Temperature dynamics during nonisothermal evaporation from drying porous surfaces



[1] The partitioning of incoming shortwave radiative energy on evaporative surfaces determines mass and energy exchange with the atmosphere, and influences measurements of various climatic and hydrologic processes. We quantified the coupling between an evaporative flux from a drying porous surface and the corresponding surface temperature dynamics. The analytical approach employs a pore-scale diffusion model for the evaporative flux from a unit cell (representing the porous surface) based on Schlünder's (1988) model. The evaporation flux from the unit cell's pore was linked with other components of the surface energy balance through heat exchange across the pore's solid walls. Model predictions for evaporative flux and associated mean thermal fields during the drying of porous surfaces were in good agreement with experimental results. The model was used to predict the so-called soil evaporation transfer coefficient (ha), yielding good agreement with measurements. Analysis shows that commonly assumed interfacial isothermal conditions (where surface and air temperatures are similar) may yield 15–40% overestimation in evaporation rates relative to nonisothermal conditions (where evaporation rates affect surface temperature). Theoretical results indicate that when shortwave radiation is significant, an evaporating porous surface may gradually warm up as it dries resulting in enhancement of evaporation rates and energy partitioning. The analytical model addresses the long standing challenge of nonlinear evaporative fluxes from drying surfaces while providing predictions for surface energy partitioning over evaporating surfaces.

1. Introduction

[2] The partitioning of shortwave radiative energy received on terrestrial surfaces and its coupling with evaporation fluxes is of central importance for hydrologic and climatic models and for various methods based on remotely sensed surface temperature [Norman and Becker, 1995; Troufleau et al., 1997; Bastiaanssen et al., 1998a]. Interactions between porous media transport properties and external boundary conditions exert significant control on surface evaporation dynamics. A relatively constant and high evaporation rate during stage 1 is sustained by capillary transport via continuous liquid pathways to vaporization plane at the surface [van De Griend and Owe, 1994; Shokri et al., 2008]. At a certain drying front depth [Lehmann et al., 2008] stage 1 evaporation abruptly ends, and the process becomes diffusion controlled (stage 2) with the vaporization plane migrating below the surface [Shokri and Or, 2011; Prat, 2002; van Brakel, 1980]. Even during the constant rate stage, numerous changes take place including the gradual drying of the surface and the continuous migration of the primary drying front into the subsurface [Lehmann et al., 2008; Shokri et al., 2008]. Shahraeeni et al. [2012] have shown that the maintenance of a constant evaporation rate involves complex diffusive adjustments that include a nonlinear vapor flux enhancement from remaining pores as the surface dries and pore spacing increases. We note that the analyses of Shahraeeni et al. [2012] were based on the implicit assumption of isothermal conditions, namely that the evaporative surface and overlaying air are at the same temperature.

[3] Evidence suggests that the temperature of an evaporative surface may be significantly different than the ambient air temperature [Shahraeeni and Or, 2010; Jacobs and Verhoef, 1997; Belhamri, 2003; Kaviany and Mittal, 1987; Lu et al., 2005], hence, accounting for surface temperature could be important for quantifying evaporation rates and modeling the components of surface energy balance [Noilhan and Planton, 1989; Mahfouf and Noilhan, 1991]. Temperature variations during drying of porous surfaces are likely to affect the vapor pressure gradient between evaporating water menisci and the air mass (relative to isothermal conditions). Such effects have been evaluated experimentally by Kaviany and Mittal [1987] and numerically by Metzger and Tsotsas [2005] demonstrating significant differences between isothermal and nonisothermal evaporation with warmer air flow increasing both surface temperature and evaporation rate. In contrast with many laboratory measurements, evaporation under natural conditions with a significant component of shortwave radiation could induce surface temperatures exceeding air temperature even for high evaporation rates [Qiu et al., 1998; Jacobs and Verhoef, 1997]. Such conditions (surfaces warmer than air) would affect vapor gradients and evaporation rates relative to isothermal conditions and deserve special attention.

[4] Various remote sensing methods have been proposed for linking surface water content with evaporation rates and energy partitioning on natural surfaces as recently reviewed by Kalma et al. [2008] and by Kustas and Anderson [2009]. An interesting application of surface temperature information for estimation of evaporative flux was proposed by Qiu et al. [1998] based on their “three temperatures” model (3T). The model uses information on air temperature, wet and dry soil temperatures to establish bounds on evaporation rates from soil surfaces [Qiu et al., 2006]. The core of the 3T model is the soil evaporation transfer coefficient (ha) used by Zhao et al. [2010] for linking such thermal measurements with surface soil water content. A logarithmic relation between ha and soil water content has been proposed which was in good agreement with field measurement data.

[5] The primary objective of this study was to analytically quantify the coupling between evaporation fluxes and surface temperature evolution during stage 1 evaporation. We then apply the analytical derivations to the empirical soil evaporation transfer coefficient (ha) proposed for remote estimation of evaporation rates of large areas [Qiu et al., 1998]. Finally, we evaluate the magnitudes of evaporation rates under isothermal and nonisothermal conditions from porous surfaces.

[6] Following this introduction, we present the theoretical background and model derivation for coupled evaporation and energy balance for a unit cell that represents an evaporating porous surface. Next, we compare model predictions for evaporation rates and surface temperature evolution with measurements including effects of surface temperature on evaporation rate from sand samples. Finally, we assess effects of shortwave radiation on evaporation dynamics for isothermal and nonisothermal conditions.

2. Theoretical Considerations

[7] The pore-scale implications of gradual drying of porous surfaces during evaporation involve the sequential invasion of surface pores by air replacing the lost liquid volume (from largest to smallest [Prat, 2002; Shahraeeni et al., 2012]), and an associated increase in spacing between remaining active pores (for uniformly distributed pore sizes on the surface). For purposes of modeling and derivation of an analytical description of a drying surface, we define a prototypic unit cell that includes an evaporating pore (of representative radius) surrounded by dry surface as shown in Figure 1 (where the proportion of the pore to total area defining surface water content). The diffusing water vapor from water menisci at the surface is transported across an air viscous sublayer (determined by mean air velocity, see Shahraeeni et al. [2012]) at a rate defined by the local concentration gradient between the surface and the air and the geometry of the diffusion field. The model for vapor diffusion from a pore across a prescribed boundary layer considers the nonlinear evolution of the vapor field above the saturated surface from a layered and uniform 1D field into a collection of 3D vapor shells as spacing between pores increases. Such adjustment in the vapor field is responsible for the enhanced per-pore evaporation rate relative to the 1D scenario and is the key to maintenance of the often-observed constant rate during stage 1 evaporation [Shahraeeni et al., 2012].

Figure 1.

Schematic sketch of an evaporating surface and the elementary unit cell that composes of a single evaporating pore and the dry surface surrounding it.

[8] Quantification of the combined thermal and diffusive processes at the pore-scale requires solution of the coupled energy and diffusion equations formulated for the unit cell in Figure 1. The results are subsequently upscaled to represent the entire surface. We investigated the coupling between heat and mass transfer at the surface of drying porous media by considering details of temperature fields forming around individual evaporating pores. This in turn permits explicit account of the water vapor concentration at the surface of water menisci for different surface water temperatures. The relatively slow dynamics of evaporation permit consideration of quasi-steady conditions in terms of variation of surface water content meaning that the governing equation is solved time independently. The key step is translating boundary conditions for a gradually drying surface into an increased dry surface region surrounding the prototypic pore described in Figure 1. The energy balance for an infinitesimal element of dry region surrounding an evaporating pore in radial coordinates is defined as

display math(1)

where T is the unit cell temperature (K) at radial coordinates r (m), k is the effective thermal conductivity of the soil (W/mK), hair is the air thermal convection coefficient (W/m2K), T is the ambient air temperature (K), σ is the Stefan-Boltzmann constant math formula, ε is the soil emissivity, Δh is the thickness of surface evaporating pore (m) (unit cell) considered to be equivalent to one grain size in the model, ΔH is the thermal decay depth below the surface (m) or the depth at which the evaporative thermal perturbation at the surface decays to prescribed value [Shahraeeni and Or, 2011], a is pore radius (m), and b is the dry surface area radius around the pore (m) (that varies with surface water content). The first term in equation (1), considers radial heat conduction within the unit cell; the second and third terms represent the longwave and shortwave radiation fluxes, respectively; the fourth term represents the convective heat transfer between surface and air flow (the sensible heat flux); and last term is the soil heat flux toward (or from) the evaporative surface.

[9] The slightly nonlinear shape of the vertical temperature distribution below an evaporating surface [Shahraeeni and Or, 2011] was linearized from the surface to the thermal decay depth (as described in detail shortly). Due to symmetry, no thermal conduction takes place between adjacent unit cells. An important element in the coupling of heat and mass transfer equations is the latent heat of vaporization that results a conductive heat flux (qcond) across the pore wall due to evaporation cooling with lower temperature within the pore. This thermal flux serves as a boundary condition for equation (1):

display math(2a)
display math(2b)

where k is the effective thermal conductivity of the soil. For clarity, we distinguish the soil heat flux considered as a vertical heat flux toward the surface (G), from the internal radial heat flux from the dry region to the water-filled pore (qcond) within the unit cell (given in equation (2a)). The value of qcond is obtained from energy balance applied to near-surface (Δh) pore water:

display math(3)

where qevap, qconv, and qrad are evaporative flux, convective flux, and radiative flux on the surface of the pore, and G is the soil heat flux. Equation (3) expresses the dependency of the radial conductive heat flux on the evaporation rate from the pore.

[10] The thermal decay depth (ΔH) is defined as the depth at which the value of the temperature reaches 0.99 of the initial soil temperature (T). The linearized temperature profile leads to the following expression for the thermal decay depth (ΔH in equation (1)):

display math(4)

[11] With k the effective soil thermal conductivity (W/mK), E is the evaporation rate (kg/m2s), L is the latent heat of vaporization for water (2.45 MJ/kg) considered as a constant, and math formula is the mean temperature of the evaporating surface (K). The thermal imaging experimental results of Shahraeeni and Or [2011] yielded values of ΔH in the range of 30–60 mm below the surface (for mean evaporation rate of 5 mm/day), these results were in general agreement with earlier measurements by Gardner and Hanks [1966] that used soil heat flux plates.

[12] A key variable in equation (3) is the evaporative heat flux (qevap) from a pore that determines the boundary condition required for solving equation (1). The evaporative flux from an individual pore is obtained from the diffusion model of Schlünder [1988] that has been used recently to determine resistance to evaporation from drying soil surfaces by Haghighi et al. [2013]. The diffusion-based model determines the evaporative flux from individual but interacting pores as a function of pore size (a), pore spacing (b), and the thickness of air viscous sublayer (δm) given in terms of an easy-to-determine mean surface water content as

display math(5)

where qevap is the evaporative heat flux from the pore surface (W/m2), D is the vapor diffusion coefficient in air (m2/s), θ is the surface water content (determined as the ratio of pore area to total unit cell area), L is the latent heat of vaporization for water (2.45 MJ/kg), and Cs and Ca are water vapor concentrations at the water meniscus surface and in the flowing air mass (kg/m3) (above the viscous boundary layer), respectively. Based on the analysis of Haghighi et al. [2013], the contribution of convective air flow to vapor transfer between the surface and the air is negligible (other than setting the boundary layer thickness), and the process is dominated by diffusion (especially for low wind speeds: v < 4 m/s).

[13] Application of equations (3) and (5) enables analytical solution of equation (1) for the radial temperature distribution around individual evaporating pores on the surface for the assumed quasi-steady conditions (solution details are provided in Appendix A):

display math(6a)

where I0 and I1, and K0 and K1 are the modified Bessel functions of the first and second kind, respectively, and the variable ψ is defined as

display math(6b)

[14] For simplicity, the nonlinear dependency of longwave radiation flux on surface temperature has been linearized according to Shahraeeni and Or [2011]:

display math(7)

[15] Details of temperature distribution over the water meniscus surface are replaced by mean water surface temperature at r = a. A central feature of the model is the explicit temperature dependency of vapor concentration math formula over the water menisci that contributes to the coupling between evaporation rate and temperature distribution of the unit cell. We thus solve the coupled energy balance and vapor diffusion equations simultaneously for prescribed values of surface water content. Equation (6a) is solved iteratively for surface temperature, but it is also solvable in closed-form for linearized values of saturated vapor concentration (Cs) with minor errors. In the following, we summarize the main assumptions made in the derivation of the analytical solution in equation (6):

[16] 1. Considering the small size of the unit cell, we assume uniform shortwave radiation flux (Rs) on the surface, and soil albedo is considered directly in the value of net Rs. Also, the air temperature (T) and sensible heat flux coefficient (hair) are constant over the surface.

[17] 2. For laminar air flow over a horizontal flat surface, the coefficient of convective heat transfer (hair) is determined based on average Nusselt number [Incropera and DeWitt, 2001] as

display math(8)

where kair is the heat conduction coefficient for air, l is the characteristic length of the surface in direction of flow, and Re and Pr are the Reynolds and Prandtl numbers, respectively. A similar relation ( math formula) for turbulent flow over flat surfaces can also be used as input to estimate convection heat transfer coefficient (hair) in equation (1).

[18] 3. The air boundary layer thickness for mass exchange is defined based on the Schmidt number as

display math(9)

that links the aerodynamic (δv) and mass (δm) boundary layers. For laminar flow over a flat surface, the thickness of the aerodynamic layer is given based on the Blasius solution as [White, 2002]

display math(10)

where x is the distance from the leading edge of the surface. Similar expressions established experimentally by Shahraeeni and Or [2012] and Hisatake et al. [1995] for thickness of viscous sublayer as a function of wind speed in laminar and turbulent flows over evaporating surfaces can be used for quantification of (δm) in equation (5).

[19] 4. The thickness of unit cell (Δh) is considered as a single grain size and is also related to the pore size, roughly three times mean pore size [Glover and Walker, 2009].

[20] 5. The evaporative heat flux is solved based on Schlünder's diffusion model (equation (5)) and provides the essential coupling in the form of a boundary condition for the temperature field solution (equation (6a)).

[21] 6. Considering the thickness of the unit cell we consider only radial heat conduction and ignore axial conduction within the unit cell (not to be confused with the macroscopic soil heat flux G).

[22] 7. We assume that the relatively slow evaporation process permits neglecting convective heat transfer due to water flow from within the porous medium.

[23] 8. The soil thermal conductivity is a strong function of soil texture and water content. As the soil surface is drying and the dry solid region surrounding a pore expands, we assume that the dry region is composed of solid particles and air-filled voids (i.e., water saturation equals to zero) hence the value of thermal conductivity is estimated using the empirical relation of Chen [2008]:

display math(11)

where n is the porosity and S is the water saturation.

[24] 9. The temperature distribution on the pore water surface is replaced by a mean value meaning that the temperature at r = a is assumed to represent pore water temperature.

[25] With the progression of evaporative drying and the concurrent increase of the dry surface area surrounding evaporating pores, we seek simultaneous solutions of equations (5) and (6) to obtain the temperature distribution over the unit cell surface and the coupled evaporation rate from the pore. The obtained temperature distribution from equation (6a) is averaged to obtain a mean value for the surface according to

display math(12)

[26] In the following, we investigate model predictions for the coupling between surface temperature and evaporation rate. Note that in this analysis we represent the entire evaporating surface by mean values of surface temperature and the evaporation rates calculated for the unit cell.

3. Materials and Methods

3.1. Experimental Drying Rates of Lu et al. [2005] and Belhamri [2003]

[27] We have used published experimental data of drying of quartz sand [Lu et al., 2005] and a porous brick [Belhamri, 2003] for comparison with model predictions of evolution of surface temperature and evaporation rate during drying of these two porous media. The first scenario considers drying of quartz sand (45 mm in diameter and 15 mm in height) with particle size ranging from 1 to 1.5 mm subjected to a prescribed air flow velocity within a wind tunnel. The air flow velocity and temperature were 1.9 m/s and 321 K, respectively, and the relative humidity was 33% [Lu et al., 2005]. The second scenario describes drying of a porous brick in laboratory dryer using hot air flowing over the brick. The temperature of the air flow was 323 K, relative humidity was 14%, and flow velocity was 3 m/s [Belhamri, 2003]. Using the reported laboratory conditions, we employ our model to predict the variation of surface temperature and coupled evaporative flux during drying of samples.

3.2. Evaluation of the 3T Method of Qiu et al. [1998]

[28] Various methods have been proposed for using surface temperature to estimate evaporation rate [Carlson, 1986; Shahraeeni and Or, 2010; Kalma et al., 2008]. Qiu et al. [1998] have proposed a method based on three-temperature values (the temperatures of the evaporating surface, the dry surface, and the air) that are used to define a coefficient for surface energy partitioning according to

display math(13)

where LE is the evaporative heat flux, Rn and Rnd are net radiation of evaporating soil surface and dry soil, respectively, with G and Gd are the corresponding soil heat fluxes for evaporating soil and dry soil surfaces, respectively. The function ha is termed the soil evaporation transfer coefficient and is defined as

display math(14)

where Ts, Tsd, and Ta are evaporating soil surface, dry soil, and air temperatures, respectively. The parameters required for equation (13) are the three temperatures, soil heat flux and net radiation flux. Soil heat fluxes (G and Gd) and temperatures (Ts, Tsd, and Ta) can be measured and net radiation fluxes (Rn and Rnd) can be measured or estimated [Qiu et al., 1998]. We evaluated our analytical solution with this temperature-based method trying to predict the soil evaporation transfer coefficient (ha) and evaporation rates with Qiu and Ben-Asher's [2010] experimental results. Their experiment was carried out under controlled laboratory conditions on a sample of coarse sand with air temperature of 25°C, relative humidity of 50%, shortwave radiation at intensity of 635 W/m2, and initial water content of 0.27 m3/m3 [Qiu and Ben-Asher, 2010]. For the conditions above, we obtained mean temperature of the surface in form of ha and latent heat flux during “drying” of the unit cell.

3.3. Drying Sand Experiments

[29] We compared the analytical model with our own evaporation and surface temperature experimental results. Data were obtained from a series of evaporation experiments using a sample of coarse sand subjected to different evaporation demands (determined by air speed over the sample). A cylindrical sample (height 300 mm and diameter 55 mm) was filled with sand (grain size 0.7–1.2 mm) to a porosity of 0.4 and was consequently fully water saturated. Figure 2 shows the measured water retention curve (WRC) of the sand sample (symbols) and fitted van Genuchten model (line). The WRC enables prediction of the transition from stage 1 evaporation to stage 2 based on the concept of evaporative characteristic length [Lehmann et al., 2008] which was determined as 80 mm. To minimize heat exchange between the sand sample and surrounding air, we isolated the exterior of the sample. Details of the experimental set up can be seen in Figure 3.

Figure 2.

Measured water retention curve of the sand sample with grain size ranging from 0.7 to 1.2 mm (dots) and fitted van Genuchten model. Note that the linearization of the water retention curve used in this study with a sketch of surface water content evolution (θsurf(t)) with primary drying front depth L(t).

Figure 3.

Evaporation experimental setup depicting the insulated sand column mounted on a digital balance (with 0.01 g accuracy) and surface temperature monitoring using an IR camera (FLIR SC6000). Two tensiometers installed at 190 and 290 mm below the surface were used to mark the invasion of air into the saturated sample and thus the onset of evaporation.

[30] Mass loss as a function of time during the evaporation experiment was monitored using a digital balance (METTLER TOLEDO, PB4002-S/FACT, Switzerland). The evolution of surface temperature was recorded using a thermal imager (FLIR SC6000, USA) at 10 min interval and at a spatial resolution of 640 × 512 (yielding thermal information at 0.01 mm2 resolution). The air temperature and relative humidity in the lab were monitored and values were in the range of 25 ± 1°C (0.1° C accuracy) and 22 ± 4% (1% accuracy), respectively (Vaisala HUMICAP®, HMT337, Finland). Mean air flow velocity was measured above the sample surface using a hot wire anemometer with 0.01 m/s accuracy (Dostmann Electronic, P600, Germany).

[31] Two tensiometers were installed at 190 and 290 mm below the surface and used to mark the initial invasion of air into the saturated sand and thus the onset of evaporation from the column. Such information coupled with linearized water retention curve for quasi-hydrostatic conditions [Lehmann et al., 2008] enable estimation of surface water content variations with time (or with cumulative evaporation). Shokri et al.'s [2008] experimental study of water content distribution above the drying front using neutron radiography clearly shows that the reduction in surface water content with drying front depth is linear [Shokri et al., 2008, Figure 7]. Hence, for a relatively constant evaporation rate during stage 1, the evaporated water depth (total evaporated water volume per surface cross section) can be expressed (approximately) as

display math(15)

where θsurf is the surface water content, θsis saturated water content of the sample, e is the evaporation rate (mm/day), t is the elapsed time since onset of evaporation (day), and L(t) is the drying front depth (mm). L(t) is linked with the evaporative characteristic length (LG) based on the assumptions above (similarity of triangles in Figure 2):

display math(16)

from equations (15) and (16), we estimate variations in surface water content with time as

display math(17)

[32] In the following section, we present experimental results and comparisons with model predictions of the evolution of surface temperature and evaporation rate.

4. Results and Discussion

4.1. Comparison of Drying Surface Temperature Dynamics With Literature Data

[33] Figure 4 depicts a comparison between the measurements of Lu et al. [2005] and Belhamri [2003] and model predictions for the variations of surface temperature and evaporation rates during drying of quartz sand subjected to air flow (relative humidity 33%, temperature 321 K, flow velocity 1.9 m/s) in a wind tunnel [Lu et al., 2005]; and drying of a porous brick in a laboratory dryer (relative humidity 14%, temperature 323 K, flow velocity 3 m/s) [Belhamri, 2003].

Figure 4.

(a) The evolution of mean evaporation rate and (b) mean surface temperature with surface water content during evaporation from quartz sand data of Lu et al. [2005] (squares) and a porous brick results of Belhamri [2003] (triangles). Model predictions (lines) were based on reported experimental information. The surface water content data were extracted from Lu et al. [2005] and Belhamri [2003].

[34] Model estimates were in reasonable agreement with measurements (with slight over prediction of surface temperature by the model for high values of surface water content). A remarkable feature of the drying process seen in the experimental results (and captured by the model) is the sharp increase in surface temperature with the reduction in evaporative flux as the surface gradually dries and the number of evaporating pores on the surface decreases significantly. The effect of high air temperature on the evaporation rate can be seen in Figure 4a where high evaporative fluxes (15 and 25 mm/day) were sustained for a wide range of surface water contents (in contrast with experimental results of Shahraeeni et al. [2012] conducted under room temperature).

[35] These limited comparisons confirm general attributes of the analytical model with the inherent links between evaporative fluxes and surface temperature dynamics (including the thermal signature marking the end of stage 1 evaporation). Next, we expand model testing to remote sensing applications that use surface temperature to estimate evaporation [Qiu et al., 1998].

4.2. Evaluation of the Concept of ha

[36] The model was used to reevaluate measurements obtained by Qiu and Ben-Asher [2010] and offers analytical estimation of the relation between ha and evaporation rate (equations (13) and (14)) from sandy surfaces. For the conditions of Qiu and Ben-Asher's [2010] experiment (listed above), we obtained simultaneous solutions of equations (5) and (6) during drying (the progression of dry area surrounding the pore with radius of 250 µm) to determine the evaporative heat flux and mean surface temperature of the unit cell. In order to compare model predictions of the coupling between surface temperature and evaporative flux with Qiu and Ben-Asher's [2010] measurements, evolution of surface temperature in form of ha and its corresponding evaporative flux (line) are plotted in Figure 5. In this figure, the measured evaporation flux (diamonds) has been obtained by using the microlysimeter and the estimated values (circles) are related to the application of 3T model (equation (13)) for estimating evaporative flux from the surface [Qiu and Ben-Asher, 2010].

Figure 5.

Variations in latent heat flux versus coefficient ha during evaporation from coarse sand. Measured evaporative flux (diamonds) and estimated values (circles) were obtained using microlysimeter and application of 3T method [Qiu and Ben-Asher, 2010], respectively, and analytical model predictions are based on the reported values (line).

[37] The results in Figure 5 illustrate that high evaporation rates during stage 1, depress the temperature of the evaporative surface [Shahraeeni and Or, 2010] and the associated ha values are also reduced. In the basis of the coefficient ha is the assumption that the reduction in evaporation rate with surface drying is associated with a concurrent increase in surface temperature. The transition from stage 1 to stage 2 evaporation is indeed accompanied by a significant reduction in evaporative flux and is expected to result in a rapid increase of ha. Model predictions (line) depicted in Figure 5 were obtained with no parameter fitting, thereby offering physically based predictive capabilities for this mostly empirical approach.

[38] The comparisons depicted in Figures 5 emphasize the inherent links between surface temperature and evaporative flux from a drying surface. In the following, this coupling has been considered for different evaporation rates during evaporation from a sand sample using IR thermography of the drying surface.

4.3. Experimental Results Using Infrared Imaging of a Drying Sand Surface

[39] We then compared model predictions with results from our own experimental studies of evaporation from sand under three different evaporative demands. We used infrared thermography to monitor the evolution of mean surface temperature during the experiments and recorded rates of mass loss and ambient conditions. The comparison between model predictions and experimental results is depicted in Figure 6 for air flow velocities of 0.3, 0.6, and 0.8 m/s. The comparison shows good agreement between measurements and model predictions for a unit cell considering mean particle size of 0.9 mm and estimated pore size as 1/3 of the mean particle size [Glover and Walker, 2009].

Figure 6.

Comparison between experimental data (symbols) and model predictions (solid lines) for (a) evaporation rate and (b) mean surface temperature from sand column and estimated surface water content for three different evaporative demands (defined by mean air velocity). The evaporative characteristic length and hydrostatic conditions were invoked to estimate evolution of surface water content with time.

Figure 7.

Model predictions for the ratio of nonisothermal to isothermal evaporation rate math formula as a function of surface water content. The increased mean surface temperature at the end of stage 1 enhances vapor concentration gradient to the isothermal condition where surface temperature depression is ignored.

[40] Increasing wind speed resulted in an increase in evaporation rate due to decreasing boundary layer thickness (and thus increasing mean vapor gradient). For a wide range of surface water contents, increasing pore spacing enhances vapor diffusion from evaporating pores and results in a nearly constant evaporation rate as described by Shahraeeni et al. [2012]. The end of stage 1 with a transition to diffusion-controlled evaporation (stage 2) is clearly marked by a precipitous decrease in evaporation rate and a concurrent increase in surface temperature as seen in Figure 6b.

4.4. Evaporation Under Isothermal and Nonisothermal Conditions

[41] We analyzed differences in evaporative fluxes for isothermal and nonisothermal conditions. The ratio of nonisothermal to isothermal evaporation rates math formula is depicted in Figure 7 for the experimental conditions in Figure 6. The key difference between these two estimates was neglecting the effect of evaporative cooling for isothermal conditions (note that the experiments were conducted with no shortwave radiation input). Consequently, the vapor pressure gradient between the evaporating surface and the air was assumed constant. In contrast, the temperature depression due to evaporation may decrease vapor concentration at the surface and thus the difference with the air mass resulting a lower evaporation rate for nonisothermal conditions. As shown in Figure 7, the evaporation rate was reduced by 20–40% for evaporation rates ranging from 3 to 10 mm/day, respectively, in comparison with isothermal conditions. The results imply that with increasing evaporation rate the deviation between isothermal and nonisothermal evaporation is expected to increase due to evaporative cooling effect on the vapor gradient between the surface and the air.

4.5. Effects of Shortwave Radiation on Temperature Dynamics and Evaporative Flux

[42] For completeness, we analyzed the potential role of shortwave radiation on surface temperature dynamics and evaporation rates during drying of porous surfaces (an ingredient missing in most laboratory drying experiments) [e.g., Shahraeeni and Or, 2010; Belhamri and Fohr, 1996]. For standardized comparison, we introduce a dimensionless surface temperature (T*) defined by mean temperature of the evaporating surface under full saturation math formula, the air flow temperature (T), and mean surface temperature during evaporative drying math formula expressed as

display math(18)

[43] Figure 8 depicts the variation of T* during evaporation from a hypothetical porous surface with a characteristic length of 100 mm and an average pore radius of 250 µm for different shortwave radiation fluxes and wind speeds. In the absence of shortwave radiation, like our experimental results for evaporation from sand, evaporative cooling decreases mean surface temperature relative to air temperature resulting a lower vapor pressure gradient in comparison with the often assumed isothermal conditions [e.g., Shahraeeni et al., 2012]. The result is math formula (Figure 9), where eNI and eI are nonisothermal and isothermal evaporation rates, respectively. As seen in Figure 8, for a wide range of surface water contents the surface temperature remains nearly constant (despite gradual drying) indicative of constant evaporation rate and evaporative flux compensation due to increased pore spacing [Shahraeeni et al., 2012]. At the end of stage 1, when the surface becomes dry and the amount of remaining active pores drops significantly, surface evaporation rate drops sharply and the variable T* approaches zero indicating that mean surface temperature gradually approaches to air temperature (no evaporative cooling).

Figure 8.

Variations in mean surface temperature math formula versus surface water content. Zero shortwave radiation (dashed lines) and shortwave radiation flux (solid lines) for two different wind speeds.

Figure 9.

The ratio of nonisothermal to isothermal evaporation rate math formula versus surface water content. In the absence of shortwave radiation flux, evaporative cooling depresses the surface temperature thereby reducing vapor pressure gradient relative to isothermal conditions. Considering shortwave radiation reverses the picture, and the temperature of the evaporating surface becomes higher than the air resulting in an increase in the ratio math formula.

[44] In the presence of significant shortwave radiation, however, the surface temperature increases relative to that of the air temperature. The increase in surface temperature increases water vapor density at the surface and enhances evaporation rates relative to isothermal conditions math formula. Figure 8 depicts enhanced difference between surface and air temperatures with the reduction in evaporation rate at the end of stage 1 (Figure 9), as a larger fraction of the radiative energy heats the surface and increases the sensible heat flux from the surface. As the temperature of the evaporating surface increases, the vapor pressure gradient between the evaporating surface and the air increases accordingly and with it, the ratio math formula (as the surface dries out).

5. Summary and Conclusions

[45] The coupling between heat and mass transfer during stage 1 evaporation from porous media was quantified using an analytical model. A representative unit cell composed of an evaporating pore surrounded by dry solid surface was used to systematically evaluate effects of shortwave radiation, wind speed, relative humidity, and air temperature on evaporation rates and associated mean surface temperature dynamics during evaporative drying. The two key features in the pore-scale model include: (1) the explicit account of nonlinear evaporative flux as the surface dries and (2) the coupling between latent heat flux and the thermal field forming around an evaporating pore. Model predictions were in good agreement with experimental results for drying porous surfaces [Lu et al., 2005; Belhamri, 2003]. The model provides a physical basis for the empirical soil evaporation transfer coefficient (ha) [Qiu et al., 1998; Qiu and Ben-Asher, 2010] used for linking remotely sensed surface temperature with evaporation rate. We conducted evaporation experiments under various boundary conditions and used infrared imaging to monitor mean surface temperature during drying [Shahraeeni and Or, 2010]. Model predictions for controlled experiments were in good agreement with measurements. The experimental results illustrated that during stage 1 evaporation (with a nearly constant flux), surface temperature remains practically constant until transition to stage 2 marked by a sharp increase in surface temperature as vaporization plane migrates below the surface [Shokri and Or, 2011].

[46] An important and underappreciated phenomenon is the critical role of naturally occurring nonisothermal (eNI) conditions on evaporation rates from porous surfaces. In contrast with many models that tacitly assume similarity between the surface and air temperatures for estimation of the vapor gradient, we capitalize on the feedback between evaporation rate and surface temperature to solve for the effective evaporative flux under nonisothermal conditions. The analysis shows that, in the absence of shortwave radiation, the cooling effect during surface drying may dramatically reduce evaporation rate relative to isothermal conditions (eI). In the absence of shortwave radiation, nonisothermal evaporation rate (eNI) was consistently lower than equivalent isothermal evaporation rate (eI) by 20–40% for evaporation rates ranging from 3 to 10 mm/day, respectively. The picture was dramatically different for evaporation under natural conditions considering shortwave radiation. For standard values of shortwave radiation fluxes, nonisothermal evaporation rate (eNI) was considerably higher than for isothermal evaporation conditions (eI), often exceeding eI by a factor of 2.0, depending on mean wind speed and shortwave radiation as shown in Figure 9.

[47] The proposed model offers a simple and physically based analytical framework that may reduce empiricism associated with several of the temperature-based algorithms for estimation of surface evaporation. Moreover, the inherent feedback embedded in the model points to a bias in our overprediction of evaporation under assumed isothermal conditions in laboratory studies (no shortwave radiation), and under estimation under natural conditions with shortwave radiation. Finally, work is underway to implement the model for the prediction of energy partitioning over evaporating surfaces and derive analytical expressions for the Bowen Ratio and the Priestley and Taylor “α” coefficient [Priestley and Taylor, 1972] for different porous surfaces and ambient conditions.

Appendix: A

[48] The radial temperature differential equation for dry region surrounding an evaporating pore is written as

display math(A1)

where k is the effective thermal conductivity of the soil (W/mK), T is the ambient air temperature (K), hair is the air thermal convection coefficient (W/m2K), σ is the Stefan-Boltzmann constant math formula, ε is the soil emissivity, Δh is the thickness of unit cell (m), ΔH is the thermal decay depth below the surface (m), Rs is the shortwave radiation flux (W/m2), a is pore radius (m), and b is the dry area radius (m) (that varies with surface water content).

[49] In order to solve the temperature equation, we linearize the longwave radiation as

display math(A2)

[50] So, equation (A1) can be written as:

display math(A3)

in which variables φ and ψ are defined as

display math

[51] The solution of equation (A3) with the mentioned boundary conditions in equation (A1) is

display math(A4)

[52] Based on the mentioned parameters, the final solution for the temperature differential equation is

display math(A5)

where I0 and I1, and K0 and K1 are the modified Bessel functions of the first and second kind, respectively. Applying the energy balance to near-surface (Δh) pore water reveals the dependency of qcond on the evaporative flux from the pore (qevap):

display math(A6)

[53] Schlünder's [1988] analytical model is used to determine evaporative flux (qevap) from the pore:

display math(A7)

where qevap is the evaporative heat flux on the surface of the pore (W/m2), D is the vapor diffusion coefficient in the air (m2/s), θ is the surface water content (determined as the ratio of pore area to total unit cell area), L is the latent heat of vaporization for water (2.45 MJ/kg), and Cs and Ca are water vapor concentrations at the surface of water meniscus and within the flowing air mass (kg/m3), respectively. We obtain simultaneous solutions of equations (A5) and (A7) to find evaporative flux and temperature distribution on the surface of unit cell during evolution of dry region surrounding the pore.


[54] The authors gratefully acknowledge funding by the Swiss National Science Foundation (200021–113442) and the generous assistance of Peter Lehmann, Hans Wunderli, and Daniel Breitenstein in various aspects of the study.