- Email: [email protected]
kinetic sorption parameters
Current Applied Physics 9 (2009) 1323–1325
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
New method for determination of equilibrium/kinetic sorption parameters Soon-Jae Lee a, Seung-Gun Chung a, Dong-Ju Kim a,*, Cheol Eui Lee b, Jae-Woo Choi c a b c
Department of Earth and Environmental Sciences, Korea University, Anam Dong 5-1, Sungbuk Ku, Seoul 136-701, Republic of Korea Department of Physics and Institute for Nano Science, Korea University, Anam Dong 5-1, Sungbuk Ku, Seoul 136-701, Republic of Korea Laboratory of Environmental Biocolloid Engineering, Program in Rural System Engineering, Seoul National University, Seoul 151-921, Republic of Korea
a r t i c l e
i n f o
Article history: Received 9 December 2008 Accepted 15 December 2008 Available online 28 February 2009 PACS: 68.10.Jy 68.45.D Keywords: Sorption kinetics Two-site sorption Three-stage kinetic model Distribution coefficient
a b s t r a c t We proposed a new method from which kinetic sorption parameters can be estimated using aqueous phase concentration versus time data commonly available from adsorption tests. The method relies on the analytical solution that was solved for two-site reversible sorption kinetics based on mass conservation law for solute in a batch reaction system. In order to validate the method, model parameters were compared with those of three-stage kinetic models available in the literature. It was revealed that model parameters of the two-site kinetics could accurately be estimated as much as other model parameters. Advantage of the new method is acquisition of additional parameter, distribution or partitioning coefficient (K d ) frequently used in the analysis of chemical transfer in the solid–liquid interface. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Adsorption–desorption reactions are important processes that affect the fate and transport of chemicals in environment. Extensive studies have been conducted on aqueous phase adsorption processes of organic pollutants using various sorbates. In our previous study, three-stage sorption model [1–3] was proposed where different stages exist in the adsorption kinetics; the first stage is a sharper portion of the adsorption curve representing the rapid decrease in aqueous phase concentration due to instantaneous adsorption on external surface, the second one is a rather gradual portion with slow rate-limiting adsorption by intraparticle diffusion and the third is a constant portion with little changes in aqueous phase concentration. For a system of kinetic irreversible adsorption where aqueous phase concentration decreases continuously as a result of instantaneous, the adsorbed concentrations for the Type-1 (instantaneous) can be given by
S1 ðtÞ S1 ð0Þ S1 ð1Þ
ð1Þ
The adsorbed concentration for the Type-2 (kinetic) can be described by the following equation:
@C S2 ðtÞ ¼ aC 1 @t bS2 ð1Þ * Corresponding author. E-mail address: [email protected] (D.-J. Kim). 1567-1739/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2008.12.016
ð2Þ
where S2(t), S2(1) are the adsorbed concentrations at Type-2 site (interior site) at time t, infinitive respectively, and a is the rate constant for mass transfer from the aqueous phase to the interior site, b is defined as a limiting factor for S2(1) and its range is 0 < b 6 1. The mass balance equation for aqueous mass in the batch system is as follows:
VCðtÞ þ ms ½S1 þ S2 ¼ VC 0
ð3Þ
where C(t), C0 is the aqueous concentrations at time t and initial stage respectively, V is solution volume, ms is mass of adsorbent. Thus S1, S2 and C are dependent variables, t is independent variable, and others are constants. The analytical solution for Eqs. (1)–(3) is given by:
CðtÞ ð1 n1 Þð1 n1 bn2 Þ ¼ C0 ð1 n1 bn2 exp½ctÞ Mq1 ð1Þ Mq2 ð1Þ ð1 n1 bn2 Þa ; n2 ¼ ; c¼ n1 ¼ VC 0 VC 0 bn2
ð4Þ ð5Þ
where the relative concentration curve (C/C0) of the aqueous phase versus time is characterized by four parameters, n1, n2, c and b. These parameters can be estimated by fitting the observed concentration curves to the Eq. (4) and graphical interpretation of sorption-related parameters are demonstrated in Fig. 1. Recently two-site sorption model with reversible equilibrium sorption in Type-1 and nonequilibrium or kinetic sorption in Type-2 were used for study of chemical transport in environment
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S.-J. Lee et al. / Current Applied Physics 9 (2009) 1323–1325
By definition of two-site sorption model [4], we have
@S2 ¼ afð1 f ÞK d C S2 g @t
ð12Þ
where a(1/h) is the sorption rate constant. The relationship between S1 and C can be given from Eq. (8):
S1 ¼ ðRb 1ÞC
ð13Þ
S2 can also be expressed in terms of C as following:
@S2 @t
Fig. 1. Schematic representation of three-stage kinetic model in the absence and presence of the limiting factor, b used in the previous study [3]. Note that dashed line represents two-stage kinetic and solid line three-stage kinetic model.
¼ Rb
@C @t
ð14Þ
s fK d where R ¼ 1 þ msVK d ; b ¼ Vþm ; fmVs K d ¼ Rb 1; f ¼ bR1 Vþms K d R1 Time-dependent sorption kinetics (Eq. (12)) in Type-2 can be given by
@S2 ¼ aðRC 1Þ @t
ð15Þ
From Eqs. (14) and (15), we have
Rb
@C ¼ aðRC 1Þ @t
ð16Þ
Solution of Eq. (16) can be given as follows: a 1 þ ð1 bÞebt R a 1 S1 ¼ ðRb 1Þ þ ð1 bÞebt R a 1 S2 ¼ 1 Rb þ ð1 bÞebt R
C ¼
Fig. 2. Illustration of conceptual two-site sorption model used in the literature [4].
[4–6]. The two-site sorption model assumes that sorption sites in adsorbents can be classified into two fractions: one fraction (Type-1) on which sorption is assumed to be instantaneous, and other fraction (Type-2) on which sorption is considered to be time-dependent (Fig. 2). In this study, we derived an analytical solution of the two-site sorption model and compared with the three-stage kinetic sorption model for the kinetic adsorption data obtained from batch experiments. 2. Analytical solution of reversible two-site sorption kinetics Total solute mass in aqueous and sorbed phase in batch condition is given by
ms S þ VC ¼ VC 0
ð6Þ
ð17Þ ð18Þ ð19Þ
where 1 b ¼ C þ0 1R, and C þ0 is the aqueous concentration after partitioning at time t = 0. The graphical solution of aqueous concentration during reversible two-site sorption is given in Fig. 3 with depiction of C þ0 . 3. Kinetic adsorption experiments Adsorbent ZAC [7] was used as a primary adsorbent to investigate the adsorption kinetics of aqueous toluene. The average grain size of ZAC was 4 mm. An aliquot of 10 g ZAC was allowed to react with 570 mL of toluene solution in 600-mL glass flask with Teflonlined septa and thus creating solid-to-liquid ratio (ms : V) equal to 1:57. The head space was less than 2.5 mL in volume to minimize the volatilization of toluene. The glass flasks were agitated in a shaking incubator (KMC-1205SW1, Vision Scientific, Bucheon, Korea) at 140 rpm while being maintained at 30 °C. Triplicate solution samples were obtained routinely from the flask using a microsy-
where C0 (mg/L) is initial concentration, S (mg/kg), C (mg/L) are sorbed and aqueous concentrations at time t (h), and ms (mg), V (L) are mass of sorbent and volume of liquid, respectively. Total sorbed concentration is the sum of sorbed concentrations in Type1 (S1) and Type-2 site (S2).
S ¼ S 1 þ S2
ð7Þ
S1 ¼ fK d C
ð8Þ
S2 ¼ ð1 f ÞK d C
ð9Þ
where f is the fraction of sorption site Type-1 and Kd (L/kg) is the distribution or partitioning coefficient. If we divide Eqs. (6) and (7) by VC 0 , then dimensionless form is as follows:
S þ C ¼ 1
ð10Þ
S ¼ S1 þ S2
ð11Þ
where C ¼
C C0
S ¼
ms S VC 0
S1 ¼
ms S1 VC 0
S2 ¼
m s S2 VC 0
Fig. 3. Graphical presentation of sorbed concentrations in two-sites and aqueous concentration at t = +0 and t = 1.
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S.-J. Lee et al. / Current Applied Physics 9 (2009) 1323–1325 1.2
Table 1 Parameters of two-site and three-stage kinetic model for aqueous toluene adsorption on ZAC.
1.00
Relative concentration of toluene
0.95 0.90
1.0
Two-site
0.85
Parameter
0.80
0.8
0.75 0.70 0.0
0.5
1.0
1.5
Three-stage
R
K d (L/kg)
a (1/h)
f
a (1/h)
n1
n2
2.3245
75.4965
0.1348
0.0686
0.2186
0.0189
0.9811
2.0
0.6
0.4
Observed Two-Site sorption Three-Stage sorption
0.2
0.0 0
20
40
60
80
100
120
140
160
Time (hr) Fig. 4. Observed and fitted data of aqueous toluene concentrations during kinetic adsorption. Data were fitted with two different kinetic models.
ringe to measure aqueous toluene concentration during adsorption. The samples contained in a 0.5-mL e-tube were centrifuged (Model J6-MC, Beckman Coulter, Fulerton, CA, USA) at 1400 rpm for 1 min, and the aqueous toluene concentrations of supernatants were analyzed using HPLC. The batch test was continued for a period of 6 days after which concentrations of aqueous toluene did not show any change. 4. Estimation of kinetic parameters Aqueous phase concentrations of toluene during adsorption onto ZAC and model fits are shown in Fig. 4. Toluene decline curve shows three phases: the instantaneous adsorption (S1 ) at the initial stage from t = 0 to t = +0, followed the slow rate-limiting adsorption at the second stage and long final constant stage. Both models fitted well the observed data of toluene adsorption. Model parameters are summarized in Table 1. It is noted that f value (0.0686) responsible for the instantaneous adsorption in the reversible two-site sorption model was higher than the value (0.0189) of n1 in three-stage model while sorption rate constant a (0.2186) of three-stage model was slightly higher than that (0.1348) of the reversible two-site sorption model. The reason for the difference of Type-1 fraction between two models can be explained by the fact that in reversible two-site sorption model f is the ratio of S1 to S whereas the fraction of instantaneous sorption site n1 in three-stage model is the ratio of sorbed solute to total solute (ms S1 =VC 0 ). Therefore care should be taken in the estimation of the Type-1 fraction for both models. The relatively lower sorption
rate constant (a) in reversible two-site sorption model difference is due to the higher value of Type-1 fraction (f) since it determines the starting value of C for each model. The advantage of reversible two-site sorption is such that the Type-1 fraction (f), retardation coefficient (R), distribution or partitioning coefficient (Kd), sorption rate coefficient (a) can be obtained simultaneously from a single data of sorption kinetics unless otherwise equilibrium batch test should additionally be performed. 5. Conclusions We derived a new analytical solution of the reversible two-site sorption kinetics based on mass conservation in static batch reaction system where aqueous phase solute shows fast initial sorption, a slow rate-limiting sorption and constant equilibrium sorption. The application of the analytical solution to the observed sorption kinetic data of toluene on adsorbent ZAC revealed that more parameters such as retardation factor (R), Type-1 fraction (f), distribution or partitioning coefficient (Kd) in addition to sorption rate coefficient (a) could be obtained when compared with the existing three-stage sorption model. Thus the method proposed in this study would offer more applicability in describing kinetic adsorption than previously developed kinetic adsorption models. Further study should be performed on the evaluation of the batch-based kinetic parameters in dynamic flow condition. Acknowledgement Authors acknowledge that this study was granted by KOSEF (Grant R01-2008-000-12439-0). References [1] S.B. Kim, I. Hwang, D.J. Kim, S. Lee, W.A. Jury, Environ. Toxicol. Chem. 22 (2003) 2306. [2] J.W. Choi, N.C. Choi, B. Mahendran, D.J. Kim, C.E. Lee, Curr. Appl. Phys. 7 (2007) 13. [3] J.W. Choi, N.C. Choi, S.J. Lee, D.J. Kim, J. Colloid Interface Sci. 314 (2007) 367. [4] M.Th. van Genuchten, R.J. Wagenet, Soil Sci. Soc. Am. J. 53 (1989) 1303. [5] D.R. Cameron, A. Klute, Water Resour. Res. 13 (1977) 183. [6] A.P. Gamerdinger, R.J. Wagenet, M.Th. Van Genuchten, Soil Sci. Soc. Am. J. 54 (1990) 957. [7] J.W. Choi, K.S. Yang, D.J. Kim, C.E. Lee, Curr. Appl. Phys. 9 (2009) 694.
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
New method for determination of equilibrium/kinetic sorption parameters Soon-Jae Lee a, Seung-Gun Chung a, Dong-Ju Kim a,*, Cheol Eui Lee b, Jae-Woo Choi c a b c
Department of Earth and Environmental Sciences, Korea University, Anam Dong 5-1, Sungbuk Ku, Seoul 136-701, Republic of Korea Department of Physics and Institute for Nano Science, Korea University, Anam Dong 5-1, Sungbuk Ku, Seoul 136-701, Republic of Korea Laboratory of Environmental Biocolloid Engineering, Program in Rural System Engineering, Seoul National University, Seoul 151-921, Republic of Korea
a r t i c l e
i n f o
Article history: Received 9 December 2008 Accepted 15 December 2008 Available online 28 February 2009 PACS: 68.10.Jy 68.45.D Keywords: Sorption kinetics Two-site sorption Three-stage kinetic model Distribution coefficient
a b s t r a c t We proposed a new method from which kinetic sorption parameters can be estimated using aqueous phase concentration versus time data commonly available from adsorption tests. The method relies on the analytical solution that was solved for two-site reversible sorption kinetics based on mass conservation law for solute in a batch reaction system. In order to validate the method, model parameters were compared with those of three-stage kinetic models available in the literature. It was revealed that model parameters of the two-site kinetics could accurately be estimated as much as other model parameters. Advantage of the new method is acquisition of additional parameter, distribution or partitioning coefficient (K d ) frequently used in the analysis of chemical transfer in the solid–liquid interface. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Adsorption–desorption reactions are important processes that affect the fate and transport of chemicals in environment. Extensive studies have been conducted on aqueous phase adsorption processes of organic pollutants using various sorbates. In our previous study, three-stage sorption model [1–3] was proposed where different stages exist in the adsorption kinetics; the first stage is a sharper portion of the adsorption curve representing the rapid decrease in aqueous phase concentration due to instantaneous adsorption on external surface, the second one is a rather gradual portion with slow rate-limiting adsorption by intraparticle diffusion and the third is a constant portion with little changes in aqueous phase concentration. For a system of kinetic irreversible adsorption where aqueous phase concentration decreases continuously as a result of instantaneous, the adsorbed concentrations for the Type-1 (instantaneous) can be given by
S1 ðtÞ S1 ð0Þ S1 ð1Þ
ð1Þ
The adsorbed concentration for the Type-2 (kinetic) can be described by the following equation:
@C S2 ðtÞ ¼ aC 1 @t bS2 ð1Þ * Corresponding author. E-mail address: [email protected] (D.-J. Kim). 1567-1739/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2008.12.016
ð2Þ
where S2(t), S2(1) are the adsorbed concentrations at Type-2 site (interior site) at time t, infinitive respectively, and a is the rate constant for mass transfer from the aqueous phase to the interior site, b is defined as a limiting factor for S2(1) and its range is 0 < b 6 1. The mass balance equation for aqueous mass in the batch system is as follows:
VCðtÞ þ ms ½S1 þ S2 ¼ VC 0
ð3Þ
where C(t), C0 is the aqueous concentrations at time t and initial stage respectively, V is solution volume, ms is mass of adsorbent. Thus S1, S2 and C are dependent variables, t is independent variable, and others are constants. The analytical solution for Eqs. (1)–(3) is given by:
CðtÞ ð1 n1 Þð1 n1 bn2 Þ ¼ C0 ð1 n1 bn2 exp½ctÞ Mq1 ð1Þ Mq2 ð1Þ ð1 n1 bn2 Þa ; n2 ¼ ; c¼ n1 ¼ VC 0 VC 0 bn2
ð4Þ ð5Þ
where the relative concentration curve (C/C0) of the aqueous phase versus time is characterized by four parameters, n1, n2, c and b. These parameters can be estimated by fitting the observed concentration curves to the Eq. (4) and graphical interpretation of sorption-related parameters are demonstrated in Fig. 1. Recently two-site sorption model with reversible equilibrium sorption in Type-1 and nonequilibrium or kinetic sorption in Type-2 were used for study of chemical transport in environment
1324
S.-J. Lee et al. / Current Applied Physics 9 (2009) 1323–1325
By definition of two-site sorption model [4], we have
@S2 ¼ afð1 f ÞK d C S2 g @t
ð12Þ
where a(1/h) is the sorption rate constant. The relationship between S1 and C can be given from Eq. (8):
S1 ¼ ðRb 1ÞC
ð13Þ
S2 can also be expressed in terms of C as following:
@S2 @t
Fig. 1. Schematic representation of three-stage kinetic model in the absence and presence of the limiting factor, b used in the previous study [3]. Note that dashed line represents two-stage kinetic and solid line three-stage kinetic model.
¼ Rb
@C @t
ð14Þ
s fK d where R ¼ 1 þ msVK d ; b ¼ Vþm ; fmVs K d ¼ Rb 1; f ¼ bR1 Vþms K d R1 Time-dependent sorption kinetics (Eq. (12)) in Type-2 can be given by
@S2 ¼ aðRC 1Þ @t
ð15Þ
From Eqs. (14) and (15), we have
Rb
@C ¼ aðRC 1Þ @t
ð16Þ
Solution of Eq. (16) can be given as follows: a 1 þ ð1 bÞebt R a 1 S1 ¼ ðRb 1Þ þ ð1 bÞebt R a 1 S2 ¼ 1 Rb þ ð1 bÞebt R
C ¼
Fig. 2. Illustration of conceptual two-site sorption model used in the literature [4].
[4–6]. The two-site sorption model assumes that sorption sites in adsorbents can be classified into two fractions: one fraction (Type-1) on which sorption is assumed to be instantaneous, and other fraction (Type-2) on which sorption is considered to be time-dependent (Fig. 2). In this study, we derived an analytical solution of the two-site sorption model and compared with the three-stage kinetic sorption model for the kinetic adsorption data obtained from batch experiments. 2. Analytical solution of reversible two-site sorption kinetics Total solute mass in aqueous and sorbed phase in batch condition is given by
ms S þ VC ¼ VC 0
ð6Þ
ð17Þ ð18Þ ð19Þ
where 1 b ¼ C þ0 1R, and C þ0 is the aqueous concentration after partitioning at time t = 0. The graphical solution of aqueous concentration during reversible two-site sorption is given in Fig. 3 with depiction of C þ0 . 3. Kinetic adsorption experiments Adsorbent ZAC [7] was used as a primary adsorbent to investigate the adsorption kinetics of aqueous toluene. The average grain size of ZAC was 4 mm. An aliquot of 10 g ZAC was allowed to react with 570 mL of toluene solution in 600-mL glass flask with Teflonlined septa and thus creating solid-to-liquid ratio (ms : V) equal to 1:57. The head space was less than 2.5 mL in volume to minimize the volatilization of toluene. The glass flasks were agitated in a shaking incubator (KMC-1205SW1, Vision Scientific, Bucheon, Korea) at 140 rpm while being maintained at 30 °C. Triplicate solution samples were obtained routinely from the flask using a microsy-
where C0 (mg/L) is initial concentration, S (mg/kg), C (mg/L) are sorbed and aqueous concentrations at time t (h), and ms (mg), V (L) are mass of sorbent and volume of liquid, respectively. Total sorbed concentration is the sum of sorbed concentrations in Type1 (S1) and Type-2 site (S2).
S ¼ S 1 þ S2
ð7Þ
S1 ¼ fK d C
ð8Þ
S2 ¼ ð1 f ÞK d C
ð9Þ
where f is the fraction of sorption site Type-1 and Kd (L/kg) is the distribution or partitioning coefficient. If we divide Eqs. (6) and (7) by VC 0 , then dimensionless form is as follows:
S þ C ¼ 1
ð10Þ
S ¼ S1 þ S2
ð11Þ
where C ¼
C C0
S ¼
ms S VC 0
S1 ¼
ms S1 VC 0
S2 ¼
m s S2 VC 0
Fig. 3. Graphical presentation of sorbed concentrations in two-sites and aqueous concentration at t = +0 and t = 1.
1325
S.-J. Lee et al. / Current Applied Physics 9 (2009) 1323–1325 1.2
Table 1 Parameters of two-site and three-stage kinetic model for aqueous toluene adsorption on ZAC.
1.00
Relative concentration of toluene
0.95 0.90
1.0
Two-site
0.85
Parameter
0.80
0.8
0.75 0.70 0.0
0.5
1.0
1.5
Three-stage
R
K d (L/kg)
a (1/h)
f
a (1/h)
n1
n2
2.3245
75.4965
0.1348
0.0686
0.2186
0.0189
0.9811
2.0
0.6
0.4
Observed Two-Site sorption Three-Stage sorption
0.2
0.0 0
20
40
60
80
100
120
140
160
Time (hr) Fig. 4. Observed and fitted data of aqueous toluene concentrations during kinetic adsorption. Data were fitted with two different kinetic models.
ringe to measure aqueous toluene concentration during adsorption. The samples contained in a 0.5-mL e-tube were centrifuged (Model J6-MC, Beckman Coulter, Fulerton, CA, USA) at 1400 rpm for 1 min, and the aqueous toluene concentrations of supernatants were analyzed using HPLC. The batch test was continued for a period of 6 days after which concentrations of aqueous toluene did not show any change. 4. Estimation of kinetic parameters Aqueous phase concentrations of toluene during adsorption onto ZAC and model fits are shown in Fig. 4. Toluene decline curve shows three phases: the instantaneous adsorption (S1 ) at the initial stage from t = 0 to t = +0, followed the slow rate-limiting adsorption at the second stage and long final constant stage. Both models fitted well the observed data of toluene adsorption. Model parameters are summarized in Table 1. It is noted that f value (0.0686) responsible for the instantaneous adsorption in the reversible two-site sorption model was higher than the value (0.0189) of n1 in three-stage model while sorption rate constant a (0.2186) of three-stage model was slightly higher than that (0.1348) of the reversible two-site sorption model. The reason for the difference of Type-1 fraction between two models can be explained by the fact that in reversible two-site sorption model f is the ratio of S1 to S whereas the fraction of instantaneous sorption site n1 in three-stage model is the ratio of sorbed solute to total solute (ms S1 =VC 0 ). Therefore care should be taken in the estimation of the Type-1 fraction for both models. The relatively lower sorption
rate constant (a) in reversible two-site sorption model difference is due to the higher value of Type-1 fraction (f) since it determines the starting value of C for each model. The advantage of reversible two-site sorption is such that the Type-1 fraction (f), retardation coefficient (R), distribution or partitioning coefficient (Kd), sorption rate coefficient (a) can be obtained simultaneously from a single data of sorption kinetics unless otherwise equilibrium batch test should additionally be performed. 5. Conclusions We derived a new analytical solution of the reversible two-site sorption kinetics based on mass conservation in static batch reaction system where aqueous phase solute shows fast initial sorption, a slow rate-limiting sorption and constant equilibrium sorption. The application of the analytical solution to the observed sorption kinetic data of toluene on adsorbent ZAC revealed that more parameters such as retardation factor (R), Type-1 fraction (f), distribution or partitioning coefficient (Kd) in addition to sorption rate coefficient (a) could be obtained when compared with the existing three-stage sorption model. Thus the method proposed in this study would offer more applicability in describing kinetic adsorption than previously developed kinetic adsorption models. Further study should be performed on the evaluation of the batch-based kinetic parameters in dynamic flow condition. Acknowledgement Authors acknowledge that this study was granted by KOSEF (Grant R01-2008-000-12439-0). References [1] S.B. Kim, I. Hwang, D.J. Kim, S. Lee, W.A. Jury, Environ. Toxicol. Chem. 22 (2003) 2306. [2] J.W. Choi, N.C. Choi, B. Mahendran, D.J. Kim, C.E. Lee, Curr. Appl. Phys. 7 (2007) 13. [3] J.W. Choi, N.C. Choi, S.J. Lee, D.J. Kim, J. Colloid Interface Sci. 314 (2007) 367. [4] M.Th. van Genuchten, R.J. Wagenet, Soil Sci. Soc. Am. J. 53 (1989) 1303. [5] D.R. Cameron, A. Klute, Water Resour. Res. 13 (1977) 183. [6] A.P. Gamerdinger, R.J. Wagenet, M.Th. Van Genuchten, Soil Sci. Soc. Am. J. 54 (1990) 957. [7] J.W. Choi, K.S. Yang, D.J. Kim, C.E. Lee, Curr. Appl. Phys. 9 (2009) 694.