**1. Introduction**

Microstructured optical fibres (MOFs) or Photonic Crystal Fibres (PCFs) are a new kind of optical fibres, appearing for the first time approximately thirty years ago [1]. This new type of fibres possesses an array of air holes arranged in a specific pattern that spans the whole fibre length. Light guidance within PCFs relies either on the index guiding or on the photonic bandgap (PBG) mechanism. If the central air capillary is removed from the structure, the electromagnetic waves are guided by a modified total internal reflection mechanism. Conversely, if the central air capillary is replaced with another one of a different size, the PBG mechanism is realized [2]. The network of air holes can be suitably designed to allow for the guidance of selected modes. This can be achieved with specific ratios between the diameter of an air capillary and the crystal lattice constant. Many advantages of this new type of fibres are represented by the high degree of flexibility and many possibilities they offer. PCFs find a large number of applications, ranging from high-power and energy transmission [3], fibre lasers [4] and amplifiers [5], Kerr-related nonlinear effects [6], Brillouin scattering [7], telecommunications [1], and optical sensors [8–12], among others.

The manufacturing process of glass optical fibres presupposes two steps. First, a fibrepreform is manufactured, and afterward it is drawn inside a high-tech furnace incorporated in a tower set-up. Fibre preforms are built by stacking silica capillary tubes and solid

**Citation:** Luzi, G.; Lee, S.; Gatternig, B.; Delgado, A. An Asymptotic Energy Equation for Modelling Thermo Fluid Dynamics in the Optical Fibre Drawing Process. *Energies* **2022**, *15*, 7922. https://doi.org/ 10.3390/en15217922

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 20 September 2022 Accepted: 20 October 2022 Published: 25 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

rods. This allows for quick, low-cost, and flexible manufacturing of preforms. After stacking, capillaries and rods are seized together by thin wires and eventually fused in an intermediate drawing process in which the structures do not achieve the required final dimensions, but are instead drawn into an intermediate preform-cane. A large number of preform-canes are usually manufactured, as they can be utilized for the development and optimization of different PCFs structures. During the drawing process, holes might experience distortion and positions and sizes might be altered. This occurs because the drawing process takes place at elevated temperatures and surface tension may lead to the collapse of the internal holes that form the air lattice. Therefore, internal pressurization is commonly applied to prevent hole collapse. An alternative to glass MOFs is represented by microstructured polymer optical fibres (MPOFs) [13]. They have several advantages compared to glass MOFs. For instance, the processing temperature of polymers is much lower than that of the glass, and the polymerization processes are easier to control. This entails the utilization of different techniques to produce polymer preforms of arbitrary crosssection arrangements, such as extrusion, polymer coating, polymerization in a mold, and injection molding [14]. Another advantage of MPOFs is that they can be drawn over a wide temperature range without significant changes in the fibre structure, unlike glass MOFs, for which temperature variations of just a few percentage points can induce significant variations in the fibre microstructure. In addition, the high temperatures involved in the fabrication of glass MOFs hinder the possibility of modifying the optical properties using dopants, as phase separation may occur [14]. Conversely, MPOFs can be easily doped with atomic species, molecular components, dispersed molecules, and phases. Moreover, the lower temperatures involved in the fabrication process of MPOFs reduce the chances of hole collapse and allow low cost production in large volumes, as both the material and the production process are cheaper [15]. Independently of the material used to fabricate MOFs, accurate control of fibre structures concerning hole dimensions and position is essential for the manufacturing of PCFs with specific properties. The inclusion or elimination of interstitial holes can have dramatic consequences on the final fibre properties. The key element in the drawing process of PCFs is the ability to maintain the highly regular structure down to the final dimensions.

Mathematical models and numerical simulations that can describe the fibre drawing process are highly desirable, as they allow for understanding and quantifying the transport phenomena and main physical quantities involved in the process. Furthermore, they represent more valid predictive and process control tools compared to expensive experiments. To this end, many theoretical and numerical studies have been carried out over the past fifty years. Early studies mainly focused on one-dimensional drawing models of solid fibres. Peak and Runk [16] derived a simplified model consisting of axial momentum and energy equations with a simplified radiation model to predict the neck shape and the temperature distribution of a silica rod during the drawing process. Myers [17] extended Glicksman's model [18] by introducing a radiative heat transfer model that considers the slope of preform surfaces, the spectral variation of the glass properties, and the dependencies of the emissivity on the fibre diameter. Fitt et al. [19] utilized asymptotic techniques to derive a model that describes the drawing process of a capillary and examined it on selected asymptotic limits to isolate and quantify the effects of main physical parameters on the drawing process. Luzi et al. [20] numerically solved the asymptotic model of Fitt et al. [19] by assuming Gaussian distributions of the temperature profile inside the furnace. The numerical results are in good accordance with the experimental ones, both for the case of an unpressurized capillary and for the case when the inner pressurization is applied. In a series of contributions, Voyce et al. [21–23] extended the previous work by Fitt et al. [19] by including the effect of preform rotation in their model. Rotation is particularly useful to control Polarization Mode Dispersion (PMD) and fibre birefringence effects, as well as to tune the capillary size. More recently, Taroni et al. [24] utilized asymptotic analysis to derive simplified momentum and energy equations to describe the drawing process of a

solid fibre while considering the heat transport within the fibre via conduction, convection, and radiative heating.

Numerical investigations of the drawing process of fibres with more complicated crosssectional structures have been initially carried out using finite element-based commercial software. In a two-series contribution, Xue et al. [25] first performed a scaling analysis of the governing equations to determine the importance of the main parameters involved in the drawing process. Afterwards, they simulated the transient drawing process of MOFs containing five holes, showing that the shape of the holes changes dramatically in the vicinity of the neck-down region. In the subsequent manuscript, Xue et al. [26] investigated the steady-state process, focusing on the effects of surface tension, viscosity, and stress redistribution within the fibre. Non-isothermal simulations revealed that the slope of the neck-down region is highly sensitive to the viscosity profile, and therefore to temperature gradients. In a different work, Xue et al. [27] scrutinized the mechanism of hole deformation for silica and polymer fibres. To this end, they simulated the drawing process of a five-hole and polarization-maintaining structure, focusing on hole deformation and hole expansion in terms of the capillary number, draw, and aspect ratio. Luzi et al. [28] modeled the drawing process of six-hole MOFs, obtaining very good agreement between numerical simulations and experiments as long as the applied inner pressurization was not too high. Even in that case, the shape of the deformed holes was in qualitatively good agreement with the experimental one. Nevertheless, solving the full three-dimensional problem proves numerically expensive, and significant computational resources are often needed.

To cope with this issue, Stokes et al. [29] presented a general mathematical framework to model the drawing process of optical fibres of general cross-sectional shape, with the only requirement that the fibre must be slender. Chen et al. [30] extended the work of Stokes et al. [29] by including channel pressurization. Buchak et al. [31] developed the generalized Elliptical Pore Model (EPM), a very efficient method for cases in which the fibre cross-section contains elliptical holes. The evolution of an inner hole is determined by the solution of a set of ordinary integrodifferential equations that determine the centroid position, orientation, area, and eccentricity along the drawing direction. In a different contribution, Buchak and Crowdy [32] employed spectral methods with conformal mapping to obtain a very accurate reconstruction of the cross-sectional shape. Chen et al. [33] utilized the numerical approach of Buchak and Crowdy [32] to model the drawing process of a sixhole MOF and compared the results with the experiments and the Finite Element Method (FEM)-based simulations by Luzi et al. [28]. The approach used by [33] allows for accurate computation of the hole-interface curvature and is in better agreement with experimental results compared with the FEM simulations, and is significantly more computationally efficient. However, in these contributions, the heat transfer between the furnace and the fibre is not modelled, and the drawing is assumed to be isothermal with an assumed fibre temperature profile. In a recent manuscript, Stokes et al. [34] derived an asymptotic energy equation for the full three-dimensional problem utilizing only asymptotic analysis based on the small fibre aspect ratio and coupled it with the generalized EPM of Buchak et al. [31]. Jasion et al. [35] proposed the MicroStructure Element Method (MSEM) for modeling the drawing process of MOFs with a high filling fraction and thin glass membranes, such as Hollow Core Photonic Crystal fibres (HC-PCFs). They used the model of Fitt et al. [19] to describe the evolution of the external jacket that surrounds the microstructured array of air-holes with a network of fluid struts linked through nodes where surface tension, viscous, and pressure force act. However, the energy equations utilized in these two contributions consider neither the heat transfer across the fibre cross-section nor the viscous diffusion.

Detailed numerical investigations concerning the conjugate heat transfer between the fibre and the furnace have been carried out by different researchers. Lee and Jaluria [36] simulated the conjugate heat transfer between the furnace and solid fibre assuming an axissymmetric geometry and a given distribution of the fibre shape. Chodhury and Jaluria [37] investigated the effects of the fibre draw speed, inert gas velocity, furnace dimensions, and gas properties on the temperature distribution within a solid glass fibre and an oven. Yin

and Jaluria [38] utilized the zonal method and the optically thick approximation to compute the radiative heat exchange between the furnace and solid glass fibre. Their numerical investigations reveal that the zonal method can predict the radiative flux with reasonable accuracy independently of the temperature distribution within the fibre, although the optically thick approximation can only predict a correct temperature distribution when the radial temperature distribution is small. In a subsequent contribution, Yin and Jaluria [39] utilized the zonal method and the optically thick approximation with a force balance to generate neck-down profiles of a solid fibre. On the one hand, the difference between the profiles generated using the zonal method and the optically thick approximation is not very large. On the other hand, other parameters have a significant influence on the thermal neck-down profiles, that is, the fibre drawing speed and the furnace wall temperature, while the purge gas velocity and gas type have only minor effects. Xue et al. [40] modeled the transient heat transfer through an eight-hole MOF and found that a MOF heats up faster than a solid one, as there is less material to heat. In addition, the inclusion of radiative heat transfer across the holes accelerates the heating in the whole fibre. However, numerical treatment of the full heat transfer problem requires significant computational efforts, even for the two-dimensional case of a solid fibre where axis symmetry can be exploited to simplify the problem.

In this work, we derive an asymptotic energy equation for the drawing process of annular capillaries by extending the work of Taroni et al. [24] and build a complete asymptotic fibre drawing model with the equations obtained by Fitt et al. [19]. In addition, we include the effects of viscous dissipation in our model. This simplified system of equations can be solved numerically quite readily compared to the full three-dimensional problem, and its predictions are in very good agreement with the experimental results by Luzi et al. [20]. The rest of this manuscript is organized as follows. In Section 2, we develop the theoretical formulation of the problem, concisely yet comprehensively providing the mass, momentum, and energy equations with the associated boundary conditions that govern the drawing process of an annular capillary. In Section 3, we derive the final asymptotic equations for the drawing process of a capillary, unifying the mass, momentum, and evolution equations of Fitt et al. [19] with a simplified energy equation. In Section 4, we compare the numerical outcomes of the asymptotic model with the experimental results of Luzi et al. [20], and in Section 5 we discuss our results, highlighting possible ways to improve the present model. Finally, in Appendix A, we derive an asymptotic energy equation for rotating capillaries and show how it can be coupled with the asymptotic model of Voyce et al. [21].
