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Film thickness in blade coating of viscous and viscoelastic liquids
Journal of Non-Newtonian Fluid Mechanics, 21 (1986) 13-38
13
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
FILM THICKNESS IN BLADE COATING OF VISCOUS AND VISCOELASTIC LIQUIDS TIMOTHY
M. SULLIVAN and STANLEY MIDDLEMAN
Department of AMES/Chemical CaIgomia 92093 (U.S.A.)
Engineering, University of Cal~omia, San Diego, La Jolla,
(Received July 30, 1985)
Coating of viscous and viscoelastic liquids is examined both theoretically and experimentally. A rigid blade, accurately positioned over a rotating roll, provides an experimental system in which coating thickness is measured as a A perturbation solution to the function of geometric parameters. Navier-Stokes equations yields a lubrication theory which shows agreement with the data to an extent depending on the specific geometry. The effect of a non-Newtonian viscosity is explored by adopting a purely viscous power-law model. The lubrication equations are solved by the method of Horowitz and Steidler [l], and predict an increase in coating thickness relative to the Newtonian case. Data for viscoelastic fluids show both an increase and a decrease in coating thickness compared with Newtonian liquids depending on the relative magnitude of shear thinning and elastic effects.
Intmduction This work is part of a series of studies aimed at elucidating the interaction of rheology and geometry in affecting the dynamics of viscous and viscoelastic liquids [2-81. The most relevant reference is to the work of Hsu et al. [8], who consider a similar geometry to study the forces generated in blade coating. Hence, their primary interest is in the pressure field generated under the blade. We are concerned here with the coating thickness and the question of how sensitive the results are to blade orientation and placement. Hence, the geometry considered here is more general than that considered in [8]. Following Hsu et al. [8], a perturbation method is used to obtain an 0377-0257/86/$03.50
6 1986 Elsevier Science Publishers B.V.
14
analytical solution for Newtonian liquids, and justification is given for retaining only the zeroth-order solution (equivalent to lubrication theory-Cameron [9]). A specially designed apparatus is used to collect coating thickness data, for comparison. Data for Newtonian liquids show good agreement with the mathematical model under some conditions, and provide insight into its limitations under others. The model is extended to purely viscous non-Newtonian liquids by incorporating a shear-dependent viscosity function into the equations. Experiments with viscoelastic liquids demonstrate that the coating thickness can differ markedly from values predicted by a theory for purely viscous fluids, thus suggesting that “elastic” effects can be significant. Lubrication theory
Newtonian liquia3
Figure 1 includes a definition sketch for the geometry and coordinates. We start with the steady state Navier-Stokes equations (neglecting gravity) in two dimensions, and the continuity equation. The details are nearly identical to those presented in [8]. The dependent and independent variables are nondimensionalized with a set of scaling factors as follows: + = u/u,
J/ = v/v,
r =p/p, ~=x/JL
(1) (2)
7)=u/X.
(3)
For the time, we leave U, z, and Lx undefined with respect to the specific system (the blade) of interest, but specify that v-
UZ/L,,
P = plJLx/712.
Fig. 1. Definition sketch of the geometry for the Blade-over-Roll coating system.
15 We define a Reynolds number (Re) as Re = pu’i2/pLx, and a geometrical parameter, CX,as a = ( h/LJ2. Then the dimensionless Navier-Stokes a7r a’+ =---i-+++a at aTJ =-*+a-+a2-
all
and continuity equations become
a’+
a2+ a$
(4 a29 at2
’
3+X()
(5) (6)
at aq -
It is useful to rewrite eqn. (6) as
(7) Boundary conditions for the coater take the forms of thefollowing (see Fig. 1): No sIip on the solid surfaces
equations
+=Oandrt,=O
on rl = P(S),
(8)
+=&,,(O
on 17= e(t)
(9)
and G=+,,g
Ambient pressure at the endr of the blade 7f=Oon
t=&,
?r=Oon[=t2
(IO) (11)
Utilizing the boundary conditions on $, we may replace eqn. (7) with an equivalent continuity expression of the form ‘(‘)+dq /a(l)
= A.
(12)
It is apparent that the constant X is a dimensionless coating thickness. The proportionality depends upon the choice of the length scale % used in defining r). The goal, then, is to solve eqns. (4), (5), and (12) for A as a function of LX, Re, and the shape factors that enter into the description of the nip region through the specific forms of a( 5) and /3(5).
16 An analytical solution to eqns. (4)-(6) may be found by a perturbation method. The perturbation parameters are [Y (accounting for curvature effects) and Re (for inertial effects). Each of the variables is expanded as follows:
Variables with superscript (00) give the zeroth-order lubrication approximation. Those with (10) and (01) give the first-order corrections for curvature and inertia, respectively. The zeroth-order (lubrication) soIution When eqns. (13) are substituted into eqns. (4), (5) and (12) and terms in a and Re are dropped, we recover the lubrication equations in the form 0
0
= -- a7rO”+ -a’cp
04)
= -- a7P a7 9
05)
at
aq2 ’
06) Boundary conditions (Q-(11) hold, but do not include any constraint on #, which does not appear in this level of approximation, The solutions may be written as follows:
where
(19)
d&
1
(20)
17
The results are general (to this level of approximation) and can be made particular upon a choice of geometry, /3(6‘)and a( I), and kinematic boundary condition &( [). Results presented by Hsu et al. [8] indicate that the higher order corrections may not be necessary for our purposes. First order corrections for curvature contributed less than 0.5% to the coating thickness, A, in the specific cases there, and the zeroth-order solution was within 1% of a numerically generated solution of the Navier-Stokes equations. (First-order effects would, however, be important if our primary concern was for information on blade loading. The analysis in [8] demonstrates that first order corrections contribute as much as 10% to the dimensionless blade loading.) Without attempting to derive the higher order solutions, we examine the results obtained from the zero&order approximation, and restrict our attention to coating thickness. The particular choices of geometry and kinematic boundary condition, as shown in Fig. 1, are Geometry
Roll surface Blade surface
u’(x) = -x2/2R; /3’(x) = H - x tan(w),
(2la) (22a)
where H = /3’(O) = H, - X0 tan(w). H, is the smallest separation (in the y-direction) between the blade and roll, i.e., H, = min{ j?‘(x) - u’(x)} = B’( X,,) - a’( X,,). X0 is the location of minimum blade separation, i.e., X0 = min{ X2, max{ Xi, R sin(w)}}. R is the radius of the roll. Kinematics
x-velocity
~lo’(x)=ul)=
u
( y-velocity
1-2
u 1a’(x)‘Uo = - u 2
In dimensionless
X2
(R )
,
1
(234
.
form,
W)
18
An obvious choice for the length scales, L, and z, is L, = R and 5 = H, where H is the blade height at x = 0. However, the results in [8] are presented with Lx = ( R%)lp and for the purpose of comparison, this is used here as well. The definition of the reference length % requires a closer look. In addition to %, three parameters are required to specify the geometry; the blade length, L, blade angle, w, and the blade displacement, X,. Hsu et al. [8] consider the particular case of w = 0 and X, = 0, for which the minimum separation between the roll and blade always occurs at the downstream edge of the blade, X = X, = 0, and the obvious choice for z is Ho. W&n X2 and/or o are not zero, an appropriate choice of reference length, %, is not obvious. Three possibilities are: (1) Separation between the blade and roll at the downstream edge of the blade. E=fl’(X,)-a’(XJ (2) Separation at X= 0. h=H
(3) Minimum separation between blade. and roll x=H,
For this study, the preferable reference length, A, is the blade height at X = 0, when X, >, 0, and at the downstream edge of the blade when X, < 0. The downstream edge is chosen when X, -z 0 to avoid negative values of $. The final expressions defining the geometry and kinematic boundary conditions are: For x2 2 0: o(t) = - +2”,
(214
&o(5)= 1 - .td2Y kdS)= -5, where now * a = (H/R), L, = ( RH)“‘, %I= H.
For X2 < 0: * In Hsu et al. [8], a =
(Ho/R)*.
19 Equations (21b), (22b), (23b), and (24b) with Lx = ( fi)“2, h=H-X,tan(w)+~.
xzz
Results
Figures 2 and 3 contain justification for considering the zeroth-order perturbation solution without higher-order contributions for curvature. The results from [8] for h as a function of Ho/R indicate that first order corrections are small relative to the sensitivity of X to variations in blade angle and displacement. In the particular geometry discussed in [8], for example, a two degree error in blade angle or 2 mm displacement of the 19 mm blade, results in up to 10% change in X compared to less than 0.5% for first-order curvature effects. The results are very sensitive to blade angle as shown in Figure 2. It is
Effect
of
Blade
L-19
mm
R=!il
mm
x2-o
mm
Angle
Fig. 2. Effect of blade angle on coating thickness. Solid lines are Lubrication Theory for a Newtonian liquid. Symbols are from Hsu et al. [8]. (0-zeroth order solution (0 deg.}, ~-first-order curvature included, A-Finite Element solution).
20 0.75 Effect
\ \
of
Blade
Dlsplocsment
L-19
mm
R-51
mm
0.60
0.55 0
0.01
0.02
0.03
H/R
Fig. 3. Effect of blade displacement on coating thickness. (Key as in Fig. 2).
unlikely, in commercial practice, that the blade angle can be set to within one degree. It is interesting to note that in the limit of zero blade height, the limiting value of h is 2/3 for a zero degree angle, and becomes A = 1 for positive blade angles, regardless of how small. Increasing the blade angle increases the coating thickness. Figure 3 shows the effect of blade displacement. The use of blade displacement as a parameter is somewhat artificial. By defining the y-axis as the radius that passes through the downstream edge of the blade, the displacement can be eliminated as a parameter and accounted for by an appropriate change in blade angle. However, experimental considerations in a fixed reference frame are simplified by using displacement as a parameter. Figure 4 shows the effect of blade length on h. The range of blade length over which the lubrication theory results are meaningful is limited by two major assumptions. Lubrication theory requires that entrance and exit effects be negligible, and that the blade and roll be nearly parallel to minimize curvature effects. When the blade is large, relative to the roll radius, the curvature of the roll is significant and the parallel flow assumption is violated. At the other extreme, solutions for small blades are only
21 0.75
] Effect
of
Blade
R-51 x2-o 0 0. 70
.'
0.55
1 0
deg
Length
mm mm angle
0.01
0.02
0.03
H/R
Fig. 4. Effect of blade length on coating thickness. (Key as in Fig. 2).
valid for very small blade heights, because as the blade height increases, the entrance length also increases and entrance effects become significant. Non-Newtonian
liquids
Many of the liquids of interest in commercial coating are polymeric solutions or melts which exhibit non-Newtonian behavior. We anticipate that the response of such solutions to the coating process is complicated by nonlinear elastic and viscous effects, which can be extremely difficult to model. As a first step in understanding these phenomena, we turn to an analysis of the role of nonlinear viscous effects by incorporating a simple shear dependent viscosity function into the analysis. We begin with the steady two-dimensional dynamic equations in the form f
au
i
us+“F
au+“& P ( ‘ax
au i ay
=
a& -- ap ax+=+53
ap =--_+---_ 1
ay
3~
ax
a%
(29
+ -al;,
along with the continuity equation.
ay
9
(26)
22
A useful model for shear viscosity is the power law, a purely viscous (i.e. non-elastic) constitutive equation that we write as T=~A,
(27)
where the rate of deformation nents
ay+ax
ax
*+ \
ay
Cartesian compo-
au au\
I 2au A=
tensor has two-dimensional
au
ax
2au ay
(28)
I
The viscosity function is written as
(29)
/.&= K(:II,)‘“-‘“2, where
(30) As in the Newtonian case, the variables are nondimensionalized and the dynamic equations display terms to zero and first order in (Yand Re. The variables are then written as in eqn. (13), and the solutions to various orders in a and Re may be obtained. The nondimensionalization is identical to that used in the Newtonian case, except that the Newtonian viscosity constant of those equations is everywhere replaced by a factor M = K( U/H,))“-‘.
(31)
We refer to M as the apparent viscosity at the (nominal) shear rate U/H,. In dimensionless form, eqn. (29) becomes
and F itself is included in the perturbation jZ=-$)o+a-‘O p + Rep” + . . . .
expansion in the form (33)
As in the Newtonian case, we concern ourselves only with the zeroth-order solution. The zeroth-order solution The equations are (Cf: eqns. (14)-(16)) (34)
23 (35) (36) where _-oo_ p-
&#p
ii
xy
2 (R-1)/2
II *
(37)
Boundary conditions (8)-( 11) hold. As noted in [8], no analytical solution of these equations may be found. Instead, we adopt a numerical procedure due to Horowitz and Steidler [l]. We rewrite eqn. (34) as a7rm 87 00 O=- --at + a7 7 where
but $‘(’ is left unspecified at this stage. We define a function f( TOO)as
(In the following, we drop all superscripts solution.) Integration of eqn. (38) yields
that denote the zeroth-order
7=r/7j+7j3,
(41)
where Q=r I q=fl(E)* Integration of eqn. (40) using eqn. (9) gives
(42) Using eqn. (41) we,write this as +=$(r)dT++&). With the boundary condition $ = 0 at q= /T!(t) we may write this as
(43)
24 where we define
With these results we may write h in the form.
(W where (47) From eqn. (41) we write (48)
li= (;I;).
0.78..
0.72..
x 0.60..
0.64..
0.d
0. 4
0.6 0.8 Powrr Lou Index
1.0
Fig. 5. Effect of power-law index on coating thickness with blade angle as a parameter.
25 Results The solution procedure is described in detail in [8]. For a power-law fluid, for which
fW= F(r)=+
17 w+17
(49)
,
71’1 (pi+
l),
(50)
one obtains results, examples of which are shown in Figs. 5 and 6. More complete results are in Sullivan [7]. Figures 5 and 6 show that the coating thickness, A, increases as the liquid becomes increasingly shear thinning (i.e. as n decreases) and is relatively unaffected by small changes in blade angle or displacement.
0.70..
0.72"
x
0.88'.
0.34..
0.601 0.4
0.8
0-e
1.0
Powal--Law Indmx
Fig. 6. Effect of power-law index on coating thickness with blade displacement as a parameter.
26 Experimental apparatus Figure 7 shows a sketch of the coating apparatus. A cylindrical steel roll is half immersed in the test fluid and driven at constant speed, U. Liquid is entrained by the moving roll and is partially wiped by the blade, while the remainder emerges downstream as residual fluid of thickness, h. The blade is held in a rig which permits accurate positioning relative to the roll. The roll has a radius of 70 mm, accurately machined to within 0.125 mm; and an axial Iength of 13 cm. Blades are made from ahrminium. Coating thickness is measured by a direct contact method (Greener [4]) using a micrometer driven needle positioned 30 degrees downstream with respect to the vertical. The contact of the needle with the surface is observed through a microscope. Detailed discussion of the experimental system and protocol are contained in Sullivan 171. Results The range over which certain experimental variables are considered is dictated by experimental limitations. In order to flood the inlet (upstream) region of the blade, a sufficient quantity of fluid must be entrained from the
Fig 7. Ekperimental blade coating apparatus.
27 reservoir. The amount entrained depends upon both the roll speed and the fluid properties (Campanella [lo]). For a given fluid, the roll speed is limited on the low end by this flooding criterion. On the high end, there is also a limit, corresponding to the onset of instabilities and/or air entrainment. Under unstable wiping conditions, the residual fluid layer is no longer smooth and often develops a “ribbed” pattern (Pearson [ll], Sullivan [6]) transverse to the direction of flow. Positive blade displacements and negative blade angles are found to promote this instability. When stability is not a problem, there is a limit on roll speed, above which air is entrained into the test fluid at the interface where the roll surface re-enters the bath (Burley and Jolly [12], Bolton and Middleman [13]). The range of roll speeds investigated is from 5 to 50 cm/s, depending on the test fluid. To obtain flooding of the blade, a lower limit on the viscosity of Newtonian test fluids is about 3 poise (0.3 Pas). Two Newtonian liquids are examined: glycerine and Karo syrup. Under conditions that the blade inlet is flooded, no effect of linear roll speed on coating thickness is observed. Figure 8 shows data for glycerine and Karo
1.0. R-70
mm
L=6.35 x2-0 0.8..
0 deg
mm mm angle
0.6..
Solid 0.2..
4
11na
-
Lubrlcatioo
Thaory
D - Glycerina
42
/ a’
x - Km-o Syrup
0.n x’
_#“‘.
0
0. 2
0. 4
0. B
0.6 H,
I. 0
1.2
Cm)
Fig. 8. Coating thickness vs. blade height for glycerine (0)
and Karo syrup (x).
28 1.0 Effect
of
R-70 L-Ii7
mm mm
x2-0
0. 0
Angle
Blade
111111
0. 6
Solid
lime noto 0 . -
-
Lvbricotion
for
Theory
Glycwine
0 deg angle 10 d-g
angle
0. 0
0:2
0:4
-
0.6 H,
O.-e
.
Lb
I:2
Fig. 9. Newtonian data and lubrication theory showing the effect of blade angle. (“large” blade, L =12.7 mm).
syrup with a 6.35 mm blade at 0 degrees and the corresponding results predicted by lubrication theory. Results for glycerine and Karo syrup are identical (within experimental uncertainty) even though their viscosities differ by almost an order of magnitude. The remainder of the Newtonian results presented are for glycerine, with the observation that they are representative of Newtonian liquids in general. In Fig 8, agreement between the data and lubrication theory is good for small blade heights, but becomes poor as H,, increases. The poor agreement with small blades at large blade heights is due to entrance effects not accounted for by lubrication theory. When Ho = 1.27 mm, L/H,, = 5 for the 6.35 mm blade. Other data indicate that L/H,, > 10 is a reasonable estimate of the requirement that entrance effects be negligible. Notice that when entrance effects are significant, data lie above the theory as expected on the basis of momentum transfer arguments. A change in the blade angle from zero to ten degrees affects the coating thickness as shown in Figs. 9 and 10. Good agreement between experimental data and the simple lubrication theory is found for the 12.7 mm blade. Blade
29
o +
0
0. 2
0. 4
0.6
0. 6 Ho
-
0 deg 10 dag
angle mgle
1.0
1.2
Fig. 10. Newtonian data and lubrication theory showing the effect of blade angle. (“small” blade, L = 1.7 mm).
angles larger than ten degrees are not discussed here because of large curvature effects, and the problem of flooding the inlet region of the blade. Some experiments performed with large blade angles (e.g. 45 degrees) produced data that fell below lubrication theory predictions, but it is not known whether this effect was due to curvature or an inadequately flooded inlet region. Figure 11 shows the effect of blade displacement, and provides an example of the above mentioned flooding problem. Data are in good agreement with theory, but only for three of the four displacements shown. The data for the largest negative displacement (X2 = - 6.35 mm) fall below the theory. In this case, not enough fluid is being entrained by the roll to flood the inlet region of the blade. As expected, the problem is exacerbated when the blade angle is increased (Sullivan [7]). Non-Newtonian
liquid
Prediction of the behavior of viscoelastic liquids is complicated by the converging-diverging nature of the coating process. We do not expect good
Effect
of
Blade R-70 L-12.7
0.8
”
0.6
.’
0 dag
Displ~,comont mm mm angle
Solid
Irnos
-
Dot0 for 0 +
0.
-
x2x2-
Lubr,cor,on
Theor;
Glycor1m 0 lnm 6.35 mm
.
-
X2--6.35
mm
x
-
X2--12.7
mm
;
0
0.2
0. 4
0.6
0.8
1.0
1.2
Fig. 11. Newtonian data and lubrication theory showing the effect of blade displacement. (“large” blade, L = 12.7 mm).
quantitative agreement between the simple power-law lubrication theory model and coating thickness data for strongly viscoelastic fluids, but rather hope to gain insight into the effect the shear dependent viscosity has on the results. Figure 12 shows viscometric data, taken with a Rheometrics Fluids Rheometer, for a shear thinning carboxymethylcellulose (CMC) solution. For a linear roll speed of 25 cm/s and blade heights ranging from 0.25 to 1.27 mils, the nominal shear rate ranges from 200 to 1000 s-l. In this range the power-law index, n, is about 0.6. Note, however, that the fluid also exhibits a finite normal stress. At a representative shear rate, p, of 500 s-i, the recoverable shear, Sa, defined by Sa=F,
W2 12
where Ni = ‘11 lrz2 , Y
(52)
31
3 2 ii xl0 2
..
I
..
2”
..
a ..
1210 1
10
lo* Shoot-
Rate
103
Cm&
Fig. 12. Viscosity and normal stress vs. shear rate for 1.25% CMC solution.
about 0.6. The recoverable shear is a parameter commonly used to estimate the relative importance of elastic and viscous effects. When. we examine data for the CMC solution, we observe the results of both shear thinning and elastic effects, while the power-law lubrication theory only accounts for the shear thinning behavior. Figure 13 shows that the coating thickness of the CMC solution is considerably greater than that for glycerine and is also greater than predicted by lubrication theory. Since shear thinning behavior is accounted for in the model, one might suspect that the increase in thickness is due to some elastic effect. To test this hypothesis, small amounts of high molecular weight polyacrylamide (PAM) are added to the CMC solution, to increase the recoverable shear without significantly affecting the power law index. Data for two such solutions are included in Fig. 13, one with a recoverable shear of 0.9 and the other with 1.4. Note that by increasing the recoverable shear, the coating thickness decreases rather than increases. The data for the solution with the largest recoverable shear fall below those of the Newtonian fluid. This is opposite to the effect predicted by the power-law lubrication is also
32 Effect
of
Shear
Thinning
and
u-70 L-6.35
IO
6”
L.
0 -
Glycwin.3
+
-
1.25%
CMC
.
-
1.25X
CHC
and
0.02X
PAM
I
-
1.252.
CHC
and
0.052
PAM
Eloct,c
4
bhov,or
111111
+
mm
x2-o
mm
deg
angle
.
+
.
0
0. 2
0.;
-
oi Ho
0.-e
1.-o
1:2
Fig. 13. Data for Newtonian and viscoelastic liquids compared to Newtonian and “power-law” lubrication theories.
theory model. Note also that the deviations from lubrication theory are observed at small values of H,. We do not believe that these are geometrical effects associated with the flooding of the entrance region. We believe that we are observing rheological effects. The dilemma of these data lies in the fact that viscoelastic data are found both above and below the purely viscous model. In an attempt to isolate elastic and shear thinning effects, the behavior of an elastic fhtid with insignificant shear thinning (i.e., a Boger fluid-Boger [14,15]) is examined. The fluid is made by adding a small amount of high molecular weight PAM to corn syrup. Figure 14 shows viscometric data for the Boger fluid as well as for a CMC solution and glycerine. Note that the viscosity of the Boger fluid is nearly constant over the shear rate range for which data were collected. The Boger fluid also exhibits large normal stresses in steady shear relative to the highly shear thinning (n = 0.6) CMC solution. Figure 15 shows a drastic decrease in coating thickness with the Boger fluid relative to glycerine. Note that although the CMC data fall above the power-law lubrication theory curve, they are in better agreement, in this case,
33
+-cct::
n:
10
: :
102
Shea-' Rota Cud’, pig. 14. Viscosity and nod * -Glycerine).
stress vs. shear rate.
(X
-Boger
fluid, O-2%
CMC,
for the larger (12.7 mm) blade than for the 6.35 mm blade (Fig. 16). This is attributed to entrance effects, which for flow in pipes are known to be considerably longer for viscoelastic fluids than those encountered with Newtonian fluids (Tung et al. [16l). In fact, if we go to a very short blade (L = 1.7 mm)-nearly a “knife coater” rather than a blade coater-we see in Fig. 17 that lubrication theory fails completely (no surprise), and that large positive deviations in X are observed for the Newtonian fluid and the 2% CMC. It would appear that entrance effects (both viscous and elastic) dominate coating with a very short blade.
Experimental results for Newtonian liquids are very useful for determining the range of validity of the lubrication theory model, but unfortunately, that range is severely limited. One obvious inadequacy of the model is the negligible curvature assumption, which is only valid for very small blade
34
ST-70 mm dog
10
x2-o L-12.7
mgle mm mm
SOlId
/ I
o-
0
‘I
:
Ilnoc
-
Lubricorton
Upperi
Power-law
Lower:
NPrtonlon
with
Theory n-O.
5
:
0. 2
0. 4
0. 6 Ho
0. a
1.0
1.2
(mm)
Fig. 15. Data for Newtonian and viscoelastic liquids compared to Newtonian and “power-law” lubrication theories. (“large” blade, L = 12.7 mm).
angles. Another is the assumption of negligible entrance effects, which can be violated for both small and large blades. For small blades, entrance effects are associated with the development of the flow field as fluid is rapidly accelerated into the nip region. Entrance effects are dependent on the rheology of the fluid, and some of the data presented above clearly demonstrate this. For large blades the problem lies in fully wetting the blade all the way to the upstream position at &, as is assumed in the mathematical model. If this condition is not met, it would be necessary to use a numerical method to calculate the position of the wetting line, and the shape of the free surface upstream from it. This would be a complex problem even for a Newtonian fluid, and would require the introduction of surface tension into the boundary conditions. The model used here fails to take into account surface tension effects by using the boundary conditions of eqns. (10) and (11). A useful estimate of the magnitude of surface tension effects is the dimensionless capillary number defined by (53) NCa= @J/u*.
35 1.0,
0.8..
0.6.. 'if 9 Jz 0. 4.
0.2.
0
0. 2
0. 4
0.6
0.8
1.0
1.2
ii, (mm)
Fig. 16. Data for newtonian and viscoelastic liquids compared to Newtonian and “power-law” lubrication theories. (L = 6.35 mm).
The fact that N, is of order unity in some of our experiments indicates that surface tension may be significant. Gravitational effects are neglected in the model as well, but the effect can be estimated with another dimensionless group, the Stokes parameter, defined by
If the blade height is large and roll speed low, St can get as large as 0.2, but in most cases gravity is negligible. Obviously a less restrictive model of the coating process is required, but since an analytical solution is not possible, it would be necessary to consider numerical techniques. Viscoelastic liquids exhibit extremely complex behavior dependent on both the magnitude and duration of deformation imposed. Nonlinear viscous behavior alone is relatively easy to model, the power-law fluid being a simple example. Difficulties arise when the viscous effects are coupled with transient, and deformation history dependent, behavior associated with the “elasticity” of the fluid. Flow problems must be solved using constitutive
36
- _~ ~.~ ~.~ ~. 0
0.
2
a.4
.
0:6 Ii,
.
0:e
.
1:o
.
1:2
(mm)
Fig. 17. Data for Newtonian and viscoelastic liquids compared to Newtonian and “power-lad’ lubrication theories. (“small” blade, L =1.7 mm).
equations chosen according to a compromise between algebraic tractability and physical reality. The coating problem is formidable because fluid particles experience rapidly changing deformations in time, which can be of considerable importance for viscoelastic fluids (Shirodkar and Middleman [5]). Elastic liquids can exhibit large transient shear stresses when undergoing the rapid changes in deformation rate that are present because of the converging nature of the flow. The resulting kinematics for viscoelastic liquids are often drastically different from those present for Newtonian liquids (Doremus and Piau [ 171, Mensah [18], Metzner and Chan Man Fong An essential step in modeling viscoelastic fluid dynamics is determining the material properties which are most closely related to observed behavior of a given fluid in a particular flow field. We have mentioned the recoverable shear, which is a steady viscometric material property providing a measure of the relative importance of elastic and viscous effects in steady shear flow. An important question is, can material properties measured in steady viscometric flows provide insight into how an elastic liquid will behave in a
37 transient flow field? It not, what are the pertinent transient fluid properties to measure and how can they provide guidelines to anticipating the behavior of a particular viscoelastic liquid in a complex flow? These questions underlie the program of research carried on in our laboratories. Notation h
z H HO
K L
Lx
k
P R Re
u u
uow V
vo(x) X
x, 4 x, Y
residual fluid thickness characteristic length in the y-direction blade height at x = 0 minimum clearance between the blade and the roll parameter in power-law fled model blade length characteristic length in x-direction power-law index primary normal stress coefficient pressure roll radius Reynolds number linear roll speed velocity in x-direction u-velocity on roll surface velocity in y-direction v-velocity on roll surface space coordinate in the primary flow direction x-coordinate at the point of minimum blade clearance x-coordinate at the upstream edge of the blade x-coordinate at the downstream edge of blade space coordinate transverse to primary flow direction geometric parameter function describing the blade surface (linear) dimensionless blade surface rate of deformation tensor dimensionless space coordinate y/l; dimensionless residual fluid thickness h/s viscosity dimensionless space coordinate x/L, dimensionless pressure second invariant of A density function describing the roll surface (quadratic) dimensionless roll surface
F&y A ? A : VT 11,
P ex) 48
38 surface tension stress tensor i, j component of the stress tensor dimensionless u-velocity dimensionless uO( x) dimensionless u-velocity dimensionless u, ( x ) blade angle shear rate References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19
H.H. Horowitz and F.F. SteidIer, ASLE Trans., 3 (1960) 124. T. Bauman, T. SulIivan, and S. Middleman, Chem. Eng. Commun., 14 (1982) 35. J. Greener and S. Middleman, Polym. Eng. Sci., 15 (1975) 1. J. Greener, Bounded Coating Flows of Viscous and Viscoelastic Fluids, Ph.D. Thesis, Univ. of Massachusetts, Amherst, MA, 1978. P. Shirodkar and S. Middleman, J. Rheol., 26 (1982) 1. T. SuBivan and S. Middleman, Chem. Eng. Commun., 3 (1979) 469. T. SuBivan, Fluid Dynamics in a Blade-Over-Roll Coater, Ph.D. Thesis, Univ. of Cahfornia at San Diego, La Jolla, CA, 1986. T. Hsu, M. Malone, R.L. Laurence, and S. Middleman J. Non-Newtonian Fluid Mech., 18 (1985) 273. A. Cameron, Principles of Lubrication, Longmans, London, 1966. O.H. CampaneIIa and R.L. Cerro, Chem. Eng. Sci., 39 (1984) 1443. J.R.A. Pearson, J. FImd Mech., 7 (1960) 481. R. Burley and R.P.S. Jolly, Chem. Eng. Sci., 39 (1984) 1357. B. Bolton and S. Middleman, Chem. Eng. Sci., 35 (1980) 597. D.V. Boger, J. Non-Newtonian Fluid Mech., 3 (1977) 87. D.V. Boger, Separation of shear thinning and elastic effects in experimental rheology. In: G. Astarita, G. Marucci and L. NicoIais @is.), Rheology I: Principles, Plenum Press, New York, 1980. T.T. Tung, K.S. Ng, and J.P..Hartnett, Lett. Heat Mass Transfer, 5 (1978) 59. P. Doremus and J.M. Piau, J. Non-Newtonian Fluid Mech., 13 (1983) 79. T.O. Mensah, Chem. Eng. Commun., 28 (1984) 143. A.B. Metzner, E.A. Uebler, and C.F. Chan Man Fong, AIChE J., 15 (1969) 750.
13
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
FILM THICKNESS IN BLADE COATING OF VISCOUS AND VISCOELASTIC LIQUIDS TIMOTHY
M. SULLIVAN and STANLEY MIDDLEMAN
Department of AMES/Chemical CaIgomia 92093 (U.S.A.)
Engineering, University of Cal~omia, San Diego, La Jolla,
(Received July 30, 1985)
Coating of viscous and viscoelastic liquids is examined both theoretically and experimentally. A rigid blade, accurately positioned over a rotating roll, provides an experimental system in which coating thickness is measured as a A perturbation solution to the function of geometric parameters. Navier-Stokes equations yields a lubrication theory which shows agreement with the data to an extent depending on the specific geometry. The effect of a non-Newtonian viscosity is explored by adopting a purely viscous power-law model. The lubrication equations are solved by the method of Horowitz and Steidler [l], and predict an increase in coating thickness relative to the Newtonian case. Data for viscoelastic fluids show both an increase and a decrease in coating thickness compared with Newtonian liquids depending on the relative magnitude of shear thinning and elastic effects.
Intmduction This work is part of a series of studies aimed at elucidating the interaction of rheology and geometry in affecting the dynamics of viscous and viscoelastic liquids [2-81. The most relevant reference is to the work of Hsu et al. [8], who consider a similar geometry to study the forces generated in blade coating. Hence, their primary interest is in the pressure field generated under the blade. We are concerned here with the coating thickness and the question of how sensitive the results are to blade orientation and placement. Hence, the geometry considered here is more general than that considered in [8]. Following Hsu et al. [8], a perturbation method is used to obtain an 0377-0257/86/$03.50
6 1986 Elsevier Science Publishers B.V.
14
analytical solution for Newtonian liquids, and justification is given for retaining only the zeroth-order solution (equivalent to lubrication theory-Cameron [9]). A specially designed apparatus is used to collect coating thickness data, for comparison. Data for Newtonian liquids show good agreement with the mathematical model under some conditions, and provide insight into its limitations under others. The model is extended to purely viscous non-Newtonian liquids by incorporating a shear-dependent viscosity function into the equations. Experiments with viscoelastic liquids demonstrate that the coating thickness can differ markedly from values predicted by a theory for purely viscous fluids, thus suggesting that “elastic” effects can be significant. Lubrication theory
Newtonian liquia3
Figure 1 includes a definition sketch for the geometry and coordinates. We start with the steady state Navier-Stokes equations (neglecting gravity) in two dimensions, and the continuity equation. The details are nearly identical to those presented in [8]. The dependent and independent variables are nondimensionalized with a set of scaling factors as follows: + = u/u,
J/ = v/v,
r =p/p, ~=x/JL
(1) (2)
7)=u/X.
(3)
For the time, we leave U, z, and Lx undefined with respect to the specific system (the blade) of interest, but specify that v-
UZ/L,,
P = plJLx/712.
Fig. 1. Definition sketch of the geometry for the Blade-over-Roll coating system.
15 We define a Reynolds number (Re) as Re = pu’i2/pLx, and a geometrical parameter, CX,as a = ( h/LJ2. Then the dimensionless Navier-Stokes a7r a’+ =---i-+++a at aTJ =-*+a-+a2-
all
and continuity equations become
a’+
a2+ a$
(4 a29 at2
’
3+X()
(5) (6)
at aq -
It is useful to rewrite eqn. (6) as
(7) Boundary conditions for the coater take the forms of thefollowing (see Fig. 1): No sIip on the solid surfaces
equations
+=Oandrt,=O
on rl = P(S),
(8)
+=&,,(O
on 17= e(t)
(9)
and G=+,,g
Ambient pressure at the endr of the blade 7f=Oon
t=&,
?r=Oon[=t2
(IO) (11)
Utilizing the boundary conditions on $, we may replace eqn. (7) with an equivalent continuity expression of the form ‘(‘)+dq /a(l)
= A.
(12)
It is apparent that the constant X is a dimensionless coating thickness. The proportionality depends upon the choice of the length scale % used in defining r). The goal, then, is to solve eqns. (4), (5), and (12) for A as a function of LX, Re, and the shape factors that enter into the description of the nip region through the specific forms of a( 5) and /3(5).
16 An analytical solution to eqns. (4)-(6) may be found by a perturbation method. The perturbation parameters are [Y (accounting for curvature effects) and Re (for inertial effects). Each of the variables is expanded as follows:
Variables with superscript (00) give the zeroth-order lubrication approximation. Those with (10) and (01) give the first-order corrections for curvature and inertia, respectively. The zeroth-order (lubrication) soIution When eqns. (13) are substituted into eqns. (4), (5) and (12) and terms in a and Re are dropped, we recover the lubrication equations in the form 0
0
= -- a7rO”+ -a’cp
04)
= -- a7P a7 9
05)
at
aq2 ’
06) Boundary conditions (Q-(11) hold, but do not include any constraint on #, which does not appear in this level of approximation, The solutions may be written as follows:
where
(19)
d&
1
(20)
17
The results are general (to this level of approximation) and can be made particular upon a choice of geometry, /3(6‘)and a( I), and kinematic boundary condition &( [). Results presented by Hsu et al. [8] indicate that the higher order corrections may not be necessary for our purposes. First order corrections for curvature contributed less than 0.5% to the coating thickness, A, in the specific cases there, and the zeroth-order solution was within 1% of a numerically generated solution of the Navier-Stokes equations. (First-order effects would, however, be important if our primary concern was for information on blade loading. The analysis in [8] demonstrates that first order corrections contribute as much as 10% to the dimensionless blade loading.) Without attempting to derive the higher order solutions, we examine the results obtained from the zero&order approximation, and restrict our attention to coating thickness. The particular choices of geometry and kinematic boundary condition, as shown in Fig. 1, are Geometry
Roll surface Blade surface
u’(x) = -x2/2R; /3’(x) = H - x tan(w),
(2la) (22a)
where H = /3’(O) = H, - X0 tan(w). H, is the smallest separation (in the y-direction) between the blade and roll, i.e., H, = min{ j?‘(x) - u’(x)} = B’( X,,) - a’( X,,). X0 is the location of minimum blade separation, i.e., X0 = min{ X2, max{ Xi, R sin(w)}}. R is the radius of the roll. Kinematics
x-velocity
~lo’(x)=ul)=
u
( y-velocity
1-2
u 1a’(x)‘Uo = - u 2
In dimensionless
X2
(R )
,
1
(234
.
form,
W)
18
An obvious choice for the length scales, L, and z, is L, = R and 5 = H, where H is the blade height at x = 0. However, the results in [8] are presented with Lx = ( R%)lp and for the purpose of comparison, this is used here as well. The definition of the reference length % requires a closer look. In addition to %, three parameters are required to specify the geometry; the blade length, L, blade angle, w, and the blade displacement, X,. Hsu et al. [8] consider the particular case of w = 0 and X, = 0, for which the minimum separation between the roll and blade always occurs at the downstream edge of the blade, X = X, = 0, and the obvious choice for z is Ho. W&n X2 and/or o are not zero, an appropriate choice of reference length, %, is not obvious. Three possibilities are: (1) Separation between the blade and roll at the downstream edge of the blade. E=fl’(X,)-a’(XJ (2) Separation at X= 0. h=H
(3) Minimum separation between blade. and roll x=H,
For this study, the preferable reference length, A, is the blade height at X = 0, when X, >, 0, and at the downstream edge of the blade when X, < 0. The downstream edge is chosen when X, -z 0 to avoid negative values of $. The final expressions defining the geometry and kinematic boundary conditions are: For x2 2 0: o(t) = - +2”,
(214
&o(5)= 1 - .td2Y kdS)= -5, where now * a = (H/R), L, = ( RH)“‘, %I= H.
For X2 < 0: * In Hsu et al. [8], a =
(Ho/R)*.
19 Equations (21b), (22b), (23b), and (24b) with Lx = ( fi)“2, h=H-X,tan(w)+~.
xzz
Results
Figures 2 and 3 contain justification for considering the zeroth-order perturbation solution without higher-order contributions for curvature. The results from [8] for h as a function of Ho/R indicate that first order corrections are small relative to the sensitivity of X to variations in blade angle and displacement. In the particular geometry discussed in [8], for example, a two degree error in blade angle or 2 mm displacement of the 19 mm blade, results in up to 10% change in X compared to less than 0.5% for first-order curvature effects. The results are very sensitive to blade angle as shown in Figure 2. It is
Effect
of
Blade
L-19
mm
R=!il
mm
x2-o
mm
Angle
Fig. 2. Effect of blade angle on coating thickness. Solid lines are Lubrication Theory for a Newtonian liquid. Symbols are from Hsu et al. [8]. (0-zeroth order solution (0 deg.}, ~-first-order curvature included, A-Finite Element solution).
20 0.75 Effect
\ \
of
Blade
Dlsplocsment
L-19
mm
R-51
mm
0.60
0.55 0
0.01
0.02
0.03
H/R
Fig. 3. Effect of blade displacement on coating thickness. (Key as in Fig. 2).
unlikely, in commercial practice, that the blade angle can be set to within one degree. It is interesting to note that in the limit of zero blade height, the limiting value of h is 2/3 for a zero degree angle, and becomes A = 1 for positive blade angles, regardless of how small. Increasing the blade angle increases the coating thickness. Figure 3 shows the effect of blade displacement. The use of blade displacement as a parameter is somewhat artificial. By defining the y-axis as the radius that passes through the downstream edge of the blade, the displacement can be eliminated as a parameter and accounted for by an appropriate change in blade angle. However, experimental considerations in a fixed reference frame are simplified by using displacement as a parameter. Figure 4 shows the effect of blade length on h. The range of blade length over which the lubrication theory results are meaningful is limited by two major assumptions. Lubrication theory requires that entrance and exit effects be negligible, and that the blade and roll be nearly parallel to minimize curvature effects. When the blade is large, relative to the roll radius, the curvature of the roll is significant and the parallel flow assumption is violated. At the other extreme, solutions for small blades are only
21 0.75
] Effect
of
Blade
R-51 x2-o 0 0. 70
.'
0.55
1 0
deg
Length
mm mm angle
0.01
0.02
0.03
H/R
Fig. 4. Effect of blade length on coating thickness. (Key as in Fig. 2).
valid for very small blade heights, because as the blade height increases, the entrance length also increases and entrance effects become significant. Non-Newtonian
liquids
Many of the liquids of interest in commercial coating are polymeric solutions or melts which exhibit non-Newtonian behavior. We anticipate that the response of such solutions to the coating process is complicated by nonlinear elastic and viscous effects, which can be extremely difficult to model. As a first step in understanding these phenomena, we turn to an analysis of the role of nonlinear viscous effects by incorporating a simple shear dependent viscosity function into the analysis. We begin with the steady two-dimensional dynamic equations in the form f
au
i
us+“F
au+“& P ( ‘ax
au i ay
=
a& -- ap ax+=+53
ap =--_+---_ 1
ay
3~
ax
a%
(29
+ -al;,
along with the continuity equation.
ay
9
(26)
22
A useful model for shear viscosity is the power law, a purely viscous (i.e. non-elastic) constitutive equation that we write as T=~A,
(27)
where the rate of deformation nents
ay+ax
ax
*+ \
ay
Cartesian compo-
au au\
I 2au A=
tensor has two-dimensional
au
ax
2au ay
(28)
I
The viscosity function is written as
(29)
/.&= K(:II,)‘“-‘“2, where
(30) As in the Newtonian case, the variables are nondimensionalized and the dynamic equations display terms to zero and first order in (Yand Re. The variables are then written as in eqn. (13), and the solutions to various orders in a and Re may be obtained. The nondimensionalization is identical to that used in the Newtonian case, except that the Newtonian viscosity constant of those equations is everywhere replaced by a factor M = K( U/H,))“-‘.
(31)
We refer to M as the apparent viscosity at the (nominal) shear rate U/H,. In dimensionless form, eqn. (29) becomes
and F itself is included in the perturbation jZ=-$)o+a-‘O p + Rep” + . . . .
expansion in the form (33)
As in the Newtonian case, we concern ourselves only with the zeroth-order solution. The zeroth-order solution The equations are (Cf: eqns. (14)-(16)) (34)
23 (35) (36) where _-oo_ p-
&#p
ii
xy
2 (R-1)/2
II *
(37)
Boundary conditions (8)-( 11) hold. As noted in [8], no analytical solution of these equations may be found. Instead, we adopt a numerical procedure due to Horowitz and Steidler [l]. We rewrite eqn. (34) as a7rm 87 00 O=- --at + a7 7 where
but $‘(’ is left unspecified at this stage. We define a function f( TOO)as
(In the following, we drop all superscripts solution.) Integration of eqn. (38) yields
that denote the zeroth-order
7=r/7j+7j3,
(41)
where Q=r I q=fl(E)* Integration of eqn. (40) using eqn. (9) gives
(42) Using eqn. (41) we,write this as +=$(r)dT++&). With the boundary condition $ = 0 at q= /T!(t) we may write this as
(43)
24 where we define
With these results we may write h in the form.
(W where (47) From eqn. (41) we write (48)
li= (;I;).
0.78..
0.72..
x 0.60..
0.64..
0.d
0. 4
0.6 0.8 Powrr Lou Index
1.0
Fig. 5. Effect of power-law index on coating thickness with blade angle as a parameter.
25 Results The solution procedure is described in detail in [8]. For a power-law fluid, for which
fW= F(r)=+
17 w+17
(49)
,
71’1 (pi+
l),
(50)
one obtains results, examples of which are shown in Figs. 5 and 6. More complete results are in Sullivan [7]. Figures 5 and 6 show that the coating thickness, A, increases as the liquid becomes increasingly shear thinning (i.e. as n decreases) and is relatively unaffected by small changes in blade angle or displacement.
0.70..
0.72"
x
0.88'.
0.34..
0.601 0.4
0.8
0-e
1.0
Powal--Law Indmx
Fig. 6. Effect of power-law index on coating thickness with blade displacement as a parameter.
26 Experimental apparatus Figure 7 shows a sketch of the coating apparatus. A cylindrical steel roll is half immersed in the test fluid and driven at constant speed, U. Liquid is entrained by the moving roll and is partially wiped by the blade, while the remainder emerges downstream as residual fluid of thickness, h. The blade is held in a rig which permits accurate positioning relative to the roll. The roll has a radius of 70 mm, accurately machined to within 0.125 mm; and an axial Iength of 13 cm. Blades are made from ahrminium. Coating thickness is measured by a direct contact method (Greener [4]) using a micrometer driven needle positioned 30 degrees downstream with respect to the vertical. The contact of the needle with the surface is observed through a microscope. Detailed discussion of the experimental system and protocol are contained in Sullivan 171. Results The range over which certain experimental variables are considered is dictated by experimental limitations. In order to flood the inlet (upstream) region of the blade, a sufficient quantity of fluid must be entrained from the
Fig 7. Ekperimental blade coating apparatus.
27 reservoir. The amount entrained depends upon both the roll speed and the fluid properties (Campanella [lo]). For a given fluid, the roll speed is limited on the low end by this flooding criterion. On the high end, there is also a limit, corresponding to the onset of instabilities and/or air entrainment. Under unstable wiping conditions, the residual fluid layer is no longer smooth and often develops a “ribbed” pattern (Pearson [ll], Sullivan [6]) transverse to the direction of flow. Positive blade displacements and negative blade angles are found to promote this instability. When stability is not a problem, there is a limit on roll speed, above which air is entrained into the test fluid at the interface where the roll surface re-enters the bath (Burley and Jolly [12], Bolton and Middleman [13]). The range of roll speeds investigated is from 5 to 50 cm/s, depending on the test fluid. To obtain flooding of the blade, a lower limit on the viscosity of Newtonian test fluids is about 3 poise (0.3 Pas). Two Newtonian liquids are examined: glycerine and Karo syrup. Under conditions that the blade inlet is flooded, no effect of linear roll speed on coating thickness is observed. Figure 8 shows data for glycerine and Karo
1.0. R-70
mm
L=6.35 x2-0 0.8..
0 deg
mm mm angle
0.6..
Solid 0.2..
4
11na
-
Lubrlcatioo
Thaory
D - Glycerina
42
/ a’
x - Km-o Syrup
0.n x’
_#“‘.
0
0. 2
0. 4
0. B
0.6 H,
I. 0
1.2
Cm)
Fig. 8. Coating thickness vs. blade height for glycerine (0)
and Karo syrup (x).
28 1.0 Effect
of
R-70 L-Ii7
mm mm
x2-0
0. 0
Angle
Blade
111111
0. 6
Solid
lime noto 0 . -
-
Lvbricotion
for
Theory
Glycwine
0 deg angle 10 d-g
angle
0. 0
0:2
0:4
-
0.6 H,
O.-e
.
Lb
I:2
Fig. 9. Newtonian data and lubrication theory showing the effect of blade angle. (“large” blade, L =12.7 mm).
syrup with a 6.35 mm blade at 0 degrees and the corresponding results predicted by lubrication theory. Results for glycerine and Karo syrup are identical (within experimental uncertainty) even though their viscosities differ by almost an order of magnitude. The remainder of the Newtonian results presented are for glycerine, with the observation that they are representative of Newtonian liquids in general. In Fig 8, agreement between the data and lubrication theory is good for small blade heights, but becomes poor as H,, increases. The poor agreement with small blades at large blade heights is due to entrance effects not accounted for by lubrication theory. When Ho = 1.27 mm, L/H,, = 5 for the 6.35 mm blade. Other data indicate that L/H,, > 10 is a reasonable estimate of the requirement that entrance effects be negligible. Notice that when entrance effects are significant, data lie above the theory as expected on the basis of momentum transfer arguments. A change in the blade angle from zero to ten degrees affects the coating thickness as shown in Figs. 9 and 10. Good agreement between experimental data and the simple lubrication theory is found for the 12.7 mm blade. Blade
29
o +
0
0. 2
0. 4
0.6
0. 6 Ho
-
0 deg 10 dag
angle mgle
1.0
1.2
Fig. 10. Newtonian data and lubrication theory showing the effect of blade angle. (“small” blade, L = 1.7 mm).
angles larger than ten degrees are not discussed here because of large curvature effects, and the problem of flooding the inlet region of the blade. Some experiments performed with large blade angles (e.g. 45 degrees) produced data that fell below lubrication theory predictions, but it is not known whether this effect was due to curvature or an inadequately flooded inlet region. Figure 11 shows the effect of blade displacement, and provides an example of the above mentioned flooding problem. Data are in good agreement with theory, but only for three of the four displacements shown. The data for the largest negative displacement (X2 = - 6.35 mm) fall below the theory. In this case, not enough fluid is being entrained by the roll to flood the inlet region of the blade. As expected, the problem is exacerbated when the blade angle is increased (Sullivan [7]). Non-Newtonian
liquid
Prediction of the behavior of viscoelastic liquids is complicated by the converging-diverging nature of the coating process. We do not expect good
Effect
of
Blade R-70 L-12.7
0.8
”
0.6
.’
0 dag
Displ~,comont mm mm angle
Solid
Irnos
-
Dot0 for 0 +
0.
-
x2x2-
Lubr,cor,on
Theor;
Glycor1m 0 lnm 6.35 mm
.
-
X2--6.35
mm
x
-
X2--12.7
mm
;
0
0.2
0. 4
0.6
0.8
1.0
1.2
Fig. 11. Newtonian data and lubrication theory showing the effect of blade displacement. (“large” blade, L = 12.7 mm).
quantitative agreement between the simple power-law lubrication theory model and coating thickness data for strongly viscoelastic fluids, but rather hope to gain insight into the effect the shear dependent viscosity has on the results. Figure 12 shows viscometric data, taken with a Rheometrics Fluids Rheometer, for a shear thinning carboxymethylcellulose (CMC) solution. For a linear roll speed of 25 cm/s and blade heights ranging from 0.25 to 1.27 mils, the nominal shear rate ranges from 200 to 1000 s-l. In this range the power-law index, n, is about 0.6. Note, however, that the fluid also exhibits a finite normal stress. At a representative shear rate, p, of 500 s-i, the recoverable shear, Sa, defined by Sa=F,
W2 12
where Ni = ‘11 lrz2 , Y
(52)
31
3 2 ii xl0 2
..
I
..
2”
..
a ..
1210 1
10
lo* Shoot-
Rate
103
Cm&
Fig. 12. Viscosity and normal stress vs. shear rate for 1.25% CMC solution.
about 0.6. The recoverable shear is a parameter commonly used to estimate the relative importance of elastic and viscous effects. When. we examine data for the CMC solution, we observe the results of both shear thinning and elastic effects, while the power-law lubrication theory only accounts for the shear thinning behavior. Figure 13 shows that the coating thickness of the CMC solution is considerably greater than that for glycerine and is also greater than predicted by lubrication theory. Since shear thinning behavior is accounted for in the model, one might suspect that the increase in thickness is due to some elastic effect. To test this hypothesis, small amounts of high molecular weight polyacrylamide (PAM) are added to the CMC solution, to increase the recoverable shear without significantly affecting the power law index. Data for two such solutions are included in Fig. 13, one with a recoverable shear of 0.9 and the other with 1.4. Note that by increasing the recoverable shear, the coating thickness decreases rather than increases. The data for the solution with the largest recoverable shear fall below those of the Newtonian fluid. This is opposite to the effect predicted by the power-law lubrication is also
32 Effect
of
Shear
Thinning
and
u-70 L-6.35
IO
6”
L.
0 -
Glycwin.3
+
-
1.25%
CMC
.
-
1.25X
CHC
and
0.02X
PAM
I
-
1.252.
CHC
and
0.052
PAM
Eloct,c
4
bhov,or
111111
+
mm
x2-o
mm
deg
angle
.
+
.
0
0. 2
0.;
-
oi Ho
0.-e
1.-o
1:2
Fig. 13. Data for Newtonian and viscoelastic liquids compared to Newtonian and “power-law” lubrication theories.
theory model. Note also that the deviations from lubrication theory are observed at small values of H,. We do not believe that these are geometrical effects associated with the flooding of the entrance region. We believe that we are observing rheological effects. The dilemma of these data lies in the fact that viscoelastic data are found both above and below the purely viscous model. In an attempt to isolate elastic and shear thinning effects, the behavior of an elastic fhtid with insignificant shear thinning (i.e., a Boger fluid-Boger [14,15]) is examined. The fluid is made by adding a small amount of high molecular weight PAM to corn syrup. Figure 14 shows viscometric data for the Boger fluid as well as for a CMC solution and glycerine. Note that the viscosity of the Boger fluid is nearly constant over the shear rate range for which data were collected. The Boger fluid also exhibits large normal stresses in steady shear relative to the highly shear thinning (n = 0.6) CMC solution. Figure 15 shows a drastic decrease in coating thickness with the Boger fluid relative to glycerine. Note that although the CMC data fall above the power-law lubrication theory curve, they are in better agreement, in this case,
33
+-cct::
n:
10
: :
102
Shea-' Rota Cud’, pig. 14. Viscosity and nod * -Glycerine).
stress vs. shear rate.
(X
-Boger
fluid, O-2%
CMC,
for the larger (12.7 mm) blade than for the 6.35 mm blade (Fig. 16). This is attributed to entrance effects, which for flow in pipes are known to be considerably longer for viscoelastic fluids than those encountered with Newtonian fluids (Tung et al. [16l). In fact, if we go to a very short blade (L = 1.7 mm)-nearly a “knife coater” rather than a blade coater-we see in Fig. 17 that lubrication theory fails completely (no surprise), and that large positive deviations in X are observed for the Newtonian fluid and the 2% CMC. It would appear that entrance effects (both viscous and elastic) dominate coating with a very short blade.
Experimental results for Newtonian liquids are very useful for determining the range of validity of the lubrication theory model, but unfortunately, that range is severely limited. One obvious inadequacy of the model is the negligible curvature assumption, which is only valid for very small blade
34
ST-70 mm dog
10
x2-o L-12.7
mgle mm mm
SOlId
/ I
o-
0
‘I
:
Ilnoc
-
Lubricorton
Upperi
Power-law
Lower:
NPrtonlon
with
Theory n-O.
5
:
0. 2
0. 4
0. 6 Ho
0. a
1.0
1.2
(mm)
Fig. 15. Data for Newtonian and viscoelastic liquids compared to Newtonian and “power-law” lubrication theories. (“large” blade, L = 12.7 mm).
angles. Another is the assumption of negligible entrance effects, which can be violated for both small and large blades. For small blades, entrance effects are associated with the development of the flow field as fluid is rapidly accelerated into the nip region. Entrance effects are dependent on the rheology of the fluid, and some of the data presented above clearly demonstrate this. For large blades the problem lies in fully wetting the blade all the way to the upstream position at &, as is assumed in the mathematical model. If this condition is not met, it would be necessary to use a numerical method to calculate the position of the wetting line, and the shape of the free surface upstream from it. This would be a complex problem even for a Newtonian fluid, and would require the introduction of surface tension into the boundary conditions. The model used here fails to take into account surface tension effects by using the boundary conditions of eqns. (10) and (11). A useful estimate of the magnitude of surface tension effects is the dimensionless capillary number defined by (53) NCa= @J/u*.
35 1.0,
0.8..
0.6.. 'if 9 Jz 0. 4.
0.2.
0
0. 2
0. 4
0.6
0.8
1.0
1.2
ii, (mm)
Fig. 16. Data for newtonian and viscoelastic liquids compared to Newtonian and “power-law” lubrication theories. (L = 6.35 mm).
The fact that N, is of order unity in some of our experiments indicates that surface tension may be significant. Gravitational effects are neglected in the model as well, but the effect can be estimated with another dimensionless group, the Stokes parameter, defined by
If the blade height is large and roll speed low, St can get as large as 0.2, but in most cases gravity is negligible. Obviously a less restrictive model of the coating process is required, but since an analytical solution is not possible, it would be necessary to consider numerical techniques. Viscoelastic liquids exhibit extremely complex behavior dependent on both the magnitude and duration of deformation imposed. Nonlinear viscous behavior alone is relatively easy to model, the power-law fluid being a simple example. Difficulties arise when the viscous effects are coupled with transient, and deformation history dependent, behavior associated with the “elasticity” of the fluid. Flow problems must be solved using constitutive
36
- _~ ~.~ ~.~ ~. 0
0.
2
a.4
.
0:6 Ii,
.
0:e
.
1:o
.
1:2
(mm)
Fig. 17. Data for Newtonian and viscoelastic liquids compared to Newtonian and “power-lad’ lubrication theories. (“small” blade, L =1.7 mm).
equations chosen according to a compromise between algebraic tractability and physical reality. The coating problem is formidable because fluid particles experience rapidly changing deformations in time, which can be of considerable importance for viscoelastic fluids (Shirodkar and Middleman [5]). Elastic liquids can exhibit large transient shear stresses when undergoing the rapid changes in deformation rate that are present because of the converging nature of the flow. The resulting kinematics for viscoelastic liquids are often drastically different from those present for Newtonian liquids (Doremus and Piau [ 171, Mensah [18], Metzner and Chan Man Fong An essential step in modeling viscoelastic fluid dynamics is determining the material properties which are most closely related to observed behavior of a given fluid in a particular flow field. We have mentioned the recoverable shear, which is a steady viscometric material property providing a measure of the relative importance of elastic and viscous effects in steady shear flow. An important question is, can material properties measured in steady viscometric flows provide insight into how an elastic liquid will behave in a
37 transient flow field? It not, what are the pertinent transient fluid properties to measure and how can they provide guidelines to anticipating the behavior of a particular viscoelastic liquid in a complex flow? These questions underlie the program of research carried on in our laboratories. Notation h
z H HO
K L
Lx
k
P R Re
u u
uow V
vo(x) X
x, 4 x, Y
residual fluid thickness characteristic length in the y-direction blade height at x = 0 minimum clearance between the blade and the roll parameter in power-law fled model blade length characteristic length in x-direction power-law index primary normal stress coefficient pressure roll radius Reynolds number linear roll speed velocity in x-direction u-velocity on roll surface velocity in y-direction v-velocity on roll surface space coordinate in the primary flow direction x-coordinate at the point of minimum blade clearance x-coordinate at the upstream edge of the blade x-coordinate at the downstream edge of blade space coordinate transverse to primary flow direction geometric parameter function describing the blade surface (linear) dimensionless blade surface rate of deformation tensor dimensionless space coordinate y/l; dimensionless residual fluid thickness h/s viscosity dimensionless space coordinate x/L, dimensionless pressure second invariant of A density function describing the roll surface (quadratic) dimensionless roll surface
F&y A ? A : VT 11,
P ex) 48
38 surface tension stress tensor i, j component of the stress tensor dimensionless u-velocity dimensionless uO( x) dimensionless u-velocity dimensionless u, ( x ) blade angle shear rate References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19
H.H. Horowitz and F.F. SteidIer, ASLE Trans., 3 (1960) 124. T. Bauman, T. SulIivan, and S. Middleman, Chem. Eng. Commun., 14 (1982) 35. J. Greener and S. Middleman, Polym. Eng. Sci., 15 (1975) 1. J. Greener, Bounded Coating Flows of Viscous and Viscoelastic Fluids, Ph.D. Thesis, Univ. of Massachusetts, Amherst, MA, 1978. P. Shirodkar and S. Middleman, J. Rheol., 26 (1982) 1. T. SuBivan and S. Middleman, Chem. Eng. Commun., 3 (1979) 469. T. SuBivan, Fluid Dynamics in a Blade-Over-Roll Coater, Ph.D. Thesis, Univ. of Cahfornia at San Diego, La Jolla, CA, 1986. T. Hsu, M. Malone, R.L. Laurence, and S. Middleman J. Non-Newtonian Fluid Mech., 18 (1985) 273. A. Cameron, Principles of Lubrication, Longmans, London, 1966. O.H. CampaneIIa and R.L. Cerro, Chem. Eng. Sci., 39 (1984) 1443. J.R.A. Pearson, J. FImd Mech., 7 (1960) 481. R. Burley and R.P.S. Jolly, Chem. Eng. Sci., 39 (1984) 1357. B. Bolton and S. Middleman, Chem. Eng. Sci., 35 (1980) 597. D.V. Boger, J. Non-Newtonian Fluid Mech., 3 (1977) 87. D.V. Boger, Separation of shear thinning and elastic effects in experimental rheology. In: G. Astarita, G. Marucci and L. NicoIais @is.), Rheology I: Principles, Plenum Press, New York, 1980. T.T. Tung, K.S. Ng, and J.P..Hartnett, Lett. Heat Mass Transfer, 5 (1978) 59. P. Doremus and J.M. Piau, J. Non-Newtonian Fluid Mech., 13 (1983) 79. T.O. Mensah, Chem. Eng. Commun., 28 (1984) 143. A.B. Metzner, E.A. Uebler, and C.F. Chan Man Fong, AIChE J., 15 (1969) 750.