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 ( Xu et al., 2009 ):

$Q_b=Q_m$ (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 ( Shakaib et al., 2011 ):

$Q_b=h_f\left(T_f-T_{fm}\right)$ (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 ( Hasani et al., 2019 ):

$\Delta H_V=2489.7-2.412\left(T_{fm}-278.15\right)$(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 ( Shokrollahi et al., 2020 ):

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

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

Local heat transfer coefficient and the temperature polarization coefficient (TPC) are as follows ( Abrofarakh et al., 2024 ) ( Yao et al., 2023 ):

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

$\text{TPC}=\frac{T_{fm}}{T_f}$(7)

The thermal efficiency of VMD can be expressed ( Vanneste et al., 2017 ):

$\eta =\frac{J\Delta H_v}{V_{f}Cp\Delta T_f}$(8)

where V_f and ΔT_f are feed velocity and temperature drop along with the fiber, respectively.

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 ( Abrofarakh et al., 2024 ):

$H=K_m\cdot p=K_m\cdot \left(P_1-P_2\right)$(9)

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 ( Soukane et al., 2017 ; Zare & Kargari, 2022 ):

$K_n=\frac{\lambda }{d}$ (10)

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 ( Choi et al., 2024 ):

$\lambda =\frac{K_{b}T}{\sqrt{2}\pi P{d}_1^2}$ (11)

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 ( Lou et al., 2022 ):

$N=\left(C_1+C_2\right)\Delta P$ (12)

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

The concentration polarization coefficient (CPC) is as follows ( Lokare & Vidic, 2019 ):

$\text{CPC}=\frac{C_{fm}}{C_f}$ (13)

C_fm is the concentration of the solution on the surface of the distillation membrane, and C_f is the concentration of the main body of the solution.

Figure 3 illustrates the comparison between the experimental data and the simulated results of the flux of the vacuum fibre membrane distillation module based on different feed temperatures (313, 323, 333, 343, 353 K). The calculated error values demonstrate that the simulated and experimental data exhibit a satisfactory agreement at different feed temperatures.

As illustrated in Figure 4 , the concentration polarisation factor demonstrates minimal variation with increasing salinity of the feed solution. Furthermore, an increase in the inlet flow rate of the feed solution results in a reduction in polarization. The gradient rate of change of CPC with feed liquid inlet velocity is 0.95, 0.42, 0.24, 0.26, indicating that when the inlet flow rate reaches 0.45 m/s, there is a notable improvement in concentration polarisation. Furthermore, the change in concentration polarisation with the increase in inlet flow rate begins to plateau.

It can be seen in Figure 5 that the concentration polarisation coefficient exhibits a notable decline with rising feed inlet temperature. This is attributed to the fact that elevated temperatures enhance the water vapour flux at the membrane interface, thereby exhibiting a more pronounced concentration polarisation ( Darman et al., 2023 ). Consequently, as temperature increases, the membrane flux also rises. However, the accompanying surge in energy consumption is a significant concern that will be elaborated upon in the thermal efficiency section.

Figure 6 displays the trend of the temperature polarisation coefficient as a function of the inlet temperature. It is evident that the inlet and outlet temperatures exert a considerable influence on the TPC. As the temperature rises, the temperature gradient of the feed solution in the hollow fibre membrane tube becomes markedly more pronounced. Furthermore, temperature polarisation results in an uneven heat transfer, which ultimately leads to a reduction in thermal efficiency. Achieving an optimal balance between thermal efficiency and membrane flux can further enhance the efficiency of vacuum membrane distillation ( Andrés-Mañas et al., 2023 ). Increasing the inlet flow rate can mitigate the impact of temperature polarisation and improve thermal efficiency.

Figure 7 presents a plot of thermal efficiency as a function of temperature and flow rate. It is evident that an increase in the inlet flow velocity has a considerable impact on the thermal efficiency. However, the improvement in thermal efficiency brought about by an increase in the inlet velocity to 0.45 m/s is less pronounced. As the temperature rises, the decline in thermal efficiency becomes less steep, occurring between 323.15 and 343.15 K. Consequently, a more detailed examination of the efficiency of vacuum membrane distillation in conjunction with membrane flux is crucial.

Figure 8 provides a visual representation of the relationship between TPC and CPC and membrane flux. At TPC = 0.985 and CPC = 0.990, the membrane flux is observed to reach 42 kg∙m⁻²∙h⁻¹. In light of the aforementioned results, it can be concluded that a high level of thermal efficiency is achieved at an inlet flow rate of 0.45 m/s and an inlet temperature within the range of 323.15 to 343.15 K. Consequently, the optimal operating parameters between thermal efficiency and membrane flux must be identified on the basis of the cost of energy supply.