A Simple Method for Estimating the Rate of Evaporation from a Dry Sand Surface. Tetsuo KOBAYASHI*, Akiyoshi MATSUDA** and MakioKAMICHIKA**

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1 (J. Agr. Met.) 44 (4): ,1989 A Simple Method for Estimating the Rate of Evaporation from a Dry Sand Surface Tetsuo KOBAYASHI*, Akiyoshi MATSUDA** and MakioKAMICHIKA** * Faculty of Agriculture, Kyushu University, Fukuoka 812 ** Sand Dune Research Institute, Faculty of Agriculture, Tottori University, Tottori 680, Japan Abstract The authors' model of a dry surface layer of sand predicts that the upward vapor flux in the dry surface layer, or the rate of evaporation from the dry sand surface is not affected by the temperature gradient developed in the surface layer ifthe gradient is not too steep (Kobayashi et al., 1987). To verify this prediction, a simple laboratory experiment was conducted. Although the temperature gradient in a sandy field grows to be more than 5 Kcm-1 on a clear day in high summer, we could make it develop only up to about 3 Kcm-1 in this experiment. However, at least under the present experimental conditions, the above prediction could be corroborated. Thus, the present model gives us a simple method for estimating the rate of evaporation from a dry sand surface, which demands only a knowledge of the surface temperature and water content, and the thickness of the dry surface layer. In a sand dune field, there always the dry surfacelayer exists except during a short period immediatelyfollowing a rainfall or irrigation; thus this method may be effective in investigating the water budget in a sandy field. 1. Introduction We formulated an isothermal and a nonisothermal steady models for a dry surface layer of sand, or a dry sand layer, and revealed the essential mechanism that grows the layer. Our nonisothermal steady model predicts that the upward vapor flux in the dry sand layer, or the rate of evaporation from the dry sand surface, is not affected by the temperature gradient developed in the surface layer if the gradient is not too steep. This prediction suggests that the evaporation rate in a sandy field where soil temperature gradients exist near the surface can be estimated from the surface temperature and water content, and the thickness of the dry sand layer (Kobayashi et al., 1986; 1987). To verify the above prediction, a simple experiment was conducted. It is very difficult to develop a steep gradient of temperature experimentally across the dry surface layer in a small test piece; Read at the Kyushu Chapter Meeting, Saga, December 5, 1987 Received July 11, 1988 also, it is not easy to measure the temperature gradient with precision. However, at least under the conditions imposed in this experiment, the above prediction was corroborated, the details of which will be discussed in this paper. We may expect it is effective to use this model in investigating the water budget in a sandy field, since there always the dry surface layer exists except during a short period immediately following a rainfall or irrigation. We wish also to use this model for estimating the evaporation rate in an arid region; we must solve a few problems before putting this method to practical use though (Kobayashi et al., 1988). 2. Derivation of basic equations In a dry sand layer, the water movement is assumed to be exclusively in the vapor phase and to occur by molecular diffusion. The effects of temperature gradients on the vapor transfer in soils is very complicated; however, in a dry sand layer at the growing stage, the upward vapor flux q (cm s1, positive upward) can be expressed as

2 q =D(ƒÆ,T) dƒæ/dz, (1) where z is the depth (cm, positive downward), the volumetric water content (cm3 cm-3 ), T the temperature (K) and D (ƒæ, T) the water vapor diffusivity in the soil (cm2 s-1) (Kobayashi et al., 1987). Although the theoretical form of the diffusivity is very complicated, this function can be expressed to a good approximation by using a polynomial. We obtained such an approximation to the diffusivity for the Tottori Dune sand: for 273 K T 333 K and 0 <ƒæ ƒæƒâ, where ƒæƒâ is the value of ƒæ at the interface between thedry surface layer and the moist underlayer (%, henceforth expressed in percent) (Kobayashi et al,1986). As a result, assuming that the state is steady and the temperature in a dry sand layer is expressed as T=T0+ƒ z (O z ƒâ), (3) and substituting equations(2) and (3) into equation (1) and integrating from zero to 8 with respect z, we get where ƒ is the temperature gradient across the dry sand layer (K cm-1), ƒâ the thickness of the layer (cm), and T0 and ƒæ0 are the soil temperature and the water content at the surface, respectively. The value of ƒæƒâ for the Tottori Dune Sand ranges from 1.5% through 2%, which range corresponds to the f ( ƒæƒâ )'s value of about When the change in temperature across the dry surface layer ƒ ƒâ is so small that ƒ ƒâ 1, i.e., ƒ ƒâ á 15K, using the átaylor series for the exponential function in the denominator, w have which is the same as the result obtained from our isothermal steady model for a dry sand layer, except that the surface temperature To is substituted for the temperature T uniform across the layer (Kobayashi et al., 1986). 3. Experimental procedure Three kinds of sample containers 5 cm long, 7.5 cm long and 10 cm long were constructed from transparent acrylic resin tubes of inside diameter 5 cm, which made it possible to distinguish the dry surface layer from the moist underlayer by the different colors. The bottom of the container was sealed except for a hole 1 cm in diameter, which hole was covered with a filter paper. We made four samples for every size of container, The Tottori Dune sand was packed into each container by water binding, the bulk density being about 1.5 g cm-3. The container was covered with an acrylic resin disk to prevent evaporation from the top surface and allowed to stand for a week to remove the gravitational water out from the hole at the bottom. The hole at the bottom of the container was sealed and the top was opened before it was placed in a chamber where the temperature and the relative humidity were maintained at 303 K and 60%, respectively. The top surfaces of sample columns were in the same level and irradiated with metalhalide lamps (30,000lx). In addition, the fans installed in the system provided horizontal air flow at a speed of about 0.5 ms-1 The temperatures at the top and bottom surfaces of the columns were measured with thermister thermometers at intervals of one hour; further, at the same time, each sample was weighed with an electronic balance to get the amount of evaporation. After the formation of the dry surface layer was confirmed, the thickness of the layer also was measured with a ruler at the outside of the sample container. Finally, as the last step of procedure, the water content in the dry surface layer of each sample was measured. 4. Results and discussion Figure 1 shows how the evaporation rate varied in the course of time. Each curve denotes the average rate of four samples with the same length. The evaporation rates were almost the same for the three kinds of samples during the first twenty

3 T. Kobayashi, et al.: A Simple Method for Estimating the Rate of Evaporation from a Dry Sand Surface Fig. 1. Time variations of the evaporation rates. The arrows attached to the curves indicate the time when the formation of the dry surface layer was confirmed by eye three hours; then, the samples 5 cm long went into the second stage of drying, or the falling-ratestage of drying. The rates of the samples 7.5 cm long began to go down after thirty five hours of evaporation. The samples 10 cm long, however, did not show the second stage of drying, because the running period of forty six hours was too short for them to go into the second stage. The arrows attached to the curves tell us when the formation of the dry surface layer was confirmed by eye. The evaporation rates at these points in time were about 0.4 to 0.5 of the rates in the first stage of drying, or the constant-rate stage of drying. The evaporation rates during the first stage did not strictly remain constant with time, but decreased slowly and steadily in this experiment; also, over all the running period, there existed shortperiod changes in them. The unstable conditions in the chamber seem to be the cause of the latter phenomenon. The major cause of the former, however, appears to be the time change in moisture profile in an upper layer of the sample column, because the three kinds of samples that were subject to different moisture conditions as a whole changed in like manner. Figure 2 shows the comparison of the evaporation rates calculated by using the present method (dashed lines) with the observed ones (solid line) for the samples 5 cm long. They depict the evaporation rates, of course, after the formation ofthe dry surface layer. The dotted line indicates the observed thickness of the dry surface layer. The dashed lines illustrate the estimates made by giving a value for ľ0 of 0.6 or 0.7%. The average water content through the dry surface layer was 0.84%. The moisture profile in a dry surface layer shows that the water content decreases progressively toward the surface; thus, the value of ľ at the surface, ľ0, must have been smaller than 0.84%. In addition, the surface water content depends on the humidity at the surface, that is, on the surface temperature and the thickness of the dry surface layer. In the present experiment, the surface temperature was in the range of 310K to 312K and the thickness was less than 1.5 cm, hence it seems reasonable to take Bo to be 0.6 `0.7% (Kobayashi et al., 1986). Thus, this extent of agreement between theory and experiment can be regarded as quite satisfactory. In the present experiment, we could not directly measure the temperature gradients developed in the dry surface layer; then we tried to estimate them on the basis of a simple model, as shown in Fig. 3. Let us assume that horizontal heat transfer can

4 Fig. 2. Comparison of the evaporation rates calculated by using the present method with the experimental ones moist underlayer, respectively. Thus where H is the length of the sample column, E is the evaporation rate and L the latent heat of vaporization of water. Since the average of water contents in the dry surface layer was about 1 % and that in the moist Fig. 3 Schematic depiction of fluxes in the energy balance at the interface between the dry surface layer and the moist underlayer be neglected and evaporation occurs only at the interface between the dry surface layer and the moist underlayer. Moreover, suppose that the heat flux through the dry surface layer downward, Q1, and the flux in the moist underlayer upward, Q2, both of which head toward the evaporating surface, are written where ƒé1 and ƒé2 are the heat conductivities in the dry surface layer and the moist underlayer, respectively; Tt, Tb and TƒÂ are the temperatures at the top surface, the bottom surface, and the interface between the dry surface layer and the underlayer was less than 4%, which decreased in the course of time, we can set the ratio ƒé2/ƒé1 à2 (Shinjow et al., 1980; Takemasa et al., 1988). Substituting this relation and H = 5 cm into equation (8), we get TƒÂ à Tb, which means that the latent heat of vaporization was supplied mostly from the top surface irradiated with metalhalide lamps. The temperature gradient occurring through the dry sand layer is large during the initial stage when the thickness of the layer is small and the evaporation rate is large. The gradient, for instance, was 2.7 K cm-1 when the thickness was 0.4 cm and was 1.2 K cm-1 when the thickness was 1.45 cm; that is, ƒ ƒâ =1.1 `1.8 K, which satisfys the inequality ƒ ƒâ á15 K. The error in the estimate of temperature gradient due to the neglect of the horizontal heat flow seems to be small, because the temperature at the side wall of sample con-

5 T. Kobayashi, et al.: A Simple Method for Estimating the Rate of Evaporation from a Dry Sand Surface tamer was nearly equal to the temperature at the bottom. The temperature gradient directed toward the surface across the dry surface layer should reduce the evaporation rate; thus, the experimental values are expected to be smaller than the estimates made by using the present method during the initial stage when the gradient is quite large. There seems to be such a tendency, but it cannot be concluded with certainty because the same phenomenon occurs also during the other period (Fig. 2), and as well the estimates of Bo and 8 are subject to errors. However, it can be concluded that the effect of temperature gradients on th evaporation rate is equal to or smaller than that due to the error in the estimates of ľ0 orĉ, at least under the present experimental conditions. The above leads to the conclusion that the rate of evaporation from a dry sand surface can be estimated using the soil temperature and water content at the surface and the thickness of the dry surface layer if the temperature gradient in the layer is not too steep. 5. Concluding remarks In a sandy field, the falling-rate stage of drying prevails except during a short period immediately following a rainfall or irrigation; thus, usingthe profiles of soil temperature and water content in the upper layer to estimate the evaporation rate, as does the present method, seems to be reasonable. According to this method, the rate of evaporation occurring at a dry sand surface can be estimated from the thickness of the dry surface layer and the soil temperature and water content at the surface. There is a likelihood that the above conditions at the surface can be estimated from the remote sensing data; therefore, if the thickness of a dry surface layer can be related to the surface conditions or can be regarded as having a particular value, we can estimate the evaporation rate in a dry sandy field by using the remote sensing data. We have every reason to expect that the present method may be applied to estimate the evaporation rate in an arid region, although there still remain a few problems that must be solved (Kobayashi et al., 1988). Our experiments on this subject were all made in a laboratory, where the speed of artificial air motion over the dry top surface was at most 0.5 ms-1. However, under field conditions, th wind will blow more hard; thus, wind-induced pressure fluctuations at the surface may expedite vapor transport through soil near the surface. T0 clarify how much influence this phenomenon exerts on the evaporation rate is a problem that must be solved. We can, however, surmise that its effect must be negligible unless the thickness of the dry surface layer is smaller than a few millimeters, since the water vapor transfer in soils due to air turbulence decreases abruptly with decreasing particle size, and the mean diameter of the Tottori Dune sand of about 0.2 mm is small enough (Scotter and Raats, 1969). Acknowledgement We would like to thank Mr. M. Matsubara for his help in carrying out the experiment. References Kobayashi, T., Li, Q. H., Motoda, Y., Matsuda, A. and Kamichika, M., 1988: A method for estimating the evaporation rate in an arid region from the soil temperature and water content at the surface. International Workshop on the Atmosphere-Land Surface Interactions Experiment at Heihe River Basin, Gansu, China, Abstract Booklet, Kobayashi, T., Matsuda, A., Kamichika, M. and Sato, T., 1986:, Studies of the dry surface layer in a sand dune field (1) Modeling of the dry surface layer of sand under isothermal steady conditions. J. Agr. Met., 42, Kobayashi, T., Matsuda, A. and Kamichika, M., 1987: Studies of the dry surface layer in a sand dune field (2) Effects of soil temperature gradients on the water content profiles in the dry surface layer. J. Agr. Met., 43, Scotter, D. R. and Raats, P. A. C., 1969: Dispersion of water vapor in soil due to air turbulence. Soil Sci., 108, Shinjow, A., Matsuda, A. and Kamichika, M., 1980: Study on the variation of the profiles of soil temperature and moisture content in a sand dune field without vegetation. Sakyu Kenkyu, 27, (in Japanese with English summary). Takemasa, T., Cho, T. and Kuroda, M., 1988: Temperature analysis of sandy soil near ground surface by the two-layer model. Trans. JSIDRE, No. 133, (in Japanese with English summary).

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