Study of the non-isothermal glass fibre drawing process

The glass fibre drawing process is simulated using a finite-element method. The two-dimensional energy and momentum equations are solved in their fully non-linear forms. These are coupled via the temperature-sensitive viscosity function. Both convective and radiative cooling mechanisms are taken into account on the filament surface. An effective emissivity of about 0.2 is found to be applicable to the drawing conditions in this paper. Even at this fairly low effective emissivity, radiation is found to be the dominant mode of cooling. The material thermal conductivity is found to have a small but definite influence on the filament profiles. Two-dimensionsl effects of the kinematic field are only significant up to a distance of about two orifice radii from the nozzle exit.The symbols in the square brackets show the dimensions of the parameters;M Mass,L Length,T Temperature,t Time. a Constant radius of a uniform cylinder L] - A Local cross-sectional area of the filament L ² ] - b _i Total tension applied on the filament boundary surface in thei ^th direction ML/t ² ] - c Specific heat L ² /t ² T] - D Local filament diameter L] - f _i i ^th component of the body-force vector L/t ² ] - h Surface convective heat transfer coefficient of the filament M/t ³ T] - H Total equivalent heat transfer coefficient due to both convection and radiation M/t ³ T] - k Thermal conductivity ML/t ³ T] - M Mass-flow rate M/t] - n Coordinate normal to the local filament surface L] - Nu Local Nusselt number –] - Average Nusselt number –] - Q Rate of heat transfer ML ² /t ³ ] - Volume-flow rate ³ /t] - r Radial coordinate L] - R Local radius of the filament L] - Re _x Reynolds number based on characteristic length scalex –] - s Coordinate along the filament surface L] - T Temperature T] - u Radial component of the velocity T/t] - U Free-stream velocity of a uniform flow L/t] - v Local speed of a fluid particle defined by v = ;L/t] - V Volume L ³ ] - v _f Constant velocity of a filament with a uniform radius L/t] - w Axial component of the velocity L/t] - Average axial velocity of the fluid inside the tube L/t] - z Axial coordinate, i.e. axial distance from the orifice exit L] - Exponential coefficient of the viscosity function T ^–1 ] - _ij Kronecker delta –] - Emissivity or total hemispherical emissivity –] - µ Viscosity M/Lt] - µ ₀ Reference viscosity defined byµ = µ ₀ e ^–T M/Lt] - Fluid density M/L ³ ] - Stefan-Boltzmann constant M/t ³ T ⁴ ] - Viscous dissipation function M/Lt ³ ] - a Of air - a Based on the (constant) filament radius - C.L. Referred to the centre line of the filament - conv Referred to convection - D Dased on the diameter - f Referred to the filament local condition - g Referred to glass - i,j Species in multi-component systems - o Quantity evaluated at the orifice exit - R Based on the radius - rad Referred to radiation - s Evaluated at the filament surface - tot Referred to the total heat transfer from the filament surface - w Evaluated at the tube wall - Ambient condition - * Refers to non-dimensional quantities - — Indicating quantities averaged over the filament cross-section