- Email: [email protected]
Time lag for transmembrane transport without interface equilibrium
Journal of Membrane Science 225 (2003) 105–114
Time lag for transmembrane transport without interface equilibrium Y.K. Zhang, N.M. Kocherginsky∗ Division of Bioengineering and Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 2 April 2003; received in revised form 17 April 2003; accepted 20 August 2003
Abstract Equilibrium at an interface is a key assumption in the traditional theory, describing a relationship of a time lag and diffusion coefficients during nonsteady-state transport through a membrane. Recent experiments demonstrated that this assumption is not always valid. Interfacial resistance has been identified in several separation processes, such as gas, vapor permeation, solvent extraction, liquid membrane, drug control release, etc. Role of non-equilibrium interface in the overall mass transfer is addressed in this paper and expanded equations for the time lag are derived. Analytical expressions allow the utilization of transient state information for the evaluation of interfacial resistance. It is demonstrated theoretically that ignorance of the interfacial resistance could decrease the estimated diffusion coefficient by a factor of 3. A concept of interfacial resistance index (IR index) is proposed, which allows the evaluation of interfacial resistance in the overall mass transfer process. For a given system the possible role of interfacial resistance in the overall mass transfer can be evaluated even before the actual interfacial transfer rate is known. © 2003 Elsevier B.V. All rights reserved. Keywords: Time lag theory; Non-equilibrium interface; Interface resistance; Mass transfer; Membrane
1. Introduction Mass transfer from one phase to another is an essential process in separation technology, such as solvent extraction [1], membrane separation [2], chemical absorption [3], etc. Generally the basic steps in these processes are (1) mass transfer through a boundary layer of a donor and acceptor phase (free diffusion); (2) mass transfer through one or two interfaces, often accompanied by chemical reactions; (3) mass transfer through a membrane. Usually it is assumed that the thickness of interface is much less then typical thickness of a ∗ Corresponding author. Tel.: +65-6874-5083; fax: +65-6779-1936. E-mail address: [email protected] (N.M. Kocherginsky).
chemical membrane or unstirred layer. Therefore, the second step is very fast in comparison to the others and can be described by equilibrium partition coefficient. One of convenient ways to investigate the process is to measure the time lag that characterizes the kinetics of approaching to a steady-state. Traditional time lag theory with equilibrium interface assumption has been intensively used for the studies of gas and vapor permeation through dense film, ceramic-based membrane and glassy layer [4–6]. Ash et al. have derived equations describing the time lag for diffusion through multiple media, based on the Fick’s second law and the equilibrium assumption at interfaces [7]. This assumption is valid when the mass transfer process at the interface is much faster than those within the two adjacent phases. It has also been used in the kinetics
0376-7388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2003.08.008
106
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
models of solvent extraction and transport through supported liquid membranes [8,9]. In some cases the interface resistance cannot be neglected, and it was described in terms of chemical reaction rates [10–14] or sorption–desorption rates [15,16]. Moreover in some cases this resistance could become a rate-limiting step [17,18]. Even though, in some cases, the role of interfacial resistance was addressed for the steady-state transport [16,19], the lack of analytic expression for the time lag has hindered the utilization of the information from the transient state and made it difficult to separate the interface kinetic effects from diffusion [20]. Numerical approach was used in some cases when non-equilibrium at interface was taken into account [15], but it is not easy to use. In this paper analytical equations for the time lag are given, and the role of non-equilibrium mass transfer at interface is taken into account. Several special cases are discussed for mass transfer through single and multi-layers.
2. Time lag for two immiscible layers
tration, time, or positional coordinate. An effective (or pseudo-) partition coefficient Kd of diffusant between the two layers could be defined based on Henry’s law [7]: Kr (1) Kd = Kf where Kf denotes the forward rate constant for the transfer through the interface between layers I and II, and Kr the rate constant of the reversed process (m/s). C1i represents the concentration of diffusant in the vicinity of the interface in the layer I, while C2i is the corresponding concentration in the layer II. For diffusion without convection and reaction, Fick’s second law is valid in both layers, so that: T > 0, 0 < X < L1 , ∂C1 (X, T) ∂2 C1 (X, T) = D1 ∂T ∂X2 L1 < X < L1 + L2 , ∂C2 (X, T) ∂2 C2 (X, T) = D2 ∂T ∂X2
(2)
(3)
The corresponding initial condition (IC) is: The schematic diagram of the model is given in Fig. 1. In this study mass transfer through two immiscible layers (with thickness of L1 and L2 , respectively) and a non-equilibrium interface is considered. Concentration of diffusant in the external region to layer I is kept constant (stirring) and the concentration of diffusant in the external region to layer II can be neglected at the initial stage. Diffusion is in the direction of increasing X and diffusion coefficients in the two layers are assumed to be independent of concen-
T = 0,
0 < X < L1 + L2 ,
C1 (X, 0) = C2 (X, 0) = 0
Three corresponding boundary conditions (BC) are: T > 0,
X = 0,
C1 (0, T) = C0
X = L1 , ∂C1i ∂C2i −D1 = −D2 = Kf (C1i − Kd C2i ) ∂X ∂X X = L1 + L2 ,
Fig. 1. Two-layer diffusion model with a non-equilibrium interface.
(4)
C2 (L1 + L2 , T) = 0
(5)
(6) (7)
Eq. (6) expresses the non-equilibrium situation at the interface. Eq. (6) indicates that if C1i and C2i were at equilibrium, i.e., C1i = Kd C2i , then mass transfer flux at interface would become zero. In fact, at the steady-state, there still exists a mass transfer from one phase to the other. However, if Kf is high enough, the difference between C1i and Kd C2i due to deviation from equilibrium is small and an equilibrium assumption becomes acceptable [16]. To obtain the expression for the time lag resulting from Eqs. (2) and (3) together with the corresponding
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
IC (Eq. (4)) and BC (Eqs. (5)–(7)), the mathematical method developed by Frisch and co-workers was used [7,21] (see Appendix A for details). The corresponding analytical expression is: 1 L21 L1 3 3Kd L2 τlag = + + 6 D 1 D1 Kf D2 L22 3L1 Js 3 K d L2 (8) + + + D2 D1 Kf D2 C0 where the steady-state flux is: Js =
C0 (L1 /D1 ) + (1/Kf ) + (Kd L2 /D2 )
(9)
The terms in the brackets in Eq. (9) show the contribution of each individual mass transfer resistance along the diffusion path. Eq. (9) demonstrates that the steady-state flux in this case is the function of three kinds of resistance, coming from the three mass transfer steps—diffusion in layer I, interfacial forward chemical reaction and diffusion in layer II, respectively. Both Js and τ lag depend on the direction of transport.
3. Discussion An inspection of Eq. (8) shows that the effect of non-equilibrium interface on the time lag can be expressed by the term of 1/Kf . It is easy to find that if Kf → ∞, Eq. (8) can be reduced to the Ash and Barrer’s equation for two-layer laminates [7], implying that equilibrium assumption is a special case of Eq. (8). 3.1. Time lag for mass transfer in a single layer with a non-equilibrium interface Mass transfer into a single homogeneous layer is the simplest case for the time lag analysis. This case corresponds, for example, to a gas or organic vapor transport into a dense film. Also this case is analogous to the situation for mass transfer from aqueous phase, where intensive stirring is used and the resistance arising from the stagnant layer can be neglected. Evidently in this case L1 = 0, and Eq. (8) becomes:
τlag =
1 L22 3(Ri /R2 ) + 1 6 D2 (Ri /R2 ) + 1
107
(10)
where Ri and R2 are the interface and the second layer resistances, respectively. Ri = 1/Kf R2 = Kd
L2 D2
(11) (12)
If interfacial resistance Ri approaches zero (Kf → ∞), Eq. (10) can be further simplified: τm,lag =
1 L2m 6 Dm
(13)
This is the well-known time lag equation for membranes developed by Daynes in 1920 [23], where the subscript m means membrane. In the opposite situation, when Ri Rm , we have similar equation, but the coefficient is 0.5. The time lag is not dependent on the Kf and Kd and is proportional to L2m . This demonstrates that simple proportionality of τ lag to L2m is not the ample evidence that the interfacial resistance can be ignored. Eq. (10) demonstrates that the existence of interfacial resistance could increase the time lag. Similar conclusion has been drawn by Barrer and also Yi [15,22]. Favre et al. have discussed the effect of experimental framework on the apparent D value for organic vapor permeation through a membrane [6]. They concluded that experimental procedure might drastically affect the calculated diffusion coefficient during transient stage resulting in a lower D value based on the time lag data. Eq. (10) gives the relationship between the apparent diffusion coefficient (Dm,app ) and the actual one: Dm,app =
(Ri /Rm ) + 1 Dm 3(Ri /Rm ) + 1
(14)
The apparent diffusion coefficient could be reduced by a factor of 3 if a very slow mass transfer occurs at the interface. This has been demonstrated experimentally by R. Wang et al. in the study of gas permeability through 6FDA–6FpDA polyimide membranes [24]. The role of interface resistance can be identified from the steady-state data if the length of diffusion path can be changed physically. For example, it is possible to change the thickness of membrane by stacking
108
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
several pieces of membrane together. When L1 = 0, Eq. (9) becomes: C0 1 Kd = + Lm Js Kf Dm
(15)
In the graph of C0 /Js versus Lm , the slope gives the information about diffusion coefficient, while the intercept with Y-axis gives the value of Kf . Several authors have used this method to determine diffusion coefficient [11,14,25–27]. In the study on stable nitroxide radicals exit from nitrocellulose membrane into aqueous solution, Kocherginsky et al. removed the radicals in water by ascorbic acid and also changed the membrane thickness. This paper demonstrated experimentally the existence of interfacial resistance, which was near 10% of the total for the 0.1 mm membrane but could become a rate-limiting step in the mass transfer through thin biomembranes [14]. 3.2. Time lag for mass transfer for two layers with a non-equilibrium interface Mass transfer through two contacting layers and interface can be important for membrane pervaporation process, in which chemical species transfer from an aqueous phase through a membrane to a gaseous phase. Eq. (8) can be rewritten as: −1 τlag Rw Ri = + +1 Rm Rm τm,lag Rw 2 Rw 3Ri 2 Dw · Kd + +3 Dm Rm Rm Rm 3Ri 3Rw + +1 (16) + Rm Rm where τ m,lag is the time lag for mass transfer through the membrane phase, which is expressed by Eq. (13). Rw /Rm is the relative mass transfer resistance of aqueous phase: Rw 1 Dm Lw = (17) Rm Kd D w L m When Rw /Rm → 0, Eq. (16) can be reduced to Eq. (10). Further, when Ri /Rm = Rw /Rm = 0, Eq. (16) can be reduced to Eq. (13). The effects of both aqueous and interfacial resistances on the time lag are illustrated in Fig. 2.
Fig. 2. Time lag as a function of Ri /Rm and Rw /Rm for Kd = 0.01, Dw /Dm = 10.
Based on Eq. (16), it can be found that an aqueous phase resistance can shield the effect of interfacial resistance on the time lag. Juang et al. have drawn the similar conclusion in the study of citric and lactic acid extraction by tri-n-octylamine in xylene with supported liquid membrane [13]. This kind of “shielding effect” has also been found in solvent extraction of metal ions [28]. From Eq. (16) one gets: −1 τlag Rw τmin = lim = +1 Ri /Rm →0 τm,lag Rm 2 Rw Rw 3Rw 2 Dw × Kd +3 + +1 Dm Rm Rm Rm (18)
τlag τm,lag Rw 2 2 Dw = 3 Kd +1 Dm Rm
τmax =
lim
Ri /Rm →∞
(19)
To evaluate the effect of interfacial resistance, an interfacial resistance index (IR index) can be proposed based on the two extreme values τ min and τ max , corresponding to Ri /Rm → 0 and ∞, respectively. This index permits us to estimate the maximum effect of Ri /Rm on the time lag even before the exact value of Ri is known. τmax IR index = − 1 × 100% (20) τmin For a given system, Dw and Dm can be estimated based on Hayduk and Laudie or Wilke-Chang
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
109
Fig. 3. The dependence of IR index on Dw /Dm and Rw /Rm for Kd = 0.01.
methods [29,30]. Kd can be obtained from equilibrium experiments. The thickness of stagnant layer can be estimated based on Sherwood number under given hydrodynamic conditions [31–33], while the membrane thickness is a measurable parameter. Therefore, the IR index can be calculated based on known physical–chemical properties and does not require the exact value of interfacial mass transfer rate. Fig. 3 shows the dependence of IR index on Dw /Dm and Rw /Rm when Kd is 0.01 (for the separation of hydrophobic species). The fact that IR index can reach 200% tells that the presence of interfacial resistance could increase the time lag by a factor of 3 comparing to the time lag for the diffusion through a single layer membrane. In this case interfacial resistance cannot be ignored except if special evidence is available. Large values of Rw /Rm and Ri /Rm are possible, for example, when the membrane thickness is very small. This situation can happen in mass transfer through biological membranes where the thickness is only several nanometers and fluid flow rate in vivo is slow [14,34]. Fig. 3 shows that when Rw /Rm > 10 IR index increases with the increase of Rw /Rm . Therefore, it could be deduced that in the mass transfer through biological membrane the interfacial resistance could be of profound importance. Both IR index and pos-
sible role of interface are even higher for hydrophilic species, where Kd > 1, for example for H+ in mitochondria membrane. When Rw is small, the shielding effect of water disappears and IR index is again increased.
3.3. Time lag for diffusion through two stagnant layers and membrane In this section the equations are further expanded to describe the situation with two different stagnant layers in contact with the membrane phase (Fig. 4). This kind of mass transfer is important, for example, in non-dispersive membrane extraction system [14,35], where organic phase containing extracted species is pumped on one side of a membrane and an aqueous stripping solution is supplied to the other side of the membrane. If a hydrophobic membrane is used, the membrane pores could be filled with organic phase. In this case the stripping process often has the possibility to become a rate-limiting step [36]. It is assumed that equilibrium always exists at the interface between the organic stagnant layer and the membrane phase impregnated by the same solvent. The partition coefficient for this interface should be 1. Under these conditions we obtain [37]:
110
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
Fig. 4. A three-layer diffusion model with a non-equilibrium interface: (I) organic stagnant layer; (II) membrane phase; (III) aqueous stagnant layer.
K Lw Lo Lm 1 + + + d 3Do Dm Kf Dw 2 Kd Lw Lm Lo Lm 1 + + + + 2Dm Do 3Dm Kf Dw
2 K Lw L Lo Lm 1 + w + + + d 2Dw Do Dm Kf 3Dw
K Lo Lm Lw L o Lm J s + + d Do Dw Do Kf C0
τlag =
L2o 2Do
K Lw C0 Lo Lm 1 = + + + d Js Do Dm Kf Dw
(21)
(22)
where Kf represents the forward interfacial mass transfer rate constant from organic phase to an aqueous phase, and Kd is the corresponding partition coefficient. In terms of individual resistance along the diffusion path, Eq. (21) can be rewritten as: τlag Ro Ri Rw −1 =3 +1+ + τm,lag Rm Rm Rm 2 Do R o Ro Ri Rw × +1+ + D m Rm 3Rm Rm Rm Ri 1 Rw Ro + + + Rm 3 Rm Rm 2 Dw Rw Ro Ri Rw + +1+ + Dm Rm Rm Rm Rm Ro Ri Rw +2 (23) + Rm Rm Rm +
Kd here is incorporated into Rw /Rm (Eq. (22)). Again, IR index can be calculated with the two extreme values of τ max and τ min , which helps to evaluate the role of interfacial resistance. In the case of nanometer-scale layers and biological membranes, interfacial resistance could become a rate-limiting step, especially in ion separation, when dehydration is involved. In some cases water in hydration shell of an ion could also penetrate into organic phase, changing the Gibbs energy and the mass transfer rate [38]. In our recent paper we have demonstrated that if the profile of standard chemical potential through the interface is a linear function of distance, transport through the interface can be described by equivalent circuit with a diode even in the case of non-electrolytes [39]. It has been also demonstrated that the presence of a surfactant on a membrane surface could significantly change the interfacial chemical potential and can even lead to fluctuations of mass transfer across the interface [40]. In this case one can assume that the profile of a standard chemical potential is more complex and has a minimum at the interface. The changes of non-electrolyte transport induced by a surfactant could be analogous to a gate regulation of electrical current from a source to a drain in field effect transistors [41]. In all the above cases, an interfacial resistance could play an important role and change the time lag due to the relatively large values of Ri /Rm . Deep understanding of these factors should be based on methods of estimation of the value and possible significance of interfacial resistance. Analytical equations given in
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
this work could be useful in these studies, especially for thin artificial and biomembranes. 4. Comment in addition When the paper was submitted, we were informed about the paper by Siegel [42], who based on a general admittance matrix method has derived the expressions for permeability, time lag, etc. as a function of process parameters. As an example, he also gave an equation, similar to the Eq. (10) in this paper. Nomenclature C D J Kd Kf Kr L q R T u w X
species concentration (mol/m3 ) diffusion coefficient (m2 /s) flux (mol/m2 s) partition coefficient, water/oil forward mass transfer rate constant (m/s) reversed mass transfer rate constant (m/s) membrane or liquid layer thickness (m) dimensionless quantity of substance that comes through outer surface mass transfer resistance (s/m) mass transfer time (s) dimensionless concentration in layer I dimensionless concentration in layer II distance (m)
Greek letters α relative diffusion coefficient β relative length of diffusion path δ dimensionless distance Φ dimensionless flux ϕ relative forward transfer rate at interface θ dimensionless time τ time lag (s) Subscripts i interface m membrane phase o organic phase s steady-state w aqueous phase Superscript 0 initial state
111
Appendix A In order to simplify the derivation process, dimensionless form of the equations can be used, based on the following definitions: u = C1 /C0 w = C2 /C0 θ = D2 T/L22 δ = X/L2 α = D1 /D2 β = L1 /L2 ϕ = Kf L2 /D2
dimensionless concentration in layer I dimensionless concentration in layer II dimensionless time dimensionless distance relative diffusion coefficient relative length of diffusion path relative forward transfer rate at the interface
After substitution the original mathematical model is changed to the following form: θ > 0,
∂u(δ, θ) ∂2 u(δ, θ) =α ∂θ ∂δ2
0 < δ < β,
(A.1) ∂w(δ, θ) = ∂θ
β < δ < 1 + β,
∂w2 (δ, θ) ∂δ
(A.2)
IC: θ = 0,
0 < δ < 1 + β,
u(δ, 0) = w(δ, 0) = 0 (A.3)
BC: θ > 0,
δ = 0,
u(0, θ) = 1
(A.4)
δ = β, ∂w(β, θ) ∂u(β, θ) =− = ϕ[u(β, θ) − Kd w(β, θ)] −α ∂δ ∂δ (A.5) δ = 1 + β,
w(1 + β, θ) = 0
(A.6)
To solve the above equations, it is necessary to find the steady-state concentration distribution inside the membrane. At the steady-state: ∂u2 (δ, θ) ∂w2 (δ, θ) ∂w(δ, θ) ∂u(δ, θ) = = α = =0 ∂θ ∂θ ∂δ2 ∂δ2 (A.7)
112
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
Therefore: Φs = −
dus (δ) dws (δ) = −α dδ dδ
(A.8)
After integration of above equations with respect to diffusion distances in layers I and II, respectively, and applying the boundary conditions, one gets: 1 dus (δ) 1 dδ 0 Φs dδ = − 0 dδ 1+β Φ dδ = −α 1+β dws (δ) dδ s 1 1 dδ α ⇒ Φs = α(1 + 1/ϕ) + Kd β
(A.15)
(A.9)
Then, the steady-state concentration distribution could be obtained: αδ us (δ) = 1 − α(1 + 1/ϕ) + Kd β ws (δ) =
1+β−δ α(1 + 1/ϕ) + Kd β
(A.10)
(A.11)
In order to obtain the expression for the time lag, we need to know the flux through the membrane Φ(1 + β, θ). Further integration of Eqs. (A.1) and (A.2) leads to: 1
1 ∂u(δ, θ) δ
0
=
1
∂θ
dδ dδ
∂δ
1+β 1+β ∂w(δ, θ) δ
1
=α
dδ dδ = 1 − Φ(1, θ) − u(1, θ)
∂θ
dδ dδ
1+β 1+β ∂w2 (δ, θ)
1
δ
Therefore, combining Eqs. (A.10) and (A.11), together with (A.5) and (A.6) one gets: 1 Φs β 2 1 q = Φs θ − 1 + + ϕ 2α 2 Φs K d Φs β 3 − + (A.17) 3 6α2 The time lag could be defined as: q = Φs (θ − θlag )
(A.18)
∂δ2
1 β2 (β/α + 3Kd ) + 3β + αKd + 3/ϕ(β2 + α) 6 β + α(1/ϕ + Kd ) (A.19)
Similar integration method was used for the derivation of the equations for three layers [37].
References dδ dδ
= −βΦ(1 + β, θ) + αw(1, θ)
(A.13)
The boundary condition at δ = 1 gives: u(1, θ) =
0
θlag =
(A.12)
The total quantity of diffusant that has come through the outer surface of layer II could be obtained by: θ q= Φ(1 + β, θ) dθ (A.16)
Comparison of Eqs. (A.12) and (A.13) gives:
1 ∂u2 (δ, θ) δ
0
Combining Eqs. (A.12)–(A.14), one gets: 1+β 1 ∂w(δ, θ) Φ(1 + β, θ) = 1− 1+ dδ Φs ϕ ∂θ 1 1 1 ∂u(δ, θ) − dδ dδ ∂θ 0 δ Kd 1+β 1+β ∂w(δ, θ) − dδ dδ α 1 ∂θ δ
Φ(1, θ) + Kd w(1, θ) ϕ
(A.14)
[1] J. Rydberg, C. Musikas, G.R. Choppin, Principles and Practices of Solvent Extraction, Marcel Dekker, New York, 1992. [2] W.S.W. Ho, K.K. Sirkar, Membrane Handbook, Van Nostrand Reinhold, New York, 1992. [3] W.L. McCabe, J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, fifth ed., McGraw-Hill, New York, 1993. [4] R.M. Barrer, Permeation, Trans. Faraday Soc. 35 (1939) 628.
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114 [5] A.S. Michaels, A sixty-year love affair with membranes. Recollections of Richard Barrer, edited and annotated by Alan S. Michaels, J. Membr. Sci. 109 (1996) 1. [6] E. Favre, N. Morliere, D. Roizard, Experimental evidence and implications of an imperfect upstream pressure step for the time-lag technique, J. Membr. Sci. 207 (2002) 59. [7] R. Ash, R.M. Barrer, D.G. Palmer, Diffusion in multiple laminates, Br. J. Appl. Phys. 16 (1965) 873. [8] P.R. Danesi, C. Cianetti, Kinetics and mechanism of the interfacial mass transfer of Eu(III) in the system: bis(2-ethylhexyl)phosphoric acid, n-dodecane-NaCl, lactic acid, polyaminocarboxylic acid, water, Sep. Sci. Technol. 17 (1982) 969. [9] S. Mukkavilli, C.K. Lee, I. Hahn, L.L. Tavlarides, Reactor analysis of metal ion extraction in liquid dispersions, Sep. Sci. Technol. 22 (1987) 395. [10] P.R. Danesi, P. Horwitz, G.F. Vandergriff, R. Chiarizia, Mass transfer rate through liquid membranes: interfacial chemical reactions and diffusion as simultaneous permeability controlling factors, Sep. Sci. Technol. 16 (1981) 201. [11] P.A. O’Hara, M.P. Bohrer, Supported liquid membrane for copper transport, J. Membr. Sci. 44 (1989) 273. [12] J. Yi, L.L. Tavlarides, Chemically active liquid membranes in organic supports for metal ion separations, Am. Inst. Chem. Eng. J. 38 (1992) 1957. [13] R.S. Juang, J.D. Chen, H.C. Huan, Dispersion-free membrane extraction: case studies for metal ion and organic acid extraction, J. Membr. Sci. 165 (2000) 59. [14] N.M. Kocherginsky, A.B. Grishchenko, A.N. Osipov, S.N. Koh, Nitroxide radicals. Controlled release from and transport through biomimetic and hollow fiber membranes, Free Rad. Res. 34 (2001) 263. [15] R.M. Barrer, Zeolites as membranes: the role of the gas–crystal interface, in: G. Ohlmann, et al. (Eds.), Catalysis and Adsorption by Zeolites, Elsevier, Amsterdam, 1991, pp. 257–273. [16] M.A. Islam, H. Buschatz, D. Paul, Non-equilibrium surface reactions—a factor in determining steady state diffusion flux, J. Membr. Sci. 204 (2002) 379. [17] T.C. Huang, R.S. Juang, Kinetics and mechanism of zinc extraction from sulfate medium with di(2-ethylhexyl) phosphoric acid, J. Chem. Eng. Jpn. 19 (1986) 379. [18] S.X. Sun, Y. Li, Y.H. Yang, J.L. Sheng, Effects of diluent polarity on the extaction kinetics, Acta Chim. Sin. 54 (1996) 1101. [19] T.C. Huang, R.S. Juang, Rate and mechanism of divalent metal transport through supported liquid membrane containing di(2-ethylhexyl) phosphoric acid as a mobile carrier, J. Chem. Technol. Biotechnol. 42 (1988) 3. [20] J.K. McCulloch, J.M. Perera, E.D. Kelly, R.W. Lee, G.W. Stevens, F. Grieser, A kinetic study of copper ion extraction by Kelex 100 at a heptane–water interface, J. Colloid Interface Sci. 184 (1996) 406. [21] H.L. Frisch, The time lag in diffusion, J. Phys. Chem. 61 (1957) 93.
113
[22] J. Yi, Effect of the boundary layer and interfacial reaction on the time lag in supported liquid membranes, Korean J. Chem. Eng. 12 (1995) 391. [23] J. Crank, The Mathematics of Diffusion, second ed., Oxford Science Publications, Oxford, 1975. [24] R. Wang, C. Cao, T.S. Chung, A critical review on diffusivity and the characterization of diffusivity of 6FDA–6FpDA polyimide membranes for gas separation, J. Membr. Sci. 198 (2002) 259. [25] W.W.S. Ho, D.C. Dalrymple, Facilitated transport of olefins in Ag+ -containing polymer membranes, J. Membr. Sci. 91 (1994) 13. [26] M.F. Paugam, J. Buffle, Comparison of carrier-facilitated copper(II) ion transport mechanisms in a supported liquid membrane and in a plasticized cellulose triacetate membrane, J. Membr. Sci. 147 (1998) 207. [27] N.M. Kocherginsky, A.B. Grishchenko, Mass transfer of long chain fatty acids through liquid–liquid interface stabilized by porous membrane, Sep. Purif. Technol. 20 (2000) 197. [28] P.R. Danesi, Solvent extraction kinetics, in: Principles and Practices of Solvent Extraction, Marcel Dekker, New York, 1992, pp. 157–207. [29] C.R. Wilke, P. Chang, Correlation of diffusion coefficients in dilute solutions, Am. Inst. Chem. Eng. J. 6 (1955) 264. [30] W.J. Lyman, W.F. Reehl, D.H. Rosenblatt, Diffusion coefficients in air and water, in: Handbook of Chemical Property Estimation Methods, McGraw-Hill, New York, 1982, Chapter 10, pp. 17–24. [31] H.S. Fogler, Elements of Chemical Reaction Engineering, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1992. [32] A.M. Urtiaga, J.A. Irabien, Internal mass transfer in hollow fiber supported liquid membranes, Am. Inst. Chem. Eng. J. 39 (1993) 521. [33] P. Schoner, P. Plucinski, W. Nitsch, U. Daiminger, Mass transfer in the shell side of cross flow hollow fiber modules, Chem. Eng. Sci. 53 (1998) 2319. [34] S.F. Timashev, Physical Chemistry of Membrane Process, Ellis Horwood, Chichester, 1991. [35] S.H. Lin, R.S. Juang, Mass transfer in hollow fiber modules for extraction and back-extraction of copper(II) with LIX64N carriers, J. Membr. Sci. 188 (2001) 251. [36] A.B. de Haan, P.V. Bartels, J. de Graauw, Extraction of metal ions from wastewater, modeling of the mass transfer in a supported-liquid-membrane process, J. Membr. Sci. 45 (1989) 281. [37] Y.K. Zhang, Kinetic and equilibrium study of strontium removal by liquid–liquid extraction and membrane separation, Master thesis, National University of Singapore, Singapore, 1999. [38] T. Osakai, Ion salvation and resolvation, in: A.G. Volkov (Ed.), Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications, Marcel Dekker, New York, 2001, pp. 23–50.
114
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
[39] N.M. Kocherginsky, Y.K. Zhang, Role of standard chemical potential in transport through anisotropic media and asymmetrical membranes, J. Phys. Chem. B 107 (2003) 7830. [40] Z. Koczorowski, Interfacial potentials and cells, in: A.G. Volkov (Ed.), Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications, Marcel Dekker, New York, 2001, pp. 1–22.
[41] J.H. Krenz, Electronic Concepts—An Introduction, Cambridge University Press, Cambridge, 2000. [42] R.A. Siegel, Algebraic, differential, and integral relations for membrane in series and other multilaminar media: permeabilities, solute consumption, lag times, and mean first passage times, J. Phys. Chem. 95 (1991) 2556.
Time lag for transmembrane transport without interface equilibrium Y.K. Zhang, N.M. Kocherginsky∗ Division of Bioengineering and Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 2 April 2003; received in revised form 17 April 2003; accepted 20 August 2003
Abstract Equilibrium at an interface is a key assumption in the traditional theory, describing a relationship of a time lag and diffusion coefficients during nonsteady-state transport through a membrane. Recent experiments demonstrated that this assumption is not always valid. Interfacial resistance has been identified in several separation processes, such as gas, vapor permeation, solvent extraction, liquid membrane, drug control release, etc. Role of non-equilibrium interface in the overall mass transfer is addressed in this paper and expanded equations for the time lag are derived. Analytical expressions allow the utilization of transient state information for the evaluation of interfacial resistance. It is demonstrated theoretically that ignorance of the interfacial resistance could decrease the estimated diffusion coefficient by a factor of 3. A concept of interfacial resistance index (IR index) is proposed, which allows the evaluation of interfacial resistance in the overall mass transfer process. For a given system the possible role of interfacial resistance in the overall mass transfer can be evaluated even before the actual interfacial transfer rate is known. © 2003 Elsevier B.V. All rights reserved. Keywords: Time lag theory; Non-equilibrium interface; Interface resistance; Mass transfer; Membrane
1. Introduction Mass transfer from one phase to another is an essential process in separation technology, such as solvent extraction [1], membrane separation [2], chemical absorption [3], etc. Generally the basic steps in these processes are (1) mass transfer through a boundary layer of a donor and acceptor phase (free diffusion); (2) mass transfer through one or two interfaces, often accompanied by chemical reactions; (3) mass transfer through a membrane. Usually it is assumed that the thickness of interface is much less then typical thickness of a ∗ Corresponding author. Tel.: +65-6874-5083; fax: +65-6779-1936. E-mail address: [email protected] (N.M. Kocherginsky).
chemical membrane or unstirred layer. Therefore, the second step is very fast in comparison to the others and can be described by equilibrium partition coefficient. One of convenient ways to investigate the process is to measure the time lag that characterizes the kinetics of approaching to a steady-state. Traditional time lag theory with equilibrium interface assumption has been intensively used for the studies of gas and vapor permeation through dense film, ceramic-based membrane and glassy layer [4–6]. Ash et al. have derived equations describing the time lag for diffusion through multiple media, based on the Fick’s second law and the equilibrium assumption at interfaces [7]. This assumption is valid when the mass transfer process at the interface is much faster than those within the two adjacent phases. It has also been used in the kinetics
0376-7388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2003.08.008
106
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
models of solvent extraction and transport through supported liquid membranes [8,9]. In some cases the interface resistance cannot be neglected, and it was described in terms of chemical reaction rates [10–14] or sorption–desorption rates [15,16]. Moreover in some cases this resistance could become a rate-limiting step [17,18]. Even though, in some cases, the role of interfacial resistance was addressed for the steady-state transport [16,19], the lack of analytic expression for the time lag has hindered the utilization of the information from the transient state and made it difficult to separate the interface kinetic effects from diffusion [20]. Numerical approach was used in some cases when non-equilibrium at interface was taken into account [15], but it is not easy to use. In this paper analytical equations for the time lag are given, and the role of non-equilibrium mass transfer at interface is taken into account. Several special cases are discussed for mass transfer through single and multi-layers.
2. Time lag for two immiscible layers
tration, time, or positional coordinate. An effective (or pseudo-) partition coefficient Kd of diffusant between the two layers could be defined based on Henry’s law [7]: Kr (1) Kd = Kf where Kf denotes the forward rate constant for the transfer through the interface between layers I and II, and Kr the rate constant of the reversed process (m/s). C1i represents the concentration of diffusant in the vicinity of the interface in the layer I, while C2i is the corresponding concentration in the layer II. For diffusion without convection and reaction, Fick’s second law is valid in both layers, so that: T > 0, 0 < X < L1 , ∂C1 (X, T) ∂2 C1 (X, T) = D1 ∂T ∂X2 L1 < X < L1 + L2 , ∂C2 (X, T) ∂2 C2 (X, T) = D2 ∂T ∂X2
(2)
(3)
The corresponding initial condition (IC) is: The schematic diagram of the model is given in Fig. 1. In this study mass transfer through two immiscible layers (with thickness of L1 and L2 , respectively) and a non-equilibrium interface is considered. Concentration of diffusant in the external region to layer I is kept constant (stirring) and the concentration of diffusant in the external region to layer II can be neglected at the initial stage. Diffusion is in the direction of increasing X and diffusion coefficients in the two layers are assumed to be independent of concen-
T = 0,
0 < X < L1 + L2 ,
C1 (X, 0) = C2 (X, 0) = 0
Three corresponding boundary conditions (BC) are: T > 0,
X = 0,
C1 (0, T) = C0
X = L1 , ∂C1i ∂C2i −D1 = −D2 = Kf (C1i − Kd C2i ) ∂X ∂X X = L1 + L2 ,
Fig. 1. Two-layer diffusion model with a non-equilibrium interface.
(4)
C2 (L1 + L2 , T) = 0
(5)
(6) (7)
Eq. (6) expresses the non-equilibrium situation at the interface. Eq. (6) indicates that if C1i and C2i were at equilibrium, i.e., C1i = Kd C2i , then mass transfer flux at interface would become zero. In fact, at the steady-state, there still exists a mass transfer from one phase to the other. However, if Kf is high enough, the difference between C1i and Kd C2i due to deviation from equilibrium is small and an equilibrium assumption becomes acceptable [16]. To obtain the expression for the time lag resulting from Eqs. (2) and (3) together with the corresponding
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
IC (Eq. (4)) and BC (Eqs. (5)–(7)), the mathematical method developed by Frisch and co-workers was used [7,21] (see Appendix A for details). The corresponding analytical expression is: 1 L21 L1 3 3Kd L2 τlag = + + 6 D 1 D1 Kf D2 L22 3L1 Js 3 K d L2 (8) + + + D2 D1 Kf D2 C0 where the steady-state flux is: Js =
C0 (L1 /D1 ) + (1/Kf ) + (Kd L2 /D2 )
(9)
The terms in the brackets in Eq. (9) show the contribution of each individual mass transfer resistance along the diffusion path. Eq. (9) demonstrates that the steady-state flux in this case is the function of three kinds of resistance, coming from the three mass transfer steps—diffusion in layer I, interfacial forward chemical reaction and diffusion in layer II, respectively. Both Js and τ lag depend on the direction of transport.
3. Discussion An inspection of Eq. (8) shows that the effect of non-equilibrium interface on the time lag can be expressed by the term of 1/Kf . It is easy to find that if Kf → ∞, Eq. (8) can be reduced to the Ash and Barrer’s equation for two-layer laminates [7], implying that equilibrium assumption is a special case of Eq. (8). 3.1. Time lag for mass transfer in a single layer with a non-equilibrium interface Mass transfer into a single homogeneous layer is the simplest case for the time lag analysis. This case corresponds, for example, to a gas or organic vapor transport into a dense film. Also this case is analogous to the situation for mass transfer from aqueous phase, where intensive stirring is used and the resistance arising from the stagnant layer can be neglected. Evidently in this case L1 = 0, and Eq. (8) becomes:
τlag =
1 L22 3(Ri /R2 ) + 1 6 D2 (Ri /R2 ) + 1
107
(10)
where Ri and R2 are the interface and the second layer resistances, respectively. Ri = 1/Kf R2 = Kd
L2 D2
(11) (12)
If interfacial resistance Ri approaches zero (Kf → ∞), Eq. (10) can be further simplified: τm,lag =
1 L2m 6 Dm
(13)
This is the well-known time lag equation for membranes developed by Daynes in 1920 [23], where the subscript m means membrane. In the opposite situation, when Ri Rm , we have similar equation, but the coefficient is 0.5. The time lag is not dependent on the Kf and Kd and is proportional to L2m . This demonstrates that simple proportionality of τ lag to L2m is not the ample evidence that the interfacial resistance can be ignored. Eq. (10) demonstrates that the existence of interfacial resistance could increase the time lag. Similar conclusion has been drawn by Barrer and also Yi [15,22]. Favre et al. have discussed the effect of experimental framework on the apparent D value for organic vapor permeation through a membrane [6]. They concluded that experimental procedure might drastically affect the calculated diffusion coefficient during transient stage resulting in a lower D value based on the time lag data. Eq. (10) gives the relationship between the apparent diffusion coefficient (Dm,app ) and the actual one: Dm,app =
(Ri /Rm ) + 1 Dm 3(Ri /Rm ) + 1
(14)
The apparent diffusion coefficient could be reduced by a factor of 3 if a very slow mass transfer occurs at the interface. This has been demonstrated experimentally by R. Wang et al. in the study of gas permeability through 6FDA–6FpDA polyimide membranes [24]. The role of interface resistance can be identified from the steady-state data if the length of diffusion path can be changed physically. For example, it is possible to change the thickness of membrane by stacking
108
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
several pieces of membrane together. When L1 = 0, Eq. (9) becomes: C0 1 Kd = + Lm Js Kf Dm
(15)
In the graph of C0 /Js versus Lm , the slope gives the information about diffusion coefficient, while the intercept with Y-axis gives the value of Kf . Several authors have used this method to determine diffusion coefficient [11,14,25–27]. In the study on stable nitroxide radicals exit from nitrocellulose membrane into aqueous solution, Kocherginsky et al. removed the radicals in water by ascorbic acid and also changed the membrane thickness. This paper demonstrated experimentally the existence of interfacial resistance, which was near 10% of the total for the 0.1 mm membrane but could become a rate-limiting step in the mass transfer through thin biomembranes [14]. 3.2. Time lag for mass transfer for two layers with a non-equilibrium interface Mass transfer through two contacting layers and interface can be important for membrane pervaporation process, in which chemical species transfer from an aqueous phase through a membrane to a gaseous phase. Eq. (8) can be rewritten as: −1 τlag Rw Ri = + +1 Rm Rm τm,lag Rw 2 Rw 3Ri 2 Dw · Kd + +3 Dm Rm Rm Rm 3Ri 3Rw + +1 (16) + Rm Rm where τ m,lag is the time lag for mass transfer through the membrane phase, which is expressed by Eq. (13). Rw /Rm is the relative mass transfer resistance of aqueous phase: Rw 1 Dm Lw = (17) Rm Kd D w L m When Rw /Rm → 0, Eq. (16) can be reduced to Eq. (10). Further, when Ri /Rm = Rw /Rm = 0, Eq. (16) can be reduced to Eq. (13). The effects of both aqueous and interfacial resistances on the time lag are illustrated in Fig. 2.
Fig. 2. Time lag as a function of Ri /Rm and Rw /Rm for Kd = 0.01, Dw /Dm = 10.
Based on Eq. (16), it can be found that an aqueous phase resistance can shield the effect of interfacial resistance on the time lag. Juang et al. have drawn the similar conclusion in the study of citric and lactic acid extraction by tri-n-octylamine in xylene with supported liquid membrane [13]. This kind of “shielding effect” has also been found in solvent extraction of metal ions [28]. From Eq. (16) one gets: −1 τlag Rw τmin = lim = +1 Ri /Rm →0 τm,lag Rm 2 Rw Rw 3Rw 2 Dw × Kd +3 + +1 Dm Rm Rm Rm (18)
τlag τm,lag Rw 2 2 Dw = 3 Kd +1 Dm Rm
τmax =
lim
Ri /Rm →∞
(19)
To evaluate the effect of interfacial resistance, an interfacial resistance index (IR index) can be proposed based on the two extreme values τ min and τ max , corresponding to Ri /Rm → 0 and ∞, respectively. This index permits us to estimate the maximum effect of Ri /Rm on the time lag even before the exact value of Ri is known. τmax IR index = − 1 × 100% (20) τmin For a given system, Dw and Dm can be estimated based on Hayduk and Laudie or Wilke-Chang
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
109
Fig. 3. The dependence of IR index on Dw /Dm and Rw /Rm for Kd = 0.01.
methods [29,30]. Kd can be obtained from equilibrium experiments. The thickness of stagnant layer can be estimated based on Sherwood number under given hydrodynamic conditions [31–33], while the membrane thickness is a measurable parameter. Therefore, the IR index can be calculated based on known physical–chemical properties and does not require the exact value of interfacial mass transfer rate. Fig. 3 shows the dependence of IR index on Dw /Dm and Rw /Rm when Kd is 0.01 (for the separation of hydrophobic species). The fact that IR index can reach 200% tells that the presence of interfacial resistance could increase the time lag by a factor of 3 comparing to the time lag for the diffusion through a single layer membrane. In this case interfacial resistance cannot be ignored except if special evidence is available. Large values of Rw /Rm and Ri /Rm are possible, for example, when the membrane thickness is very small. This situation can happen in mass transfer through biological membranes where the thickness is only several nanometers and fluid flow rate in vivo is slow [14,34]. Fig. 3 shows that when Rw /Rm > 10 IR index increases with the increase of Rw /Rm . Therefore, it could be deduced that in the mass transfer through biological membrane the interfacial resistance could be of profound importance. Both IR index and pos-
sible role of interface are even higher for hydrophilic species, where Kd > 1, for example for H+ in mitochondria membrane. When Rw is small, the shielding effect of water disappears and IR index is again increased.
3.3. Time lag for diffusion through two stagnant layers and membrane In this section the equations are further expanded to describe the situation with two different stagnant layers in contact with the membrane phase (Fig. 4). This kind of mass transfer is important, for example, in non-dispersive membrane extraction system [14,35], where organic phase containing extracted species is pumped on one side of a membrane and an aqueous stripping solution is supplied to the other side of the membrane. If a hydrophobic membrane is used, the membrane pores could be filled with organic phase. In this case the stripping process often has the possibility to become a rate-limiting step [36]. It is assumed that equilibrium always exists at the interface between the organic stagnant layer and the membrane phase impregnated by the same solvent. The partition coefficient for this interface should be 1. Under these conditions we obtain [37]:
110
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
Fig. 4. A three-layer diffusion model with a non-equilibrium interface: (I) organic stagnant layer; (II) membrane phase; (III) aqueous stagnant layer.
K Lw Lo Lm 1 + + + d 3Do Dm Kf Dw 2 Kd Lw Lm Lo Lm 1 + + + + 2Dm Do 3Dm Kf Dw
2 K Lw L Lo Lm 1 + w + + + d 2Dw Do Dm Kf 3Dw
K Lo Lm Lw L o Lm J s + + d Do Dw Do Kf C0
τlag =
L2o 2Do
K Lw C0 Lo Lm 1 = + + + d Js Do Dm Kf Dw
(21)
(22)
where Kf represents the forward interfacial mass transfer rate constant from organic phase to an aqueous phase, and Kd is the corresponding partition coefficient. In terms of individual resistance along the diffusion path, Eq. (21) can be rewritten as: τlag Ro Ri Rw −1 =3 +1+ + τm,lag Rm Rm Rm 2 Do R o Ro Ri Rw × +1+ + D m Rm 3Rm Rm Rm Ri 1 Rw Ro + + + Rm 3 Rm Rm 2 Dw Rw Ro Ri Rw + +1+ + Dm Rm Rm Rm Rm Ro Ri Rw +2 (23) + Rm Rm Rm +
Kd here is incorporated into Rw /Rm (Eq. (22)). Again, IR index can be calculated with the two extreme values of τ max and τ min , which helps to evaluate the role of interfacial resistance. In the case of nanometer-scale layers and biological membranes, interfacial resistance could become a rate-limiting step, especially in ion separation, when dehydration is involved. In some cases water in hydration shell of an ion could also penetrate into organic phase, changing the Gibbs energy and the mass transfer rate [38]. In our recent paper we have demonstrated that if the profile of standard chemical potential through the interface is a linear function of distance, transport through the interface can be described by equivalent circuit with a diode even in the case of non-electrolytes [39]. It has been also demonstrated that the presence of a surfactant on a membrane surface could significantly change the interfacial chemical potential and can even lead to fluctuations of mass transfer across the interface [40]. In this case one can assume that the profile of a standard chemical potential is more complex and has a minimum at the interface. The changes of non-electrolyte transport induced by a surfactant could be analogous to a gate regulation of electrical current from a source to a drain in field effect transistors [41]. In all the above cases, an interfacial resistance could play an important role and change the time lag due to the relatively large values of Ri /Rm . Deep understanding of these factors should be based on methods of estimation of the value and possible significance of interfacial resistance. Analytical equations given in
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
this work could be useful in these studies, especially for thin artificial and biomembranes. 4. Comment in addition When the paper was submitted, we were informed about the paper by Siegel [42], who based on a general admittance matrix method has derived the expressions for permeability, time lag, etc. as a function of process parameters. As an example, he also gave an equation, similar to the Eq. (10) in this paper. Nomenclature C D J Kd Kf Kr L q R T u w X
species concentration (mol/m3 ) diffusion coefficient (m2 /s) flux (mol/m2 s) partition coefficient, water/oil forward mass transfer rate constant (m/s) reversed mass transfer rate constant (m/s) membrane or liquid layer thickness (m) dimensionless quantity of substance that comes through outer surface mass transfer resistance (s/m) mass transfer time (s) dimensionless concentration in layer I dimensionless concentration in layer II distance (m)
Greek letters α relative diffusion coefficient β relative length of diffusion path δ dimensionless distance Φ dimensionless flux ϕ relative forward transfer rate at interface θ dimensionless time τ time lag (s) Subscripts i interface m membrane phase o organic phase s steady-state w aqueous phase Superscript 0 initial state
111
Appendix A In order to simplify the derivation process, dimensionless form of the equations can be used, based on the following definitions: u = C1 /C0 w = C2 /C0 θ = D2 T/L22 δ = X/L2 α = D1 /D2 β = L1 /L2 ϕ = Kf L2 /D2
dimensionless concentration in layer I dimensionless concentration in layer II dimensionless time dimensionless distance relative diffusion coefficient relative length of diffusion path relative forward transfer rate at the interface
After substitution the original mathematical model is changed to the following form: θ > 0,
∂u(δ, θ) ∂2 u(δ, θ) =α ∂θ ∂δ2
0 < δ < β,
(A.1) ∂w(δ, θ) = ∂θ
β < δ < 1 + β,
∂w2 (δ, θ) ∂δ
(A.2)
IC: θ = 0,
0 < δ < 1 + β,
u(δ, 0) = w(δ, 0) = 0 (A.3)
BC: θ > 0,
δ = 0,
u(0, θ) = 1
(A.4)
δ = β, ∂w(β, θ) ∂u(β, θ) =− = ϕ[u(β, θ) − Kd w(β, θ)] −α ∂δ ∂δ (A.5) δ = 1 + β,
w(1 + β, θ) = 0
(A.6)
To solve the above equations, it is necessary to find the steady-state concentration distribution inside the membrane. At the steady-state: ∂u2 (δ, θ) ∂w2 (δ, θ) ∂w(δ, θ) ∂u(δ, θ) = = α = =0 ∂θ ∂θ ∂δ2 ∂δ2 (A.7)
112
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
Therefore: Φs = −
dus (δ) dws (δ) = −α dδ dδ
(A.8)
After integration of above equations with respect to diffusion distances in layers I and II, respectively, and applying the boundary conditions, one gets: 1 dus (δ) 1 dδ 0 Φs dδ = − 0 dδ 1+β Φ dδ = −α 1+β dws (δ) dδ s 1 1 dδ α ⇒ Φs = α(1 + 1/ϕ) + Kd β
(A.15)
(A.9)
Then, the steady-state concentration distribution could be obtained: αδ us (δ) = 1 − α(1 + 1/ϕ) + Kd β ws (δ) =
1+β−δ α(1 + 1/ϕ) + Kd β
(A.10)
(A.11)
In order to obtain the expression for the time lag, we need to know the flux through the membrane Φ(1 + β, θ). Further integration of Eqs. (A.1) and (A.2) leads to: 1
1 ∂u(δ, θ) δ
0
=
1
∂θ
dδ dδ
∂δ
1+β 1+β ∂w(δ, θ) δ
1
=α
dδ dδ = 1 − Φ(1, θ) − u(1, θ)
∂θ
dδ dδ
1+β 1+β ∂w2 (δ, θ)
1
δ
Therefore, combining Eqs. (A.10) and (A.11), together with (A.5) and (A.6) one gets: 1 Φs β 2 1 q = Φs θ − 1 + + ϕ 2α 2 Φs K d Φs β 3 − + (A.17) 3 6α2 The time lag could be defined as: q = Φs (θ − θlag )
(A.18)
∂δ2
1 β2 (β/α + 3Kd ) + 3β + αKd + 3/ϕ(β2 + α) 6 β + α(1/ϕ + Kd ) (A.19)
Similar integration method was used for the derivation of the equations for three layers [37].
References dδ dδ
= −βΦ(1 + β, θ) + αw(1, θ)
(A.13)
The boundary condition at δ = 1 gives: u(1, θ) =
0
θlag =
(A.12)
The total quantity of diffusant that has come through the outer surface of layer II could be obtained by: θ q= Φ(1 + β, θ) dθ (A.16)
Comparison of Eqs. (A.12) and (A.13) gives:
1 ∂u2 (δ, θ) δ
0
Combining Eqs. (A.12)–(A.14), one gets: 1+β 1 ∂w(δ, θ) Φ(1 + β, θ) = 1− 1+ dδ Φs ϕ ∂θ 1 1 1 ∂u(δ, θ) − dδ dδ ∂θ 0 δ Kd 1+β 1+β ∂w(δ, θ) − dδ dδ α 1 ∂θ δ
Φ(1, θ) + Kd w(1, θ) ϕ
(A.14)
[1] J. Rydberg, C. Musikas, G.R. Choppin, Principles and Practices of Solvent Extraction, Marcel Dekker, New York, 1992. [2] W.S.W. Ho, K.K. Sirkar, Membrane Handbook, Van Nostrand Reinhold, New York, 1992. [3] W.L. McCabe, J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, fifth ed., McGraw-Hill, New York, 1993. [4] R.M. Barrer, Permeation, Trans. Faraday Soc. 35 (1939) 628.
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114 [5] A.S. Michaels, A sixty-year love affair with membranes. Recollections of Richard Barrer, edited and annotated by Alan S. Michaels, J. Membr. Sci. 109 (1996) 1. [6] E. Favre, N. Morliere, D. Roizard, Experimental evidence and implications of an imperfect upstream pressure step for the time-lag technique, J. Membr. Sci. 207 (2002) 59. [7] R. Ash, R.M. Barrer, D.G. Palmer, Diffusion in multiple laminates, Br. J. Appl. Phys. 16 (1965) 873. [8] P.R. Danesi, C. Cianetti, Kinetics and mechanism of the interfacial mass transfer of Eu(III) in the system: bis(2-ethylhexyl)phosphoric acid, n-dodecane-NaCl, lactic acid, polyaminocarboxylic acid, water, Sep. Sci. Technol. 17 (1982) 969. [9] S. Mukkavilli, C.K. Lee, I. Hahn, L.L. Tavlarides, Reactor analysis of metal ion extraction in liquid dispersions, Sep. Sci. Technol. 22 (1987) 395. [10] P.R. Danesi, P. Horwitz, G.F. Vandergriff, R. Chiarizia, Mass transfer rate through liquid membranes: interfacial chemical reactions and diffusion as simultaneous permeability controlling factors, Sep. Sci. Technol. 16 (1981) 201. [11] P.A. O’Hara, M.P. Bohrer, Supported liquid membrane for copper transport, J. Membr. Sci. 44 (1989) 273. [12] J. Yi, L.L. Tavlarides, Chemically active liquid membranes in organic supports for metal ion separations, Am. Inst. Chem. Eng. J. 38 (1992) 1957. [13] R.S. Juang, J.D. Chen, H.C. Huan, Dispersion-free membrane extraction: case studies for metal ion and organic acid extraction, J. Membr. Sci. 165 (2000) 59. [14] N.M. Kocherginsky, A.B. Grishchenko, A.N. Osipov, S.N. Koh, Nitroxide radicals. Controlled release from and transport through biomimetic and hollow fiber membranes, Free Rad. Res. 34 (2001) 263. [15] R.M. Barrer, Zeolites as membranes: the role of the gas–crystal interface, in: G. Ohlmann, et al. (Eds.), Catalysis and Adsorption by Zeolites, Elsevier, Amsterdam, 1991, pp. 257–273. [16] M.A. Islam, H. Buschatz, D. Paul, Non-equilibrium surface reactions—a factor in determining steady state diffusion flux, J. Membr. Sci. 204 (2002) 379. [17] T.C. Huang, R.S. Juang, Kinetics and mechanism of zinc extraction from sulfate medium with di(2-ethylhexyl) phosphoric acid, J. Chem. Eng. Jpn. 19 (1986) 379. [18] S.X. Sun, Y. Li, Y.H. Yang, J.L. Sheng, Effects of diluent polarity on the extaction kinetics, Acta Chim. Sin. 54 (1996) 1101. [19] T.C. Huang, R.S. Juang, Rate and mechanism of divalent metal transport through supported liquid membrane containing di(2-ethylhexyl) phosphoric acid as a mobile carrier, J. Chem. Technol. Biotechnol. 42 (1988) 3. [20] J.K. McCulloch, J.M. Perera, E.D. Kelly, R.W. Lee, G.W. Stevens, F. Grieser, A kinetic study of copper ion extraction by Kelex 100 at a heptane–water interface, J. Colloid Interface Sci. 184 (1996) 406. [21] H.L. Frisch, The time lag in diffusion, J. Phys. Chem. 61 (1957) 93.
113
[22] J. Yi, Effect of the boundary layer and interfacial reaction on the time lag in supported liquid membranes, Korean J. Chem. Eng. 12 (1995) 391. [23] J. Crank, The Mathematics of Diffusion, second ed., Oxford Science Publications, Oxford, 1975. [24] R. Wang, C. Cao, T.S. Chung, A critical review on diffusivity and the characterization of diffusivity of 6FDA–6FpDA polyimide membranes for gas separation, J. Membr. Sci. 198 (2002) 259. [25] W.W.S. Ho, D.C. Dalrymple, Facilitated transport of olefins in Ag+ -containing polymer membranes, J. Membr. Sci. 91 (1994) 13. [26] M.F. Paugam, J. Buffle, Comparison of carrier-facilitated copper(II) ion transport mechanisms in a supported liquid membrane and in a plasticized cellulose triacetate membrane, J. Membr. Sci. 147 (1998) 207. [27] N.M. Kocherginsky, A.B. Grishchenko, Mass transfer of long chain fatty acids through liquid–liquid interface stabilized by porous membrane, Sep. Purif. Technol. 20 (2000) 197. [28] P.R. Danesi, Solvent extraction kinetics, in: Principles and Practices of Solvent Extraction, Marcel Dekker, New York, 1992, pp. 157–207. [29] C.R. Wilke, P. Chang, Correlation of diffusion coefficients in dilute solutions, Am. Inst. Chem. Eng. J. 6 (1955) 264. [30] W.J. Lyman, W.F. Reehl, D.H. Rosenblatt, Diffusion coefficients in air and water, in: Handbook of Chemical Property Estimation Methods, McGraw-Hill, New York, 1982, Chapter 10, pp. 17–24. [31] H.S. Fogler, Elements of Chemical Reaction Engineering, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1992. [32] A.M. Urtiaga, J.A. Irabien, Internal mass transfer in hollow fiber supported liquid membranes, Am. Inst. Chem. Eng. J. 39 (1993) 521. [33] P. Schoner, P. Plucinski, W. Nitsch, U. Daiminger, Mass transfer in the shell side of cross flow hollow fiber modules, Chem. Eng. Sci. 53 (1998) 2319. [34] S.F. Timashev, Physical Chemistry of Membrane Process, Ellis Horwood, Chichester, 1991. [35] S.H. Lin, R.S. Juang, Mass transfer in hollow fiber modules for extraction and back-extraction of copper(II) with LIX64N carriers, J. Membr. Sci. 188 (2001) 251. [36] A.B. de Haan, P.V. Bartels, J. de Graauw, Extraction of metal ions from wastewater, modeling of the mass transfer in a supported-liquid-membrane process, J. Membr. Sci. 45 (1989) 281. [37] Y.K. Zhang, Kinetic and equilibrium study of strontium removal by liquid–liquid extraction and membrane separation, Master thesis, National University of Singapore, Singapore, 1999. [38] T. Osakai, Ion salvation and resolvation, in: A.G. Volkov (Ed.), Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications, Marcel Dekker, New York, 2001, pp. 23–50.
114
Y.K. Zhang, N.M. Kocherginsky / Journal of Membrane Science 225 (2003) 105–114
[39] N.M. Kocherginsky, Y.K. Zhang, Role of standard chemical potential in transport through anisotropic media and asymmetrical membranes, J. Phys. Chem. B 107 (2003) 7830. [40] Z. Koczorowski, Interfacial potentials and cells, in: A.G. Volkov (Ed.), Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications, Marcel Dekker, New York, 2001, pp. 1–22.
[41] J.H. Krenz, Electronic Concepts—An Introduction, Cambridge University Press, Cambridge, 2000. [42] R.A. Siegel, Algebraic, differential, and integral relations for membrane in series and other multilaminar media: permeabilities, solute consumption, lag times, and mean first passage times, J. Phys. Chem. 95 (1991) 2556.