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Eng. Chem. Fundam. 1906, 25, 360-367
A Model for Bipolar Membranes in an Acid-Base Environment Wllllam H. Rose and Irving F. Miller’ Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60680
A mathematical model has been developed to describe the behavior of a bipolar membrane system separating either two salt solutions or an acid and a base. The model is based on the Nernst-Planck transport equations and includes such effects as imposed potential gradients, ion and water flow, nonideal membrane permeability, nonzero space charge, membrane polarization, and interfaces which are of finite thickness and are not at thermodynamic equilibrium. The model has been tested against experimental data for both highly and poorly selective membrane systems in a computer simulation and had been found to fit experimental behavior.
Introduction The bipolar membrane is an ion-exchanging membrane composed of two ion-exchange regions of opposite polarity in series. Such bipolar membrane systems have been studied under both static and dynamic conditions, and dynamic studies have shown that such membranes exhibit P-N junctionlike behavior (Frilette, 1956). We consider the response of bipolar membrane systems operated under static conditions only and, in particular, the electrical response of such systems when they act as a separator between two salt solutions or between acids and bases. The review of Nagasubramanian et al. (1977) describes bipolar membranes operated under a variety of static conditions. When the membrane separates salt solutions of different concentrations and an electric current is passed in a direction such that the anion is driven toward the cation-anion interface through the anion-exchange side, water is generated at the cation-anion interface. If, however, the current is passed in the opposite direction, water is split at the interface and acid and base are generated, which migrate in opposite directions into the two reservoirs. Conversely, the bipolar membrane could separate acid and base. This situation is roughly the reverse of that described above and would result in the development of an electrical potential gradient in the membranes. If the two reservoirs were connected through an external circuit, current would flow. The use of bipolar membranes as water splitters and as electrical power generators has some practical potential, especially since the advent of highly selective bipolar membrane systems such as those described by Dege and Liu (1977). However, such systems suffer from the same limitations as other ion-exchange membrane systems, such as limited operating range with regard to electrical potential difference and temperature. In addition, there is the problem of minimizing the interfacial resistance between the cationic and anionic phase, as well as controlling the water flow rate. In general, the key problems are control of water flow, co-ion leakage, electrical resistance, and polarization at interfaces, all of which are interrelated. The use of computer modeling has some obvious advantages over a purely experimental approach to a system as complex as a bipolar membrane. First of all, one cannot control membrane parameters (e.g., porosity, resistance, charge density) either precisely or independently in an experimental system; however, in a computer simulation, one can investigate their independent relative importance to overall system behavior. Second, one can regulate such independent parameters as water flow or polarization over a wider range than can be done experimentally, so that their effects can be studied in great detail. Third, one can investigate systems which cannot be constructed in the 0196-4313/86/1025-0360$0 1.50/0
laboratory to study the effects of different ions or different membrane structures on the properties of the system. Fourth, when compared to an experimental system, a computer simulation is fast and inexpensive. This allows one to investigate many systems and, thus, can be used as a screening device preliminary to an experimental program. Description of Model. The model described herein uses a hydrodynamic approach and is an outgrowth of the work of Patel et al. (1977). Parameters used in the model were obtained from the experimental work of Patel et al. (1977), de Korosy and Zeigerson (1971),and Keusch (1980). The model is based on the Nernst-Planck equations and other related transport and balance equations and describes the system shown in Figure l. Although it is an outgrowth of the work of Patel et al., the model is not as limited. Patel considered only a single-phase membrane, while the membrane in this work is bipolar. However, there are many more improvements in this work over that of Patel that make this model a much better representation of reality. The membrane phases are not assumed to be ideally permselective, and co-ion flow is allowed. Water flow by ordinary diffusion and by electroosmosis is included in the model. There is no assumption of thermodynamic equilibrium at phase interfaces, and the interfaces themselves are not assumed to be of zero thickness. The system is essentially treated as a single phase in which such local and ion properties as porosity, fixed charge density, diffusion coefficient, etc., can vary from point to point in the direction of flow in the one-dimensional space. That is, rather than assuming the membrane surface is a sharp boundary across which thermodynamic equilibrium must be assumed, as with other membrane models in the literature, we define a “fuzzy interface” region of thickness comparable to the actual interfacial region, in which system properties are allowed to vary continuously from one set of pure phase values to another set. The same fuzzy interface concept is applied to the zone connecting the anion-exchange region to the cation-exchange region. In this case, the thickness of the fuzzy interface can vary from atomic dimensions to most of the thickness of the membrane, depending on the particular characteristics of the system being modeled. Although this interface concept complicates the solution algorithm and increases the computation time substantially, it makes the model much more versatile and a much better approximation of reality than other models, particularly for those systems operating far from equilibrium. Previous models of membrane transport have invariably assumed sharp boundaries between phases. This assumption is often coupled with an assumption of equilibB 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 361
1
Phase 1
I
Phase2
I
Phase3
I
Phase4
1
C5 = [PA- C(CiMJ1/18
(5)
Water equilibrium (see Patel et al. (1977)) follows the equation C3C4 = K,A,G
(6)
where
l / A m = (1- cos a ) / 2 - brm2sin2 a/8q2
(7)
Here q is the Bjerrum radius of bound ions (= 3.56 X cm), a is the angle of allowable approach of co-ions to counterions, given by Figure 1. Bipolar membrane with defined regions of interest: the repulsion zones are in black, and the fuzzy interfaces are crosshatched. Region widths are not to scale.
rium at the interface, in spite of the fact that current is flowing. These assumptions lead to the physically unrealizable conclusion that such variables as concentration, potential, etc., change instantaneously over distance when one crosses a phase boundary. In fact, all real phase boundaries must be “fuzzy” to a certain extent on a microscopic scale. The fuzzy interface model acknowledges this fact and eliminates the need to make any assumptions about the phase boundaries, with the exception of its thickness. One of the results of the study reported herein is that system behavior is only weakly dependent on the assumed thickness of the interface. This fact strengthens our belief that this interface approach is a major improvement over previous approaches to membrane modeling. The advances of Patel over previous work are also in the model, including polarization effects through the use of a repulsion zone adjacent to the membrane interface and the use of the Poisson equation to account for the possible existence of a space charge. The repulsion zone is a region adjacent to the membrane face where co-ion-fixed charge repulsion dominates. In this zone, the recombination rate of hydrogen and hydroxide ions is therefore strongly affected by the local electric field, as opposed to what occurs in homogeneous solution. Limitations which remain in the model include the ignoring of temperature and pressure gradients, the assumptions that all solutions are ideal and all transport properties are constant, and the assumption that all flows are at steady state and normal to the membrane surface. One additional change has been made. Patel computed the thickness of the repulsion zone to fit the transport data best. Since these computed thicknesses are all in the range of the size of the Debye-Huckel diffuse double layer for this system, the computed size of the flat double layer was used for the thickness of the repulsion zone (Vervey and Overbeek, 1948). The model, then, has been developed from the following series of transport and balance equations: The Nernst-Planck equations are Ji = -Dj(dCi/dx) - DiCizi(d@/dx) + CiW (1) where i refers to the ions in the system (1 = cation, 2 = anion, 3 = hydrogen ion, and 4 = hydroxide ion), and the dimensionless potential is given by 9 = -EF/RT (2) The equation used for water flow is W = elJl + e& + e3J3+ e4J4- 18D5(dC5/dx) (3) The total current is represented by I = ~1Jl + z Z J Z + ZSJ~+ Z q J 4 Water concentration is calculated from
(4)
a = cos-1 ( x r / r m ) (8) r, is the mean distance of approach of ion pairs (cm), and x , is the distance of co-ion from edge of repulsion zone
(4. G= 1
+ b + b2/3 + b3/18 + b4/180 + b5/2700 + b6/56700 (9)
for b < 3, and G = 0.797885eUu-’.5[1- 3/(8u) - 15/(128u2) 105/(1024u3)] (10) for b 1 3 where b = 3.546 X 10-8(EF/RT(
(11)
and u = (8b)1/2
The Poisson equation is d29/dX2 -(4T/C)(Z1c1 + zZCZ + z3c3 + Z q c 4
(12)
+ zACA) (13)
Water stoichiometry is expressed by dJ3/dx = dJ4/dx
(14)
and (d/dx)[(W- JIM1 - J2M2 - J3M3 - J&f4)/18] = - d J 3 / d ~ (15) In addition, each experiment has fixed values of total current and the concentrations of acid or base in each of the two reservoirs. The equations listed above, along with the specified boundary conditions on total current and reservoir concentration, can then be used to solve for the 11unknowns at each position, namely, 5 concentrations, 5 fluxes, and the local electric potential. Parameters such as diffusion coefficient, valence, molar volume, and electroosmotic coefficient for each species in each region and such membrane parameters as phase thickness, porosity, and fixed charges density must also be specified. Algorithm. The algorithm was developed by using FORTRAN H EXTENDED with double precision arithmetic. The integration of the model’s flux equations is accomplished by a library program, entitled ODE, called from the FORTRAN program. ODE implements the Adams method via the predictor-estimator-corrector-estimator (PECE) formulas of order k and step size h. This implies the utilization of the Adams-Bashforth predictor of order k and the Adams-Moulton corrector of order k + 1. Upon successful completion of PECE, the values of k and h are recalculated, based on local error and stability considerations. The algorithm is expressed in four basic programming levels and termed the “elementary solution sequence”. The second and third levels control the x range to be integrated by ODE within the boundary layers and the membrane
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Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
phases. The fourth level involves the calculation and communication of the derivatives of ion concentration to ODE. On the first level, the algorithm measures the success of a full system integration relative to a predetermined goal, makes adjustments in the input parameters, and then restarts the integration process in levels two through four. The “elementary solution sequence” is the basic building block for simulations of actual bipolar membrane systems under different external loads (current or concentration boundary conditions). It is also used to study the behavior of the algorithm when system definitions are changed. The first step in simulating an experimental membrane system is the determination of the system parameters from design features and experimental data. The simulation is successful when its predictions match experimental results under various experimentally realizable external loads. Once the model parameters are defined, the simulation can be used to test the system’s response to variations in the parameters, some of which may not be amenable to experiment. The results of some of these simulations are presented below.
Results The model was used to simulate bipolar membrane systems under conditions where membrane selectivity, boundary layer concentration, and total current density were controlled variables. Dege and Liu (1977) describe a highly selective membrane system and provide experimental results obtained from it. de Korosy and Zeigerson (1971) describe a relatively poorly selective membrane system and provide curves of electrical potential vs. total current for it. These systems were chosen for the simulation, and in particular, the highly selective membrane system was used to investigate the effects of varying system parameters. The system constants used in the simulations came from a variety of sources. The diffusion coefficients came primarily from ion conductance data in the literature. In the case of the boundary layers, diffusion coefficients at infinite dilution were used. For the membrane phases, ion conductance data for similar single-phase ion-exchange membranes were employed (Breslau et al., 1970). Although the use of infinite dilution values of diffusion coefficients for the boundary layers may be questionable, it should be remembered that these values are more accurately known than those at higher concentration. In addition, the process is certainly under membrane control and errors resulting from inaccurate values for diffusion coefficients in the boundary layers are likely to be negligible. The electroosmotic coefficients were estimated from the data of Breslau and Miller (1971),while the molar volumes were calculated from handbook data on crystal radii. The constants used in the simulation are shown in Tables I and 11. Highly Selective Bipolar Membrane System. Each elementary solution produces a set of concentration, potential gradient, and flux profiles, as shown in Figures 2, 3, and 4. The plots use different distance scales for the system boundary layers, repulsion zones, exterior fuzzy interfaces, and interior fuzzy interfaces and show vertical lines separating the various regions. Each of these lines is labeled with an identifier and its location (in cm) from the beginning of the leftmost boundary layer. The number of points plotted in each region is a function of the relative change that occurs in that variable within that region and has no other significance. The concentration profiles for a typical highly selective membrane simulation are shown in Figure 2 (data from
Table I. System Constants for Highly Selective Membrane Simulation Potassium Ion D (except cation exchanger) = 2.03 X cm2/s D (cation exchanger) = 1.96 X 10” cm2/s e (boundary layers) = 84.4 cm3/mol e (anion exchanger) = 99.9 cm3 mol e (cation exchanger) = 80.0 cm /mol M = 5.93 cm3/mol
d
Hydrogen Ion
D (except cation exchanger) = 9.31 x 10” cm2/s D (cation exchanger) = 2.86 X 10” cm2/s e (boundary layers) = 56.6 cm3/mol e (anion exchanger) = 52.2 cm3/mol e (cation exchanger) = 41.8 cm3/mol
M=O Fluoride Ion
D (except anion exchanger) = 4.06 X cm2/s D (anion exchanger) = 3.92 X IO“ cmz/s e (boundary layers) = 191 cm3/mol e (anion exchanger) = 377 cm3/mol e (cation exchanger) = 301 cm3/mol M = 6.35 cm3/mol Hydroxide Ion
D (except anion exchanger) = 5.26 X cmz/s D (anion exchanger) = 4.09 x 10” cm2/s e (boundary layers) = 104 cm3/mol e (anion exchanger) = 169 cm3 mol e (cation exchanger) = 135 cm /mol M = 18.06 cm3/mol
d
D = 2.44 X
Water cm2/s
Table 11. System Constants for Poorly Selective Membrane Simulation Sodium Ion D (boundary layers) = 3.18 X cm2/s D (anion exchanger) = 5.56 X IO” cm2/s D (cation exchanger) = 6.7 X IO-’ cm2/s e (boundary layers) = 114 cm3/mol e (anion exchanger) = 134 cm3/mol e (cation exchanger) = 108 cm3/mol M = 2.16 cm3/mol Hydrogen Ion
D (boundary layers) = 9.31 X D (anion exchanger) = 3.89 X D (cation exchanger) = 2.86 X
cm2/s cm2 s 10” cm / s e (boundary layers) = 56.6 cm3/mol e (anion exchanger) = 52.2 cm3/mol e (cation exchanger) = 41.8 cm3/mol M=O
1
Chloride Ion
D (boundary layers) = 2.03 X D (anion exchanger) = 1.96 X
cm2/s 10” cm2/s
D (cation exchanger) = 6.5 X lo4 cm2/s e (boundary layers) = 85.6 cm3/mol e (anion exchanger) = 169 cm3 mol e (cation exchanger) = 135 cm /mol M = 14.96 cm3/mol
d
Hydroxide Ion
D (boundary layers) = 5.26 X
cm2/s
D (anion exchanger) = 4.09 X 10” cm2/s D (cation exchanger) = 1.68 X 10” cmz/s e (boundary layers) = 104 cm3/mol e (anion exchanger) = 169 cm3/mol e (cation exchanger) = 135 cm3/mol M = 18.06 cm3/mol
D
= 2.44 X
Water cm2/s
Dege and Liu (1977)). This simulation had an open external circuit and a total membrane thickness of 0.020 32 cm and was 80% cation exchanger. The cation-exchange
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 363
0.06
0.17
-$
0.15 h
,-
ex
X
u
0.13
Y
(1
6 E
-8
. -
0.11
0.02
u)
al
-
0.08
z
0.00
0
Q
8 0 8z
0.03
al
Y
1
0.05
-
0.06
3
-0.02
ii
0.04
-0.03
u
X
2
0.02
-0.06
0.00 m uuu
a
A U
$fez
2
2 E F BY
DISTANCE, cm ( x 102) Figure 2. Concentration profiles for a typical run. Open circuit; KOH concentration in the left reservoir is 0.001 mol/cm3, HF concentration in the right reservoir is 0.002 mol/cm3, and fixed charge density in the anion and cation exchange membranes is +0.015 and -0.015 mol/cm3, respectively. Membrane is 0.02032 cm thick and is 80% cation exchanger. Region designations at the bottom of the figure are at the left boundary of the region: BL is boundary layer, R is repulsion zone, F is fuzzy interface, AEM is anion-exchange membrane, and CEM is cation-exchange membrane. The designations L and R refer to left and right side, respectively, and the numbers are zone identifiers. Regions are not drawn to scale. Distances from the leftmost boundary of the system are indicated at the top.
3.00
2.25
3
:P:
1.50
X
0.75
n
$
I
0.00
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F 5
A -0.75
A
26
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B
-0.05
-2.25
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A&;$
$$&E$
deea
DISTANCE, cm ( x 102) Figure 3. Potential profile for the same run as Figure 2.
fixed charge density was -0.015 mol/cm3, and the pore fraction was 0.2. The reservoir concentrations were 0.001 mol/cm3 for KOH and 0.002 mol/cm3 for HF, while the
A U
JELL
1
A!!
2$
E BE
ifJz:
DISTANCE, cm ( x 102) Figure 4. Flux profiles for hydrogen ion, hydroxide ion, and water for the same run as Figure 2.
potassium and fluoride ion fluxes were found to be 4.6 X and -1.3 X 10" mol/(s cm2), respectively. For this run, the concentration profiles for potassium and hydroxide ions tend to drop as one moves through the system from left to right, while those for hydrogen and flouride ions rise. Most of the significant changes occur within the membrane and its fuzzy interfaces. The hydroxide and fluoride ions dominate the anion-exchange phase of the membrane, while hydrogen ion alone dominates the cation-exchange phase with the concentrations of other ions remaining small. Within fuzzy interface 1 at the left, the concentration gradients for the anions increase, while those for the cations decrease; hydroxide ion has the largest gradient. In the anion-exchange phase, the concentration gradient for fluoride ion is large and positive, while that for hydroxide ion is large and negative. Fuzzy interface 2 appears to be a region of very low total ion concentration. Fluoride ion dominates the left half of this fuzzy interface while potassium ion dominates the right half. The cation-exchange phase has large gradients for the cations and a small positive gradient for fluoride. Fuzzy interface 3 at the right is dominated by a large gradient for hydrogen ion as its concentration drops to meet the rising concentration of fluoride. The boundary layers and the repulsion zone regions contain very small concentration gradients. The electric potential for this run is plotted in Figure 3, and the fluxes for hydrogen ion, hydroxide ion, and water are shown in Figure 4. Several runs were made which differed from this particular case only in the total current density and in the fluxes for potassium and fluoride ions. The total current densities ranged from -9.7 X lo4 to 9.7 X lo4 mol/(s cm2). One particular case had a total current density of 9.7 X IO4 mol/ (s cm2)and potassium and fluoride ion fluxes of -1.2 X lo4 and -1.8 X mol/(s cm2),respectively. This change resulted in some concentration changes within the first two fuzzy interfaces and in the anion-exchange membrane, but the profiles were otherwise very similar. Other runs were made in which the acid concentration in the right reservoir was increased at constant total
364
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
current. In general, the systems behaved similarly in the boundary layers, but within the membrane itself, the dominant counterion concentration was a strong function of the acid concentration in the right boundary layer. When the acid concentration was reduced, the dominant counterions were hydroxide in the anion-exchange phase and potassium in the cation-exchange phase. When the acid concentration was high, the dominant counterions were fluoride in the anion-exchange phase and hydrogen in the cation-exchange phase. In all cases, the potential gradients in the boundary layers and fuzzy interfaces showed little dependence on changes in concentration of acid. However, the potential gradients in the pure membrane phases were strongly affected by changes in acid concentration. The ion-exchange membrane thickness and fixed charge density are controllable parameters in the model. When the relative width of the cation-exchange phase was increased at constant total membrane width, fluoride ion did not reach the same maximum reached previously and hydroxide ion did not approach zero, as it had done. In addition, the total ion concentration in the second fuzzy interface was reduced. The magnitudes of hydrogen and hydroxide ion fluxes were unchanged; however, XHoH shifted to the right edge of the second fuzzy interface. In addition, the water flux in the left side of the second fuzzy interface increased substantially. In another simulation, the fixed charge density was set at 0.0135 mol/cm3, instead of the value of 0.015 mol/cm3 used previously; otherwise, all system parameters were identical. The concentration profiles for this run show that the total free ion concentration within the membrane was reduced. The ratio of fluoride to hydroxide ion within the anion-exchangephase was about the same; however, XHoH shifted to the right edge of the anion-exchange phase. The potential gradient decreased approximtely 90% in the second fuzzy interface and in the cation-exchange phase. In fact, the potential profile in the cation-exchange phase was essentially flat and the total system potential drop was -0.05 V, as opposed to -0.61 V for the previous run. The flux profiles showed approximately an 80% reduction in magnitude. Poorly Selective Bipolar Membrane System. de Korosy and Zeigerson (1971) gave experimental results for a poorly selective bipolar membrane system. Three of these experimental situations were modeled, differing only in total current density. In these simulations, the total membrane thickness was taken to be 0.005 em, with the anion-exchangeregion covering 0.0033 cm of the total. The fixed charge density was 1.6 X lom4mol/cm3 in the anion-exchange phase and -1.8 X lom4mol/cm3 in the cation-exchange phase. The solution in the left boundary layer contained 1 X mol/cm3 NaOH, while that at the right contained 1 X lo4 mol/cm3 HC1. The pore fractions for the anion- and cation-exchange regions were 0.418 and 0.320, respectively. The open-circuit run a t zero total current density had simulation ion fluxes for sodium and chloride ions of 7.83 X lo-* and -1.55 X mol/(s cm2),respectively. A run at a total current density of -8.62 x lo4 mol/(s cm2)required ion fluxes of -1.87 X and -6.18 x lom9mol/(s cm2)for sodium and chloride ion, respectively. When the total current was 8.62 X lo4 mol/(s cm2), the required fluxes for sodium and chloride ions were respectively 3.20 x and -5.79 x mol/(s cm2). In the open-circuit run, there was infiltration of co-ions into each membrane region and no significant region of ion depletion at all. Hydroxide ion dominated the first
fuzzy interface, while chloride ion dominated the anionexchange phase and the left side of the second fuzzy interface. On the right side of the second fuzzy interface, sodium ion was present at highest concentration, while hydrogen ion dominated the cation-exchange phase and the third fuzzy interface. As the total current density decreased from its highest positive value through zero and finally to its negative value, the role of sodium and chloride ions as the primary counterions in their respective membrane phases was replaced by hydrogen and hydroxide ion, respectively. However, even at the lowest current density investigated, chloride ion continued to dominate the left side of the second fuzzy interface. In the open-circuit run, the potential profile rose in the first fuzzy interface and showed a small decline in the anion-exchange phase. In the second fuzzy interface, the potential gradient was negative but became positive in the third fuzzy interface, resulting in an overall potential difference of -0.05 V. As the total current density decreased from its highest to its lowest value, the potential gradients in the first and third fuzzy interfaces remained unchanged; however, the gradients in the ion-exchange phases increased while that in the second fuzzy interface became more steeply negative. For the open-circuit run, the fluxes for both hydrogen and hydroxide ion were near zero, and XHoHwas at 0.0033 cm. The water flux had positive peaks on the right side of the first and third fuzzy interfaces and negative peaks on both sides of the second fuzzy interface. In order of decreasing total current density, XHOH moved from 0.0031 to 0.0033 cm and finally to 0.0037 cm. The water equilibrium constant profiles for all runs in the series showed positive peaks in all three fuzzy interfaces. Of course, these are the regions of steepest potential gradient. When the results from this series are compared to those obtained for the highly selective membrane series, it can be seen that all the profiles are qualitatively similar. However, the highly selective membrane exhibits much larger maxima and minima and much steeper gradients.
Discussion The bipolar membrane model has been used to calculate internal profiles for both a water splitter and a fuel cell while varying the membrane’s properties from poorly to highly selective. As indicated by the results in the previous section, the performance of these various membrane configurations has been evaluated over a range of external conditions, including total current and boundary layer acid concentration. The results produced have made it possible to judge the performance of the model against two typical bipolar membrane systems and to further our theoretical understanding of the mechanism underlying this transport process. Also, it is now possible to make recommendations for improvements in the design of future bipolar and ion-exchange membrane systems. Highly Selective Membrane Model. The bipolar membrane system discussed by Dege and Liu (1977) provides an example of a highly selective bipolar membrane with a low electrical resistance and high selectivity over a wide range of total current densities. The open-circuit potential for this system approaches that predicted from thermodynamic equilibrium. The model parameters were chosen to mimic the design characteristics of the experimental bipolar membrane with slight variations, such as the assumption that the membrane was 80% cation exchanger, rather than 94%, as in the experimental system. The electrical potential vs. total current density data for the experiment and the model are shown in Table 111.
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 365
0’641
Table 111. Potential Gradient vs. Total Current Density ~
~
current, mA/cm2 Eelpt,V ElimulatiOIl, v Highly Selective Membranea 0.0 0.75 0.62 36.2 0.87 0.63 72.5 0.96 0.64 109.0 1.07 0.65 0.0 2.0 5.0 10.0
Poorly Selective Membraneb 0.12 0.17 0.24 0.35
Data of Dege and Liu.
2.90
N
Data of Korosy and Ziegerson.
r1
3.30
0.11 0.11 0.12 0.13
9
X
0.48t t
B
0.40 0.32
E
0.24
X
0
0.16
0
0
0.08
” I
-
6
0
1
2.50
0.56
1 0.00 - 10.08
2.10
0.12
0.16
0.20
0.24
0.28
Conc HF IN BLR, moles/cma ( x 102) Figure 6. XHoH vs. acid concentration in right boundary layer for highly selective membrane system.
0.90
which the electrical potentials were reported and because different ions were used. The electrical potential reported by de Korosy is called “potential drop” because of the polarity of the membrane phase
I-
E’= E
0.10
0.08
0.12
0.16
0.20
0.24
0.28
Conc HF IN BLR, moles/cm3 ( x 102) Figure 5. Open-circuit potential vs. acid concentration in right boundary layer for highly selective membrane system. Base concentration in left boundary layer is 0.001 mol/cm3.
The data compare favorably on a qualitative basis, with open-circuit potentials of 0.62 and 0.75 V for model and experiment, respectively. Dege and Liu indicated that the selectivity of their system was very good when measured in terms of boundary layer contamination while that of the model indicated some leakage of fluoride ion into the right boundary layer. The model for the highly selective membrane system was used for calculations at various acid concentrations not reported in the work of Dege and Liu. The open-circuit potential vs. acid concentration plots are shown in Figure 5 for a base concentration of 1.0 M. The E o / C Afor the highly selective membrane is linear for low concentration of acid with a slope of 88 V/(mol/cm3) for concentrations above 1.9 mol/cm3. Of particular interest at CA = 1.9 M is that the position of XHoH is at the interface between the anion-exchange phase and the cation-exchange phase (see Figure 6). For extreme values of C,, the concentration profiles for these runs indicate that KOH dominates the membrane phase when CAis low and HF dominates when CA is large. Poorly Selective Membrane Model. Work by de Korosy and Zeigerson (1971) provided an example of a poorly selective bipolar membrane. This system was somewhat difficult to analyze because of the manner in
- E(AEM) - E(CEM)
(16)
where E is the total potential drop, E(AEM) is the total potential drop of equivalent anionic phase alone, and E(CEM) is the total potential drop of equivalent cationic phase alone. Modeling the system, therefore, required knowledge of the experimental open-circuit potential across the system, information that was unavailable. It was assumed, therefore, for the purposes of this comparison that E’is approximately equal to E. The poorly selective membrane system model definition has been compared to de Korosy’s results in terms of E vs. I curves, transport of salt ions, and ion content. E vs. I for de Korosy and the model are presented in Table 111. The open-circuit potential values are 0.12 V for de Korosy and 0.11 V for the model, with slopes of 22 X lo5 and 1.4 X lo6 (V s cm2)/mol for de Korosy and the model, respectively. The ion transport of the experiment and the model can be represented by the ratios of salt ions transported. The ion ratios of Na+ to C1- (model) at -53 X lo4 and -8.8 X lo4 mol/(s cm2)were 1:3.03 and 15.36, as compared to those for the experiment at 11.3 X and 4.7 X mol/(s cm2),which were 1:3.05 and 1:3.16, respectively. (The sign difference on the total currents is due to different system configurations.) These seem to be in good qualitative agreement. The experimental and model results can also be compared based upon co-ion content within the two membrane phases, anionic and cationic. de Korosy’s results indicate that, after 15 min of run time (approximately steady state), 120% of the bipolar membrane’s capacity for salt has been used. This implies that 20% excess salt has accumulated in the neighborhood of the median plane where de Korosy concludes that it swamps Donnan exclusion forces; however, because of the nature of the experiments, the exact location of the salt cannot be determined. In order to
366 Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
compare the model results to de Korosy’s for co-ion content, one must consider that the model simulation was run in an acid-base environment and not in a salt environment as in the experiment. Therefore, the model’s percent capacity will be the co-ion content of interest divided by the total counterion content. The Na+ capacity for the model bipolar membrane is 117% and is 126% for the C1- ion. These capacities compare well with the 120% reported by de Korosy. The model thus appears to be a good representation of a poorly selective bipolar membrane such as de Korosy’s, at least in terms of E / I , ionic transport, and co-ion content. Model Parameter Changes and Limitations. The system diffusion coefficients control the rate at which an ion can transport through a region in response to a concentration gradient. A series of simulations were run in which diffusion Coefficients were increased by 10% within both the anionic and the cationic phases. The majority of the effects of this change were minor; however, changing the diffusion coefficients of the counterions in the anionic phase led to a 28% drop in its open-circuit potential and a 690 increase in the open-circuit potential of the cationic phase. The subroutine which funnels results of each integration step back into the concentration and potential variables also checks newly calculated concentrations for negative values. Such negative concentrations are, of course, invalid; however, if the absolute value of the concentration is less than lo-” mol/cm3, the subroutine is set to ignore the new concentration. This procedure can cause slight errors in the concentration derivatives calculated from the new concentrations. Fortunately, the only negative concentration detected by this mechanism is that of OH- when it is very small (near mol/cm3). This is also near the minimum level of significance of the algorithm; therefore, since the appearance of this condition seems to be random, this appears to be a minor limitation. Other Simulations. One highly selective membrane simulation was run in which the cation-exchange phase fraction was increased from 0.8 to 0.9. This caused XHOH to shift to the right side of the second fuzzy interface, and, in addition, there was an increase in the open-circuit potential from 0.60 to 0.68 V. This increase is consistent with results reported by Dege and Liu. A second simulation was run in which the total membrane thickness was increased; this led to a shift of XHoHto the left of the second fuzzy interface, the replacement of hydrogen ion by potassium ion as the dominant ion in the second fuzzy right interface, and a reduction in water flow away from the center of the second fuzzy interface. The open-circuit potential drop decreased from 0.60 to 0.54 V, and in general, the counterion and co-ion concentrations became closer. In other simulations, the fixed charge density was reduced from 0.015 to 0.0135 mol/cm3, with results comparable to those produced by increasing membrane thickness. Experiments conducted by Dege and Liu show that decreasing CA should reduce the bipolar membrane ion exclusion capacity, thus confirming the observed leakage of co-ions and the reduced open-circuit potential.
Conclusions Based on the results of this study, the following observations and conclusions were developed: 1. The model behaves qualitatively and, in most cases, quantitatively in a manner very similar to the experimental system that it is supposed to represent. This is the first bipolar membrane model that can do this. It is, however, subject to the limitations inherent in its construction. That
is, it is limited to steady-state flow in one dimension perpendicular to the membrane face. In addition, since it assumes that all solutions are ideal and that there are no temperature or pressure gradients, the model is limited to situations where the water concentration is at least 0.001 mol/cm3 everywhere. 2. The highly selective membrane series shows that a given model definition can have a E , less than the theoretically predicted value and still have a wide current range where its operation is linear. However, this series shows a significant value for Wand a large transport of F- from the left boundary layer to the right boundary layer. This decreased selectivity results in co-ions infiltrating the membrane and allows the system to approach homogeneity. 3. As the model membrane becomes more and more selective, there is an associated tendency for XHoH to approach the center of the second fuzzy interface and for W to approach zero. 4. Selectivity is a characteristic of bipolar membranes which is controlled experimentally by the degree of cross-linking and the concentration of fixed ion-exchange groups. In the model, the equivalent parameters are pore fraction and fixed charge density. However, other parameters, such as region width, also have an effect on membrane selectivity. 5. As the fraction cation exchanger approaches 1.0, the selectivity of the bipolar membrane increases. It appears that the most highly selective membranes will be over 90% cation exchanger. 6. Increasing the fixed charge density increases the system selectivity, as does increasing the ratio of counterion to co-ion diffusion coefficients. 7. The width of the second fuzzy interface needs to be minimized to keep interfacial resistance low. Experimentally, this can be done by intimate bonding of the cation and anion exchangers, or if the bipolar membrane is made from a single base membrane, then the functional groups must be brought into very close contact without overlapping. Future work on the model will include its application to single ion-exchange membrane systems, in order to gain access to more experimental data for simulations. In addition, the model will be pushed to much higher current densities, so that increased polarization effects can be simulated. Attempts will also be made to simulate a perfectly selective bipolar membrane to be used as a reference standard.
Acknowledgment We thank Dr. Preston Keusch for originally suggesting this problem and for his many valuable insights, particularly on the experimental systems. This work is based on a dissertation submitted by William H. Rose, in partial fulfillment of the requirements for the Ph.D. in Bioengineering at the University of Illinois at Chicago.
Nomenclature A,,, = water equilibrium factor, defined by eq 7 a = angle of allowable approach of co-ions to counterions, defined by eq 8 b = 3.546 X IEFIRTI, dimensionless C = concentration, mol/cm3 D = diffusion coefficient, cm2/s E = electrical potential, V e = electroosmotic coefficient, cm3 of solution/mol of ion 6 = electrical permittivity = 78.54 X 10-18/(36a) C2/(ergcm) F = Faraday’s constant = 96500 C/mol G = water equilibrium factor, defined by eq 9 and 10
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Ind. Eng. Chem. Fundam. 1986, 25, 367-372
I = current, mol/(s cm2) J = ion flux, mol/(s cm2) K, = dissociation constant for water, (mol/~m~)~ M = molar volume (crystal), cm3/mol PA = pore fraction, dimensionless (PA = 1for solution phases) CP = dimensionless potential, defined by eq 2 q = Bjerrum radius of bound ions = 3.56 X cm R = universal gas constant = 8.3143 V C/(K mol) r,,, = mean distance of approach of ion pairs, cm T = temperature, 298 K u = (8b)1/2 W = solution flux, cm3/(s cm2) XHoH = location of hydrogen-hydroxide shift, cm x = distance, cm x , = distance of co-ion from edge of repulsion zone, cm z = valence, dimensionless
Subscripts 1 = cation 2 = anion
3 = hydrogen ion 4 = hydroxide ion 5 = water A = fixed charge group in membrane L i t e r a t u r e Cited Breslau, B. R.; Miller, I. F. Ind. Eng. Chem. Fundam. 1071, 70, 554. Breslau, B. R.; Miller, I . F.; Gryte, C.; Gregor, H. P. Preprint Volume, MESD Biennial Conference; AIChE: New York, 1970; p 363. Dege, G. J.; Liu, K.J. U S . Patent 4024043, 1977. de Korosy, F.; Zelgerson, E. Isr. J. Chem. 1071, 9 , 483. Frllette, V. J. J. Phys. Chem. 1056, 60, 435. Keusch, P., RAI Research Corp., private communication, 1980. Nagasubramanlan, K.; Chlanda, F. P.; Llu, K . J . J. Membr. Sci. 1077, 2 , 109. Patel, R. D.; Lang, K. C.; Miller, I.F. I n d . Eng. Chem. Fundam. 1077, 76, 340. Vervey, E. J. W.; Overbeek, J. Th. G. Theory of the Stabilw of Lyophobic Colloids;Elsevler: New York, 1948; p 22.
Received for review August 1, 1984 Revised manuscript received July 29, 1985 Accepted October 24, 1985
A Study of the Separation Efficiency of the Batch-Type Thermal Diffusion Column with an Impermeable Barrier Inserted between the Plates Shau-We1 Tsal and Ho-Mlng Yeh’ Chemical Engineering Department, National Cheng Kung Universiv, Talnan, Taiwan, Republic of China
Instailation of an impermeable barrier between the plates substantially increases the separation efficiency by reducing the remixing effect. Theoretical considerations show that when the barrier is installed at the best position, maximum separation, maximum concentration of top product, or minimum concentration of bottom product may be obtained. Considerable improvement is obtained when the column is operated at the conditions leading to the best performance.
Introduction
The thermogravitational thermal diffusion column, introduced by Clusius and Dickel (1938), can be used to separate the mixtures which are difficult to separate by means of conventional methods such as adsorption, distillation, etc. The first complete presentation of the theory of the Clusius-Dickel column was that of Furry et al. (1939). A more detailed study of the mechanism of the separation shows that convective currents have two conflicting effects: a desirable “cascading” effect, owing to the countercurrent action of the thermally driven flow in the column, and an undesirable “remixing” effect, owing to diffusion along the column axis and across the column. At steady state, a dynamic equilibrium is established between these at which no further separation takes place. Thus, it is evident that any improvement in the equilibrium separation must be associated with either a suppression of the remixing effect and/or an enhancement of the cascading effect. Based on this concept, some improved columns have been developed in the literature, such as inclined columns (Powers and Wilke, 1957; Chueh and Yeh, 1967),wired columns (Washall and Molpolder, 1962; Yeh and Ward, 1971), inclined moving-wall columns (Ramser, 1957; Yeh and Tsai, 1972), rotary columns 0196-43131861 l025-Q367$01.5OlO
(Sullivan et al., 1957; Yeh and Cheng, 1973; Yeh and Tsai, 1981b, 1982), packed columns (Lorenz and Emery, 1959; Yeh and Chu, 1974),and rotated wired columns (Yeh and Ho, 1975; Yeh and Tsai, 1981a). Installation of an impermeable barrier between the plates may decrease the strength of natural convection, and therefore, both cascading and remixing effects are reduced. It is believed that properly reducing the convective flow may effectively suppress the remixing effect while still preserving the cascading effect and thereby lead to improved separation, such as in inclined, wired, and packed columns. The barrier should resist flow of liquid between the channels but must not permit molecular diffusion under the influence of the temperature difference between vertical walls. The purpose of this work is to develop the separation theory and investigate the separation efficiency for such an improved thermal diffusion column. Column Theory 1. T h e Open Column. Consider a flat-plate thermo-
gravitational thermal diffusion column filled with a binary mixture. The distance between the plates is WA + wB. At steady state, the horizontal mass flux of component 1, Jx, is related to the velocity, V,, by the differential mass balance equation 0 1986 American Chemical Society
