# Investigating the Dialysis Treatment Using Hollow Fiber Membrane: A New Approach by CFD

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{®}Fluent software. The predicted results were compared with results reported in the literature and a good concordance was obtained. The simulation results showed that the proposed model can predict the fluid behavior inside the hollow fiber membrane adequately. In addition, it was found that the clearance decreases with increasing radial viscous resistance, with greater permeations in the vicinity of the lumen inlet region, as well as the emergence of the retrofiltration phenomenon, characteristic of this type of process. Herein, velocity, pressure, and volumetric fraction fields are presented and analyzed.

## 1. Introduction

## 2. Methodology

#### 2.1. Problem Description

#### 2.2. Computational Domain

_{1}, M

_{2}, and M

_{3}) were built using the Ansys

^{®}Designer Modeler and Meshing 15.0 software (Canonsburg, PA, USA), as illustrated in Figure 3.

#### 2.3. Mathematical Modeling

- Newtonian fluids;
- Flow in a laminar, incompressible, isothermal, and transient regime;
- Constant thermophysical and chemical properties;
- Anisotropic porous medium;
- Negligible gravitational effect;
- The proteins present in the blood were disregarded;
- Adsorption of urea on the membrane contact surface, blockage of membrane pores, formation of concentration polarization layer, and chemical reactions are disregarded;
- Only one section of the hollow fiber membrane is considered, due to the angular symmetry presented by the geometry;
- The Eulerian–Eulerian approach was adopted for multiphase flow.

- Mass conservation equation for the non-porous media

- Linear momentum equation

^{®}software uses an interaction term between the forces, described by:

- Linear momentum equation for the porous medium

#### Conditions Used in Simulations

- (a)
- Initial and boundary conditions

- Initial conditions

_{0}, before starting the process.

- Boundary conditions

- (b)
- Thermophysical parameters of membrane and fluids

#### 2.4. Studied Cases

#### 2.5. Procedures Used

- (a)
- Mesh evaluation

- (b)
- Validation of the mathematical model

^{®}software. With these data, a new simulation was made, where a new Clearance was obtained, which was compared with the one obtained experimentally by Liao et al. [18] under the same operating conditions. Once the error of this comparison was verified, the value of $1/{\alpha}_{y}$ was corrected and the process was repeated until a minimum error was obtained (trial and error method).

## 3. Results and Discussion

#### 3.1. Mesh Quality Assessment

^{®}. This refinement ratio is per the methodology proposed by Roache [26]. The number of elements of the meshes used can be seen in Table 3. The meshes were made in a structured way, with a standardized refinement throughout the domain; details can be seen in Figure 3 and Figure 5.

#### 3.2. Hollow Fiber Membrane Analysis

#### 3.2.1. Clearance

^{2}= 0.84.

#### 3.2.2. Volume Fraction

#### 3.2.3. Flow Lines and Velocity Vectors

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sesso, R.C.; Lopes, A.A.; Thomé, F.S.; Lugon, J.R.; Martins, C.T. Brazilian Chronic Dialysis Survey 2016. Braz. J. Nephrol.
**2017**, 39, 261–266. (In Portuguese) [Google Scholar] [CrossRef] [PubMed] - Ribeiro, R.D.C.H.M.; Oliveira, G.A.S.A.; Ribeiro, D.F.; Bertolin, D.C.; Cesarino, C.B.; Lima, L.C.E.Q.; Oliveira, S.M. Characterization and etiology of chronic renal failure in a nephrology unit in the interior of the State of São Paulo. ACTA Paul. Enferm.
**2008**, 21, 207–211. (In Portuguese) [Google Scholar] [CrossRef] - Kalra, S.; McBryde, C.W.; Lawrence, T. Intracapsular hip fractures in end-stage renal failure. Injury
**2006**, 37, 175–184. [Google Scholar] [CrossRef] - Vanholder, R.; Smet, R.; Glorieux, G.; Argilés, A.; Baurmeister, U.; Brunet, P.; Clark, W.; Cohen, G.; Deyn, P.P.; Deppisch, R.; et al. Review on uremic toxins: Classification, concentration, and interindividual variability. Kidney Int.
**2003**, 63, 1934–1943. [Google Scholar] [CrossRef] [PubMed] [Green Version] - De Rosa, S.; Prowle, J.R.; Samoni, S.; Villa, G.; Ronco, C. Acute kidney injury in patients with chronic kidney disease. In Critical Care Nephrology, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 2019; pp. 85–89. [Google Scholar]
- Lu, J.; Lu, W.Q. A numerical simulation for mass transfer through the porous membrane of parallel straight channels. Int. J. Heat Mass Transf.
**2010**, 53, 2404–2413. [Google Scholar] [CrossRef] - Lu, J.; Lu, W.-Q. Blood flow velocity and ultra-filtration velocity measured by CT imaging system inside a densely bundled hollow fiber dialyzer. Int. J. Heat Mass Transf.
**2010**, 53, 1844–1850. [Google Scholar] [CrossRef] - Clark, W.R.; Gao, D.; Neri, M.; Ronco, C. Solute transport in hemodialysis: Advances and limitations of current membrane technology. In Expanded Hemodialysis; Karger Publishers: Basel, Switzerland, 2017; Volume 191, pp. 84–99. [Google Scholar]
- Ding, W.; Li, W.; Sun, S.; Zhou, X.; Hardy, P.A.; Ahmad, S.; Gao, D. Three-Dimensional Simulation of Mass Transfer in Artificial Kidneys. Artif. Organs
**2015**, 39, E79–E89. [Google Scholar] [CrossRef] - Klein, E.; Holland, F.; Lebeouf, A.; Donnaud, A.; Smith, J.K. Transport and mechanical properties of hemodialysis hollow fibers. J. Membr. Sci.
**1976**, 1, 371–396. [Google Scholar] [CrossRef] - Liao, Z.; Klein, E.; Poh, C.K.; Huang, Z.; Lu, J.; Hardy, P.A.; Gao, D. Measurement of hollow fiber membrane transport properties in hemodialyzers. J. Membr. Sci.
**2005**, 256, 176–183. [Google Scholar] [CrossRef] - Lu, J.; Lu, W. An approximate analytical solution to the ultra-filtration profile in a hemodialysis process between parallel porous plates. Chin. Sci. Bull.
**2008**, 53, 3402–3408. [Google Scholar] [CrossRef] [Green Version] - Pstras, L.; Stachowska-Pietka, J.; Debowska, M.; Pietribiasi, M.; Poleszczuk, J.; Waniewski, J. Dialysis therapies: Investigation of transport and regulatory processes using mathematical modelling. Biocybern. Biomed. Eng.
**2022**, 42, 60–78. [Google Scholar] [CrossRef] - Cancilla, N.; Gurreri, L.; Marotta, G.; Ciofalo, M.; Cipollina, A.; Tamburini, A.; Micale, G. A porous media CFD model for the simulation of hemodialysis in hollow fiber membrane modules. J. Membr. Sci.
**2022**, 646, 120219. [Google Scholar] [CrossRef] - Gostoli, C.; Gatta, A. Mass transfer in a hollow fiber dialyzer. J. Membr. Sci.
**1980**, 6, 133–148. [Google Scholar] [CrossRef] - Ding, W.; He, L.; Zhao, G.; Shu, Z.; Cheng, S.; Gao, D. A novel theoretical model for mass transfer of hollow fiber hemodialyzers. Chin. Sci. Bull.
**2003**, 48, 2386–2390. [Google Scholar] [CrossRef] - Kanchan, M.; Maniyeri, R. Computational Study of Fluid Flow in Wavy Channels Using Immersed Boundary Method. In Soft Computing for Problem Solving; Springer: Singapore, 2019; pp. 283–293. [Google Scholar]
- Liao, Z.; Poh, C.K.; Huang, Z.; Hardy, P.A.; Clark, W.R.; Gao, D. A numerical and experimental study of mass transfer in the artificial kidney. J. Biomech. Eng.
**2003**, 125, 472–480. [Google Scholar] [CrossRef] [PubMed] - Donato, D.; Boschetti-de-Fierro, A.; Zweigart, C.; Kolb, M.; Eloot, S.; Storr, M.; Krause, B.; Leypoldt, K.; Segers, P. Optimization of dialyzer design to maximize solute removal with a two-dimensional transport model. J. Membr. Sci.
**2017**, 541, 519–528. [Google Scholar] [CrossRef] - Choi, Y.K.; Kim, J.T.; Ryou, H.S. Investigation on the effect of hematocrit on unsteady hemodynamic characteristics in arteriovenous graft using the multiphase blood model. J. Mech. Sci. Technol.
**2015**, 29, 2565–2571. [Google Scholar] [CrossRef] - Kim, J.C.; Cruz, D.; Garzotto, F.; Kaushik, M.; Teixeria, C.; Baldwin, M.; Baldwin, I.; Nalesso, F.; Kim, J.H.; Kang, E. Effects of dialysate flow configurations in continuous renal replacement therapy on solute removal: Computational modeling. Blood Purif.
**2013**, 35, 106–111. [Google Scholar] [CrossRef] - ANSYS INC. ANSYS FLUENT Theory Guide; Release 15.0; Ansys Inc.: Canonsburg, PA, USA, 2013; Volume 15317. [Google Scholar]
- Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng. -Trans. ASME
**2008**, 130, 1–4. [Google Scholar] - Lira, D.S. Textile Industry Effluent Microfiltration Process Using Hollow Fiber Membrane-Modeling and Simulation. Master’s Thesis, Federal University of Campina Grande, Campina Grande, PB, Brazil, 2018. (In Portuguese). [Google Scholar]
- Paudel, S.; Saenger, N. Grid refinement study for three dimensional CFD model involving incompressible free surface flow and rotating object. Comput. Fluids
**2017**, 143, 134–140. [Google Scholar] [CrossRef] - Roache, P.J. Perspective: A Method for Uniform Reporting of Grid Refinement Studies. J. Fluids Eng.
**1994**, 116, 405–413. [Google Scholar] [CrossRef] - Celik, I.B.; Karatekin, O. Numerical experiments on application of Richardson extrapolation with nonuniform grids. ASME J. Fluids Eng.
**1997**, 119, 584–590. [Google Scholar] [CrossRef] - Eloot, S. Experimental and Numerical Modeling of Dialysis. Ph.D. Thesis, Ghent University, Gent, Belgium, 2004. [Google Scholar]

**Figure 3.**Two-dimensional mesh of the tubular membrane cross-section: (

**a**) Mesh ${M}_{1}$, (

**b**) Mesh ${M}_{2}$, and (

**c**) Mesh ${M}_{3}$.

**Figure 5.**Details of the meshes used in the simulations (

**a**) ${M}_{1}$, (

**b**) ${M}_{2}$, and (

**c**) ${M}_{3}$.

**Figure 7.**The urea velocity profile in (

**a**) 20.0 mm, (

**b**) 106.6 mm, and (

**c**) 183.2 mm axial positions for different meshes.

**Figure 9.**Pressure profile at (

**a**) 10.0 mm, (

**b**) 106.6 mm, and (

**c**) 193.2 mm axial positions for different meshes.

**Figure 11.**Distribution of the volume fraction of urea in the XY plane at Z = 0 m and different process times: (

**a**) 500 s, (

**b**) 1000 s, (

**c**) 1500 s, (

**d**) 2000 s, and (

**e**) 2500 s.

**Figure 12.**Distribution of the volume fraction of urea in the XY plane at Z = 0 m and at time t = 6200 s.

**Figure 13.**Volume fraction profile of urea inside the membrane, in the axial positions of 20 mm, 101.6 mm, and 183.2 mm, at t = 6200 s (Case 9).

**Figure 14.**Distribution of the local velocity of urea in the XY plane at Z = 0 m and at time t = 6200 s.

**Figure 15.**Velocity profile of urea inside the membrane, in the axial positions 20.0 mm, 101.6 mm, and 183.2 mm, at t = 6200 s (Case 9).

**Figure 17.**Pressure profile inside the membrane, in the axial positions of 20.0 mm, 101.6 mm, and 183.2 mm, at t = 6200 s (Case 9).

**Table 1.**Dimensions of the hollow fiber membrane [18].

Equipment Dimensions (mm) | |
---|---|

Length $\left(\mathrm{L}\right)$ | 203.2 |

Section thickness $\left(\mathrm{E}\right)$ | 0.0208962 |

The thickness of the dialysate flow region $\left({\mathrm{E}}_{\mathrm{d}}\right)$ | 0.04475 |

Membrane thickness $\left({\mathrm{E}}_{\mathrm{m}}\right)$ | 0.015 |

Blood flow thickness $\left({\mathrm{E}}_{\mathrm{b}}\right)$ | 0.1 |

**Table 2.**Thermo-physical properties and parameters of fluids and membrane [18].

Fluids | Density $\mathit{\rho}$ (kg/m ^{3})
| Viscosity $\mathit{\mu}$ (kg/m·s) | Viscous Resistance Axial
$1/{\mathit{\alpha}}_{\mathit{x}}$ (m ^{−2})
| Porosity | |
---|---|---|---|---|---|

Dialysate | 998.2 | 0.001003 | - | - | |

Blood | Water | 998.2 | 0.001003 | - | - |

Urea | 1280.0 | 0.002300 | - | - | |

Membrane | - | - | $7.75\times {10}^{8}$ | 0.2 |

Case | Number of Mesh Elements $\left({\mathit{N}}_{\mathit{m}}\right)$ |
---|---|

01 | 718.920 |

02 | 344.267 |

03 | 147.785 |

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

Lumen feed flux (mL/min) | ${Q}_{Bin}$ | 300 |

Shell feed flux (mL/min) | ${Q}_{Din}$ | 300 |

Axial viscous resistance (m^{−2}) | $1/{\alpha}_{x}$ | $7.75\times {10}^{8}$ |

Radial viscous resistance (m^{−2}) | $1/{\alpha}_{y}$ | $2.15\times {10}^{14}$ |

Urea concentration in the lumen feed (kg/m^{3}) | ${C}_{in}$ | 0.7 |

Case | $1/{\mathit{\alpha}}_{\mathit{y}}\left({\mathbf{m}}^{-2}\right)$ |
---|---|

04 | $2.40\times {10}^{10}$ |

05 | $2.40\times {10}^{11}$ |

06 | $2.40\times {10}^{13}$ |

07 | $2.40\times {10}^{14}$ |

08 | $2.40\times {10}^{15}$ |

09 | $2.15\times {10}^{14}$ |

**Table 6.**Parameters obtained from the study of the Grid Convergence Index for urea velocity as response variable (y = 0.159 m).

Parameter | Axial Position | |||
---|---|---|---|---|

${\mathit{x}}_{1}$ = 20 mm | ${\mathit{x}}_{2}$ = 101.6 mm | ${\mathit{x}}_{3}$ = 183.2 mm | ||

Urea velocity (m/s) | Mesh M_{1} | $3.709\times {10}^{-3}$ | $2.422\times {10}^{-3}$ | $4.036\times {10}^{-3}$ |

Mesh M_{2} | $3.706\times {10}^{-3}$ | $2.399\times {10}^{-3}$ | $4.034\times {10}^{-3}$ | |

Mesh M_{3} | $3.698\times {10}^{-3}$ | $2.340\times {10}^{-3}$ | $4.030\times {10}^{-3}$ | |

$p$ | $1.384$ | $1.509$ | $1.831$ | |

${\varphi}_{ext}^{21}={M}_{ext}$ (m/s) | $3.712\times {10}^{-3}$ | $2.411\times {10}^{-3}$ | $4.037\times {10}^{-3}$ | |

$IC{M}_{21}$ | $1.067\times {10}^{-3}$ | $1.031\times {10}^{-2}$ | $2.778\times {10}^{-4}$ | |

$IC{M}_{32}$ | $2.104\times {10}^{-3}$ | $2.179\times {10}^{-2}$ | $6.815\times {10}^{-4}$ | |

$c$ | $0.394$ | $0.368$ | $0.308$ | |

${r}_{p}IC{M}_{21}$ | $2.102\times {10}^{-3}$ | $2.159\times {10}^{-2}$ | $6.813\times {10}^{-4}$ |

Mesh | Mean Relative Error (%) | ||
---|---|---|---|

${\mathit{x}}_{1}$ (20 mm) | ${\mathit{x}}_{2}$ (101.6 mm) | ${\mathit{x}}_{3}$ (183.2 mm) | |

${M}_{1}$ | 1.44 | 0.82 | 0.2 |

${M}_{2}$ | 2.04 | 1.66 | 0.36 |

${M}_{3}$ | 1.50 | 3.86 | 0.59 |

**Table 8.**Parameters obtained from the study of the Grid Convergence Index for pressure as response variable (y = 0.159 m).

Parameter | Axial Position | |||
---|---|---|---|---|

${\mathit{x}}_{1}$ = 10.0 mm | ${\mathit{x}}_{2}$ = 101.6 mm | ${\mathit{x}}_{3}$ = 193.2 mm | ||

Pressure (Pa) | Mesh M_{1} | 188.42 | 135.71 | 80.58 |

Mesh M_{2} | 185.58 | 134.78 | 81.96 | |

Mesh M_{3} | 177.54 | 132.85 | 87.39 | |

$p$ | 1.598 | 1.030 | 2.205 | |

${\varphi}_{ext}^{21}={M}_{ext}$ (Pa) | 190.81 | 137.14 | 79.88 | |

$IC{M}_{21}$ | $1.58\times {10}^{-2}$ | $1.31\times {10}^{-2}$ | $1.09\times {10}^{-2}$ | |

$IC{M}_{32}$ | $3.52\times {10}^{-2}$ | $2.19\times {10}^{-2}$ | $3.16\times {10}^{-2}$ | |

$c$ | 0.350 | 0.483 | 0.252 | |

${r}_{p}IC{M}_{21}$ | $3.47\times {10}^{-2}$ | $2.17\times {10}^{-2}$ | $3.22\times {10}^{-2}$ |

Mesh | Mean Relative Error (%) | ||
---|---|---|---|

${\mathit{x}}_{1}$ (10.0 mm) | ${\mathit{x}}_{2}$ (101.6 mm) | ${\mathit{x}}_{3}$ (193.2 mm) | |

M_{1} | 1.46 | 1.04 | 1.17 |

M_{2} | 2.30 | 1.72 | 1.56 |

M_{3} | 4.00 | 3.13 | 1.90 |

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

© 2022 by the authors. 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**

Magalhães, H.L.F.; Gomez, R.S.; Leite, B.E.; Nascimento, J.B.S.; Brito, M.K.T.; Araújo, M.V.; Cavalcante, D.C.M.; Lima, E.S.; Lima, A.G.B.; Farias Neto, S.R.
Investigating the Dialysis Treatment Using Hollow Fiber Membrane: A New Approach by CFD. *Membranes* **2022**, *12*, 710.
https://doi.org/10.3390/membranes12070710

**AMA Style**

Magalhães HLF, Gomez RS, Leite BE, Nascimento JBS, Brito MKT, Araújo MV, Cavalcante DCM, Lima ES, Lima AGB, Farias Neto SR.
Investigating the Dialysis Treatment Using Hollow Fiber Membrane: A New Approach by CFD. *Membranes*. 2022; 12(7):710.
https://doi.org/10.3390/membranes12070710

**Chicago/Turabian Style**

Magalhães, Hortência L. F., Ricardo S. Gomez, Boniek E. Leite, Jéssica B. S. Nascimento, Mirenia K. T. Brito, Morgana V. Araújo, Daniel C. M. Cavalcante, Elisiane S. Lima, Antonio G. B. Lima, and Severino R. Farias Neto.
2022. "Investigating the Dialysis Treatment Using Hollow Fiber Membrane: A New Approach by CFD" *Membranes* 12, no. 7: 710.
https://doi.org/10.3390/membranes12070710