# Analysis of the Shear Stresses in a Filling Line of Parenteral Products: The Role of Fittings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}) [7].

^{−1}). In addition, they found that the duration of stress application ($\tau $) played a determining role. Therefore, they introduced a quantity, later identified as “shear history” ($SH$), to quantify shear stress as a function of residence time [7]:

^{−1}[10,11]. Therefore, they raised the hypothesis that in previous works, deactivation was more due to interfacial phenomena than shear stress. Indeed, it is difficult to isolate the contribution of shear stress from interfacial stress, which has been extensively studied instead [12,13,14]. Furthermore, in a study conducted by Murphy et al. [15], higher shear rates were investigated, reaching up to ${10}^{6}$ s

^{−1}. It was observed that under these conditions, aggregation of monoclonal antibodies was detected. Another recently popular theory holds that product instability is more due to the combined effect of interfacial and shear stress, and it is unlikely that shear stress alone causes protein aggregation [16,17]. While interfaces play an important role in protein adsorption, unfolding, and film formation, shear forces could release aggregates in the bulk solution and, hence, renew the available surface [3]. In this controversial landscape, it becomes necessary to characterize the shear stress distribution in bioprocessing units.

## 2. Governing Equations and Theoretical Background

^{−1}), $R$ is the tubing radius, and $Q$ is the volumetric flowrate (m

^{3}s

^{−1}).

## 3. Numerical Set Up

^{−1}were arbitrarily chosen among the ranges of possible flowrates industrially used for these systems in order to explore both laminar and turbulent conditions, respectively [27]. Under laminar conditions, for example, a low velocity was deliberately chosen to ensure that, after flow deviation, no turbulence would develop due to an increase in velocity.

## 4. Results

#### 4.1. Velocity and Residence Time Study

^{−1}), the majority of particles take between 1 and 6 s to cross the domain, while some particles take substantially longer—between 6 and 40 s. These high residence time values are caused by particles passing through the dead zone, indicated by the black solid circles in Figure 5b. Additionally, particles that deviate from their original direction and come close to the wall experience the no-slip condition, entering the lower velocity boundary layer and are indicated by the dotted circles. For the sake of clarity, particles passing through the dead zone with a residence time lower than 6 s are also identified by the red solid circles in the same figure. The upper limit used for this analysis, i.e., 6 s, was identified using the Tukey test [34].

^{−1}. The average velocity in Figure 6a is slightly higher because particles were released onto a surface that was smaller than the actual tubing surface, thus ignoring some of the slowest particles.

#### 4.2. First Case Study: Laminar Flow

#### 4.2.1. Shear Stress Distribution

#### Approach 1—Maximum Shear Stress per Streamline

#### Approach 2—Damage Factor

#### Approach 3—Damage Fitting Factor

#### Approach 4—Damage Critical Factor

#### Approach 5—Time-Averaged Shear Stress

#### Approach 6—Time-Averaged Shear Stress weighted on Flowrates

#### 4.3. Second Case Study: Turbulent Flow

#### Shear Stress Distribution

#### 4.4. Comparison with Shear Stress in Straight Tubing

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A$ | streamline area of influence, m^{2} |

${C}_{f}$ | skin friction factor, - |

$CV$ | coefficient of variation, - |

$DCF$ | damage critical factor, - |

$DF$ | damage factor, - |

$DFF$ | damage fitting factor, - |

$k$ | turbulent kinetic energy, m^{2} s^{−2} |

$m$ | number of streamlines, - |

$n$ | factor, - |

$p$ | fluid pressure, Pa |

$Q$ | volumetric flowrate, m^{3} s^{−1} |

$r$ | distance from the center, m |

$R$ | tubing radius, m |

$Re$ | Reynolds number, - |

$St$ | Stokes number, - |

$SH$ | shear history, - |

$t$ | time, s |

$u$ | fluid velocity, m s^{−1} |

$\overline{u}$ | average fluid velocity, m s^{−1} |

${u}^{t}$ | friction velocity, m s^{−1} |

$w$ | volumetric flowrate weight, m^{3} s^{−1} |

$x$ | spatial coordinate, m |

$y$ | absolute distance from the wall, m |

${y}^{+}$ | sub-layer scaled distance, - |

${y}_{w}$ | distance to the first cell center normal to the wall, m |

Greek letters | |

$\beta $ | turbulent model constant, - |

$\gamma $ | shear rate, s^{−1} |

$\mathsf{\Delta}$ | difference, - |

$\epsilon $ | turbulent dissipation rate, m^{2} s^{−3} |

$\kappa $ | turbulent kinetic energy, m^{2} s^{−2} |

$\mu $ | dynamic fluid viscosity, kg m^{−1} s^{−1} |

$\nu $ | kinematic viscosity, m^{2} s^{−1} |

${\nu}_{T}$ | turbulent kinematic viscosity, m^{2} s^{−1} |

$\rho $ | fluid density, m^{3} kg^{−1} |

$\sigma $ | shear stress, Pa |

$\tau $ | residence time, s |

$\Gamma $ | filtered shear rate, s^{−1} |

${\tau}_{f}$ | fluid time scale, s |

${\tau}_{p}$ | particle response time, s |

$\omega $ | specific turbulent dissipation rate, s^{−1} |

Subscripts | |

$fit$ | fitting |

$i$ | index |

$j$ | index |

$lam$ | laminar |

$max$ | maximum |

$tot$ | total |

$tub$ | tubing |

$turb$ | turbulent |

$wall$ | wall |

Abbreviations | |

CDF | Cumulative Distribution Function |

CFD | Computational Fluid Dynamics |

GAMG | Geometric Agglomerated Algebraic Multigrid |

Probability Distribution Function | |

RANS | Reynolds Averaged Navier–Stokes |

SST | Shear Stress Transport |

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**Figure 1.**Geometries analyzed. (

**a**,

**b**) display the real geometries focused on in the present study for T- and Y-fittings with an internal diameter of 9.53 mm. (

**c**,

**d**) represent the modeled geometries obtained, where the clamped outlets are colored in red.

**Figure 2.**Numerical verification and validation analyses for T-fitting under laminar conditions: grid independence analysis (

**a**), the percent error between the numerical and analytical shear stress in the inlet branch of the fitting (where the fluid dynamics develop as in a tubing) is monitored by varying the number of cells (

**b**).

**Figure 3.**Details of the mesh for T-fitting under laminar conditions: outer mesh (

**a**) and inlet patch mesh (

**b**).

**Figure 4.**A total of 520 particles are followed through their streamlines, and their trajectories and velocities are displayed for T-fitting (

**a**) and Y-fitting (

**b**) under laminar conditions.

**Figure 5.**(

**a**) A total of 520 seeds were randomly released at the inlet of the connector on a surface with a diameter equal to 90% of the tubing diameter. (

**b**) Particles’ residence times. As visible from the x scale, 520 particles were released from the inlet of the connector.

**Figure 6.**Average velocity calculated based on streamline information (

**a**). The related coefficients of variation ($CV$) are reported in (

**b**).

**Figure 7.**Frequency histograms of the maximum shear stress for T- (

**a**) and Y- (

**c**) fittings under laminar conditions. PDF (continuous line) and CDF (dotted line) are also presented for T- (

**b**) and Y- (

**d**) fittings.

**Figure 8.**Frequency histograms of the $DF$ for T- (

**a**) and Y- (

**c**) fittings under laminar conditions. $PDF$ (continuous line) and $CDF$ (dotted line) are also presented for T- (

**b**) and Y- (

**d**) fittings.

**Figure 9.**Average shear stress according to Approach 5 (

**a**) and Approach 6 (

**b**) is monitored by varying the number of streamlines and the number of tests per streamline for T-fitting under laminar conditions. (

**c**,

**d**) refer to Y-fitting.

**Figure 10.**Velocity profile in the cross-section of the straight part of the connector is shown in panel (

**a**). Trends of turbulent properties $\kappa $, $\omega $ and ${\nu}_{T}$ are presented in panels (

**b**–

**d**), respectively.

**Figure 11.**Average shear stress according to Approach 5 (

**a**) and Approach 6 (

**b**) is monitored by varying the number of streamlines and the number of tests per streamline for T-fitting under turbulent conditions. Panels (

**c**,

**d**) refer to Y-fitting.

**Table 1.**Boundary conditions (at the wall and the inflow patch) are expressed in terms of OpenFOAM classes.

Turbulence Property | Boundary Condition at the Wall | Estimation |
---|---|---|

$\kappa $ | fixedValue or kLowReWallFunction | $1\times {10}^{-10}$ |

$\omega $ | omegaWallFunction | $10\frac{6\nu}{\beta {y}_{w}^{2}}$ |

${\nu}_{T}$ | nutLowReWallFunction | $0$ |

Turbulence Property | Boundary Condition at the Inflow Patch | Estimation |

$\kappa $ | fixedValue | $1\times {10}^{-10}$ |

$\omega $ | fixedValue | $10\frac{6\nu}{\beta {y}_{w}^{2}}$ |

${\nu}_{T}$ | calculated | $0$ |

Boundary Condition | ||
---|---|---|

Patch | $\mathit{p}$ | $\mathit{u}$ |

inlet | zeroGradient | $\mathrm{fixedValue}(\overline{u}$) |

outlet | uniformValue (0) | zeroGradient |

wall | zeroGradient | noSlip |

end | symmetry | symmetry |

**Table 3.**Comparison in $\overline{DF}$, $\overline{DFF}$, and $\overline{DCF}$ for T- and Y-fittings under laminar conditions.

Case | $\mathit{v},\mathbf{m}{\mathbf{s}}^{-1}$ | $\overline{\mathit{D}\mathit{F}},-$ | $\overline{\mathit{D}\mathit{F}\mathit{F}},-$ | $\overline{\mathit{D}\mathit{C}\mathit{F}},-$ |
---|---|---|---|---|

T- | $0.042$ | $97.30$ | $33.93$ | $2.89$ |

Y- | $0.042$ | $105.10$ | $44.84$ | $0.16$ |

**Table 4.**Comparison in the shear stress distribution between fitting and tubing [21] under laminar and turbulent conditions.

Case | Regime | $\overline{\mathit{u}},\mathbf{m}{\mathbf{s}}^{-1}$ | ${\overline{\mathit{\sigma}}}_{\mathit{t}\mathit{u}\mathit{b}},\text{}\mathbf{Pa}$ | ${\overline{\mathit{\sigma}}}_{5,\mathit{f}\mathit{i}\mathit{t}},\text{}\mathbf{Pa}$ | ${\overline{\mathit{\sigma}}}_{6,\mathit{f}\mathit{i}\mathit{t}},\text{}\mathbf{Pa}$ | ${\mathbf{\Delta}}_{\mathit{t}\mathit{u}\mathit{b},\mathit{f}\mathit{i}\mathit{t}}$ |
---|---|---|---|---|---|---|

T- | Lam | $0.042$ | $1.88\times {10}^{-2}$ | $2.10\times {10}^{-2}$ | $1.92\times {10}^{-2}$ | $2\%$ |

Y- | Lam | $0.042$ | $1.88\times {10}^{-2}$ | $2.08\times {10}^{-2}$ | $1.89\times {10}^{-2}$ | $1\%$ |

T- | Turb | $0.500$ | $7.69\times {10}^{-1}$ | $12.3\times {10}^{-1}$ | $12.1\times {10}^{-1}$ | $57\%$ |

Y- | Turb | $0.500$ | $7.69\times {10}^{-1}$ | $8.13\times {10}^{-1}$ | $7.84\times {10}^{-1}$ | $2\%$ |

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© 2023 by GlaxoSmithKline Biologicals SA. Submitted for possible open access publication 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**

Moino, C.; Scutellà, B.; Bellini, M.; Bourlès, E.; Boccardo, G.; Pisano, R.
Analysis of the Shear Stresses in a Filling Line of Parenteral Products: The Role of Fittings. *Processes* **2023**, *11*, 1797.
https://doi.org/10.3390/pr11061797

**AMA Style**

Moino C, Scutellà B, Bellini M, Bourlès E, Boccardo G, Pisano R.
Analysis of the Shear Stresses in a Filling Line of Parenteral Products: The Role of Fittings. *Processes*. 2023; 11(6):1797.
https://doi.org/10.3390/pr11061797

**Chicago/Turabian Style**

Moino, Camilla, Bernadette Scutellà, Marco Bellini, Erwan Bourlès, Gianluca Boccardo, and Roberto Pisano.
2023. "Analysis of the Shear Stresses in a Filling Line of Parenteral Products: The Role of Fittings" *Processes* 11, no. 6: 1797.
https://doi.org/10.3390/pr11061797