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High Speed Wire Coating by Withdrawal From a Bath of
Viscoelastic Liquid
STANLEY MIDDLE MAN
Departments of Chemical Engineering and
Polymer Science and Engineering University of Massachusetts
Amherst, Massachusetts 01 003
Data are presented for the thickness H , of liquid coating
entrained by continuous withdrawal at speed U of a wire of radius R
from the free surface of a large bath. For Newtonian fluids of
viscosity p, density p, and surface tension u, the data are carried
out to coating speeds beyond the applicability of current theories,
to Capillary numbers of nearly one hundred. In the high speed range
the data, which cover several orders of magnitude in viscosity, can
be well represented by the equa- tion
for Uplm = N c a > 3. All data presented are at an
essentially constant Goucher number of 0.08, where N,, = R ( p g 1
2 ~ ) ~ . Data for viscoelastic fluids show phenomena quite
distinct, qualitatively and quantitatively, from Newtonian
observations. In particular, strongly elastic fluids show a
markedly reduced ability to be entrained onto the wire. Further,
the coating thickness appears to become independent ofcapillary
number at high speed.
INTRODUCTION
his paper is concerned with the thickness of liquid T c o a t i
n g which can be entrained by a wire or filament withdrawn
vertically through the free surface of a large quiescent liquid
bath. Figure 1 shows the features of the system of interest, which
might form a model of indus- trial systems for use in producing
solid plastic coatings
meniscus I r e g i o n
I - 4 c o a t i n g b a t h
on wires, or for putting finish solutions onto textile fibers
prior to dyeing and/or processing into fabric.
Prior studies of this and related problems exist (I), which
provide background and motivation for the present work reported
here. The major contributions of this paper are experimental and
extend previous studies (2, 3) to higher speed coating (higher
Capillary number) including coating of viscoelastic solutions. I t
is shown that elasticity leads to order of magnitude reductions in
coating thickness by comparison to Newtonian fluids.
EXPERIMENTAL The coating tank was a box 10 x 10 x 9 cm.
Nylon
monofilament of diameter 0.0524 cm was drawn through the tank
from a reel, onto a takeup roll driven by a constant speed motor.
Figure 2 shows a sketch of the sys tern .
Physical properties of the fluids studied are given in Table 1.
Rheological data for the polymer solutions are presented in Figs. 3
and 4 .
One set of data was obtained with glycerine at subam- bient
temperatures (down to 3.3C), producing vis- cosities as high as 83
poise.
Monofilament speeds were in the range of 3 to 86 cm/s (6 to 170
ft/min).
Coating thickness was measured photographically. High speed
flash photographs were taken using a 35 mm
POLYMER ENGINEERING A N D SCIENCE, APRIL, 1978, V o l . 18, N o
. 5 355
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Stanley Middleman
feed reel
c
It
C n : : p r e e l
coated wire
a camera
I coating bath I
Table 1. Physical Properties of the Fluids Studied (25C)
I4 P9 (T, Symbol on poise glcm3 dynelcm Fig. 5
Paraffin oil 0.46 0.86 26.5 Glycerine F 10 1.26 64.9 Glycerine E
8.6 1.26 64.9 80% Glycerine 0.61 1.21 70.7 Polyhall 295 0.1% 10 1
72.5
0.25% 48 1 72.5 0.5% 160 1 68.6 0.75% 350 1 66.5 1.5% 820 1
67.0
POlyOX 301 0.5% 1.1 1 62.4 0.75% 6.6 1 62.0 1.0% 23 1 62.0
Polyhall is a water soluble polyacrylamide. Polyox is a water
soluble polyethylene oxide. Viscosity for the polymer solutions is
the zero-shear viscosity.
0 v A
0 n U
v 0
iLi A
e
Nikon camera with close-up attachments. Negatives were projected
and measurements taken from the en- larged projected images. All
liquids used were transpar- ent, and the nylon monofilament could
be clearly seen in each picture. From the known diameter of the
mono- filament the measured enlarged coating diameter could be
converted to the actual coating thickness. The coating thickness H
, is shown in Fig. 1 .
RESULTS
Data are correlated in dimensionless terms using the
Coating thickness D , = H,(pg/a) (1)
following definitions:
Capillary number NCa = p U / a (2) Figure 5 shows the
theoretical curves of White and
Tallmadge (3) , believed to be the best available theory
relating coating thickness to speed of withdrawal. The theory is
limited in two respects. It is, first of all, a low
100
10
m u)
0 a Ly
.-
- I
0. 10 I00
I 10 I o2 I o3 i (s-1)
F i g . 4 . Pri rn (1 r!/ ti o m i a/ .c tress coeffic irti t -s
l i eci r r(i te dn I (I f o r , S ( J I , I ~ of the . f l i t i d
s strtdietf. Same ,s!ynihl ! iv!/ ( 1 , ~ i t 1 Fig . , 3 ,
speed (non-inertial) theory, which cannot be extrapo- lated with
any confidence to high speed coating beyond a Capillary number of
approximately 0.5. All of the data presented herein, and the region
of dominant commer-
356 POLYMER ENGINEERING AND SCIENCE, APRIL, 1978, Vol. 18, No.
5
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High Speed Wire Coating by Withdruwul From ci Buth of
Viscoelustic Liquid
0.01 L 0. I I 10 100 1000
= * Nca 0-
Fig. 5 . Dimelisionless correlation ofcoating thickness as a
func- tion of wire speed. See Tuble 1 f o r key to .symbols.
cia1 interest, are at high Capillary numbers. The second
limitation of the theory is with regard to rheological properties:
it is strictly a Newtonian theory.
The theory predicts a dependence on the Goucher number, defined
as
N G o = R(pg/2a)l2 (3) This is the only parameter containing the
filament radius. All of our data are at nearly constant values
ofNGo in the range of 0.07 to 0.08. For the p a r f i n oil the
Goucher number is 0.1
It can be seen, from Fig. 5, that the Newtonian data are in
reasonably good correspondence with the theory at the smallest
Capillary numbers. The data obtained with the polymer solutions,
particularly the more elastic polyacrylamide solutions, depart
markedly from the Newtonian data.
DISCUSSION Newtonian Data
As the Capillary number gets large the data show a power
dependence which may be written as
D , = 0.15N:S (N(.,, > 3) (4) No other wire coating data are
available for compari-
son to or confirmation of this result. Data are available,
however, for coating entrained by withdrawal of a flat film from a
bath (4) which also suggest a similar square root dependence ofD,
on Capillary number, and which may be represented by
D, = 0.8Nii,2 (Nc, > 3) (5) A flat film may be regarded as a
cylinder of very small
curvature (large radius), and the Goucher number pro- vides a
dimensionless ratio of curvature of the meniscus (given
appro,;mately by (2a/pg)-*) to the curvature of the cylinder (Mi).
Thus Ey 5 may be regarded as the upper limit on wire coating, for
large wires, i .e., the infinite Goucher number limit. All of the
data for which E q 4 serves were obtained at a Goucher number of
0.08.
There is some evidence, both in theory and experi- ment, that an
inertial effect can produce a reduction in the dependence ofD,
onWr.rr, to an extent depending on a parameter y defined as
y = a(p/p4g)3 (6)
Both the theory and supporting data (5) are for the flat
film case, but one would presume that the idea pertains to wire
coating as well. We see no evidence of inertial effects in our
data, presumably because of the relatively high viscosity of the
fluids used. At the highest Capillary numbers we calculate a value
of y of 0.02 which is too small, according to theory and evidence,
to exhibit this inertial effect.
It would appear, then, that the Newtonian data presented are
consistent with available related obser- vations. The data are
limited in two respects:
Only a single filament diameter was investigated, giving a
Goucher number of 0.08.
The high Capillary number data were obtained with high viscosity
fluids, and so show no inertial effects.
Viscoelastic Data
From the normal stress data ofFig. 4 , it is apparent that the
polyacrylamide solutions are generally more elastic than the
polyethylene oxide solutions. Normal stresses were not measurable
in the 0.5 percent polyethylene oxide solution, which may also be
seen (Fig. 3 ) to be only mildly non-Newtonian. We note in Fig. 5
that the 0.5 percent polyethylene oxide data fall in with the
Newtonian data, when the zero-shear viscosity is used in
calculating N c n .
One of the difficulties in presenting Fig. 5 for non- Newtonian
fluids lies in the lack of a well-defined shear rate for this flow
field. It is not clear how to define the viscosity appropriate to
this flow, and so the magnitude of the Capillary number is subject
to some arbitrary definition of the shear rate at which the
viscosity should be evaluated. In Fig. 5, all non-Newtonian data
are plotted using the zero-shear viscosity.
The use of any other viscosity would shift any particu- lar set
of viscoelastic data horizontally and toward the left along the Nca
axis. Since viscosity does not appear in D , there would be no
vertical shift of the data. Thus, while it might be possible to
displace some sets of vis- coelastic data into coincidence with the
Newtonian data, it is not clear that this could be done in an
unequivocal manner, based on an a priori rational choice of
viscosity for each fluid. Indeed, it is clear that no shift in the
data for 1.5 percent polyacrylamide will cause these data to
resemble a Newtonian fluid, especially if the data are shifted
toward the left.
It would appear, then, that the effect of significant
viscoelasticity is to reduce entrainment of liquid by the filament.
The extent of reduction can be as much as an order of magnitude.
Perhaps the most interesting aspect of the behavior of viscoelastic
fluids is the attainment, at high speed, of an essentially
constant, limiting, value of coating thickness, independent o f f l
amen t speed. Ifsuch a result turns out to be generally observed in
other viscoelastic fluids, it has significant implications for
process operability.
Of course, the presence of an asymptotic upper limit to coating
thickness has obvious consequences in proc- ess design.
Independence ofD, from N ( . , , however, has additional
consequences. For example, one would ex- pect the process to be
insensitive to fluctuations in line speed U and, to a large extent,
insensitive to modest
POLYMER ENGINEERING AND SCIENCE, APRIL, 1978, Vol. 18, No. 5
357
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Stanley Middleman
changes in temperature. Normally, temperature changes would
affect viscosity strongly, altering the Capillary number. For a
Newtonian fluid this would lead to significant changes in coating
thickness.
Despite the fact that the polyacrylamide solutions cover a wide
range of viscous and elastic properties, the data suggest an
asymptotic value of D, within a very narrow range, nearly
independent of polymer concen- tration. More data will be required
before this feature can be elucidated clearly and quantitatively.
The qual- itative implications and features are apparent,
however.
One might speculate, at this point, on the mechanism responsible
for the observed results. It may lie in the character of the
entrainment region, which may be re- garded as a stretching flow
with a short timescale La- grangian transient. Over a distance of
the order of the meniscus height, fluid elements are accelerated
from near rest to the filament speed. We may estimate an order of
magnitude of the stretch rates that the fluids studied have been
subjected to. The static meniscus height is given roughly by the
value of
1 /2
H u = (2) For these data H u is approximately 0.2 cm. The
dynamic meniscus is larger than this, a reasonable estimate being
about 0.5 cm. At 50 cm/s wire speed (a median value for these
data), an approximate stretch rate would be given
i = U / H u = 100s-
The inverse of this stretch rate gives a rough measure of the
time scale of the deformation imposed on the fluid, or
by
(8)
t d = 10-2s (9)
The question of whether this deformation is short depends on the
ratio oftn to some appropriate relaxation time for the fluid, A. In
effect this comparison introduces the Deborah number,
N i m = utd (10) A large Deborah number process is one in which
the deformation has a very short time scale in comparison to the
characteristic fluid time. X. It is well established that
a viscoelastic fluid will show essentially elastic behavior in a
large Deborah number process, and viscous be- havior at the
opposite extreme.
The relaxation times of the polyacrylamide solutions are of the
order of one second, and larger. Hence we see that our experiments
in the region of asymptotic coating thickness correspond to a very
large Deborah number process. Thus the observation that relatively
little vis- cous entrainment occurs seems explicable in these
terms.
ACKNOWLEDGMENT The bulk of the data presented here was obtained
in
undergraduate research projects carried out by Thomas Mumley and
Samuel November. Financial support was provide by the Materials
Research Laboratory of the Polymer Science and Engineering
Department, and by Eastman Kodak Company.
D ,
NOMENCLATURE = dimensionless coating thickness on wire =
acceleration of gravity (cm/s2) = (m/pg) l@ length scale for
meniscus height (cm) = equilibrium coating thickness on wire (cm) =
p U / u Capillary number = h/tD Deborah number = R(pg/2u)lR Goucher
number = wire radius (cm) = time scale of deformation (s) = wire
speed (cm/s) = 0-/(p/p4g)lG inertial parameter = stretch rate (s-l)
= fluid relaxation time (s) = viscosity (poise) = density (g/cm3) =
surface tension (dyne/cm)
REFERENCES 1. J. A . Tallrnaclge and C. Gutfinger, Znd. E n g .
Chem., 59, 19
2. D. A. White and J . A. Tallmadge, AZChEJ. , 12, 333 (1966).
3. D. A. White and J . A. Tallmadge, AZChEJ., 13, 745 (1967). 4. R.
P. Spiers, C. V. Subbaraman, and W. L. Wilkinson, Cheni.
5. M. N. Esmail arid R. L. Hrimmel, AZChEJ., 21, 958 (1975).
(1967).
E n g . Sci., 29, 389 (1974); 30, 379 (1975).
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