J Polym Res (2012) 19:10 DOI 10.1007/s10965-012-0010-7
ORIGINAL PAPER
Dual-mode transport of inorganic acids through polybenzimidazole (PBI) membrane Morteza Sadeghi & Somaieh Khatti & Mohammad R. Shafiei & Elham Ameri
Received: 13 February 2012 / Accepted: 12 October 2012 / Published online: 6 November 2012 # Springer Science+Business Media Dordrecht 2012
Abstract The diffusion and permeation of sulfuric, perchloric, and phosphoric acids through the polybenzimidazole (PBI) membrane were evaluated by using the timelag method. The results showed that the permeation of these acids in the polymeric film is due to normal Fickian diffusion. Increasing the acidity and concentration of each inorganic acid upstream increased the diffusion coefficients. Permeation results confirmed that the dual-mobility model was applicable. The dual-mobility model constants were found by using a least square program via fitting experimental data to dual-mobility equation. The results have indicated that the partial immobilization of Langmuir sorption coefficient, F, decreases with molecular size of acids. Besides, the diffusion coefficient of acid molecules in Henry’s law, DD, increases with increasing acidity of the acids studied. Comparison of permeability coefficients of acids in PBI membrane indicates that permeability increases with acidity. Therefore, diffusion is considered the dominant factor in this transport mechanism. Keywords Dual-mobility . Glassy polymer . Permeation . Polybenzimidazole M. Sadeghi (*) : S. Khatti Department of Chemical Engineering, Isfahan University of Technology, Isfahan 84156-8311, Iran e-mail:
[email protected] M. R. Shafiei Department of Chemical Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran E. Ameri Department of Chemical Engineering, Shahreza branch, Islamic Azad University, Shahreza, Iran
Introduction The dual-mode sorption and mobility models have been widely used to simulate sorption and diffusion behavior of gases in glassy polymer membranes. In the original dualmode mobility model, the diffusivities of both Henry’s law and Langmuir species were assumed to be constant. Vieth and Sladek [1] assumed the immobilization of penetrants in the Langmuir mode to describe the transport behavior of penetrants in glassy polymers in their proposed dual-mode model reported in 1965. In 1969, Paul developed a model based on total immobilization [2]. In 1976, Paul and Koros developed a model that assumes only partial immobilization of sorbed penetrant in the Langmuir mode [3]. The partial immobilization model is developed assuming independent dual-diffusion of the Henry and Langmuir modes. Partial immobilization means that the Langmuir mode species can mobilize partly in glassy polymeric membranes, in contrast to the basic dual-mode model in which the Langmuir mode species is considered not to be mobile at all. If the gas sorption isotherm is represented via the dualmode sorption model, the total penetrant concentration C is given by 0
C ¼ CD þ CH ¼ KD :p þ
CH :b:p 1 þ b:p
ð1Þ
where p is the upstream gas concentration, kD is the 0 Henry’s solubility coefficient, CH is the Langmuir saturation constant, and b is the Langmuir affinity constant. Thus the basic equation describing this model, known as the dual-mobility or partial immobilization model, is " # ¯CD ¯CH F:K ¯CD J ¼ DD DH ¼ DD 1 þ 2 ¯x ¯x ¯x ð1 þ b:pÞ
ð2Þ
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J CD, CH
J Polym Res (2012) 19:10
Diffusive flux Concentrations corresponding to Henry’s law and Langmuir population Diffusion coefficients of molecules in Henry’s law and Langmuir mode, respectively
DD, DH
0
K ¼ CH :b=kD F ¼ DH =DD 0
kD ; C H ; b p
(partial immobilization of Langmuir sorption) Parameters in the dual-mode sorption model Gas pressure in the feed phase
In this approach, the diffusion coefficients, DD and DH, are assumed to be independent of concentration. From Eqs. (1) and (2) we can deduce " J ¼ DD
FK # 1 þ ð1þbpÞ 2 ¯C K 1 þ ð1þbpÞ2 ¯x
ð3Þ
This treatment of penetrant transport in dual-mode model leads one to define a phenomenological diffusion coefficient Deff as the effective diffusivity, which express the flux in terms of the total concentration, C: J ¼ Deff
¯C ¯x
ð4Þ
It can be seen that the concentration-dependent diffusion coefficient, D eff , is related to D D and D H in Eqs. (3) and (4). Thus, it can be described by the following equation: " FK # 1 þ ð1þbpÞ 2 Deff ¼ DD ð5Þ K 1 þ ð1þbpÞ 2 The parameter F in Eq. (5) represents the physical significance of the mobile fraction of the Langmuir mode species. Assuming F00 reduces Eq. (5) to the immobilization model which only represents the diffusion of the Henry mode species. In such cases, P0kD.DD corresponds to the permeability coefficient of the rubbery polymeric membranes. It is the so-called solutiondiffusion concept of permeability. The concentration dependency of the permeability coefficients in the dualmobility model can be expressed in the case of partly mobile Langmuir sorption as follows: FK P ¼ DD :kD 1 þ ð6Þ 1 þ bp Equation (6) is known as the partial immobilization model, and this form characterizes the glassy polymeric membranes with dual-mode sorption [4–12]. In the partial
immobilization model, the permeability coefficients of gases decrease gradually when there is an increase in their pressure. This increase in the permeability coefficients at lower pressures is caused by the contribution of the diffusion due to the Langmuir mode species. The permeability coefficient is enhanced notably by the increase in the value of the partial immobilization of Langmuir sorption coefficient, F, [3]. In the present study, we try to model acid transport through polybenzimidazole (PBI) glassy polymer. For this reason, we adopt the dual-mode mobility model for showing transport behavior of acids through the polymer. In our simultaneous study [13], we have shown that the dual-mode sorption isotherm model can be applied to evaluate sorption of sulfuric, perchloric, and phosphoric acids through a PBI glassy membrane. We assumed that the acid concentration (mole fraction) can be used instead of gas pressure, p, in the dual-mode sorption equation. In addition, we replaced Henry’s solubility coefficient by a general form of the Nernst distribution coefficient. The partial immobilization model for modeling the acid transport through the polymer can be written as follows: FK P ¼ DD :kD 1 þ ð7Þ 1 þ bc where c is the mole fraction of acid in the liquid phase. We have replaced pressure, p, with the mole fraction, c, on the assumption that the Nernst distribution law and Langmuir mode in our dual-mode sorption model must hold. Furthermore the mole fraction of acids must be replaced by gas pressure in the partial immobilization model, and the acid sorption behavior must also follow the dual-mode sorption. We chose sulfuric, perchloric, and phosphoric acids as the liquid penetrant for testing the model for acid transport through glassy polymer membrane. PBI membrane was chosen as the glassy polymer because of its good physical properties and great potential for use as an ion exchange system.
Theory According to Fick’s first law, the flux can be represented as F ¼ DðC1 C2 Þ=L
L c1 and c2
ð8Þ
Membrane thickness Surface concentration of penetrant on either side of the membrane
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The concentration of solution can be substituted via C1 ¼ Cf :kD
ð9Þ
C2 ¼ Cr :kD
ð10Þ
Qt (cc STP)
J Polym Res (2012) 19:10
P Transient state Steady State
Cr Cf kD
Receiving cell acid concentration (mole fraction) Feed cell acid concentration (mole fraction) Partition coefficient
Time (s)
Tg
Fig. 1 Typical result of a transient permeation experiment. Qt the volume of the permeant transported across the membrane
Thus F ¼ DkD ðCf Cr Þ=L
ð11Þ
Experimental
or F ¼ PðCf Cr Þ=L
ð12Þ
from the definition of flux F¼
dQ Vdc ¼ Adt Adt
ð13Þ
Because of the finite volume of the feed and receiving cells, a correction was made for the change of concentrations with permeation. Combining Eqs. (12) and (13) and integrating the result with the boundary condition Cr 00 at t≤tg (lag time), the following equation results: lnð1 Cr =C Þ ¼ ðt tg Þ
tg V A P L Cr dQ C
PA VL
ð14Þ
Lag time (s) Volume of received cell (cm3) Effective membrane area (cm2) Permeability coefficient (cm2/s) Membrane thickness (cm) Receiving cell acid concentration (mole fraction) Amount of penetrant transported during the time interval dt Initial feed cell acid concentration (mole fraction)
By plotting the term in the left-hand side of Eq. (14) versus time, the permeability coefficient is deduced from the slope of the line and the lag time, tg, from intercept [14]. The lag time is related to the diffusion coefficient through the equation
Materials PBI polymer (inherent viscosity01, measured in dimethylsulfoxide) was provided by Hoechst Celanese. Poly(2,2-(m-phenylene)-5,5-bibenzimidazole) or polybenzimidazole is a thermally stable polymer which is typically condensed from aromatic bis-o-diamines and dicarboxylates. PBI is a thermoplastic glassy polymer with a Tg of 435 °C. As shown in Fig. 2, the repeat unit of PBI has two basic sites (imidazole groups). Dimethylacetamide (DMAc) was purchased from Merck. Sulfuric, perchloric, and phosphoric acids required for evaluation of acid transport in PBI membrane were also provided from Merck. Their acidity constants are listed in Table 1. Membrane preparation The 7 wt% PBI solution in DMAc was prepared by dissolving PBI powder in DMAc at 200 psig and 100 °C in a high pressure reactor for 2 h. Then, the solution was agitated for 24 h at 40 °C and 20 psig to completely dissolve the PBI. The prepared solution was filtered through a 20-μm ceramic filter. The PBI solution was concentrated to 20 wt% in a rotary evaporator. The PBI films were then cast on a clean glass sheet by doctor blading and heated at 80 °C for 2 h. The prepared film was kept at 100 °C for 24 h for complete removal of the solvent. After heat treatment, the PBI membranes were removed from the glass surface by immersing them in deionized water. The obtained films were used for H N
N
N
N
*
L2 tg ¼ 6D
ð15Þ
The conventional form of the lag time curve is shown in Fig. 1.
H
n
Fig. 2 Repeat unit of poly(2,2-(m-phenylene)-5,5-bibenzimidazole) (PBI)
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J Polym Res (2012) 19:10
Table 1 Acidity constant and decomposition reactions of sulfuric, perchloric, and phosphoric acids Acid
Decomposition reaction
Acidity constant
Sulfuric acid
H2 SO4 1HSO4 þ H þ HSO4 2SO4 2 þ H þ H3 PO4 1H2 PO4 þ H þ H2 PO4 2HPO4 2 þ H þ HPO4 2 3PO4 3 þ H þ HC1O4 kaH þ þ C1O4
Ka1 ≫1 Ka2 00.1023 Ka1 ¼ 6:92 103 Ka2 ¼ 1 107 Ka3 ¼ 4:78 1013 Ka 039.8
Phosphoric acid
Perchloric acid
investigating the transport of acids. The thicknesses of the prepared membranes were 40 μm. Equipment The permeability of PBI films toward H2SO4, HClO4, and H3PO4 solutions at room temperature were studied by the lag-time method. Figure 3 shows a schematic diagram of the apparatus used. Acid solution was placed in one reservoir and water in the other reservoir; a water-conditioned film separated the two reservoirs. The volume of acid solution feed and water in both reservoirs was 600 mL. The agitation with stirring bars is enough to reduce boundary layer effects [15]. The flow rate of acid transported through the PBI membrane was calculated by recording the variation of acid concentration in the receiver reservoir.
Results and discussion The room temperature permeabilities of PBI films were studied at five different concentrations of sulfuric and perchloric acids and four different concentrations of phosphoric acid solutions. The required times to reach the steady state permeation of acids through PBI films were determined by the lag time, tg, and the concentration-dependent diffusion coefficients or D were measured by using these data. The value of permeability coefficient, P, was also
measured from the slope of the steady state transport of acids through PBI films. The results shown in Figs. 4, 5, and 6 indicate that acid permeation in PBI films is normal Fickian diffusion. According to the lag-time model for the case of dual-mode sorption, the permeability coefficient of the examined inorganic acids through PBI film decreases with increasing acid concentration upstream. The concentration dependency of the permeability coefficient is plotted in Figs. 7, 8, and 9. The decrease in permeability coefficient with increasing acid concentration is in complete agreement with Eq. 7. It is in accordance with the dual-mobility model for transport of gases through glassy polymers. According to the partial immobilization model, some penetrant could be trapped in the polymer network in holes or binding to active sites. In this case two basic groups in PBI’s repeat unit interact with some of the inorganic acid molecules to form ionic complexes. It is suggested that these portions of acid molecules that are partially mobile can be represented by the Langmuir mode. The rest of the acid molecules that are not bound to the polymer but are dissolved in it can be represented by the Henry’s law model. The diffusion coefficient constant of the Henry’s law portion, DD, and partial mobility of Langmuir sorption coefficient, F, of sulfuric, perchloric, and phosphoric acids were calculated by fitting Eq. 7 to experimental permeation data and dual-mode sorption constant [13]. The results are presented in Tables 2 and 3. Our results show that partial mobility of bound H3PO4, F, is very small, and it is an order of magnitude smaller than F of the other two acids. The partial mobilities of sulfuric and perchloric acids are about 1/5 times lower than the values reported for the mobilities of gases and water in glassy polymers [16–18]. This is due to the strong interaction between the polymer and the inorganic acid molecules in this system. The sequence of the partial
0.11 x= 0.000293
0.1
x= 0.000326
0.09
x= 0.00065
0.08
x= 0.00105
Feed
Membrane
Downstream
-Ln (1-Cr/Co)
x= 0.002621
0.07 0.06 0.05 0.04 0.03 0.02 0.01
Magnet
0 0
Stirrer
Fig. 3 Apparatus used for transport of acid through PBI membranes
2000
4000
6000
8000
10000
12000
14000
time (min)
Fig. 4 Transport of sulfuric acid through PBI membrane for different mole fractions of acid in feed
J Polym Res (2012) 19:10
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0.03
65
0.025
Permeability ×10-9 (cm2/s)
x=0.000285 x= 0.00043 x= 0.00108 x= 0.00478 x= 0.01
-Ln (1-Cr/Co )
0.02
0.015
0.01
55
45
35
25
15 0
0.005
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
Sulfuric acid mole fraction 0 0
5000
10000
15000 time (min)
20000
25000
30000
Fig. 5 Transport of perchloric acid through PBI membrane for different mole fractions of acid in feed
mobility coefficients of the inorganic acids, F, in PBI membrane is H3 PO4 The order of the partial mobility coefficients decreases with increasing molecular size of the inorganic acids. The partial mobility coefficient represents the ability of Langmuir moieties to be mobile and transport across the membrane. The molecular sizes of these three inorganic acids were calculated by HyperChem software. The results are listed in Table 4. This means the mobility of large (size) acid molecules that form complexes with basic groups in PBI repeat units is much lower than the smaller ones. By increasing the size of the acid molecules, more space is
needed for them to move; thus, the mobilization of Langmuir sorption decreased when the molecular size of acids was increased. It means that the inorganic acid with larger molecular size has lower motion in the polymer matrix. Using the results of F and DD, we can calculate the effective diffusion coefficient, Deff, from Eq. (5). In this equation acid concentration (mole fraction) is used instead of gas pressure. The variations of Deff versus acid concentrations for the three examined inorganic acids are plotted in Fig. 10. The obtained curve for Deff has an S shape that conforms with literature precedent [19–21]. The S shape curve of Deff versus acid concentration shows the critical concentration region that affects the diffusivity coefficient and diffusion mechanism in the membrane. The results plotted in Figs. 7, 8, and 9 indicate that the permeability coefficient obtained for sulfuric acid decreased from 56.6×10−9 cm2/s at 0.000293 mol fraction to 23.2× 3
(a)
1
-ln (1-Cr/Co)×10-2
-ln (1-Cr/Co)×10-2
1.2
0.8 0.6 0.4 0.2
(b)
2.5 2 1.5 1 0.5 0
0 0
20000 40000 time (min)
0
60000
0.25
(c)
0.2
-ln (1-Cr/Co)×10-2
-ln (1-Cr/Co)×10-2
Fig. 6 Transport of phosphoric acid through PBI membrane for a 0.000295, b 0.000743, c 0.005419, d 0.01395 mol fraction of acid in feed
Fig. 7 Permeation variation of sulfuric acid versus acid mole fraction due to dual-mobility model. The data are from experiments and the curve is plotted according to the dual-mobility model
0.15 0.1 0.05
0.16
20000
40000 time (min)
60000
80000
(d)
0.12 0.08 0.04 0
0 0
5000 10000 time (min)
15000
0
2000
4000 time (min)
6000
8000
Page 6 of 8
J Polym Res (2012) 19:10
Permeability ×10-9 (cm2/s)
14 12
Table 2 Dual-mode parameters calculated for acid sorption in PBI membrane [13]
10
Acid
KD
b
CH′
Sulfuric acid Perchloric acid Phosphoric acid
6.88 3.98 8.228
3,450 2,503 1,490
1.251 2 1.97
8 6 4 2 0 0
0.002
0.004
0.006
0.008
0.01
0.012
Perchloric acid mole fraction
Fig. 8 Permeation variation of perchloric acid versus acid mole fraction due to dual mobility model. The data are from experiments and the curve is plotted according to the dual-mobility model
10−9 cm2/s at 0.00262 mol fraction. That of perchloric acid decreased from 11×10−9 cm2/s at 0.000285 mol fraction to 1.79×10−9 cm2/s at 0.01 mol fraction. Finally, phosphoric acid’s permeability coefficient decreased from 1.45 × 10−9 cm2/s at 0000295 mol fraction to 0.717×10−9 cm2/s at 0.0138 mol fraction of acid in feed. By using Eq. 7 and the parameters reported in Tables 2 and 3, we calculated the permeability coefficient of sulfuric, perchloric, and phosphoric acids in 0.00262 mol fraction of acid in feed. The permeability coefficient of sulfuric, perchloric, and phosphoric acids is 18.3 × 10 −9 , 2.75 × 10 −9 , and 0.885 × 10−9 cm2/s, respectively. The decline in permeability of sulfuric, perchloric, and phosphoric acids from lower concentrations of each acid to the 0.00262 mol fraction of acid in feed is approximately 67.6, 75, and 39 %, respectively. The obtained permeability coefficient of sulfuric acid is one order of magnitude greater than that of perchloric and the perchloric acid permeability coefficient is one order of magnitude greater than that of phosphoric acid. Similar
Permeability ×10-9 (cm2/s)
1.6
1.4
1.2
results for Deff are observed in Fig. 10, which shows that effective diffusion coefficients of sulfuric, perchloric, and phosphoric acids at 0.01 mol fraction are 100×10−11, 8× 10−11, and 3×10−11 cm2/s, respectively. These results indicate that despite the reverse trends that are obtained for Henry’s coefficients, KD, which are reported in Table 2 for each acid, the permeability coefficient is affected by the diffusion coefficient. Moreover, the results show that with an increase in the acidity, permeability will increase. The reason may lie in the fact that more interaction between the strong inorganic acids and the PBI film leads to easier motion of the penetrant in the polymer medium and, therefore, more permeation of the acid through the membrane. It can be postulated from these results that the diffusion mechanism is the dominant one in the permeation of acid through the PBI membrane. This phenomenon is obvious in the transport of gases through glassy membranes. On the other hand, the results obtained for effective diffusion coefficient of these inorganic acids indicate that the flat region of the S shape curve for Deff began at 0.01, 0.017, and 0.02 mol fractions of sulfuric, perchloric, and phosphoric acid, respectively. These results indicate more interaction between the stronger acids and the membrane. The more acidic acid can interact with the polymer more and the basic sites would be saturated at lower concentrations of acid in the polymer and so the mobility of the acid molecules in Henry’s mode and partial mobility of the acid molecules in Langmuir’s mode would occur rapidly and so the diffusion coefficient increased. According to the dualmode mobility model, the portion of molecules which are sorbed in Henry’s mode is quite mobile and the Langmuir’s one is partially mobile. Owing to the presence of active sites in Langmuir mode, molecules are first sorbed in the polymer in this mode. Then the sorption occurred in Henry’s mode.
1
0.8
0.6 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Phosphoric acid mole fraction
Fig. 9 Permeation variation of phosphoric acid versus acid mole fraction due to dual mobility model. The data are from experiments and the curve is plotted according to the dual-mobility model
Table 3 Diffusion coefficients of molecules in Henry’s law mode and partial immobilization of Langmuir sorption for acids calculated from dual-mode mobility model 2 DD cmS Acid F Sulfuric acid Perchloric acid Phosphoric acid
0.0126 0.0130 0.0049
1.49×10−9 2.19×10−10 7.93×10−11
J Polym Res (2012) 19:10
Page 7 of 8 1.E+05
Acid
Molecular volume
Perchloric acid Sulfuric acid Phosphoric acid
238.54 250.53 269.17
Sulfuric acid Perchloric acid Phosphoric acid
lag time (min)
Table 4 Molecular size of acids used in this study calculated by HyperChem software
Therefore, the acid molecules after sorption would transport through the membrane and in the case of the more acidic ones the mobility of the acid molecules occurred rapidly. The effects of acid concentration on lag time are plotted in Fig. 11. The lag times decrease with increasing acid concentration upstream and decrease with increasing acidity of the three acids. Paul and Koros [20] observed similar results and postulated that lag time predicted from the immobilization model yields a function dependent on the upstream concentrations and the sorption constants. In the present study, the decrease of the lag time with increasing upstream concentration of each acid and the variation of lag time for these three different acids confirm the dependency of lag time on the upstream concentrations and the sorption constants. The dependency of lag time on sorption constants can be explained by the variation in lag time with acid type. This variation occurs owing to different behavior of each acid in the hole filling and the quality of its bonding strength in active sites and the normal solubility of the permeant in glassy polymers. The decrease in lag time with acidity confirms the higher diffusivity of stronger acids in the polymer. The variations of diffusion coefficient of each acid through the PBI membrane calculated from lag-time data are plotted versus upstream acid concentration in Fig. 12. In the concentration range shown, the diffusion coefficient increases linearly with increasing upstream concentration for sulfuric and phosphoric acids. However, in the case of perchloric acid, the linearity of the data is weak and there is a kink at mole fraction 0.00108. Thus, the diffusion
1.E+04
1.E+03
1.E+02 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
acid concentration (mole fraction)
Fig. 11 Variation of lag time of sulfuric, perchloric, and phosphoric acids versus acid concentration
coefficients of sulfuric, perchloric, and phosphoric acids in 0.0026 mol fraction are 25 × 10 −11 , 9 × 10 −11 , and 1 × 10−11 cm2/s, respectively. Moreover, evaluation of the gathered data shows that the diffusion coefficient of sulfuric acid is two times greater than that of perchloric acid. The diffusion coefficient of perchloric acid is greater than that of phosphoric acid by the same order of magnitude. Comparing these results with the data reported for the diffusion coefficient of acids reveals that, regardless of the molecular size of the acids, the diffusion coefficients of acids increase when their acidity increases. This is in complete agreement with the results reported in Table 3 for the diffusion coefficient constant of Henry’s law, DD, calculated from Eq. (7). As mentioned before, by increasing the acidity the molecules can be sorbed in the polymer more. And by increasing the fraction of small acid molecules in the polymer, the space created between polymer chains increases, and more suitable and available spaces needed for transportation of acid molecules in a polymer matrix increase in number and this lead to higher diffusion coefficients of the more acidic inorganic acids in PBI membrane.
1.E-09
(a)
3E-11
2
Effective Diffusivity (cm /s)
4E-11 3.5E-11
1.E-09
2.5E-11
1.E-09
2E-11 1.5E-11
1.6E-10
1E-11
8.E-10
(b)
1.4E-10
5E-12
1.2E-10
0 0
0.0001
0.0002 0.0003
0.0004 0.0005
6.E-10
0.0006 0.0007
0.0008
1E-10 8E-11 6E-11 4E-11
4.E-10
2E-11 0
2.E-10
0
0.005
0.01
0.015
0.02
0.E+00 0
0.005
0.01
0.015
0.02
acid concentration (mole fraction)
Fig. 10 Calculated Deff for sulfuric ( ), perchloric ( ), and ) acids by using dual-mode sorption and transport phosphoric ( constants. a Deff of acids at the start. b Expanded Deff curve of perchloric and sulfuric acids
Fig. 12 Diffusion coefficient of sulfuric, perchloric, and phosphoric acids versus acid concentration upstream
Page 8 of 8
Conclusion The transportation of three inorganic acids through the glassy polybenzimidazole (PBI) membrane was investigated. The results confirm that the partial immobilization model accounts for the transportation of these acids in this membrane which has basic groups in its repeat units. This leads to trapping of the inorganic acid molecules in the polymer media. The diffusion coefficient constant of Henry’s law, DD, and partial mobility of Langmuir sorption coefficient, F, were calculated by using the dual-mobility model equation and dual-mode sorption constants. The results show that the partial mobility of Langmuir sorption species decreases with increasing molecular size of the inorganic acids. Increasing the acidity led to increasing calculated permeability and diffusion constants. Sulfuric acid, the strongest acid among those examined, shows the greatest permeability constant. Furthermore, the results indicate that the diffusion mode is the dominant mechanism in acid permeation through PBI membranes.
References 1. Vieth WR, Vieth WR, Sladek KJ (1965) J Colloid Sci 20:1014– 1033
J Polym Res (2012) 19:10 2. Paul DR (1969) J Polym Sci Part A-2 7:1811–1818 3. Paul DR, Koros WJ (1976) J Polym Sci Polym Phys 14:675–685 4. Koros WJ, Paul DR, Rocha AA (1976) J Polym Sci Polym Phys 14:687–702 5. Koros WJ, Paul DR, Fujii M, Hopfenberg HB, Stannett VJ (1977) Appl Polym Sci 21:2899–2904 6. Stannett VJ (1978) Membr Sci 3:97–115 7. Koros WJ, Paul DR (1978) J Polym Sci Polym Phys 16:2171– 2187 8. Felder RM, Patton CJ, Koros WJ (1981) J Polym Sci Polym Phys 19:1895–1909 9. Chern RT, Koros WJ, Sanders ES, Yui B (1983) J Membr Sci 15:157–169 10. Koros WJ, Chern RT, Stannett VT, Hopfenberg HB (1981) J Polym Sci Polym Phys 19:1513–1530 11. Chern RT, Koros WJ, Hopfenberg HB, Stannett VT (1983) J Polym Sci Polym Phys 21:753 12. Sanders FS, Koros WJ, Hopfenberg HB, Stannett VT (1984) J Membr Sci 18:53–74 13. Sadeghi M, Moaddel H, Khati S, Ghalei B (2010) Dual mode sorption of inorganic acids in polybenzimidazole membrane. J Macr Sci Polym Phys 49:1128–1135 14. Barrer RM (1939) Trans Farad Soc 35:628–635 15. Naylor T (1989) Compr Polym Sci 2:643 16. Tsujita Y (2003) Prog Polym Sci 28:1377–1401 17. Hayashi Y, Sugiyama S, Kawanishi T, Shimizu N (1999) J Membr Sci 156:11–16 18. Thundyil MJ, Jois YH, Koros WJ (1999) J Membr Sci 152:29–40 19. Ranade GR, Chandler R, Plank CA, Laukhuf WLS (1985) Polym Eng Sci 25:164–169 20. Wolf RV (1991) Diffusion in and through polymers: principles and applications. Hanser Gardner, Munich 21. Koros WJ, Chan AH, Paul DR (1977) J Membr Sci 2:165–190
