1416
I n d . Eng. C h e m . Res. 1990,29, 1416-1419
A Simple Model of Reverse Roll Coating Dennis J. Coyle,* Christopher W. Macosko, and L. E. Scriven Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
The key feature of a reverse roll coater is the fluid flow in the gap between two parallel rigid cylinders rotating with opposed surface velocities. A simplified model system studied by Greener and Middleman has the two rolls side-by-side and half-submerged in the coating liquid. Over a certain range of roll surface speed ratio, their experimental measurements of the flow rate through the gap deviated from predictions of a simple lubrication theory and recirculations were observed. This model configuration is examined experimentally and by finite element solutions of the Navier-Stokes equations in two dimensions. The results indicate that the film-transfer free surface and the recirculations under it do not significantly influence the flow rate through the gap. Deviations from lubrication theory occur only under conditions of low speed ratio and large gap when the effect of gravity becomes appreciable. Under such conditions, experimental error is introduced if the end effects of a third component of flow off the ends of the rolls is not accounted for or is not eliminated.
1. Introduction Reverse roll coating is a technique to coat a uniform liquid layer onto a continuous solid substrate and is widely used to make such products as magnetic tape, adhesive tape, and coated paper and paperboard (Booth, 1970; Higgins, 1965). A reverse roll coater is able to produce uniform films of less than 25-pm (l-mil) thickness at speeds of 0.5-9 m/s (100-1800 ft/min), using liquids ranging in viscosity from 0.001 to 50 Pa s (1 to 50000 cP). This makes it an important method of high-speed precision coating. A typical reverse roll coater is depicted in Figure la. The area of interest in this work is the metering gap, between the applicator and metering rolls, where excess coating is wiped off of the applicator roll by the reverseacting metering roll. Of interest is the flow rate through the gap, which determines the thickness of the metered film, which is then transferred to the substrate to be coated. Recent work (Coyle et al., 1990a,b) examined the steady operation of reverse roll coaters and the various parameter regions where fluid flow instabilities are encountered. A conclusion was that the key feature of the process was the flow in the region near the dynamic wetting line and the way in which this flow could interact with the main flow in the narrow gap between the rolls. In an earlier work by Greener and Middleman (1981), the system depicted in Figure l b was proposed as a simple model of reverse roll coating. The applicator and metering rolls are arranged horizontally and are half-submerged in a bath of coating liquid. The applicator roll dips a coating out of the bath and transfers it to the metering roll, which is the same as the upstream side of the metering gap of Figure la. There is also a high-shear flow in the narrow gap between the rolls, but a key difference is that there is no dynamic wetting line to interact with this flow and no instabilities are observed. In spite of this simplification, their experimental results are intriguing. A t low speed ratios ( Vm/V , l), they observed strong recirculations in the flow and their measured flow rate through the gap deviated strongly from predictions of a model based on lubrication theory. The results from the investigations of the complete metering gap flow (Ho and Holland, 1978; Benkreira et al., 1981; Coyle et al., 1990a) indicate that
* Author to whom all correspondence should be addressed at
General Electric Company, Corporate Research and Development, Schenectady, NY 12301. 0888-5885/90/2629-1416$02.50/0
lubrication theory should be accurate in this region. The apparent discrepancy motivated a reexamination of this simple model reverse roll coating system. New experiments, along with a computation of the two-dimensional, steady-state, free-surface flow, were undertaken in order to understand the reasons for the earlier results and to evaluate the usefulness of the simple system as a laboratory model of a reverse roll coater. 2. Analysis of the Flow Field
It is straightforward to develop a mathematical model, based on lubrication theory, of the half-submergedreverse roll flow of Figure l b (Ho and Holland, 1978; Greener and Middleman, 1981). If x is the streamwise coordinate whose origin is at the center of the gap and that increases in the direction of the film-transfer free surface, the dimensionless pressure is, according to the lubrication approximation,
where
is the dimensionlessflow rate through the gap and 0 is the dimensionlessx coordinate (-a < x < w , -x/2 < 0 C x/2). The rolls are half-submerged, i.e., p(-7r/2) = 0, and their
surface profiles are approximated by parabolas. Another pressure boundary condition is needed to determine the flow rate. Ho and Holland (1978) used p(a/2) = 0 (submerged), whereas Greener and Middleman (1981) used both (1)p = 0 at the first stagnation point (Hopkins, 1957) and (2) the Reynolds (1886) hypothesis that at some point both the pressure and its gradient vanish. The flow rate predictions for these three cases are, respectively, 4 = 0.667(1 - V,/V,) (2) 9 = 0.650(1 - V m /V,) (3) q = 0.613(1 - V,/V,) (4) These equations are simply generalizations of theories developed for symmetric forward roll coating where V,/ V, = -1 and q = 1.333,1.30, and 1.226, respectively. The data of several authors are in the 1.28-1.35 range over all capillary numbers investigated (Coyle et al., 1986). In eqs 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1417 Coated Web
W
f
Back-up Roil Film-Transfer Nip (Rubbercovered)
Figure 2. Finite element discretizationsof film-transfer flow using nine node quadratic elements at (a) low speed ratio and (b) high speed ratio.
Transferred Film
Incoming Film
Figure 1. Typical reverse roll coater (a, top) and a simplified half-submerged model configuration (b, bottom). The metering gap is the dominant factor controlling the thickness and uniformity of the coating.
1-4, the flow rate is so defined that it is positive when the net flow is in the negative x direction, i.e. into the liquid bath. This convention implies that in the actual metering gap the flow rate must always be positive if there is any metered film at all. (In fact, q is simply the metered film thickness divided by the metering gap thickness.) A negative value of q implies a net flow up through the gap; such a result is possible only in this model flow and not in the actual metering gap flow of Figure la. Comparing the prediction of lubrication theory (eq 2) with the measured flow rates, Greener and Middleman found good agreement at high speed ratios (1.0,1.75),but not at low ones (0.57). The data indicated that much more liquid flows down through the gap than the theory predicts. As a result, they postulated that the large recirculation under the free surface that they observed at low speed ratios perturbs the flow in the gap. The lubrication theory assumes virtually one-dimensional flow and would break down were there recirculations where both the x and y components of velocity are of the same order of magnitude. In sharp contrast, Ho and Holland (1978) obtained excellent agreement between theory and their experiments on an actual reverse roll coater; experiments limited to low speed ratios (0.07 < V J V , < 0.37). Benkreira et al. (1981) also found good agreement with their metering gap system, though with a coefficient of 0.63 rather than 2/3 in eq 2. To investigate the results of Greener and Middleman, the Navier-Stokes system that governs the two-dimensional free surface flow was solved by the Galerkin/finite element method (see Kistler and Scriven, 1983; Coyle, 1984). As in the analysis of film splitting in forward roll coating described by Coyle et al. (1986),the finite element domain was truncated by locating a boundary at the center
Figure 3. Predicted streamlines of film-transfer flow as a function of speed ratio (RIH,, = 77, St = 0.1,Re = 0.6,Ca = 0.23, H J H , = 1.9; (a) V J V , = 0.57, (b) V J V , = 1.0, (c) V,/V, = 1.75). The conditions attempt to duplicate the experiments of Greener and Middleman (19811, which showed a deviation from lubrication theory at the low speed ratio.
of the roll gap, as shown at the bottom of Figure 2, and imposing there a lubrication flow as the boundary condition. In particular, there the y velocity is zero and the x velocity is parabolic and proportional to q, which is an extra unknown in the equation set. The needed equation which is added requires the pressure at the boundary of the finite element domain to match the pressure given by lubrication theory for the submerged part of the flow (eq 1). The results of the computations were tested to determine that they were insensitive to further mesh refinement. At the same time, experiments were conducted on a specially designed roll coating machine, which consisted of two horizontally mounted, equal-diameter, polished, hard chrome plated steel rolls driven via timing belts by fractional horsepower dc motors with feedback control. The runout of the rolls (TIR) was less than 2.5 pm (0.0001 in.) and the surface roughness (RMS) was approximately 0.2 pm (8 pin.). The rolls were 20 cm in diameter by 30 cm in length and could be driven at 0-500 cm/s surface speeds. The gap between roll surfaces was typically set at 125-1000 pm (5-40 mils). A micrometer-driven needle was used to measure the coating thickness directly. A glycerin/water solution of approximately 1.4-P viscosity was used as the coating liquid. Further details of the apparatus and methods are given elsewhere (Coyle, 1984). 3. Results
Calculations were made with dimensionless parameters estimated to correspond to Greener and Middleman’s data (1978, 1981). Figure 3 shows how the calculated streamlines depend on the speed ratio. As emphasized by the dashed line, the recirculation does not penetrate further into the gap region and perturb the flow at low speed ratios in the way postulated by Greener and Middleman. Instead, the minimum x value of the recirculation is constant,
1418 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 f
0.8 r
I
I
P
0
0
.e.
- 0.6
1
Data (Greener R Middleman1 Data (no R Holland1 Data (This Work) Lubrication tEan 21
Lubrlcatlon (Eqn --- Finite Element
-0.4
1
1
0
4)
1 0
I
1
1
0.5
1.o
1.5
2.0
Speed Ratio (V,/V,)
c
Figure 4. Photographs of the meniscus position in reverse roll film-transfer flow (a) without and (b) with end dams. The view is along the roll axes from a slight elevation. The far end dam is centered in the background while the near end dam is not visible.
and as the speed ratio is decreased, the free surface simply rises higher above the gap to accommodate the recirculation that grows beneath. This recirculation can be made clearly visible by injecting small air bubbles into it. With gravity neglected, the computed flow rate varies with speed ratio in the way predicted by lubrication theory (approximately eq 3). Thus, the presence of the recirculation and the free surface does not perturb the flow rate through the gap* The computations, along with the results of the new experiments, provide an explanation for why the flow rate data depart from the theoretical prediction. The flow field a t low speed ratio (Figure 3a) is highly susceptible to three-dimensional effects a t either end of the roll pair. There is a large body of slowly recirculating liquid high above the gap, and thus high above the liquid in the bath. This sets up a flow in the transverse direction, toward each end of the roll pair where a "waterfall" plunges into the liquid bath below. This is seen in Figure 4a, where the meniscus is much higher a t the roll center than near the edges as the fluid flows along the roll axes toward each end of the roll pair (from the center toward the foreground and background). The meniscus is low in the gap and has appreciable curvature in the plane of the photo (the xy plane), especially a t the far end. Figure 4b clearly shows that, when this leakage a t the roll's end is blocked by using end dams (metal plates pressed against the roll's end), the meniscus position rises dramatically and becomes flat in the transverse direction, i.e., that of the roll axes. The meniscus is now high in the gap and has a flat central
Figure 5. Measured flow rate as a function of speed ratio compared to predictions, showing agreement of computations that include gravity with experiments performed by using end dams ( R / H o = 182, St = 0.16, Re = 0.9, Ca = 0.36, H , / H o = 15). When gravity is negligible, the model based on lubrication theory is accurate.
region in the plane of the photo. The transverse flow due to these end effects is significant since one uses a mass balance to determine the flow rate through the gap. Thus, if one measures the film thickness on each roll near its middle and assumes that all of the liquid mass flux not in these films must flow through the gap, one is adding the side leakage to the actual flow rate and thereby arriving a t a flow rate that is falsely high. There is a second source of deviation from the above analyses of the flow. In cases such as those of Figures 3a and 4b where the roll speeds are low, the gap is very large, and the body of liquid above the gap is large, gravity can play a significant role in the purely two-dimensional flow field. This effect is measured by the Stokes number, the ratio of gravity to viscous forces (St = pgH,2/pVa), which for these conditions takes on a value of 0.1 or greater. The flow rate (with gravity accounted for), as predicted by finite element solutions of the Navier-Stokes equations, is shown alongside the results of approximate theories and experimental data in Figure 5. The computed predictions for St = 0 lie between the lines representing the lubrication theories, which are in themselves close together. But in the new experiments reported here, gravity is important (St = 0.16), and including it in the computations shifts the predicted flow rate curve so that it deviates more and more from the lubrication approximation as the speed ratio falls. This is because the effect of gravity tends to increase the flow down through the gap, especially when the average roll speed is low. The prediction is in excellent agreement with the new measurements made with end dams in place. The experimental system of Renkreira e t al. (1981) had the rolls oriented in the vertical direction (gravity acting in the negative y direction in Figure lb), so gravity had no effect on the flow rate through the gap. Ho and Holland (1978) used large rolls and a small gap setting (20-25-cm diameter, 0.305mm gap), whereas Greener and Middleman (1981) used small rolls and large gap settings (5-cm diameter, 0.65-1.5-mm gap). Thus, the value of the Stokes number was typically 1-2 orders of magnitude lower for the experiments of Ho and Holland, so gravity was not a factor (as they correctly argue). In contrast, the experiments of Greener and Middleman appear to be dominated by gravity effects: the excessively high measured flow rates result from neglecting both the side leakage present when
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1419 end dams are not used and the high values of the Stokes number. Including the force of gravity in the lubrication model is a simple matter, but the lack of appropriate boundary conditions for this model at the free surface results in predictions of free surface position that are relatively insensitive to speed ratio. Moreover, the predicted position is close to that of high speed ratios where gravity has very little effect. Because the recirculations a t low speed ratio are not accounted for, gravity has no opportunity to play a significant role in this model, and the predictions are virtually identical with eq 2. In a reverse roll coater, there is always a metered film remaining on the applicator roll downstream of the metering gap, so in an actual metering gap flow, the flow rate as defined here must always be positive. Since the filmtransfer flow model presented here admits negative flow rates a t speed ratios greater than unity, it can plainly be an unrealistic approximation to a reverse roll coater.
4. Conclusions A simple half-submerged model reverse roll coating system first proposed by Greener and Middleman (1981) has been examined both theoretically and experimentally. Solving the Navier-Stokes system that governs the twodimensional viscous free surface flow field produces predictions of a large recirculation high above the gap at low speed ratios, but the presence of this recirculation and the free surface (with surface tension) does not cause the flow rate through the gap to deviate significantly from that predicted by a simple lubrication model for conditions where the effect of gravity is negligible. The computations and experiments demonstrate that there are two sources of deviation from the simply theory. First, the two-dimensional flow field is sensitive to the effect of gravity at the parameter values of the experiments reported here. The influence of gravity on the larger recirculating body of liquid above the gap induces a hydrostatic pressure gradient that increases the flow rate down through the gap. Lubrication theory cannot capture this effect because it does not correctly predict the high position of the free surface and the recirculation under it. The second effect is the possibly strong three-dimensionality of the flow. Experiments showed that there is a large component of the flow along the roll axes as fluid flows off each end of the roll a t speed ratios low enough that the free surface is high above the level of the bath. When end dams are used to stop this axial flow and gravity is included in the computations, experiment and computation are in excellent agreement. The excessively high flow rates a t low speed ratios reported by Greener and Middleman are the result of ignoring this axial flow. This model system, and the simple lubrication theory, are accurate only for the flow rate in a reverse roll metering gap and then only in cases where gravity is negligible and the speed ratio is less than about 0.5. Even in these cases, the simple theory does not correctly predict other aspects of the flow field such as the free surface position and the large recirculation. At higher speed ratios, this model system results in a linear decrease in flow rate, which becomes negative at speed ratios greater than unity. This behavior is markedly different than that in an actual metering gap where as the speed ratio increases the flow rate goes through a minimum and can never become negative. Experimentally this model flow appears to be always stable; that is, no irregularities or defects such a ribbing or cascade are seen in the free surface. Thus, this
model flow, though it is simple, is not a good representation of flow in the metering gap of a reverse roll coater since it displays none of the critical-and interestingphenomena observed in such a coater, phenomena associated with the wetting line and its interaction with the flow field in the gap. Acknowledgment We thank S. F. Kistler for much useful advice. This work was supported by the 3M Company, Westvaco Co., and the University of Minnesota Computer Center. Nomenclature H , = minimum gap half-width P = pressure p = dimensionless pressure (PHJpV,) q = dimensionless flow rate (-(su dy/2H0V,)) R = average roll radius ( 2 / ( 1 / R n+ l/R,,J) R, = radius of applicator roll R, = radius of metering roll u = fluid velocity in flow direction V , = applicator roll surface speed V , = metering roll surface speed x = flow direction coordinate y = cross-flow direction coordinate Ca = capillary number (viscous/surface tension force ratio, /J V,/ a) Re = Reynolds number (inertial/viscousforce ratio, pVJfo/p) St = Stokes number (gravity/viscous force ratio, pgH?/FV,) = surface tension p = viscosity 0 = dimensionless flow direction coordinate ( X / ( R H , ) ' /= ~ Z1I2 tan 0) Literature Cited Benkreira, H.; Edwards, M. F.; Wilkinson, W. L. Roll coating of purely viscous liquids. Chem. Eng. Sci. 1981,36, 429. Booth, G. L. Coating Equipment and Processes; Lockwood Publishing Company: New York, 1970. Coyle, D. J. The Fluid Mechanics of Roll Coating: Steady Flows, Stability, and Rheology. PhD Dissertation, University of Minnesota a t Minneapolis, 1984. Coyle, D. J.; Macosko, C. W.; Scriven, L. E. Film-splitting flows in forward roll coating. J. Fluid Mech. 1986,171, 183. Coyle, D. J.; Macosko, C. W.; Scriven, L. E. The fluid dynamics of reverse roll coating. AIChE J. 1990a,36, 161. Coyle, D. J.; Macosko, C. W.; Scriven, L. E. Reverse roll coating of non-Newtonian liquids. J. Rheol. 1990b,34, 615. Greener, J.; Middleman, S. Theoretical and experimental studies of the fluid dynamics of a two-roll coater. Ind. Eng. Chem. Fundam. 1978,18,35. Greener, J.; Middleman, S. Reverse roll coating of viscous and viscoelastic liquids. Ind. Eng. Chem. Fundam. 1981,20, 63. Encycl. Polym. Sci. Higgins, D. G. Coating methods-survey. Technol. 1965,3, 765. Ho, W. S.; Holland, F. M. Between-roll metering coating technique, a theoretical and experimental study. TAPPI 1978,61, 53. Hopkins, M. R. Viscous flow between rotating cylinders and a sheet moving between them. BF. J. Appl. Phys. 1957,8,443. Kistler, S. F. The Fluid Mechanics of Curtain Coating and Related Viscous Free Surface Flows. Ph.D. Dissertation, University of Minnesota a t Minneapolis, 1983. Kistler, S. F.; Scriven, L. E. Coating Flows. In Computational Analysis of Polymer Processing; Pearson, J. R. A., Richardson, S., Eds.; Applied Science Publishers: London and New York, 1983. Reynolds, 0. On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments. Philos. Trans. R . SOC.London 1886,A1 77,157. Received for reuiew November 9, 1989 Revised manuscript received March 7, 1990 Accepted March 14, 1990
