1. art and entertainment
2. visual art and design
3. design

Ind. Eng. Chem. Res. 1992,31, 2222-2231
2222
ditional information to suppport the performance difference.
Nomenclature
A: constant used in eq 3
P: adsorption pressure, Torr
P,: saturated vapor pressure, Torr
R: gas constant, cal/(mol.K)
T: adsorption temperature, K
V: molar volume of adsorbate, cm3/mol
W adsorption capacity, cm3/g
Wd limiting volume of microporous adsorption space, cm3/g
Greek Letters
E:
adsorption potential energy, cal/mol
6: affinity coefficient, cal/mol
7:
conetant used in eq 3
Registry No. C, 7440-44-0; HzO, 7732-18-5.
Literature Cited
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to the Adsorption of Hydrocarbon Vapors on Silica Gel. 2nd. Eng.
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Amett, E. M. Acid-Base Properties of Coals and Other Solids; First
Year Report, 9/1/85-8/30/86, DOE Grant: DE-FG2285PC80521, 1986.
Dubinin, M. M. The Potential Theory of Adsorption of Gases and
Vapors for Adsorbenta with Energetically Non-Uniform Surfaces.
Chem. Rev. 1960,60,235.
Dubinin, M. M.; et al. Adsorption of Water Vapor and Microporous
Structures of Carbon Adsorbents. Akad. Nauk SSSR, Ser. Khim.
1991, 1, 31.
Frusawa, T.; Smith, J. M. Fluid-Particle and Intraparticle Mass
Transport Rates in Slurries. Znd. Eng. Chem. Fundam. 1973,12,
197.
Grant, R. J.; et al. Adsorption of Normal Paraffii and Sulfur Compounds on Activated Carbon. AZChE J. 1962,8,403.
Grant, R. J.; et al. The Effect of Relative Humidity on the Adsorption of Water-Immiscible Organic Vapors on Activated Carbon.
Fundamental of Adsorption; Myers, A. L., Belfort, G., Eds.; Engineering Foundation: New York, 1984; p 219.
Greenspan, L. Humidity Fixed Points of Binary Saturated Aqueous
Solution. J. Res. Natl. Bur. Stand., Sect. A 1976,81A, 89.
Gregg, S. T.; Sing, K. S. W. Adsorption, Surface Area and Porosity;
Academic Press: New York, 1982.
Konnert, J. H.; D’Antonio, P. Diffraction Evidence for Distorted
Graphite-like Ribbons in an Activated Carbon of Very Large
Surface Area. Carbon 1983,21, 193.
Levin, J. 0.;Carleborg, L. Evaluation of Solid Sorbents for Sampling
Ketones in Work-Room Air. Ann. Occup. Hyg. 1987,31,31.
Lewis, W. K.; et al. Pure Gas Isotherms. Znd. Eng. Chem. 1950,42,
1326.
Manes, M. The Polanyi Adsorption Potential Theory and Its Applications to Adsorption from Water Solution onto Activated
Carbon. Activated Carbon Adsorption of Organics from the
Aqueous Phase; Suffet, I. H., McGuire, M. J., Eds.; Ann Arbor
Science: Ann Arbor, MI, 1980; Vol. 1.
Marsh, H.; et al. Carbons of High Surface Area. A Study of Adsorption and High Resolution Electron Microscopy. Carbon 1982,
20, 419.
Marsh, H.; et al. Formation of Active Carbons from Cokes Using
Potassium Hydroxide. Carbon 1984,22, 603.
Naujokas, A. A. Spontaneous Combustion of Carbon Beds. Plant/
Oper. Prog. 1985, 4 (2), 120.
Neely, J. W. A Model of the Removal of Trihalomethanes from
Water by Ambersorb XE-340.Activated Carbon Adsorption of
Organics from the Aqueous Phase; Suffet, I. H., McGuire, M. S.,
Eds.; Ann Arbor Science: Ann Arbor, MI, 1980; Vol. 2, p 417.
Neely, J. W.; Isacoff, E. G. Carbonaceous Adsorbents for the
Treatment of Ground and Surface Waters: Marcel Dekker: New
York, 1982.
O’Grady, T. M.; Wennerberg, A. N. High Surface Area Active Carbon. Presented at the 187th National Meeting of the ACS. St.
Louis, MO, April 8, 1984.
Reich, R.; et al. Adsorption of Methane, Ethane and Ethylene Gases
and Their Binary and Ternary Mixtures and Carbon Dioxide on
Activated Carbon at 212-301 K and Pressure to 36 Atm. Znd. Eng.
Chem. Process Des. Dev. 1980, 19, 336.
Rohm and Haas Company. Ambersorb Carbonaceous Adsorbents;
Philadelphia, PA, 1980?
Wigmans, T. Industrial Aspects of Production and Use of Activated
Carbons. Carbon 1989,27, 13-22.
I
Received for reuiew December 3, 1991
Accepted May 14, 1992
Interfacial Viscosity in Viscous Free Surface Flows. A Sample Case
Maria D. Giavedoni and Fernando A. Saita*
Zmtituto de Desarrollo Tecnoltigico para la Industria Quimica (ZNTEC), Universidad Nacional del
Litoral-Consejo Nacional de Inuestigaciones Cientificas y Tgcnicas, Giiemes 3450, 3000 Santa Fe, Argentina
This work presents a technique for introducing the surface viscosity terms into the numerical codes
employed in the analysis of viscous free surface flows,namely, coating flows. Also, it qualitatively
describes the effects produced by the surface viscosity on the interfacial dynamic behavior of an
incompressible Newtonian liquid in a steady-state flow. The teat problem chosen is the flow occurring
in the slot coating process, particularly in the region where one end of a curved free surface (a
meniscus) detaches from a stationary solid wall while the other end contacts a moving substrate.
The governing equations are solved by the finite element method, and a limited set of computed
predictions is presented to illustrate how the surface viscosity affects the flow domain.
Introduction
There exist a wide variety of processes where the
fluid-fluid interfacial dynamics plays an important role,
and, without doubt, coating processes are relevant among
them.
To a first approximation, these processes can be analyzed considering a two-dimensional steady flow of an
incompressible liquid which is mathematically represented
by the equation of conservation of h e a r momentum-the
* Author to whom correspondence should be addressed.
Osaa-5sa5/92/2631-2222$03.00/0
Navier-Stokes equation for a Newtonian fluid-and also
by the equation of conservation of mass. The mathematical description is completed with the appropriate
boundary conditions; among them are extremely important
those which are applied to the free surface and relate the
interfacial properties (physical and geometric) to the
difference between the fluid tractions on either side of the
interface.
A general formulation of this boundary condition was
presented by Scriven (1960) (see also Aris (1962), and, for
a more general treatment including mass transport, see
Slattery (1990)) who derived the equation of motion for
0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992 2223
a Newtonian surface fluid and then connected this equation with the flows occurring in the adjacent bulk phases.
By so doing, various interfacial properties enter into the
formulation of the dynamic boundary condition: the
surface tension, the surface density, and the surface viscosity.
Admittedly, the surface tension is the most relevant
interfacial property and is the only one that counts for
systems in equilibrium and also in many flows with free
boundaries. However, there exist occasions when the dynamic behavior of an interface cannot be explained on the
basis of surface tension alone, viz., the stability of films
under the presence of surface active agents (Whitaker,
1964,1966; Sparling and Sedlak, 1989; Ruschak, 19871, and
additional interfacial properties must be accounted for.
Surface active agents are employed in many industrial
coating processes. As an example, we might mention the
'emulsions" used in the manufacturing of photographic
films; there, the liquid to be deposited-actually a dispersion of particles of silver halides in gelatin-contains
additives such as antifoggants, hardeners, stabilizers, and
surfactants. Since surfactants reduce the surface tension,
they work as spreading agents facilitating the coating operation; however, they also alter the interfacial properties
which in turn may affect the shape and location of the free
boundaries. In addition to viscous properties, the interface
might show elastic properties which are not necessarily
related to the interfacial structure but to the presence of
a nonhomogeneous concentration of surfactants along the
interface caused by a dynamic interfacial state (Malhotra
and Wasan, 1987);i.e. the elastic behavior is of a compositional nature.
When a coating liquid detaches from the coating device,
it is accelerated to match the speed of the moving
boundary; that is, the free surface suffers elongational
strains, and the local concentration of surfactants diminishes. Consequently, certain amounts of the surface active
solute should be transferred-from the bulk and/or from
the interface-in order to restore the interfacial equilibrium; however, in most cases this equilibrium is not
reached, and the system steadily operates in a nonequilibrium condition. A rigorous analysis of this problem
requires that one solves the combined fluid mechanics and
mass-transport processes in an unknown domain whose
limits depend on both the shape and the location of the
gas-liquid interface; additionally, the normal and the
tangential stress balances on the interface will present
locally varying coefficients since the interfacial properties
(surface tension, viscosity, and elasticity) will depend upon
the local concentration of surfactants.
The problem just described is rather complicated and
still awaits solution; nonetheless, under certain conditions
we can attempt an approximate solution. The changes in
the interfacial concentration produced by the dynamic
state of the interface trigger several mechanisms of mass
transport that tend to reestablish the state of equilibrium.
Those mechanism are interfacial diffusion, bulk diffusion
followed by interfacial adsorption, and bulk convection
followed by interfacial adsorption.
Since the flow in the coating bead is intense, the mass
transport by convection in the bulk must be much more
important than the diffusional processes; thus, we might
expect a homogeneous concentration of solutes in the bulk
phase and a concentration jump in the thin interfacial
sublayer where the adsorption step takes place. At this
point we can consider two limit cases: (i) low concentration
of surfactants, small rate of interfacial adsorption, and
large values of interfacial acceleration; (ii) high concen-
Figure 1. Schematic representation of the slot coating and the
coordinate system adopted.
tration of surfactants in the coating liquid, high rate of
interfacial adsorption, and relatively small interfacial fluid
acceleration.
In the first limit (case i) the variation of concentration
of the adsorbed species should be nearly proportional to
the interfacial deformation (see Whitaker (1966)),while
in the second limit (case ii) the interfacial concentration
of adsorbed species must approach the equilibrium value
(see Jaycock and Parfitt (1981)).
In this work, we study the simpler case (i.e. constant
concentration of surfactants) and we establish the procedure for introducing the interfacial properties, viz., the
interfacial viscosity, into the finite element coating flow
simulators. To this end, we analyze the flow occurring in
the rear part of the slot coater; this device is sketched in
Figure 1,and it consists of a solid die through which the
liquid is extruded at a metered rate.
Three different quasi 2-D flow regions can be identified
in the gap shaped between the coating die and the substrate; one is the central or metering zone where the liquid
turns direction as it enters the flow channel. The other
two regions contain gas-liquid interfaces. One of them is
the film forming zone which begins at a short distance
upstream from the exit-where the flow is still 1-D-and
ends far downstream where the plug-flow condition is
reached asymptotically; the flow in this zone has been
extensively analyzed by Saito and Scriven (1981). The
second one occurs at the other end of the coating device
and is delimited by the Couette-Poiseuille flow that develops upstream of the metering zone and the meniscus
that bridges the gap existing between the die face and the
substrate.
The rear meniscus, and its neighborhood, is the region
we chose to study the changes introduced by the surface
viscosity; we restrict our observations to the cases where
the meniscus not only is pinned at the rear edge of the
coating die but also contacts the substrate with an angle
OD (measured through the liquid phase) equal to 180'.
The numerical technique we employed was formerly
developed by Ruschak (1980); additional improvements
were later incorporated by Saito and Scriven (1981) and
by Kistler (1983). This technique uses the finite element
method combined with a suitable parametrization of the
free surface; a key feature is the manner in which the
interfacial boundary conditions enter into the numerical
code, they are easily introduced even if the surface viscosity
terms are considered.
In the section that follows we present the equations and
boundary conditions to be solved; we also give a short
summary of the numerical procedure employed. The way
2224 Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992
the surface viscosity terms are treated in the dynamic
boundary condition is particularly emphasized.
Mathematical Formulation
Governing Equations. We consider the steady, isothermal flow of an incompressible Newtonian fluid in the
domain shown in Figure 1. The dimensionless governing
equations are the continuity equation
v-v = 0
(1)
y Dw
--F=
Dt
+ + e)V*,(V*,*w)+ e[2H(w - mew) +
X w) + 2(n X V*,n X n).V*,(n.w)] +
n[2%u + 2 % ( +
~ e)V*,*w- 2 4 1 X V*,n X n):V*,w]
V*,u
(K
n X V*,(n.V*,
(8)
and the Navier-Stokes equation
(Re)v-(Vv)- V.T = 0
(2)
+
where T = -PI + [Vv ( V V ) ~ ]Since
.
the characterisic
length is small, the effect of gravity can be neglected;
therefore, the body forces are not included in eq 2.
The foregoing equations were put into their dimensionless form using the following scales: the substrate
speed (v) and the clearance between the die face and the
substrate (d). Pressure and stresses are measured in units
of the characteristic viscous forces ( p U / d ) ,Re = pUd/p
is the Reynolds number, and p and p are the density and
the viscosity of the liquid in the bulk phase, respectively.
Boundary Conditions. The mathematical representation is completed with the appropriate boundary conditions; they are as follows:
(a) On solid surfaces, the nonslip condition holds; then,
on the stationary solid wall the components of the velocity
vector are
u=o, u=o,
y=l, O I X I X S
(3)
and on the moving web they are
u = - I , u = 0,
y =o, 0 I x I
Interfacial Dynamic Boundary Condition. A general
formulation of the interfacial dynamic boundary condition
was presented by Scriven (1960)who derived the following
equation:
XD
(4)
(b) Far enough from the meniscus, the flow is unidirectional, entailing a parabolic velocity distribution that
must accomplish with a zero net flow rate and with the
boundary conditions just stated; thus,
u = -3y2+ 4y- 1, u = 0,
x = 0, 0 I y I1
(5)
(c) For the gas-liquid interface we have to apply
boundary conditions at the points where the meniscus
intersects the solid boundaries; either the location of the
contact line or the value of the contact angle has to be
prescribed there. In this work the gas-liquid interface is
aasumed to be pinned at the rear edge of the die face (xs,l)
and to contact the moving substrate at a given angle OD,
i.e.
n,m COS 8D
x xD, y = 0
(6)
where n, and n are the unit normal vectors to the substrate and to the interface, respectively. For the sake of
simplicity we chose 8D = 180° so that the free surface
velocity is continuous at the dynamic contact line and
match- the web speed smoothly. Though the actual value
of the dynamic contact angle is not known, and it might
change depending on the dynamic conditions and on the
physical properties of the phases that come into contact,
some authors adduce evidences in support of our choice
(Pismen and Nir, 1982; Mues et al., 1989).
Two additional conditions must be specified at the free
surface: one is the kinematic condition which expresses
that no mass is transferred across the interface, that is,
v.n = 0
(7)
and the other is the dynamic boundary condition which
relates the interfacial properties to the tractions on either
side of the free surface.
where w is the interfacial velocity, K is the coefficient of
dilatational surface viscosity, e is the coefficient of shear
surface viscosity, H is the total curvature, 7f is the mean
curvature, u is the surface tension, and y is the surface
density. The external forces acting on the interface are
represented by F n and t are the normal and tangent unit
vectors to the free surface, and V*, is the surface gradient
operator. It must be remarked that eq 8 was derived on
the assumption that the coefficients of surface viscosity
are constant.
In this work we make three additional simplifications:
the surface tension is constant, the surface density is small
enough so that the inertia terms can be neglected, and the
gas phase is inviscid and inertialess. Thus, the left side
of eq 8 simplifies to
Dw
y- F = T-n + &n
(9)
Dt
where Tm is the traction vector in the liquid side of the
interface and aG is the gas pressure which is set equal to
zero.
The right side of eq 8 also simplifieswhen one considers
the characteristics of the flow we are studying. Since the
free surface is a cylindrical interface, the total curvature
(H)
is zero, and because the flow is assumed to be 2-D, the
interfacial velocity w can be written as wt. Consequently,
it is straightforward to show that
(n X V*,n X n):V*,w = 0
(n X V*,n
n
X
X
n).V*,(n.w) = 0
V*s(n.V*sX w) = 0
(10)
and eq 8 (now in dimensionless form) reduces to
Ca = p U / u is the capillary number, 6 = (K + c)/pd is the
dimensionless surface viscosity, and dt/ds is the mean
curvature with ds being the differential arc length along
the interface.
Equation 11 is the interfacial force balance that completes the set of boundary conditions to be applied to the
present problem; notice that the frequently used version
of the dynamic boundary condition is recovered when the
surface viscosity is neglected.
Numerical Procedure. The system of equations was
discretized using the finite element method; this method
combines subdomains-where simple basis functions are
defined-with the Galerkin method of weighted residuals.
Since the application of the finite element method to free
surface flow problems has been thoroughly explained by
Kistler and Scriven (1984), only a short summary of the
fundamentals is given here.
When the governing equations (1,2, and 7) are weighted
with suitable trial functions (a’s and W s ) and are then
Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992 2225
integrated in the flow domain (eqs 1 and 2) and along the
free surface (eq 7), the following vanishing residuals result
R i = sII.’V.v ds2 = 0
(12)
n
RMk = &(ak(Re)v.Vv
W
+ Vak.T) ds2 X a k ( n - T ) ds = 0 (13)
RKi = X@’([=O,&-n
ds = 0
The boundary integral appearing in eq 13 arises from the
second term of eq 2 when the divergence theorem is applied to the weighted residual of the Navier-Stokes
equation.
In the present problem the weighted residuals of the
momentum equation are discarded on every portion of the
boundary (except for the gas-liquid interface) in favor of
the essential boundary conditions given by eqs 3-5.
Therefore, we concentrate our analysis of the boundary
integral of eq 13 on the nodes pertaining to the free surface;
in such a case the vector traction (Tsn) is substituted by
the dynamic boundary condition ( l l ) , yielding
1 h - T ds =
1
+ V,(V,.W)
t
(14)
1
ds (15)
Following the procedure suggested by Ruschak (19801, the
right side of eq 15 is integrated by parts. Recalling that
the flow is assumed to be 2-D,
the last term in the second
integral can be written as
d
V,(V,.W) = -(V,*W)t
(16)
ds
then, once the surface divergence theorem is applied, eq
15 transforms to
The last two terms in eq 17 are the interfacial tangential
forces acting at both ends of the free surface, i.e. at the
static contact line and at the dynamic contact line.
However, those terms need not be computed because at
the contact lines the weighted residuals (13) are replaced
by essential boundary conditions (eqs 3 and 4). Additionally, if k is the node at the contact line, the basis
function akis zero at any other free surface node.
The preceding paragraphs show that the weighted residuals employed in analyzing free surface flows when
surface tension is the only interfacial property considered
can be readily extended to include the surface viscosity
effects. Actually, the weighted residuals of both the
kinematic condition and the continuity equation remain
unaltered, while a new term appears in the boundary integral of the weighted residual of momentum; this new
term accounts for the surface viscosity forces.
The introduction of surface viscosity terms into the
governing equations of the problem produces no additional
features worth mentioning; thus, in what follows we briefly
describe details of the numerical procedure employed in
this work. (For a more exhaustive description see Kistler
(19831.)
The flow domain is tesselated into quadrilaterals;
nine-node biquadratic basis functions 3and fourth-node
bilinear basis functions \ k j are defined in each quadrila-
X
Figure 2. Spine parametrization employed.
teral. Those basis functions are used to expand the velocity and the pressure fields:
K
v = Cvk@k([,o)
k=l
J
p =
CPW,,)
(18)
j=1
where vkand p’ are the nodal values of the velocity vector
and the pressure, respectively; K is the total number of
velocity nodes and J is the totaJ number of pressure nodes.
The basis function 9’([=0,~)employed in the kinematic
weighted residual is a one-dimensional simplified version
of the biquadratic basis function ah([,?)
when [ = 0. The
polynomial basis functions are built in fixed Cartesian
coordinates (5,~) on a standard square. Every quadrilateral
element is mapped onto the standard square using an
isoparametric transformation (Strang and Fix,1973). The
free surface is represented by a curve of [ = 0; consequently, all geometric properties of the free surface can
be easily obtained from the transformation. Thus, the
normal and the tangent unit vectors to the free surface are
y,i - x , j
x,i y j
+
The free surface is parametrized by conveniently located
‘spines” which are defined (see Figure 2) by a base point
xBi and a direction e’ (Kistler and Scriven, 1984). The
distribution of the base points located on y = y1 and x =
x1 is chosen so as to refine the mesh near the dynamic
contact line. Figure 2 shows that some spines can change
direction by pivoting on their base points; when the dynamic contact line displaces, the spine passing through X D
turns about its base point and changes the angle it defines
with the y-axis. The direction vectors e’ of the remaining
spines which are free to rotate depend upon the value of
this angle, which is an unknown of the problem to be
determine together with the velocity and the pressure fields
and the interfacial coordinates.
The tesselation employed is shown in Figure 3; it has
56 quadrilateral elements, 250 velocity nodes, 70 pressure
nodes, and 11 nodes which are used to determine the free
surface location. The isoparametric transformation
changes the real domain into the computational domain
depicted in Figure 3. It should be noted that the computational domain becomes open along the segment D E
i.e. this segment appears duplicated: therefore, the node
numbering along DE must be duplicated too in order to
preserve both the total number of nodes and the original
connectivity among the elements.
The final set of nonlinear equations was simultaneously
solved using Newton’s method and a quasi Newton iteration, The latter was introduced when a suitable measure
of the difference between two consecutive Newton’s steps
was smaller than a certain prescribed value (lo-*in our
case); we found this strategy to be more cost effective than
2226 Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992
D
C
G
10
15
2.0
x
Figure 4. Effect of Reynolds number on free surface profiles.
A
C
D
A
F
/
tl
F
/
m/
/
G
Figure 3. Finite element tessellation of the flow domain and the
corresponding computational domain.
using full Newton iteration exclusively. The system of
equations was solved using a computer, VAX 11/780,
where a complete Newton iteration took 3 min of CPU
time while a quasi Newton step needed 9 s of CPU time.
Also, with the purpose of reducing the computing time,
the Gauss elimination was performed taking into account
the great number of zeros appearing in the Jacobian matrix.
Newton’s method needs good initial estimates to achieve
convergence; once a solution is obtained, it is easy to
compute solutions for other values of the dimensionless
parameters using continuation methods. Here we employed zero-order continuation when either the Reynolds
number or the dimensionless surface viscosity was changed
and first-order continuation when the capillary number
was varied.
The convergence criterion employed was that the norm
of the difference between two consecutive vector approximations should be equal to or smaller than 1 O P
Results
The results to be presented in this section were obtained
for constant interfacial properties: i.e. we assume that the
conditions leading to the limit case ii hold; therefore, the
validity of the predictions-which illustrate the changes
produced when the characteristic parameters (Re, Ca, and
6 ) are modified-should be considered under these restrictions.
The discretization employed was shown in the previous
section and, we believe, is suitable for our purpose which
is a qualitative description of the behavior of the system.
A more accurate picture would require that we solve the
problem with a finer mesh, particularly near the dynamic
contact line.
When surface active agents are present, the interfacial
viscosity can reach values close to 1 superficial P (dyn
s/cm); however, in most of the cases the concentration of
surface active agents determines values of the surface
viscosity smaller than 0.01 P (Davies and Rideal, 1963).
Considering the typical operating values used in slot
coating, the values of Re and Ca explored in this work
imply surface viscosities smaller than 0.01 P, except for
the case when the capillary number is 1; in that case pa z
0.015 P.
In order to solve the equations and boundary conditions
that govern the flow in the rear part of the slot coater, a
reference pressure (Po)has to be set at some location in
the flow domain; for that purpose we fixed Po at x = 0, y
= 1. Since we consider the region under study to be independent of both the metering and the film forming
zones, the values of the reference pressure can be arbitrarily chosen as long as the resulting gas-liquid interface
can be suitably resolved with the discretization adopted.
The value of Po was chosen to be about 12 because this
would be the pressure drop between x = 0 and x = x s if
the fully developed l-D flow of Q = 0 were to extend up
to x s and the capillary pressure were neglected.
Figure 4 shows that Po= 12.17 is appropriate since the
location and the shape of the free surface are properly
solved at different values of Re; also, it shows for the
nonviscous case (6 = 0) that the interface moves out of the
slot as the inertia forces grow. This movement increases
the value of the static contact angle (6,) and displaces the
location of the wetting line toward greater values of the
x-coordinate.
In the results that follow we analyze how the interface
and the bulk phase are affected by the surface viscosity.
First, we concentrate our study on the neighborhood of the
dynamic contact line; then, we extend our observations to
the whole free surface and to the velocity field in the flow
domain.
Analysis near the Dynamic Contact Line. The way
the dimensionless surface viscosity enters into the mathematical description of the problem is given by eq 17. This
equation shows that 6 is multiplied by the surface divergence term which should be significant in the region where
the liquid accelerates to match the web speed. Thus, we
expect the presence of surface active agents to produce
important changes on the free surface near the dynamic
contact line.
In order to illustrate those changes we solved the
problem for certain values of Re, Ca, and 6 ; those values
define a domain of dimensionless parameters where this
coating device is usually operated. Thus, the Reynolds
number was varied between 10 and 70, the capillary number was varied between 0.16 and 0.50, and the range of
dimensionless surface viscosity was 0-1.
Figure 5 shows the location of the dynamic contact line
vs 6 for different values of the Reynolds number and the
capillary number. A t low values of Re or Ca, the dynamic
contact line steadily moves into the slot as the surface
viscosity increases. However, when the dimensionless
Ind. Eng. Chem. Res., Vol. 31, No. 9,1992 2227
I
I
c
148
t
xD
1.85
1.80
1.4
1I
ao
1
,75
-
170
-
165
-
c
46
4
1
Po
- 136
=io
P o = 12.171
160
0.5
2)
QS
i44
I I !
'
,,I
"I'
124
10
Figure 5. Effect of dimeneionlese surface viscosity on the location
of the dynamic contact line for different values of Re and Ca.
parameters reach some intermediate values (Re = 45 or
Ca 0.32),the movement of xD becomes nonmonotonic;
i.e. the dynamic contact line moves out of the slot first,
but then the direction of the motion is reversed. This
behavior is clearly depicted by the curve of Re = 60 and
the m e of Ca = 0.50; both show a maximum value of xD.
The maximum just described occurs at some value (a*)
which depends on the values of Re and Ca; that is, if 6 <
6*(Re,Ca)the value of xD increases as 6 increases and the
opposite happens when 6 > G*(Re,Ca).
Figure 6 presents the movement of xD vs 6 when Re =
70; from a comparison of this curve with the curve of Re
= 60, our observation is that at greater values of the
Reynolds number the maximum displacement of the dynamic contact line occurs at greater values of the surface
viscosity. Thus, for the curves we referred to above, the
fact is that 6*(Re=70) > 6*(Re=60); a similar situation
comes to pass with the capillary number.
The existence of a maximum in xD depends not only on
the parameters just examined but also on the value of the
reference pressure (Po). Assuming that the distance between the metering zone and the meniscus (in the rear part
of the coating device) is large enough so that a CouettePoiseuille flow develops between them, the reference
pressure Po is the only variable that transfers information
toward the region under study. For example, a change in
the reference pressure might indicate that the metered flow
rate has been changed, thus, the shape and location of the
rear meniscus and the flow field near it would change
accordingly. Figure 6 shows that the free surface behavior
is strongly affected by the reference pressure to such a
point that the nonmonotonic displacement of the dynamic
contact line practically disappears when Po is reduced.
Although the results presented do not elucidate the
'
mechamam
by which the surface vi&oeityaffects the shape
and location of the free surface and, under certain conditions,produces a maximum displacement of the dynamic
contact line, we may conclude that it significantly modifies
the interfacial location tending to move the meniscus into
the slot.
The surface viscosity also alters the free surface curvature, particularly near xD; close to the wetting line the
=
T,, = 2H(
Ca
+ 6V8-W)
-
Near the dynamic contact line (s 1) the derivative of
V,*W with respect to 6 is negative and large (see Figure
8); however, it is straightforward to verify that the product
of the surface divergence times 6 increases as the surface
viscosity does so. Thus, if the normal stress component
were to remain constant for increasing values of 6, the
surface curvature would decrease as indicated by eq 20.
This is what actually happens but, presumably because of
the curvature reduction, the flow becomes more parallel
to the substrate producing an additional reduction on the
absolute value of T,, (see Figure 9); this reduction generates a feedback effect, reducing even more the curvature
of the free surface.
Regardless of the explanation given above the fact is that
R increases with increasing values of 6; in other words it
means that, in the vicinities of the dynamic contact line,
the free surface flattens against the web.
As the interface approachea the wetting line, the existing
air gap between the substrate and the free surface acquires
a wedgelike shape; since the interfacial viscosity reduces
the free surface curvature, one might presume that a
sharpening of the air gap also occurs. This presumption
is confirmed by the results presented in Tables I and 11;
there we summarize the values of the interfacial coordinates and the surface velocities that were obtained for the
following cases: Re = 10, Re = 70, Ca = 0.16, and Ca =
0.50 (each case was solved considering 6 = 0 and 6 = 1).
The sharpening of the air gap is better visualized in
Figure 10a,b which portrays the results for Re = 10 and
Ca = 0.16, respectively; the behavior presented by the
other two cases (Re = 70, Cu = 0.50)is qualitatively similar,
though the relative sharpening is less pronounced. In
addition, the free surface velocity is considerably altered
when the interfacial viscosity is accounted for; parta a and
b of Figure 10 portray this effect; there the modulus of the
2228 Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992
t
R
R
1.0
1.2
0.9
1.0
't
8 0.25 0
0.1
0
8 0.25
0.1
Re=50'
0
0.5
8 '
Figure 7. Effect of the dimensionless surface viscosity on the local radius of curvature of the free surface near the dynamic contact line: (a,
left) 0.0 Id I0.25; (b, right) 0.25 < d 5 1.
-A 1.0
A
IWI
0.5
IS.N
-
0.4
0.5
0.3
10
0.2
0.05
0.0
O io
(X-XD)
8
0.1
I
1
I
0.5
ClO
s
'
I
I
I
0.96
s
I
1.00
Figure 8. Variation of the velocity surface divergence near the
contact lines, for different values of 6.
1.00
0.96
05
0.92
Re 50
-1 2
-16
Tnn
v
000
Po =12.17
t
Figure 9. Values of T,,,,
near the dynamic contact line for different
values of 6.
surface velocity has been plotted for 6 = 0 and 6 = 1. Both
figures show a noticeable increment in the surface velocity
when 6 = 1;this increment is even more remarkable when
the Reynolds number increases (see Table I).
The resulta presented heretofore can be summarized as
follows: the computed predictions demonstrate that when
the surface viscosity terms are introduced into the governing equations, the results obtained clearly differ from
0 05
O '0
( x - XD)
n
"n
"
Figure 10. Values of the y-coordinate of the free surface and interfacial velocity modulus near the dynamic contact line: (a, top) Ca
= 0.32,Re = 10; (b, bottom) Ca = 0.16, Re = 50.
those which do not account for this parameter. When the
dimensionless surface viscosity is larger than a certain
value 6*(Re,Ca) the dynamic contact line moves in the
down-web direction and the existing wedgelike gap
sharpens. Besides, the higher interfacial velocities observed should increase the dragging action that tends to
move the gas phase into the narrowest part of the air gap.
All these events might favor the appearence of "air
entrainment", a highly undesirable phenomenon in any
coating process. Nevertheless, we must remark that the
possible relationship between surface viscosity and air
entrainment by no means explains the still unknown
Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992 2229
Table I. Free Surface Coordinates and Free Surface Velocity Components (Ca = 0.32 - P o = 12.17)
b=O
6=1
Re = 10
Re = 70
XES
U
u
xps
U
u
(2,l)
(2.041,0.647)
(1.923,0.324)
(1.810,0.185)
(1.669,0.084)
(1.585,0.045)
(1.500,0.018)
(1.454,O.m)
(1.408,0.003)
(1.383,0.0007)
(1.359,O)
(2J)
(2.286,0.725)
(2.332,0.439)
(2.292,0.292)
(2.209,0.162)
(2.145,O.lOl)
(2.064,0.050)
(2.012,0.027)
(1.951,0.0098)
(1.915,0.0024)
(1.870,O)
0
-0.007
-0.0871
-0,176
-0,295
-0.378
-0.494
-0.573
-0.663
-0.749
-1
0
-0.020
0.00368
-0.0413
-0.074
-0.132
-0,202
-0.267
-0.340
-0.449
-1
0
-0.064
-0.140
-0.163
-0.161
-0.146
-0.119
-0,094
-0.069
-0.047
0
0
0.0407
-0.123
-0.0813
-0.0963
-0.101
-0.106
-0.093
-0.083
-0.056
0
(291)
(1.964,0.596)
(1.774,0.261)
(1.626,0.135)
(1.470,0.054)
(1.387,0.026)
(1.310,O.OlO)
(1.270,0.004)
(1.233,0.001)
(1.214,0.0003)
(1.195,O)
(291)
(2.288,0.725)
(2.342,0.436)
(2.302,0.288)
(2.218,0.158)
(2.152,0.097)
(2.070,0.046)
(2.016,0.024)
(1.953,0.0076)
(1.913,0.002)
(1.867,O)
0
-0.042
-0.229
-0,396
-0.592
-0.700
-0.814
-0.874
-0.934
-0.967
-1
0
0.047
-0.0323
-0.124
-0.272
-0.395
-0.543
-0,657
-0.789
-0.885
-1
0
-0.138
-0.255
-0.268
-0.225
-0.192
-0.129
-0.097
-0.053
-0.030
0
0
-0.0807
-0.226
-0.280
-0.312
-0.299
-0.274
-0,217
-0.157
-0.078
0
Table 11. Free Surface Coordinates and Free Surface Velocity Components (Re= 50 - P o = 12.17)
6=0
6-1
Ca = 0.16
Ca = 0.50
XFS
U
u
(2s)
(2.032,0.636)
(1.900,0.311)
(1.776,0.174)
(1.629,0.077)
(1.545,0.041)
(1.461,0.016)
(1.416,0.008)
(1.372,0.0026)
(1.348,0.0006)
(1.325,O)
(291)
(2.353,0.733)
(2.422,0.451)
(2.392,0.305)
(2.319,0.173)
(2.259,0.109)
(2.181,0.055)
(2.129,0.031)
(2.067,O.Oll)
(2.031,0.003)
(1.982,O)
0
-0.0018
-0.0393
-0.0734
-0.146
-0.199
-0.299
-0.375
-0.475
-0.583
-1
0
-0.0029
-0.0040
-0.0542
-0.110
-0.176
-0.256
-0.330
-0.405
-0.519
0
-0.010
-0.0612
-0.0626
-0.0751
-0.0710
-0.0660
-0.0570
-0,046
-0.034
0
0
-0.0027
-0.165
-0.142
-0.151
-0.150
-0.149
-0.125
-0.113
-0.073
0
-1
mechanism that causes the phenomenon.
Air-Gas Interface and the Velocity Field in the
Liquid Phase. When the surface viscosity is very low, the
free surface appears to be affected in the neighborhood of
the dynamic contact line only; however, as the values of
6 increase, the shape and location of the whole free surface
are modified. Figure 11portrays the free surface profiles
for two sets of values of Re, Ca, and Po; in each set the
values of 6 were fixed at 0.0, 0.1, and 1.0, respectively.
When the capillary number is 0.16 (Re = 50,Po= 12.17),
the free surface proflea for 6 = 0 and 6 = 0.1 are practically
coincident except for a small displacement of the dynamic
contact line; this result can be explained observing the
variations of the surface divergence (V,*W) along the free
surface (Figure 8). If the surface viscosity is small (6 I
O.l), close to the dynamic contact line the surface gradient
of V;W results large; in fact at s = 1the surface divergence
presents values that are 5-10 times greater than the values
it presents at s = 0.95. Since the surface viscosity enters
into the mathematical formulation affected by the surface
divergence and also by the surface gradient of the surface
divergence (see eq 15), and both are significant close to zD,
U
u
0
-0.023
-0.156
-0,292
-0.481
-0.601
-0.740
-0.820
-0.904
-0.951
-1
0
0.078
-0.032
-0.131
-0.292
-0.421
-0.567
-0.680
-0.805
-0.894
-1
0
-0.087
-0.180
4.203
-0.187
-0.166
-0.116
xFS
(2,l)
(1.987,0.602)
(1.807,0.268)
(1.661,0.139)
(1.505,0.055)
(1.420,0.027)
(1.340,0.0099)
(1.299,0.0043)
(1.260,0.0012)
(1.240,0.~3)
(1.221,O)
(291)
(2.325,0.731)
(2.399,0.445)
(2.366,0.298)
(2.289,0.166)
(2.227,0.103)
(2.147,0.050)
(2.094,0.026)
(2.031,0.009)
(1.990,0.002)
(1.941,O)
-0.090
-0.049
-0.028
0
0
-0.112
-0.277
-0.335
-0.364
-0.345
-0.312
-0.246
-0.178
-0.088
0
(Po. 12.171
Figure 11. Effect of surface viscosity on free surface profiles.
the surface viscosity should modify the interface in that
region.
However, when one compares the free surface profiles
obtained for 6 = 1and 6 = 0, the differences are noticeable;
this behavior can be related again to the variations produced in the surface divergence. As the value of 6 increases, the surface divergence decreases near the dynamic
2230 Ind. Eng. Chem. Res., Vol. 31, No. 9, 1992
e..;Fl
0 06
n 04
o
1
L-
128
.-
f
...”
_,_.e-
&,-
t35
1
I
145
x
.,L
118
./ .-
’
/.+ ,
125
‘35
x
Figure 12. Dependence of free surface shape and velocity field on
surface viscosity.
contact line; nonetheless, this trend is reversed at a short
distance from the dynamic contact line. Figure 8 shows
that the increments of V,-W with 6 are significant for s I
0.5, particularly near the static contact line (ns)where the
surface divergence increaaea by 2 orders of magnitude when
6 is varied between 0 and 1. Thus,when 6 is large, the term
SV,*W turns out to be of the same order of magnitude for
any arc length value, and as Figure 11displays for Ca =
0.16 and 6 = 1, the shape of the free surface becomes
modified all along its extension.
The second set of results depicted in Figure 11 (Ca =
1, Re = 50, Po= 8) does not present new relevant features
except for a more marked separation among the different
interfacial profiles shown there; now, the curves of 6 = 0.1
and 6 = 0 are clearly distinct, and the differences between
the curvea of 6 = 1and 6 = 0 are considerably more decided
than the differences obtained in the previous cases. Theae
results seem to be in accord with eq 17, which indicates
that the effect of surface viscosity should become more
evident at high capillary values, i.e. when the capillary
forces are weak. However, we cannot assert that the
changes o b s e ~ e dare only caused by the relative weakening
of the capillary forces since the reference pressure has been
reduced too.
When the whole interface is modified due to the action
of the surface viscosity, the static contact angle also appears modified. The results presented show that the static
contact angle diminishes when the dynamic contact angle
moves toward the slot interior and vice versa. Therefore,
if the dynamic contact angle presents a maximum displacement, a similar maximum is observed in the value of
the static contact angle, as Figure 6 portrays. Also, this
figure shows that both maxima disappear when the refis reduced.
erence pressure (Po)
The surface viscosity not only modifies the interfacial
velocities but it modifies the flow field in the liquid phase
too, Figure 12 exemplifies those changes when 6 is varied
from zero to one. The most salient feature is that the
changes caused by the surface viscosity are conveyed from
the free surface to the bulk of liquid; thus, near the interface the liquid layers augment their velocitiea in the flow
direction while their velocity gradients in the same direction diminish.
To conclude our description, we must point out that
under certain conditions the model predicts the existence
of a stagnation point located in the upper part of the
meniscus; one of these cases occurs at Re = 70 and 6 = 0
and is detected because the y-component of the free surface velocity is positive (see Table I). When the surface
viscosity is increased (6 = O.l), no significant changes are
observed in the free surface location; however the stagnation point disappears (or moves upward becoming undetected by the discretization employed), suggesting that
substantial changes are occurring in the flow field. Though
these observations seem to be real, we acknowledge that
further analysis, with a finer discretization, must be performed in order to have conclusive evidences.
The foregoing results, which are restricted to the conditions stated at the beginning of this section, can be
summarized as follows:
(a) The surface viscosity clearly affects the location of
the air-liquid interface and also affects the flow field in
the neighborhood of the free surface. The changes observed are prominent considering that the values explored
are smaller (except for just one case) than 0.01 interfacial
poise.
(b) When the values of 6 are small, the gas-liquid interface is affected near the dynamic contact line mainly;
however, as the surface viscosity increases the whole interface changes. If 6 is larger than a minimum value, i.e.
6 > 6*(Re,Ca),the static contact angle decreases and the
wetting line moves in the down-web direction as the surface
viscasity increases. Furthermore, the free surface curvature
diminishes, particularly near the dynamic contact line.
(c) Both effecta, the displacement of X D into the slot and
the curvature reduction, sharpen the air channel shaped
by the substrate and the liquid, especially in the vicinities
of the dynamic contact line. In the same region, if we
assume the lubrication approximation to be valid for the
gas phase, the air pressure should increase due to a higher
value of the free surface velocity. The features just
described-ning of the wedgelike gap and increasing
air pressure-might set conditions favoring the air-entrainment phenomenon.
(d) The flow field in the coating bead might be affected
by the surface viscosity too; nonetheless, more extensive
analysis should be performed to confirm this prediction.
Conclusion
This work is a first step in analyzing a coating flow
process when surface active agents are present; a general
analysis of this problem is rather difficult because the
momentum and the mass-transport equations must be
simultaneously solved in an unknown domain which is
determined by a gas-liquid interface whose physical
properties change locally. However, there are two limit
cases which are less difficult to study; they occur when the
surfactants are transported to the interface at extremely
high or extremely low rates. When the rate of mass
transport is extremely high, the interfacial concentration
of solute approaches the equilibrium value, while for the
second case the interfacial concentration of surfactants
should be inversely proportional to the local deformation
suffered by the free surface.
In this work we establish the procedures for introducing
the interfacial properties into the numerical codes used
in coating flow analysis and we study the limit case of
nearly constant interfacial composition; in this way we set
the stage for analyzing the second limit situation. This
analysis is now being undertaken.
Acknowledgment
We thank the Consejo Nacional de Investigaciones
Cientificas y TBcnicas of Argentina (CONICET) and the
Universidad Nacional del Litoral (UNL), Santa Fe, Argentina, for financial support.
Nomenclature
Ca = capillary number, Ca = p U / p
d = gap width, cm
F = external force acting on the free surface, dyn/cm2
7f = mean curvature, cm-’
H = dimensionless mean curvature, H = 7fd
Yi = totd curvature, cm-2
Ind. Eng. Chem. Res., Vol. 31, No. 9,1992 2231
n = unit normal vector to the interface
n, = unit normal vector to the substrate
p = dimensionless pressure, measured in units of r U / d
Po= dimensionless reference pressure for the liquid phase,
measured in units of p U / d
R = dimensionless radius of curvature of the free surface, R
= 1/H
Re = Reynolds number, Re = p U d / p
s = arc length
t = unit tangent vector
T = dimensionless stress tensor, T = T d / r U
T,, = dimensionless normal component of the stress tensor,
measured in units of pU/d
u = dimensionless x-velocity component, measured in units
of u
U = web speed, cm/s
v = dimensionless velocity vector, measured in units of U
u = dimensionless y-velocity component, measured in units
of u
w = surface velocity vector, cm/s
W = dimensionless surface velocity, W = w/U
x = dimensionless x-coordinate, measured in units of d
xFS = free surface location measured in units of d
~ c D= location of the dynamic contact line measured in units
of d
x s = location of the static contact line measured in units of
d
y = dimensionless y-coordinate, measured in units of d
Greek Letters
y = surface mass density, g/cm2
6=
t = shear surface viscosity, (dyn s)/cm
OD = dynamic contact angle
Os = static contact angle
K = dilatational surface viscosity, (dyn s)/cm
p = bulk liquid viscosity, (dyn s)/cmz
= K + e, (dyn s)/cm
rC = gas pressure, dyn/cm2
p = bulk fluid density g/cm3
u = surface tension, dyn/cm
l
'= stress tensor, dyn/cm2
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Received for review December 16,1991
Revised manuscript received May 26, 1992
Accepted June 5, 1992