Meniscus Shapes in Withdrawal of Flat Sheets from Liquid Baths. 11. A Quasi-One-Dimensional Flow Model for Low Capillary Numbers Chie Y . Lee’ and John A. Tallmadge” Department o f Chemical Engineering. Drexel University. Philadelphia. Pennsylvania 19104
The size and shape of meniscus profiles, which were enlarged by flow, have been predicted by a quasi-onedimensional flow model for a range of flow conditions. The geometry considered was the free coating of flat sheets withdrawn vertically and continuously from a pool of wetting liquids. The thick portions of the deformed profiles were described with an analytical expression, involving a slope. Data confirm the predicted effect of capillary numbers, Ca, on thick slope for cases where surface tension forces are larger than viscous forces, namely for capillary numbers below one. The predicted analytical parameters may be used by interpolation to determine the influence of Ca, coating speed, surface tension, viscosity, and density on the profile size and shape. Other aspects of the prediction are discussed.
Introduction The shape of a static meniscus on a flat sheet is well known, having been solved analytically using a balance between pressure and gravitational forces, with pressure described using liquid-gas interfacial tension. The shape of a meniscus deformed by fluid motion is generally not well understood quantitatively, however, even though such dynamic menisci are present in many experimental tests and in many industrial processes. The authors are interested in describing and understanding the deformed shapes of these dynamic meniscus profiles. The geometry chosen here is that of free coating, primarily because of the extensive work published on this geometry by the authors and by Landau-Levich (1942), Van Rossum (1958), Groenveld (1970), and others. A diagram of a continuous belt device, used previously to obtain free-coating data, is given in Soroka and Tallmadge (1971). In particular, this paper is concerned with the shape of the meniscus near the point of emergence of a flat sheet as it is withdrawn from a pool of liquid. The withdrawal is vertical, steady, and continuous and the bath is large. See Figure 1A. This meniscus shape has been found to have a significant influence on the coating thickness which results, to affect the flow field inside the meniscus, as well as the resultant vortices and surface stagnation points, and is believed to play a role in the onset of film nonuniformities. It is for these reasons that the study of dynamic meniscus profiles was begun. One use of such meniscus shapes is in two-dimensional studies of flow fields (Lee and Tallmadge 1972b) and the properties of these flow fields. The general problem of describing the size and shape of dynamic menisci (for use with flow field studies) can be restated as predicting the profiles in terms of a meniscus thickness h as a function of laboratory position x and speed u. See Figure 1A. In a recent study of free coating (Lee and Tallmadge 1972c, 1973a), profile data with large deformations were obtained with a high viscosity oil of 1.31 N s/m2 (1310 cP). In general, the resulting capillary numbers were above 1, where Ca = u ( g / u ) . Although many theories have been proposed for predicting the amount of entrainment in free coating, there is no simple method for predicting the size and shape of the complete profile of the meniscus. The greatest need is for a prediction of size and shape in the lower region of the meniscus where the film is thick.
’ DeLaval Co.. Pleasant Valley, N Y . 120
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
The main purpose of this paper is to present a simple method for predicting meniscus profiles, using a quasione-dimensional model, and to compare the prediction with data. A secondary purpose of this study is to determine whether the quasi-one-dimensional model has any usefulness in describing dynamic menisci, such as those in free coating. Small deviations from static profiles are considered here. We take “small” to mean that the viscous force is smaller than the surface force of capillarity. Since the capillary number Ca is the ratio of viscous to surface forces, we are thus considering capillary numbers less than one. Suitable quantitative descriptions of the small deviations would be useful for calculating the magnitude of flow effects in tests where a slow motion is used to approach a static case, such as in the measurement of advancing and receding contact angles. Previous Work (Data at High Ca) A typical meniscus profile for free coating in rectangular coordinates is shown in Figure 1A. The meniscus thickness h increases with speed and liquid viscosity. Figure 1B shows the same profile in nondimensional form based on the film thickness ho which occurs in the constant thickness region. Here height X = n/ho and meniscus thickness L E h/ho. Using the stagnation point which occurs near L of 2 (Lee and Tallmadge 1973b), we consider the meniscus as having two regions-the thick region a t L larger than 2 and the thin region a t L smaller than 2. In this paper, we are concerned primarily with the thick region of the profile. Earlier entrainment studies have shown that the film thickness ho is described in normalized, nondimensional form by To as follows.
To
ho/h,
3
h0(pg/pu)l’2
Some understanding of the dynamic profiles has been given by experimental observations of a viscous oil, in which the size of several profiles was measured photographically. The thick profile data were found to be linear on a semilog plot of X us. log ( L - l), except very near the bath surface where X = 0. The linear portion of the thick region was described by TOand two nondimensional profile parameters M Z and Bz (Lee and Tallmadge 1973a) given by the following analytical expression
L - 1 = B, e-(-
x/M~)
(2)
The profile given by eq 2, when plotted in semilog form,
shows that Mz is the slope of the meniscus interface and is the extrapolated intercept of the profile a t X = 0. We associate the intercept B2 with meniscus size and the slope Ma with meniscus shape. See Figure 1C. It was also noted that the thin profile was linear on the semilog plot but had a different slope. The thin profile region was described by To and two other values for slope and intercept, namely MIand B1. There is need for describing and understanding the influence of speed and viscosity on both meniscus slopes ( M 2 and MI) and intercepts (B2 and B1) for a wide range of Ca At large Ca, such as for Ca above 1, previous work has shown that the Bz intercept is primarily a function of bath depth for shallow baths but becomes relatively constant a t larger bath depths. Previous experimental results with deeper baths a t Ca from 3 to 12 indicated that the M I and M 2 slopes were approximately constant at 1.5 and 2.0, respectively. B2
Specific Purpose This paper is concerned a t first with the thick region slope M z Specifically, the main objective of this paper is to predict the M2 slope as a function of Ca for Ca below 1, using a quasi-one-dimensional flow model, and to compare the prediction with data. Also predicted are the Bz and To parameters as secondary objectives. To do this, we focus attention on the prediction of the following function
121, = f(ca)
(at C a
