutions reveals that the numerical method and grid system are accurate and reliable. 3. Results and discussion The emphasis of the present study is on CO2 capture by single droplets in sprays where the surrounding temperature and pressure of the droplets are 25 °C and 1 atm, respectively. Accordingly, the important properties of the mass transfer such as the gas and liquid diffusivities, Henry’s law constant and mass diffusion number for CO2 absorbed by water droplets are 1.66  105 m2 s1, 1.96  109 m2 s1, 1.2 M gas (M aqueous)1 and 9.839  105, respectively [33]. In reviewing past studies [35–37], as shown in

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W.-H. Chen et al. / Applied Energy 92 (2012) 185–193

1.02

Table 2 A list of droplet number density and droplet radius.

3.02  10 3.14  105 1.54  105 2.43  105 8.86  103 1.06  104 5.11  103

Droplet radius (lm)

Ref.

6 19 58 33 52.5 28.65 40

Sazhin et al. [35] Pikkula et al. [36] Pikkula et al. [36] Pikkula et al. [36] Chen [33] Chen [33] Chen [33]

1

0.98 1.02

3

n=10 4 n=10 5 n=10 6 n=10

0.96

1

us

6

us

Number density (cm3)

0.98

0.94

Table 2, the number densities and radii of droplet in sprays are approximately in the ranges of 8.8  103–3  106 cm3 and 6–60 lm, respectively. For this reason, four different orders of magnitude of the number density (i.e. 103, 104, 105 and 106 cm3), accounting for the interaction of droplets upon the uptake process, and the droplet radii of 10–50 lm are taken into account. All physical phenomena in the following discussion are predicted in terms of dimensionless scales, the ambient CO2 concentration is thus not assigned. 3.1. Influence distance and concentration distribution Based on the investigated range of number density, the distributions of bubble radius (Ra) and the ratio of bubble radius to droplet radius (Ra/Rd) are sketched in Fig. 3 where the droplet radius is 30 lm. As can be seen in the figure, when the number density n increases, the bubble radius, namely, the influence distance, decreases dramatically. Specifically, for the number density of 103 cm3, the bubble radius is 561 lm. Once the number density increases to 106 cm3, the bubble radius becomes 56 lm. As a result, the ratio of bubble radius to droplet radius withers from 18.7 to 1.87. Seeing that the influence distance becomes 1.87 folds of the droplet radius at the number density of 106 cm3, it is inferred that the droplet–droplet interaction is pronounced at this number density. 3.2. Temporal concentration distribution Temporal distributions of CO2 concentration at the gas–liquid interface at various number densities are presented in Fig. 4. By virtue of CO2 absorption characterized by a small mass diffusion number [33], for the number density ranging from 103 to

0.96

0.92

0.9

Ra Ra / Rd

τl

0.6

0.015

0.8

1

1

20

18.7

ubs

0.98

Ra / Rd

R a (μm)

0.4

0.01

30

200

0.96 3

n=10 4 n=10 n=10 5 n=10 6

0.94 10

56μm 100

0 3 10

0.2

τl

1.02

400

300

0

0.005

105 cm3, Fig. 4 depicts that the interfacial concentration jumps to the place near unity soon after the droplet is exposed to the environment filled with CO2. With regard to the case of n = 106 cm3, the droplet mutual interaction becomes notable so that the interfacial concentration jumps to 0.99 initially rather than 1. As a whole, Fig. 4 indicates that the interfacial concentration is close to unity all the time in the cases of n = 103 and 104 cm3, implying that the droplet–droplet interaction is almost ignorable. This can be explained by the large influence distances (Fig. 3) in that the values of Ra/Rd in the former and the latter are 18.7 and 8.68, respectively. With the condition of n = 105 cm3, a weak interaction is exhibited because the interfacial concentration departs from unity a bit. In regard to the case of n = 106 cm3, the concentration decays progressively and its decaying extent is pronounced compared to the other three cases. It is thus recognized that the absorption process is significantly affected by droplet mutual interaction, as a consequence of the interfacial concentration below 1 to a certain extent. Fig. 5 shows the temporal distributions of CO2 concentration at the bubble surface along with the four number densities. It is evident that the decaying paces of the curves are fairly similar to those shown in Fig. 4, regardless of what the number density is.

561μm 500

0

Fig. 4. Temporal distributions of nondimensional CO2 concentration at the droplet surface under various number densities with the condition of Rd = 30 lm.

40

600

0.94

0.92 1.87 4

5

10

10

0

6

10

-3

n (cm ) Fig. 3. Distributions of bubble radius and the ratio of bubble radius to droplet radius with respect to number density.

0.9

0

0.2

0.4

τl

0.6

0.8

1

Fig. 5. Temporal distributions of nondimensional CO2 concentration at the bubble surface under various number densities with the condition of Rd = 30 lm.

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W.-H. Chen et al. / Applied Energy 92 (2012) 185–193

3.3. Spatial concentration distribution

(a)

With attention shifted to the concentration distributions in the bubble and the droplet, Fig. 6 demonstrates the three-dimensional profiles of temporal and spatial CO2 concentrations at the condition of n = 105 cm3. Conceptually, the droplet can be thought of as a CO2 sink so that partial gaseous CO2 contained in the bubble is transported into the droplet via absorption. It can be seen that the gaseous concentration in the vicinity of the droplet surface is featured by a spatially uniform distribution, whereas it drops from unity and approaches 0.99 when the absorption time is sufficiently long (Fig. 6a). Unlike the distributions observed above, the penetration of CO2 from the droplet surface toward the droplet center results in the aqueous concentration being spatially non-uniform in the initial state of absorption. Contrary to the behavior observed in Fig. 6a, a progressive growth in aqueous CO2 concentration with time is exhibited. When the absorption time sl is as long as 0.5, the variation in the concentrations of the gas phase and the liquid phase becomes slight. To proceed further into the recognition of CO2 transport phenomena, spatial CO2 concentrations in the gas phase and the liquid phase at various absorption times along with the condition of n = 106 cm3 are plotted in Fig. 7. It is apparent that the spatial distribution of gaseous concentration is almost uniform (Fig. 7a), regardless of what the absorption time is. This is attributed to the solute characterized by a low mass diffusion number [33]

1.05

1

ug

0.95

0.9 τ l =0 τ l =0.25 τ l =0.5 τ l =0.75 τ l =1

0.85

0.8

1

1.2

1.4

1.6

1.8

x

(b) 0.95 0.9

ul

0.85

0.8

(a) ug 1.000 0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.990

1 0.998 0.996

ug

0.994 0.992 0.994

2 3.4

1.5 2.8

x

2.2

1 1.6

τl

0.5 10

(b) ul 1.00 0.91 0.82 0.73 0.64 0.55 0.45 0.36 0.27 0.18 0.09 0.00

1 0.8 0.6

ul 0.4 0.2 01

2 0.8

x

1.5 0.6

1 0.4

0.2

0.5

τl

00

Fig. 6. Three-dimensional profiles of (a) gaseous and (b) aqueous CO2 concentrations with the conditions of Rd = 30 lm and n = 105 cm3.

τ l =0.25 τ l =0.5 τ l =0.75 τ l =1

0.75

0.7

0

0.2

0.4

0.6

0.8

1

x Fig. 7. Spatial distributions of (a) gaseous and (b) aqueous CO2 concentrations at various absorption times with the conditions Rd = 30 lm and n = 106 cm3.

where its value is 9.839  105. Similar to the observations in Fig. 6, the gaseous concentration is almost invariant as the absorption time reaches 0.5. Considering the mass transfer process in the droplet, Fig. 7b clearly indicates the behavior of CO2 penetration from the droplet surface toward the droplet center at sl = 0.25, stemming from the concentration gradient exhibited inside the droplet. Only when the absorption time is larger than or equal to 0.75, the aqueous CO2 concentration changes no more, implying that the droplet is in the saturated state. 3.4. Absorption process and absorption rate Temporal distributions of dimensionless CO2 absorption amount (m) at various number densities are examined in Fig. 8 where the physical scale is defined by

R 2p R p R Rd m¼

0

0

0

ul r 2 sin hdrdhdw R3d =3

4p

¼3

Z

1

ul x2 dx

ð27Þ

0

As can be seen in the figure, the curves with n = 103 and 104 cm3 almost overlap completely. It follows that the droplet–droplet interaction on the absorption process can be disregarded under the aforementioned conditions. With the condition of n = 105 cm3, the absorption process departs from the previous two curves slightly. This reveals that the CO2 capture process is affected by

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W.-H. Chen et al. / Applied Energy 92 (2012) 185–193

1.2

(a)

1

1

0.42

0.98 0.41

0.6

0.4

0.94

τl,qss

0.96

mqss

m

0.8

3

n=10 n=104 5 n=10 n=106

0.4

0.2

0.39

mqss τl,qss

0.9

0.2

0.4

τl

0.6

0.8

0.88 3 10

1

Fig. 8. Temporal distributions of nondimensional CO2 absorption amount by a droplet at various number densities with the condition of Rd = 30 lm.

the droplet mutual interaction to a small extent. With further increasing the number density to 106 cm3, the influence of interaction on the initial uptake process is not perceived. However, with time marching the effect of interaction on the capture process tends to be exhibited. The curve with n = 106 cm3 suggests that the steady-state absorption amount (m) is 0.90. Accordingly, it should be emphasized that the droplet–droplet interaction abates the CO2 uptake amount of a droplet around 10%, in contrast to the droplet without interaction. Based on the distributions shown in Fig. 8, the profiles of quasisaturated CO2 absorption amount mqss and the quasi-saturated time sl,qss are provided in Fig. 9a. The quasi-saturated time is identified at the moment when the absorption time reaches the state of m = mqss = 0.99mss. The subscript ss represents the saturated state. It is evident that the higher the number density, the lower the quasi-saturated absorption amount and the quasi-saturated time. In particular, once the number density is increased from 105 to 106 cm3, a pronounced drop in mqss and sl,qss occurs. Furthermore, the distribution of the droplet absorption rate is presented in Fig. 9b where the absorption rate is defined as mqss divided by sl,qss. The declining extent in mqss is beyond that in sl,qss; it is thus found that the absorption rate decreases with increasing number density. Fig. 10 shows the profiles of relative mass of CO2 contained in droplets, bubbles and the space outside the bubbles at the four number densities. When the number density is in the range of 103–105 cm3, most of the CO2 is retained in the bubbles which accounts for 73–74% of the total mass. Alternatively, around 26% of CO2 is contained in the space outside the bubbles. Only around 0.01–1% of CO2 is absorbed by the droplets. Once the number density is enlarged to 106 cm3, approximately 10% of CO2 is absorbed into the droplets. Accordingly, from the perspective of CO2 capture using sprays, the number density of 106 cm3 is recommended, as the result of a large portion of CO2 absorbed by droplets.

0.38 106

105

(b) 2.38 2.36

2.34

2.32

2.3

2.28 3 10

4

5

10

6

10

10

n Fig. 9. Profiles of (a) quasi-saturated CO2 absorption amount and the quasisaturated time as well as (b) absorption with the condition of Rd = 30um.

120

100

Outside bubble In bubble In droplet

80

60

40

20

0

3.5. Influence of droplet radius

104

n

Absorption percentage (%)

0

Absorptionrate (=m qss / τl,qss)

0

0.92

3

10

4

5

10

10

6

10

n The influence of droplet size on the absorption process is evaluated in Figs. 11 and 12 where the number density is fixed at 106 cm3. With a fixed number density, more absorbent (i.e. water) is injected when the droplet size is increased. As a consequence, it is not surprising that the solute concentrations at the droplet surface and the bubble surface are lower as the droplet size is larger. For the droplet radius of 10 lm, the mutual interaction affecting

Fig. 10. Profiles of relative mass of CO2 contained in droplets, bubbles and the space outside the bubbles at various number densities under the condition of Rd = 30um.

the uptake process is slight, even though the number density is as high as 106 cm3. In contrast, once the droplet radius is increased to 50 lm, the interaction is so strong that the

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(a) 1 0.9 1.1

0.8

1

R a=10 μm R a=20 μm R a=30 μm R a=40 μm R a=50 μm

0.7

0.6

0.5

us

us

0.9 0.8 0.7 0.6 0.5

0

0.2

0.4

0

0.05

0.6

τl

0.1

τl

0.15

0.8

0.2

1

(b) 1 0.9

ubs

0.8

0.7

R a=10 μm R a=20 μm R a=30 μm R a=40 μm R a=50 μm

0.6

0.5

0

0.2

0.4

0.6

τl

0.8

1

Fig. 11. Temporal distributions of nondimensional CO2 concentration at (a) the droplet surface and (b) the bubble surface under various droplet radii with the condition of n = 106 cm3.

Absorption percentage (%)

120

100

Outside bubble In bubble In droplet

80

60

40

20

0

10

20

30

40

concentrations at the droplet and the bubble surfaces approach 0.52, much lower than those of Rd = 10–30 lm. This means that the CO2 uptake capacity of a droplet is reduced by 48% in comparison with that without droplet–droplet interaction. Upon inspection of the relative mass of CO2 contained in droplets, bubbles and the space outside the bubbles shown in Fig. 12, it can be seen that the absorption percentage of CO2 by the droplets grows markedly with increasing droplet radius. Specifically, the mass percentage of CO2 transported into droplets is lifted from 0.35% to 47.8% as the droplet radius increases from 10 to 50 lm. Alternatively, it is decreased from 73.6% to 23.7% for CO2 in bubbles. With regard to the mass of CO2 outside the bubbles, its variation is slight, altering from 26% to 28.5%. In summary, with the droplet radius of 50 lm and the number density of 106 cm3, around 50% of CO2 can be captured from the gas phase, even though the droplet mutual interaction is strong.

4. Conclusions Conceiving single droplets individually enveloped by the spheres of influence, a method of predicting carbon dioxide capture by single droplets in sprays under the impact of droplet– droplet interaction has been successfully developed in the current study. Typical droplet radii and number densities in sprays have been taken into consideration where the former and the latter are in the ranges of 10–50 lm and 103–106 cm3, respectively. With the droplet radius of 30 lm and the number densities between 103 and 105 cm3, the predictions suggest that the impact of droplet–droplet interaction on CO2 capture process is slight so that the absorption process is close to that in the absence of droplet mutual interaction. Under these situations, relatively few droplets are injected so that the total amount of CO2 transported from the gas phase to the liquid phase is less than 1%. When the number density is promoted to 106 cm3, the droplet–droplet interaction makes the CO2 absorption amount by a droplet be reduced by 10%. Nevertheless, around 10% of CO2 in the gas phase can be removed by droplets. As a whole, an increase in number density or droplet mutual interaction decreases CO2 capture amount, uptake period and absorption rate of a droplet. With the number density of 106 cm3, when the droplet radius is as large as 50 lm, the droplet mutual interaction is so pronounced that the absorption capacity of a droplet is reduced by 48%. However, CO2 removal from the gas phase to the liquid phase is lifted from 0.35% to 47.8% when the droplet radius is increased from 10 to 50 lm. Accordingly, from the viewpoint of CO2 capture in scrubbers, the number density of droplet and the droplet radius in a spray at least should be controlled at 106 cm3 and 50 lm, respectively, so as to effectively remove around 50% of CO2 from the gas phase to the liquid phase. Besides, a higher number density will decrease the absorption period of a droplet, rendering that the length or the size of the scrubber can be reduced. This means that the cost down of the facility can be achieved. Though the effect of convective flow is ignored in the present study, the Froesslling equation can be used to account for the enhancement of convective flow or Reynolds number on the mass transfer when the convective flow is regarded. The momentum equations have to be taken into account if CO2 captured by convecting droplets is investigated and this topic deserves further investigation in the future.

50

R d (μm) Fig. 12. Profiles of relative mass of CO2 contained in droplets, bubbles and the space outside the bubbles at various droplet radii with the condition of n = 106 cm3.

Acknowledgment The authors gratefully acknowledge the financial support of the National Science Council, Taiwan, ROC, on this study.

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