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Ind. Eng. Chem. Res. 2002, 41, 4626-4631
SEPARATIONS Approximate Method for the Solution of Facilitated Transport Problems in Liquid Membranes M. A. Morales-Cabrera, E. S. Pe´ rez-Cisneros, and J. A. Ochoa-Tapia* Area de Ingenierı´a Quı´mica, Universidad Auto´ noma Metropolitana, Unidad Iztapalapa, Me´ xico D.F. 09340
An approximate analytical solution to solve the nonlinear diffusion-reaction transport problem has been developed. The system considered is a liquid membrane where the facilitated transport or carrier-mediated transport occurs. The approximate analytical solution methodology is based on the Taylor series expansion of the reaction rate around the two limiting surfaces of the liquid membrane. The methodology leads to analytical expressions for the concentration profiles of the species in the membrane. Predictions for the facilitation factor have been obtained for a wide range of Damko¨hler number values, from the physical diffusion regime to the equilibrium chemical reaction regime, considering cases of equal and unequal carrier and complex diffusivities and cases of zero and nonzero downstream solute concentration. For the example presented here, the difference between the approximate predictions and those obtained from the numerical solution is not greater than 5.0%. Introduction The separation of high-profit compounds from a mixture through liquid membranes has received considerable attention because of the specific characteristics of the process such as low energy consumption and high selectivity. In general, a liquid membrane may be regarded as a semipermeable barrier. When it is placed between two aqueous phases, chemical species (solutes) can move through the membrane from a region of high solute concentration to a region of low solute concentration by means of only a diffusional process. However, mass transport of a solute through a liquid membrane can be affected by the presence of a chemical reaction. If a mobile carrier is contained inside of a liquid membrane, it may react with the solute to produce a complex, increasing the solute permeation across the membrane. This mechanism is known as facilitated transport. The application of this transport process is currently extended to branches of science such as physiology and biochemistry and to industrial separation processes.1-3 Recently, the facilitated transport has been recognized as a potentially valuable technology for the selective separation of precious metals4 and toxic metals.5 The chemical reaction that has mainly been considered to study the facilitated transport problem is that where a solute A reacts with a carrier B forming a complex P. The treatment of the nonlinear diffusionreaction (NLDR) problem, that is, the solution of the differential equations raised from the material balances in the liquid membrane, has been developed considering approximate analytical and/or numerical solutions. Despite the advance in the numerical techniques and computational power, the approximate analytical solu* To whom all correspondence should be addressed. Phone: +(52) 5804-4648. Fax: +(52) 5804-4900. E-mail: jaot@xanum. uam.mx.
tions are still useful because they are more flexible and reliable for extreme cases such as the chemical equilibrium regime. However, most of the approximate solutions have been restricted to the steady state and to the case where the diffusivities of the carrier and of the complex are the same. Among the approximate solutions previously developed, several can be mentioned. Goddard et al.6 considered the reaction under chemical equilibrium conditions. Ward7 assumed that the reaction is slow enough such that the concentrations of the carrier and complex are constant through the membrane. Smith et al.8 developed two approximate analytical solutions: (a) purely physical diffusion (membranes with thin thickness) and (b) equilibrium reaction condition (membranes with thick thickness). To obtain these solutions, they employed different techniques of matched asymptotic expansions. In the analysis of Kreuzer and Hoofd,9 the existence of two nonequilibrium marginal regions and a core very near chemical equilibrium is assumed. Smith and Quinn10 considered chemical equilibrium through the whole membrane and that the carrier concentration is the same. Basaran et al.11 used a perturbation analysis to derive an approximate solution which is applicable for equal and unequal carrier and complex diffusivity cases. In one of the latest works related to the theoretical analysis of the facilitated transport problem, Teramoto12 presented a methodology to obtain approximate solutions. The approach, in principle, can be used to solve diffusion-reaction problems that involve a variety of kinetic schemes and considers unequal carrier and complex diffusivities. Teramoto’s approximation assumes, to linearize the kinetic expressions, that one or more of the species concentrations are constant in certain regions of the liquid membrane. This linearization leads to important simplifications in the solution procedure.
10.1021/ie010923i CCC: $22.00 © 2002 American Chemical Society Published on Web 08/08/2002
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However, this restricts the possibility of correctly reproducing the concentration profiles. Also, it introduces restrictions in the form that the total carrier amount in the membrane must be taken into account. Most of the accuracy tests for the approximate methods have been carried out by comparison of the predicted facilitation factors with those obtained by an exact method. The objective of this work is to develop a general approximate analytical solution for the calculation of the facilitation factor. In particular, the analysis is applied to the solution of the diffusion-reaction problem of oxygen through hemoglobin solutions.1 The reaction for this system follows the stoichiometry oxygen + carrier h complex.14 The solution considers the following: (a) the whole range of Damko¨hler numbers, that is, from the purely physical diffusion regime to the chemical equilibrium regime; (b) the cases of equal and unequal carrier and complex diffusivities; and (c) the cases of zero and nonzero downstream solute concentration. Governing Equations
conditions given by eq 6 are independent. Therefore, an additional boundary condition is required. The extra independent boundary condition that may be considered is the carrier mass conservation relation:
∫-1+1(UB + UP) dX ) UBT
1 2
where UBT is the dimensionless total carrier concentration. In eqs 2-7, the dimensionless variables and parameters used are defined in the Nomenclature section. The use of eq 2 for A and i ()B or P) leads to a differential equation for Ui in terms of UA. The integration of this, result leads to the following expression for the concentration Ui in terms of UA:
Ui(X) )
A + B {\ }P k -1
(1)
where k1 and k-1 are the forward and reverse reaction rate constants, respectively. The steady-state dimensionless governing diffusionreaction equations, for one-directional transport in flatplate geometry with thickness l, are
νi ) - Φ2RA ri dX
for i ) A, B, P
(2)
In this equation, Φ2 is the Damko¨hler number, ri are the diffusion ratio coefficients, νi are the stoichiometric coefficients (νA ) -1, νB ) -1, and νP ) +1), and RA is the dimensionless nonlinear reaction rate expression given by
RA ) UAUB - UP/K
(3)
JA ) -
|
dUA dX
X ) +1 X ) -1 and +1
UA ) 1
(4)
UA ) UA|X)+1
(5)
dUi )0 dX
)-
X)-1
(8)
|
dUA dx
X)+1
(9)
ri ri Ti ) 1 - JA + Ui|X)-1 ) UA|X)+1 + JA + Ui|X)+1 νi νi for i ) B and P (10) To obtain eq 8, the zero flux boundary conditions for B and P have been imposed along with the concentrations at the boundaries for B and P, which still are unknown. At this point, eq 8 can be used to express the original boundary value problem only in terms of the concentration UA. Method of Solution Following Marroquı´n de la Rosa et al.,13 the reaction rate term can be linearized by neglecting the secondorder and higher order terms of the Taylor series expansion to obtain i)nc
where K is a dimensionless equilibrium constant. The solution of eq 2 is subjected to the following boundary conditions:
X ) -1
for i ) B and P
and
k1
2
νi 1 νi νi U - J X + Ti νA ri A ri A ri
where JA is the permeate flux at the boundaries given by
The reaction scheme considers the permeate A reacting with the carrier B to form the complex P according to the following stoichiometry:
d2Ui
(7)
for i ) B and P
(6)
The boundary conditions given by eqs 4 and 5 indicate that the permeate concentrations at the membrane boundaries are known, and the boundary conditions in eq 6 represent the fact that the carrier B and the complex P are constrained to remain inside the membrane. Nevertheless, with the above conditions the problem is not completely defined, and they are not enough to solve the boundary value problem. Goddard et al.6 have shown that only three of the four boundary
RA ≈ RA|Xs +
∑ i)1
|
∂RA ∂Ui
(Ui - Ui|Xs)
(11)
Xs
where nc ()3 in this case) is the total number of species participating in the reaction system and Xs is the position where the concentration and its derivatives is evaluated. Note that eq 8 allows one to replace easily UB and UP in terms of UA in the reaction term. Next, the Taylor series expansion is evaluated around the two limiting surfaces of the liquid membrane system, X ) (1. The reason for considering the two limiting surfaces is that, under steady-state conditions, as indicated by eq 9, the flux of A should be the same on both surfaces, and it is necessary to evaluate the facilitated transport across of the liquid membrane. Therefore, JA should be calculated with the minimum error in order to obtain satisfactory facilitation factor predictions. The evaluation of the reaction term on the two limiting surfaces leads to representation of the entire domain for the membrane in two regions and, as a
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consequence of that, to one solute concentration (UA) governing equation for each region. Therefore, two additional boundary conditions for UA are needed, and these are the concentration and flux continuity conditions in the middle of the membrane. After some straightforward algebraic manipulation, which involves eqs 2, 8, and 11, the following governing differential equations, for the approximate UA boundary value problem, are obtained:
dX 2
- φL2UA,L ) RLX + βL, for -1 e X e 0
d2UA,R dX
- φR2UA,R ) RRX + βR, for 0 e X e +1
]
(24)
It is clear from eq 10 that TB is a function of UB|X)-1 and JA and that TP is a function of UP|X)-1 and JA. The exact solution for eqs 12 and 13 is given by the following:
X ) +1
UA,L ) 1
(15)
UA,L ) UA,R
2
∫0
1 (UB,L + UP,L) dX + -1 2
+1
[ |
φL2 ) Φ2 UB
[ |
φR2 ) Φ2 UB
X)-1
[
AL )
φR
φL tanh(φR) + φR tanh(φL) βR
φR
(17)
)
X)+1
+
(
RR ) Φ2
[(
2
AR )
1 1 + rB KrP
]
(19)
-
]
) ]
(20)
(21)
UA|X)+1 1 + J rB KrP A
(22)
)]
]
TB TP + βL ) -Φ2 UB X)-1 + rB KrP
BL )
R
)
X)+1
L
RR
+
φR2
R
+
[
(28)
R
(
RL βL 1 AL sinh(φL) + 1 - 2 + 2 cosh(φL) φL φL
[
]
φL RR 1 RL AL + φR φR φ 2 φ 3
)]
1 -AR sinh(φR) + cosh(φR)
(
|
X)+1
+
RR φR
2
+
βR φR2
(29)
)]
(30)
Equations 25 and 26 together with eq 8 for B and P allow one to reduce the total carrier concentration condition given by eq 18 to the following formula:
) ( ){[ ( ( )( ) ) ( | [( ( ) ]}
UBT )
1 1 1 2 rB rP
]
RL βL tanh(φL) + φL φL φL2 2φL2 φL2 RR βR AR 1 1- UA X)+1 + 2 + 2 φR cosh(φR) φR φR RR βR TB TP tanh(φR) + + + (31) φR rB rP 2φ 2 φ 2 1-
RL
AL 1 1+ φL cosh(φL)
+ 2
βL
R
(23)
|
RR tanh(φR) βL βR + 2 - 2 (27) 2 φ φ φ φ R
UA
UA|X)+1 1 + rB KrP
)
L
BR )
1 1 + J rB KrP A
[( [ |
RL
φL
From eq 10 for i ) B, it is found that UB|X)+1 can be represented as a function of UA|X)+1, UB|X)-1, and JA. The constants RL, RR, βL, and βR in the approximate model differential equations, eqs 12 and 13, are defined as follows:
RL ) Φ2
(
][(
UA
RL βL 1 1 - 1- 2+ 2 + cosh(φR) φL φL cosh(φL)
(UB,R + UP,R) dX ) UBT (18)
+
2
where AR, AL, BR, and BL are integration constants. These constants are evaluated using eqs 14-17, and they are
2
In eqs 12 and 13, φL2 and φR2 are the modified Damko¨hler numbers, which are defined as follows:
βR
X-
, φR φR2 for 0 e X e +1 (26)
and to the total carrier restriction 0
RR
UA,R ) AR sinh(φRX) + BR cosh(φRX) -
(16)
dUA,L dUA,R ) dX dX
X)0
Right region
βL
X-
, φL φL2 for -1 e X e 0 (25)
(14)
UA,R ) UA|X)+1
X)0
Left region
RL
UA,L ) AL sinh(φLX) + BL cosh(φLX) -
(13)
X ) -1
∫
UA|X)+1TB TP + rB KrP
Right region
These must be solved subjected to the boundary conditions
1 2
+
(12)
Right region 2
|
X)+1UB X)+1
Left region
Left region d2UA,L
[ |
βR ) -Φ2 UA
)
R
The definition of the facilitation factor, F, for the permeate transport is given by
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F+1)
2JA UA|X)-1 - UA|X)+1
(32)
The permeate flux JA, defined by eq 9, can be obtained by derivation of eq 25 or eq 26. Therefore, to predict the facilitation factor, it is necessary to evaluate the integration constants given by eqs 27-30. Evaluation of the Solution To evaluate the solution (eqs 25 and 26), it is necessary to specify values for the following variables: UA|X)+1, UBT, K, rB, rP, and Φ2. However, the values of UB|X)-1, UP|X)-1, and JA must be calculated recursively because these variables appear in an implicit form in the solution. The calculation of these variables can be accomplished by considering the constraint given by eqs 8 and 31. A Newton-Raphson-like method should be used to determine such variables; in the present work the algorithm presented by Ochoa-Tapia15 has been employed. Results and Discussion The predictions of the facilitation factors with the developed analytical solution were compared with numerical solutions.14 The results are presented for four cases on the basis of the oxygen hemoglobin system properties.1,14 Facilitation Factors. Case I. Damko1 hler Number Effect on the Facilitation Factor for DB ) DP and UA|X)+1 g 0. Figure 1 shows the effect of varying the Damko¨hler number on the facilitation factor when the carrier and complex diffusivities are equal. In these results, zero and nonzero downstream solute concentrations are considered. The approximate solution predictions are compared with the results from the numerical solution reported by Kutchai et al.14 For cases where the downstream solute concentration is 0 and 0.4, it is observed that the predictions from the approximate solution are satisfactory for any Damko¨hler number value. The numerical results shown in Figure 1 were taken from Tables 6 and 7 of Kutchai et al.14 The maximum error obtained with respect to the numerical solution is 1.63%. For the cases where the downstream solute concentration is 0.1, 0.2, 0.3, and 0.5, it can be noticed that when the Damko¨hler number increases, the approximate predictions reproduce satisfactorily their corresponding equilibrium facilitation factors. This value was obtained from the exact solution for the equilibrium case, when DB ) DP, and is given by the following expression:7
F + 1|equilibrium ) 1 +
rPKUBT (1 + K)(1 + KUA|X)+1)
(33)
Case II. Damko1 hler Number Effect on the Facilitation Factor for DB * DP and UA|X)+1 ) 0. Figure 2 shows the effect of varying the Damko¨hler number on the facilitation factor for the case in which the carrier and complex diffusivities are different. The downstream solute concentration is zero. This figure only shows approximate results, but it is possible to notice that, in all cases, when the Damko¨hler number increases, the approximate method derived in this work reproduces satisfactorily the corresponding equilibrium facilitation factor. Because the total carrier concentra-
Figure 1. Effect of the Damko¨hler number on the facilitation factor for DB ) DP and UA|X)+1 g 0. Other parameters are K ) 4.55 and UBT ) 117.65. (b) Numerical solution.14 (s) Approximate analytical solution (this work). (- - -) Exact equilibrium.16
Figure 2. Effect of the Damko¨hler number on the facilitation factor for r ) DB/DP e 1 and UA|X)+1 ) 0. Other parameters are K ) 4.55 and UBT ) 117.65. (s) Approximate analytical solution (this work); (- - -) Exact equilibrium.16 (‚‚‚) Approximate equilibrium.16 (- ‚ -) Teramoto’s eq 35.12
tion restriction is given by eq 31, the predictions of the facilitation factor only require the solution of algebraic nonlinear equations. In Figure 2, it is also clear that when the complex diffusivity decreases with respect to the carrier diffusivity, that is, when DB/DP diminishes, the membrane effectiveness to permeate A diminishes as well. This occurs because when the complex diffusivity decreases, the permeability across its membrane also decreases, and this reduces the facilitated transport. For Damko¨hler numbers near the equilibrium regime, this effect is more significant because the diffusivity effects are more important at the equilibrium condition. It is important to mention that, when DB * DP, an exact analytical expression for the equilibrium facilitation factor cannot be obtained because of the carrier mass conservation constraint. However, Teramoto12 has proposed an approximate analytical expression for this situation. Teramoto’s approximation is based on the assumption that the carrier and the complex concentra-
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Figure 3. Effect of the equilibrium constant on the facilitation factor for DB ) DP and UA|X)+1 ) 0. Other parameters are rB ) DB/DA ) 0.1 and UBT ) 500. (b) Numerical solution.11 (s) Approximate analytical solution (this work).
tions are constant in each region. Therefore, the carrier mass conservation restriction is given by an expression equivalent to the rectangle area formula for each half of the membrane. In this work, exact equilibrium values of the facilitation factor were obtained by using a numerical solution involving nonlinear algebraic equations and the numerical integration of the total carrier restriction.16 Also, on the basis of the approximate method presented in this work, an analytical solution for the equilibrium problem, which does not require numerical integration, was derived.16 The equilibrium facilitation factors obtained with the two approximate solutions and those predicted from the numerical solution are shown in Figure 2. It is clear that for the unequal carrier and complex diffusivities case there are differences between the results obtained in this work and those obtained with the equation proposed by Teramoto.12 A more precise comparison between the predictions from the three methods shows that the facilitation factors obtained by the approximate solution of this work are closer to the exact values than those obtained with the Teramoto12 approximate solution. Case III. Equilibrium Constant Effect on the Facilitation Factor for DB ) DP and UA|X)+1 ) 0. In Figure 3, the effect of the variation of the equilibrium constant on the facilitation factor for the case in which the carrier and complex diffusivities are equal is shown. In these results, the downstream solute concentration is zero. The comparison between our approximate solution and the numerical solution reported by Basaran et al.11 shows a maximum error of 4.89%. The results are shown for three Damko¨hler number values, from the physical diffusion regime to the equilibrium chemical reaction regime. It should be noticed that the Damko¨hler number has been defined as k-1l 2/ DB because this expression was employed by Basaran et al.11 It is important to observe that these results agree with the predictions shown in the previous cases. That is, large enough Damko¨hler numbers offer large facilitation factors. It is also possible to see that there is an optimal facilitation factor for each Damko¨hler number.
Figure 4. Effect of the equilibrium constant on the facilitation factor for DB/DP ) 2.0 and UA|X)+1 ) 0. Other parameters are rB ) DB/DA ) 0.1 and UBT ) 500. (b) Numerical solution.11 (s) Approximate analytical solution (this work).
Case IV. Equilibrium Constant Effect on the Facilitation Factor for DB * DP and UA|X)+1 ) 0. In Figure 4 is shown the equilibrium constant effect on the facilitation factor for the case when carrier and complex diffusivities are unequal and when the downstream solute concentration is zero. The approximate predictions of the facilitation factors are compared with the numerical results of Basaran et al.11 Figure 4 shows the predictions for DB/DP ) 2, and in this case, the largest error was 1.27%. The presence of a maximum facilitation factor in the curve for a given Damko¨hler number in Figures 3 and 4 has a physical explanation.11 When the equilibrium constant K is small, it means that the rate of the forward reaction is small with respect to the backward reaction rate. Therefore, the carrier is not totally consumed near the high permeate concentration membrane boundary. When K is large, it means that the rate of the reverse reaction is slow, and this causes the saturation of the carrier in the right-hand-side boundary of the membrane. Both extreme cases for K produce small facilitation factors. As a consequence, there is a maximum for the facilitation factor that is a function of the K value and the Damko¨hler number. Concentration Profiles. To complete the evaluation and to test the robustness of the approximate method, we compare the approximate concentration profiles with those obtained from the numerical solution. The comparison was carried out for three Damko¨hler numbers that include values from the diffusion regime to the equilibrium regime.16 In the regime where the reaction rate is small in comparison with the diffusion rate, there is not a noticeable difference, as expected, between the concentration profiles for the three species (permeate, carrier, and complex) from the approximate method and from the numerical solution. At intermediate Damko¨hler number, the difference of the approximate facilitation factor predicted with respect to the one obtained from the exact solution is almost negligible (
