# Numerical Modeling of a Short-Dwell Coater for Bio-Based Coating Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. Numerical Method

^{T})/2, and μ is the viscosity of the color, which is characterized as a constant or zero-shear-rate viscosity for model 1:

_{o}and μ

_{∞}are the viscosities at zero and infinite shear rates, respectively.

_{p}and a purely viscous component τ

_{s}:

_{s}is computed from

_{s}is the viscosity factor for the Newtonian (i.e., purely viscous) component of the extra-stress tensor.

_{p}is computed from

_{p}is the polymer contribution to the viscosity, α is an empirical constant (0 ≤ α ≤ 1) descriptive to shear-thinning phenomena, and ${\tau}_{p}^{\nabla}$ is the upper-convected time derivative of the viscoelastic extra stress defined as

#### 2.2. Boundary Conditions

#### 2.3. Computational Domain

_{min}/H

_{2}and Δy

_{min}/H

_{2}) and the total number of cells (NC) for each mesh are described in Table 1. Similarly, a refined mesh was developed to simulate the flow of viscoelastic fluids in 12:1 axisymmetric contraction flows.

#### 2.4. Solver Formulation

^{−6}. In all the simulation cases, the average velocity at the exit of the narrow channel with imposed open boundary condition was monitored to ensure convergence and consistency of the solutions. Figure 4 shows how the average velocity in the Newtonian models attained absolute steady-state, while with the non-Newtonian viscoelastic cases, the average velocity presented minor fluctuations with flow time at the exit of the blade channel.

## 3. Results and Discussion

_{2})/H

_{2}) was used, where U

_{2}and H

_{2}are the average velocity and the height in the downstream channel, respectively. For the viscoelastic models, two relaxation time constants, i.e., 3 × 10

^{−3}and 3 × 10

^{−5}s, were used to calculate the Deborah number De = 200 and De = 2, respectively.

_{s}is the density of the solvent, and ρ

_{p}is the density of CNC.

_{∞}.

^{−5}to 10

^{−7}depending on the Deborah number. Because an implicit formulation was used for the time integration, the values of time steps were varied based on the convergence test, namely, the residual sum and point monitoring. Note that the flow patterns in Figure 7 correspond to a flow time of 6 s; beyond this time, the flow pattern changed to its early times, revealing a periodic flow.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Computational grid and boundary conditions used in the SDC model, (

**b**) magnified view of the mesh 4 at the entry of the narrow channel.

**Figure 4.**Average velocity at the exit of the blade channel for (

**a**) Newtonian and (

**b**) non-Newtonian viscoelastic fluids.

**Figure 6.**Flow patterns with (

**a**) model 1, (

**b**) model 3, and (

**c**) model 4 in the axisymmetric contraction.

**Figure 7.**Velocity streamlines for (

**a**) model 1, (

**b**) model 2, (

**c**) model 3 (De = 2), (

**d**) model 4 (De = 2), (

**e**) model 3 (De = 200), and (

**f**) model 4 (De = 200).

**Figure 8.**Strain rate distribution for (

**a**) model 1, (

**b**) model 2, (

**c**) model 3 (De = 2), (

**d**) model 4 (De = 2), (

**e**) model 3 (De = 200), and (

**f**) model 4 (De = 200).

**Figure 10.**Stresses along the surface of the paper: (

**a**) normal stress in the x-direction (τ

_{xx}), (

**b**) shear stress (τ

_{xy}), (

**c**) normal stress in the y-direction (τ

_{yy}).

Mesh | NC | Δx_{min}/H_{2} | Δy_{min}/H_{2} |
---|---|---|---|

1 | 9216 | 5 | 0.25 |

2 | 92,980 | 1 | 0.1 |

3 | 181,860 | 1 | 0.1 |

4 | 282,780 | 1 | 0.1 |

**Table 2.**Parameters used in the models [26].

Parameter | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

ρ_{sus} (kg/m^{3}) | 1024.8 | 1024.8 | 1024.8 | 1024.8 |

C (%) | 5 | 5 | 5 | 5 |

μ_{0} (Pas) | 0.3 | 0.3 | - | - |

Μ_{∞} (Pas) | - | 0.05 | - | - |

n | - | 0.5 | - | - |

ξ | - | 1 | - | - |

μ_{s} (Pas) | - | - | 0.001 | 0.001 |

μ_{p} (Pas) | - | - | 0.299 | 0.299 |

α | - | - | - | 0.001 |

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**MDPI and ACS Style**

Gozali, E.; Järnström, L.; Papadikis, K.; Idris, A.
Numerical Modeling of a Short-Dwell Coater for Bio-Based Coating Applications. *Coatings* **2021**, *11*, 13.
https://doi.org/10.3390/coatings11010013

**AMA Style**

Gozali E, Järnström L, Papadikis K, Idris A.
Numerical Modeling of a Short-Dwell Coater for Bio-Based Coating Applications. *Coatings*. 2021; 11(1):13.
https://doi.org/10.3390/coatings11010013

**Chicago/Turabian Style**

Gozali, Ebrahim, Lars Järnström, Konstantinos Papadikis, and Alamin Idris.
2021. "Numerical Modeling of a Short-Dwell Coater for Bio-Based Coating Applications" *Coatings* 11, no. 1: 13.
https://doi.org/10.3390/coatings11010013