flow. For the only hydrocarbon-nitrogen run reported, the pressure drop reduction was approximately 25 to 30%. It is interesting to note that during two-phase slug flow, it has been proposed by Hubbard (1965) and Hubbard and Dukler (1968) that the total pressure drop is composed of frictional losses (at the wall) and acceleration losses at the noise and tail of the slug. From the limited amount of data collected in this study, it is observed (Figure 1) that when 50 wppm Polyox is present in the liquid, the pressure drop reduction tends t o level off at approximately 40’%. These results suggest that the friction losses tend t o level off and thus support the predictions of Hubbard (1965). The contradiction between Hubbard’s results and the results of Dascher (1968) may be easily explained using the data presented here. Conclusions
The experimental data presented here have conclusively demonstrated the effectiveness of dilute polymer solution in reducing the pressure drop in two-phase flow, presumably by reducing frictional losses. For fundamental studies and modeling of two-phase flows, addition of dilute polymer solutions to the flow can be extremely helpful in assessing the pressure losses due t o friction. I n addition, large scale uses of dilute polymer solutions may exist over relatively short distances where polymer degradation is limited. Rhenever the results of these studies become available, more useful correlations may be obtained from theoretical modeling.
Acknowledgment
The authors wish t o express appreciation t o Jack Martin for the collection of data and Esso Research and Engineering Co. for releasing data contained in this paper. literature Cited
Dascher, R. E., h1.S. Thesis, University of Houston, Houston, Tex., 1968. Greskovich, E. J., Shrier, A. L., Bonnecaze, R. B., IND.ENG. CHEW,FCNDAM. 8, 591 (1969). Greskovich, E. J., Shrier, A. L., A.Z:Ch.E. J . 17, 1214 (1971). Hubbard, h i . G., Ph.D. Thesis, University of Houston, Houston, Tex.. 1966. Patterson, G. K., (1969). Toms, B. A,!, “Proceedings of the 1st International Congress on Rheology, Yol. 11, p 135, North-Holland Publishing Co., Amsterdam, 1949. EUGENE J. GRESKOVICH*
Bucknell University Lewisburg, Pa. 17837
ADAM L. S H R I E R Esso Research and Engineering Co. Linden, iV. J . RECEIVED for review January 27, 1971 ACCEPTEDJuly 9, 1971
A Postwithdrawal Expression for Drainage on Flat Plates A theoretical expression for predicting the film thickness, obtained in a two-step process of withdrawal and drainage, is derived by extending the Denson equation. The theory i s verified by measurements of postwithdrawal film thickness of a 19-Poil in the top 5 to 20y0of the 18-cm film over a 7-85-sec range of withdrawal time. The theory is shown to reduce to the Jeffreys theory of drainage at long drain times.
w h e n a plate is withdrawn from a liquid bath and then motion is stopped, the entrained film thins by drainage. This condition of postwithdrawal drainage occurs in many applications, including dipcoating, measuring contact angle, and rinsing to minimize water pollution. Although drainage has been described by Jeffreys (1930) for times long enough t o neglect the initial conditions, the study of postwithdrawal drainage requires consideration of short time effects. Consider a vertical plate immersed in a quiescent bath of a wetting Sewtonian liquid. The solid is then withdrawn from the liquid bath a t a constant speed (U,) for a distance L and stopped (Figure 1). The resultant thin film is then allowed t o drain under the influence of gravity. The location of the liquid-solid-gas junction remains fixed with respect to the solid support. The problem in question is the prediction of film thickness ( h ) as a function of drain time ( t D ) , distance (2) from the top of the film, and other relevant parameters, or h(2,fD). The Jeffreys (1930) expression for film thickness in drainage is
648
Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
As will be shown, eq 1 is inaccurate for the data obtained herein. The deficiency of eq 1 for this case is that it does not indicate the influence of withdrawal speed or other withdrawal variables. I n order to do this one must consider the entire history of the film. This might be done in a t least b o ways, both starting with the liquid a t rest. One method would be to use partial differential equations and describe flow for both types of solid motion, including the proper discontinuities when withdraival was stopped. The second approach, which is used below, splits the process into two parts and uses the first-stage withdrawal result as a boundary condition (initial profile) for the second stage of drainage. I n developing eq 1, Jeffreys (1930) considered the different case of liquid lowering, which involves a fixed solid support and a moving liquid. H e then supposed a film of constant thickness and concluded that the resultant profile would be given by eq 1 after a short time. The assumption of a constant initial thickness is not suitable for Figure 1 films, however. Many other authors have derived and discussed equivalent forms of eq 1, some of which start from more complex equations. For example, see eq 88 of Rhitaker (1966). However,
most authors have not described the initial period of film formation, which is the matter of concern here. In this communication, we shall develop and experimentally verify a n expression which describes the influence of withdrawal variables on bhickness for postwithdrawal drainage.
I
Basic Eq 4
The momentum equation of viscous motion u(y,tn) for a liquid film on a flat plate is given, for negligible inertial and capillary effects, b y
+
d 2u I.( -
dY2
du -
dY
Pg = 0
Figure 1 . Sketch of
(h,tn) = 0
The boundary conditions are the usual ones of negligible slip at y = 0 and negligible shear a t y = h. Integration of the momentum equation leads to the well known Nusselt velocity profile and, from it, the volume flow rate per unit width, Le., flux Q
Q
=
pgha/&
w postwithdrawal
drainage
m Z
0
(2)
From a control volume mass balance for a slice d x thick, we obtain the continuity expression for eq 2
bh _ _ dtD
pgh2 bh P ax
_ _bQbX
(3)
Using the method of characteristics (Hildebrand, 1963), Lang (1969) has shown that eq 3 leads to
(y) dx
dtD dh - - = -
-__
l
DRAIN TIME
t o ,SFC
Figure 2. Effect of drain time (the solid line is Lang eq 8 and the dashed line is eq 1 )
Here t , is the withdrawal time and 2 is the distance from the bath surface. Noting that f,. = L / C , where L is the total length withdrawn, eq 5 becomes
o
and, since h is not constant, to
Equation 5a describes the film profile a t the instant withdrawal stops. Using x = L - Z, the profile is given by
Discussion of Eq 4
Equation 4 is the basic equation of interest here. It has also been derived b y Denson (1969, 1970) using the identity given below and integration btn
(E)
=(E)
-(%)
btn
5
hZ = (pU,x/pgL)
(5b)
We take eq 5b as the initial profile for drainage; thus
tD
h
The fixed junction h = 0 and x = 0 a t the top of the film indicates that h(0,tn) = 0, but this is not sufficient to determine $(h) in eq 4. The functioii $ ( h ) is related to the profile of the film a t the onset of drainage, i.e., when tn is zero. Thus to obtain the solution to eq 4,the initial profile must be specified. Denson did evaluate $ for the case of a large, flat sessile drop on a horizontal surface but did not describe poqtwithdrawal drainage.
Drainage Expression
Now we evaluate the $ function in eq 4 by substituting eq 6 as a boundary condition. Thus
Substituting the $ function into eq 4, n e obtain the deqired h(x,tn) expression, which is (Lang and Tallmadge, 1969)
Initial Profile
The profile a t tn = 0 for the case considered here is that resulting from withdrawal. The Chase-Gutfinger (1967) expressioii for transient withdrawal of a short object is
h2 =
PS
:]
(5)
Equation 8 is a new equation and is the postn-ithdraival theory of drainage. Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
649
Film thickness was measured spectrophotometrically as a function of drain time with a 19.32-P oil having p = 0.887 g/ml and u = 32.6 dyn/cm (Lang, 1969). Figure 2 shows that eq 8 corresponds well with data. Discussion of the Jeffreys Equation
Jeffreys (1930) reasoned that the h ( x , t ~ )function was expected to have the form A
0.2
80
2t.4
1
40 20
(9)
100
TOTAL TIME,
where a iare positive constants. Substitution of this assumption into differential eq 3 leads to the Jeffreys theory of drainage (eq 1). Jeffreys did not mention the additional or $ function term in eq 4.I n hindsight, however, we see that his assumption of eq 9 implies that $ ( h ) = 0 and that the initial profile is
h(x,O) =
co
(10)
It is of interest to note that eq 10 is consistent with the profile shown pictorially in Bird, et al. (1960), in order to describe a physically obtainable initial condition from which Jeffreys’ expression might be approached. However, Bird, et al., chose the liquid lowering case of a fixed plate and a moving liquid, not the withdrawal case considered here. Further Data
Since withdrawal time t, = L / C W ,the speed-explicit eq 8 may be written in the time-explicit form as
1
200
tw+tD,SEC
Figure 3. Effect of total time (the solid line is Lang eq 1 1 )
and Van Rossum (1958), but none has provided a clear insight regarding the variables affecting critical time. Capillary Effects
At low capillary numbers ( U w p / u 50
(12)
Experimental studies of postwithdrawal drainage have been reported previously, such as by Satterly and Givens (1933)
650 Ind. Eng. Chem. Fundom., Vol. 10, No. 4, 1971
RECEIVED for review August 6, 1970 ACCEPTEDJuly 20, 1971 This work was supported in part by National Science Foundation Grant GK-1206. Parag Rele assisted in the preparation of the drawings.
