- Email: [email protected]
Flow of a laminar liquid film down a vertical surface
FLOW
OF A LAMINAR
LIQUID FILM SURFACE
DOWN
A VERTICAL
A. DONIEC Institute of Chemical Engineering, Lodz Technical University, Poland (Received
9 July 1986; accepted
in revised form
11 December
1986)
Abstract-A theory of laminar film flow-down of an arbitrary width is presented taking into account the conditions occurring at the film edges. Using solutions of the minimum energy equations the film shape is described. The shape consists of a central rectangular segment and two side segments with a curved surface. Conditions are determined under which the film of minimum thickness is formed. Dependences determining linear wetting density (specific wetting rate) and mean liquid velocity in the minimum film of a given width as well as simplified relations for infinitely wide film are presented. Numerical calculations in the range 10m3 to 4 x lo* mm-s of the parameter characterizing the liquid and for contact angle ranging from 5” to 90” have been performed. Results of calculations are presented on graphs. For films much wider than the minimum rivulet the results of simplified calculations (infinite width) differ insignificantly from exact calculations.
INTRODUCTION
necessary for breaking the film (Hobler, 1964; Mikielewicz and Moszyriski, 1976). In this approach energy of the system is taken into account only at the flow-down of rivulets whose cross-section is assumed to be a segment of a circle (Bankoff, 1971; Mikielewicz and Moszyhski, 1976). A reverse can be also done, i.e. finding an equilibrium shape of liquid cross-section for a determined amount of liquid flowing down a vertical surface (Don&, 1984). This shape is defined by the minimum total energy of the system. A more precise mathematical approach enabled us to draw conclusions on the basis of the results obtained for rivulets also for the flow-down of the laminar film (Doniec, 1986).
Flow of a liquid down a vertical solid surface occurs in many industrial processes often being of a paramount importance for their efficiency. For a correct operation of such apparatus as thin layer evaporators, film heat exchangers, wetted wall columns etc. it is necessary to maintain a liquid film flowing uniformly down during the operation. This very important problem comprises two aspects. One is the determination of a minimum wetting rate with a corresponding film thickness. The other is film stability. The minimum wetting rate is the minimum amount of liquid that should be provided in a time unit to ensure that the surface being wetted is covered completely with a liquid. The liquid forms a film whose
FLOW-DOWN
thickness is therefore a minimum and critical one since a decrease in the wetting rate will cause breaking of the film. The problem of stability is connected with the determination of conditions under which the film is either broken or maintained. These problems were investigated both experimentally and theoretically. A theoretical analysis of the uniform laminar film most often refers to the case when limits are not defined (Brauer, 1956; Nusselt, 1916) which corresponds to the indefinitely wide liquid film with free surface parallel to the solid surface. A change in general
OF A SINGLE RIVULET
The liquid flowing down a vertical surface, flat on the solid side, forms a free surface on the other side. The shape of this surface is a result of forces acting in the system. For a small amount of liquid flowing the free surface comes into contact with the solid surface and thus a rivulet is formed. The curve created at the cross section of the surface perpendicular to flow direction with the free surface is the rivulet profile (Fig. 1). The point of contact of the profile with the flow surface is the point of contact of three phases. A tangent to the profile at that point together with the flow surface forms an angle being a measure of interrelationships of surface forces of a solid, liquid and gas. This angle, called the contact angle, is a
flow-down conditions may result in breaking of the film at some point and formation of a dry spot. From that point on the liquid will flow down as rivulets. The rivulet flow-down as an alternative of the film flowdown is also a basis for determining the critical film thickness, minimum wetting rate and stability conditions. In some theoretical studies the authors start with the equilibrium of forces at the point of liquid stagnation over the dry spot (Hartley and Murgatroyd, 1964; Munakata et al., 1975; Ponter et ul., 1967). In other studies the energy of the system in a film and rivulet flow-down is compared (Bankoff, 1971) and the minimum energy of the rivulet pattern is assumed to be
Fig. 1. Single rivulet profile. 847
A.
848
DONIEC
property of a three-phase system determining the relation between interfacial energies of liquid-gas, solid-liquid and solid-gas systems, which is illustrated by Young and Dupre’s equation OS1- CS” ~ = case. Cl”
principle of minimum energy was employed to determine the profile z(x) of flowing Liquid. Considering the flow-down of a laminar fully developed rivulet it was assumed that its profile was symmetrical and that there was no mass and heat transfer in the system. Total energy of such a three-phase system consists of kinetic and interfacial energies: E = Ek+E,,+Ef,.
neglected as it is assumed to be used wholly for overcoming the forces of viscosity of resistance. Thus, the total energy of the rivulet segment of
E = 2o,,Ay
by
r(W,zS+,/~+cosB)dx. s Cl
(3)
The volumetric flow rate corresponds rium shape of the flowing rivulet
to the equilib-
Q = ;FIz3dx.
(4)
Hence, searching for the profile z(x), from the curves connecting the axes z and x this one is chosen which ensures that the system has a minimum energy at a determined flow rate. This is a so-called isoperimetric problem. After completing the operations of variational calculus adequate for this type of problem, the ordinary differential equation of the second order is obtained
(Doniec,
2” (1 + z’212’3
For
the physical
the dependence ness follows S=
3 WZ lOWI
J
conditions
considered,
determining
the laminar
(7) from
eq. (7)
rivulet thick-
~)1’5(~)1’5(l-cos8)1”.
(8)
Hence, the rivulet thickness depends only on the physical parameters of the system, i.e. on the type of liquid and substrate material. Therefore, the fraction including physicochemical parameters of the liquid can be taken determined
as a value characterizing
the liquid and
by eq. (9)
(2)
The kinetic energy is determined by a parabolic velocity distribution in the rivulet. Surface energy is a sum of energies on the solid-liquid interface and on the liquid free surface. The potential energy of gravity is
Ay is expressed
z=
(1)
The equilibrium, stationary free surface of the flowing liquid is formed in such a way as the condition of minimum total energy of the system be fulfilled. In the theory of laminar flow-down described earlier (Doniec, 1984) using the variational calculus the
length
such points in which the curve z(x) attains the maximum or minimum values. For x = 0 the curve z(x) has the smallest curvature and the function reaches then the maximum value
In the problems of liquid motion with free surface, the parameter M has a similar importance as a capillary constant in static capillary phenomena. A result of numerical integration (Doniec, eq. (6) is the rivulet profile z(x) and width
1986) of x = z(r).
Thus, it is possible to calculate mean liquid velocity, flow rate and Reynolds number. The calculations were made for various values of the parameter M characterizing the liquid and for the contact angle not exceeding 90”. Figures 2 and 3 present curves for the respectively. velocity, width and mean rivulet Moreover, from the analysis of eq. (6) it follows that the rivulet described by the curves is the smallest one that can be formed in a given system. Hence, values calculated for the rivulet are the minimum values. This means that for a solid-liquid-gas system there is a
b
mm
1984)
= 5w,z4
-
3w*z2
z=o z’=
--go
>
x = r. (5)
Equation (5) cannot be solved analytically. possibility is a single integration dz -= dx
1J
(W,z3
(W,z3-
-
w,z5
The
+ cos 8)2
w,z5+cos0)+
.
only
(6)
However, it appeared advantageous that the right hand side of eq. (5) was a curvature of the curve z(x) describing the cross-section profile of the rivulet. The investigation of this curvature and thus of the function expressed by the right hand side of eq. (5) leads to the determination of the vertices of the curve z(x), i.e. of
Fig. 2. Rivulet width depending on contact angle and type of liquid.
Laminar liquid film flow
849
(Muszynski and Myszkis, 1984). The solution composed of the segment of the straight line z = 6 and
2L Wm
integral
F 20
curve z(x) is displayed z=
(O, r - r*in)
6
XE
z(x)
XE(r-rminrr)
-I
(10)
In terms of flow conditions this means that the shape of the rivulet cross-section comprises the straight-line
72
segment and calculated profile of half of the minimum rivulet. There can be an infinite number of such solutions and each of them corresponds to a determined value of r for a given system. For r’s large
8
0
in Fig. 4.
20
LO
80 9
Fig. 3. Mean liquid velocity in a rivulet as a function of contact angle and liquid type.
minimum flow rate, Q,,,, necessary to form a uniform rivulet of the width bmi,. For Q smaller than Qmin instead of a rivulet one should expect single droplets flowing down. A question arises how the rivulet geometry will change when the flow rate is bigger than the minimum one. It was proved that in such a case the rivulet width would increase and its maximum thickness (height) would not change. This conclusion is in close agreement with others resulting from the discussion of solution of eq. (6). FLOW-DOWN
OF A LAMINAR
FILM
Numerical calculations provide particular solutions of eq. (6), each being determined in some interval such that for an arbitrary x from this interval z(x) # 6. The set of these solutions is a general solution and it is contained in a strip limited by the axis Ox, segment 06 of the axis Oz and half-line z = d that half-line excluded. This strip is a set of explicitness points of eq. (6). The half-line z = 6 is a singular solution of eq. (6) and, at the same time, it is a set of uniqueness points. Both these solutions satisfy the initial condition z(0) = 6. There can be also solutions “composed” of the particular solution and part of the singular solution
enough (in popular meaning) the rivulet transforms into a film. Therefore, the theory presented earlier and the resulting equations refer to both the minimum rivulet and to the film. When the wetting of a vertical plate occurs by a liquid emerging from a vertical orifice a sufficiently small amount of liquid flows out tiom the nozzle, single droplets slide down the surface. With an increase in liquid feed rate the frequency of droplet outflow grows until the moment when the droplets form a rivulet. This is the minimum rivulet. A further increase in the amount of liquid causes a widening of the smallest rivulet which becomes wider but is ofthe same thickness. Finally, the rivulet is transformed into a film consisting of a cuboidal segment and two extreme ones which are halves of the minimum rivulet (Fig. 5). This rivulet has been “cut” along the flow axis (symmetry axis) and between symmetrical parts an excess of liquid over the amount required for the minimum rivulet formation has been placed. The thickness of film in its central part is therefore determined by eq. (8). This is
Fig. 4. Solutions of eq. (3); a “composed”
Fig. 5. Laminar film.
solution.
A. DONIEC
850
the thickness of the thinnest liquid film that can be formed in a given system at constant flow conditions and physico-chemical parameters of the system. Thus, it is also the thickness (height) of the minimum rivulet. MINIMUM
Wetting of by mass flow wetting rate. down in time surface. This
WE’MXNG
RATE
a vertical surface is usually rate per unit width I called T is defined as the mass of unit vs. the unit of width can be written as
r2c.
characterized also a specific liquid flowing of the wetting
(11)
b
The film flow rate Q is the sum of flow rates in the cuboidal segment for which all dependences of the Nusselt theory are valid, and in side segments. For a given three-phase system, i.e. for a given M and @ there is only one rivulet of the thickness 6 and width b,,, = 2rmi,,. Hence, the minimum flow rate necessary to form a film of minimum thickness and width b, or to cover the surface of a given width b, with a uniform film, is equal to Q = Qmin + which
as a specific
$6’
wetting
(b - b,,,J
(12)
rate has the form (13)
It is known that irrespective of the surface condition, for the same solid the values of minimum wetting rate can be different. This refers also to the case when the wetting surface is wet already. Equation (13) encompasses all these cases since Qminr bmin and 6 depend on the contact angle which for a given liquid is determined by the state of the solid surface. Mean velocity of liquid in the film can be determined from the volume flow rate (12)
Qmin+~~3(b-bmin) ,
wnl= 2
s0
z(x) dx + 6 (b -
(14)
b,,,J
Characteristic values for the minimum rivulet in eqs (12)(14) can be calculated upon integration of eq. (6). This, however, requires the use of a computer. For films wide enough these values are less important and the respective expressions are simplified. Then, the film mean velocity is approximately the same as in the cuboidai segment. Thus, after taking into account the film thickness (8) the formulas for film mean velocity and minimum wetting rate have the forms
W,
=
;(‘i’)l’pJso2/1
This refers strictly to the film of an infinite width. Equations (13)-(16) can be used to calculate the Reynolds number for a continuous film of an arbitrary width b. Hence, this will be the lowest Reynolds number for a given three-phase system.
(1
-eoso)2/5
(15)
CALCULATIONS
AND
DISCUSSION
In order to investigate and generalize the properties of the proposed description of laminar film flow calculations were made for various values of the parameter M from the interval 1O-3 to 4 x IO4 mrnm5. This interval has been determined on the basis of the value M for glycerol and diethyl ether. These liquids, being much different as far as their physico-chemical properties are concerned, have the M values which can be taken as extreme ones for a practically useful interval. At 20°C M = 1.372x lop3 mm-5 for gIycero1 and M = 3.779 x lo4 mm ’ for diethyl ether. For each value of M calculations were made using full and simplified equations and changing the contact angle from 5” to 90”. Differences between exact and approximate results are insignificant (the biggest one is for M = 1O-3 mmP5 and does not exceed 4%) both for the film 1000 mm wide and for a narrower one (100 mm). For very small contact angles (0 = 1”) and width of the wetting surface smaller than 100 mm the deviation is bigger and can reach even 100/o. Results obtained for extreme values of M (Table 1) determine the range of critical film thickness and the interval 1.74 x 10m3 to 6.24 x lo- 1 kg/(ms) in which the minimum values of wetting rate for most wetting systems should be contained (Fig. 6). The difference between flow of a single rivulet and film was observed in the run of curves illustrating the dependence of Re on 8 (Fig. 7). It is characteristic that the ratio of mean velocity of infinitely wide film and minimum rivulet is constant for a given contact angle irrespective of M in the interval 1.5095-1.242. To verify the proposed theory a comprehensive comparison with experimental data should be performed. Such a comparison, however, exceeds the scope of the present work and will be carried out separately. In an attempt of initial evaluation two classical studies on the flow down a flat surface may be used. Dukler and Bergelin (1952) wetted a vertical plate made from stainless steel with water at 25°C thus obtaining f
= 0.1083
kg/(ms);
6 = 0.305
mm.
The authors did not determine the contact angle in the system considered. Using eqs (8) and (lo), calculations can be performed for various values of the contact angle. Results most close to experimental ones are obtained for 0 = 75”: I- = 0.1085
kg/(ms);
S = 0.310
mm.
This value of the contact angle is within the range of 54” to 89” determined (Krell, 1967) for stainless steel, depending on roughness, and it is closer to those for
Laminar Table M (mm - ‘) 4x
lo4
10
3
851
liquid film flow
1. Results of calculations for a film 1000 mm wide for extreme values of M 0 (degree)
(mm)
6
5 90 5 90
0.057 0.174 1.895 5.775
%I (m/s) 3.34 3.1 0.8 7.44
x 10-Z X lo-’ X lo-* x 10-Z
lkg/(ms) 1.75 4.94 2.12 6.24
X x x x
RtZ
10-J lo-* 1o-2 IO-’
23.8 674.3 0.04 1.17
who determined
minimum wetting rates for various surfaces wetted with water and glycerol-water mixtures the comparison yields worse results. Calculations, in which the contact angle measured by these authors was used, differ significantly from experimental ones. In the case of stainless steel, for instance,
for which they gave 0 = 36” it is as follows:
Fig. 6. Dependence of minimum wetting rate on constant M characterizing the liquid (------water at 20°C).
l’- = 0.1764 kg/(ms)
experimental
I = 0.0525 kg/(ms)
calculated.
To make the experimental and calculated values of I similar, the contact angle should be much bigger than that determined experimentally and should exceed 90” It is difficult to specify clearly the reason for such a big discrepancy. In these experiments a significant role is played by the way in which the momentum corresponding to the minimum wetting rate is defined, the determination of the state of solid surface and correct determination of the contact angle. It should be noted that the values of contact angle determined by Hobler and Czajka (1968) for all systems investigated differ significantly from those presented in the literature. The above considerations do not depreciate the theory but it becomes more evident that the experiment should be performed in strictly determined conditions at explicitly defined measurement criteria. DIMENSIONLESS In considerations surface flow is numbers of Bond,
APPROACH
of hydrodynamic problems “free” characterized by dimensionless Froude, Reynolds and Weber. A
quadruple film thickness is assumed as an equivalent linear dimension. Note that the combination of Fr, Re and We numbers -z
FF.2
P20
Re' We
p3g2(4W
(17)
has at the right hand side the expression identical to that obtained after transformation of eq. (8) for the minimum film thickness Fig. 7. Reynolds number for the minimum rivulet and film as a function of contact angle (-~~~~ ~ film).
smooth surfaces. In this case the calculated and experimental results are in especially good agreement. For the data presented by Hobler and Czajka (1968)
7
1
45 x 45 1 -case
PZW p3gZ(46)5.
(18)
Comparing the left hand side of eqs (17) and (18) and taking into account that for laminar motion
A. DONIEC
852
and hence Re/Fr
= 48, one obtains We = :(I
-cost+
(19)
The Weber number is a measure of the rate of inertia force to surface tension force. In the system considered the surface tension force, as a resultant of three interfacial tensions, should be expressed together with the contact angle. In such an approach the Weber number can be modified in the way which takes into account interactions of three phases We =
wit&p
cr(1 -ccosf?)’
(20)
Thus, from eq. (19) We=:.
(21)
Also in eq. (18) the surface tension and contact angle should be handled jointly. Then, comparing both dimensionless fractions, two complementary conditions are obtained: W:6P a(1 -cos@)
5 7
(22)
The latter, let it be called the thickness number (of the rivulet and film), determines the conditions under which the film of the minimum thickness is formed. That condition has a constant value for any threephase system. This means that for a system whose physical parameters are determined, there can be a film of a minimum thickness such that eq. (23) be fulfilled. This also means that any change of any value in fraction (23) should be followed by an equivalent change of another value. In fact, only the change of thickness as a result of changes in other physical values is encountered here. It is rather improbable that a local change in surface tension could cause a change of density or viscosity to satisfy condition (23). Thickness is the element which should be altered. Among the reasons of changes in surface tension (but also in density and viscosity) mass and heat transfer phenomena are most frequent. Concentration and thermal fields on liquid surface are not homogeneous then. If at any point of the film the surface tension is decreased, the thickness also decreases at that point. Reduction of the film thickness promotes its breakdown but is not sufficient to cause it. Assume, however, that just at this point the film has been ruptured. A newly formed configuration (a “dry patch”, rivulet) has still lower surface tension at the edges and should satisfy condition (23) from which it follows that the contact angle should be now bigger at the thickness not lower than the minimum one for a given system. Therefore, the change of contact angle must be related to the change of components of force
which determine this angle. Hence, film breakdown occurs when on the solid surface conditions are created to form an increased contact angle on the edges of the dry patch. Then, despite of decreased surface tension at the edges, the film “contracting” forces are larger than “spreading” forces and tend to form an equilibrium configuration
In other words, in the place where the dry patch is formed the solid has smaller surface energy and the contact angle at the edge of the dry patch is higher than that corresponding to a homogeneous surface. Naturally, it is also possible that for thin films the surface nonhomogeneity alone is the factor sufficient for breaking up the films (Fig. 8). This is confirmed by common observations from which it follows that on the flow surface there are “privileged” spots that usually remain unwetted or are exposed when the wetting rate is altered. An increase of surface tension should be followed by a local increase of film thickness and as a result, the film should be reinforced and after levelling the liquid properties (balancing of concentration) the thickness of the whole film increases. Such an interpretation of condition (23t_despite a reverse of the cause and result-is in good agreement with the explanation of liquid film behaviour during mass transfer given by Zuiderweg and Harmens (1958). A result of condition (22) generated from a modified Weber number is the relationship between mean film velocity and film thickness, the consequence of which becomes evident in the light of the above considerations concerning condition (23). The combination of both these equalities produces a Nusselt equation for mean liquid velocity in an infinitely wide film which is expressed by eq. (15) for the film of minimum thickness. For these conditions the minimum Reynolds number can be obtained using eq. (15) or (16) Re = i(fj5)15(&)1’5d15t1
-cosB)315.
(24)
The dimensionless fraction
9P4 --y =
(25)
Ca
PO
is known as the capillarity buoyancy number. Introduction of the function of the contact angle to the capillarity number allows the Reynolds number to be written concisely as Re = 4.072Ca,-
‘I’.
(26)
Fig. 8. Film breakdown (------the case corresponding to unchanged surface energy of a solid body).
iaminar
iiquid film flow
Using the capillarity number whose reverse was called the film number by Brauer, a critical value of the
Reynolds number determining the transition from a laminar to turbulent motion was written as Re, =
140CK
“lo.
(27)
From eqs (26) and (27) a relationship between the critical and minimum Reynolds numbers can be obtained Re
2
Re
= 34.4Ca’/‘0
(1 -cos8)“5.
(28)
The latter should be always smaller than the critical number. For the calculation interval assumed in this study none of the Reynolds number exceeded the critical value for a given M. The maximum one was 2.7 times smaller than the critical value. CONCLUSIONS
The rivulet and film are alternative forms of liquid flow down a vertical surface. For a given solid-liquid-gas system the minimum rivulet is the smallest continuous form of flow-down whose shape, height and width result from the solution of the equation for minimum system energy (5). This equation refers also to the film which is an extension of the minimum rivulet. The film profile is a solution composed of the minimum energy equation consisting of a straight line segment and two curvilinear segments that correspond to halves of the minimum rivulet profile. Flow-down in the central, flat part of the film is determined irrespective of the minimum energy condition. At the edges of the film, where the film free surface intersects the solid surface, results of surface forces of the three-phase system are observed. Therefore, the theory presented above determines boundary conditions of the film of width larger than b,i,. In this respect this theory is complementary to that presented by Nusselt which refers to flat film with undefined edges. The minimum film thickness depends only on physico-chemical properties of the system, i.e. on the contact angle and constant M characteristic for liquids 6 = 145085(l
-;“)‘:‘.
Formulas for mean velocity and minimum wetting rate of a film of the width b, contain parameters referring to the minimum rivulet, which were obtained from numerical calculations. For an infinitely wide film these formulas are reduced to dependences taken from Nusselt theory in which the critical thickness formula was taken into account. Being the simplified formulas for films of definite width, they give good approximation for most calculations. In the case of not very wide films (below 100 mm) of viscous liquids and for small contact angles (below 5”) exact equations should be applied since discrepancies may exceed 10%. Dimensionless fraction (23), called the thickness
853
number, has a constant value. An uncompensated change of any value contained in the fraction causes its increase and, therefore, film breakage. This condition can be taken as a criterion of formation and maintenance of a laminar film with minimum thickness. From the analysis of condition (23) it follows that the only reason for the rupture of a film with a critical thickness can be a local decrease of surface energy of the solid surface. NOTATION
film width minimum rivulet width
b bmin
Ca
= 3,
Gas d, 9 E
modified capillary buoyancy number = 45 equivalent linear dimension acceleration of gravity energy
Fr
= 2,
:
Qmin r I ITll” Re Re, W, WI,
w2
W,
w2
capillary buoyancy number
Froude number
consiant characterizing the liquid (9) volume flow rate volume flow rate of minimum rivulet half film width half minimum rivulet width w,d,p 4l= ~ = Reynolds number P P’ critical Reynolds number mean velocity constants = E constant in eq. (5) 2 25 1’5 = _-( 1 iV3j5(1 - cos tJ)3’5 constant in 3 147 eq. (5)
n&&p
’ Weber number
We
= -,
Weti x,y,=
modi&d Weber number (20) coordinates rivulet profile
z(x) Greek I-
8 6 P P fs
letters
specific mass flow rate contact angle film thickness viscosity density surface tension REFERENCES
Bankoff,S. G., 1971, Minimum thicknessof a drainingliquid film. Znt.J. Heat Muss Transfer 14, 2143-2146. Elrauer, H., 1956, Striimung und WzXrmelbergang bei
Rieselfilmen. VDI-Forschungsheft 457, Diisseldorf. Don&, A., 1984, Lamjnar flow of a liquid down a vertical solid surface. Maximum thickness of liquid rivulet. Phys. Chem. Nydrodyn. 5, 143-152. Don&, A., 1986, Laminar flow of a liquid down a vertical solid surface. Integration of minimum system energy equation, submitted to Phys. Chem. Hydrodyn.
854
A. DONIEC
Dukler, A. E. and Bergelin, 0. P., 1952, Characteristics of flow in falling liquid films. Chem. Engng Prog. 48, 557-563. Hartley, D. F. and Murgatroyd, W., 1964, Criteria for the break-up of thin liquid layers flowing isothermally over a solid surface. Int. J. Heat Mass Transfer 7, 1003 1015. Hobler, T., 1964, Minimum surface wetting (in Polish). Chemia Stosow. 2B, 145 159. Hobler. T. and Czaika. J.. 1968. Minimum wettine of a flat surface (in Polish). dhemia S&w. 2B. 169-186 Krell, E., 1967, The solid-liquid boundary. British Chem. Engng It, 562-567. Mikielewicz, J. and Moszynski, J. R., 1976, Minimum thickness of a liquid film flowing vertically down a solid surface. In?. J. Heat Mass Transfer 19. 771-775.
Munakata, T., Watanabe, K. and Miyashita, K., 1975, Minimum wetting rate on wetted wall columncorrelation over wide range of liquid viscosity. J. Chem. Engng Japan 8, 440-444. Muszynski, J. and Myszkis, A. D., 1984, Ordinary Diflerentinl Eyuurions (in Polish). PWN, Warszawa. 1916. Die Oberftichenkondensation des Nusselt, W., Wasserdamofes. Z. Ver. dtsch. Ina. 60. 541. Ponter, A. B., -Davies, G. A., Ross, T-K. and Thornley, T. G., 1967. The influence of mass transfer on liquid film breakdown. Int. J. EIeat Mass Transfer 10, 3491359. Zuiderweg. F. J. and Harmens, A., 1958, The influence of surface phenomena on the performance of distillation columns. Chem. Engng Sci. 9, 89-103.
OF A LAMINAR
LIQUID FILM SURFACE
DOWN
A VERTICAL
A. DONIEC Institute of Chemical Engineering, Lodz Technical University, Poland (Received
9 July 1986; accepted
in revised form
11 December
1986)
Abstract-A theory of laminar film flow-down of an arbitrary width is presented taking into account the conditions occurring at the film edges. Using solutions of the minimum energy equations the film shape is described. The shape consists of a central rectangular segment and two side segments with a curved surface. Conditions are determined under which the film of minimum thickness is formed. Dependences determining linear wetting density (specific wetting rate) and mean liquid velocity in the minimum film of a given width as well as simplified relations for infinitely wide film are presented. Numerical calculations in the range 10m3 to 4 x lo* mm-s of the parameter characterizing the liquid and for contact angle ranging from 5” to 90” have been performed. Results of calculations are presented on graphs. For films much wider than the minimum rivulet the results of simplified calculations (infinite width) differ insignificantly from exact calculations.
INTRODUCTION
necessary for breaking the film (Hobler, 1964; Mikielewicz and Moszyriski, 1976). In this approach energy of the system is taken into account only at the flow-down of rivulets whose cross-section is assumed to be a segment of a circle (Bankoff, 1971; Mikielewicz and Moszyhski, 1976). A reverse can be also done, i.e. finding an equilibrium shape of liquid cross-section for a determined amount of liquid flowing down a vertical surface (Don&, 1984). This shape is defined by the minimum total energy of the system. A more precise mathematical approach enabled us to draw conclusions on the basis of the results obtained for rivulets also for the flow-down of the laminar film (Doniec, 1986).
Flow of a liquid down a vertical solid surface occurs in many industrial processes often being of a paramount importance for their efficiency. For a correct operation of such apparatus as thin layer evaporators, film heat exchangers, wetted wall columns etc. it is necessary to maintain a liquid film flowing uniformly down during the operation. This very important problem comprises two aspects. One is the determination of a minimum wetting rate with a corresponding film thickness. The other is film stability. The minimum wetting rate is the minimum amount of liquid that should be provided in a time unit to ensure that the surface being wetted is covered completely with a liquid. The liquid forms a film whose
FLOW-DOWN
thickness is therefore a minimum and critical one since a decrease in the wetting rate will cause breaking of the film. The problem of stability is connected with the determination of conditions under which the film is either broken or maintained. These problems were investigated both experimentally and theoretically. A theoretical analysis of the uniform laminar film most often refers to the case when limits are not defined (Brauer, 1956; Nusselt, 1916) which corresponds to the indefinitely wide liquid film with free surface parallel to the solid surface. A change in general
OF A SINGLE RIVULET
The liquid flowing down a vertical surface, flat on the solid side, forms a free surface on the other side. The shape of this surface is a result of forces acting in the system. For a small amount of liquid flowing the free surface comes into contact with the solid surface and thus a rivulet is formed. The curve created at the cross section of the surface perpendicular to flow direction with the free surface is the rivulet profile (Fig. 1). The point of contact of the profile with the flow surface is the point of contact of three phases. A tangent to the profile at that point together with the flow surface forms an angle being a measure of interrelationships of surface forces of a solid, liquid and gas. This angle, called the contact angle, is a
flow-down conditions may result in breaking of the film at some point and formation of a dry spot. From that point on the liquid will flow down as rivulets. The rivulet flow-down as an alternative of the film flowdown is also a basis for determining the critical film thickness, minimum wetting rate and stability conditions. In some theoretical studies the authors start with the equilibrium of forces at the point of liquid stagnation over the dry spot (Hartley and Murgatroyd, 1964; Munakata et al., 1975; Ponter et ul., 1967). In other studies the energy of the system in a film and rivulet flow-down is compared (Bankoff, 1971) and the minimum energy of the rivulet pattern is assumed to be
Fig. 1. Single rivulet profile. 847
A.
848
DONIEC
property of a three-phase system determining the relation between interfacial energies of liquid-gas, solid-liquid and solid-gas systems, which is illustrated by Young and Dupre’s equation OS1- CS” ~ = case. Cl”
principle of minimum energy was employed to determine the profile z(x) of flowing Liquid. Considering the flow-down of a laminar fully developed rivulet it was assumed that its profile was symmetrical and that there was no mass and heat transfer in the system. Total energy of such a three-phase system consists of kinetic and interfacial energies: E = Ek+E,,+Ef,.
neglected as it is assumed to be used wholly for overcoming the forces of viscosity of resistance. Thus, the total energy of the rivulet segment of
E = 2o,,Ay
by
r(W,zS+,/~+cosB)dx. s Cl
(3)
The volumetric flow rate corresponds rium shape of the flowing rivulet
to the equilib-
Q = ;FIz3dx.
(4)
Hence, searching for the profile z(x), from the curves connecting the axes z and x this one is chosen which ensures that the system has a minimum energy at a determined flow rate. This is a so-called isoperimetric problem. After completing the operations of variational calculus adequate for this type of problem, the ordinary differential equation of the second order is obtained
(Doniec,
2” (1 + z’212’3
For
the physical
the dependence ness follows S=
3 WZ lOWI
J
conditions
considered,
determining
the laminar
(7) from
eq. (7)
rivulet thick-
~)1’5(~)1’5(l-cos8)1”.
(8)
Hence, the rivulet thickness depends only on the physical parameters of the system, i.e. on the type of liquid and substrate material. Therefore, the fraction including physicochemical parameters of the liquid can be taken determined
as a value characterizing
the liquid and
by eq. (9)
(2)
The kinetic energy is determined by a parabolic velocity distribution in the rivulet. Surface energy is a sum of energies on the solid-liquid interface and on the liquid free surface. The potential energy of gravity is
Ay is expressed
z=
(1)
The equilibrium, stationary free surface of the flowing liquid is formed in such a way as the condition of minimum total energy of the system be fulfilled. In the theory of laminar flow-down described earlier (Doniec, 1984) using the variational calculus the
length
such points in which the curve z(x) attains the maximum or minimum values. For x = 0 the curve z(x) has the smallest curvature and the function reaches then the maximum value
In the problems of liquid motion with free surface, the parameter M has a similar importance as a capillary constant in static capillary phenomena. A result of numerical integration (Doniec, eq. (6) is the rivulet profile z(x) and width
1986) of x = z(r).
Thus, it is possible to calculate mean liquid velocity, flow rate and Reynolds number. The calculations were made for various values of the parameter M characterizing the liquid and for the contact angle not exceeding 90”. Figures 2 and 3 present curves for the respectively. velocity, width and mean rivulet Moreover, from the analysis of eq. (6) it follows that the rivulet described by the curves is the smallest one that can be formed in a given system. Hence, values calculated for the rivulet are the minimum values. This means that for a solid-liquid-gas system there is a
b
mm
1984)
= 5w,z4
-
3w*z2
z=o z’=
--go
>
x = r. (5)
Equation (5) cannot be solved analytically. possibility is a single integration dz -= dx
1J
(W,z3
(W,z3-
-
w,z5
The
+ cos 8)2
w,z5+cos0)+
.
only
(6)
However, it appeared advantageous that the right hand side of eq. (5) was a curvature of the curve z(x) describing the cross-section profile of the rivulet. The investigation of this curvature and thus of the function expressed by the right hand side of eq. (5) leads to the determination of the vertices of the curve z(x), i.e. of
Fig. 2. Rivulet width depending on contact angle and type of liquid.
Laminar liquid film flow
849
(Muszynski and Myszkis, 1984). The solution composed of the segment of the straight line z = 6 and
2L Wm
integral
F 20
curve z(x) is displayed z=
(O, r - r*in)
6
XE
z(x)
XE(r-rminrr)
-I
(10)
In terms of flow conditions this means that the shape of the rivulet cross-section comprises the straight-line
72
segment and calculated profile of half of the minimum rivulet. There can be an infinite number of such solutions and each of them corresponds to a determined value of r for a given system. For r’s large
8
0
in Fig. 4.
20
LO
80 9
Fig. 3. Mean liquid velocity in a rivulet as a function of contact angle and liquid type.
minimum flow rate, Q,,,, necessary to form a uniform rivulet of the width bmi,. For Q smaller than Qmin instead of a rivulet one should expect single droplets flowing down. A question arises how the rivulet geometry will change when the flow rate is bigger than the minimum one. It was proved that in such a case the rivulet width would increase and its maximum thickness (height) would not change. This conclusion is in close agreement with others resulting from the discussion of solution of eq. (6). FLOW-DOWN
OF A LAMINAR
FILM
Numerical calculations provide particular solutions of eq. (6), each being determined in some interval such that for an arbitrary x from this interval z(x) # 6. The set of these solutions is a general solution and it is contained in a strip limited by the axis Ox, segment 06 of the axis Oz and half-line z = d that half-line excluded. This strip is a set of explicitness points of eq. (6). The half-line z = 6 is a singular solution of eq. (6) and, at the same time, it is a set of uniqueness points. Both these solutions satisfy the initial condition z(0) = 6. There can be also solutions “composed” of the particular solution and part of the singular solution
enough (in popular meaning) the rivulet transforms into a film. Therefore, the theory presented earlier and the resulting equations refer to both the minimum rivulet and to the film. When the wetting of a vertical plate occurs by a liquid emerging from a vertical orifice a sufficiently small amount of liquid flows out tiom the nozzle, single droplets slide down the surface. With an increase in liquid feed rate the frequency of droplet outflow grows until the moment when the droplets form a rivulet. This is the minimum rivulet. A further increase in the amount of liquid causes a widening of the smallest rivulet which becomes wider but is ofthe same thickness. Finally, the rivulet is transformed into a film consisting of a cuboidal segment and two extreme ones which are halves of the minimum rivulet (Fig. 5). This rivulet has been “cut” along the flow axis (symmetry axis) and between symmetrical parts an excess of liquid over the amount required for the minimum rivulet formation has been placed. The thickness of film in its central part is therefore determined by eq. (8). This is
Fig. 4. Solutions of eq. (3); a “composed”
Fig. 5. Laminar film.
solution.
A. DONIEC
850
the thickness of the thinnest liquid film that can be formed in a given system at constant flow conditions and physico-chemical parameters of the system. Thus, it is also the thickness (height) of the minimum rivulet. MINIMUM
Wetting of by mass flow wetting rate. down in time surface. This
WE’MXNG
RATE
a vertical surface is usually rate per unit width I called T is defined as the mass of unit vs. the unit of width can be written as
r2c.
characterized also a specific liquid flowing of the wetting
(11)
b
The film flow rate Q is the sum of flow rates in the cuboidal segment for which all dependences of the Nusselt theory are valid, and in side segments. For a given three-phase system, i.e. for a given M and @ there is only one rivulet of the thickness 6 and width b,,, = 2rmi,,. Hence, the minimum flow rate necessary to form a film of minimum thickness and width b, or to cover the surface of a given width b, with a uniform film, is equal to Q = Qmin + which
as a specific
$6’
wetting
(b - b,,,J
(12)
rate has the form (13)
It is known that irrespective of the surface condition, for the same solid the values of minimum wetting rate can be different. This refers also to the case when the wetting surface is wet already. Equation (13) encompasses all these cases since Qminr bmin and 6 depend on the contact angle which for a given liquid is determined by the state of the solid surface. Mean velocity of liquid in the film can be determined from the volume flow rate (12)
Qmin+~~3(b-bmin) ,
wnl= 2
s0
z(x) dx + 6 (b -
(14)
b,,,J
Characteristic values for the minimum rivulet in eqs (12)(14) can be calculated upon integration of eq. (6). This, however, requires the use of a computer. For films wide enough these values are less important and the respective expressions are simplified. Then, the film mean velocity is approximately the same as in the cuboidai segment. Thus, after taking into account the film thickness (8) the formulas for film mean velocity and minimum wetting rate have the forms
W,
=
;(‘i’)l’pJso2/1
This refers strictly to the film of an infinite width. Equations (13)-(16) can be used to calculate the Reynolds number for a continuous film of an arbitrary width b. Hence, this will be the lowest Reynolds number for a given three-phase system.
(1
-eoso)2/5
(15)
CALCULATIONS
AND
DISCUSSION
In order to investigate and generalize the properties of the proposed description of laminar film flow calculations were made for various values of the parameter M from the interval 1O-3 to 4 x IO4 mrnm5. This interval has been determined on the basis of the value M for glycerol and diethyl ether. These liquids, being much different as far as their physico-chemical properties are concerned, have the M values which can be taken as extreme ones for a practically useful interval. At 20°C M = 1.372x lop3 mm-5 for gIycero1 and M = 3.779 x lo4 mm ’ for diethyl ether. For each value of M calculations were made using full and simplified equations and changing the contact angle from 5” to 90”. Differences between exact and approximate results are insignificant (the biggest one is for M = 1O-3 mmP5 and does not exceed 4%) both for the film 1000 mm wide and for a narrower one (100 mm). For very small contact angles (0 = 1”) and width of the wetting surface smaller than 100 mm the deviation is bigger and can reach even 100/o. Results obtained for extreme values of M (Table 1) determine the range of critical film thickness and the interval 1.74 x 10m3 to 6.24 x lo- 1 kg/(ms) in which the minimum values of wetting rate for most wetting systems should be contained (Fig. 6). The difference between flow of a single rivulet and film was observed in the run of curves illustrating the dependence of Re on 8 (Fig. 7). It is characteristic that the ratio of mean velocity of infinitely wide film and minimum rivulet is constant for a given contact angle irrespective of M in the interval 1.5095-1.242. To verify the proposed theory a comprehensive comparison with experimental data should be performed. Such a comparison, however, exceeds the scope of the present work and will be carried out separately. In an attempt of initial evaluation two classical studies on the flow down a flat surface may be used. Dukler and Bergelin (1952) wetted a vertical plate made from stainless steel with water at 25°C thus obtaining f
= 0.1083
kg/(ms);
6 = 0.305
mm.
The authors did not determine the contact angle in the system considered. Using eqs (8) and (lo), calculations can be performed for various values of the contact angle. Results most close to experimental ones are obtained for 0 = 75”: I- = 0.1085
kg/(ms);
S = 0.310
mm.
This value of the contact angle is within the range of 54” to 89” determined (Krell, 1967) for stainless steel, depending on roughness, and it is closer to those for
Laminar Table M (mm - ‘) 4x
lo4
10
3
851
liquid film flow
1. Results of calculations for a film 1000 mm wide for extreme values of M 0 (degree)
(mm)
6
5 90 5 90
0.057 0.174 1.895 5.775
%I (m/s) 3.34 3.1 0.8 7.44
x 10-Z X lo-’ X lo-* x 10-Z
lkg/(ms) 1.75 4.94 2.12 6.24
X x x x
RtZ
10-J lo-* 1o-2 IO-’
23.8 674.3 0.04 1.17
who determined
minimum wetting rates for various surfaces wetted with water and glycerol-water mixtures the comparison yields worse results. Calculations, in which the contact angle measured by these authors was used, differ significantly from experimental ones. In the case of stainless steel, for instance,
for which they gave 0 = 36” it is as follows:
Fig. 6. Dependence of minimum wetting rate on constant M characterizing the liquid (------water at 20°C).
l’- = 0.1764 kg/(ms)
experimental
I = 0.0525 kg/(ms)
calculated.
To make the experimental and calculated values of I similar, the contact angle should be much bigger than that determined experimentally and should exceed 90” It is difficult to specify clearly the reason for such a big discrepancy. In these experiments a significant role is played by the way in which the momentum corresponding to the minimum wetting rate is defined, the determination of the state of solid surface and correct determination of the contact angle. It should be noted that the values of contact angle determined by Hobler and Czajka (1968) for all systems investigated differ significantly from those presented in the literature. The above considerations do not depreciate the theory but it becomes more evident that the experiment should be performed in strictly determined conditions at explicitly defined measurement criteria. DIMENSIONLESS In considerations surface flow is numbers of Bond,
APPROACH
of hydrodynamic problems “free” characterized by dimensionless Froude, Reynolds and Weber. A
quadruple film thickness is assumed as an equivalent linear dimension. Note that the combination of Fr, Re and We numbers -z
FF.2
P20
Re' We
p3g2(4W
(17)
has at the right hand side the expression identical to that obtained after transformation of eq. (8) for the minimum film thickness Fig. 7. Reynolds number for the minimum rivulet and film as a function of contact angle (-~~~~ ~ film).
smooth surfaces. In this case the calculated and experimental results are in especially good agreement. For the data presented by Hobler and Czajka (1968)
7
1
45 x 45 1 -case
PZW p3gZ(46)5.
(18)
Comparing the left hand side of eqs (17) and (18) and taking into account that for laminar motion
A. DONIEC
852
and hence Re/Fr
= 48, one obtains We = :(I
-cost+
(19)
The Weber number is a measure of the rate of inertia force to surface tension force. In the system considered the surface tension force, as a resultant of three interfacial tensions, should be expressed together with the contact angle. In such an approach the Weber number can be modified in the way which takes into account interactions of three phases We =
wit&p
cr(1 -ccosf?)’
(20)
Thus, from eq. (19) We=:.
(21)
Also in eq. (18) the surface tension and contact angle should be handled jointly. Then, comparing both dimensionless fractions, two complementary conditions are obtained: W:6P a(1 -cos@)
5 7
(22)
The latter, let it be called the thickness number (of the rivulet and film), determines the conditions under which the film of the minimum thickness is formed. That condition has a constant value for any threephase system. This means that for a system whose physical parameters are determined, there can be a film of a minimum thickness such that eq. (23) be fulfilled. This also means that any change of any value in fraction (23) should be followed by an equivalent change of another value. In fact, only the change of thickness as a result of changes in other physical values is encountered here. It is rather improbable that a local change in surface tension could cause a change of density or viscosity to satisfy condition (23). Thickness is the element which should be altered. Among the reasons of changes in surface tension (but also in density and viscosity) mass and heat transfer phenomena are most frequent. Concentration and thermal fields on liquid surface are not homogeneous then. If at any point of the film the surface tension is decreased, the thickness also decreases at that point. Reduction of the film thickness promotes its breakdown but is not sufficient to cause it. Assume, however, that just at this point the film has been ruptured. A newly formed configuration (a “dry patch”, rivulet) has still lower surface tension at the edges and should satisfy condition (23) from which it follows that the contact angle should be now bigger at the thickness not lower than the minimum one for a given system. Therefore, the change of contact angle must be related to the change of components of force
which determine this angle. Hence, film breakdown occurs when on the solid surface conditions are created to form an increased contact angle on the edges of the dry patch. Then, despite of decreased surface tension at the edges, the film “contracting” forces are larger than “spreading” forces and tend to form an equilibrium configuration
In other words, in the place where the dry patch is formed the solid has smaller surface energy and the contact angle at the edge of the dry patch is higher than that corresponding to a homogeneous surface. Naturally, it is also possible that for thin films the surface nonhomogeneity alone is the factor sufficient for breaking up the films (Fig. 8). This is confirmed by common observations from which it follows that on the flow surface there are “privileged” spots that usually remain unwetted or are exposed when the wetting rate is altered. An increase of surface tension should be followed by a local increase of film thickness and as a result, the film should be reinforced and after levelling the liquid properties (balancing of concentration) the thickness of the whole film increases. Such an interpretation of condition (23t_despite a reverse of the cause and result-is in good agreement with the explanation of liquid film behaviour during mass transfer given by Zuiderweg and Harmens (1958). A result of condition (22) generated from a modified Weber number is the relationship between mean film velocity and film thickness, the consequence of which becomes evident in the light of the above considerations concerning condition (23). The combination of both these equalities produces a Nusselt equation for mean liquid velocity in an infinitely wide film which is expressed by eq. (15) for the film of minimum thickness. For these conditions the minimum Reynolds number can be obtained using eq. (15) or (16) Re = i(fj5)15(&)1’5d15t1
-cosB)315.
(24)
The dimensionless fraction
9P4 --y =
(25)
Ca
PO
is known as the capillarity buoyancy number. Introduction of the function of the contact angle to the capillarity number allows the Reynolds number to be written concisely as Re = 4.072Ca,-
‘I’.
(26)
Fig. 8. Film breakdown (------the case corresponding to unchanged surface energy of a solid body).
iaminar
iiquid film flow
Using the capillarity number whose reverse was called the film number by Brauer, a critical value of the
Reynolds number determining the transition from a laminar to turbulent motion was written as Re, =
140CK
“lo.
(27)
From eqs (26) and (27) a relationship between the critical and minimum Reynolds numbers can be obtained Re
2
Re
= 34.4Ca’/‘0
(1 -cos8)“5.
(28)
The latter should be always smaller than the critical number. For the calculation interval assumed in this study none of the Reynolds number exceeded the critical value for a given M. The maximum one was 2.7 times smaller than the critical value. CONCLUSIONS
The rivulet and film are alternative forms of liquid flow down a vertical surface. For a given solid-liquid-gas system the minimum rivulet is the smallest continuous form of flow-down whose shape, height and width result from the solution of the equation for minimum system energy (5). This equation refers also to the film which is an extension of the minimum rivulet. The film profile is a solution composed of the minimum energy equation consisting of a straight line segment and two curvilinear segments that correspond to halves of the minimum rivulet profile. Flow-down in the central, flat part of the film is determined irrespective of the minimum energy condition. At the edges of the film, where the film free surface intersects the solid surface, results of surface forces of the three-phase system are observed. Therefore, the theory presented above determines boundary conditions of the film of width larger than b,i,. In this respect this theory is complementary to that presented by Nusselt which refers to flat film with undefined edges. The minimum film thickness depends only on physico-chemical properties of the system, i.e. on the contact angle and constant M characteristic for liquids 6 = 145085(l
-;“)‘:‘.
Formulas for mean velocity and minimum wetting rate of a film of the width b, contain parameters referring to the minimum rivulet, which were obtained from numerical calculations. For an infinitely wide film these formulas are reduced to dependences taken from Nusselt theory in which the critical thickness formula was taken into account. Being the simplified formulas for films of definite width, they give good approximation for most calculations. In the case of not very wide films (below 100 mm) of viscous liquids and for small contact angles (below 5”) exact equations should be applied since discrepancies may exceed 10%. Dimensionless fraction (23), called the thickness
853
number, has a constant value. An uncompensated change of any value contained in the fraction causes its increase and, therefore, film breakage. This condition can be taken as a criterion of formation and maintenance of a laminar film with minimum thickness. From the analysis of condition (23) it follows that the only reason for the rupture of a film with a critical thickness can be a local decrease of surface energy of the solid surface. NOTATION
film width minimum rivulet width
b bmin
Ca
= 3,
Gas d, 9 E
modified capillary buoyancy number = 45 equivalent linear dimension acceleration of gravity energy
Fr
= 2,
:
Qmin r I ITll” Re Re, W, WI,
w2
W,
w2
capillary buoyancy number
Froude number
consiant characterizing the liquid (9) volume flow rate volume flow rate of minimum rivulet half film width half minimum rivulet width w,d,p 4l= ~ = Reynolds number P P’ critical Reynolds number mean velocity constants = E constant in eq. (5) 2 25 1’5 = _-( 1 iV3j5(1 - cos tJ)3’5 constant in 3 147 eq. (5)
n&&p
’ Weber number
We
= -,
Weti x,y,=
modi&d Weber number (20) coordinates rivulet profile
z(x) Greek I-
8 6 P P fs
letters
specific mass flow rate contact angle film thickness viscosity density surface tension REFERENCES
Bankoff,S. G., 1971, Minimum thicknessof a drainingliquid film. Znt.J. Heat Muss Transfer 14, 2143-2146. Elrauer, H., 1956, Striimung und WzXrmelbergang bei
Rieselfilmen. VDI-Forschungsheft 457, Diisseldorf. Don&, A., 1984, Lamjnar flow of a liquid down a vertical solid surface. Maximum thickness of liquid rivulet. Phys. Chem. Nydrodyn. 5, 143-152. Don&, A., 1986, Laminar flow of a liquid down a vertical solid surface. Integration of minimum system energy equation, submitted to Phys. Chem. Hydrodyn.
854
A. DONIEC
Dukler, A. E. and Bergelin, 0. P., 1952, Characteristics of flow in falling liquid films. Chem. Engng Prog. 48, 557-563. Hartley, D. F. and Murgatroyd, W., 1964, Criteria for the break-up of thin liquid layers flowing isothermally over a solid surface. Int. J. Heat Mass Transfer 7, 1003 1015. Hobler, T., 1964, Minimum surface wetting (in Polish). Chemia Stosow. 2B, 145 159. Hobler. T. and Czaika. J.. 1968. Minimum wettine of a flat surface (in Polish). dhemia S&w. 2B. 169-186 Krell, E., 1967, The solid-liquid boundary. British Chem. Engng It, 562-567. Mikielewicz, J. and Moszynski, J. R., 1976, Minimum thickness of a liquid film flowing vertically down a solid surface. In?. J. Heat Mass Transfer 19. 771-775.
Munakata, T., Watanabe, K. and Miyashita, K., 1975, Minimum wetting rate on wetted wall columncorrelation over wide range of liquid viscosity. J. Chem. Engng Japan 8, 440-444. Muszynski, J. and Myszkis, A. D., 1984, Ordinary Diflerentinl Eyuurions (in Polish). PWN, Warszawa. 1916. Die Oberftichenkondensation des Nusselt, W., Wasserdamofes. Z. Ver. dtsch. Ina. 60. 541. Ponter, A. B., -Davies, G. A., Ross, T-K. and Thornley, T. G., 1967. The influence of mass transfer on liquid film breakdown. Int. J. EIeat Mass Transfer 10, 3491359. Zuiderweg. F. J. and Harmens, A., 1958, The influence of surface phenomena on the performance of distillation columns. Chem. Engng Sci. 9, 89-103.