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SURFACE COATINGS SURFACE COATINGS
Chemical Engineering 347
Department of Chemical Engineering
Princeton University
Princeton, NJ 08544-5263
August 1999 TABLE OF CONTENTS
OBJECTIVES.................................................................................................................................. 1
BACKGROUND............................................................................................................................. 2
Lubrication Theory........................................................................................................................ 2
Constrained Coating Process: Uniform Film.................................................................................. 3
Constrained Coating Process: Ribbing ....................................................................................... 5
Air Entrainment ............................................................................................................................. 6
EXPERIMENTAL APPARATUS.................................................................................................... 6
Fluid Properties............................................................................................................................. 6
Roller Coating Apparatus .............................................................................................................. 9
Week-by-Week Breakdown of Experimentation......................................................................... 11
ANALYSIS AND DISCUSSION ................................................................................................. 11
BIBLIOGRAPHY.......................................................................................................................... 12
APPENDIX A: Surface Coatings Laboratory Equipment Inventory................................................. 13 1
OBJECTIVES
The continuous coating of a layer of fluid onto a moving web, which runs around a roller-coating
device, is of considerable practical interest and is applied in a host of industrial operations. Three
examples of coating devices, taken from Middleman (1977), are given below:

WEB
WEB
Figure 1. Free coating.
Figure 2. Reverse roll coating.
WEB
Figure 3. Constrained coating.
The most desirable properties of the coated layer are controllable thickness and uniformity. The purpose
of this experiment is to determine what parameters control these properties in the constrained coating
system schematized in Figure 3. Towards this end, the properties of a suitable Newtonian fluid will be
measured, and the fluid used in the coating process where two process variables will be explored: the
distance between roller and constraint, and roller speed. The results will be compared with theories and
previous experimental results given in the literature. 2
BACKGROUND
Lubrication Theory
It is apparent from Figure 3 that in order to understand the coating process we must investigate
fluid flow in the region of closest approach between the roller and the wall. This region is shown in
more detail in Figure 4.
x
y
H
Roller
Wall
o
H
f
U
Figure 4. Definition sketch for the analysis
In the figure above U refers to the linear speed of the roller surface, H
0
is the gap width, and H
f
is the
thickness of the coated layer. For simplicity, and more straightforward comparison with theory, we will
not use a web in our experiments.
We might expect that the film thickness, H
f
, would depend on the properties of the fluid, the gap
width H
0
, the radius of the roller R, and the roller speed U. In addition, the film may become non-
uniform or "ribbed" under certain conditions. For example, as the roller speed is increased, at some
critical rotation rate the film will no longer be smooth. Two common instabilities are observed at high
rotation rates: 1) the fluid near the roller surface may entrain air bubbles (entrainment, discussed
further below), or 2) the coating surface may exhibit waviness (ribbing), as in illustrated in Figure 5.
In this experiment, you will determine the controlling factors for the film thickness H
f
and the
critical conditions for the onset of ribbing. It is worth pointing out at this stage that the uniform film
thickness is most likely dependent only on the bulk properties of the fluid but that the instability or
ribbing observed at the air/fluid interface will also depend on the surface tension.
The flow of the fluid in the roller-coating device is described by the Navier-Stokes equation,
which for the steady-state flow of an incompressible fluid is given by: v

(
)
v
[
]
=
p
+ µ
2
v
(1)
where p is the gradient of the pressure, the gradient of some potential field such as that due to
gravity, and v is the fluid velocity. To simplify this exact equation of motion we can apply 3
approximations developed by Reynolds in 1886 for lubrication theory. This simplification is based on
the assumption of nearly parallel flow in the region of interest, that is, in the fluid region between the
roller and the wall. Denn (1980) provides a good chapter on lubrication theory.
Figure 5. "Ribbing" instability observed in constrained coating (no web).
Constrained Coating Process: Uniform Film
For the two-dimensional, planar flow case of interest, a good treatment was worked out by
Sullivan and Middleman (T.M. Sullivan and S. Middleman, Roll Coating in the Presence of a Fixed
Constraining Boundary, Chem. Eng. Commun., 3,(1979) 469-482.). The overall basis and first-
order results from their calculation are given below; these are sufficient background to read and
understand the rest of the handout. However, once you have finished this handout, read through the
Sullivan and Middleman article thoroughly to ensure that you understand all the assumptions and
approximations made in their derivation. Some of these may turn out to be poor for part of the process
range you will investigate, and when analyzing your data you may question their validity.
The following variables are convenient to use for the nondimensionalization of equation (1):
x x/(RH
0
)
1/2
y y/H
0
u x
u
x
/U
p (p-p
0
) H
0
2
/
µ
U(H
0
R)
1/2
When the Navier-Stokes equation (x-component) is recast in these variables, the following is obtained:
0
=
p x
+
2
u
x y
2

gH
0
2
µ
U
(2) 4
where the grouping gH
0
2
/
µ
U, which is the nondimensional gravity term, is conveniently given the symbol
F and termed the gravity parameter. Two boundary conditions result simply from applying the no-slip
condition at the constraining wall and the roller:
boundary condition 1:
u x
= 0 at y = 0
boundary condition 2:
u x
= 0 at the roller surface
Equation (2) may be integrated to give u x
, which can then be used to find the film thickness H
f
.
However, the presence of the free surface in the problem demands an additional boundary condition to
describe its location (for example, on the height of the separation point, where the fluid comes away
from the constraining wall, above the fluid surface in the bath). The appropriate condition is not
obvious, and Sullivan and Middleman discuss this issue in some detail (Appendix C). Whatever the
boundary condition assumed, the desired result is an equation which will relate the dimensionless film
thickness H
f
/H
0
to parameters governing the coating process. From equation (2), one parameter is
F; additional parameters may enter when the free surface boundary condition is specified, depending on
what is chosen.
One possible assumption for the free-surface boundary condition is that the pressure gradient in
the x direction becomes zero at the separation point. As discussed by Sullivan and Middleman, this
leads to uniquely dependent on F, with the functional form shown in Figure 6. You should be able to
generate this figure easily from Sullivan and Middlemans equations (12) and (15) using a spreadsheet;
this will be very helpful in comparing the theory with your data.
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.01 0.03 0.05
0.1
0.3
0.5
1
3
5
10
30
50
100
Gravity Paramter (F)
Dimensionless thickness (Hf/Ho)
Figure 6. Calculated dimensionless film thickness as a function of the dimensionless gravity
parameter F (adapted from Sullivan and Middleman (1979)). 5
With this boundary condition, the dimensionless film thickness has a high-F asymptote of ½
and a low-F asymptote of 0.6129. (Note that Sullivan and Middleman quote the latter number as
0.605, apparently due to a small calculational error.) Perhaps the most important result of this theory is
not the exact functional form of the curve, but the prediction that the coating thickness will be 56% (
±
6)
of the gap width, for any process condition (any fluid, any gap width, any roller speed giving a uniform
coating, etc.). In other words, very tight control of the coating thickness should be achievable simply by
adjusting the gap. Your intuition should suggest that this result must break down in at least some limits
(e.g., if the wall were six feet from the roller, would a three-foot film be obtained?), and you should
keep this possibility in mind as you analyze your data.
For comparison