Analysis of Drying Front Propagation and Coupled Heat and Mass Transfer During Evaporation From Additively Manufactured Porous Structures Under a Solar Flux

Evaporation from porous media coupled with heat and mass transfer under influence of external heat flux is a complex phenomenon relevant in sectors such as electronics cooling [1–3], heat pipes [4,5], fuel cells [6,7], and conservation of water in soils [8–11]. Different parameters such as pore sizes or porous structure formation [11–13], characteristic length [8,11,14], liquid island formation [15–18], hydraulic connection [17,19–22], and coupled heat/mass transport [6,23–25] influence evaporation rates and mechanisms.

Evaporation from porous media can be categorized into three distinct periods: constant rate, falling rate, and slower rate periods [8–11,14,16,17,26–28]. In a fully saturated porous media, liquid is evaporated from the top surface at a constant rate. The constant rate of evaporation continues while a hydraulic linkage is maintained through the saturated and unsaturated portion by means of capillary action. When this hydraulic linkage breaks down, evaporation experiences a sharp decrease and enters into the falling rate period. The maximum distance between the saturated and unsaturated portion at the end of constant rate period is indicated as the characteristic length, and it plays a vital role in evaporation. The constant rate period elongates for larger characteristic lengths [8,11]. In the falling rate period, the continuous hydraulic linkages break down and small liquid islands are formed between the particles of porous structure, which leads to enhanced vapor diffusion. At the later stage (i.e., slower rate period), the liquid islands start to break up and the evaporation is dominated by diffusion. During the slower rate period, evaporation rate significantly decreases (e.g., 0–1 mm/day) [9,11,26].

Evaporation from porous structures depends primarily on pore sizes and its subsequent effects on capillary action. In the study of Lehmann et al. [11], liquid was transported from interconnected porous structure of larger pores to smaller pores due to capillary action and subsequently created a continuous hydraulic chain through the saturated and unsaturated parts. Nachshon et al. [12] observed an elongated constant rate period (∼7 days) in porous media consisting of smaller particles (i.e., 150–300 μm) compared to larger particles (i.e., 1–2 mm), as smaller particles create smaller pore sizes. The constant rate period elongated more in a porous matrix of smaller pore sizes than larger pores (i.e., particle sizes ranging from 6 nm to 45 μm) [13]. The constant rate of evaporation was more dominant with porous structure consisting of mixed pore sizes (i.e., larger and smaller pores) than uniform porous configuration due to higher capillary potential differences [12].

Formation of liquid islands between multiple particles across the porous structure affects evaporation mechanisms [29,30]. Liquid islands across the structure create a continuous hydraulic linkage between saturated and unsaturated portions. In a study of evaporation from heterogeneous sand column (260 mm of height, 75 mm of width, and 11 mm of thickness), liquid islands were formed between adjacent particles, leading to a supply of water to the evaporative surface with continuous liquid chain [9,16,17]. Shokri et al. [17] noted the presence of two evaporative fronts (i.e., primary and secondary), where a liquid connection was maintained between both fronts created with liquid islands. The term “secondary capillary effect” was mentioned by Chen et al. [19,20], where the liquid islands were formed in cylindrical porous structure and created a hydraulic effect toward the evaporative front. For evaporation from 2D-networks (i.e., where the thickness is negligible compared to length and width) [18,31,32], the liquid films contributed significantly to form hydraulic connection leading to a stable evaporation rate.

Evaporation from porous media under the influence of external heat source is a coupled heat/mass transfer phenomenon that has been studied along with the effects of porous structure, capillary action, and evaporation stages [6,15,23–25]. In a fuel cell diffusion media, natural-convection-induced evaporation was experimentally investigated and modeled, along with mass transport consideration [6]. Cho and Mench [6] theoretically and experimentally proved the association of longer constant rate period with formation of liquid islands leading to continuous liquid islands. Chakraborty et al. [15] experimented with porous media consists of similar sized (2.38-mm-diameter) glass and Teflon spheres under influence of a solar-simulated heat flux (i.e., 1000 W/m²). The coupled heat and mass transport were influenced by capillary action and hydraulic linkage during evaporation from hydrophilic and hydrophobic media. Horri et al. [23] experimented with solar evaporation in light-absorbing porous material (i.e., carbon-Fe₃O₄ composite) and the model included contributions of solar irradiance along with natural convection during evaporation; the model matched the experiment within 4.67–13.2%. Zannouni et al. [24] modeled diffusive evaporation numerically along with mass transport, and significant agreement was noticed between theoretical and experimental analysis.

Evaporation from porous media is a coupled heat and mass transfer phenomenon that is influenced by parameters such as pore size, capillary action, hydraulic linkage, liquid islands, and applied heat fluxes. In this study, evaporation was studied from three additively manufactured porous structures. The advantages of using these structures compared to natural porous media (e.g., sand or soil) are the controllable pore sizes and structures due to precision of the manufacturing process (i.e., ±0.1 mm of thickness resolution), compared to natural porous media with structures that vary depending on porosity, bulk density, and other parameters [8]. In this study, the pore sizes were kept constant to reduce variability of complex porous media and the steady-state solution of mass transfer during evaporation was modeled. In addition, in a controllable porous structure, the water movement during evaporation as well as hydraulic linkages can be analyzed properly and these fundamental findings are useful in solving complex transient heat/mass transfer in real porous media with complex microstructures. The main objectives of this study are to (1) analyze evaporation phenomena from three different additively manufactured structures with different pore sizes and understand water transport due to lateral movement restriction, (2) understand and study the effects of pore sizes, capillary actions, and subsequent formation of liquid islands and its effects on different stages of evaporation, (3) formulate steady-state coupled heat and mass transfer model to analyze evaporation under the influence of heat flux, and (4) model transient heat and mass transfer during the falling rate period of evaporation.

In this study, de-ionized water was evaporated from three additively manufactured porous structures. The discussion of experimental apparatus is divided into three parts: Sec. 2.1 manufacturing process and specifications of additively manufactured porous structures, Sec. 2.2 wettability measurements of the additively manufactured material, and Sec. 2.3 experimental procedures.

The porous structures were designed in solidworks 2021 and were manufactured by Protolabs using stereolithography technology (i.e.,±0.1 mm of thickness resolution). All structures were built with translucent acrylonitrile butadiene styrene (ABS). The front, top, and side views of 3D structures 1, 2, and 3 are pictured in Fig. 1. In 3D-structure-1, 360 1.2-mm-diameter spheres of were stacked in 10 layers with center-to-center spacings of 1.43 mm. The spheres were connected using 0.2-mm-diameter cylinders in the x, y, and z directions; each layer consisted of 36 spheres. The design specifications of 3D-structure 1 and 2 were similar, except that two layers in 3D-structure 2 in the y-z direction were filled with solid material (i.e., ABS translucent) to restrict lateral interaction of water in x-y direction. As a result, 3D-structure 2 had a higher mass (i.e., 0.61 g) and lower porosity (i.e., 31%) than 3D structure 1. Figure 1 and Table 1 include the dimensional specifications of all additively manufactured porous structures. In 3D-structure 3, 1.2-mm-diameter spheres were used and the center-to-center distances were 2 mm (i.e., a larger spacing than structures 1 and 2). The spheres were connected with 0.2-mm-diameter cylinders in x, y, and z directions. To vary the pore size in z-direction, a horizontal cross shaped structure (built with two cylinders of 0.2-mm-diameter) was incorporated in the middle of each layer. Figure 2 includes the zoomed-in images of unit cell of (a) 3D-structure 1 and 2, and (b) 3D-structure 3. The effective pore radius is measured based on the radius of a circle inscribed inside a pore. Figure 2 also demonstrates the dimension of inscribed circles inside the pores of 3D-structure 1, 2, and 3. For the horizontal cross-shaped structure, the effective pore radius was smaller in 3D-structure 3 (i.e., 0.16 mm) than the 3D-structure 1 and 2 (i.e., 0.41 mm). The effective pore radius was measured following the inscribed circle method described by Takeuchi et al. [33,34].

Translucent ABS material was selected for manufacturing the additively manufactured structures to enable visualization of water during evaporation with a high-speed camera. The material had density of 1.25 g/cm³ with 55±10 MPa of tensile strength [35]. To measure the wettability, a small flat plate (L = 30 mm, W = 30 mm, H = 3 mm) was manufactured from the same material and a 1-μL de-ionized water droplet was placed on the top of ABS flat plate with a 0.2–2 μL pipette (Fisherbrand Elite, Waltham, MA). A goniometer was used to measure the contact angle; Fig. 2(c) shows the contact angle of 1-μL droplet on flat ABS surface; the material is hydrophilic with a static contact angle of 36.93 deg. Figure 3 includes the top-view and side-view microscopic images of the three additively manufactured structures. An LSM-5 PASCAL (Zeiss, Oberkochen, Baden-Württemberg, Germany) confocal microscope was used to capture the microscopic images of the 3D-printed structures with 4× magnification.

Evaporation of water from additively manufactured structures was conducted at atmosphere pressure with T_air = 22.2 °C, 13–17% relative humidity (RH). Figure 4 represents the schematic of experimental setup. Initially, the de-ionized water was placed in a glass beaker (i.e., 2-cm-diameter, 3-cm-height) and equilibrated to room temperature over thirty minutes. Then, the additively manufactured structure was fully submerged in the de-ionized water with tweezers and remained there until all the pores were invaded with water, as determined by visual inspection and mass measurements. Subsequently, the structure with water was placed on the top of a FX-1200i (A&D; Ann Arbor, MI) scale (sensitivity of ±0.01 g) to record the evaporative mass loss at 15 min time intervals. Five sides (i.e., all sides except the bottom) of the structure were exposed to natural convection, as all of the experiments were conducted without a container. A solar simulator (ABET LS-10500; Milford, CT) was used to apply 1000 W/m² heat flux and a 90 deg beam tuner was used to apply the simulated solar light on the top of the structure. The heat flux was measured with a LI-200R pyranometer (LI-COR Biosciences, Lincoln, NE) (sensitivity of 75 μA per 1000 W/m²). The evaporation phenomenon and drying front propagation during evaporation were visualized and recorded with a Fastec-IL3 high-speed camera (Fastec Imaging, San Diego, CA) at 24 FPS (frame per second) and the total evaporation time was recorded. To observe the temperature profile of the structure during evaporation, a Teledyne FLIR E8-XT infrared camera (Wilsonville, OR) (320 $×$ 240 pixels resolution) was used to capture images at 15 min interval. The mass losses were recorded with rs-multisoftware and each experiment was replicated three times (N = 3).

Evaporation phenomena were observed and investigated from three differently designed additively manufactured porous structures. The investigations included evaporative mass loss, drying front propagation during evaporation, analysis of transient saturation, thermal gradient analysis due to evaporation and incident heat flux, evaporation rate, and heat and mass transfer analysis.

where $φ$ is porosity, $VE$ is the empty volume, $VS$ is the solid volume, and $VT$ is the total volume of the 3D printed structure. The porosity was calculated using two methods: (a) based on mass or volume (i.e., Eq. (1)) and (b) using “mass properties” toolbox of solidworks. The empty volume was estimated using the mass of the invaded water and the total volume was calculated from the total length, height, and width of the 3D-prinmted structures. Both the values are presented in Table 3. The porosity calculated with Eq. (1) and with solidworks, matched well with minimum of 0% to maximum of 3.5% percentage difference.

Figure 5 represents the evaporative mass loss from the structures; each experiment was continued until at least 80% of the total water was evaporated. 3D-structure 3 demonstrated the sharpest decrease, as water evaporated from 0.8 g to approximately 0.1 g within 180 min. 3D-structure 1 and 2 experienced similar trends in evaporative mass loss due to similarity in design specifications (i.e., similar number of spheres, layers, and same pore sizes). Comparing structure 1 and 2, it took more time in structure 1 (i.e., 225 min) to evaporate 0.38 g of water (i.e., 86% of total water) due to greater water holding capacity while 0.28 g (i.e., 88% of total water) was evaporated in 150 min from 3D-structure 2. In previous studies, Shokri et al. [9,10,14,16,17,26] and Lehmann et al. [11] postulated the relationship between pore size and faster evaporation rates. In general, if a porous media has two different pore sizes (⁠$r1$ and $r2$⁠), and if the pores are interconnected, the liquid will travel from the larger pore to the smaller pore due to capillary action. In a regular cylindrical-shaped porous system, the liquid level drops in the larger pore while it increases in the smaller pore due to capillary action, until the drying front reaches to a certain level called characteristic length. Due to faster movement into smaller pore, the liquid is more susceptible to faster evaporation rate. On the other hand, if a porous media consists of similar sized pores, liquid cannot travel through capillary action, and liquid is evaporated at same rate from all pores due to absence of characteristic length (i.e., $Lc=0$⁠).

The faster evaporation rate from 3D-structure 3 may be explained by this phenomenon. For the presence of two different-sized pores, more liquid from a larger pore traveled toward a smaller pore, leading to a sharp increase in evaporation rate [Fig. 5(c)]. On the other hand, 3D structures 1 and 2 consisted of similar pore sizes along all three axes, leading to an absence of capillary movement of water. As a result, the capillary potential was limited, resulting in the absence of characteristic length (i.e., $Lc=0$⁠), and the liquid was evaporated at a same rate from all pores [Fig. 5(b)]. Comparing the pore matrix of all three structures, the faster evaporation from structure 3 compared to structures 1 and 2 is theoretically explained and practically observed.

where $Vi$ is the initial water volume and $Vt$ is the transient volume of water. The transient saturation of water for three structures for all three replications was plotted against time (Fig. 6). The transient saturation decreased gradually for all three additively manufactured structures with time, but 3D-structure 2 experienced the sharpest decrease due to lower initial water volume.

where $LD$ is the drying front depth, $H$ is the height of the porous structure, and $S$ is the transient saturation of the porous medium. The drying front depths, calculated using the previous equation as a function of transient saturation, were plotted against time for three replications for all three additively manufactured structures (Figs. 7–9). For all three structures, the drying front depths increased with time as the portion of unsaturated areas increased and encapsulated the whole structure. Drying front depths were also measured with images captured with the high-speed camera. The depths were measured with imagej after complete evaporation of water from a layer and that depths were also plotted in the same graph (Figs. 7–9). The drying front depths measured as a function of transient saturation and measured from the images matched significantly for all three 3D-structures.

The evaporation rates (i.e., $dmdt$⁠) were calculated from the mass loss data (Fig. 5). Consequently, a layer-by-layer coupled heat and mass transfer model was formulated based on different layers of the porous structures. The following assumptions were made to simplify the predictive model:

• Steady-state.

• Energy balance was considered only in the vertical direction of the porous structure.

• In each layer, the additively manufactured material and water were assumed to be in isothermal state.

• Energy and mass loss from the sides were assumed negligible.

• The temperature of the porous structure during evaporation was measured and recorded with a thermal camera and the subsequent layer temperature was estimated from the thermal images and since, all the structures were almost similar in design specifications, similar thermal gradients were assumed for all three structures during evaporation.

Figure 10 demonstrates the energy balance, assuming the steady-state in a single layer of the porous structure. Here,$Q˙evap$ is the evaporative heat flux, $Q˙solar$ is the incident solar heat flux, $Q˙conv$ is the convective heat flux, $Q˙cond$ is conductive heat flux between layers, $Q˙abs$ is the absorbed heat flux and $Q˙ref$ is reflected heat flux.

where $Nu$ is the Nusselt number, $hconv$ is the convective heat transfer coefficient, and $Ka$ is the thermal conductivity of air. Since all the 3D structures were manufactured with same material (i.e., ABS plastic), the common parameters to estimate steady-state evaporation rate are presented in Table 4.

For evaporation rate calculations, the temperature of each layer was measured from the images captured with the thermal camera at 15 min intervals. For the first layer, the top layer of the porous structure was considered as the surface temperature, $Ts$⁠. Subsequently, after the evaporation of water from first layer, the temperature of the second layer was considered as the surface temperature (i.e., $Ts$⁠) and they were determined from the thermal images Fig. 11). The similar procedures were followed until the water reached to the last layer (i.e., 10th layer). Since all the three structures were manufactured with similar material (i.e., translucent ABS) and the design specifications were almost similar, the thermal gradients during evaporation exhibited similar trends. For each layer, the material and water were assumed to be in thermal equilibrium (i.e., $Ts=Tw$⁠), and for each layer the subsequent parameters such as $Keff$⁠, $Ra, k, Nu$⁠, and $hconv$ were calculated. In different layers, the heat flux, the surface temperature, water temperature, and bottom temperature changed. At the last two layers (i.e., 9th and 10th layers), the temperatures were similar (e.g., with approximately 0.5 K temperature difference), and thus they were considered to be in equilibrium with eighth layer's temperature. Figure 11 demonstrates the layer-by-layer thermal gradient during evaporation from 3D structure-1.

where i is layer 1 and n is layer 10. Due to the small distance between top and bottom surface of each layer (i.e., 1.43 mm for 3D structure-1 and -2, and 2 mm for 3D structure-3), the temperature difference (i.e., $Ts−TB$⁠) was measured as 0.5 K for each layer.

Evaporation rates for all three 3D structures were plotted against time and are demonstrated in Figs. 12–14(a). The normalized evaporation rate was calculated by dividing the transient evaporation rate by the maximum evaporation rate [Figs. 12–14(b)]. All the three structures experienced a certain time (i.e., approximately 60–80 min) of constant rate period evaporation and then experienced a decrease in evaporation rate (i.e., falling rate of evaporation). The highest evaporation rate was found in 3D-structure 3 (approximately 0.005 g/min) and it continued until 80 min of total evaporation time. The larger evaporation rate was associated with larger surface area (i.e., 125.21 mm²) and smaller effective pore size (i.e., 0.16 mm), which resulted in extended period of constant evaporation rate. The evaporation rate in constant rate period was similar for 3D-structure 1 and 2 (i.e., 0.0025–0.003 g/min). Due to larger void fraction, 3D-structure 1 held more water than the others resulting in longest evaporation time (i.e., approximately 250 min).

The analytical evaporation rate calculated from the steady-state energy balance equation was plotted in the same graph (Figs. 12–14). The analytical and experimental results matched well for all three 3D-structres during the constant period of evaporation. Evaporative mass loss with application of external heat flux is a complex transient phenomenon that was not considered in the model assumption. So, at the later stages of evaporation (i.e., the falling rate of evaporation), the data did not match with the experimental results as the falling rate period of evaporation is a transient process and the model had limitations in predicting the transient phenomena. The falling rate period of evaporation was predicted by another model in Sec. 3.4.

Though 3D-structure 1 and 2 had similar initial evaporation trends, the overall evaporation trends were not similar. After approximately 60 min, in 3D-structure-2, evaporation experienced a significant decrease compared to 3D-structure-1. The restriction of lateral water movement could be one of the contributing factors leading to decrease in evaporation rate in 3D-structure 2. Previous research [6,9,11,14,17,37] postulated that the presence of thin liquid films or liquid islands contributes to formation of hydraulic linkages throughout the porous structure, which leads to extended period of constant rate period or smaller decrease in initial evaporation rates. In 3D-structure 2, thin liquid islands created in partially saturated parts could not create linkage with water laterally which resulted in steep decrease in evaporation rate.

Figure 15 demonstrates the 2D view of all 3D structures and illustration of evaporation front and water movement. In 3D-structure-1 [Figs. 15(a)–15(c)], the pore sizes are uniform, and all the pores are interconnected. While evaporating, water created a continuous liquid linkage from the saturated part to the unsaturated part by forming tiny liquid islands. Shokri et al. [17,26] demonstrated the presence of two evaporation fronts: primary and secondary while evaporating from porous media where liquid islands are formed. Though the primary water level drops down with primary evaporation front, the liquid still maintains a hydraulic linkage with the unsaturated part, and water evaporates from the secondary evaporation front exposed to the air. In 3D-structure 1, it is postulated that due to formation of liquid islands, the evaporation rate was almost constant throughout the entire period. After maintaining a constant period (i.e., 0–60 min), the evaporation rate dropped down and again maintained a constant rate (i.e., ∼90 to 180 min), irrespective of other two structures due to presence of two drying fronts simultaneously.

In 3D-structure-2 [Figs. 15(d)–15(f)], the lateral liquid movement was restricted with a solid fill between two layers in the y–z direction, and it is postulated that this lead to the formation of fewer liquid islands and an absence of hydraulic linkages. When the water level dropped down due to evaporative loss, the primary evaporation front dropped down as well, and the evaporation experienced a sharp decrease due to absence of secondary evaporation front. In 3D structure-3 [Figs. 15(g)–15(i)], the movement of water is dominated by capillary action due to presence of two different-sized pores [Fig. 15(c)]. The upward movement of water is obstructed by gravity acting in the opposite direction. When the capillary force was dominated by gravity, the upward water movement stopped, and the evaporation experienced a sharp decrease similar to 3D structure-2.

where $φ$ is the porosity. The model is based on the following assumptions:

1. The model is valid when evaporation takes place while forming of liquid islands by rupture of continuous hydraulic linkages

2. Evaporation takes place on the intersection of saturated and unsaturated parts, i.e., the drying front

3. The unsaturated part is considered to be completely dry irrespective of presence of tiny liquid islands

where $enorm$ is the experimental normalized evaporation rate, $(dmdt)i$ is the evaporation rate at a particular time and $(dmdt)max$ is the maximum evaporation rate throughout the evaporation process. The calculated normalized evaporation rate from the experimental data and the modeled normalized evaporation rate (⁠$e˙$⁠) were plotted against (a) time and (b) drying front depths for both 3D-structure 2 (Fig. 16) and 3 (Fig. 17). For both the 3D-structures, the experimental evaporation rate matched significantly with the falling rate period model.

Three additively manufactured porous structures of different design specifications were manufactured with translucent ABS materials to study evaporation and drying front propagation with application of external source of heat flux. To visualize the drying front propagation and thermal gradients, a high-speed camera and a high-resolution thermal camera were used to capture images at 15 min time intervals. The evaporative mass loss from the porous structures was recorded with a sensitive scale and the evaporation rate was calculated as a function of evaporative mass loss. A steady-state heat-mass transfer model was established to predict the evaporation rates. 3D-structure 3 experienced the sharpest decrease in the mass loss as the water evaporated from 0.8 g to 0.1 g within 180 min. Though structures 1 and 2 were almost similar in design, the inclusion of lateral liquid interaction due to solid fill in y–z direction in structure 2 made it significantly different than structure 1. Due to lateral motion of water, more liquid islands were formed in structure 1 leading to creation of continuous hydraulic linkage which resulted in extended period of constant rate period. As a result, after the surface evaporation (i.e., 0–60 min), the evaporation rate dropped down a bit when liquid islands started to form in upper layers and again maintained a constant rate period from 90 to 180 min due to formation of hydraulic linkage contrary to structure 2. In structure 2, constant rate of evaporation was dominated by surface evaporation (until ∼75 min) and the evaporation dropped (i.e., entered into falling rate period) when the hydraulic linkage broke due to restriction in lateral water movement. In 3D-structure 3, the movement of water was dominated by capillary action as water transported from larger pores to smaller pores and the constant rate of evaporation lasted for ∼75 min. When the water level dropped down due to evaporative loss, the capillary action was restricted by reverse acting gravity and after 75 min, falling rate period of evaporation started.

The drying front depths were calculated as a function of transient saturation and were also measured from the high-resolution evaporation images and plotted against time. The calculated and measured drying front depths matched well representing the correlation between transient saturation and evaporation rate. The steady-state heat and mass transfer coupled modeling matched well in 3D-structure 2 and 3, when the evaporation was in constant rate period (i.e., ∼0 to 75 min). In 3D-structure 1, due to continuous liquid linkage and breakup of liquid islands, there were two part of constant rate period: one due to surface evaporation (0–60 min), then the evaporation rate dropped and again maintained a constant rate (i.e., 90–180 min) due to continuous hydraulic linkages. The steady-state model matched well with second part of constant rate period. For predicting the falling rate of evaporation, normalized evaporation rate model was applied in this study and during falling rate of evaporation (i.e., from 75 min to rest of evaporation), the calculated evaporation rate (i.e., the analytical modeling) matched well with experimental results for 3D-structure 2 and 3. Due to prolonged constant rate of evaporation, 3D-structure 1 experienced very short period of falling rate of evaporation and was not considered for transient modeling. The future works could include establishing a transient model of heat and mass transfer that can predict all the evaporation stages from porous structures. Also, the contribution of liquid island formation during evaporative drying from additively manufactured structures could be another important phenomenon to investigate to understand the water dynamics.

We would like to acknowledge Emily Stallbaumer-Cyr for assistance in capturing microscopic images of the 3D-printed structures.

• Directorate for Engineering, National Science Foundation (Grant No. 1651451; Funder ID: 10.13039/100000084).

• Division of Graduate Education, National Science Foundation (Grants No. 1828571; Funder ID: 10.13039/100000082).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

A =

surface area (mm²)

D =

diffusion coefficient (m²/s)

g =

acceleration due to gravity (m/s²)

H =

height of porous structure (mm)

h_conv =

convective heat transfer coefficient (W/m²K)

h_lg =

latent heat of vaporization (KJ/kg)

k =

intrinsic permeability (mm²)

K =

thermal conductivity (W/m K)

K_eff =

effective thermal conductivity (W/m K)

K_s =

thermal conductivity of water (W/m K)

K_w =

thermal conductivity of solid structure (W/m K)

L =

single layer thickness (mm)

L_c =

characteristic length (mm)

L_e =

Lewis number

LD =

drying front depth (mm)

$m˙$ =

evaporation rate (g/min)

Nu =

Nusselt number

$Q˙$ =

heat flux (W/m²)

r₁, r₂ =

pore radius (mm)

Ra =

Rayleigh number

Ra_x =

Darcy-modified Rayleigh number

RH =

relative humidity (%)

S =

transient saturation

Sh =

Sherwood number

T_w =

water temperature (°C)

T_∞ =

ambient temperature (°C)

V =

volume (mL or mm³)

α =

absorptivity

α_m =

thermal diffusivity (m²/s)

β =

thermal expansion coefficient (K^–1)

δ =

boundary layer thickness

ε =

emissivity

ρ =

density (Kg/m³)

σ =

Stefan–Boltzman constant (⁠$5.67×$ 10^–8 W/m²K⁴)

υ =

kinematic viscosity (m²/s)

φ =

porosity (%)