# Modeling of Nanofiltration Process Using DSPM-DE Model for Purification of Amine Solution

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{i,p}= f (C

_{i,f}) was used in modeling. The results showed that the calculated data from the model provided a good agreement with experimental results (R

^{2}= 0.90–0.75). Also, the effect of operating conditions (including feed pressure and feed flow rate on ions rejection and recovery ratio across the flat-sheet membrane) was studied. The results showed that the recovery and rejection ratios of the NF membrane depend on the driving pressure across the membrane. While the driving pressure is affected by the feed flow conditions and feed pressure.

## 1. Introduction

_{2}S), and carbon dioxide (CO

_{2}), which as well as being harmful to human health cause problems such as corrosion, plugging, freezing, erosion, and environmental hazards [1,2,3,4,5,6]. Therefore, the removal of undesirable species is an important part of natural gas sweetening industries. Usually, this is done using an amine-based solvent such as methyl diethanolamine (MDEA) [7,8,9,10,11,12]. The reaction between MDEA, H

_{2}S, and CO

_{2}forms heat stable salts (HSS) that cannot be regenerated through heating. The maximum concentrtions of the dominant anionic components in amine solvents are as follows: sulfate—500 ppm; acetate—1000 ppm; glycolate—500 ppm; oxalate—250 ppm; formate—500 ppm; chloride—500 ppm respectively [13,14]. The accumulation of these salts in amine solution leads to a reduction in the efficiency of the CO

_{2}absorption process and operational problems such as corrosion, fouling, foaming, high viscosity, capacity reduction [15,16,17,18]. Therefore, the removal of HSS from the amine solvent is an indispensable part of gas sweetening industries. Various technologies that can be used to remove HSS ions from amine solvent include electrodialysis, thermal reclamation, ion exchange and distillation [19,20,21,22].

## 2. Materials and Methods

#### 2.1. Experimental Set-Up

_{2}H

_{3}NaO

_{2}, CH

_{2}O

_{2}, and Na

_{2}SO

_{4}in 45 wt. % MDEA solution, which was purchased from Ghatran Shimi Tajhiz, Tehran, Iran. MDEA was supplied by Ilam Gas Treating Co., Ilam, Iran.

#### 2.2. Process Modeling

- the gradient of the concentration is neglected in the width and length direction along feed side;
- flow rate profile along the module is obtained by total mass balance equation;
- mass transfer by diffusion in the axial direction is neglected due to the high flow rate of solvent.

_{f}and Q

_{P}are volumetric flow rates (L/h) in the feed side and the permeate side, respectively. Also, C

_{i,f}and C

_{i,P}are “i” component concentration (mg/L) on the feed and permeate sides, respectively, J

_{V}is permeating flux (L/h.m

^{2}) and P is perimeter (m

^{2}).

_{i,P}was calculated from the quadratic function which was obtained by fitting the experimental values of C

_{i,P}versus C

_{i,f}. The set of differential equations was solved by the 4th order Runge-Kutta method using MATLAB software (MATLAB R2014a, Available online: www.mathworks.com/products/matlab/ (accessed on 11 December 2020).

^{2}, are applied to evaluate the validity of the results. AARE and R

^{2}were determinate as follows [57,58,59,60]:

_{i}

^{exp}, R

_{i}

^{calc}, $\overline{{R}^{exp}}$, n are the rejection of the species “i” for the experimental data, the rejection predicted for the species “i”, the average value of the rejection and the number of total data, respectively.

_{i,pore}) is independent of ions position within pores in steady-state and is given by [48,54]:

_{obs}), intrinsic rejection (R

_{int}) and permeate flux (J

_{V}) were calculated by the following equations [41,61]:

^{2}), V (L) and t (h) are the effective membrane area, permeate volume, and time, respectively. C

_{i,m}is concentrations at the membrane surface for “i” component, which can be calculated with the concentration polarization equation.

_{h}is the hydraulic diameter of feed channel, u is the bulk velocity of flow, L is the channel length, and D

_{i,∞}is the diffusion coefficient of ion “i”. The diffusion coefficient of ions in the electrolyte solution was calculated by molecular simulation software.

_{loss}) along the feed flow direction is given by the following equation,

_{f}and P

_{P}and ∆π are pressure in feed bulk, permeate pressure, and osmotic pressure, respectively. μ is solvent viscosity. r

_{pore}, ∆x, and A

_{k}are pore radius of the membrane, membrane active layer thickness, and porosity of membrane, respectively.

## 3. Results and Discussion

#### 3.1. Correlation for C_{i,P} = f(C_{i,f})

_{i,P}= f(C

_{i,f})). Therefore, the effect of the feed concentration on the permeate concentration in 45 wt. % MDEA solution by NF-3 membrane for ternary salts was investigated at 70 bar and pH = 10, and the results were illustrated in Figure 3 Also, the correlation of permeate concentration and feed concentration in the amine solution was obtained as a quadratic function for each ion by curve fitting C

_{i,P}versus C

_{i,f}using experimental results in Figure 3. Since the NF-3 membrane has a negative charge, solutions with a pH of 10 lead to a strong negative charge of the membrane surface. Moreover, solutions with pH > 10 create scaling and fouling problems. On the other hand, the osmotic pressure of 45% wt. MDEA solution was high according to the Van’t Hoff equation (π = CRT, where π is the osmotic pressure, R is constant of proportionality also called general solution constant or gas constant, C is the concentration of the solution and T is the temperature [67]), and the rejection of MDEA by NF-3 was 1.2% at 70 bar. For the reasons above illustrated, the tests were performed at 70 bar and pH = As can be seen, the permeate concentration increased with increasing feed concentrations for all ions. This is due to the fact that in NF and RO, the solute flux is described by ${J}_{s}=B.\Delta {c}_{s}$ where B is the solute permeability coefficient. Therefore, the higher concentration of ions in the feed will lead to the lower quality of the permeate (since the solute leakage through the membrane is directly proportional to the solute concentration at the membrane feed side surface). Also, with the increase in the concentration of ions, the fixed negative charge on the membrane surface was partially neutralized by the counter ions leading to a decrease in the electrostatic repulsion between the ions and the membranes [53]. Hence, the concentration of ions in the permeate increases with increasing feed concentrations. On the other hand, the effect of concentration polarization (the ions accumulation in the boundary layer) can increase the concentration of ions in the permeate and leads to a reduction in rejection. The increase of the concentration polarization by increasing the feed concentration can be better investigated by evaluating the observed and intrinsic rejection, which were calculated using Equations (10)–(15) and plotted as a function of the feed concentration in Figure 3. As Figure 4 shows, the ion rejection decreased with increasing the ion concentration at the same operating pressure. Moreover, the difference between the observed rejection and intrinsic rejection increases when the concentration of ions rises. These indicate that the concentration polarization layer on the membrane surface increases when the ion concentration enhances [68].

#### 3.2. DSPM-DE Validation on Experimental Data

^{2}≈ 0.90–0.75). On the other hand, these correlations were exploited as the first guesses to compute the DSPM–DE model’s data in Equations (T1)–(T10). The parameter of the model and physical properties are mentioned in Table 2. The results of the model are shown in Figure 6 and are compared with experimental results. A glance at Figure 6 reveals that there is acceptable compatibility between the results of the model and the experiments.

#### 3.3. Ion Diffusion Coefficients Calculation

#### 3.4. Analysis and Model Description

#### 3.4.1. Effect of Feed Flow Rate on Feed Pressure Variation along the Feed Flow Direction

#### 3.4.2. Effect of Feed Flow Rate on Ions Concentration along Flow Direction

#### 3.4.3. Effect of Inlet Feed Pressure on the Ions Concentration along Flow Direction

#### 3.4.4. Effect of Inlet Feed Pressure and Feed Flow Rate on Rejection and Recovery Ratios

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 3.**Effect of feed concentration on permeate concentration for ternary salts of 45 wt. % MDEA solution by NF membrane at 70 bar (T = 35 °C, pH = 10).

**Figure 4.**Comparison of observed and intrinsic rejection for acetate, formate, and sulfate in the NF-3 membrane at 35 °C and 70 bar.

**Figure 5.**Comparison of the experimental data with the obtained results of the mathematical equations reported for acetate, formate, and sulfate in the NF-3 membrane at 35 °C and 70 bar.

**Figure 6.**Comparison between experimental data and model results for the ternary salts in 45 wt. % MDEA solution by NF membrane at 70 bar, 35 °C, pH = 10 and feed flow rate of 0.693 L/h.

**Figure 7.**The mass transfer coefficient as a function of feed flow rate for different ions in 45 wt. % MDEA solution at 70 bar and 35 °C.

**Figure 8.**Feed pressure along the feed flow direction at different feed flow rates and 70 bar inlet feed pressure.

**Figure 9.**Ions concentration on the feed side for (

**a**) formate, (

**b**) acetate, (

**c**) sulfate of 45 wt. % MDEA solution at different feed flow rates and 70 bar inlet feed pressure.

**Figure 10.**Feed concentration profile on membrane surface along the feed flow direction for (

**a**) formate, (

**b**) acetate, (

**c**) sulfate at 70 bar inlet feed pressure and different feed flow rates.

**Figure 11.**Permeate concentration variations along the feed flow direction for (

**a**) formate, (

**b**) acetate, (

**c**) sulfate at different feed flow rates and 70 bar inlet feed pressure.

**Figure 12.**Feed bulk concentration profile during feed side for (

**a**) formate, (

**b**) acetate, (

**c**) sulfate of 45 wt. % MDEA solution at 100 L/h feed flow rate and different inlet feed pressure.

**Figure 13.**Rejection rate for (

**a**) formate, (

**b**) acetate, (

**c**) sulfat, of 45 wt. % MDEA solution at different inlet feed pressures and feed flow rates.

**Figure 14.**Recovery ratio for 45 wt. % MDEA solution at different inlet feed pressures and feed flow rates.

**Table 1.**Characteristics of nanofiltration membrane [56].

NF-3 Membrane | ||||||
---|---|---|---|---|---|---|

MWCO (Da) | Pore Radius (nm) | Membrane Thickness (μm) | (∆x/A_{k}) (μm) | Pure Water Permeate (L m^{−2} h^{−1} bar^{−1}) | Operation Limits | Rejection (%) |

250–300 | 0.55 | 0.9 | 0.51 | 8.86 | 50 °C, 83 bar, 3–10 pH | NaCl: 60% MgSO _{4}: 98% |

Parameters | Values | References |
---|---|---|

Feed flow rate (L h^{−1}) | 0.693 | |

Cross flow velocity (m s^{−1} 1 × 10^{6}) | 7.67 | |

Temperature maintained in units (K) | 308 | |

Solute radius of HCO_{2}^{−} ion (r_{s} nm) | 0.00738 | |

Solute radius of C_{2}H_{3}O_{2}^{−} ion (r_{s} nm) | 0.00832 | |

Solute radius of SO_{4}^{2−} ion (r_{s} nm) | 0.0102 | |

Solute radius of Na^{+} ion (r_{s} nm) | 0.116 | [53] |

Solute radius of H^{+} (r_{s} nm) | 0.025 | [56] |

Bulk diffusivity of HCO_{2}^{−} ion (D_{i,∞} × 10^{9} m^{2}/s) | 1.33 | |

Bulk diffusivity of C_{2}H_{3}O_{2}^{−} ion (D_{i,∞} × 10^{9} m^{2}/s) | 1.18 | |

Bulk diffusivity of SO_{4}^{2−} ion (D_{i,∞} × 10^{9} m^{2}/s) | 0.96 | |

Bulk diffusivity of Na^{+} ion (D_{i,∞} × 10^{9} m^{2} s^{−1}) | 1.9 | [53] |

Bulk diffusivity of H^{+} ion (D_{i,∞} × 10^{9} m^{2} s^{−1}) | 9.3 | [56] |

Mass transfer coefficient of HCO_{2}^{−} ion (m s^{−1} × 10^{5}) | 1.44 | |

Mass transfer coefficient of C_{2}H_{3}O_{2}^{−} ion (m s^{−1} × 10^{5}) | 1.33 | |

Mass transfer coefficient of SO_{4}^{2−} ion (m s^{−1} × 10^{5}) | 1.16 | |

Boltzmann constant (k) (J K^{−1} × 10^{−25}) | 1.38066 | [71] |

Faraday constant | 96,500 | [71] |

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**MDPI and ACS Style**

Ghorbani, A.; Bayati, B.; Drioli, E.; Macedonio, F.; Kikhavani, T.; Frappa, M.
Modeling of Nanofiltration Process Using DSPM-DE Model for Purification of Amine Solution. *Membranes* **2021**, *11*, 230.
https://doi.org/10.3390/membranes11040230

**AMA Style**

Ghorbani A, Bayati B, Drioli E, Macedonio F, Kikhavani T, Frappa M.
Modeling of Nanofiltration Process Using DSPM-DE Model for Purification of Amine Solution. *Membranes*. 2021; 11(4):230.
https://doi.org/10.3390/membranes11040230

**Chicago/Turabian Style**

Ghorbani, Asma, Behrouz Bayati, Enrico Drioli, Francesca Macedonio, Tavan Kikhavani, and Mirko Frappa.
2021. "Modeling of Nanofiltration Process Using DSPM-DE Model for Purification of Amine Solution" *Membranes* 11, no. 4: 230.
https://doi.org/10.3390/membranes11040230