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Kinetics of sorption and permeation of water in glassy polyimide
Journal of Membrane Science 156 (1999) 11±16
Kinetics of sorption and permeation of water in glassy polyimide Yoshishige Hayashi, Shigehiro Sugiyama*, Takuya Kawanishi, Nobuaki Shimizu Department of Chemistry and Chemical Engineering, Faculty of Engineering Kanazawa University, 2-40-20 Kodatsuno, Kanazawa 920-8667, Japan Received 15 August 1998; accepted 24 August 1998
Abstract A vapor permeation experiment for water±ethanol mixtures was carried out using asymmetric Ube polyimide hollow-®ber membranes, which exhibit high selective permeability for water vapor, under the conditions of T413 K, upstream gas pressure Ph1.51052.95105 Pa and downstream gas pressure Pl400 Pa. To represent gas separation properties of the Ube polyimide membrane with a high transition temperature (570 K), the contribution of Henry's law part and Langmuir part modes on the diffusion through the membrane is studied on the basis of the dual-mode transport models. The results show that Henry's law penetrant controls the diffusion in the membrane. For the separation of water±ethanol mixtures by permeation through Ube polyimide membranes, the water trapped in microcavities can be assumed to be totally immobilized under the operating conditions applied here. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Dual mobility; Glassy polymer; Polyimide; Vapor permeation; Membrane
1. Introduction
2. Theory
The penetration of gases through nonporous polymeric membranes depends on whether the polymer is in its rubbery or glassy state. Ube polyimide membranes have a high glass transition temperature (570 K), and for the general case of gas separation polyimide membranes are used in a glassy state. In this paper, the experiment for the vapor permeation separation of water±ethanol mixtures was carried out using a module comprised of asymmetric polyimide hollow ®ber membranes with high selective permeability for water. On the basis of the data obtained, the gas separation behavior is analyzed in terms of the dual sorption and dual mobility models.
2.1. Dual sorption model
*Corresponding author. Tel.: +81-76-2344806; fax: +81-762344811; e-mail: [email protected]
On the basis of the dual sorption theory, the following equations for the isotherm well-describe the transport of gases through membranes [1]. @CD @ 2 CD Deff @t @x2
(1)
and Deff
DD
K= 1 bPh 2 1
K CH0 b=kD ;
;
(2) (3)
where Deff is the effective diffusion coef®cient of the penetrant, DD is the diffusion coef®cient of the pene-
0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00273-7
12
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
trant in the Henry law environment, K is a useful collection of the sorption parameters, and kD, CH0 and b are generally referred to as the Henry law constant, the Langmuir capacity constant and the Langmuir af®nity constant, respectively. The boundary conditions are as follows: CD 0 CD CD0 kD Ph CD 0
at t 0; at x 0; at x ;
(4)
where Ph is the upstream pressure and is the membrane thickness. The solution of Eq. (1) is given by [2] 1 CD x 2X 1 nx Deff n2 2 t sin exp ÿ 1ÿ ÿ : 1 n 2 CD0 (5) In this case, permeability Perm is given by [1] Perm kD DD :
(6)
t which has At any time t, the total volume of gas N penetrated the membrane is given by the following equation: 1 t N Deff t 1 2X ÿ1n ÿ ÿ Af CD0 2 6 2 1 n2 Deff n2 2 t exp ÿ ; (7) 2
in the Langmuir environment. The form of Eq. (8) is identical to that of Eq. (1) and the boundary conditions are also the same as Eq. (4). The permeability is given by [3] FK Perm kD DD 1 : (11) 1 bPh t is as The total volume of the diffusing penetrant N follows: !" t K Deff t 1 N 1 ÿ 2 2 Af CD0 6 1 bPh # n 1 2 X ÿ1 ÿDeff n2 2 t exp ÿ 2 : (12) 1 n2 2 3. Experimental Figs. 1 and 2 show the schematic diagram of experimental apparatus and permeation module, respectively. The asymmetric polyimide membrane with high selective permeability for water vapor, developed by Ube industry Co. Ltd., was used. Ube polyimide membranes have a high glass transition temperature of around 570 K and its permeation rate constant for water is 1.2910ÿ8 m3-STP/(m2 s Pa). The outer and inner diameters of hollow ®ber membrane were approximately 490 and 300 mm, respectively. The
where Af is the membrane surface area. 2.2. Partial immobilization model Penetrant molecules in the two distinct molecular environments (Henry's law and Langmuir) have different inherent mobilities. The so-called ``dual mobility'' or ``partial immobilization'' model accounts for this fact as follows [3]: @CD @ 2 CD Deff ; @t @x2
(8)
and Deff DD
1 FK= 1 bPh 2 1 K= 1 bPh 2
F DH =DD ;
;
(9) (10)
where DH is the diffusion coef®cient of the penetrant
Fig. 1. Schematic diagram of experimental appartus.
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
13
Fig. 2. Details of module.
module was placed in a constant temperature air bath to keep the temperature at 413 K. The used gas was the steam with ethanol vapor (0±4 mol%) and the separation time was 60 min. The upstream pressure was in the range 1.5105± 2.95105 Pa, and a vacuum pump on the downstream side operated at 400 Pa to provide the necessary driving force. The permeate and nonpermeate vapors were condensed by a heat exchanger and recovered as a liquid. The compositions of the feed, permeate and nonpermeate were determined by gas chromatography. 4. Results and discussion A typical example for the experimental results is plotted in Fig. 3 as a parameter of ethanol concentrations in the feed. The molar ¯ux of penetrated water is independent of the upstream partial pressure of ethanol and it is clear the ethanol does not cause signi®cant differences in the molar ¯ux of penetrated water under the operating conditions applied here. Table 1 shows the model parameters estimated by the simplex method using Eq. (7) or Eq. (12) for the cases of complete immobilization of the Langmuir species and partially immobilization, which are the mean of estimated values from 60 data sets. In the calculation the outer skin thickness, which was esti-
Fig. 3. Relationship between molar flux of penetrated water and partial pressure of water in the feed for no. 1 fiber. Ethanol concentration in the feed (mol%): (~) 0, (&) 2, and (*) 4.
mated by applying image processing technique to the SEM photographs of the skin region with high magni®cation (Fig. 4), was used as a characteristic dimension of . It was 0.092 and 0.094 mm for no. 1 and no. 2 ®bers, respectively. In Fig. 3, the results calculated by Eqs. (A.1) and (A.3) using the model parameters listed in Table 1 are also shown. The calculated results agree well with those obtained experimentally and it can be considered that the estimated model parameters represent characteristics of Ube polyimide membranes adequately.
14
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
Table 1 Sorption and transport parameters for water in Ube polyimide membranes at 413 K Parameters
Partial immobilization
Dual sorption
FK (dimensionless) F (dimensionless) b (Paÿ1) K (dimensionless) kD (m3-vapor (STP)/(m3-polymer Pa)) CH0 (m3-vapor (STP)/(m3-polymer)) DD (m2/s) Deff (m2/s)
3.01 0.05 1.8910ÿ4 60.1 4.0010ÿ4 127.2 2.8810ÿ12 2.70±2.8310ÿ12
± ± 1.8910ÿ4 60.4 4.0010ÿ4 127.8 2.9710ÿ12 2.78±2.9210ÿ12
The results indicate the accuracy of the estimation procedure1. Water vapor is highly condensable, and the values of parameters show its high af®nity for the Langmuir sorption. Ranade et al. [4] have reported that the af®nity constant for water in polyacrylonitrile (PAN) is 2.4110ÿ3 Paÿ1 at 308 K and 5.9410ÿ4 Paÿ1 at 333 K. The af®nity constant decreases with an increase in the temperature. In general, the value of the af®nity constant of a given penetrant is not a strong function of the polymer type, and then, it can be considered that b1.8910ÿ4 Paÿ1 in Ube polyimide membranes at 413 K is reasonable for water. The apparent Henry law coef®cient at low pressures, kD CH0 b, is considerably larger than kD itself because of a signi®cant contribution from the Langmuir sorption. 4.2. Transport parameter The transport parameters for water show that the mobility of the Langmuir species in Ube polyimide membranes is considerably less than that of the Henry law species (F0.05); the diffusivity for the Langmuir mode is almost 5% of that for the Henry law mode. 4.3. Concentration of penetrant Fig. 4. SEM photographs of the cross section near the upper surface of a polyimide hollow fiber membrane (no. 1 fiber).
4.1. Sorption parameters Dual mode parameters for water kD, CH0 and b at 413 K show almost the same values in both the cases and the mean values are 4.0010ÿ4 m3-vapor/(m3polymer Pa), 127.5 and 1.8910ÿ4 Paÿ1, respectively.
In Table 2, the concentration ratio CD/CT, CH/CT, Cm/CT and CD/Cm, which were calculated using the parameters in Table 1, are listed. Here CD and CH are the local concentrations of water molecules in the Henry law and Langmuir environments, respectively; CT is the total concentration of water molecules (CDCH) in the membrane; Cm is the concentration 1
See Appendix.
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16 Table 2 Calculated results on the basis of estimated parameters for Ube polyimide membranes at 413 K (partial immobilization) Parameters (dimensionless)
Ph1.5105 (Pa)
Ph3.0105 (Pa)
CD/CT CH/CT Cm/CT CD/CH CD/Cm Deff/DD CH =CH0
0.33 0.67 0.36 0.49 0.91 0.93 0.97
0.49 0.51 0.52 0.96 0.95 0.98 0.98
of water molecules contributed to the diffusion (CDFCH) in the membrane. The fractional saturation of the Langmuir isotherm CH =CH0 is 0.97 at Ph1.5105 Pa and 0.98 at Ph3.0105 Pa. The results show that the microvoids in Ube polyimide membranes are nearly ®lled with the water sorbed by the Langmuir contribution at lower upstream pressure; though the upstream gas pressure is 2.95105 Pa at the maximum in the experiment, a highly condensable nature of H2O is presumably responsible for its high af®nity for the Langmuir sorption. As the water molecules in the Henry law environment occupies 91±95% of those contributed to the diffusion in Ube polyimide membranes, that is CD/Cm0.91 and 0.95. It could be considered that almost all water molecules in the Langmuir environment is trapped in a microvoid, even in the presence of a large dual sorption effect. As a result, it could be concluded that the permeation behavior in Ube polyimide membranes follows essentially only the diffusion of Henry's law species, with the microvoid acting as sinks to trap the diffusing penetrant. 5. Conclusions Kinetics of the sorption and transport of water vapor in Ube glassy polyimide membranes are investigated on the basis of the data obtained from the vapor permeation experiment for ethanol±water mixtures. The results are as follows: 1. Almost all microcavaties are ®lled with the Langmuir species.
15
2. The diffusion process is controlled by the Henry law species (FDH/DD0.05 and Deff/DD0.93± 0.98). 3. The separation of water by permeation in Ube polyimide membranes could be analyzed using the dual sorption model with complete immobilization. Appendix The third term in Eqs. (7) and (12) is essentially not t in this experiment. contributed to N (a) Dual sorption mode. Eq. (7) is rearranged by using Eqs. (2) and (6) " # Af Ph Perm 1 2 tÿ : Nt 6Deff K= 1 bPh 2 1 (A.1) As is evident from Eq. (A.1), unknown parameters are Deff, K and b. These values are estimated by the parameter ®tting with simplex method. On the basis of the results estimated, DD and kD are calculated by Eqs. (2) and (6), respectively. (b) Partial immobilization mode. First step. Eq. (12) is rearranged by using Eqs. (9) and (11) " # Af Ph Perm 1 FK= 1 bPh 2 2 tÿ : Nt 1 FK= 1 bPh 6Deff (A.2) In this case unknown parameters are Deff, FK and b. These values are estimated by the parameter ®tting with simplex method. Second step. At the ®rst step, the values for Deff, FK and b have been determined and by using the following equation which is identical to Eq. (A.2), parameters kD and K are estimated by the simplex method: ! A P k D K 2 f h D eff 1 tÿ Nt : 6Deff 1 bPh 2 (A.3) Using the above procedures, ®ve unknown parameters Deff, F, b, kD and K could be estimated precisely.
16
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
References [1] W.R. Vieth, K.J. Sladek, A model for diffusion in a glassy polymer, J. Colloid Sci. 20 (1965) 1014. [2] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959, p. 99.
[3] D.R. Paul, W.J. Koros, Effect of partially immobilization sorption on permeability and the diffusion time lag, J. Polym. Sci., Polym. Phys. Ed. 14 (1976) 675. [4] G. Ranade, V. Stannett, W.J. Koros, Temperature dependence and energetics of the equilibrium sorption of water vapor in glassy polyacrylonitorile, J. Appl. Sci. 25 (1980) 2179.
Kinetics of sorption and permeation of water in glassy polyimide Yoshishige Hayashi, Shigehiro Sugiyama*, Takuya Kawanishi, Nobuaki Shimizu Department of Chemistry and Chemical Engineering, Faculty of Engineering Kanazawa University, 2-40-20 Kodatsuno, Kanazawa 920-8667, Japan Received 15 August 1998; accepted 24 August 1998
Abstract A vapor permeation experiment for water±ethanol mixtures was carried out using asymmetric Ube polyimide hollow-®ber membranes, which exhibit high selective permeability for water vapor, under the conditions of T413 K, upstream gas pressure Ph1.51052.95105 Pa and downstream gas pressure Pl400 Pa. To represent gas separation properties of the Ube polyimide membrane with a high transition temperature (570 K), the contribution of Henry's law part and Langmuir part modes on the diffusion through the membrane is studied on the basis of the dual-mode transport models. The results show that Henry's law penetrant controls the diffusion in the membrane. For the separation of water±ethanol mixtures by permeation through Ube polyimide membranes, the water trapped in microcavities can be assumed to be totally immobilized under the operating conditions applied here. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Dual mobility; Glassy polymer; Polyimide; Vapor permeation; Membrane
1. Introduction
2. Theory
The penetration of gases through nonporous polymeric membranes depends on whether the polymer is in its rubbery or glassy state. Ube polyimide membranes have a high glass transition temperature (570 K), and for the general case of gas separation polyimide membranes are used in a glassy state. In this paper, the experiment for the vapor permeation separation of water±ethanol mixtures was carried out using a module comprised of asymmetric polyimide hollow ®ber membranes with high selective permeability for water. On the basis of the data obtained, the gas separation behavior is analyzed in terms of the dual sorption and dual mobility models.
2.1. Dual sorption model
*Corresponding author. Tel.: +81-76-2344806; fax: +81-762344811; e-mail: [email protected]
On the basis of the dual sorption theory, the following equations for the isotherm well-describe the transport of gases through membranes [1]. @CD @ 2 CD Deff @t @x2
(1)
and Deff
DD
K= 1 bPh 2 1
K CH0 b=kD ;
;
(2) (3)
where Deff is the effective diffusion coef®cient of the penetrant, DD is the diffusion coef®cient of the pene-
0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00273-7
12
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
trant in the Henry law environment, K is a useful collection of the sorption parameters, and kD, CH0 and b are generally referred to as the Henry law constant, the Langmuir capacity constant and the Langmuir af®nity constant, respectively. The boundary conditions are as follows: CD 0 CD CD0 kD Ph CD 0
at t 0; at x 0; at x ;
(4)
where Ph is the upstream pressure and is the membrane thickness. The solution of Eq. (1) is given by [2] 1 CD x 2X 1 nx Deff n2 2 t sin exp ÿ 1ÿ ÿ : 1 n 2 CD0 (5) In this case, permeability Perm is given by [1] Perm kD DD :
(6)
t which has At any time t, the total volume of gas N penetrated the membrane is given by the following equation: 1 t N Deff t 1 2X ÿ1n ÿ ÿ Af CD0 2 6 2 1 n2 Deff n2 2 t exp ÿ ; (7) 2
in the Langmuir environment. The form of Eq. (8) is identical to that of Eq. (1) and the boundary conditions are also the same as Eq. (4). The permeability is given by [3] FK Perm kD DD 1 : (11) 1 bPh t is as The total volume of the diffusing penetrant N follows: !" t K Deff t 1 N 1 ÿ 2 2 Af CD0 6 1 bPh # n 1 2 X ÿ1 ÿDeff n2 2 t exp ÿ 2 : (12) 1 n2 2 3. Experimental Figs. 1 and 2 show the schematic diagram of experimental apparatus and permeation module, respectively. The asymmetric polyimide membrane with high selective permeability for water vapor, developed by Ube industry Co. Ltd., was used. Ube polyimide membranes have a high glass transition temperature of around 570 K and its permeation rate constant for water is 1.2910ÿ8 m3-STP/(m2 s Pa). The outer and inner diameters of hollow ®ber membrane were approximately 490 and 300 mm, respectively. The
where Af is the membrane surface area. 2.2. Partial immobilization model Penetrant molecules in the two distinct molecular environments (Henry's law and Langmuir) have different inherent mobilities. The so-called ``dual mobility'' or ``partial immobilization'' model accounts for this fact as follows [3]: @CD @ 2 CD Deff ; @t @x2
(8)
and Deff DD
1 FK= 1 bPh 2 1 K= 1 bPh 2
F DH =DD ;
;
(9) (10)
where DH is the diffusion coef®cient of the penetrant
Fig. 1. Schematic diagram of experimental appartus.
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
13
Fig. 2. Details of module.
module was placed in a constant temperature air bath to keep the temperature at 413 K. The used gas was the steam with ethanol vapor (0±4 mol%) and the separation time was 60 min. The upstream pressure was in the range 1.5105± 2.95105 Pa, and a vacuum pump on the downstream side operated at 400 Pa to provide the necessary driving force. The permeate and nonpermeate vapors were condensed by a heat exchanger and recovered as a liquid. The compositions of the feed, permeate and nonpermeate were determined by gas chromatography. 4. Results and discussion A typical example for the experimental results is plotted in Fig. 3 as a parameter of ethanol concentrations in the feed. The molar ¯ux of penetrated water is independent of the upstream partial pressure of ethanol and it is clear the ethanol does not cause signi®cant differences in the molar ¯ux of penetrated water under the operating conditions applied here. Table 1 shows the model parameters estimated by the simplex method using Eq. (7) or Eq. (12) for the cases of complete immobilization of the Langmuir species and partially immobilization, which are the mean of estimated values from 60 data sets. In the calculation the outer skin thickness, which was esti-
Fig. 3. Relationship between molar flux of penetrated water and partial pressure of water in the feed for no. 1 fiber. Ethanol concentration in the feed (mol%): (~) 0, (&) 2, and (*) 4.
mated by applying image processing technique to the SEM photographs of the skin region with high magni®cation (Fig. 4), was used as a characteristic dimension of . It was 0.092 and 0.094 mm for no. 1 and no. 2 ®bers, respectively. In Fig. 3, the results calculated by Eqs. (A.1) and (A.3) using the model parameters listed in Table 1 are also shown. The calculated results agree well with those obtained experimentally and it can be considered that the estimated model parameters represent characteristics of Ube polyimide membranes adequately.
14
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
Table 1 Sorption and transport parameters for water in Ube polyimide membranes at 413 K Parameters
Partial immobilization
Dual sorption
FK (dimensionless) F (dimensionless) b (Paÿ1) K (dimensionless) kD (m3-vapor (STP)/(m3-polymer Pa)) CH0 (m3-vapor (STP)/(m3-polymer)) DD (m2/s) Deff (m2/s)
3.01 0.05 1.8910ÿ4 60.1 4.0010ÿ4 127.2 2.8810ÿ12 2.70±2.8310ÿ12
± ± 1.8910ÿ4 60.4 4.0010ÿ4 127.8 2.9710ÿ12 2.78±2.9210ÿ12
The results indicate the accuracy of the estimation procedure1. Water vapor is highly condensable, and the values of parameters show its high af®nity for the Langmuir sorption. Ranade et al. [4] have reported that the af®nity constant for water in polyacrylonitrile (PAN) is 2.4110ÿ3 Paÿ1 at 308 K and 5.9410ÿ4 Paÿ1 at 333 K. The af®nity constant decreases with an increase in the temperature. In general, the value of the af®nity constant of a given penetrant is not a strong function of the polymer type, and then, it can be considered that b1.8910ÿ4 Paÿ1 in Ube polyimide membranes at 413 K is reasonable for water. The apparent Henry law coef®cient at low pressures, kD CH0 b, is considerably larger than kD itself because of a signi®cant contribution from the Langmuir sorption. 4.2. Transport parameter The transport parameters for water show that the mobility of the Langmuir species in Ube polyimide membranes is considerably less than that of the Henry law species (F0.05); the diffusivity for the Langmuir mode is almost 5% of that for the Henry law mode. 4.3. Concentration of penetrant Fig. 4. SEM photographs of the cross section near the upper surface of a polyimide hollow fiber membrane (no. 1 fiber).
4.1. Sorption parameters Dual mode parameters for water kD, CH0 and b at 413 K show almost the same values in both the cases and the mean values are 4.0010ÿ4 m3-vapor/(m3polymer Pa), 127.5 and 1.8910ÿ4 Paÿ1, respectively.
In Table 2, the concentration ratio CD/CT, CH/CT, Cm/CT and CD/Cm, which were calculated using the parameters in Table 1, are listed. Here CD and CH are the local concentrations of water molecules in the Henry law and Langmuir environments, respectively; CT is the total concentration of water molecules (CDCH) in the membrane; Cm is the concentration 1
See Appendix.
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16 Table 2 Calculated results on the basis of estimated parameters for Ube polyimide membranes at 413 K (partial immobilization) Parameters (dimensionless)
Ph1.5105 (Pa)
Ph3.0105 (Pa)
CD/CT CH/CT Cm/CT CD/CH CD/Cm Deff/DD CH =CH0
0.33 0.67 0.36 0.49 0.91 0.93 0.97
0.49 0.51 0.52 0.96 0.95 0.98 0.98
of water molecules contributed to the diffusion (CDFCH) in the membrane. The fractional saturation of the Langmuir isotherm CH =CH0 is 0.97 at Ph1.5105 Pa and 0.98 at Ph3.0105 Pa. The results show that the microvoids in Ube polyimide membranes are nearly ®lled with the water sorbed by the Langmuir contribution at lower upstream pressure; though the upstream gas pressure is 2.95105 Pa at the maximum in the experiment, a highly condensable nature of H2O is presumably responsible for its high af®nity for the Langmuir sorption. As the water molecules in the Henry law environment occupies 91±95% of those contributed to the diffusion in Ube polyimide membranes, that is CD/Cm0.91 and 0.95. It could be considered that almost all water molecules in the Langmuir environment is trapped in a microvoid, even in the presence of a large dual sorption effect. As a result, it could be concluded that the permeation behavior in Ube polyimide membranes follows essentially only the diffusion of Henry's law species, with the microvoid acting as sinks to trap the diffusing penetrant. 5. Conclusions Kinetics of the sorption and transport of water vapor in Ube glassy polyimide membranes are investigated on the basis of the data obtained from the vapor permeation experiment for ethanol±water mixtures. The results are as follows: 1. Almost all microcavaties are ®lled with the Langmuir species.
15
2. The diffusion process is controlled by the Henry law species (FDH/DD0.05 and Deff/DD0.93± 0.98). 3. The separation of water by permeation in Ube polyimide membranes could be analyzed using the dual sorption model with complete immobilization. Appendix The third term in Eqs. (7) and (12) is essentially not t in this experiment. contributed to N (a) Dual sorption mode. Eq. (7) is rearranged by using Eqs. (2) and (6) " # Af Ph Perm 1 2 tÿ : Nt 6Deff K= 1 bPh 2 1 (A.1) As is evident from Eq. (A.1), unknown parameters are Deff, K and b. These values are estimated by the parameter ®tting with simplex method. On the basis of the results estimated, DD and kD are calculated by Eqs. (2) and (6), respectively. (b) Partial immobilization mode. First step. Eq. (12) is rearranged by using Eqs. (9) and (11) " # Af Ph Perm 1 FK= 1 bPh 2 2 tÿ : Nt 1 FK= 1 bPh 6Deff (A.2) In this case unknown parameters are Deff, FK and b. These values are estimated by the parameter ®tting with simplex method. Second step. At the ®rst step, the values for Deff, FK and b have been determined and by using the following equation which is identical to Eq. (A.2), parameters kD and K are estimated by the simplex method: ! A P k D K 2 f h D eff 1 tÿ Nt : 6Deff 1 bPh 2 (A.3) Using the above procedures, ®ve unknown parameters Deff, F, b, kD and K could be estimated precisely.
16
Y. Hayashi et al. / Journal of Membrane Science 156 (1999) 11±16
References [1] W.R. Vieth, K.J. Sladek, A model for diffusion in a glassy polymer, J. Colloid Sci. 20 (1965) 1014. [2] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959, p. 99.
[3] D.R. Paul, W.J. Koros, Effect of partially immobilization sorption on permeability and the diffusion time lag, J. Polym. Sci., Polym. Phys. Ed. 14 (1976) 675. [4] G. Ranade, V. Stannett, W.J. Koros, Temperature dependence and energetics of the equilibrium sorption of water vapor in glassy polyacrylonitorile, J. Appl. Sci. 25 (1980) 2179.