Air entrainment in a roll coating system

Ckcmrcal

ErQwnmg

sxnce Vol

Pergamon Press Lid

35 pp 597401

I!?80 Pnnted m Great Bntam

AIR ENTRAINMENT BARRY Department of Chemical

Engmeenng

IN A ROLL

BOLTON

and STANLEY

COATING MIDDLEMAN

and Polymer Science and Engineering, Amherst, MA 01003, U S A (Recewed

for publrcahon

SYSTEM

25 June

Uruversity

of Massachusetts,

1979)

Abstract-Experiments are performed to determme the crlttcal speed at which atr IS entrained by a roll, rotating partially submerged about an axls parallel to the free surface Data for Newtoman fluids correlate m terms of a cr!tical CapdIary number, Ca* = V* p/a, which IS found to depend on a parameter y = c~(pl~‘g)“~ over four decades m y A model based on simple force balances at the dynamic meniscus reflects these observations Data obtained with viscoefastic polymer solutions indicate that elastlctty IS a strong stabkmg factor

In this paper we report data for au entramment by a wet roll, using hqulds with vlscosltles rangmg over four orders of magmtude In addltlon, data were obtamed m ddute polymer solutions which suggest that elastic@ 10~ may play a dommant stablllzmg role as surprlsmgly levels of polymer concentration

INTRODUCTION

#en a sohd surface plunges across a gaslhquld mterface mto the hquld It may drag or entrain a film of gas with It Such a phenomenon occurs, and plays an lmportant role, when a tape or roll 1s coated with hquld, as m Fig 1 This paper concerns Itself w&h experiments amed at estabhshmg cntlcal condltlons at which the interface becomes unstable, and IS dragged mto the hquld Previous studies of such entramment phenomena have helped to mdlcate certam general features Invermty [ 11, for example, has mvestlgated an entramment by glass fibers, while Perry[2] has studied the same problem usmg magnetic tape Burley and Kennedy [3] measured crItIca speeds for a movmg tape, and found that the speed decreased with about a two-thuds power dependence on hquld vlscoslty, and increased as (approximately) the cube root of mterfaclal tension Their studies were carned out on fhuds of vtscosltles up to about 5 poise In another paper Burley and Kennedy[4] show data on entrainment which suggest that the crItIcal speed mcreases with surface tenslon to a power slightly less than umty In each paper Burley and Kennedy give a different correlation of their data, and neither correlation 1s m a useful dImensIonless format Wdkmson[5], m work most closely related to ours, performed experiments using a cylmdncal roll The roll surface was wiped m such a way that the entermg surface had a thin film of hquld on it One might antlclpate that entrainment could depend upon whether the plungmg surface IS dry or wet, and mdeed Wdkmson found that the cntlcal speed decreased hnearly w&h vlscoslty and increased lmearly with mterfaclal tension, results somewhat different from those of Burley and Kennedy Wllkmson’s data were obtamed ustng a sertes of aqueous glycerine solutions

EXPERIME~AL

Figure 2 shows a diagram of the apparatus used A stamless steel roll, 5cm m dla and 15 cm long, was mounted on bearmgs m a rectangular reservoir made from acryhc sheet Liquid level m the reservou came to the center of the roll under static condltlons The roll was rotated about Its axis by a variable speed motor A hquld was placed m the reservoir, roil rotation was begun at low speed, and the speed was slowly Increased until the air interface was contmuously drawn mto the bath The cntlcal rotational speed at that pomt was noted, and denoted fl* (rpm) The onset of am entramment was found to be quite sharp Just before entramment, I e at a speed about ten percent below fl*, one could sometlmes observe a set of cusps to form along the zur-llquld interface Occaslonally, tmy iu~ bubbles would break off of the cusps and be dragged along with the thud near the roll surface With a modest increase m speed, beyond that pomt, the mterface would enter the bath along most of the length of the roll axis Vlscositles of the hqulds used were measured m an Epprecht coaxial cylmder vlscometer Interfacml tenslons were measured by the pendant drop method Densltles were measured with a pycnometer Table 1 shows relevant properties of the Newtonian flulds studied Vlsoelastlc solutions were prepared by addmg polyacrylamlde to water, or to aqueous glycerme solutions The polyacrylamlde was Polyhall M-195 (Stem-Hall) Solutions were prepared by heatmg the solvent to 310K and addmg polymer, after which the mvcture was stured for two hours The resultmg solution was allowed to cool down and stand overmght before use All rheologrcal properties of the polyacrylarmde solutions were measured w&h a Rhometics Mechamcal Spectrometer, and relevant data are shown m Fig 3 The solutions show varymg degrees of non-Newtonmn character, as evidenced by the degree of dependence of

Ftg 1 AII entranunent by a plungmg tape or roll CES Vol 35 No f-E

597

598

B BOLTONand S MIDDLEMAN

vlscos~ty on shear rate Most of these polymer solutions exhibited measurable normal stresses, and Fu 3 shows elasticity as given by the recoverable shear SRIdefined as[6]

zap]

where TII-T~~IS the pnmary normal stress dfierence, and 7121s the shear stress, at a given shear rate y ROLL

RESULT!3

Newtonran jhrds Data for Newtoman fluids are shown m Fig 4, plotted as iI* vs vlscoslty p Consistent with results cited above from related studies, a* decreases with mcreasmg VIScosity The small vanation m surface tension CTfor the fhuds used (relative to the varlatlon in viscosity JL) makes it ddlicult to gve a precise estunate of the dependence of a* on u at fixed P It IS possible to model some features exhibited by these data by mtroducmg a sunpie force balance around the meniscus remon Figure 5 shows a definition sketch

II

I

rl

MOTOR

Fw

2 Ehagram of the apparatus

Table I PropertIes of Newtonian flmds Huld Motor 011 Karo syrup Glycerme Glycermelwater

I

solutions

Denstty Wcm3)

Surface Tenslon

Vlscoslty

088 14 12 l&12

35 70 63 63-70

1 35 80 10 8 0 l-10

poise

(dyne/cm)

t

J A

1.

A

0

0

A

A

”

* 0

SR _

0

0

,,

0

II

-

0 0

Cl

01-1-

005

-

9 1

1

IILIII

I

I 10

,

0

v

I,,,,

I

-

I

100 x (s-1)

solutrons 0 ppm PA m 90/10 glycermelwater 10 ppm, V 20 ppm, 17 30 ppm, 0 50 ppm. A 60 ppm

Fa, 3 Rheolog~cal data for polyacrylanude

(22 5°C). 0

Au entramment m a roll coatmg system 103.

0

0

0 0

00

0

0

-z : 102

1

B

“so

(In.0

o

2

IO 10-B

10-Z

1

10-l

QJO 0

IO

8

I02

)L (POISG)

Ftg

4

Entramment data for Newtoman flmds

where

(6) IS an medal parameter for this system Ca 1s the Capdlary number For very low coating speeds, or high vlscosrty, we m&t speculate that VISCOUSforces exceed mertlal forces, and that entramment occurs when the vscous shear force exerted by the roll on the meniscus exceeds to the force F, The VISCOUSshear force IS wcult assess, but symbohcally we may wnte It as F,=TWL

FE 5 Defimtlon sketch for a force balance around tbe memscus

The memscus exerts a force

is stabllzed

by surface tension, which

where r 1s the (unknown) shear stress, and L 1s the length over which It acts (Fig 5) L IS unknown, also A model which IS crude, but testable, may be constructed m the followmg way For the shear stress 7 we w1u use T =

F, = 2uW

(2)

along the width W of the meniscus Two forces act to drag the memscus mto the hqurd, and entram au thereby One force ISdue to the metia of the layer of liquid that coats the roll, and plunges mto the bath The inertial force would be gwen by

H = 0 sq&&ulpg)“2

(8)

~rlth H gwen by eqn (4) For L we use the analogy to the problem of velocity profile rearrangement m a VISCOUS let lssumg from a cap&try, as suggested m Ftg 6

(3)

Fr = pU2HW Previous studles[7] have shown thickness H IS well described by

pUlH

that

the coatmg

(4)

tn the remon of parameters used here

If we regard a~ entrammest to occur at the pomt that F, then eqns (2)-(4) yeld, after algebrruc mampulatlon mto a dunenslonless format, a cntical speed U* gven by Fr Just exceeds

Fe

6 Velocity profile relaxation m a lammar Jet emergmg from acap&

B BOLT~Nand S

MIDDLEMAN

Co*

lo-‘-

l#Y?z Inertlol

Theory

0

Fig 7 Tests of eqns (5) and (10) I

I POLYACR

YL A MlDE

I

SOLUTf Offs

BOO

n” (rpml

700

600

IO

c (ppm) FE 8 Cntnxl

entramment speed as a ftmcbon of polymer concentration m parts per mdhon (ppm) a 90/10 glycermelwater solution, V, or a 50150 glycerme/water solution, 0

Mddleman[%l has analyzed this flow, and presented expenmental data which may be represented m the followmg manner +8(~~* Here L IS the ax~al distance

(9) over which

the profile

The solvent LS

flattens to wlthm 1% of umformlty, as defined m Ref [83 D IS the uutlal Jet diameter Equation (9) IS taken over, replacmg D mth 2H and CJ (the average Jet veto&y, and therefore half the maxlmum velocity at the capillary exit) by l/2 U, I e half the linear speed of the roll surface The analogy IS very tenuous, and when F, and F, are equated, the result IS expressible as

601

AK entrammentm a roll coatmg system Ca* = 1 237-O 23

(10)

Wlule we do not expect the coefficient 1 23 to be much better than an order of magrutude estimate, we do have some hope that the dependence of Ca* on y m&t be modeled by thus analogy Figure 7 tests eqns (5) and (10) As expected, the data correlate wtth the smgle parameter y over the entire range (nearly five decades in y) The predlcted slope of -0 6 m the Inertial reDon ISm good agreement with the data, and the lme ISbelow the data (underestunates Ca*) by a factor of exactly 2 0 In the viscous region the predicted slope agam agrees quite well with the observed behavior, and the line IS below the data by a factor of about 3 In view of the tenuous nature of carrymg eqn (9) over to this probIem wlthout modlficatlon, the agreement seems quite reasonable The dependence of U* on u and J.JIS observed (and predlcted) to be

elastlclty of the solution Further studies, m progress, should ad m clanfymg this pomt, and wdl hopefully lead to a better understanding of ths phenomenon Achowledgements-Bay Bolton was supported by the Mater& Research Laboratory (NSF) of the Polymer Science and Engmeermg Department We alsb wsh to thank Eastman Kodak Company and 3M Company for support of thrs program of study NOTATION

Ca = @U/U) D F, FI, Fv

L SR

u* _ u*-rr

477CL-0’59

04/.L-02

v~.cous reDon (y < 10) Inertial remon ( y > 10)

u

W

(11)

vIscoELAsTrcFMJILS Au entramment data were obtamed usmg several polymer solutions Fgure 8 shows typical sets of data for various concentrations of polyacrylanude m a gylcenne/water solvent From FQ 3 we see that for less than 100 ppm of polymer the solutions are only mtidly non-Newtoman Modest mcrements of vlscoslty attend the addltlon of polymer m ths low concentration reDon If these solutions behaved as Newtoman flulds one would expect to see a decrease m a* as the polymer concentration Increased, smce a* IS a decreasmg functlon of viscosity (We note that surface tension ISessentlally independent of polymer concentration) For the 90110 glycennelwater solutions, a shght decrease m 0* IS observed, relatwe to the solvent wth no polymer, up to about 50 ppm Thereafter, there IS a preclpltous nse m fi* SImllar behavior IS seen m solutions made with a SO/SOglycennelwater solvent, except that the decrease m a* at small poIymer concentrations IS not observed Agam, the preclpltlous nse m a* occurs around 50 ppm polymer Fme 3 shows that while vlscoslty Increases unrformly (and slowly) w&h mcreasmg polymer concentration m the 90/10 glycerme/water solvent, the recoverable shear mcreases rapldIy between 50 and 60 ppm Thus leads us to speculate that the greatly enhanced stab&atlon of the entramment meniscus 1s associated w&h the

Cap&uy number Jet diameter, cm forces actmg on memscus due to surface tenslon (a), mertla (I), and vlscous shear (v), dynes gravItational constant, 980 dynes/g film thickness, cm length over which shear stress acts, cm recoverable shear hnear roll speed, cm/s axial width of meniscus, cm

Greek symbols Y = 4dPW” Y

property number shear rate m vlscometnc flow, s-’ 17 non-Newtonian vlscoslty, poise c1 Newtoman vlscoslty, poise P fluld density, g/cm’ surface tenslon, dyne/cm u shear stress acting on entramment 7 region, dyne/cm’ TV stress components measured m a VIScometnc flow, dynes/cm* R angular speed of roll, rpm

Superscnpt *

denotes cntlcal condltlon for entramment REFERENCES

111 Inverantv G . Bnt Polvm 1 I%9 1 245 Perry R- T ,’ Ph D tiesIs, Umverslty of Mnnesota I%9 (Umverslty ticrofilms 67-14,639) Burley R and Kennedy B S , Chem Engng Scr 1976 31 901 Burley R and Kennedy B S , Bnt Polym J 1976 8 140 Wdkmson W , Chem Engng Sa 1975 30 1227 tiddleman S , Fundamentals of Polymer Processmg, (Chap 3) McGraw-H111.New York 1977 Mddleman S , Polym Engng SCI 1978 18 734 Mddleman S , I E C Fundls , 19643 1I8