Meniscus Shapes in Withdrawal of Flat Sheets. 3. A Quasi-One-Dimensional Flow Model Using a Stretch Boundary Condition Chie Y. Lee and John A. Tallmadge' Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19 104
The size and shape of flow-enlarged meniscus profiles are predicted by an improved quasi-one-dimensional flow model. The geometry considered is the free coating of flat sheets withdrawn vertically and continuously from a pool of wetting liquids. The new model is based on extending the normal-stress boundary condition to include a stretch term in addition to the Laplace-Young term used in the earlier theory. The new model provides improved predictions of meniscus size and extends the useful range to higher capillary numbers. The predicted parameters for an analytical expression may be used by interpolation to determine the influence of Ca, coating speed, surface tension, viscosity, and density on the profile size and shape.
Introduction This paper is concerned with the size of the meniscus near the point of emergence of a flat sheet as it is withdrawn from a pool of liquid. The geometry is called free coating. The withdrawal is vertical, steady, and continuous and the bath is large. See Figure 1A. This meniscus has been found to have a significant influence on the coating thickness which results. Meniscus size also affects the flow field below the meniscus, including a vortex cell and two surface stagnation points, and is believed to play a role in the onset of film non-uniformities. These and other reasons for predicting dynamic meniscus profiles are discussed in papers describing a two-dimensional flow theory (Lee and Tallmadge, 1974a) and a one-dimensional theory (Lee and Tallmadge, 1975). The problem involves the prediction of meniscus thickness h as a function of laboratory position z, speed u w ,and fluid properties. In nondimensional form, this is prediction of meniscus thickness L hlho as a function of height X E x l h o and capillary number Ca = uw(p/a).Experimental work has shown that the complete L(X)profile has two large linear regions in semilog coordinates (a thin, upper region and a thick, lower region) and two small, curved zones in the transition and near the bath surface (Lee and Tallmadge, 1973a, 1974b). See Figure 2. The shape of the complete profile has been predicted by only one theory, a quasi-one-dimensional model (Lee and Tallmadge, 1975). However this simple model underestimates the size of the meniscus. There is, therefore, a need for better prediction of size. This need is greatest in the lower region of meniscus where the film is thick. The main purposes of this paper are to extend the onedimensional model for predicting meniscus profiles and to compare the prediction with data. A secondary purpose of this study is to show the improvement due to the introduction of the stretch term in the normal stress boundary condition. Previous Work (Model) The governing equation used previously to predict meniscus profiles was a third-order, ordinary differential equation (Lee and Tallmadge, 1975) d2L/dX2
Here To is the normalized form of film thickness ho in the constant thickness region. The term in brackets is equal to the meniscus curvature C
Ca
d2L/dX2
[I
+ (dL/dX)2]3/2
The problem was one of predicting the meniscus profile L (A) as a function of Ca. The meniscus profile decreases from a very large L at the bottom or bath surface to L of 1a t a large height or the top, as shown in Figure 1B. The three boundary conditions used were (1)a flat surface a t the top, (2) no curvature a t the top, and (3) no curvature a t the bottom, or BC 1: dL/dX = 0
258
- 1) - To2(L3 - 1)
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
(1)
(3)
BC 2:
C=0
( a t L = 1)
(4)
BC3:
C=O
(atL-m)
(5)
Equation 1was solved numerically by a trial-and-error technique in TOfor each Ca using integration from the thin region a t L of 1down through the thick region of large L . One result was a prediction of film thickness TOfor each Ca; for example To = 0.41302 a t Ca = Plots of L(X)at each Ca showed that the model predicted two small nonlinear regions-one and one benear the bath surface (A below 3 for Ca of tween the thick and thin regions ( A near 30 and L near 2 for Ca of lO-+-and two large linear regions which may be described with constant slopes M I and M2. Because the presence of these four regions agreed with that of data both a t high Ca (Lee and Tallmadge, 1973a) and a t low Ca (Lee and Tallmadge, 1974b), the model was found to predict the proper functional form of experimental profiles. The thick regions occurred a t L larger than 2. The linear portion of the thick region was described by To and two nondimensional profile parameters M2 and E 2 given by L
- 1 = E2 exp(-h/Ma)
(6)
The profile given by eq 6, when plotted in semilog form, indicated that M2 is the slope of the meniscus interface and B P is the extrapolated intercept of the profile at h = 0. Hereafter we will associate the intercept E2 with meniscus size. The thin profile region was described by To and two other values for slope and intercept, namely M I and E l .
L - 1 = B 1 exp(-h/MI) = 3(L
(at L = 1)
(at L
< 2)
(7)
However, thin profile eq 7 is not as useful as eq 6 in describing
v. k M O V l N G BEL?
GAS THICKNESS Lih/h, MENISCUS
THICKNESS
Figure 1. Sketch of meniscus shapes in free coating (laboratory coordinates): A, dimensional h ( x ) ;B, nondimensional L(X).
meniscus profiles-for several reasons. For example, the intercept B1 does not really have much physical meaning (whereas B 2 does-& implies size observed in photographs). More importantly, eq 6 accurately describes, in rectangular form, the complete profiles from the thick region to the constant thickness region, whereas eq 7 is only accurate for small
ADJUSTED MENISCUS THICKNESS, t-1 or (h-hJ/ho
Figure 2. Typical meniscus shape (semilog coordinateiadjusted thickness); predicted profile for Ca of 0.01 (Lee and Tallmadge, 19751.
AP = crc. See Appendix D.
L. The model of eq 1was used to predict the influence of speed and viscosity on both meniscus slopes ( A 4 2 and MI) and intercepts ( B 2and B1) for a wide range of Ca. It was reported to be most precise for low Ca. Comparison with data in the thick region indicated that the model properly predicted the effect of Ca on slope M 2 but underestimated the meniscus size, as given by B P .
Specific Purpose This paper is primarily concerned with meniscus size and slopes. Specifically, the main objective of this paper is to predict the meniscus size as a function of Ca, using an improved quasi-one-dimensional flow model, and to compare the prediction with data. Also predicted are the slopes, intercepts, and film thickness as secondary objectives. The improvement is based on eq 10. Presented below are results which may be used to predict the profiles as a function of speed, viscosity, surface tension, density, and gravity. All calculations are for creep flow (negligible Re effects) in baths of infinite extent (negligible wall effects). Theoretical Model (Dimensional Form) At the outset, let us repeat that it is clear that any onedimensional flow method is an approximation to the twodimensional flow (Lee and Tallmadge, 1974a) that actually occurs. Equation 1 was developed by considering viscous, gravity, and pressure forces in the one-dimensional equation of motion for the vertical velocity component of a Navier-Stokes fluid, neglecting inertia terms, and integrating in the LandauLevich manner. Thus one obtains the following differential equation for the h ( x ) function (Lee and Tallmadge, 1973b, 19'75).
po
-p
= crc
+ 211-dvs ds
Thus
Here us is the tangential velocity of the surface in the direction of the surface and s is the length along the interface. Equation 10 holds for the case of negligible surface tension gradients, negligible surface viscosity and elasticity, and negligible mass flux across the interface (Scriven, 19601, which the case considered here. Substituting the pressure gradient dpldx from eq 11 into eq 9 leads to
du, ---2h3 d pu,ho dx
ds
)
(I2)
We now wish to evaluate the c and du,/ds terms in eq 12 in terms of x, h, and derivatives of h. As before, the curvature is given by geometry. Thus c is given by c =
d2h/dx2 11 + ( d h / d ~ ) ~ ] ~ / ~
Now consider the velocity term du,/ds. This appears as part of the last term in eq 12, which we call the stretch term. Since the stretch term is taken along the streamline along the surface, we note that (ds2) = (dx)2 (dy)2and that
+
dx = cos 0 ds
(14)
As a consequence the derivative in the stretch term becomes du _ du, - cos 8 -2
or after rearrangement
To evaluate the pressure profile p ( x ) , we invoke an improved form of the normal stress boundary condition across the liquid-gas interface, instead of the expression used earlier,
(15) ds dx In keeping with the pseudo-one-dimensional assumption of starting eq 8, we approximate the two-dimensional surface velocity u s by that of the one-dimensional estimate of surface velocity us. Thus du du, 2 cos 0ds
dx
To evaluate the surface velocity term, we use the one-diInd. Eng. Chern., Fundam., Vol. 15, No. 4, 1976
259
VISCOUS DEFORMATION IN FREE COATING
Upon differentiation, eq 25 becomes
FLOW THEORY . b P = bC+2p (dU,/dS)
Substitution of eq 26 into eq 22 leads directly to the desired equation, which is
L:’ dC - L:’ d = 3(L - 1) - T&L3
- 1) - (3 - To21
L”L - 2(L’)2 3
l
2
3
Y = h / c , *“HERE
4
h
5
- MENISCUS
6
7
5
TiilCKNESS
F i g a r e 3. Predicted meniscus profiles (stretch theory, eq 27 of this work).
mensional prediction of Lee and Tallmadge (1973a) which is
We note parenthetically that this expression for surface velocity (eq 17) was obtained using equations resulting from the x component equation of motion. (See Appendix B.) The cos B is given by geometry as cos 8 =
1
[l
+ (dh/d~)2]~/~
(18)
Substitution of eq 16,17, and 18 into differential eq 12 leads to the desired expression for integration. It is the new, onedimensional approximation used in this paper and called the stretch theory. For integration purposes, it is convenient to place the expression in non-dimensional form.
Theoretical Model (Dimensionless Form) In non-dimensional terms of L and X, eq 9, 11,and 12 becume
L:j dC - 3(L - 1) - T&L3 - 1) Ca dX 2L“ho d (dos) u , dX ds
and (23) Rewriting approximate eq 16 in non-dimensional form leads to
As shown in Appendix A (without any further approximation), the stretch term becomes
Ind. Eng. Chern., Fundarn., Vol. 15, No. 4, 1976
Predicted Profiles Profiles were predicted using new eq 27, together with the same boundary conditions (eq 3,4, and 5 ) and the same numerical technique of Lee and Tallmadge (1975). Results are shown in laboratory coordinate form in Figure 3, for a range of Ca (and thus for a range of speed and fluid properties). In semilog form, the meniscus shapes were all similar to that of Figure 2, having two large linear regions and two small curved regions. As in the previous work, the slopes and intercepts were determined by graphically fit to eq 6 and 7. The resultant parameters are given in Table I. As compared to the gravity theory (Table I), the stretch parameters are similar to the previous work a t Ca of 10+ and lower. At Ca of 10’ and higher, there are large differences between the two predictions. Now consider comparison of the two predictions a t Ca above Table I shows that the stretch equation predicts larger B2 intercepts and thus larger meniscus size. The predicted increase in B2 is about 20?h at Ca of loo and about 300?? a t Ca of 102. The stretch equation also predicts larger slopes, namely an increase of about 60% a t Ca of loo and 400% a t Ca of 102. Specifically, the predicted “stretch” slopes are 1.5,1.43, and 1.35 for capillary number of I, 10, and 100, whereas the “gravity” slopes are 0.93,0.48, and 0.28, respectively, a t these Ca values.
(21)
Equation 21 may be written in simplified form by defining a stretch term F,. Thus
260
Equation 27 is a new theoretical expression (Lee and Tallmadge, 1972;Lee, 1974).The third term on the right-hand side is new. When the stretch term in the pressure jump boundary condition is neglected (so that eq 10 reduces to Ap = uc of the well-known Laplace-Young equation of capillary statics), eq 27 reduces to eq 1.
Comparison of Profiles with Data Stretch theory profiles were compared with data taken previously using two oils of different viscosity. The less viscous oil (Fluid M) had a viscosity of 0.194 N-s/m2 (194 cP) (Lee and Tallmadge, 1974b).The more viscous oil (Fluid B) had a viscosity of 1.31 N-s/m2 (1310 cP) (Lee and Tallmadge, 1973a). Other properties of the oils were similar; a t 26.7 “C, for M and B, respectively, they were densities of 874 and 885 kg/m:j, surface tensions of 31.5 and 32.7 N/km (31.5 and 32.7 dynjcm), and capillary lengths ( a ) of 2.72 and 2.76 mm. Figure 4 shows a typical comparison of both theories with data on a size-sensitive, adjusted-thickness plot, for a Ca near 1. It is seen that the stretch theory thicknesses are closer to the data than the gravity theory-at all heights. This result was also noted in all twelve runs studied; the runs involved both oils over a Ca range from 0.08 to 24. Thus the stretch model is clearly a better prediction of meniscus size than the gravity model. T o investigate predicted shapes, the slopes were also calculated from the models a t experimental Ca. This provides
Table I. Comparison of Profiles Predicted by Stretch Theory of This Work with the Gravity Theory of Previous Work Thick meniscusa Basis
Coating speed, Ca
Stretch Theory Eq 27 (This work)
10-4
Gravity Theory Eq 1
(Previous work)
a
10-3 10-2 10-1 100 10’ 102 10-4 10-3 10-2 10-1 100 10’ 102
Slope Mz Stretch Predictions 176 39 8.7 2.7 1.5 1.43 1:35 Gravity Predictions 176 39 8.7 2.6 0.93 0.48 0.28
Thin meniscusa
Intercept B2
Slope M1
Intercept B1
1300 280 70 17 5.3 2.6 2.6
15.6 7.4 3.7 2.1 1.5 1.43 1.35
10+23 10+9 3000 35 5.3 2.6 2.6
1300 280 70 16 4.3 1.7 0.7
15.6 7.4 3.7 1.9 0.93 0.48 0.28
10+23 10+9 3000 37 4.3 1.7 0.7
See Table IV for film thickness.
a direct comparison with data. The resultant slos are shown in Tables I1 and 111. As shown in Table 11, the top slopes ( M I )indicate that the stretch model slopes are appreciable closer to the data than the gravity model. For example, the median deviation of the five high-viscosity oil runs (Fluid B) is 12% for the stretch model vs. 51% for the gravity case. For the four low-viscosity runs the comparison of median deviation is 10%vs. 33%. Runs taken a t Re of 4 or above were not considered in the median because inertial effects become important a t these Re values; these effects were neglected in these one-dimensional theories. Table I11 shows the lower slope ( M 2 )results. Although the improvement over the gravity model is smaller than with the top slope, the stretch model does offer substantial improvement. For example, the median deviation for the higher viscosity runs is 28% vs. 64% and is 31% vs. 44% for the lower viscosity runs. Overall, it appears that the stretch theory is good for predicting the top slope (M1),but slightly underpredicts on the bottom slope (M2).The stretch theory al,o underpredicts the intercept B2 somewhat and thus the size of lower meniscus. In summary, the stretch theory is appreciably better than the gravity theory in terms of meniscus slope and meniscus size. Furthermore, the improvement is larger a t higher Ca. Figure 5 shows a comparison of four of the low-viscosity runs with the stretch theory, using a laboratory coordinate type plot. These data show that agreement is best a t low Ca, as expected. Thus the theoretical profile is good a t the speed of Ca near 0.08 and 0.2 and fair a t a Ca near 0.5. The largest deviations appear to be midway in height and near the stagnation point. At the highest speed shown (Ca of 2), the experimental meniscus deformation is larger than that predicted by stretch theory. It is believed that this disagreement is caused mainly by the limitations imposed by the one-dimensional model. The two-dimensional model has been shown to agree with data (Lee and Tallmadge, 1974a), but it is very costly and complex; furthermore, only simple one-slope profiles have been tested. The one-dimensional models do the best job to date of predicting complete meniscus shapes. In conclusion, the stretch theory is a useful model for predicting meniscus shapes.
Film Thickness, To Predicted values of To are shown in Table IV. As with the gravity theory, they agree closely (within 20%) with data re-
ADJUSTED MENISCUS THICKNESS,hho,mm
Figure 4. Comparison of both one-dimensional theories with profile data (high sensitivity plot) (see Appendix C) S (eq 27), G (eq l),N (no flow), P (near stagnation point). ported earlier for a wide range of Ca from to 10+l and with previous To theories. However, To is not a very sensitive parameter to test meniscus profiles. In summary, the film thickness values agree in general with previous work. The stretch theory predictions of film thickness are nearly equal to the gravity theory predictions a t a condition of Cu less than 0.1. Film thickness values predicted by stretch theory are slightly less than those predicted by the gravity theory a t high capillary numbers. For example, film thicknesses To are 0.720 and 0.727 by stretch theory and are 0.806 and 0.883 by gravity theory, a t conditions of Ca equal to 10 and 100, respectively.
Discussion The important assumption of quasi-one-dimensional flow in the thick meniscus region is a difficult one to understand. I t was discussed a t length by Lee and Tallmadge (1975). One aspect worth noting is that eq 1 and 27 are both expressions for changes in the derivatives. I t appears that the centerportion deviations found in Figure 5 are due to deviations in the third derivative (L”’),which are largest and most serious Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
261
Table 11. Comparison of Top Slopes with Data Top slope M I
Fluid conditions Run no.
Ca
Re
Gravity theory
Exptl data"
% Dev. from data M I
Stretch theory
Gravity theory
Stretch theory ~
0.083 0.20 0.45 0.95
219M 213M 210M 208M
0.041 0.15 0.47 1.44
2.6 2.0 2.0 1.7
Lower Viscosity Oil Data (Fluid M) 2.38 2.00 1.55 1.86 1.68 1.20 1.52 0.98
2.2 1.7 1.4 1.5 1.7
Higher Viscosity Oil Data (Fluid B) 1.21 1.68 0.88 1.50 0.68 1.48 0.55 1.45 0.44 1.42
-25 -23 -40 -42
-8 -7 -16 -11
Median 33% 0.43 1.17 2.75 5.66 11.89
312B 309B 306B 304B 302B
0.013 0.053 0.18 0.57 1.67
-45 -48 -51 -63 -75
14% -24 -12
+6 -3 - 17
Median 51% 23.90 1.98 4.02
301B 206M 205M
4.76 4.21 12.02
Higher Re Data 1.40 0.38 1.49 0.75 1.47 0.60
1.8
2.5 3.3
-78 -70 -82
12% -22
-40 -55
Fluid B data from Table I of Lee and Tallmad'ge (1973a). Fluid M data from Table 7-2 of Lee (1974).
Table 111. Comparison of Bottom Slopes with Data Fluid conditions
Bottom slope M2
Ca
Run no.
Re
0.083 0.20 0.45 0.95
219M 213M 210M 208M
0.041 0.15 0.47 1.44
Exptl datan
Gravity theory
% Dev. from data M2
Stretch theory
4.5 3.4 2.3 2.1
Lower Viscosity Oil Data (Fluid M) 2.80 2.90 1.85 2.12 1.36 1.78 0.98 1.55
3.2 2.5 1.9 2.0 1.9
Higher Viscosity Oil Data (Fluid B) 1.40 1.78 0.88 1.52 0.68 1.48 0.55 1.45 0.44 1.42
Gravity theory -38 -46 -41 -54
Stretch theory -36 -38 -23 -26
Median 44% 0.43 1.17 2.75 5.66 11.89
312B 309B 306B 304B 302B
0.013 0.053 0.18 0.57 1.67
-56 -64 -64 -73 -77
31% -44 -39 -22
-28 -26
Median 64% 23.90 1.98 4.02 a
301B 206M 205M
4.76 4.21 12.02
2.7 2.5 5.0
Higher Re Data 1.40 0.38 1.49 0.75 1.47 0.60
-48 -40 -71
Fluid B data from Table I of Lee and Tallmadge (1973a). Fluid M data from Table I11 of Lee and Tallmadge (1974b).
in the stagnation point (center) region near L of 2 and 3. The quasi-one-dimensional approximation is weakest near the stagnation point ( L near 2) and use for the thick region is also uncertain. At this stage of the analysis, it appears that eq 27 is a better predictor of slope in the thin region than in the thick region. The normal stress boundary condition considered here involves the stretch term. The two-dimensional form given by eq 10 is well accepted and often cited. Use for one-dimensional models is less frequent, but one successful use is the entrainment model of Spiers et al. (1974). The approximation given by eq 16 is, of course, not precise, but is probably the best to use here as a first approximation. The stretch-term, profile predictions were first presented in 1972 (Lee and Tallmadge, 1972). It is of interest to note that an equiualent way of evaluating 262
-87 -70 -88
28%
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
du,/ds in the stretch term (in addition to the cos method of eq 15) is to use the sine component, which is
The authors recognize that eq 1 and 27 are not the only choices for profiles and that other approximations can be developed. In fact, the authors have tested other equations, but found they result in smaller profiles than those from eq 27. One such example is use of the approximation of du,/ds by dusldx (eq 16 without the cosine term), but this led to smaller thicknesses and profiles. See Appendix 11 in Lee (1974). The stretch theory allows (for the first time) simple and rapid prediction of the size of each of three forces in the normal-stress boundary condition, namely surface tension, vis-
20
Table IV. Predicted Entrainment Thickness
MENISCUS PROFILES IN FREE COATING ’ ’ - - - S (STATIC THEORY)
Film thickness, To
. A P = S C + 2 p (dlq/dS) .ONE
DIM. FLOW
DATA 0
x A t
+
OIL
SPEED C, 0.08 0.20 0.45 2.0
uu
HU/SEC
.
I5 35 77 334
~
p = l . 9 4 FWSE
THICKNESS, h , m
Stretch theory”
Gravity theory ~~
+ MENISCUS
Speed, Ca
m
Figure 5. Comparison of stretch theory with profile data (viscosity of 0.194 N-s/m2) (Oil M/194 cP).
cous stress, and pressure jump. As expected, the surface tension (and pressure jump) terms are dominant terms among the three forces whern the capillary number is less than 0.1; the viscous stress (and pressure jump) are dominant terms when the capillary number is greater than 1. Two examples of these predictions (when Ca equals 0.1 and 10) are presented in Appendix 13 of Lee (1974). The flow (or stretch) term in the normal stress equation tends to be negligible for very small Ca (Ca less than 0.1) and dominant a t very large Ca (Ca greater than 1).Therefore, the stretch theory improves the profile prediction primarily a t higher capillary number. The top slope M I predicted by stretch theory agrees very well with experimental data for the entire capillary range studied. However, the bottom slope Mz and the intercept B Z are still slightly lower values than the experimental data on Ca > 1. This stretch theory, in spite of some deviations from experimental data, has substantially improved its profile predictions (as compared to the gravity theory) by introducing the flow term. The inertial term is neglected in both the stretch theory and the gravity theory. As a result, some deviation occurs a t higher Reynolds numbers, as expected. Some further improvement has also been made using an inertial term and a Re approximation (Appendix 14 in Lee, 1974). The result was an improved profile, but still not a precise center-region description of meniscus size. Other properties of the stretch and gravity theories were presented by Lee et al. (1972). The flow in the upper, thin portion of meniscus is dominated by a velocity in the upward, x-direction; because of this, the one-dimensional approximation is a reasonable assumption. However, the flow on the bottom (thick) portion of the meniscus is a combination of upward and downward flow. In lower regions the velocity is equally important in both vertical, x-direction and the horizontal, y-direction. Because of this, a one-dimensional model certainly cannot describe this meniscus flow property accurately. It is apparent that a twodimensional model is needed to describe this meniscus flow with particular emphasis on the bottom of the meniscus. One form of the latter model has been published (Lee and Tallmadge, 1974a). The five predicted parameters (To,M I , MS, B1, Bz) vary a great deal in their sensitivity to changes in Ca. Most constant is the film thickness To. The slope values ( M I ,M z ) are quite constant a t Ca above 1, but vary a good deal below Ca of 1. There is an effect of Re a t Re above 4. The size intercept Bz is a more sensitive function; it varies with both depth as well as Ca at Ca below 1.At large depths and Ca > 1,Bz was found to be quite constant. The most complex and sensitive parameter is B 1. See Tables 1-111, Lee and Tallmadge (1973a, 1974b), and Lee (1974). The values of the datahheory parameters in Tables I1 and 111 are only approximate due to the simple graphical curve-
~~~
1 x 10-5 0.138 ... 3 x 10-5 0.169 ... 1 x 10-4 0.202 0.202 3 x 10-4 0.245 ... 1 x 10-3 0.293 0.292 3 x 10-3 0.350 ... 1 x 10-2 0.416 0.413 3 x 10-2 0.485 ... 1 x 10-1 0.561 0.555 3 x 10-1 0.624 ... 1 x 100 0.677 0.694 3 x 100 0.705 ... 1 x 10’ 0.720 0.806 3 x 10’ 0.725 ... 1 x 102 0.727 0.883 From eq 27 which is based on a quasi-one-dimensional flow, (eq 8) and the complete, two-term pressure for normal stress (eq 10). From eq 1, which is based on eq 8 and A p = U C .
fitting method used. More precise values would require careful determination of the L (A) boundaries for each of the four regions, together with an analytical fit within each region.
Summary This work was concerned with the following question about predicting profiles free coating: “Does the addition of the stretch (or flow term) in the normal-stress boundary condition improve one-dimensional-model predictions?” In addition, this paper has also provided an answer to a second question: “What, if any, is the size of the improvement in terms of slopes and intercept?” The answer t o the first question is clearly yes for Ca above lo-’ or so. The answer t o the second question is that the size of the improvement (over the gravity theory) is large for Ca above loo or so, as shown in Figure 4, and in Tables I to 111. We also conclude that eq 27 properly predicts the functional form of the L (A) profile and approximates the effect of Ca on the size ( B )and slope ( M )of both the thick and thin regions of the meniscus. Emphasis was placed on the semilog plot (Figures 2 and 4, Tables I to 111) and on the complete profiles, as shown in all the figures. The stretch term in the pressure jump equation tends to be negligible for very small Ca and dominant a t very large Ca. Use of the stretch term improves the profile prediction primarily a t the lower portion of the coating meniscus. Tables I1 and I11 show that a larger deviation in both slopes occurs a t larger Reynolds numbers. The reason for this deviation is that the inertial term is neglected in the stretch theory. The profiles of this work are apparently the second a priori predictions of the complete L(A, Ca) functions. They allow predictions of the quantitative effect of five parameters on profile size. The five parameters are coating speed, viscosity, surface tension, density, and gravitational field. Thus this paper shows that the one-dimensional, stretchflow model is useful for providing approximate but theoretical answers and correlations for meniscus shapes. Acknowledgment This work was supported in part by the Eastman Kodak Co. The authors record, with appreciation, discussions with Charles B. Weinberger regarding the nature of the stretch term approximations. The theory of this work was first preInd. Eng. Chern., Fundam., Vol. 15, No. 4, 1976
263
sented a t the 46th National A.C.S. Colloid Symposium a t Amherst. Mass. in June 1972.
Appendix A. Development of Stretch Eq 25 The purpose of this section is to present some intermediate algebraic steps used to derive eq 25. Combining eq 23 and 24 leads to
F, = 2
~
dX
0
dh)
1 [ l (L’)2]1’* U 1 U , E 2 = 1 - - [3(L - 1) u, 2L
+
+ To21
(A31
Thus
Substituting eq A2 and A5 into eq A1 leads to eq 25 in the text. No approximation is used in this appendix.
Appendix B. Development of Surface Eq 17 The purpose of this appendix is to summarize how eq 17 for surface velocity was derived (Lee and Tallmadge, 1973b). Using the one-dimensional, x -component motion equation for viscous, gravity, and pressure terms, one can obtain dp d y 2 - i (dr
-+pg
+ A($
)
- hy)
--
h, =
]
(1.94 P)(16.3cm/s) = [(0.874 g/cm3)(980cm/s2)
112
=0.192cm
(Cl)
-yDw
yF
+ a‘vn-
~ e n
-
Based on the stretch theory, the predicted film thickness was calculated as follows
T$((Ca) = T$(O.%) = 0.675 (stretch theory)
Ind. Eng. Chem., Fundam., Vol. 15, No. 4 , 1976
+ V,u + C$ (t,K)
(D1)
Equation D1 is a force (or momentum) balance, where y is the density of the interface; w is the velocity of the substrate; F is the external field term; T and T’ are the stress tensors (in each phase); n is the unit vector normal to the surface (in the direction of the T’ phase-the prime phase); u is the surface tension; V, is the surface gradient operator; C$ is the function of surface viscosities, as given by Scriven; and t,K are surface viscosities (shear and dilatational). If we consider the conditions where the density (7)of the interface is sufficiently small (so that the first two terms drop), where the surface tension gradient (V,u) is negligible and where the two coefficients of surface viscosity (K,t) are negligible so that the 4 term drops, then eq D1 simplifies to
r ’ . n - T * n = un.V,n
(D2)
T o make stress more explicit, it is usually convenient to consider separately the components of the stress vector which are normal and tangential to the surface. Our interest is in the normal stress. The normal component of (Ten) is nvrn, so that the normal component of eq D2 is
n * i r ’ . n- n - r * n= -ac
(D3)
Here c is the curvature and is equal to the sum of the principal curvatures ( c l and c 2 in any orthogonal planes containing n). The curvature may also be expressed as a function of the principal radii of curvature ( r : and r 2 ) as follows
thus h , = 1.92 mm
ho(stretch) = T$h, = 0.675(1.92) = 1.30 mm
(C5)
Appendix D. Development of Eq 10 (Normal Stress BC) The purpose of this section is to develop and discuss the normal stress boundary condition given by eq 10, because of its major importance in this paper. The derivation is presented in two sections: first, the general case of a two-fluid interface and second, the specific case of liquid-gas interface for twodimensional flow. The comparable tangential forms are given in section 3. Because these equations were originally used and applied in terms of different notational systems, namely those of Emmons (1958) and Batchelor (1962), section 4 presents a discussion of the original notation used. The notation in this Appendix D is self-contained and differs in some cases from that listed in the nomenclature section of this paper. 1. Two-Fluid Interface (Stress and Deformation). We start with a general expression for interfacial boundary conditions for two “immiscible” fluids in motion. One form has been given by Scriven (1960) as the “dynamic condition” relating differences in stress; his eq 29 is
= an V, n
Appendix C. Calculations for Figure 4 The purpose of this appendix is to report the three film thicknesses used to compare two theories with data. The method of calculation is also given because it is probably not obvious to new readers. Figure 4 is based on experimental conditions, namely the fluid properties of oil M and withdrawal speed for Run 208M. From these parameters the characteristic thickness h , and capillary number Ca were calculated as follows
264
Based on direct measurement for Run 208M, the data value was
Dt
=A
Ah2 2 Equation 17 was derived by eliminating A (and thus pressure) from eq 8 using eq B3. us = u ,
(C4)
Equations C3, C4, and C5 show the three ho values used in calculating the adjusted thickness of ( h - ho),the latter of which are plotted in Figure 4.
The stretch derivative can be evaluated as follows
u = u,
ho(gravity) = ToGh, = 0.695(1.92) = 1.33 mm
ho(data) = 1.20 mm
From eq 17 and 18,we see that
d2u - 1 _
ToC(Ca)= Toc(0.95) = 0.695 (gravity theory)
d Us
d3 -
cos 0 =
Based on the gravity theory, ho was calculated as follows
(‘23)
c = c 1 + cp =
1 -+ rl
1 -
r:!
Now consider the constitutive equations for a Newtonian fluid. The boundary condition given above is in stress form.
Now we change to the rate of strain deformation tensor, E, such as given by Batchelor (1962). As a result, viscosity and pressure are introduced. The normal stress component is given by
+ 2p [ n * E * n- (A/3)]
n * r . n= - p
(D5)
Here the symbol A denotes the rate of expansion, V.V. Substituting constitutive eq D5 into the expression for the normal component, eq D3, leads to
+ 2p ( n . E . n - +) 3
-p
Equation D13 is identical with the desired expression, which is given as eq 10 in the text of this paper. 3. Tangential Terms. For comparison, the tangential expressions for the boundary condition arising from eq D2 are summarized here. Comparable to (D3) and (D6), we have, for two fluids
t - r - n= t - s ‘ e n
(Dl41
t.wE.n = t.p’E‘.n
(Dl5)
Comparable to (D7), (D9), and (DlO), we have for incompressible flow near a two dimension, liquid-gas interface. (D6)
t . E * n= 0
(D16)
2. Liquid-Gas Interface (Two-Dimensional, Incompressible Flow). Where one of the fluids is a gas, the equations simplify because the density and viscosity of a gas are usually much smaller than those of a liquid under normal conditions. For example, Batchelor (1962) states that “. . . as an approximation, the stress a t any point in the gas may be taken as -PO 6,, where p o is the uniform gas pressure”. Therefore, the normal stress boundary condition reduces to
~ 1 =2 -2fiE12
(D17)
=
PO - p =
-GC
A + uc - 2p ( n . E . n - -) 3
(D7)
Equation D7 is an expression describing the air-liquid surface for the three-dimensional case of compressible flow. If we restrict consideration to incompressible flow, then 1 = 0. Thus
+ uc - 2p (n . E * n )
po - p =
(D8)
Now we further restrict the flow to two-dimensional; for this case, the free surface is a stream line described by the stream function. Let us also call the normal direction the 1-direction and the tangential direction the 2-direction. In this case and notation, we have Ell = dul/dxl and XI1
= -p
po-p=
+ %El,
+UC
(D9)
- 2pEll
ac 1
(DlOa) ax I From the equation of continuity, we note for two-dimensional that du1
0 . v = 0 =ax1
au., +-ax2
(Dll)
=0
(D18)
4. Other Notation. The above equations are not new to some people. When we first started, however, they were new to us. The important equation, (D13),was first derived by Lee using only the textbook of Emmons (1958).Subsequently, in his Chapter 6, Lee (1974) described his development using (a) Batchelor (1962) for equations equivalent to (D5) to (D8) and (b) Emmons for equations equivalent (D9) to (D12a). Because of the difference in notational systems used above and in Lee, Batchelor, and Emmons, we note the equivalent equations below for future reference. The expressions of Batchelor (1962) are similar to those given above, but use an indicia1 notation. The expression equivalent to normal stress eq D6 is his eq 3-3-18, which is
= -uc - p
+ 2 p (e,,n,nJ - 4) 3
(D20)
= F‘e’,,t,n,
The expressions of Emmons also utilize indices. Lee (1974) used the expressions for stress tensor components given on page 57 of Emmons (1958) to write the equivalent of eq D19 and D10 as -rnn = p = 2penn = p o
- GC
i)Cl du2 _ - -ax1 ax>
(Dlla)
Combining the equation of continuity result ( D l l a ) into the boundary condition (DlOa) leads to
po - p =
du + uc + 2 p ax
‘>
(D12)
Now we let the tangential direction (2-direction) be denoted as the s-direction. Thus po-p=+uc+2p-
dV
as
(D21)
Tangential stress eq D l 7 and D18 are given by Lee (1974) as -r n t= p e n t = 0
So that
(~19)
Tangential stress eq D15 is given as eq 3-3-17 in Batchelor (1962) as fieLJtini
or po-p=+uc-2p-
El2
0322)
For the special case of steady, two-dimensional flow, the intrinsic coordinates along streamlines are given on page 62 of Emmons (1958).The Lee-Emmons notation is as follows. Let s be the length along the stream line and n the length normal to it. The velocity vector v is specified by its magnitude of velocity along the surface u, and the inclination of 0 of v with respect to the x-axis. In this n, s intrinsic coordinate notation, the normal component of the “rate of strain” (consistent with eq D21) is given by dB
(D12a)
Since u s is a function only of distance s along the streamline (or interface) in two-dimensional flow, the partial derivative du,/ds is identical with the total derivative dc,/ds. Thus
enn = u , -
an
The equation of continuity, in this notation, is given by Emmons (1958) as
c * pv
dpL‘ =5
as + pus -
as an Lee evaluated eq D24 for an incompressible fluid to show Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
265
that
u s = surface velocity, one-dimensional, mm/s (parallel to
Lee then derived eq D12a by substitution of eq D23 and D25 into eq D21. It is also of some interest to note that the curvature c is given by d0/ds.
belt) u , = belt velocity, mm/s u s = surface velocity, two-dimensional, mm/s, eq 10 x = vertical coordinate upward from bath, meniscus height, mm X = meniscus height, nondimensional, n/a y = horizontal coordinate, distance from belt, mm
Nomenclature a = capillary length of liquid, (2u/pg)'/*, mm A = parameter, Appendix B B 1 = upper profile intercept, nondimensional, eq 7 B 2 = lower profile intercept, nondimensional, eq 6 c = curvature of meniscus, eq 13, mm-l C = curvature of meniscus, nondimensional, eq 2 Ca = capillary number, u,(p/u), nondimensional F , = stretch term, nondimensional, eq 23 g = acceleration of gravity h = meniscus thickness a t any point, mm h , = characteristic thickness, (pu,/pg)1/2,mm ho = film thickness, constant thickness region, mm H = meniscus thickness, nondimensional, h/a L = meniscus thickness, nondimensional, h/ho L' = dL/dX (Also L" and L"' are used for higher derivatives) M1 = slope of upper profile, nondimensional, eq 7 M 2 = slope of lower profile, nondimensional, eq 6 p = pressure in liquid phase po = pressure in gas phase Re = Reynolds number, hcu,p/p s = the length along the interface, mm T o = film thickness, nondimensional, ho(pg/puw,)1/2 or ho h c
Greek Letters liquid viscosity, N-s/m? liquid density, kg/m" X = meniscus height, nondimensional, x/ho u = surface tension of the liquid-air interface, N/km
p = p =
Literature Cited Batchelor. G. K.. "An Introduction to Fluid Dynamics," ChaDter 3, Cambridae University Press, 1962. Ernrnons, H W., "Fundamentals of G a s Dynamics," pp 57, 62, Princeton University Press, Princeton, N.J., 1958. Lee, C. Y., "Meniscus Flow Fields", Ph.D. Dissertation, Drexel University, Philadelphia, Pa., 1974. Lee, C. Y., Braccilli, J. J., Tallrnadge, J. A., Annual AlChE Meeting, New York. N.Y., 1972. Lee, C. Y., Tallmadge, J. A . , 46th National Colloid Symposium, ACS, Amherst, Mass., 1972. Lee, C. Y., Tallrnadge, J. A., AiChEJ., 19, 403 (1973a). Lee, C. Y., Tallrnadge, J. A,, AlChEJ., 19, 865 (1973b). Lee, C. Y., Tallmadge. J. A., AiChEJ., 20, 1079 (1974a). Lee, C. Y., Tallrnadge, J. A., h d . Eng. Chem., Fundam., 13, 356 (1974b). Lee, C. Y., Tallrnadge, J. A., ind. Eng. Chem., Fundam., 14, 120 (1975). Scriven, L. E., Chem. Eng. Sci., 12, 98 (1960). Spiers. R. P.. Subbaraman. C. V.. Wilkinson, W. L., Chem. Eng. Sci., 29, 389 ( 19 74).
Receiced for reuieu September 2 5 , 1975 Accepted June 7,1976
Decomposition of Carbon Dioxide in an Induction-Coupled Argon Plasma Jet Yukio Nishimura" and Takeyoshi Takenouchi Research Institute of industrial Science, Kyushu University, Fukuoka 8 12, Japan
The decomposition of C 0 2 was studied in an induction-coupled argon plasma jet at 1 atm in order to obtain information on high-temperature reactions. The major products were CO and 02.As the inside diameter of the probe which was used to quench the decomposed products decreased, the mole fractions of CO and 02 in the quenched gas increased, whereas the gas temperature at the probe inlet was constant within the limits of experimental error. An increase in the flow rate of argon also increased the mole fractions of CO and O2 in the quenched gas. The product distribution was different when a CO-O2 mixture (molar ratio of C 0 : 0 2 = 2:l)was used instead of C02 as the feed.
Introduction In hot or arc plasma chemistry, the quenching rate of hot gas streams has an important influence on the distribution of the final products. One of the techniques used to quench reacting high-temperature gas streams is placing them in contact with a cold wall by, for example, passing them through a small-diameter, water-cooled tube. Baddour and Iwasyk (1962) have studied the reactions of elemental carbon with hydrogen above 2800 K, using this quenching method. A similar method of quenching has been also applied to the synthesis of nitrogen fluorides (Bronfin and Hazlett, 1966), to that of hydrogen cyanide (Bronfin, 1969), and to the decomposition of CO (Nishimura et al., 1974). 266
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
The purpose of this work was to obtain information on the high-temperature reactions in the carbon-oxygen system in the plasma jet and on these reactions during very rapid cooling.
Experimental Section Materials. Argon used in this work was high-purity (>99.99%) cylinder gas which contained N2 (
