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Preparation of Polymeric Membranes 81
where s ¼ 1=n, r ¼ r=R, l can be obtained by solving Eq. (32) numerically:
ð l
k
l 2
r
r
s
dr ¼
ð 1
r
r
l 2
r
s
dr: ð32Þ
The volumetric flow rate of the dope solution in an annulus can be calculated by
s ð1 3 DPR
Qp ¼ pR l
2mL k
2
r 2 sþ1 r s dr; ð33Þ
where Qp is the volumetric flux of the dope solution and R is the outer radius of the spinneret.
The shear rate induced at the outer wall of a spinneret is expressed as:
dvz DPR
dr ¼ r¼R 2mL ð1 l2Þ s ;
¼ Qp
pR 3
ð1 l 2 Þ s
R 1
k l2 r2 j j sþ1 r s dr :
The velocity distribution, shear rate of a dope solution with 21% copolyimide P84 are
illustrated in Fig. 2.28 (98). The maximum shear rates were obtained at the outer and inner
edges of the annular dope flow in the spinneret. When the outer surface of the hollow fiber
membranes was the selective skin layer, the shear rate of the dope solution on the outer edge
of the annular dope flow in the spinneret represented the shear rate of dope solution within
Shear rate (s –1 )
0
2000
4000
6000
8000
10000
12000
OD=1mm
ID=0.44mm
A
B
Fig. 2.28 Velocity and shear rate distributions of dope solution within the spinneret for different
spinning conditions (polymer solution: 21 wt% P84; H 2O/NMP: 3.6/96.4 (wt/wt); a = 0.45. Spinneret:
OD = 1 mm; ID = 0.44 mm; extruding velocity: A: 1.6 mL/min; B: 16 mL/min. Power law; s: shear
stress, _g: shear rate) [adapted from ref. (98)].
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
velocity (m.s –1 )
ð34Þ