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An Unsteady Heterogeneous Mass Transfer Model for Gas Absorption Enhanced by Dispersed Third Phase Droplets
SEPARATION SCIENCE AND ENGINEERING Chinese Journal of Chemical Engineering, 17(4) 602ü607 (2009)
An Unsteady Heterogeneous Mass Transfer Model for Gas Absorption Enhanced by Dispersed Third Phase Droplets* SHEN Shuhua (ಋ೮)ܟ, MA Youguang (৴ဖ**)ڛ, LU Sumin (ഭਖ਼) and ZHU Chunying (ᅋҝ)
School of Chemical Engineering and Technology, State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300072, China Abstract A model for one-dimensional unsteady heterogeneous mass transfer was developed based on Danckwerts’ surface renewal theory in order to describe the mass transfer enhancement of absorption process for a slightly soluble gas in a gas-liquid-liquid system. The model accounts for the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity on mass transfer. An analytical solution for enhancement factor was obtained by Laplace domain transformation. The absorption rates of carbon dioxide in the dodecane-in-water and castor oil-in-water systems were measured in a thermostatic reactor, and the enhancement factors were calculated at different volume fractions of dispersed phase and stirrer speeds. The model predictions agree well with the experimental data. Keywords enhancement factor, mass transfer, absorption, gas-liquid-liquid
1
INTRODUCTION
Mass transfer in gas-liquid-liquid system is of increasing interest due to its widely application in bioprocess industry and homogeneous catalysis systems [1]. Experiments have shown that the gas-liquid mass transfer rate can be enhanced by adding a dispersed liquid phase in which gas is more soluble than in the continuous phase [25]. The mechanism of the enhancement was mostly explained by the “shuttle effect” [6, 7], in which the dispersed droplets were supposed to transfer an additional amount of gas to the liquid bulk through the adsorption in the gas-liquid diffusion layer and desorption in the bulk, consequently, resulting in an enhanced concentration gradient of the gas solute near the interface. Many theoretical models were developed for describing the effect of the dispersed phase droplets on gas absorption enhancement and could be mainly classified as “stationary” or “instationary” and “pseudo-homogeneous” or “heterogeneous” models, which could be further subdivided as 1-D, 2-D and 3-D models [812]. The advantages of the homogeneous models are their numerical simplicity and shorter computation time. Although these models can predict the changing trend of the enhancement factor with operation conditions, such as gas-liquid contact time and relative solubility, some of the assumptions are obviously unreasonable. For example, the mass transfer resistance within the dispersed phase is neglected, this may be suitable for solid particles with high diffusion coefficients, but for droplets, it will bring about marked deviation due to the relatively lower diffusion coefficients. Heterogeneous mass transfer models consider the local geometry of the dispersed phase near the gas-liquid interface, the mass transfer resis-
tance within the dispersed phase etc., so that it is more rational to describe the gas absorption, especially from 2-D or 3-D perspectives. Unfortunately, the computation is much more complex and it is usually difficult to attain an analytical solution. In the present wok, the effects of dispersed dodecane and castor oil on carbon dioxide absorption are investigated experimentally. A model for one-dimensional unsteady heterogeneous mass transfer is developed based on Danckwerts’ surface renewal theory by considering the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity. The model is solved analytically by Laplace domain transformation [13]. 2
EXPERIMENTAL
The experiments were carried out in a thermostatic reactor [(298.15±0.1)K] for CO2 (>99.5% mass fraction) absorption in dodecane-in-water emulsions [consisted of dodecane in water stabilized by 0.4% (volume fraction) Tween 20] and castor oil-in-water emulsions [consisted of castor oil in water stabilized by 3% (volume fraction) Tween 20]. The initial absorption pressure was 2.0265×105 Pa. As shown in Fig. 1, two stirrers were located centrally in the gas phase and the liquid phase for mixing uniformity. Four stainless steel baffles were mounted symmetrically to prevent tangential flows and increase the mixing effectiveness. A pressure difference transmitter was connected with the computer for real-time automatic data collection. Before each experiment, the emulsion was degassed by sparging nitrogen and renewed by carbon dioxide; the reactor was washed with acetone then distilled water. ˉ Tween 20, dodecane (viscosity 1.93×10 3 Pa·s) and castor oil (viscosity 0.715 Pa·s) were purchased
Received 2008-11-12, accepted 2009-03-11. * Supported by the National Natural Science Foundation of China (20176036). ** To whom correspondence should be addressed. E-mail: [email protected]
Chin. J. Chem. Eng., Vol. 17, No. 4, August 2009
603
Figure 1 Schematic set-up of the absorption in stainless stirred tank 1—N2 inlet valve; 2—CO2 inlet valve; 3—balance tank; 4—junction valve; 5—pressure difference transmitter connected with the computer; 6—gas outlet valve
from Tianjin Kermel Chemical Reagent Co., Ltd. For all the emulsion fractions applied, the mean droplets ˉ size (dp) of dodecane and castor oil are 4×10 6 m (1.0× ˉ6 ˉ5 ˉ5 ˉ6 ˉ 10 1.0×10 m) and 1.7×10 m (1.0×10 1.0×10 4 m), respectively, measured by Mastersizer S (Malvern Instruments Ltd, UK). The viscosities of dodecanein-water and castor oil-in-water emulsions were measured by the DV-III Ultra Programmable Rheometer (Brookfield Engineering Laboratories, Inc., USA). As shown in Figs. 2 and 3, the relative viscosity of emulsion to pure water can be expressed as a function of the volume fraction of dispersed phase as follows. Dodecane-water:
Pemulsion P water
21232Id5 9744.1Id4 1538.1Id3 2.12Id2 4.3161Id 1.0016
(1)
Castor oil-water:
Pemulsion Pwater
Figure 3 Effect of castor oil volume fraction on relative viscosity of emulsion to pure water Ʒexperimental data; üü fitting curve
3081.8Id4 1434.3Id3 205.16Id2 14.537Id 0.9999
(2)
decane and pure water was 1.93 calculated by the data in Ref. [15]. Lacking of the solubility of carbon dioxide in castor oil in the literature, the distribution coefficient of carbon dioxide in castor oil and pure water was measured as 1.73. 3 3.1
Figure 2 Effect of dodecane volume fraction on relative viscosity of emulsion to pure water Ʒexperimental data; üü fitting curve
The relative diffusion coefficients of carbon dioxide in dodecane and castor oil to pure water are 1.28 and 0.0058 respectively by Wilke-Chang Equation [14]. The distribution coefficient of carbon dioxide in do-
THEORETICAL MODEL Assumptions
As shown in Fig.4, we make following assumptions for modeling of gas absorption enhanced by dispersed phase droplets. (1) The gas-liquid interface in the reactor is planar in the range of stirrer speeds applied. (2) The dispersed phase droplets are small so that they are accommodated in the mass transfer region and are cubic ones with identical volumes [10]. (3) There is no contact between the gas and the dispersed phase. (4) The thickness of the continuous phase layer near the gas-liquid interface (į1) is dp/4 [13, 16]. (5) The distribution of the dispersed droplets near the gas-liquid interface is the same as that in the bulk phase [11].
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0, x ı 0, CAII
t
t ! 0, x
0 A*
0, CAII
t ! 0, x G1 , CAII
(9)
AII
Equations (8) and (9), transformed into the Laplace domain ( P S ), yield the following solution Figure 4
Sketch of mass transfer in a surface element
(6) The effect of the droplets adhering to the gas-liquid interface on mass transfer rate is dominating, and the effect of droplets next to them is negligible. (7) The relationship of mass transfer coefficient kl and viscosity ȝ is kl v P Z (Ȧ>0), where the parameter Ȧ is defined as the influence factor of emulsion viscosity on mass transfer coefficient. Due to the addition of the dispersed liquid phase, the emulsion viscosity is usually higher than that of pure water, which decreases the diffusion coefficient and increases the gas-liquid contact time [17, 18]. 3.2
CAII
sinh c1 x A* sinh ¬ªc1 G1 x ¼º AII S sinh c1G1 sinh c1G1
For the dispersed droplets adhering to the gas-liquid interface, the differential mass balance equation is wCAd wt
wCAI wC 2 DA AI wt wx 2 The initial and boundary conditions are:
t
0, x ı 0, CAI
t ! 0, x
0, CAI
t
0, x ı 0, CAd
t ! 0, x G1 , CAd
CAd
wCAII wx
Dd
0
wCAd wx (12)
m AII
e c2G1
e c2 x
(13)
where
(3)
c2
S Dd
(14)
Transformed into the Laplace domain (P S), Eq. (12) leads to
0 A*
A exp c1 x S
(5)
DA
wCAII wx
S DA
(6)
k12 (7) S DA For the continuous phase in region II, the differential mass balance equation is
(8)
Dd
wCAd wx
(15)
With Eqs. (10) and (13), Eq.(15) gives:
AII
where
wCAII wC 2 DA AII wt wx 2 with the following conditions:
mAII ; DA
0
Equations (11) and (12), transformed into the Laplace domain (P S), yield the following solution
c1 DA A* S sinh c1G1
*
c1
(11)
wx 2
t ! 0, x o f, CAd
t ! 0, x o f, CAI Cb | 0 (4) Equations (3) and (4), transformed into the Laplace domain ( P S ), yield the following solution CAI
2 wCAd
Dd
The initial and boundary conditions are:
Governing equations and solutions
For the convenience of calculation, the continuous phase close to the gas-liquid interface is divided into two regions: without droplets adhering (region I) and with droplets adhering (region II), as shown in Fig. 4. For the continuous phase in region I, the differential mass balance equation is
(10)
DA c1ctgh c1G1 Dd c2 m
(16)
For the continuous phase near the gas-liquid interface, the overall solute concentration AO can be fitted as: AO
AII] AI 1 ]
Transformed into the Laplace domain ( P (17) becomes
AO
AII] AI 1 ]
(17) S ), Eq.
(18)
where AI
CAI
x G1
A* exp c1G1 S
(19)
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and ȗ is the volume fraction of the droplets near the gasliquid interface. Based on assumption (5), ȗ is given as: ] Id (20) For the continuous phase near the gas-liquid interface, the overall differential mass balance equation is wCAO wC 2 DA AO (21) wt wx 2 with the following conditions: t 0, x ı 0, CAO 0 t ! 0, x
A*
0, CAO
t ! 0, x G1 , CAO AO (22) Equations (21) and (22), transformed into the Laplace domain ( P S ), yield the following solution CAO
sinh c1 x A* sinh ª¬c1 G1 x º¼ AO S sinh c1G1 sinh c1G1
(23)
With the surface renewal theory [13], the absorption rate with the dispersed phase, Jemulsion, can be described as J emulsion
f
³0 DA
wCAO wx
Se St dt
DA S
x 0
wCAO wx
x 0
(24) The absorption rate in pure water Jwater is J water
f
³0 DA
wCAI wx
Se St dt
DA S
x 0
wCAI wx
x 0
(25) 3.3
The enhancement factor based on “shuttle effect” is defined as § gas absorption flux · ¨ ¸ ¨ with dispersed phase ¸ ¨ gas absorption flux ¸ ¨ ¸ under similar © without dispersed phase ¹hydrodynamic conditions
(26) Substitution of Eqs. (16), (19), (23), (24) and (25) into Eq. (26) gives ESE
· 1 § k1 § k · ° G1 ¸ ctgh ¨ 1 G1 ¸ ®Id sinh ¨ © DA ¹ © DA ¹ °¯ ª º § k1 · G1 ¸ m Dr » «ctgh ¨ «¬ »¼ © DA ¹ § k · sinh 1 ¨ 1 G1 ¸ © DA ¹
Dr
1
1 Id e
Dd DA
(28)
To our knowledge, the influence of emulsion viscosity on mass transfer was seldom considered in the literature. Evidently, the emulsion viscosity affects the hydrodynamic behavior of the system, increasing the gas-liquid contact time and decreasing the diffusion coefficient [17, 18]. According to assumption (7), the relative mass transfer coefficient of the emulsion system to the pure water can be described as Z
§ kl,emulsion · § P water · (29) ¨ ¸ ¨ ¸ © kl,water ¹ © Pemulsion ¹ Combining Eqs. (27) and (29), we obtain the overall enhancement factor Z
EOE
§ k1 · § P water · ¨P ¸ ®ctgh ¨ D G1 ¸ © A ¹ © emulsion ¹ ¯
ª º § k · sinh 1 ¨ 1 G1 ¸ k « 1 G1 » D © A ¹ «Id 1 Id e DA » « » § k1 · « ctgh ¨ D G1 ¸ m Dr » © A ¹ ¬ ¼ § k · °½ sinh 1 ¨ 1 G1 ¸ ¾ © DA ¹ °¿ (30) where kl is calculated according to Ref. [19]. 4
Enhancement factor
ESE
where
RESULTS AND DISCUSSION
The models proposed in the literature are mostly solved numerically and usually contain many unknown parameters, so that the precise and rapid prediction of enhancement factors is considerably difficult. It is always desirable to have an analytical solution for both theoretical study and practical applications. In this work, the model proposed was solved analytically by Laplace domain transformation, and especially, the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity on mass transfer were taken into account. Only one parameter Ȧ needs to be determined for calculating the enhancement factor. 4.1 Effect of dispersed phase on gas absorption
k1 G1 DA
°½ ¾ °¿
(27)
As shown in Fig. 5, the addition of dodecane enhances the carbon dioxide absorption substantially. The enhancement factor first decreases with the increase of dodecane volume fraction at the beginning and then increases gradually. The initial decrease in the mass transfer rate may be explained in the view of surface chemistry, for instance, the formation of a
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oxide in the dispersed phase to pure water (Dr>1). The fitted values of influence factor Ȧ are shown in Table 1. Table 1
Model parameter Ȧ at different stirrer speeds ˉ1
Stirrer speeds/s
Ȧ
4
0.150
5
0.106
6
0.034
ˉ1
(a) N 4 s
(b) N 5 sˉ1
As shown in Fig. 6, the addition of castor oil decreases the absorption rate of carbon dioxide, although the distribution coefficient of carbon dioxide in castor oil and pure water is higher ( m ! 1 ), and the enhancement factor ( E 1 ) decreases with the increase of castor oil volume fraction. With lower relative diffusion coefficient of carbon dioxide in castor oil to pure water ( Dr 1 ), the mass transfer resistance within the castor oil is much higher than that in water, so that the presence of castor oil results in an additional mass transfer resistance near the gas-liquid interface. Fig. 6 shows that the present model is more accurate for lower relative diffusion coefficient of carbon dioxide in the dispersed phase to pure water ( Dr 1 , Ȧ 0.346).
ˉ (c) N 6 s 1
Figure 5 Influence of dodecane volume fraction on enhancement factor at different stirrer speeds Ʒexperimental data; üü calculated value with the effect of emulsion viscosity; calculated value without the effect of emulsion viscosity
rigid or semi-rigid surface structure may dramatically impede the liquid flow near the surface [20]. With the increase of dodecane volume fraction, more droplets are accommodated in the mass transfer layer. Since the mass transfer resistance of carbon dioxide in dodecane is lower than that in pure water ( Dr ! 1 ) and the distribution coefficient of carbon dioxide in dodecane and water is higher ( m ! 1 ), the droplets absorb more dissolved gas solute and then return to the bulk phase. As a result, the solute concentration gradient near the interface increases. On the other hand, the increase of emulsion viscosity will decrease the mass transfer coefficient. Therefore, enhancement or decline of gas absorption in emulsion depends on the competition of above two aspects. Compared with the experimental data, the present model is more accurate for higher relative diffusion coefficient of carbon di-
Figure 6 Influence of castor oil volume fraction on enhancement factor (stirrer speed 2 sˉ1) Ÿexperimental data; üü calculated value with the effect of emulsion viscosity; calculated value without the effect of emulsion viscosity
4.2 Effect of emulsion viscosity on enhancement factor
The comparison of predicted values of enhancement factor between the two models shows that the model taking into account the influence of emulsion viscosity on mass transfer is more accurate, as shown in Figs. 5 and 6. The effect of emulsion viscosity on enhancement factor is significant and should not be neglected. However, Ȧ decreases as the stirring intensity increases, so that its influence on enhancement factor decreases accordingly. It could be ascribed to the collision of droplets to the boundary layer in higher agitation intensity, resulting in higher surface renewal frequency, hence, the negative effect of emulsion
Chin. J. Chem. Eng., Vol. 17, No. 4, August 2009
viscosity on mass transfer becomes inconspicuous even negligible. Based on the discussion above, the enhancement of gas absorption with the dispersed phase is due to the enhanced solute concentration gradient near the interface. In addition, the effect of emulsion viscosity on mass transfer coefficient should be considered for better prediction of the enhancement factors. 5
(1) A model for one-dimensional unsteady heterogeneous mass transfer was developed considering the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity. An analytical solution for the enhancement factor was obtained by Laplace domain transformation. (2) The effects of dodecane ( Dr ! 1 ) and castor oil ( Dr 1 ) on carbon dioxide absorption were investigated experimentally. The addition of dodecane enhances the absorption of carbon dioxide, while the addition of castor oil tends to decrease the mass transfer. (3) The emulsion viscosity has an obvious effect on mass transfer at lower stirrer speeds. The present model is in good agreement with the experimental results for both dodecane ( Dr ! 1 ) and castor oil ( Dr 1 ) as the dispersed phase droplets added. NOMENCLATURE
Ȧ
̝
Laplace transformation
b d emulsion O OE
bulk phase dispersed phase emulsion system overall overall enhancement factor
dp E J kl m N S į1 ȗ ȝ
Id
ˉ
Superscripts Subscripts
1
3 4 5 6 7 8 9 10 11
solute concentration in the continuous phase, mol·m 3 ˉ solute concentration at the gas-liquid interface, mol·m 3 ˉ solute concentration, mol·m 3 ˉ diffusion coefficient, m2·s 1 relative diffusion coefficient of solute in dispersed phase to pure water mean droplets size, m enhancement factor ˉ ˉ absorption rate of gas, mol·m 2·s 1 ˉ mass transfer coefficient, m·s 1 distribution coefficient of solute in dispersed phase and pure water ˉ stirrer speed, s 1 ˉ surface renewal frequency, s 1 thickness of continuous phase close to the gas-liquid interface, m volume fraction of dispersed phase near the gas-liquid interface viscosity, Pa·s volume fraction of dispersed phase influence factor of viscosity on mass transfer coefficient
enhancement factor based on “shuttle effect” water system continuous phase in region I continuous phase in region II
REFERENCES
2
CONCLUSIONS
A A* CA D Dr
SE water I II
607
12 13 14 15 16 17 18 19 20
Dumont, E., Delmas, H., “Mass transfer enhancement of gas absorption in oil-in-water systems: A review”, Chem. Eng. Pro., 42 (6), 419438 (2003). Bruining, W.J., Joosten, G.E.H., Beenackers, A.A.C.M., Hofman, H., “Enhancement of gas-liquid mass transfer by a dispersed second liquid phase”, Chem. Eng. Sci., 41 (7), 18731877 (1986). Littel, R.J., Versteeg, G.F.A., van Swaaij, W.P.M., “Physical absorption of CO2 and propene into toluene/water emulsions”, AIChE J., 40 (10), 16291638 (1994). van Ede, C.J., van Houten, R., Beenackers, A.A.C.M., “Enhancement of gas to water mass transfer rates by a dispersed organic phase”, Chem. Eng. Sci., 50 (18), 29112922 (1995). Junker, B.H., Hatton, T.A., Wang, D.I.C., “Oxygen transfer enhancement in aqueous/perfluorocarbon fermention systems: I. Experimental abservation”, Biotechnol. Bioeng., 35 (6), 578585 (1990). Kars, R.L., Best, R.J., Drinkenburg, A.A.H., “The sorption of propane in slurries of active carbon in water”, Chem. Eng. J., 17 (3), 201210 (1979). Alper, E., Wichtendahl, B., Deckwer, W.D., “Gas absorption mechanism in catalytic slurry reactors”, Chem. Eng. Sci., 35 (1/2), 217222 (1980). Holstvoogd, R.D., van Swaaij, W.P.M., “The influence of adsorption capacity on enhanced gas absorption in activated carbon slurries”, Chem. Eng. Sci., 45 (1), 151162 (1990). Karve, S., Juvekar, V.A., “Gas absorption into slurries containing fine catalyst particles”, Chem. Eng. Sci., 45 (3), 587594 (1990). Nagy, E., “Three-phase mass transfer: One-dimensional heterogeneous model”, Chem. Eng. Sci., 50 (5), 827836 (1995). Lin, C., Zhou, M., Xu, C.J., “Axisymmetrical two-dimensional heterogeneous mass transfer model for the absorption gas into liquid-liquid dispersions”, Chem. Eng. Sci., 54 (3), 389399 (1999). Brilman, D.W.F., Goldschmidt, M.J.V., Versteeg, G.F., van Swaaij, W.P.M., “Heterogeneous mass transfer models for gas absorption in multiphase systems”, Chem. Eng. Sci., 55 (15), 27932812 (2000). Demmink, J.F., Mehra, A., Beenackers, A.A.C.M., “Gas absorption in the presence of particles showing interfacial affinity: case of the fine sulfur precipitates”, Chem. Eng. Sci., 53 (16), 28852902 (1998). Wilke, C.R., Chang, P., “Correlation of diffusion coefficients in dilute solutions”, AIChE J., 1 (2), 264270 (1955). Walter, H., Edgar, B.W., Philip, S., “Solubility of propane and carbon dioxide in heptane, dodecane, and hexadecane”, J. Chem. Eng. data., 17 (1), 5961 (1972). Vinke, H., Hamersma, P.J., Fortuin, J.M.H., “Enhancement of the gas-absorption rate in agitated slurry reactors by gas-adsorbing particles adhering to the gas bubbles”, Chem. Eng. Sci., 48 (12), 21972210 (1993). Zhang, J.M., Xu, C.J., Zhou, M., “The mechanism model of gas-liquid mass transfer enhancement by fine catalyst particles”, Chem. Eng. J., 120 (3), 149156 (2006). Zhang, G.D., Cai, W.F., Xu, C.J., Zhou, M., “A general enhancement factor model of the physical absorption of gases in multiphase systems”, Chem. Eng. Sci., 61 (2), 558568 (2006). Lekhal, A., Chaudhari, R.V., Wilhelm, A.M., Delmas, H., “Gas-liquid mass transfer in gas-lqiuid-liquid dispersions”, Chem. Eng. Sci., 52 (21/22), 40694077 (1997). Eckenfelder, W.W., Barnhart, E.L., “The effect of organic substances on the transfer of oxygen from air bubble in water”, AIChE J., 7 (4), 631634 (1961).
An Unsteady Heterogeneous Mass Transfer Model for Gas Absorption Enhanced by Dispersed Third Phase Droplets* SHEN Shuhua (ಋ೮)ܟ, MA Youguang (৴ဖ**)ڛ, LU Sumin (ഭਖ਼) and ZHU Chunying (ᅋҝ)
School of Chemical Engineering and Technology, State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300072, China Abstract A model for one-dimensional unsteady heterogeneous mass transfer was developed based on Danckwerts’ surface renewal theory in order to describe the mass transfer enhancement of absorption process for a slightly soluble gas in a gas-liquid-liquid system. The model accounts for the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity on mass transfer. An analytical solution for enhancement factor was obtained by Laplace domain transformation. The absorption rates of carbon dioxide in the dodecane-in-water and castor oil-in-water systems were measured in a thermostatic reactor, and the enhancement factors were calculated at different volume fractions of dispersed phase and stirrer speeds. The model predictions agree well with the experimental data. Keywords enhancement factor, mass transfer, absorption, gas-liquid-liquid
1
INTRODUCTION
Mass transfer in gas-liquid-liquid system is of increasing interest due to its widely application in bioprocess industry and homogeneous catalysis systems [1]. Experiments have shown that the gas-liquid mass transfer rate can be enhanced by adding a dispersed liquid phase in which gas is more soluble than in the continuous phase [25]. The mechanism of the enhancement was mostly explained by the “shuttle effect” [6, 7], in which the dispersed droplets were supposed to transfer an additional amount of gas to the liquid bulk through the adsorption in the gas-liquid diffusion layer and desorption in the bulk, consequently, resulting in an enhanced concentration gradient of the gas solute near the interface. Many theoretical models were developed for describing the effect of the dispersed phase droplets on gas absorption enhancement and could be mainly classified as “stationary” or “instationary” and “pseudo-homogeneous” or “heterogeneous” models, which could be further subdivided as 1-D, 2-D and 3-D models [812]. The advantages of the homogeneous models are their numerical simplicity and shorter computation time. Although these models can predict the changing trend of the enhancement factor with operation conditions, such as gas-liquid contact time and relative solubility, some of the assumptions are obviously unreasonable. For example, the mass transfer resistance within the dispersed phase is neglected, this may be suitable for solid particles with high diffusion coefficients, but for droplets, it will bring about marked deviation due to the relatively lower diffusion coefficients. Heterogeneous mass transfer models consider the local geometry of the dispersed phase near the gas-liquid interface, the mass transfer resis-
tance within the dispersed phase etc., so that it is more rational to describe the gas absorption, especially from 2-D or 3-D perspectives. Unfortunately, the computation is much more complex and it is usually difficult to attain an analytical solution. In the present wok, the effects of dispersed dodecane and castor oil on carbon dioxide absorption are investigated experimentally. A model for one-dimensional unsteady heterogeneous mass transfer is developed based on Danckwerts’ surface renewal theory by considering the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity. The model is solved analytically by Laplace domain transformation [13]. 2
EXPERIMENTAL
The experiments were carried out in a thermostatic reactor [(298.15±0.1)K] for CO2 (>99.5% mass fraction) absorption in dodecane-in-water emulsions [consisted of dodecane in water stabilized by 0.4% (volume fraction) Tween 20] and castor oil-in-water emulsions [consisted of castor oil in water stabilized by 3% (volume fraction) Tween 20]. The initial absorption pressure was 2.0265×105 Pa. As shown in Fig. 1, two stirrers were located centrally in the gas phase and the liquid phase for mixing uniformity. Four stainless steel baffles were mounted symmetrically to prevent tangential flows and increase the mixing effectiveness. A pressure difference transmitter was connected with the computer for real-time automatic data collection. Before each experiment, the emulsion was degassed by sparging nitrogen and renewed by carbon dioxide; the reactor was washed with acetone then distilled water. ˉ Tween 20, dodecane (viscosity 1.93×10 3 Pa·s) and castor oil (viscosity 0.715 Pa·s) were purchased
Received 2008-11-12, accepted 2009-03-11. * Supported by the National Natural Science Foundation of China (20176036). ** To whom correspondence should be addressed. E-mail: [email protected]
Chin. J. Chem. Eng., Vol. 17, No. 4, August 2009
603
Figure 1 Schematic set-up of the absorption in stainless stirred tank 1—N2 inlet valve; 2—CO2 inlet valve; 3—balance tank; 4—junction valve; 5—pressure difference transmitter connected with the computer; 6—gas outlet valve
from Tianjin Kermel Chemical Reagent Co., Ltd. For all the emulsion fractions applied, the mean droplets ˉ size (dp) of dodecane and castor oil are 4×10 6 m (1.0× ˉ6 ˉ5 ˉ5 ˉ6 ˉ 10 1.0×10 m) and 1.7×10 m (1.0×10 1.0×10 4 m), respectively, measured by Mastersizer S (Malvern Instruments Ltd, UK). The viscosities of dodecanein-water and castor oil-in-water emulsions were measured by the DV-III Ultra Programmable Rheometer (Brookfield Engineering Laboratories, Inc., USA). As shown in Figs. 2 and 3, the relative viscosity of emulsion to pure water can be expressed as a function of the volume fraction of dispersed phase as follows. Dodecane-water:
Pemulsion P water
21232Id5 9744.1Id4 1538.1Id3 2.12Id2 4.3161Id 1.0016
(1)
Castor oil-water:
Pemulsion Pwater
Figure 3 Effect of castor oil volume fraction on relative viscosity of emulsion to pure water Ʒexperimental data; üü fitting curve
3081.8Id4 1434.3Id3 205.16Id2 14.537Id 0.9999
(2)
decane and pure water was 1.93 calculated by the data in Ref. [15]. Lacking of the solubility of carbon dioxide in castor oil in the literature, the distribution coefficient of carbon dioxide in castor oil and pure water was measured as 1.73. 3 3.1
Figure 2 Effect of dodecane volume fraction on relative viscosity of emulsion to pure water Ʒexperimental data; üü fitting curve
The relative diffusion coefficients of carbon dioxide in dodecane and castor oil to pure water are 1.28 and 0.0058 respectively by Wilke-Chang Equation [14]. The distribution coefficient of carbon dioxide in do-
THEORETICAL MODEL Assumptions
As shown in Fig.4, we make following assumptions for modeling of gas absorption enhanced by dispersed phase droplets. (1) The gas-liquid interface in the reactor is planar in the range of stirrer speeds applied. (2) The dispersed phase droplets are small so that they are accommodated in the mass transfer region and are cubic ones with identical volumes [10]. (3) There is no contact between the gas and the dispersed phase. (4) The thickness of the continuous phase layer near the gas-liquid interface (į1) is dp/4 [13, 16]. (5) The distribution of the dispersed droplets near the gas-liquid interface is the same as that in the bulk phase [11].
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Chin. J. Chem. Eng., Vol. 17, No. 4, August 2009
0, x ı 0, CAII
t
t ! 0, x
0 A*
0, CAII
t ! 0, x G1 , CAII
(9)
AII
Equations (8) and (9), transformed into the Laplace domain ( P S ), yield the following solution Figure 4
Sketch of mass transfer in a surface element
(6) The effect of the droplets adhering to the gas-liquid interface on mass transfer rate is dominating, and the effect of droplets next to them is negligible. (7) The relationship of mass transfer coefficient kl and viscosity ȝ is kl v P Z (Ȧ>0), where the parameter Ȧ is defined as the influence factor of emulsion viscosity on mass transfer coefficient. Due to the addition of the dispersed liquid phase, the emulsion viscosity is usually higher than that of pure water, which decreases the diffusion coefficient and increases the gas-liquid contact time [17, 18]. 3.2
CAII
sinh c1 x A* sinh ¬ªc1 G1 x ¼º AII S sinh c1G1 sinh c1G1
For the dispersed droplets adhering to the gas-liquid interface, the differential mass balance equation is wCAd wt
wCAI wC 2 DA AI wt wx 2 The initial and boundary conditions are:
t
0, x ı 0, CAI
t ! 0, x
0, CAI
t
0, x ı 0, CAd
t ! 0, x G1 , CAd
CAd
wCAII wx
Dd
0
wCAd wx (12)
m AII
e c2G1
e c2 x
(13)
where
(3)
c2
S Dd
(14)
Transformed into the Laplace domain (P S), Eq. (12) leads to
0 A*
A exp c1 x S
(5)
DA
wCAII wx
S DA
(6)
k12 (7) S DA For the continuous phase in region II, the differential mass balance equation is
(8)
Dd
wCAd wx
(15)
With Eqs. (10) and (13), Eq.(15) gives:
AII
where
wCAII wC 2 DA AII wt wx 2 with the following conditions:
mAII ; DA
0
Equations (11) and (12), transformed into the Laplace domain (P S), yield the following solution
c1 DA A* S sinh c1G1
*
c1
(11)
wx 2
t ! 0, x o f, CAd
t ! 0, x o f, CAI Cb | 0 (4) Equations (3) and (4), transformed into the Laplace domain ( P S ), yield the following solution CAI
2 wCAd
Dd
The initial and boundary conditions are:
Governing equations and solutions
For the convenience of calculation, the continuous phase close to the gas-liquid interface is divided into two regions: without droplets adhering (region I) and with droplets adhering (region II), as shown in Fig. 4. For the continuous phase in region I, the differential mass balance equation is
(10)
DA c1ctgh c1G1 Dd c2 m
(16)
For the continuous phase near the gas-liquid interface, the overall solute concentration AO can be fitted as: AO
AII] AI 1 ]
Transformed into the Laplace domain ( P (17) becomes
AO
AII] AI 1 ]
(17) S ), Eq.
(18)
where AI
CAI
x G1
A* exp c1G1 S
(19)
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and ȗ is the volume fraction of the droplets near the gasliquid interface. Based on assumption (5), ȗ is given as: ] Id (20) For the continuous phase near the gas-liquid interface, the overall differential mass balance equation is wCAO wC 2 DA AO (21) wt wx 2 with the following conditions: t 0, x ı 0, CAO 0 t ! 0, x
A*
0, CAO
t ! 0, x G1 , CAO AO (22) Equations (21) and (22), transformed into the Laplace domain ( P S ), yield the following solution CAO
sinh c1 x A* sinh ª¬c1 G1 x º¼ AO S sinh c1G1 sinh c1G1
(23)
With the surface renewal theory [13], the absorption rate with the dispersed phase, Jemulsion, can be described as J emulsion
f
³0 DA
wCAO wx
Se St dt
DA S
x 0
wCAO wx
x 0
(24) The absorption rate in pure water Jwater is J water
f
³0 DA
wCAI wx
Se St dt
DA S
x 0
wCAI wx
x 0
(25) 3.3
The enhancement factor based on “shuttle effect” is defined as § gas absorption flux · ¨ ¸ ¨ with dispersed phase ¸ ¨ gas absorption flux ¸ ¨ ¸ under similar © without dispersed phase ¹hydrodynamic conditions
(26) Substitution of Eqs. (16), (19), (23), (24) and (25) into Eq. (26) gives ESE
· 1 § k1 § k · ° G1 ¸ ctgh ¨ 1 G1 ¸ ®Id sinh ¨ © DA ¹ © DA ¹ °¯ ª º § k1 · G1 ¸ m Dr » «ctgh ¨ «¬ »¼ © DA ¹ § k · sinh 1 ¨ 1 G1 ¸ © DA ¹
Dr
1
1 Id e
Dd DA
(28)
To our knowledge, the influence of emulsion viscosity on mass transfer was seldom considered in the literature. Evidently, the emulsion viscosity affects the hydrodynamic behavior of the system, increasing the gas-liquid contact time and decreasing the diffusion coefficient [17, 18]. According to assumption (7), the relative mass transfer coefficient of the emulsion system to the pure water can be described as Z
§ kl,emulsion · § P water · (29) ¨ ¸ ¨ ¸ © kl,water ¹ © Pemulsion ¹ Combining Eqs. (27) and (29), we obtain the overall enhancement factor Z
EOE
§ k1 · § P water · ¨P ¸ ®ctgh ¨ D G1 ¸ © A ¹ © emulsion ¹ ¯
ª º § k · sinh 1 ¨ 1 G1 ¸ k « 1 G1 » D © A ¹ «Id 1 Id e DA » « » § k1 · « ctgh ¨ D G1 ¸ m Dr » © A ¹ ¬ ¼ § k · °½ sinh 1 ¨ 1 G1 ¸ ¾ © DA ¹ °¿ (30) where kl is calculated according to Ref. [19]. 4
Enhancement factor
ESE
where
RESULTS AND DISCUSSION
The models proposed in the literature are mostly solved numerically and usually contain many unknown parameters, so that the precise and rapid prediction of enhancement factors is considerably difficult. It is always desirable to have an analytical solution for both theoretical study and practical applications. In this work, the model proposed was solved analytically by Laplace domain transformation, and especially, the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity on mass transfer were taken into account. Only one parameter Ȧ needs to be determined for calculating the enhancement factor. 4.1 Effect of dispersed phase on gas absorption
k1 G1 DA
°½ ¾ °¿
(27)
As shown in Fig. 5, the addition of dodecane enhances the carbon dioxide absorption substantially. The enhancement factor first decreases with the increase of dodecane volume fraction at the beginning and then increases gradually. The initial decrease in the mass transfer rate may be explained in the view of surface chemistry, for instance, the formation of a
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oxide in the dispersed phase to pure water (Dr>1). The fitted values of influence factor Ȧ are shown in Table 1. Table 1
Model parameter Ȧ at different stirrer speeds ˉ1
Stirrer speeds/s
Ȧ
4
0.150
5
0.106
6
0.034
ˉ1
(a) N 4 s
(b) N 5 sˉ1
As shown in Fig. 6, the addition of castor oil decreases the absorption rate of carbon dioxide, although the distribution coefficient of carbon dioxide in castor oil and pure water is higher ( m ! 1 ), and the enhancement factor ( E 1 ) decreases with the increase of castor oil volume fraction. With lower relative diffusion coefficient of carbon dioxide in castor oil to pure water ( Dr 1 ), the mass transfer resistance within the castor oil is much higher than that in water, so that the presence of castor oil results in an additional mass transfer resistance near the gas-liquid interface. Fig. 6 shows that the present model is more accurate for lower relative diffusion coefficient of carbon dioxide in the dispersed phase to pure water ( Dr 1 , Ȧ 0.346).
ˉ (c) N 6 s 1
Figure 5 Influence of dodecane volume fraction on enhancement factor at different stirrer speeds Ʒexperimental data; üü calculated value with the effect of emulsion viscosity; calculated value without the effect of emulsion viscosity
rigid or semi-rigid surface structure may dramatically impede the liquid flow near the surface [20]. With the increase of dodecane volume fraction, more droplets are accommodated in the mass transfer layer. Since the mass transfer resistance of carbon dioxide in dodecane is lower than that in pure water ( Dr ! 1 ) and the distribution coefficient of carbon dioxide in dodecane and water is higher ( m ! 1 ), the droplets absorb more dissolved gas solute and then return to the bulk phase. As a result, the solute concentration gradient near the interface increases. On the other hand, the increase of emulsion viscosity will decrease the mass transfer coefficient. Therefore, enhancement or decline of gas absorption in emulsion depends on the competition of above two aspects. Compared with the experimental data, the present model is more accurate for higher relative diffusion coefficient of carbon di-
Figure 6 Influence of castor oil volume fraction on enhancement factor (stirrer speed 2 sˉ1) Ÿexperimental data; üü calculated value with the effect of emulsion viscosity; calculated value without the effect of emulsion viscosity
4.2 Effect of emulsion viscosity on enhancement factor
The comparison of predicted values of enhancement factor between the two models shows that the model taking into account the influence of emulsion viscosity on mass transfer is more accurate, as shown in Figs. 5 and 6. The effect of emulsion viscosity on enhancement factor is significant and should not be neglected. However, Ȧ decreases as the stirring intensity increases, so that its influence on enhancement factor decreases accordingly. It could be ascribed to the collision of droplets to the boundary layer in higher agitation intensity, resulting in higher surface renewal frequency, hence, the negative effect of emulsion
Chin. J. Chem. Eng., Vol. 17, No. 4, August 2009
viscosity on mass transfer becomes inconspicuous even negligible. Based on the discussion above, the enhancement of gas absorption with the dispersed phase is due to the enhanced solute concentration gradient near the interface. In addition, the effect of emulsion viscosity on mass transfer coefficient should be considered for better prediction of the enhancement factors. 5
(1) A model for one-dimensional unsteady heterogeneous mass transfer was developed considering the mass transfer resistance within the dispersed phase and the effect of emulsion viscosity. An analytical solution for the enhancement factor was obtained by Laplace domain transformation. (2) The effects of dodecane ( Dr ! 1 ) and castor oil ( Dr 1 ) on carbon dioxide absorption were investigated experimentally. The addition of dodecane enhances the absorption of carbon dioxide, while the addition of castor oil tends to decrease the mass transfer. (3) The emulsion viscosity has an obvious effect on mass transfer at lower stirrer speeds. The present model is in good agreement with the experimental results for both dodecane ( Dr ! 1 ) and castor oil ( Dr 1 ) as the dispersed phase droplets added. NOMENCLATURE
Ȧ
̝
Laplace transformation
b d emulsion O OE
bulk phase dispersed phase emulsion system overall overall enhancement factor
dp E J kl m N S į1 ȗ ȝ
Id
ˉ
Superscripts Subscripts
1
3 4 5 6 7 8 9 10 11
solute concentration in the continuous phase, mol·m 3 ˉ solute concentration at the gas-liquid interface, mol·m 3 ˉ solute concentration, mol·m 3 ˉ diffusion coefficient, m2·s 1 relative diffusion coefficient of solute in dispersed phase to pure water mean droplets size, m enhancement factor ˉ ˉ absorption rate of gas, mol·m 2·s 1 ˉ mass transfer coefficient, m·s 1 distribution coefficient of solute in dispersed phase and pure water ˉ stirrer speed, s 1 ˉ surface renewal frequency, s 1 thickness of continuous phase close to the gas-liquid interface, m volume fraction of dispersed phase near the gas-liquid interface viscosity, Pa·s volume fraction of dispersed phase influence factor of viscosity on mass transfer coefficient
enhancement factor based on “shuttle effect” water system continuous phase in region I continuous phase in region II
REFERENCES
2
CONCLUSIONS
A A* CA D Dr
SE water I II
607
12 13 14 15 16 17 18 19 20
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