MD is a complex physical process involving coupled mass and heat transfer phenomena. In this work, the steady-state theoretical model for DCMD through the biporous anisotropic membrane is developed. The upstream (feed) side of the anisotropic membrane with hydrophobic nano-sized pores, hereinafter referred to as the “active layer”, is contacted with the warm saline feed solution, while the downstream (permeate) side is provided with much larger pores, hereinafter Called the “supporting layer” that is in contact with the colder deionized water. Due to the hydrophobicity of the active layer, a water-air interface forms at the entrance to the nanoscale pores (see Fig. 1), at which the saturation vapor pressure is significantly increased according to the Kelvin equation (Eq. 1).^{14}which in turn leads to a significant improvement in water vapor transport through the membrane^{fifteen}.

$$ frac{{p_{s,r} }}{{p_{s} }} = {text{exp}}left( { – frac{{2sigma V_{m} cos theta }}{rRT}} right) $$

(1)

Where *p*_{s, r} is the vapor pressure in a capillary with a radius *right*, *p*_{s} is the vapor pressure at the flat surface, *σ* is the surface tension, *v*_{m} is the molar volume of liquid water, *θ* is the contact angle, *R* is the ideal gas constant, and *T* is the absolute temperature. Under these, *p*_{s,}* σ,* and *v*_{m} are a function of temperature as listed in Supplementary Material Section S3, Table S1. Equation (1) shows that if the membrane is hydrophobic, (theta) is greater than 90°, resulting in (p_{s,r} > p_{s}). The data presented in Table S2 or Section S3 clearly show that the capillary effect significantly increases the driving force caused by the vapor pressure increase at the curved meniscus.

In Table S2, when the feed water of 25 °C contacts the active layer with a pore radius of 1 nm, the vapor pressure in the pore becomes equal to that of 43 °C of the flat meniscus, which corresponds to an enhancement of 18 °C (shown in Table S2 as (Delta T)). The situation is similar when the pore radius is reduced to 0.5 nm (Delta T) becomes 41 °C. Therefore, the basic concept of the capillary effect on vapor pressure leads us to design a membrane with the biporous anisotropic structure that we patented earlier^{16}1, ie a thin active layer with a large number of pores in the nano or sub-nanometer range is supported by a thick layer with much larger pores, possibly in the micron range. It is desirable that the active layer be superhydrophobic to prevent liquid water from entering the pore and also to form a liquid/gas interface with a meniscus large enough to allow a significant increase in vapor pressure enable. The support layer, on the other hand, provides mechanical strength. It is also desirable to keep the support layer hydrophilic to draw water into the pore so that we can take advantage of the shorter vapor path length and fast liquid transport via viscous flow, as discussed in more detail in Section S4. Thus, mass transport is mainly controlled by vapor transport through the active layer. The membrane so constructed can significantly reduce the energy consumption in MD since heating of the feed solution can be minimized, aided by the capillary action of the nano-sized pores.

With respect to vapor transport through the active layer, the vapor flux (*J*_{W}) is proportional to the vapor pressure difference (the driving force), as in Eq. (2)^{17}.

$$ J_{w} = B_{m} left( {p_{f,m} – p_{p,m} } right) $$

(2)

Where *B*_{m} is the membrane mass transfer coefficient, *p*_{f, m} and *p*_{watch} are the vapor pressure at the pore entrance and exit of the active layer. The heat transfer resistance at the feed and permeate boundary (including the heat transfer through the support layer due to the high thermal conductivity of the support material) is neglected. This assumption serves to simplify the model equation and in particular to show the capillary effect on vapor transport more clearly. According to the assumption above (p_{f,m}) is assumed to be the flow temperature (p_{p,m}) at permeate temperature.

It can also be easily assumed that the mass transfer takes place through *nano size* Poring of the active layer occurs via the Knudsen flow mechanism.

Then,

$$ B_{m} = frac{2}{3}frac{{varepsilon _{a} r_{a} }}{{tau _{a} delta _{a} }}left( {frac {8M}{{pi RT}}} right)^{1/2} $$

(3)

Where (varepsilon, r, tau)and (Delta) are porosity, radius, tortuosity or pore length and subscript *a* is for the active layer, and (M) is the molecular weight of water. *T* is the temperature in the pore and the average of the feed and permeate temperatures is used.

Then,

$$ J_{w} = frac{2}{3}frac{{varepsilon_{a} r_{a} }}{{tau_{a} delta_{a} }}left( {frac {8M}{{pi RT}}} right)^{1/2} times left( {p_{s,1} exp left( { – frac{{2sigma_{1} V_ {m1} cos theta_{1} }}{{r_{a} RT_{1} }}} right) – p_{s,2} } right) $$

(4)

where subscripts 1 and 2 stand for feed and permeate, respectively. Note that the capillary effect on the permeate side of the pore in Eq. (4), which is justified in Section S4.

In addition, together with the Antoine equation

$$ p_{s,i} = expleft( {23.1964 – frac{3816.44}{{T_{i} – 46.13}}} right)quad i = 1;{text{or}} ;2$$

(5)

(J_{w}) can be obtained as a function of (T_{1}) for a given set of membrane structure parameters, surface tension, contact angle and permeate temperature.

(J_{w}) is further normalized with respect to the flow at 25 °C

$$ NJ = frac{{J_{w,t} }}{{J_{w,25} }} $$

(6)

Where (J_{w,t}) and (J_{w,25}) are (J_{w}) at temperature *t* (°C) or 25 °C to express the effect of temperature for a given value (r_{a}). (Note that in *NJ*the ramped parameter (frac{{varepsilon_{a} r_{a} }}{{tau_{a} delta_{a} }}) is canceled and *NJ* just depends *right*_{a} and *T*_{1})*.*