Meniscus Shapes in Withdrawal of Flat Sheets from Liquid Baths. Dynamic Profile Data at Low Capillary Numbers Chie Y . Lee’ and John A. Tallmadge* Department of Chemical Engineering. Drexel University, Philadelphia, Pennsylvania 79704
The size and shape of meniscus profiles, which were enlarged by flow, have been measured experimentaliy and photographically for a range of flow conditions. The geometry used was the free coating of flat sheets withdrawn from a pool of wettinq liquids. The withdrawal meeds were varied over several capillary numbers ( C a ) below 1, using an oil having a viscosity of 0.194 N s/m2 (194 cP). The deformed profiles were described with a three-parameter analytical expression. The 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. An influence of Reynolds number on the profiles was noted at Re above 2.
Introduction Static menisci, when deformed by fluid velocities, become enlarged and distorted. 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, Van Rossum, Groenveld, and others. A diagram of the continuous belt device, used here to obtain free coating, is given in‘the paper by 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particular, this paper is concerned with the shape of the meniscus near the point of emergence of a flat sheet as it is withdrawn from a pool of liquid. See Figure 1. 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. This paper reports the meniscus shapes measured photographically for a viscous oil of 0.194 N s/m2 (194 cP) a t sheet coating speeds of 15 to 163 mm/sec. The resultant profiles are fit to existing analytical expressions and the resultant expressions are compared. The general problem of describing the size and shape of dynamic menisci (for use with flow field studies) can be restated as measuring the profiles in terms of a meniscus thickness h as a function of laboratory position x and speed u. In a recent study of free coating (Lee and Tallmadge 1972, 1973a), profile data with large deformations were obtained with a high viscosity oil of 1.31 Ns/m2 (1310 cP). In general, the resulting capillary numbers were above 1,where Ca u ( p / a ) . The main purpose of this paper is present profile data obtained with another fluid-of lower viscosity-under conditions such that the capillary numbers were below 1. These data are needed to test theories developed for low Ca. These data are also needed for describing and understanding the influence of Ca on profiles, in terms of the effect on meniscus slopes and intercepts. 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 1. Suitable quantitative descriptions of the small deviations
’ DeLaval Ca, Pleasant Valley, N.Y.
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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 (High Ca) Flow in the free coating case considered here is caused by the steady, continuous, vertical withdrawal of a flat sheet through a large bath of a wetting liquid at sheet speed u. A typical meniscus profile in rectangular coordinates is shown in Figure 1. The meniscus thickness h also increases with speed and liquid viscosity. Figure 1 is given in nondimensional form based on the film thickness ho which occurs in the constant thickness h/ho. region. Here height X = x/ho and meniscus L Using the stagnation point which occurs between L of 2 to 3 (Lee and Tallmadge 1973b), we can consider the meniscus as having two regions-the thick region a t L larger than 3 and the thin region a t L smaller than 2. Earlier entrainment studies have shown that the film thickness ho is described in normalized, nondimensional form by To as Some understanding of the dynamic profiles has been given by experimental observations of a viscous oil a t Ca above 1, 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 - 1).See Figure 2. The linear portion of the thick region was described by TO and two nondimensional profile parameters Mz and Bz (Lee and Tallmadge, 1973a) given by the following analytical expression
L - 1 = B~ exp (- A . / M ~ )
(2)
The profile given by eq 2, when plotted in semilog form, shows that M2 is the slope of the meniscus interface and BZ is the extrapolated intercept of the profile at X = 0. We associate the intercept Bz with meniscus size and the slope Mz with meniscus shape. (See Figure 2.) 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 TOand two other values for slope and intercept, namely M I and B1 given by
L - 1 = B, e-(h / M 1 ) for L < 2 (3) Equation 3 is also shown in Figure 2. There is need for describing and understanding the influence of speed and viscosity on both meniscus slopes ( M 2 and M1) and intercepts (Bz and B1) for a wide range
,6
EFFECT
OF DEPTH ON SIZE
VlSCOUS OIL ,u= 1.3 N - j / m Z (1310 cp)
a
a 3 1
w
/
> o / m
BATH DEPTH,
d /ha
(LOG SCALE) MENISCUS THICKNESS, L= h /
ho
Figure 1. Sketch of the free coating profile, L ( X ) .
Figure 3. Effect of bath depth on meniscus size. The solid line is from data of Lee and Tallmadge (1973a).The dashed line is from data of Lee (1974).
Table I. Run Conditions and Profile Parameters Speed, C(i 0.083
4 k
\ SLOPE M 2 0 F THICK REGION \(LOW E R MEN I SCUS)
W
a L
-
I
E
I
1
1
\
EXTRAPOLATED INTERCEPTS = ADJ. THICKNESS,
\',& In
\ '
',.-BI
(L- I)
Figure 2. The two linearized profiles observed in free coating.
of Ca. At large Ca, such as for Ca above 1, previous work has shown that the B2 intercept is primarily a function of bath depth for shallow baths but becomes relatively constant at larger bath depths. See Figure 3 for depth results obtained with the higher viscosity oil. Previous experimental results with deeper baths a t Ca from 3 to 12 indicated that the M I and M2 slopes were approximately constant a t 1.5 and 2.0, respectively. Another purpose of this paper is to present, for comparison purposes, measured values of slopes and intercepts for lower Ca (below 1).
Speed, 21, mm/sec Liquid temperature,"C Bath depth, mm Run no.
Conditions 14.9 35.1 28.3 28.0 20 40 219M 213M
Entrainment 0.310 Thickness, k,, mm Thickness, To 0.561 Flow rate, w , g/sec 0.216 Flow rate, Q," 0.482 Flow rate, Q o a 0.502 Difference in Q +4% Slope Mi Intercept B , L Range X Range
0.20
Upper Meniscus 2.64 55 1.3/3 14/9
Data 0.531 0.621 0.856 0.522 0.541 +4%
0.45
0.95
77
163 27.5 88 208M
27.7 56 210M 0.837 0.65 2.90 0.540 0.561 +4%
(Small L ) 1.99 1.99 27 10 1.2/2 1.2/2 11/5 8/3
1.20 0.64 9.00 0.546 0.553 +1R
1.73 14 1.112 8/5
Lower Meniscus (Large L ) Slope M 2 4.51 3.38 2.30 2.12 Intercept B, 13.0 9.2 8.4 7.0 See the text in the last paragraph of "Profile Data" for discussion of Qw and Qo.
Experimental Conditions Data were obtained using a continuous belt apparatus described by Soroka and Tallmadge (1971) using techniques described by Lee (1974) as summarized by Lee and Tallmadge (1972, 1973a). As before, the belt was 62 mm wide and about 2500 mm long and the liquid bath was located between the two pulleys. The choice of liquid viscosity and withdrawal speeds was based on obtaining capillary numbers between 10-2 and lo", in order that Ca was less than 1 but large enough to provide easily measurable deformations. The liquid used in all runs was a lubricating oil (Fluid M) having the following properties a t 26.7"C: viscosity of 0.194 N s/m2 (194 cP), surface tension of 31.5 N s/mm, and a density of 874 kg/m3. Thus the capillary length a is 2.72 mm for this ( 2 ~ / p g ) ~ /and 2 a equals the maximum fluid; here a static rise. Meniscus profiles were photographed at several speeds. The conditions for each run are shown in Table I, together with the film thickness ho in the constant thickness region. The film thickness ho was measured independently using a micrometer a t a fixed position of 545 mm above
the bottom of the bath. The distance from the top of the liquid level to the micrometer was in the range of 460 f 65 mm. Deep baths were maintained by selecting bath depths which were 60 to 80 times as large as the film thickness ho. Flow rate data and consistency checks are also shown in the table. Table I shows the range of Ca obtained, from 0.083 to 0.95. These Ca values were calculated using the measured speed, measured surface tension, and adjusted viscosity. The air temperature in the room was held at 26.7 f 0.1"C. Small changes in viscosity, due to the small changes in oil temperature from run to run, were calculated using the equation (P/PJ = exp[5840(t0 - W(WO)l (4) Here t is "K, the reference viscosity K O is 0.194 N s/m2, and the reference temperature to is 299.7"K. The constant in eq 4 was determined from the linear slope of viscosity data plotted as the logarithm of viscosity us. reciprocal temperature. The viscosity data were taken at four temperatures, namely the reference temperature plus about Ind. Eng. Chem., Fundam., Vol. 1 3 , No. 4 , 1974
357
1 1
Figure 4. Meniscus photograph at Co of 0.083 at a withdrawal speed of 15 rnrnlsee; run 219M; taken at f3.5 and $50 sec with a 55-mrn lens, an ASA film speed of 400, ail viseasityof 0.194 N s i
Figure 6. Meniscus photograph at Co of 0.20 at I7 mrnjsec; run
210M.
mz.
onaph at Ca of 0.95 at 163 rnrnjsec; run Figure 5. Meniscus photograph at Ca of 0.20 at 35 mmjsec; run
213M.
l n e prome u a ~ awere uu~aineuuy prujec~mgLIE
___
15, 21, and 33°C. Liquid temperatures were n the return liquid. The small temperature risL ,-.marily to viscous heating; heating from an illuminating lamp was secondary or negligible. Profile Data The meniscus profiles were determined photographically, using a Nikon camera with a 55-mm lens a t a distance of about 200 mm. They were taken at 53.5 and %SO see using a 35-mm film with ASA speed of 400. The photographs obtained are shown in Figures 4-7. These photographs were taken perpendicular to the belt so the belt appears only as a line in the film. The free coating film of interest here appears on the right side of the belt in the photographs. The perpendicular distance from the belt to the wall on the free withdrawal side was about 200 mm. Illumination was provided by two 75-W lamps placed to the right of the camera. The lamps were placed ahout 30" above the horizontal plane of the bath level and above each comer of the 325 X 200 mm bath container. This position was superior to lighting from the back, side, or below the hath. The film on the left of the photograph was in restricted (obstructed) flow and is not of interest here: this film was influenced by the wiper used on the back side of the belt which resulted. as expfcted, in the larger obstructed thickness. Tests showei1 that flow between the two sides was negligible at the con ditions studied here. 358
Ind. Eng. Chem.. Fundam., VoI. 13,No. 4,1974
mga-
tive image on a screen. The expanded profile thickness was traced on a sheet of paper, from which h and x values were measured using the scale included in the photograph. The resultant h(z) profiles are shown in Table 11. Comparison of the ho valile from the micrometer measurement (Table I) confirms the asymptoptic values of h obtained photographically (Table E). Comparison of the measured mass flow rate (Qw in Table I) with that calculated using the film thickness (Qoin Table I) shows agreement within 4%, which indicates good accuracy of the ho value. Analytical Parameters The experimentally obtained profiles of Table U were transformed into L(h) profiles by use of the film thickness ho and plotted on semilog coordinates of X us. In ( L - 1). Two linear regions were observed. The presence of these two regions is consistent with Figure 3, eq 2 and 3, and the results reported for Ca above 1 for the other liquid of higher viscosity. The slopes ( M I and Mz)and intercepts (BI and B d were evaluated, using the new data of Table II, and are given in Table I. It is seen that all four parameters tend to decrease with increasing Ca and the M Z slope appears to he larger than the MI slope for the same run: the sizes of these parameters are consistent with those observed with the more viscous oil (Lee and Tallmadge 1973a). Figure 8 shows the influence of capillary number on slopes for the data of this work
-
Table 11. Experimental Meniscus Profiles Ca number:
Run number: h , thickness, mm:
O(static)b 0.083 ... M219 None 0.310
Height x , mm
EFFECT
€ , , , , I
OF CA ON SLOPES
I
0.199 0.45 0.95 M213 M210 M208 0.531 0.837 1.200
Meniscus thickness h , mm -
0.0"
oc
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8 9
2.45 1.20 0.55 0.19 0.02 -0-0-
10
20 30 40 50
5.36 3.27 2.24 1.68 1.32 0.96 0.75 0.55 0.45 0.38 0.34 0.32
... ...
... ...
... ... 0.31 0.31 0.31
5.50 4.10 3.25 2.60 2.15 1.75 1.40 1.10 0.85 0.75 0.67 0.61
... 0.56
... ...
... ...
7.75 6.10 5.00 4.10 3.50 2.80 2.30 1.90 1.65 1.40 1.25 1.12 1.05
... ...
...
0.88
...
.., ...
0.53 0.53 0.53
0.84 0.84 0.84
9.75 7.75 6.45 5.70 4.95 4.25 3.75 3.25 3.00 2.70 2.45 2.20 2.00 1.85 1.72 1.60 1.50 1.40 1.30
...
1.20 ... 1.20 ... . . . . . . 1.20 0.31 0.53 0.84 1.20 a Actually at r = 0.05 f 0.05 mm. Thicknesses at this height appear finite for finite baths and are near the location of the second stagnation point. b Static values from theory. c Presented by Lee and Tallmadge at the National ACS Colloid Conference, Amherst, Mass., 1972. x)
Comparison of Eq 2 and 3 Before comparing slopes of this work with those reported for the more viscous oil, we state why we now consider eq 2 to be more useful than eq 3. In an earlier paper (Lee and Tallmadge, 1973a), eq 2 and 3 were presented as co-equal in the sense that they were taken as applicable to only one of two portions of the meniscus: If one is interested in describing the parameter ( L - 1) or ( h - ho) precisely, then they are of relatively similar or coequal in importance. However, if the criterion of precision is the size L (or h ) , then eq 2 is clearly superior to eq 3. This is because eq 2 describes the h profile data (such as that in Table 11) very precisely for the thin region of L values from 1 to 3 as well as in the thick region; the agreement is within 5 to 10% for all the data in Table 11. On the other hand, eq 3 is in considerable error in L for all the data in the thick region (with errors ranging up to 1 0 % and 400%) and is thus able to predict L values precisely only for the thin region, where L is between 1 and 2. Neither equation is precise at the bath surface, but eq 2 intercepts are closer to the experimental values than the intercepts of eq 3. In summary, eq 2 is the best simple description of the complete profiles in free coating in terms of size and location. With three parameters (TO, M2, B2), it describes the entire L(X) and h ( r ) profiles from near the surface to the constant thickness region with good precision. Therefore, eq 2 should be considered henceforth as a general profile for L and h calculations, whereas eq 3 is restricted to use for the thin region.
W
2
OL6''d.I
012
0:3
'd.6
";
i2
CAPILLARY NUMBES, C A S U( H ' ~ )
Figure 8. Effect of capillary number on meniscus slopes. The solid line is for Ma. The dashed line is for M I .
The Slope M2 of the General Profile In the earlier study of the more viscous oil (Lee and Tallmadge, 1973a), it was found that the slope M2 decreased to a constant value a t increasing Ca. This constant value (or lower limit) may be taken as the M 2 value of 2.0 reported for Ca of 3, 6, and 12. Thus we conclude that the lower limit Mz* is M2*
+
2.0
(at i n c r e a s i n g l y l a r g e C a )
(5)
The data 0; this work (Figure S j appear to approach this limiting value asymptotically. Similar asymptotic behavior is noted for the B2 intercept as compared to the constant value of B2* = 6 reported for Ca of 3,6, and 12. An exception to the lower limit on slope M2 is the M2 value of 2.5 previously reported for Ca of 24. This led to the consideration of inertial forces and calculation of Reynolds numbers. Higher Re were obtained using fluid M of this work at two higher speeds. See results at Ca of 2 and 4 in Table 111. Results for M2 slopes, as shown in Table 111, indicate that eq 5 holds for low Re, specifically for Re below 2. In other words, slope M2 approaches a lower limit at increasing Ca only when inertial effects are small or negligible. At higher Re of 4 and above, inertial effects tend to increase values of slope M2. Table I11 shows that in order to study the influence of both Ca and Re it is necessary to the study the influence of viscosity independently of speed. This shows an additional benefit of studying the profile behavior for a second fluid of different viscosity, as is done in this paper. Similar behavior was noted for the B2 intercept and for the M I and B1 parameters of the thin-region eq 3. These results are of particular interest for testing the influence of Ca which can be predicted by one-dimensional-flow models and for testing the influence of Re (and Ca) which can be predicted by two-dimensional-flow models (Lee and Tallmadge, papers submitted for publication).
Discussion of Precision The influence of random variations is considered first in terms of the independent variables and then for dependent variables. Systematic errors are then discussed. The largest error in measurement of independent variables was for viscosity. The variation in viscosity due to the bath temperature change was sizeable, so the fluid temperature was measured during each run and the viscosity corrected to that temperature. The uncertainty in viscosity was usually about 0.8% (maximum of 1.5%) for a temperature measuring error of 0.1"C. The next largest error was in speed, so the withdrawal velocity was measured in each run. The speed fluctuation during a run was usually about 0.6%. The uncertainty was about 0.6% for surface tension and smaller for density. The largest errors in dependent variables occurred in the determination of position x in the lower part of the Ind. Eng. Chem.,
Fundam., Vol. 13, No. 4 , 1974
359
Table 111. The Influence of Reynolds Number on Slope M Z Re
CO
0.083 0.20 0.45 0.95 2.0 4.0 0.43 1.2 2.8 5.7
12 24 a
0.041 0.15 0.47 1.4 4.2" 12. oa 0.013 0.053 0.18 0.57 1.7 4.8"
Liquid (viscosity)
Slope .V2
M (low) M
4.5 3.4 2.3 2.1 2.5"
M
M M M B (high) B B B B B
This work This work This work This work This work This work Lee and Tallmadge L e e and Tallmadge L e e and Tallmadge Lee and Tallmadge Lee and Tallmadge Lee and Tallmadge
(1973a) (1973a) (1973a) (1973a) (1973a) (1973a)
Deviations n o t e d a t Re of 4 a n d above
meniscus. The film thickness ho was measured by micrometer and was most precise. The average reading to reading variation in ho was 0.5%; a n extreme of 1.3% was noted with the thin films a t the slowest speed. Mass flow rates varied within 0.1 to 0.5% for most runs; larger uncertainties were noted a t high flows (up to 1% a t the highest speed) due simply to the short sample times used a t high flow rates. The thickness h and position x were both measured to within 0.05 mm, which gave precise results for large x (4% or less) and large h (less than 1'70). Although small thickness h (near the constant region) has a larger error, the micrometer values of ho indicated that these errors were small. The uncertainty in determining the position of the liquid level line and thus the origin in the x direction is fairly large (0.2 mm or more), which leads to large uncertainties in x a t the lower part of the meniscus. Systematic errors are more difficult to evaluate quantitatively than the repeatability test results noted above. However, exploratory tests with variable micrometer locations indicated that backflow was not noticeable for these runs. Tests with shallow bath depths indicated considerable influence on intercepts, some slight influence on film thickness, and little or no influence on the profile slopes; these results were, in general, consistent with those for the more viscous oil (Lee and Tallmadge, 1973a). In summary, the largest uncertainty is in the position x: near the bath level, because of the difficulty in locating the origin. Summary New profile data for another liquid, a lower viscosity oil, and for several Ca below 1 are presented in Table I1 based on the photographic evidence shown in Figures 4-7. These h ( x ) data were described by three analytical parameters, 7'0, Mz, and &, as given in Table I for use in eq 2. Parameters for estimating ( h - ho) values precisely in the thin film region are given by M I and B1 in Table I. The influence of Ca is shown in Figure 8. These profile data are useful for testing theoretical models and for studying flow fields in menisci.
360
5.0a
3.1 2.5 2.0 2.0 2.0 2.5"
Reference
Ind. Eng. Chem., Fundam., Vol. 13, No.4, 1974
Study of the influence of Ca on slope for two liquids led to the recognition of the influence of Re (and inertial forces) on meniscus profiles, as shown in Table III. Acknowledgments This work was supported in part by the Eastman Kodak Company. The figures were drawn by William Wasylenko. Nomenclature a = capillary length, (2u/pg)1'2, mm b = unit width of plate, mm B1 = intercept, eq 3 B2 = intercept, eq 2 Ca = capillary number, u ( g / p ) g = acceleration of gravity h = meniscus thickness a t any point, mm hc = characteristic thickness, (gu/pg)l/Z,mm ho = film thickness, constant thickness region, mm L = meniscus thickness, dimensionless, h/ho MI = slope, eq 3 Mz = slope, eq 2 Q = mass flow rate, dimensionless, w/(pubhc) Re = Reynolds number, hcup/p To = film thickness, dimensionless, ho/h, u = velocity of belt, mm/sec x = upward coordinate, meniscus height, mm w = mass flow rate, g/sec
Greek Letters liquidviscosity p = liquiddensity X = meniscus height, dimensionless x/ho = surface tension of the liquid-air interface g =
Literature Cited Lee, C . Y.. Ph. D. Dissertation, Drexel University, Philadelphia, Pa., 1974. Lee, C . Y., Tallmadge, J. A,. A l C h E J . , 18, 1077 (1972). Lee, C . Y., Tallrnadge, J. A , , A l C h E J . , 19, 403 (1973a). Lee, C . Y., Tallrnadge. J. A , , AlChE J., 19, 865 (1973b). Soroka, A. J.. Tallrnadge. J. A,, A I C h E J . . 17, 505 (1971).
Received for review November 7, 1973 Accepted M a y 21,1974
