An efficient numerical framework for fiber spinning scenarios with evaporation effects in airflows

Abstract

In many spinning processes, as for example in dry spinning, solvent evaporates out of the spun jets and leads to thinning and solidification of the produced fibers. Such production processes are significantly driven by the interaction of the fibers with the surrounding airflow. Faced with industrial applications producing up to several hundred fibers simultaneously, the direct numerical simulation of the three-dimensional multiphase, multiscale problem is computationally extremely demanding and thus in general not possible. In this paper, we hence propose a dimensionally reduced, efficiently evaluable fiber model that enables the realization of fiberair interactions in a two-way coupling with airflow computations. For viscous dry spinning of an uni-axial two-phase flow, we deduce one-dimensional equations for fiber velocity and stress from cross-sectional averaging and combine them with two-dimensional advection-diffusion equations for polymer mass fraction and temperature revealing the radial effects that are observably present in experiments. For the numerical treatment of the resulting parametric boundary value problem composed of one-dimensional ordinary differential equations and two-dimensional partial differential equations we develop an iterative coupling algorithm. Thereby, the solution of the advection-diffusion equations is implicitly given in terms of Green's functions and leads for the surface values to Volterra integral equations of second kind with singular kernel, which we can solve very efficiently by the product integration method. For the ordinary differential equations a suitable collocation-continuation procedure is presented. Compared with the referential solution of a three-dimensional setting, the numerical results are very convincing. They provide a good approximation while drastically reducing the computational time. This efficiency allows the twoway coupled simulation of industrial dry spinning in airflows for which we present results for the first time in literature.

Modeling and simulation of curved fibers in dry spinning scenarios

“…The model combines onedimensional rod equations with two-dimensional advection-diffusion equations covering the radial effects of mass fraction and temperature. Adopting the efficient numerical framework from the uniaxial case [10] allows simulations of industrial dry spinning setups with two-way coupled fiber-air interactions, where lateral fiber movements take place. The efficiency of the proposed modelsimulation framework builds a good basis for optimization and optimal design of dry spinning processes.…”

“…with the associated dimensionless function N : R 3 → R and the model for the diluent density in the air at the fiber surface given in [10]. For an appropriate closing of the System 1 we need rheological laws for the viscosity µ, the evaporation enthaply δ, the conductivity C, the diffusivity D, as well as the material densities ρ 0 p , ρ 0 d and the material specific heat capacities q 0 p , q 0 d , which we will specify in the industrial setup.…”

“…Concerning industrial devices up to several hundred fibers are spun simultaneously such that mutual fiber-air interaction effects have to be incorporated into the dry spinning models. In [10] we developed an efficient model simulation framework, which made a fully two-way coupled simulation of dry spun fibers immersed in an airflow feasible. Proceeding from a three-dimensional free boundary value problem for a viscous uni-axial radially symmetric stationary two-phase flow, we deduced a dimensionally reduced fiber model.…”

“…Considering industrial dry spinning devices with lateral air inflow into the spinning duct, i.e., the airflow velocity is meanly perpendicular to the fiber direction, the assumption of straight fibers is no longer tenable due to expected lateral deflections. Therefore, we extend the model and simulation framework presented in [10] to the description of curved fibers in such processes. There are two classes of fiber models: string and more sophisticated Cosserat rod models.…”

Melt-blowing of viscoelastic jets in turbulent airflows: Stochastic modeling and simulation

“…This model follows from System 3 by a re-parameterization into Euler (spatial) description, transition to steady-state, and the limit De → 0. The resulting boundary value problem of ordinary differential equations is solved by a continuation-collocation method, which we successfully employed in studies on glass wool manufacturing [3], electrospinning [2] and dry spinning [31]. Further details on the model and its numerical treatment are given in Appendix C. Note that the use of the viscous fiber model is not only physically reasonable, but it also simplifies crucially the numerical treatment.…”