Asymptotic solutions of glass temperature profiles during steady optical fibre drawing

Abstract

In this paper we derive realistic simplified models for the high-speed drawing of glass optical fibres via the downdraw method that capture the fluid dynamics and heat transport in the fibre via conduction, convection and radiative heating. We exploit the small aspect ratio of the fibre and the relative orders of magnitude of the dimensionless parameters that characterize the heat transfer to reduce the problem to one- or two-dimensional systems via asymptotic analysis. The resulting equations may be readily solved numerically and in many cases admit exact analytic solutions. The systematic asymptotic breakdown presented is used to elucidate the relative importance of furnace temperature profile, convection, surface radiation and conduction in each portion of the furnace and the role of each in controlling the glass temperature. The models derived predict many of the qualitative features observed in real industrial processes, such as the glass temperature profile within the furnace and the sharp transition in fibre thickness. The models thus offer a desirable route to quick scenario testing, providing valuable practical information about the dependencies of the solution on the parameters and the dominant heat-transport mechanism.

Appendix

1.1 Phase plane analysis at constant furnace temperature

As noted in the main text, system (18a,b) is amenable to a phase plane analysis in the case where the furnace temperature $T_\mathrm{f}$ is assumed constant (for consistency with the assumption of §2.1.4 the ambient temperature $T_\mathrm{a}$ is then also constant). Dropping all leading-order $^{(0)}$ superscripts to simplify notation, it proves convenient to consider the phase plane in $(w,\mu )$-space. With $T(\mu )$ given by inverting relation (14), the autonomous system (18a,b) then becomes

$$\begin{aligned} \frac{{\text{ d}}w}{{\text{ d}}z}&=\frac{Fw}{3\mu }, \end{aligned}$$

(36)

$$\begin{aligned} \frac{{\text{ d}}\mu }{{\text{ d}}z}&=\frac{64\mu }{ \text{ Pe}^* T(\mu )^2 \sqrt{w}} \left( \alpha (T(\mu ) ^4-T_\mathrm{f}^4) +\beta (T (\mu )-T_\mathrm{a}) \right) , \end{aligned}$$

(37)

with $T(\mu )=(1+\log (\mu ) /32)^{-1}.$ The phase plane for this system is the solution trajectories of the ordinary differential equation

$$\begin{aligned} \frac{{\text{ d}}w}{{\text{ d}}\mu } = \frac{\text{ Pe}^* F T(\mu )^2 w^{3/2}}{192 \mu ^2 \left( \alpha (T(\mu ) ^4-T_\mathrm{f}^4) +\beta (T (\mu )-T_\mathrm{a}) \right) }. \end{aligned}$$

(38)

As in the main text, we can consider either constant $\alpha , \,\beta ,$ or $\alpha (w), \,\beta (w)$ as in, for example, Eq. (22). Here we present two representative example phase planes for the case in which $\alpha$ and $\beta$ are functions of $w.$ A key factor influencing the qualitative features of the phase diagram is whether the ambient temperature $T_\mathrm{a}$ is greater than or less than the glass-softening temperature. In Fig. 11 we show the phase plane for $\alpha =\alpha _0 (1-\exp (-10/\sqrt{w})), \,\beta =\beta _0 w^{2/3},$ and with $\alpha _0=160, \beta _0=1, \,T_\mathrm{f}=1.57,\,T_\mathrm{a}=0.75$ (lower than the glass-softening temperature, which normalizes to 1), and $\text{ Pe}^* F =3840.$ The solid curves are the phase trajectories, while the dashed curve is the nullcline on which $\alpha (T(\mu ) ^4-T_\mathrm{f}^4) +\beta (T (\mu )-T_\mathrm{a}) =0.$ Only a limited range of the $\mu$-axis is shown for simplicity, but it is clear what the evolution would be for such a fibre. The fibre enters at a small value of $w,$ the input velocity. At this stage its viscosity is extremely large. As the fibre enters further into the furnace it gains axial velocity and its viscosity drops precipitously before levelling off. Depending on the draw ratio (ratio of exit velocity to input velocity), the viscosity may reach a minimum value (phase path crossing a nullcline) and then increase again before the fibre exits the furnace.

Fig. 11

Phase plane for ambient temperature below glass-cooling temperature. If the draw ratio is sufficiently high, then the fibre viscosity may increase again before the fibre exits the furnace

In the case where the ambient temperature $T_\mathrm{a}$ is higher than the softening temperature, the evolution is less dramatic. An example is shown in Fig. 12, with all parameters as before except $T_\mathrm{a} =1.18 >1.$ In this case, the viscosity again drops rapidly as the fibre enters the furnace, but then stays at a low value until exit.

Fig. 12

Phase plane for ambient temperature above glass-softening temperature. The fibre viscosity decreases monotonically as the fibre passes through the furnace for all draw ratios

Bearing in mind the comments of §4.1.2 about the unrealistic nature of the proposed functional dependence of $\beta ,$ we also show a phase plane for the case of constant $\beta =50$ (Fig. 13). In this case the phase plane shows no qualitative difference between the cases where the ambient temperature is above or below the softening temperature, so we show only the case considered throughout most of the paper, where $T_\mathrm{a} = 3T_\mathrm{f}/4=1.18 >1$ (and all other parameters are as above). The fibre viscosity now decreases monotonically as the fibre passes through the furnace, for all draw ratios.

Fig. 13

Phase plane for ambient temperature above glass softening temperature, and parameter $\beta$ taken to be constant. The fibre viscosity decreases monotonically as the fibre passes through the furnace for all draw ratios