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DIVISION S-1 - SOIL PHYSICS

Osmotically Driven Water Vapor Transport in Unsaturated Soils

Department of Bioresource Engineering, Gilmore Hall, Oregon State Univ., Corvallis, OR 97331

* Corresponding author (shaun{at}orst.edu)

	ABSTRACT

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Knowledge of the transport processes of bulk fertilizers and other solutes in soils and porous media is vital to understanding the environmental and economic impacts of many agricultural and waste disposal practices. Previous experimental research shows that significant water movement can occur towards regions of high salt concentrations in unsaturated porous media. It is hypothesized that in unsaturated media, large gradients in aqueous osmotic potentials cause significant water-vapor fluxes towards regions of high solute concentrations. A physically based model to describe this process is developed by combining existing theories of liquid-water and water-vapor transport with aqueous electrolyte theory. A simplified solution, demonstrating these water-vapor transport processes, is compared with experimental results from discontinuous soil column experiments previously published by Wheeting (1925). The results confirm that large osmotic potentials created in the presence of high solute concentrations in unsaturated porous media, can account for the quantities of water vapor movement observed. This redistribution of pure water vapor complicates modeling of ion diffusion in these systems and is an important process to include in solute transport models when high solute concentrations exist in unsaturated soils.

	INTRODUCTION

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THE PRESENCE OF SOLUTES in soil-pore water creates an osmotic potential. Gradients in osmotic pressure can induce significant flow of water in these soils under conditions of high solute concentrations and low water content. This has been verified experimentally by a number of researchers (Wheeting, 1925; Letey et al., 1969; Raats, 1969; Scotter and Raats, 1970; Scotter, 1974a; Nassar and Horton, 1989a; Nassar et al., 1992b; Kelly et al., 1997; and Kelly, 1998). For osmotic water transport to occur in soils, a semipermeable membrane is required where ions are excluded yet water can cross freely. Osmotic potentials can give rise to variations in fluid pressure in the presence of such a semipermeable membrane. This is most commonly seen across plant cell walls, which excludes certain ions and allows water to move freely across the membrane (Nobel, 1983).

Two mechanisms are identified in the literature where an osmotic gradient because of varying solute concentrations in soil can cause water to move in either liquid or vapor phase.

1. Liquid-water transport can occur when anions are excluded from negatively charged soil particles, which act as a semipermeable membrane. The resulting driving force because of the osmotic pressure is quantified with an osmotic efficiency coefficient expressed as a function of solute concentration and soil water content (Kemper and Rollins, 1966). The osmotic efficiency coefficient is a function of the distribution of adsorbed ions and the thickness of the soil solution, which is calculated from diffuse double-layer theory (Bresler, 1973).
   
2. Water-vapor transport can occur in unsaturated soils when differing solute concentrations exist across gas-filled pores separating soil solution. The gas–liquid interfaces act as semipermeable membranes across which water vapor is transported freely and ions are excluded. The resulting osmotic potential gradient causes water vapor to be transported from regions of lower solute concentration to areas of higher solute concentration.
   

Theoretical analyses of water flow have been carried out by several researchers (Letey et al., 1969; Parlange, 1973; Scotter, 1974b; Nassar and Horton, 1989b; Nassar et al., 1992a; and Kelly, 1998), who considered the complex phenomena involving coupled transport of liquid, vapor, solute, and heat. This paper considers the water-vapor transport mechanism and focuses on the special case where water vapor can be transported exclusively in the presence of high solute concentrations.

Wheeting (1925) conducted experiments, where salts were added to columns of unsaturated soil, demonstrating significant water transport from unsaturated salt-free soil towards unsaturated salinized soil. Relationships between added salts and the moisture of soils in column experiments were investigated using salinized soil separated from solute-free soil by an air gap. In one set of experiments, soil columns were set up such that a 50.8-mm length soil was treated with a 1% (wt./wt.) salt solution to a constant initial water content and a 152.4-mm length of soil was wet to the same initial constant water content but without the salt. The initial water content and solute concentrations of the columns are provided in Table 1. Four treatments were tested using two soils with different textures (medium sand or clay loam) and two salts (KCl or Na₂CO₃). The two soil sections were then arranged in the column such that a 12.7-mm space existed between them. The columns were left at 18°C and determinations of water content in 25.4-mm sections were made after intervals of 5 and 15 d. It was observed that water had moved rapidly from the solute-free soil towards the side of the column with salinized soil (Fig. 1 and 2) . The rate of water-vapor transport was slower in the clay loam treatments as compared with the medium sand treatments of the same salt. After 15 d, nearly twice the quantity of water was transported in both the KCl treatments as compared with the Na₂CO₃ treatments of the same soil texture (Fig. 3 and 4) . Wheeting (1925) concluded that water-vapor transport was dependent on both soil texture, salt type, and salt concentration. This conclusion was later supported by experiments conducted by Scotter (1974b) and Kelly et al. (1997).

View larger version (25K): [in this window] [in a new window]   Fig. 1. Gravimetric water content distributions for (a) KCl and (b) Na2CO3 in medium sand from the broken column data of Wheeting (1925).

View larger version (24K): [in this window] [in a new window]   Fig. 2. Gravimetric water content distributions for (a) KCl and (b) Na2CO3 in clay loam from the broken column data of Wheeting (1925).

View larger version (19K): [in this window] [in a new window]   Fig. 3. Observed and predicted cumulative water vapor transport across a 12.7-mm air gap from a salt-free section to the saline section of the broken column for (a) KCl and (b) Na2CO3 in medium sand.

View larger version (18K): [in this window] [in a new window]   Fig. 4. Observed and predicted cumulative water vapor transport across a 12.7-mm air gap from a salt-free section to the saline section of the broken column for (a) KCl and (b) Na2CO3 in clay loam.

	THEORY

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Model Development
In the section below, we derive the set of governing differential equations describing simultaneous movement of water in the vapor and liquid phase and consequent transport of ions in unsaturated porous media. The equations are developed in one-dimension, assuming isothermal conditions. It is also assumed that movement of water in both the vapor and the liquid phase are first-order phenomena described by Fickian diffusion (Jackson, 1964) and Darcy's law, as follows:

[1]

[2]

[3]

where x is the spatial coordinate, q_w is the total water flux, ql_w is the liquid-water flux, and q^v_w is the water-vapor flux, all with units of mass per length squared per time. The variables, K_l and D_v, are first-order transport coefficients for liquid and vapor phase flow, respectively, with units of length per time. The variables, _l and p_v (vapor pressure), are the driving potentials for water in the liquid and vapor phase, respectively, with units of mass per length squared.

Osmotic Potential, _s
The osmotic potential of a dilute solution can be calculated using Rauolt's Law, which relates the vapor pressure of a solvent above a solution to the quantity of solute dissolved in the solution (Alberty, 1987).

[4]

Where P_i is the partial pressure of the i^th component of the solution, is the saturated vapor pressure of the pure component, and N_i is the mole fraction of that component. The following equation is used for calculating the mole fraction of an aqueous solution, N_w.

[5]

Where P^°_w is the pressure of pure water, P_w is the vapor pressure of the solution. The term, N_w, can also be calculated from the following equation:

[6]

Where n_j is the number of moles of component j. The water potential because of the presence of solutes, _s (MPa), can then be calculated from the Van't Hoff relation (Nobel, 1983).

[7]

Where is the partial molal volume of water (18.0 x 10^-6 m³ mol^-1), T is the temperature in K, and R is the gas constant (8.3143 x 10^-6 J K^-1 mol^-1). It is sometimes convenient to calculate osmotic potential in terms of the concentration of the solute, c_j (mol m^-3). In this case, Eq. [7] can be reduced to (Nobel, 1983).

[8]

Equation [8] is valid for dilute ideal solutions.

Aqueous solutions with solute concentrations exceeding a few millimoles depart from the ideality of the Van't Hoff relation. To accurately represent the mole fraction of the water in solutions, which approach unity as the dilution is increased, it is common to tabulate data in terms of the osmotic coefficient, . The osmotic coefficient for any aqueous solution can be obtained from the following equation (Robinson and Stokes, 1959).

[9]

Where a_w is the activity of water, v_i is the stoichiometric number of the solute, m_i is the molality (mol kg^-1), M_w is the molecular weight of water (mol kg^-1) and i indicates the solute of interest. Tabular data of molality and osmotic coefficients for a number of solutes are presented in Robinson and Stokes (1959).

Alternatively, one can calculate the osmotic coefficient for many solutes over a wide range of concentrations using a semi-empirical formula. One of the most widely used formula for estimation of the osmotic coefficient is Pitzer's method presented in a series of papers beginning in 1973 (Horvath, 1985; Zemaitis et al., 1986). For 1:1 and 1:2 electrolytes Pitzer's equation is:

[10]

ß₀ (kg mol^-1), ß₁ (kg mol^-1) and C (kg² mol^-2) are parameters specific to each electrolyte (Zemaitis et al., 1986), ₁ for 1:1 and 2:1 electrolytes is 2.0 kg^1/2 mol^-1/2, B is 1.2 kg^1/2 mol^-1/2 for all electrolytes, and m₂ is the molality of the electrolyte with units of (mol kg^-1) (equivalent to the ionic strength). The osmotic Debye-Hückel coefficient, A_m (kg^1/2 mol^-1/2) is computed as follows:

[11]

where N₀ is Avagodro's number (mol^-1), _w is the density of pure water (kg m^-3), T is the absolute temperature (K), ₀ is the permittivity of a vacuum [C² (N^-1 m^-2)^-1], _r is the relative permittivity water at T [unitless], k is Boltzmann's constant (J K^-1), and e is the elementary charge (C). Once the activity of the water in solution is known, it is possible to calculate the osmotic potential of the soil water solution using the Kelvin equation (Alberty, 1987),

[12]

At low molalities, most solutions behave ideally and follow Raoult's law, while significant deviations often occur at higher molalities. The shape of the osmotic potential vs. solution molality is strongly dependent on the ionic species under consideration, which is accounted for by using Eq. [10]. To account for temperature effects on the activity coefficient, Pitzer suggested corrections be made through the relative permittivity of water, the density of water, and the temperature in the Debye-Hückel coefficient (Zemaitis et al., 1986). The activity coefficient for multicomponent mixtures of salts may also be calculated using Pitzer's method which is outlined in Zemaitis et al. (1986).

Transport Equations
For both liquid and vapor, the governing equations have similar form. We will derive these results applicable for either phase. From Eq. [1] or [2] the mass movement is governed by:

[13]

Where K (length time^-1) is a general first-order coefficient and q is a flux (mass length^-2 time^-1). Applying mass conservation principles at a point, one obtains the following equation:

[14]

Where m is the mass and (mass length^-2 time^-1) is the mass source and sink term. Substituting Eq. [13] into Eq. [14] we obtain the following equation:

[15]

Equation [15] is a generalized partial differential equation that is modified to account for liquid, vapor and ion transport.

Liquid Transport
Equation [15] can be written for liquid mass transport.

[16]

Where m_l is the mass of the liquid and _l is a source or sink term. The source or sink term is used to couple the mass transfer of water between the gas phase and the liquid phase, which assumes that the thermodynamic equilibrium occurs much more rapidly than the transport processes. The potential term driving the liquid mass flow, _l, is composed of several individual potentials that can be calculated at any point in the flow field. Neglecting the effect of anion exclusion as previously noted, osmotic potential, _s, does not contribute to the driving potential of liquid flow. Assembly of the remaining potentials results in the following equation:

[17]

Where _g is the gravitational potential, and _p represents external pressure potentials applied, which is equal to zero everywhere in this system. The term, , is the matric potential and can be related in an approximate form to the volumetric water content of the soil using a soil water retention function such as provided by van Genuchten (1980).

[18]

Where g is the gravitational constant (m s^-2), is the inverse air entry pressure (m^-1), is the soil water content (m³ m^-3), _r is the residual soil water content (m³ m^-3), _s is the saturated soil water content (m³ m^-3), n is a dimensionless fitting parameter and m = 1 - . By substituting Eq. [17] into Eq. [16], the liquid water mass transport equation then becomes.

[19]

Vapor Transport
Similarly, Eq. [15] can be written for vapor mass transport.

[20]

Where m_v is the mass of vapor, and _v is a source or sink term for the water vapor. The potential term driving the vapor phase flow is the vapor pressure of water, which varies with temperature, osmotic potential and the matric potential. The osmotic potential can be calculated using Eq. [9] through [12], or Eq. [8] at low molalities. The vapor pressure or water vapor density, p_v, above the solution can then be calculated using the following equation:

[21]

Where p⁰_v is the saturation water vapor density at temperature, T (Hillel, 1998). From Eq. [21] it can readily be shown that the vapor pressure above a solution can be decreased by increasing solute concentrations or decreasing matric potentials. Figure 5 shows the effect of solute concentration on water vapor density as calculated using Eq. [10], [11], [12], and [21]. The maximum concentration for each salt is its solubility, which is shown as the endpoint of each line in Fig. 5. Figure 5 shows that at high molalities, there can be significant differences in water vapor density between the different salt solutions at the same molality.

View larger version (19K): [in this window] [in a new window]   Fig. 5. Water vapor density of salt solutions at 25°C calculated using Eq. [21].

	MATERIALS AND METHODS

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[22]

Under these assumptions the flux of water vapor between the two sections is described by the following equation.

[23]

Where ₀ is the volumetric water content of the saline section, ₁ is the volumetric water content of the salt-free section and K_v is the water vapor diffusion coefficient (Nobel, 1983) in the airspace , between the sections. The water vapor density in the salinized segment, p_v (₀), and the salt-free segment, p_v (₁) is calculated using the method described in the section above. We solved Eq. [23] numerically using the Runge-Kutta method as implemented in standard procedures provided in Mathcad PLUS 6.0 (Mathsoft Inc., Cambridge, MA).

The initial water content and molalities of the saline and salt-free sections used in the model were calculated from the data of Wheeting and shown in Table 1. Other than the textural classification of the soils used by Wheeting, no other soil properties were given. The soil water properties were initially estimated using other soils with similar textural classifications for which the parameters in Eq. [18] were known. These initial parameters were adjusted incrementally about these initial estimates in the model to subjectively achieve the best fit results shown in Fig. 3 and 4. The same soil parameters were used for both salt treatments in the same soil (Table 1). All other parameters are physically based, depending on the particular salt and temperature, are summarized in Table 1.

	DISCUSSION

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The modeling and observed cumulative water transport, shown in Fig. 3 and 4, confirm that significant water-vapor transport occurs across the air gap can be attributed to gradients in osmotic potential. The theory developed shows water-vapor transport in the same magnitude as seen in Wheeting's original experiment. The model overestimates the vapor transport for the Na₂CO₃, medium sand treatment at 15 d (Fig. 3b). This may be attributed to the noninstantaneous redistribution of water in the nonsaline section in Wheeting's columns. Among all of Wheeting's experiments (Fig. 1 and 2) only Fig. 1b shows a dramatic decrease in water content of the soil adjoining the air gap after 15 d, with water distributed unevenly throughout the column violating this model assumption. The side of the column with no salt showed decreased water content at the interface, lowering the water potential and decreasing the vapor pressure gradient across the air gap, which decreased the vapor transport. Therefore, one would expect the present model, which assumes instantaneous redistribution of water in the soil to show higher vapor transport rates than observed by Wheeting.

Sodium carbonate caused less water vapor to be transported across the interface initially than the KCl in the medium sand (Fig. 3). The initial concentration of KCl in the sand was greater than the initial concentration of Na₂CO₃ (Table 1) consistent with the higher solubility of the KCl. Even though the Na₂CO₃ produces a slightly larger water vapor deficit than KCl at equal molalities, the increased solubility of the KCl allowed a greater vapor deficit than the Na₂CO₃ experiment column (Fig. 5).

Soil texture affected the rate of water-vapor transport because of the different matric potentials that could be developed in the soils (Fig. 3 and 4). This is masked somewhat because of the different initial water contents of the medium sand and the clay loam soils. Because salt was added to the columns on a dry weight basis, more dilution occurred with the clay loam columns as compared with the columns with the medium sand. This dilution caused the vapor pressure gradient to be much lower in the clay loam treatments resulting in a slower transport rate as compared with the drier medium sand columns.

To illustrate the effect of different salts on the water-vapor transport, the model was executed using a single set of initial conditions, with four different salts. The initial conditions of the medium sand at the volumetric moisture content of 0.045 m³ m^-3 and the hydraulic properties as shown in Table 1 were used with different salts. The concentration of salt in the saline section was treated with 1% (wt./wt.) salt solution of NaCl, KCl, NaBr, or KBr. Cumulative transport rates increase with decreasing molecular weight of the salt (Fig. 6) . Because salt concentrations were added on a percentage of weight bases, the molality of NaCl is greater than KCl, NaBr, and KBr, respectively. Maximum water-vapor transport rates across the 12.7-mm air gap during the first 5 d ranged from about 0.15 to 0.25 mm of water per day on an equivalent depth basis. These rates may be expected to be somewhat less in the soil without an air gap becuase of the tortuosity, but it can be expected to represent the magnitude of water vapor transport that can occur in unsaturated soils in the presence of salts at high concentrations. This can be significant in many situations such as dissolution of fertilizer in agricultural situations or movement of contaminants in the vadose zone.

View larger version (17K): [in this window] [in a new window]   Fig. 6. Predicted cumulative water vapor transport across a 12.7-mm air gap from salt-free section to saline section of a broken column in a medium sand.

	SUMMARY

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	ACKNOWLEDGMENTS

 
We acknowledge the Oregon State University Agricultural Research Foundation for financial support of this research.

Received for publication July 29, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS DISCUSSION SUMMARY REFERENCES

 

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This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via ISI Web of Science (4) Citing Articles via Google Scholar Google Scholar Articles by Kelly, S. F. Articles by Selker, J. S. Search for Related Content PubMed Articles by Kelly, S. F. Articles by Selker, J. S. GeoRef GeoRef Citation Agricola Articles by Kelly, S. F. Articles by Selker, J. S.