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ORIGINAL RESEARCH PAPERS

A Single Source–Cylindrical Soil Domain Model for Studying Simultaneous Controlled-Release and Mixing Processes

^a School of Mathematics and System Science, Shandong Univ., Jinan, 250100, P. R. China
^b Dep. of Mathematics and Computer Sci., Bar-Ilan Univ., Ramat Gan, 52900
^c Institute of Soil, Water, and Environmental Sciences, ARO, the Volcani Center, Bet Dagan 50250, Israel
^* Corresponding author (VWSFRIED{at}AGRI.GOV.IL).

Received 30 January 2003.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

 
Favorable application of controlled-release agrochemicals requires an understanding of the two simultaneous processes of release from the controlled-release sources into the surrounding soil, and the subsequent spread, uptake, and degradation in the soil profile. In previous research the release process was conceptually approximated by artificial decoupling of the diffusional release and vertical, convective–dispersive transport of the released material. The present article describes a new physical model that avoids this decoupling approximation and conceptualizes the complete three-dimensional, axisymmetric problem of diffusional controlled release and convective–dispersive vertical transport of the agrochemical in the soil. A numerical finite-element scheme was coded for solving the model equations, and it was employed to investigate the effects of the various quantifiers of the controlled-release source and the soil on the release process and, in particular, to study the interplay between the processes of release into the cylindrical soil domain and the convective–dispersive spread in the soil. The actual computations refer to the three-dimensional water velocity filed around an impermeable capsule and the resulting spreading mechanisms, but for the sake of brevity, the outline of the model equations refers to vertical water velocities throughout the cylindrical domain, disregarding radial water fluxes and the resulting vertical dispersive fluxes of the released material. In general, the release rate is expected to increase with increasing membrane conductance, capsule radius, capsule inner concentration, water flux, and the soil longitudinal and, especially, transverse dispersivities. A dimensional analysis of the problem provided a deeper understanding of the dependence of the release rates on these factors, and provided a distinction between the two extremes of slow capsule-controlled release and fast soil-controlled release.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

 
APPLICATION of agricultural chemicals (fertilizers and pesticides) often results in unintended pollution of precious soil and groundwater resources, and one way of coping with this problem is to use controlled-release formulations. The optimal design and application of controlled-release agrochemicals requires an understanding of the two simultaneous processes involved in the dispersion of the active compound in the soil: (i) the release of the compound into the surrounding soil and (ii) its subsequent dispersion, uptake, and degradation within the soil profile. The information derived from experimental studies of these processes is rather limited, as the spread of the released chemicals in the soil cannot be measured nondestructively and because some factors affecting the process cannot be controlled. Therefore, the complex problem of the interplay between the controlled release from the source and the spread of released material has to be studied via modeling and simulations as well. Previous modeling efforts either assumed that the rate of release was not limited by the spread of the agrochemicals in the soil (Jarrell and Boersma, 1979, 1980; Kochba et al., 1990; Hassan et al., 1992), neglected convection processes in assuming that the spread of the released material occurred solely by diffusion in the stagnant soil solution (Friedman and Mualem, 1994; Wang et al., 1998a), or assumed a macroscopic two-dimensional flow region and disregarded the three-dimensional distribution of the controlled-release sources in the soil (Bear et al., 1998; Wang et al., 1998b). In a previous article (Friedman, 1997), a general formulation of the release process was presented; it also accounted for the dispersion of the released agrochemical in the soil profile that results from downward (infiltration) and upward (evaporation) movement of water. The problem of vertical transport and degradation of the released material was solved numerically, and this was coupled to an analytical solution of the local release process in an equivalent spherical soil domain, which surrounds the membrane-controlled capsule. Thus, the real problem of simultaneous controlled release and spread of the agrochemical in the soil profile was conceptualized by artificial decoupling of the diffusional release and vertical, convection–dispersion transport processes at each time step of the numerical solution to the overall vertical transport problem. Namely, for calculating the release rate at each depth, Friedman (1997) neglected the effect of the vertical water flow and also assumed an approximately quasi–steady state diffusion process, so that the calculated release rate served as a source term in the one-dimensional, convection–dispersion transport equation (which also included a first-order decay term). This approach is somewhat similar to the solution of the problem of release from solubility-limited sources distributed on the microscale, with their release rate approximated by an equivalent, first-order chemical reaction (Jia et al., 1999), the principal difference being that the release rate in the Friedman's (1997) model depends also on the macroscopic characteristic of the mean distance between the release sources.

Our objectives are (i) to present a model that avoids the decoupling approximation applied by Friedman (1997), conceptualizing the complete (not decoupled) three-dimensional (axisymmetric) problem of a diffusional controlled-release and convective–dispersive vertical transport of the agrochemical in the soil, and (ii) to use this model to investigate the complex interplay between the release and spread processes. Related, and somewhat similar problems have been analyzed previously; one example is the spreading of material from a sphere (or a slab) dissolving in a packed bed (Coelho and Guedes de Carvalho, 1988a, 1988b; Guedes de Carvalho and Delgado, 1999; Jia et al., 1999), which involves the processes of free dissolution and convective spread of the dissolved materials. However, to the best of our knowledge, the present problem of release from a controlled-release source and the convective spread of the released material into a porous medium has not yet been formulated and analyzed systematically. Although the controlled-release problem analyzed here is not fully analogous to other environmental problems of release and leaching of pollutants from, for example, landfills and deteriorated, buried waste containers, some of the insight gained here is also relevant to the latter problems.

	MODEL EQUATIONS AND BOUNDARY CONDITIONS

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

 
The Cylindrical Release Domain of a Single Capsule
We refer to the problem of agrochemicals that are released from capsules uniformly distributed and arranged in a simple-cubic array in the soil (Fig. 1). To study the interplay between the release and spread processes we assume that there are enough release sources incorporated down the soil profile to enable us to refer to a release domain (left cube in Fig. 1) that is sufficiently deep for it to be reasonable to assume that the downward efflux of released material from that domain into its lower neighboring cube is similar to the influx coming from its upper neighbor. The problem of release and spread in the cubic soil domain (of side b_c) can be formulated and solved numerically in Cartesian coordinates. Nevertheless, to facilitate the formulation and solution of the problem and the presentation of results and analysis we approximate the actual cube by a confined cylindrical soil domain of radius R_s, in which R_s is determined by the average distance between adjacent capsules (Fig. 2). (See the Appendix for a complete list of variables used in this discussion.) We relate the cylinder radius (also half-length), R_s, to the cube side, b_c, as 2R_s = (2)^-1/3b_c 1.08b_c by assuming a cylinder whose diameter and length are equal (2R_s = 2X_s) and conserving the volume of the cube surrounding each capsule . We will use a cylindrical coordinate system, with its origin located in the center of the capsule, with the axial axis, x, considered to be positive upward, and the top and bottom faces of the cylinder located at x = +X_s and x = -X_s, respectively (Fig. 2).

View larger version (49K): [in this window] [in a new window]   Fig. 1. The simple cubic array distribution of the controlled-release sources in the soil, and a representative lower release domain of approximately similar influx and efflux of the released material.

View larger version (34K): [in this window] [in a new window]   Fig. 2. The assumed cylindrical release domain with boundary conditions of no flux through its vertical wall and periodic flux through its horizontal planes (right) and released material fluxes out of the capsule and in the soil (left).

The surface of the capsule is defined by (Fig. 2)

[1]

The local release rate (flux) out of the capsule (defined positive), q_c, is given by

[2]

where k_D is the overall membrane conductance of the capsule, determined by the thickness of the membrane, the diffusion coefficient through the membrane, and its external surface area. The boundary condition on the surface of the capsule has to maintain the continuity of the material flux on each side of the capsule membrane. The release flux in the soil domain J_c(x) is not normal to the capsule surface (because of water flow), but is directed at an angle (x) to the normal (Fig. 2), namely

[3]

in which the radial and vertical components of J are given by Eq. [8] and [9] below. The overall discharge of the released material at any given time is defined as the rate of decrease of the remaining encapsulated mass, which, for the case of no sinks or sources in the soil, also equals the rate of increase of the material spread in the soil domain:

[4]

This overall release rate at any given time can be evaluated by integrating the normal component of the local release flux [q_c(x) = J_c(x)cos()] over the capsule surface from the bottom to the top (as the release rates differ between the upstream and downstream surfaces). The mass of agrochemical released from the capsule, M_c(0) - M_c, is conserved by the continuity equation (Eq. [13]) of the spread problem described below.

Spread of the Released Material in the Cylindrical Soil Domain
At the beginning of the release process all the mass of the material to be released, M_c(0), is in the capsule core (M_s = 0), and the concentration in the cylindrical soil domain is zero:

[5]

It is assumed that the released material is spread in the soil domain by a combination of the mechanisms of molecular diffusion, mechanical dispersion, and convection. The cylindrical cell represents a relatively small fraction of the vertical soil profile (depending on the distribution density, N_c, of the capsules). Therefore, in the following treatment we regard the water content and water flux prevailing in the majority of the cylindrical soil domain as uniform and do not attempt to provide a solution to the water-flow problem. The water flow is regarded as an input of uniform water content and vertical water flux to the release problem, which also takes into account the disturbance by the small capsule of the water flow field around it. The general formulation of the model outlined here is for water contents that are uniform in space and can vary with time:

[6]

and similarly for the vertical water fluxes:

[7]

However, since the main objective of the present article is to investigate the interplay between the release and spread processes, we will restrict ourselves to a steady water flow in the form of a constant, vertically downward water flux. The steady and uniform volumetric water content may represent, for example, conditions of a unit gradient flow. It is assumed that the released material is spread in the cylindrical soil domain by a combination of the mechanisms of molecular diffusion and dispersive convection. The latter comprises convection by the mean water flux and mechanical dispersion (assumed to be Fickian, by analogy with molecular diffusion), stemming from the local, microscale distribution of pore-water velocities. Thus, the radial flux of the released material, J_r, is composed of both mechanical dispersion and molecular diffusion terms that combine as a hydrodynamical diffusive flux:

[8]

where the coefficient of hydrodynamic dispersion in the radial direction, D_r, accounts for the radial (transverse to the vertical direction of water flow) mechanical dispersion and molecular diffusion (which are assumed to be additive). The vertical flux, J_x, of the released material also comprises these two terms of mechanical dispersion (this time parallel to the direction of water flow) and molecular diffusion, and also the contribution of the mean vertical water convection:

[9]

The longitudinal (D_L) and transverse (D_T) mechanical dispersion coefficients are both assumed to depend linearly on the vertical water velocity, v_x (=q_x/), according to

[10]

where _L and _T are the longitudinal (parallel to the mean direction of the water flow) and transverse (perpendicular to this direction) dispersivities (mixing lengths), characterizing the soil. Because of the radial symmetry of the problem (axisymmetric), there are no planar (azimuthal) fluxes. The apparent molecular diffusion coefficient depends mainly on the water content of the soil, D_a = D_a(), and on some other soil properties. Here we will use the general expression of Friedman (1993):

[11]

D₀ denotes the molecular diffusion coefficient in free solution and _sat the porosity of the soil.

By inserting the radial and vertical material fluxes (Eq. [8] and [9]) into the continuity (mass conservation, without sources and sinks) equation:

[12]

we obtain the following equation that describes the spread of the released material in the soil:

[13]

Since the soil is assumed to be homogeneous (and isotropic), and since the vertical water velocity, v_x (flux, q_x = v_x) is space invariable in most of the release domain, all the parameters of the above second-order parabolic partial differential equation are space-invariant.

The cylindrical domain is supposed to represent the release volume of a single capsule. Therefore, the boundary condition on the external radial surface of the cylindrical cell is that the radial flux is zero:

[14]

In studying the effect of convection on the release rate, we do not wish to add the (real) effects of the entry of additional material, released in upper soil layers, through the top face of the cylinder, or of the leaching of some of the material released in the given cylindrical domain, through the bottom circular face, into lower layers. Therefore, we will impose periodical boundary conditions on the vertical fluxes, which means that the same material flux that exits through the bottom surface at a given radial location re-enters via the corresponding location in the top surface, as if the cylinder in question is placed between two neighboring cylinders in which the same conditions prevail. The vertical flux is both diffusive and convective (Eq. [9]), resulting in a periodical boundary condition of a third (Cauchy) type:

[15a]

Yet, because of the periodicity, the boundary condition on either the concentration

[15b]

or its derivative

[15c]

is sufficient.

For the sake of brevity, the above model equations were outlined for a vertical water velocity, v_x, uniform throughout the cylindrical domain (i.e., on the assumption of no radial water flow, v_r = 0 and disregarding the vertical dispersive fluxes of the released material that would result from such radial convection). This is a satisfactory approximation for the major part of the cylindrical soil volume, in which the distance from the capsule is sufficiently large. However, the presence of the capsule, assumed to be impermeable to viscous water flow, causes a disturbance to the vertical field, as depicted in Fig. 3. The velocity field (arrow lengths and directions) results from a solution of the Laplace equation for a distant uniform potential flow (according to Darcy's Law in our case) passing near a sphere of zero conductivity. It can be conveniently represented in spherical coordinates: u_r = -u₀cos()[1 - (R_c/r)³]; u = u₀sin()[1 + 0.5(R_c/r)³]; u = ^1/2, where r is the radial distance from the capsule's center, is the angle from the downward vertical line, and u₀ is the prescribed upward vertical flux far away from the capsule). At a sufficient distance from the impermeable capsule the disturbance to the flow is negligible. In the horizontal mid plane and at a distance, R_C from the capsule surface (R_R r/R_c = 2, X_X x/R_c = 0), for example, the velocity is vertical and is 6.25% higher than the background velocity. However, closer to the capsule, the disturbance to the local velocity field is more significant: On the capsule surface the velocity rises from zero (stagnation point) at the top of the capsule (R_R r/R_c = 0, X_X x/R_c = 1) to a maximal velocity, 1.5 times higher than the background velocity, at the equator (R_R = 1, X_X = 0) and diminishes back to zero at the bottom of the capsule (R_R = 0, X_X = -1). The average velocity on the capsule surface is 3/ times the background vertical velocity; that is, it is quite similar to it. Nevertheless, the velocity distribution and its lateral components affect the release process: directly by the convective diversion of the released material, and "indirectly" by the vertical, dispersive fluxes that result from the lateral convective water fluxes. All these fluxes are taken into account in the actual calculations of the model, but, solely for the sake of clarity and brevity, the above listing of the model equations omitted the corresponding cross terms and mixed derivatives.

View larger version (27K): [in this window] [in a new window]   Fig. 3. The potential flow velocity field around the impermeable capsule (arrow lengths and directions) for conditions of prescribed uniform vertical downward flow far from the capsule.

	DIMENSIONAL ANALYSIS

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

[16]

The resulting nondimensional variables, denoted by upper-case letters, are

[17]

the continuity equation for the spreading in the soil takes the form

[18]

in which the nondimensional parameters ( numbers; Bird et al., 1960) describing the radial (S_R) and longitudinal (S_L) spreading of the released material are defined as

[19]

(diverging for the case of stagnant water and solely diffusive spread).

The radius and half-length of the cylinder are defined to be R_S = X_S = 1/2, by the choice of the scaling length, and the rest of the nondimensional parameters (appearing in the boundary conditions) are

[20]

The dimensional analysis reveals that for the case of a constant vertical water flux (steady-state water flow) and a zero-order release mechanism [C_c(t) = C_c(0) = C_s), the controlled release and spreading problem, C(X,R,T) is characterized by six numbers: S_R, S_L, R_C, Q_X, K_D, and C_S. The radial spreading parameter (S_R) depends on the water flux, the volumetric water content, the apparent molecular diffusion coefficient, and the transverse dispersivity of the soil. The longitudinal spreading parameter (S_L, its reciprocal is usually termed the column Peclet number) is usually larger than the radial one, especially for high water fluxes, because the longitudinal dispersivity (_L) is usually larger than the transverse one (_T) by a factor of 3 to 10. The nondimensional water flux takes the simple form of the volumetric water content, , because the nondimensional time, T, is what is usually termed pore volumes, the number of times the soil solution was replaced by the inflowing solution. The nondimensional membrane conductance, K_D = , reflects the characteristics of a relative release rate, which increases as the residence time of the water in the column increases (t* = 2R_s/v_x) and decreases for larger release volumes .

On the basis of the ranges of possible orders of magnitudes of the dimensional parameters (Table 1), the ranges of possible orders of magnitudes of the nondimensional parameters are those listed in Table 2, which also presents the default values used for most of the computations described below.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

The dimensional version of the problem (Eq. [1]–[15]) was used for the actual computations, and the resulting dimensional outputs were retransformed to the nondimensional presentation mode. The following arbitrary set of dimensional parameters were used in all simulations, of which the results are presented in Fig. 4 to 14: X_s = 2 cm; R_s = 2 cm; _s = 0.5; D₀ = 10^-5 cm² s^-1; M_c = 0.01 g; v_x = 0.001 cm s^-1 (except for the case of low dispersion–diffusion numbers in Fig. 12 where v_x = 2 x 10^-6 cm s^-1). Then, after presetting the desired nondimensional parameters—R_C, K_D, C_C, Q_X, S_L, and S_R—the required additional dimensional parameters are computed from (Eq. [19] and [20]):

[21]

View larger version (19K): [in this window] [in a new window]   Fig. 4. Zero-order release curves (fraction released as a function of the number of the pore volumes that have passed through the soil) for two membrane conductance values, KD = 0.1 (left) and 0.0005 (right), and five capsule radii, RC = 0.0025, 0.0125, 0.025, 0.05 and 0.1, right to left for each KD group.

View larger version (77K): [in this window] [in a new window]   Fig. 14. The concentration fields for the time corresponding to release of 100% of the encapsulated material. The central concentration plot is for the default case of dispersion parameters of SL = 1 and SR = 0.25, the three vertical plots illustrate the effects of changes to SL, the three horizontal plots illustrate the effects of changes to SR, and the three plots along the diagonal illustrate the effects of changes to both SL and SR. The plotted contour lines represent concentrations of 1.6, 1.4, 1.2, and 1.0, respectively, progressing outwards from the capsule.

View larger version (15K): [in this window] [in a new window]   Fig. 5. First-order release curves for four membrane conductance values.

View larger version (17K): [in this window] [in a new window]   Fig. 6. Release curves for two values of membrane conductance, KD = 0.01 (left) and 0.001 (right), and four inner capsule concentrations, CC = 1, 2, 5, and 10, right to left for each KD group.

View larger version (25K): [in this window] [in a new window]   Fig. 7. Release time (nondimensional time required for the complete release of the encapsulated material) as a function of membrane conductance for various values of inner capsule concentration. The straight solid line represents the slope corresponding to a K-1D dependence.

View larger version (31K): [in this window] [in a new window]   Fig. 8. Release time as a function of capsule radius for various finite values of membrane conductance and for the asymptotic case of infinite membrane conductance. The straight solid line represents the slope corresponding to a R-1C dependence.

View larger version (58K): [in this window] [in a new window]   Fig. 9. Successive concentration fields in the cylindrical soil domain for a fast-release case. The four upper concentration plots are for release periods before the completion of the release process and the two lower plots for periods after the release had ceased. The values of plotted contour lines are 1.12 and 0.98 outward from the capsules for all the concentration charts.

View larger version (55K): [in this window] [in a new window]   Fig. 10. The concentration plots in the cylindrical soil domain during the course of the release process for a slow-release case.

View larger version (17K): [in this window] [in a new window]   Fig. 11. Release curves for five values of nondimensional water flux, QX, (volumetric water content), under conditions of slow release (KD = 0.001).

View larger version (18K): [in this window] [in a new window]   Fig. 12. Release curves for different values of nondimensional water flux, QX, under conditions of fast release (KD = 0.1). The solid lines represent cases of high dispersion/diffusion ratios (vx = 0.001 cm s-1) and the dashed lines low dispersion/diffusion ratios (vx = 2 x 10-6 cm s-1).

View larger version (17K): [in this window] [in a new window]   Fig. 13. Release curves for the default case of dispersion parameters of SL = 1 and SR = 0.25 (thick solid line), and the release curves for the cases of changes to SL (dashed lines), SR (dotted lines), and both SL and SR (thin solid line).

Zero-Order and First-Order Release Processes
The concentration inside the capsule, C_C, varies in one of the three following ways, depending on the agrochemical solubility in aqueous solution (Friedman and Mualem, 1994; Friedman 1997):

1. In the case of low solubility, the concentration of the agrochemical is always equal to its saturation concentration, C_C = C_S, which dictates a zero-order release.
   
2. In the other extreme case of very high solubility, when all the active ingredient is dissolved in the capsule core, its concentration, C_C(t), starts at a maximum, C_C(0), and diminishes with time, resulting in a first-order release.
   
3. The general intermediate case is that of limited solubility, characterized by a two-stage release: initially, the concentration of the agrochemical in the solution of the core is constant, C_S, and the release is zero-order as long as solid phase agrochemical is present in the core; later, when all the solid phase has been dissolved, the concentration in the capsule starts to decline, and the release process is first order.
   

Figure 4 presents zero-order release curves for two values of membrane conductance, K_D—0.1 and 0.0005—and five capsule radii, R_C—0.0025, 0.0125, 0.025, 0.05, and 0.1—for each K_D group. The values of the other four numbers that define the problem are C_C = 10, Q_X = 0.2, S_L = 1, and S_R = 0.25. The release curves are almost straight lines, indicating that the release rates are almost constant during the course of the release process; this is because of the relatively high concentration within the capsule, which is 10 times as high as the final concentration in soil after the release of all the material.

Figure 5 demonstrates the release pattern of the other extreme case of a first-order release process. In this case the initial capsule concentration C_C(0) = 10⁴, and it diminishes to 1 in the course of the release process, which is why the release curves have negative curvature. Figure 5 is the only representation of first-order release curves, and all the figures discussed below and the discussions themselves address only zero-order release processes.

Effect of Capsule Factors (R_C, K_D, C_C) on the Release Process
The release rates, represented by the slopes of the release curves of Fig. 4, for example, obviously increase as the membrane conductance, K_D, and the capsule radius, R_C increase. The effect of the membrane conductance, whose value ranges over two orders of magnitude, 10^-7 to 10^-5, is also clearly seen in Fig. 5. Figure 6 presents the effect of the third capsule quantifier, namely the within-capsule concentration, C_C, which is time-invariable for zero-order release. The release rates increase and change less in the course of the release process (straighter lines) as C_C increases. As C_C approaches unity, the time required for the release of all the encapsulated material approaches infinity, which is why these lines approach the completion of the release process asymptotically. For all other C_C values the release time is finite. Figure 7 presents the dependence of these finite release times on K_D for various C_C values, with R_C held constant at 0.05. The data were taken from some of the zero-order release curves of Fig. 4 and 6 and from other simulated release processes not presented here. The release time obviously decreases with increasing membrane conductance and within-capsule concentration, and the effect of K_D is similar for all C_C values. Smaller values of nondimensional membrane conductance represent cases of slow release processes in which the release rate is controlled by the capsule and not by the subsequent spreading in the soil. Under these conditions the release rate, which for sufficiently high C_C values is related approximately reciprocally to the release time, seems to be directly proportional to the membrane conductance, K¹_D (parallel to the straight solid line). In the other extreme cases of large values of K_D, corresponding to "fast release" that is controlled by the spreading in the soil profile, the dependence of the release time on K_D involves a smaller power of K_D, about K^2/3_D for K_D = 0.1 (Fig. 7). Plotting the data of Fig. 7 as release times against C_C for various K_D values would reveal that for any K_D the release time diverges to infinity as C_C approaches 1 and should, in principle, diminish to 0 as C_C approaches infinity. As C_C increases, its relative effect on the increasing release rates diminishes, as can be seen by, for example, comparing the effect of increasing C_C from 2 to 5 with that of increasing it from 5 to 10 (Fig. 7).

The effect of the capsule radius, R_C, on the release rate is shown in Fig. 8, which presents the nondimensional release time as functions of capsule radius for various values of membrane conductance, K_D. K_D itself already contains the dependence of the release rate on the capsule surface area, which is proportional to R²_C. Therefore, since each line in Fig. 8 represents the variation of release rate with R_C for a particular value of K_D (held constant), the varying R_C value reflects only the ability of the soil to absorb the material released from a spherical source of radius R_C and not the R²_C dependence of the membrane conductance. The release time seems to decrease with increasing R_C until it reaches a finite value, which depends on K_D, when R_C approaches its maximum possible value of R_C = 0.5 (when the capsule radius equals that of the cylindrical soil domain, the dimensional r is scaled by 2R_s). For slow release (e.g., K_D = 0.0001), the dependence of the release rate on R_C is very weak since it is the membrane conductance (held constant) that controls the release and not the spreading in the soil. The pattern is different for the case of infinite membrane conductance represented here by the cases of K_D = 10 and K_D = 100 (lowest line in Fig. 8); since these yield the same results, they can be safely regarded as infinite for practical purposes. In the case of infinite K_D and small R_C, the release time is inversely proportional to the capsule radius; that is, R^-1_C (the release time line is parallel to the straight solid R¹_C line), and as R_C increases toward 0.5, the release time seems to diminish to 0, as opposed to a finite value for the finite K_D cases. A release time proportional to R^-1_C means a release rate (M T^-1) proportional to R¹_C. This asymptotic R¹_C–dependent release rate is similar to the case of diffusional release into a stagnant soil solution from an encapsulated (Friedman, 1997) or a freely soluble spherical source. For the case of negligible longitudinal dispersion, as compared with the lateral dispersion, when R_C >> _L (not analyzed in this discussion), the release rate has been found to depend on a higher power of R_C, that is, R^3/2_C (Coelho and Guedes de Carvalho, 1988b).

In addition to presenting and discussing the effects of the various influential factors on the release curves (Fig. 4, 5, and 6), it is also worthwhile to analyze the way these quantifiers affect the concentration fields during the course of the release process. The dynamic concentration distribution in the soil domain is also of practical importance, since the efficacy of agrochemicals (mostly pesticides and fertilizers) depends not only on their average concentrations in the soil (which are related to the amounts released); it also depends on the maintenance of sufficiently high concentrations in the whole volume of the root zone through the whole of the growing season (Shaviv and Mikkelsen, 1993; Scher, 1999). From both the agricultural and the environmental points of view, it is also sometimes required that those concentrations do not exceed designated upper limits. The concentration plots in Fig. 9 and 10 are meant to demonstrate the different concentration fields resulting from the two extreme cases of (nondimensionally) fast (Fig. 9) and slow (Fig. 10) release processes. The six numbers, chosen in combination with K_D = 0.01, for the simulation presented in Fig. 9, represent a relatively fast release process, in which the release from the capsule to the surrounding soil is fast and the rate-limiting process is the convective–dispersive spreading of the released material in the soil. In this situation a major proportion of the released material is located around the capsule and there are appreciable concentration gradients from the capsule vicinity outwards into the cylindrical soil domain. This can be seen in the four upper concentration plots, which show the distributions at four successive time points (T = 0.48, 0.96, 1.48, 1.99) up to the completion of the release process at T = 1.99. After completion of the release process the material continues to spread in the soil until a uniform concentration field, in which C(X,R) = 1 by definition, fills the whole of the cylindrical release domain, as illustrated in the two lowest concentration plots. The concentration fields in the other extreme case, slow release, are illustrated in Fig. 10. In this case, the controlled release from the capsule is slow compared with the spreading in the soil; therefore, the material released from the capsule immediately spreads in the soil, and the resulting concentrations are uniform in space with very small outwards gradients, and they increase as time progresses, to match the relative amount released.

Effects of Soil Convective (Q_X) and Dispersive (S_L, S_R) Factors on the Release Process
Increasing the prescribed (dimensional) water flux, v_x, will, of course, increase the (dimensional) release rate, because of the enhanced outward spread of the released material by convective–dispersion mechanisms, as illustrated below in the discussion of the effects of S_L and S_R. However, since the chosen scaling time is the convection residence time, 2R_s/v_x, this trivial effect of v_x is obscured when referring to the nondimensional release curves and concentration fields. In the nondimensional mode of analysis, the water flux, Q_X, has the meaning of the volumetric water content of the soil, and its expected effect is, therefore, different. Figure 11 illustrates this effect of the volumetric water content for conditions of relatively slow release (K_D = 0.001) and when the mechanical dispersion spread (_Tv_x and _Lv_x) is significant compared with the molecular diffusion spreading (D_a/). In this case, as the volumetric water content increases, the dimensional capsule concentration, c_c must decrease (Eq. [20]) to keep C_C constant. Therefore, the release rate slows down, approximately in inverses proportion to (Q_X) for the present example of C_C = 5. The effect of Q_X is different for the other extreme conditions of fast release (K_D = 0.1), as illustrated in Fig. 12. In this case, the release is controlled by the spread in the soil and not by the capsule. Therefore, the tenfold decrease of the within-capsule concentration, c_c, while Q_X increases from 0.05 to 0.5 (with C_C held constant) slows down the release rate by a factor of only 1.5, as compared with a factor of about 9 for the case of slow release (Fig. 11). A second feature observed in fast-release (soil-controlled) conditions is the effect of two additional numbers that reflect the ratios of dispersive to diffusional longitudinal [_Lv_x/(D_a/)] and transverse [_Tv_x/(D_a/)] spreading of the released material. The actual problem of nonvertical water flow around the capsule is now defined by eight rather than six numbers (Eq. [19] and [20]), as a result of the concise, approximated vertical water flow formulation presented above. The effect of these two additional numbers of mechanical dispersion/molecular diffusion flux ratios, mostly that of the transverse one [_Tv_x/(D_a/)] is reflected in the nondimensional presentation (Fig. 12) as follows: as Q_X () increases, D_a/ must decrease (Eq. [11]) and, since S_L and S_R are held constant, this means that _L and _T decrease (Eq. [21]), slowing down the release rates, in addition to the above-mentioned effect of decreased capsule concentrations. For high dispersion/diffusion numbers (high water velocities, v_x = 0.001, solid lines) and low water contents, this effect is negligible, but for low dispersion/diffusion numbers and high water contents, when _Tv_x (and _Lv_x) are comparable with D_a/ (low water velocities, v_x = 2 x 10^-6, dashed lines) this effect is observable. It should be noted here that the release simulations presented in the other figures all refer to conditions of high dispersion/diffusion numbers.

In all the release simulations discussed above, the nondimensional spreading parameters were held constant at S_L = 1 and S_R = 0.25. Now we are going to demonstrate the effects of altering these two spreading parameters, which represent both diffusive and dispersive fluxes (Eq. [19]), on the release process. It should be recalled that S_L and S_R are defined in the cylindrical release domain scale in the sense that they relate to the assumption of vertical water flow, as manifested in the approximate form of the transport equation (Eq. [18]), and that in the near vicinity of the capsule, the existence of radial flow components (Fig. 3) necessitates, in principle, a more detailed analysis of the effects of the dimensional spreading parameters, _L, _T, and D_a. Figure 13 presents the release curves for the default case of dispersion parameters S_L = 1 and S_R = 0.25 and for three sets of simulations in which, respectively, just S_L, just S_R, and both S_L and S_R were changed. Decreasing S_L to 0.25 or increasing it to 2.5, with S_R (and R_C, K_D, C_C, Q_X) unchanged, caused smaller effects on the release rates (dashed lines) than decreasing S_R to 0.1 or increasing it to 1, with S_L constant (dotted lines). The reason for this is that since the water-flow streamlines are tangential to the capsule surfaces (Fig. 3), the lateral (radial) dispersion (_T) is more effective in removing the released material from the capsule than the longitudinal dispersion (_L); the latter spreads the material along the capsule surfaces and not perpendicularly to them. The more significant effect of the lateral dispersion was previously demonstrated by Coelho and Guedes de Carvalho (1988b), who analyzed free spreading from a dissolving spherical source. Decreasing both parameters by a factor of 10, to S_L = 0.1 and S_T = 0.025, or increasing them tenfold, to S_L = 10 and S_T = 2.5, has a significant effect on the release rates (Fig. 13). The simulations referred to in Fig. 13 are of fast-release processes, completed within less than or very few pore volumes passing through the soil. In the other extreme situation of slow release, S_L and S_R are expected to have negligible effects on the release process, since the release and spreading are controlled by the capsule membrane. However, in intermediate cases, not illustrated here, when a greater number of pore volumes passes through the release domain, the differences in the effects of S_L and S_R are expected to be even higher, since the material which spreads vertically (longitudinally) reaches the vicinity of the capsule again in the next cycle (after a period, T = 1), as a result of the assumed cubic array distribution of source capsules and the prescribed periodic boundary conditions (Fig. 1 and 2). A more visual insight into the effects of altering S_L and S_R is given in Fig. 14, which presents the concentration fields at times corresponding to complete release of the encapsulated material for the above seven S_L–S_T cases. The central concentration plot is for the default case of S_L = 1 and S_R = 0.25; it shows the large concentration gradients typical of the present fast-release case. The series of three vertical plots of concentration distribution illustrate how changes in S_L extend (S_L = 2.5) or restrict (S_L = 0.25) the plume of released material. The series of three horizontal plots illustrate the more significant removal (S_R = 1) or accumulation (S_R = 0.1) of the released material caused by increasing or decreasing S_R. The series of three plots along the diagonal illustrate the combined effects of altering both S_L and S_R. For the low spreading case (S_L = 0.1, S_T = 0.025), after 5.93 pore volumes have passed through the cylindrical soil domain, most of the released material is still enclosed within a small volume surrounding the capsule, with a downward bias because of the vertical water flow. In contrast to this, for the intensive dispersion case (S_L = 10, S_T = 2.5), the release is already complete after T = 0.425, and the released material is distributed almost homogeneously in the soil domain.

	CONCLUSION

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

 
A new physical model describing the controlled release of agrochemicals from a diffusional, membrane-controlled source into the soil was formulated, and a numerical finite-element scheme was coded for solving its equations. This release model was employed to investigate the effects of the various quantifiers of the properties of the controlled-release source and of the soil on the release process and, in particular, to study the interplay between the processes of release of material into the cylindrical soil domain and its convective–dispersive spreading in the soil. In general, the release rate is expected to increase as each of the following parameters increases (with the other parameters held constant): membrane conductance, capsule radius, within-capsule concentration, water flux, and the longitudinal and, especially, transverse dispersivities of the soil. A dimensional analysis of the problem provided a deeper understanding of the dependence of the release rates on these factors and facilitated the distinction between the two extreme cases of slow, capsule-controlled release and fast, soil-controlled release. For controlled-release formulations of low membrane conductance ("slow" release) the release rate is proportional to the membrane conductance, but for relatively high membrane conductance ("fast" release), the diffusional resistance of the soil also impedes the release processes. In the latter, fast release, conditions the release rate (for a given membrane conductance) also increases with the capsule radius. It should be noted that the single capsule–cylindrical soil domain model presented here is not meant to describe the transient controlled release and leaching processes occurring in the soil profile, but solely the interplay between local processes of diffusional release and convective–dispersive spread. To model the long-range soil profile processes a local-temporal release rate calculated by the model presented here should serve as a source term in a vertical transport equation, similar to the model proposed by Friedman (1997). In subsequent work we intend to present extensions to the present model and release simulations that demonstrate the effects of adsorption and degradation of the released material and of periodic water flow on the release process.

	APPENDIX

TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

 

• b_c distance between two adjacent capsules (L)
  
• c concentration of the agrochemical in the solution of the soil cylinder (M L^-3)
  
• c_c agrochemical concentration in the capsule core (M L^-3)
  
• c_s saturation concentration of the agrochemical in aqueous solution (M L^-3)
  
• D₀ molecular diffusion coefficient of the agrochemical in free solution (L² T^-1)
  
• D_a apparent molecular diffusion coefficient of the agrochemical in the soil (L² T^-1)
  
• D_L longitudinal mechanical dispersion coefficient (L² T^-1)
  
• D_T transverse mechanical dispersion coefficient (L² T^-1)
  
• D_x vertical hydrodynamic dispersion coefficient (L² T^-1)
  
• D_r radial hydrodynamic dispersion coefficient (L² T^-1)
  
• J_x vertical flux of the agrochemical in the soil cylindrical cell domain (M L^-2 T^-1)
  
• J_r radial flux of the agrochemical in the soil cylindrical cell domain (M L^-2 T^-1)
  
• k_D membrane conductance (L³ T^-1)
  
• M_c mass of agrochemical encapsulated in each capsule (M)
  
• M_s mass of agrochemical in the solution of the soil cylindrical domain (M)
  
• N_c capsule distribution density in the soil (number of capsules per unit volume) (L^-3)
  
• Q_c release rate from the capsule (M T^-1)
  
• q release flux from the capsule (M L^-2 T^-1)
  
• q_c normal component of the release flux from the capsule (M L^-2 T^-1)
  
• q_x vertical water flux in the soil cylindrical domain (L T^-1)
  
• R_s radius of the cylindrical soil cell domain (L)
  
• R_c radius of the capsule (L)
  
• r radial distance from the axis of the cylinder (L)
  
• t time since the beginning of the release process (T)
  
• v water velocity in the soil (L T^-1)
  
• X_s half-length of the cylindrical cell domain (L)
  
• x upward distance from center of capsule (L)
  
• volumetric water content
  
• _sat saturated volumetric water content (porosity)
  
• _L longitudinal mechanical dispersivity (L)
  
• _T transverse mechanical dispersivity (L)
  

	ACKNOWLEDGMENTS

 
This study was supported in part by grant 1317 from the Israeli Ministry of Science, Culture, and Sport. Aijie Cheng is also grateful to the Fred and Barbara Kort Sino-Israel Postdoctoral Fellowship Program of the Bar-Ilan University for supporting his stay in Israel. Contribution no. 606/03 from the Agricultural Research Organization.

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TOP ABSTRACT INTRODUCTION MODEL EQUATIONS AND BOUNDARY... DIMENSIONAL ANALYSIS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

 

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