Linear stability analysis of nonisothermal glass fiber drawing

The draw resonance effect appears in fiber drawing processes when the draw ratio, defined as the ratio between the take-up and the inlet velocities, exceeds a critical value. In many cases, inertia, gravity, and surface tension cannot be neglected, and a model combining all these effects is necessary in order to correctly describe the physics of the phenomenon. Additionally, it is also known that cooling can have a highly stabilizing effect on the draw resonance instability. However, a detailed analysis encompassing the effect of inertia, gravity, surface tension, and temperature is still lacking. Due to a destabilizing effect induced by geometry in the heat equation, we first show that the maximum critical draw ratio for fiber drawing can be two orders of magnitude lower than the one for the film casting problem when the heat transfer coefficient is assumed constant. By introducing a scaling making the fiber aspect ratio an independent parameter, we next show that the high value of the critical draw ratio encountered in industrial applications could be rationalized only if we consider that the heat transfer coefficient is not constant but depends on both the velocity and the cross-section area of the fiber. Within this framework, we show how the practical stability window is affected by the five control parameters: the draw ratio, the fiber aspect ratio, the inlet temperature, the convective heat transfer coefficient, and the stiffness of the non-homogeneous ambient temperature. We finally discuss the influence of radiative heat transfer on the stability.

Figure 1

Left: Sketch of the fiber drawing process. Right: One-dimensional nonisothermal fiber model.

Figure 2

Critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ for the purely viscous model with convective heat transfer only (${\text{St}}_r^{*}=0$) and a VFT viscosity law. The curve is computed for a constant heat transfer coefficient, an aspect ratio $\mathrm{\Lambda }={10}^3$, an inlet temperature ${\theta }_0=1573.15$ K, ${\theta }_{h_{\infty }}=0.22$, and $k=1$. The inset represent the steady temperature ${\theta }_s\left(x\right)$ for ${\text{St}}^{*}=2.5\times {10}^{-6}$ (blue curve) and ${10}^{-3}$ (red curve) and characterizes the advection-dominated and heat transfer-dominated regimes, respectively.

Figure 3

Neutral stability curves for the purely viscous model with convective heat transfer only (${\text{St}}_r^{*}=0$) and a VFT viscosity law. Each curve is computed for a fiber of aspect ratio $\mathrm{\Lambda }={10}^3$, an inlet temperature ${\theta }_0=1573.15$ K and ${\theta }_{h_{\infty }}=0.22$. The black dashed line represents the classical value ${\text{Dr}}_c=20.218$ encountered in the isothermal case. (a) Influence of the non-homogeneous ambient temperature: critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ with $k=1,2,5,10,20$, and 30 for a constant heat transfer coefficient. The inset represents the ambient temperature ${\theta }_h\left(x\right)$ for $k=1,10$, and 30. (b) Influence of the heat transfer coefficient governing law: critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ with $k=1,2,5,10,20$, and 30 for a heat transfer coefficient following the Kase and Matsuo model (12).

Figure 4

Influence of the fluidity $F^{*}$: critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ computed for various values of the fluidity $F^{*}$, with an aspect ratio $\mathrm{\Lambda }=1000$ and a stiffness parameter $k=30$. The dimensionless inlet velocity is fixed to its value corresponding to ${\theta }_0=1603.15$ K. The black dashed line represents the classical value ${\text{Dr}}_c=20.218$ encountered in the purely viscous isothermal case.

Figure 5

Influence of the fiber aspect ratio: critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ computed for fibers of different values of the aspect ratio $\mathrm{\Lambda }$, with an inlet temperature ${\theta }_0=1603.15$ K and a stiffness parameter $k=30$. The range of industrial applications is represented by a black box centered around the realistic Stanton number ${\text{St}}^{*}=5\times {10}^{-3}$.

Figure 6

(a) Influence of the inlet temperature on the stability of the system for the full model with convective heat transfer only (${\text{St}}_r^{*}=0$). Critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ computed with $F^{*}=1.2\times {10}^{-3},1.9\times {10}^{-3}$, and $2.9\times {10}^{-3}$, respectively corresponding to inlet temperatures ${\theta }_0=1543.15,1573.15$, and 1603.15 K. Each curve is obtained for a fiber of aspect ratio $\mathrm{\Lambda }=120$ and a stiffness parameter $k=30$. (b) Critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ computed for stiffness parameter $k=10,20$, and 30 with an inlet temperature ${\theta }_0=1603.15$ K and an aspect ratio $\mathrm{\Lambda }=100$.

Figure 7

Influence of radiative heat transfer on the stability of the system. Neutral stability curves for the full model represented in the case of a pure convective heat transfer (continuous line) and convective and radiative heat transfer (dashed line) for $\mathrm{\Lambda }=25$ (a) and 120 (b). Each curve is computed for an inlet temperature ${\theta }_0=1603.15$ K and $k=30$. The radiative Stanton number is fixed to ${\text{St}}_r^{*}=5.8\times {10}^{-5}$. The viscosity and the heat transfer coefficient respectively follow a VFT law and the Kase and Matsuo model.

Figure 8

Neutral stability curves for the purely viscous model with convective heat transfer only (${\text{St}}_r^{*}=0$) and constant heat transfer coefficient. Each curve is computed for an aspect ratio $\mathrm{\Lambda }={\mathrm{\Lambda }}_{\text{film}}={10}^3$ and an inlet temperature ${\theta }_0=1573.15$ K. (a) Comparison with film casting: critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ for film casting (black dashed line) and fiber drawing (blue line) with ${\theta }_{h_{\infty }}=0$ and $k=1$. The viscosity is given by the Arrhenius law with $\mu =3$. (b) Influence of the viscosity law for fiber drawing: critical draw ratio as a function of the Stanton number ${\text{St}}^{*}$ for a temperature-dependent viscosity following an Arrhenius law with $\mu =B^{*}$ (blue line) and a VFT law (yellow dashed line) with ${\theta }_{h_{\infty }}=0.22$ and $k=1$. The black dashed line represents the classical value ${\text{Dr}}_c=20.218$ encountered in the purely viscous isothermal case.