Fluidfiberinteractions in rotational spinning process of glass wool production
 Walter Arne^{1, 2},
 Nicole Marheineke^{3}Email author,
 Johannes Schnebele^{1} and
 Raimund Wegener^{1}
DOI: 10.1186/2190598312
© Arne et al.; licensee Springer 2011
Received: 9 December 2010
Accepted: 3 June 2011
Published: 3 June 2011
Abstract
The optimal design of rotational production processes for glass wool manufacturing poses severe computational challenges to mathematicians, natural scientists and engineers. In this paper we focus exclusively on the spinning regime where thousands of viscous thermal glass jets are formed by fast air streams. Homogeneity and slenderness of the spun fibers are the quality features of the final fabric. Their prediction requires the computation of the fluidfiberinteractions which involves the solving of a complex threedimensional multiphase problem with appropriate interface conditions. But this is practically impossible due to the needed high resolution and adaptive grid refinement. Therefore, we propose an asymptotic coupling concept. Treating the glass jets as viscous thermal Cosserat rods, we tackle the multiscale problem by help of momentum (drag) and heat exchange models that are derived on basis of slenderbody theory and homogenization. A weak iterative coupling algorithm that is based on the combination of commercial software and selfimplemented code for flow and rod solvers, respectively, makes then the simulation of the industrial process possible. For the boundary value problem of the rod we particularly suggest an adapted collocationcontinuation method. Consequently, this work establishes a promising basis for future optimization strategies.
Keywords
Rotational spinning process viscous thermal jets fluidfiber interactions twoway coupling slenderbody theory Cosserat rods drag models boundary value problem continuation method Mathematics Subject Classification: 76xx 34B08 41A60 65L10 65Z051 Introduction
Temperature  Velocity  Diameter  

Burner air flow in channel  ${T}_{\mathit{air}1}$ 1,773 K  ${V}_{\mathit{air}1}$ 1.2⋅10^{2} m/s  ${W}_{1}$ 1.0⋅10^{−2} m 
Turbulent air stream at injector  ${T}_{\mathit{air}2}$ 303 K  ${V}_{\mathit{air}2}$ 3.0⋅10^{2} m/s  ${W}_{2}$ 2.0⋅10^{−4} m 
Centrifugal disk  ${T}_{\mathit{melt}}$ 1,323 K  Ω 2.3⋅10^{2} 1/s  2R 4.0⋅10^{−1} m 
Glass jets at spinning holes  θ 1,323 K  U 6.7⋅10^{−3} m/s  D 7.4⋅10^{−4} m 
The dynamics of curved viscous inertial jets is of interest in many industrial applications (apart from glass wool manufacturing), for example, in nonwoven production [1, 2], pellet manufacturing [3, 4] or jet ink design, and has been subject of numerous theoretical, numerical and experimental investigations, see [5] and references within. In the terminology of [6], there are two classes of asymptotic onedimensional models for a jet, that is, string and rod models. Whereas the string models consist of balance equations for mass and linear momentum, the more complex rod models contain also an angular momentum balance, [7, 8]. A string model for the jet dynamics was derived in a slenderbody asymptotics from the threedimensional free boundary value problem given by the incompressible NavierStokes equations in [5]. Accounting for inner viscous transport, surface tension and placing no restrictions on either the motion or the shape of the jet’s centerline, it generalizes the previously developed string models for straight [9–11] and curved [12–14] centerlines. However, already in the stationary case the applicability of the string model turns out to be restricted to certain parameter ranges [15, 16] because of a nonremovable singularity that comes from the deduced boundary conditions. These limitations can be overcome by a modification of the boundary conditions, that is, the release of the condition for the jet tangent at the nozzle in favor of an appropriate interface condition, [17–19]. This involves two string models that exclusively differ in the closure conditions. For gravitational spinning scenarios they cover the whole parameter range, but in the presence of rotations there exist small parameter regimes where none of them works. A rod model that allows for stretching, bending and twisting was proposed and analyzed in [20, 21] for the coiling of a viscous jet falling on a rigid substrate. Based on these studies and embedded in the special Cosserat theory a modified incompressible isothermal rod model for rotational spinning was developed and investigated in [16, 19]. It allows for simulations in the whole (Re, Rb, Fr)range and shows its superiority to the string models. These observations correspond to studies on a fluidmechanical ‘sewing machine’, [22, 23]. By containing the slenderness parameter δ explicitely in the angular momentum balance, the rod model is no asymptotic model of zeroth order. Since its solutions converge to the respective string solutions in the slenderness limit $\delta \to 0$, it can be considered as δregularized model, [19]. In this paper we extend the rod model by incorporating the practically relevant temperature dependencies and aerodynamic forces. Thereby, we use the air drag model F of [24] that combines Oseen and Stokes theory [25–27], Taylor heuristic [28] and numerical simulations. Being validated with experimental data [29–32], it is applicable for all air flow regimes and incident flow directions. Transferring this strategy, we model a similar aerodynamic heat source for the jet that is based on the Nusselt number Nu [33]. Our coupling between glass jets and air flow follows then the principle that action equals reaction. By inserting the corresponding homogenized source terms induced by F and Nu in the balance equations of the air flow, we make the proper momentum and energy exchange within this slenderbody framework possible.
The paper is structured as follows. We start with the general coupling concept for slender bodies and fluid flows. Therefore, we introduce the viscous thermal Cosserat rod system and the compressible NavierStokes equations for glass jets and air flow, respectively, and present the models for the momentum and energy exchange: drag F and Nusselt function Nu. The special setup of the industrial rotational spinning process allows for the simplification of the model framework, that is, transition to stationarity and assumption of rotational invariance as we discuss in detail. It follows the section about the numerical treatment. To realize the fiberflow interactions we use a weak iterative coupling algorithm, which is adequate for the problem and has the advantage that we can combine commercial software and selfimplemented code. Special attention is paid to the collocation and continuation method for solving the boundary value problem of the rod. Convergence of the coupling algorithm and simulation results are shown for a specific spinning adjustment. This illustrates the applicability of our coupling framework as one of the basic tools for the optimal design of the whole manufacturing process. Finally, we conclude with some remarks to the process.
2 General coupling concept for slender bodies and fluid flows
We are interested in the spinning of ten thousands of slender glass jets by fast air streams, $MN=26\text{,}950$. The glass jets form a kind of curtain that interact and crucially affect the surrounding air. The determination of the fluidfiberinteractions requires in principle the simulation of the threedimensional multiphase problem with appropriate interface conditions. However, regarding the complexity and enormous computational effort, this is practically impossible. Therefore, we propose a coupling concept for slender bodies and fluid flows that is based on drag force and heat exchange models. In this section we first present the twoway coupling of a single viscous thermal Cosserat rod and the compressible NavierStokes equations and then generalize the concept to many rods. Thereby, we choose an invariant formulation in the threedimensional Euclidian space ${\mathbb{E}}^{3}$.
Note that we mark all quantities associated to the air flow by the subscript _{⋆} throughout the paper. Moreover, to facilitate the readability of the coupling concept, we introduce the abbreviations Ψ and ${\mathbf{\Psi}}_{\star}$ that represent all quantities of the glass jets and the air flow, respectively.
2.1 Models for glass jets and air flows
2.1.1 Cosserat rod
2.1.2 NavierStokes equations
with pressure ${p}_{\star}$ and inner energy ${e}_{\star}$. The specific gas constant for air is denoted by ${R}_{\star}$. The temperaturedependent viscosities ${\mu}_{\star}$${\lambda}_{\star}$, heat capacity ${c}_{p\star}$ and heat conductivity ${k}_{\star}$ can be modeled by standard polynomial laws, see, for example, [33, 35]. External loads rise from gravity ${\rho}_{\star}g{\mathbf{e}}_{\mathbf{g}}$ and forces due to the immersed fiber jets ${\mathbf{f}}_{\mathit{jets}}$. These fiber jets also imply a heat source ${q}_{\mathit{jets}}$ in the energy equation. Appropriate initial and boundary conditions close the system.
2.2 Models for momentum and energy exchange
The coupling of the Cosserat rod and the NavierStokes equations is performed by help of drag forces and heat sources. Taking into account the conservation of momentum and energy, ${\mathbf{f}}_{\mathit{air}}$ and ${\mathbf{f}}_{\mathit{jets}}$ as well as ${q}_{\mathit{air}}$ and ${q}_{\mathit{jets}}$ satisfy the principle that action equals reaction and obey common underlying relations. Hence, we can handle the delicate fluidfiberinteractions by help of two surrogate models, socalled exchange functions, that is, a dimensionless drag force F (inducing ${\mathbf{f}}_{\mathit{air}}$${\mathbf{f}}_{\mathit{jets}}$) and Nusselt number Nu (inducing ${q}_{\mathit{air}}$${q}_{\mathit{jets}}$). For a flow around a slender long cylinder with circular crosssections there exist plenty of theoretical, numerical and experimental investigations to these relations in literature, for an overview see [24] as well as, for example, [29, 30, 33, 36] and references within. We use this knowledge locally and globalize the models by superposition to apply them to our curved moving Cosserat rod. This strategy follows a GlobalfromLocal concept [37] that turned out to be very satisfying in the derivation and validation of a stochastic drag force in a oneway coupling of fibers in turbulent flows [24].
2.2.1 Drag forces  ${\mathbf{f}}_{\mathit{air}}$ vs ${\mathbf{f}}_{\mathit{jets}}$
for every orthonormal basis $\{{\mathbf{e}}_{\mathbf{i}}\}$.
with $S({w}_{n})=2.0022ln{w}_{n}$, transition points ${w}_{1}=0.1$${w}_{2}=100$, amplitude $\gamma =2$. The regularity involves the parameters ${p}_{n,0}=1.6911$${p}_{n,1}=6.7222\cdot {10}^{1}$${p}_{n,2}=3.3287\cdot {10}^{2}$${p}_{n,3}=3.5015\cdot {10}^{3}$ and ${p}_{\tau ,0}=1.1552$${p}_{\tau ,1}=6.8479\cdot {10}^{1}$${p}_{\tau ,2}=1.4884\cdot {10}^{2}$${p}_{\tau ,3}=7.4966\cdot {10}^{4}$. To be also applicable in the special case of a transversal incident flow $\mathbf{w}\parallel \mathit{\tau}$ and to ensure a realistic smooth force F, the drag is modified for ${w}_{n}\to 0$. A regularization based on the slenderness parameter δ matches the associated resistance functions ${r}_{n}$${r}_{\tau}$ (3) to Stokes resistance coefficients of higher order for ${w}_{n}\ll 1$, for details see [24].
2.2.2 Heat sources  ${q}_{\mathit{air}}$ vs ${q}_{\mathit{jets}}$
The (line) heat source $\mathcal{Q}$ acting on a slender body also depends on several material and geometrical properties as well as on the specific inflow situation. The number of dependencies can be reduced to three by help of nondimensionalizing which yields the dimensionless Nusselt number Nu in dependence of the cosine of the angle of attack, Reynolds and Prandtl numbers. The Reynolds number corresponds to the relative velocity between air flow and glass jet, the typical length is the half jet circumference.
2.3 Generalization to many rods
3 Models for special setup of rotational spinning process
In the rotational spinning process under consideration the centrifugal disk is perforated by M rows of N equidistantly placed holes each ($M=35$, $N=770$). The spinning conditions (hole size, velocities, temperatures) are thereby identical for each row, see Figures 1 and 2. The special setup allows for the simplification of the general model framework. We introduce the rotating outer orthonormal basis $\{{\mathbf{a}}_{\mathbf{1}}(t),{\mathbf{a}}_{\mathbf{2}}(t),{\mathbf{a}}_{\mathbf{3}}(t)\}$ satisfying ${\partial}_{t}{\mathbf{a}}_{\mathbf{i}}=\mathbf{\Omega}\times {\mathbf{a}}_{\mathbf{i}}$, $i=1,2,3$, where Ω is the angular frequency of the centrifugal disk. In particular, $\mathbf{\Omega}=\Omega {\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{e}}_{\mathbf{g}}={\mathbf{a}}_{\mathbf{1}}$ (gravity direction) hold. Then, glass jets and air flow become stationary, presupposing that we consider spun fiber jets of certain length. In particular, we assume the stresses to be vanished at this length. Moreover, the glass jets emerging from the rotating device form dense curtains for every spinning row. As a result of homogenization, we can treat the air flow as rotationally invariant and each curtain can be represented by one jet. This yields an enormous complexity reduction of the problem. The homogenization together with the slenderbody theory makes the numerical simulation possible.
3.1 Transition to stationarity
3.1.1 Representative spun jet of certain length
(cf. Table 1). Considering the jet as representative of one spinning row, we choose the nozzle position to be $(H,R,0)$ with respective height H R is here the disk radius. The initialization $\mathsf{R}(0)$ prescribes the jet direction at the nozzle as $({\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}})(0)=({\mathbf{a}}_{\mathbf{1}},{\mathbf{a}}_{\mathbf{3}},{\mathbf{a}}_{\mathbf{2}})$.
3.1.2 Rotationally invariant air flow
with $\nabla \cdot {\stackrel{\u02c6}{\mathsf{v}}}_{\star}={\partial}_{x}{v}_{x\star}+({\partial}_{r}(r{v}_{r\star}))/r$ and equipped with appropriate inflow, outflow and wall boundary conditions, cf. Figures 1 and 2.
3.2 Exchange functions
4 Numerical treatment
The numerical simulation of the glass jets dynamic in the air flow is performed by an algorithm that weakly couples glass jet calculation and air flow computation via iterations. This procedure is adequate for the problem and has the advantage that we can combine commercial software and selfimplemented code. We use FLUENT, a commercial finite volumebased software by ANSYS, that contains the broad physical modeling capabilities needed to describe air flow, turbulence and heat transfer for the industrial glass wool manufacturing process. In particular, a pressurebased solver is applied in the computation of (6). To restrict the computational effort in grid refinement needed for the resolution of the turbulent air streams we consider alternatively a stochastic kω turbulence model. (For details on the commercial software FLUENT, its models and solvers we refer to http://www.fluent.com.) Note that the modification of the model equations has no effect on our coupling framework, where the exchange functions are incorporated by UDFs (user defined functions). For the boundary value problem of the stationary Cosserat rod (5), systems of nonlinear equations are set up via a RungeKutta collocation method and solved by a Newton method in MATLAB 7.4. The convergence of the Newton method depends thereby crucially on the initial guess. To improve the computational performance we adapt the initial guess iteratively by solving a sequence of boundary value problems with slightly changed parameters. The developed continuation method is presented in the following. Moreover, to get a balanced numerics we use the dimensionless rod system that is scaled with the respective conditions at the nozzle. The M glass jet representative are computed in parallel. The exchange of flow and fiber data between the solvers is based on interpolation and averaging, as we explain in the weak iterative coupling algorithm.
4.1 Collocationcontinuation method for dimensionless rod boundary value problem
Here, ${T}_{\mathit{ref}}$ and the air flow associated ${T}_{\star}$ and ${\stackrel{\u02d8}{\mathsf{v}}}_{\mathit{rel}}$ are scaled with θ and U, respectively. System (7) contains the slenderness parameter ϵ ($\u03f5\ll 1$) explicity in the equation for the couple $\mathsf{m}$ and is hence no asymptotic model of zeroth order. In the slenderness limit $\u03f5\to 0$, the rod model reduces to a string system and their solutions for $(\stackrel{\u02d8}{\mathsf{r}},\stackrel{\u02d8}{\tau},u,N={n}_{3},T)$ coincide. Only these jet quantities are relevant for the twoway coupling, as they enter in the exchange functions. However, the simpler string system is not wellposed for all parameter ranges, [15, 16]. Thus, it makes sense to consider (7) as ϵregularized string system, [19]. We treat ϵ as moderate fixed regularization parameter in the following to stabilize the numerics, in particular we set $\u03f5=0.1$.
for $i=0,\dots ,N1$. The convergence and hence the computational performance of the Newton method depends crucially on the initial guess. Thus, we adapt the initial guess iteratively by help of a continuation strategy. We scale the drag function F with the factor ${\mathrm{C}}_{\mathrm{F}}^{2}$ and the righthand side of the temperature equation with ${\mathrm{C}}_{T}$ and treat Re, Rb, Fr, ℓ${\mathrm{C}}_{\mathrm{F}}$ and ${\mathrm{C}}_{T}$ as continuation parameters. We start from the solution for $(\mathrm{Re},\mathrm{Rb},\mathrm{Fr},\ell ,{\mathrm{C}}_{\mathrm{F}},{\mathrm{C}}_{T})=(1,1,1,0.15,\infty ,0)$ which corresponds to an isothermal rod without aerodynamic forces that has been intensively numerically investigated in [19]. Its determination is straight forward using the related string model as initial guess. Note that we choose ℓ so small to ensure that the glass jet lies in the air flow domain. The actual continuation is then divided into three parts. First, $(\mathrm{Re},\mathrm{Rb},\mathrm{Fr},\mathrm{CF})$ are adjusted, then ${\mathrm{C}}_{T}$ and finally ℓ. In the continuation we use an adaptive step size control. Thereby, we always compute the interim solutions by help of one step and two half steps and decide with regard to certain quality criteria whether the step size should be increased or decreased.
4.2 Weak iterative coupling algorithm
The ratio ${I}_{V}/V$ can be considered as the jet length density for the cell V. In case of M jet representatives, we deal with ${I}_{\Delta ,V,i}$ and ${I}_{V,i}$ for $i=1,\dots ,M$. Consequently, we have ${I}_{\Delta ,V}={\bigcup}_{i=1}^{M}{I}_{\Delta ,V,i}$ and ${I}_{V}={\sum}_{i=1}^{M}{I}_{V,i}$. Note, that the interpolation and averaging approximation strategies have the disadvantage that they are qualitatively different. Thus, momentum and energy conservation are only ensured for very fine resolutions.
Summing up, the algorithm that we use to couple glass jet ${\mathcal{S}}_{\mathit{jets}}$ and air flow ${\mathcal{S}}_{\mathit{air}}$ computations has the form:
Algorithm 1 Generate flow mesh ${\Omega}_{h}$
Perform flow simulation ${\mathcal{S}}_{\mathit{air}}$ without jets to obtain ${\Psi}_{\star}^{(0)}$
Initialize $k=0$

Compute: ${\Psi}_{i}^{(k)}={\mathcal{S}}_{\mathit{jets}}({\Psi}_{\star}^{(k)})$ for $i=1,\dots ,M$ where flow data is linearly interpolated on ${I}_{h}$

Interpolate jet data on equidistant grid ${I}_{\Delta}$

Find for every cell V in ${\Omega}_{h}$ the relevant rod points ${I}_{\Delta ,V}$ and average the respective data

Compute: ${\Psi}_{\star}^{(k+1)}={\mathcal{S}}_{\mathit{air}}({\Psi}^{(k)})$

Update: $k=k+1$
while $\parallel {\Psi}^{(k)}{\Psi}^{(k1)}\parallel >\mathit{tol}$
Remark 2 From the technical point of view, the efficient management of the simulation and coupling routines is quite demanding. In a preprocessing step we generate the finite volume mesh${\Omega}_{h}$via the software Gambit and save it in a file that is available for FLUENT and MATLAB. The program of Algorithm 1 is then realized with FLUENT as master tool. After the air flow simulation FLUENT starts MATLAB. MATLAB governs the parallelization of the jets computation via MATLAB executables. Collecting the jets information, it provides the averaged jets data on${\Omega}_{h}$in a file. FLUENT reads in this data and performs a new air flow simulation with immersed jets.
5 Results
In this section we illustrate the applicability of our asymptotic coupling framework to the given rotational spinning process. We show the convergence of the weak iterative coupling algorithm and discuss the effects of the fluidfiberinteractions.
6 Conclusion
The optimal design of rotational spinning processes for glass wool manufacturing involves the simulation of ten thousands of slender viscous thermal glass jets in fast air streams. This is a computational challenge where direct numerical methods fail. In this paper we have established an asymptotic modeling concept for the fluidfiber interactions. Based on slenderbody theory and homogenization it reduces the complexity of the problem enormously and makes numerical simulations possible. Adequate to problem and model we have proposed an algorithm that weakly couples air flow and glass jets computations via iterations. It turns out to be very robust and converges to reasonable results within few iterations. Moreover, the possibility of combining commercial software and selfimplemented code yields satisfying efficiency offtheshelf. The performance might certainly be improved even more by help of future studies. Summing up, our developed asymptotic coupling framework provides a very promising basis for future optimization strategies.
In view of the design of the whole production process the melting regime must be taken into account in modeling and simulation. Melting and spinning regimes influence each other. On one hand the conditions at the spinning rows are crucially affected by the melt distribution in the centrifugal disk and the burner air flow, regarding, for example, cooling by mixing inside, aerodynamic heating outside. On the other hand the burner flow and the arising heat distortion of the disk are affected by the spun jet curtains. This obviously demands a further coupling procedure.
Declarations
Acknowledgements
The authors would like to acknowledge their industrial partner, the company Woltz GmbH in Wertheim, for the interesting and challenging problem. This work has been supported by German Bundesministerium für Bildung und Forschung, Schwerpunkt ‘Mathematik für Innovationen in Industrie und Dienstleistungen’, Projekt 03MS606 and by German Bundesministerium für Wirtschaft und Technologie, Förderprogramm ZIM, Projekt AUROFA 114626.
Authors’ Affiliations
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