# Exergy Analysis of a Direct Contact Membrane Distillation (DCMD) System Based on Computational Fluid Dynamics (CFD)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

- Both the feed and permeate have a laminar flow regime and are in a steady state.
- Heat loss to the ambient environment is negligible.
- No chemical reaction occurs.
- Convective transport of water vapor via the membrane pore is negligible.

#### 2.1. Computational Fluid Dynamics Model Equation

^{3}), u is the velocity vector (m/s), $p$ is the dynamic pressure (Pa), $\mathsf{\mu}$ is the viscosity (N·s/m

^{2}), and T is the inlet temperature (K) in both channels. I, F, and $\nabla $ represent the unit tensor, the volume force vector, and the dell operator, respectively. The density of both channels was calculated according to Equation (3), and the viscosity of both channels was calculated using Equation (4).

^{2}), Q is the heat source (W/m

^{3}), and ${k}_{m}$ is the thermal conductivity of the membrane (W/[m·K]), respectively. $\mathsf{\epsilon}$ is the membrane’s porosity. ${k}_{m}$ was calculated using Equation (7). ${k}_{g}$ and ${k}_{s}$ refer to the thermal conductivity coefficients of the vapor within the membrane’s pores and solids, respectively, as in Equation (8), and ${T}_{m}$ is the mean temperature.

^{3}), ${D}_{k}$ is the Knudsen diffusion coefficient (m

^{2}/s), ${D}_{p}$ is the Poiseuille flow coefficient (m

^{2}/s), and $\mathsf{\epsilon}$ and $\mathsf{\tau}$ are the porosity and tortuosity of the membrane, respectively. ${D}_{k}$ $\mathrm{and}{D}_{p}$ are calculated using Equations (11) and (12), respectively. ${d}_{p}$ is the membrane pore size, and M is the water (vapor) molecular weight (g/mol). In the Poiseuille flow equation, P is the mean pressure (Pa), and $\mathsf{\mu}$ is the water (vapor) viscosity (N·s/m

^{2}).

#### 2.2. Temperature Polarization Coefficient

#### 2.3. Analysis of Exergy Destruction

_{1}and T

_{2}, respectively, and the resultant irreversibility can be calculated according to [41], as follows:

_{T}is the total heat transferred by the membrane and T

_{0}is the temperature of the environment. Equation (17) above indicates that the exergy destruction is related to the temperature difference. On the feed side, the TP phenomenon causes the temperature on the membrane to be lower than that in the bulk phase. A further reduction in temperature occurs in the membrane because of its heat transfer resistance. On the permeate side, the temperature on the membrane is lower than that in the bulk phase owing to both TP and cooling phenomena. Therefore, the magnitudes of exergy destruction in the feed, membrane, and permeate can be assessed using Equations (18)–(20), respectively, as follows:

## 3. Materials and Methods

^{2}, its pore size was 0.22 μm, its porosity was 0.75%, and its tortuosity was 2.

^{®}Multiphysics 5.6 commercial software. A grid independence test was conducted to assess the optimal grid resolution. Based on this test, the number of meshes for the CFD model was determined to be 14,000. Figure 2c illustrates the model geometry and meshes used for the CFD simulation. Table 1 presents a summary of the key parameters for the model. After the model was verified, a series of CFD simulations were conducted under different operating conditions (see Table 2 for details).

## 4. Results

#### 4.1. Verification of the Computational Fluid Dynamics Model

#### 4.2. Velocity Distribution

#### 4.3. Temperature Distribution

#### 4.4. Vapor Pressure and Flux

#### 4.5. Temperature Polairzation (TP) Phenomenon

#### 4.6. Exergy Destruction Profiles

#### 4.7. Exergy Flow Analysis

^{2}in Figure 11a. The feed, membrane, and permeate account for 24.1%, 43.0%, and 32.9% of exergy destruction levels, respectively. Conversely, the total exergy in Figure 11d is 3.288 kW/m

^{2}. The exergy destruction contributions from the feed, membrane, and permeate are 26.7%, 32.4%, and 40.8%, respectively. These results indicate that the relative importance of the exergy destruction mechanism is affected by different operating conditions. It is expected that the CFD-based approach for calculating exergy destruction will be a useful tool for analyzing and optimizing MD systems.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${a}_{w}$ | Water activity in NaCl solution |

c | Concentration of water, mol/m^{3} |

${C}_{fm}$ | Feed-membrane concentration, mol/m^{3} |

${C}_{mp}$ | Membrane-permeate concentration, mol/m^{3} |

${C}_{p}$ | Heat capacity, J/[kg·K] |

${d}_{p}$ | Membrane pore size, m |

${D}_{k}$ | Knudsen diffusion coefficient, m^{2}/s |

${D}_{p}$ | Poiseuille flow coefficient, m^{2}/s |

E_{D} | Energy destruction, kW/m^{2} |

E_{feed} | Exergy destruction on the feed side, kW/m^{2} |

E_{membrane} | Exergy destruction in the membrane, kW/m^{2} |

E_{permeate} | Exergy destruction on the permeate side, kW/m^{2} |

F | Volume force vector |

I | Unit tensor |

${k}_{m}$ | Thermal conductivity coefficients of membrane, W/[m·K] |

${k}_{g}$ | Thermal conductivity coefficients of vapor, W/[m·K] |

${k}_{s}$ | Thermal conductivity coefficients of solid, W/[m·K] |

M | Water(vapor) molecular weight, g/mol |

$p$ | Dynamic pressure, Pa |

P | Mean pressure, Pa |

$\mathrm{q}$ | Heat flux, W/m^{2} |

Q | Heat source, W/m^{3} |

Q_{T} | Total heat transferred by the membrane, kW |

R | Gas constant, J/[mol·K] |

T | Temperature, K |

T_{0} | Temperature of the environment, °C |

${T}_{m}$ | Mean temperature, K |

${T}_{fm}$ | Feed temperature on the membrane surface, °C |

${T}_{pm}$ | Permeate temperature on the membrane surface, °C |

${T}_{fb}$ | Feed temperature in bulk fluid, °C |

${T}_{pb}$ | Permeated temperature in bulk fluid, °C |

$\mathrm{u}$ | Velocity vector, m/s |

${x}_{w}$ | Liquid mole fraction of water |

${x}_{NaCl}$ | Liquid mole fraction of NaCl solution |

$\mathsf{\rho}$ | Density, kg/m^{3} |

$\mathsf{\mu}$ | Viscosity, N·s/m^{2} |

$\mathsf{\epsilon}$ | Porosity of the membrane |

$\mathsf{\tau}$ | Tortuosity of the membrane |

$\nabla $ | Dell operator |

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**Figure 1.**Schematic of heat/mass transfer through a hydrophobic porous membrane used in a DCMD process: (

**a**) evaporation in the feed side, (

**b**) transportation of water vapor across the membrane pore, and (

**c**) condensation of water vapor in the permeate side.

**Figure 2.**(

**a**) Acrylic DCMD module, (

**b**) model geometry and meshes used for the CFD simulation, and (

**c**) schematic diagram of lab-scale experimental setup.

**Figure 3.**Validation of CFD model, experimental water flux results, and error rate: (

**a**) difference in feed temperatures and (

**b**) difference in feed flow rates.

**Figure 4.**Flow velocity distribution inside the module: (

**a**) velocity contour in the feed and permeate sides, (

**b**) feed side, and (

**c**) permeate side. Conditions: feed temperature = 60 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min.

**Figure 5.**(

**a**) Temperature distribution inside the DCMD module and (

**b**) temperature variation according to channel height in the middle of the module (y = 0.03 m).

**Figure 6.**(

**a**) Vapor pressure (kPa) inside the DCMD module in the membrane side, and (

**b**) flux distribution. Conditions: feed temperature = 60 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min.

**Figure 7.**TP ratio inside the DCMD module: (

**a**) effect of feed temperature, (

**b**) effect of feed flow rate, (

**c**) effect of permeate flow rate, and (

**d**) comparison of average TP ratios under various conditions.

**Figure 8.**Profiles of exergy destruction in the DCMD module: (

**a**) Feed temperature = 40 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min; (

**b**) Feed temperature = 50 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min; (

**c**) Feed temperature = 60 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min; (

**d**) Feed temperature = 70 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min.

**Figure 9.**Profiles of exergy destruction in the DCMD module: (

**a**) Feed temperature = 60 °C, feed flow rate = 0.24 L/min, and permeate flow rate = 0.4 L/min; (

**b**) Feed temperature = 60 °C, feed flow rate = 0.48 L/min, and permeate flow rate = 0.4 L/min.

**Figure 10.**Profiles of exergy destruction in the DCMD module: (

**a**) Feed temperature = 60 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.2 L/min; (

**b**) Feed temperature = 60 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.6 L/min.

**Figure 11.**Exergy flow analysis for the DCMD module: (

**a**) Feed temperature = 40 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min; (

**b**) Feed temperature = 50 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min; (

**c**) Feed temperature = 60 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min; (

**d**) Feed temperature = 70 °C, feed flow rate = 0.6 L/min, and permeate flow rate = 0.4 L/min.

Parameter | Value |
---|---|

Feed channel height | 0.003 m |

Permeate channel height | 0.003 m |

Channel length | 0.06 m |

Heat of evaporation | 2.333 × 10^{6} [W·s]/kg |

Feed thermal conductivity | 0.64 W/[m·K] |

Permeate thermal conductivity | 0.6 W/[m·K] |

Membrane thermal conductivity | 0.04 W/[m·K] |

Membrane pore size | 0.22 μm |

Membrane thickness | 100 μm |

Membrane tortuosity | 2 |

Case | Feed Side | Permeate Side | ||
---|---|---|---|---|

Temperature (°C) | Flow Rate (L/min) | Temperature (°C) | Flow Rate (L/min) | |

1 | 40 | 0.6 | 20 | 0.4 |

2 | 50 | |||

3 | 60 | |||

4 | 70 | |||

5 | 60 | 0.24 | ||

6 | 0.48 | |||

7 | 0.6 | 0.2 | ||

8 | 0.6 | 0.6 |

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**MDPI and ACS Style**

Choi, J.; Choi, Y.; Lee, J.; Kim, Y.; Lee, S.
Exergy Analysis of a Direct Contact Membrane Distillation (DCMD) System Based on Computational Fluid Dynamics (CFD). *Membranes* **2021**, *11*, 525.
https://doi.org/10.3390/membranes11070525

**AMA Style**

Choi J, Choi Y, Lee J, Kim Y, Lee S.
Exergy Analysis of a Direct Contact Membrane Distillation (DCMD) System Based on Computational Fluid Dynamics (CFD). *Membranes*. 2021; 11(7):525.
https://doi.org/10.3390/membranes11070525

**Chicago/Turabian Style**

Choi, Jihyeok, Yongjun Choi, Juyoung Lee, Yusik Kim, and Sangho Lee.
2021. "Exergy Analysis of a Direct Contact Membrane Distillation (DCMD) System Based on Computational Fluid Dynamics (CFD)" *Membranes* 11, no. 7: 525.
https://doi.org/10.3390/membranes11070525