PUBLICATIONS. Water Resources Research. Modeling and analysis of evaporation processes from porous media on the REV scale

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1 PUBLICATIONS RESEARCH ARTICLE Key Points: Coupled REV-scale modeling of drying processes Concept for the saturationdependency of the drying rate Analysis of the influence of porous-medium properties on the drying dynamics Correspondence to: K. Mosthaf, Citation: Mosthaf, K., R. Helmig, and D. Or (2014), Modeling and analysis of evaporation processes from porous media on the REV scale, Water Resour. Res., 50, , doi:. Received 18 JULY 2013 Accepted 8 JAN 2014 Accepted article online 15 JAN 2014 Published online 7 FEB 2014 Modeling and analysis of evaporation processes from porous media on the REV scale Klaus Mosthaf 1, Rainer Helmig 1, and Dani Or 2 1 Department of Hydromechanics and Modelling of Hydrosystems, Institute for Modelling Hydraulic and Environmental Systems, Universit at Stuttgart, Stuttgart, Germany, 2 Soil and Terrestrial Environmental Physics, Institute of Terrestrial Ecosystems, Department of Environmental Systems Science, ETH Z urich, Z urich, Switzerland Abstract The dynamics of drying processes from porous media are critically influenced by the intensity of an adjacent free flow and by processes at the interface between free flow and the porous medium. In this paper, the influence of hydraulic properties of a porous medium and of the interaction between fluids and porous medium on the drying dynamics during the capillary-flow dominated stage-1 and transition to the diffusion-dominated stage-2 are studied using a coupled free-flow porous-medium flow model on the REV scale. We present a detailed model concept that considers mass balance equations, an energy balance equation, and the coupling to the adjacent free flow. Key microscale processes are identified and incorporated in the macroscale description of the evaporation process. Own experimental results are used to illustrate main features of the modeling framework. We demonstrate that the use of a homogeneous distribution of soil parameters without consideration of pore-scale induced nonlinearities in the numerical simulations results in a rather constant drying rate in stage-1, which was not observed for the high evaporative demand in the experiments. To account for the dependency of the drying rate on the surface moisture content, special conditions based on the work of Haghighi et al. (2013) and Schl under (1988) are analyzed for their applicability on the REV scale. Typical features of a drying process, such as different stages of the drying rate, could be reproduced. 1. Introduction The dynamics of drying processes from porous media are critically influenced by the intensity of an adjacent free flow that shapes the boundary conditions for the exchange processes. A representation of evaporation processes from porous media on the REV scale is required for various practical applications [Smits et al., 2013; Defraeye et al., 2012; Ghezzehei et al., 2004; Kondo and Saigusa, 1994; Maroulis et al., 1995; Schl under, 1988], such as the modeling of evaporative fluxes in arid regions for water resource management or the optimization of industrial drying processes, such as food drying, drying of concrete, chemical engineering, or the optimization of proton exchange membrane fuel cells (PEM-FC) [Gurau and Mann, 2009]. A representation on the Darcy scale, however, may present several challenges and some pitfalls for the modeler, as the process involves interactions between different domains, internal transport mechanisms, media properties, coupled energy, and mass exchange, all occurring simultaneously and involving subtle nonlinearities. One such nonlinear process involves the dependency of the drying rate on the surface water content [van de Griend and Owe, 1994] and even on the distribution of pores on the drying surface [Haghighi et al., 2013]. Standard parameterizations of the hydraulic soil properties may lead to an overestimation of the capillary pressure especially in the dry region, as will be shown later. Thus, the choice of the model concept, soil property parameterization, and the boundary conditions must account for and meet these challenges for reliable estimates of drying dynamics. The objective of this paper is to study the influence of hydraulic properties of a porous medium, such as intrinsic permeability, porosity, thermal conductivity, and the two-phase relations expressed by capillary pressure and relative permeability, on the transition of an evaporation process from the capillary-flow dominated stage-1 to the diffusion-dominated stage-2 of a drying process using a process formulation on the REV scale, where macroscopic quantities and equations like Darcy s law are employed. We make use of own experimental evaporation rates under controlled laboratory conditions to illustrate some of the key features of the modeling framework. Furthermore, special boundary conditions at the evaporating surface, which may account for a dependency of the drying rate on the surface moisture content of the porous medium, are analyzed. They are based on the theoretical work by Schl under [1988], Suzuki and Maeda [1968], and MOSTHAF ET AL. VC American Geophysical Union. 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2 Figure 1. Qualitative evolution of the drying rate from a porous medium showing different characteristic stages. For thin boundary layers and a large distance between evaporating pores, the drying rate may decrease already from the beginning on and show different drying dynamics, as indicated by the dashed line. Shahraeeni et al. [2012], making use of the diffusion solution by Schl under [1988] as upscaled in Haghighi et al. [2013]. For the modeling, a detailed concept is presented, which considers mass balance equations formulated for the fluid components, and one energy balance equation based on the assumption of a local thermal equilibrium. This is coupled to a free-flow model, as described in Baber et al. [2012]. The paper is structured as follows: in section 2, key features of evaporation processes are introduced. The typical evolution of a drying rate from a saturated porous medium is discussed. This is followed by an explanation of the evaporation model, which is applied to study the influence of various soil properties on the drying rate. For this purpose, a coupled model was chosen which considers a porous-medium flow coupled to a free flow. The employed coupling and boundary conditions are explained. Then, on the basis of an evaporation experiment, a reference case is chosen for the numerical simulations and the results are presented. A parameter study sheds light on the influence of several parameters of the porous material on the computed drying rates. The importance of the two-phase parameters on the transition from stage-1 to stage-2 is pointed out. Deviations from the typical shape of the drying curve with a constant rate at the beginning, which have been observed in several drying experiments, are considered and possible modeling strategies based on pore-scale considerations are implemented on the REV scale and analyzed. 2. Evaporation Processes and Stages Evaporation processes from porous media are strongly influenced by the conditions and the flow regime of the ambient air. Often, the surface of a drying porous medium is exposed to an air stream, where even at relatively low flow velocities turbulent conditions may prevail. This introduces an enhanced mixing in the turbulent region and the formation of boundary layers for velocity, temperature, and vapor at the surface of the porous medium, which may have a decisive influence on the evaporative exchange particularly in the externally controlled stage-1 of the drying process, where both fluid phases are present at the porousmedium surface. The laminar part of the boundary layer defines a primary distance, over which gradients form and which introduces an effective diffusive resistance, because advective flow and transport within this layer happens relatively orderly and mainly parallel to the interface, if the main flow is also acting laterally to the porous-medium surface [Bird et al., 2007]. Thus, the vertical transfer of vapor and heat through the viscous sublayer can be assumed to happen predominantly due to diffusive and conductive processes [Haghighi et al., 2013]. Outside this layer, turbulent eddies lead to strong mixing and greatly enhance the vertical transport into the free flow above. These eddies may also penetrate into the boundary layer and lead to pressure fluctuations at the top of the porous medium [Scotter and Raats, 1969; Farrell et al., 1966]. The pressure fluctuations may significantly enhance the mixing in the porous medium and contribute to the net fluxes across the interface [Ishihara et al., 1992; Kimball and Lemon, 1972], especially when the porosity and the air saturation at the interface are high [Maier et al., 2012] Stages of Drying Processes Drying processes from porous media (like quartz sand) are typically subdivided into three different stages [Lehmann et al., 2008; Yiotis et al., 2007; Hillel, 2004; van Brakel, 1980], as schematically shown in Figure 1: (I) externally controlled period or stage-1 with sufficient water supply at the surface of the porous medium; in the case of a relatively constant drying rate this stage is also termed constant-rate period, (IIa) profile-controlled stage, which is controlled by a combination of transport processes within the porous medium and at the surface; opposed to the constant-rate period it is often termed falling rate period and marks the transition to stage-2, MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1060

3 Figure 2. Pore-scale consideration of the boundary layer with orderly flow parallel to the interface: a high surface water content and a short distance between evaporating pores leads to a quasi one-dimensional diffusion of the vapor normal to the boundary layer and relatively constant evaporation rates in stage-1. When the saturation decreases and the boundary layer is thin, the diffusion from evaporating pores lateral to the interface may become important. The evaporation rate can decrease already in stage-1, particularly if the diffusion length between these pores is large in relation to the mass boundary-layer thickness [Suzuki and Maeda, 1968; Shahraeeni et al., 2012]. (IIb) a slow-rate period or vapordiffusion stage (stage-2) without direct evaporation at the porous surface. The evaporation rate from an initially saturated porous medium tends to be high and (for natural conditions) often remains relatively constant. This stage, at which the primary vaporization plane is at the porous-medium surface, is known as stage-1 or constant-rate evaporation period [van Brakel, 1980; Yiotis et al., 2007; Lehmann et al., 2008]. It is characterized by water flow in connected liquid pathways, which provide a great deal of the water supply to the evaporating pores at the surface of the porous medium. In stage-1, the similarity of evaporative drying to drainage processes [Shahraeeni et al., 2012] is exhibited by the sequential pore invasion at the porous-medium surface according to the prevailing capillary forces (the largest pores empty first). For conditions, where the internal capillary flow to the evaporation front is not limiting (stage-1), details of vapor exchange from the surface across the boundary layer exert a large influence on the drying rate. As long as many pores at the interface are water-filled, the adjacent air immediately above remains fully vapor saturated (saturated vapor pressure can be assumed) and an almost onedimensional vertical vapor diffusion through the laminar part of the concentration boundary layer at the interface between free flow and the porous medium with a relatively constant concentration gradient controls the evaporation rate. During the drying process, the surface area of water-filled pores contributing to the direct evaporation from the water phase decreases. Then, if the space between evaporating pores becomes larger, the concentration gradients from these pores especially in the lateral direction become stronger. In combination with an energy input which does not limit the evaporation process, this enhances the evaporation rate per pore. This may lead to partly or even complete compensation of the decreasing wet surface area by a stronger evaporation per water-filled pore, as discussed in Shahraeeni et al. [2012], resulting in a relatively constant drying rate for a certain saturation range. Note, however, that experimental evidence suggests that this may not always be the case, and the evaporation rate may drop from the onset of the drying process on, as indicated by the dashed line in Figure 1. This gradual drop in the evaporation rate during stage-1, when the vaporization plane remains anchored at the surface, is often associated with high wind velocities with relatively thin boundary layers, large pores [Shahraeeni et al., 2012], and a high atmospheric demand. The gradual decrease of the drying rate may be due to a limited supply to the evaporating surface from the porous-medium side (e.g., because of limited permeability), or, more likely, due to limitations to the flux compensation at the drying surface, which was described before [see Shahraeeni et al., 2012]. As the surface gradually dries, the air at the surface of the porous medium may no longer be fully vapor-saturated everywhere and a one-dimensional gradient is not established, if the mass boundary layer is relatively thin. If the distance between remaining evaporating pores or pore clusters increases, the quasi one-dimensional concentration gradient through the laminar part of the boundary layer may change its character and become three-dimensional, as illustrated in Figure 2. Depending on the distribution of evaporating pores or pore clusters and on the thickness of the boundary layer, the lateral diffusion from these pores within the (mass) boundary layer gains importance and may MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1061

4 lead to decreasing drying rates [Shahraeeni et al., 2012]. An overview of the variety of possible shapes of drying curves is given in van Brakel [1980]. When the liquid connections supplying the interface cannot meet the evaporative demand at the surface anymore, the porous surface dries and the vaporization plane migrates below the surface as described in Shokri and Or [2011]. This transition from stage-1 to stage-2 of the evaporation process is indicated as IIa in Figure 1 and also called profile-controlled stage [Hillel, 2004]. It is determined by a combination of transport processes happening within the porous medium and at the surface. When the surface water saturation decreases further and the water phase becomes immobile and finally disappears, the vapor transport in the air phase through the tortuous porous medium becomes the limiting process. The secondary evaporation front recedes into the porous medium and the diffusion distance gradually increases, resulting in a decrease of the evaporative flux during stage-2 evaporation. The transition from the externally controlled stage-1 to the slow-rate period (stage-2) is defined here by the following criteria: 1. The water saturation at the surface drops to zero and the porous-medium surface dries out, 2. The temperature at the surface increases in response to the dry interface without direct evaporation and without the cooling effect of the vaporization enthalpy. During stage-2, the restrictions for the drying rate are usually within the porous medium and the boundary layer is less important for the vapor exchange to the free flow. On the pore scale, the occurring processes and phase interfaces are directly resolvable and can be modeled, for example, with direct numerical simulations (DNS), or by using a representative pore network enabling the representation of processes. However, due to constraints of computational power and data availability, practical applications like evaporation from bare soils often require averaging and modeling on the representative elementary volume (REV) scale. The subsequent part discusses how it is possible to incorporate relevant pore-scale processes into an REV-scale framework. 3. Model Concept and Setup For the modeling of drying processes from porous media, which are subject to a laminar air stream, a coupled model has been developed in Mosthaf et al. [2011]. A nonisothermal two-phase two-component porous-medium model is combined with a nonisothermal single-phase two-component free flow, allowing a detailed analysis of the interplay of the porous-medium and the free-flow domain on the averaged scale of representative element volumes (REV). The numerical concept and algorithm have been described in Baber et al. [2012]. Thus, we provide only a brief overview, but emphasize expansions of the coupled model and highlight innovations and their effect on the computed results in the sequel. In particular, the quantities influencing the onset of the transition from the relatively high evaporation rate in stage-1 to a considerably lower evaporation rate when the interface is drying out are analyzed using a two-dimensional setup throughout the simulations. The focus is clearly on the influence of the porous-medium properties on the drying rate. The drying process is assumed to be dominated by diffusion through the concentration boundary layer and by the limitations provided by the porous medium. Consequently, the feedback into the free flow and an ideal representation of the external flow field is less critical. Evidence from a series of simulations and from a theoretical consideration suggests that a laminar and irrotational flow field leads to an overestimation of the boundary layer thickness. Hence, the coupling condition for the vapor transfer is slightly modified to the one presented in Mosthaf et al. [2011]. A solution-dependent condition at the coupling interface which depends on a predefined boundary layer thickness is employed to compute the diffusive vapor flux across the boundary layer during stage-1. This leads to a constant and comparable evaporative demand at the surface and, hence, differences in the drying rates are solely due to the change of the soil properties. In the simulations that follow, the conditions in the free flow are kept constant while varying single parameters of the soil. In the externally controlled stage-1, we assume that the drying rate is not influenced much by the porous-medium properties, but it is mostly determined by the flow field with boundary layers and by the conditions at the surface of the porous medium. Naturally, this assumption depends on the porous MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1062

5 Table 1. Porous-Medium Equations and Associated Primary Variables Balance Equations Mass Balance for Component j {w, a}: X ð. axa jsaþ 1$ F j 2 X q j a 50 a2fl;gg F j 5 X. a v a Xa j 2Dj a;pm. a;mol $xa j Mj a2fl;gg a2fl;gg Total Mass Balance X X ð. a SaÞ 1$ ð. a v a Þ2 a2fl;gg a2fl;gg q a 50 Primary Variable Darcy s Law for Multiple Phases v a 52 kra l K ð$p a 2. a a gþ; a 2 fl; gg Energy Balance X ð. a 1ð12/ ð. s 1$ F T 2q T 50 T a2fl;gg F T 5 X. a h a v a 2k pm $T a2fl;gg S l or X j a p g material considered, particularly on the pore sizes at the interface [Shahraeeni et al., 2012; Haghighi et al., 2013]. For porous media with large pores at the interface, this assumption may be violated and the turbulent flow may also penetrate into the upper region of the porous medium [Farrell et al., 1966]. The porousmedium properties become more important as the water saturation at the surface decreases. They may govern the onset and even concretize the shape of the drying curve at the transition from stage-1 to stage Porous-Medium Model The porous medium is simulated with a two-phase two-component porous-medium model using Darcy s law extended for multiple phases, as described, for example, in Class et al. [2002]. The setup and the porous-medium parameters are chosen according to physical evaporation experiments in a wind tunnel, which have been performed at the laboratory of the ETH Z urich as will be briefly described in section 4.1. A mass balance for the component vapor and a total mass balance (sum of the two-component mass balances) are solved, employing the gas phase pressure p g and the water saturation S l as primary variables. The balance equations, the material laws, and equations of state for the dependent variables are listed in Tables 1 and2. Local thermodynamic equilibrium is assumed, including local mechanical, chemical, and thermal equilibria. Such an assumption is justified for low flow velocities within the porous medium and sufficient time for the fluids to equilibrate. The conditions under which this assumption is violated, especially in the vicinity of the soil surface, remain subject of additional research. Due to the assumption of a local thermal equilibrium, only a single energy balance for all phases is required with the temperature as primary variable. Table 2. Material Laws and Equations of State Dependent on Primary Variables Equations of State References Parameters Density. a. a 5fð. j ; xa j; p a; TÞ IAPWS [2007] Component density. j Incompressible fluid and ideal gas Reid et al. [1987] Diffusion coefficient D a,pm D a;pm 5fðS l Þ Capillary pressure p c p c 5f ðs l Þ Brooks and Corey [1964] or Relative permeability k ra k ra 5fðS l Þ van Genuchten [1980] and Mualem [1976] Eff. thermal conductivity k pm k pm 5fðS l Þ Johansen [1977] Internal energy u a u a 5h a 2p a =. a Enthalpy h Component air h a ðt2273:15 KÞ Component water h w 5fðp a ; TÞ IAPWS [2007] Water phase h l 5 h w IAPWS [2007] Gas phase h g 5Xg whw 1Xg aha Secondary Variables Saturation S g S g 5 1 S l Liquid-phase pressure p l p c 5 p g p l Vapor in gas phase xg w xg w5pw sat =p g Mass fractions Xa w1xa a 5xw a 1xa a 51 Xa j5xj a Mj =ðxa wmw 1xa ama Þ MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1063

6 We employ an energy balance based on phase enthalpies and internal energies, where the contributions of all phases are summed up. The energy transfer due to evaporation (vaporization enthalpy) is accounted for due to the computation of the gas phase enthalpy as the sum of the component enthalpies, h g 5X w g hw 1X a g h a (see Table 2). A diffusive component flux in the gas phase is manifested as a change of the phase composition, i.e., the mass fractions will change, leading to an alteration of the gas phase enthalpy. However, the energy contribution of diffusive vapor fluxes is not explicitly considered in the energy balance equation here. The amount of the component air which can be dissolved in the water phase is very limited and its effect on the fluid properties is small. Thus, the effect of dissolved air in the water phase on the water enthalpy is neglected in this analysis. The vapor concentration is computed as a function of pressure and temperature via the saturated vapor pressure as long as both fluids are present. At residual water saturation, the water phase becomes immobile and vapor can be transported only via the gas phase. Still, the vapor concentration is computed as secondary variable as function of pressure and temperature. When the water phase vanishes and the water saturation becomes zero, a variable switch is triggered and the mass fraction of vapor in the gas phase is used as primary variable instead. Then, the air is not fully water-saturated anymore and the transition to stage-2 evaporation happens. Further details of the porous-medium model and its implementation into the numerical simulator DuMu x [Flemisch et al., 2011] can be found in Baber et al. [2012]. The computation of the vapor diffusion and the thermal conduction in the porous medium is facilitated by effective models for the diffusivity and the thermal conductivity. The effective diffusion coefficient is computed in dependency on the binary diffusion coefficient D a, the porosity /, and the phase saturation S a : D a;pm 5D a s/s a : (1) An approximation of the tortuosity s is given in Millington and Quirk [1961] as s5ð/s a Þ 7=3 =/ 2. This approximation gives an adequate representation for structureless fine sand [Kristensen et al., 2010]. Clearly, the effective gas diffusivity exhibits a strong dependency on saturation. The effective thermal conductivity k pm is usually also expressed as function of the water saturation. We have chosen a parameterization according to Johansen [1977]: k pm 5k dry 1K e ðk sat 2k dry Þ: (2) k dry and k sat are the effective thermal conductivity of a completely dry and an entirely wet porous medium, K e is usually a nonlinear function of the water saturation and can be approximated according to C^ote and Konrad [2005] and Smits et al. [2010] as: K e 5jS l =ð11ðj21þs l Þ; (3) where j is an empirical fitting parameter accounting for the influence of the grain size distribution and S l is the water saturation (liquid). In this analysis, we choose j , a value which was fitted to a similar sand (30/40 loose) in Smits et al. [2010]. Table 3. Free-Flow Equations and Associated Primary Variables Balance Equation Primary Variable Mass Balance for Component ð. g Xg j 1$ F j 2q j g 50; Fj 5. g v g Xg j2dj g. g;mol $xg j X j Mj a Total Mass 1$ F m2q g 50; F m 5. g v p g g Momentum Balance ð. g vgþ 1$ F v 2. g g50 g F v 5. g v g v g 1p g I2s Energy g 1$ F T 2q T 50; F T 5. g h g v g 2k g $T T MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1064

7 3.2. Laminar Free-Flow Model and Coupling Concept Table 3 provides an overview of the equations, which are employed in the adjoining free flow. Laminar and irrotational flow conditions are assumed. Based on the assumption of Newtonian fluids, Newton s law is used for the shear stress in the momentum flux: s5l g ð$v g 1$v T g Þ, with the gas phase viscosity l g and velocity v g. For the sake of consistency, the material laws are identical to the ones used in the porous domain (Table 2). The coupling concept developed in Mosthaf et al. [2011] guarantees the continuity of mass, momentum, and energy fluxes across the interface between the domains. It is based on a local thermodynamic equilibrium at the interface. In particular, the mechanical equilibrium is given by: 1. the continuity of the normal stresses resulting in a possible jump in the gas-phase pressure: 2. the continuity of the normal mass fluxes: n ½ðð. g v g v g 1p g I2sÞnÞŠ ff 5½p g Š pm ; (4) ½. g v g nš ff 52½ð. g v g 1. l v l ÞnŠ pm ; (5) 3. the Beavers-Joseph-Saffman condition for the tangential component of the free-flow velocity [Beavers and Joseph, 1967; Saffman, 1971]: " pffiffiffi! # ff k i v g 1 sn t i 50; i 2 f1;...; d21g; (6) a BJ l g where a BJ is the Beavers and Joseph coefficient, k i 5t i ðkt i Þ is the tangential component of the permeability tensor. The thermal equilibrium is given by Alazmi and Vafai [2001]: 1. the continuity of the temperature: 2. the continuity of the heat fluxes: ½TŠ ff 5½TŠ pm ; (7) ½ð. g h g v g 2k g $TÞnŠ ff 52½ð. g h g v g 1. l h l v l 2k pm $TÞnŠ pm : (8) Due to the use of different model concepts in the two domains, the gas phase pressure may have a small jump at the interface. Hence, the chemical equilibrium cannot be established completely. The conditions which are as close to the chemical equilibrium as possible are chosen, namely: 1. the continuity of mass fractions: 2. the continuity of the component fluxes across the interface: ½X j g Šff 5½X j g Špm ; (9) h i ff h i pm 2. g v g Xg j n1jvap 5.g v g Xg j2d g;pm. g;mol $xg jmj 1. l v l Xl j 2D l;pm. l;mol $xl j M j Þn ; j 2fw; ag; (10) where J vap are the diffusive fluxes at the free-flow side of the interface. Based on the choice in the submodels, two of the three conditions 5, 10 are employed. We use one total mass balance and a component mass balance for vapor. MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1065

8 Figure 3. Schematic overview of the evolution of a boundary layer above a flat and smooth plate. The boundary layer starts as laminar. After a certain distance, a transition to a turbulent boundary layer occurs. The turbulent boundary layer is composed of a laminar part, namely the viscous sublayer, and a part where already turbulent eddies occur. The boundary-layer thicknesses for concentration, velocity, and temperature may differ from each other. In summary, the underlying assumptions of the coupled model are as follows: 1. rigid porous medium with creeping flow (applicability of Darcy s law), 2. local thermodynamic equilibrium, 3. mass boundary layer provides main limitation of evaporative fluxes in stage-1, 4. no adsorption of water to the porous material, 5. negligible influence of dispersion, 6. flat surface without roughness, 7. laminar, irrotational air flow mainly parallel to the interface, 8. water and air as Newtonian fluids. The evaporative fluxes are computed based on the conditions and parameters of the soil, such as fluid saturation, permeability, porosity, the capillary-pressure, and relativepermeability curves, and the exterior boundary conditions, such as the boundary layer thickness, air humidity, and air temperature. The transition from stage-1 to stage-2 is achieved by a variable switch as soon as the water saturation becomes zero and a cell dries out. Then, chemical equilibrium between the gas phase and water phase can no longer be assumed, and the mass fraction of vapor in the gas phase is used as primary variable. The porous-medium model allows the partitioning of the water supply to the interface into capillary water fluxes and diffusive vapor fluxes in the gas phase. Several simulations showed a strong domination of the advective water fluxes in the capillary dominated regime in stage-1. However, this relies on the idealization of a flat and smooth surface. The inclusion of surface roughness and geometric features of the surface may lead to stronger normal advective fluxes. Naturally, for a high water saturation in the porous medium, air cannot penetrate across the interface because pores are blocked with water Modified Computation of Diffusive Fluxes With the given air velocity, turbulent conditions can be expected to develop within the free air stream, as previously mentioned. Hence, a coupling of the porous-medium model to a laminar free flow leads to an underestimation of the computed evaporation rate, because the thickness of the laminar part of the boundary layer for velocity, vapor concentration, and temperature is thinner for a turbulent flow as schematically shown in Figure 3, resulting in a larger concentration and temperature gradient at the interface. Note that the boundary layer thickness for concentration, velocity, and temperature may differ from one another; they are often related using the dimensionless Schmidt number for concentration and the Prandtl number for temperature [Bird et al., 2007]. An option to improve the approximation quality is to employ a Reynoldsaveraged turbulence model (RANS), such as an algebraic turbulent model (mixing length approach) as described in Fetzer [2012] or a k- model, in the free flow. However, this analysis is beyond the scope of this paper, where the main focus is on the influence of the soil parameters on the computed drying rates. To simplify the system, the influence of the turbulent flow field is accounted for by substituting the computation of the diffusive vapor fluxes at the free-flow side of the interface by a concentration difference and a simple boundary layer approximation, which guarantees a comparable evaporative demand at the porousmedium surface during stage-1. Therefore, an averaged concentration boundary layer thickness is estimated based on the physical experiments using the measured evaporation rate, the vapor concentration in MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1066

9 the center of the wind tunnel, the saturated vapor pressure at the interface and thus the concentration of the fully saturated air and the diffusion coefficient of vapor in air: d m 2D w g. g;mol Dxg w J vap Mw : (11) For a free flow velocity of 3.5 m/s, this leads to an estimated average thickness of d m mm. This value is Figure 4. Setting of wind tunnel, insulated sand sample, and measurement devices. slightly higher to the values stated in Shahraeeni et al. [2012], which were determined for a free water surface. With that, the diffusive vapor flux across the boundary layer at the free-flow side of the interface in equation (10) can be approximated as a function of the difference between the vapor mole fraction in the external free flow xg;ext w and the vapor mole fraction at the surface of the porous medium xg;pm w J vap 52D w g. xg;ext w ff 2xw g;pm g;mol M w d : (12) m Further implications of turbulent conditions in the free flow, like viscous energy dissipation, the heat transfer across the turbulent thermal boundary layer, eddy dispersivity, and eddy thermal conductivity, are not considered here. 4. Parameter Study The model described in the previous chapter is employed to study the influence of different soil parameters on the computed drying rate in the sequel. A reference case is chosen according to a series of physical drying experiments in a wind tunnel, which are briefly described in the following Experimental Setup A thermally insulated glass vessel ð0:25 m 3 0:25 m 3 0:08 mþ was filled with quartz sand (grain size distribution: mm), saturated with water, and then the surface was exposed to air flow of a wind tunnel with a diameter of 0.5 m, as illustrated in Figure 4. The setup and the employed sand is the same as used in Shahraeeni et al. [2012]. Furthermore, the same sand has been analyzed and used in several experiments from the DFG (German Research Foundation) research unit MUSIS ( and in the drying experiments presented in Lehmann et al. [2008] and Shokri et al. [2008]. The key parameters for the sand are listed in Table 4. The evaporation rate was evaluated from mass loss measurements determined by a digital balance, which records the weight of the sample at predefined intervals. Within the wind tunnel, the flow velocity can be controlled in the range of 0.75 to 4.1 m/s. A relatively high maximum velocity of 3.5 m/s has been selected Table 4. Soil Parameters of the Quartz Sand and of the Reference Case, With the Grain Diameter d, the Intrinsic Permeability K, the Porosity /, and the Two van Genuchten Parameters a VgN and n VgN d (mm) K (m 2 ) / a VgN (1/Pa) n VgN Fine sand e e MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1067

10 Figure 5. Section of the wind tunnel with insulated sand sample, thermocouples, hot wire anemometers, heat and moisture sensor, and infrared camera. for the drying experiment which is used for the analysis and the comparison with computed drying rates in the later part of this chapter. This choice leads to a relatively high and continuously decreasing drying rate, which shows no constant rate in stage-1. At the end of this section, a case with a lower velocity of 1.5 m/s will be considered and compared to experimental data [Shahraeeni et al., 2012]. Two hot wire anemometers were placed along the center line of the tunnel with which the maximum flow velocity above the sample can be measured. Furthermore, the mean air humidity and temperature is determined above the porous sample at the center of the wind tunnel. The experiments were performed at an ambient temperature of C and a relative humidity of approximately 40%. The heat transfer was monitored with an array of stacked thermocouples above and below the surface. The thermocouples below the sand surface show clear temperature gradients within the sand toward the surface. This reflects the effect of evaporative cooling and heat conduction, as long as the evaporation process remained in stage-1 and with the vaporization plane at the surface of the sample. The thermocouples above the sample show also a vertical gradient as long as water was evaporating, indicating the thermal boundary layer. An infrared camera was placed above the porous surface to give detailed information about the surface temperature evolution during the drying experiments. As shown in Figure 6, the temperature data can be used as further indicator for the transition from stage-1 to stage-2 and to obtain information about heat fluxes and the local evaporation [Shahraeeni and Or, 2011]. For the free stream velocity of 3.5 m/s, a constant drying rate could not be observed; instead the evaporation rate dropped continuously from the onset of the experiment. However, with help of the temperature evolution stage-1 and the transition to stage-2 can still be identified and occurred approximately after 2 days. This transition was confirmed by temperature data from the infrared camera and the thermocouple elements in the surroundings of the interface. With the setup according to the described physical experiment, a reference case was simulated. Then, the porosity, intrinsic permeability, and the van Genuchten parameters for the two-phase relations are varied while keeping all other simulation parameters constant. Table 5 provides an overview of the simulation parameter ranges Description of the Reference Case A two-dimensional porous medium with an extent of 0:25 m 3 0:25 m was considered, according to the physical setup of the experiment. Adjoining to it is the equally sized free flow compartment, with air flowing from the left to the right with the highest velocity of 3.5 m/s in the center of the domain. Therefore, a grid with cells was generated and subdivided in a free-flow and a porous-medium domain. The height of the grid cells was refined toward the interface, as can be seen in Figure 5. The overall temperature in the inflowing air and in the porous medium is chosen as 25 C. The mass fraction of vapor at the inflow is set to kg/ kg, which corresponds to a relative humidity of the air of approximately 40%. The porous-medium properties used for the simulations (reference case), which were based on measured values from the drying experiment, are presented in Table 4. Figure 5 provides an overview of the employed boundary conditions. Figure 6 depicts a comparison of measured and computed drying rates. It illustrates that a laminar free flow underestimates the drying rates already at the beginning of the drying process. The computed rates using MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1068

11 Figure 6. (left) Model setup with grid, which is refined toward the domain interface. (right) Comparison of a measured drying curve and the rates computed with the porous-medium model coupled to a laminar free flow, to a laminar free flow with boundary-layer approximation, and to a free flow modeled with an algebraic turbulence model. A laminar free flow underestimates the drying rate. The initial rate can be met with the turbulence model. The use of a boundary-layer approximation in the laminar free-flow model leads to similar results as the algebraic turbulence model. an algebraic turbulence model [e.g., Pope, 2000; Bird et al., 2007] in the free flow as described in Fetzer [2012] and using a laminar free flow with a boundary-layer approximation lead to comparable results and meet the initial drying rate. The computed drying rate in stage-1 is decreasing at the beginning of the simulation. This is mainly due to the cooling effect of the evaporating water, which leads to a lowering of the saturated vapor pressure at the surface of the porous medium. However, in contrast to the measured drying curve, the initial decrease is followed by a period with a relatively constant drying rate, which gradually decreases due to the decreasing water saturation. The thermal conductivity of the gas phase is orders of magnitude lower than the thermal conductivity of water, thus a decrease of the water saturation results in a reduction of the energy supply from the porous medium to the surface. The wetting behavior of water leads to a nonlinear relationship between the effective thermal conductivity and saturation, as described in equations (2) and (3). The effective diffusion coefficient of vapor in the gas phase is also affected by a change in saturation (equation (1)). When the air saturation increases, the effective vapor diffusion coefficient increases as well and the vapor fluxes in the porous medium become stronger. The drying of the surface with zero saturation marks the transition to stage-2 with a precipitous decrease in the evaporation rate. Diffusive transport across the tortuous pore space gradually slows down as the thickness of the dry region increases. Figures 6 8 demonstrate that a uniform distribution of the soil parameters leads to the development of a constant evaporation plateau in stage-1 of the computed drying rates and to a relatively sharp transition to stage-2, if the drying rate is not already limited by the supply in the porous medium. An explanation and possible ways to handle that are pointed out in section The Influence of Porosity A comparison of computed drying rates for porosities between 30% and 50% is shown in Figure 7. A higher porosity provides more space for the two fluids; if the porous medium is initially water saturated, more Table 5. Simulated Parameter Ranges Parameter (mm) Range Unit Porosity Permeability m 2 van Genuchten a 1/1000 1/2000 1/Pa van Genuchten n 4 12 Solid thermal conductivity W/(m K) MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1069

12 Figure 7. Influence of the porosity on the computed drying rates. The transition from stage-1 to stage-2 happens later for a higher porosity because more water is in place and has to evaporate. For high porosities, the drying rates have a slightly lower level during stage-1 due to the parameterizations of the effective thermal conductivity and the effective diffusivity. water is in place and can evaporate and the transition to stage-2 happens later. Moreover, porosity affects the values of the effective thermal conductivity and the effective diffusivity. These two effective quantities are altered especially at the evaporating surface by a change of porosity and the water saturation. A higher porosity leads to a lower effective thermal conductivity and effective diffusivity at decreasing saturations, as dictated by equations (1) and (2). Due to the lower value for the effective thermal conductivity, less heat is conducted from the interior of the porous medium to the porous-medium surface. This leads to a lower temperature and a lower saturated vapor pressure there and thus to a lower evaporation rate and a later transition to stage-2. In contrast, the drying rate is higher in stage-2 for a higher porosity due to more pore space and a larger effective diffusivity of the gas phase in the porous medium The Influence of Intrinsic Permeability The intrinsic permeability is the proportionality factor between potential gradient and the averaged phase velocity (Darcy velocity) in the porous medium. Its effect on the drying process is illustrated in Figure 8 with numerical simulations using different homogeneous intrinsic permeabilities in a range between and m 2, while keeping other parameters constant. Here the evaporation rate in stage-1 has the same level for all permeability values, but the transition from stage-1 to stage-2 happens later for a higher intrinsic permeability. High intrinsic permeability delays the transition from stage-1 to stage-2 due to a lower resistance of the porous medium and a delayed breakup of the liquid connections to the interface. Figure 8. Influence of the intrinsic permeability (m 2 ) on the computed drying rates. The transition from stage-1 to stage-2 happens later for a higher permeability due to a lower viscous resistance in the porous medium and a better water supply at the evaporating pores The Influence of the Solid Thermal Conductivity and Temperature At the beginning of the drying process, the porous medium and the two fluid phases have a uniform temperature. When water evaporates, the latent heat of vaporization cools the surface of the porous medium, leading to a decrease of the saturated vapor pressure at the surface and subsequently to a lowering of the drying rate. The surface temperature will decrease until an equilibrium between the latent heat of vaporization and the heat supply is established. The incoming heat fluxes are composed of the heat transfer from the free flow due to thermal conduction and convection in the ambient air, radiative heat input, and of the fluxes from the interior of the porous medium, such as thermal conduction through the porous medium (water, air, and soil) and convective heat fluxes. Presumably, the heat of vaporization is withdrawn partly from the water phase, which is assumed to be nearly in equilibrium with the solid MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1070

13 material, and from the surrounding air. Radiative heat transfer within the porous medium is not considered here because it is usually accounted for in the effective thermal conduction. The equilibrium temperature at the surface will set depending on the cumulative heat demand and supply at the interface. The general surface energy balance can be formulated as [Brutsaert, 1982]: Figure 9. Influence of different values for the thermal conductivity (W/(mK)) of the solid phase on the computed drying rates. A higher thermal conductivity provides a better energy supply due to heat conduction within the porous medium. This counteracts the cooling effect of evaporation and leads to higher vapor concentrations at the surface during stage-1 evaporation. R n 5L e E1H1G (13) with the net radiation R n, the latent heat of vaporization L e E, the sensible heat flux into the atmosphere H, and the ground heat flux G. When an interface is considered, equation (13) can be subdivided into fluxes from the porous medium, the free flow and the latent heat of vaporization. This leads to following balance equation: 2 3ff 6 R n 5. g ðh w Xg w 1ha Xg a Þv 7 4 g2k g $T5 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} H g ðh w Xg w 1ha Xg a Þv 7 4 g1. l h l v l 2k pm $T 5 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} L ee1g pm (14) The latent heat of vaporization L e E is inherently in equation (14). The total internal energy is composed of the enthalpies of the different phases, because only one energy balance is used. When water evaporates, the water saturation decreases while the vapor concentration in the gas phase increases, leading to an overall decrease of temperature. Due to the coupling of the heat fluxes at the interface, the latent heat of vaporization is properly accounted for. Furthermore, the gas phase enthalpy is composed of the enthalpy of the pseudocomponent air and of the component vapor. Evaporation from the liquid phase or a diffusive component transport affecting the vapor concentration in the gas phase will lead to a change in the gas phase enthalpy. In the simulated cases, the decrease of temperature at the surface due to the cooling effect of evaporation was approximately 3 4 K. Similar values have been observed during stage-1 of the drying experiment. An estimation of the heat supply from the porous medium to the surface is possible, if the heat input across the boundaries of the porous medium is quantifiable. To reduce the external heat input, the vessel for the sand sample in the drying experiment has been insulated with Styrofoam, which has a considerably lower thermal conductivity as the sand and the glass walls of the vessel. The external heat input requires special care when choosing the boundary conditions in the porous medium. No-flow boundaries are chosen at all sides but the bottom, where a constant temperature is set. The influence of the solid thermal conductivity has been examined with three simulation runs, varying the thermal conductivity of the solid material in the range of W/(mK). As can be seen in Figure 9, a lower thermal conductivity of the solid material leads to a weaker supply of conducted heat from the porous medium to the evaporating surface. Thus, the initial decrease of the drying rate is stronger for a lower thermal conductivity in stage-1. However, the overall effect of the thermal conductivity of the solid material on the drying rate is comparably small. In order to examine the influence of the porous-medium temperature, a series of simulations with different temperatures in the porous medium was carried out. For that purpose, the inflow temperature was kept constant and the temperature at the bottom boundary and the initial temperature in the porous medium was varied. The resulting drying rates are depicted in Figure 10. The temperature in the porous medium has MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1071

14 Figure 10. Effect of a different temperature in the porous medium as in the free flow. The inflow temperature was kept constant while the initial temperature and the temperature at the lower boundary of the porous medium were changed. a strong influence on the drying dynamics, because it affects a variety of temperature-dependent parameters like density and viscosity and saturated vapor pressure. Furthermore, the effect of thermal conduction through the porous medium can be seen especially at the beginning of the drying process. For a much lower temperature in the porous medium, the cooling effect due to the vaporization enthalpy is weaker as the heat contributions from the free flow and from the porous medium (mainly by thermal conduction), which may even lead to increasing drying rates during stage-1. A high temperature of the porous medium leads to high saturated vapor pressures particularly at the surface of the porous medium and thus to a much higher drying rate. Figure 11. Capillary-pressure curves for different van Genuchten parameters a [1/ Pa] The Influence of Parameters for the Two-Phase Relations The two van Genuchten parameters n (shape parameter) and a, which are often used for the nonlinear capillarypressure-saturation and the relative permeability-saturation relationship, have been varied in the range between 4 and 12 for n, and between and /Pa for a. The parameter n is related to the width of the pore-size distribution and has a high value for a porous medium with relatively uniform pore sizes. a is related to the entry pressure of a medium. The entry pressure, known from the Brooks-Corey model, can be estimated as 1/a. Both parameters also influence the endpoint value of the capillary-pressure curve, as can be seen in Figures 11 and 13. The numerical simulations reveal that the maximum capillary pressure, or more precisely its gradient in the dry region close to residual water saturation, in combination with the relative permeability is most relevant for the transition from stage-1 to stage-2. Lenhard et al. [1989] provides a transformation of the van Genuchten parameters to the Brooks- Corey parameterization. Tests have been performed employing the Brooks-Corey model, which show a similar behavior as with the van Genuchten model. Figure 12. Influence of the van Genuchten parameter a [1/Pa] (related to the entry pressure) on the computed drying rates. As shown in Figure 12, higher values for the van Genuchten a (see Figure 11) provide lower capillary-pressure gradients especially in the dry region of the curve. Thus, the transition to stage-2 happens earlier as for high values. The influence of the shape parameter n on the resulting capillarypressure-saturation curve is relatively MOSTHAF ET AL. VC American Geophysical Union. All Rights Reserved. 1072