During the VMD process, the permeate side region of the membrane is always in a high vacuum environment and the membrane material itself has poor thermal conductivity, so the heat loss is generally ignored in the modelling process According to the principle of energy conservation, the following equation can be obtained [12] :

$Q_b=Q_m$. (1)

where $Q_b$ is the heat of the feed liquid and $Q_m$ is the latent heat of evaporation transfer.

The heat and latent heat of vapour transferred between the membrane surface and the feed liquid is calculated using the following equation [13] :

$Q_b=h_f\left(T_f-T_{fm}\right)$ (2)

$Q_m=N\Delta H_V+\frac{K_m}{\delta }\left(T_{fm}-T_{pm}\right)$ (3)

where δ is the membrane thickness (m), $T_{fm}$ is the temperature of the membrane surface on the feed side (K), $T_{pm}$ is the temperature of the membrane surface on the permeate side (K), N is the membrane flux per unit time (kg·m⁻²·h⁻¹), and the equation of the relationship between the temperature of the membrane surface and the latent heat of vapour is as follows [14] :

$\Delta H_V=2489.7-2.412\left(T_{fm}-278.15\right)$ (4)

In the existing research literature, the calculation of heat transfer coefficients is based on the Nussell number. Although the models used in the various studies are different, the theory and procedure are basically the same. Based on these empirical formulas, the following Nussell’s formula can be derived [15] :

$N_u=\frac{h_f\cdot d}{k}$ (5)

where $d$ is the film thickness (m) and k is the thermal conductivity (W∙m⁻¹∙K⁻¹).

Local heat transfer coefficient on the material-liquid side [16] :

$h_f=\frac{Q_b}{\left(T_f-T_{fm}\right)}=\frac{Q_m}{\left(T_f-T_{fm}\right)}$ (6)

In membrane distillation operation, material transfer and heat transfer are interrelated processes. Thus, the membrane flux of VMD can be calculated by the following equation [17] :

$H=K_m\cdot p=K_m\cdot \left(P_1-P_2\right)$ (7)

where P₁ is the membrane surface pressure (Pa) on the feed side and P₂ is the membrane surface pressure (Pa) on the permeate side.

On this basis, three main mass transfer mechanisms were proposed: molecular diffusion, Knudsen diffusion and viscous flow. In VMD, due to the negative pressure on the cold side, there is only a small amount of water vapour in the pore space of the membrane, and the resistance caused by the collision between molecules is negligible, and the driving force for the transfer is the saturated vapour pressure on the inlet side and the absolute pressure difference on the permeate side. The Knudsen number ( $K_n$) is [17] [18] :

$K_n=\frac{\lambda }{d}$ (8)

where λ is the mean free range of water vapour (m) and d is the mean pore diameter of the membrane (m).

The calculated expression for the mean free range of water vapour through the membrane is as follows [19] :

$\lambda =\frac{K_{b}T}{\sqrt{2}\pi P{d}_1^2}$ (9)

where $K_b$ has a value of 1.38 × 10⁻²³ J∙K⁻¹ and d₁ is the collision diameter value of 2.641 × 10⁻¹⁰ m. The mass transfer mechanism in this study is a viscous flow-Nussen diffuser.

The mass transfer mechanism in this study is a viscous flow-Nussen diffusion machine, so the equation for the transmembrane mass transfer flux is [20] :

$N=\left(C_1+C_2\right)\Delta P$ (10)

where C₁ is the viscous flow coefficient, C₂ is the Knudsen coefficient, and ∆P is the vapour partial pressure difference.