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Ind. Eng. Chem. Res. 2002, 41, 456-463
Analysis of Diffusion and Reversible Reaction in Spatially Confined Systems Jerry H. Meldon† Chemical and Biological Engineering Department, Tufts University, Medford, Massachusetts 02155
Facilitated gas transport in membranes is a diffusion/reaction system in which at least one reactant is spatially confined. Despite longstanding interest in this process, the consequences of spatial constraint have generally been overlooked. They are examined here in the cases of transport of the acid gases CO2, H2S, and SO2 in solutions of alkali and of ethylene in solutions of silver salts. Diffusion potentials are shown to be significant in each case, playing a role that complements spatial constraints. Nonetheless, at least in the ethylene case, fluxes calculated using an analysis that enforces spatial constraints but neglects ionic interactions are indistinguishable from those generated by an analysis that accounts for both. Introduction Confinement of reactants or catalysts in membranes,1 zeolites,2 and gels3 can promote selective reaction and separation. When an immobilized liquid membrane confines nonvolatile solute B that reacts reversibly with dissolved gas A (see Figure 1), B “facilitates” the flux of A, ΦA, while the global reaction rate remains zero.4 Developers of separation processes have sought to exploit the selectivity of facilitated transport.5,6 Others have elucidated the underlying phenomena7 and determined parameters such as kinetic constants from transport measurements.8 Most analyses have neglected the potentially significant consequences of differences (∆D) between diffusivities of confined reactants. When confined reactants are electrolytes, ∆D causes diffusion potentials9 that ensure the absence of local current in electrically floating systems. For example, facilitated carbon dioxide transport in aqueous media involves counterdiffusion of bicarbonate and slower carbonate ions. Diffusion potentials equalize opposing charge fluxes while forcing inert cations into Nernstian distributions.10 Much larger electrical effects, including measurable potentials of up to 40 mV,11 mediate bicarbonatefacilitated CO2 transport in protein solutions.12 Confined solutes need not react directly with permeants or carry charge for ∆D to mediate fluxes. Consider hemoglobin (Hb)-facilitated oxygen transport.13 Human red blood cells confine both Hb and the metabolic intermediate 2,3-diphosphoglycerate (DPG). Binding of DPG to Hb reduces the protein’s O2 affinity. Because DDPG > DDPGHb, [DPG]T (≡[DPG] + [DPGHb]) increases in the direction of O2 diffusion. On the basis of a theoretical analysis that neglects electrical effects, it has been estimated that skewing of the [DPG]T distribution increases O2 fluxes by as much as 50%.14 There has also been considerable interest in using membranes containing Ag+ to separate olefins, particularly ethylene, from alkanes.15,16 Yet, the open literature apparently contains no analysis of silver-facilitated olefin transport that explicitly accounts for diffusion potentials. The following section includes an examination of coupled spatial constraint and electrical effects on † Phone: 617-627-3570. Fax:
[email protected].
617-627-3991.
E-mail:
Figure 1. Schematic diagram of the membrane including concentration profiles when the membrane is exposed to gradients in partial pressure of gas A, the permeation of which is facilitated via a reversible reaction with nonvolatile solute B to form AB.
facilitated transport of CO2, H2S, and SO2. This is followed by two analyses of Ag+-facilitated olefin transport: one analogous to that applied to acid gas transport and another that neglects electrical effects. Unexpectedly, the two analyses yield essentially identical flux estimates. Facilitated Transport of Acid Gases We first consider steady-state transport of dissolved acid gas A ()H2S, CO2, or SO2) in a film of an aqueous solution that may contain alkali-metal ion M+. Complexes such as NaCO3- are neglected; the metal is treated as a free cation. The following reactions mediate gas transport:
A + nH2O S B- + H+
(1)
where n ) 0 when A ) H2S and n ) 1 when A ) CO2 or SO2.
B- S C2- + H+
(2)
H2O S OH- + H+
(3)
where (B-, C2-) denotes (HS-, S2-), (HCO3-, CO32-), or (HSO3-, SO32-).
10.1021/ie0102706 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/27/2001
Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 457
Reactions (2) and (3) are effectively instantaneous. In the absence of a suitable catalyst, reaction (1) kinetics are CO2-flux-limiting.17 For simplicity, we assume such a catalyst is present and treat all reactions as instantaneous, i.e.,
[B-][H+] ) K1 [A] [C2-][H+] [B-]
) K2
[H+][OH-] ) Kw
(4)
(5) (6)
We also treat K’s as constants for a given gas and temperature and implicitly incorporate the activity of water in Kw and in K1 when n ) 1. Steady-state species balances are expressed by
dNi ) ri dx
(7)
while flux is given a Nernst-Planck formulation, i.e.,
Ni ) -Di
F dV + z [i] (d[i] dx RT dx ) i
(8)
Because alkali-metal ions are both inert and confined by nonvolatility, eq 7 implies that NM is uniformly zero. In light of eq 8, this further implies that
d[M+] F dV ) -[M+] dx RT dx
(9)
Thus, metal ions achieve uniform electrochemical potentials consistent with their overall concentration, i.e.,
∫0L[M+] dx ) CML
(10)
When eq 7 is applied to A, B-, and C2- and the results are summed to eliminate indeterminate reaction terms, one obtains
d (N + NB- + NC2-) ) 0 dx A
(11)
or after integration
NA + NB- + NC2- ) ΦA
(12)
Fluxes of B- and C2- sum to zero at the film’s boundaries (with the assumption of equilibrium of reaction (2) having precluded enforcement of boundary conditions for both B- and C2-); ΦA may be identified with the transmembrane flux of A. The remaining constraints are the absence both of electrical current:
∑ziNi ) 0
(13)
∑zi[i] ) 0
(14)
and of net charge
neglecting minute local net charges responsible for diffusion potentials.18
Table 1. Thermodynamic and Transport Parameters at 25 °Ca parameter
carbon dioxide
hydrogen sulfide
sulfur dioxide
pK1 pK2 pKW R (mol m-3 atm-1) Da × 109 (m2/s) DB × 109 (m2/s) DC × 109 (m2/s) DH × 109 (m2/s) DOH × 109 (m2/s)
6.3521 10.321 14 3424 1.9224 1.19b,26 0.92b,26 9.0b,27 5.1b,27
7.0522 14.922 same 10022 1.4422 1.19c 0.82c same same
1.8123 6.9123 same 116025 1.7625 1.19b,25 0.7223 same same
a Infinite dilution values, corrected for small temperature differences as needed. b Di ) RTΛi/F2|zi|, where Λ denotes equivalent ionic conductance. c Estimate.
Equation 14 expands to
[M+] + [H+] ) [B-] + 2[C2-] + [OH-]
(15)
Insertion of eq 8 into eq 11, differentiation of eq 15 to eliminate d[C2-]/dx, insertion of the results into eq 13, and rearrangement yield
[
d[B-] d[OH-] F dV ) (DB - DC) + (DOH - DC) RT dx dx dx d[H+] /[DB[B-] + DC(4[C2-] + [M+]) + (DH - DC) dx DOH[OH-] + DH[H+]] (16)
]
Equation 16 explicitly links diffusion potentials to differences, ∆D, in diffusion coefficients of reactive ions. Together with concentration gradients, ∆D’s determine the sign of the electrical field, -dV/dx, and thus (a) that of d[M+]/dx (see eq 9) and (b) whether migration of Band C2- promotes or reduces gas transport (see eq 8). Although eq 16 is complicated, generally one or more concentration gradients is negligble. Appendix I outlines an exact solution to the general problem. At 25 °C and with [M+] ) 2 M, the reactive ions with significant gradients in a membrane may be deduced from the equilibrium compositions shown in Figure 2ac, which were derived from the thermodynamic parameters in Table 1, eqs 4-6 and 16, and the simple solubility relationship
[A] ) RP
(17)
Parts a and c of Figure 2 elucidate the difficulty of regenerating hydroxide from solutions preequilibrated with even minute partial pressures P of either CO2 or SO2. When P is as low as 10-6 atm, the predominant anions are B- and C2- and solutions contain 0.5-1 mol of reacted gas/equiv of alkali. On the other hand, the second dissociation of H2S is so weak that there is no range of P in which sulfide predominates (see Figure 2b). Furthermore, although H2S is more easily stripped, its solutions too require large reductions in P to promote its release. To examine facilitated transport of the acid gases, fluxes were calculated over practical ranges of upstream partial pressure, P0, with [M+] ) 2 M, PL/P0 ) 0.1, and T ) 25 °C. Note that hydrolyzed SO2 is such a strong acid that bisulfite concentrations in alkali-free water
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Figure 3. Enhancement factor calculated assuming diffusivities of all reactive ions equal to DC vs upstream partial pressure of permeating acid gas; T ) 25 °C.
for CO2. The total amounts of SO2 and H2S level off at lower partial pressures, because of the very high solubility of SO2 and negligible second dissociation of H2S. Because of the latter limitation, at practical partial pressures a second weak acid must be present for there to be significant facilitation of H2S transport.20 Electrical effects are depicted in Figure 4 as ratios of enhancement factors (and therefore fluxes) calculated by including and neglecting them. For SO2, the effects are modest because similarly mobile HSO3- and SO32predominate over the entire range of P. Large effects for CO2 are attributable to highly mobile OH-. In the case of H2S, such effects are first seen at intermediate P values. In general, ion fluxes are determined primarily by the diffusivity of the more dilute of two ions.10,28 Facilitated C2H4 Transport in AgNO3 Solutions We next consider transport of ethylene (A) in solutions containing silver ions (S+) that complex C2H4 via the reaction
A + S+ S C+
(18)
Proceeding as before, the governing relationships are eq 8 plus
[C+] [A][S+] Figure 2. Liquid-phase equilibrium composition vs gas-phase partial pressure of acid gas at 25 °C and [M+] ) 2 M: (a) CO2, (b) H2S, (c) SO2.
promote substantial facilitation.19 With CO2 and H2S, alkali must be present for there to be significant facilitation of transport. Diffusivities used in the calculations are listed in Table 1. Results are expressed in terms of enhancement factor E, the ratio of fluxes with and without reaction. Maximal transmembrane potential differences were of order 1 µV. Values of E0, calculated with all ionic diffusivities set at DC (which eliminates electrostatic effects), are depicted in Figure 3. Enhancement is great when P is small, reflecting relatively large contributions of carrier-mediated transport. As P0 increases, E0 values for SO2 and H2S approach unity more rapidly than that
) KE
(19)
d[X-] F dV ) [X-] dx RT dx
(20)
∫0L[X-] dx ) CML
(21)
N A + N C+ ) Φ A
(22)
[S+] + [C+] ) [Χ-]
(23)
D S - DC d[C+] F dV ) (24) RT dx D ([X-] + [S+]) + D [C+] dx S C with counterion Χ- typically nitrate. A solution is outlined in appendix II. In an attempt to decouple the consequences of electrical and spatial constraints and thereby compare impacts
Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 459
Figure 4. Calculated ratio of rigorously calculated enhancement factor and E0 from Figure 3 vs upstream partial pressure of permeating acid gas; T ) 25 °C.
upon ethylene transport, an analysis was derived that treats all species as electrically neutral. Thus, eq 8 reduces to
Ni ) -Di
d[i] dx
(25)
and eqs 19 and 22 remain unchanged while eqs 20, 23, and 24 are not applicable. Applying eq 7 to S and C, we obtain
d (N + NC) ) 0 dx S
(26)
NS + NC ) 0
(27)
and so
Equation 21 is replaced by
∫0L([S] + [C]) dx ) CML
(27)
Insertion of eq 25 into eqs 22 and 26 and integration yield
DA[A] + DC[C] ) -ΦAx + Φ2
(28)
DS[S] + DC[C] ) Φ3
(29)
where Φ2 and Φ3 are constants. The solution is outlined in appendix III. Both analyses were used to calculate ethylene fluxes in membranes containing aqueous AgNO3 at 25 °C, with PL ) 0. Teramoto et al.16 measured ethylene fluxes under these conditions with CM ) 1 M and reported that KE ) 0.11 m3/mol, DA ) 1.87 × 10-9 m2/s, DS ) 1.66 × 10-9 m2/s, and DC ) 1.1 × 10-9 m2/s. We have calculated enhancement factors using these parameters and plotted them versus P0 in Figure 5a. Remarkably, results based on a rigorous accounting for electrical and spatial constraints (solid line) differ imperceptibly from those derived from the nonionic analysis (circles) that accounts only for the latter.
Figure 5. (a) Theoretical enhancement factors vs upstream partial pressure of ethylene; T ) 25 °C. Comparison of results obtained rigorously with those calculated by setting DS equal to DC, setting DC equal to DS, and neglecting electrical effects. (b) Calculated enhancement factors vs KE; T ) 25 °C and P0 ) 1 atm. Comparison of the results obtained as in Figure 5a (results of rigorous and nonionic analyses again overlap).
Additional results depicted in Figure 5a as the broken line and squares were respectively calculated by assigning DC the value of DB and vice versa. In each such case, mobility equality eliminates electrical effects as well as the basis for nonuniformity in the total silver concentration. More generally, in an analysis that neglects electrostatic interactions but will, nonetheless, accurately predict fluxes, the effective diffusion coefficient that should be assigned to each member of a pair of reactive ions varies between the respective D values.10,28 Under the conditions of Teramoto and co-workers,16 most silver ions were not complexed. Consequently, the effective D for S+ and C+ was ≈DC, consistent with the results depicted in Figure 5a, where results obtained with DS set equal to DC (squares) lie only slightly below the rigorously derived curve. Figure 5b compares E values calculated the same three ways: rigorously and with both ions assigned either DS or DC, as reaction equilibrium constant KE (see eq 19) was hypothetically varied from its reported value to much higher values such that nearly all of the silver is comlexed with ethylene; to further promote the latter condition, PL/P0 was set at 0.1 instead of zero. Even with the relatively small ∆D, it is clear that the effective D varies from DC to DS as KE increases. Interestingly, Teramoto et al. recognized the significance of electrical effects, yet they developed an accurate
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as electrically neutral but enforces spatial constraints correctly predicts transmembrane fluxes. This unanticipated result is attributable to the larger transmembrane gradients in the total carrier concentration that are calculated in the absence of electrical interactions. Why the mathematics is so forgiving remains a mystery. Appendix I: Solution to Eqs 4-6, 8-10, 12, 15, and 16 For conciseness, dimensionless variables are defined as follows:
i≡
[Izi] i ) a, b, c, ... and I ) A, B, C, ... [A]0
(A1-1)
F dV x y≡ ; ψ≡ L RT dy
(A1-2)
Figure 6. Ratio of up- and downstream total silver concentrations vs DC/DS, calculated rigorously and via nonionic analyses.
analysis in which inequality of DB and DC was retained, without including electrical terms. Applying instead the simplifying approximation of linear [S] and [C] profiles, they calculated fluxes with P0 ) 1 atm that were indistinguishable from what they obtained from an exact numerical analysis of a set of equations equivalent to those solved in our nonionic analysis. The value they reported for DC was obtained from a fit to their experimental data. In an attempt to determine why there is essentially exact agreement between the results of rigorous and “nonionic” analyses, we calculated ratios of up- and downstream total silver concentrations, [S+] + [C+]. This ratio deviates from unity whenever DC * DS, whether or not species are charged. Because DS > DC, a membrane’s downstream surface actually assumes a relatively negative potential. Accordingly, [X-] is greater at x ) 0 than at x ) L, and electroneutrality requires the same of total silver. Enhanced silver concentrations at x ) 0 translate into larger d[C+]/dx values. In addition, electrical migration complements C+ transport driven by its concentration gradient. Migration naturally vanishes without electrostatic interactions. However, the total silver concentrations remain higher at x ) 0 because DS > DC (as DS increases, the [S+] profile flattens while d[C+]/dx remains negative). Having eliminated one of the positive effects on ethylene transport, one might anticipate lower ethylene fluxes, which is contradicted by the behavior in Figure 5a. The explanation, shown in Figure 6, is the larger silver gradients calculated using the nonionic analysis; in fact, they are just large enough to compensate for the absence of migration. Conclusions Electrical effects and spatial constraints are potentially important determinants of acid gas fluxes in membrane-immobilized solutions of alkali-metal ions. Calculations indicate modest effects when the conjugate bases formed by first and second acid dissociations are the predominant anions. Effects are considerably larger when OH- diffusion is significant. The same phenomena mediate ethylene transport in silver salt solutions. An analysis that treats all species
It follows from eqs 4-6 that
θ1a θwb θ2b2 , h) , oh ) θ1a b θ1 a
c)
(A1-3)
where
θ1 ≡
K1
, θ2 ≡
[A]0
K2
, θw ≡
[A]0
Kw [A]02
(A1-4)
Equation 15 becomes
θ2b2 θwb θ1a )b+2 + m+ b θ1 a θ1a
(A1-5)
from which b may be calculated, given a and m. Equations 9 and 10 become
dm ) -mΨ dy
(A1-6)
∫01m dy ) F
(A1-7)
while eq 16 transforms to
ψ)
(λB - λC)
db doh dh + (λOH - λC) - (λH - λC) dy dy dy δ (A1-7)
where
λi ≡
D iF CM , F≡ , DA [A]0 δ ≡ λBb + λC(4c + m) + λOHoh + λHh (A1-8)
Differentiation of the second and third terms of eq A1-3 and combination of the results with eqs A1-5 and A1-6 yield
db 1 da ) (f - mabg5) dy f1 2 dy Ψ ) g5 where
da dy
(A1-9) (A1-10)
Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 461
6θ2b2 2θwb θ1 θ1
f1 ≡ m - 2ab 2
f2 ≡ b - mb - 2θ1a
g3 ≡ (λB - λC) + g4 ≡
Equations 20 and 21 become
(A1-11) (A1-12)
f1 mab g1 ≡ , g 2 ≡ f2 f1
(A1-13)
θ2 oh (λ - λC) + (λH - λC) b OH c
(A1-14)
θ1 oh (λOH - λC) + (λH - λC) a b
(A1-15)
g1g3 - g4 ∆ - g2g3
(A1-16)
g5 ≡
da φ )dy g6
∫01χ dy ) F
(A2-3)
s+c)χ
(A2-4)
while eq 23 becomes
From eqs A2-1 and A2-4, it follows that
s)
χ 1 + θEa
(A2-5)
Equation 24 becomes
dc dy
(A2-6)
where
µ1 ≡
(A1-17)
λ S - λC λS(χ + s) + λCc
(A2-7)
Proceeding as in appendix I, we obtain the following differential equations:
or
dζ ) -g6 da
(A1-18)
where
Φ AL
(A2-2)
ψ ) µ1
Combination of eqs 8 and 12, conversion to dimensionless variables, and insertion of eqs A1-9 and A1-10 yield
φ≡
dχ ) χψ dy
dζ ) -µ3 da
(A2-8)
dχ ) χµ1µ2 da
(A2-9)
where
, ζ ≡ φy
(A1-19)
DA[A]0
(
g6 ≡ 1 + λBg7 + λCc
2g7 1 b a
g7 ≡ g1 + g5(g2 - b)
)
µ2 ≡ (A1-20) (A1-21)
Finally, combination of eqs A1-6, A1-10, and A1-17 yields
θ Es 1 - θEa(µ1χ - 1)
(A2-10)
µ3 ≡ 1 + λCµ2(1 + µ1c)
(A2-11)
With a guess of the χ value where a ) 1, we may proceed to solve eqs A2-8 and A2-9; dimensionless ethylene flux φE ) ζ(γ); y(a) ) ζ(a)/φE. Equation A2-3 tests the guess of the initial χ. Appendix III: Solutions to Eqs 19 and 27-29
dm ) -mg5 da
(A1-22)
Equations A1-17 and A1-22 are easily solved (e.g., using Runge-Kutta methods) to obtain ζ(a) and m(a). The value of ζ where a ) 1 is 0; that of m requires a trial-and-error experiment. Because the y value where a ) γ must be 1, φ may be equated with ζ(γ), after which each y(a) may be obtained by dividing ζ(a) by φ. The correct initial m ensures satisfaction of eq A1-7.
Upon transformation to dimensionless variables as before, eq A2-1 remains unchanged, while eq A2-3 is replaced (see eq 27) by
∫01(s + c) dy ) F Equations 28 and 29 become
Appendix II: Solution to Eqs 8 and 19-24 When dimensionless variables are defined as in eqs A1-1 and A1-2, eq 19 becomes
c ) θEas where θE ≡ KE[A]0.
(A3-1)
a + λCc ) -φEy + φ2
(A3-2)
λSs + λCc ) φ3
(A3-3)
where φ2 and φ3 are constants. Combining eqs A2-1 and A3-3, we obtain
c)
(A2-1) where
θEφ3a u
(A3-4)
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Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002
u(a) ≡ λS + λCθEa
(A3-5)
Equations A3-2-4 transform eq A3-1 to
φ3 λC
∫01
[
]
u + λC - λS u2
dy ) 1
(A3-6)
Combining eqs A3-2 and A3-4, evaluating the result at y ) 0 and 1, and subtracting one from the other yield
[
φE ) (1 - γ) 1 +
µ4φ3 u(1) u(γ)
]
(A3-7)
where
µ4 ≡ λSλCFθE
(A3-8)
Differentiating eq A3-6 and the relationship derived from eqs A3-2 and A3-4 and combining the results yield
dy ) -
(
)
µ4φ3 1 1 + 2 du θEφEλC u
(A3-9)
After inserting eqs A3-8 and A3-9 into eq A3-6 and integrating, one may solve for φ3 and, in turn, φE. Nomenclature a, b, c, h, m, oh, s ) defined by eq A1-1 CM ) overall metal ion concentration (mol/m3) Di ) diffusion coefficient of species I (m2/s) F ) Faraday’s constant (96 500 J V-1 equiv-1) f1, f2 ) defined by eqs A1-11 and A1-12 g1, g2 ) defined by eq A1-13 g3, g4, g5, g6, g7 ) defined by eqs A1-14-16, A1-20, and A1-21 K1, K2 ) equilibrium constants of reactions 1 and 2 (mol/m3) KE ) equilibrium constant of reaction (18) (m3/mol) Kw ) equilibrium constant of reaction (3) (mol2/m6) L ) membrane thickness (m) P ) partial pressure (atm) r ) rate of production by recation (mol/m3/s) R ) gas constant (8.314 J mol-1 K-1) S+ ) silver ion u ) defined by eq A3-5 V ) electrical potential (V) X- ) anion in the silver salt x ) distance from the upstream membrane surface (m) y ) defined by eq A1-2 zi ) charge of species i Greek Symbols R ) solubility coefficient (mol m-3 atm-1) δ ) defined by eq A1-8 σ ) defined by eq A1-19 θ1, θ2, θw ) defined by eq A1-4 θE ) defined following eq A2-1 F ) defined by eq A1-8 λ ) defined by eq A1-8 µ1, µ2, µ3, µ4 ) defined by eqs A2-7, A2-10, A2-11, and A3-8 ΦA ) transmembrane flux of A (mol m-2 s-1) Φ2, Φ3 ) constants of integration (mol m-1 s-1) φ ) dimensionless flux defined by eq A1-19
φE ) dimensionless ethylene flux φ2, φ3 ) constants of integration χ ) [X-]/[A]0 ψ ) defined by eq A1-2
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Received for review March 23, 2001 Revised manuscript received October 19, 2001 Accepted October 31, 2001 IE0102706
