In this study, we developed a comprehensive two-dimensional computational fluid dynamics (CFD) model using COMSOL Multiphysics to describe and simulate heat transfer, mass transfer and fluid flow in the flat sheet vacuum membrane distillation (VMD) under laminar flow conditions. A combination of Knudsen and Poiseuille flow was applied to study mass transfer across the membrane. The effect of variation of Reynolds number, inlet feed temperature and degree of vacuum on different parameters (mass flux, temperature polarization coefficient- TPC, concentration polarisation, heat transfer coefficient) was studied. There was a positive impact of the Reynolds number (50–200) on mass flux (13.15%), heat transfer coefficient (2.64%) and TPC (1.42%), while CPC decreased by 56.63%. The increment in the heat transfer coefficient was due to fluid mixing on the feed side, while the increment in the TPC was due to a higher temperature gradient across the membrane surfaces. The increment in the feed temperature (323–343 K) resulted in an increase in mass flux by 132.9%, while TPC decreased from 0.98 to 0.90. The degree of vacuum (640–750 mm Hg) increased mass flux and heat transfer coefficient by 72.52 and 425.83%, respectively, while the TPC decreased by 8.81%. The feed temperature was the most sensitive parameter with respect to mass flux. The developed CFD model was validated with in-house experimental results with reasonable accuracy.

  • Comprehensive 2D CFD model developed using COMSOL to simulate heat transfer, mass transfer and fluid flow in VMD.

  • CFD model used to determine the temperature and concentration polarization at the membrane surface.

  • The effect of Reynolds number, feed temperature, and vacuum degree on VMD performance parameters were studied.

  • Combination of Knudsen and Poiseuille flow was applied to study mass transfer across the membrane.

Graphical Abstract

Graphical Abstract
Graphical Abstract

Symbol

     
  • Water activity

  •  
  • C

    Solute concentration (%)

  •  
  • Specific heat (J kg−1 k−1)

  •  
  • Diffusivity of water (m2 s−1)

  •  
  • d

    Pore diameter of membrane (m)

  •  
  • h

    Heat transfer coefficient (W m2 k−1)

  •  
  • J

    Mass flux (kg m2 h)

  •  
  • Thermal conductivity (W m−1 K−1)

  •  
  • Molecular weight of water (kg mol−1)

  •  
  • p

    Pressure (N m−2)

  •  
  • Heat transfer rate (W m−2)

  •  
  • Gas constant (J mol−1 K−1)

  •  
  • Pore radius (m)

  •  
  • T

    Temperature (K)

  •  
  • Membrane thickness (m)

  •  
  • u

    Velocity (m s−1)

  •  
  • x

    Salt mole fraction

Greek symbol

     
  • η

    Water vapour viscosity (Pa s)

  •  
  • Density (kg m−3)

  •  
  • Dynamic viscosity of water (Pa s)

  •  
  • Tortuosity

  •  
  • Porosity

Subscript

     
  • b

    Bulk

  •  
  • f

    Feed side

  •  
  • fm

    Feed side membrane surface

  •  
  • g

    Gas (air)

  •  
  • in

    Inlet

  •  
  • p

    Permeate

  •  
  • pm

    Permeate side membrane surface

  •  
  • s

    Solid

  •  
  • v

    vacuum

  •  
  • w

    Water

Membrane distillation (MD) is a thermal separation process that involves transportation across the hydrophobic microporous membrane of the volatile feed components, with the difference in vapour pressure on both sides of the membrane acting as the driving force (Yadav et al. 2021b, 2022e). MD can be categorized into several configurations based on the condensation on the permeate side and removal strategies for volatile components (Alkhudhiri et al. 2012; Yadav et al. 2022a). Direct contact membrane distillation employs a condensing pure liquid (typically a pure volatile component) in direct contact with the membrane surface on the permeate side (Yadav et al. 2021c); air-gap membrane distillation uses a condensing surface separated from the membrane by an air gap (Duong et al. 2015), and sweeping gas membrane distillation uses a sweeping gas to remove the vapour from the distillate side (Karanikola et al. 2015); vacuum membrane distillation (VMD) uses vacuum in the permeate channel to collect the vapour (Yadav et al. 2021f, 2022d). VMD provides the highest distillate flux, negligible conductive heat loss across the membrane, and lowest thermal and concentration boundary layers over the other configurations of the MD (Drioli et al. 2015). Describing the VMD model is complicated due to the combination of all transport events, i.e. mass and heat transfer, occurring simultaneously (Yadav et al. 2021e). The operating parameters that govern the VMD process are pressure, temperature, concentration, flow rate, vacuum degree, etc. For optimal apparatus design and operating condition optimization, a thorough understanding of mass and heat transfer mechanism along with fluid dynamics in industrial membrane separation processes is essential (Coroneo et al. 2009).

The mathematical modelling and empirical correlation technique are the two methods for quantifying the separation performance of membrane modules in VMD. However, mathematical modelling is based on a simplified geometry and unrealistic assumptions rendering inaccurate predictions (Staszak 2017; Al-Obaidi & Alhamid 2021). Another possibility is to develop a membrane module-specific empirical correlation. While the application reach of the correlations in the literature is quite limited in terms of geometric aspects of the membrane, this results in substantial constraints in their use. In various applications, CFD is a helpful and powerful approach for investigating many intricate flow phenomena in complex form geometries (Al-Obaidi & Chaer 2021; Al-Obaidi et al. 2021). Computational fluid dynamics (CFD) simulations are an alternate method for numerically solving interrelated physical phenomena such as fluid flow and heat conduction using computational numerical calculations and graphic display (Guillen & Hoek 2009). CFD has been utilized as a modelling approach for simulating complicated geometries in steady and unsteady state modes since the late 1990s. The durability and efficiency of numerical approaches for getting comprehensive solutions for VMD module and membranes problems (e.g., concentration and temperature polarization and mass flux) have led to the widespread use of CFD to study transport phenomena in VMD processes (Liu et al. 2004; Kougoulos et al. 2005; Chen & Wu 2021; Al-Obaidi 2022).

There have been a few computational simulation-based studies in VMD. However, the models suffered from various drawbacks addressed in this study. Few researchers (Yu et al. 2011, 2012) have developed models to simulate the mass and heat transfer coefficients. As a first step, they computed the distributions of velocity and temperature in the modules, and then they calculated the available temperatures based on mass fluxes. As a result, mass transfer's influence on momentum and heat transfer was ignored. Tang et al. (2011) used Darcy's model to study a hollow fibre VMD module with a PVDF membrane for desalination to predict mass transfer in the VMD membrane; however, their model did not fit the experimental data satisfactorily. Wu et al. (2015) designed air-bubbling VMD to boost the heat and mass transfer and studied the effect of flow patterns on the performance of the process using an analytical model. The model was limited to studying the effect of process parameters on temperature polarization coefficient (TPC) and concentration polarization coefficient (CPC). For flat sheet membranes, Upadhyaya et al. (2016) developed a 2D VMD model to study the effect of various parameters on mass flux and thermal field. In another study by the same group, a 3D CFD model of the VMD was established to predict the mass flux and thermal field (Baghel et al. 2020). 3D transient CFD simulation was performed for VMD by Anqi et al. (2020) coupled with the water flux concentration and temperature at the spacer incorporated membrane; the spacer filled VMD showed the increased temperature and feed flow. Qi et al. (2020) performed a 3D CFD simulation to verify their experimental findings. Moreover, they used response surface methodology for the effects of various input parameters. The effect of feed concentration, vacuum pressure, and thickness of the membrane was studied by Parakala et al. (2019) in the VMD process via 3D CFD simulation, and they developed a 3D CFD model for the prediction of the liquid entry pressure in the case of dye solution as the feed. However, these researchers did not simulate fluid flow, a critical parameter in VMD.

As discussed earlier, there have been very few CFD studies in the VMD process involving flat sheet membranes (Upadhyaya et al. 2016; Baghel et al. 2020). Moreover, very few CFD studies have incorporated fluid flow in their model, which greatly influences the VMD process's transport mechanisms. Against this backdrop, we developed a comprehensive 2D CFD model using COMSOL™ Multiphysics to describe and simulate heat transfer, mass transfer and fluid flow in the flat sheet vacuum membrane distillation (VMD) under laminar flow conditions. We also studied the effect of varied Reynolds number, inlet feed temperature and degree of vacuum temperature profiles on different parameters (mass flux, TPC, CPC, heat transfer coefficient) in VMD.

A comprehensive 2D mathematical model was developed for estimating the mass flux of saline water and other critical performance parameters such as heat transfer rates, mass flux, temperature polarization coefficient and concentration polarization coefficient through a flat-sheet membrane for desalination via the VMD process. The computational domain and the discretized mesh are shown in Figure 1. The computational domain is divided into three sections: feed, membrane, and permeate. A vacuum was used to capture water vapour in the permeate channel from the hot feedwater flowing in the feed channel. The feed and permeate channels were both 150 × 3 mm in size. A 120 μm thick and 150 mm long porous membrane was sandwiched between the feed and permeate channels. The model was solved using the commercial COMSOL™ Multiphysics software package. The computational domain was discretized using a structured quadratic mesh refined at the top and bottom boundary of each zone to capture the kinetic and thermal boundary layers. The Galerkin finite element method with linear shape functions with stabilization methods based on streamline diffusion and crosswind diffusion was used (Griffiths & Lorenz 1978). To allow for proper resolution of the mass transfer boundary layer, computational meshes consisting of orthogonal elements were created, with local mesh refinement around sharp corners where edges were in direct contact with membrane walls and the proximity of the membrane walls. During a mesh sensitivity test, the number of mesh elements was increased until the difference in average water flux between two consecutive meshes was less than 0.15%. The computational domain's discretized mesh comprises 2,80,500 orthogonal computational cells with perfect orthogonal quality of one and a maximum aspect ratio of 5. The governing equations and boundary conditions are discussed in more detail in the subsequent subsections. The assumptions that were used to construct the CFD model are listed below:

  • (a)

    VMD models were run in a steady-state environment;

  • (b)

    The outer wall of the module was assumed to be insulated. Thus there was no heat loss into the atmosphere;

  • (c)

    In the feed and permeate channels, the flow was laminar;

  • (d)

    In the membrane pore, there was no pore wetting. From the feed to the permeate channel, only vapour molecules were transported.

Figure 1

(a) Schematic and (b) Discretized mesh of computational domain in VMD model.

Figure 1

(a) Schematic and (b) Discretized mesh of computational domain in VMD model.

Close modal

Governing equations

The overall work of VMD can be described by evaporation on the membrane surface in the feed channel, vapour crossing the membrane to the permeate channel and condensation in permeate channel. Laminar flow is considered for the fluid flow in the feed and permeate channels. The continuity and Navier-Stokes equations can be stated as follows for laminar flow in the feed and permeate channels (Ansari et al. 2021):
formula
(1)
formula
(2)
where is viscosity of the fluid, p is pressure, is density of the fluid, uvelocity of the fluid. Heat transfer occurs from the hot channel to the cold channel during the fluid flow. Heat transfer can be calculated using the energy balance equation, which includes conduction and convection, and can be represented as follows in the feed and permeate channels (Aliabadi et al. 2021):
formula
(3)
where T is temperature k is thermal conductivity of the fluid, is heat capacity of the fluid. The vapour partial pressure difference across the membrane is a driving force for mass transfer in the VMD process. The following describes the diffusive water vapour flux (J) (Yadav et al. 2022c):
formula
(4)
where of water vapour, is diffusivity of water vapour, is of water vapour.
In the MD process, molecular diffusion, Knudsen diffusion, and Poiseuille diffusion are the most common modes of water vapour transfer over the membrane (Shirazi et al. 2016). The collisions of water vapour molecules with each other give a barrier to mass transfer in molecular diffusion, whereas the collisions of water vapour molecules with the membrane pore walls offer resistance to mass transfer in Knudsen diffusion. As a result, Knudsen diffusion is critical in low pressure and/or narrow pores. In Poiseuille flow, mass transport resistance is provided by the membrane matrix (Mohammadi & Akbarabadi 2005; Xu et al. 2006). The molecular diffusion in the VMD process is minimal due to the low partial pressure of the air inside the pores. When the pore size is larger than the molecule size, the Poiseuille flow occurs. When the input temperature was adjusted from 313 to 373 K, the Knudsen number was found to be more than 1. As a result, molecule-to-wall collisions are the most common. As a result, employing the Knudsen and Poiseuille flow diffusion approaches yields satisfactory outcomes (Schofield et al. 1987):
formula
(5)
where and represents the diffusion coefficient for Poiseuille and Knudsen flow, respectively. The following equation can be used to compute the Poiseuille diffusion coefficient (Schofield et al. 1987):
formula
(5rma)
where is membrane surface temperature on the feed side, is universal gas constant (8.315 J mol−1 K−1), η is water vapour viscosity, is pore radius, is membrane porosity, is thickness of the membrane, is inside average pressure inside the membrane's pores, is water vapour diffusivity and is tortuosity. The following equation can be used to compute the Knudsen diffusion coefficient (Li & Sirkar 2017):
formula
(5rmb)
where is molecular weight of water. On the feed side of the membrane, the partial pressure of water vapour can be determined by (Anqi et al. 2020):
formula
(6)
where E, F, G are constants, and their values are 23.1964, 3,815.44 and 46.18, respectively, is water activity coefficient and is salt mole fraction. The Lawson equation (Lawson & Lloyd 1997; Khayet 2011) can compute the water activity coefficient (b):
formula
(6rma)
Convection heat transfer takes place in the feed channel of the VMD model. The given equation can be used to calculate convective heat transfer (Abu-Zeid et al. 2015):
formula
(7)
where is feed channel's convective heat transfer, is bulk feed temperature, is feed channel's heat transfer coefficient. In the VMD process, a vacuum is applied to the permeate side of the membrane, which allows water vapour molecules to pass through the pores. Heat is transferred through both conduction and convection in this area. The following equation can calculate the total heat transfer over the membrane (Aliabadi et al. 2021).
formula
(8)
where membrane surface temperature, is membrane surface's mean temperature, is membrane's heat transfer coefficient, and is total heat transfer through the membrane. In the permeate channel, water vapour molecules condense and turn into liquid. The following equation can be used to compute the amount of heat released as a result of vapour condensation (Abu-Zeid et al. 2015):
formula
(9)
where is bulk permeate temperature, convective heat transfer, heat transfer coefficient.

Calculation of physical and transport properties

The following equations were used to compute the membrane's thermophysical parameters, including density (ρm), thermal conductivity (km), and specific heat (Cp,m) (Yadav et al. 2017; Patel et al. 2020):
formula
(10rma)
formula
(10rmb)
formula
(10rmc)
where kg is thermal conductivity of air, ρg is density of air, Cp,g heat capacity, ks is thermal conductivity of membrane material Cp,s heat capacity, ε is porosity of membrane, ρs is density of membrane material. The thermo-physical material properties are given in Table 1.
Table 1

Thermo-physical properties of materials

MaterialsConductivity (W m−1 K−1)Viscosity (kg m−1 s−1)Specific Heat (J Kg−1 K−1)Density (kg m−3)
PTFE 0.30 – 1000 2170 
Vapour 0.0261 – 2014 0.554 
Pure water 0.613 1.003 × 10−3 4182 995 
Seawater 0.642 4.14 × 10−4 4064 1013 
MaterialsConductivity (W m−1 K−1)Viscosity (kg m−1 s−1)Specific Heat (J Kg−1 K−1)Density (kg m−3)
PTFE 0.30 – 1000 2170 
Vapour 0.0261 – 2014 0.554 
Pure water 0.613 1.003 × 10−3 4182 995 
Seawater 0.642 4.14 × 10−4 4064 1013 

The boundary conditions for the computational domain are shown in Table 2. It may be noted that simultaneous heat and mass transfer occurs during the transfer of water vapour through the membrane.

Table 2

Boundary conditions used in the CFD model

PositionFeed inletFeed outletMembrane surface
Permeate outlet
TopBottom
Fluid flow in feed channel   No slip condition – – 
Heat transfer in feed channel  outflow – – – 
Mass transfer – –   – 
Fluid flow in permeate channel – – – –  
Heat transfer in permeate channel – – – – Outflow 
PositionFeed inletFeed outletMembrane surface
Permeate outlet
TopBottom
Fluid flow in feed channel   No slip condition – – 
Heat transfer in feed channel  outflow – – – 
Mass transfer – –   – 
Fluid flow in permeate channel – – – –  
Heat transfer in permeate channel – – – – Outflow 

Rein = Reynolds number at the feed inlet of (50, 100,150 and 200).

Tin = Temperatures at the inlet of feed (313, 323, 333 and 343 K).

Cfeed = concentration of solute on the membrane at feed side (4%).

Pv = Vacuum pressure at permeate side (640, 680, 720, 750 mm Hg).

Temperature and concentration polarization coefficient

Measuring temperature polarization coefficient (TPC) is a common method for determining the magnitude of thermal boundary layer resistance proportion to overall heat transfer resistance. TPC is the ratio of heat transfer resistance through the membrane to bulk heat transfer resistance and is represented by (Xu et al. 2021; Yadav et al. 2022c):
formula
(11)
and are very close to each other for the VMD process. Hence, we used the modified Bandit equation (Bandini et al. 1992) to calculate TPC:
formula
(12)
where is saturation temperature of water vapour at the given pressure.
The concentration polarization coefficient (CPC) indicates the degree of driving force lost compared to the total force between feed and permeate. Mathematically, CPC can be defined as the ratio of the difference between the solute concentration near the membrane wall and bulk feed concentration to the bulk feed concentration at the inlet (Olatunji & Camacho 2018):
formula
(13)
where is concentration in bulk feed and is concentration at the membrane surface on the feed side.

Experimental test rig

The developed CFD model was validated with in-house experiments. A lab-scale VMD apparatus was used in this investigation (Yadav et al. 2021a, 2021d). The experimental setup and its schematic used in the present study are presented in Figure 2. The test kit consisted of 5 L jacketed storage vessels for feed solution. For heating the feed solution, hot water was circulated on the outer surface of the feed storage vessel. SS316 material was used to fabricate the reservoir (10 L capacity). The external heater (2 kW) consisted of K type temperature sensor fitted with an electrical control panel. The feed was prepared and fed to the membrane module through a feed tank and a feed pump (Kempflow). The feed solution flow rate was assessed by flow meters, and pressure was assessed by pressure gauges at the membrane inlet. The vacuum pump (Tarson Rockyvac) was connected to the permeate side of the membrane module. Polytetrafluoroethylene (PTFE) membrane (effective area of 0.10 × 0.15 m) was clamped in the VMD cell (Figure 2(b)). Parametric studies were performed by varying feed temperature (313–343 K), Reynolds number (50–200) and degree of vacuum (640–750 mm Hg). On the permeate side, the permeate water vapour was condensed and measured. To obtain mass flux following equation was used:
formula
(14)
Figure 2

(a) Actual setup, (b) MD cell (c) and schematic diagram of VMD system.

Figure 2

(a) Actual setup, (b) MD cell (c) and schematic diagram of VMD system.

Close modal

Effect of reynolds number

Figure 3 illustrates the thermal field in the computational domain at different Re (50, 100, 150 and 200). The thermal boundary layer was thinner at higher Re values since the temperature at the membrane surface on the feed side is higher at lower Re and vice versa (Ali et al. 2013; Ansari et al. 2021). The decrease in the thermal boundary layer was attributed to increased mixing on the feed side due to an increase in Re, hence the decrease in heat transfer resistance.

Figure 3

Thermal field in the computational domain at different Reynolds numbers (a) 50, (b) 100, (c) 150, and (d) 200 (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 3

Thermal field in the computational domain at different Reynolds numbers (a) 50, (b) 100, (c) 150, and (d) 200 (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 4 illustrates the mass flux at different Reynolds numbers (50, 100, 150, 200) (inlet feed temperature – 333 K and degree of vacuum – 750 mm Hg). The mass flux increased with increasing Reynolds number. This increment in the mass flux was due to a decrease in the mass transfer resistance on the feed side (Alanezi et al. 2021). The mass flux increased by 13.15% when the Reynolds number was increased from 50 to 200. A similar trend in mass flux with Re was also reported by others (Li et al. 2003; Mengual et al. 2004). The experimental mass flux was in good agreement with the theoretically obtained mass flux.

Figure 4

Variation of mass flux with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 4

Variation of mass flux with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 5 illustrates the heat transfer coefficient at different Reynolds numbers (50, 100, 150, and 200) (inlet feed temperature - 333 K and degree of vacuum – 750 mm Hg). The heat-transfer coefficient showed an increasing trend with the Reynolds number. The increase in heat transfer coefficient was due to increased convective heat transfer because the fluid mixing on the feed side increases (heat transfer from bulk feed to membrane surface increased) with the Reynolds number (Alanezi et al. 2020). The heat-transfer coefficient increased by more than five times when the Reynolds number was increased from 50 to 200.

Figure 5

Variation of heat transfer coefficient with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 5

Variation of heat transfer coefficient with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 6 illustrates the TPC at different Reynolds numbers (50, 100, 150, and 200) (inlet feed temperature – 333 K and degree of vacuum – 750 mm Hg). TPC increased from 0.985 to 0.997 when the Reynolds number was increased from 50 to 200. The high TPC indicated the high-temperature gradient across the membrane surfaces, increasing the vapour transport across the membrane. The increased feed circulation velocity, i.e. Reynolds numbers, reduced the temperature and concentration polarization effects (El-Bourawi et al. 2006; Bahmanyar et al. 2012). This trend was such because increasing Reynolds number led to a high heat transfer. Hence, Tmf and Tmp became closer to the temperatures of feed and permeate bulk solutions, respectively. This caused a higher temperature difference and increased TPC (Alanezi et al. 2016).

Figure 6

Variation of TPC with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mmHg, feed concentration – 4%).

Figure 6

Variation of TPC with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mmHg, feed concentration – 4%).

Close modal

Figure 7 illustrates the flow field at different Reynolds numbers (50, 100, 150, and 200) (inlet feed temperature – 333 K and degree of vacuum – 750 mm Hg). The velocity vector in the computational domain increased with the Re. The increment in Re was achieved by increasing the feed velocity, which was the reason for the increased velocity vector in the computational domain.

Figure 7

Flow field in the computational domain at different Reynolds number (a) 50, (b)100 (c) 150, and (d) 200 (inlet feed temperature –333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 7

Flow field in the computational domain at different Reynolds number (a) 50, (b)100 (c) 150, and (d) 200 (inlet feed temperature –333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 8 illustrates concentration fields at different Reynolds numbers (50, 100, 150, 200) (inlet feed temperature – 333 K and degree of vacuum – 750 mm Hg). The concentration differential between the bulk flow and the membrane-fluid surface can lead to CP. The concentration differential formed because the salt particles accumulated on the membrane surface, creating a higher concentration layer on the membrane surface than bulk feed. This accumulation hinders the transmembrane mass transfer resulting in a decline in mass flux (Yun et al. 2006; Olatunji & Camacho 2018). The accumulation of salts on the membrane surface on the feed side was higher at lower Re (Figure 8(a)) than other Re (Figure 8(b)–8(d)). The increase in Re caused the mixing in the feed, which decreased the concertation polarization (taking accumulated salts from the membrane surface). Therefore, the magnitude of the concentration field decreased with Re.

Figure 8

Concentration field in the feed channel at different Reynolds number (a) 50, (b) 100 (c) 150, and (d) 200 (inlet feed temperature –333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 8

Concentration field in the feed channel at different Reynolds number (a) 50, (b) 100 (c) 150, and (d) 200 (inlet feed temperature –333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 9 illustrates the CPC at different Reynolds numbers (50, 100, 150, and 200) (inlet feed temperature – 333 K and degree of vacuum – 750 mm Hg). The CPC decreased from 1.36 to 0.60 (56.63%), increasing Reynolds number from 50 to 200. The thickness of the concentration boundary layer was lowered by increased shear stress on the membrane surface (Wu et al. 2015). A decrease in CPC with Re confirmed that increasing Re could greatly intensify the heat and mass transfer process while diminishing the boundary layer's temperature and concentration polarization effect. At lower Re CPC, more than 1 indicates that the degree of scale formation is high.

Figure 9

Variation of CPC with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 9

Variation of CPC with Reynolds number (inlet feed temperature – 333 K, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Effect of feed temperature

Figure 10 illustrates the mass flux at different feed inlet temperatures (313, 323, 333, 343 K) (Reynolds number – 150 and degree of vacuum – 750 mm Hg). The mass flux in the VMD process increased exponentially with increasing feed inlet temperature. The mass flux increased by 54.8% when the temperature was increased from 313 to 323 K, and when increased from 323 to 343 K the flux increased by 132.9%. The reason was that saturated vapour pressure on the hot side increased with the increase in the feed temperature, and the mass transfer was enhanced (Yadav et al. 2022c). Moreover, the increment in mass flux was more rapid on increasing the inlet feed temperature than the Reynolds number. Other researchers reported similar findings (Mericq et al. 2009; Qusay et al. 2017; Yadav et al. 2022b).

Figure 10

Variation of mass flux with inlet feed temperature (Reynolds number –150, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 10

Variation of mass flux with inlet feed temperature (Reynolds number –150, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 11 illustrates the mass flux at different feed inlet temperatures (313, 323, 333, and 343 K) (Reynolds number – 150 and degree of vacuum – 750 mm Hg). The heat transfer coefficient increased rapidly with increasing feed inlet temperature. In the VMD process, conductive heat transfer is negligible compared to convective heat transfer (Zhang et al. 2016). As established earlier, when the inlet feeds temperature increases, the mass flux increases, enhancing the convective heat transfer on the membrane surface.

Figure 11

Variation of heat transfer coefficient with inlet feed temperature (Reynolds number – 150, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 11

Variation of heat transfer coefficient with inlet feed temperature (Reynolds number – 150, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Figure 12 illustrates the TPC at different feed inlet temperatures (313, 323, 333, and 343 K) (Reynolds number – 150 and degree of vacuum – 750 mm Hg). TPC decreased from 0.997 to 0.966 with the increase in Tf from 313 to 343 K. This could be due to the high energy consumption associated with vaporization at higher temperatures. When Tf increased, the flux increased exponentially. This higher mass flux leads to a higher heat flux in the liquid phase; hence the temperature across the membrane is decreased, resulting in the TPC being reduced by increasing the inlet feed temperature (Martínez-Díez et al. 1998).

Figure 12

Variation of TPC with inlet feed temperature (Reynolds number – 150, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Figure 12

Variation of TPC with inlet feed temperature (Reynolds number – 150, degree of vacuum – 750 mm Hg, feed concentration – 4%).

Close modal

Effect of degree of vacuum

Figure 13 illustrates the thermal field in the computational domain at different degree of vacuum (640, 680, 720, and 750 mm Hg) (Reynolds number – 150 and feed inlet temperature – 333 K). As the degree of vacuum increases, there is an increase in transmembrane pressure, which results in increased mass transfer driving force across the membrane. Therefore, the flux increased; consequently, the temperature of the permeate side increased because of the thermal energy released during the condensation of the water vapour passed through the membrane pores. The feed side membrane's surface temperature was less than the feed channel. This was because of the evaporation of the water molecules. As the evaporation occurs, the water molecules take the energy (the latent heat of vaporization), resulting in decreases in membranes surface temperature and nearby region (Baghel et al. 2020).

Figure 13

Thermal field in the computational domain at different degree of vacuum (Reynolds number – 150, temperature – 333 K, feed concentration – 4%).

Figure 13

Thermal field in the computational domain at different degree of vacuum (Reynolds number – 150, temperature – 333 K, feed concentration – 4%).

Close modal

The difference in vapour pressure across the membrane is the primary driving force for VMD. Figure 14 illustrates the mass flux at different degrees of vacuum (640, 680, 720, and 750 mm Hg) (Reynolds number – 150 and feed inlet temperature – 333 K). The mass flux showed an increasing trend with an increase in the degree of vacuum. The mass flux varied from 25.99 kg m−2 h−1 (degree of vacuum 640 mm Hg) to 44.84 kg m−2 h−1 (degree of vacuum 750 mm Hg). This increase in mass flux was due to decreased mass transfer resistance across the membrane pore (Rauter et al. 2021). Other researchers have also reported a positive increment in the mass flux when vacuum pressure was increased (Banat et al. 2003; Mericq et al. 2009).

Figure 14

Variation of mass flux with inlet degree of vacuum (Reynolds number – 150, temperature – 343 K, feed concentration – 4%).

Figure 14

Variation of mass flux with inlet degree of vacuum (Reynolds number – 150, temperature – 343 K, feed concentration – 4%).

Close modal

Figure 15 illustrates the heat transfer coefficient at different degrees of vacuum (640, 680, 720, 750 mm Hg) (Reynolds number – 150 and feed inlet temperature – 333 K). The heat transfer coefficient increased with the degree of vacuum in permeate channel. This was attributed to higher convective heat transfer resulting in increased mass flux. The increase in the vacuum pressure increased the driving force for the vapour across the membrane pores, hence the heat transfer (Qi et al. 2020).

Figure 15

Variation of heat transfer coefficient with degree of vacuum (Reynolds number – 150, temperature – 333 K, feed concentration – 4%).

Figure 15

Variation of heat transfer coefficient with degree of vacuum (Reynolds number – 150, temperature – 333 K, feed concentration – 4%).

Close modal

Figure 16 illustrates the TPC at different degree of vacuum (640, 680, 720, and 750 mm Hg) (Reynolds number – 150 and feed inlet temperature – 333 K). The TPC decreased with the degree of vacuum in the permeate channel. The TPC varied from 0.983 (degree of vacuum 750 mm Hg) to 0.9 W m−2 k−1 (degree of vacuum 750 mm Hg). Due to higher mass flux, more amount of heat (latent) was consumed (evaporation) at the membrane surface on the feed side, whereas released (condensation) at the membrane surface on the permeate side. This causes the temperature difference across the membrane to decrease, leading to a decrement in TPC.

Figure 16

Variation of TPC with the degree of vacuum (Reynolds number – 150, temperature – 333 K, feed concentration – 4%).

Figure 16

Variation of TPC with the degree of vacuum (Reynolds number – 150, temperature – 333 K, feed concentration – 4%).

Close modal

This study was limited to 2D analysis and involved some assumptions. The same model can be extended to 3D and varied environments to improve the accuracy of the model. From this research, we hope to understand the heat and mass transfer phenomena in the VMD process, which will guide future work on the design of flat sheet membrane modules, module scale-up, and process optimization to expedite VMD commercialization.

This study developed a comprehensive two-dimensional computational fluid dynamics (CFD) model using COMSOL Multiphysics software to simulate heat transfer, mass transfer, and fluid flow in the flat sheet VMD under laminar flow conditions. The effect of Reynolds number, feed temperature, and vacuum degree was studied on VMD performance parameters such as mass flux, heat transfer coefficient, concentration polarization and temperature polarization. The mass flux increased with increasing Reynolds number. This was because a smaller temperature and concentration boundary layer resulted in lower resistance to water vapour molecules through the membranes. The heat-transfer coefficient showed an increasing trend with the Reynolds number. This was because the convective heat transfer increases when the Reynolds number increases. The high TPC indicated the high temperature gradient across the membrane surfaces, hence increasing the vapour transport across the membrane. CPC reduced as the Reynolds number increased because the thickness of the concentration boundary layer was reduced due to higher shear stress acting on the membrane surface. The mass flux in the VMD process increased exponentially with increasing feed inlet temperature. The heat transfer coefficient increased rapidly with increasing feed inlet temperature. This could be due to the high energy consumption of vaporization energy at higher temperatures. The mass flux showed an increasing trend with an increase in the degree of vacuum. This increase in mass flux was due to decreased mass transfer resistance across the membrane pore. From this research, we hope to understand the heat and mass transfer phenomena in the VMD process, which will guide future work on the design of flat sheet membrane modules, module scale-up, and process optimization expedite VMD commercialization.

The CSIR-CSMCRI PRIS number for this manuscript is 051/2022. The authors are grateful for partial funding support from the Council of Scientific and Industrial Research, India (MLP-0065). The authors acknowledge AED&CIF, CSMCRI for providing instrumental facilities. The comments from anonymous reviewers and the editor have greatly improved the content.

We certify that the authors are not affiliated with or involved with any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this paper.

All relevant data are included in the paper or its Supplementary Information.

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