Hollow fiber membranes (HFM) bundles are incorporated into devices used as blood oxygenators for cardiopulmonary bypass for cardiac surgery and in next generation respiratory assist devices used as artificial lungs for patients with failing lungs.1–5 Computational fluid dynamics (CFD) is often used as a tool to design these devices.6–10 Modeling the flow at the fiber level of the bundle can be computationally difficult as the bundles are composed of thousands of individual fibers. Instead modeling approaches typically treat the fiber bundle as a packed bed or porous medium in which the effect of local fluid drag from the individual fibers is incorporated using a Darcy permeability coefficient for the fiber bundle.11–14 The Darcy permeability, k, is estimated using the empirical Blake–Kozeny (BK) equation:
in which ε is the bundle porosity and Dp2 is the effective fiber diameter.15
We recently published a study assessing how well the Darcy permeability of hollow fiber bundles can be predicted using the BK equation. We determined that the prediction of Darcy permeability can be significantly improved whether the constant in Equation (1),15A = 150, is empirically correlated to fiber bundle porosity using A = 542ε − 128. These studies were done using seven different fiber bundles constructed from Celgard microporous polypropylene hollow fiber membrane mats with several different fiber sizes, fiber spacing in the mats, and fiber orientation between adjacent fiber mat layers within the fiber bundle. Increasingly, HFM bundles used in clinical blood oxygenators and in respiratory assist devices under development are using Membrana polymethylpentene (PMP) hollow fiber mats.1–3,16–18 The Membrana PMP fiber has an asymmetric membrane wall with closed surface pores, rather than a microporous membrane wall, to prevent blood plasma wetting, which can adversely affect gas exchange and device function.3,19 The Membrana fiber mat has larger fibers than the largest Celgard fiber used in our previous study (380 μm vs. 300 μm outer diameter) and a different fiber density (44 vs. 51 fibers per inch). Because the differences in fiber arrangement may affect flow at the fiber level, we wanted to assess how well our empirical correction to the BK constant could predict the Darcy permeability of fiber bundles made from the Membrana PMP fiber mats. This brief report summarizes the findings of our study.
All manufacturing and test methods were the same as previously reported15 with the exception that the circular fiber swatches used in our experimental apparatus were constructed from Membrana GmbH (Wuppertal, Germany) Oxyplus PMP hollow fiber mats (380 μm outer diameter, 44 fibers/inch density). High porosity swatches were created by increasing fiber spacing in the mats by removing every other fiber. The circular swatches were mounted at the bottom of a plastic tube and a pure glycerol solution (average nominal kinematic viscosity ν = 400 cSt) was poured into the tube above the swatches. Measuring the time interval, Δt, for the glycerol solution to flow from an initial height, hi, to a final height, hf, in the tube provided the Darcy permeability of the fiber swatch using the relation
derived previously, where δ is the thickness of the fiber bundle swatch, and g is gravitational acceleration.15 Values of Δt versus ln were averaged over two runs for each height ratio used, and a linear regression to these data provided the Darcy permeability from Equation (2). This flow-through test is a controlled and simple test setup in which flow is driven by the force of gravity. The setup also works best for a very viscous fluid like glycerol, which ensures that the net pressure force per unit volume in the fiber swatch is predominantly overcoming viscous forces per unit volume has given by Darcy’s law.
Results and Discussion
Darcy permeability values measured for each fiber swatch tested are shown in Table 1 along with their individual coefficients of variation (CV). Coefficients of variation values ranged from 3.1% to 15% with the maximum CV occurring in the parallel-arranged fiber bundles, consistent with the findings of Pacella et al.15 Darcy permeability values predicted based on Equation (1) and the BK constant given by A = 542ε − 128 are shown in Table 1 for comparison. The percent difference between measured and predicted Darcy permeability ranged from −8.3% to 6.7%.
The BK constant was determined for each of the fiber swatches in this study and a linear regression of these data versus porosity, combined with the data from Pacella et al.15 yielded the new correlation: A = 497ε − 103, which has an R2 value of 0.9 compared with 0.8 in the Pacella article. The percent difference between measured and predicted Darcy permeability using this new correlation ranged from −5.7% to 3.6%. Furthermore, the percent difference between this new correlation and the Pacella correlation is shown in Figure 1 over a relevant range of porosity from 0.4 to 0.8. The percent difference ranged from 8.0% to −3.5 %. Conceivably, our permeability correlation may not work as well if high porosities were achieved by removing alternating fiber layers from a swatch. We believe, however, that manufacturing devices of high porosities by removing fiber layers is challenging and we have not seen clinical or experimental devices created in this manner. Fiber spacing however is increased to increase porosity as mats with different fiber spacing are commercially available. Removing every other fiber in our experiment represents an extreme of this. In addition, we would expect our correlation to predict permeability of commercial Membrana GmbH Oxyphan 50/280 type PP fiber membranes as well owing, to the small percentage difference between the old and new correlations.
One should note whether blood or glycerol was used, permeability measured would be unchanged as long as the appropriate viscosity is used in Darcy’s law, since permeability is a material property of a given porous medium.
The Darcy permeability of hollow fiber bundles made from commonly used commercial Membrana PMP hollow fiber mats used in blood oxygenation devices can be predicted within ±6% if the constant in the BK equation (Eq. 1), A = 150, is empirically correlated to fiber bundle porosity using A = 497ε − 103 with a larger R2 value, as opposed to within ±8% using the previous correlation.
This publication was made possible by Grant Number 5R01HL117637-03 from the National Institutes of Health, National Heart, Lung and Blood Institute. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of NIH.
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