Estimating Molecular Diffusion Coefficients of Urea in Unsaturated SoilA. M. Sadeghi, D. E. Kissel,* and M. L. Cabrera

ABSTRACTA correct value for the molecular diffusion coefficient of urea in

soil (D,) is required to accurately predict urea movement in soil bymolecular diffusion. In previous work, to estimate D, in a simulationmodel of urea diffusion, we used an empirical equation of Papendickand Campbell that describes D, for a dissolved species in soil to bea product of a tortuosity factor squared, the species diffusion coef-ficient in water (/>„), and the volumetric water content. Comparisonsof measured and computed urea concentration with depth indicatedthat this equation was not adequately general over a wide range ofsoils. The objective of this study was to modify the parameters inthe equation and, if necessary, develop a new relationship to estimatethe value of Z), in soils. Laboratory studies were conducted on sevensoils in which the clay content ranged from 10 to 51%. Urea con-centrations with depth at 48 h following surface-application weremeasured and also computed using numerical techniques with aninitial estimate for D, instead of computing it using Papendick andCampbell's equation. The D, was modified incrementally, until thedifference between computed and measured concentrations was min-imized. In all seven soils, good agreement was obtained betweenmeasured and computed urea concentrations with depth. The max-imum depth of urea movement occurred in Kahola soil (approx. 3.5-cm deep), whereas least movement occurred in Crete soil (approx.2.6-cm deep). Nonlinear regression analysis gave a better relation-ship (D, = 0.18 X Dw (©./porosity)2•", R2 = 0.88) when relative watercontent (®,/porosity) of the seven soils was substituted for the vol-umetric water content (D, = 0.73 X />„ (®v)258, R2 = 0.66) in Pa-pendick and Campbell's equation.

Dep. of Agronomy, Kansas State Univ., Manhattan, KS 66506.Contribution no. 88-359-J, from the Kansas Agric. Exp. Stn. Thisstudy was supported by a grant from Farmland Industries, KansasCity, MO. Received 12 Feb. 1988. "Corresponding author.

Published in Soil Sci. Soc. Am. J. 53:15-18 (1989).

THE MOVEMENT of surface-applied urea fertilizer bymolecular diffusion in a homogeneous soil, like

that of other non-adsorbed solutes in soil, can bemathematically described by,

dC/dt = DSX d (dC/dx)/dx [1]where dC/dt is the change in urea concentration withtime, (dC/dx) is the urea concentration gradient, andDs is the diffusion coefficient of urea in soil. The valueof Ds varies with the physical properties of soils.Several investigators have attempted to develop asimple mathematical relationship to estimate the val-ues of molecular diffusion coefficients of chemical spe-cies in soil (Porter et al, 1960; Romkens and Bruce,1964; Olsen and Kemper, 1968; Mahtab et al., 1971;Paetzold and Scott, 1978). To estimate Ds, we used anempirical equation of Papendick and Campbell (1980)

A = (L/Le)2 X Dw X 0V [2]where Dw is the diffusion coefficient of urea in water(Sadeghi et al., 1988), ©v is the volumetric soil watercontent, and L/Le is the tortousity factor (L being theminimum and Le the actual distances between twopoints through which the solute molecules diffuse). Thetortuosity factor in Eq. [2] is squared, since it affectsthe diffusion in two ways. The gradient (driving force)is less because of the actual path (Le) being longer,and the actual pathway is narrower than the area timeswater content by a factor of (L/Le), since the actualpath is generally at an angle to the minimum pathway(Olsen and Kemper, 1968).

From the work published by Brooks and Corey(1966), Papendick and Campbell (1980) concluded that

16 SOIL SCI. SOC. AM. J., VOL. 53, JANUARY-FEBRUARY 1989

the tortuosity factor is linearly related to the soil watercontent, so that

(L/Le)2 = k X 0? [3]where k is a constant, suggested to have a value of2.8. Replacement of the tortuosity factor in Eq. [2]with the corresponding term from Eq. [3] resulted inthe empirical equation suggested by Papendick andCampbell (1980)

A = 2.8 X Dw X 0;! [4]We solved Eq. [4] to obtain Ds and then using theNewton-Raphson finite difference method we solvedEq. [1] to compute urea concentrations with soil depthat 48 h following surface application of urea to severalsoils (Sadeghi et al., 1988). The results of these com-putations indicated less urea movement into soil thanwas measured, suggesting a possible error in estimat-ing Ds with Eq. [4].

The objective of this study was to modify the pa-rameters in Eq. [4] or, if necessary, develop a newrelationship to estimate the value of Ds for soils thatvary widely in clay and water content.

MATERIALS AND METHODSLaboratory studies were conducted on seven soils in which

clay content ranged from 10.3 to 51.1% (Table 1). The Hay-nie, Crete, Richfield, Kahola, and Smolan soils were fromKansas, whereas Cecil soil was obtained from Georgia andHouston Black soil from Texas. All seven soils were air-dried, ground, passed through a 2-mm sieve, and then heatedat 110 °C for 48 h to destroy urease enzymes. Urease activitywas considered to be sufficiently destroyed if >90% of theapplied urea could be extracted from the column by 48 hafter urea application. To meet this criterion the Kahola,Crete, and Smolan soils needed further treatment with pro-pylene oxide vapors in a dessicator for 48 h.

Water was then added uniformly to the soil with a spraybottle and thoroughly mixed until the desired water contentwas achieved. The wetted soils were packed into soil col-umns to a depth of 5 cm. The soil packing was done witha tamping device, a plexiglass rod with an o.d. slightly smallerthan the i.d. of the soil column. Known amounts of soilequivalent to 0.5-cm depths of soil were added sequentiallywith careful tamping between soil additions to obtain a uni-form density throughout the soil column. The columns, madeof plexiglass, had an i.d. of 5 cm and consisted of threesections, the base with a movable piston that allowed thesoil to be pushed out the top for sampling, the cylinder to

Table 1. Classification and particle size distribution of soils studied.Series Family classification Sand Silt Clay

Cecil

Houston Black

Crete

Smolan

Richfield

Kahola

Haynie

Clayey, kaolinitic, thermicTypic HapludultsFine, montmorillonitic, thermicUdic PellustertsFine, montmorillonitic, mesicPachic ArgiustollsFine, montmorillonitic, mesicCumulic HapludollsFine, montmorillonitic, mesicAridic ArgiustollsFine, silty, mixed, mesicCumulic HapludollsCoarse, silty, mixed, mesicTypic Udifluvents

33

11

9

13

20

11

41

— TO

16

40

53

52

46

67

49

51

49

38

35

34

22

10

hold the soil, and the top to cover the soil (Fig. 1 in Sadeghiet al., 1988).

After the columns were packed with soil, they were placedin an incubator at 25 °C for several hours prior to urea ap-plication to allow the packed soil columns to reach 25 °C.Urea then was applied as a fine powder uniformly to thesoil surface at a rate equivalent to 200 kg N/ha (43 mg urea-N/chamber). The powdered urea was generally totally dis-solved within 2 min. Following urea application, the col-umns were returned to the incubator at 25 °C for 48 h. Thecolumns then were removed from the incubator, and thesoil in each column was sectioned into several depth incre-ments: one 2-mm increment, six 3-mm increments from 2to 20 mm, two 5-mm increments from 20 to 30 mm, andtwo 10-mm increments from 30 to 50 mm. A subsamplewas removed from each section to determine water content.The remaining soil was extracted with 2 M KC1, and theleachate was analyzed for urea using an autoanalyzer method(Technicon Industrial Systems, Tarrytown, NY, 1980).Briefly, in an acidic medium, diacetylmonoxime is hydro-lyzed to diacetyl, which, in turn, reacts with urea. The re-sulting complex is read at 530-nm wavelength in a spectro-photometer.

Urea concentrations with depth at 48 h also were com-puted for each soil column using numerical techniques tosolve Eq. [1] as described earlier but with an initial estimateof Ds_ instead of a computed value from Eq. [4]. The Ds wasmodified incrementally with the computer program until theroot mean square error (RMSE) of the difference betweenthe measured and computed urea concentrations with depthwas minimized. The root mean square error was calculatedas M

RMSE =7-1

- US(I))2/Af\ 05 [5]

where 110(1) is observed urea-N concentration, US(I) is thesimulated urea-N concentration at depth /, and M is thenumber of depths. For each soil column, the boundary andinitial conditions and other input values required by thesimulation model were those used in the laboratory exper-iments. That is, the depth to the midpoint of our first soildepth increment was given a value of Z(l), and so on forthe rest of the soil depth increments.

Finally, an SAS (SAS Institute Inc., 1985) nonlinearregression was used to reexamine the parameter values (2.8and 3) in Eq. [4], using the values of D, and 0V from theexperimental soil columns. Similarly, nonlinear regressionwas used to obtain the parameter values (a and b) for thegeneral equation

D, = a X Dw X 0*, [6]where ®tr is the relative water content (0v/porosity).

RESULTS AND DISCUSSIONThe maximum depth of urea movement and urea

distribution with depth varied according to soil type.The actual urea concentrations in the 0 to 2-mm depthranged from a maximum of 2472 to a minimum of1702 g N/m3 for Crete and Kahola soils, respectively.The maximum depth of urea movement occurred inKahola soil (approx. 3.5-cm deep), whereas the leastdepth of movement occurred in Crete soil (approx.2.6-cm deep). Other soils were intermediate in con-centrations at the surface and in depth of movement.

The selected diffusion coefficients and the associ-ated root mean square errors (RMSE) describing thefit between measured and modeled concentration withdepth, along with the related physical properties for

SADEOHI ET AL.: ESTIMATING MOLECULAR DIFFUSION COEFFICIENTS OF UREA 17

Table 2. Selected D, and their corresponding RMSE values alongwith several physical properties of seven soils.

BulkSoil series D.XIQ-" density

Cecil

Houston Black

Crete

Smolan

Richfield

Kahola

Haynie

m2/s Mg/m3

6.202.604.552.053.853.153.553.403.200.968.057.802.400.87

.22

.18

.10

.16

.26

.24

.23

.35

.20

.20

.37

.37

.47

.39

Gravimetricwater content 9,,,t

kg/kg0.2300.1740.2730.2080.2000.1970.1930.1870.1850.1230.2060.2030.1020.068

0.5200.3690.5130.4290.4810.4580.4420.5140.4060.2700.5860.5750.3370.200

RMSE

2152392956

1489364285050373726

t Relative water content (9v/porosity).

2 500 if

2000

g 1500w

1000

500

1500

OH

CC

IIIO

oot 1000UIcc3 500

D8=3.5X10"11( m2 /s )'RMSE = 88

CRETE SOIL(a)

D8 = 7.95X10"11( m2/s )RMSE = 40

KAHOLA SOIL -(b)

10 20 30 40 50

SOIL DEPTH (mm)Fig. 1. Comparison between measured (squares) and computed (curve)

urea-N concentration for Crete soil (a) and Kahola soil (b).

the seven soils, are given in Table 2. The diffusioncoefficients selected at these RMSE values provided agood agreement between the urea concentrations mea-sured in the laboratory studies and the values pre-dicted by the model simulations. Results of this com-parison for Crete and Kahola soils are presented inFig. 1.

The values of Ds computed with the model wereespecially sensitive to soil water content. For four ofthe seven soils, paired columns were packed at sub-stantially different water contents. The computed val-ues of Ds were greatly reduced at the lower water con-tents in each of these soils. Reductions in watercontents of 24 to 34% for the Cecil, Houston Black,Richfield, and Haynie soils caused values of Ds to be2.2 to 3.3 times lower. We felt the range of water con-tents within and between soils would provide a val-uable data set for testing the general form of Eq. [4].

Our first intention was to test the goodness of andpossibly modify the parameter values for the tortuos-ity factor k and the power of the volumetric water

UJyu.IL111 -T-O f

2$§ *

ui

9.0

7.5

6.0

4.5

3.0

1.5

0.0

ALL SEVEN SOILSD8 = 0.73Dw(ey)2.58

:0.66

0.05 0.15 0.25 0.35

VOLUMETRIC SOIL WATER CONTENT(m3/n?)Fig. 2. Urea diffusion coefficient as a function of volumetric water

content for seven soils.

9.0

7.5"oiZ

S«; 6-°2*E 4.5I'o» ~ 3.0

£O

UJoc

1.5

0.0

ALL SEVEN SOILS2.98

D8=0.18 Dw(9v/por08ity)

R =0.88

• •.

0.1 0.2 0.3 0.4 0.5 0.6

RELATIVE WATER CONTENT ( 8y /porosity )Fig. 3. Urea diffusion coefficient as a function of relative water con-

tent for seven soils.

content in Papendick and Campbell's equation for es-timating values of Ds for soils. The parameter valuesobtained from nonlinear regression varied substan-tially from those given in Eq. [4]. The value for thetortuosity factor given by Papendick and Campbell'sequation was 2.8, but the value estimated from ournonlinear regression analysis was 0.73. The parametervalue for the power of the volumetric water contentfrom nonlinear regression was 2.58, somewhat less thanthe value of 3 given in Eq. [4]. In addition to the lackof agreement of parameter values, the fit between thedata values of Ds and the model was not especiallygood (R2 = 0.66) (Fig. 2).

Since diffusive flux in the liquid phase would beproportional to the cross sectional area of a soil col-umn filled with water, we substituted relative watercontent (0v/porosity) for volumetric water content inEq. [4] and repeated the nonlinear regression. Usingthis equation as a model, we found a better relation-ship (R2 = 0.88) with the new model, as shown inFig. 3. The parameter values were 0.18 for a and 2.98for b, as used in Eq. [6].

ACKNOWLEDGMENTSSupport for this research by Farmland Industries, Inc., is

gratefully acknowledged. We are also indebted to MarthaBlocker for help with chemical analysis.

18 SOIL SCI. SOC. AM. J., VOL. 53, JANUARY-FEBRUARY 1989