Published online November 20, 2006

The Dual Gravimetric Hot-Air Method for Measuring Soil Water Diffusivity J. S. Tyner,* L. M. Arya, and W. C. Wright content as a function of distance from the drying surface [u(x)]. A smooth line is fitted through the measured u(x) data points, and the hydraulic diffusivity is calculated from

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

ABSTRACT The hot-air method provides rapid measurement of a soil’s unsaturated hydraulic diffusivity function. The original method consists of blowing hot air across one end of a soil column for a short period, and then quickly extruding, dissecting, and oven drying the soil to provide the soil water content profile, which is used to calculate the soil’s unsaturated hydraulic diffusivity. This research presents a novel approach to measuring the soil water content profile during a hot-air test. During the drying process, a soil column is suspended in a horizontal position by a load cell attached at each end. The measured change in force from the two load cells enables calculation of the soil water content profile, without soil extrusion, dissection, and oven drying. Because the test is nondestructive, it permits estimation of the water content profile and calculation of the unsaturated hydraulic diffusivity function at multiple times. Similarity of the diffusivity functions at various times during the drying process provides evidence of proper testing conditions, a utility not available with the destructive approach. We applied the method to a silt loam and a clay loam and both soils achieved a RMSE of 0.012 compared with the traditionally measured water content profile. We also modeled the performance of the new method on previously published hot-air water content profiles and achieved similar results.

ui

D(u) 5

1 dx 2t du

# x du

[1]

u

where D is the hydraulic diffusivity. A thorough review of the hot-air method is presented in Arya (2002). The thermal gradient and decreased viscosity within a soil core due to application of a hot-air stream can potentially alter the results of testing. Fortunately, the thermal gradient and decreased viscosity act in opposite directions; the former decreases the water flux toward the drying surface, and the latter increases the water flux. Arya et al. (1975) estimated that only 2% of water fluxes within a sandy loam core were due to temperature gradients created by using a 90 to 1008C hot-air stream. Following the introduction of the hot-air method, other practitioners have increased the temperature of the hotair stream applied to the soil cores (e.g., 1808C, Van den Berg and Louters, 1986; 2008C, Stolte et al., 1994). Van Grinsven et al. (1985) conducted a study to determine the effect of temperature using a 2408C hot-air stream. They concluded that “differences between actual, temperature-affected flux densities and assumed isothermal flux densities appear to be limited by mutual compensation of temperature effects.” Errors associated with hand-drawn curves through measured u(x) data prompted Van den Berg and Louters (1986) to fit standardized continuous functions through measured u(x) data. In the process, they proposed the following empirical algorithm:

T

HE HOT-AIR METHOD (Arya et al., 1975) enables rapid measurement of the hydraulic diffusivity function in a drying soil. It consists of rapid drying of a soil core by directing hot air toward one end while the opposite end remains sealed. The following initial and boundary conditions must be met:

u 5 ui , x . 0, t 5 0 u 5 u0 , x 5 0, t . 0 u 5 ui , x 5 ¥, t . 0 where x is the distance from the drying surface, t is elapsed time, u is the volumetric water content, ui is the initial volumetric water content, and u0 is the volumetric water content at the drying surface. If these conditions are properly maintained, then the cumulative evaporation should remain proportional to the square root of time. The boundary conditions are similar to those described by Bruce and Klute (1956) except that water is evaporated (instead of imbibed) from the open end of the column. After drying a soil core under these conditions, it is extruded using a suitably designed piston, dissected into small sections, and oven dried to determine the water

1

u(x) 5 ui 2 (ui 2 u0 )

b x1b

2

n

[2]

where b and n are fitting parameters. Gieske and de Vries (1990) noted that for some u(x) curves, b and n are not independent and a single-parameter equation should suffice, particularly if u(x) resembles an exponential curve. In such cases, Eq. [2] can be replaced with u(x) 5 ui 2 (ui 2 u0 ) exp (2ax)

[3]

where a is a fitting parameter. One of the primary difficulties of the hot-air method is maintaining the semi-infinite boundary condition within the column during the drying process. To ensure that the boundary condition is maintained, the soil column must be relatively long or the testing duration must be short. Since measurement of u(x) requires destruction of the soil column following testing, often several preliminarily tests on a soil are required to determine an appropriate amount of time to apply hot air before dissection. If the testing duration is too short, u(x) has a very steep slope

J.S. Tyner and W.C. Wright, Dep. of Biosystems Engineering and Soil Science, Univ. of Tennessee, 2506 E.J. Chapman Dr., Knoxville, TN 37996; L.M. Arya, Agricultural Experiment Station, Univ. of California–Riverside, 770 El Caballo Dr., Oceanside, CA 92057. Received 9 June 2006. *Corresponding author ([email protected]). Published in Vadose Zone Journal 5:1281–1286 (2006). Technical Note doi:10.2136/vzj2006.0079 ª Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA

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near the drying face and a very shallow slope near the sealed end of the column. Very steep or shallow u(x) slopes make calculation of D(u) using Eq. [1] problematic. In addition, the linear relationship between cumulative evaporation and the square root of time is normally achieved a few minutes after the initiation of the drying process, thus further shortening the valid portion of the drying period. Alternatively, if the testing duration is too long, the semi-infinite boundary condition (u 5 ui, x 5 ¥, t . 0) may fail, which ruins the test. Simply using longer soil columns is also problematic because the soil must be rapidly extruded and dissected after drying ceases. The goal of this research was to develop an improved data collection system for the hot-air method that addresses the primary difficulties of the original method. MATERIALS AND METHODS Measuring Soil Water Content Profiles during Hot-air Drying with Load Cells The method we devised and present here does not require extrusion, sectioning, and oven drying of the soil core. It was deemed necessary, however, to validate computed water content profiles generated by the new method with traditionally measured data. Therefore, we constructed soil core casings that would permit easy slicing of soil cores into small sections for oven drying and experimental determination of water content profiles. We prepared 15-cm-long soil columns by hand packing air-dry soil into a series of 4.0-cm inner diameter by 1.0-cm-long polyvinyl chloride rings (1.5-inch nominal Schedule 40 PVC). Using PVC rings enabled rapid dissection of the soil for gravimetric determination of u(x) following hot-air testing. Standard brass soil sampling tubes (5.1-cm diameter by 15.2-cm length) are a better option for routine hot-air testing because the brass walls are thin and have a large thermal conductivity, which minimizes thermal gradients imposed on the soil. Tests were conducted on a silt loam and a clay loam. Soil columns were packed by placing a 1-cm layer of soil, applying 20 blows to the surface of the layer with a 2-kg metal rod, disturbing the upper 1.5 cm of soil with a knife, and repeating until the column was full. Water was applied to the soil using a hanging water column attached to a Buchner funnel with ceramic disk at a tension of 10 cm. After wetting a soil column, both ends were sealed with tape, and the column was placed horizontally for a minimum of 48 h to more evenly distribute the water. During this period, the column should be rotated periodically to minimize the formation of a vertical water content gradient. Just before testing, a 1-cm slice of soil was removed from the soil column for gravimetric measurement of ui. The soil column was hung from two load cells (YB6–1.2, Sentran, Ontario, CA) as shown in Fig. 1. A hot-air stream was blown against one end of the soil column; the other end of the column remained sealed. As with the original hot-air method, the specified initial and boundary conditions had to be maintained to ensurepthat cumulative evaporation remained linffi early related to t. We generated our hot-air stream by using a small pump (SA55NXGTC-4143, Emerson, St. Louis, MO) to drive air through a Cu pipe wrapped with a 210-W resistive heating tape (FGS101–020, Omega Engineering, Stamford, CT). The electrical current through the heating tape was controlled with a dimmer switch, enabling good control of the air temperature. All tests were conducted at an air temperature of

Fig. 1. Photograph of gravimetric hot-air system. A soil column is suspended from two logging load cells as hot air is blown across the open left end.

90 6 38C. As water was evaporated from the column, a datalogger (21X, Campbell Scientific, Logan, UT) recorded the cumulative change of force measured at the column’s drying end, DF0, and at the column’s sealed end, DF1. Following evaporation of approximately 8 g of water, and with the hot air still in place, approximately 1 mm of soil was removed from the drying face for gravimetric measurement of u0. This practice eliminates any undesirable redistribution of water at the drying face after the hot air has been removed and does not require extrusion of the soil from its casing. The cumulative volume of water evaporated, E, was calculated from

E5

DF0 1 DF1 grw

[4]

where g is acceleration due to gravity and rw is p the ffi density of water. A plot of cumulative evaporation, E vs. t , enables verification of linearity, and conformance to the required initial and boundary conditions. The time at which the boundary condition, u 5 ui, x 5 ¥, t . 0, fails can be pffi readily deduced from the location on the plot where dE/d t decreases from the steady-state value. Figure 2 shows an example where the hot air was applied beyond failure of the semi-infinite boundary condition, which occurred at 1.8 h. Additionally, Fig. 2 shows that linearity began approximately 3 min after the initiation of the hot-air drying. During this initial 3-min period, the water content at the drying surface was decreasing from ui to u0. The centroid of the cumulative change in water content along the x axis, x, was calculated by a moments analysis:

x5

L (DF0 /DF1 Þ 1 1

[5]

where L is the length of the soil column. Figure 3 shows a graphical representation of the air-drying apparatus. The gray area represents the change in water content, and the crosshatched circle marks the location of x(t). At 15- or 30-min intervals, u(x) was estimated from measured E, x, ui, u0, and an assumed form of u(x) given by

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pffi where l 5 x/ t , should result in a single curve that is independent of time. Lack of agreement between u(l) curves from different times indicates failure of the initial or boundary conditions, or excessive soil heterogeneity. Using the normalized plot, D(u) is calculated from

16

8

ui

becomes nonlinear after 1.8 hours due to failure of aft boundary condition

4

D(u) 5

1 2 # l du

1 dl 2 du

[9]

u

Application to Previously Published Data

0 0

25

50

75

100

125

t 1/2 [s1/2] Fig. 2. Relationship between cumulative evaporation (E) and the square root of time. Data show failure of aft boundary conditions at 1.8 h.

Eq. [2]. The optimization of b and n necessary to predict u(x) was accomplished by minimizing the least squares error between measured and fitted, E and x such that

{Emeasured 2 Efitted }2 1 {xmeasured 2 xfitted }2 ! 0

[6]

where Emeasured and x measured were calculated from Eq. [4] and [5] using measured data. Starting with an initial guess for b and n (1.0 is often suitable for both b and n), u(x) was predicted with Eq. [2]. From this prediction of u(x), xfitted was calculated from Eq. [5], where predicted DF0 and DF1 are given by L

DF0 5

pd 2 gr 4

# 11 2 Lx 2Du(x)dx

[7]

0

L

pd 2 gr DF1 5 4

# 1 Lx 2Du(x)dx

[8]

0

where d is the diameter of the soil column and Du(x) is the change in water content or ui 2 u(x). The values of b and n were subsequently modified, and the process was repeated until the left side of Eq. [6] approached zero. If during the optimization process b and n tend toward large values (.25), Eq. [3], which has only a single fitted parameter (a), can be substituted for Eq. [2] to describe u(x). Applying Eq. [3] frees the method to estimate an additional parameter, either ui or u0. Following optimization of u(x), the hydraulic diffusivity can be calculated from Eq. [1]. Alternatively, a plot of u vs. l,

We also applied this method (Eq. [4]–[9]) to previously published hot-air u(x) curves. In each case, u0 was set to the measured water content at the outlet, and ui was set equal to the measured initial water content. By applying Eq. [7] and [8] to the measured u(x) curve, we determined what the measured values of DF0 and DF1 would have been if load cells had been present. Using these four values (u0, ui, DF0, and DF1), we applied the previously described procedure to estimate u(x) by optimizing b and n of Eq. [2]. We plotted the predicted and measured u(x) curves to demonstrate how well our method would work on other soil types and hot-air testing protocols.

RESULTS AND DISCUSSION Application of Method to New Test Data Figure 4 presents a graph of cumulative evaporation from the silt loam soil column vs. the square root of time, and a line is laid atop the data to show its linearity. The data show that approximately 3 min elapsed before pffi linearity between cumulative evaporation and t was established. The measured initial and outlet water contents were ui 5 0.33 and u0 5 0.076. Figure 5 shows a comparison of the predicted u(x) curves at multiple times and the measured u(x) data points at 92 min. At 15 min, the slope at the outlet is very steep, and the rest of the curve is almost horizontal. As additional time passes, the slopes toward the outlet and the sealed end both become more moderate, which in turn makes interpretation of D(u) via Eq. [1] less sensitive to experimental errors in predicting u(x) (van Grinsven et al., 1985). At 92 min, the predicted and 8

t = 75 min t = 45 min

6 E [ml]

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

E [mL]

12

t = 92 min

4 t = 15 min 2

t = 3 min

t = 60 min t = 30 min

0 0 Fig. 3. A soil column is suspended at the drying end, F0, and at the sealed end, F1, by a load cell. Imposed on the soil column is a graph of water content vs. distance from the drying face, u(x, t). The gray area represents the change in water content. The cross-hatched circle denotes the centroid of the change in water content along the x axis, x(t).

20

40

60

80

t [s1/2] Fig. 4. Cumulative evaporation (E) from the silt loam core vs. the square root of time (t). From 0 to 3 min, the outlet water content (x 5 0) was reducing from the initial water content, ui, to the water content at the drying surface, u0.

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0.35

0.35

0.30

0.30

0.25

0.25

0.15 0.10 0.05

θ

15 min 30 min 45 min 60 min 75 min 92 min Measured at 92 min

θ

0.15 0.10 0.05

0.00 0

5

10

15

0.00 0.00

0.05

0.10

x [cm]

0.15

0.20

0.25

-1/2

λ [cm s

Fig. 5. Silt loam predicted water contents (u) at 15, 30, 45, 60, 75, and 92 min. Note that the 15-min curve is quite square, which would make interpretation of the hydraulic diffusivity, D(u), via Eq. [1] difficult. The data points were measured for validation purposes by oven drying the dissected soil column.

measured u(x) curves are similar, with a RMSE of 0.012. The optimized values of b and n at various times are presented in Table 1. Figure 6 presents the same p data ffi as Fig. 5, but with the x axis normalized to l 5 x/ t . Ideally, all six curves should lie atop one another. The top curve (15 min), and to a lesser degree the second curve (30 min), are slightly different than the nearly identical 45-, 60-, 75-, and 92-min curves. As shown in Fig. 5, almost all the change during the first 15 min takes place within 0 to 1 cm of the drying surface, which can cause slope sensitivity problems. Also, the 3-min delay in initially establishing u0 becomes less significant as additional time elapses. For example, at 15 min of elapsed time, the 3-min nonlinearity period accounts for one-fifth of the test. The silt loam D(u) curves calculated using Eq. [9] are presented in Fig. 7. Predictably, the 15- and 30-min curves are different from the four later curves, which represent better estimates of D(u). Figure 8 presents the predicted and measured u(l) curves for the clay loam. The clayploam required 6 min to ffi achieve linearity between E and t . The measured initial and outlet water contents were ui 5 0.38 and u0 5 0.04. The u(l) curves do not fall atop one another as they should, particularly at early times, which indicates that one of the initial or boundary conditions was not fully achieved. The trend present in Fig. 8 is reflected in the D(u) curves presented in Fig. 9. The benefit of collecting time series data is that a comparison of estimates from

]

Fig. 6. Silt loam predicted water contents (u) vs. the normalized x axis (l). Ideally, all four plots should lie atop one another. The top curves (15 and 30 min) are slightly different than the nearly identical 45-, 60-, 75-, and 92-min curves.

different times can be conducted, allowing at least a qualitative judgment of the data. In the case of Fig. 9, it appears that the curves are moderately similar, with the exception of the early time data, which is easy to dismiss for reasons discussed above.

Application of Method to Previously Published Data Figure 10 presents a comparison of estimated and measured u(x) curves for several measured u(x) data sets obtained from the literature. The measured data sets were previously published by Arya et al. (1975), van Grinsven et al. (1985), Van den Berg and Louters (1986, 1988), and Arya (2002). Table 2 presents a description of the soil descriptions, water contents, fitted b and n values, the RMSE for this method, and the RMSE for directly measured values. The RMSE for this method compares predicted and measured water content profiles after optimizing b and n using the procedure pre10-1

D (θ ) [cm2 s-1]

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15 min 30 min 45 min 60 min 75 min 92 min Measured at 92 min

0.20

0.20

15 min 30 min 45 min 60 min 75 min 92 min

10-2

10-3

Table 1. Optimized values of the fitting parameters b and n at various times for the silt loam sample. Elapsed time min 15 30 45 60 75 92

b 0.41 0.77 1.11 1.53 1.53 1.86

n 1.50 1.54 1.59 1.72 1.56 1.58

10-4 0.0

0.1

θ

0.2

0.3

Fig. 7. Plots of the silt loam hydraulic diffusivity [D(u)] vs. water content (u). Note that the 15- and 30-min D(u) curves are different from those measured at later times. The 45-, 60-, 75-, and 92-min D(u) curves are very similar.

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0.4

-1

10

15 min 30 min 45 min 60 min 75 min Measured at 75 min

0.2

0.1

0.0 0.00

D (θ ) [cm2 s-1]

10

0.05

0.10

0.15

λ [cm s

-1/2

0.20

0.25

15 min 30 min 45 min 60 min 75 min

-2

10-3

10

-4

10

-5

0.0

]

Fig. 8. Clay loam water content (u) vs. the normalized x axis (l).

sented here. The RMSE for directly measured values also compares predicted and measured water content profiles, but b and n were optimized using the measured u(x) data. Although the RMSE are smaller for some of the cores when u(x) is known a priori, the RMSE values for this method match those for the measured data quite well.

0.1

0.2 0.3 0.4 θ Fig. 9. Plots of clay loam hydraulic diffusivity [D(u)] vs. water content (u). As with the silt loam (Fig. 7), the curves measured at later times are very similar.

CONCLUDING REMARKS This research presents a new technique to collect and analyze data from a hot-air soil test, which provides soil water diffusivity. Dissection and oven drying of the

0.5 0.4

θ

0.3 0.2 0.1

a)

0.0 0

2

4 x [cm]

6

8 0.4

0.4

0.3

θ

θ

0.3 0.2

0.2 0.1

0.1

b) 0.0

d) 0.0 0

2

4

6

8

10

0

2

4

6

8

10

x [cm]

x [cm] 0.4

0.4

0.3

0.3

0.2

0.2

θ

θ

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

θ

0.3

0.1

0.1

e) 0.0

c) 0.0 0

2

4

6 x [cm]

8

10

0

2

4

6

x [cm]

Fig. 10. Measured u(x) and modeled curves fit by applying this method to data sets originally published by: (a) Arya et al., 1975; (b) van Grinsven et al., 1985; (c) Van den Berg and Louters, 1986; (d) Van den Berg and Louters, 1988; and (e) Arya, 2002.

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Table 2. Optimization results of previously published and this study’s hot-air profile water content [u(x)] curves.

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

Fitting parameters Source of measured u(x)

Soil description

b

n

RMSE (this method†)

RMSE (direct‡)

Arya et al. (1975) van Grinsven et al. (1985) Van den Berg and Louters (1986) Van den Berg and Louters (1988) Arya (2002) This study This study

loam loamy, fine sand weathered marl loamy loam silt loam clay loam

0.50 9.21 0.70 0.68 0.57 1.86 1.28

1.12 5.73 1.72 1.64 1.73 1.58 1.45

0.021 0.005 0.011 0.026 0.014 0.012 0.012

0.013 0.005 0.010 0.015 0.007 0.009 0.008

† The RMSE between measured and modeled values were calculated after optimizing b and n via Eq. [6]. ‡ The RMSE between measured and modeled values were minimized by varying b and n without regard to Eq. [6].

soil column is not required, which enables the use of longer columns with more moderate water content profiles that ease the estimation of hydraulic diffusivity. Instead, the soil column is hung from two load cells, which along with the initial and outlet water content, enable estimation of the water content profile as a function of time. Multiple estimates of the water content curve through time can be normalized to what should be a single curve. Confirming that the normalized curves lie atop one another, and that a linear relationship exists between pffi cumulative evaporation and t provides feedback that the initial and boundary conditions were properly maintained. These assessments of the testing quality are not available without the time series data provided by the new method. A silt loam and a clay loam were tested. The RMSE values between the measured and estimated water content profiles were 0.012 for both soils. In addition, we modeled the performance of the new testing procedure on five previously published hot-air water content profiles and achieved similar performance. In view of the many sources of error in measuring water content profiles in a hot-air dried core, we believe our load-cell technique maintains consistency and produces reasonable results. It overcomes many of the prob-

lems associated with the original method, which requires extrusion and sectioning of the core. REFERENCES Arya, L.M. 2002. Wind and hot-air methods. p. 916–926. In J.H. Dane and G.C. Topp (ed.) Methods of soil analysis. Part 4: Physical methods. SSSA Book Ser. 5. SSSA, Madison, WI. Arya, L.M., D.A. Farrell, and G.R. Blake. 1975. A field study of soil water depletion in presence of growing soybean roots: I. Determination of hydraulic properties of the soil. Soil Sci. Soc. Am. J. 39:424–430. Bruce, R.R., and A. Klute. 1956. The measurement of soil diffusivity. Soil Sci. Soc. Am. Proc. 20:458–462. Gieske, A., and J.J. de Vries. 1990. Note on the analysis of moisture– depth curves obtained by the hot-air method for the determination of soil moisture diffusivity. J. Hydrol. 115:261–268. Stolte, J., J.I. Freijer, W. Bouten, C. Dirksen, J.M. Halbertsma, J.C. Van Dam, J.A. Van den Berg, G.J. Veerman, and J.H.M. Wo¨sten. 1994. Comparison of six methods to determine unsaturated soil hydraulic conductivity. Soil Sci. Soc. Am. J. 58:1596–1603. Van den Berg, J.A., and T. Louters. 1986. An algorithm for computing the relationship between diffusivity and soil moisture content from the hot air method. J. Hydrol. 83:149–159. Van den Berg, J.A., and T. Louters. 1988. The variability of soil moisture diffusivity of loamy to silty soils on marl, determined by the hot air method. J. Hydrol. 97:235–250. van Grinsven, J.J.M., C. Dirksen, and W. Bouten. 1985. Evaluation of the hot air method for measuring soil water diffusivity. Soil Sci. Soc. Am. J. 49:1093–1099.