# Understanding the Residence Time Distribution in a Transient Inline Spiking System: Modeling, Experiments, and Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Experimental

#### 2.1. Transient Inline Spiking System

#### 2.2. Concentration Easurement

## 3. Modeling and Simulation

#### 3.1. Modeling

#### 3.2. CFD Simulation

^{3}and $\mu =0.898$ mPa·s at the values of 0.1 M salt concentration at 25 °C (see Figure S1 in Supporting Information). Readers may refer to previous numerical studies on laminar mixing in static mixers [17,18,19,20].

**u**= 0) was applied to the remaining solid boundaries.

^{2}, an average of the diffusion coefficients of Na+ and Cl− [21]). As the solute was treated as an inert tracer, a reaction term was not considered in this equation.

## 4. Results and Discussion

#### 4.1. RTD Curve

#### 4.2. Static Mixer

#### 4.3. Filter Holder

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the inline spiking system. The syringe needle penetrates the bottom of the tee to directly discharge the salt spike at the center of the feed stream. The circled numbers indicate the processing point to measure the holdup time.

**Figure 2.**Processing units used in the transient inline spiking system and simulation domains for CFD simulation: (

**a**) the geometry of the static mixer, (

**b**) the periodic unit of the static mixer, and (

**c**) the static mixer (SM) simulation domain in three dimensions composed of six periodic units of the static mixer and two buffer regions at the inlet and outlet. (

**d**) The geometry of the filter holder, and (

**e**) the filter holder (FH) in two dimensions with axisymmetry. A red single dash-dotted line in the simulation domain of the filter holder represents the axis of rotation.

**Figure 3.**Mesh convergence test results for the CFD simulations of (

**a**) the static mixer and (

**b**) the filter holder.

**Figure 4.**RTD curve from three repeated runs of the salt spiking experiments. The feed flow rate was 0.41 mL/min, and the spiking duration was 1 min at a flow rate of 0.041 mL/min.

**Figure 6.**Streamlines evaluated from the center of the inlet where the salt spike was discharged through the syringe needle. The color of the streamline represents the velocity magnitude divided by ${\overline{u}}_{in,SM}$.

**Figure 7.**Concentration distribution in the static mixer. (

**a**) Concentration distribution (${C}_{n}$) at the cross-section of the n-th mixing period. When c ≥ 20 mM, the color of the contour is represented in red because the maximum concentration is much larger than the concentration at the mixed state. (

**b**) The RTD curve in the static mixer when the concentration at the tip of the syringe needle is kept constant at 1 M.

**Figure 8.**(

**a**) Pressure field within the filter holder, (

**b**) velocity magnitude in the filter holder, and (

**c**) trajectory of passive tracers under the flow that develops within the filter holder, where ${t}_{0}$ is the time when the completely mixed stream of the feed solution reaches the inlet port of the filter holder.

**Figure 9.**Evolution of solute concentration within the filter holder at varying spiking duration. Blue boxes represent the time duration before the end of each spike.

**Figure 10.**The RTD curves from the CFD simulation compared to those from the experiments and the PFR-2CSTR model.

No. | Interval | Holdup Time [min] | Description |
---|---|---|---|

1 | ①–② | 0.98 ± 0.02 | Tubing (spiking station → static mixer) |

2 | ②–③ | 2.18 ± 0.29 | Static mixer |

3 | ③–④ | 0.98 ± 0.07 | Tubing (static mixer → filter holder) |

4 | ④–⑤ | 1.05 ± 0.57 | Filter holder |

5 | ⑤–⑥ | 0.34 ± 0.01 | Tubing (filter holder → collection tube) |

Static Mixer (SM) Domain | |||
---|---|---|---|

Symbol | Value | Unit | Description |

${\overline{u}}_{in,SM}$ | $3.8\times {10}^{-4}$ | m/s | Average inlet velocity into the static mixer |

${D}_{SM}$ | 4.76 | mm | Diameter of the static mixer |

${D}_{s,SM}$ | 0.337 | mm | Diameter of the syringe needle |

${L}_{p,SM}$ | 10.06 | mm | Length of the periodic unit of the static mixer |

${h}_{SM}$ | 1.00 | mm | Thickness of the static mixer |

Filter Holder (FH) domain | |||

Symbol | Value | Unit | Description |

${\overline{u}}_{in,FH}$ | $4.3\times {10}^{-4}$ | m/s | Average inlet velocity into the filter holder |

${D}_{i,FH}$ | 4.3 | mm | Diameter of the inlet of the filter holder |

${D}_{o,FH}$ | 2.3 | mm | Diameter of the outlet of the filter holder |

${D}_{m}$ | 24 | mm | Diameter of the membrane |

$\Delta {P}_{m}$ | 1380 | Pa | Transmembrane pressure (TMP) |

${h}_{m}$ | 140 | μm | Membrane thickness |

$\kappa $ | $1.27\times {10}^{-15}$ | m^{2} | Membrane permeability |

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**MDPI and ACS Style**

Hwang, M.; Wang, J.; Jung, S.Y. Understanding the Residence Time Distribution in a Transient Inline Spiking System: Modeling, Experiments, and Simulations. *Membranes* **2023**, *13*, 375.
https://doi.org/10.3390/membranes13040375

**AMA Style**

Hwang M, Wang J, Jung SY. Understanding the Residence Time Distribution in a Transient Inline Spiking System: Modeling, Experiments, and Simulations. *Membranes*. 2023; 13(4):375.
https://doi.org/10.3390/membranes13040375

**Chicago/Turabian Style**

Hwang, Minsun, Junsuk Wang, and Seon Yeop Jung. 2023. "Understanding the Residence Time Distribution in a Transient Inline Spiking System: Modeling, Experiments, and Simulations" *Membranes* 13, no. 4: 375.
https://doi.org/10.3390/membranes13040375