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# Shape design for polymer spin packs: modeling, optimization and validation

- Christian Leithäuser
^{1}Email authorView ORCID ID profile, - René Pinnau
^{2}and - Robert Feßler
^{1}

**8**:13

https://doi.org/10.1186/s13362-018-0055-2

© The Author(s) 2018

**Received:**12 February 2018**Accepted:**27 November 2018**Published:**4 December 2018

## Abstract

A shape optimization approach for the design of cavities with a specified wall shear stress profile is presented. Applications are the design of spin pack geometries with low and uniform residence times and without dead spaces to prevent polymer degradation for sensitive materials. The optimization uses a Surrogate Model based on the Newtonian Stokes equation as a simplification. An indirect objective based on wall shear stress is used to improve the residence time. The results are then validated on a realistic spin pack with the Full Model based on the non-Newtonian Navier–Stokes equation.

## Keywords

- Shape optimization
- Computational fluid dynamics
- Navier–Stokes equation
- Non-Newtonian fluid
- Approximate controllability

## 1 Introduction

Polymer spin packs are widely used for the production of synthetic fibers and non-woven materials. Polymer melt is extruded through a pipe into the spin pack geometry where it is first distributed along the whole cross-sectional area before it passes several layers of filter material and is finally spun into fibers by the spinneret plate which consists of a large number of very small nozzles. The whole spin pack is heated in order to prevent premature solidification of the melt. However, the influence of heat can lead to polymer degradation if the residence times are too long. For sensitive polymers with interesting properties this issue can be the limiting factor which prevents innovations due to the fact that spinning is not possible. This can be resolved by designing special spin packs with low residence time profiles. An important part in the spin pack design is the cavity which distributes polymer from the inlet pipe onto the whole cross-sectional area. This part of the geometry is in particular vulnerable for dead spaces and regions with slow flow velocities where degradation can take place. An indirect objective based on the wall shear stress is used to improve the residence time. The idea is that problematic regions usually occur in close proximity to the walls. In general a low wall shear stress coincides with a slow velocity zone close to the wall. On the other hand being able to design cavities with a sufficiently high level of stress throughout its wall is an effective tool against dead spaces and large residence times and thus against polymer degradation.

In Sect. 2 the two models are introduced: The Full Model is used for validation and is based on the non-Newtonian Navier–Stokes equation. It considers the whole geometry of the spin pack and pathlines are traced to evaluate the distribution of residence times. The Surrogate Model on the other hand is based on the Newtonian Stokes equation. It only considers the geometry of the spin pack cavity and wall shear stress is used as an indirect objective instead of residence time. The numerical approach for solving the shape optimization problem based on the Surrogate Model is derived in Sect. 2.3 and optimized cavities with uniform wall shear stress are presented in Sect. 3.1. These cavities are then validated with the Full Model in Sect. 3.2 in a realistic setting.

Further numerical and theoretical results on the shape optimization of polymer spin packs can be found in [1–4]. Another interesting application of a similar problem is studied in [5, 6]. The authors use shape optimization to design aorto-coronaric bypasses and use wall shear stress as an optimization criterion.

## 2 Methods

A typical spin pack used for fiber production is shown in Fig. 1. Polymer enters through the Inlet, passes a short Tube and is distributed in Cavity 1 onto the Filter. Polymer passes the Breaker plate, which is basically a metal plate with a number of holes and its main purpose is to support the Filter. There is a second cavity (Cavity 2) before the material enters the Nozzles and is spun into fibers. The Nozzles consist of larger counterbores and then very small capillaries where the actual spinning takes place.

Comparison between Full and Surrogate Model

Full Model | Surrogate Model | |
---|---|---|

PDE | Navier–Stokes | Stokes |

Viscosity | non-Newtonian | Newtonian |

Geometry | whole spin pack | Cavity 1 (Filter in boundary condition) |

Objective | residence time | wall shear stress |

### 2.1 Full model: non-Newtonian Navier–Stokes

**u**, pressure

*p*and density

*ρ*. The viscosity \(\eta(\dot{\gamma}, T)\) depends on the shear rate

*γ̇*and temperature

*T*. The temperature is assumed to be constant due to the controlled heating of the spin pack block. The source term

**F**is set to zero everywhere except in the Filter where it is used to model a porous medium.

*D̄*by (see [7])

*λ*and power-law index

*n*. The temperature dependence is modeled through an Arrhenius law by

*α*and reference temperature \(T_{\alpha}\).

**u**is negative because the pathline is traced in reverse direction from outlet to inlet. The residence time for this individual pathline is then the time

*t*at which the pathline reaches the inlet boundary.

### 2.2 Surrogate model: Newtonian Stokes

The goal is now to optimize the geometry such that the residence time distribution is short and with small deviations. To do this a number of simplifications leading to the Surrogate Model are introduced in the following.

*Simplification 1: geometry*. Instead of the full spin pack geometry depicted in Fig. 1 we only consider Cavity 1 for our Surrogate Model. This is signified because typically most of the residence time is spent in this cavity. The boundary decomposed into an inlet part \(\varGamma ^{\mathrm{in}}\), a wall part \(\varGamma^{\mathrm{wall}}\) and an outlet part \(\varGamma^{\mathrm{out}}\) (see Fig. 2).

*Simplification 2: Newtonian viscosity*. Typical polymers used for fiber production are non-Newtonian and shear thinning plays a role, i.e., higher shear rates lead to a lower viscosity. However, for most applications the shear thinning only occurs in the fine capillaries. The shear rates in the distributor cavities usually lie in the zero shear rate limit of the Cross model [8, Ch. 3.6] used to represent the viscosity (cf. Fig. 6). Thus we use a constant viscosity

*Simplification 3: Stokes equation*. Inertia does not play a role for the flow within the spin pack cavity, due to the high viscosity compared to the low velocities. Therefore, the Navier–Stokes equation can be replaced by the Stokes equation. The previous simplification lead to the following problem:

**n**the outward pointing normal vector. With

*Simplification 4: wall shear stress objective*. We use a cost function based on the wall shear stress as an indirect criterion to optimize the residence time. The reason is that the wall shear stress

The idea for using the wall shear stress is the following: Dead spaces and regions with slow flow velocities where degradation can take place usually occur in close proximity to the walls. A low wall shear stress coincides with a slow velocity zone. On the other hand a sufficiently high level of stress throughout the wall prevents dead spaces and reduces the residence time.

### 2.3 Numerical shape optimization for the surrogate model

Let us derive the shape optimization approach for the Surrogate Model (8).

*Geometry variations*. Starting from any admissible domain \(\varOmega_{0} \subset\mathbb{R}^{3}\) of class \(C^{1,1}\) we want to compute a gradient \(\mathbf{ V}_{g}\) which enables us to use a gradient descent approach for solving the shape optimization problem. Knowing the gradient we can perform a descent step towards the domain \(\varOmega_{-s\mathbf{ V}_{g}} = (\mathrm{Id}-s\mathbf{ V}_{g})(\varOmega_{0})\) for some step size \(s>0\). Let

### Remark 1

For \(\theta\in\varTheta^{1}\) it is implied by [11] that \(\mathrm{Id}+ \theta: \mathbb{R}^{3} \rightarrow\mathbb{R} ^{3}\) is an invertible \((1,1)\)-diffeomorphism and thus \(\varOmega_{\theta}= (\mathrm{Id}+\theta)(\varOmega_{0})\) is also of class \(C^{1,1}\). Then, a regularity argument similar to [12] would yield \(\mathbf{ u}(\theta) \in[H^{2}(\varOmega_{\theta})]^{3}\), thus \(\sigma (\theta) \in L^{2}(\varGamma_{\theta}^{w})\) by the Trace Theorem [13, Thm. 8.7] and the objective (8) is well-defined. However, the focus of the current paper lies in the application and we will not derive the regularity result for our specific set of boundary conditions (10). We will rather use Assumption 1 and derive the gradient in a purely formal way.

### Remark 2

*Sensitivity analysis*. In order to compute the sensitivity we follow the optimize then discretize approach. Therefore, we need to differentiate the cost functional and the partial differential equation with respect to the shape, which requires the existence of the corresponding shape derivatives. However, the focus of this part is to derive a numerical method suited to solve the industrial problem, therefore, we omit the existence and regularity proofs for the shape derivatives and make the following assumption:

### Assumption 1

Assume the existence of \(\mathbf{ u}(\theta) \in H^{2}(\varOmega_{\theta}, \mathbb{R} ^{3})\), \(p(\theta) \in H^{1}(\varOmega_{\theta})\) and \(\sigma(\theta) \in L^{2}(\varGamma_{\theta}^{w})\) for \(\theta\in\varTheta^{1}\) and the existence of the shape derivatives \(\mathbf{ u}'(\varOmega_{0};\mathbf{ V}) \in H^{2}(\varOmega_{0}, \mathbb{R} ^{3})\), \(p'(\varOmega_{0};\mathbf{ V}) \in H^{1}(\varOmega_{0})\) and \(\sigma '(\varGamma _{0};\mathbf{ V}) \in L^{2}(\varGamma_{0}^{w})\) for \(\mathbf{ V}\in\mathcal{V}^{1}\).

A crucial point in the proof would be to show the regularity of the solution of the Stokes problem which is not trivial due to the mixed boundary condition. If this regularity is provided by some means we can use the implicit function theorem to show the existence of the material derivatives which provide the existence of the shape derivatives. This general approach to proof the existence of shape derivatives for partial differential equations is shown in [11]. In [3] we have applied this approach to a problem similar to the Surrogate Model considered here.

### Definition 1

(Shape derivative [14])

*y*in direction

**V**is defined by

*y*restricted to the boundary is

*θ*. The derivative of a boundary integral is given in [14]. Applying this to (12) yields for the derivative in direction \(\mathbf{ V}\in\mathcal{V}^{1}\)

*κ*denotes the curvature (see [14]).

*Discretization*. Following the optimize then discretize approach we us Taylor–Hood finite elements to discretize the partial differential equations. The implementation is done in COMSOL Multiphysics.

*Gradient descent method*. In the last section the gradient of the cost function was derived, which enables us to apply the gradient descent method to solve the shape optimization problem. A small change of notation is performed: In the following let \(\varOmega_{k}\) denote the domain of iteration

*k*of the gradient descent algorithm and the gradient at \(\varOmega_{k}\) is denoted by \(\mathbf{ V}_{k}\). Using this notation the gradient descent method is given in Algorithm 1. An Armijo rule (cf. [16]) was used to determine the step length, where \(\beta, \gamma>0\) are proper constants. Note, that the \(L^{2}\)-norm was used for step size control and stopping criterion. However, with the state space \(\mathbf{ V}_{k} \in \mathcal{H}_{k}^{2}\) the \(H^{2}\)-norm might be a better choice. We do not expect that this would significantly change the results, but it might improve the convergence.

*Moving the mesh*. This section explains how the mesh is moved and what smoothing operator \(S_{\mathcal{T}}\) is used. An important question is how the shape deformation \(\varOmega_{k+1} = (\mathrm{Id}- \beta^{j} \mathbf{ V}_{k})(\varOmega_{k})\) is carried out. In the current setup we work with a triangulation \(\mathcal{T}_{k}\) of \(\varOmega_{k}\) by tetrahedral elements. Then we move every vertex \(\xi\in\mathcal {T}_{k}\) to \((\mathrm{Id}- \beta^{j} \mathbf{ V}_{k})(\xi)\) to generate the new mesh \(\mathcal{T}_{k+1}\). In the case that any elements of the mesh are inverted we regenerate \(\mathcal{T}_{k+1}\) while keeping the boundary triangulation.

However, a problem is that the mesh tends to loose smoothness and becomes irregular. This has the effect that already after very few iterations the gradient fails to provide a proper descent direction and the algorithm stops. This is a common problem in shape optimization, as discussed in [15]. To overcome this we apply a smoothing operator \(S_{\mathcal{T}}\) to the deformed mesh. The operator is applied to the boundary mesh and is able to recover a boundary shape with more regularity. Of course the in- and outflow boundaries are not effected by the smoothing. We use the implementation [17] which relies on [18]. Using the smoothing step leads to a good quality of the gradient and a stable gradient descent algorithm. See [2, Theorem 6.5] where we have shown the existence of an optimal control for a similar problem.

## 3 Results and discussion

In the following Algorithm 1 bases on the Surrogate Model is used to compute optimized cavities. These cavities are then tested with the Full Model. A dimensionless setting was used for the optimization, while the validation was done in a realistic setting with units given later.

### 3.1 Shape optimization based on the surrogate model

### 3.2 Validation based on the full model

While we have seen that the proposed shape optimization algorithm is able to design distributor cavities with a very uniform wall shear stress it remains to show how these cavities perform in a realistic application. In this section a typical reference spin pack design is compared to optimized designs. The reference design is shown in Fig. 1. In the reference design Cavity 1 is just a flat rectangular-shaped space which is quite common for many spin packs still in use today. However, this space is very vulnerable to dead spots which can then encourage polymer degradation due to long residence times. Therefore in the following Cavity 1 will be replaced by the two optimized cavities computed in Sect. 3.1 and results regarding residence time, wall shear stress and pressure drop are compared.

The commercial CFD software ANSYS® Fluent is applied to model the spin pack in this validation step.

*Geometric setup, data and boundary conditions*. The geometry of our reference spin pack with 279 nozzles is depicted in Fig. 1. The spacial dimensions of the bounding box are: 286 mm x 146 mm x 96 mm (length x width x height). In the following three geometries are compared:

Cross model coefficients to model the viscosity of Polypropylene (PP) in Fig. 6

zero-shear-rate viscosity | \(\eta_{0}\) | 6421.8 Pa s |

time constant |
| 0.6304 s |

power-law index |
| 0.4276 |

activation energy |
| 3742.5 K |

reference temperature | \(T_{\mathrm{alpha}}\) | 473.16 K |

*Numerical results*. Simulations for the given setup for the Full Model were performed with ANSYS® Fluent for the reference design, as well as for the two optimized designs (Optimized A and Optimized B). Results for the wall shear stress in Cavity 1 are shown in Fig. 7. The wall shear stress in the reference design Fig. 7(a) is very low, especially in the outer regions which indicates stagnation zones. The wall shear stress for the optimized designs in Fig. 7(b) (Optimized A) and Fig. 7(c) (Optimized B) is significantly higher. As intended by the shape optimization approach the wall shear stress is very uniform. This shows that the shapes obtained with the Surrogate Model are still valid with the Full Model. Note, that the absolute level of the wall shear stress differs between the optimization step (Figs. 3 and 4) and the validation step (Fig. 7). The reason is that the optimization step was carried out in a dimensionless setting, while typical real world values were used for the validation step.

## 4 Conclusion

The presented shape optimization approach is able to generate cavities with specific wall shear stress profiles. This has proven to be an effective tool in the optimization of polymer spin packs and filter devices for sensitive applications. It is now possible to design geometries without dead spaces and with short and more uniform residence times. The optimization approach uses the Surrogate Model as a simplification. However, the validation has shown that the cavities optimized with the simplified model are still valid in the more realistic setting of the Full Model. We have further seen that cavities with optimized wall shear stress lead to reduced and more uniform residence times within the spin pack while the increase in pressure stays minimal. Our ongoing and future research on this matter includes the use of parameterized geometries to improve the regularity as well as optimizing the residence time directly without the use of wall shear stress as an indirect criterion.

## Declarations

### Acknowledgements

Not applicable.

### Availability of data and materials

All spin pack geometries depicted in Fig. 7 are provided.

### Funding

This work was supported by the German Federal Ministry of Education and Research (BMBF) grant no. 03MS606F and by the German Federal Ministry for Economic Affairs and Energy (BMWI) grant no. IGF 17629 N.

### Authors’ contributions

All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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