A mathematical analysis of hollow fiber spinning : bore and dope velocity profiles in the air gap

Abstract

Hollow fiber membranes are the most applicable form of membranes in laboratory and industrial scale with their large surface to volume ratios. Hollow fiber membranes usually are fabricated by dry-wet solution spinning, where a polymer solution is co-extruded through an annular region with a bore fluid and the nascent hollow fiber passes through an air gap and then enters a liquid coagulation bath. The spinneret dimension, dope and bore fluid flow rates, air gap length, bore fluid and dope compositions and their physical properties, coagulant composition and condition, shear stress within a spinneret, the ratio of dope to bore fluid volumetric flow rate, and the take-up-to-initial velocity ratio (draw ratio) are the primary factors that determine the final hollow fiber morphology and separation properties. All of the aforementioned parameters are not independent and some of them are functions of others. The objective of this paper is to elucidate the interdependence between some of the above parameters. For this purpose, dope and bore fluid axial velocity in the air gap was assumed as a function of the axial distance from the spinneret outlet, z, and radial velocity as a function of the radial distance from the center of the hollow fiber, r, and z. Dimensionless equations of motion and continuity were then simplified and solved simultaneously based on preceding assumptions. It was found from the analysis that the dimensionless axial velocity of dope was a function of the Reynolds number, Capillary number and Stokes number indicating that the axial velocity acceleration in the air gap is determined by viscous, capillary, and gravity force gradients. It was also concluded that the optimum value of the ratio of bore liquid flow rate to dope flow rate is equal to 0.8 of the ratio of the cross-sectional area of the bore fluid to that of the dope at the spinneret outlet.