# Numerical Simulation of Convective Diffusion of Point Particles in a Laminar Flow Past a Row of Profiled Hollow Fibers

^{1}

^{2}

## Abstract

**:**

^{5}for Sc = 1, 10, 1000 and Re ≤ 100.

## 1. Introduction

## 2. Governing Equations and Simulation Methods

**n**—the outer normal vector to the surface, $d\mathsf{\Sigma}$—the surface element, ${S}_{g}$—the fiber surface area.

## 3. Simulation Results and Discussion

## 4. Conclusions

^{5}for Schmidt numbers $\mathrm{Sc}$ = 1, 10, 1000, and Reynolds numbers $\mathrm{Re}\le 50$. The effect of the protrusion/groove on the flow and convective diffusion has been investigated. The efficiency of supplying a substance from an external flow to the absorbing fibers is determined, and it is shown that the profiling of the fibers improves the fiber retention (absorption, collection) efficiency. The effect of external protrusions for a fiber of a given radius is greater than that from the grooves (dimples). The depth of the groove has little impact, while the altering the width of the groove has a bigger impact. However, grooves or dimples made on the surface of the fiber might enhance trans-membrane diffusion transport, since the membrane thickness becomes locally smaller, while the rest of the material serves as a stiffener. Overall, the design of novel sorption (filtration) materials composed from profiled fibers is a promising direction of investigations. The results obtained may be of interest in solving practical problems of liquid (gas) filtration, sorption, catalysis, membrane separation, electrochemistry, and convective heat transfer (heat exchangers). The future investigations should involve not only 2D transverse flow case, but also parallel flow, including inner flow in a hollow-fiber membrane with the internal surface profiling.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Distribution of the dimensionless concentration of the point particles in the transverse flow past a row of parallel profiled fibers with protrusions for different Reynolds numbers; the concentration values are marked on the curves; Sc = 1; a/h = 0.5; flow direction from left to right.

**Figure 2.**Dimensionless drag forces per unit length of profiled fibers with parallel rectangular protrusion obstacles (curves 2–5) with a protrusion thickness of 0.1 and a protrusion height of 0.1 (2), 0.2 (3), 0.3 (4), 0.4 (5), solid curve 1—smooth fiber; dashed-dotted curve 6—a fiber with parallel grooves with width and depth equal to 0.1 and 0.4; a/h = 0.5.

**Figure 3.**Retention efficiencies for a profiled fiber with protrusion obstacles vs. Reynolds number at different values of the Schmidt number (given in the Figure); the thickness of the protrusion is 0.1 and the heights of the protrusions are 0.1 (1), 0.2 (2), 0.3 (3), 0.4 (4); a/h = 0.5.

**Figure 4.**Retention efficiencies for fibers with profiling (with protrusions of rectangular cross section) vs. Peclet number $\mathrm{Pe}=\mathrm{Re}\mathrm{Sc}$ at $\mathrm{Sc}$ = 1 (

**a**), 1000 (

**b**): (curves 1–3)—deposition on external protrusions, (1’–3’)—deposition on the part of the fiber surface of radius $a$. Calculations for different protrusion heights 0.1 (1, 1’), 0.2 (2, 2’), 0.3 (3, 3’).

**Figure 5.**Retention efficiencies of profiled hollow fibers with parallel grooves (solid lines) and smooth circular hollow fibers with the same outer radius (dotted lines) on the Reynolds number: pairs of curves 1—K = 0 (solid and dotted curves merge), 2—K = 0.1, 3—K = 1, 4—K = 10; Sc = 1.

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**MDPI and ACS Style**

Kirsch, V.A.
Numerical Simulation of Convective Diffusion of Point Particles in a Laminar Flow Past a Row of Profiled Hollow Fibers. *Fibers* **2022**, *10*, 77.
https://doi.org/10.3390/fib10090077

**AMA Style**

Kirsch VA.
Numerical Simulation of Convective Diffusion of Point Particles in a Laminar Flow Past a Row of Profiled Hollow Fibers. *Fibers*. 2022; 10(9):77.
https://doi.org/10.3390/fib10090077

**Chicago/Turabian Style**

Kirsch, Vasily A.
2022. "Numerical Simulation of Convective Diffusion of Point Particles in a Laminar Flow Past a Row of Profiled Hollow Fibers" *Fibers* 10, no. 9: 77.
https://doi.org/10.3390/fib10090077