# Mathematical modelling of C[O.sub.2] facilitated transport through liquid membranes containing amines as carrier.

INTRODUCTIONRemoval of C[O.sub.2] from sour natural gas, refinery gases, and synthesis gas by absorption into aqueous solutions of alkanolamines has become a well-established process and is of major industrial importance.

Facilitated transport is a well-known process in which the reversible complexation reactions between solutes with carrier are carried out on the feed side, then diffuses through a liquid membrane and releases the solute on the permeate side. Due to mass transfer accompanied by chemical reaction, the permeability and selectivity of liquid membranes are enhanced as compared to just physical transport. Because of very high permselectivity of facilitated transport membrane in comparison to conventional polymeric membranes on one hand and relatively very low energy consumption with respect to other conventional processes on the other hand, facilitated transport membranes have attracted the attention of many industries and researchers (Schultz et al., 1974; Saha et al., 1977; Niiya and Noble, 1985; Sengupta and Sirkar, 1986; Way et al., 1987; Way and Noble, 1989; Kreulen et al., 1993; Langevin et al., 1993; Saha and Chakma, 1995; Ito et al., 2001; Al Marzouqi et al., 2005; Bao and Trachtenberg, 2006).

Many studies have been carried out to calculate the theoretical permeation rates and predict the facilitation factor, which is generally defined as the ratio between the facilitated transport flux to the flux without carrier or purely molecular diffusive. Many numerical and approximate solutions have been carried out in order to predict the facilitation factor for the transport of a single component facilitated transport in liquid membranes (Ward, 1970; Smith and Quinn, 1979; Kemena et al., 1983; Noble et al., 1986; Basaran et al., 1989; Guha et al., 1990; Dindi et al., 1992; Jemaa and Noble, 1992; Davis and Sandall, 1993; Teramoto, 1994, 1995; Teramoto et al., 1996, 1997; Al-Marzoughi et al., 2002; Morales-Cabrera et al., 2005).

However, in literature not much information can be found on analytical solution of the non-linear diffusion-reaction transport problem in case of the following reaction occurs.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Guha et al. (1990) proposed a simulation for facilitated transport of C[O.sub.2] through an immobilized liquid membrane containing diethanolamine. They adopted a numerical solution procedure for integration of simplified equations in order to calculate the facilitation factor corresponding to their experimental conditions. They showed that the proposed model for facilitated transport of C[O.sub.2] through an ILM containing aqueous DEA solution provides an excellent estimate of the C[O.sub.2]/[N.sub.2] separation factor (and accordingly facilitation factor) over a wide range of [DELTA][MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Davis and Sandall (1993) studied the separation of carbon dioxide from methane by a supported liquid membrane containing secondary amines such as diethanolamine (DEA) and diisopropanolamine (DIPA) dissolved in polyethylene glycol. They also developed a mathematical model for calculating the facilitation factor. Their model which employs the zwitterions mechanism for the C[O.sub.2]-amine reaction, constant global concentration of amine based species and equal diffusion coefficients was numerically solved.

Teramoto (1995) developed an approximate solution for the facilitation factors in case of reaction (1). He assumed separate and constant concentrations of the carrier and the protonated amine at the two boundaries of the membrane for evaluating the flux of the permeant species and compared his results with numerical results in the literature which were in good agreement.

Yamaguchi et al. (1995) developed a new facilitated transport model for C[O.sub.2] through ion-exchange membranes containing a diamine complexing agent. They considered electrical double layer and friction effects for facilitated transport in ion-exchange membranes and showed that the numerical results of model agreed well with C[O.sub.2] transport experiments through both HN117-EDA and N117-EDA membranes.

The purpose of this study is to derive an approximate analytical solution for the prediction of the facilitation factor and permeation rate through liquid membranes for the facilitated transport of C[O.sub.2] where the reaction rate for C[O.sub.2] is expressed by the zwitterions reaction mechanism of Danckwerts (1979) and over a range from the moderate chemical reaction rate region to the chemical equilibrium region. In this analysis, both equal and unequal diffusivities of carrier and complexes throughout the membrane, zero and nonzero concentration of C[O.sub.2] at permeate side are considered.

Finally, model predictions are compared with the present numerical solution and also the numerical results which had been obtained by previous models in literature.

THEORY

Danckwerts (1979) described the forward reaction of C[O.sub.2] with primary and secondary alkanolamines with the zwitterion mechanism. The reaction steps successively involve the formation and reaction of a "zwitterions." The first step is the formation of the zwitterions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

And the second step is the subsequent removal of the proton from the zwitterions by a base, which in case of an amine gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Here [R.sub.1][R.sub.2]NH is amine in which [R.sub.1] and [R.sub.2] are functional groups or hydrogen and [k.sub.1], [k.sub.3] and [k.sub.2], [k.sub.4] are the forward and reverse reaction rate constants. The overall reaction is expressed by

C[O.sub.2](A) + 2[R.sub.1][R.sub.2]NH(B) [??] [R.sub.1][R.sub.2]NCO[O.sup.-](AB) + [R.sub.1][R.sub.2]N[H.sup.+.sub.2](BH) (4)

Application of the steady-state approximation to the zwitterions [R.sub.1][R.sub.2]N[H.sup.+]CO[O.sup.-] and the assumption that deprotonation of the zwitterions is carried out by an amine gives the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The reactive species, A, associates with the carrier, B, at the high-pressure side forming the complexes AB and BH. The complexes AB and BH diffuse through the membrane and are regenerated at the low-pressure side of the membrane by releasing carbon dioxide. This reaction enhanced diffusion which can lead to a high permeability and a high selectivity.

The reaction rate for the above reaction scheme may be expressed as:

[r.sub.A]= - d[C.sub.A]/dt = [k.sub.1]([C.sub.A] [C.sub.B] - [C.sub.AB] [C.sub.BH]/[K.sub.eq][C.sub.B])/ (1 + [k.sub.2]/[k.sub.3][C.sub.B]) = -[r.sub.AB] = -[r.sub.BH] (6)

[r.sub.B]= - d[C.sub.B]/dt = 2[k.sub.1]([C.sub.A] [C.sub.B] - [C.sub.AB] [C.sub.BH]/[K.sub.eq][C.sub.B])/ (1 + [k.sub.2]/[k.sub.3][C.sub.B]) (7)

The governing differential mass balance equations for the steady-state, one dimensional mass transport in a rectangular geometry are as below. It should be noted that the mass transfer resistance in the gas phase at the gas-liquid interface on both sides has been neglected.

[D.sub.eA] [d.sup.2] [C.sub.A]/[dx.sup.2] = [epsilon] [r.sub.A] (8)

[D.sub.eAB] [d.sup.2] [C.sub.AB]/d[x.sup.2] = -[epsilon] [r.sub.AB] (9)

[D.sub.eBH] [d.sup.2] [C.sub.BH]/d[x.sup.2] = -[epsilon] [r.sub.BH] (10)

[D.sub.eB] [d.sup.2] [C.sub.B]/d[x.sup.2] = 2 [epsilon] [r.sub.B] (11)

where, [D.sub.ei] is effective diffusivity which is defined by the following equation:

[D.sub.ei] = [D.sub.i] [epsilon/[tau]] (12)

Boundary conditions are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Equations (8)-(11) and boundary conditions are transformed to a dimensionless form to obtain the following equations. Dimensionless variables are defined in Table 1.

[d.sup.2][C.sup.*.sub.A]/d[X.sup.2] = [[zeta].sup.2] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*.sub.AB] [C.sup.*.sub.BH]/[K.sub.A] [C.sup.*.sub.B]/(1 + [gamma]/[C.sup.*.sub.B]) (15)

[d.sup.2][C.sup.*.sub.AB]/d[X.sup.2] = [-[zeta].sup.2]/[[alpha].sub.A][[beta].sub.AB] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*.sub.AB] [C.sup.*.sub.BH]/[K.sub.A] [C.sup.*.sub.B]/(1 + [gamma]/ [C.sup.*.sub.B]) (16)

[d.sup.2][C.sup.*.sub.BH]/d[X.sup.2] = [-[zeta].sup.2]/[[alpha].sub.A][[beta].sub.AB] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*.sub.AB] [C.sup.*.sub.BH]/[K.sub.A] [C.sup.*.sub.B]/(1 + [gamma]/ [C.sup.*.sub.B]) (17)

[d.sup.2][C.sup.*.sub.B]/d[X.sup.2] = 2[[zeta].sup.2]/[[alpha].sub.A] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*.sub.AB] [C.sup.*.sub.BH]/[K.sub.A] [C.sup.*.sub.B]/(1 + [gamma]/ [C.sup.*.sub.B]) (18)

At X = 0 [right arrow] [C.sup.*.sub.A] = 1, d[C.sup.*.sub.B]/dx = d[C.sup.*.sub.AB]/dx = d[C.sup.*.sub.BH]/dx = 0 (19)

At X = 1 [right arrow] [C.sup.*.sub.A] = [a.sub.L], d[C.sup.*.sub.B]/dx = d[C.sup.*.sub.B]/dx = d[C.sup.*.sub.BH]/dx = 0 (20)

with assuming that the reaction reaches equilibrium, the overall chemical equilibrium constant is expressed as

[C.sub.AB][C.sub.BH] = [K.sub.eq][C.sub.A][C.sup.2.sub.B] [??] [C.sup.*.sub.AB][C.sup.*.sub.BH] = [K.sub.A][C.sup.*.sub.A][C.sup.*2.sub.B] (21)

The electroneutrality condition at anywhere inside liquid membrane is resulted in

[C.sup.*.sub.AB] = [C.sup.*.sub.BH] = [square root of [K.sub.A][C.sup.*.sub.A][C.sup.*.sub.B]] (22)

Approximate Analytical Solution

If Equations (15) and (16), Equations (15)-(17), Equations (16)-(18) are added and their sum is integrated twice, the following expressions are derived.

[C.sup.*.sub.A] + [[alpha].sub.A] [[beta].sub.AB][C.sup.*.sub.AB] = [a'.sub.1]X + [a'.sub.2] (23)

2[C.sup.*.sub.A] + [[alpha].sub.A] [[beta].sub.AB][C.sup.*.sub.AB] + [[alpha].sub.A] [[beta].sub.BH] [C.sup.*.sub.BH] = [a.sub.1]X + [a.sub.2] (25)

[C.sup.*.sub.B] + [[beta].sub.AB] [C.sup.*.sub.AB] + [[beta].sub.BH] [C.sup.*.sub.BH] = [a.sub.3]X + [a.sub.4] (25)

where [a'.sub.1], [a'.sub.2], [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4] are the integration constants. From boundary conditions, Equations (19) and (20), it is obtained that [a.sub.3] = 0 and therefore

[C.sup.*.sub.BO] + [[beta].sub.AB] [C.sup.*.sub.ABO] + [[beta].sub.BH] [C.sup.*.sub.BHO] = [C.sup.*.sub.BL] + [[beta].sub.AB] [C.sup.*.sub.ABL] + [[beta].sub.BH [C.sup.*.sub.BHL]

= [a.sub.4]

= constant

= [C.sup.*] (26)

Equations (15)-(18) were integrated analytically by assuming a constant free carrier concentration, [C.sup.*.sub.Bavg], and constant protonated amine concentration, [C.sup.*.sub.BHavg], throughout the membrane phase. Since boundary conditions on A (C[O.sub.2]) and accordingly BH ([R.sub.1][R.sub.2][NH.sup.+.sub.2]) represent the maximum and minimum concentration at feed and permeate side respectively, the average concentrations of solute, complex and carrier by assuming reaction equilibrium at each membrane interface was derived as below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Due to assuming reaction equilibrium at each membrane interface and because of high dependency of facilitation factor and permeation rate to C[O.sub.2] concentration at permeate side and also physicochemical properties especially at high value of [zeta] and [K.sub.A], the below assumptions is used in approximate solution. (1) [a.sub.L] should be considered to negligible value of 0.001 in case of zero solute concentration of permeate side. (2) In case of [gamma] [not equal to] 0 and very high value of a, the (1 - [a.sub.L])/4 is used as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to calculate the concentration gradients across the membrane, the arithmetic averages of amine (B) and protonated amine (BH) at two sides of membrane are used.

By substituting of Equations (28) and (29) for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Equation (15) is simplified and leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

and reduced to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Equation (31) was analytically solved to obtain [C.sup.*.sub.A] and accordingly [C.sup.*.sub.AB] and [C.sup.*.sub.B] as below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

[C.sup.*.sub.B] = [C.sup.*] - ([[beta].sub.AB] + [[beta].sub.BH])[C.sup.*.sub.AB] = [C.sup.*] - [[beta].sub.A] [C.sup.*.sub.AB] (37)

where

[[lambda].sub.3] = [[lambda].sub.2]/[K.sub.A] [C.sup.*2.sub.B]

[m.sub.1], [m.sub.2], [m.sub.3], and [m.sub.4] are the integration constants and can be solved by implementation of boundary conditions, Equations (19) and (20), as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[m.sub.3] = 2[[lambda].sub.A]/[[lambda].sub.2] ([m.sub.2] - [m.sub.1]) (40)

[m.sub.4] = 1 - [m.sub.1] - [m.sub.2] (41)

The following additional condition is required for derivation of [C.sup.*]. The conservation of carrier B in the membrane is expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

We know from Equation (26) that [C.sup.*.sub.B] + [[beta].sub.AB] [C.sup.*.sub.AB] + [[beta].sub.BH] [C.sup.*.sub.BH] = [C.sup.*] and therefore by considering of Equation (22):

[C.sup.*.sub.B] + [[beta].sub.A] [square root of [K.sub.A] [C.sup.*.sub.A]] [C.sup.*.sub.B] = [C.sup.*] [??] [C.sup.*.sub.AB] = [C.sup.*.sub.BH] = [C.sup.*] - [C.sup.*.sub.B]/[[beta].sub.A]

By substituting in Equation (42)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

In order to obtain constant [C.sup.*], one constant value is supposed for [C.sup.*] and by using it, the concentration of A, B, AB and BH is obtained for each point of membrane thickness. Simpson's Rule was used to accomplish numerical integration, which consists of computing the value of a definite integral from the set of values of the integrand.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

Then the calculated value of above definite integral should satisfy the condition of Equation (44) and this procedure is repeated again by an iterative method.

The facilitation factor, F, is defined as the ratio of the permeation flux in the presence of carrier in liquid membrane to that of permeation in the absence of carrier, that is, physical diffusion, and is expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

Facilitation factor F was calculated for C[O.sub.2] using Equation (34) as below:

F = 1 + [[lambda].sub.2]/1 + ([[lambda].sub.2]/[[lambda].sub.A])tan h [[lambda].sub.A] (47)

And the solutes permeation flux are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

Reaction Equilibrium Limit

With assuming that the reaction is essentially instantaneous or the membrane thickness is high, the C[O.sub.2] reaction can be supposed to be at equilibrium everywhere in the immobilized liquid membrane. The facilitation factor and concentration gradients for equilibrium case is calculated as following procedure. Also the reaction species were constrained to be at equilibrium. So, By using Equations (22) and (26) the following expression is derived for dimensionless carrier concentration.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

Equations (24) and (26) at equilibrium condition are expressed as below:

2[C.sup.*.sub.A] + [[alpha].sub.A] [[beta].sub.A] [square root of [K.sub.A] [C.sup.*.sub.A]] [C.sup.*.sub.B] = [a.sup.1] X + [a.sup.2] (50)

[C.sup.*.sub.B] + [[beta].sub.A] [square root of [K.sub.A] [C.sup.*.sub.A]] [C.sup.*.sub.B] = [C.sup.*] (51)

And by using boundary conditions, results in:

At X = 0 [??] [C.sup.*.sub.A] = 1 [??] 2 + [[alpha].sub.A] [[beta].sub.A] [C.sup.*] [square root of [K.sub.A]] / 1 + [[beta].sub.A] [square root of [K.sub.A]] = [a.sub.2] (52)

At X = 0 [??] [C.sup.*.sub.A] = [a.sub.L] [??] [2a.sub.L] + [[alpha].sub.A] [[beta].sub.A] [C.sup.*] [square root of [K.sub.A] [a.sub.L]] / 1 + [[beta].sub.A] [square root of [K.sub.A] [a.sub.L]] - [a.sub.2] = [a.sub.1] (53)

Combination of Equations (22), (49), and (50) gives equation:

[[beta].sub.A] [C.sup.*.sub.A] [square root of [K.sub.A] [C.sup.*.sub.A]] + [C.sup.*.sub.A] + [[beta].sub.A] [[[alpha].sub.A] [C.sup.*] - ([a.sub.1] X + [a.sub.2])] [square root of [K.sub.A] [C.sup.*.sub.A]] - ([a.sub.1] X + [a.sub.2]]

= 0 (54)

and consequently

[C.sup.*(3/2).sub.A] + 1 / [[beta].sub.A] [square root of [K.sub.a]] [C.sup.*.sub.A] + [[[alpha].sub.A] [C.sup.*] - ([a.sub.1] X + [a.sub.2])] [C.sup.*(1/2).sub.A]

- 1 / [[beta].sub.A] [square root of [K.sub.A]] ([a.sub.1] X + [a.sub.2]) = 0 (55)

Equation (55) is simplified to a cubic equation respect to [C.sup.*.sub.A] and solved analytically as below:

[Z.sup.3] + [A.sub.1] [Z.sup.2] + [B.sub.1]Z + [C.sub.1] = 0 (56)

[Y.sub.3] + [P.sub.1]Y + [P.sub.2] = 0 (57)

[C.sup.*.sub.A] = [(([M.sub.1] + [M.sub.2]) - [A.sub.1]/3).sup.2] (58)

where

[A.sub.1] = 1/[[beta].sub.A] [square root of [K.sub.A]] (59)

[B.sub.1] = [[alpha].sub.A] [C.sup.*] - ([a.sub.1]X + [a.sub.2]) (60)

[C.sub.1] = [a.sub.1]X + [a.sub.2]/[[beta].sub.A] [square root of [K.sub.A]] (61)

[C.sup.*(1/2).sub.A] = Z = Y - [A.sub.1]/3 = ([M.sub.1] + [M.sub.2]) - [A.sub.1]/3 (62)

[P.sub.1] = 1/3 (3[B.sub.1] - [A.sup.2.sub.1]) (63)

[P.sub.2] = 1/27 (27[C.sub.1] - 9[A.sub.1][B.sub.1] + 2[A.sup.3.sub.1]) (64)

R = [([P.sub.1]/3).sup.2] + [([P.sub.2]/2).sup.2] (65)

[M.sub.1] = [cube root of - [P.sub.2]/2 + [square root of R] (66)

[M.sub.2] = [cube root of - [P.sub.2]/2 + [square root of R] (67)

Considering of obtained [C.sup.*.sub.A], the concentration of carrier, B, is calculated by Equation (49). In order to calculate the constant [C.sup.*], the same procedure as non-equilibrium case is followed. Facilitation factor F was calculated for C[O.sub.2] at equilibrium condition as below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68)

Numerical Solution

In order to evaluate the proposed analytical solution, the nonlinear boundary value problem, Equations (15)-(18) are numerically solved.

By assuming electroneutrality condition anywhere inside liquid membrane, the following main equations are derived which are numerically solved using boundary conditions in Equations (19) and (20).

[d.sub.2] [C.sup.*.sub.A]/d[X.sup.2] = [[zeta].sup.2] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*2.sub.AB]/[K.sub.A] [C.sup.*.sub.B])/(1 + [gamma]/[C.sup.*.sub.B]) (69)

[d.sup.2] [C.sup.*.sub.AB] / d[X.sup.2] = -[[zeta].sup.2] / [[alpha].sub.A] [[beta].sub.AB] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*2.sub.AB] / [K.sub.A] [C.sup.*.sub.B] / (1 + [gamma]/ [C.sup.*.sub.B]) (70)

[d.sup.2] [C.sup.*.sub.B]/d[X.sup.2] = 2[[zeta].sup.2]/[[alpha].sub.A] ([C.sup.*.sub.A] [C.sup.*.sub.B] - [C.sup.*2.sub.AB]/[K.sub.A] [C.sup.*.sub.B]/(1 + [gamma]/ [C.sup.*.sub.B]) (71)

RESULTS AND DISCUSSION

Analysis of the Analytical Solution and Comparison With Present Numerical Solution

Effects of physiochemical properties and system parameter dependencies on the calculated facilitation factor by both analytical and numerical solutions are presented. As it can be seen in Figures 1-5, present model results are in well agreement with numerical solutions.

Effect of equilibrium constant ([K.sub.A]) on the facilitation factor for different values of [zeta] has been shown in Figure 1. As it is shown in Figure 1, there is a maximum value of F as the equilibrium constant is varied, in other hand the facilitation factor starts from unity and goes up to maximum value, then it drops down to unity. This can be explained as follows: For low equilibrium constant, facilitation factor and permeation rate are low due to slow reaction rate and in case of very high equilibrium constant, facilitation factor and also permeation rate are decreased due to slow release of permeate in sweep side of membrane howbeit the reaction rate of permeate with carrier is high and fast. As expected, the facilitation factor increases as [zeta] increases. The dimensionless parameter [zeta] is proportional with the membrane thickness (L), forward reaction rate ([k.sub.1]), carrier concentration ([C.sub.T]), and tortusity ([tau]), and inversely varies with solute diffusion coefficient ([D.sub.A]). Therefore as it is expected the facilitation factor increase with increasing L, [k.sub.1], [C.sub.T], and [tau] and decreasing [D.sub.A].

Figure 2a and b shows the effect of [zeta] on the facilitation factor for different values of [alpha] in cases zero and nonzero value of [gamma]. It clearly presents that the facilitation factor increases as the parameter [zeta] is increased from physical diffusion up to chemical equilibrium region. As Figure 2 presents increase of [alpha] is resulted in higher facilitation factors. This behaviour can be described by physicochemical interpretations:

As defined in Table 1, the dimensionless parameter [alpha] is proportional with carrier concentration ([C.sub.T]) and diffusion coefficient ([D.sub.B]) and inversely proportional with solute concentration ([C.sub.A]) and diffusion coefficient ([D.sub.A]). The parameter [gamma] is proportional with [k.sub.2]/[k.sub.3] ratio and inversely varies with carrier concentration ([C.sub.T]). Therefore as shown in Figure 2a and b, facilitation factor increases with increase of [C.sub.T] and [D.sub.B] and decrease of [C.sub.A] and [D.sub.A]. Comparing Figure 2a and b shows that the facilitation factor decreases with increasing [k.sub.2]/[k.sub.3] or decreasing [C.sub.T].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Figure 3 presents the effect of [zeta] on facilitation factor for different values of complexes to carrier diffusion coefficients ratio, [[beta].sub.AB] and [[beta].sub.BH]. As it is expected, the facilitation factor increases with increasing of [[beta].sub.AB] and [[beta].sub.BH]. In other hand, the facilitation factor increases with increasing of solute-carrier complexes diffusion coefficients.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Figure 4 presents the effect of permeate side solute concentration on the facilitation factor. As shown in figure, the facilitation factor of solute decreases with increasing [a.sub.L]. The present model prediction in case of zero solute concentration at permeate side is not in a good agreement with numerical results because of assumed reaction equilibrium at each membrane interface.

Figure 5 shows the effect of equilibrium constant ([K.sub.A]) on the facilitation factor for different values of a in case of instantaneous reaction of C[O.sub.2] and amine inside liquid membrane. As can be seen, the facilitation factors obtained from the analytical model for equilibrium region (Equation 68) are in the good agreement with numerical results.

Comparison of the Model Predictions With Literatures Numerical Results

The facilitation factors predicted by the present approximate analytical model were compared and tested with the numerical solutions which had been reported by Guha et al. (1990) and Davis and Sandall (1993).

Guha et al. (1990) demonstrated C[O.sub.2] facilitation in an immobilized liquid membrane containing 20 wt% diethanolamine solution over a wide range of C[O.sub.2] partial pressures.

They developed a model for facilitated transport of C[O.sub.2] through an amine-containing liquid membrane. A complete set of equations relating simultaneous diffusion and reaction were proposed and solved numerically to estimate the facilitated flux of C[O.sub.2] corresponding to their experimental results.

Comparison of their experimental data and also numerical results with the present analytical solution is shown in Table 2. Physicochemical properties of the liquid membrane system were obtained using their article (Guha et al., 1990). It can be seen that approximate analytical results are in relatively good agreement with their numerical solutions.

Davis and Sandall (1993) studied the separation of C[O.sub.2] from C[H.sub.4] by supported liquid membrane containing secondary amine solutions such as diethanolamine and diisopropanolamine in polyethylene glycol 400. They also developed a mathematical model to describe the transport process and some experiments were carried out to evaluate the model predictions.

Their numerical results were compared with the present model predictions as shown in Table 3 which are in good agreement. [F.sub.eq] which are calculated by Equation (G8) assumes that reaction of C[O.sub.2] with amine can be supported at equilibrium. Comparison of their reported values and the present model predictions for equilibrium facilitation factors shown in Table 3 are in absolute agreement. Physicochemical properties of the liquid membrane system were obtained using their paper (Davis and Sandall, 1993).

CONCLUSION

An approximate analytical model is developed for C[O.sub.2] facilitated transport in the case where the reversible reaction (1) is occurred inside the liquid membranes. This mathematical model considers the effect of carrier diffusivity too, and can be successfully used for unequal permeates and complexes diffusivities and cases of zero and nonzero permeate side solute concentration.

This theoretical model can provide a quick and reasonable prediction of facilitation factor in comparison with numerical results. The model agreed well with C[O.sub.2] facilitated transport experiment through liquid membranes, present and literature numerical results.

This model can also be used for calculating the concentration profile of solute, carrier and complexes across liquid membrane.

NOMENCLATURE a concentration ratio of A at two sides of membrane ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) [A.sub.1], [B.sub.1], [C.sub.1], constant parameters of cubic equation [a'.sub.1], [a'.sub.2], [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4], and [C.sup.*] integration constants C concentration D diffusion coefficient F facilitation factor J permeation flux [k.sub.1] forward reaction rate constants for first reaction [k.sub.3] forward reaction rate constants for second reaction [k.sub.2] reverse reaction rate constants for first reaction [k.sub.4] reverse reaction rate constants for second reaction K dimensionless equilibrium constant [K.sub.eq] equilibrium constant ([k.sub.i]/[k.sub.-i]) L membrane thickness r reaction rate t time x distance parameter X dimensionless distance parameter Greek Letters [alpha] mobility ratio of carrier to solute ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) [beta] dimensionless diffusivity ratio of the carrier to complex [epsilon] porosity [tau] tortuosity [gamma] [k.sub.2]/[k.sub.3][C.sub.T] [zeta] [square root of [L.sup.2] [k.sub.1] [C.sub.T] [tau] / [D.sub.A] Superscripts * refers to dimensionless concentration Subscripts A chemical solutes AB, BH complexes B carrier e effective i general chemical species 0 value at x = 0 L value at x = L T total carrier (concentration)

ACKNOWLEDGEMENTS

Research support from the Petrochemical Research and Technology Company of Iran is gratefully acknowledged.

Manuscript received January 5, 2008; revised manuscript received March 15, 2008; accepted for publication April 14, 2008

REFERENCES

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Aliakbar Heydari Gorji and Tahereh Kaghazchi * Department of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), No. 424 Hafez Avenue, Tehran 15875-4413, Iran

* Author to whom correspondence may be addressed. E-mail address: kaghazch@a aut.ac.ir

Table 1. Dimensions variables [MATHEMATICAL EXPRESSION NOT Dimensionless concentration of REPRODUCIBLE IN ASCII] solute A [C.sup.*.sub.AB] = [C.sub.AB]/ Dimensionless concentration of [C.sub.T] complex AB [C.sup.*.sub.BH] = [C.sub.BH]/ Dimensionless concentration of [C.sub.T] complex BH [C.sup.*.sub.B] = [C.sub.B]/ Dimensionless concentration of [C.sub.T] free carrier B [MATHEMATICAL EXPRESSION NOT Dimensionless concentration of REPRODUCIBLE IN ASCII] solute A at X=1 X = X/L Dimensionless membrane thickness [[beta].sub.AB] = [D.sub.AB]/ Dimensionless ratio of the complex [D.sub.B] AB and carrier diffusivity [[beta].sub.BH] = [D.sub.BH]/ Dimensionless ratio of the complex [D.sub.B] BH and carrier diffusivity [MATHEMATICAL EXPRESSION NOT Mobility ratio of carrier to REPRODUCIBLE IN ASCII] solute A for the first reaction [MATHEMATICAL EXPRESSION NOT Dimensionless equilibrium constant REPRODUCIBLE IN ASCII] for the first reaction [zeta] = [square root of [L.sup.2][k.sub.1] [C.sub.T][tau]/[D.sub.A]] [gamma] = [k.sub.2]/[k.sub.3] [C.sub.T] Table 2. Comparison of present model results with numerical solutions by Guha et al. (1990) [P.sub.Af] [MATHEMATICAL [[alpha]. [zeta] (cm Hg) EXPRESSION NOT sub.A] REPRODUCIBLE IN ASCII] 12.73 1.08 152 117 12.73 1.08 92.8 91.4 16.22 1.07 120 117 16.22 1.07 72.9 91.4 19.71 1.07 98.3 117 19.71 1.07 60 91.4 23.20 1.07 83.6 117 23.20 1.07 50.9 91.4 57.79 2.64 33.6 117 57.79 2.64 20.4 91.4 70.71 2.79 27.4 117 70.71 2.79 16.7 91.4 90.11 2.26 21.5 117 90.11 2.26 13.1 91.4 109.5 2.27 17.7 117 109.5 2.27 10.8 91.4 128.89 2.36 15 117 128.89 2.36 8.94 91.4 [P.sub.Af] [a.sub.L] [F.sup.a] [F.sup.b] [F.sup.c] (cm Hg) 12.73 0.0848 7.34 7.63 5.02 12.73 0.0848 -- 5.4 3.78 16.22 0.066 5.78 6.43 4.66 16.22 0.066 -- 4.57 3.51 19.71 0.0543 4.78 5.7 4.36 19.71 0.0543 -- 4.07 3.29 23.20 0.0461 4.08 5.05 4.12 23.20 0.0461 -- 3.64 3.11 57.79 0.0457 2.34 2.53 2.32 57.79 0.0457 -- 1.95 1.85 70.71 0.0395 2.10 2.25 2.13 70.71 0.0395 -- 1.77 1.72 90.11 0.025 2.01 2.01 2.05 90.11 0.025 -- 1.62 1.67 109.5 0.0207 1.70 1.83 1.91 109.5 0.0207 -- 1.5 1.58 128.89 0.0183 1.48 1.68 1.8 128.89 0.0183 -- 1.41 1.5 Table 3. Comparison of present model results with numerical solutions by Davis and Sandall (1993) [P.sub.Af] (atm) Amine conc. (wt%) [alpha]A [zeta] 1 10% DEA 1.91 152 1 20% DEA 6.62 250 1 30% DEA 21.5 377 0.005 10% DEA 382 152 0.005 20% DEA 1324 250 0.005 30% DEA 4300 377 1 10% DIPA 1.26 72.4 1 20% DIPA 3.86 113 1 30% DIPA 11.6 165 0.005 10% DIPA 253 72.4 0.005 20% DIPA 773 113 0.005 30% DIPA 2320 165 [P.sub.Af] (atm) [K.sub.A] [gamma] [F.sup.a] 1 93.9 2.07 1.14 1 65.7 1.04 1.71 1 40.1 0.692 3.57 0.005 0.469 2.07 17.1 0.005 0.328 1.04 42.2 0.005 0.201 0.692 85.2 1 58.1 1.23 1.09 1 41.9 0.62 1.41 1 25.6 0.418 2.44 0.005 0.294 1.23 11 0.005 0.21 0.62 23.9 0.005 0.128 0.418 44.3 [P.sub.Af] (atm) [F.sup.b] [F.sup.a. [F.sup.b. sub.eq] sub.eq] 1 1.21 1.91 1.91 1 1.98 4.12 4.12 1 4.35 10.96 10.96 0.005 18.73 111.67 111.4 0.005 44.58 354.54 354.44 0.005 86.28 1016.9 1016.9 1 1.15 1.59 1.59 1 1.62 2.79 2.79 1 3 6.28 6.28 0.005 12.20 66.64 66.81 0.005 25.78 185.83 185.83 0.005 45.86 484.94 484.83 Physicochemical properties and accordingly the value of parameters are obtained from Davis and Sandall (1993) paper. [[beta].sub.AB] = [[beta].sub.NH] = 1, [a.sub.L] = 0, [F.sup.a] = [F(numerical from Davis and Sandall paper) + 1], [F.sup.b] = F(Present model).

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Author: | Gorji, Aliakbar Heydari; Kaghazchi, Tahereh |
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Publication: | Canadian Journal of Chemical Engineering |

Date: | Dec 1, 2008 |

Words: | 6386 |

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