Membrane and Desalination Technologies - TCE Moodle Website

80 J. Ren and R. Wang
5.1.1. Rheology of the Polymer Solution Inside the Spinneret
For the shear flow inside the annular channel of the spinneret in Fig. 2.27, the velocity
changes in r direction perpendicular to the direction of the motion (z direction), but it remains
constant along the z direction of the flow as long as one stays on the same streamline. In
contrast, for the nascent hollow fiber membranes in the air gap, a uniaxial elongational field
exists, the fluid velocity changes in the z direction of the flow, but it remains constant in the
perpendicular r direction. The shear flow generates a rotational flow component by the
perpendicular velocity gradient. And the macromolecules of the dope solution are partly
compressed and extended, which causes some orientations imparted on the parts of the
molecules. But for an uniaxial elongational flow field, only a velocity gradient in the flow
direction exists and no rotational flow component is generated. And the macromolecules only
experience elongation. Compared with the shear flow, the elongation flow is more efficient in
imparting molecular orientation provided that the velocity gradients and resident times in the
elongational and shear fields are comparable, especially for hollow fiber membranes spun at a
very high drawing ratio.
When a dope solution is extruded through a spinneret, its rheological properties could be
correlated with shear flow induced by the shear stress within the spinneret, which has an
effect on chain conformation and induces molecular orientation in the skin layer during the
phase inversion process of hollow fiber formation.
During the hollow fiber spinning, when a non-Newtonian polymer solution is pumped
through an annular spinneret, the z-component of momentum equation may be written in
cylindrical coordinates in terms of shear rate as follows (89, 97):
1 d
r dr rtrz ð Þ ¼ dp
; ð28Þ
dz
where r and trzare the radius axis of the annular spinneret and the shear stress, respectively.
dp
dz is a pressure gradient along z-direction, and thus, can be replaced by DP=L, where L is
the length of the annulus of the spinneret. Polymer solutions used for spinning hollow fiber
membranes are usually non-Newtonian fluids, showing shear-thinning behavior. For a powerlaw
fluid of spinning solutions, shear stress trz can be described as follows:
trz ¼ m dvz
dr
n 1 dvz
dr
; ð29Þ
where dvz
dr is the shear rate along the r direction, m and n are the viscosity coefficient and
power number of Eq. (29), respectively. At r ¼ lR, vz is maximum and trz ¼ 0. The velocity
profile with the boundary conditions of vz ¼ 0atr ¼ kR and r ¼ R is as follows:
vz ¼ R DPR
2mL
vz ¼ R DPR
2mL
sð r
k
s ð 1
r
l 2
r
r
r
l 2
r
s
s
dr; k r l ð30Þ
dr; k r l ð31Þ