braic calculation u p to f = F ; in the graphical procedure we followed Henry’s law only a t zero pressure. I t appears reasonable to assign a n uncertainty of 1500 to the values of 9 reported here. If this is a constant quantity as our calculations indicated, this uncertainty will have no effect on the values of the derivatives presented in Figure 9. However, in the calculation of heat of aclsorption b y Equation 16, there is a direct contribution dule to 9 . The magnitude of this uncertainty can be illustrated with the conditions of Example 1, where c = 41.7 cc(XPI’) per gram.
This amounts to about 1% of the heat of adsorption calculated in Example l.
GREEKLETTERS = surface area of adsorbent CY 4 = spreading pressure as usually defined 9 = spreading pressure, redefined in Equation 15 I.( = chemical potential SUPERSCRIPTS -21 = a mixing quantity = per mole of fluid
-
SUBSCRIPTS Ad = adsorption E = ethane g = gas i = component i -11 = methane o = pure i adsorbed a t spreading pressure of mixture s = adsorbed phase T = total
Nomenclature
A c
= constants in Equation 12 = quantity adsorbed, cc(NPT)/g f = fugacity F = constant in Equations 12 and 13 H = Henry’s constant H = enthalpy n = number of moles P = pressure Q = heat absorbed by system R = universal gas constant entropy T = absolute temperature hT = internal energy volume TV = work done b y system x = mole fraction in adsorbed phase z = compressibility factor
literature Cited
American Petroleum Institute, New York, “Technical Data Rnnk - ___ ” 1966 I
s =
Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” Part 11, 2nd ed., p. 595, Wiley, Xew York, 1959. Kidnav. A. J.. hlvers. A. L.. A.I.Ch.E. J . 12. 981 (1966). hlyer; A . L.,‘Ind. Eng. Chem. 60, No. 5 , 45‘(1968). hlyers, A. L., Prausnitz, J. A I . , A.I.Ch.E. J. 11, 121 (1965). Ray, G. C., Box, E. O., Ind. Eng. Chem. 42, 1315 (19j0). Young, D. hl., Crowell, A. D., “Physical Adsorption,” p. 72, Butterworths, London, 1962.
v =
RECEIVED for review September 18, 1969 ACCEPTEDMay %5,1970 Division of Industrial and Engineering Chemistry, lb9th hIeeting, ACS, Houston, Tex., February 1970.
The Drainage of Newtonian Liquids Entrained on a Vertical Surface Costel D. Denson’ General Electric Co., Schenectady, N . Y .
The drainage of Newtonian liquids entrained on a vertical flat plate was studied under conditions when the profile of the film at the onset of drainage i s similar to the one assumed by a large sessile drop formed on a horizontal surface. A theoretical expression for predicting the film thicknesses obtained during drainage was derived by extending Jeffreys’ free-drainage theory to include effects introduced b y the initial shape of the film. Experimental values for the film thicknesses were in good agreement with the theoretical predictions. It i s concluded that Jeffreys’ free-drainage theory can be used to predict the film thickness of a liquid draining from a vertical flat plate when the drainage time exceeds a critical time that depends on the film thickness prior to the onset of drainage, the viscosity, the capillary length, and the distance from the upper edge of the film.
WHEX
a liquid film drains from a vertically disposed surface under conditions such that the gravitational force is much larger than the pressure gradient introduced b y variations in the curvature of the film, the drainage process is referred to as “free drainage” (Tallmadge and Gutfinger, Present address, Major Appliance Laboratories, General Electric Co., Louisville, Ky. 40225
1967). I n this type of drainage the flow rate is unknown and varies wit’h time as well as with distance from the upper edge of the film (Figure 1). The research reported herein is concerned with free drainage of Yewtonian liquids entrained on a vertically disposed, smooth flat plate. The first reported theoretical analysis that dealt with free drainage was made b y Jeffreys (1930). By assuming that Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
443
in conjunction with the continuity equation a- u+
TL 7Y
ax
- =bvo
(3)
by
such that the conditions u = o u
=
t = o ; x >o,y
>o
(4)
0;x> O , t
>0
(5)
0 y
=
(6) are satisfied in the solution of Equation 2 and the conditions
y=O;t>O,x>O
v = o
Figure 1. film
(7)
Profile and coordinate system for draining
acceleration effects were negligible, Jeffreys obtained an expression, Equation 1,
are satisfied in the solution of Equation 3 (Whitaker, 1966). Because the boundary condition expressed in Equation 8 is in the form of a differential equation, an additional restraint-an initial condition-must be specified. Thus, when the profile of the film a t the onset of drainage is specified, h
which was considered to be valid when drainage had proceeded for a long time. I n an effort to obtain an evpression that would be valid for the early as well as the late stages of drainage, Greene (1936), and later Wyllie (1945), extended Jeffreys' analysis by retaining the acceleration term
a,.
in the momenbt tum equation. The expressions obtained by these investigators, however, did not satisfy the continuity equation and consequently were of little value. I n a subsequent treatment, Gutfinger and Tallmadge (1964) corrected the analyses of Greene and Wyllie and concluded that Equation 1 could be used to predict film thickness values after the drainage time had exceeded a critical time equal to 1.95
p-
c
5. These investi-
gators overlooked the fact that two initial conditions must be specified in order to obtain a particular solution to the boundary value problem. Thus, the authors' conclusion t h a t the critical time is independent of fluid properties is erroneous. Experimental investigations concerned with free drainage were conducted by Satterly et al. (1931, 1932, 1933) and Van Rossum (1958). ?Sone of these investigations led to a clear verification of the free drainage theory, nor provided insight with regard to the variables which affect critical time. I n this research we have derived and experimentally verified a theoretical expression which describes the early as well as the late stages of drainage, when the initial shape of the film is the one obtained when a large sessile drop is formed on a flat horizontal surface. Film thicknesses mere measured a t various times for varying values of distance from the leading edge, initial film thickness, and liquid viscosity. Theory
=
h(s), t
=
0
(9)
an expression for predicting the film thickness during the early as well as the late stages of drainage-when Jeffreys' law applies-can be derived. Implicit in the above equations are the following assumptions. Inertia effects are negligible. The axial pressure gradient introduced by variations in curvature of the film is negligible. Tangential stresses resulting from surface tension gradients a t the gas-liquid interface are negligible. These assumptions, as well as the limitations which they impose on the analysis, are discussed in detail below. Equations 2, 4, 5, and 6 can be solved using well-known procedures to obtain the velocity profile, Equation 10.
After substituting Equation 10 into Equation 3, integrating, and using Equations 7 and 8, one obtains
Equation 11 can be simplified by noting that within a few milliseconds after drainage has been initiated, the bracketed term does not differ significantly from unity. Thus,
An expression for predicting the film thickness of a liquid draining from a flat vertical surface (Figure 1) can be obtained b y solving the momentum equation for creeping flow p
444
bu
at
= P
b2U
by2
+ PS
Ind. Eng. Chem. Fundom., Vol. 9, No. 3, 1970
That this is the case can be seen by using typical values for h ( = lO-km), v ( = 100 cm2 per second), and t ( = 10 milliseconds) to evaluate the bracketed term in Equation 11.
Combining Equation 1 l a with the identity
and integrating once more yields
Figure 2. Profile and coordinate system for sessile drop prior to onset of drainage
The undefined function $ ( h ) appearing in Equation 12 is related to the profile of' the film a t the onset of drainage. Thus, b y setting t = 0 in Equation 12, one obtains
x
=
IC.@)
where
e
=
sin-'
(T) h
- h,
(13)
If the initial film profile is the one for a large sessile drop on a horizontal wrface, then $ ( h ) = 2a(cos el
81/2 + a In [ ttan ___ a n e12 ]
- cos e)
When Equation 19 is compared with Equation 13, it can be seen that +(h)is (13')
X complete description o'f the profile of the film during drainage is given by Equations 12 and 13a. I n the limit of long drainage times, Equat,ions 12 and 13a reduce to the result obtained by Jeffreys' Equation 1. I n this regard, it will be useful t'o define a critical time, 7, as the time required after the onset of drainage when the film thickness predicted b y Equations 12 and 13a is just 99% of the film thickness predicted b y Equation I . I n the subsequent development it is shown that the critical time depends on several system parameters. Initial Profile
The initial profile considered in this research is the profile obtained when a large sessile drop i. formed on a horizontal flat plate, as shown in Figure 2. The mathematical form for thiq profile can be derived b y solving Equation 14,
d2h ax2
1
_ -
-h
a2
+ co
which results from the application of elementary principles of capillary >tatic.;. COi n Equation is gireri by
c o = - - ho a2 since h
=
1 ho a t - =
R
for a large sessile drop. Equation 14 can be tjolved in a straightforward manner using the boundary conditions dh dx
=
tan a , h
h=0,
=
x =
to obtain Equation 19, an expression for the initial profile,
x
=
2a(cos el
- cos e)
+ a In
[;;I ~
(19)
Experimental
The liquid film thicknesses which obtain during the course of drainage were esperimentally measured using a technique based on the ability of a given filrii to absorb light. Thus, a moiiochroniatic beam of light of known wavelength was directed a t the draining film, the fractional light, absorption, ZjIo, was measured, and the corresponding film thicknesses were calculated using a calibration curve found to obey Lambert's law, Equation 20.
Apparatus
Light' absorpt'ion measurements were made using the Model 14 Cary recording spectrophotometer manufactured by the Applied Physics Corp. This instrument consists of a light source, a n optical system which produces a monochromatic light beam, a means for selecting a specific wavelength throughout a range extending from 1800 to 26,000 -1,sample and reference compartments, a multiplier phototube, and a recorder. The light beam produced by t,he Cary is rectangular and has a vertical dimension of 5 cm and a horizontal dimension of 0.1 cm. The substrate used to support' the draiiiiiig liquid films in all experimental runs was a smooth, flat quartz plate having dimensions of 2 X 2 x 1/8 inch. This plate was attached to a specially designed metal stand and could be rotated from a horizontal to a vertical position. Thus, by positioning the plate horizontally, a sessile drop could be formed using the liquid to be studied, and, by positioning the plate vertically, drainage could be initiated. The design of the metal stand referred to above was such that it could be placed in the sample compart'ment of the spectrophotometer aiid positioned so that the plane of the incident light beam aiid the plane consisting of the quartz plate and the entrained draining film would be mutually perpendicular. A vertically disposed metal shield containing a horizontal slit 5 cm wide by 0.1 cm high was placed between the incoming light beam and the draining film and was used to reduce the size of the beam impinging on the film from 6 by 0.1 cm to 0.1 by 0.1 em. The inclusion of this shield thus permitted making point measurements of the film thickness a t known distances from the upper edge of the draining film. T o compensate for the decreased intensity of the light beam passing through the sample compartment, a second shield, geometrically similar to the one used in the sample compartment, was placed in the reference compartment . Ind. Eng. Chem. Fundam., Val. 9, No. 3, 1970
445
Table 1. Material
Dioctyl phthalate Aroclor 1254. Piccolyte-mineral oil solutionb
48 9.0
2975 3525
The capillary length, a, was calculated using the relation after accurately measuring the height of rise of a liquid in a precision bore capillary tube of known radius. Viscosities were measured using a Cannon-Fenske U-tube viscometer and a Weissenberg rheogoniometer, the latter being used for the high viscosity liquid.
4.4
3450
Procedure
u2 = u / p g = I/*&-,
Physical Properties p / p , Cmz/Sec
a, Cm
... ... 0,187
X,A
/3,Cm-'
0.585 51.0 261
Chlorinated biphenyl. Piccolyte ( 2 5 ) , P-pinene dissolved in mineral oil t o yield 70% solution. a
b
Calibration
Because the value of the proportionality constant, p, appearing in Equation 20 depends on the chemical structure of a given liquid and the wavelength of light passing through the liquid, this constant had to be esperimentally evaluated for each liquid studied. A small quantity of liquid was placed in a rectangular quartz cell of known path length h and the absorption, Z/IQ was measured. Errors due to the reflection of light a t the airquartz and quartz-liquid interfaces were minimized by using two cells. Thus, one cell filled with liquid was placed in the light path in the sample compartment, while a second cell filled with the same liquid and having a shorter liquid path length was placed in the reference compartment. The actual film thickness a t the measured absorbance then was equal to the difference in thickness of the light paths in the two cells. The calibration cells, made by the Pyrocell Manufacturing Co., consisted of a hollow rectangular container (2 X l/4.X 1/4 inch) open a t one end and a solid rectangular insert which could be placed inside the container. A small quantity of the liquid to be calibrated was poured into the rectangular container and the insert carefully lowered into the liquid, forcing it to fill the annular space between the walls of the container and the insert. By using inserts of different thicknesses, the effective path length of liquid through which the beam traveled could be varied. Typically, the wavelength to be used with a given liquid was selected by finding the wavelength that gave a value of Z/Io = 0.15 for an effective liquid path length of 0.1 cm. After selecting a wavelength, cells with differing path lengths were inserted in the instrument and Z/Zo values noted. Materials and Physical Properties
The physical properties of the liquids used in the course of this study are given in Table I.
When a n experimental run was to be made, the quartz plate was thoroughly cleaned with a commercial cleaning solution containing sulfuric acid, then sequentially washed with distilled water, acetone, and methanol, and subsequently dried with a soft tissue. The quartz plate was positioned horizontally and a small quantity of the liquid to be studied was placed on the plate. The entire apparatus was placed in the sample compartment of the spectrophotometer and enough time allowed to elapse for the liquid to flow out and cover the entire plate. After the desired wavelength had been selected, the recorder was started and the plate rotated through 90' to the vertical position to initiate drainage. When the run was completed, values of I/Zo yere read from the recorder trace a t various times and film thicknesses calculated using the calibration curves. All runs were made in duplicate. Discussion
The theoretical and experimental results obtained during this study are shown graphically in Figures 3 through 6. Esamination of Figures 3, 4, 5 ( L = 2 em. only) shows that the agreement between the theoretical and esperimental results is good-to within @-and suggests that the assumptions in the theoretical analysis did not introduce large errors. The poor agreement between theory and esperiment for the data shown in Figure 5 a t L = 1 em is believed to he due t o the fact that near the upper edge of the film surface tension and film curvature effects are dominant and cannot be neglected, as was done in our analysis. Thus, Bascom et nl. (1967), while studying the behavior of the upper edge of the film draining from a vertical flat plate, found that an extremely thin layer of liquid estends above the apparent edge of the draining film, and that the profile of this estended film is greatly different from that predicted by Jeffreys' equation. These investigators noted that the upper edge of the main body of the film appeared to obey Jeffreys' equation. Probably the most significant feature of this work is associated with the manner in which the critical time depends on the initial film thickness, viscosity, distance from t'he upper edge of the film, and capillary length. Thus, esamination of Figure 3 shows that the critical time decreases with an increase
-
THEORY EXPERIMENTAL h o ~0.169CM ho ~ 0 . 1 4 3C M h o ~ 0 . 2 1 6C M
0 0 0
IC2
IO'
IO0
IO2
lo3
t (SEC)
Figure 3. ness
Effect on drainage characteristics due to changes in initial film thickL = 2.9 CM
446
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
r
I
I
l
-
l
I
I
I
1
I
I
I l l
I
I
I
1
1
--
-
--
-
EXPERIMENTAL THEORY -0- JEFFREYS t THIS WORK JEFFREYS t
I
1
I l l
I
I
I
0 DIOCTYL PHTHALATE
A I
I
I
"
I
~
I l l
I
I
I
I
I
I
-
AROCLOR 1254 , , I
I l l l l
,
I
I
I
I
l
I
l
1
1
1 It2
IO'
102 t (SEC) Figure 5. Effect on drainage characteristics due to changes in distance from upper edge 100
in the initial film thickness. This effect is to be expected, however, since the velocity of the liquid in the film increases with increased film thickness. An example of the reproducibility obtained during the course of this work is also shown in this graph. The effect on critical time due to changes in viscosity is displayed in Figure 4.Increases in viscosity lead to increases in critical time-for example, the critical time for dioctyl phthalate (Y = 0.59 cm2 per second) is about 6 seconds, whereas that for the piccolyte solution ( Y = 261 cm2 per second) is about 20 seconds. The fact that the initial film thickness in these two experiments was not equivalent is inconsequential, since, as pointed out above, the smaller initial film thickness for the dioctyl phthalate would yield a longer critical time. Figure 5 demonstrates the effect of changes in the distance between the upper edge of the draining film and the point of measurement on the critical time. Despite the difference in initial film thickness for these runs, one can deduce that
3. 0.4
loo
I00
IO'
102
103
104
v (CM~ISEC) Figure 6. Dependence of critical time on viscosity, capillary length, and initial film thickness Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
447
the critical time increases as the distance from the upper edge increases. The data suggest that the critical times for the two runs are nearly equal; however, since the critical time is decreased with increased film thickness, the deduction is true. Approximate values for the critical time, T , can be calculated using Equation 21 :
This equation was derived by analytically locating the intersection of the two straight-line segments of the drainage curve --for example, Figure 3. Thus, substitution of Equation 22
Nomenclature
a = capillary length, em g = gravitational constant, cm per sec2
h ho
I = lo=
L = n = R = r = t = u = v = z =
y = LY
6 '6
el X p u p
u
# Conclusions
We have shown how the initial profile of the film affects "free drainage" behavior. I n particular, the time interval between the initiation of drainage and the time when Jeffreys' free-drainage expression is valid varies directly with the viscosity of the liquid, the distance from the upper edge of the film, and the capillary length, and varies inversely with the initial film thickness. Although our results obtain only when the initial profile is similar to the one assumed b y a large sessile drop, the theoretical approach developed herein should be valid regardless of the particular choice for the initial profile. The functional relationship between the critical time and system parameters is expected to be different, however.
448
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
liquid film thickness, em maximum thickness or height of sessile drop, em intensity of attenuated light beam intensity of unattenuated light beam specified distance from upper edge of film, cm integer radius of curvature, cm radius of capillary tube, cm drainage time, seconds film velocity, parallel to plate, cm per second film velocity, perpendicular to plate, cni per second coordinate distance, parallel to plate, cni coordinate distance, perpendicular to plate, cm
GREEKSYMBOLS
p into Equation 1 yields Equation 21 when t is replaced by T. To obtain an expression for the critical time in terms of the capillary length and the maximum film height, ho, one would have to find values of h which hatisfy Equation 19 for given values of z ( = L ) . This has been done numerically (Figure 6).
= =
7
= contact angle, degrees = Lambert's law constant, cm-l
height of rise, cm parameter, defined in Equation 19 parameter, defined in Equation 19 wavelength, A dynamic viscosity, poises kinematic viscosity, em2 per second density, grams per cm3 surface tension, dynes per cm = function, defined in Equation 13a = critical time, seconds
= = = = = = = =
literature Cited
Bascom, W. D., Cottington, R. L., Singleterry, C. R., Advan. Chcm. Sm.. No. - - 26. 355 (19671. Greene, G.: Phil. Maq.-22, 730 (1936). Gutfinger, C., Tallmadge, J. A., A.I.Ch.E. J . 10, 5, 776 (1964). Jeffreys, H., Cambridge Phil. SOC.Proc. 26, 204 (1930). Satterly,, J., Collingwood, L. H., Trans. Roy. SOC.Can. 25, 213 .I
~
~~
\ - - - - ,
,Tfi",
(IYal).
Satterly, J., Givens, H., Trans. Roy. SOC.Can. 27, 143 (1933). Satterly, J., Stuckey, E. L., Trans. Roy. SOC.Can. 26, 131 (1932). ENG.CHEWFUNDAM. 59, Tallmadge, J. A,, Gutfinger, C., IND. 19 (1967). Van Rossum, J. J., Appl. Sci. Res. A7, 121 (1958). 5, 379 (1966). Whitaker, S., IXD.ENG.CHEM.FCNDAM. Wyllie, G., Phil. Mag. 36, ,581 (1943). RECEIVED for review September 11, 1969 ACCEPTED June 8, 1970
