Published in Soil Sci. Soc. Am. J. 68:481-488 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-2—SOIL CHEMISTRY
Deriving Boron Adsorption Isotherms from Soil Column Displacement Experiments
G. Communara,
R. Keren*,a and
FaHu Lia,b
a Institute of Soil, Water and Environmental Sciences, the Volcani Center, Agricultural Research Organization (ARO), P.O. Box 6, Bet Dagan 50250, Israel
b Dep. of Environmental Engineering, Chinese Agricultural Univ., Beijing, 100083, China
* Corresponding author (rkeren{at}agri.gov.il).
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ABSTRACT
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This study was conducted to elucidate the applicability of the column technique to obtain B adsorption isotherms. A column technique based on miscible displacement was used to investigate B equilibrium adsorption and B transport in soil at various pHs and B concentrations. Boron adsorption by soil at equilibrium was described by the Langmuir equation with the pH-dependent apparent adsorption coefficient. This equation, combined with one-dimensional convection–dispersion equation (CDE), was used to simulate B transport in soil columns. The results indicated that adsorption equilibrium was reached during the B transport in the homogeneously packed soil columns. The adsorption isotherms that were obtained from the column breakthrough curves (BTCs) and the batch experiments were identical at any given pH. This indicates that under equilibrium conditions, the column technique can be used for calculating B adsorption isotherm parameters for any given pH.
Abbreviations: BTC, breakthrough curve • CDE, convection–dispersion equation • SAR, sodium adsorption ratio
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INTRODUCTION
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BORON IS AN essential element that is required for normal plant growth, and its uptake by plants depends on B concentration in the soil solution. Since the range of B soil solution concentrations between those causing deficiency or toxicity symptoms in plants is relatively narrow, the prediction of B concentration in the soil solution is particularly important. Such a prediction is usually based on the assumption that B adsorption-desorption is a reversible process and that the B distribution between the solid and the liquid phases at equilibrium can be described by an adsorption isotherm. The parameters of such an isotherm depend on the mineral composition of the soil as well as the soil texture and pH (Keren and Bingham, 1985; Goldberg, 1993; Goldberg, 1997).
Adsorption isotherms are usually determined by using a standard batch technique or a column technique. Each of these techniques has advantages and disadvantages. The disadvantages of batch techniques have been discussed by several investigators (Schweich et al., 1983, Sparks, 1985; Bond and Phillips, 1990a; Barnett et al., 2000). The breakdown of soil aggregates during sample agitation, the relatively small soil/solution ratio and differences in mass-transfer and hydrodynamic conditions often result in inappropriate estimates of the degree of adsorption. Even for the same a given soil, the shapes of the batch-measured isotherms can vary. The column technique, however, overcomes these limitations. This technique uses solutions of the CDE to obtain an adsorption isotherm from BTCs, either by fitting (Schweich et al., 1983) or by mathematical inversion (Kool et al., 1989; van Veldhuizen et al., 1995). Thus, isotherm parameters obtained from displacement experiments reflect the average adsorption characteristics of a given soil. The major limitation of such a column study is that physical and chemical equilibrium are not always attained during the displacement experiments. Water-flow velocity has been found to be an important variable that affects the validity of the assumption of a local equilibrium. James and Rubin (1979) found that the BTCs might deviate from equilibrium even at a relatively low water flow rate. However, Schweich et al. (1983) obtained good agreement between exchange isotherms obtained from soil column measurements and those from bath experiments and similar findings were also reported by Bond and Phillips (1990a) and Griffioen et al. (1992). The disadvantage of the column technique is the absence of an accurate solution for a nonlinear CDE with equilibrium adsorption. The data-fitting procedure based on the numerical solutions is not trivial (van Veldhuizen et al., 1995) and considerable computer time is required to obtain accurate model parameters. However, this disadvantage can be partly overcome by application of semi-analytical solutions of a nonlinear transport CDE (Bond and Phillips, 1990b) and this approach was applied in the present study.
The objective of the present study was to test the applicability of the column technique to the derivation of B adsorption isotherms and to compare the isotherms so obtained with those obtained from a standard batch experiments at various soil pHs.
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MATERIALS AND METHODS
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The soil used in this study was loamy sand and Hamra (Rhodoxeralf) soil from the coastal plain of Israel. The predominant clay mineral type in the soil is montmorillonite. Some characteristics of the soil are presented in Table 1. The soil samples were packed in a column and leached with distilled water, and no extractable B was observed in the leachate. The soil was then dried in an oven at 40°C and passed through a 2-mm sieve before use.
Batch Experiments
Boron adsorption by the soil was studied in 50-mL polypropylene centrifuge tubes containing 15-g soil samples and 35 mL of background solution at total CaCl2 + NaCl concentration of 20 mmolc L–1 and sodium adsorption ratio (SAR) of 6, pH of 7, 8.5, and 10, and at a temperature of 25 ± 2°C.
The suspensions pH was adjusted before B addition by successive washings with CaCl2 + NaCl background solution at the appropriate pH, which was obtained by adding 0.5 M NaOH solution. pH adjustments were repeated until there was no pH change during 1 d of shaking. Then, the soils in the tubes were leached again with the background solution at the appropriate pH. After centrifugation and removal of the supernatant, the background solutions at appropriate pHs were added to the soil samples at initial B concentrations ranging from 0.5 to 25 mg L–1. The batch experiments for each B concentration and each pH were performed in triplicate. The average pH drift over the course of the experiments was <0.1 unit. The suspensions were shaken for 5 d before centrifugation. One repetition at pH 7 was equilibrated for 7 d. These times were selected despite the fact that Mezuman and Keren (1981) found 1 d sufficient for a similar soil. The longer time was used to compare B adsorption parameters obtained from batch and column experiments. Similarly, Corwin et al. (1999) equilibrated soil samples with B for 7 and 14 d. Following equilibration, the samples were removed from the shaker table, centrifuged, and filtered through a 0.45-µm membrane filter. Separate aliquots of the supernatant were analyzed for B using the inductively coupled plasma atomic emission spectroscopy (ICP-AES).
Transport Experiments
The transport experiments were conducted in plastic columns with an inside diameter of 5.2 cm and length (l0) of 10 and 40 cm, at room temperature 25 ± 2°C. The columns were uniformly packed with the soil in 1-cm increments. Each increment was tapped firmly to achieve homogeneous packing and uniform bulk density and the columns were then slowly saturated from the bottom using a peristaltic pump. During the displacement experiments, water saturation and steady-state flow conditions were satisfied. The effluents were collected by a fraction collector, the solution volume was measured and solution samples were analyzed.
First, the columns were leached with 100 mmolc L–1 (CaCl2 + NaCl) solution at SAR 6 and pH 7. When the composition and electrolyte concentrations in the effluent samples reached a steady state, this solution was replaced with the background solution that was used for the batch experiments. The columns of Series A, B, and C1 and those of Series C2 were adjusted to pH 7, 9.3, and 10, respectively. During pH adjustment the effluents were analyzed for Na and Ca using a flame photometer and atomic adsorption, respectively. The pH was determined during the leaching, and the soil pH adjustment was continued until a steady state with respect to the leaching solution composition, electrolyte concentration, and pH was reached.
The BTCs for Br were obtained to determine the dispersion characteristics of the packed soil columns: the Br concentrations in the effluents were determined by ion chromatography. For the B adsorption-desorption experiments, the background solution (at pH 7) containing B was introduced into the columns of Series A and Series B till the B concentration in the effluent reached the inlet B concentration. At this point in time, the leaching solution was replaced with the B-free background solution (pH 7) to observe B desorption. Pulses of B-containing solution were injected into the columns of Series-C at pH 7, 9.3, and 10. Once the pulse was injected, the columns were leached with the background solutions at the appropriate pH. The column experiments for each B concentration and each pH were performed in triplicate.
DATA ANALYSIS
Boron Adsorption Equilibrium Study (Batch Experiment)
Adsorbed B was calculated as the difference between the amount added and that found in the equilibrium solution. A B adsorption model (Keren et al., 1983) was used to describe B adsorption on the soil
 | [1] |
where cHB, cB–, and cOH are the solution concentration of the species [B(OH)3]0, [B(OH)4]–, and [OH]–, respectively, kHB, kB–, and kOH are the adsorption coefficients for these species, respectively, b = bHB + bB– is the total amount of adsorbed B and bm is the maximum B adsorption. The concentrations of B species in solution (Keren and Sparks, 1994) were specified as
 | [2] |
and
 | [3] |
where A 
Kh(104cOH) is the coefficient dependent on the solution OH– concentration, Kh is the B hydrolysis constant (Kh = 5.9 x 10–10 at 298 K), and c = cHB + cB. Elimination of concentrations cHB and cB– from Eq. [1] gives the Langmuir-type isotherm
 | [4] |
with the pH-dependent apparent adsorption coefficient
 | [5] |
The Langmuir Eq. [4] describes B adsorption on soils as a function of concentration for a given pH. The apparent adsorption coefficient, k in this equation depends on the adsorption coefficients, kHB, kB–, and KOH (that are considered to be constant for a particular soil/solution system) and pH (since Eq. [5] includes the pH-dependent coefficient A and the concentration of OH– ion). In fact, Eq. [4], with the apparent adsorption coefficient k defined according to Eq. [5], represents Keren's model, written in terms of total B concentrations, c = cHB + cB and b = bB–. Note that, Eq. [4] and [5] are written in terms of B solution concentrations, that is, with no correction for solution-phase activity. These equations were used to describe the B adsorption isotherms at various pH values. Values of bm and k were determined by fitting Eq. [4] to the batch adsorption data for a particular pH, and then for known sets of values of k – pH, the magnitudes of the adsorption coefficients kBH, kB–, and kOH were calculated using Eq. [5].
Boron Transport Simulations (Column Experiment)
Under physical and chemical equilibrium, the transport of B in a homogeneous soil column (for a given pH) can be simulated using the one-dimensional CDE
 | [6] |
where u is the mean pore-water velocity, D is the solute dispersion coefficient, 
is the column bulk density, 
is the volumetric water content, z is the distance along the column, and t is the time. The term 
b/t = (db/dc)(c/t) in Eq. [6] was evaluated by using Eq. [4]. By letting
 | [7] |
the model equations were reduced to the dimensionless form
 | [8] |
where c0 is the input concentration of solute, b0 = f(c0) is the B capacity of the soil that is reached at saturation with the concentration c0
 | [9] |
where k0 = kc0, R
is the dimensionless retardation factor and
 | [10] |
The concentration 
in Eq. [8] is defined as
 | [11] |
Simulations of B transport in soil columns were conducted by using a semi-analytical solution for a nonlinear CDE. The B adsorption isotherm was approximated by a piecewise linear equation
 | [12] |
where i and 
i represent the concentrations, and , respectively, for i = 1,2,...,n linear pieces, 
·i–1 = 
c· and ·i = i· are the concentrations at which the slope of the isotherm changes, · = n–1 and 
i are empirical coefficients. The applicability of such an approach to the solution of the CDE with nonlinear adsorption was discussed by Zolotarev (1968) and Bond and Phillips (1990b). Equations [8] and [12] were solved analytically (see the Appendix) in a manner described by Moench (1973). The final solution for i (
i) is
 | [13a] |
where i = (Pe/4)1/2
(RiT)–1/2 is the dimensionless argument of the error function, = ZRi –T, j = 1,2, ..., n – 1 is the label of the moving boundaries, and Ri = (1 + p0i) is the local retardation coefficient. For a solute adsorption k = (n + 1 – j), i = 
and sign (–), and for desorption = (j – 1), i = j and sign (+). The positions of moving boundaries, required for applying Eq. [13a] are determined from the following set of (n – 1) transcendental equations
 | [13b] |
Where Rh = Rn+1–j, Rg = Rn–j for a solute adsorption and Rh = Rj, Rg = Rj+1 for a solute desorption. Equations [13a] and [13b] were used to simulate the continuous injection experiments. The principle of superposition (Eq. [14]) was used to simulate solute transport under pulse injection conditions
 | [14] |
Here, i(i)ads represents the solution for a continuous solute input during time T, H(T) is the Heavy-side unit function, T· is the pulse size in pore volumes and i
ads is calculated for time T – T· using solution for adsorption in the parameters of concave isotherm.
In the present study, the velocity, bulk density, and volumetric water content were measured or were calculated from direct physical measurements. The unknown parameters Pe and bm, k were determined from the least-squares minimization of the function 
(Kool et al., 1989; van Veldhuizen et al., 1995)
 | [15] |
where l and (1,Tl) are the measured and simulated values of (relative) concentration at time Tl, and N is the number of the measured data points. These parameters were estimated independently; the column Peclet numbers were determined from the BTCs for Br, and the values of bm and k were obtained from the BTCs for B. To save computation time, the Br BTCs obtained from continuous or pulse injection experiments were simulated by using Eq. [13] or [14], respectively, at n = 1. The B transport simulations were performed by using the same equations at n = 10; since in the case n < 10, an incomplete description of the B elution curves for a pulse injection was observed. Models containing different combinations of adjustable parameters bm and k were compared to select the one that best fitted the experimental BTCs for B. The reduced B adsorption isotherm (Eq. [11]) was approximated by Eq. [12]. Since this isotherm has a convex shape, the coefficients i were in the range of i–1 > i 
i+1 for all i = 1, 2, ..., n. The value p0 (Eq. [10]) and the fitted coefficients i were used in calculating the local retardation factors, Ri = (1 + p0i).
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RESULTS AND DISCUSSION
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pH Dependent Equilibrium Boron Adsorption (Batch Experiments)
Boron adsorption isotherms for the soil at several different pHs are shown in Fig. 1a
. Good agreement between the experimental and the calculated values was obtained by using the Langmuir equation plotted as c/b versus c (Fig. 1b). The best fit values for the Langmuir parameters bm and k are given in Table 2. The slopes of the three lines in Fig. 1b were similar. This indicates that the B adsorption capacity of this soil is independent of pH in the studied range. The average value of bm for the soil was 17.24 mg kg–1. This value was close to that obtained by Mezuman and Keren (1981) for a similar soil (23.3 mg kg–1) when the equilibration time was 1 d. The small difference in bm values is due to the differences in clay contents (9.5% formerly vs. 7% in the soil used in the present study). It is important to note that the values of the adsorption coefficients obtained after 7 d of equilibration (bm = 17.158 mg kg–1 and k = 0.0548 L mg–1) at pH 7 were similar to given in Table 2. Thus, it is concluded that no B diffusion occurred in the clay mineral lattice during longer-term batch experiments (for 5 and 7 d).
The adsorption coefficients, kBH, kB–, and kOH, were calculated from the apparent k values given in Table 2 for the various pH values using Eq. [16]
 | [16] |
where Aj, 
j = [k(1 + A)]j and ßj = (cOH)j labeled by the index j = 1,2,3 corresponds to pH of 7, 8.5, 10, respectively. One can see (Table 2) that the affinity coefficient of B
–4 was greater than that of B(OH)3 while the affinity coefficient of OH– was greater than those of the former two species. This sequence is in accordance with that reported by Keren and Bingham (1985) for various soils and clays.
Transport Experiments
The BTCs for transport of Br (Fig. 2)
are essentially symmetrical, indicating ideal transport behavior. The retardation factors for all the soil columns were close to 1 and the Pe values are given in Table 3. The solid lines in this figure represent the best-fit BTCs simulated by Eq. [13] or [14] for continuous and pulse injection of Br solutions. All subsequent simulations of B transport in the soil columns were performed with , , u, and Pe held at fixed values (Table 3).
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Fig. 2. Measured and fitted breakthrough curves for Br in columns: 1, Series-A; 2, Series-B; and 3, Series-C. Measured data-symbol and result of fitting-solid line.
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Series-A.
In this series the average pore-water velocity was 7.58 cm h–1. The experimental results and the simulated curves for B adsorption and desorption for Columns A1 and A2 at B input concentrations of 5 and 1 mg L–1, respectively, are presented in Fig. 3 . Although the equilibrium model (solid line in Fig. 3) fitted the observed data well, the values of the isotherm parameters, bm and k, for B adsorption were different from those obtained for B desorption (Table 4). This indicates that the isotherms were explicitly hysteretic (Fig. 4)
. Moreover, these values did not match those obtained from the batch experiment at pH 7 (Table 2). The calculated BTCs for the batch adsorption parameters (bm = 17.24 mg kg–1 and k = 0.0549 L mg–1) are shown in Fig. 3 (dashed line). If a local equilibrium is attained during the displacement experiment, the BTCs for adsorption and desorption at various initial B concentrations should converge to a single adsorption isotherm. Thus, the validity of the assumption of local equilibrium can be checked by comparing the adsorption and desorption isotherms obtained from column BTCs. Since B adsorption on a given soil is considered to be reversible and rapid (Mezuman and Keren, 1981), the failure to reach equilibrium in these series of experiments was probably due to the short contact time of B solutes with the solid phase of the soil.
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Fig. 3. Measured, predicted and fitted breakthrough curves for B adsorption and desorption on columns Series-A for the input B concentration (a) 5 mg L–1 and (b) 1 mg L–1. Predictions were made using the batch-measured isotherm parameters, bm = 17.24 mg kg–1, k = 0.0549 L mg–1, and Pe = 21.
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Fig. 4. Hysteretic B adsorption and desorption isotherms obtained from columns Series-A. 1, adsorption isotherm; 2, desorption isotherm at the input B concentration of 5 mg L–1; 3, desorption isotherm at the input B concentration of 1 mg L–1.
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Series-B.
In this series, the average pore-water velocity was about half (3.85 cm h–1) of that used for Series-A. The BTCs for the initial B concentrations of 5 and 1 mg L–1 are presented in Fig. 5a and 5b
, respectively, in which the solid lines represent the fitted BTCs for B. The obtained isotherm parameters, bm and k, for B adsorption and for desorption were identical (Table 4) and they well matched the values determined from the batch experiments at pH 7 (Table 2). This similarity in isotherm parameters indicates that B adsorption equilibrium was reached at this water flow velocity. These two consecutive experiments on B adsorption and desorption can also be simulated as a pulse injection experiment. This requires that we specify the pulse duration for the initial B injection at which the soil columns are fully saturated with the inlet B concentration. The BTCs simulating the pulse technique are presented in Fig. 6
. These curves were obtained by combining the data given in Fig. 5. For B concentrations of 5 and 1 mg L–1, a pulse of duration, T·, was obtained equal to 4.31 and 5.8 pore volumes, respectively. The predicted curves (solid line) were simulated for Pe = 41 and the average values of isotherm parameters given in Table 4 (bm = 15.10 mg kg–1 and k = 0.061 L mg–1). The retardation for the high initial B concentration (5 mg L–1) was slightly less than that for the low initial B concentration (1 mg L–1). This difference was due to the shape of the B adsorption isotherm; the greater the nonlinearity (which is a function of B concentration), the greater the retardation.
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Fig. 5. Measured and fitted breakthrough curves for B adsorption and desorption on columns Series-B for the input B concentration (a) 5 mg L–1 and (b) 1 mg L–1. Simulations were made using Pe = 41. Best fitted isotherm parameters, bm and k, are given in Table 4 (B1 and B2).
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Fig. 6. Measured and predicted breakthrough curves for the transport of B in column Series-B simulated under pulse injection conditions. Observed data points were taken from Fig. 5a (solid circles), and from Fig. 5b (open circles). Predictions were made using Pe = 41, and average values for the isotherm parameters bm = 15.10 mg kg–1 and k = 0.061 L mg–1 of Table 4. A pulse durations T· = 4.31 and T· = 5.8 were used to simulate the input B concentrations of 5 mg L–1 and 1 mg L–1, respectively.
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Series-C.
The displacement experiments of Columns C1, C2, and C3 were conducted at u = 0.36 cm h–1, T· = 0.85, and c0 = 20 mg L–1. The only difference between these three columns was the soil pH (7, 9.3, and 10). The BTCs at these pH values are presented in Fig. 7
. The retardation of B in soil leached with solution at pH 7 was less than that observed for the other two pHs. At pH 7 the dominant B species was B(OH)3 that has the lowest affinity coefficient. The B retardation in the soil increased only slightly when the pH increased to 10, despite the fact that the dominant B species at this pH is B–4 whose affinity coefficient is much greater than that of boric acid (Table 2). The small deviation in B retardation can be explained by the increase in competition between the hydroxyl ions and the B species on the common adsorption sites. The B transport in the soil at pH 9.3 was retarded further in comparison with that at pH 7 and 10 (Fig. 7), because of the reduction in the hydroxyl ion concentration in the soil solution.
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Fig. 7. Measured and fitted breakthrough curves for the transport of the B species in columns Series-C. Simulations were made using Pe = 26 and T· = 0.85. Best fitted isotherm parameters, bm and k, are given in Table 4 (C1, C2, and C3).
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The best-fitted curves (solid lines, Fig. 7) were derived with the isotherm parameters given in Table 4 for pHs 7, 9.3, and 10 (Columns C1, C2, and C3, respectively). The values of adsorption capacity obtained from the BTCs (Table 4) were similar to those obtained from the batch experiment (Table 2), independent of pH. Therefore, the apparent adsorption coefficients (as expressed above) can be used to compare the B adsorption isotherms at different pHs. The relationship between k and pH is displayed in Fig. 8
. The solid line was calculated from Eq. [5] using the values of the affinity coefficients kBH, kB–, and kOH (Table 2) and the points represent the k values calculated from column data. It is concluded that B adsorption isotherms obtained from the displacement experiments conducted at low water flow velocity were identical to those measured by the batch technique.
The use of the Langmuir equation with the apparent adsorption coefficient is based on the simplifying assumption that the pH remains constant. Because of the reasonably high buffer capacity of the soil, this assumption was valid. Applying this assumption essentially simplifies the B transport simulations, since only two Langmuir parameters, bm and k, are the subject of definition at each particular pH, and batch or column experiments conducted at various pHs enable the parameters of the Keren model (Eq. [1]) to be calculated from these data.
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CONCLUSION
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Boron adsorption and transport in the soil were investigated using both the batch and column techniques. The transport model was based on the assumptions that the soil pH remained constant and that B adsorption by soil was in equilibrium. Under these assumptions, the transport of B species in the soil column can be described by the simple one-dimensional CDE and the Langmuir equation with the apparent adsorption coefficient. The absence of equilibrium during displacement experiments leads to hysteretic phenomenon, but the validity of the local equilibrium assumption can be checked by comparing the isotherms obtained from the column experiments where B adsorption is alternated with B desorption. The adsorption isotherms that were obtained from the column BTCs and the batch experiments were identical at any given pH.
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APPENDIX
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When a piecewise linear isotherm (Eq. [12])
 | [A1] |
in substituted into the transport equation (Eq. [8])
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| [A2] |
the result, after redefinition of variables, is the parabolic diffusion equation
 | [A3] |
which is written for a particular concentration i
. Here, = ZRi – T is the new axial coordinate, where labeled by subscripts i refers to the moving boundaries Z·i reflecting the transposition of the concentration point ·i, and Ri = (1 + p0i) represents the local retardation coefficient. The number of moving boundaries is n –1 and the total number of boundaries is n + 1. The moving boundary conditions are stated as
 | [A4] |
 | [A5] |
The solution to Eq. [A3] to [A5] is constructed for an infinite column when the boundary conditions corresponding to the case of a solute adsorption are stated as
 | [A6] |
The problem of desorption from a saturated soil column is simulated by using the conditions
 | [A7] |
Integrating Eq. [A3] yields
 | [A8] |
where erf is the standard denotation of the error function, i is a variable defined as
 | [A9] |
and i, ßi are constants, which are determined by applying conditions [A4], [A6], or [A4], [A7]. After definition of the constant, the general solution for i
in written as (Eq. [13a])
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| [A10a] |
Where j = 1, 2, ..., n – 1 is the label of the moving boundaries reflecting transposition of concentration points ·i = i· and ·i=1 = c·. A solute adsorption is simulated by using k = (n + 1 – j), i = and sign (–), and the problem of desorption can be generated by applying = (j – 1), i = j and sign (+). Applying conditions [A5] gives the following set of (n – 1) transcendental equations (Eq. [13b])
 | [A10b] |
that is used to define the position of the moving boundaries.
The following iterative procedure enables Z·j to be obtained from the coupled system of Eq. [10b]. The mth approximation to Z·j at any temporal interval T is denoted by Zm·j and the initial values for Zm·j at m = 1 are defined for the domain Zm·j–1 
Zm·j 
, where at j = 1, Zm·j = –. For m > 1 the values of Zm·j are evaluated for domain Zm·j–1 Zm·j Zm+1·j where at j = (n – 1), Zm–1·j+1 = , and on each iterative step the value Zm·j is compared with Zm–1·j; the procedure is repeated until further change in the computed Zm·j becomes negligible. When Z·i(T) are known, Eq. [A10a] allows obtaining the continuous concentration distribution Z·j(T) that passes through all points c·1 of piecewise linear isotherm. The solution given by Eq. [10a] and [10b] describes the frontal solute behavior in an infinite column. For such conditions it is exact for any n and for n = 1 or when the values of i are identical, it is resulted in the simplest solutions for linear CDEs, which for the case of adsorption is = 0.5
and for desorption it is = 0.5
.
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ACKNOWLEDGMENTS
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This research was supported in part by a grant from the Ministry of Science, Culture and Sports and a grant from the Chief Scientist, Ministry of Agriculture and Rural Development, Israel.
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NOTES
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Contribution from the Institute of Soil, Water and Environmental Sciences, the Volcani Center, Agricultural Research Organization (ARO), P.O. Box 6, Bet Dagan 50250, Israel, No. 608/02, 2002 Series.
Received for publication January 28, 2003.
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INTRODUCTION
