The MD transport model in a single pore is developed for the steady state at which the liquid–vapor phase boundary remains inside the pore (Figure 2).

In DCMD, both ends of the pore are in contact with liquid, and it seems possible that liquid enters from both sides. However, for the reason given in the introduction, it is assumed that the liquid enters only from the pore entrance that is in contact with the feed stream.

There is some evidence to support the water entry from the feed side. For example, Gryta (2007) reported that water partially filled the pore from the feed side to the distance of 15 µm in the total 400 µm of membrane thickness by showing the Mg and Ca content profile in the longitudinal direction of the pore by EDX, wherein DCMD was performed by polypropylene membrane using tap water as a feed. Jiang and co-workers also showed the presence of Na on the feed side by EDX when DCMD was performed by their hydrophilic–hydrophobic hybrid membrane (Zhu et al., 2015; Feng et al., 2017). Based on this assumption, the model is also applicable for air gap membrane distillation (AGMD) with some changes in the transport parameters.

For the heat transfer in the liquid phase, the following differential equation is used at the steady state: d 2 T d x 2 + a d T d x = 0 , (18)

where T is the temperature in the liquid phase (K) and x is the distance from the pore inlet (m) in the longitudinal direction, and α = k l ρ c p , (19)

where k l and c p are thermal conductivity (W/m K) and specific heat capacity (J/kg K) of the liquid, respectively.

The general solution of the differential equation is T = C 1 + C 2 ⁡ exp − a x , (20)

where C 1 and C 2 are the integration constants.

Using the boundary conditions B 1 : T = T f  at  x = 0 , (21) B 2 : d T d x = − N ∆ H v k l π r 2 − h g T i − T p  at  x = δ g , (22)

where ∆ H v is the heat of evaporation of liquid (J/kg) and h g is the heat transfer coefficient of the gas phase (W/m² K).

C 1 and C 2 can be calculated from the above boundary conditions, resulting in the specific solution of the differential equation, T = T f + − α u − N ∆ H v k l π r 2 − h g T i − T p exp − u δ l α − ⁡ exp − u α δ l − x , (23) where u = N ρ π r 2 . (24)

Furthermore, because T = T i at x = δ l , T i = T f + − α u − N ∆ H v k l π r 2 − h g T i − T p exp − u δ l α − 1 . (25)

In Eq. 25, the temperature dependence of ∆ H v is considered as ∆ H v = 8.314 × 647.3 5.6297   τ r 1 3 + 13.962   τ r 2 3 − 11.673 τ r + 2.1784   τ r 2 − 0.31666   τ r 6 × 1000 / 18 . 02 , (26) where τ r is τ r = 1 − T i 647.3 . (27) N l = N ,   T i  and  δ l can be given by solving Eqs 1a, 1b, (2), (3), (7), and (25) simultaneously.The following equations (Eqs 8–17) are used in Eq. 7:

The simulation was performed using Microsoft Excel, and the algorithm is presented in Figure 3.

Figure 4 illustrates how varying the pore radius (r) affects δ l , N, and T i , while maintaining fixed parameters: T f = 353.2 K, T p = 293.2 K, p f = 1 × 10⁵ Pa, p p = 1 × 10⁵ Pa, θ = 60°, and δ = 5 × 10⁻⁵ m. The pore radius, r, was adjusted from 0.01 to 1 × 10⁻⁶ m.

In Figure 4, the influence of pore radius (r) on wetting characteristics, when the pore radius r = 0.01 × 10⁻⁶ m, δ l reaches 4.92 × 10⁻⁵ m, signifying that over 90% of the pore space is occupied by the liquid phase due to the pronounced capillary force in such a small pore. As r increases to 1 × 10⁻⁶ m, δ l progressively decreases to 1.92 × 10⁻⁵, accompanied by a significant rise in N from 6.69 × 10⁻¹⁶ kg/s to 1.73 × 10⁻¹² kg/s at the larger radius (r = 1 × 10⁻⁶ m, the red bars in Figure 4). This underscores the substantial impact of the r on N. Despite T i being nearly 40°C lower than T _f due to significant temperature polarization caused by the liquid phase in the pore, T i remains relatively constant with varying r. This constancy is attributed to the compensatory effect of the decrease in N countering the increase in δ l . In summary, the intricate interplay between pore size, capillary forces, and temperature dynamics shapes the wetting behavior within porous media.

Figure 5 illustrates the impact of feed temperature T _f on δ l , N and T i , while maintaining fixed parameters: T p = 293.2 K, p f = 1 × 10⁵ Pa, p p = 1 × 10⁵ Pa, r = 1 × 10⁻⁶ m, θ = 60°, and δ = 5 × 10⁻⁵ m.

In Figure 5, the effect of the impact of feed temperature (T _f ) on wetting characteristics (blue bar), δ l (blue bar) decreases with rising T _f , maintaining approximately 20% pore filling of the entire pore length with liquid. However, in Figure 5 (red bar), N deviates from the expected exponential increase tied to vapor pressure elevation with increasing feed temperature (T _f ). This unexpected behavior is attributed to heightened temperature polarization at higher T _f . In Figure 5, ( T i ) reveals the dynamics of temperature change, showing a mere 2% decrease (from 303.2 K to 297 K) when T _f is 303.2 K but a more significant 11% drop (from 353.2 K to 315 K) when T _f is 353.2 K. This contextualizes the observed deviation in N, underscoring the impact of temperature polarization on the anticipated exponential relationship between N and vapor pressure as T _f increases.

Figure 6 shows the effect of T _p on δ l , N, and T i   when T f , p f , p p , r, θ , and δ are fixed to 353.2 K, 1 × 10⁵ Pa, 1 × 10⁵ Pa, 1 × 10⁻⁶ m, 60^o, and 5 × 10⁻⁵ m, respectively.

As depicted in Figure 6, the liquid intrusion length δ l decreases as the permeate temperature T _p increases. This trend can be attributed to the wettability of the membrane surface. At higher permeate temperatures, the membrane becomes more hydrophobic, leading to reduced liquid intrusion into the pores.

Temperature polarization (T _p ): T _p represents the difference between the bulk feed temperature T _f and the temperature at the membrane/solution interface T _mf where the vapor–liquid transition occurs. As T _p increases, the temperature polarization, T f − T i , decreases. This decrease in temperature polarization is noteworthy and can be linked to the heat transfer dynamics within the system.

Impact on mass flux (N): although temperature polarization decreases, there is also a decrease in mass flux (N) with increasing T _p . However, it is essential to recognize that the impact of T _p on mass flux is quantitatively smaller than the influence of feed temperature T f   For instance, a 40 K increase in T _p results in a measured decrease in N of 0.59 × 10⁻¹² kg/s. In contrast, a similar increase in T f   leads to a more substantial increase in N, specifically 1.17 × 10⁻¹² kg/s.

The discrepancy between the effects of T _p and T f suggests that the system is more sensitive to variations in T f . The nuanced interplay between temperature parameters and mass flux underscores the complexity of the system dynamics. In summary, understanding the intricate balance between temperature, wettability, and mass transport is crucial for optimizing membrane-based processes. Further investigations will help uncover the underlying mechanisms governing these observed trends.

Figure 7 shows the effect of p _f on δ l , N, and T i when T f , T p , p p , r, θ , and δ are fixed to 353.2 K, 293.2 K, 1 × 10⁵ Pa, 0.1 × 10⁻⁶ m, 60^o, and 5 × 10⁻⁵ m, respectively.

In Figure 7, a compelling pattern emerges: the parameter δ _l displays a decreasing trend as feed pressure (p _f ) increases. This seemingly paradoxical observation results from various factors. Despite higher feed pressure propelling liquid deeper into the pore, a simultaneous rise in temperature at the liquid–gas interface (T _i ) amplifies capillary pressure in the gas phase (p _g ), diverting the liquid back toward the pore entrance. The findings in Figure 7 (blue bar) underscore that the influence of p _g on δ _l outweighs that of p _f . This pattern is also evident in the reduction of another parameter, N (mass flux of water through liquid and gas phases), as p _f increases (Figure 7, red bar).

The initially counterintuitive decrease in δ _l with increasing p _f becomes clear upon examining the role of capillary pressure (p _g ). As p _f rises, more liquid infiltrates the pore structure, but the pivotal factor is the impact of p _g , controlled by T _i . Analyzing the relationship between T _i , we observe its increase with p _f , leading to an elevated saturation vapor pressure (ψpsi) (Eq. 16), subsequently influencing p _g (Eq. 10). The essence lies in the dominance of p _g over p _f in influencing δ _l . Essentially, capillary forces play a predominant role in determining δ _l , and a similar phenomenon is observed for parameter N. With an increase in p _f , T _i rises, causing an escalation in p _g , which, in turn, alters flow behavior and results in a decrease in N.

The core interaction driving the observed phenomena lies in the intricate relationship between capillary pressure in the gas phase (p _g ) and feed pressure (p _f ). The augmentation of p _f facilitates enhanced liquid entry into the pore space, yet the prevailing influence of capillary forces, particularly governed by T _i , ultimately dictates the behavior of the liquid phase. Eq. 10 and (16) are likely instrumental in comprehending this intricate interplay, encapsulating the dynamic involvement of surface tension, capillary pressure, and porosity.

To synthesize the findings, it becomes evident that the increase in T _i (and subsequently p _g ) eclipses the impact of p _f , shedding light on the discernible trends in δ _l and N. This interdependence is graphically depicted in Figures 5–7 (as depicted by the blue bars), where an escalation in T _i , T _p , and pf precipitates a swift reduction in δ _l .

The decrease in δ _l is particularly conspicuous when T _f rises, as this prompts a rapid decline in wetting characteristics. Various factors related to wettability properties come into play as the temperature increases. Elevated temperatures induce a reduction in hydroxyl groups within cellulose chains, leading to diminished moisture uptake and decreased water adsorption, thereby influencing wettability properties (Sipahutar et al., 2021). Furthermore, heightened temperatures result in the expansion of membrane pores, escalating the risk of membrane wetting and enlarging pore sizes. This expansion is more pronounced at elevated temperatures due to an amplified temperature gradient and increased heat transfer rates. Additionally, the rise in temperature concurrently diminishes the surface tension and contact angle, further contributing to membrane wetting (Gryta, 2020). In summary, the collective impact of these factors at heightened temperatures manifests in a precipitous decline in wetting characteristics (δ _l ).

Figure 8 elucidates the impact of feed pressure (p _f ) on key parameters— δ l , N, and T i , while maintaining specific conditions: T f , T p , p p , r, and δ are fixed to 353.2 K, 293.2 K, 1 × 10⁵ Pa, 1 × 10⁻⁶ m, and 5 × 10⁻⁵ m, respectively. Two distinct contact angles, θ = 60° (blue shades) and θ = 82° (red shades), are considered. Notably, the pore radius (r) has been adjusted from 0.1 × 10⁻⁶ m to 0.1 × 10^–6 m in Figure 7 to 1 × 10^–6 m in Figure 8. Despite the different pressure ranges studied for 60° (0.5–1.5 × 10⁵ Pa) and 82° (1.4–2.0 × 10⁵ Pa), both contact angles exhibit consistent trends. A notable observation is the reversal in the trend of T i in Figure 8 compared to Figure 7. Specifically, T i decreases as p _f increases, resulting in an upswing in δ l (Figure 8). In contrast, the flux, N, follows the same decreasing trend with increasing p _f in both Figures 7, 8.

Here, the contact angle values of 60° and 82° are commonly used in studies to represent different wetting properties of surfaces. A contact angle above 90° indicates hydrophobicity, where the surface repels water, while an angle below 90° signifies hydrophilicity, indicating good wetting by water (Danish, 2020). These specific angles are chosen to represent these distinct wetting behaviors for experimental and analytical purposes. Similarly, the values of 1.0×10⁵ Pa and 1.5 × 10⁵ Pa for p _f (pressure) are set to explore the impact of varying pressures on wetting characteristics and interfacial tension in fluid–rock systems. By using different pressure values within this range, researchers can analyze how changes in pressure affect contact angles and interfacial tension, providing insights into the behavior of fluids interacting with rock surfaces under different conditions. These specific pressure values are selected to study a range of scenarios and understand the nuances of fluid–rock interactions comprehensively (Taetz et al., 2016).

This reversal in T i   behavior underscores the system’s sensitivity to variations in p _f , a shift that can be influenced by adjusting parameters such as pore radius and contact angle. The cohesion between fixed conditions and altered experimental factors in Figure 8 emphasizes the intricate dynamics governing the observed trends in δ l , N, and T i . These findings contribute to a nuanced understanding of how the system responds to changes in feed pressure (p _f ) under controlled conditions.

Figure 9 shows the effect of p _p on δ l , N, and T i when parameters T f , T p , p f , and δ are fixed to 353.2   K , 293.2 K, 1.5 × 10⁵ Pa, and 5 × 10⁻⁵ m, respectively. Again, θ is either 60^o (blue shades) or 82^o (red shades), and r is 1 × 10⁻⁶ m. Comparing Figures 8, 9, it is found that the decrease in p p has the same effect as the increase in p f . As p f   − p p increases by either an increase in p f or a decrease in p p , δ l experiences an increase, while N and T i exhibit a decrease.

In fluid flow, the significance of maintaining a consistent pressure difference ( p f   − p p ) is evident in the relationship between flux (N) and permeate pressure ( p p ). As permeate pressure increases, the flux also increases, indicating a direct correlation between these two parameters (Naidu et al., 2015). This consistency in pressure difference ensures predictable and controlled flux rates, which are crucial for optimizing system performance and efficiency in various applications like reverse osmosis, gas separation, and membrane processes (Stewart, 2014). By regulating this pressure difference, engineers can manipulate the flux of substances through membranes, leading to improved separation efficiency and overall system effectiveness.

Figure 10 shows the effect of p _f on δ l , N, and T i when T f , T p , p f − p p , r, θ , and δ are fixed to 353.2 K, 293.2 K, 0.5 × 10⁵ Pa, 1 × 10⁻⁶ m, 60^o, and 5 × 10⁻⁵ m, respectively. In this specific scenario, an escalation in p _f leads to a concurrent decrease in δ l , a subtle augmentation in N, and an elevation in T i . Remarkably, this dynamic unfolds while maintaining a consistent ( p f − p p ).

Understanding the relationship between liquid intrusion length ( δ l ), feed pressure ( p f ), mass flux (N), and liquid–gas interface temperature ( T i ) is crucial in various engineering applications. As p f increases, a decrease in δ l suggests improved heat transfer efficiency, with smaller values indicating more effective heat transfer. The subtle increase in mass flux (N) signifies a rise in fluid flow or mass flow rate. Additionally, the elevation in T i indicates that the system’s internal temperature rises with escalating p f , which is essential for designing and optimizing temperature-controlled processes like those in chemical reactors or electronic devices (Rouquerol et al., 2011). In this case, we must balance the benefits of enhanced heat transfer and fluid flow against drawbacks like increased energy consumption or system complexity (Rouquerol et al., 2011).

Figure 11 shows the effect of θ on δ l , N, and T i , when T f , T p , p f , p p , r, and δ are fixed to 353.2   K , 293.2 K, 1 × 10⁵ Pa, 1 × 10⁵ Pa, 1 × 10⁻⁶ m, and 5 × 10⁻⁵ m, respectively.

In Figure 11 (blue bar), δ l decreases as θ increases because less water is driven into the pore as the hydrophobicity of the membrane material increases. In Figure 11, the line shows that T _i increases as θ increases because of the decrease in temperature polarization with the decrease in the length of the liquid phase. A decrease in δ l enhances the flow rate of the liquid phase (Eq. 1), and as a result, N increases (red bar). At θ = 81^o, δ l and N become 0.326 × 10⁻⁵ m and 4 × 10⁻¹² kg/s, respectively. Even though it is not shown in the figure, θ may increase until it becomes 85.6^o, where δ l becomes 0, and the pore is filled only with the gas phase. N becomes as high as 7.79 × 10⁻¹² kg/s. This scenario is more explicitly illustrated in Figure 13.

Figure 12 illustrates the impact of contact angle (θ) on δ l , N, and T i with fixed parameters ( T f = 353.2 K, T p = 293.2 K, p f = 1.5 × 10⁵ Pa, p p = 1 × 10⁵ Pa, r = 1 × 10⁻⁶ m, and δ = 5 × 10⁻⁵ m). It is important to note that p f has been increased to 1.5 × 10⁵ Pa from the 1 × 10⁵ Pa used in Figure 11. The observed patterns in Figure 12 align with those in Figure 11, where δ l decreases, and N and T i increase with higher θ values.

However, a key distinction is the use of larger θ values in Figure 12. This adjustment is necessitated by the higher hydrophobicity required in the membrane material to counteract the elevated feed pressure ( p f ) and maintain the interface position. Remarkably, even with θ values exceeding 90°, indicating hydrophobicity in the pore, there is partial liquid filling, according to Figure 12. Notably, the maximum θ is 92° in this scenario, where δ l and N are reported as 0.201 × 10⁻⁵ m and 4.73 × 10⁻¹² kg/s, respectively. It is worth mentioning that θ may further increase until it reaches 107.9°, leading to δ l becoming 0 and N reaching 7.79 × 10⁻¹² kg/s, as revealed in the data.

In Figure 13, N is graphed against δ l for two different feed pressures: p _f = 1.0 × 10⁵ Pa (blue) and p _f = 1.5 × 10⁵ Pa (red). Notably, the data for both feed pressures overlap, and there is a consistent increase in N as δ l decreases. Particularly, N exhibits a steep increase when δ l is below 0.5 × 10⁻⁵ m, reaching N = 7.79 × 10⁻¹² kg/s when δ l = 0. This signifies a scenario where there is no liquid in the pore, and transport occurs solely in the vapor phase.