# Computation of Global and Local Mass Transfer in Hollow Fiber Membrane Modules

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}removal in a prototype hollow fiber membrane oxygenator.

## 1. Introduction

_{2}and O

_{2}in an oxygenator has no significant impact on hydrodynamics, numerical complexity is, to some extent, reduced. Hence, substantial progress has been achieved in this field. For instance, simulations of transmembrane CO

_{2}transport are not limited to two-dimensional flat sheet models, but are extended towards periodic fiber arrangements [3] or parts of packings [19]. Kaesler et al. [20] used an Eulerian-Eulerian approach to model the O

_{2}transfer in an prototype oxygenator and treated blood as two phases, red blood cells and blood plasma, to account for the heterogenous distribution of hemoglobin. As the concentration polarization layer in blood is the main transport resistance [21], most simulations include only the blood side of the oxygenator. Taskin et al. [22] additionally resolved the membrane wall and simulated the oxygen transport in a small packing of an experimental oxygenator. Harasek et al. [23] extended this approach and included the hollow fiber lumen to the simulation domain. Deviation of the real packing from an ideal packing states a challenge when generating computational meshes. D’Onofrio et al. [24] proposed a workflow to convert microcomputer tomography data of an oxygenator into a computational grid. To conclude, CFD simulations of oxygenator membrane modules are relatively advanced. Nevertheless, a significant gap in geometric scale and complexity can be observed in the literature when comparing flow and species transport simulations of oxygenator membrane modules.

- Flow simulations for computation of the velocity field in the complete geometry,
- Identification of characteristic velocity components with significant influence on the transmembrane species transport,
- Development of a reduced packing geometry based on the velocity distribution,
- Numerical conversion of the characteristic velocity to an inlet velocity for the reduced geometry,
- Species transport simulations of the reduced geometry to predict transmembrane flux,
- Upscaling of the transmembrane flux to predict the total transmembrane transport of the whole module.

_{2}removal process of an oxygenator prototype.

## 2. Experimental and Numerical Methods

#### 2.1. Ex Vivo Tests

_{2}removal performance of a prototype oxygenator (see Figure 2) was determined. As test animals, two pigs were chosen, which were provided by the teaching and research farm of the University of Veterinary Medicine, Vienna. All tests were performed under the ethics proposal ZI. 8/115-97/98. For stable experimental conditions, the animals were sedated and mechanically ventilated via an endotracheal tube. Oxygen saturation and CO

_{2}partial pressure were controlled via the ventilator (Servo 900C, Siemens). To maintain physiological blood pressure, Ringer’s solution (mixture of sodium chloride, sodium lactate, potassium chloride, and calcium chloride in water) was administered. To prevent blood coagulation, which could cause clotting of the hollow fiber bundles, heparin was injected intravenously. Blood pressure, cardiac output (blood flow rate provided by the heart), body temperature (approximately 37 °C), and heart rate were monitored (PiCCO plus, Pulsion Medical System). Blood was pumped (BPX-80, Medtronic) from the vena femoralis (largest vein of the leg) out of the pig to the prototype oxygenator and back in, via the vena jugularis (largest vein in the neck, Figure 2a). These veins were chosen because they are capable of providing and taking over the necessary blood flow rates. Blood flow rate was measured with a clamp-on ultrasonic flow probe (SONOFLOW CO.55/080). CO

_{2}was removed from blood flowing at the shell side of the prototype oxygenator. For each measuring point, three blood samples (BG, Figure 2a) were taken before and after the prototype module and analyzed with a blood gas analyzer (ABL500 FLEX, Radiometer Medical A/S). The device allows the measurement of CO

_{2}partial pressure as well as Hematocrit (volume percentage of red blood cells in blood) and other blood parameters relevant for clinical monitoring. The prototype module fiber lumen were swept with pure O

_{2}(1 L STP/min) to remove the CO

_{2}from the circuit. Sweep gas flow of the prototype oxygenator was regulated by a mass flow controller (GF40, Brooks). Flow rate of the outgoing sweep gas flow was recorded using a piston stroke volumetric measurement device (Defender 510, Bios DryCal). Furthermore, CO

_{2}concentration of the sweep gas flow exiting the prototype module was measured (BINOS 100 M, Emerson). By combining volumetric and concentration measurement, the CO

_{2}removal was determined. A parameter study was conducted to examine the influence of CO

_{2}partial pressure and blood flow on the CO

_{2}removal. CO

_{2}partial pressure of blood entering the prototype oxygenator was set to three different levels (50, 70, 100 mmHg). For each partial pressure, three blood flows (1000, 1300, 1600 mL/min) were tested. Pressure was measured before and after the modules on blood and gas side (PR, Figure 2a), using miniaturized pressure transmitters (AMS 4711, Analog Microelectronics).

#### 2.2. Computational Fluid Dynamics

_{2}removal of the prototype oxygenator described in Section 2.1. However, the method can be used generically for membrane processes where there is no significant influence of transmembrane transport on hydrodynamic transport. All CFD simulations were carried out using the opensource toolbox OpenFOAM

^{®}4.1 (ESI Group). The simulations were run on server nodes equipped with 32 core CPUs (16 cores in two physical modules, EPYC 7351, AMD).

#### 2.2.1. Flow Simulation of the Complete Membrane Module

^{®}. The final computational mesh used for flow simulations contained 32 Mio. cells. Mesh resolution within the packing was matched to the cell refinement of the reduced geometry (Section 2.2.3). Average cell size outside the packing was significantly larger to reduce computational costs. The cell length in the longitudinal direction was set uniformly to 0.35 mm. Cell size in the radial and tangential direction variated from 0.087 (cells close to walls or the packing) to 0.175 mm (cells within the main flow), depending on the position in the module. Flow within the packing can be assumed laminar based on the Reynolds number (Re

_{packing}= u·L/ν ≈ 0.1 m/s · 200 µm / 2.3E-6 m

^{2}/s ≈ 8.7). In contrast, the Reynolds number of the inlet pipe is slightly elevated at maximum blood flow rates (1600 mL/min, Re

_{inlet}= u·L/ν ≈ 2.1 m/s · 4 mm / 2.3E-6 m

^{2}/s ≈ 3650). Hence, additional flow simulations were conducted utilizing the k-ω turbulence model. These turbulent flow simulations showed similar flow distribution and pressure loss when compared to laminar flow simulations. Consequently, the numerically less expensive method (laminar flow simulation) was chosen. The governing equations were discretized utilizing second order schemes (Van Leer). A uniform inlet velocity at the beginning of the inlet pipe was matched to inlet flow rates ranging from 1000 to 1600 mL/min. At all walls, including membrane surfaces, a no-slip velocity boundary condition was applied. At the end of the outlet pipe, a fixed uniform value of 0 Pa was set for relative pressure. The remaining boundary conditions for velocity and pressure were set to zero gradient (Neumann conditions). Blood was modelled as a single phase fluid. To account for the shear thinning rheological behavior of porcine blood, a power law viscosity model (Equation(1)) was utilized [26] and fitted to experimental data (Section 3.1). The power law was extended by an upper and lower limiter. At high shear rates, the viscosity converges towards its minimum, the Newtonian viscosity (µ

_{min}= µ

_{Newtonian}). To improve numerical stability during the first solver iterations, viscosity is limited to max 100 Pa s at low shear rates.

_{min}, min(µ

_{max}, µ

_{0}× ɣ̇

^{η}

^{-1}))

#### 2.2.2. Derivation of the Reduced Geometry, Computation of Inlet Velocities

**x**

_{j}) and the velocity magnitude |

**U**|:

**x**

_{j}·

**U**÷ |

**U**|

_{radial},

**x**

_{j}=

**x**

_{radial}) returns values close to one within the packing (Section 3.1). In this study, the radial velocity component was therefore considered as the characteristic velocity for calculation of the cross flow in the reduced geometry. It can be calculated by projecting the velocity field on a polar coordinate system with the coordinate plane perpendicular to the module length axis (Figure 3a).

_{radial}).

_{max}= u

_{radial}) between two fibers. The uniform inlet velocity of the reduced geometry can be adapted iteratively to produce velocity profiles within the reduced geometry, which fit to the maximum velocities of the complete geometry. In Figure 5a, the velocity distribution between fibers is illustrated. The presented profiles were gained by the quasi two-dimensional CFD simulations of the reduced geometry (full lines, Figure 5a). They represent the distribution of radial velocities in the packing of the complete geometry. In addition to the profiles, their average is inserted into the graph (ū, dashed lines, Figure 5a). The average is calculated based on the arithmetic mean of the velocity profiles between the fibers. As can be seen in Figure 5a, following correlation between maximum (u

_{max}) and mean (ū) velocity was found to be in good agreement with the CFD simulations of the reduced geometry:

_{max}÷ (2)

^{0.5}= u

_{radial}÷ (2)

^{0.5}

_{inlet}, dotted and dashed line, see Figure 5b) for the reduced packing segment can be calculated using the mean velocity, the inlet boundary area (A

_{inlet}), and the cross-section between two fibers (A

_{spacing}). Utilizing Equation (3) and Equation (4) allowed the avoidance of the iterative determination of the uniform inlet velocity.

_{inlet}= ū × A

_{spacing}÷ A

_{inlet}

#### 2.2.3. Species Transport Simulations of the Reduced Geometry

^{®}. It comprises 8000 hexahedron cells in the cross-section. At the membrane surface, the mesh was refined to adequately resolve the concentration polarization. Therefore, ten cell layers were applied. The cell layer thickness was decreased successively towards the membrane with an expansion ratio of 5 (ratio of most outer layer thickness to most inner layer thickness). The thickness of the most inner layer measured 1.4 µm. Grid convergence index (GCI) [27], determined for the used mesh, predicts an error due to the discretization of about 3%.

^{®}. The transport equations of the single regions, which are divided by the membrane surface, are computed separately but are coupled by transmembrane transport (J

_{i}). The later is implemented as a volumetric source term for cells attached to the membrane surface in all solved transport equations. It is calculated based on membrane permeance (P) and membrane surface area (A) of the computational cell. Since membrane resistance was found to be increased as a result of the ex vivo trials, pure gas permeances of used fibers were applied for the CO

_{2}transport simulation. As a driving force, the partial pressure difference (Δp

_{i}) between the computational cell and its neighbor cell, on the other side of the membrane, is utilized. If there is no direct neighbor cell, the partial pressure can be interpolated based on nearby values using various schemes.

_{CO}

_{2}= P × A × Δp

_{CO}

_{2}

_{2}partial pressure (the driving force of transmembrane transport) is located at the blood side and the outer selective membrane layer. Hence, the porous sub-structure and the lumen of the hollow fiber membrane were neglected and not added as an additional region to the computational domain. The multi-region solver membraneFoam was therefore utilized in a single region mode. Based on previous findings [23], the partial pressure on the gas side of the selective layer was assumed to be uniformly 0 mmHg. This reduces Eq(5) for CO

_{2}to:

_{CO2}= P × A × (p

_{CO2}

_{,Blood}− 0)

_{2}transport, the different CO

_{2}-related species, dissolved CO

_{2}and bicarbonate (HCO3

^{−}), were summarized to one total CO

_{2}species. The CO

_{2}partial pressure (p

_{CO2}) was then calculated based on the concentration of this single species (c

_{CO2}

_{,total}) [29].

_{CO2}

_{,total}= q × p

_{CO2}

^{t}with q = 0.128 and t = 0.369

_{2}(D

_{CO2}

_{,total}) was derived by enforcing the diffusion of total CO

_{2}to be equal to the combined diffusion of dissolved CO

_{2}and bicarbonate. D

_{CO2}

_{,total}can then be expressed as a function of the dissociation slope (λ = (δ c

_{CO2}

_{,total}/ δ p

_{CO2}

_{,Blood})), the solubility of dissolved CO

_{2}(α

_{CO2}), and the diffusion coefficients of dissolved CO

_{2}(D

_{CO2}) and bicarbonate (D

_{HCO3}

^{-}). Thereby, λ can be assumed constant at clinically relevant CO

_{2}partial pressures above 50 mmHg. All parameter values, necessary for the calculation of D

_{CO2}

_{,total}, are summarized in Table 1.

_{CO2}

_{,total}= D

_{HCO3}

^{-}+ (D

_{CO2}− D

_{HCO3}

^{-}) × α

_{CO2}÷ λ

_{2}transport simulations, the finite volume formulation of the transient, laminar, and incompressible Navier–Stokes equation was solved by applying the Pressure Implicit Method for Pressure-Linked Equations (PIMPLE) algorithm. Discretization schemes and the viscosity model were set equally to the flow simulations described above (Section 2.2.1). The inlet velocity was calculated based on Equation (3–4) and was applied uniformly on the inlet patch. Mass fractions of blood and CO

_{2}were set to match the CO

_{2}partial pressure at the blood inlet of the prototype oxygenator during the ex vivo trials. To model the influence of neighboring fibers, symmetry conditions were applied at the sides of the geometry. The remaining boundary conditions were set analogous to the flow simulation.

_{2}removal (J

_{CO2}

_{,reduced geometry}) can be up-scaled relatively simply to the complete geometry (J

_{CO2}

_{,complete geometry}) by the use of the membrane area of complete and reduced geometry (A

_{membrane,i}):

_{CO2}

_{,complete geometry}= J

_{CO2}

_{,reduced geometry}× A

_{membrane,complete geometry}÷ A

_{membrane,reduced geometry}

_{2}is transported from the sweep gas into the blood via the membrane; this study solely focuses on the CO

_{2}transport. The dependency of the CO

_{2}solubility from the O

_{2}saturation (Haldane effect), or any other coupling between transmembrane CO

_{2}and O

_{2}transport, was neglected.

## 3. Results and Discussion

_{2}removal performance of a prototype oxygenator was determined in ex vivo trials and predicted by our upscaling method. The experimentally and numerically determined results are compared in the following section. While the presented results are specific, the proposed method itself is generic and can be applied to a wide range of hollow fiber membrane separation processes where hydrodynamic transport is not influenced by transmembrane transport.

#### 3.1. Hydrodynamic Results

_{0}= 8.81 mPa s and n = 0.792, with an acceptable coefficient of determination (R

^{2}= 0.96). The whole blood viscosity, which converges towards Newtonian viscosity at high shear rates, amounts to 2.38 mPa s (µ

_{min}). A maximal viscosity of 19.4 mPa s was measured at shear rates of 1.0 s

^{−1}. In Figure 6, the correlation between shear rate and dynamic viscosity given by the power law model is compared to the experimental data.

_{2}transport simulations).

_{max}), averaged for multiple positions (Figure 3a). The maximum radial velocities were computed based on the flow simulation results of the complete geometry. The inlet velocity for the reduced geometry was calculated by using Eq(3) and Eq(4). It increases approximately linear (R

^{2}= 0.98) with the blood flow rate (Figure 9).

#### 3.2. Species Transport Results

_{2}removal determined in the ex vivo tests is illustrated. Transmembrane CO

_{2}flux rises slightly with higher blood flow rates. With higher CO

_{2}partial pressures at the blood inlet, this dependency gets stronger. This can be seen by comparing the slopes of the linear regressions shown in Figure 10. They approximate the dependency of the specific CO

_{2}removal from the blood flow rate. The slope for an inlet pCO

_{2}of 100 mmHg is 15 times higher compared to the slope at 50 mmHg. In general, the specific CO

_{2}removal is more sensitive to the inlet pCO

_{2}than to the blood flow rate. The CO

_{2}removal performance predicted by CFD species transport simulations were compared to experimental data (Figure 10). Transmembrane flux is predicted most accurately (within the error margins) at flow rates of 1300 mL/min or at an inlet pCO

_{2}of 70 mmHg. In general, the accuracy of the numerical model is satisfactory. The highest deviation (approximately 7%) occurs at a pCO

_{2}of 100 mmHg and a blood flow rate of 1600 mL/min. On average, the error is low, at 3.2%. The largest errors, and therefore a systemic deviation, can be observed at high pCO

_{2}(100 mmHg). At this stage of research, it cannot be differentiated if these deviations are caused by the CO

_{2}solubility model or by the proposed upscaling method. The upscaling is done relatively simply by multiplying the specific transmembrane CO

_{2}removal of the reduced geometry with the outer membrane area of the complete module.

_{2}removal suggests that the computed inlet velocities for the reduced geometry are representative for the different blood flow rates. Hence, the presented method is able to account for complex flow phenomena. This includes fluctuations of radial velocities along a sample line, with deviations of up to ±84% from the average. Furthermore, the dependency from the radial position in the packing can be considered reasonably. This is important because the radial velocities at the first fiber mat winding are 2.7 times higher compared to the radial velocities in the last winding. While in the reduced geometry blood flow is modelled only as a single pass, the method remains stable at higher blood flow rates, where back flow can be detected in parts of the complete geometry.

_{2}pure gas permeance determined experimentally after the ex vivo tests amounts to 0.7 mL STP/min/bar/m

^{2}.

#### 3.3. Computational Costs

_{2}species balance was normalized with the transmembrane CO

_{2}transport. This relative error (ε) amounted to 2.46% and 0.28% at 2 and 4 s of simulated time, respectively (Figure 11, ε—porcine). As a test case, a species transport simulation was conducted for bovine blood on the mesh of the flow simulation. Due to the less pronounced shear thinning behavior of bovine blood (see Table 3) and shear rates significantly higher than 1.0 s

^{-1}, a Newtonian viscosity model (µ = 5.8 mPa s) was chosen [30]. Membrane permeances were set to pure gas permeances measured for unused membranes (4.23 mL/min/bar/m

^{2}). The remaining settings were defined equally to the species transport simulations for porcine blood.

_{2}removal (j

_{CO2}) of bovine and porcine blood cannot be compared directly. This is due to the different viscosities and the chosen permeances. Nevertheless, the analogous decline of the relative species balance error (Figure 11) indicates that the numerical effort can be considered similar. To compute 2 s of simulated time for the complete geometry, it took 62,244 CPU hours. At this point, the relative species balance error amounted to 16.1%. To reach this level of convergence, the species transport simulation on the reduced geometry takes significantly less computation time. Approximately 200 CPU hours were needed for 0.7 s of simulated time (ε = 16.6%). This equals a reduction of computation time by 99.3% when considering the time effort of both, the species simulation, and the corresponding flow simulation necessary for the prediction of the inlet velocity. The number of cells in the numerical grid was reduced from 32 Mio. cells for the whole module (complete geometry) to 202,000 cells for the reduced packing (cell number decrease of 99.4%).

_{2}removal differs from the converged solution by maximally 10%. The CO

_{2}removal for bovine blood, determined by the simulations of the complete and reduced geometry, deviates by approximately 10% (Figure 11). The error introduced by the reduction of the geometry can therefore be regarded as acceptable. The difference in transmembrane CO

_{2}flux for bovine and porcine blood (j

_{CO}

_{2}, Figure 11) can be explained by the chosen permeances (new fibers—bovine simulation versus used fibers—porcine simulations).

_{2}species balance error below 1%, 2.7 s of simulated time (4.7 times the maximum residence time) have to be calculated on the reduced geometry. As multiples of the maximum residence time have to be simulated to gain simulation results with a high level of convergence (ε < 1%), even more tremendous numerical effort for the complete geometry would arise.

#### 3.4. Radial Dependency of Transmembrane Transport

_{2}/min/m

^{2}), transmembrane transport reduces only slightly (14%) with increasing layer number. As more fibers are accommodated in the outer than in the inner fiber mat layers, multiple averaging methods for transmembrane flux could be considered. In this study, two averaging methods, given in Eq(10), were compared. The arithmetic average (j

_{CO}

_{2,arithmetic average}= 710 mL STP CO

_{2}/min/m

^{2}) weights the transmembrane flux (j

_{CO}

_{2}

_{,i}) of each fiber layer (i) equally. The weighted average (j

_{CO}

_{2,weighted average}= 704 mL STP CO

_{2}/min/m

^{2}) accounts for the fact that the total number of fibers (n) is unequally distributed among the fiber mat layers, which accommodate varying numbers of fibers (n

_{i}). Due to small variations of transmembrane flux between the fiber layers, the two averaging methods give similar results.

_{CO2}

_{,arithmetic average}= (∑ j

_{CO2}

_{,i}) / (∑ i), j

_{CO2}

_{,weighted average}= (∑ (j

_{CO2}

_{,i}· n

_{i})) / (∑ n

_{i}), i … layer number

^{st}section: 0.0–22.5 mm, 2

^{nd}section: 22.5–45.0 mm, 3

^{rd}section: 55.0–77.5 mm, 4

^{th}section: 77.5–100.0 mm) can be tracked in Figure 4. Similar to the complete geometry, the different longitudinal sections show only a small dependency of transmembrane flux from the radial position. The highest standard deviation between fiber layers (10% of average) occurs in the first section, closest to the blood inlet. Nevertheless, weighted averaging, which considers the unequal distribution of fibers among the different fiber layers, could play a crucial role if a high radial dependency of the transmembrane flux is observed.

## 4. Conclusions

- Flow simulations of a complete module to gain the velocity distribution,
- Identification of velocity components characteristic for transmembrane species transport,
- Development of a simplified hollow fiber packing geometry based on flow simulation results,
- Calculation of matching inlet velocities for the reduced geometry, to account for different flow rates in the complete geometry,
- Species transport simulations of the simplified (reduced) geometry for different flow rates and species compositions,
- Upscaling of the transmembrane transport to the complete geometry.

_{2}removal performance of a prototype oxygenator in ex vivo trials. The prediction performance of the proposed method is satisfactory. A mean deviation of 3% between experimental and numerically determined CO

_{2}removal was recorded. The saving of computational costs is significant. For the same level of convergence, the upscaling method reduces the necessary CPU hours to conduct the CFD simulations by 99.3% (factor of 150). The proposed method can therefore be considered as a significant reduction in computational effort for CFD species transport simulations of hollow fiber membrane modules and similar units.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Acronyms | |

CFD | Computational Fluid Dynamics |

CO_{2} | Carbon dioxide |

HCO3^{−} | Bicarbonate |

Latin Symbols | |

A | Membrane area of a computational cell attached to the membrane surface |

A_{membrane,i} | Membrane area of reduced (i = reduced) or complete geometry (i = complete) |

A_{inlet} | Flow cross-section at inlet |

A_{spacing} | Area between two fibers |

c_{CO2,total} | Total CO_{2} concentration (dissolved CO_{2} and bicarbonate) |

D_{CO2} | Diffusivity of CO_{2} in blood |

D_{CO2,total} | Diffusivity of total CO_{2} in blood |

D_{HCO3}^{-} | Diffusivity of HCO3^{−} in blood |

E(t) | Residence time distribution |

F(t) | Cumulative distribution function |

J_{CO2} | Transmembrane CO_{2} transport |

j_{CO2} | Transmembrane CO_{2} flux (Transmembrane CO_{2} Transport per membrane area) |

L | Characteristic length of Reynolds number |

n | Total number of fibers |

n_{i} | Number of fibers in fiber layer (fiber mat winding) i |

P | Permeance |

p_{CO2} | CO_{2} partial pressure |

p_{i} | Partial pressure of component i |

q | Empirical coefficient of CO_{2} solubility model |

Re | Reynolds number |

t | Empirical exponent of CO_{2} solubility model |

u | Characteristic velocity of Reynolds number |

U | Velocity vector field |

ū | Mean velocity between two fibers |

u_{inlet} | Uniform inlet velocity |

u_{max} | Maximum velocity between two fibers |

u_{radial} | Velocity component in radial direction |

x_{j} | Unit vector in direction j |

Greek Symbols | |

α_{CO2} | Solubility of CO_{2} in blood |

ɣ̇ | Shear rate |

Δp_{CO2} | CO_{2} partial pressure difference |

Δp_{i} | Partial pressure difference of component i |

ε | CO_{2} species balance error normalized with transmembrane CO_{2} transport |

η | Empirical exponent of power law model |

λ | Slope of CO_{2} dissociation curve |

µ | Dynamic viscosity |

µ_{0} | Empirical coefficient of power law model |

µ_{max} | Maximum viscosity of whole blood at low shear rates |

µ_{min} | Minimum viscosity of whole blood at high shear rates |

µ_{Newtonian} | Newtonian viscosity of whole blood (at high shear rates) |

ν | Kinematic viscosity |

τ | Mean residence time |

ψ_{j} | Fraction of velocity component in direction j and velocity magnitude |

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**Figure 1.**Workflow of the presented upscaling method for prediction of hydraulic and transmembrane transport on a macroscopic scale.

**Figure 2.**Ex vivo test set up: (

**a**) Scheme of ex vivo loop with pig, prototype oxygenator, blood pump, pressure sensors (PR), flow rate sensors (FR), and CO

_{2}concentration sensor (QR); (

**b**) Blood guiding parts of the prototype oxygenator with cut through the membrane packing.

**Figure 3.**Packing geometry: (

**a**) Scheme of the prototype oxygenator fiber arrangement, including sampling line positions and polar coordinate system; (

**b**) Slice of fiber potting to illustrate real geometry;

**(c)**Scheme of reduced geometry for CO

_{2}transport simulations.

**Figure 4.**Blood guiding parts of the prototype oxygenator (complete geometry) with a cut through the membrane packing. Red dotted lines: line sources at the angular position ϕ = 0° used for sampling of radial velocity profiles, shown in Section 3.1. Turquoise dot-and-dashed line: cross-sections at five different longitudinal positions (20, 40, 50, 60, and 80 mm) used for velocity magnitude and radial velocity fraction contour plots, shown in Section 3.1.

**Figure 5.**(

**a**) Velocity distribution and velocity average between fibers gained by computational fluid dynamics (CFD) simulations of reduced geometry; (

**b**) Uniform inlet velocity condition (reduced geometry) in front of the fibers. Fibers are inserted schematically into the graph.

**Figure 6.**Shear rate dependency of dynamic viscosity given by the power law model fitted to porcine blood. The Newtonian viscosity (2.38 mPa s) is entered in the graph for comparison.

**Figure 7.**(

**a**) Pressure drop in the prototype module in dependency of blood flow rate, determined by experiments (

**+**) and CFD (▲, flow simulations complete geometry); (

**b**) Radial velocity profiles determined by CFD flow simulations of the complete geometry.

**Figure 8.**Contour plots at five different longitudinal positions of the complete geometry (Figure 4): (

**a**) Velocity magnitude; (

**b**) Fraction of radial velocity component and velocity magnitude.

**Figure 9.**Uniform inlet velocity for reduced geometry and maximum radial velocity between two fibers averaged for multiple positions in the complete geometry.

**Figure 10.**Experimentally and numerically determined specific CO

_{2}removal at different flow rates and pCO

_{2}at the blood inlet of the prototype oxygenator.

**Figure 11.**Convergence of specific CO

_{2}removal (j

_{CO}

_{2}) and decline of relative CO

_{2}species balance error (ε) with increasing simulated time.

**Figure 12.**Residence time distribution, E(t), and cumulative distribution function, F(t), for (

**a**) reduced geometry; (

**b**) complete geometry.

**Figure 13.**Average specific CO

_{2}removal of the eight different fiber mat layers (complete geometry) or fiber positions (reduced geometry). Average specific CO

_{2}removal (complete geometry) is additionally given for four different longitudinal sections (1

^{st}: 0.0–22.5 mm, 2

^{nd}: 22.5–45.0 mm, 3

^{rd}: 55.0–77.5 mm, 4

^{th}: 77.5–100.0 mm, see Figure 4).

**Table 1.**Summary of diffusion model parameters. Values were taken from [29].

Notation | Description | Value | Units |
---|---|---|---|

α_{CO2} | Solubility of CO_{2} in blood | 6.62 E-04 | mL CO_{2} STP/mL/mmHg |

D_{CO2} | Diffusivity of CO_{2} in blood | 4.62 E-10 | m^{2}/s |

D_{HCO3}^{-} | Diffusivity of HCO_{3}^{-} in blood | 7.39 E-10 | m^{2}/s |

λ | Slope of CO_{2} dissociation curve | 4.25 E-03 | mL CO_{2} STP/mL/mmHg |

**Table 2.**Maximum radial velocity between two fibers averaged for multiple positions in the complete module and inlet velocity for the reduced geometry at different blood flow rates.

Blood Flow Rate | Maximum Radial Velocity (u_{max}) | Inlet Velocity (u_{inlet}) |
---|---|---|

1000 mL/min | 0.061 m/s | 0.015 m/s |

1300 mL/min | 0.098 m/s | 0.024 m/s |

1600 mL/min | 0.126 m/s | 0.031 m/s |

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## Share and Cite

**MDPI and ACS Style**

Lukitsch, B.; Ecker, P.; Elenkov, M.; Janeczek, C.; Haddadi, B.; Jordan, C.; Krenn, C.; Ullrich, R.; Gfoehler, M.; Harasek, M.
Computation of Global and Local Mass Transfer in Hollow Fiber Membrane Modules. *Sustainability* **2020**, *12*, 2207.
https://doi.org/10.3390/su12062207

**AMA Style**

Lukitsch B, Ecker P, Elenkov M, Janeczek C, Haddadi B, Jordan C, Krenn C, Ullrich R, Gfoehler M, Harasek M.
Computation of Global and Local Mass Transfer in Hollow Fiber Membrane Modules. *Sustainability*. 2020; 12(6):2207.
https://doi.org/10.3390/su12062207

**Chicago/Turabian Style**

Lukitsch, Benjamin, Paul Ecker, Martin Elenkov, Christoph Janeczek, Bahram Haddadi, Christian Jordan, Claus Krenn, Roman Ullrich, Margit Gfoehler, and Michael Harasek.
2020. "Computation of Global and Local Mass Transfer in Hollow Fiber Membrane Modules" *Sustainability* 12, no. 6: 2207.
https://doi.org/10.3390/su12062207