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DIVISION S-3—SOIL BIOLOGY & BIOCHEMISTRY

Modeling the Effects of Diffusion Limitations on Nitrogen-15 Isotope Dilution Experiments with Soil Aggregates

Dep. of Geosciences, Oregon State Univ., Corvallis, OR 97331

* Corresponding author (john.cliff{at}oregonstate.edu)

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
An assumption inherent in isotope dilution methodologies is that of homogeneous distribution of label. This assumption may not hold, however, because of mass transfer limitations in most soil systems. The effects of mass transfer limitations on isotope dilution in soil aggregates with radii up to 0.36 cm were examined using spherical diffusion-reaction models designed to simulate ¹⁵NH⁺₄ isotope dilution experiments measuring gross production and consumption of NH⁺₄ across a 24-h period. Equations that described transport and reaction of NH⁺₄ assumed Fickian diffusion, linear, equilibrium adsorption, zero-order production of natural abundance ¹⁵N, and either pseudo-first-order or zero-order consumption of NH⁺₄. In the case of pseudo-first-order consumption, rate calculations were sensitive to the adsorption coefficient (emphasizing the need to interpret results as apparent rates), but not to other transport parameters. In the case of zero-order consumption, both production and consumption rates were always underestimated. Errors increased as aggregate size increased and as effective diffusivity decreased. Increasing the consumption to production rate ratio increased the error in production rate estimates. Allowing the applied label to diffuse into soil aggregates for 24 h prior to initial time sampling decreased errors by a factor of about three (to <-8% relative error) in the largest aggregate size class. These simulations reemphasize the need to optimize experimental protocol when designing isotope dilution experiments in structured soils and suggest an equilibration period prior to initial time sampling will improve accuracy of reaction rate estimates obtained from isotope dilution experiments.

Abbreviations: CENIT, Central Nitrogen Experimental site

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
THE USE OF ISOTOPE DILUTION METHODOLOGY to study N cycling has been available to soil scientists for almost 50 yr (Kirkham and Bartholomew, 1954, 1955); however, widespread use of this technology awaited technical advances to reduce cost and increase the precision of ¹⁵N analyses. During the last 15 yr, these advances have fueled insightful research about N cycling in soils. These studies have shown that: (i) gross rates of production and consumption of NH⁺₄ and NO^-₃ are often much higher than net rates, (ii) production and consumption are often nearly balanced, resulting in rapid turnover of the inorganic pools, and (iii) competing processes, such as NH⁺₄ immobilization and nitrification or NO^-₃ and NH⁺₄ assimilation, may occur concurrently in the same soil volume (Davidson et al., 1990, 1992; Hart et al., 1994a; Chen and Stark, 2000).

Isotope dilution experiments typically involve adding isotopically-labeled product to a chemical pool and tracking both pool size and isotope ratio of the pool as a function of time (Kirkham and Bartholomew, 1954, 1955; Hart et al., 1994b). This approach is currently preferred over tracer experiments to estimate reaction rates because it is thought that inflating the product pool may have less effect on observed rates than inflating the reactant pool (Hart et al., 1994b). Although users of the isotope pool dilution procedure assume that label is homogeneously distributed throughout the soil, there are differences of opinion about the importance of this matter. For example, Davidson et al. (1991) showed that homogeneous distribution of label is not necessary for accurate experimental results provided that the label is present in areas of microbial activity. Monaghan (1995), however, combined actual measurements of N isotope distribution in soil cores with mathematical modeling of isotope dilution experiments to show that heterogeneous distribution of label could be a significant source of error in these experiments. On the basis of an initial ¹⁵NH⁺₄:¹⁴NH⁺₄ ratio and assuming zero-order consumption of NH⁺₄ in gross mineralization assays, Watson et al. (2000) measured a disproportionately high amount of label applied as ¹⁵N in the NO^-₃ pool. They suggested that this would lead to erroneously high gross rate estimates, and attributed this phenomenon to greater availability of ¹⁵NH⁺₄ label than the native NH⁺₄ pool to microorganisms.

Under many circumstances, diffusion limits the availability of substrate to microorganisms in soils (Greenwood, 1961; Myrold and Tiedje, 1985; Priesack, 1991; Focht, 1992), and may prevent homogeneous labeling of soils with ¹⁵N. Although the use of gaseous forms of ¹⁵N has been proposed as a strategy to facilitate label distribution through the soil (Stark and Firestone, 1995; Murphy et al., 1997, 1999), these methods are probably most effective in relatively dry soils. Thus, aqueous labeling is still the dominant method reported in most studies. Because of the uncertainties associated with diffusional constraints on label distribution and calculation of gross rates of N transformations, our objective was to apply a modeling approach and estimate the error associated with diffusional constraints on aqueous ¹⁵N labeling of soil aggregates and its impact on gross rate estimates of N transformations.

	METHODS

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
The models presented here are meant to represent a bench-scale experiment in which aggregated soil is spread out and misted with an aqueous solution containing ¹⁵N. For simplicity, we consider the special case of adding highly labeled ¹⁵NH⁺₄ to measure gross rates of organic N mineralization and NH⁺₄ consumption. We consider two extreme cases where N consumption may be described by zero-order kinetics or by pseudo-first-order (hereafter referred to as first-order) kinetics. In both cases, NH⁺₄ production is assumed to be zero-order. For first-order consumption of NH⁺₄, equations that describe the rate of change of ¹⁵N label and NH⁺₄ pool size and that assume homogeneous distribution of label are

[1]

and

[2]

with initial conditions that

[3]

and

[4]

Nomenclature for these and subsequent equations is provided below. These equations differ from those of Kirkham and Bartholemew (1955) because Eq. [1] and [2] consider mixed first-order and zero-order kinetics and set the isotope signature of mineralizing N at a fixed value. Solutions to Eq. [1] and [2] as well as subsequent equations are provided in the Appendix. For zero-order consumption, the equations are

[5]

and

[6]

with identical initial conditions to those for Eq. [1] and [2]. Equations [5] and [6] differ from those of Kirkham and Bartholemew (1954) only because they allow mineralizing N to have a fixed atom % ¹⁵N greater than zero.

In this analysis, we assume that highly labeled ¹⁵NH⁺₄ is applied uniformly to the outside of a spherical aggregate (Fig. 1) . Once applied, the ¹⁵NH⁺₄ undergoes radial diffusion, consumption, and isotope dilution by NH⁺₄ mineralizing from organic N. The NH⁺₄ diffusion rate is modified by the effects of adsorption, production and consumption, and volumetric water content.

View larger version (18K): [in this window] [in a new window]   Fig. 1. Graphical depiction of a composite spherical soil aggregate which has been labeled with 15NH+4. The total aggregate radius (r) is represented by b. At the moment of labeling, the unlabeled portion of the aggregate exists at 0 < r < a, and the labeled portion of the aggregate exists at a < r < b.

Under these circumstances, the equations describing isotope transport as a function of time and aggregate radius take the form for first-order consumption of NH⁺₄ of:

[7]

and

[8]

and for zero-order consumption of NH⁺₄ of:

[9]

and

[10]

Volumetric water content () modifies the first-order consumption rate constant in Eq. [7] and [8] because the reaction occurs in the solution phase; it does not modify the zero-order consumption reaction rate constants in Eq. [9] and [10] because we chose to define those constants in relation to soil volume.

Initial and boundary conditions are similar for the reactive-transport equations for either N isotope and for either set of kinetic assumptions.

Initial conditions are given by constant solute concentration in the outer and inner spheres (Fig. 1):

[11]

and

[12]

with boundary conditions

[13]

[14]

[15]

and

[16]

Base model parameters used in the following simulations are presented in Table 1.

Case 2: Zero-Order Consumption
In the case of zero-order consumption, solutions for ¹⁴NH⁺₄ and ¹⁵NH⁺₄ concentrations were arrived at using the Crank-Nicolson method of finite-difference approximation (Crank, 1975). A value of r was chosen to provide a realistic value for total label mass at the appropriate concentration of label added (Table 1). Values of t were chosen so that

Sensitivity analysis showed this value produced an integrated mass error of <±0.7% between finite-difference approximations and special cases in which no reaction occurred or where k_cA = 0 µg cm^-3 d^-1 and k_pA = 1 µg cm^-3 d^-1 (data not shown). Under these conditions, this mass error produced a maximum relative error in k_cA of +2.1 x 10^-5 µg cm^-3 d^-1 and in k_pA of +1.5%.

Error Estimates
Error estimates in simulations of isotope dilution experiments due to heterogeneous distribution of label were produced for both production and consumption terms for both first-order and zero-order cases. The appropriate diffusion reaction equations were solved for ¹⁴NH⁺₄ and ¹⁵NH⁺₄ at appropriate times and radii. At appropriate times, masses of inorganic ¹⁴NH⁺₄ and ¹⁵NH⁺₄ (both adsorbed and aqueous phases) were integrated across the entire volume of the aggregate. This had the effect of simulating extraction of NH⁺₄ at t = 0 and t = final time points in an isotope dilution experiment. Two hypothetical, 24-h experimental procedures were simulated. Most simulations mimicked the effect of allowing label to diffuse for 24 h before the t= 0 extraction so that masses of ¹⁴NH⁺₄ and ¹⁵NH⁺₄ were integrated at t = 24 and t = 48 h. Some experiments were also simulated from t = 1 h to t = 25 h. The 1-h initial time point was chosen to simulate the length of time needed for laboratory manipulations before the initial extraction of inorganic N. After ¹⁵NH⁺₄ and total NH⁺₄ masses were ascertained at appropriate times, these values were substituted into the solutions to the appropriate equations assuming homogeneous labeling (see Appendix). The resulting rate constants were back-calculated and were compared with the diffusion-reaction simulation input values.

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
In this analysis, we purposely made the model input parameters and assumptions either as realistic as possible or to bracket the parameters likely to be found in soils. Although the assumption of a perfectly spherical aggregate is unrealistic, it is not necessarily a prerequisite to valid approximations using diffusion models. For example, Rao et al. (1982) showed that solute diffusion out of cube shaped aggregates could be approximated by a spherical diffusion model.

In our laboratory experiments, we often wet up soils for a week prior to labeling. At this time, we typically label with enough solution to increase water content from about 50% to 60% water-holding capacity. Pre-wetting is often desirable because rapid increase of soil moisture may cause substantial increase in mineralization rates (Seneviratne and Wild, 1985; Cabrera and Kissel, 1988). The exclusive use of diffusion as the primary mechanism of mass transfer limits the models employed here. For example, adding aqueous label to unsaturated soils will invariably cause a gradient in water potential and advective transport along the water potential gradient will increase label movement into the aggregate. Thus, the use of diffusion as the sole transport mechanism in this model will exaggerate label heterogeneity.

Although we have tried to use realistic parameters for factors that modify diffusion rate in soils, effective diffusion coefficients for NH⁺₄ or NO^-₃ at small scale in undisturbed aggregates have not been measured. In lieu of using empirically obtained effective diffusion coefficients, we chose to modify the effective diffusion coefficient by considering the effects of impedance and linear equilibrium adsorption. We used an impedance factor that is consistent with values found in the literature (Nye and Tinker, 1977; Olesen et al., 1996). Using concentration ranges from 0 mg L^-1 to 114 mg L^-1, we have measured linear adsorption coefficient (K_d) values of 5 cm³ g^-1 for a Pachic Ultic Argixeroll (33% sand, 51% silt, and 16% clay) and 13 cm³ g^-1 for an Andic Haplumbrept (54% sand, 19% silt, and 27% clay). The impact of varying K_d values on concentration profiles of a 0.356-cm-radius aggregate is presented in Fig. 2 . Increasing K_d has the effect of slowing down diffusion and even relatively small values for K_d preclude homogeneous distribution of label after 48 h. Therefore, assuming that diffusion is the primary mechanism of transport, label is not expected to be homogeneously distributed in large aggregates within 48 h.

View larger version (16K): [in this window] [in a new window]   Fig. 2. Effect of adsorption without reaction on concentration profiles of 15NH+4 48 h after labeling an aggregate with a 0.356-cm radius. Numbers on graph represent Kd values.

View this table: [in this window] [in a new window]   Table 2. Comparison of errors associated with retardation of label transport into a 0.356-cm-radius aggregate under conditions of first-order consumption. Errors are calculated for model simulations in which kcA consumption constant, s-1 for first-order consumption) is not adjusted for retardation, or adjusted for retardation by multiplying by volumetric water content x retardation factor (R). Model parameters not listed are reported in Table 1.

Adsorption Coefficient
Figure 3A illustrates the effect of K_d on the error associated with k_cA and k_pA during an experiment lasting from t = 24 to 48 h in a 0.356-cm-radius aggregate. As K_d increases, effective diffusion rate decreases, and the relative errors in both k_cA and k_pA increase. Because the highest adsorption coefficient we measured in our soils was 13 cm³ g^-1, our model predicts that error due to diffusion limitation should be minimal under conditions of zero-order consumption. In all cases, except where K_d = 0 cm³ g^-1, the relative error due to diffusion limitation is negative. Because simulations show both ¹⁴NH⁺₄ and ¹⁵NH⁺₄ to be homogeneously distributed in concentration profiles throughout the aggregate at t = 24 h when K_d = 0 (data not shown), the small positive error in the estimate of k_cA must be due to numerical error associated with taking 1.08 x 10⁷ time steps necessary to keep

View larger version (32K): [in this window] [in a new window]   Fig. 3. Effect of Kd (A), aggregate radius (B), initial incubation time (C), and kcA:kpA ratio (D) on errors in zero-order consumption and production estimates in isotope dilution experiments. Unless noted otherwise, experiments lasting from 24 to 48 h after labeling were simulated and used base model parameters presented in Table 1.

Pre-Incubation of Label
Figure 3C illustrates the effect of pre-incubation on the relative error of k_cA and k_pA caused by diffusion limitations. These results suggest that incubation prior to taking t = 0 samples may greatly improve reliability of rate estimates, however, pre-incubation protocols must be balanced with the need to keep the target pool labeled highly enough for statistically different results from background values as well as with the need to minimize recycling of label through the organic pool.

k_cA:k_pA Ratio
Figure 3D illustrates the effect of the ratio of rates of consumption and production on the relative error of k_cA and k_pA due to diffusion constraints in an experiment lasting from 24 to 48 h. An increase in the ratio of k_cA:k_pA greatly increases the error associated with NH⁺₄ production. This is caused by consumption of ¹⁵NH⁺₄ lowering the atom% ¹⁵N of the NH⁺₄ pool, and because k_pA is constant, this causes a net decrease in the overall pool size.

Although from the standpoint of mathematics, adsorption has no effect on overall consumption rate in the case of zero-order kinetics, it has an indirect effect of limiting diffusion and thus altering the ¹⁵NH⁺₄:¹⁴NH⁺₄ ratio as a function of aggregate radius. Figure 4 compares simulated concentration profiles of ¹⁴NH⁺₄ and ¹⁵NH⁺₄ in a 0.356-cm-radius aggregate in which diffusion and adsorption act alone (without reaction) or diffusion and adsorption act in concert with zero-order production and consumption. As can be seen, the amounts of ¹⁴NH⁺₄ and ¹⁵NH⁺₄ produced and consumed are not homogeneous throughout the aggregate. This is a consequence of the concentration terms which modify k_cA to adjust for the ratio of ¹⁵NH⁺₄:¹⁴NH⁺₄ in Eq. [7] and [8], that is

and

View larger version (15K): [in this window] [in a new window]   Fig. 4. Effect of zero-order consumption and reaction on concentration profiles of 15NH+4 and 14NH+4 in a 0.356-cm-radius aggregate 24 h after labeling. Base model parameters are presented in Table 1.

View larger version (12K): [in this window] [in a new window]   Fig. 5. Simulated reaction rate profiles in a 0.356-cm-radius aggregate 48 h after labeling with 99.9 atom % 15NH+4. Reaction rate profiles were calculated from the time period of 24 to 48 h at each spatial node in the zero-order consumption, finite difference model. Dashed line represents simulation input reaction rate of kcA = kpA = 1 µg N cm-3 d-1, Kd = 13 cm3 g-1.

In contrast to our modeling results and the results of Monaghan et al. (1995), Watson et al. (2000) suggested that heterogeneous distribution of label would lead to overestimation of gross mineralization rates. Watson et al. (2000) tested the assumption of zero-order production and consumption of NH⁺₄ by calculating five rate measurements using isotope dilution within a 24-h period. They showed that a disproportionately higher percentage of ¹⁵NH⁺₄ label appeared in the NO^-₃ pool than would be expected from zero-order consumption of the ¹⁵NH⁺₄ and ¹⁴NH⁺₄ pools alone. They attributed this phenomenon to greater availability of the ¹⁵NH⁺₄ label than of the native NH⁺₄ to microorganisms and suggested that diffusion limitations may be responsible. They interpreted these data to mean that gross mineralization rates would be overestimated due to greater decline in the ¹⁵NH⁺₄ enrichment than should occur under conditions of homogeneous distribution of label. It is possible that a combination of high consumption of ¹⁵NH⁺₄ near the outside of an aggregate (perhaps due to nitrification or NH⁺₄ assimilation) with or without relatively high consumption of ¹⁴NO^-₃ at microsites near the inside of the aggregate (perhaps due to denitrification or NO^-₃ assimilation) could have caused the observed pool size changes. Davidson et al. (1991) showed through modeling that it is possible to achieve positive errors in rate estimates if native NH⁺₄, ¹⁵NH⁺₄ label, and microbial activity are not homogeneously distributed throughout the soil volume being interrogated.

Other hypotheses exist, however, that could lead to the observed discrepancy in the atom% ¹⁵N of the NO^-₃ pool. If zero-order consumption of NO^-₃ applies, ¹⁴NO^-₃ will be preferentially consumed if it is present at higher concentration than ¹⁵NO^-₃. This possibility was explored by Stark and Schimel (2001) and Watson et al. (2002). These researchers disagreed about the potential importance of zero-order consumption of ¹⁴NO^-₃ in explaining the apparent disproportionate consumption of ¹⁵NH⁺₄ observed by Watson et al. (2000).

The hourly gross rate estimates for four soils presented by Watson et al. (2000)(Tables 4 and 5) suggest that a variety of kinetic models may describe their data. It is possible to obtain positive errors in both production and consumption rates if the wrong kinetic model is assumed. In order to illustrate this, we chose to model the CENIT (Central Nitrogen Experimental site) soil in which 15 mg ¹⁵NH⁺₄ kg^-1 of soil was applied (Watson et al., 2000). The hourly zero-order NH⁺₄ consumption rates calculated by Watson et al. (2000) decrease with time, suggesting that first-order consumption kinetics may be a more appropriate model. In order to compare rate estimates obtained using different kinetic models, we modeled the CENIT data of Watson et al. (2000) assuming zero-order production and either zero-order consumption or first-order consumption of NH⁺₄, and zero-order consumption of NO^-₃. Rate constants were chosen by finding the least-squares difference between our modeled time final data and the time final data of Watson et al. (2000)(Table 2). We were able to match their time final data to within <±0.1% error for all inorganic soil N pools using either model. Assuming that first-order consumption occurred in this case, the use of a zero-order consumption model lead to an overestimation of NH⁺₄ consumption of 4.5% and an overestimation of NH⁺₄ production of 6.3%. These errors occur because of the nonlinear nature of first-order consumption and occur despite the fact that the same net quantity of N is produced and consumed. On the basis of modeling exercises, Nason and Myrold (1991) also found that errors in gross rate calculations due to the use of an inappropriate kinetic model were modest. Despite the fact that using inappropriate kinetic assumptions probably lead to small errors in rate estimates, understanding the reaction kinetics may provide insight into controls of the system. For example, first-order consumption of NH⁺₄ may indicate that the soil system is N limited and zero-order consumption may indicate that the system is C limited.

For simplicity, we chose to model the effects of diffusional limitations on isotope dilution experiments designed to estimate NH⁺₄ production and consumption rates. These results may be extrapolated to an isotope dilution experiment designed to estimate gross NO^-₃ production and consumption after adding ¹⁵NO^-₃ by ignoring adsorption in the model. In this case, our zero-order consumption model suggests that relative errors in production and consumption rate estimates due to diffusional constraints would be <-2% in the 0.36-cm-radius aggregate even if a 24-h equilibration period is not employed (data not shown).

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
We have demonstrated that mass transfer limitations have the potential to cause errors in isotope dilution experiments used to estimate gross NH⁺₄ production and consumption in soils. Under the conditions of zero-order consumption, this error was always negative and increased as effective diffusion coefficient decreased and as aggregate size increased. Errors in production rate estimates were generally comparable with those of consumption rate estimates except where the k_cA:k_pA ratio was 3:1, in which case errors in production rate estimates were higher. Errors could be reduced by allowing the label to equilibrate for 24 h before taking an initial time sample. In the case of true pseudo-first-order consumption, adsorption affects absolute rate estimates but mass transfer limitations do not affect apparent rates. Our analyses show that analysis of isotope dilution data based on inappropriate kinetic assumptions may cause only small errors in rate estimates; however, the determination of the correct kinetic model might prove invaluable in determining controls on system processes.

	NOMENCLATURE

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
Where subscripts _out and _in replace _x in the following variables indicate concentrations of outer and inner regions of a composite sphere (Fig. 1), respectively: ¹⁴A_x = concentration of NH⁺₄ in solution, µg N cm^-3; ¹⁵A_x = concentration of ¹⁵NH⁺₄ in solution, µg N cm^-3; a = radius of inner region of composite sphere, cm; b = radius of outer region of composite sphere, cm; D_w = diffusion coefficient of the solute in water, cm² s^-1; D_p = pore diffusivity, cm² s^-1, given by

[17]

f = impedance factor, cm cm^-1; i = spatial node for finite-difference approximation; j = temporal node for finite-difference approximation; K_d = linear adsorption coefficient, cm³ g^-1; k_cA = NH⁺₄ consumption constant s^-1 for first-order consumption, µg N cm^-3 soil s^-1 for zero-order consumption; k_pA = zero-order NH⁺₄ production rate constant (µg N cm^-3 soil s^-1); M = total NH⁺₄, given by

[18]

M* = total NH⁺₄ at t = 0, µg N cm^-3; ¹⁵A* = concentration of ¹⁵NH⁺₄ in solution at t = 0, µg N cm^-3; n = total number of spatial nodes used in finite-difference approximation; R = retardation factor, given by

[19]

r = radial distance from the center of the sphere, cm; s = Laplace parameter s^-1; t = time, s; = proportion of NH⁺₄ produced that is ¹⁵NH⁺₄ (assumed to be 0.3663%); ß = proportion of NH⁺₄ produced that is NH⁺₄ (assumed to be 99.6337%); = volumetric water content, cm³ cm^-3; = bulk density, g cm^-3.

APPENDIX

Equations [1] and [2] do not need to be solved simultaneously, but two equations are needed to solve for both k_cA and k_pA. The solution to Eq. [1] subject to the initial condition Eq. [3] is

[20]

and the solution to Eq. [2] subject to Eq. [4] is

[21]

So that

[22]

and

[23]

The solution to Eq. [5] and Eq. [6] subject to Eq. [3] and Eq. [4] is for k_cA k_pA:

[24]

So that

[25]

or

[26]

The rate constants k_cA or k_pA must be solved for implicitly in this case; however, once either k_pA or k_cA is obtained, the solution to Eq. [6],

[27]

may be used to obtain the missing constant. For the case in which k_cA = k_pA,

[28]

The solutions to Eq. [7] and [8] were obtained semianalytically. The solution procedure in the Laplace domain follows closely the works of Carslaw and Jaeger (1959) and Priesack (1991). Because the solutions to Eq. [7] and [8] are similar, only the solution to Eq. [7] will be given here. Solutions for the inner and outer spherical regions for Eq. [8] may be obtained by analogy.

Initial conditions are given by constant solute concentration in the outer and inner concentric spheres in Eq. [11] and [12], with boundary conditions given by Eq. [13] through [16]. The sequential variable substitutions

[29]

and

[30]

are performed. Subsequently, the Laplace transform is applied to obtain the transform u_x = u_x (s, r) of U_x = U_x (t, r), leaving

with boundary conditions

[31]

[32]

[33]

and

[34]

The solutions to these equations in the Laplace domain for the outer spherical shell is

[35]

and for the inner sphere:

[36]

where

[37]

[38]

[39]

and

[40]

The inverted time domain solution must be substituted back into Eq. [30] and [29] to obtain the correct solution to Eq. [7]. For the special case where k_cA = 0, a very small number may be substituted for k_cA as a suitable approximation.

The solutions to Eq. [9] and [10] subject to Eq. [11] to [16] were arrived at by the Crank-Nicolson method of finite-difference approximation (Crank, 1975). The resulting approximations for ¹⁵A are

[41]

[42]

and

[43]

where

[44]

[45]

[46]

[47]

and

[48]

	ACKNOWLEDGMENTS

 
The work described in this paper was supported primarily by USDA-CSREES Grant no. NRI 97-35107-4357. Special thanks go to Pat Welch for help with LINUX administration and with FORTRAN code.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 
Contribution from the Oregon Agric. Exp. Stn., Tech. paper no. 11813.

Received for publication October 31, 2001.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION METHODS RESULTS AND DISCUSSION CONCLUSIONS NOMENCLATURE REFERENCES

 

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