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Analysis of CO2 removal by hollow fiber membrane contactors
Journal of Membrane Science 194 (2001) 57–67
Analysis of CO2 removal by hollow fiber membrane contactors Yongtaek Lee a,∗ , Richard D. Noble a , Bong-Yeol Yeom b , You-In Park b , Kew-Ho Lee b b
a Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424, USA Membrane and Separation Center, Korea Research Institute of Chemical Technology, P.O. Box 107, Daedeok-Danji, Taejon 305-606, South Korea
Received 29 September 2000; received in revised form 22 May 2001; accepted 29 May 2001
Abstract A numerical analysis was performed to investigate the removal behavior of carbon dioxide using a hollow fiber membrane contactor in which an aqueous potassium carbonate solution flows as an absorbent. An important feature is the inclusion of the nonlinear reversible reaction terms that result from the reactions between the absorbent solution and carbon dioxide. The coupled nonlinear partial differential equations were derived and solved numerically, which describe either the absorption or desorption of carbon dioxide in a membrane contactor. In a numerical analysis, the absorbent liquid always flows through the inner side of the hollow fiber in both absorption and desorption steps. The analysis includes the effects of the pressure of feed gases and the extent of vacuum, the absorbent flow rates, the concentration of potassium carbonate in an absorbent solution, the diameter and length of the fibers. One important result is the determination of the optimal absorbent flow rate such that it is saturated just at the exit of the contactor. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Facilitated transport; Gas separations; Modules; Membrane contactors; Numerical simulation
1. Introduction Gas mixtures are usually dispersed and contacted with liquids in conventional gas absorption processes where several types of devices are used: a packed tower, a spray tower, a venturi scrubber or a bubble column. Dispersion in these contacting devices has many drawbacks. Membrane-based contacting devices could overcome these shortcomings by preventing gas dispersion in the contacting phase. The absorbent liquid flows on one side and the gas flows on the other side of the membrane.
∗ Corresponding author. On leave from Chungnam National University, South Korea. Tel.: +82-42-821-5686; fax: +82-42-822-8995. E-mail addresses: [email protected], [email protected] (Y. Lee).
Qi and Cussler developed the idea of the hollowfiber contactor using a microporous non-wetted polypropylene hollow fiber membrane for absorption of carbon dioxide where aqueous sodium hydroxide solution was used as an absorbent [1,2]. Teramoto et al. also proposed a similar technique and applied it for separation of ethylene from ethane using an aqueous silver nitrate solution [3]. These membrane contactors were used by several authors in a shape of either a flat membrane or a hollow fiber membrane to absorb the various gases such as carbon monoxide, carbon dioxide, sulfur dioxide [4–8]. Sirkar recently reviewed the gas absorption processes based on the membrane contactors [9]. Separation of carbon dioxide from nitrogen was extensively explored by Karoor and Sirkar with either pure water or an aqueous amine solution [10]. They carried out comprehensive absorption experiments on gas separation including pure
0376-7388/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 1 ) 0 0 5 2 4 - 5
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Nomenclature C¯ i DAt DBt DK A DK B G
H k1 k−1 k2 k−2 K1 K2 K3 K4 L m PeA PeB r rt r¯ vl z z¯
dimensionless concentration of component ‘i’ diffusion coefficient of CO2 (m2 s−1 ) diffusion coefficient of HCO3 −2 (m2 s−1 ) dimensionless parameter as defined in Eq. (14a) dimensionless parameter as defined in Eq. (14a) dimensionless ratio of CO2 concentration over HCO3 2− concentration at wall solubility of CO2 in absorbent solution (mol dm−3 atm−1 ) forward rate of constant of reaction (1) (s−1 ) reverse rate of constant of reaction (1) (dm mol−1 s−1 ) forward rate of constant of reaction (2) (dm3 mol−1 s−1 ) reverse rate of constant of reaction (2) (s−1 ) equilibrium constant of Eq. (3) (mol dm−3 ) equilibrium constant of Eq. (4) (mol2 dm−6 ) equilibrium constant of Eq. (1) (mol dm−3 ) equilibrium constant of Eq. (2) (mol−1 dm3 ) length of fiber (m) initial molar concentration of K2 CO3 (mol dm−3 ) Peclet number as defined in Eq. (14a) Peclet number as defined in Eq. (14a) radial coordinate (m) radius of fiber (m) dimensionless radial coordinate average liquid velocity in tube side (m s−1 ) axial coordinate (m) dimensionless axial coordinate
Greek letters α parameter as defined in Eq. (10) β parameter as defined in Eq. (10)
φ γ τ
dimensionless parameter as defined in Eq. (13) dimensionless parameter as defined in Eq. (14) dimensionless parameter as defined in Eq. (14)
carbon dioxide, pure sulfur dioxide, and its mixture with nitrogen or air using a microporous hydrophobic hollow-fiber module in a parallel flow configuration. They also proposed a theoretical model to describe a system with an absorbent of pure water resulting in partial differential equations; solved them numerically and compared the results with the experimental data. On the other hand, a non-porous hollow fiber membrane made of polydimethylsiloxane was used as a membrane contactor to absorb CO2 selectively. Since the permeation rate was mainly controlled by the membrane resistance, a better performance was expected by a less thicker membrane wall [11]. Teramoto et al. showed that the permeation rates of CO2 was simulated by a theory of facilitated transport through a supported liquid membrane of aqueous amine solutions [12]. The experimental data could be successfully predicted by their theoretical approximation method. Al-Safar et al. conducted a separation of CO2 using porous and non-porous hollow fiber membrane contactors with an aqueous diethanolamine solution as an absorbent [13]. Even if good CO2 removal efficiencies were obtained for both membranes, the lower permeation rate was achieved in the non-porous membrane. However, they proposed that the improved CO2 selectivity in the non-porous membrane might compensate for the relatively low permeation rate. Experimental absorption studies were carried out using a membrane contactor with several absorbent liquids: pure water, aqueous sodium hydroxide solution, aqueous potassium carbonate solution or aqueous amine solution [9]. When the absorbent liquid contacts the gas, it reacts with the gas to enhance the absorption rates into the absorbent. However, few theoretical analyses have been performed to describe the membrane contactors, in which the important chemical reaction steps were included without any simplification [3,4,10,14]. If an aqueous potassium carbonate solution is used as an absorbent, complicated reactions are involved and it makes the theoretical analysis
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
more difficult than with other types of absorbent. Thus, many investigators utilized macroscopic overall mass transfer coefficients to describe their membrane contactor systems instead of microscopic partial differential equations [5–8]. In this work, a numerical analysis was carried out to analyze the performance of a hollow fiber membrane contactor for the removal of carbon dioxide where aqueous potassium carbonate solution was used as an absorbent. The complete reversible reactions were included in the microscopic theoretical model equations as opposed to the use of overall mass transfer coefficients to describe the CO2 uptake and release. A numerical analysis was conducted for the system which consisted of an absorption contactor and a desorption contactor where the absorbent flows through the bore of the hollow fibers in both contactors.
2. Theory
Fig. 1. Flow configuration for absorption and desorption of carbon dioxide in potassium carbonate aqueous solution flowing in the tube side.
The overall reaction is written as [15] CO2 + CO3 2− + H2 O 2HCO3 −
2.1. Reaction mechanism of carbon dioxide When potassium carbonate is dissolved in water, it is ionized into the potassium ion (K+ ) and carbonate ion (CO3 2− ). The bicarbonate ion (HCO3 − ) and hydroxyl ion (OH− ) are then generated by the hydrolysis of carbonate ion. In the potassium carbonate aqueous solution, the rate controlling reactions are given as [15] k1
CO2 + H2 O H+ + HCO3 − k−1 k2
CO2 + OH− HCO3 − k−2
(1)
(2)
where k1 , k−1 , k2 and k−2 are the reaction constants of Eqs. (1) and (2). K3 and K4 are designated as equilibrium constants of Eqs. (1) and (2), respectively. The following two more reactions are involved; they are very fast reactions so that they are assumed to be at equilibrium: K1
HCO3 − H+ + CO3 − K2
59
H2 OH+ + OH−
(3) (4)
where K1 and K2 are the equilibrium constants of Eqs. (3) and (4).
(5)
2.2. The flow configuration and the coordinates of the membrane contactor The flow configuration is shown in Fig. 1. The aqueous potassium carbonate solution flows through the bore of the hollow fiber in each contactor. Since the liquid side pressure is kept higher than the gas pressure, the gas will not mix directly with the liquid. The liquid also cannot penetrate to the bulk gas phase because the surface tension force is maintained high enough due to the hydrophobic property of the porous membrane. Therefore, the pores of the membrane are filled with gas, which is referred as a non-wetted mode of operation. The gas–liquid interface can be maintained at the pore mouth of the fiber in the absorbing liquid side. The absorbent liquid is fed into the desorption membrane contactor where the carbon dioxide is desorbed using a vacuum in the recovery phase. The desorbed absorbent liquid is used as a feed solution for the absorption contactor. The coordinates of a membrane contactor is shown in Fig. 2. The liquid flows through the tube side and the gas is fed on the shell side. The coordinates of r and z are shown for the numericial analysis. The radial position of r = 0 denotes the center of a fiber and the radial distance of r = rt
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and
RHCO3 − = 2k1
2[CO3 2− ] 1+α [HCO3 − ]
[HCO3 − ] × [CO2 ] − β [CO3 2− ]
(9)
where α and β are defined as α= Fig. 2. Schematic coordinates of a membrane contactor: A = CO2 , B = HCO3 2− .
represents the wall of a fiber. The axial distance of z = 0 means the inlet position of a fiber and the axial distance of z = L represents the outlet position of a fiber. 2.3. System controlling equations for absorption Differential mass balances on carbon dioxide and bicarbonate ion in the absorbent solution are derived for a membrane contactor. These model equations are based on the following assumptions: (1) steady state and isothermal conditions; (2) no axial diffusion; (3) fully developed parabolic liquid velocity profile in the tube side; (4) ideal gas behavior; (5) application of Henry’s law. The differential mass balance in a hollow fiber for a component ‘i’ may be written using a cylindrical coordinate as ∂Ci 1 ∂ ∂Ci vi = Di (6) r + Ri ∂z r ∂r ∂r where Ri represents the generation rate of a component ‘i’ and vi denotes the velocity profile in a hollow fiber. When the velocity is fully developed in a laminar flow, the axial velocity can be written as 2 r vi (r) = 2vz 1 − (7) rt The reaction rates of CO2 and HCO3 − are given by [15] 2[CO3 2− ] RCO2 = −k1 1 + α [HCO3 − ] [HCO3 − ] × [CO2 ] − β (8) [CO3 2− ]
k 2 K2 2k1 K1
and
β=
k−1 K1 k1
(10)
Eq. (9) can be obtained from the facts of the local conservation of carbon and the local electroneutrality [16]. After Eqs. (7)–(9) are substituted in Eq. (6) and these partial differential equations are further manipulated into dimensionless equations as given by Eqs. (11) and (12) for CO2 and HCO3 − , respectively. All concentrations are expressed in a dimensionless form. The subscript ‘A’ denotes carbon dioxide and the subscript ‘B’ indicates bicarbonate ion. ¯ ∂ 2 C¯ A φ 1 ∂ C¯ A 2 ∂ CA = + 2(1 − r¯ ) ∂ z¯ PeA r ∂ r¯ ∂ r¯ 2 τ − C¯ B −DKA 1 + α C¯ B C¯ A2 × C¯ A − γ (11) τ − C¯ B φ ∂ C¯ B 2(1 − r¯ 2 ) = ∂ z¯ PeB
∂ 2 C¯ B 1 ∂ C¯ B + r ∂ r¯ ∂ r¯ 2
τ − C¯ B +GDKB 1 + α C¯ B C¯ A2 ¯ × CA − γ τ − C¯ B
(12)
where the dimensionless variables and coordinates are defined as follows: [CA ]l,t [CB ]l,t , C¯ B = , [CA ]wall,abs [CB ]wall,abs z r L z¯ = , r¯ = and φ = L rt rt
C¯ A =
(13)
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
Also, the following dimensionless parameters are defined as v l rt v l rt k1 L , PeB = , DKA = , DAt DBt vl 2k1 L [CA ]wall,abs = , G= vl [CB ]wall,abs
PeA = DKB
(14a)
and γ =
2β[CB ]wall,abs , [CA ]wall,abs
τ=
2[CO3 2− ]initial [CB ]wall,abs
C¯ B,abs = C¯ B,des,outlet ,
for all r¯ , z¯ = 0 ∂ C¯ A = 0, ∂ r¯ C¯ A = 1,
(16) ∂ C¯ B = 0, ∂ r¯
C¯ B = 1,
for r¯ = 0
for r¯ = 1
the contacting interfacial area and controls the contact time between the gas and liquid phases depending on the porosity of the porous hollow fibers. Therefore, in this study, the simple boundary conditions are adopted as given in Eq. (18) and it might reduce the great effort for solving the nonlinear coupled partial differential equations. Our next prospective works will include all of the mass transfer resistances in the calculation in terms of the boundary condition.
(14b)
The following initial and boundary conditions are subjected ((K2 /K1 )[CO3 2− ]initial ) ¯ ¯ CA = 0.0, CB = , [CB ]wall for all r¯ , z¯ = 0 (15) C¯ A,abs = C¯ A,des,outlet ,
61
(17) (18)
The initial conditions for the absorption process are given in Eq. (15). The bicarbonate ion concentration could be calculated using the equilibrium relationship of potassium carbonate and water. As the iteration is continued, the initial concentrations are replaced with the concentrations of carbon dioxide and bicarbonate ion of the aqueous solution circulated by the desorption membrane contactor as shown in Eq. (16). The concentration profile is symmetrical with respect to the center of the hollow fiber tube, resulting in Eq. (17). Even if the mass transfer resistance is the function of the liquid side resistance, the membrane side and the gas side resistance in a practical operation as indicated by Iversen et al. [17], sometimes a mass transfer resistance in gas and membrane phases plays a minor role: the reaction rate of CO2 is relatively slow compared to the other gases such as NH3 , SO2 , H2 S and so on, resulting in a minor gas and membrane resistance [2]. Then a constant gas phase concentration over the fiver length can be assumed and it leads to the mass transfer analogy to the Graetz–Lévêgue problem, which means that the porous hollow fiber plays a role that it prepares
2.4. Flow diagram of computer program The calculation steps are shown in Fig. 3. The method of lines was used with a finite differential discretization. A 4th order Runge–Kutta routine was used to handle the systems of coupled ordinary differential equations to find the concentrations in the hollow fiber. The total absorption or desorption rates are obtained from a mass balance between the inlet concentrations and the outlet concentrations of CO2 and HCO3 − through the hollow fiber. Since only half of HCO3 − contributes to the CO2 permeation rates according to Eq. (5), the total absorption rate is calculated to be the sum of the pure CO2 absorption rate and the half of the generation rate of HCO3 − . Both radial concentration profiles of CO2 and HCO3 − are known after the numerical analysis along the fiber length, either the input rate or the output rate of CO2 and HCO3 − at the inlet point or the outlet point of a fiber can be evaluated as follows input or output rate =
discritized node number
[(concentration)r=ri
i=1
(velocity)r=ri (discretized area)r=ri ]
(18)
Then, the total absorption rate of CO2 is calculated using a following relation total absorption rate of CO2 = (number of fibers)[(output rate − input rate)CO2 + 21 (output rate − input rate)HCO3 − ]
(19)
The total desorption rate is also analyzed with a similar method.
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Fig. 3. Flow diagram for solving nonlinear coupled partial differential equations.
3. Operating conditions and parameters The operating conditions are shown in Table 1. Unless otherwise stated, the underlined conditions are used as the typical operating conditions. When the absorbent liquid flow rates are 0.17, 0.83 and 1.67 ml s−1 , the linear mean velocities are 0.021, 0.105 and 0.21 m s−1 , respectively. Since the
Table 1 Operating conditions for numerical analysis Variables
Conditionsa
Temperature (K) Gas pressure in absorber (× 10−5 Pa) Gas pressure in desorber (× 10−3 Pa) Absorbent flow rate (mL s−1 ) Concentration of K2 CO3 (wt.%) Concentration of K2 CO3 (mol dm−3 ) Fiber i.d. (m) Fiber length (m) Number of fibers
298 1.7, 2.39, 3.08 1.33, 6.67, 13.3 0.17, 0.83, 1.67 0, 5, 10 0, 0.3623, 0.7246 700, 840, 1000 0.1, 0.2, 0.3 10
a The underlined conditions are used as the typical operating conditions.
corresponding Reynolds numbers are given by 21, 105 and 210, all the liquid flows are in the laminar flow regime. The parameters used in the calculation are tabulated in Table 2.
4. Results and discussion In Fig. 4, typical dimensionless concentrations of carbon dioxide are shown as a function of dimensionless axial distance with various dimensionless radial positions. The numerical results clearly show that the carbon dioxide concentration increases as the axial distance increases. Also it is shown that the carbon dioxide concentration near the wall of the fiber is much higher than that of the center of the fiber at the same axial distance. At the outlet, the concentration is constant with respect to radial position. This indicates that the solution is fully saturated with carbon dioxide. In Fig. 5, the bicarbonate ion concentrations are shown with the same coordinates as used in Fig. 4. It can be recognized that the bicarbonate ion concentration increases and is almost saturated as the axial distance
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
63
Table 2 Reaction rates and equilibrium constants Constants Equilibrium constant, K1 Equilibrium constant, K2 Equilibrium constant, K3 Equilibrium constant, K4 Reaction constant, k1 Reaction constant, k2 Diffusion coefficient Solubility
Equation
Unit
2902.4 log10 K1 = − + 6.498 − 0.0238T T log10 K2 = −23.5325 + 0.03184T
K1 : mol dm−3 , T: K K2 :
Reference
mol2
dm−6 ,
T: K
[18] [19]
3404.7 log10 K3 = − + 14.843 − 0.03279T T log10 K4 = log10 K3 − log10 K2
K3 : mol dm−3 , T: K
17265.4 log10 k1 = 329.850 − 110.541 log10 T − T 2895 log10 k2 = 13.653 − T 2.35 × 10−2 exp(−2119/T ) DCO2 = (1 + 0.354m)0.82
k1 : s−1 , T: K
[18]
k2 : dm3 mol−1 s−1 , T: K
[18]
DCO2 : cm2 s−1
[20]
H: mol dm−3 atm−1 , T: K
[18]
log10 H =
1140 − 5.30 − 0.125m T
K4 :
dm3
mol−1 ,
T: K
[18] [18]
increases. Also it can be seen that the bicarbonate ion concentration near the wall of the fiber is also higher than that of the center of the fiber at the same axial distance. The bicarbonate ion concentration at the inlet is not 0, but about 0.7 which is the same as the concentration of the outlet absorbent liquid from the desorption membrane contactor. This is probably due
to the incomplete dissociation of bicarbonate ion into carbon dioxide in the desorption mode. In Figs. 6 and 7, the dimensionless concentrations of both components are represented for the desorption mode. The dimensionless concentration of carbon dioxide rapidly decreases with increasing dimensionless radial distance, which means that the most of
Fig. 4. Dimensionless concentration of CO2 vs. dimensionless axial distance with various dimensionless radial distance for absorption mode.
Fig. 5. Dimensionless concentration of HCO3 − vs. dimensionless axial distance with various dimensionless radial distance for absorption mode.
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Fig. 6. Dimensionless concentration of CO2 vs. dimensionless axial distance with various dimensionless radial distance for desorption mode.
desorption occurs before the absorbent flow reaches the mid point of the desorption fiber in this particular case. The concentration of bicarbonate ion decreases with increasing dimensionless radial distance. However, it approaches to the constant. This means that
Fig. 7. Dimensionless concentration of HCO3 − vs. dimensionless axial distance with various dimensionless radial distance for desorption mode.
Fig. 8. Permeation rate of CO2 vs. absorbent flow rate with various absorbent concentrations.
the bicarbonate ion is not completely dissociated. This concentration is used as an input concentration for the absorption contactor. In Fig. 8, the permeation rate of carbon dioxide is plotted as a function of absorbent flow rate for several values of the potassium carbonate concentration. As the absorbent flow rate increases, the permeation rate of carbon dioxide increases because the liquid boundary layer resistance decreases with increasing liquid velocity. Also, the transfer rate of carbon dioxide into the liquid increases as the concentration of potassium carbonate increases since the facilitated absorption ratio also increases. If the concentration of potassium carbonate increases from 0 to 5 wt.%, the permeation rate increases almost 10 times. If it increases to 10 wt.%, then the absorption rate increases two times, resulting in 20 times higher absorption rates than those with pure water. Fig. 9 shows how the permeation rate of carbon dioxide depends on the concentration of the carbonate ions in the absorbent liquid. The transfer rate of carbon dioxide continuously increases as the concentration of potassium carbonate increases up to 1.0 mol dm−3 . Fig. 10 shows the permeation rate of carbon dioxide versus absorbent flow rate for various values of gas pressure. The permeation rate of carbon dioxide increases as the pressure of the gas mixture increases.
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
Fig. 9. Permeation rate of CO2 vs. concentration of carbonate ion.
This appears to be due to the high carbon dioxide concentration in the liquid interface at which the absorbent solution is in equilibrium with the gas. If the 5 wt.% potassium carbonate solution is used as an absorbent, there is no large difference in the permeation rate regardless of the feed gas pressure because the potassium carbonate is completely saturated with the carbon dioxide at the exit of the fiber.
Fig. 10. Permeation rate of CO2 vs. absorbent flow rate with various pressures of gas mixture for absorption mode.
65
Fig. 11. Permeation rate of CO2 vs. absorbent flow rate with various pressures of gas mixture for desorption mode.
In Fig. 11, the desorption rate of carbon dioxide is plotted as a function of the absorbent flow rate with an operating variable of the vacuum pressure of gas phase in the desorption membrane contactor. The desorption rate of carbon dioxide increases as the absorbent flow rate increases. This is due to the high mass transfer rates of carbon dioxide and the bicarbonate ion in the liquid phase because the higher flow rate makes the boundary layer thinner. When the 5 wt.% potassium carbonate solution is used as an absorbent, it is found that there is a large difference in the permeation rate with respect to the degree of gas pressure in a desorber. For example, if the gas pressure decreases from 13.3 × 103 (10 cm Hg) to 6.67 × 103 Pa (5 cm Hg), then the permeation rate increases by a factor of two. These results indicate that the dissociation reaction rate of bicarbonate ions in the liquid phase is greatly affected by the carbon dioxide in the gas phase. However, when the pure water is used as an absorbent, the vacuum condition does not have large effect on the permeation rate because the physical desorption rate is only dependent of the pressure of the gas, the difference of which is only limited to the order of 103 Pa (several cm Hg) of pressure. Therefore, it can be said that the extent of vacuum is one of the most important factors for the design of the operation of the desorption membrane contactor.
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Fig. 12. Dimensionless axial distance where the solution is saturated.
The absorbent liquid may be saturated in between the fiber length, which means that a large portion of the fiber may not be used for absorption. In Fig. 12, the dimensionless axial distance is plotted as a function of the absorbent flow rate with different diameter of fibers. As shown in this figure, the saturation point moves toward the direction of the outlet of fiber as the liquid flow rate increases. It is thought that since it takes the same contact time for the liquid to be saturated independent of the flow rate, the saturation point will be moved to the outlet as the velocity increases. When the radius of fiber is small such as 350 m, the saturation point moves toward the exit of fiber as the flow rate increases, resulting in an increment of the permeation rate, even if the absorbent is not completely saturated. This is simply because the permeation rate is equal to the concentration multiplied by the flow rate. Also it is found that the liquid absorbents are saturated just at the exit position at the flow rate of 2.6 and 1.4 ml s−1 for the 420 and 500 m radii of the fibers, respectively. It means that the capacity of absorption for the liquid absorbent could be fully utilized for the given operating conditions. It can be said that the optimal flow rate for the contactor consisted of either the 420 m radius fibers or 500 m radius fibers is 2.6 or 1.4 ml s−1 , respectively, in terms of the efficiency of absorbent usage.
Fig. 13. Permeation rate of CO2 vs. absorbent flow rate with various module lengths.
In Fig. 13, the permeation rate is plotted as a function of the absorbent flow rate with various module lengths. The permeation rate increases as the flow rate increases regardless of the module length. If the flow rate is, for example, lower than 0.8 ml s−1 , there is no difference in the total permeation rate of carbon dioxide between three different module lengths. This might be due to the saturation of the absorption before it leaves the fiber in all three different module length. However, if the flow rate increases further, the permeation rate becomes larger for the longer module. It can be thought that the module length is not so long that the absorption liquid cannot be saturated and leaves as in an unsaturated. However, the absorbent liquid leaves in a more saturated condition at the longer module, simply because of longer retention time resulting in higher permeation rate.
5. Conclusions The coupled nonlinear partial differential equations were derived using the reversible reaction rates of the potassium carbonate with carbon dioxide to describe the absorption and the desorption of carbon dioxide in a hollow fiber membrane contactor. The equations
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
were solved numerically and the concentration profiles of carbon dioxide and bicarbonate ions in the absorbent solution were successfully obtained as a function of the operating parameters. It was found that the degree of the gas pressure in a desorption module has a large effect on the permeation rate of carbon dioxide when an aqueous potassium carbonate solution was used as an absorbent. One of the important results provides the optimal absorbent flow rate such that it is saturated just at the exit of the contactor. Since there is a lack of experimental validation so as to compare the numerical analysis, our next prospective work will include the experimental data as well as the improvement of the numerical analysis by taking into account the resistances of the porous membrane in terms of the boundary condition.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Acknowledgements We gratefully acknowledge Mr. Ingi Cho and Mr. Hyo-Sung Ahn for preparation of the manuscript including the drawings.
[13]
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Analysis of CO2 removal by hollow fiber membrane contactors Yongtaek Lee a,∗ , Richard D. Noble a , Bong-Yeol Yeom b , You-In Park b , Kew-Ho Lee b b
a Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424, USA Membrane and Separation Center, Korea Research Institute of Chemical Technology, P.O. Box 107, Daedeok-Danji, Taejon 305-606, South Korea
Received 29 September 2000; received in revised form 22 May 2001; accepted 29 May 2001
Abstract A numerical analysis was performed to investigate the removal behavior of carbon dioxide using a hollow fiber membrane contactor in which an aqueous potassium carbonate solution flows as an absorbent. An important feature is the inclusion of the nonlinear reversible reaction terms that result from the reactions between the absorbent solution and carbon dioxide. The coupled nonlinear partial differential equations were derived and solved numerically, which describe either the absorption or desorption of carbon dioxide in a membrane contactor. In a numerical analysis, the absorbent liquid always flows through the inner side of the hollow fiber in both absorption and desorption steps. The analysis includes the effects of the pressure of feed gases and the extent of vacuum, the absorbent flow rates, the concentration of potassium carbonate in an absorbent solution, the diameter and length of the fibers. One important result is the determination of the optimal absorbent flow rate such that it is saturated just at the exit of the contactor. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Facilitated transport; Gas separations; Modules; Membrane contactors; Numerical simulation
1. Introduction Gas mixtures are usually dispersed and contacted with liquids in conventional gas absorption processes where several types of devices are used: a packed tower, a spray tower, a venturi scrubber or a bubble column. Dispersion in these contacting devices has many drawbacks. Membrane-based contacting devices could overcome these shortcomings by preventing gas dispersion in the contacting phase. The absorbent liquid flows on one side and the gas flows on the other side of the membrane.
∗ Corresponding author. On leave from Chungnam National University, South Korea. Tel.: +82-42-821-5686; fax: +82-42-822-8995. E-mail addresses: [email protected], [email protected] (Y. Lee).
Qi and Cussler developed the idea of the hollowfiber contactor using a microporous non-wetted polypropylene hollow fiber membrane for absorption of carbon dioxide where aqueous sodium hydroxide solution was used as an absorbent [1,2]. Teramoto et al. also proposed a similar technique and applied it for separation of ethylene from ethane using an aqueous silver nitrate solution [3]. These membrane contactors were used by several authors in a shape of either a flat membrane or a hollow fiber membrane to absorb the various gases such as carbon monoxide, carbon dioxide, sulfur dioxide [4–8]. Sirkar recently reviewed the gas absorption processes based on the membrane contactors [9]. Separation of carbon dioxide from nitrogen was extensively explored by Karoor and Sirkar with either pure water or an aqueous amine solution [10]. They carried out comprehensive absorption experiments on gas separation including pure
0376-7388/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 1 ) 0 0 5 2 4 - 5
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Nomenclature C¯ i DAt DBt DK A DK B G
H k1 k−1 k2 k−2 K1 K2 K3 K4 L m PeA PeB r rt r¯ vl z z¯
dimensionless concentration of component ‘i’ diffusion coefficient of CO2 (m2 s−1 ) diffusion coefficient of HCO3 −2 (m2 s−1 ) dimensionless parameter as defined in Eq. (14a) dimensionless parameter as defined in Eq. (14a) dimensionless ratio of CO2 concentration over HCO3 2− concentration at wall solubility of CO2 in absorbent solution (mol dm−3 atm−1 ) forward rate of constant of reaction (1) (s−1 ) reverse rate of constant of reaction (1) (dm mol−1 s−1 ) forward rate of constant of reaction (2) (dm3 mol−1 s−1 ) reverse rate of constant of reaction (2) (s−1 ) equilibrium constant of Eq. (3) (mol dm−3 ) equilibrium constant of Eq. (4) (mol2 dm−6 ) equilibrium constant of Eq. (1) (mol dm−3 ) equilibrium constant of Eq. (2) (mol−1 dm3 ) length of fiber (m) initial molar concentration of K2 CO3 (mol dm−3 ) Peclet number as defined in Eq. (14a) Peclet number as defined in Eq. (14a) radial coordinate (m) radius of fiber (m) dimensionless radial coordinate average liquid velocity in tube side (m s−1 ) axial coordinate (m) dimensionless axial coordinate
Greek letters α parameter as defined in Eq. (10) β parameter as defined in Eq. (10)
φ γ τ
dimensionless parameter as defined in Eq. (13) dimensionless parameter as defined in Eq. (14) dimensionless parameter as defined in Eq. (14)
carbon dioxide, pure sulfur dioxide, and its mixture with nitrogen or air using a microporous hydrophobic hollow-fiber module in a parallel flow configuration. They also proposed a theoretical model to describe a system with an absorbent of pure water resulting in partial differential equations; solved them numerically and compared the results with the experimental data. On the other hand, a non-porous hollow fiber membrane made of polydimethylsiloxane was used as a membrane contactor to absorb CO2 selectively. Since the permeation rate was mainly controlled by the membrane resistance, a better performance was expected by a less thicker membrane wall [11]. Teramoto et al. showed that the permeation rates of CO2 was simulated by a theory of facilitated transport through a supported liquid membrane of aqueous amine solutions [12]. The experimental data could be successfully predicted by their theoretical approximation method. Al-Safar et al. conducted a separation of CO2 using porous and non-porous hollow fiber membrane contactors with an aqueous diethanolamine solution as an absorbent [13]. Even if good CO2 removal efficiencies were obtained for both membranes, the lower permeation rate was achieved in the non-porous membrane. However, they proposed that the improved CO2 selectivity in the non-porous membrane might compensate for the relatively low permeation rate. Experimental absorption studies were carried out using a membrane contactor with several absorbent liquids: pure water, aqueous sodium hydroxide solution, aqueous potassium carbonate solution or aqueous amine solution [9]. When the absorbent liquid contacts the gas, it reacts with the gas to enhance the absorption rates into the absorbent. However, few theoretical analyses have been performed to describe the membrane contactors, in which the important chemical reaction steps were included without any simplification [3,4,10,14]. If an aqueous potassium carbonate solution is used as an absorbent, complicated reactions are involved and it makes the theoretical analysis
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
more difficult than with other types of absorbent. Thus, many investigators utilized macroscopic overall mass transfer coefficients to describe their membrane contactor systems instead of microscopic partial differential equations [5–8]. In this work, a numerical analysis was carried out to analyze the performance of a hollow fiber membrane contactor for the removal of carbon dioxide where aqueous potassium carbonate solution was used as an absorbent. The complete reversible reactions were included in the microscopic theoretical model equations as opposed to the use of overall mass transfer coefficients to describe the CO2 uptake and release. A numerical analysis was conducted for the system which consisted of an absorption contactor and a desorption contactor where the absorbent flows through the bore of the hollow fibers in both contactors.
2. Theory
Fig. 1. Flow configuration for absorption and desorption of carbon dioxide in potassium carbonate aqueous solution flowing in the tube side.
The overall reaction is written as [15] CO2 + CO3 2− + H2 O 2HCO3 −
2.1. Reaction mechanism of carbon dioxide When potassium carbonate is dissolved in water, it is ionized into the potassium ion (K+ ) and carbonate ion (CO3 2− ). The bicarbonate ion (HCO3 − ) and hydroxyl ion (OH− ) are then generated by the hydrolysis of carbonate ion. In the potassium carbonate aqueous solution, the rate controlling reactions are given as [15] k1
CO2 + H2 O H+ + HCO3 − k−1 k2
CO2 + OH− HCO3 − k−2
(1)
(2)
where k1 , k−1 , k2 and k−2 are the reaction constants of Eqs. (1) and (2). K3 and K4 are designated as equilibrium constants of Eqs. (1) and (2), respectively. The following two more reactions are involved; they are very fast reactions so that they are assumed to be at equilibrium: K1
HCO3 − H+ + CO3 − K2
59
H2 OH+ + OH−
(3) (4)
where K1 and K2 are the equilibrium constants of Eqs. (3) and (4).
(5)
2.2. The flow configuration and the coordinates of the membrane contactor The flow configuration is shown in Fig. 1. The aqueous potassium carbonate solution flows through the bore of the hollow fiber in each contactor. Since the liquid side pressure is kept higher than the gas pressure, the gas will not mix directly with the liquid. The liquid also cannot penetrate to the bulk gas phase because the surface tension force is maintained high enough due to the hydrophobic property of the porous membrane. Therefore, the pores of the membrane are filled with gas, which is referred as a non-wetted mode of operation. The gas–liquid interface can be maintained at the pore mouth of the fiber in the absorbing liquid side. The absorbent liquid is fed into the desorption membrane contactor where the carbon dioxide is desorbed using a vacuum in the recovery phase. The desorbed absorbent liquid is used as a feed solution for the absorption contactor. The coordinates of a membrane contactor is shown in Fig. 2. The liquid flows through the tube side and the gas is fed on the shell side. The coordinates of r and z are shown for the numericial analysis. The radial position of r = 0 denotes the center of a fiber and the radial distance of r = rt
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and
RHCO3 − = 2k1
2[CO3 2− ] 1+α [HCO3 − ]
[HCO3 − ] × [CO2 ] − β [CO3 2− ]
(9)
where α and β are defined as α= Fig. 2. Schematic coordinates of a membrane contactor: A = CO2 , B = HCO3 2− .
represents the wall of a fiber. The axial distance of z = 0 means the inlet position of a fiber and the axial distance of z = L represents the outlet position of a fiber. 2.3. System controlling equations for absorption Differential mass balances on carbon dioxide and bicarbonate ion in the absorbent solution are derived for a membrane contactor. These model equations are based on the following assumptions: (1) steady state and isothermal conditions; (2) no axial diffusion; (3) fully developed parabolic liquid velocity profile in the tube side; (4) ideal gas behavior; (5) application of Henry’s law. The differential mass balance in a hollow fiber for a component ‘i’ may be written using a cylindrical coordinate as ∂Ci 1 ∂ ∂Ci vi = Di (6) r + Ri ∂z r ∂r ∂r where Ri represents the generation rate of a component ‘i’ and vi denotes the velocity profile in a hollow fiber. When the velocity is fully developed in a laminar flow, the axial velocity can be written as 2 r vi (r) = 2vz 1 − (7) rt The reaction rates of CO2 and HCO3 − are given by [15] 2[CO3 2− ] RCO2 = −k1 1 + α [HCO3 − ] [HCO3 − ] × [CO2 ] − β (8) [CO3 2− ]
k 2 K2 2k1 K1
and
β=
k−1 K1 k1
(10)
Eq. (9) can be obtained from the facts of the local conservation of carbon and the local electroneutrality [16]. After Eqs. (7)–(9) are substituted in Eq. (6) and these partial differential equations are further manipulated into dimensionless equations as given by Eqs. (11) and (12) for CO2 and HCO3 − , respectively. All concentrations are expressed in a dimensionless form. The subscript ‘A’ denotes carbon dioxide and the subscript ‘B’ indicates bicarbonate ion. ¯ ∂ 2 C¯ A φ 1 ∂ C¯ A 2 ∂ CA = + 2(1 − r¯ ) ∂ z¯ PeA r ∂ r¯ ∂ r¯ 2 τ − C¯ B −DKA 1 + α C¯ B C¯ A2 × C¯ A − γ (11) τ − C¯ B φ ∂ C¯ B 2(1 − r¯ 2 ) = ∂ z¯ PeB
∂ 2 C¯ B 1 ∂ C¯ B + r ∂ r¯ ∂ r¯ 2
τ − C¯ B +GDKB 1 + α C¯ B C¯ A2 ¯ × CA − γ τ − C¯ B
(12)
where the dimensionless variables and coordinates are defined as follows: [CA ]l,t [CB ]l,t , C¯ B = , [CA ]wall,abs [CB ]wall,abs z r L z¯ = , r¯ = and φ = L rt rt
C¯ A =
(13)
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
Also, the following dimensionless parameters are defined as v l rt v l rt k1 L , PeB = , DKA = , DAt DBt vl 2k1 L [CA ]wall,abs = , G= vl [CB ]wall,abs
PeA = DKB
(14a)
and γ =
2β[CB ]wall,abs , [CA ]wall,abs
τ=
2[CO3 2− ]initial [CB ]wall,abs
C¯ B,abs = C¯ B,des,outlet ,
for all r¯ , z¯ = 0 ∂ C¯ A = 0, ∂ r¯ C¯ A = 1,
(16) ∂ C¯ B = 0, ∂ r¯
C¯ B = 1,
for r¯ = 0
for r¯ = 1
the contacting interfacial area and controls the contact time between the gas and liquid phases depending on the porosity of the porous hollow fibers. Therefore, in this study, the simple boundary conditions are adopted as given in Eq. (18) and it might reduce the great effort for solving the nonlinear coupled partial differential equations. Our next prospective works will include all of the mass transfer resistances in the calculation in terms of the boundary condition.
(14b)
The following initial and boundary conditions are subjected ((K2 /K1 )[CO3 2− ]initial ) ¯ ¯ CA = 0.0, CB = , [CB ]wall for all r¯ , z¯ = 0 (15) C¯ A,abs = C¯ A,des,outlet ,
61
(17) (18)
The initial conditions for the absorption process are given in Eq. (15). The bicarbonate ion concentration could be calculated using the equilibrium relationship of potassium carbonate and water. As the iteration is continued, the initial concentrations are replaced with the concentrations of carbon dioxide and bicarbonate ion of the aqueous solution circulated by the desorption membrane contactor as shown in Eq. (16). The concentration profile is symmetrical with respect to the center of the hollow fiber tube, resulting in Eq. (17). Even if the mass transfer resistance is the function of the liquid side resistance, the membrane side and the gas side resistance in a practical operation as indicated by Iversen et al. [17], sometimes a mass transfer resistance in gas and membrane phases plays a minor role: the reaction rate of CO2 is relatively slow compared to the other gases such as NH3 , SO2 , H2 S and so on, resulting in a minor gas and membrane resistance [2]. Then a constant gas phase concentration over the fiver length can be assumed and it leads to the mass transfer analogy to the Graetz–Lévêgue problem, which means that the porous hollow fiber plays a role that it prepares
2.4. Flow diagram of computer program The calculation steps are shown in Fig. 3. The method of lines was used with a finite differential discretization. A 4th order Runge–Kutta routine was used to handle the systems of coupled ordinary differential equations to find the concentrations in the hollow fiber. The total absorption or desorption rates are obtained from a mass balance between the inlet concentrations and the outlet concentrations of CO2 and HCO3 − through the hollow fiber. Since only half of HCO3 − contributes to the CO2 permeation rates according to Eq. (5), the total absorption rate is calculated to be the sum of the pure CO2 absorption rate and the half of the generation rate of HCO3 − . Both radial concentration profiles of CO2 and HCO3 − are known after the numerical analysis along the fiber length, either the input rate or the output rate of CO2 and HCO3 − at the inlet point or the outlet point of a fiber can be evaluated as follows input or output rate =
discritized node number
[(concentration)r=ri
i=1
(velocity)r=ri (discretized area)r=ri ]
(18)
Then, the total absorption rate of CO2 is calculated using a following relation total absorption rate of CO2 = (number of fibers)[(output rate − input rate)CO2 + 21 (output rate − input rate)HCO3 − ]
(19)
The total desorption rate is also analyzed with a similar method.
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Fig. 3. Flow diagram for solving nonlinear coupled partial differential equations.
3. Operating conditions and parameters The operating conditions are shown in Table 1. Unless otherwise stated, the underlined conditions are used as the typical operating conditions. When the absorbent liquid flow rates are 0.17, 0.83 and 1.67 ml s−1 , the linear mean velocities are 0.021, 0.105 and 0.21 m s−1 , respectively. Since the
Table 1 Operating conditions for numerical analysis Variables
Conditionsa
Temperature (K) Gas pressure in absorber (× 10−5 Pa) Gas pressure in desorber (× 10−3 Pa) Absorbent flow rate (mL s−1 ) Concentration of K2 CO3 (wt.%) Concentration of K2 CO3 (mol dm−3 ) Fiber i.d. (m) Fiber length (m) Number of fibers
298 1.7, 2.39, 3.08 1.33, 6.67, 13.3 0.17, 0.83, 1.67 0, 5, 10 0, 0.3623, 0.7246 700, 840, 1000 0.1, 0.2, 0.3 10
a The underlined conditions are used as the typical operating conditions.
corresponding Reynolds numbers are given by 21, 105 and 210, all the liquid flows are in the laminar flow regime. The parameters used in the calculation are tabulated in Table 2.
4. Results and discussion In Fig. 4, typical dimensionless concentrations of carbon dioxide are shown as a function of dimensionless axial distance with various dimensionless radial positions. The numerical results clearly show that the carbon dioxide concentration increases as the axial distance increases. Also it is shown that the carbon dioxide concentration near the wall of the fiber is much higher than that of the center of the fiber at the same axial distance. At the outlet, the concentration is constant with respect to radial position. This indicates that the solution is fully saturated with carbon dioxide. In Fig. 5, the bicarbonate ion concentrations are shown with the same coordinates as used in Fig. 4. It can be recognized that the bicarbonate ion concentration increases and is almost saturated as the axial distance
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
63
Table 2 Reaction rates and equilibrium constants Constants Equilibrium constant, K1 Equilibrium constant, K2 Equilibrium constant, K3 Equilibrium constant, K4 Reaction constant, k1 Reaction constant, k2 Diffusion coefficient Solubility
Equation
Unit
2902.4 log10 K1 = − + 6.498 − 0.0238T T log10 K2 = −23.5325 + 0.03184T
K1 : mol dm−3 , T: K K2 :
Reference
mol2
dm−6 ,
T: K
[18] [19]
3404.7 log10 K3 = − + 14.843 − 0.03279T T log10 K4 = log10 K3 − log10 K2
K3 : mol dm−3 , T: K
17265.4 log10 k1 = 329.850 − 110.541 log10 T − T 2895 log10 k2 = 13.653 − T 2.35 × 10−2 exp(−2119/T ) DCO2 = (1 + 0.354m)0.82
k1 : s−1 , T: K
[18]
k2 : dm3 mol−1 s−1 , T: K
[18]
DCO2 : cm2 s−1
[20]
H: mol dm−3 atm−1 , T: K
[18]
log10 H =
1140 − 5.30 − 0.125m T
K4 :
dm3
mol−1 ,
T: K
[18] [18]
increases. Also it can be seen that the bicarbonate ion concentration near the wall of the fiber is also higher than that of the center of the fiber at the same axial distance. The bicarbonate ion concentration at the inlet is not 0, but about 0.7 which is the same as the concentration of the outlet absorbent liquid from the desorption membrane contactor. This is probably due
to the incomplete dissociation of bicarbonate ion into carbon dioxide in the desorption mode. In Figs. 6 and 7, the dimensionless concentrations of both components are represented for the desorption mode. The dimensionless concentration of carbon dioxide rapidly decreases with increasing dimensionless radial distance, which means that the most of
Fig. 4. Dimensionless concentration of CO2 vs. dimensionless axial distance with various dimensionless radial distance for absorption mode.
Fig. 5. Dimensionless concentration of HCO3 − vs. dimensionless axial distance with various dimensionless radial distance for absorption mode.
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Fig. 6. Dimensionless concentration of CO2 vs. dimensionless axial distance with various dimensionless radial distance for desorption mode.
desorption occurs before the absorbent flow reaches the mid point of the desorption fiber in this particular case. The concentration of bicarbonate ion decreases with increasing dimensionless radial distance. However, it approaches to the constant. This means that
Fig. 7. Dimensionless concentration of HCO3 − vs. dimensionless axial distance with various dimensionless radial distance for desorption mode.
Fig. 8. Permeation rate of CO2 vs. absorbent flow rate with various absorbent concentrations.
the bicarbonate ion is not completely dissociated. This concentration is used as an input concentration for the absorption contactor. In Fig. 8, the permeation rate of carbon dioxide is plotted as a function of absorbent flow rate for several values of the potassium carbonate concentration. As the absorbent flow rate increases, the permeation rate of carbon dioxide increases because the liquid boundary layer resistance decreases with increasing liquid velocity. Also, the transfer rate of carbon dioxide into the liquid increases as the concentration of potassium carbonate increases since the facilitated absorption ratio also increases. If the concentration of potassium carbonate increases from 0 to 5 wt.%, the permeation rate increases almost 10 times. If it increases to 10 wt.%, then the absorption rate increases two times, resulting in 20 times higher absorption rates than those with pure water. Fig. 9 shows how the permeation rate of carbon dioxide depends on the concentration of the carbonate ions in the absorbent liquid. The transfer rate of carbon dioxide continuously increases as the concentration of potassium carbonate increases up to 1.0 mol dm−3 . Fig. 10 shows the permeation rate of carbon dioxide versus absorbent flow rate for various values of gas pressure. The permeation rate of carbon dioxide increases as the pressure of the gas mixture increases.
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
Fig. 9. Permeation rate of CO2 vs. concentration of carbonate ion.
This appears to be due to the high carbon dioxide concentration in the liquid interface at which the absorbent solution is in equilibrium with the gas. If the 5 wt.% potassium carbonate solution is used as an absorbent, there is no large difference in the permeation rate regardless of the feed gas pressure because the potassium carbonate is completely saturated with the carbon dioxide at the exit of the fiber.
Fig. 10. Permeation rate of CO2 vs. absorbent flow rate with various pressures of gas mixture for absorption mode.
65
Fig. 11. Permeation rate of CO2 vs. absorbent flow rate with various pressures of gas mixture for desorption mode.
In Fig. 11, the desorption rate of carbon dioxide is plotted as a function of the absorbent flow rate with an operating variable of the vacuum pressure of gas phase in the desorption membrane contactor. The desorption rate of carbon dioxide increases as the absorbent flow rate increases. This is due to the high mass transfer rates of carbon dioxide and the bicarbonate ion in the liquid phase because the higher flow rate makes the boundary layer thinner. When the 5 wt.% potassium carbonate solution is used as an absorbent, it is found that there is a large difference in the permeation rate with respect to the degree of gas pressure in a desorber. For example, if the gas pressure decreases from 13.3 × 103 (10 cm Hg) to 6.67 × 103 Pa (5 cm Hg), then the permeation rate increases by a factor of two. These results indicate that the dissociation reaction rate of bicarbonate ions in the liquid phase is greatly affected by the carbon dioxide in the gas phase. However, when the pure water is used as an absorbent, the vacuum condition does not have large effect on the permeation rate because the physical desorption rate is only dependent of the pressure of the gas, the difference of which is only limited to the order of 103 Pa (several cm Hg) of pressure. Therefore, it can be said that the extent of vacuum is one of the most important factors for the design of the operation of the desorption membrane contactor.
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Fig. 12. Dimensionless axial distance where the solution is saturated.
The absorbent liquid may be saturated in between the fiber length, which means that a large portion of the fiber may not be used for absorption. In Fig. 12, the dimensionless axial distance is plotted as a function of the absorbent flow rate with different diameter of fibers. As shown in this figure, the saturation point moves toward the direction of the outlet of fiber as the liquid flow rate increases. It is thought that since it takes the same contact time for the liquid to be saturated independent of the flow rate, the saturation point will be moved to the outlet as the velocity increases. When the radius of fiber is small such as 350 m, the saturation point moves toward the exit of fiber as the flow rate increases, resulting in an increment of the permeation rate, even if the absorbent is not completely saturated. This is simply because the permeation rate is equal to the concentration multiplied by the flow rate. Also it is found that the liquid absorbents are saturated just at the exit position at the flow rate of 2.6 and 1.4 ml s−1 for the 420 and 500 m radii of the fibers, respectively. It means that the capacity of absorption for the liquid absorbent could be fully utilized for the given operating conditions. It can be said that the optimal flow rate for the contactor consisted of either the 420 m radius fibers or 500 m radius fibers is 2.6 or 1.4 ml s−1 , respectively, in terms of the efficiency of absorbent usage.
Fig. 13. Permeation rate of CO2 vs. absorbent flow rate with various module lengths.
In Fig. 13, the permeation rate is plotted as a function of the absorbent flow rate with various module lengths. The permeation rate increases as the flow rate increases regardless of the module length. If the flow rate is, for example, lower than 0.8 ml s−1 , there is no difference in the total permeation rate of carbon dioxide between three different module lengths. This might be due to the saturation of the absorption before it leaves the fiber in all three different module length. However, if the flow rate increases further, the permeation rate becomes larger for the longer module. It can be thought that the module length is not so long that the absorption liquid cannot be saturated and leaves as in an unsaturated. However, the absorbent liquid leaves in a more saturated condition at the longer module, simply because of longer retention time resulting in higher permeation rate.
5. Conclusions The coupled nonlinear partial differential equations were derived using the reversible reaction rates of the potassium carbonate with carbon dioxide to describe the absorption and the desorption of carbon dioxide in a hollow fiber membrane contactor. The equations
Y. Lee et al. / Journal of Membrane Science 194 (2001) 57–67
were solved numerically and the concentration profiles of carbon dioxide and bicarbonate ions in the absorbent solution were successfully obtained as a function of the operating parameters. It was found that the degree of the gas pressure in a desorption module has a large effect on the permeation rate of carbon dioxide when an aqueous potassium carbonate solution was used as an absorbent. One of the important results provides the optimal absorbent flow rate such that it is saturated just at the exit of the contactor. Since there is a lack of experimental validation so as to compare the numerical analysis, our next prospective work will include the experimental data as well as the improvement of the numerical analysis by taking into account the resistances of the porous membrane in terms of the boundary condition.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Acknowledgements We gratefully acknowledge Mr. Ingi Cho and Mr. Hyo-Sung Ahn for preparation of the manuscript including the drawings.
[13]
[14]
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