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Preparation of Polymeric Membranes 89
where F, r and g are the elongation force, density of the polymer solution and the gravity
constant, respectively. And the force in the air gap decreases from the exit of the spinneret
(z = 0) to the bottom of the air gap (z = L). The elongation stress (tzzðzÞ) at point z can be
described as:
tzzðzÞ ¼ F
ðL 1 dvzðzÞ
¼ rgvzðzÞ dz ¼ k
A vzðzÞ dz
z
n
; ð47Þ
where A is the cross-section area of the nascent hollow fiber membrane. With the low
deformation at air gap, the elongation viscosity can be assumed to be a constant. And
n = 1, the rheology equation can be obtained in the air gap:
d 2 vzðzÞ
dz2 1 dvzðzÞ rg
þ ¼ 0:
vzðzÞ dz k
ð48Þ
The boundary conditions are as follows:
vzðzÞ ¼v0 at z ¼ 0; (48a)
dvzðzÞ
¼ 0
dz
at z ¼ L: (48b)
To spin ultrafiltration and microfiltration hollow fiber membranes with a small air gap, the
spinline velocity does not change greatly, and the extension stress can be described as:
tzzðzÞ ¼ F
A
rgðL zÞ: ð49Þ
And the velocity distribution in the air gap is written as:
vzðzÞ ¼vzð0Þþ rg
k
The velocity at the bottom of the air gap:
1
n
L nþ1
n 1 1
vzðLÞ ¼vzð0Þþ rg
k
z
L
nþ1
n
: ð50Þ
1
n
L nþ1
n : ð51Þ
For the free-fall spinning process at the same air gap, with the increase of power number n,
the velocity of the nascent hollow fiber membranes in the air gap will decrease. And the
velocity at the bottom of the air gap increases as a power number (n þ 1/n) of the air gap
distance (L). That is to say, for the free-falling process, the spinning velocity and the air gap
are coupled each other, which is different from the drawing process. When the factors of
solvent evaporation, temperature decrease, solvent and nonsolvent exchange between the
polymeric solution and the bore fluid, the phase inversion and solidification are considered,
the velocity of the nascent fibers will decrease.
For the free-fall wet spinning process, there is almost no elongation stress existing and the
spinning velocity is almost the same as that of the spinneret exit (die-swell). A small drawing
velocity change caused by the spinning roll or drum will cause a greatly increase of the tensile