SCIENCE SOCIETY OF AMERICA

PROCEEDINGSVOL. 34 MARCH-APRIL 1970 No. 2

DIVISION S-l—SOIL PHYSICSSoil Water Evaporation, Isothermal Diffusion, and Heat and Water Transfer1

D. D. FRITTON, DON KIRKHAM, AND R. H. SHAW2

ABSTRACTTemperature and water distribution data were taken for a

9- by 11- by 20-cm soil column where wind was the evaporationagent and for a similar soil column where radiation was theevaporation agent. A Webster silty clay loam soil characterizedby a soil-water retention curve and an inflow diffusivity versuswater content curve was used in the experiments. The experi-mental results were compared with results calculated from asolution of the diffusion equation normally used to describeisothermal movement of soil water and from a heat and masstransfer solution which accounted for temperature effects. Itwas found that the isothermal diffusion equation would de-scribe the cumulative evaporation for the wind and for theradiation treatments. The isothermal diffusion equation did notpredict the formation of a surface layer of dry soil, and thus,did not describe the water distributions for the radiation treat-ment for times greater than 20 hours or for the wind treatmentfor times greater than 200 hours. A heat and mass transferequation did predict the development of a dry surface layerand did describe the water distribution where temperaturegradients were important.

Additional Key Words for Indexing: inflow and outflow dif-fusivities, water distribution, temperature distribution.

EVAPORATION of water from a bare soil surface occurs innature under a variety of environments, many of which

are nonisothermal. The purpose of this study was to testtwo unsaturated soil-water-flow equations for their abilityto predict soil water content versus depth for near isother-mal and nonisothermal soil-water evaporation in laboratorysoil columns.

The isothermal diffusion equation is usually given as

1 Journal Paper no. J-6272 of the Iowa Agr. & Home Econ.Exp. Stat., Ames, Iowa. This research was supported by U. S.Atomic Energy Commission Contract At(ll-l)-1269, Officeof Water Resources project A-003-IA, and the Iowa Agr. &Home Econ. Exp. Sta. projects 998, 1003, 1235, 1276, and1653. Received June 9, 1969. Approved Nov. 18, 1969.2 Formerly Research Associate, Iowa State University, nowAssistant Professor of Soil Physics, Cornell University, Ithaca,New York; Professor of Soils and Physics; and Professor ofAgricultural Climatology, respectively.

dt 3z dz [1]

where 8 is volumetric water content, t is time, z is depth,and D (8) is water diffusivity. This equation has been usedto analyze soil-water evaporation experiments in severalpapers. Gardner (1959) was able to fit experimentalcumulative evaporation curves by using a diffusivity func-tion measured with the pressure plate outflow method.Jackson (1964) calculated desorption diffusivities fromexperimental water content data for horizontal distancesin a soil column. Wiegand (1960) found experimentalcurves of water content versus depth of what he calls para-bolic form. The shape of the curves in both Jackson(1964) and Wiegand (1960) are similar to the theoreticalcurves shown by Gardner (1959). Gardner and Hillel(1962) compared cumulative evaporation curves underdifferent evaporation potentials with a solution to the iso-thermal diffusion equation. They found that the worst fitof the cumulative evaporation curves was for the casewhere the assumption of isothermal conditions was notmet. Hanks and Gardner (1965) used the isothermal dif-fusion equation to investigate the influence of the variationof the diffusivity function on the evaporation of soil water.They found that changes in the diffusivity at low watercontents, of less than 10% for their soil, made no signifi-cant difference in the cumulative evaporation or the watercontent versus depth profiles. Since only a very small depthof their soil was actually below 10% water content, thisconclusion may not hold for soil columns that have a con-siderable portion of their depth at low water contents.Variation of the diffusivity function at water contentsgreater than 10% for their soil gave large differences incumulative evaporation and in the water content versusdepth profiles, but the change in shape of the water profilesdue to variation in diffusivity was not significant.

Equations which can be used to analyze nonisothermalevaporation of water from soil have been the subject ofseveral papers. Philip and DeVries (1957) derived twosimultaneous flow equations for liquid and vapor water

183

184 SOIL SCI. SOC. AMER. PROC., VOL. 34, 1970

transfer under simultaneous temperature and water gradi-ents. The equations are

and

=-X ~pLDfidz ^w 6 dz [3]

where q is water flux density; Pwis density of water; DTis thermal water diffusivity; T is temperature; z is depth;De = D(S) as in equation [11; 6 is volumetric water con-tent; K is hydraulic conductivity; Q is soil heat flux density;A is thermal conductivity, which includes that portion dueto vapor movement; L is latent heat of evaporation; and£>i,vap is vapor water diffusivity, where D(6) = Deliq +D0vap and £>miq is liquid water diffusivity. Philip (1957)eliminated the dT/dz term from the two simultaneous equa-tions of Philip and DeVries (1957) and obtained theequation

Z =

-I9Cz)

{9(0)

>-DTQ]}d6

[4]

where E is evaporation rate, q/pw, in equation [2]. Tay-lor and Gary (1964) derived equations for the simulta-neous flow of water and heat in a moist soil under a tem-perature gradient from the theory of irreversible thermo-dynamics. Their equations are of the same general formas those derived by Philip and DeVries (1957).

Several researchers have used either the analysis ofPhilip and DeVfies (1957) or Taylor and Gary (1964)for various purposes. And both theories have found someexperimental verification in several studies in the literature.Hanks, Gardner, and Fairbourn (1967) determined ex-perimental water and temperature profiles for a silt loam,a loamy sand, and a sand for both wind- and radiation-induced evaporation potentials. From the analysis ofPhilip and DeVries (1957), they showed that the vaporcomponent of water flow is insignificant in the lower depthswhere temperature gradients were small, but became sig-nificant in the upper 5 cm for the same wind treatment.For the radiation treatment, where temperature gradientswere large, the vapor component was important for alldepths except for early times. Gee (1966) used both theanalysis of Philip and DeVries (1957) and Taylor andGary (1964) to analyze water flow in a closed soil samplewith an imposed temperature gradient. He found that theanalysis of Philip and DeVries (1957), modified to calcu-late the vapor water diffusivity according to a technique ofJackson (1964), underestimated the flow by an averagefactor of 2.7. The analysis of Taylor and Gary (1964) asused by Gee (1966) overestimated the water flow by anaverage factor of 1.3. Dirksen and Miller (1966) usedthe analysis of Philip and DeVries (1957) to calculate the

movement of water due to thermal gradients for freezingsoils. They concluded that the analysis gave results in theright order of magnitude for the movement of water in soilsubject to a thermal gradient.

In the previously mentioned studies, Gee (1966) andDirksen and Miller (1966) calculated one experimentalflow independent of the experimental flow data. Hankset al. (1967), on the other hand, used all their experimen-tal data to calculate the flow due to vapor and liquid water.In our study, we use independent measurements to calcu-late the water content distribution and then use the experi-mental water distribution to provide a check on the validityof the theory when it is applied to evaporation of waterfrom columns of soil not at constant temperature.

PROCEDUREWebster silty clay loam surface soil (19.9% sand, 48.0%

silt, and 32.1% clay) was sieved with a 5-mm screen, spreadto dry, passed through a 2-mm screen, mixed thoroughly, andstored at room temperature and humidity until needed. Theparticle density was 2.58 g/cm3 and the organic carbon per-centage was 3.85. A water-retention curve was determined witha hanging water column for Tempe cells and with air pressurefor pressure cooker and pressure plate. Soil was packed to abulk density of 1.2 g/cm3 and initially saturated in the Tempecell and pressure cooker samples. The gamma-ray water deter-mination apparatus is the same as that used in a previous study(Kirkham, Rolston, and Fritton, 1967). All counting rateswere corrected for an experimentally determined resolving timeof 5 ^see/count (Fritton, 1969). Water diffusivity for Web-ster silty clay loam was determined by using a gamma-raytechnique described by Whisler, Klute, and Peters (1968)

Two soil columns were constructed with 1-cm thick plexi-glas with inside dimensions of 11 by 9 by 20 cm, packed withair dry soil, and insulated with a layer of aluminum foil and a2.5-cm thickness of styrofoam. Four copper-constantan ther-mocouples were located perpendicular to the gamma-ray di-rection at the surface of the soil and at 0.5- and 1.0-cm depths;three thermocouples were located at the 2.0- and 3.0-cmdepths; and two thermocouples were located at 1-cm intervalsfrom 4.0 to 10.0 cm and at 2-cm intervals from 10.0 to 18.0cm. The system for temperature measurement, a temperature-calibrated potentiometer, was capable of measuring 0.025Cdifferences. The center of the gamma-ray beam, 5 mm in diam-eter, was located for density and water sampling at depths of0.55, 0.75, and 1.25 cm, and at 1.0-cm intervals from 1.55 to18.55 cm. Measurements were taken 1.5, 4.5, and 7.5 cm fromone vertical side of each column.

Two, 250-watt reflector heat lamps were placed 60 cmabove the radiation-treated column and adjusted in height andangle until a uniform area of 1.5 cal/cm2 min radiation coveredthe surface area of the column. A table fan with base locatedat the level of the column top, but 30 cm from the column,was used for the wind-treated soil. The wind velocity immedi-ately above the column was measured with a 10-cm diameteranemometer. The evaporation potential of each treatment wasdetermined by the weight loss of each water-filled columnafter 15.33 hours of evaporation.

The soil in each column was wet from the bottom with dis-tilled water with a positive head of 50 cm with reference tothe column base until the surface was visually wet and let setfor 3 days. A 5- by 10- by 20-cm lead brick was placed on aspecial plexiglas top that laid on the soil during wetting toretard swelling. The lead brick was removed 12 hours beforeevaporation started. The soil swelled vertically 3 to 5 mm de-spite precautions.

Water and temperature distributions and column weightwere determined after approximately 0, 1, 2, 3, 6, 9, 15, and28 days of evaporation. During the experiment, the room tern-

FRITTON ET AL.: EVAPORATION, ISOTHERMAL DIFFUSION, HEAT AND WATER TRANSFER 185

s02

WEBSTER SILTY CLAY LOAMBULK DENSITY = l .2g. /cm.3TEMPERATURE-27 ± 2'c

•••TEMPE CELL•AAPRESSURE COOKER>o ..PRESSURE PLATE

A CALCULATED TOTAL POROSITY

TENSION (MILLIBARS)

Fig. 1—Soil water retention curve for a Webster silty clayloam soil.

perature was 27 ± 2C and the relative humidity was 35 ±7%. At the end of the experiment, water distribution was de-termined gravimetrically by sectioning the column in 1-cmintervals and drying the soil in an oven at 112C for 24 hours.Soluble salt content was determined by the technique ofBower and Wilcox (1965) and showed accumulation at thesurface but only in insignificant quantities. More detailed pro-cedures can be found in Fritton (1968).

RESULTSThe water retention curve and the water diffusivity curve

for Webster silty clay loam soil are shown in Fig. 1 and 2,respectively.

Initial values of gamma-ray determined bulk density at57 positions in the soil columns averaged 1.26 ± 0.04and 1.29 ± 0.04 g/cm3 for the wind and radiation treat-ments, respectively. Initial values of gamma-ray deter-mined water content at 57 positions in the soil columnsaveraged 42.8 ± 2.2 and 40.9 ± 2.4% for the wind andradiation treatments, respectively. In all water determina-tions, a greater variation of water content occurred nearthe surface of a column than at other depths. This is attrib-uted to soil swelling, which is recorded by the gamma-raytechnique as a change in water content. The variationgiven for initial values of water content was similar to thevariation found in determinations of water content forlater times.

Figure 3 shows the distribution of average soil tempera-

•••10.Ocm.•••I 9. Ocm

AAA25.0 cm.00035.0 cm

15 30VOLUMETRIC WATER

45CONTENT (%)

Fig. 2—Water diffusivity plotted as a function of volumetricwater content for four locations in a horizontal soil slab.

ture at each depth at times of 0.00, 21.67, 66.25, 206.92,and 676.92 hours for the wind-treated soil column (Fig.3a) and at times of 0.00, 20.58, 64.25, 206.10, and 657.32hours for the radiation-treated soil column (Fig. 36). InFig. 3a, the increase in temperature shown in curves 3 and4 over that for curve O was due to an increase in the roomtemperature, and variation in temperature at any one depthwas less than 0.2C. For Fig. 3b, the variation in tempera-ture among the four measurement positions was frequently5C for the 0.0- and 0.5-cm depths and was usually lessthan 1C for two or three measurement positions at all otherdepths. During the 2-hr sampling period for gamma-ray

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Fig. 3—Initial and subsequent average values of temperature for the wind-treated (a) and radiation-treated (b) soil, plotted as afunction of depth for several times.

CUMULATIVE EVAPORATION (cm.)ro u .̂ in

O_____O_____O_____O_ 6 ————————————r———|———r-

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FRITTON ET AL.: EVAPORATION, ISOTHERMAL DIFFUSION, HEAT AND WATER TRANSFER 187

measurements when the radiation-treated soil column wasremoved from the heat lamps, the temperatures droppedfrom 80 to 40C, from 55 to 49C, and from 50 to 43C forthe surface, middle, and lower layers of the soil column,respectively.

Figure 4 in parts a and b shows the curve of cumulativeevaporation for the wind- and radiation-treated soil col-umns, respectively. The points plotted in the graphs areexperimental points calculated from the weight of the soilcolumn at various times during evaporation. The smoothcurve was calculated by using a numerical solution to theisothermal diffusion equation (Hanks and Gardner, 1965).To obtain the fit of the experimental points with the theo-retical curves as calculated from the isothermal diffusionequation, the values of the water diffusivity, Dg, shownin Fig. 2 had to be divided by a factor (see discussion) of35 for the wind- and 10 for the radiation-treated soilcolumn.

Figure 5 in parts a and b shows the experimental watercontent distribution for wind- and radiation-treated soilcolumns compared with those calculated from the iso-thermal diffusion equation using the adjusted diffusivityfunction, De, for various times of evaporation as shown.The curves are labeled 0, 1, 2, 3, and 4 and correspondwith evaporation time where each number is associatedwith a gamma-ray determined curve, indicated by the lineconnecting the points, and a theoretical curve, indicatedby the heavy smooth line, except curve O where both arethe same. Each point on the experimental curve determinedwith gamma-ray attenuation is the average of three samp-ling locations. The points on the gravimetrically determinedexperimental curve are for 1-cm thick sections obtained atthe end of the experiment. The negative water contentsnear the soil surface are a result of shrinking and swelling,which changed the bulk densities and thus the water con-tent calculated from gamma-ray data.

Figure 6 shows the experimental and theoretical waterprofiles for the 657.32 hour time for the radiation-treatedsoil column. The experimental curves are the same experi-mental curves shown in Fig. 5b for the 657.32-hour time.The curve for the theory of heat and mass transfer was

calculated by using equation [4]. Values of evaporation rate(E) and heat flux (Q) were calculated from the rate ofevaporation between the 510- and 657.32-hour time fromFig. 4b. For the calculation, the surface water content( 6 ( O ) ) was set equal to 2%. Water diffusivity from Fig.2 was used as liquid water diffusivity (Deliq), divided by35, and adjusted for temperature by using the viscosityratio. The soil water retention curve, needed to calculateDT, £>9vap, and K, was adjusted for temperature by using30 times the temperature coefficient of surface tensionwhich was used without a factor by Philip and DeVries(1957). This factor of 30 was obtained by fitting data inGardner (1955) and Taylor, Evans, and Kemper (1961).Thermal conductivity was calculated by the techniquedescribed by DeVries (1963) using suggestions of Cochran,Boersma, and Youngberg (1967) and starting with thethermal conductivity of dry soil taken from Jackson andKirkham (1958). A value of the atmospheric diffusioncoefficient, needed to calculate D6vap, was taken fromDeVries and Kruger (1967).

DISCUSSION

Data in Fig. 3a show that at any one time the wind-treated column was near an isothermal condition. On theother hand, data in Fig. 3b show the large temperaturegradient that existed in the radiation-treated soil column.The surface temperatures in Fig. 3b are very high, rangingfrom 75 to 88C, and exemplify the conditions that wouldexist if a net radiation of approximately 1.2 cal/cm2 minfell continuously on a soil with little wind movement.Since the vertical temperature gradients as seen from theslope of the curves in Fig. 3b were much greater than hori-zontal temperature gradients (less than IC/cm in the sur-face 3 cm), horizontal temperature gradients were neglectedin the heat and water transfer model.

Figure 4 shows the fit of the experimental cumulativeevaporation data with the theoretical curves calculatedfrom isothermal diffusion theory for both the wind (Fig.4a) and radiation (Fig. 4b) treatment. The only problemis that the diffusivity function needed to fit the cumulative

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RADIATION TREATMENT• "-EXFTL. GAMMA-RAY(657.32hrs)ooooEXPTL.GRAVIMETRIC (657.32hrs)

———HEAT AND MASS TRANSFER THEORY(657.32hrs)

-10VOLUMETRIC

10WATER

20CONTENT (%)

30

Fig. 6— Experimental and theoretical curves of water content versus depth for the radiation-treated soil for the 657.32-hr, timewhere the experimental curves are the same as those numbered 4 in Fig. 5b,

188 SOIL SCI. SOC. AMER. PROC., VOL. 34, 1970

evaporation curve was smaller than the measured diffusiv-ity by a factor of 35 for the wind-treated soil and by afactor of 10 for the radiation-treated soil. The factors, 35and 10, were found by trial and error. Since the wind-treated soil was drying and the soil used in the determina-tion of diffusivity was wetting and both were at the sametemperature, the factor of 35 must result from hysteresisin the diffusivity versus water content relationship. Severalpapers (Klute, Whisler, and Scott, 1964; Jackson, 1964;Gardner, 1959; and Wiegand and Taylor, 1961) haveindicated that inflow diffusivities can be larger than outflowdiffusivities by factors ranging from O to 100, but this factwas not appreciated until the experimental results wereanalyzed.

The difference in the two resistance factors, a factor of3.5, can be explained by the difference in soil temperaturefor the wind and radiation treatments. To see the effectof the temperature difference, consider the definition ofthe diffusivity in terms of the hydraulic conductivity andthe slope of the soil-water retention curve, De = K(d\p/d8),where K is unsaturated hydraulic conductivity, t// is soil-water tension, and 8 is water content. Here, the hydraulicconductivity K is temperature dependent because it in-volves the viscosity of water as given by K = pgk/r/, wherep is the density of water, g is the acceleration of gravity, kis the intrinsic permeability, and ^ is the viscosity. The vis-cosity of water at 25C is 0.00891 poises and at 75C is0.00380 poises. This viscosity decrease results in an in-crease by a factor of (0.00891/0.00380) — 2.34 in thehydraulic conductivity and thus, in the diffusivity as com-pared with the diffusivity used in the wind treatment. Thefactor 2.34 is smaller than the factor of 3.5 which would beneeded, and the difference will be left as an experimentalfactor arising from the difference in packing of the wind-and radiation-treated soil columns.

In considering the deviation in Fig. 5a of the experimen-tal curves from the calculated, it is recalled that the varia-tion of water content from the average at any one depthexcept near the surface was approximately 2.3%. Thissame value of variation (2.3%) can be assumed to applyto the curves of Fig. 5a except in the upper layers. Withthis amount of water content variation in the experimentaldata, the deviation of points on the experimental curvesfrom the theoretical curves would not be significantly dif-ferent for the curves 1 and 2 of 21.67 and 66.25 hours.In curve 4 (676.92 hours in Fig. Sa), the theoretical curvefails to match the shape of either of the experimental curvesover the full 19-cm depth of the vertical soil column. Forthe surface depths from O to 7 cm, the theoretical curveneeds to be moved to the left from 5 to 0%. For the 10-to 19-cm depths, the theoretical curve needs to be movedto the right from 3 to 2%. From the minimum shown inthe temperature curves of Fig. 3a, it can be concluded thatwater escaping from the soil column originated at the 4- to5-cm depth for the 676.92 hour time. It can then be specu-lated that the reason for the incorrect shape of the theoreti-cal curves is an inadequate diffusivity versus water contentrelationship. It would seem from Fig. 5a that the diffusivityat water contents of less than 20% should be greater, how-

ever, Hanks and Gardner (1965) indicate that a changein diffusivity at low water contents does not change theshape of the water profile curves. Whatever the explana-tion, the isothermal diffusion equation as it has been usedin this study predicts an inadequate water distribution whena dry layer exists on the surface of the soil column eventhough it predicts the cumulative evaporation.

Figure 5b is a graph to show how poor a theoretical fitof our experimental water data is achieved if temperatureinfluences other than the linear change in the diffusivityjust discussed are not considered. The factor used to adjustthe diffusivity is the same as that used to calculate thecumulative evaporation curve (Fig. 4b) for this sameradiation-treated soil column. The fit is poor except for thecurves labeled T because the diffusion theory as used inthis study does not predict the formation of a dry surfacelayer.

In Fig. 6, the difference between the gamma-ray andgravimetric water contents was probably due to the prob-lem of shrinking of the soil during drying, and the gravi-metric curve is probably the most reliable. The calculationof the theoretical curve from equation [4] involves manyassumptions (Philip and DeVries, 1957), especially in thecalculation of the values of the thermal conductivity, thethermal vapor water diffusivity, and the slope of the waterretention curve for various temperatures. All three of thesefunctions are approximate, but they correspond to experi-mental observations in DeVries (1963), Philip and De-Vries (1957), and Gardner (1955) and Taylor et al.(1961). Details of these calculations can be found inFritton (1968).

The cause of the deviation of the theoretical water pro-file from the experimental water profiles for water contentsabove 20% at depths below 12 cm is not known. One pos-sibility is that the liquid water diffusivity should decreasemore rapidly in the region of 20 to 30% water contentsimilar to the curve of Kunze and Kirkham (1962) andthat this caused the observed deviation. In any case, thereis little question that the heat and mass transfer theoreticalcurve gives a much better fit than the isothermal diffusionequation of both the shape and magnitude of the experi-mental points where temperature gradients are important.

FINK: WATER REPELLENCY AND INFILTRATION OF ORGANIC-FILM-COATED SOILS 189