2.1.5. Determination of the Furnace and Ambient Temperature Profiles

As our furnace is very similar to the one used by Voyce et al. [22], we begin the determination of the temperature profiles of the furnace by extracting their experimental values. Voyce et al. [22] measured the temperature profile of the oven by inserting a thermocouple into the air hole of a capillary tube that was slowly introduced into the oven, allowing the thermal equilibrium between the fibre and the furnace to be assumed. They measured the temperature profile of the furnace at three peak temperatures, say, 1300 ◦C, 1600 ◦C, and 1760 ◦C, reporting that it is very challenging to measure temperature profiles at peak temperatures higher than 1700 ◦C as thermocouples become inaccurate and may melt. They measured the temperature distribution only in a restricted portion of the furnace, in the vicinity of the peak value where the temperature is high enough to permit the drawing process to occur. The length of this portion of the oven is approximately 12 mm. We extract the experimental data of Voyce et al. [22] and interpolate them with a functional form similar to the one proposed by Taroni et al. [24]:

$$\overline{T}\_f(\overline{z}) = \overline{T}\_M \left( \frac{1}{5} + \frac{4}{5} \exp\left( c(\overline{T}\_M)(\overline{z} - 0.5)^2 \right) \right) \tag{26}$$

see Figure 2, where *TM* = *TM Ts* and

$$c(\overline{T}\_M) = 2.1053\overline{T}\_M - 4.6316\tag{27}$$

With this selection of the parameters, we keep the temperature distribution centred in the middle of the "hot zone" and let the width of the distribution depend on the peak temperature. Our choice is motivated by the fact that the temperature peaks shift only slightly towards the left with respect to the furnace center when the peak temperature increases. On the other hand, the width of the temperature distribution is affected by the peak temperature. Using Equation (26), we always obtain interpolation curves with *R*<sup>2</sup> > 0.9 for the three temperature distributions measured by Taroni et al. [24]. We accept this result and do not try to obtain a closer match between interpolated and experimental data. In addition, we choose *Ta*(*z*) = <sup>3</sup> <sup>4</sup>*Tf* , in accordance with Taroni et al. [24].

**Figure 2.** Furnace temperature profiles measured at three different peak temperatures by Voyce et al. [22] and interpolated temperature profiles according to the relation in (26).

2.1.6. The Functional Form of *r*, *kh*, *α*, and *β*

In general, values of the specific emissivity *<sup>r</sup>* and the convective heat transfer coefficient *kh* depend on fibre thickness, temperature, and properties of the material. Here, we follow the approach of Taroni et al. [24] by selecting

$$
\epsilon\_r = 1 - e^{-2.5\chi(h\_2 - h\_1)}\tag{28}
$$

where we replace the fibre radius with the capillary thickness. Therefore, the specific emissivity *<sup>r</sup>* decreases with decreasing capillary thickness. Concerning the convective heat transfer coefficient *kh*, we utilize the form suggested by Geyling and Homsy [42]

$$k\_{\rm li} = \frac{w^{\frac{1}{3}}}{\left(h\_2 - h\_1\right)^{\frac{2}{3}}}\tag{29}$$

for the case of slow drawing ratios, while in the case of high drawing ratios we employ the correlation from Patel et al. [43] in the form suggested by Xue et al. [25]

$$k\_h = 128.27w^{0.574} \tag{30}$$

In Equation (29), we consider the capillary thickness instead of the radius. Finally, following the work of Taroni et al. [24], we let *α* vary with the capillary thickness:

$$\alpha = \frac{\sigma \epsilon\_r T\_s^3 L}{k\_c} \left( 1 - e^{-2.5 \chi (h\_2 - h\_1)} \right) \tag{31}$$

However, in contrast to Taroni et al. [24] we utilize variable Nusselt numbers *β*, with *kh* defined by Equations (29) and (30):

$$\beta = \frac{w^{\frac{1}{3}}L}{k\_c(h\_2 - h\_1)^{\frac{2}{3}}}\tag{32}$$

in the case of slow drawing ratios and

$$\beta = \frac{128.27w^{0.574}L}{k\_c} \tag{33}$$

in the case of high drawing ratios.
