gas interface

(.3eaid

l!4l&&g

science, 1976. Vol. 31, pp. 901311.

Perpmon Presa. Printed in Great Britain

AN EXPERIMENTAL STUDY OF AIR ENTRAINMENT AT A SOLID/LIQUID/GAS INTERFACE R. BURLEY and B. S. KENNEDY? Departmentof ChemicalBi Process Engineering,Heriot-WattUniversity,ChambersStreet, EdinburghEHl IHX, Scotland (Received21January 1976) Abstract-Some data from an experimentalstudy of air entrainmentinto a fluidbath by a continuousmovingplane tape is presented.The separateeffects of surface tension and viscosity are describedand the variousmodes of air entrainmentare given in the context of fluidproperties.The velocityof ah entrainmentis found to be a function of surface tension and viscosity for viscosities less than 4.65 poise. For viscosities greaterthan this value, the air entrainmentvelocity tendedto a constantvalue of 9.5 cm see-’ independentof surfacetension.Relationshipsof the form: We = k Re” and We = c (Bo t l)Red, are suggestedby analogyto describeair entrainmentdata without and with buoyancyeffects. Data from studieson four tapes and nine fluidsgave a very high correlationwhen plottedin the above form. The data is in substantialagreementwith that from similarstudies, and shows that the condition Ca = We/Re = constant is not a global criteriafor air entraimnentby a plungingsurface. The experimentaldata shows that air entrainmentvelocitv mav be estimatedfrom the relationshipV, = 67.679 (p~(g/p~))-~~~‘* for the normallyplungingplane tapes stodied.~ INTRODUCTION

ships between Weber, We, and Reynolds Numbers, Re of the form

In a typical process where a continuous surface enters an initially quiescent tank of treatment liquor, the limiting process speed may be controlled by the onset of entrainment of air which is characterised by the dynamic contact angle tending to a value of 180”.The movement of the plunging solid surface also has a secondary effect which is to deform the interface at the plunge point of entry to the fluid bath. In short residence time contact processes such as one finds in the textile industry [l] this deformation deleteriously effects the expected liquid/solid contact time, so that in this particular case, there may be also a further criteria for the maximum processing speed. The number of studies carried out on systems of continuous surfaces are relatively few and have recently been extensively reviewed by Kennedy[2]. Of the important studies, historically speaking, it was Ablett[3] in 1923,who first considered the variation of contact angle with velocity at the surface of a rotating wax-coated drum semi-immersed in water. This topic did not appear to receive further serious attention until the entrainment of air by plunging liquid jets was studied by Shiiley[4] in 1950. Since then, the subject, from the aeration of fluids point of view, has gained momentum with the presentation of work by Lin and Do~elly[S], Cibrowski and Bin[6] and more recently Van de Sande and Smith [7], amongst others. In 1%7, Perry [Sl, also studied systems of plunging liquid jets, con6rming the dam of Lin and Donnelly[S], and extending his observations to a system where the plunging surface was a magnetic tape. Perry also attempted a theoretical and experimental analysis of the forces involved when a continuous solid surface entered a quiescent fluid, which culminated in several relation-

We = k,R

(1)

where a and kl were constants that were also found to depend on Reynolds Number. In 1%9 Inverarity[9] completed a study of the dynamic wetting of glass fibre f&rents (dia RI&200 pm) up to contact angles of 180”, where he observed the phenomena of “meniscus collapse” and intermittent entrainment of air. Inverarity suggested that the description of dynamic contact angle was mainly viscosity dependent and that surface tension was relatively unimportant. More recently, Burley and Brady[l], have reported a study on the depth of depression of a fluid interface caused by the passage of wool fibres into liquid and found that the data was satisfactorily represented by the relationship of the form given by eqn (l), where the Capillary Number, Cc = WelRe, varied in the range 10m3
tPresent address:c/o, Amoco ResearchCenter,P.O. Box 400, Napervihe,IL 60540,U.S.A.

50RCHACTING ATTEEINTERFACE Dimensional analysis has shown that an important dimensionless group for characterisation of air entrainment and interfacial displacement, where buoyancy forces may be ignored, is the Capillary Number, Cu = (@J/o) [ 101. This is the ratio of Weber to Reynolds Number, as defined above and represents the ratio of viscous to surface tension forces. Where the interface is signi6cantly displaced, however, buoyancy forces may also play a part in any equilibrium force balance created

901

902

R.

BURLEY

and

Pi. 1.Displacedinterface(1) due to movingsolidsurface,(2). as a result of solid surface motion. This is shown in Fig. 1 where the principal forces of interest are given in two-dimensional representation. The buoyancy force will give rise to a further dimensionless group which will give the ratio of buoyancy to surface tension force. This group has been called alternatively the Bond or Eiitvos Number and is denoted by the symbol, Bo. This extension of the functional argument gives

we=j,(Re,

Bo, e).

(2)

If there are no significant buoyancy effects an empirical relationship of the form given by eqn (1) might be anticipated from investigation of the actual form of (2), but the condition

B.

S. KENNEDY

region; and the undefined viscous length over which the drag force is developed. It is reasonable to suggest, however, that a drag coefficient relationship of the type: Cd = k2Re-“” would be representative. As the dynamic contact angle tends to 1800,the surface tension force will also tend to a constant value. The buoyancy force is distributed along the displaced fluid interface, (1) in Fig. 1, which hydrostatic theory shows to be equivalent to a force acting at the centre of pressure of the curved interface. This force will have a normal component. The buoyancy force will be characterised by the depth of displacement, xb, measured from the free surface, as shown in Fig. 1, adjacent to the plane surface (2). Although the three forces operate in different planes, one way to demonstrate the type of relationship which might be expected between the dimensionless groups given in eqn (2) is to consider a mechanical analogy and resolve the normal forces directly. If we consider the viscous force F,, the normal buoyancy force, Fb, and the force due to surface tension, Fsr, we obtain

UT + Fst)= Fb.

(5)

The buoyancy force will be proportional to xzpg and the surface tension force F,, is given by u cos 0 per unit width. If, for the sake of the argument, we let the viscous length to be proportional to, xb, then eqn (5) becomes, using the dimensionless groups We = k,(Bo t 1) Re””

(6)

at air entrainment, when 0 + 180”, U = VA, and k, is a constant. Ca = WelRe = constant This analogy also indicates that the condition Ca = (3) constant arises when the dependence of drag coefficient only arises when the value of a in any such eqn (1) is on Reynolds Number is linear and the buoyancy forces unity. This suggests that the results due to Wilkinson [ 101, become constant. It is unlikely that either of these that air entrainment is characterised by a value of conditions would be met at entrainment for a wide range Ca = 1.2, is strictly limited to the type of system of fluids, and it is therefore reasonable to anticipate that studied-air entrainment caused by a pre-wet curved the data for air entrainment would be given by a surface. correlation of the type Dimensional considerations also show that one simple dimensionless group that may be formed from fluid We,,,,,= & = k,Re”” properties and tape velocity alone is given by [Re*/&,]“*

=

f(y)“*.

This expression is used later to evaluate air entrainment velocity directly from fluid properties. The forces which act at the plunge point do so in different planes of reference as is shown in Fig. 1. The viscous force has been shown by Perry[8] to be completely generated below the plunge point by the surface in contact with the fluid. The usual relationship for drag coefficient for a continuously moving plane surface has been evaluated by Sakiadis[l l] as Cd = 0.888/d(Re), but this solution to the boundary layer equations will not be strictly applicable in this case due to the development of flow by entrainment in the interfacial

and that Ca will vary depending on the nature of fluid viscosity and surface tension. EWERIMENTAL The general layout of the apparatus employed in this investigation is shown in Fig. 2. Full details of the apparatus may be found elsewhere[2]. A continuous tape was drawn through the twin-sectioned tank by engaging the tape between a drive and buffer roller. The buffer roller was attached to a swinging plate, tensioned by a spring, so that the take-up reel and the buffer roller moved in concert. The take-up reel was fitted with a specially designed slip clutch mechanism to allow for increasing spool diameter during runs. After passing over a guide roller the continuous tape entered normally into the fluid

903

An experimental study of air entrainment at a solid/liquid/gas interface

sion of vibration. Several continuous tapes which were employed, mainly manufactured from polyester, are detailed in Table 1. Different fluids and solutions were also used to give as wide a range as possible to surface tension and viscosity values as given in Table 2. PIIVSICALOBSERVATIONS OF TEE INTERFACE

effect of the moving continuous tape was in all cases to displace the fluid interface. The effect of increasing viscosity was found not only to increase the depth of entrainment and the dynamic contact angle but also to increase the extent to which the local surface was affected by the continuous motion of the solid material. That is to say that the interface of more viscous fluids was displaced over an extended area. During the first instants of solid surface motion, the small capillary rise adjacent to the surface was displaced in a downwards direction with no relative movement of the contact line. Later, depending on fluid properties, a point was reached where the solid surface moved past the contact line. With low surface tension fluids this occurred before 0 + 90”, whilst with others, this commenced at values greater than 90”. From that point the effect of the drag force was observed to displace the position of the contact line to successive dynamic equilibrium positions until the dynamic contact angle reached 180”and air entrainment ensued. The characteristic behaviour of the onset of air entrainment was found to vary markedly with fluid viscosity. As may be seen in Fig. 3 the mode of air entrainment changes in four broad bands of viscosity. The

unit-

Fig 2.

Apparatusfor investigation of dynamic wetting and air entrainment.

in the top section of the right hand tank. The plunge point was photographed directly using a 35mm camera mounted on the photographic arm which was capable of movement through 90”. The whole apparatus was supported on a thick metal plate cushioned from the bench and floor by rubber pads to prevent the transmis-

c

Table 1. Properties of tapes used in this work Yield

liokness

cn?/grh

micron

qvlar

Polpster

75

Polyester

125

1.92

565

150

2.58

705

50

1.27

Lllnex

Ionolmr cast

bgnetic

Tape

(a)

(b)

95

(cl

0.7

(d)

Level surface

Very fine bubMes

.

. .

p-z O.lpoise

~=(O.l-1O)paise

p=(1.5-P.O)poise

pc 2.0poke

Fii. 3. The nature of air entrainment as a function of fluid viscosity.

904

R. BURLEY and B. S. KENNEDY

Diagram 3(b) (p =O.l-1.0 poise) differs from that shown for a lower viscosity fluid, diagram 3(a), in that at some distance down the continuous tape, air bubbles broke off from the tip of the “vee shape” and were entrained into the bulk fluid. With increase in viscosity up to 1.0 poise, the air film was pulled down to a lesser depth before air bubbles were seen to be entrained into the bulk fluid. An increase in velocity of the tape from that at which bubble entrainment was initially observed, produced a lengthened “vee shape” as shown by the lower position in Fig. 3(b). A larger rate of bubble formation and entrainment was observed in this case. Photographs taken normal to the plane of the tape of the air film development are shown in Fig. 4 for a 0.456 poise glycerine/water solution. These indicate very clearly the development of the “vee shaped” interface. The f‘vee shape” results as the effect of hydrostatic pressure displacing the vertical fluid/air interface at the side of the moving surface, forming a symmetrical triangular shaped intrusion into the fluid. This behaviour is modified with increasing viscosity and with extended wider surfaces the pattern is repeated along the length of the interface. For fluids of viscosity in the range 1.5-2.0 poise, the air 8lm formed a number of smaller “vee shapes” extending along the tape as shown in Fig. 3(c). Bubbles smaller than those observed for lower viscosity solutions, were entrained from the points of the small “vee shape”. An increase tape velocity lengthened these “vee shapes” only slightly, the main effect being to increase the rate of fine bubble entrainment. For viscosities between 1.0-1.5 poise, intermediate air film characteristics of the two described were obtained. Several small “vee shapes” were formed initially which, with increase in tape velocity, joined together. Unstable oscillations resulted between the single and the several ftlm. For viscosities small “vee shape” modes of greater than 2 poise, Fig. 3(d), thin 7 treaks emerging from the three phase contact line were the only indication of fact that the contact angle had reached 180”.Tiny bubbles, broken off from the ends of these streaks, travelled down with the tape. Entrainment into the bulk liquid resulted when the bubbles were detached from the tape surface as it passed through the tank base. With pure glycerine

(CL= 19.34 poise) these streaks were not observed, the

only indication of air entrainment being the cloud of very small bubbles rising slowly from the bottom of the tank. The formation of the air film for solutions of ditTerent surface tensions appeared in character as described for the glycerine/water solutions. The only difference noted was that for the lower viscosity solutions the air tilm broke into bubbles more readily the lower the value of surface tension. In other words, the air film formed with a 0.2 poise glycerine/water solution (u = 64.0 dyne cm-‘) had to be pulled a greater distance down the surface before an air bubble broke off when compared with a 0.2 poise polypropylene glycol/iso-propyl alcohol solution (o = 29.0 dyne cm-‘). The entrainment process at higher viscosities, (i.e. ~=4.62 poise), was the same as that shown for a glycerine/water solution in Fig. 3.

The maximum depth of displacement of the three phase contact line is plotted against tape velocity in Fig. 5. The graphs shown are for glycerine/water solutions with viscosity in the range 0.1-1.12 poise using “Melinex” polyester film as the solid surface. It is seen that in region (l), the displacement depth increases slowly with velocity. For the solutions shown the dynamic contact angle tends to 180” at a displacement depth X, of 0.4cm s, corresponding to tape velocities V, of 130, 110,55 and 30 cm set-’ respectively. The sharp increase in displacement depth (region 2) in Fig. 5 corresponds to the initial formation of the “vee shaped” air film. Further increase in tape velocity leads to a continued displacement of the three phase contact junction as shown by region (3) in Fig. 5. With increase in viscosity to 1.18 poise, the sharp increase in region (2) diminishes corresponding to the transition of air film characteristics from (b) to (c) in Fig. 3. (b) Dynamic contact angle and the effect of surface tension The behaviour of motor oils, liquid paralfin, and pure polyethylene glycol 400 were investigated along with several solutions as detailed in Table 1. The variation of dynamic contact angle with tape velocity is plotted in Fig.

Fii. 4. Air6lmdevelopmentwith increasingtapevelocity,~1= 0.465poise. VP = 72,90,110cm set-’ .

905

An experimentalstudyof airentrainmentat a solid/liquid/gasinterface

oq 0

20

40

60

80

DO

I

L

120

140

160

i80 200

22U

Tape velocity, cm S’

Fig. 5. Displacementdepth - tape velocity. “Melinex”polyester tape in glycerine/watersolutions. x + 0 v

50%glycerine/water 60%glycerine/water 70%glycerine/water 80%glycerine/water

&oise) 0.1045 0.18 0.456 1.18

6 for several solutions which were chosen within a small range of viscosity and a wide range of surface tension. The plot shows an initial fast rate of change of dynamic contact angle followed by a more gradual change as the angle tends to 180”.As might be expected a decrease in surface tension at constant viscosity lowers the velocity of air entrainment. If the experimental data, using different fluids for studies on several tapes, is amalgamated in the form presented in Fii. 7, then the effect of surface tension may be clearly seen. Surface tension is seen to have its greatest effect on VApat low values of viscosity. With increase in

90

”

0

5

IO

”

15

20

25

viscosity to 4.65 poise the value of VAFis less alfected by surface tension, until a point is reached where the curves merge asymptotically at a constant value of VAp of 9.5 cm set-‘. This suggests that irrespective of the value of surface tension, air entrainment will not result below a velocity of 9.5 cm set-’ for viscosities greater than 4.62 poise in the system studied of a plunging plane surface. Apart from lowering the value of V,, surface tension had a further small but sign&ant effect. It was found that the value of displacement depth, X,, on commencement of formation of the air film, decreased with decreasing surface tension. As can be seen from Table 5, the values of X, decreased from 0.41 cm at u =65dynecm-’ to 0.3 cm and at u = 25.5 dyne cm-’ for viscosities up to 6.7 poise. Only at very high viscosities (Jo= 19.34 poise) was there an decrease in displacement depth and then only to a value of 0.61 cm s. The figures do show, however, that the buoyancy force will vary in the range considered and that these forces will be more significant with high viscosity fluids. (c) Data evaluation in terms of semi-empirical equations The data from forty separate runs using nine fluids and three tapes were evaluated. When the Weber and buoyancy moditied Weber Numbers were plotted against Reynolds Number with Bond Numbers lying in the range 2.0~ Bo < 8.0, for the conditions of incipient air tilm formation, the data appeared as given in Fig. 8. This figure also plots the Capillary Number, Cu against the Reynolds Number showing a clear variation of Ca from a value of 0.834 at Re = 1 (high viscosity fluid) to 0.209 at Re = l@ (low viscosity fluid). It is clear that the data are well correlated by a relationship of the form of eqn (7), and that the expected

*

a

30

35

1’ 40

45

Iti 50

55

Tape velocity I+. cm s-I Fii. 6. Dynamiccontactangle-

0 0 A 0 x

tapevelocity. “Mylar”polyestertape.

a(dynes cm-‘) glycerine/water 62.3 glycerinelwaterlteepol 49.0 polyethyleneglycol/Iso-propylalcohol/water 40.3 glycerine/Is*propylalcohol/water 31.6 polypropyleneglycol/Iso-propylalcohol 28.2

Apoise) 0.456 0.456 0.472 0.458 0.465

J 60

906

R. BURLEY and B. S.

0. 0.1

0.2

0.4

0.6

KENNEDY

1.0 Viscosity, poise

2.0

4.0

6.0

I 10.0

Fig.7. Velocity of incipient air entrainment as a function of viscosity. 0 X 0 A + * 0 7

glycerine/water polypropylene glycol/Iso-propyl alcohol polyethylene glycol/Iso-propyl alcohol/water glycerine/water/teepol glycerine/Isa-propyl alcohol/water motor oils 100%polyethylene glycol liquid pa&in

u(dynes cm-‘) (62-65) (25-35) (41.5-40) (46.5-51) (31-33) (32.9-34.9) 45.5 31.6

The slopes of the lines given in Fig. 8 show that either Sakiadis’ drag coefficient expression CD= 0.888 Re-“* is not appropriate to the problem or, if it is, then the hydrodynamic length to be used for the evaluation of Reynolds Number is itself an as yet undetermined function of velocity. The fact that the slopes of each line are in such close agreement suggests that by comparison with eqn (7) the effective drag coefficient is given by the expression

1000

400

C, = 1.199Re10.798

I

4

400 40 CO Reynolds number

IO

1000 4OOom

Fii.

8. Weber numbers - Reynolds number and Reynolds number - Capillary number, for the data of air entrainment. 1. We = 0.834 Ream model:eqn (8).2. We/(Bo + 1) = 0.192Re”.812, model: eqn (9). 3. Ca = 0.834Re-“.zm.

slope of the graphs is not 1.0 as would be required for eqn (3) to be satisfied. The best regression curves (r > 0.99) are, commencing with the upper line We = 0.834 ReO.m

&

=0.192 Re0.8’2

(8)

where the Reynolds Number is evaluated on a displacement depth basis. At low Reynolds Number, Re < 1, the air film entrainment Reynolds Number tends to a constant value. This region corresponds to systems of high viscosity. It was shown previously that this range gives a constant value of VAp, of the order of 9.5 cm set-‘. This region is also outside the quoted range of validity of Sakiadis’ [I l] work which was given as Re > 100. CALCULATION

OF AIR W

Using eqn (8) in conjunction with the data from Table 4 for fluids of specific gravity in the range 0.79-1.26 we may evaluate the expression

(9) VA, = 0.585 (F)““(k)““’

for 0.1 < We < 200; 1 < Re < l(r and 0.8 < Cu < 0.2. It may be noted that eqn (8) is in very good agreement with Perry’s [ 1,8] data which gave We = 0.1% Re0.83.

(10)

(11)

t12)

for air entrainment velocity V,. These values are given in the final column of Table 4 for comparison. Alternatively, the air entrainment velocity may be calculated from the relationship suggested by eqn (4)

An experimentalstudyof airentrainmentat a solid/liquid/gasinterface

907

Table2. Rangeof fluidproperties

Pclyprcpgene

gQm1

2am/iso-pmpyl alcchcl.

0.1 -

6.772

0.1 -

I.36

Pclyet.&lene glyccl &m/iCaprcyPl zSl.c&cl/vater

Qlycerine/iso-prcpyl alcchcL'vater

0.1 - 5.09

0.1 - IS.62

Ebtcr oils

Liquid

1.855 - IL.99

par&fin

0.319

Qlycerine

19.339

Table3. Characteristicsof airentrainmentwith fluidviscosity

vticcdty

observaticn:cf llquid/vapcur

mJid

o.osg*

interrace

0.1

3phasejunctionlim 'vee'sbape

0.1 + 1.0

Jphass

formed

atvF-vLp

jmctionlinsfcrmed

'vw'sbnpeatVF.vLp

bubbles of air entrained for v** at 'vee' 5' shape peak

1.5*2.0

seveml

small

rclTmdat3

‘VW’

shapes

phase junction

from peaks Of 'vee' shapes atVp.

lim

> 2.0

bubbles given off

vu

sevarslthinstreakaerarged

air bubbles travellad

fYcintbrwphSeccntact1ine

wit21 the tape end sheared off

at the

tank bottom

COMPARLWN WITUOTEFBDATA

which gives: V,, = 67.679 [ p($~]+“”

(13)

for the best fit regression curve (r < 0.99) to the data of Table 4 shown in Fig. 9. Values of V. calculated from eqn (13) and experimentally determined values of air entrainment velocity are given in Table 5. This table also gives the error involved in prediction of the air entrainment velocity from eqns (12) and (13). These figures are only indicative of absolute error however as the accuracy of measurement of V, in particular would be subject to random observational error. The average error over the whole range is 7.73% and 10.73% for eqn (12) and (13) respectively.

Data of V, values from several studies of types of air entrainment is shown along with results from this Work in Fig. 10. Perry’s [8] data from one side of a magnetic tape are in reasonable agreement although slightly lower. This side was characterised by a 20”static angle for glycerine/water solutions. A disparity with the 70” static angle case is shown but this may be attributed to the diierent nature of the surface, and in any case, the data shows the same tendency to a constant value of V.. Because such a large difference in V,, values was obtained for the two static angles of 70” and 20”, Perry concluded that the meniscus height and static contact angle above the level surface was of great importance in predicting dynamic contact angle behaviour below the

908

R. BURLEY and B. S. KENNEDY Table 4. Displacementdepth data for a series of fluids of ditTerentpropertiesat air entrainment;estimates of air entrainmentvelocityfor Re > 1

-

P

0

dyne.cn

-1

vdJ_,

SF

r&see

cm

h (12)

1.1

0.059

6lb.O

180

0.11

205.25

1.128

0.1045

65.0

120

122

125

O.&

lb.20

1.156

0.18

63.0

90.0

94.0

100

0.4

95.79

l.lell

O.L56

63.0

58.0

53.0

55

0.h

52.99 27.15

1.211

1.188

63.0

31.0

27.0

28.0

O.hl

1.238

h.61

63.0

10.0

10.0

10.0

0.b

10.99

0.090

0.1055

25.0

77.0

0.395

69.1

0.919

0.213

26.1

47.0

0.3%

Ida.76

0.915

O.M5

20.2

25.0

0.317

28.1

0.97

1 .16

30.9

17.0

0.33

16.93

0.909

3.026

311.6

10.0

0.3&

l.olr6

0.1055

31.5

65 ')

0.33

@a33

1.07

o.us7

31.6

32.0

0.33

30.32

1.092

1.1907

33.4

17.0

0.335

16.76

l.olr6

0.1ou

LO.3

97.0

0.35

9s.u

1.0727 O.L72

LO.3

36.0

0.36

35.89 20.05

9.ll2

1.08

1.175

Ill.5

18.0

0.36

1.091

1.36

Is.5

18.0

0.37

1.128

0.103

46.5

102.0

0.375

I.156

0.21

49.1

76.0

0.38

72.08

l.lsq

0.156

L9.0

50.0

0.38

b2.49

1.211

1.16

lr9.0

23.0

0.38

22.32

I.24

lr.61

51.0

10.0

3.385

I.916

1.855

32.9

12.0

m& -

Fig.9. Airentrainmentvelocityas a functionof fluid properties.

19.52 109.2

9.3 12.lr7

An experimental study of a&entrainment at a solid/liquid/gas interface Table 5. Estimation of air entrainment velocity directly from hid properties using eqn (13)

YkP

P

gm.cc’

cm.

..30-1

x

x

VAF ieviatio ievlation

5s

VAF

cm

a, (12.

eg (13)

RI (12)

s, (13)

10.9

l.CUt6

o.rc&

40.X0

97.m

0.360

98.U

107.60

1 .w

1.073

O.ll72

IrO.303

3uxxl

0.360

35.89

39.27

0.31

9.o8

1.080

1.175

l&l .5co

18.CCG

0.39

20.05

21.53

1.02

l.%

1.123

0.103

16.500

lCQ.CCG

0.380

109.17

116.55

7.03

9.9

1.1%

0.210

h9.100

76.m

0.380

70.88

74.15

6.7L

2.w

1.1%

oA50

ll9.ooo

SO.WO

0.380

lG?.L9

44.76

15.02

1.211

1.180

49.ow

23.CUJ

0.380

22.32

23.60

2.%

0.96

1.885

32.500

12.cco

0.350

12.b7

13.72

3.92

III.33

0.098

0.106

25.ow

77.oM)

0.300

69.10

35.96

10.26

Il.61,

0.919

0.213

26.100

ll7.cco

0.305

LA.76

a.99

0.9u

oh65

28.2oo

25.wo

0.317

28.11

33.71

l2.U

14.8

0.989

3.026

3lt.6~

lO.ow

0.3&O

10.42

5.8

h.2

1.046

0.106

31.500

85.ooO

0.330

80.33

97.79

5.119

5.05

1.070

oA59

31.6~

32.ooO

0.330

30.32

%.8l1

5.25

5.13

l.O92

1.191

33.400

17.wo

0.335

16.76

19.91

1.u

7.12

1.123

0.105

65.~~0

13O.CCXl

o.l!w

lll1.20

128.76

8.62

0.95 9.85

9.112

lOA 2.61

It.77

17.0

1.184l

0.456

62.300

60.~0

o.liw

50.99

la.09

15.02

1.211

1.188

63.WQ

28.WO

O.&W

27.15

25.56

3.01

8.71

1.128

0.1%

65.000

12h.000

O.hW

l&l.20

128.76

13.87

3.8I1

1.156

0.180

63.000

lW.cmJ

o.lbw

95.79

89.&3

Ii.21

0.57

1.18b

oh56

62.300

55.ow

OAW

so.59

48.09

7.29

2.56

1.100

0.059

6Low

18O.ooO

O.&W

205.25

157.10

lb.03

3.9b

1.125

O.lc5

65.0~

12o.cco

O.&W

llrl.20

123.76

7.67

7.3

1.18Il

DA56

62.300

58.Mx,

OAW

50.99

Ir8.09

2.09

7.09

1.211

1.188

63.0X

31.m

o.lhw

27.15

25.56

2.42

7.55

1.211

1.188

63.300

1.128

0.105

65.0~

0.1

0.2

27.OW

o.lbw

122.wo

0.4

O.LW

0.6

27.26

25.60

l10.20

123.76

1.0

2.0

O.%

5.19

15.74

4.0

6.0

5.w

IO.<

pohe

Viscosity.

Fig. 10. Comparison of data from several studied on air entrahment. (a) This work glycerine/water, 0, 0. = 35”;0, 4 = 45”; A, 0. = 53”; X, 0. = 60”. @) PerryPI glycerine/water, 0, = 20“. (c) Perry glycerine/water, e. = 70”. (d) Wilkinson[8] glycerine/water. D = 68-70 dynes &I-’ a-, &‘= 78.5”; 0, e. = 90”. (e) Wilkinson various oils LT = 30-36,dynes cm-’ +, 0,. = lu)“.

910

R. BURLEY and B. S. KENNEDY

surface. This conclusion does not follow from the results of this work, where it has been shown that the static contact angle for the hydrophilic surfaces studied is of minor importance in predicting values of V, Wilkinson[lOl reported air entrainment data from rotating a cylinder through a liquid interface. Different initial static contact angles, &, were created by varying the initial depth of immersion of the cylinder in the bulk fluid. The values of V.., thus obtained were defined not as the velocity at which the air 8hn commenced but by some velocity in excess of it where the air film became unstable. This could explain the higher values of V, obtained when compared with data from this work. What is significant, however, is the same asymptotic trend in curves (d) and (e) of Fig. 9 towards a constant value of Vm at a higher viscosity as obtained with the results of this work shown by curve (a). The results in this work show that values of V,, are independent of the static contact angle made above the level surface, substantially confirming the data of Perry[8] from a study of plunging magnetic tapes, and the observations of Wilkinson[ lo]. The data from Inverarity’s[9] experiments with glycerine/water solutions on glass fibre filaments was presented as a plot between cos 8 and log (pVp). The data below the level surface formed a single master curve. Figure 11 shows a plot of cos 0 against In (pVplu) for

Q 8 ”

boundary was observed by the appearance of a “triangle shape” on attaining 180”at the critical wetting velocity. This “triangle shape” appears to be equivalent to the “vee shape” observed in this work, whilst the “stick and slip” movement of the three phase junction observed by Burley and Brady [l] and Huh and Scriven[l3] has also been observed.

CONCLUSIONS ANDSUMMARY

An experimental investigation into the nature of air entrainment at a plunging solid surface/liquid interface has shown the limiting effect of the 8uid viscosity and surface tension on (1) the mode of air entrainment, (2) the depth of interfacial displacement, (3) the variation of contact angle, (4) the onset of air entrainment for particular velocities. The characteristics of the air film are summarised in Table 3 and the collected data for displacement depth and velocity of air entrainment from liquids of several viscosities and surface tension given in Table 4. It has been found that empirical relationships eqns (8) and (9), between Reynolds, Weber and Bond Numbers represent the experimental data satisfactorily, and that the Capillary Number varies in the range 0.8 > Ca > 0.2 for the system studied. Further the air entrainment velocity may be estimated directly from fluid properties via eqn (13), as given in Table 5.

-0.6 _ -0.5 -0.4 -0.3

Fig. 11. Comparison of data due to Inverarity[Z] with this work in the form of a plot of cos 13against Capillary number. 0 100%glycerine 0 70%glycerine/water 0 60% glycerine/water A Inverarity’s results glycerine/water

Inverarity’s data and that in this work from which good agreement results for the high viscosity data, although as the viscosity decreases the agreement is not so definite. However, considering the geometrical difference alone, the agreement is substantial. The plot also puts the data of Wilkinson[lO] into context in that the Capillary Number of 1.2 lies in the range of higher viscosity fluids approximated at air entrainment shown as the horizontal part of the curves. Deryagin and Levi[l2] observed the displacement of the wetting boundary during the coating of photographic emulsions on flexible supports. A shift of the wetting

&oise) 19.335 0.456 0.18

Further investigation is needed into the nature of action of hydrodynamic forces at the interface to elucidate the concept of hydrodynamic length in this context. More experimental data from fluids of widely varying properties is also required so that sound prediction of air entrainment velocity from fluid properties alone may be attempted. It is hoped that this work may continue. Acknowledgements-The aid of Mr. C. McLeod is gratefully appreciated in respect of apparatus design and construction, and the award of a University Scholarship to one author (B. S. Kennedy) is acknowledged.

An experimental study of air entrainment at a solid/liquid/gas interface NOTATION

force due to viscous drag, M L T2 force due to surface tension, M L TV2 force due to buoyancy, M L T-* velocity causing air entrainment, L T-i solid surface velocity, L T’ depth of interfacial displacement, L depth of interfacial displacement at incipient air entrainment (6 + 180‘7,L acceleration due to gravity, LTV2 Greek symbols p fluid density, M Le3 u fluid surface tension, M L-* p fluid viscosity, M L-’ T-i 0 dynamic contact angle 0, static contact augle defined in (10) Dimensionless groups Re = (xbpu/p), Reynolds number We = (xbpu*/u), Weber number BO = (xb2pg/a), Bond number (Eiitvos number)

CESVd.31No.I&D

911

Ca= WelRe =&U/u), Webn

=

&lOJQUlCy

Capillary number modified Weber number, defined in eon

= We/(Bo + 1)

REFERFACES

Ul Burley R. and Brady P. R., I. Colllnter. Sci. 197342(l) 131. PI Kennedy B. S., Ph.D. Thesis, Heriot-Watt University, Edinburgh, 1975. [31 Ablett R., Phil. Mag. 192346(6) 244. 141Shirley R. hf., MS. Thesis, University of Iowa, 1950. PI Lm J. J. and Donnellv H.. A.Z.Ch.E.J. 196612 563. [61 Ciborowski J. and Bid A., inst. ZnyChem. Polite& 1972l(3) 247. [71 Van de Sande E. and Smith J. M., C/rem.Engng Sci. 197228 1161. [81 Perry R. T., Ph.D. Thesis, University of Minnesota. University Microfilms 67-14,639, 1%7. [91 Inverarity G., &it. Polvm. Z. 19691 245. (101 Wilkinson WI, Chem. &gng Sci. 197538 1227. 1111Sakiadis B. C., A.Z.Ch.E.J. l%l 7(i) ?a(ii) (iii) 467. w1 Deryagin B. V. and Levi S. hf., Film Coating Theory. Focal Press, N.Y. 1964. Ll31Huh C. and Striven L. E., J. Coil. Inter. Sci. 197135 85.