# 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

## References

- van der Waal, M.J.; Racz, L.G. Mass transfer in corrugated-plate membrane modules. I. Hyperfiltration experiments. J. Membr. Sci.
**1989**, 40, 243–260. [Google Scholar] [CrossRef] - Nijdam, W.; de Jong, J.; van Rijn, C.J.M.; Visser, T.; Versteeg, L.; Kapantaidakis, G.; Koops, G.-H.; Wessling, M. High performance micro-engineered hollow fiber membranes by smart spinneret design. J. Membr. Sci.
**2005**, 256, 209–215. [Google Scholar] [CrossRef] - Heinz, O.; Aghajani, M.; Greenberg, A.R.; Ding, Y. Surface-patterning of polymeric membranes: Fabrication and performance. Curr. Opin. Chem. Eng.
**2018**, 20, 1–12. [Google Scholar] [CrossRef] - Culfaz, P.Z.; Wessling, M.; Lammertink, R.G.H. Hollow Fiber Ultrafiltration Membranes With Microstructured Inner Skin. J. Membr. Sci.
**2011**, 369, 221–227. [Google Scholar] [CrossRef] - García-Fernández, L.; García-Payo, C.; Khayet, M. Hollow fiber membranes with different external corrugated surfaces for desalination by membrane distillation. Appl. Surf. Sci.
**2017**, 416, 932–946. [Google Scholar] [CrossRef] - Chwojnowski, A.; Wojciechowski, C.; Dudzinski, K.; Lukowska, E. Polysulphone and Polyethersulphone Hollow Fiber Membranes with Developed Inner Surface as Material for Bio-medical Applications. Biocybern. Biomed. Eng.
**2009**, 29, 47–59. Available online: http://www.pup.prudnik.ibib.pl/images/ibib/grupy/Wydawnictwa-Tomy/dokumenty/2009/BBE_29_3_047_FT.pdf (accessed on 1 September 2022). - Culfaz, P.Z.; Rolevink, E.; van Rijn, C.J.M.; Lammertink, R.G.H.; Wessling, M. Microstructured Hollow Fibers for Ultrafiltration. J. Membr. Sci.
**2010**, 347, 32–41. [Google Scholar] [CrossRef] - Brewers, J.M.; Goren, S.L. Evaluation of metal oxide whiskers grown on screens for use as aerosol filtration medium. Aerosol. Sci. Techn.
**1984**, 3, 411–429. [Google Scholar] [CrossRef] - Kirsh, V.A. Aerosol filters made of porous fibers. Colloid. J.
**1996**, 58, 786–790. [Google Scholar] - Kirsh, V.A. Deposition of aerosol nanoparticles in filters composed of fibers with porous shells. Colloid J.
**2007**, 69, 615–619. [Google Scholar] [CrossRef] - Kirsch, A.A.; Stechkina, I.B. The theory of aerosol filtration with fibrous filters. In Fundamentals of Aerosol Science; Shaw, D.T., Ed.; John Wiley & Sons: New York, NY, USA, 1978; Chapter 4, pp. 165–256; ISBN 0471029491. [Google Scholar]
- Slezkin, N.A. Dynamics of Viscous Incompressible Fluids; Gostekhizdat: Moscow, Russia, 1955. (In Russian) [Google Scholar]
- Happel, J. Viscous flow relative to arrays of cylinders. J. Amer. Inst. Chem. Eng.
**1959**, 5, 174–177. [Google Scholar] [CrossRef] - Kuwabara, S. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds number. J. Phys. Soc. Jpn.
**1959**, 14, 527–532. [Google Scholar] [CrossRef] - Kvashnin, A.G. Cell model of suspension of spherical particles. Fluid Dyn.
**1979**, 14, 598–602. [Google Scholar] [CrossRef] - Golovin, A.M.; Lopatin, V.A. The flow of a viscous fluid through a doubly periodic rows of cylinders (English transl.). J. Appl. Mech. Tech. Phys.
**1968**, 9, 198–201. [Google Scholar] [CrossRef] - Kirsh, V.A. Deposition of aerosol nanoparticles in fibrous filters. Colloid J.
**2003**, 65, 726–732. [Google Scholar] [CrossRef] - Tamada, K.; Fujikawa, H. The steady two-dimensional flow of viscous fluid at low Re numbers passing through an infinite row of equal parallel circular cylinders. Quart. J. Mech. Appl. Math.
**1957**, 10, 425–432. [Google Scholar] [CrossRef] - Miyagi, T. Viscous flow at low Reynolds numbers past an infinite row of equal circular cylinders. J. Phys. Soc. Jpn.
**1958**, 13, 493–496. [Google Scholar] [CrossRef] - Sangani, A.; Acrivos, A. Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiph. Flow
**1982**, 8, 193–206. [Google Scholar] [CrossRef] - Wang, W.; Sangani, A.S. Nusselt number for flow perpendicular to arrays of cylinders in the limit of small Reynolds and large Peclet numbers. Phys. Fluids
**1997**, 9, 1529–1539. [Google Scholar] [CrossRef] - Wang, C.Y. Stokes flow through a rectangular array of circular cylinders. Fluid Dyn. Res.
**2001**, 29, 65–80. [Google Scholar] [CrossRef] - Kirsch, V.A.; Roldugin, V.I.; Bildyukevich, A.V.; Volkov, V.V. Simulation of convective-diffusional processes in hollow fiber membrane contactors. Sep. Purif. Technol.
**2016**, 167, 63–69. [Google Scholar] [CrossRef] - Kirsch, V.A.; Bildyukevich, A.V.; Bazhenov, S.D. Simulation of convection-diffusion transport in a laminar flow past a row of parallel absorbing fibers. Fibers
**2018**, 6, 90–100. [Google Scholar] [CrossRef][Green Version] - Kirsch, V.A.; Bazhenov, S.D. Numerical simulation of solute removal from a cross-flow past a row of parallel hollow-fiber membranes. Sep. Purif. Technol.
**2020**, 242, 116834. [Google Scholar] [CrossRef] - Launder, B.E.; Massey, T.H. The Numerical Prediction of Viscous Flow and Heat Transfer in Tube Banks. ASME J. Heat Transfer.
**1978**, 100, 565–571. [Google Scholar] [CrossRef] - Samarskii, A.A.; Vabishchevich, P.N. Computational Heat Transfer, Volume 2: The Finite Difference Methodology, 2nd ed.; John Wiley & Sons: Chichester, UK, 1995; ISBN 0471956600/9780471956600. [Google Scholar]
- Berkovskii, B.M.; Polevikov, V.K. Effect of the Prandtl number on the convection field and the heat transfer during natural convection (English translation). J. Eng. Phys.
**1973**, 24, 598–603. [Google Scholar] [CrossRef] - Allen, D.N.; Southwell, R.V. Relaxation methods applied to determine the motion, in two dimensions of a viscous fluid past a fixed cylinder. Quart. J. Mech. Appl. Math.
**1955**, 8, 129–143. [Google Scholar] [CrossRef] - Fletcher, C.A.J. Computational Techniques for Fluid Dynamics 2, Specific Techniques for Different Flow Categorie; Springer: Berlin/Heidelberg, Germany, 1991; ISBN 1434-8322. [Google Scholar] [CrossRef]
- Berkovskii, B.M.; Nogotov, E.F. Difference Methods for Investigating Heat Exchange Problems; Nauka i Tekhnika: Minsk, Belarus, 1976. (In Russian) [Google Scholar]
- Roos, H.-G.; Stynes, M.; Tobiska, L. Numerical Methods for Singularly Perturbed Differential Equations. In Convection-Diffusion and Flow Problems; Springer: Berlin/Heidelberg, Germany, 1996; ISBN 3-540-60718-8. [Google Scholar]
- Miller, J.J.H.; O’Riordan, E.; Shishkin, G.I. Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, Revised Edition; World Scientific Publishing Company: Singapore, 2012; ISBN 9814390739. [Google Scholar] [CrossRef]
- Yu, Y.; Luo, X.; Zhang, H.Y.; Zhang, Q. The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient. Mathematics
**2019**, 7, 388–404. [Google Scholar] [CrossRef] - Savović, S.; Drljača, B.; Djordjevich, A. A comparative study of two different finite difference methods for solving advection–diffusion reaction equation for modeling exponential traveling wave. Ric. Di Mat.
**2022**, 71, 245–252. [Google Scholar] [CrossRef] - Ivanovic, M.; Svicevic, M.; Savovic, S. Numerical solution of Stefan problem with variable space grid method based on mixed finite element/finite difference approach. Int. J. Numer. Methods Heat Fluid Flow
**2017**, 27, 2682–2695. [Google Scholar] [CrossRef] - Natanson, G.L. Diffusional precipitation of aerosols on a streamlined cylinder with a small capture coefficient (English translation, Dokl. Akad. Nauk SSSR). Proc. Acad. Sci. USSR Phys. Chem. Sect.
**1957**, 112, 21–25. Available online: http://mi.mathnet.ru/eng/dan21504 (accessed on 1 September 2022). - Polyanin, A.D.; Kutepov, A.M.; Kazenin, D.A.; Vyazmin, A.V. Hydrodynamics, Mass and Heat Transfer in Chemical Engineering. Series: Topics in Chemical Engineering (Book 14), 1st ed.; CRC Press: Boca Raton, FL, USA, 2001; ISBN 0415272378. [Google Scholar]

**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.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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