I n d . Eng. Chem. Res. 1995,34, 1267-1272
1267
SEPARATIONS Approximate Solution of Facilitation Factors for the Transport of C02 through a Liquid Membrane of Amine Solution Masaaki Teramoto* Department of Chemistry and Materials Technology, Faculty of Engineering and Design, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606, Japan
An approximate solution of the facilitation factors in the facilitated transport of CO2 through a liquid membrane containing primary or secondary amine has been developed for the case where a reversible reaction, A (C02) 2B (amine) E (carbamate) F (protonated amine), occurs in the membrane, and the reaction rate is expressed by ~ ~ ( C A-CCECF/KeqCB)/( B 1 K ’ / ~ B ) In . the present approximation method, separate constant concentrations of the carrier B and the protonated amine F are assumed at the two boundaries of the membrane for evaluating the influx and outflux of the permeant species, and these concentrations are determined so t h a t these fluxes agree with each other in the steady state. It was confirmed that the facilitation factors calculated by the present approximation method agree with the results which had been obtained previously by numerical integration of the governing differential equations. Calculated result on the system parameter dependencies of the facilitation factor are also presented.
+
+
Introduction
CO,
Facilitated transport membranes containing carriers which react reversibly and selectively with permeant species have been attracting attention since they have much higher selectivity compared with polymer membranes with no carriers (Noble and Way, 1987). In developing such functional membranes and also for optimizing the efficiency of the membranes, it is important to predict the permeation rates theoretically using the physicochemical properties such as reaction rate constant, chemical equilibrium constant, diffusivities of the chemical species, and membrane thickness. Since the governing differential equations for the facilitated transport systems are generally nonlinear due to nonlinear kinetics, general analytical solutions are not available. Recently, an author (Teramoto, 1994) developed an approximate solution of facilitation factors in the facilitated transport membrane where the reaction A (permeate) B (carrier) == C (complex) occurs. Here, the facilitation factor is defined as the ratio of facilitated transport flux to the flux with no carriers. In the analysis, separate concentrations of the carrier were assumed a t the two boundaries of the membrane in evaluating the influx and outflux of the permeant species, and these boundary concentrations were determined so that these fluxes might agree with each other in the steady state. The approximate solutions were found to agree very well with the solutions obtained by integrating the governing equation numerically. In the transport of C02 through a membrane containing an aqueous solution of primary amine such as monoethanolamine (MEA, RNHz), C02 is considered to react with amine as follows (Danckwerts and McNeil, 1967).
+
* Fax:
075-711-9483.
+
+ RNH, ==kl RNHCOO- + H+ H+ + RNH, RNH,’
(a) (b)
--L
The overall reaction is CO, (A)
+ BRNH, (B) -RNHCOO- (E)
+ RNH,’
(F) (c)
where R represents the functional groups on the amine. The forward reaction rate constant of reaction a is much greater than the reverse reaction rate constant. Since reaction b is very fast, the overall reaction is limited by reaction a. It was reported that the forward reaction is first order with respect to both C02 and amine (Hikita et al., 1977; Alvarez-Fuster et al., 1980; Laddha and Danckwerts, 1981; Sada et al., 1985). If reaction b is assumed to reach equilibrium the reaction rate is expressed as
r = -dC,/dt
= -(‘/,)dCB/dt = dCE/dt = dCddt =
k,(CACB - C&p’KeqCB) (1) where Keqis the overall chemical equilibrium constant.
On the other hand, concerning the reaction between COz and diethanolamine (DEA, RzNH), a secondary amine, the order of reaction with respect to DEA reported so far ranges from 1 to 2 (Hikita et al., 1977; Alvarez-Fuster et al., 1980; Laddha and Danckwerts, 1981; Sada et al., 1985). Danckwerts (1979) proposed a zwitterion mechanism to explain amine concentration dependence of the reaction rate. CO,
+ R,NH k,R,NH+COOkl
0888-5885/95/2634-1267$09.00/00 1995 American Chemical Society
(d)
1268 Ind. Eng. Chem. Res., Vol. 34,No. 4,1995
R2NH+COO-
membrane
I
+ R,NH k, R,NCOO- + R2NH2+ (e) k3
The overall reaction is expressed by reaction f. CO, (A)
+ 2R,NH (B) =+
+
R,NCOO- (E) R2NH2+(F)(0 Application of the steady-state approximation to the zwitterion R,NH+COO- gives the following rate expression (Laddha and Danckwerts, 1981).
r=
kl(CAcB - CECdKeqcB) (1+ k&,CB)
x=o > Figure 1. Concentration profiles in the membrane.
(3)
When kdk3C~is much smaller than unity, eq 3 reduces to eq 1. Therefore, eq 1can be considered as a special case of eq 3. In this work, the facilitation factor of C02 in the facilitated transport where the reaction rate is expressed by eq 3 is analyzed approximatelyby the similar method employed in the previous work (Teramoto, 1994),and the approximate solutions are compared with the numerical solutions which have been reported by Guha et al. (1990) and Davis and Sandal1 (1993).
Basic Equations The concentration profiles in the membrane are schematically shown in Figure 1. The steady-state differential mass balance equations in the membrane are
DA(d2CA/dX2)=
(4)
DB(d2CB/dX2)= 27
(5)
d2e dy2
d2 a b - ef/(Kb) rEq
+
1 mlb
d2 a b - ef/(Kb) d2f - - rfl 14-mlb dy2 y = 0:
a = 1,dbldy = delay = dfldy = 0
y = 1:
a = aL,dbldy = deldy = Vldy = 0
Here the facilitation factor F is defined as follows:
It is seen from the above equations that F is a function of seven dimensionless parameters: K = KeqcAO, m = k d k d h , q = DBCBTIDACAO, m = W D B ,r~ = DFIDB,6 =L and a L = CAIJCAO.
Jmi,
Approximate Solution In the analysis of facilitation factor for the case where the reaction A (permeate) B (carrier) == C (complex) occurs in the membrane, it was confirmed that for the given values Of CAOand CAL,the influx of the permeant species A at the feed side boundary is mostly determined by CBOand CCO,the concentrations of CBand Cc a t x = 0, respectively, and is little influenced by CBLand CCL, the concentrations at x = L (Teramoto, 1994). When this approximation method is applied to the present facilitated transport, it seems reasonable to consider that the influx can be approximately analyzed by assuming that the concentrations CBand CFare uniform a t CBOand CFO,respectively, throughout the membrane and also by using the following boundary conditions instead of eqs 8 and 9 or 15 and 16.
+
subject to x = 0:
C A = CAO,dCBIdX = dCEIdX = d C j & = O (8) x=L: CA= C a , dCBIdX = dCEIdX = dCF/dX = O (9) The boundary conditions on B, E, and F represent that these species are nonvolatile and constrained t o stay within the membrane. The conservation of the carrier B in the membrane is expressed as
a t x = 0:
or a t y = 0: where L is the membrane thickness and CBTis the total carrier concentration. The above equations and boundary conditions are transformed in a dimensionless form: d2a = g2 a b -ef/(Kb) dY2
1
+ mlb
d2b - 26, a b - ef/(Kb) dy2 q 1 +m/b
(11)
(12)
atx=L: or a t y = 1
C A = CAO, CE = CEO, dCd& = 0 a = 1,e = e,, deldy = 0
(19)
c, = c, a = aL
(20)
Here, CBO,CEO,and CFOare unknown concentrations that can be determined as will be described later. Although the boundary condition dCd& = 0 a t x = L, i.e., eq 9, is not satisfied, this may not lead to a serious error in the evaluation of the influx.
Ind. Eng. Chem. Res., Vol. 34, No. 4,1995 1269 The concentration profiles of A and E can be obtained by solving eqs 11 and 13 after linearization by replacing b and f with bo (= CB&'BT) and fo (= CFD/CBT), respectively, and the facilitation factor FOis derived as follows in the similar manner described in the previous paper (Teramoto, 1994):
+
+
where yo = 62/[b, (f,/rEqKb,)Y(l mlb,). The outflux of the permeate a t x = L can be analyzed similarly by using the following boundary conditions and also by assuming that the concentration of B and F are uniform throughout the membrane at CBLand CFL,respectively. cA = cAO
a=l
or a t y = 0:
or a t y = 1:
(22)
CA= C u , CE = C,
a = aL,e = eL, deldy = 0
(23)
=
(24)
+
+
where y~ = 62/[bL (fL/rEqKbL)Y(l m/bL). From eqs 11-16 and 18, the following equations are derived without any approximation.
+
F = 1 - U L - q(bo - bJ/2 = 1 - U L rEq(e,- e,> = 1 - aL + r.Fq(fo- f L ) (25) The following equation holds for steady-state permeation.
F=Fo=FL
(26)
According to the previous works (Teramoto et al., 1986; Teramoto, 1994), eq 17 is approximated as follows: (Cgo
12.73 12.73 16.22 16.22 19.71 19.71 23.20 23.20 57.79 57.79 70.71 70.71 90.11 90.11 109.50 109.50 128.89 128.89
1.08 1.08 1.07 1.07 1.07 1.07 1.07 1.07 2.64 2.64 2.79 2.79 2.26 2.26 2.27 2.27 2.36 2.36
6
K
q
aL
117 91.4 117 91.4 117 91.4 117 91.4 117 91.4 117 91.4 117 91.4 117 91.4 117 91.4
22.3 22.3 28.4 28.4 34.5 34.5 40.6 40.6 101 101 124 124 158 158 191 191 225 225
152 92.8 120 72.9 98.3 60.0 83.6 50.9 33.6 20.4 27.4 16.7 21.5 13.1 17.7 10.8 15.0 8.94
0.0848 0.0848 0.0660 0.0660 0.0543 0.0543 0.0461 0.0461 0.0457 0.0457 0.0395 0.0395 0.0250 0.0250 0.0207 0.0207 0.0183 0.0183
hum) (approx) 6.98 4.94 6.01 4.29 5.39 3.85 4.82 3.47 2.41 1.86 2.16 1.70 1.96 1.58 1.79 1.47 1.65 1.38
6.91 4.87 6.02 4.25 5.34 3.80 4.82 3.45 2.39 1.84 2.13 1.68 1.95 1.56 1.77 1.45 1.64 1.37
%
-1.06 -1.55 0.13
-0.34 -0.98 -1.37 0.00
-0.74 -0.87 -1.20 -1.20 -1.44 -0.75 -1.22 -1.16 -1.22 -0.47 -0.97
Physicochemical properties: HA = 3.20 x mol/(dm3.atm), 400 dm3/(mol-s),Keq = 4.17 x lo3 dm3/mol, D A = 0.914 x or 1.50 x cm2/s, respectively, according to eq 28 or 27 of the literature (Guha et al., 19901, and D B = 0.383 x cm2/s. Experimental conditions: CBT= 1.9429 moVdm3,L = 127 pm. m = 0, ?'E = T-F = 1, and ar,= pAS/pm. F (this work) = [F (Guha et al.) l I ( 1 - aL). a
+
dC$& = O
Here, CBL,CEL,and CFL are unknown concentrations t o be determined later. The concentration profiles of A and E can be derived by solving eqs 11 and 13 with b and f replaced by bL (= C ~ C B Tand ) f ~ ( CFIJCBT), = respectively, and the facilitation factor FL at y = 1 is obtained as follows:
FL = -(da/dy),,,
cmHg cmHg
kl =
a t x = 0:
atx=L:
Table 1. Comparison of Approximate Solutions with Numerical Solutions by Guha et al. (1990)" PAF, PAS, F F error,
+ CBL + CEO + Cm + CFO + CFJ2 = CB, or (bo+ b, + e, + eL + fo + fL)/2 = 1 (27)
It can be shown from eqs 12-17 that eq 27 holds strictly for the case of Dg = DE = DF,i.e., IE: = qi = 1. Electrical neutrality in the membrane is expressed by the following equation.
For given combinations of the seven dimensionless parameters, six concentrations at the interfaces bo, bL, eo, eL, f o , and f~ and the facilitation factor F can be determined by the trial and error procedure using eqs 21 and 24-28.
Results and Discussion Case of m = 0. Guha et al. (1990) simulated the permeation rate of COS through a supported liquid membrane containing diethanolamine by assuming that the reaction rate is expressed by eq 1, that is, eq 3 with m = k2/Jt&BT = 0. They obtained numerical solutions for the facilitation factor corresponding to their experimental conditions by numerical integration of the basic equations. Physicochemical properties of the liquid membrane system such as reaction rate constant, equilibrium constant, Henry's constant of CO2, and diffusivity of C02 and DEA in the membrane solutions were obtained using Tables IV and V together with eqs 2729 of their paper and are shown in the lower part of Table 1. The effective membrane thickness L was taken as tL' where the tortuosity factor t is 5 and the membrane thickness L' is 25.4 pm. From these values, the parameters 6, K,q, and U L were calculated. The values of rE and rF are unity. Table 1 shows the comparison between their numerical solutions and the present approximate solutions of the facilitation factor. It can be seen that the present approximate solutions agree with the numerical solutions within an error of 1.6%. The effects of some dimensionless parameters on the facilitation factor were investigated using the present approximation method. Figure 2 shows the effect of 6 on F with U L as the parameter when m = 0. As 6 decreases, the value of F approaches an asymptotic value, 1 - UL, corresponding to the physical diffusion. On the other hand, as 6 increases, F approaches another asymptotic value, Fes, for each value of U L . This asymptotic value corresponds to the facilitation factor when the reaction reaches equilibrium everywhere in the membrane. In this case, Feq is approximately
1270 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 20
'
"""'1
'"""'I
'
"'"''1
' "'''''t
q=50,m=0,rE=rF=l,K=l
'
"'
aL=O
rigorously. As shown in Figure 2, the present approximation method gives correct asymptotic values at both small and large values of 6. This fact together with Table 1 which shows satisfactory agreement between the approximate solutions and the numerical solutions at the intermediate values of 6 suggests that the present method gives reasonable estimates of F over the entire range from physical diffusion region (6 0)to chemical equilibrium region (6 -1. Figure 3 shows the effect of K on F with 6 as the parameter when m = 0. In the region of small K, F increases with increasing K. However, as K becomes larger than some critical value, the reversibility of the reaction decreases, which results in low releasing rate of the permeant molecule at the receiving side interface. Therefore, F becomes low at very large values of K, and each F vs K curve has a maximum as has been already suggested by Schultz et al. (1974). Case of m > 0. Davis and Sandall (1993)performed a series of experiments on the permeation of COZ through a supported liquid membrane containing diethanolamine (DEA) or diisopropanolamine(DIPA) dissolved in poly(ethy1ene glycol). They also calculated the facilitation factor corresponding to each experimental condition by numerical integration of eqs 4-7. Comparison of their numerical results with the present approximate solutions for the facilitation factor is shown in Table 2. The values of dimentionless parameters shown in Tables 6 and 7 of their paper were used in the calculation. The approximate solutions agree with the numerical solutions within an error of 2.3%. Figures 4 and 5 represent the comparison of the present approximate solutions (lines) with the numerical solutions (open circles) by Davis and Sandall (1993) for COZ-DEA and C02-DIPA systems, respectively. The values of parameters K, m,and q shown in Tables 6 and 7 of their paper were used in the calculation. Agreement between the numerical and approximate solutions are fairly well over the entire range of 6. Figure 6 shows the effect of 6 on F with m as the parameter. As m increases, the reaction rate r decreases (see eq 3). This results in the decrease in F as can be seen in this figure. Recently, many softwares for solving nonlinear ordinary differential equations such as eqs 11-14 have become available. However, since the programs are very complicated, a computer with large memories is necessary. Furthermore, it is generally difficult to obtain accurate and reliable numerical solutions when the value of 6 s very large. On the other hand, in the case of the present approximation method, a personal
-
Figure 2. Effect of 6 on F with a L as parameter in the case of m = 0.
-03
-""I ""'-1 "'"'I
! -
"1
q = 5 0 , m = O , r ~ = r ~,aL=O =l
I
u
LL
K [-I Figure 3. Effect of K on F with 6 as parameter in the case of m = 0.
expressed by the following equation irrespective of the value of m.
Fes = (1 - a,)
+ x r E q ( f i - &)
+
(1 2rEJm(1 f 2 s )
+ (1+ 2TE&)(
1 f 2Cm
(29) This equation is derived in a similar manner proposed by the present author (Teramoto et al., 1986) by using two chemical equilibrium equations at the two interfaces and eqs 25,27,and 28 with m = n. In the case of ?'E = TF = 1, eq 29 reduces to the equation presented by Davis and Sandall (1993).In this case, eq 29 holds
-
Table 2. Comparison of Approximate Solutions with Numerical Solutionsby Davis and Sandall (1993)O amine DAF, atm [aminel, wt % 6 K 4 m F bum) F (amrox) error, 5% DEA 1 10 152 93.9 1.91 2.07 1.14 1.11 -2.22 1 1.04 1.71 1.68 -0.90 20 250 65.7 6.62 1 21.5 40.1 0.692 3.57 3.54 -0.90 30 377 0.005 10 0.469 382 2.07 17.1 17.0 152 -0.47 0.005 20 0.328 1324 1.04 42.2 250 42.1 -0.17 0.005 30 0.692 4300 377 0.692 85.2 85.1 -0.073 DIPA 1 72.4 58.1 1.26 1.07 10 1.23 1.09 -1.73 1 113 41.9 3.86 0.620 1.41 20 1.39 -1.77 1 11.6 165 25.6 0.418 2.44 30 2.41 -1.10 0.005 72.4 0.294 253 1.23 11.0 10 10.9 -0.67 20 0.005 23.8 0.620 23.9 113 0.210 773 -0.29 44.2 0.418 44.3 165 0.128 2320 30 -0.11 0.005 The values of parameters 6, m, and q for each experimental condition were obtained from Tables 6 and 7 of the literature (Davis and Sandall, 1993). aL = 0,and TE = TF = 1. F (this work) = [F (Davis and Sandall) 11(1 - a L ) .
+
Davis and Sandall(num.) pAF=0.005atm, F,=1017
Even in such a case, the trial and error procedure for calculating PAO and PAL, the partial pressures at the boundaries, by the present method is much simpler than that by the numerical integration method. The present approximation method may be easily extended to other facilitated transport systems with different types of chemical reaction.
Conclusion Figure 4. Comparison of present approximate solutions with numerical solutions by Davis and Sandal1 (1993) for the COZdiethanolamine system. Open circles represent the numerical solution shown in Figure 10 of their paper, and lines represent the present approximate solutions. The values of parameters K , m, and q are given in Table 2. aL = 0, and rr: = r F = 1.
A n approximate solution of the facilitation factors in the facilitated transport has been developed for the case where a reversible reaction, A ((202)2B (amine) E (carbamate) F (protonated amine), occurs in the membrane, and the reaction rate r is expressed by r = kl[CACB - CECF/(KeqCB)Y(l f K'ICB). It was confirmed that the facilitation factors calculated by the present approximation method agree with the previously reported results which had been obtained by numerical integration of the governing differential equations. It was also confirmed that the present method gives correct asymptotic values both at very small values of 6 (physical permeation region) and a t very large values of 6 (chemical equilibrium region).
+
+
Nomenclature a = CA/CAO -
.o--
1006
,
'
'
'
'
'
'
10'0
'
'
'
'
'
'
'
[-I
' ' ' 1020
'
'
I
Figure 6. Comparison of present approximate solutions with numerical solutions by Davis and Sandal1 (1993) for the COzdiisopropanolamine system. Open circles represent the numerical solution shown in Figure 11 of their paper, and lines represent the present approximate solutions. The values of parameters K , m ,and q are given in Table 2. aL = 0, and r~ = r~ = 1.
201
t
'
"""'I
' """'I
' " " ' " 1
'
"""'I
'
"'1...1
q=50, rE=rF=l K=l ,aL=O
7 -1
aL = C ~ C A O b = C$CBT
C = concentration, mol/" CBT = total carrier concentration, m0l/m3 D = diffusivity, m2/s e = CE/CBT F = facilitation factor defined by eq 18 Feq= facilitation factor at chemical equilibrium condition f = CF~CBT HA = Henry's constant of A, mol/(m3-atm) K = KeqcAO Keq = chemical equilibrium constant (klkdkd4 = (CECF/ CACB2)eq),m3h0l K' = kdk3, mol/m3 k G = gas phase mass transfer coefficient, mol/(m2.s.atm) k l = forward reaction rate constant, m3/(mol.s) ki = reaction rate constant of reactions c and d L = membrane thickness, m m = kdk&BT
LL
PAF = partial pressure of A in feed phase, cmHg or atm PAS = partial pressure of A in receiving phase, cmHg or
atm p ~= o partial pressure of A at feed side interface, cmHg or
atm ?Lo'
""""
102
' """"
'
""""
6
[-I
' ,,,d
Io4
Figure 6. Effect of 6 on F with m as parameter.
computer is sufficient to get the facilitation factors very quickly and accurately without any problem of convergence. Therefore, it may be concluded that the present method is useful for the analysis of facilitated transport. If the external mass transfer resistance is important, the boundary conditions on A (eqs 8 and 9) must be modified as follows:
= partial pressure of A at receiving side interface, cmHg or atm
PAL
q = DBCBTIDACAO rE = DE/DB rF = DF/DB x = distance, m y = x/L
Subscripts
A = permeant species ( ( 2 0 2 ) B = carrier (amine) E = component E (carbamate) F = component F (protonated amine)
1272 Ind. Eng. Chem. Res., Vol. 34,No. 4, 1995
Literature Cited Alvarez-Fuster, C.; Midoux, N.; Laurent, A.; Charpentier, J. C. Chemical Kinetics of the Reaction of Carbon Dioxide with Amines in Pseudo m-th Order Conditions in Aqueous and Organic Solutions. Chem. Eng. Sci. 1980,35, 1717-1723. Danckwerts, P. V. Reaction of COz with Ethanolamines. Chem. Eng. Sci. 1979, 34, 443-446. Danckwerts, P. V.; McNeil, K. M. The Absorption of Carbon Dioxide into Aqueous Amine Solutions and the Effects of Catalysis. Trans. Znst. Chem. Eng. 1967,45, T32-T49. Davis, R. A.; Sandall, 0. C. COdCH4 Separation by Facilitated Transport in Amine-Polyethylene Glycol Mixtures. AZChE J . 1993,39, 1135-1145. Guha, A. K.; Majumdar, S.; Sirkar, K. K. Facilitated Transport of COz through an Immobilized Liquid Membrane of Aqueous Diethanolamine. Znd. Eng. Chem. Res. 1990,29, 2093-2100. Hikita, H.; Asai, S.; Ishikawa, H.; Honda, M. The Kinetics of Reactions of Carbon Dioxide with Monoethanolamine, Dietha-
nolamine and Triethanolamine by a Rapid Mixing Method. Chem. Eng. J. 1977,13, 7-12. Laddha, S. S.; Danckwerts, P. V. Reaction of COZwith Ethanolamines: Kinetics from Gas-Absorption. Chem. Eng. Sei. 1981, 36,479-482. Noble, R. D.; Way, J. D. Liquid Membranes: Theory and Applications; ACS Symposium Series 347; Americal Chemical Society: Washington, DC, 1987. Sada, E.; Kumazawa, H.; Han, Z. Q.; Matsuyama, H. Chemical Kinetics of the Reaction of Carbon Dioxide with Ethanolamines in Nonaqueous Solvents. AIChE J . 1985,31, 1297-1303. Schultz, J. S.; Goddard, J. D.; Suchdeo, S. R. Facilitated Transport via Carrier-Mediated Diffusion in Membranes. AJChE J . 1974, 20,417-445. Teramoto, M. Approximate Solution of Facilitation Factors in Facilitated Transport. Znd. Eng. Chem. Res. 1994,33, 21612167. Teramoto, M.; Matsuyama, H.; Yamashiro, T.; Katayama, Y. Separation of Ethylene from Ethane by Supported Liquid Membranes Containing Silver Nitrate as a Carrier. J . Chem. Eng. Jpn. 1986,19,419-424.
Received for review September 7, 1994 Revised manuscript received December 21, 1994 Accepted January 4, 1995" 139405347 Abstract published i n Advance ACS Abstracts, March 1, 1995. @
