# Simulation and Experimental Investigation of the Vacuum-Enhanced Direct Membrane Distillation Driven by a Low-Grade Heat Source

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical and CFD Model

#### 2.1. Theoretical Model

#### 2.1.1. Mass Transfer

_{f,W}(Pa) and p

_{p,W}(Pa) are the partial pressures of water vapor on the feed side and permeate side, respectively. K

_{M}is the mass transfer coefficient. The subscripts “f” and “p” indicate parameters at the feed and permeate side, respectively. The subscripts “W” and “M” indicate the parameters in the flow channel and on the membrane surface, respectively. p

_{f,W}(Pa) can be calculated according to Equation (2), where γ

_{w}is the activity coefficient of water as shown in Equation (3) [28]. The subscripts “w” indicate the parameter of water. X

_{NaCl}is the molar fraction of sodium chloride in feed solution as calculated by Equation (4). The saturation vapor pressure of pure water ${p}_{v}^{S}$ (T) (Pa) at different temperatures T (K) can be obtained by the Antoine Equation (5).

_{n}can be calculated by Equation (6).

_{B}is the Boltzmann constant (1.381 × 10

^{−23}J·mol

^{−1}·K

^{−1}) and T

_{m}(K) is the average temperature of both sides of the membrane. The subscript “a” indicates the parameter of air. β

_{a}(2.641 × 10

^{−4}μm) and β

_{w}(3.711 × 10

^{−4}μm) are the collision diameters of air and water molecules, respectively. p (Pa) is the absolute pressure, and M

_{w}and M

_{a}are the molecular weight of water and air, respectively. According to the Knudsen number, the transmembrane mass transfer caused by diffusion can be classified into four patterns.

_{n}> 1, the collision between molecules becomes the main form of diffusion, and the mass transfer coefficient is calculated by Equations (8) and (9) [30]:

_{h}is the surface porosity, δ (mm) is the thickness of the membrane, p

_{t}is the gas pressure inside the membrane pore space, D

_{wa}(m

^{2}·s

^{−1}) is the diffusion coefficient of water in air, τ is the membrane tortuosity factor, and R (J·mol

^{−1}·k

^{−1}) is the gas constant.

_{n}< 0.01, the pore size will primarily affect the transmembrane motion of molecules. The collision between molecules and the wall of the membrane pore becomes the major obstruction. In this case, the mass transfer coefficient is calculated by Equation (10) [30].

_{n}< 1, molecular diffusion and Knudsen diffusion both become the primary forms of molecular mass transfer across the membrane. In this case, using the Knudsen-molecular diffusion model, the mass transfer coefficient can be expressed as Equation (11).

_{PO}(kg·m

^{−2}·s

^{−1}·pa

^{−1}) can be expressed as Equation (12) [30].

_{0}(pa·s) and Su (K) are the kinetic viscosity and Susland constant of the gas at 0 °C, respectively.

_{f}(Pa) and p

_{p}(Pa) are the total pressure of the feed channel and permeate channel. K

_{D}is the mass transfer coefficient caused by the diffusion, and can be calculated by Equations (8)–(11) according to the K

_{n}number.

#### 2.1.2. Heat Transfer

- (1)
- Heat is transferred from the main body of feed seawater flow to the membrane surface on the feed side.
- (2)
- A part of the heat is taken away from the feed side and passes through the membrane by heat conduction and vaporization.
- (3)
- The vapor condenses on the permeate side, together with the conducted heat of the membrane, raising the temperature on the permeate side.

_{M}(w·m

^{−2}) can be calculated by Equation (15) [28].

_{H}(w·m

^{−2}) is the latent heat of vaporization through the membrane and q

_{C}(w·m

^{−2}) is the conducted heat through the membrane, considered as heat loss. The evaporation enthalpy of water ΔH

_{V}(kJ·kg

^{−1}) can be calculated by Equation (16). Since the membrane is porous, there is a gap in the membrane pores, and the thermal conductivity of the membrane k

_{m}(J·m

^{−1}·k

^{−1}) can be calculated by Equation (17).

_{a}(J·m

^{−1}·k

^{−1}) is the thermal conductivity of air, and k

_{s}(J·m

^{−1}·k

^{−1}) is the thermal conductivity of the membrane material.

#### 2.2. CFD Model for Permeate Flux Prediction

#### 2.2.1. Governing Equations

_{m}(kg·m

^{−3}·s

^{−1}) is given by Equation (25), and the heat transfer source S

_{h}(kW·m

^{−3}) is given by Equation (26). These source terms only exist in the flow at the first grid layer away from the membrane surface. In other flow areas, they are equal to 0. In the feed flow, the source term S

_{m}is negative while in the permeate flow it is positive.

_{eff}(W·m

^{−1}·k

^{−1}) is the thermal conductivity of fluid. h

_{j}is the enthalpy of the component j. ${\stackrel{=}{\tau}}_{eff}$ (kg·m

^{−1}·s

^{−1}) is the stress tensor related to the viscous force. $\rho \overrightarrow{g}$ and $\overrightarrow{F}$ are the gravitational and external body forces, respectively. G

_{k}and G

_{b}represent the generation of turbulence kinetic energy due to the mean velocity gradients and buoyancy, respectively. Y

_{M}is the contribution of the fluctuating dilatation in the compressible turbulence to the overall dissipation rate. C

_{1ε}and C

_{2}are constants. σ

_{k}and σ

_{ε}are the turbulent Prandtl numbers for k and ε, respectively. S

_{k}and S

_{ε}are user defined source terms.

#### 2.2.2. Simplified Geometrical Model

#### 2.2.3. Boundary Conditions

_{p,out}= p

_{f,out}= 101,325 Pa,). As turbulence is involved, turbulence intensity and hydraulic diameter need to be calculated at the velocity inlet, and reasonable parameter values need to be set to ensure the accuracy of the simulation results. The membrane surface of both feed and permeate side are non-slip walls(u

_{g}= 0). The transmembrane mass and heat then spread to other fluid domain with the flow. The upper and lower wall of the membrane module are assumed to be adiabatic.

## 3. Experimental Setup

^{2}. The module is made of PMMA (polymethyl methacrylate) so that the fluid flow in the channels and the membrane state can be observed.

## 4. Results and Discussion

#### 4.1. Experimental Verification of the Proposed Transmembrane Transfer Model

#### 4.2. Flow Field along the Membrane Surface

_{f}equal to 318 K as an example).

## 5. Conclusions

- (1)
- Compared with the model introducing a semi-empirical coefficient, all the parameters of the model proposed in this paper are independent of the operating condition, and thus the model is easier for use and has better adaptability to the fluctuating operating conditions.
- (2)
- The simulation results based on the proposed model have good agreement with the experimental data. In the VEDCMD desalination, the permeate flux is significantly enhanced by decreasing the permeate side pressure. When the absolute pressure of the permeate side reaches 30 kPa, the permeate flux can be improved by nearly 200%.
- (3)
- The flow fields of the flow along each side of the membrane are revealed under different pressure of the permeate side. The permeate flux rising caused by the vacuum enhancement leads to increasing temperature rise/drop and salinity, while it can hardly influence the velocity distribution.
- (4)
- The CFD simulation results are helpful for guiding the VEDCMD operation and module structure improvement. Increasing the temperature of the feed flow and decreasing the pressure of the permeate flow can contribute to larger permeate fluxes. The length of the module should be controlled to avoid excessive heat conduction between the two sides of the membrane. In addition, the turbulence of the flow near the membrane should be enhanced for better mass transfer and smaller concentration polarization, by inserting or mounting some small obstacles near the membrane surface.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | |

J | Permeation flux (kg·m^{−2}·s^{−1}) |

K | Membrane distillation coefficient (kg·m^{−2}·s^{−1}·pa^{−1}) |

p | Pressure (Pa) |

X_{NaCl} | Molar fraction of solute |

T | Temperature (K) |

γ | Activity coefficient |

W | Mass fraction |

M | Molar mass (kg·mol^{−1}) |

K_{n} | Knudsen number |

λ | Average free path of water vapor (μm) |

d | Diameter of membrane pore (μm) |

K_{B} | Boltzmann constant (J·mol^{−1}·K^{−1}) |

β | Collision diameters(μm) |

D_{Wa} | Diffusion coefficient of water vapor (m^{2}·s^{−1}) |

p_{t} | Gas pressure in the pore of the membrane |

ε_{h} | Surface porosity |

τ | Membrane tortuosity factor |

δ | Membrane thickness (mm) |

R | Gas constant (J·mol^{−1}·k^{−1}) |

r | Membrane pore radius (μm) |

μ | Viscosity (pa·s) |

q | Heat flux (w·m^{−2}) |

ΔH_{V} | Enthalpy of evaporation (kJ·kg^{−1}) |

κ | Thermal conductivity (J·m^{−1}·k^{−1}) |

ρ | Density (kg·m^{−3}) |

S | Source term |

b | Thickness of the first grid layer (mm) |

$\stackrel{=}{\tau}$ | Stress tensor (kg·m^{−1}·s^{−1}) |

$\rho \overrightarrow{g}$ | Gravitational body force |

$\overrightarrow{F}$ | External body force |

k | Turbulent kinetic energy |

σ | Turbulent Prandtl number |

G_{b} | Turbulence kinetic energy (buoyancy) |

G_{k} | Turbulence kinetic energy (the mean velocity gradients) |

Y_{M} | Contribution of the fluctuating dilatation |

C_{1} | constant |

C_{2} | constant |

C_{1e} | constant |

C_{3e} | constant |

Superscript | |

S | Saturation properties |

Subscripts | |

m | Membrane |

f | Feed side |

p | Permeate side |

W | Flow channel |

v | Vapor |

w | Water |

MD | Molecular diffusion |

K | Knudsen diffusion |

K-MD | Knudsen-molecular diffusion |

PO | Poiseuille flow |

g | Gas |

D | DCMD |

H | Latent heat |

C | Heat conduction |

a | Air |

s | Solid |

eff | Effective |

j | Component |

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**Figure 3.**Schematic diagram of VEDCMD experiment. 1—Heater, 2—Seawater tank, 3—Micro-circulation pump, 4—Control valve, 5—Temperature sensor (seawater), 6—Liquid flow meter, 7—DCMD module, 8—Temperature sensor (freshwater), 9—Freshwater pressure stabilization tank, 10—Vacuum gauge, 11—Vacuum pump.

**Figure 5.**Comparison of simulating and experimental values of permeate flux of VEDCMD. V

_{f}= 0.56 L (min), p

_{f}= 101,325 Pa, V

_{p}= 0.29 L (min), T

_{p}= 303 K, and W

_{NaCl}= 3.5%.

**Figure 7.**Temperature distribution of the flow along the permeate side of the membrane surface. T

_{f}= 318 K.

**Figure 8.**Velocity distribution of the flow along the feed side of the membrane surface. T

_{f}= 318 K.

**Figure 9.**Velocity distribution of flow along the permeate side of the membrane surface. T

_{f}= 318 K.

**Figure 10.**Salinity distribution of flow along the feed side of the membrane surface. T

_{f}= 318 K.

VEDCMD Module Property | Value |
---|---|

Membrane material | PVDF |

Porosity | 0.75 |

Membrane nominal pore size | 0.22 μm |

Membrane thickness | 0.12 mm |

Length L | 125 mm |

Hot channel height H | 12 mm |

Cold channel height H | 12 mm |

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## Share and Cite

**MDPI and ACS Style**

Ma, Q.; Tong, L.; Wang, C.; Cao, G.; Lu, H.; Li, J.; Liu, X.; Feng, X.; Wu, Z.
Simulation and Experimental Investigation of the Vacuum-Enhanced Direct Membrane Distillation Driven by a Low-Grade Heat Source. *Membranes* **2022**, *12*, 842.
https://doi.org/10.3390/membranes12090842

**AMA Style**

Ma Q, Tong L, Wang C, Cao G, Lu H, Li J, Liu X, Feng X, Wu Z.
Simulation and Experimental Investigation of the Vacuum-Enhanced Direct Membrane Distillation Driven by a Low-Grade Heat Source. *Membranes*. 2022; 12(9):842.
https://doi.org/10.3390/membranes12090842

**Chicago/Turabian Style**

Ma, Qingfen, Liang Tong, Chengpeng Wang, Guangfu Cao, Hui Lu, Jingru Li, Xuejin Liu, Xin Feng, and Zhongye Wu.
2022. "Simulation and Experimental Investigation of the Vacuum-Enhanced Direct Membrane Distillation Driven by a Low-Grade Heat Source" *Membranes* 12, no. 9: 842.
https://doi.org/10.3390/membranes12090842