Membrane and Desalination Technologies - TCE Moodle Website

86 J. Ren and R. Wang
where A is the cross-section area of nascent hollow fiber membrane in the air gap, which can
be described as:
combining with the following equation
A ¼ Qp
; ð40Þ
vzðzÞ
tzzðzÞ
vzðzÞ
tzzð0Þ
¼ ; ð41Þ
vzð0Þ
here tzzð0Þ and vzð0Þ are the tension stress and the velocity of the die-swell at the exit of the
spinneret.
The tensile force F can be obtained at any point in the air gap (z) or at the air gap distance L:
F ¼ k n
n 1
¼ k pðD2 0 d2 0 Þ
4
n vzðzÞ
vzð0Þ
n
n 1
n 1
n
1
n
vzð0Þ
z
n
A0;
n DR n 1
n 1 n vzð0Þ
L
where, A0 is the cross-section area of the nascent hollow fiber membranes at the die-swell.
DR, D0 and d0 are defined in Eq. (35). The velocity distribution of the nascent hollow fiber in
the air gap can be described as:
h
n 1 z
i n
n 1
vzðzÞ ¼vzð0Þ DR n 1 þ 1 : ð43Þ
L
And the elongation rate inside the air gap can be written as:
!
vzðzÞ n
h
1 z
i 1 n 1
n 1 DR n 1
¼ vzð0Þ DRn n 1 þ 1
; ð44Þ
dt n 1 L L
where vzðzÞ
is the elongation rate at the point z in the air gap. And the tensile stress at the
dt
bottom of the air gap (tzzðLÞ) is described as:
tzzðLÞ ¼kDR
n
n 1
n
n 1
DR n 1 n vzð0Þ
L
n
:
ð42Þ
n
: ð45Þ
The elongation rate inside the air gap and the tensile stress at the bottom of the air gap are
strongly controlled by DR, the air gap distance (L), and the velocity of the polymer solution at
the die-swell.
The theoretical velocity distribution in the air gap at different power numbers (with n < 1,
n = 1 and n > 1) is shown in Fig. 2.32. In fact, when the hollow fibers are spun into the air gap
from the spinneret, the solvent evaporation, temperature change, the exchange between the
polymer solution and the bore fluid, the phase inversion and solidification strongly influence
the spinline dynamics in the air gap. The actual spinline dynamics can be divided into three
regions, which is shown in Fig. 2.32: