A A
J O H N A. TALLMADGE C H A M GUTFINGER
ENTRAINMENT OF LIQUID FILMS DRAINAGE WITHDRAWAL AND REMOVAL Predicting drainage rates of liquidsfrom solid surfaces has a wide variety of applications. The relevant theories are noted, simplfied design equations are proposed, and areas offiture research are indicated in a unijied description of the processes of drainage, withdrawal, and removal
Figure 1. The interrelationships among the processes of DRAINAGE, WITHDRAWAL, AND REMOVAL provide a basis for the establishment of a general classification system within which the different types of processes can be defined. Using this classification system the available theoretical and experimental investigations for a wide range of geometrical, dynomical and rheological conditions are available.
any applications in engineering practice involve
M the entrainment of liquid films on solid objects.
T o simplify the large number of systems, these applications may be described as problems in drainage, withdrawal, and removal (Figure 1). In drainage the wet support is stationary in space, while the liquid flows downward by gravity. In withdrawal the upward motion of an object from a liquid bath causes entrainment of the wetting liquid. Withdrawal is limited to the case where a part of the solid support remains in contact with the liquid bath. In removal, an object of finite length is entirely removed from a liquid bath so that the entrained film is separated from the bath. These three problems are closely related. In these problems, the mass flow rate or flux of the liquid is unknown. Thus the class of problems considered here is quite different from that of liquid film situations where the flux is known, such as in flow down wetted wall columns and in many of the thin film 9ystems discussed previously (72). The flux is closely related to the entrainment, the film distribution, and the total amount of liquid removed. To the designer, the most important parameter is the flux. Other parameters of interest are the thickness of the film, the surface velocity, and the velocity distribution in the film. The designer would like to have predictions of these quantities as functions of position, time, and physical properties of the liquid. Knowledge of the dynamics of these films is useful in many diverse fields, four of which are coating, cleaning, draining, and lubrication. Some important references on these applications are shown in Table I. VOL 59
NO. 1 1
NOVEMBER 1967
19
Application
A.
Country
Authors
ClassiJca-
Year
liana
Coating Iliithdrawal
I n the past, there has been little communication between groups of workers, between countries, between one application and another, or between empiricists and theoreticians. The most striking example of this lack of communication is the Russian theoretical work and the Dutch empirical studies in the period 1942 to 1958; apparently neither group knew of the other group’s activities. As a result of this communication problem, the literature is considerably fragmented. Some progress has been made in relating drainage to withdrawal, but explicit interrelationships among drainage, withdrawal, and removal have not been described. The communication aspect has improved since 1962. The two most significant developments are the appearance of English translations of Russian books by Levich (30) and Deryagin and Levi ( I 7) and new results obtained in U. S. (77, 57, 63). However, the primary concern of both groups has been the continuous withdrawal problem. Thus, a unified description of drainage, withdrawal, and removal is definitely needed.
Ward
Withdrawal film B.
Removal
4. Rinsing
20
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
1
5. Burets
~
Herschel
6. Vessels
Jeffreys
7. Gas h o l d e r s k 8. Oil strata
1
Walker & USA Tall-
(27)
Bondarenko
11
1959 Removal
USA
Drainage
Ensland
1922 Drainage
USA
1940 M’ithdrawal
USSR
1948 Drainage
D . Lubrication 9. Gears 10. Bearings
Purpose
The primary objective of this review is to describe the approaches, solutions, and interrelationships of drainage, withdrawal, and removal in a unified way. Two further objectives are to place equations in forms useful for design and to discuss their relevance to practical problems from as general a viewpoint as is currently possible. Where feasible, we shall relate the two types of approaches (empirical and theoretical) which have been used. Another objective is to emphasize that there is a wide range of applications where these problems arise. Because of the complexities of the general problem, the early part of this review will be restricted to the conditions listed in Table 11. Each condition is discussed individually in the section, Extensions to Other Systems, with emphasis on relaxing each condition. The wave free restriction in Table I1 does not restrict the discussion of low speeds because waves d o not occur at the speeds and shears involved. The laminar restriction is implied by the wave free restriction since it is generally accepted that turbulent flow only occurs at speeds in excess of those for wave flow. The restriction to simple objects excludes objects of a nonuniform cross section, such as a sphere, The predominant forces present in these processes are viscous, gravitational, and those of interfacial surface tension. I n some cases, inertial or acceleration forces may also be important. The phrase “short object” is used frequently. Whether an object is short or long is a relative question which depends on time as well as length. Thus the relationships among tinie, length, and speed are especially important in drainage, withdrawal, and removal.
I
Cleaning
I Holland I 1951 1 M’ithdrawal
1 g’.ulll I
IBlok ( 2 ) Blok &
Holland
~
E.
’
I
Measurements
L
L
p
j
1948 Withdrawal
p
p
I I I l a M a n y u j theie papers are interrelated. Fur example, remounl tests h a m jreq u e n t l j been used to test wzthdrowal theories.
TABLE I I .
RESTRICTIVE CONDITIONS of
S#ecific Restriction
Restriction
1. \Yave free and laminar flow 2. Simple axially uniform objects, such as cylinders and flat plates 3. Vertical removal 4. Unobstructed flow 5. Homogeneous fluid 6. Newtonian fluid 7. Complete wetting and no slip 8. No shear a t the liquid-gas interface 9. Constant temperature 10. Constant uressure
Flow regime Solid support geometry
Flow geometry Flow geornetry Liquid material Liquid inaterial Surface property Surface property State parameter State parameter
Delaikd Classiflcdion
To distinguish among the many kinds of theoretical and experimental approaches which have been used, further subclassifications are needed. Types of Drainage. We divide drainage into two classes, free and restricted. Let free drainage be a model where the effect of curvature is everywhere negligible (Figure 1A or 2A) and let restricted drainage be those real cases where curvature is not negligible. One influence of curvature is due to surface tension. There are many kinds of restricted drainage. The two examples in Figures 2B and 2C are called immersed and postremoval drainage. By comparison, we see that free drainage is a mathematical model in which surface tension effects at the top and bottom are neglected; thus only viscous and gravitational forces are considered. Cases of restricted drainage are more complex than free drainage, but under some conditions they may be closely related. Types of Withdrawal and Flux. By definition, all withdrawal occurs in immersed systems, but there are three types based on differences in flux, Q. We define variable flux as that where flux varies with time and position, or
bQ -#0 at
and
aQ - # O ax
Constant flux is that condition where flux is constant over a large range of positions:
3 = =0 at ax
The constant flux case is, in a sense, a mathematical model and is somewhat artificial. We define and distinguish it because it has been found useful for discussing the relationships among the three types of withdrawal shown in Figure 3. W e see from Figure 3C or 1B that the frequently observed case of unsteady withdrawal is an example of the variable flux case for very short length objects. The continuous, steady state withdrawal of an infinitely long object is a case of steady state flux, as shown in Figure 3A. I n unsteady withdrawal of the finite length object shown in Figure 3B, the region between the top and bottom menisci (MTand MB)is an example of constant flux. Based on Figure 3, we see that, in general, the condition of constant flux is more restrictive than unsteady withdrawal and less restrictive than continuous withdrawal. T h e Mising Constraints. Consider the region in Figure 38 which lies between the two menisci and assume a negligible change in thickness with x. For constant flux and constant thickness, the Navier-Stokes terms for surface tension and inertial forces vanish and only gravity and viscous terms remain. Thus, for a flat plate, we have the familiar Nussclt (Nu) expression:
for x ,
0.3 5. Interpolation at Go > 0.3 Lengths 6 . Derive and ev aIuatc theory for cylinders of finite radius Lengths 7. Report theory for cylinders of finite radius 8. Compare theory with recise data thin wires and small cylinders of Go < 0.3 9. Interpolation at Go < 0.3
L
introduces an error of only 5 to 10% where Ca < 1. Use of a Go of 10, or above, results in errors of 2% or less. However, the theory for this error calculation was not available until 1966 (60). Data for short cylinders likewise require correction for end effects before comparisons with continuous withdrawal theory can be made. Morey's data were corrected by Van Rossum using the intercept method and are available in graphical (56) and tabuiar form (57, 59). Deryagin corrected for his end effects by a difference method using measurements of removal mass a t two immersion lengths. Another method for correcting for end effects is the White dope method, as described in the cylinders section. It was developed for small cylinders at conditions where the Van Rossum intercept extrapolation was found to be nonlinear. Because Deryagin in effect selected a slope (on a mass versus length plot of two.points) as the corrected value, the Deryagin differ-
T h e gravity corrected theory has been verified within experimental precision ( 5 7 0 ) for all capillary numbers described by Van Rossum and Morey’s data-i.e., from Ca = 0.0001 to 5 (62). Because of the nature of the assumptions, it is believed that the theory may be applied to much smaller Ca than that (0.0001) which has been verified by experiment. However, the validity of the theory above a capillary number of 5 cannot be ascertained without experimental data. Numerical comparison with data indicates that, below a capillary number of 5, the medium speed theory is not valid within 570, Since no data are available above this speed, the medium speed theory is considered a n untested theory. As noted previously, the low speed theory is a special degenerate case of the gravity corrected theory. Since comparison of these theories indicates a 5% agreement a t Ca = 0.03 and better agreement a t lower speeds, we conclude that the low speed theory is valid for Ca < 0.03. Historically, of course, the low speed theory was developed first. It was verified by Deryagin using his own data on a plot (log D 0s. log Ca) which is less sensitive than that of T us. log Ca. The gravity corrected theory has been retested over a range of capillary number from 0.05 to 5 using thickness and entrainment data of other fluids (20). Scattering of the order of 5 to 10% was noted, but no explanation was given. I n summary, we conclude that all continuous withdrawal data have agreed closely with theory for all fluids tested. Relationship with Removal. There appear to be no general theories for removal or its degenerate problems of separation and drippage. The influential parameters are not well understood. Drippage has been observed and accounted for in many experiments (20), but no empirical correlation has been attempted. I t
ence method may be considered a precursor to the White slope method. Of all the data in Tables V, VI, and VII, only those by Deryagin-Titiyevskaya ( 70) and by Gutfinger-Tallmadge (20) were taken to verify theoretical solutions. Although Blok and Van Rossum (2, 3, 56) were unaware of the Russian theoretical work of 1942 to 1945, they were able to develop a n empirical correlation which was general for several fluids and speeds; the correlation agrees closely with the gravity corrected theory. Comparison of Theory a n d Data. One of the first assumptions to be made for constant flux withdrawal is a region in which the film thickness does not vary with height. The existence of this region has been verified u p to the maximum heights available in two withdrawal experiments-those by Van Rossum (57) and those for a 100-cm. height by Gutfinger and Tallmadge (20). Proof of this constant thickness region is further substantiated by comparison of simultaneous measurements of entrainment flow rates and film thicknesses in withdrawal (20,57), using the relationship of Equation 14A. Similar data were obtained in removal tests. The results indicate that either flow thickness or film thickness may be used for verification of the continuous withdrawal theory. The most sensitive plot for comparing thickness data for flat plates with theory is a graph of T us. log Ca. Based on the scatter of corrected data on this plot, the most precise T data are those taken by Morey and the least precise are those of Deryagin. Morey’s data have been cited as verification by Deryagin for the low speed theory, by Van Rossum for his empirical correlations, and by White-Tallmadge for the gravity corrected theory. It is clear that the parameters of Ca and dimensionless thickness properly describe the influence of fluid properties.
Authors Year Measurement method C a range Speed range, cm./sec. Fluid properties range p , poise p , g./cu. cm. Q, dyne/cm. a, cm. Number of fluids
Goucher and Ward ( 7 5 ) 1922 Mass; freeze and weigh 0.1-0.3b 2.5-8
Van Rossum (56) 1958 Thickness; micrometera 0.3-2 0.4-3.2
Gutfinger and Tallmadge (20) 1965 Thickness; micrometera 0.4-5 0.4-80
0.12-0.32 0.8-1.2 Unknown6 Unknown 2 waxes
2.7-24 0.88-0.90 32-34 0.27-0.28 2 oils
0.5-6.2 0.88-1.26 32-62 0.27-0.31 5 aqueous glycerin 1 oil 48
Number of points Belt Material Width, cm. Height, cm. Side effect correction
3
8
Copper 2.5 Unknown Averaging (thinner a t edge)
h, cm.
0.008-0.03
Brass 15.5 Unknown Central 8 cm. for flow rate 0.02-0.2
a
Stainless steel 6.5 100 Midpoint measure (thicker a t edge)
.
0 01-0 .35
Also uerrfied by flow rate measurements of mass scraped off and time.
~
~~~~~~~~
VOL. 5 9
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T A B L E VI. Authors Year Data also given in System used Measurement method Ca range Speed range, cm. /sec. Fluid property range p, poise P , g./cc. U, dyne/cm. a, cm. Number of fluids Number of points Immersion lengths Number Range, cm. Plate widths, cm. Correction method Side effects End effects
UNSTEADY FLAT PLATE DATA
Van Rossum (56) 1958
Van Rossum (56) 1958
Gutfinger and Tallmadge (20) 1965
(57, 59) Withdrawal Thickness; photograph 3-1 1 1.3-3.8
(57, 59) Removal Mass; gravimetric 0.04-0.09 1.3
(77) Removal Mass; gravimetric 0.04-1.3 0.4-12
25-300 0.90-0.95 34-35 0.27-0.28 2 oils
6.2 1.26 59 0.31
L
0.9-24 0.87-0.89 32-34 0.27-0.28 2 oils 4
6 9-1 8 12 (glass)
Unknown 1-20 20 (brass)
3 5 , 1 0 , 20 1 3 . 8 (stainless)
Beveled edge Intercept
Averaged Intercept
Averaged Intercept
r)
appears that there are no predictive equations for removal, separation, or the occurrence of drippage. Probably the most important practical problem here is that of predicting the removal mass. Early studies were reported by the electroplating industry, but were very approximate and were limited to the conditions applicable to that industry. For example, Soderberg (43) reported that for well drained vertical surfaces one might expect a drag-out of 0.4 ga1./1000 sq. ft. and that for very poorly drained surfaces one might expect 2 gal./l000 sq. ft. for vertical parts and 10 ga1./1000 sq. ft. for horizontal parts. Removal masses are related to constant flux withdrawal as noted above. Because of uncertainties in end effects, however, constant flux withdrawal theories cannot be used to predict removal masses with known accuracy. I n one preliminary study, a relative end effect of 50% has been found; this result shows that an assumption of negligible end effect is not generally valid. Prediction of the magnitude of the end effects might be approached by considering them to be due to deviations both at the top (meniscus) and a t the bottom (separation). Although little is known about the factors which influence these effects, there are some indications. For example, it appears that the ratio of end effect mass to the mass predicted by constant flux withdrawal varies inversely with capillary number and directly with Goucher number. I t also appears that some negative end effects occur and that other factors, such as motor acceleration, might be important. Because of the low precision of the end effect data, however, the indications are considered to be quite tentative. Most removal data have been obtained with oily liquids, such as mineral oils, petroleum oils, and glycerin; no anomalies have been noted. There is some indication, however (49, 52), that water behaves paradoxically in removal (the removal mass of water is much 28
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1 (glycerin) 7
larger than one might expect from other considerations). Relationship with Drainage. The free drainage theory has not been satisfactorily verified by drainage experiments. The reasons are not well understood, but may be due to meniscus effects at the top of the film (7, 53). However, several withdrawal implications have been drawn from free drainage using transformations between position and time parameters (6, 7 7 , 20, 57). One valid implication is that free drainage theory can be used to predict the medium speed withdrawal theory. However, the medium speed theory can also be developed directly by neglecting the effect of surface tension ( I , 32). I t is true that the medium speed theory is followed approximately in withdrawal above Ca of 1, but this evidence does not necessarily indicate that drainage in practice follows free drainage theory. A second implication is that the film thickness predicted in continuous withdrawal is related to the film thickness in free drainage at the free surface of the liquid. This has been suggested (32), for example, where one considers the wall to the stationary and the fluid to be receding in free drainage ( I ) , where meniscus effects are neglected. The problem with the analysis is that there is no initial distribution specified. This difficulty is probably the most important deficiency with the free drainage approach (7). The one distribution that has been postulated (7) has not led to any new results. A third implication is the prediction of unsteady withdrawal properties from free drainage equations (6). Presently unverified, the equations d o offer a possible way of predicting restricted drainage after withdrawal. Restricted drainage is easier to state in terms of initial conditions than is free drainage, but more difficult to
describe quantitatively. Fbr example, consider the case of restricted drainage which occurs after removal (Figure 4). Here the initial distribution, which is not uniform at the onset of drainage, will influence subsequent profiles. The nonuniformities are largest at the top and the bottom of the film. Factors influencing postwithdrawal drainage at the top of the meniscus have recently been studied by Denson (7). It seems possible that restricted drainage near the top of a solid object may be tractable by theoretical approaches. The overall film distribution for an object in restricted drainage is probably also sensitive to bottom separation effects. I n one empirical study with water over a limited range of conditions (52),it was found that the mass at the plate bottom represented as much as half of the total mass shortly after removal. Further studies of the effect of time on restricted drainage have been reported for similar flat plate water systems (37) but general conclusions from these limited studies would be premature. Cylinders
The study of cylinders introduces a new parameterthe radius. I n dimensionless form this independent parameter is called the Goucher number. Any general description for cylinders of all sizes would be a description for Goucher numbers from infinity to those approaching zero. The effect of surface tension for cylinders is more complex than for flat plates because there are two principal radii of curvature. For this case, the pressure drop across the interface of a static meniscus is given by: AP _ u
d2s/dx2
[l
+ ( d s / d ~ ) ~ ] ~s[l/ ~ + +
1 ( d ~ / d x ) ~ ] l /(1~9)
TABLE V I I . Author Year Data also given in Method used Geometry used Measurement method Ca range Go range Speed range, cm./sec. Radius range, cm. Fluid property range p , poise P , g./cc. U, dyne/cm. a , cm. Type of fluids Number of points Immersion lengths Number Range, cm.
Stott ( 4 5 ) 1923 (56) Drainage (long times) Inside buret Mass gravity 0.00005-0.0005 1.5 0.55 0.01 1 72 0.38 1 (water)
7
... ...
Where s is the distance from the center line to the edge of the film so that s equals the film thickness plus the radius (s = h I?). Examination of Equation 1 9 indicates a curvature in the vertical plane (first term on the right) and a curvature in the nearly horizontal radial plane. Thus, the second right-hand term is an additional one which arises for cylinders. This equation has been solved using the boundary conditions for a static meniscus, where the first derivative is zero at the top of the meniscus ( x = 6) and the second derivative is taken as infinite a t the free surface ( x = 0). Static profiles, maximum heights, and top curvatures have been given in numerical form (63). For use in the withdrawal theory, we need only the curvature at the top of the static meniscus (Cnd
+
2
em= 1 + 2.4 GooGS5 + GO 4.8
1
Where Go is less than 3, C, by Equation 20 is accurate within 1% and becomes more accurate at smaller Goucher numbers. Use of Equation 20 in the withdrawal theory for cylinders includes the influence of both radial and vertical curvatures of the film. Before 1965, only two limited ranges of cylinder radii were studied. The two special cases were a study of small radii (75) and the infinite radii studied mentioned above for flat plates. Only recently have any descriptions appeared which attempted to be general for all radii: the empirical description in 1965 (50) and the first theory in 1966 (60), the low speed theory. For purposes of this review, however, we start immediately with the more general gravity corrected theory. Gravity Corrected Theory of Continuous Withdrawal. The withdrawal theory for cylinders was developed in a similar fashion to that for flat plates
LARGE CYLINDER DATA
Morey (34) 1940 (33, 39,57, 59) Controlled liquid lowering Hollow cylinder Mass gravity 0.00009-0.03 11 .9-12.3 0.1-1.4 3.25
Deryagin and Titiyevskaya (70) 1945
Van Rossum (56) 1958
(77,591
Controlled liquid iowering
Drainage (long times)
Outside test tube Mass gravity 0 00002-6 13 (est) 0.0008-0.3 3.5
Inside buret Mass gravity 0.001-0.07 1-3 ... 0.24-0.82
0.03-1.1 0.8-1.4 26-43 0.25-0.27 7 oils 2 organics 64
1-900 Unknown 35 0,26(est) 4 oils 5 oil mixtures 19
0.03-0.68 0.83-0.86 28-29 0.27 2 oils
5 3-25
2 2 and 12
7
... ...
VOL. 5 9
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I
using all five assumptions listed previously. I n addition, two more simplifying assumptions were made in the dynamic meniscus region (67) 6. Negligible radial curvature 7. Thin meniscus, ( h / R
