BIBLIOTHEEK TU Delft P 1986 5291
"^
DIP-COATING BY WITHDRAWAL
OF LIQUID FILMS
PROEFSCHRIFT
TER VERKRIJGINÜ VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN
OP WOENSDAG 18 FEBRUARI 1970 TE 14.00 UUR
DOOR PIETER GROENVELD natuurkundig ingenieur geboren te Amsterdam Uj
(|001979^|
<':^„D0tLEf(SlR.1Ci-r.-, 'JJl Li.DELFTSCHE UITGEVERS MAATSCHAPPIJ N.V. DELFT 1970
F
Dit proefschrift is goedgekeurd door de promotor PROF. DR. IR. W. J. BEEK.
Photograph of the streamlines during withdrawal of a vis-cous liquid
Bij het tot stand brengen van het werk dat in dit proef-schrift resulteerde zijn velen betrokken geweest. De studen-ten wil ik hiervoor met name bedanken, te westuden-ten:
J. Bac
R.A. van Dortmond
C. van Lookeren Campagne
H.A. van Oostveen
P.G. Scargo
P.G. Stoutjesdijk
5 CONTENTS Summary Introduction 1.1 Introduction 11 1.2 Industrial applications 12
1.3 Contribution of this work 1^ A theory for withdrawal without capillary and
inertial forces
2.1 Newtonian liquids I7
1 Sheets I7
2 Wires 2k
2.2 Power law liquids on sheets 27 A theory for withdrawal with capillary and without
inertial forces
3.1 Newtonian liquids 3I
1 Sheets 3I 2 Wires 36 3 Capillary tubes 39
3.2 Power law liquids ]^Q
1 Sheets llO 2 Wires and tubes 1^2
A theory for laminar withdrawal with appreciable inertial forces
i+.1 Newtonian liquids 1^5
1 Sheets 1^5 2 Wires l^j
k.2 Power law liquids on sheets ll8
k.3 Regions of applicability of the three
with-drawal theories 1^8
1 T h e t r a n s i t i o n f r o m c r e e p i n g w i t h d r a w a l w i t h c a p i l l a r y f o r c e s t o w i t h d r a w a l w i t h o u t c a p i l
-l a r y f o r c e s ( c h a p t e r s 2 a n d 3 ) I49
2 The transition from withdrawal with inertial forces to withdrawal without inertial forces in the absence of capillary forces (chapters
3 The transition from withdrawal under the in-fluence of capillary forces with inertial forces to withdrawal without inertial forces
(chapters 3 and h) 5I
_5 Experimental, fluid properties
5.1 Viscosity, surface tension and density 53
5.2 Non-newtonian rheology 53
6_ Experimental apparatus and procedures
6.1 Introduction 55 6.2 Film thickness measurements on plates gg
6.3 Flux on wires 62
6,h Visualisation of the streamlines 62
6.5 Determination of the position of the
stagna-tion point and the form of the interface gl^
_7 Experimental results
7.1 Film thickness on sheets 67
7.2 Flux on wires 68
7.3 Curvatures of streamlines 69
f . k The shape of the interface 73
_8 Conclusions 75
I Obstructed filjn flow 79 II Relation between drainage and withdrawal 82
III Driving a viscous fluid out of a tube Qk
IV Rol coating theory 86 V Influence of surface impurities 92
VI Withdrawal of other non-newtonian liquids 97 VII Instabilities in suspended liquid films 98 VIII Theory relating interface shape and fluid flux 101
References 107 Notation 113
7
Summary
DIP-COATING BY WITHDRAWAL OF LIQUID FILMS
The main goal of this thesis is to predict the thickness of the liquid layer that remains on objects that are removed from a liquid bath. Knowledge of this thickness is important, for example, in the photographic and the painting industry. The thickness of this liquid film is determined in the inter-mediate region where the liquid film reduces in thickness from the diameter of the pool of liquid from which it is formed to the constant thickness film on the object, further away from the pool.
The four forces that can influence the flow in this re-gion are viscous, gravitational, capillary, and inertial forces. The thickness is mainly dependent on the viscous and gravitational forces:
h Af^
Therefore we introduce a dimensionless thickness, T:
which is relatively constant (0.2 < T < 0.6). The two dimen-sionless numbers that show the relative importance of the other forces are the capillary number:
_, nv viscous forces a capillary forces
and the Reynolds number:
pvh inertial forces
Re = - — = —^ .
n viscous forces
For dimensional reasons the relation for the thickness during withdrawal on a flat sheet is therefore:
T = f(Ca, Re) .
It is convenient to distinguish between three regions of Ca-and Re-numbers for which this relationship has been estab-lished.
8
a) Firstly: a region where only viscous and gravitational forces are predominant (chapter 2 ) . That means:
Ca >> 1 Re << 1 .
This is the case for very viscous fluids. In the bath the moving object can draw along more liquid than it can remove out of the pool. Therefore, there is a stagnation point on the interface where the two flows up the plate and back into the bath separate. The liquid flux that passes the meniscus region is calculated in chapter 2 using a simplified Navier-Stokes equation and the known presence of a stagnation point. This resulted in the prediction of a lower flux than proposed thus far in the literature.
b) Secondly: a regime where viscous and gravitational as well as capillary forces apply (chapter 3 ) . In this case:
Ca » 1 Re << 1 .
The problems of the region of low viscosity and low speed withdrawal has been solved by Landau and Levioh by numerical integration of a modified Navier-Stokes equation. In chapter 3 we obtain the same result in a less strict but physically more comprehensive way.
c) Thirdly: a region where inertial forces dominate over viscous, gravitational, and capillary forces (chapter \):
Re > 1 .
This is the case in high speed withdrawal. To our knowledge, this is the first time that inertial forces are taken into account. Inertial forces only exert a minor influence on the flux, although they considerably modify the flow in the me-niscus region.
The regimes where the various theories apply are shown in fig. i+-5 and i|-6.
.f( • .
The three theories, mentioned above which pertain to flat sheets are modified in order to predict the film thick-ness in withdrawal of wires and tubes. In this case the ra-dius of the wire or tube has to be considered as a variable influencing the film thickness.
Many fluids used for coating exhibit pseudoplastic be-haviour. The three theories are therefore also adapted to
9
predict the film thickness when entraining these liquids.
Measurements of the film thickness on flat plates are reported; use is made of a newly developed method. The trained fluxes on wires were measured by weighing the en-trained liquid. We photographed the streamlines and the posi-tion of the stagnaposi-tion point on the interface in the menis-cus region in order to check the validity of the simplifica-tions introduced in the Navier-Stokes equation while dealing theoretically with regime a ) . The shape of the interface is correlated in this case a) with the eventual film thickness.
The thicknesses measured by us and others were generally in agreement with our theories. Therefore these theories solve the existing discrepancies between theory and experiment in, for example: the coating of plates of limited length, wire coating, and coating with pseudoplastic liquids.
§1.1 11
Chapter 1 INTRODUCTION
1.1 Introduction
A widely used operation in industry is the withdrawal of sheets and wires out of baths filled with liquid. These ob-jects entrain a liquid film out of the bath. The thickness of this fluid film decreases continuously from the diameter of the pool to a much smaller constant value further away from the bath. Knowledge of this eventual thickness, or of the re-lated flux of liquid being withdrawn from the pool, is im-portant in many applications. The phenomena taking place in this jacket of liquid between the constant thickness film and the pool of liquid will prove to be of crucial importance to understand the withdrawal process.
For a newtonian liquid the thickness, h, of the liquid film will be a function of the following parameters: p, g, n, a, the speed of withdrawal, v, and radius of the object, r. We can deduce with dimensional analysis that the
relation-ship should be:
AIRF = f(I]V pvh ££ ) ^ « (^)
y nv a ' n 2o
If we introduce dimensionless numbers we can write instead of this equation:
T = f(Ca, Re, Go) . (l) whereby: ^__^
T = h\[£?. (2)
is the dimensionless thickness. The thickness h is made di-mensionless with the viscous length U — . This is
advanta-pieous because the thickness will prove to be dependent prin-The symbols have the meaning given to them m the notation.
12 §1.1
cipally on this viscous length. Therefore T does not vary much (T = 0,2 - 0,6).
The Goucher number :
~ (3) 2a
is the radius of the object made dimensionless with the
capil-2p
lary length y Pg
The capillary number, Ca, shows the relative importance of viscous (n 7") and capillary forces (r-) .
Ca = ^ . (ll)
0
The Reynolds number shows the relative importance of inertial (pv^) and viscous forces (n X.).
h'
Re = £ ^ . (5)
We can divide the variables in the withdrawal process into three groups.
The first group is related to the flow conditions. They determine whether the viscous, capillary, or inertial forces take part in the withdrawal operation (Ca, Re).
The second group of variables matches the withdrawal geometry.
Withdrawal can be achieved with sheets or with curved objects such as wires and tubes (Go).
The object can be withdrawn under an angle.
The meniscus, from which the object is removed, can be mani-pulated through capillary and gravitational forces, etc.
The third group is the rheology of the liquid. The liquid can be newtonian, or can exhibit pseudoplastic or elastic properties.
We will now consider several applications in order to select the variables of practical importance.
1.2 Industrial Applications
- A well-known application is the covering of metal ob-jects with coloured varnish and laquer layers by dipping them in a bath (household appliances, car bodies, metal sheets
§1.2 13
for cans and electrical wires (26^ 27)).
Glass bottles and plates are dipcoated with p.v.c. in order
to prevent splintering (25).
Electrical compounds are insulated and protected against
moisture by dipcoating (24).
The well known Edam cheeses are covered with red paraffin by dipcoating.
It is important in all these applications to get a good cov-ering by the film. The process does not need much investment
and can be made continuously by using a conveyor (26), but
disadvantages are the large amount of liquid required, fur-thermore ageing may occur, and the fire hazard is consider-able.
- The oil needed for lubrication of bearings is sometimes lifted from an oil reservoir by means of a riding ring. Some of the older research in dipcoating was undertaken with this
application in mind {Blok and van Rosswn (2j 28)).
- The thickness of a dipped coating of light sensitive
gelatine for photographic film or paper has to remain
con-stant within very narrow tolerances (29^ 40, 41). Other
photo-graphic materials are dipcoated on paper, for example
diazo-type materials (44). The liquid meniscus is sometimes
modi-fied with knifes to give better thickness control and a
thin-ner film at a higher speed (42, 42).
- A moderen development in high speed processing of
pho-tographic film is to dipcoat the film with a viscous layer
of a developing solution (45, 46, 49). The process is easy
to control, and the amount of chemicals needed is reduced. Another way to develop photographic film, used specially to process immedeately after expose, is to pull the film through a slit filled with low viscosity developer kept in place by
capillary forces (47, 48); the photographic film removes a
layer of liquid from the developer solution.
- A liquid film is left on the wall when draining tanks and burettes. A similar phenomenon happens when an oil field is emptied by a gas drive. The oil is replaced by gas, and a thin layer of oil remains on the sand when the oil recedes
(54).
- Freezing rolls can be used to make flakes. These cooled rolls rotate half submerged in a melt and freeze a layer in the pan. This layer entrains an adhering liquid layer out of the pan. This liquid layer is subsequently frozen and removed from the roll (for example flaking of ammoniimi nitrate). The
l i t §1.2
filmthickness has to be known as a function of the processing variables in order to design the roll. Liquid can also be dried on heated drying rolls to obtain a flaky product (e.g. milkpowder and cereals).
- In many paper coating operations viscous liquids with a high solid content (Ti02) are first picked up by a roll rotating in a pan. The thickness control and the equalizing of the coating are achieved by various other rolls over which the liquid film is transferred before being applied to the paper (52). In a similar fashion ink is removed from a pan by a rotating roll in printing operations (52). The liquid is generally withdrawn out of the pan under a small angle.
- When moving objects from one bath to another the liquid left on the object after withdrawal is sometimes im-portant because it represents a loss (so electrolyte is lost when pulling electrodes in electroplating) or it pollutes the next bath (cleaning and pickling operations for instance in the textile industries).
- Copper sheet and wire is dipcoated with tin, and steel is dipcoated with zinc to improve corrosion resistance or conductivity (55). Rubber gloves are made by dipcoating form^. in rubber solutions, and stripping the dried film from the forms. Easily damaged objects (gears) can be protected during transport and storage by a thick transparant dipcoated layer of plastic. The coating can be stripped off when the part has to be used.
This list of applications is given to show the wide vari-ation in requirements and conditions met in practice. Applica-tions operate over a wide range of withdrawal speed, viscosi-ty and filmthickness and therefore of Ca- and Re-numbers. The objects can be continuous or limited in length. The surface can be flat or curved, as in the case of tubes and wires. And sometimes withdrawal occurs under an angle smaller than 90°.
There are very few experimental or theoretical data on dipcoating, in spite of the widespread use of the method. We will end this introduction by mentioning the contribution of this work to the known coating theories.
1.3 Contribution of this Work
A new theory is given in chapter 2.1 for withdrawal in those cases where inertial and capillary forces can be ne-glected (table 1-1). The thickness proves to be lower than
§1.3 15 Re«1 Re»1 Ca>>1 chapter 2 Ca«1 chapter 3 chapter h
Table 1-1 The three with-drawal theories
would be expected from existing
theories (e.g. Deryagin (40)).
This effect is even more pro-nounced for pseudoplastic liq-uids (chapter 2.2), and enables us to explain the existing dis-agreement between theory and experiment in coating with power
law liquids {Gutfinger and Tatlmadge (12-17)).
The well-known result of Landau and Levioh (2) for
with-drawal of newtonian liquids under influence of the surface tension is obtained in a less strict but physically more understandable way (chapter 3.1). This enables us to better evaluate the influence of variations in geometry and to esti-mate the limits of application of the theory. It is relative-ly easy to adapt this simplified theory to the case of coating a power law liquid (chapter 3.2).
A theory for withdrawal when inertial forces dominate
is given in chapter k. It is shown that it is not always
jus-tified to neglect the influence of these forces.
New measuring techniques were developed to verify these theories experimentally (chapter 6). The streamlines, the form of the air interface, and the location of the stagna-tion point on the interface were determined. A new, more ac-curate and less time consuming method was developed to mea-sure the film thickness on flat plates. In general, the re-sults (chapter 7) agree with the theories of chapter 2, 3
and h.
Results for some special geometries and for other liquid properties are mentioned in the appendices.
§2
17
Chapter 2
A THEORY FOR WITHDRAWAL WITHOUT CAPILLARY AND INERTIAL FORCES
In this chapter we will develop a theory for the with-drawal flux without capillary and inertial forces. The flux in withdrawal is determined for newtonian liquids on sheets (2.1.1) and on wires (2.1.2). The flux of a fluid with a power law rheology is presented in chapter 2.2.
2.1 Newtonian Liquids
2.1.1 Sheets
We will first describe withdrawal of newtonian liquids on flat sheets in the absence of capillary (Ca>>1) and in-ertial forces (Re<<l). In the next chapters this theory will be adapted for wires and power law liquids.
A sheet being withdrawn out of a liquid entrains a film with a certain thickness. There is a gradual transition from the liquid in the bath to a film of constant thickness (Fig. 2-1). In this case of withdrawal without capillary and in-ertial forces we can reduce the general
equation (l) giving the dimensionless thickness:
to:
T = f(Ca, Re, Go)
T = c o n s t a n t , or h = cw —
(1)
Thus we already know from dimensional analysis that T is constant. In order to determine this constant, we will first look at the flow under the in-fluence of gravity of a film of con-stant thickness on a vertical flat sheet. In this case of one dimensional flow the Navier-Stokes equation is re-duced to: / / / / / / / / / / / / / / Fig. 2-1 Diagram of the withdrawal process
18 §2.1.1
Fig. 2-2 Velocity distribu-tions in flat liquid films
Fig. 2-3 Dimensionless thick-ness versus dimensionless flux in flat film flow, with cor-responding velocity profiles
3^v
2
3y2
(6)
with X taken in the direction of the movement of the plate and y perpendicular to the plate. The boundary conditions are X 3v X 3y for y = 0 = 0 for y = h
This results in a parabolic velocity profile in the film. These profiles are shown in Fig. 2-2 for various film thick-nesses. This equation was solved by Blok and Van Rosswn {2, Fig. 2-3) resulting in:
= T 1 T 3 3
(7)
with T h \ ^ nv and Q V v nvT is the dimensionless thickness and Q is the dimensionless flux. There are two physically feasable thicknesses with the same flux." Thicknesses with a flux higher than:
§2.1.1 19
cannot be realised. Therefore
T = 1 , Q = 1 , (8)
gives the thickness for the maximum realisable flux on a moving sheet.
Deryagin (40) and Blok and van Rossum (2) predicted
that this maximum flux will actually be realised in high Ca, low Re number withdrawal. They obtained this result by com-paring the thickness of a film on a draining plate, which was
calculated by Jeffreys (20), with the thickness in withdrawal
by the transformation:
f - V . (9) A draining plate is a plate which is pulled out of a bath
in a short time, and then left to drain, v is the velocity of withdrawal, x the distance from the top of the draining plate and t the time at which we measure the thickness on
the draining plate. The result: T = 1 and Ql = — corresponds
with the maximum realisable flux of liquid that can be en-trained by a plate.
This analogy between drainage and withdrawal does not hold because during withdrawal different effects take place in the bath. The withdrawn plate or sheet starts building a boundary layer of liquid moving with it after entering the
bath. According to Sohliahting (59), the thickness of this
boundary layer after traveling a distance x in the bath will be:
%
% 5 S . (10)
W-As the flux Q will always be below the maximum possible flux of eq. (8) the following holds:
h
<=\l^ .
(11)
VPS
The film thickness h will be smaller than the boundary layer thickness 6, if:
X >> "" (12)
20 §2.1.1
This holds true for most cases. The amount of liquid entrained in the bath is therefore large compared to the amount of liquid entrained on the plate. There-fore, somewhere on the inter-face must be a stagnation point with zero velocity (Fig. 2-k).
The Navier-Stokes equations for vertical withdrawal without capillary and inertial forces become:
n(-3^v 32v 8y 32v ^) = = Pi dxFig. 2-U The flowlines in the withdrawal process
3x
r
3y^^ dyX) =l£ .
(13) The boundary conditions are no slip at the wall and no tangen-tial force and a constant pressure at the gas interface. These equations cannot be solved analytically because the form of the interface is not known beforehand.The flow above the stagnation point.will be mainly in the vertical direction, therefore we can assume that above this point the following holds:
3^v 32v (-3x^ 3y ^ ) ( • )2v X 3^v X -) (1U)
The pressure in the region of constant thickness will equal the pressure along the interface. Therefore we can assimie also that in the region above the stagnation point holds:
dx
«ay
15)
If we introduce these estimations into the Navier-Stokes eq. (13) we can reduce those to:
l(-S^v ^^^
dx'' 3y ^) = pg
§2.1.1 21
Only the term
distin-Fig. 2-5 Dimensionless thickness versus dimensionless flux in a film formed by withdrawal guishes this equation from
equation (6) for the case of a film of constant thickness. It is this term that has been neglec1> ed by other workers on withdrawal theories.
?2^ The term
measure for the curva-ture of the streamlines, is not only zero in the constant thickness region, but also on the line B in
fig. 2-k connecting the in-flection points of the
streamlines. The thickness-flux relationship in B is given by equation (7) and is shown in fig. 2-3 when we,simplify B to a horizontal line x = constant. The flux through all the hori zontal cross sections A, B, C and D in figure 2-k is the same although the thickness varies continuously. A flat film on the contrary can only exhibit two feasible thicknesses for one flux. This paradox can be explained by the fact that the flux in A compared to the f]ux through a flat film with the same velocity profile is raised by the curvatures of the streamlines. The flux in C is lowered by these curvatures that have changed sign in B. The curvatures ° ^^ can
there-3x^
fore explain that the film has various thicknesses with the same flux (Fig. 2-5).
We will now estimate that flux. Since:
(17)
3x
3'vx
"IF
we can assume that the velocity profile in the cross section containing the stagnation point will approximate a parabole, with zero velocity at the free interface. In the constant thickness region we can regard the velocity profile as ap-proximate plug flow. Therefore:
3T^ . (18)
A = 3h D or
h =
DAssuming furthermore that the flowlines at some distance from the free surface show an inflection point about half way the
22 §2.1.1
planes A and D, we may write for h
or
\ 2 = ^ \ '
Tg = 2Tj^ . (19) As the same flux goes through B and D we can deduce from
equation (7) (Fig. 2-3) that the only two points that satis-fy this relation are:
Tg = 2 ^ ; ^ 1.31 Tj^ = ^ ^ 0.66
with
-m
Q =
^\ii ^ 0.56
.
The flux and the ultimate thickness in absence of capillary forces will therefore be:
Q = 0.56 and T^ = 0,66 , ' (20) or written differently:
h = 1 = 0.56\/^ and hn=0.66\fÖ?. V Vpg D ypg
This result is relatively insensitive to the estimate Tg = 2TD. An inaccuracy of 15^ in this estimate results only in an inaccuracy of 5^ in the final dimensionless flux Q.
We will now adapt this theory predicting the flux in withdrawal of endless plates to withdrawal of finite plates. The boundary layer in a withdrawal operation of a continuous sheet develops in a time which is short compared to the time needed to establish the flow in the meniscus region on the
plate (i.e. the region between A and D in Fig. 2-k). The
time for creating the meniscus region is about equal to the ratio of the volume of the meniscus region and the pumping capacity of the plate. This occurs when the plate has moved a distance equal to several times the ultimate thickness of the film and is also quite short. The flux Q at the meniscus of the bath during withdrawal of an endless sheet will there-fore be the same as the flux on a plate of finite length moving with the same velocity. However the filmthickness h
§2.1.1 23
Fig. 2-6 Withdrawal under an angle
0.2 04 06 OS
Fig. 2-7 Dimensionless thickness versus dimensionless flux during withdrawal, and dimensionless flux versus the dimensionless flux di-vided by the dimensionless thick-ness reduction distance K
on an endless sheet will be constant, while the thickness on a finite plate will vary with the height.
The maximum flux theory of Deryagin (40) predicts there-fore film thicknesses on endless plates which are 51^ too high and fluxes on endless and finite plates that are 18^ above what we would expect according to our theory.
When the plate is withdrawn at an angle a. we have to replace pg by pg sin a giving for the eventual thickness and the flux: T = 0.66 /sin a 0.56 (21)
/si
m a or: h = 0.66 \ JUL ^ = 0.56 Vif
nv pg s m a V y Pg s m a The angle a is defined as the angle between the direction of movement (the positive x-axis) and the positive y-axis(di-recting towards the interface in the pool) (Fig. 2-6).
It is interesting to know, apart from the flux in with-drawal, the length along the plate which it takes to estab-lish the eventual film thickness. The ultimate thickness-flux relation on the plate is determined by equation (7) for flat films. There are two film thicknesses possible for a flat film at one liquid flux. The ultimate film thickness is
2k §2.1.1
approached by a film with a higher dimensionless thickness but with the same dimensionless flux. A flat film with this higher dimensionless thickness should have a higher dimen-sionless flux (see Fig. 2-7). However, this flux is lowered. according to equation (l6) by the curvatures ^ "^x of the
9x2
flow lines. The magnitude of this curvature term close to the ultimate film thickness region will depend on the gradient dQ/dT in Fig. 2-7. The curvatures have to lower the flux only a small amount when dQ/dT ^ 0 (i.e. for Q !fe 0.67). How-ever, the curvatures have to lower the fluxes more when dQ/dT ^ 1 (i.e. for Q < 0.1+).
Therefore there is a relation between the curvatures of the flow lines close to the constant thickness film and the dimensionless flux of the film. The curvatures of the flow lines in the film are related to the curvature of the air interface. This air interface can be described by an exponen-tial function: 2c h " k h. = " ' or: X T T ^ = e ^ (22) D
when h, x and k are made dimensionless with the viscous length:
X = x l / ^ and K = k\/£i' . y nv y nv
Experimentally, this exponential f-unction for the shape of the interface will prove to be a valid description in the viscinity of the eventual thickness region. In the Appendix the relation is derived between the dimensionless flux and the dimensionless length K over which the excess film thick-ness decreases 1/e (Fig. 2-7). Our theoretical flux in withdrawal Q = O.56 should correspond to a dimensionless re-duction distance K = 1,1. The results obtained from photo-graphs of the interface (chapter 'J.k) are in agreement with this value of K.
2.1.2 Newtonian Liquids on Wires
(19-§2.1.2 25
-^•f^^
Fig. 2-8 Symbols in wire withdrawal
Fig. 2-9 Thickness-flux quotient in constant film thickness flow with zero interface velocity on wires
22) for coating wires are based on the maximum flux theory of Deryagin (40) for the flow regime where the inertial and capillary forces can be neglected. We will now introduce a new theory based on the presence of a stagnation point on the gas
interface and on the curvature of the streamlines.
The radius r of the wire can be made dimensionless with the capillary length:
Go = - = r a
or with the viscous length:
R = r\P .
V nv
(3)
(23)
In withdrawal without capillary forces the viscous length is to be preferred. The relationship between R and Go is
Go = R = R \ / ^ (2U)
Like in the case of a flat plate it is possible to calcu-late the relationship between the flux and the thickness of a. film of constant thickness on a wire. The result is given by Tallmadge, Labine and Wood (20):
~1 — 2. -R with: 1 +
»V! - Ina)] ,
(Fi£ 2-t a = 1 + — , c r(corresponding to eq. (7) for sheets)
26 §2.1.2
This is a far more complex relationship than the one for flat plates. For the velocity of the liquid in the film can be written:
^ = ^ -ÏÏ L^ " "r ^ 2a2lna^J . (26)
For the flux and thickness of a film with zero velocity at the interface this results in:
^^ = 1 + «2(2 Ina- l) • ^27)
Which corresponds with T = v^ for flat plates. With this equa-tion (25) we can calculate the quotient of the film thickness in a constant thickness film with zero interface velocity and the thickness the film would have with the same flux in ab-sence of gravity forces (flux thickness) (Fig. 2-9). We see that the film thickness in the plane of the stagnation point divided by the ultimate thickness does not differ much from the value for flat plates:
^B h, (-) = 3 , (28) as long as D h Go R = /2 -^^ > 5 10-2 ^ (2p) /Ca
which is true for most cases of wire withdrawal.
Knowing this we can determine the points where the curvatures of the streamlines are zero:
32v
- 3 ^ = 0 . (30)
Like in the case of a flat plate it can be assumed that this is the situation for the two thicknesses h and h where:
a Ü
h ^ = 2 h ^ . (31)
The flux and thickness in these points are given by equation (25). The corresponding flux can be calculated in the same manner as shown in section 2.1.1; the result is shown in Fig.
2-10 as a function of the dimensionless wire radius, R, and as afiLiction of the Ca-number with the Go-mimber as a
para-§2.1.2 27
meter. Notice the drop in flux with increasing Ca-number.
In-creasing Ca-numbers correspond with an increasing viscous l e n g t h y — and a decreasing dimensionless radius R. With decreasing R the curvature of the film and hence its more than increasing weight with increasing ultimate film thickness becomes important,
resulting in lower Q-values than predicted for plates. These fluxes are lower (+ 20^) than the maximum fluxes
expect-ed by White (19).
2.2 Power Law Liquids on Sheets
The relation between stress and shear rate is for a
n-1 ,,/a.v> T 10' 0.4 Q2 0 '0 i „ R.^
^ ' °
^
1 ' i 1 --_______^ ^ , 1 1 ^ _ o o^-^-3r~~~~
Go .0.01 1Fig. 2-10 Flux on wires in high Ca-, low Re-number with-drawal; fl\ix on wires as a function of the Ca-number with the Go-number as a para-meter
power law liquid;
= -
«S'
dv dx (32)A power law liquid is characterised by two constants K and n. A newtonian liquid is determined by only one, n. The apparent viscosity of a power law liquid depends on the shear rate. For n < 1 it decreases with increasing shear rate (pseudo-plastic behaviour). The characteristic shear rate in coating is:
fÉX) = Z
•' characteristic
(33)
When we replace in the newtonian definitions of the dimension-less thickness T and the capillary number Ca:
n -
U~)" ,
(3U)
we obtain: T' = h n+1 2 (P£^; Kv" (35)28 §2.1.2
and
^ ,1-n n K Ca = h V —
a (36;
The dimensionless thickness T' is not directly proportional to the physical thickness h if we define T' in this way. To achieve this we redefine T:
2 1 (T-)^-^^ = h ( £ ^ ) n+1 Kv n (37) and Kv n+1 l/Pg Kv 1 n+1 Fig. 2-11 Dimension-less flux versus di-mensionless thickness
in flat film flow of a power law liquid
For flow in a constant thickness film we rewrite equation (6):
- % ^ = pg . (38)
Gutfinger and Tallmadge (12) solved this equation for a power law fluid with the boundary conditions: V = V X T = 0 xy for for y = 0 , y = h .
The resulting relationship between the flux and thickness is in dimensionless form:
« = -['-?ST ""']
(39) For n=1 this equation reduces to the newtonian case (equation (7)). Figure 2-11 shows this relation graphically for various values of the flow index n.We will now calculate the coating flux on a continuous flat sheet in the same way as for newtonian liquids. The thickness of the film with a zero interface velocity is for newtonian liquids 3 times the corresponding flux thickness.
§2.1.2 29
* Kv
a<eav
0.2 0 4 0.6 0.B 1.0
Fig. 2-12 Thickness-flux relation during coating with a power law liquid
Fig. 2-13 Flux in high Ca and low Re niomber with-drawal of power law li-quids as a fiinction of the flow index n; Q is the ac-tual realised flux; Q^ is the maximum possible flux on a moving plate
For a power law fluid the result is obtained with the aid of equation (39) and the extra boundary condition (Fig, 2-12):
V X ^A ^D = 0 f o r y = h 2n+1 (1+0)
We can assume in analogy with the newtonian case that the in-flection points:
32v
^ = 0
occur when the thickness is half way between the plane D of constant film thickness and the thickness in plane A con-taining the stagnation point:
T^
1^-.
3n + 1
30 §2.1.2
The thickness-flux relation in Tp and Tg is described by equation (39), and the same flux goes through both. With this information it is possible to determine the flux Q for a given value of n. The result is presented in Fig. 2-13 to-gether with the maximum possible flux Q^ that can be entrain-ed by the plate when the stagnation point and the streamlines are not taken into account.
Gutfinger and Tallmadge expected that the flux Q during withdrawal would equal Q^^^. The fluxes they measured were how-ever, considerably lower than Q^. They tried to solve this discrepancy by describing the liquid as an ellis liquid, and by using new mathematical apporoximations (11+-17). But, ac-cording to our theory the explanation lies in the fact that the maximum possible flux is not realised, and that the dif-ference between Qi^ and Q increases with an increasing pseudo-plasticity (i.e. with a decreasing value of n) of the liquid.
§3 31
Chapter 3
A THEORY FOR WITHDRAWAL WITH CAPILLARY AND WITHOUT INERTIAL FORCES
In this chapter we will develop a theory for the flux in withdrawal under the influence of capillary forces with-out inertial forces. The solution for the withdrawal of new-tonian fluids on sheets (3.1.1) can easily be adapted to wires (3.1.2) and capillary tubes (3.1.3). The withdrawal of power law liquids is also solved for sheets (3.2.1) and wires and tubes (3.2.2).
3.1 Newtonian Liquids
3.1.1 Sheets
In this case of withdrawal, capillary forces play an important part. The capillary forces lower the pressure in the meniscus region. The liquid has to pass the pressure gradient between this negative capillary pressure and the zero (atmospheric) pressure in the constantthickness film. This will reduce the resulting coating thickness. The equa-tion describing the thickness T is in this case, for
dimen-sional reasons:
T = f(Ca) . (1+2)
Landau and Levioh (2) were the
first to determine this relation. They divided the liquid jacket into three parts (Fig. 3-1):
a) A part of constant thickness film, where only viscous and gravita-tional forces govern the flow:
32v^
^~3F='S- (6)
b) A transition region, where
viscous and gravitational as well as the withdrawal pro-capillary forces have to be considered: cess
32 § 3 . 1 . 1 O- 0.6 av/fï V f I V 0.4 ^htghCa^lowRs ,'THfTlD«r ^ '^thtory «quation 2 0 •nowCo-.lowRe numbar th«ory
•quQtlon 4 7
O 10
+ a 4 % = 0. (1+3)
dx''
c) A stationary meniscus where only gravitational and capillary forces are important;
R' = Pgx . ikk)
Fig. 3-'2 The dimensionless flux Q as a f\inction of the Ca-number for low
Re-num-R' is the local radius of the curved interface, and x is the bérs vertical distance above the
liquid level in the bath. The flow in the transition region will determine the flux. The boundary conditions of this region are: no slip at the w a l l , no tangential forces on the interface, no capil-lary pressure at the transition to the flat film, and a cer-tain negative pressure at the transition to the stationary meniscus. This negative pressure will b e equal to the pres-sure at the top of the stationary meniscus. Equation (1+3) was first solved by Landau and Levioh with the boundary
con-ditions mentioned. The result of their numerical calculation is:
h = O.9I+I+ / 2 - ( ^ ) R' . (^5)
The meniscus radius R' at the top of the meniscus on a ver-tical flat plate is:
R' = - = É \ / ^ 2 U pg
giving the well known result (Fig. 3 - 2 ) ;
T = 0.91+1+ Ca^ .
(1+6)
(1+7)
The same result has been derived by Deryagin (40) and Levioh
(6).
We will now derive this solution in a less strict, but physically more understandable way. The top of a stationary meniscus along a vertical plate will exhibit an infinetely high pressure gradient. As soon as the plate starts moving this pressure difference a/R', will be distributed over a
§ 3 . 1 . 1 33
stationary plats
- y 0 0 5 1.0
Changing «P/dx Constant ''"/ax
Fig. 3-3 Pressure distribu-tions in a stationary meniscus and during withdrawal of a vertical plate
Fig. 3-1+ Film thickness and flux in low Ca-number with-drawal
larger distance (Fig. 3 - 3 ) . If we manage to calculate the distance over which the pressure difference works we are also able to solve the problem. The film thickness decreases tinuously when going in the direction of the region of a stant film thickness (Fig. 3-1+), while the flux remains con-stant. This is made possible by the varying pressure gradient. For the velocity profile with the maximum flux at a certain pressure gradient dp/dx we can deduce that the mean velocity is 2/3 of the velocity of the plate (eq, ( 7 ) ) , or:
h h 1 2
.5
(i+8) A l l o t h e r t h i c k n e s s - f l u x r a t i o s w i l l g i v e a l o w e r n e t f l u x a t t h a t d p / d x . T o g e t h e r w i t h t h e f a c t t h a t t h e f l u x i s t h e same f o r a l l p r o f i l e s , t h i s e n a b l e s us t o s t a t e : ÊR = dx max f o r h ii 1.5 (^9)(i.e. for cross section C in Fig. 3-1+). From this we may con-clude that the whole pressure drop works in the area where the thickness of the film decreases from:
3^ §3.1.1
h ^
pg 3x
Fig. 3-5 The simplified trans- Fig. 3-6 Capillary pressure ition region drop in the transition region
as a function of the Ca-num-ber for vertical withdrawal of a flat plate
The radius of the interface is R' (= a/2 for vertical plates) for a thickness 3h and is infinite at 1h. If we suppose that the average radius of curvature of the interface between the planes where the film thickness is 3h and h respectively, is 3R' we can calculate the distance between these two cross sections. That distance is according to Fig. 3-5:
as long as h < -r-;r R ^ 10
1 = /2.3R' .2fi = /l2R'h
1 or when R ' = - | i f C a < 2 . 1 0 2,
gives for the maximum pressure gradient (Fig. 3-6!
f^) . R ' " cc a
Mx^
max /l2R'h R'/l2R'h ; 5 i ) This (52)When we neglect gravity, just like Landau and Levioh did, the ultimate thickness of the film will be:
h % h ^ 0 . 6 7 U ^ = 0.67VCaR'/l2R'h , - 1+ \ ^^"^
h ;^ ^ Ca3 R' ,
For f l a t s h e e t s withdrawn v e r t i c a l l y out of a l i q u i d t h i s equation can be r e w r i t t e n as follows:
T^ ;v Q = I /2 Ca^/^ = O.9I+3 Ca^/^ .
:53)
§3.1.1 35
which is the same as Landau and Levioh found for low
capil-lary nimibers. We can equate the dimensionless flux, Q, and the dimensionless thickness, Tp, because:
^D < •'^
for the conditions where this theory applies.
There are several assumptions that limit the application, of this theory to relatively low Ca-numbers. This solution will break down at higher Ca-numbers because we took the pressure at the top of a stationary meniscus as the condition for the lower boundary of the transition region. But this pressure has to be lowered when the transition region ex-pands at the cost of the stationary meniscus in higher Ca-number withdrawal. This will not occur as long as:
1 = /12R'h < sR' , when R' = | : Ca < 0.2x10-3 . (55)
Another point which we have neglected thusfar is the curvature term 32vjj./3x2, In retrospect we can show this to be a valid assumption. The velocity profiles in the film change over a distance 1 which is large compared to the film thickness h when:
1 ^ /l2R'h > 20h , when R' = ^l : Ca < 1,6x10"^ , (56)
The curvatures of the flowlines are therefore very small in the meniscus region. But at higher capillary numbers we will have to take them into account,
The solution given by equation (5I+) holds only for the case that capillary forces are dominant over gravitational forces. It is therefore irrelevant for this solution if the sheet is withdrawn horizontally or vertically, as long as it moves out of a region with a low capillary pressure a/R', But at higher Ca-numbers when the capillary forces decrease, gravity forces become important in the transition region and have to be taken into account. They can be neglected as long as:
| £ = — > 5pg , when R' = I : Ca < 10"3 . (57)
36 §3.1.1
The theory of Landau and Levioh was adapted for higher
Ca-numbers by White and Tallmadge (7) by only taking gravity
into account. They neglected the curvatures and the raise in capillary pressures at the transition between the sta-tionary meniscus and the transition region.
For withdrawal under an angle a between the positive X- and y-axes, we can write for the radius at the top of the stationary meniscus (63):
R = ^ (58)
2/2 cosja
This gives for the ultimate thickness in withdrawal under an angle:
T^ = Q = | - ^ . (59) D 3 cos2a
3.1.2 Newtonian Liquids on Wires
For the thickness of a film on a wire in low Re- and low Ca-number withdrawal, we can write for dimensional rea-sons:
T = f(Ca, Go) . (60) The theory we have for flat sheet withdrawal (3.1.1) can be
modified for wire withdrawal. Hereto we have to replace the radius at the top of the meniscus on a flat plate by the ra-dius in a vertical cross section at the top of the stationary meniscus around a wire:
(61) a 2 ' a 2 1 2.1+GoO-85 ^ 1+2.1+GoO-85 0 . 5 Go
This radius on a wire was measured and calculated by White
(19). Apart from the pressure gradient caused by the change in the radius in a vertical cross section, we have to intro-duce the pressure gradient caused by the change in the radius in a horizontal cross section (Fig. 3-7). This contribution to the pressure drop in the meniscus region has been neglected
§3.1. 2 37
by others (19-22)_. It is allowed to do this as long as h << r, because the change in pressure in the transition region caused by the horizontal radius can be neglected compared to the pres-sure drop caused by the vertical radi-us , when:
0, 0
^Ph '^7^ r+3E < - 2: Ap (62) when h << r
Fig. 3-7 Meniscus In this case only the change in radius region in wire with-in the vertical cross section has to be drawal
considered (see Fig. 3-8).
We will first calculate the thickness for the case that only the change in radius in the vertical cross section con-tributes to the pressure distribution. When we introduce the modified radius in equation (53) for the flat sheet we get for the film thickness on a wire:
T^ = Q = O.9I+3 Ca 1/6 2.1+Go°-85 1+2.1+GoO-85 ;63) 0.5 Go
For very small wires this will be reduced to:
Q = 1.89 Ca'^/SQQ ^ Go << 1 .
For very large wires, or sheets this will be reduced to the solution of the preceeding chapter:
Q = O.9I+3 Ca}/^ , Go >> 1 .
White and Tallmadge (21) replaced the radius of the wire r by r+h in the calculation of the radius at the top of the stationary meniscus. This correction becomes important when r Iv h, but in that case the pressure gradient caused by the horizontal cross section radius is more important.
We will now calculate the contribution of the horizontal cross section radius to the capillary pressure gradient. The capillary forces due to a change of the radius in the verti-cal plane are dominating, as we have seen, for very low
Ca-38
§3.1.2medium Ca.
numbers. The capillary forces due to a change in the horizontal radius will operate in what we may call the intermediate Ca-number region, as there are no capillary forces for high Ca-number with-drawal (Fig. 3 - 8 ) . We have the choice between intro-ducing this extra pressure drop in the low Ca-n\jmber theory of this chapter, or in the high Ca-number the-ory of chapter 2.1.2. The easiest is to introduce it into the high Ca-number theory. A general relationship for all Ca-nimibers can then be ob-tained by graphically connecting this theory with the low Ca-number theory.
The pressure caused by the horizontal radius is zero in the bath but,
Fig. 3-8 The interface and pressure gradients in low, m e -dium and high Ca-nimiber wire withdrawal
Ap =
r+h (61+)
on the wire. The vertical distance over which h decreases from 2h to h will approximate 3h for flat plates in high Ca-number withdrawal. If we suppose the same to hold true for wires we can make an estimate of the pressure gradient caused by the horizontal cross section radius in high Ca-number withdrawal. Hence: a a (^) dx , r+E r+2H 3S pg
3(/2Go + T/Ca)(v^Go + 2T/Ca )
(65)
This pressure drop has to be added to the gravity force pg giving for the dimensionless flux Qj^:
% =
pg (^) dx
(66)
The value of Q,the dimensionless flux in withdrawal without capillary forces can be found in Fig. 2-10. The solution to
§ 3 . 1 . 2
low Ca-numb«r rcglm* m«dli/ri Co- r>umb«r r«gima high Ca- numbar ragim*
J
Fig. 3-9 Withdrawal flux on wires
-T-T-rv-rT-v-TTy
Fig. 3-10 Moving air bubble in a capillary tube
this equation is presented in Fig, 3-9 together with the values of the low Ca-number theory. In the intermediate region both horizontal and vertical cross section capillary pressures contribute. Both theories were joined graphically, resulting in a relation applicable over the whole range of Ca-numbers,
This theory explains the deviations between the measure-ments and calculations of White and Tallmadge (21, 22). Their measurements for medium and high Ca-numbers gave thicknesses considerably below their theoretical expectations,
3,1,3 Newtonian. Liquids in capillary Tubes
Because of the cylindrical shape of the interface (Fig 3-10), the capillary pressure in a film inside a tube will be:
a „ , 1 Ap = a_
r for h < 10 and r < a (68; Because of the two directions of curvature of the spherical cap the pressure in the bulk liquid is:
Ap
r (69;During the film forming the liquid has to pass a pressure difference of 2 — - — = — . This compares with a pressure
r r r
difference of 2a/a which has to be overcome in film forming on a vertical flat plate. To adapt the solution for a flat plate to flow inside small tubes we have to replace:
1+0 § 3 . 1 . 3
f ^ r . (70)
Introduced i n equation (53) t h i s r e s u l t s i n :
T = 1.89 Ca^Go for Ca < lO'^ . (71) In r e s e a r c h on moving bubbles i t i s customary t o use t h e
f r a c t i o n W of t h e tube f i l l e d with s o l u t i o n :
W= 2jrrh / 2 h _ ^
•nr^ r
If we write equation (71) with the fraction W we obtain
W = 2.67 Ca3 , (71) This is highly similar to:
2
W = 2,68 Ca3 , (73)
obtained by Bretherton (26) from basis principles. He was
un-aware of the similarity of his conditions with those of Landau
and Levioh. Experimental verifications by Bretherton, and
Maroheasault and Mason (24) show general agreement with the theory. Deviations occur at extremely low Ca-numbers. In the Appendix this is explained by assimiing the presence of a low concentration of a surface active impurity.
3.2.1 Power Law Liquids on Sheets
Analogous to the newtonian case we can calculate the film thickness when coating with a power law liquid. From equation (39) and Fig. 2-11 we know that we can write for the velocity profile with the maximum pressure gradient:
h _ 2n+1 „ dp ,„,\
— = for -r^ = max .li n+1 dx [Jk)
The whole pressure difference will therefore operate between the thicknesses:
2 - ^ h and h . (75) n+1
If we assume again 3R' to be the average of the radius of the interface in the transition region, we can calculate the
§3.2.1 1+1
= ^ 3 H [ . ^ - , ] 5 ' = V 7
n+1 (76) Knowing the distance over which the pressure drop operates we can write: a_ a_i £ = R
dxf
3n+1 (77) n+1 RhThe maximum possible fliix on a moving sheet is (Fig. 2-12): ^ a x 2n+1 •
We know the dimensionless flux to be (eq. (53)):
1 1 1
« = ««ax <i>
dx n+1 n+1(M)
2n+1 ^d£^ dx n+1 (78)When we combine these two equations we get for the flux
(Fig.3-11): k 2 1 3 n+1 3 3n+1 2n+1 ^ ^7f n+1 3(n+1) 3(n+1 ) Ca (79)
For a newtonian liquid, where n=1, this equation will be re-duced to:
Q = O.9I+ Ca^ .
For a power law liquid with power n=g we get: 1
2 • Q = 1.19 Ca^ , n = And for the liquid with n=0, which is a bingham liquid with
a yield value 0 = K and zero viscosity (n=0) the flux will be:
Q = 1.65 Ca^ , n = 0 , where Q and Ca stand for:
1 n+1
Q = E (£\) = h M ,
Kv"^ 0 Q. 0 6 f > ^ ' - ' 0 4 1 0.2 0 -^^ n> , y — ^^•^ n.O n.1-"4.
n.O 10 10 10 10 10 1 10 ^.. h''"v"KFig. 3-11 Flux in low Ca-number withdrawal of a jiower law liquid on flat plates as a function of the Ca-number with the flow index n as a parameter
1+2 §3.2.1
and
, 1-n n K He Ca = h V — = — ,
a 0
Written in the symbols of a bingham liquid this results in:
02
h = 2.12 V - T . (80) (pg)2a^
Deryagin (40) solved the case of withdrawal of a bingham liq-uid by numerical integration and found:
h = 1+.58 ^ . (81) (Pg)2a^
Although these relations are the same in principle, Deryagin predicts a thickness of twice the value given by equation
(80). These formulas only hold when Ca < 10"3, that is if:
M < 10-3 or ^ < 0.3x10-2 . (82) a Pga
At these low Ca-numbers no data are available in order to discriminate between the two theories.
3.2.2 Power Law Liquids on Wires and_in Tubes
We will now consider withdrawal at those low Ca-numbers where only the capillary pressure caused by the change in radius in a longitudinal cross section is of importance, Like in the newtonian case we have to replace the radius at the top of a flat plate by the radius on a wire:
a a 1 2 ^ 2 2,1+ G o ^ - ^ ^ OTT * 1+2,^ GoU.y^ "*• Go I n t r o d u c e d i n e q u a t i o n (79) t h i s w i l l r e s u l t i n : - ^ 1 3 o o ^1 3 ( n + l ) 3 ( n + l ) (61 n+1s I 3 3n+1 n - (-ttLL\ ( -> 21^Z±\ Ca r2,HGo°-Q5 ^ 0,5"]
Ll+2,l+Go"-«^ GoJ
n+1§3.2.2
U3
and for: - 2 1 2 .. 3 ^ . . . 3(n+1) 3(n+1) n+1 n = 1 to Q = 1.88 Ca^ Go , 2 i n = g to Q = 3.0 Ca^ Go3 , n = 0 to Q = 6.6 Ca^ Go2 .The film thickness inside of a tube is the same as on the outside of a wire for these low Ca-number conditions. So equation (83) also represents the solution for the case in which an air bubble moves in a capillary filled with a power law liquid. Hence, the fraction W of the tube filled with liquid will be:
2 _!+• 2 5+3n 1-n
" ' ' • _ ^ ^ ' r 3 3 n + l . ^ ^ ^ ^ ^ ^ ^ ^ ""'' '2n+r ^72 n+1
W = / 2 2 ( ^ ) ( 4 ^ ) ' - • Ca • -Go
which reduces,
for n = 1 to: W = 2.67 Ca3 ,
for n = 5 to: W = 1+.21+ Ca"'"^ G o 3
§1+ 1+5
C h a p t e r 1+
A THEORY FOR LAMINAR WITHDRAWAL WITH APPRECIABLE INERTIAL FORCES
In this chapter we will first develop a theory for the liquid flux during withdrawal, of newtonian liquids on sheets
(l+.l.l) and wires (1+.1.2), with appreciable inertial forces. The theory describing the withdrawal of newtonian liquids on flat plates is adapted for power law liquids in section 1+.2.
In the last part of this chapter (1+.3), we will deter-mine the various regions of applicability of the theories of the chapters 2 and 3 and of this chapter.
1+.1.1 Newtonian Liquids on Sheets
During high speed withdrawal of low viscosity liquids, inertial forces become important. These forces have been neglected in all withdrawal theories known to the author. We will therefore derive a theory describing laminar film flow for the case that inertial forces become appreciable relative to viscous and capillary forces. The applicability of this theory will be limited to a certain range of Re-numbers, Re = pvh/n. At very low Re-numbers (Re < 1 ), the theories of chapters 2 and 3 will apply. Because of wave formation the solution of this chapter will not hold for re-latively high Re-numbers (Re > 150). However, the flow is then still laminar. The transition from laminar to turbu-lent flow takes place at even higher Re-numbers (Re % lOOO). Interpolation between this high Re-number theory and the low Re-nimiber theories of chapters 2 and 3 is possible.
We know that the amount of liquid entrained by the plate in the bath will be larger than the amount of liquid entrained out of the bath (equation 12). A stagnation point is there-fore situated on the interface where the flows up the plate and back into the bath separate. The velocity profiles of the fluid are shown in fig. 1+-1. Impuls has to be transferred in two areas:firstly, to slow the liquid down below the stagna-tion point, and secondly, above that point to accelerate it to the speed of the sheet. The Fourier-number representing
1+6 § 1 + . 1 . 1 h ^ " ^ c 0 , ^ ' • / -t h e p e n e -t r a -t i o n dep-th of momentimi as compared t o t h e d i s t a n c e over which t h e impuls has t o be t r a n s -f e r r e d , i s : Fo = v t
Here t is the transfer time through the meniscus region and h the film thickness. The impuls will have to penetrate first until the velocity becomes zero at the stagnation point, and after that, un-til the liquid is accelerated to the speed of the sheet. Hence the Fourier-number must be of the order 1:
y 0 05 1.0
— o.a\^
Fig. 1+-1 Dimensionless thick-ness versus dimensionless flux in high Re-number coating
Fo = vt
1
(85)We introduce: n = 1
h (86)
where 1 is the vertical distance required to change the veloc-ity profiles. Consequently equation 85 becomes:
Fo = pvh
n
Re or n ^ Re (87)
The dimensionless vertical distance (n) required to change the velocity profiles is therefore proportional to the Re-number. From this we may conclude that the curvatures in the flow lines can be neglected for high Re-number withdrawal. Furthermore, near the stagnation point the fluid acceleration changes sign. Therefore, there will be no inertial forces on the liquid in that area. The general Navier-Stokes equations:
3v. 3v^ py X 3x + pv y 8y
t?
"
" ^ ^13^ " ^ F
X ^ 3^^^ • ) -3v pv X 3x X + 3v, pv, y 3y dy 3x'^ 3y^ (88)§1+,1.1
k7
may therefore be simplified in this stagnation point area to:32v
n - ^ = pg • (6)
In this case the boiondary conditions and the condition that determines the position of the free boundary are: the no slip condition on the wall, the no tangential force and the zero velocity condition at the interface. Integration yields:
T^ = /2 = 1,^1 , Q = ^/2 = 0,1+7 . (89)
For the constant film thickness on the plate, equation 7 can be used; this results in:
T^ = 0.52 and Q = ^/2 = 0.1+7 , (90) or:
h = 0 . 5 2 l / ^ and ^ = 0.kl\j^.
D V pg V y p g
These are lower values than the low Re-number and high Ca-number theory of chapter 2 predicts:
Tj^ = 0.66 , Q = 0.56 . (20)
In this case the flux through the stagnation point cross section is raised by the curvatures of the flow lines (fig. 2.5) compared to the flux in creeping flow through a constant thickness film with zero interface velocity. However, in high Re-number withdrawal, the same t'lux passes the stagnation point cross section "A" as in creeping flat film flow with zero interface velocity. The liquid has to be accelerated in C thereby lowering the flux for that velocity profile com-pared to creeping flat film flow (fig. I+-I).
On the other hand, this theory for high Re-number with-drawal predicts thicker films than in the case of the low Re-and low Ca-number withdrawal theory of chapter 3. This occurs because the capillary pressure drop at the top of the menis-cus is distributed over a larger distance by the inertial forces. This results in a thicker film.
1+. 1 .2 Ne^2nian_Li^uids_on_Wires
1+8 §1+.1 . 2
B ^ ' low Re-high Co-numb«r
Ra; high Co numb«r wlthorawal th«ory
0 5 6 0.47
Fig. 1+-2 Withdrawal flux of a newtonian liquid in high Re-nimiber withdrawal
Fig. 1+-3 Withdrawal flux of a non-newtonian liquid in high Re-number withdrawal
wires. The flux of the constant thickness film with zero interface velocity is shown in fig. 1+-2. This theory for high Re-number wire withdrawal again yields lower fluxes than the low Re- and high Ca-number theory (fig. 2-9), because of the absence of the curvatures of the streamlines. And again the theory predicts higher fluxes than the low Re-, low Ca-number theory (fig. 3-9) because of extension over a larger distance of the capillary pressure drop by inertial forces.
I4.2 Power Law Liquids on Sheets
For a power law liquid we also may assume that the flux is the same as the flux through a flat film with zero inter-face velocity. The equation describing the flow of a non-newtonian liquid in a flat film is the same as in chapter 2:
- ^ = P g . (38)
Here again the boundary conditions and the condition that determines the boundary are: the no slip condition at the wall, no tagential force and zero velocity at the interface. The flux is shown in fig. 1+-3 together with the flux for low Re- and high Ca-number withdrawal.
I+.3 Regions of Applicability of the three Withdrawal The-ories
§l+.3
k9
for which the three theories for the vertical withdrawal of a flat plate out of a newtonian liquid apply. There are three intermediate Ca- and Re-number regions between the three with-drawal theories, where none of the theories hold exactly. We will first estimate those regions for which the two bordering theories give equally reliable predictions of the withdrawal rate. After that we will determine the Ca- and Re-nimibers for which the three theories have a high validity.
1+.3.1 The Transition from Creeping Withdrawal with Capillary Forces to Withdrawal without Capillary Forces TChap- ~ ters 2 and 3")
Since we may assume that in high Ca-number withdrawal both the interface radius (R') and the distance (l) along the plate over which the interface radius changes from this value to a flat interface approximate 5h (fig. 2-1+), we can write for the capillary pressure gradient in withdrawal without capillary forces: _a_
^ 5h _ a
(?)
dx^ ''^ 5h 25h2cap
(91)
The capillary pressure gradient in withdrawal under the influ-ence of capillary forces (chapter 3, eq. 52) is:
f = - ^ ^ . (92) a/6ah
Both theories will hold about equally well when the capillary pressure drop is the same in both theories:
° "• ^° , or when Ca ^^ 0.2 . (93) a/6 ah
We may assume that the theory for withdrawal without capil-lary forces of chapter 2 holds, when the capilcapil-lary pressure gradient can be ne;^lected compared to gravity:
| £ = 2 ^ i ^pg, or when Ca è 1 . (9I+)
The theory for withdrawal under the influence of capillary forces of chapter 3 will hold if the capillary pressure gra-dient is much higher than gravity:
50 §1+,3.1 o. aUÊï V ï i v 0.4
1
1 02 0 -^„.y^ ^'NghCcHumticr / th«ory y;>'^GBn«ral low R«-^^>^'^numb«r theory _ • bw Ca-number thaory 1 1 1 1 1 10 - Ca.Fig. 1+-1+ Low Re-number withdrawal of a newtonian liquid on a flat plate
<ip - 2q „
dï
-
—^
>
5pg
,
a/6ah
or when Ca £ 10"3.
(95)
Both theories are shown in fig. 1+_U together with the limits of their applicability. The general relation for the withdrawal flux Q and the Ca-number is ob-tained through a graphical in-terpolation.
1+.3.2 The Trans it ion_ from Withdrawal with Inert ial_Forces tö~WÏthdrawai~withöüt~ïnëHial~FÖrcës~in~t^
öf~Capiiïary~FÖrces Ichaptërs 2 and 57
The inertial and viscous forces have to be the same in the middle of the transition region between the two theories for withdrawal with and without inertial forces:
pv2h _ 1
^¥
h 1 = - ^ = R e v = ^ ^ 1 Fo Re n (96)In withdrawal without inertial forces the ratio n approxi-mates (chapter 2, fig. 2-1+):
n = i Ri 10 . h
(97)
The transition from the high to the low Re-number theory will therefore occur when:
Re pvh ^ ^ 10 (98)
We may assume that the theory for withdrawal without inertial forces of chapter 2 will hold when:
Re Y < 0.1 , or Re è 1 . (99)
Viscous forces will then be able to change the velocity pro-files within the required distance 1 along the plate.
Fur-§l+.3.2 51
thermore we may assume that the theory of chapter 1+ for with-drawal with appreciable inertial forces will hold when:
Re Y è 10 , or Re è 100 . (lOO)
Inertial forces will then determine the length needed to change the velocity profiles in the meniscus region. The for-mation of waves at these high Re-nimibers will probably
re-strict the applicability of this theory considerable.
1+.3.2 The Transition from Withdrawal under the Influence of Capillary Forces wïtïi ïnertiaï Forces to Withdrawal withoüt~ïnirtiai Forces Tchapters 3 and"^7~
The transition from the withdrawal theory with inertial forces to the theory without inertial forces will occur when the inertial and viscous forces are the same:
pv2h 1 „ h - Re = •^s- = Re T ~ — V T Fo 1 n
(101)
In this case of withdrawal under the influence of the capil-lary forces the ratio n is (eq. 51):
n= 1 =
J : M. (102)
h h
Therefore, in this case the transition from withdrawal under the influence of inertial forces to withdrawal without in-ertial forces will occur when:
Re = J ^ = - 3 ^ . (103) h Ca3
We may assmne that the withdrawal theory of chapter 3 ne-glecting inertial forces will apply for:
Re é ^ , (101+) Ca3
and that the high Re-number theory of chapter 1+ will apply for:
Re è -^2 • (105) Ca3
52 §l+.3.2 R « . 10 pvh 1 , f 10 1 ,A* 10 \ \ low Co tow Re-number 10 10 \ h i g h He-number . \ theory ^ \ ch. 4 ^ \ V 1 1 1 low Re-high Ca-number theory " ch. 2 I • a 2 10 1 10 IC C a . " - ^ "/, 1ö' . 2 10 . 3 10 i n * highRé-numbérlheoj>^^^ ' ^ "' c h 4 ^^^""^ ^ ^ J^ t ^ - ^ ^ ^ ' ^ g h Ca-1 1 X <1ow Renumber A. ^ \ theory - \ ^ \ c h . 2 i \ "^ ^\ \ ^ ^\ -\ \ ^ >. \ k>wCa-lowRe\ ^ \ -number \ ^ N theory \ ^ ch.3 \ \ Ö^ 1Ö* 1Ö' 1 1 0 10 • Ns/m'
Fig. 1+-5 Regions of Fig. 1+-6 Regions of applica-bility of the three withdrawal applica-bility of the three withdrawal theories in a Re- versus Ca- theories on a n versus v plot number plot for p = 1000 kg/m^ and a =
0.07 N/m
Fig. 1+-5 shows the regions where the three withdrawal theories apply in a Ca- versus Re- number plot. The disad-vantage of this plot is that we have to know the film thick-ness in advance in order to determine the Re-number. There-fore the regions are shown as well on a viscosity (n) versus withdrawal speed (v) plot, for specified other fluid proper-ties (a, pg; fig. 1+-6).
