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Fluidfiberinteractions in rotational spinning process of glass wool production
Journal of Mathematics in Industry volume 1, Article number: 2 (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.
1 Introduction
Glass wool manufacturing requires a rigorous understanding of the rotational spinning of viscous thermal jets exposed to aerodynamic forces. Rotational spinning processes consist in general of two regimes: melting and spinning. The plant of our industrial partner, Woltz GmbH in Wertheim, is illustrated in Figures 1 and 2. Glass is heated upto temperatures of 1,050°C in a stove from which the melt is led to a centrifugal disk. The walls of the disk are perforated by 35 rows over height with 770 equidistantly placed small holes per row. Emerging from the rotating disk via continuous extrusion, the liquid jets grow and move due to viscosity, surface tension, gravity and aerodynamic forces. There are in particular two different air flows that interact with the arising glass fiber curtain: a downwardsdirected hot burner flow of 1,500°C that keeps the jets near the nozzles warm and thus extremely viscous and shapeable as well as a highly turbulent crossstream of 30°C that stretches and finally cools them down such that the glass fibers become hardened. Laying down onto a conveyor belt they yield the basic fabric for the final glass wool product. For the quality assessment of the fabrics the properties of the single spun fibers, that is, homogeneity and slenderness, play an important role. A longterm objective in industry is the optimal design of the manufacturing process with respect to desired product specifications and low production costs. Therefore, it is necessary to model, simulate and control the whole process.
Up to now, the numerical simulation of the whole manufacturing process is impossible because of its enormous complexity. In fact, we do not long for an uniform numerical treatment of the whole process, but have the idea to derive adequate models and methods for the separate regimes and couple them appropriately, for a similar strategy for technical textiles manufacturing see [1]. In this content, the melting regime dealing with the creeping highly viscous melt flow from the stove to the holes of the centrifugal disk might be certainly handled by standard models and methods from the field of fluid dynamics. It yields the information about the melt velocity and temperature distribution at the nozzles which is of main importance for the ongoing spinning regime. However, be aware that for their determination not only the melt behavior in the centrifugal disk but also the effect of the burner flow, that is, aerodynamic heating and heat distortion of disk walls and nozzles, have to be taken into account. In this paper we assume the conditions at the nozzles to be given and focus exclusively on the spinning regime which is the challenging core of the problem. For an overview of the specific temperature, velocity and length values we refer to Table 1. In the spinning regime the liquid viscous glass jets are formed, in particular they are stretched by a factor 10,000. Their geometry is characterized by a typical slenderness ratio \delta =d/l\approx {10}^{4} of jet diameter d and length l. The resulting fiber properties (characteristics) depend essentially on the jets behavior in the surrounding air flow. To predict them, the interactions, that is, momentum and energy exchange, of air flow and fiber curtain consisting of MN single jets (M=35N=770) have to be considered. Their computation requires in principle a coupling of fiber jets and flow with appropriate interface conditions. However, the needed high resolution and adaptive grid refinement make the direct numerical simulation of the threedimensional multiphase problem for ten thousands of slender glass jets and fast air streams not only extremely costly and complex, but also practically impossible. Therefore, we tackle the multiscale problem by help of drag models that are derived on basis of slenderbody theory and homogenization, and a weak iterative coupling algorithm.
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
A glass jet is a slender body, that is, a rod in the context of threedimensional continuum mechanics. Because of its slender geometry, its dynamics might be reduced to a onedimensional description by averaging the underlying balance laws over its crosssections. This procedure is based on the assumption that the displacement field in each crosssection can be expressed in terms of a finite number of vector and tensorvalued quantities. In the special Cosserat rod theory, there are only two constitutive elements: a curve specifying the position \mathbf{r}:Q\to {\mathbb{E}}^{3} and an orthonormal director triad \{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}}\}:Q\to {\mathbb{E}}^{3} characterizing the orientation of the crosssections, where Q=\{(s,t)\in {\mathbb{R}}^{2}s\in I(t)=[0,l(t)],t>0\} with arclength parameter s and time t. For a schematic sketch of a Cosserat rod see Figure 3, for more details on the Cosserat theory we refer to [6]. In the following we use an incompressible viscous Cosserat rod model that was derived on basis of the work [20, 34] on viscous rope coiling and investigated for isothermal curved inertial jets in rotational spinning processes [16, 19]. We extend the model by incorporating temperature effects and aerodynamic forces. The rod system describes the variables of jet curve r, orthonormal triad \{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}}\}, generalized curvature κ, convective speed u, crosssection A, linear velocity v, angular velocity ω, temperature T and normal contact forces \mathbf{n}\cdot {\mathbf{d}}_{\mathit{\alpha}}\alpha =1,2. It consists of four kinematic and four dynamic equations, that is, balance laws for mass (crosssection), linear and angular momentum and temperature,
supplemented with an incompressible geometrical model of circular crosssections with diameter d
as well as viscous material laws for the tangential contact force \mathbf{n}\cdot {\mathbf{d}}_{\mathbf{3}} and contact couple m
Rod density ρ and heat capacity {c}_{p} are assumed to be constant. The temperaturedependent dynamic viscosity μ is modeled according to the VogelFulcherTamman relation, that is, \mu (T)={10}^{{p}_{1}+{p}_{2}/(T{p}_{3})} Pa s where we use the parameters {p}_{1}=2.56{p}_{2}=4\text{,}289.18 K and {p}_{3}=(150.74+273.15) K, [33]. The external loads rise from gravity \rho Ag{\mathbf{e}}_{\mathbf{g}} with gravitational acceleration g and aerodynamic forces {\mathbf{f}}_{\mathit{air}}. In the temperature equation we neglect inner friction and heat conduction and focus exclusively on radiation {q}_{\mathit{rad}} and aerodynamic heat sources {q}_{\mathit{air}}. The radiation effect depends on the geometry of the plant and is incorporated in the system by help of the simple model
with emissivity {\epsilon}_{R}, StefanBoltzmann constant σ and reference temperature {T}_{\mathit{ref}}. Appropriate initial and boundary conditions close the rod system.
2.1.2 NavierStokes equations
A compressible air flow in the spacetime domain {\Omega}_{t}=\{(\mathbf{x},t)\mathbf{x}\in \Omega \subset {\mathbb{E}}^{3},t>0\} is described by density {\rho}_{\star}, velocity {\mathbf{v}}_{\star}, temperature {T}_{\star}. Its model equations consist of the balance laws for mass, momentum and energy,
supplemented with the Newtonian stress tensor {\mathbf{S}}_{\star}, the Fourier law for heat conduction {\mathbf{q}}_{\star}
as well as thermal and caloric equations of state of a ideal gas
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}}
Let Ψ and {\mathbf{\Psi}}_{\star} represent all glass jet and air flow quantities, respectively. Thereby, {\mathbf{\Psi}}_{\star} is the spatially averaged solution of (2). This delocation is necessary to avoid singularities in the twoway coupling. Then, the drag forces are given by
where δ is the Dirac distribution. By construction, they fulfill the principle that action equals reaction and hence the momentum is conserved, that is,
for an arbitrary domain V and {I}_{V}(t)=\{s\in I(t)\mathbf{r}(s,t)\in V\}. The (line) force \mathcal{F} acting on a slender body is caused by friction and inertia. It depends on material and geometrical properties as well as on the specific inflow situation. The number of dependencies can be reduced to two by help of nondimensionalizing which yields the dimensionless drag force F in dependence on the jet orientation (tangent) and the dimensionless relative velocity between air flow and glass jet. Due to the rotational invariance of the force, the function
can be associated with its component tuple F for every representation in an orthonormal basis, that is,
for every orthonormal basis \{{\mathbf{e}}_{\mathbf{i}}\}.
For F we use the drag model [24] that was developed on top of Oseen and Stokes theory [25–27], Taylor heuristic [28] and numerical simulations and validated with measurements [29–32]. It shows to be applicable for all air flow regimes and incident flow directions. Let \{\mathbf{n},\mathbf{b},\mathit{\tau}\} be the orthonormal basis induced by the specific inflow situation (\mathit{\tau},\mathbf{w}) with orientation τ and velocity w, assuming \mathbf{w}\nparallel \mathit{\tau}
Then, the force is given by
according to the Independence Principle [38]. The differentiable normal and tangential drag functions {c}_{n}{c}_{\tau} are
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}}
Analogously to the drag forces, the heat sources are given by
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.
For Nu we use a heuristic model. It originates in the studies of a perpendicular flow around a cylinder [33] and is modified for different inflow directions (angles of attack) with regard to experimental data. A regularization ensures the smooth limit for a transversal incident flow in analogon to the drag model for F in (3). We apply
2.3 Generalization to many rods
In case of k slender bodies in the air flow, we have {\mathbf{\Psi}}_{\mathbf{i}}, i=1,\dots ,k, representing the quantities of each Cosserat rod, here k=MN. Assuming no contact between neighboring fiber jets, every single jet can be described by the stated rod system (1). Their multiple effect on the air flow is reflected in {\mathbf{f}}_{\mathit{jets}} and {q}_{\mathit{jets}}. The source terms in the momentum and energy equations of the air flow (2) become
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
For the viscous Cosserat rods (1), the mass flux Q is constant in the stationarity, that is, uA=Q/\rho =\mathit{const}. We deal with Ωadapted linear and angular velocities, {\mathbf{v}}_{\Omega}=\mathbf{v}\mathbf{\Omega}\times \mathbf{r} and {\mathit{\omega}}_{\Omega}=\mathit{\omega}\mathbf{\Omega}, which fulfill the explicit stationarity relations
resulting from the first two equations of (1). Moreover, fictitious Coriolis and centrifugal forces and associated couples enter the linear and angular momentum equations. Using the material laws we can formulate the stationary rod model in terms of a boundary value problem of first order differential equations. Thereby, we present it in the director basis \{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}}\} for convenience (see (5) and compare to [19] except for the temperature equation). Note that to an arbitrary vector field \mathbf{z}={\sum}_{i=1}^{3}{\stackrel{\u02d8}{z}}_{i}{\mathbf{a}}_{\mathbf{i}}={\sum}_{i=1}^{3}{z}_{i}{\mathbf{d}}_{\mathbf{i}}\in {\mathbb{E}}^{3}, we indicate the component tuples corresponding to the rotating outer basis and the director basis by \stackrel{\u02d8}{\mathsf{z}}=({\stackrel{\u02d8}{z}}_{1},{\stackrel{\u02d8}{z}}_{2},{\stackrel{\u02d8}{z}}_{3})\in {\mathbb{R}}^{3} and \mathsf{z}=({z}_{1},{z}_{2},{z}_{3})\in {\mathbb{R}}^{3}, respectively. The director basis can be transformed into the rotating outer basis by the tensorvalued rotation R, that is, \mathbf{R}={\mathbf{a}}_{\mathbf{i}}\otimes {\mathbf{d}}_{\mathbf{i}}={R}_{ij}{\mathbf{a}}_{\mathbf{i}}\otimes {\mathbf{a}}_{\mathbf{j}}\in {\mathbb{E}}^{3}\otimes {\mathbb{E}}^{3} with associated orthogonal matrix \mathsf{R}=({R}_{ij})=({\mathbf{d}}_{\mathbf{i}}\cdot {\mathbf{a}}_{\mathbf{j}})\in SO(3). Its transpose and inverse matrix is denoted by {\mathsf{R}}^{\mathrm{T}}. For the components, \mathsf{z}=\mathsf{R}\cdot \stackrel{\u02d8}{\mathsf{z}} holds. The crossproduct \mathsf{z}\times \mathsf{R} is defined as mapping (\mathsf{z}\times \mathsf{R}):{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}\mathsf{y}\mapsto \mathsf{z}\times (\mathsf{R}\cdot \mathsf{y}). Moreover, canonical basis vectors in {\mathbb{R}}^{3} are denoted by {\mathsf{e}}_{i}i=1,2,3, for example, {\mathsf{e}}_{\mathsf{1}}=(1,0,0). Then, the stationary Cosserat rod model stated in the director basis for a spun glass jet reads
with {q}_{\mathit{rad}}=2\sqrt{\pi}{\epsilon}_{R}\sigma \sqrt{Q/\rho}({T}_{\mathit{ref}}^{4}{T}^{4})/\sqrt{u} and diagonal matrix {\mathsf{P}}_{k}=diag(1,1,k)k\in \mathbb{R}. For a spun jet emerging from the centrifugal disk at s=0 with stressfree end at s=L, the equations are supplemented with
(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}}).
Remark 1 The rotations\mathsf{R}\in SO(3)can be parameterized for example in Euler angles or unit quaternions[39]. The last variant offers a very elegant way of rewriting the second equation of (5). Define
with\parallel \mathsf{q}\parallel =1then we have{\partial}_{s}\mathsf{q}=\mathcal{A}(\kappa )\cdot \mathsf{q}with skewsymmetric matrix
3.1.2 Rotationally invariant air flow
Due to the spinning setup the jets emerging from the rotating device form rowwise dense curtains. As a consequence of a rowwise homogenization, the air flow (2) can be treated as stationary not only in the rotating outer basis \{{\mathbf{a}}_{\mathbf{1}}(t),{\mathbf{a}}_{\mathbf{2}}(t),{\mathbf{a}}_{\mathbf{3}}(t)\}, but also in a fixed outer one. Because of the symmetry with respect to the rotation axis, it is convenient to introduce cylindrical coordinates (x,r,\varphi )\in \mathbb{R}\times {\mathbb{R}}^{+}\times [0,2\pi ) for the space and to attach a cylindrical basis \{{\mathbf{e}}_{\mathbf{x}},{\mathbf{e}}_{\mathbf{r}},{\mathbf{e}}_{\mathit{\varphi}}\} with {\mathbf{e}}_{\mathbf{x}}={\mathbf{a}}_{\mathbf{1}} to each space point. The components to an arbitrary vector field \mathbf{z}\in {\mathbb{E}}^{3} are indicated by \stackrel{\u02c6}{\mathsf{z}}=({z}_{x},{z}_{r},{z}_{\varphi})\in {\mathbb{R}}^{3}. Then, taking advantage of the rotational invariance, the stationary NavierStokes equations in (x,r) simplify to
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
To perform the coupling between (5) and (6), we have to compute the exchange functions in the appropriate coordinates. These calculations are simplified by the rotational invariance of the problem. As introduced, we use the subscripts \phantom{\rule{0.2em}{0ex}}\stackrel{\u02d8}{}\phantom{\rule{0.2em}{0ex}} and \stackrel{\u02c6}{} to indicate the component tuples corresponding to the rotating outer basis \{{\mathbf{a}}_{\mathbf{1}}(t),{\mathbf{a}}_{\mathbf{2}}(t),{\mathbf{a}}_{\mathbf{3}}(t)\} and the cylindrical basis \{{\mathbf{e}}_{\mathbf{x}},{\mathbf{e}}_{\mathbf{r}},{\mathbf{e}}_{\mathit{\varphi}}\}, respectively. Essentially for the coupling are the jet tangent and the relative velocity between air flow and glass jet, they are
and
Then, the drag forces are
and the heat sources
Here, {\stackrel{\u02c6}{\mathsf{f}}}_{\mathit{jets}} and {q}_{\mathit{jets}} represent the homogenized effect of the N glass jets emerging from the equidistantly placed holes in an arbitrary spinning row. Correspondingly, system (5) with {\stackrel{\u02d8}{\mathsf{f}}}_{\mathit{air}} and {q}_{\mathit{air}} describes one representative glass jet for this row. To simulate the full problem with all MN glass jets in the air, jet representatives {\Psi}_{i}, i=1,\dots ,M for all M spinning rows with the respective boundary and air flow conditions have to be determined. Their common effect on the air flow is
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
The computing of the glass jets is based on a dimensionless rod system. For this purpose, we scale the dimensional equations (5) with the spinning conditions of the respective row. Apart from the air flow data, (5) contains thirteen physical parameters, that is, jet density ρ, heat capacity {c}_{p}, emissivity {\epsilon}_{R}, typical length L, velocity U and temperature θ at the spinning hole as well as hole diameter D and height H, centrifugal disk radius R, rotational frequency Ω, reference temperature for radiation {T}_{\mathit{ref}} and gravitational acceleration g. The typical jet viscosity is chosen to be {\mu}_{0}=\mu (\theta ). These induce various dimensionless numbers characterizing the fiber spinning, that is, Reynolds number Re as ratio between inertia and viscosity, Rossby number Rb as ratio between inertia and rotation, Froude number Fr as ratio between inertia and gravity and Ra as ratio between radiation and heat advection as well as ℓ h and ϵ as length ratios between jet length, hole height, diameter and disk radius, respectively,
In addition, we introduce dimensionless quantities that also depend on local air flow data, similarly to the Nusselt number in (4)
Here, {\mathrm{A}}_{4} is the Prandtl number of the air flow. To make (5) dimensionless we use the following reference values:
We choose the disk radius R as macroscopic length scale in the scalings, since it is well known by the setup. As for L, we consider jet lengths where the stresses are supposed to be vanished. In general, R and L are of same order such that the parameter ϵ can be identified with the slenderness ratio δ of the jets, cf. Introduction. The last two scalings for {n}_{0} and {m}_{0} are motivated by the material laws and the fact that the mass flux is Q=\pi \rho U{D}^{2}/4. Then, the dimensionless system for the stationary viscous rod has the form
with
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 the numerical treatment of (7), systems of nonlinear equations are set up via a RungeKutta collocation method and solved by a Newton method. The RungeKutta collocation method is an integration scheme of fourth order for boundary value problems, that is, {\partial}_{s}\mathsf{z}=\mathsf{f}(s,\mathsf{z})\mathsf{f}:[a,b]\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n} with \mathsf{g}(\mathsf{z}(a),\mathsf{z}(b))=\mathsf{0}. It is a standard routine in MATLAB 7.4 with adaptive grid refinement (solver bvp4c.m). Let a={s}_{0}<{s}_{1}<\cdots <{s}_{N}=b be the collocation points in [a,b] with {h}_{i}={s}_{i}{s}_{i1} and denote {\mathsf{z}}_{\mathsf{i}}=\mathsf{z}({s}_{i}). Then, the nonlinear system of (N+1) equations, \mathsf{S}({\mathsf{z}}^{h})=\mathsf{0}, for the discrete solution {\mathsf{z}}^{h}={({\mathsf{z}}_{\mathsf{i}})}_{i=0,\dots ,N} is set up via
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 numerical difficulty of the coupling of glass jet and air flow computations, {\mathcal{S}}_{\mathit{jets}} and {\mathcal{S}}_{\mathit{air}}, results from the different underlying discretizations. Let {I}_{h} denote the rod grid used in the continuation method and {I}_{\Delta} be an equidistant grid of step size Δs with respective jet data Ψ_{Δ} for data exchange. Moreover, let {\Omega}_{h} denote the finite volume mesh with the flow data {\Psi}_{\star ,V} for the cell V, (so the chosen mesh realizes the necessary averaging). For the air associated exchange functions, the flow data is linearly interpolated on {I}_{h}. Precisely, the linear interpolation \mathbb{L} with respect to \stackrel{\u02d8}{\mathsf{r}}({s}_{j}), {s}_{j}\in {I}_{h} is performed over all V\in \mathcal{N}({s}_{j}), where \mathcal{N}({s}_{j}) is the set of the cell containing \stackrel{\u02d8}{\mathsf{r}}({s}_{j}) and its direct neighbor cells,
For the jet associated exchange functions entering the finite volume scheme, we need the averaged jet information for every cell V\in {\Omega}_{h}. We introduce {I}_{\Delta ,V}=\{{s}_{j}\in {I}_{\Delta}\stackrel{\u02d8}{\mathsf{r}}({s}_{j})\in V\} and {I}_{V}=\Delta s{I}_{\Delta ,V}, then the averaging \mathbb{E} with respect to V is performed over the {I}_{\Delta ,V}associated data,
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
Do

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.
For all air flow simulations we use the same finite volume mesh {\Omega}_{h} whose refinement levels are initially chosen according to the unperturbed flow structure, independently of the glass jets. This implies a very fine resolution at the injector of the turbulent cross flow which is coarsen towards the centrifugal disk. For mesh details see Figure 4. The turbulent intensity is visualized in Figure 5. As expected it is high at the injector and moderate in the remaining flow domain. In particular, it is less than 2% in the region near the centrifugal disk where the glass jets will be presumably located. Thus, we neglect turbulence effects on the jets dynamics in the following. However, note that such effects can be easily incorporated by help of stochastic drag models [24, 37, 40] that are based on RANS turbulence descriptions (for example, kϵ model or kω model). For the jet computations the grid {I}_{h} is automatically generated and adapted by the continuation method in every iteration. To ensure that sufficient jet points lie in each flow cell and a proper data exchange is given we use an equidistant grid {I}_{\Delta} with appropriate step size Δs (at minimum 2 jet points per interacting flow cell).
The weak iterative coupling algorithm is fully automated. Each iteration starts with the same initialization. There is no parameter adjustment. The algorithm turns out to be very robust and reliable in spite of coarse flow meshes. For our setup an air flow simulation takes around 30 minutes CPUtime, and the computation of a single jet takes approximatively just as long. The algorithm converges within 1214 iterations. Figure 6 shows the relative {\mathcal{L}}^{2}error of all jet curve components over the number of iterations k, that is,
The effects of the fluidfiber interactions and the necessity of the twoway coupling procedure for the rotational spinning process can be concluded from the following results. Figure 7 shows the swirl velocity of the air flow and the location of the immersed glass jets over the iterations. In the unperturbed flow without the glass jets there is no swirl velocity. In fact, the presence of the jets cause the swirl velocity, since the jets pull the flow with them. Moreover, the jets deflect the downwards directed burner flow, as seen in Figures 8 and 9. The jets behavior looks very reasonable. Trajectories and positions are as expected. Furthermore, their properties, that is, velocity u and temperature T, correspond to the axial flow velocity and flow temperature, which implies a proper momentum and heat exchange. For jet details we refer to Figures 10, 11, 12 and 13. They show the influence of the spinning rows. The jet representative of the highest spinning row is warmer than the one of the lowest row which implies better stretching capabilities. It is also faster and hence thinner (A={u}^{1}). This certainly comes from the fact that the highest jet is longer affected by the fast hot burner flow. However, in view of quality assessment, slenderness and homogeneity of the spun fiber jets play an important role. This requires the optimal design of the spinning conditions, for example, different nozzle diameters or various distances between spinning rows. But for this purpose, also the melting regime has to be taken into account in modeling and simulation which is left to future research.
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.
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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.
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The success of this study is due to the strong and fruitful collaboration of all authors. Even in details it is a joint work. However, special merits go to WA for the numerical analysis of Cosserat rods; to NM for modeling, investigating the asymptotic coupling concept and drafting the manuscript; to JS for conceptualizing and implementing the weak coupling software, performing the simulations and designing the visualizations; and to RW for developing the model framework, investigating the asymptotic coupling concept and implementing the continuation method for the jets. All authors read and approved the final manuscript.
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Arne, W., Marheineke, N., Schnebele, J. et al. Fluidfiberinteractions in rotational spinning process of glass wool production. J.Math.Industry 1, 2 (2011). https://doi.org/10.1186/2190598312
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DOI: https://doi.org/10.1186/2190598312