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Theses and Dissertations

1984

A study of fluid flow in screen process printingShiaw-Min ChenLehigh University

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Recommended CitationChen, Shiaw-Min, "A study of fluid flow in screen process printing" (1984). Theses and Dissertations. 5163.https://preserve.lehigh.edu/etd/5163

A STUDY OF FLUID FLOW

IN SCREEN PROCESS PRINTING

by

Shi aw-Min Chen·

A Thesis

Present in Partial Fulfillment

of the Requirement for the Degree of

Master of Science

in

Chemical Engineering

Lehigh University

1984

ACKNOWLEDGEMENT

The author expresses his sincere appreciation to Professor

A. C. Zettlemoyer for his many helpful criticisms and inspiring dis­

cussions on the research project. Many thanks are due· to Professor

F. J. Micale for suggesting this project and for his many contribu­

tions as supervisor of this research.

Most of all, the author wishes to ·thank the M. Lowenstein Corp.

for their financial support this study, without which this work

would not have ·been possible.

Also, the author would like to thank the Lehigh University Com­

puting Center for their facilities and helps.

Acknowledgements are also given to Dr. K. Chaing, Dr. J •. L. Lin

and Dr. S. Kiatkamjornwong who offerred a number of very helpful

suggestions.

The author also wishes to express his appreciation to Ms.

J. Lavelle for her help in -this study.

Finally, the greatest appreciation must be given to my parents,

Mr. and Mrs. J. N. Chen, for their constant enc~uragement at all

times.

11

CERTIFICATE OF APPROVAL

This thesis is accepted and approved in partial fulfillment of

the requirements for the degree of Master of Science in Chemical En-

gineering.

Date

~t;r~ Professor in Charge

/f_.c.~~ Dr~.Tz:ttl~er

Professor in Charge

r#£? c. c/~ Dr. J. C. Chen

Chairman Department of Chemical ·Engineering

iii

/ \

\

Table or Contents

ACKNOWLEDGF.MENT Abstract

CHAPTER· I Introduction 1.1 Approach to the problem

CHAPTER II Mathematical modelling 2. 1 Introduction 2.2 Region !--flow pattern in the ink bank 2.3 Region II --Flow in the screen holes 2.4 Summary of Region I and Region II 2.5 Limitation

2.5.1 Limitation in Region I "_2.s.1.1 The creeping flow

2 -~ 5 .1 • 2 The singular poi~ t 2.5.2 -Limitation in Region II

2.5.2.1 A long tube 2.5.2.2 Steady state

CHAPTER III Rheology or the ink 3.1 Viscosity of test inks 3.2 Rheology of test inks

CHAPTER IV Results and discussion 4.1 Criteria for calculations 4.2 Rate of deformation

4.2.1 Rate of shear 4.2.2 Rate of normal strain

4.3 Pressure drop build up by ink flow in region I 4.4 The flow rate through the screen 4.5 Volume ·of ink delivered

CHAPTER V Conclusions and future work 5.1 conclutions 5.2 Future research references Appendex A Appendex B

iv

ii

1

2 3 4 4 5

15 16 18 18 18 19 20 20 21 22 22 31 39 39 40 41 41 46 59 55 63 63 64 65 67

71

Figure 2-1: Figure 2-2: Figure 2-3:

Figure 2-4:

Figure 3-1:

Figure 3-2:

Figure 3-3:

Figure 3-4:

Figure 3-5:

Figure 3-6:

Figure 3-'.7:

Figure 3-8: Figure 3-9: Figure 4-1:

Figure 4-2:.

Figure 4-3:

Figure 4~4:

Figure 4-5:

Figure 4-6 :_

Figure 4-7:

Figure 4-8:

List or Figures

Schematic model of rotary screen printer. 4 Simplified model of the ink bank. 6

Schematic or blade and screen where the fluid 9 may move from the ink bank through the holes of the moving screen. Flow in a long cylindrical tube where R is the 15 radius of the tube, Lis the length of the tube.

Apparent viscosity as a function of RPM for six 24 test inks on the Brookfield Viscometer.

Pseudo-stress as a function of RPM for six test 25 inks on the Brookfield Viscometer.

Apparent viscosity as a function of shear rate 26 for six test inks on the Band Viscometer. Shear stress as a function of shear rate for six 27

test inks on the Band Viscometer. Apparent viscosity as a function of shear rate 28

for inks and clears on the Weissenburg Vis­cometer.

shear stress as a function of shear rate for 29 inks and clears on the Weissenburg Viscometer.

Log viscosity as a function of· log shear rate 32 (replot from Fig. 3-5) A fully developed fluid flow in a circul~ tube. 35 Shear stress distribution in the circular tube. 36

a, b, c Shear rate as a function of r, 80

and U, 42 calculated from eq.2-17(c). a, b and c., Rate of normal strain as a function 44

of r, 80 and U, calculated from. eq. 2-17(b). a, b, c, d and e, The pressure drop build up in ~7;

the ink bank acting on the screen as a function of m, n, r, 8~ and U.

a and b, The pressure drop exerting a lifting 51 force on the squeegee as a function of 8

0 and U;

a, b, c; d and e, The flow rate through a single 52 hole in the screen as a function· of R, n, 8

0, r

and u. · a, b and c, Volume ot ink delivered as a function· 56 of R, n, and 8

0•

Volume of ink delivered as a function of upper 61 limit of integration, RU. Volume of ink delivered as a function of lower 62 limit of integration, RL.

V

Table 3-1: Table 3-2:

List of Tables

Formulation of the test inks Constant m and n calculated from Figure 3-7

........

vi

23 33

I.

Abstr:-act

A mathematical model of screen process printing to predict the

hydraulic pressure and the volume of ink delivered through the

screen was developed. The model deals with both fluid dynamics of

the screen process and the rheological properties of the printing

inks.

Calculations from this mathematical model agre~ qualitatively

with the published experimental data in terms · of hydraulic pressure

build-up and volume of ink transferred with respect to squeegee

angle and speed of the moving screen.

..

CHAPTER I

Introduction

The screen process printing, for a variety of applications,

date! bapk many years and is ·older than conventional printing .it-

self. Since the advent of the screen process printing, which

brought forth the capability of producing large volumes of printe~

material at relatively low cost, screen process printing had been

used primarily for low volume, high cost application. It is only in

more rece.nt times that screen process printing has been used for

commercial. operations, and usually on irregular shaped or fll­

defined surfaces such as textiles.(1,2]

The challenge for screen process printing in textiles is two

fold: first, to increase the printing speed, and second, to increase

the quality of the printing. The invention of the rotary screen

printer was a major breakthrough for increasing printing speed and . . '

rendering this technique economically feasible for many prin~~g ap-

plications. Work has also progres~ed on the materials used for· the

screen and on method~ for preparing the holes for imaging. Toe

results of these advances for textile printing applications have

been improved image quality and line definition, and reduced costs

which have led to an expaned volume of screen printing.(3,4,5,6,7]

Al though many advances have been made in terms of the mechanics

2

of rotary screen printing and _quality of the screen, very little

work has been accomplished in terms of refining the process vari­

ables from knowledge of the mechanisms involved in the process. [ 8]

So, the objective .of this work is to study a model of the screen

process printing.

1.1 Approach to the problem

The fundamental mechanism of screen process printing is that a

stencil bearing an image or. a design is attached to a screen. When

stock is placed directly und.er the screen, process ink is forced

through the open mesh the image area of the screen, with a squeegee

allowing, thus, the ink penetrates through to the stock. In other

words, screen process printing involves the principle of actually

printing through a plate.[1l

Based on this fundamental mechanism, a mathematical model is

presented in Chapter 2, and the results of this mathematical model

are presented in Chapter 4, along with a comparism of the.model with

wxperimental results published in 11 terature.

Chapter 3 includes a preliminary study of the rheologiqal

properties of the process inks, which is necessary in order to es­

tablish the relationship among the printing variables as establis_hed

in the modelling.

3

\

2. 1 Introduction

CHAPTER II

Mathematical modelling

A schematic presentat~on of the rotary screen printer is shown

in Figure 2-1.

Roller Bar

Textile

Figure 2-1: Schematic model of rotary screen printer

Some applications utilize a blade in place of the roller bar. The

pressure of the roller bar, or blade, on the screen cylinder is con­

trolled by an electro-magnetic bar under the substrate.

The flow pattern in this process contains three different

regions which are Region I, U~e ink bank contained between the

roller bar and the screen cylinder; Region II, the ink "flow through

the holes on the screen cylinder; Region III, the ink absorbed by

4

the substrate. The Reg~on III is being eliminated from this modell­

ing because of the complexity of the forces involved in the

mechanism of absorption on the substrate. The model will be based

on Region I and Region II only. Assuming that no fprce exists

beyond Region II, which means that once the ink passes through the

screen cylinder, there are no effects due to surface tension, capil­

lary force or graviational force.

2.2 Region !--flow pattern in the ink bank

Region I, represented in Figure 2-2, has been ~implified due to

the complicated nature of the mathematics. Figure 2-2 illustrates

the stationary blade, represented as the roller bar, and the moving

plate, represented as the screen cylinder without holes, which moves

with the· velocity of U in the direction opposite to th~ direction. of

r.

The· ink bank is further assumed to be· infini tly long in the Z

direction, i.e. there is no edge effect, which reduces this system

into a two dimensional problem. Also, it is assumed that the ink is

a newtonian, incompressible fluid. A c_or.rection f~r a generalized

newtonian fluid will be discussed later. Therefore, ~e problem

fits the criteria for the stream function as expressed .in Equation

(2.1).(9,.10]

:-. 1 ~('\/21/J)+-ch r

a<I/J , '\J 2"')

<Y( r, OJ

5

( 2.1)

where

plate

r is the distance from the point of contact of the blade with the plate.

80 is the angle.of the blade with the plate.

9 is the angle measured from the plate.

Figure 2-2: Simplified model of the ink bank.

where 1/1 is the stream function defined by F.quation, (.2 .2) and

(2.3).

(2.2)

and

6

and

'v 2 is Laplace oper~ tor

v is kinematic viscosity, and

t is time.

Vr, v8 are velocity in r, 80

direction, respectively.

ac "', 'v 2

1/1 >

fY ( r, 8 ) is the Jacob operator.

(2.3)

The first term on the left hand side in Eq~ation (2.1) represents

the time d·ependency of the stream ·runction. And, the second term on

the leftr.hand side represents. the inertia force of the fluid.

For further simpUfication, the following assumptions are made:

1. steady st·ate, and

2. creeping flow.

If a steady state is assumed, the first term in Equation

( 2 .1) cancels. If creeping flow is assumed, the second term

represents the· inertia force whic.h becomes very small. Therefore,

F.quation (2.1) is reduced to Equation (2.4).(11]

7·

'v 4 t/J'= 0.

The boundary conditions are:

Vr(r,8=0) = - u

V8(r,8:0) = 0

Vr(r,9:80

) = 0

v8(r,8=80 ) = 0

By solving Equations (2.2), (2.3) and (2.4) (see appendex A) we

get the stream funoUon and the velocity profile in the ink bank can

be expressed by Equations (2.5), (2.6) and (2.7), respectively.

"' =

(2.5)

where A is an integration constant.

u Vr = - 2 2 . ·[ ( sin( 90 ) cos( 80 ) - 80 80- sin (80 )

+ 9 sin2(80 )) sin(9) + ( 9~ ~ sin2(80 } + sin(80

) cos(90

)

+ sin{ 80 ) cos( 80 ) - 8 80 ) cos( 9)], (2.6)

8

u v9 = [ 00

2 sin(9) - e sin2(e0

) cos(e) 92

- sin2 (8) 0 0

(2. 7)

In the next step, replace the plate in Figure 2-2 with a screen

plate as shown in Figure 2-3.

Figure 2-3:.

Schematic of blade and screen

where the fluid may move from the ink bank

through the holes of the moving screen.

Once the holes in the screen are parti.ally depleted as the ink

is printed, the layer of ink immediately above the plate flows in a

9

r

manner to refill the holes. Consequently, the velocity .of this ink

layer is no longer the same as the velocity of the plate. Then, the

boundary conditions change to:

' Vr(r,8:0) = - kU

' . v8(r,8:0) = - 0

' Vr(r,8=80 ) = O

' v8(r ,8=80 ) = 0

Again, solving ·Equations (2.2), (2.3) and (2.4) by using these new

boundary condi .tions yields the new stream function, 1/1' , velocity

profile, v; and V~ , as in Equations (2.8), (2.9) and (2.10).

k r U ,/,

1 = [( a

02 + 8 sin(8

0) cos(8

0)

'¥ 8~ - sin2(80

)

( 2. 8)

k u 2 2 · [( sin(9

0) cos(8

0) - 9

0 8 - sin ( 8 ) 0 0

(2. 9)

10

' Vg = k u

2 2 I 8~ sin(8) - 9·sin2(80

) cos(8) 80 - sin (8

0)

(2.10)

In reality, the flow rate of the ink through a single hole on

the screen is very small so that 'the proportionality constant, k, is

very close to one. Therefore, Equ~tion (2.8), (2.9) and (2.10) ar.e

approxima.tely equal to Equatiqns (2.5), (2.6) and (2.7), respec­

tively.

The most. interesting parameter in Regi~n I is the normal pres­

sure distribution along the plate, which is generated by the ink

flow in the ink bank.

The total stress tensor, or pressure tensor, is defined in

Equation (2.11) ~[ 12]

!J = ! + 6 p =· (2.11)

where

!J is total stress tensor , or pressure tensor.

T is stress tensor. =·

d is unit tensor.

P is the surrounding pressure.

11

If one considers the ink bank, the only surrounding fore~. ac­

ting on the system is the atmospheric pressure. Therefore, Equation

( 2. 11) can be simplified to Equa ~ion ( 2. 12) •

fl"=T+6P, = = = a

where Pa.is atmospheric pressure.

The stress built up by the flow, 1. e. the pressure drop, or dif­

ference between surrounding pressure and the pressure in the system

in Region I can be discribed by Equation (2.13),

n= T • ( 2. 13)

The· stress tensor is def_ined in Equation ( 2 .14). [ 13]

! = T/a y • (2.14)

where

~a is apparent viscosity.

Y is rate of deformation tensor. =

The rate of strain tensor is defined in Equation ( 2 .15) •

12

(2.15)

where

is "del" c;,pera tor .•

is velocity vector.

is transpose of ('\7 V).

The rate of strain tensor can be ~xpressed by Equation ( 2. 16) ,

(2.17) and (2.18).

Yrr ryVr

= 2-' ~r

Yee 1 ave vr

) , = 2-(-+-r as r

'Yre . ~ ( ~.) 1 avr

= Yer= r °a + -·--. r r r a 9

where

Yrr· is the normal rate of strain in r direction.

Yee is the normal rate of strain in 9 direction.

'Yre is the shear ra.te.

(2.16)

(2.17)

(2.18)

By substituting for Vr and v8 in Equation (2.16) to·(.2.18), Equation

(2.19), (2.20) and (2.21) is obtained.

Yrr = a, (2.19)

(2.20)

13

• Vg Yre = - -;- (2.21)

The. stress tensor built up by Region I is defined by Equation

( 2 • 2 2) , ( 2 • 2 3) and ( 2 ~ 2 4 ) •

flee= ( 2 ( ave + V )) a r ae r

llre = a< V9 f c3Vr -+--) r r ae

where

flrr

flee

is the normal stress in r direction.

is the normal stress in 9 direction, also called Pr in this case.

fl re is the shear stress.

(2.22)

(2.23)

(2.24)

The only pressure which is exerted normally to the ~creen plate

is .D.Pr and is expressed by Equation (2.25).

where .D.P1 is the pressure drop which is exerted normal to the screen plate and which is generated by the flow in the Region.I.

14

(2.25)

2.3 Region II --Flow in the screen boles

Consider a single screen hole as a cylindrical tube as shown in

Figure 2-4.

·I

L R

I I

I

f

Figure 2-4: Flow· in a long cylindrical tube

where· R is the radius of the tube,

Lis the length of the tube.

Assuma.._1h~ink in the tube is a newtoniari fluid, and the ink

flow is well developed, so there is no edge effect. Then, by Hagen­

Poiseuille law, the volume flow rate is expressed by Equation

(2.26) and (2.27).(14]

. 4 ~PIIR

Q = ----8 L T/ a

,. (2.26)

15

\/

where

Q is the volume flow rate.

~P11 is the pressure/'reqUired to produce flow at a flow rate of Qin this tube.

A is the internal area of the tube.

2.4 Summary of Region I and Region II

(2.27)

Because effects of. Region Ill are not been considered, so,

there is no additional _force applied to this system. Therefore, the

pressure. drop /Uired in Region II to produce ink flow must be

equal to the pressure drop created in Region I, as expressed by

Equation (.2.28).

Therefore ·the .6PII term in Equati (2.26) is replaced by ~PI" as

defined in Equation (2.25), then the ow rate in the tube is dis-

cribed by Equation (2.29). \

A2 2 oV8 Q = - ( C - + vr )) , 8 L r O 8

(2.29)

16

Aocording to Equation (2.29) the flow rate in the tube is not a

function of the viscosity of the. ink. The equatj.on is applicable,

if the ink is a newtonian fluid, but, in most cases, the ink is a

non-newtonian fluid .• [7, 15] Therefore, the effective visco~ity for a

particular ink will vary from Region I to Region II because rate of

shear applied in each region· is differe~t. To compensate for this,

two terms designated as rJ aI and T?arI are inserted in Equations

(2.29) to yiel~ Equation (2.30).

A2 T/ar 2 (YV9 . Q=--(-(-+V))

8 L T] a II r (Y 9 r (2.30)

(

The volume of the ink transferred through a single hole can be

expressed ~Y Eq ua ti on ( 2. 31) •

V = J Q dt

where

V is the volume. of ink transfer.red through a single hole.

t is the 'time a single hole travels during. ink transfer through the hole.

Substituting dr/U for dt in Equation (2.31) yields Equation

(2.32) which discribes the volume of ink transfer.

17

The calculation can be d.one at this point, that once the

rheological properties of the ink and the operating parameters of

the syst-em are provided, then, th.e volume of ink transferred as well

as the flow rate in a single hole and the hydrodynamic pressure drop

can be calculated· by Equations (2.25}, (2.30) and (2.32).

(_-) -"

2.5 Limitation

2.5.1 Limitation in Region I

2.5.1.1 The creeping flow

In Region I, the assumption. of creeping flow was made for the

purpose of simplification. However, this assumptiom restricts the

application of this model to a small region. )

Since the flow is creeping flow, then, the Reynold's number

must be very small. [ H, 12] By defirii tion, Reynold's number is ex­

pressed by Equation ~2.33).

pr U R = - ' e 7/a

(2.33)

where

18

Re is the Reynold's number.

Consequently, in order for Re to remain very small, the limitation

for r in F.quation (2.33) is expressed in (2.34).

r « V -u (2.34)

where is kinematic viscosity.

In conclusion, this model is valid only when r .is much smaller

then the ratio of the kinematic viscosity to the screen velocity.

2.s.1.2 The singular point

If one considers the extreme case that if r approaches zero in

Region I, it is very obvious that there are two different boundary

conditions which apply to this single point such as:

Vr(r=8, 8=0) = - U,

and

Obviously, in the neighborh9od of r equal to zero, this model is

divergent.

19

2.5.2 Limitation in Region II

2.s.2.1 A long tube

The Hagen-Poiseuille law applies only if the edge effect is

negligible which is accomplished by assuming that the ink is. new­

tonian and ink flow is well developed.

An energy loss due to the edge effect can be caused by a sudden

contraction at a entrance to a pipe line. This energy loss is less

for a laminar flow than for turbulent flow.[16)

Actually, an entrance length of a tube must be in the order of

where Le is entrance.length.[16)

in order to produce the characteristic parabolic velocity profiles.

Eisenstadt and Kline et.aL, suggested that in order to main­

tain Poiseuille-flow the dimension must be:

L --> 1

In addition, for a rounded entrance and for laminar flow, the

entrance loss is negligible.

20

2.5.2.2 Steady state

The Hagen-Poiseuille .law is based on steady state flow However,

examination of Equation (2.30), reveals that the flow rate of the

ink varies with a change in r. This means that during a s.ingle pass

of the .holes through the ink bank, the flow rate of ink through the

screen is not constant, and therefore, it is not steady state.

J

..

21

\

,,.---

CHAPTER III

Rheology of the ink

Al though, it was assumed in the theoretical approac~, that the

ink is a newtonian fluid in reality, most inks exhibit non-newtonian

behavior. [ 7, 15]

3. 1 Viscosity of test inks

As shown in Table 3-1, a series of test inks were formulated

for investigation as ·model inks where, in one set., the pig~ent

remained constant and the binder system was systematically changed,

to produce inks 1, 3, 5 and .6. In another set, the pigment was sys­

tematically changed, while attempting to maintain the same shade of

blue, and the binder remained constant producing inks 1, 2· and 4.

The viscosity of the test inks were measured on the Brookfield

Model LV Viscometer with a 114 spindle, Band Viscometer, and the

Weissenburg Viscometer witn a cone and plate configuration. The

results on the Brookfield Viscometer are presented in Figures 3-i in

which apparent viscosity is plotted as a function of spindle speed

and in 3-2 illustrating apparent viscosity times RPM, i.e. a pseudo­

stress, as a function of RPM, approximating the classical plot of

stress versus rate of shear.

Figures 3-3 and 3-4 repr·"sent the results from the Band Vis­

cometer illustrating apparent vi~cosi ty· and shear stress, respec-

22

Table 3-1:

Formulation of the test inks

1 2 3 4 5 . 6 Blue 3G 80.0 54.0 80.0 82.5 80.0 80.0 Valet 4BN 50.0 50.0 50.0 50.0

9.0 4.0 9 .. 0 5.0 9·.0 9.0 27.0

Orange C

Scarlet FDLN 24.0. 282C Clear 661.0 715.0 688.~

200.0 200.0 200.0 200.0

661.0 70-SE Clear 661.0 Binder TP989 200."0 200.0

661.0

tively, as a function of the shear rate.

The results from Weissenburg Viscometer are shown in Figures

3-5 and 3-6 in which apparent viscosity and shear stress, respec­

tively, are plotted as a function of shear rate for six .inks and

clears, i.e. the unpigmented portion of the ink.

Comparison of these results indicate that the Brookfield Vis­

cometer is measuring the viscosity only at low rates of shear as

evidenced by th~ very slow spindle speeds and the corresponding high

23

40

,..., 32 Cl)

1/)

0 a. ._,

(\J

0 24 .-X

>--+'

1/)

0 l 6 (.)

1/)

>

8

0 0 3 6

RPM

Figure 3-1:

9 12

Apparent viscosity as a function of RPM

for six test inks on the Brookfield Viscometer.

24

0

0

I::,.

'iJ

<> lS1

l.NK 1

1 N-K 2 1NK 3 1NK 4 1NK 5 ]NK 6

so

'""' 40 :r: CL

0:: 'lJ

1/)

0

0. 30 ....,

(\J

0 .-

X

L 20 0... a:::

X

(.)

1/)

1 0 >

0

••' ·~'' '~---'Y", . .-,,~·• - • ,.._.r•, ..... ....,- -_,.• __ .• ·••·• -··-. ~ ,

0 3 6

RPM·

Figure 3-2:

9

Pseudo-stress as a function of RPM

1 2

for six test inks on the Brookfield Viscometer.

25

D INK 1 0 INK 2 6 INK 3 \J· lNK 4 O· lNK S lSI ·1 NK 6

1 0

D INK 8 0 INK

.!J. INK 'v INK ·O lNK

6 1SI 1NK

4

2

0 0 2500 soon 7500 10000

Shear Rate (sec- 1).

Figure 3-3:

Apparent viscosity as a function of shear r:-ate for six test inks

on the Band Viscometer.

26

1

2

3

4 5

6

,-...

(\J

E l)

' C >,

-0 ._,

If)

(/)

V

'--+'

U)

'-(0

\)

..c U)

8600

7200

5800

4400

3000 0 2500 5000 ?500

Sh c a r Ra t e ( s e ·c - l )

Figure 3-4:

0

0

6

V

<> 1S1

10000

Shear stress as a function of shear rate for six test inks

on the Ba.nd Viscometer.

27

lNK 1

1NK 2 1NK 3 1NK 4 1 NK .5 lNK 6

. ,-..,

Cl.)

IJ)

0 a..

-.J

>-......

IJ)

0 (.)

IJ)

>

so

40

30

20

10

0 0

0 1NK 1 0 lNK 2 ~ INK 3 "v lNK 4 0 1NK 5 ISl JNK 6 Q) NT4A ~ 282C w 70SE1

250 500 750

Sh e a r Ra t e ( s e c - l ) ·/

Figure 3-5:

1000

Apparent viscosity as a function of shear rate for

inks and clears on the.Weissenburg Viscometer.

28

7500

6000

........ (\J

E c..)

'-C 4500 ;:,,,.

-0 ......, If)

If)

V

'--+"' 3000 U)

'-(0

V

...c U)

1500

0 0 1500 3000 4500 6000

Shear Rate (s·ec- 1) ,·

Figure 3-6:

shear stress as a function of shear rate for

inks and· clears. on the Weissenburg Viscometer.'

29

D lNK 1 0 INK 2 ~ lNK 3

'7 INK 4 () INK s ISi INK 6 Q) NT4A & 282C w 7 OS"E 1

apparent viscosity values, Th.e Band Viscometer is capable of

measuring viscosity only at relatively high rates of shear.

However, the Weissenburg Viscometer is capable of taking measure­

ments over both lower and higher rates of .shear range.

There is a lack of agreement in the results from the Band Vis­

cometer and Weissenburg Viscometer which may be related to the in­

complete wetting of the mylar film on the Band Viscometer by these

essentially water based inks~ These results suggest that the .Band

Viscometer is not the best instrument for measuring these test inks

which vary as to their ability to wet the mylar film which fn turn

would significantly effect the rheological measurements.

The results from the Brookfield Viscosity are inadequate for

complete rheologi~al charaterizat1011 of the inks because the rate of

shear is not well defined. However, this instrument is very easy to

operate and is capable of differentiating the inks based on their

relative apparent viscosity in -terms of RPM.

These results indicate that the _most c~mprehensive and reliable

viscosity measurement are obtained with the Weissenburg Viscometer.

It should be mentioned, however, that the measurements are t.i,me con­

suming particularly at.lower rates of shear.

J.

30

3,2 Rheology ot test inks

Plotting the apparent viscosity data from the Weissenburg Vis­

cometer on a log-log scale. as shown in Figure 3-7 illustrates that

the apparent viscosity of the inks decreases linearly as the shear

rate is increased. This relationship ~uggests that the rheological

properties of the inks can be described by the Ostwald-deWaele power

law which is defined by equation (3.1) .[ 17]

where

T is the shear stress.

t is the shear rate.

The constant m, with dimension of N sn

M2 constant n are characteristic of each fluid.

(3.1)

and the dimensionless

While n equal to one,

then, mis apparent viscosity and is newtonian fluid.

Since a generalized newtonian fluid is defined by equation

(3.2).

T : "a 'Y ,_

where "a is apparent viscosity.

So, the apparent viscosity for the power law defined in equation

31

D lNK 1 0 1NK 2

1 02 ~ lNK 3

,....._ 'v 1NK 4 (l.l

0 lNK 5 11)

0 ISi 1 NK. 6 a.

(D NT4A .....,

>- l O 1 £ 282C -t-'

11) w 70SE1 0 (.)

11)

>

1 OO

1 0 - l ----.............. -~~...u.----..J..--+~UJ.:IJ-J.-.J..-1...J..UJJ.I

10° 10 1 1 o3 10 4

Shear Rate (sec-1).

Figure 3-7:

Log viscosity as a function of log she~r rate (replot from Fig. 3-5)

32

(3.1) is:

· n-1 '1a = m >' , (3.3)

or

log 17 a = log m + ( n-1) log >' (3.4)

Thus, the constant (n-1) and ( log m) represent as slope and inter­

cept, respectively,in equation (3.4).

The constant m and n are calculated from Figure 3-7 and i.s

shown in Table 3-2.

Table 3-2:

Constant m and n calculated from Figure 3-7

Sample No

rn

n

Inks

1 2 3 4 5 6

379 513 423 402 247 151

0;2a4 0.23a 0~311 o.306 0.301 o;3aa

Clears

282C NT4A 70-:-SE

92 186 214

· 0. 384 0. 381 0. 302

Substitute equation (3.3) and (2.21) into equation (2.25),

then, the pressure .drop build up in region I is calculated by equa­

tion (3.5).

33

V 1 av n-1 2· av = m (- ...:.i + - ---!'.:..) (- (---.i. + V ))

r r <Y e r .a e r (3.5)

Also, by the same substitution to equation (2.JO), the flow

rate in region II may be calculated by equation (3.6).

A2 1 V 1 oV n-1 Q = - - (- ~ + - ---!'.:..)

8 L TJ all r r () 8

( ~ ( ()V9 + V )) ,. r CY 9 r

( 3. 6)

But, in equation (3.6), the viscosity in region II, >'n , is still

not known because the shear rate.in region II has not been defined.

In order to determine the shear _rate in region II, consider a

circular tube as shown in· Figure 3-8.

A momentum balance over this tube is described by equation

(3.7).

d 'r'z dr'

where

~p = (-) L .

r I ,

, r, z i_s shear stress.

~P is pressure drop over the tube.

34

( 3. 7)

where

L

I

J.

r' and z are cylindrical coordinates.

R is the radius of the tube.

Lis the length of the tube.

Figure 3-8:

A fully. developed fluid flow in a circular tube.

solving the differential eqation (3.7) for shear stress yields equa­

tion (3.8).

T r'z 6P

= (-) 2 L

r' . , (3". 8)

According to this equation, the shear stress distribution in

this tube can be described by Figure 3-9.

Because the radius of the holes in the screen is very small the

shear stress at any point along the radius may assumed to be a

constant, r 8, as described _by equation (3.9).

35

where

R

T 1 (r' = 0) = Q r z

L__ T ( r' = R) = ...----- r' z

TR is the shear stress at r' equal to R.

Figure 3-9:

Shear stress distribution in t~e circular tube.

T: T r'z

8P = TR = (- ) R ,

2L ( 3.9)

Then, substitute the pressure drop in equation (3~9) by equation

(2.27) and solve for the shear rate in the tube as described in

(3.10).

4Q ?' II =-;jp

where >' II is the rate of shear in region II •.

(3,10)

In equation (3,6), express the value of 7JaII in terms of the

shear rate as ex;pressed in equations (3,3) and (3.10) to obtain the

flow rate defined by equation (3.11).

36

[ 1 A2 ( ave ) Q = --- ae + vr 4 ,rr L

1

(- ..!§ + ~ avr n-1 n

) l r r ae (3. U)

. .

. 4

~

It is noted in equation (3. 11) that the flow rate, Q, is not a

function of the rheological parameter m.

In the next step, consider the values for the. velocity

profiles, Vr and v8 , described in equations (2.6) and (2.7). Ex­

tract the the screen velocity ter·m,: U, from the equations and ex­

press the remainder a·s fr( 8 ,80

} and f 8( 8 ,80

) which are functions of.

8 and 80 only, as represented in equation (3.12) and (3.13).

(3.12)

(3.13)

Replacing the V r and V 8 in equation ( 3. 11 ) with their cor­

responding terms as given iri equation (3.12) and (3.13) yields equa-

tion (3.14) describing the flow rate ·or ink.

37

·~

..

Q = u

1

(

- ~ + -.1 ~ ) n-1 l r r ae . 4

~

n

(3.14)

If in equation (2.32) the Q term is replaced by the description

of Q given in equation (3.14), the volume of ink transferred through

the screen can be expressed by equation (3.15).

V = J [ 1

dr , (3.15)

According to equation (3.15), the volume of ink transf~rred

through the screen is not a function of the speed of the screen.

38

CHAPTER IV

Results and discussion

4.1 Criteria for calculations

In order to test the validity of the ma thema ti.cal model, values

for the pressure drop , .6P, ink flow rate , Q, and volume of ink

transferred through ·the screen , V, were calculated by computer for

comparison with corresponding values reported in the literature.

The independent variables carefully selected for these calculations

are:

r 10-6 to 1.0 cm.

80 5° to 25°.

u 25 to 125 cm/sec.

m 100 to 500 N(s)n/M2.

n 0.2 to LO.

L 0.01 cm.

R 0.002 to 0.01 cm.

The lower limit for distance, r, is arbi trari.ly chosen t.o be

cm. According to the limitations discussed in sectlon 2 .5 .1.2

when r approaches zero this model approaches a singular point. This

means that as r approaches zero all the dependent variables will ap-­

proaoh infinity. The lower limit, therefore, cannot be zero. The

upper limit is obtained by setting the · maximum Reynold's number

39

equal to one in Equation 2-26 in section 2.5.1.1, where the cal­

culated maximum distance, rmax• is equal to:

V rmax =-

u

According to calculations based on a newtonian ink, r varies · max

from 1 • O cm to 20 cm, and 1 • O cm is assigned as the upper limit. 90

and U are arbitrarily chosen to vary from s0 to 25° and from 25 to

125 cm/s, respectively. Rheological parameters, m and n, are

selected based on the values- obtained from the test inks. Screen

parameters, L and R, are assigned values which meet th~ specifica­

tions discussed in section 2.5.2.1. If the Reynold's number is set

equal to one, then .according to Eisenstadt's Equation, the ratio of

L to R must be greater than or equal to one.

4.2 Rate of deformation

The rate of deformation, which is a function of the rate of

shear and the rate of normal strain, is very important in this math-

ematical modelling. The fluid to which the model applies is assumed

to be non-Newtonian or shear thinning, and its effective vfscosi ty

would depend on the rate of shear applied. In addition, the pres-

sure build up in the ink bank, the ink flow rate, and the amount of

ink transfer through the screen are dependent on the rate of nor.ma!

strain.

40

4.2.1 Rate or shear

The rate of shear chosen for the calculations was calculated

based on Equation. 2-17(0), and the data are shown in Figure 4-1.

Figures 4-1 (a) ano (b), which: represents shear rate as a func­

tion of r and 80 , respectively, indicates that r and 80

have a

similar relationship to shear rate. As the d~stance, r, or squeegee

angle, 90 , i_ncreases,. the shear rate initially decreases rapidly,

and then gradually starts to level off. Figure 4~1 (c), shows that

the shear rate increases linearly with an increase in screen speed.

It should be noted in Figure 4-1 (a) that the shear rate increases

to very high values for small values of r.

4.2.2 Rate of normal strain

The rate of normal strain is the strain which is operation nor­

mally to the moving surface, i.e. screen, and which is responsible

for the pressure normal to that surface. Figure 4-2 represents the

relationship between rate of normal strain and screen process print­

ing parameters r, 90 and Figures 4-2(a) and (b), show that the rate

of normal strain decreases with increasing r or 90

•. Ffgure 4-2 (c)

shows that the normal strain decreases linearly with increasing

printing speed u.

41

106

:'105 I 0 Q) a)

U=75cm/s e =5°

0

I I

103 -ti----+-, -------J,......_ __ ....,;._+,---~1

0 0.3 0~6 0.9 1.2

12 t('\ I 0 ..... >< ,....._

"'j 8 0 Q) a) .._,

Q)

+> Ill ~

~ QJ

..c: U)

0

r (cm) (a) Shear rate as a function of distance r.

U=.75cm/s i i

r=1 .Ocm I

I I I

--------2 10 18 26 34

00

(degree) (b) Shear rate as a function of squeegee angle 90.

Figure 4-1: a, b, C

Shear rate as a function of r, 80

and U,

calculated frora eq.2-17(c).

42

10

"' I 0 ... >< 7

-... I 0 4) rn ....... 4) .µ a, 4 s.. s.. a, (1l

..c: en

0

9 •5° 0

r=1 • 'Jcm

1.5 U (cm/sec) X 1 o'"'

15

( c) Shear ·rate as a func ·ti on of scr e.en speed U.

Figure 4-1, continued.

43

0

~ -300 I

() Q) O')

Q) .µ al S..

S--4 -600 al Q)

.r:: Cl) . z

-900

+1

-3

-.... I () Q) O')

....... Q) .µ

-7 al S--4

S--4 al Q)

.r:: Cl)

• z -11

(b)

0 0.6 r (cm)

·u=75cm/s 9 =5°

0

0.9 1.2

(a) Rate of normal strain as a function of distance r.

2

Rate or 10 '( 18 ) 9 degree

U=75cm/s

r=1 .Ocm

26 34 0 . . ..

normal strain as a function of squeegee angle 90

,

Figure 4-2: a, b and c,

Rate of normal strain as a runctioh of r, 90

and U,

calculated from cq. 2-17(b).

44

0

-- -4 I C) Q) r1l

Q) .µ al S..

S.. -8 al Q)

..c: U)

• z

-12 0 75

U ( cm/sec )

9o=5o

r=1 .Qc~

I !

·' i I I

l I

150

( c) Rate of normal strain as a function of screen spead U.

Figure 4-2, continued

45

4.3 Pressure drop build up by ink tlow in region I

The pressure drop which results from ink flow in region I is a

significant driving force regulating both the ink flow through the

screen holes and, ev.enturally, the printing of the ink on the sub­

strate. It is also possible from the proposed model to calculate

the lifting force, i.e., the force acting on the squeegee blade,

which would be required actually to lift the squeegee blad·e from the

screen. Figur·es 4-3 a, b, c, d and e 111 us tra te as a result of the

computer calculation based on the proposed model how the pressure

drop in region ·I of the system is affected by systematic changes in

one of the independent variables, m, n, r, 80

or U.

It is evident from the results presented in Figure 4-3 (a) and

(b) that the pressure drop is greater when the rheological

properties of the ink on the press are characterized by large m and

n values. The increase in ~P is linear with respect to m and ex­

ponential with respect to n. The advantage of this enhanced pressure

drop is the fact that more ink will be delivered through the holes

of the screen. As anticipated, the pressure drop decreases as the

value of r increases, i.e. as the distance from the "blade-screen nip

gets larger, as shown in Figure 4-3(c).

When the squeegee angle is increased from 5° to 25°, the pres­

sure drop in the system decreases significantly as represented in

Figure 4-3(d). This trend in· pressure drop by varing 80

is similar

46

14

-N

Us75cm/s

eo=50

r=1 .Ocm

n=0.3

e ~ 10 ~ 'tj

........

2 100 200 300

m 400

i

I

I I I i j I

! I I

I I I

500

(a) Pressure drop on the screen as a function of m. 104

U=75cm/s

g- ,02

r=1 .Ocm

9 =5° 0 M 'tj

Q) M ::s [1)

~ 10 M

p...

m=300

0 0.8 1.2 1 • 6 n

(b) Pressure drop on the screen as a function of n.

Figure 4-3: a, b, c, d and e,

The pres-sure drop build up in the ink bank acting on

the screen as a functioh of n, n, r, 80

and U.

47

45 ·-·-··- ·-·-.. - ---·-·-·--·

U=75cm/s I

I 9o=5o

i I - ! N I e m=300 I

30 I C, I

I ....... i c:: n=o.3 >,

'O ........ p. 0 s...

'O

(I) 15 s... ·, ::s a,

"----------a, (I) S... p..

0 n 0 0.3 0.6 0.9 1.2

r (cm) (c) Pressure drop on the screen as a function of r.

90

30

U=75cm/s

r-=1.0cm

m=300

n=0."5

2 JO 18 26 34 e (degree)

(d) Pressure drop on \he screen as a function of 80

•

Fi;tire 4-3, c and d, continued.

48

'

-N e t.)

9

'= 6 >, 'O .....,

0 0

··--- .. ·--... -.. ,-·-------·--····-1 I

7.5

9 =5° 0

r=1 .Ocm

m=300

n=0.3

U (cm/sec} X 10~1

i

15

(e) Pressure drop on the scr~en as a function of U.

Figure 4-3, e, continued.

49

r

to that reported by G. Boycigiller and L •. w. c. Miles.[8, 18] The

model also predicts, as shown in Figure 4-3(e), that as the screen

velocity is increased, the corresponding pressure dr·op increases in

a nonlinear mode. G. Boycigiller and L. W. c. Miles also reported a

similar relationship between .6 P and U. [ 8, 18] The ·validity of the

mathematical model is substant;f.ated by its ability to predict the

results obtained experfmentally as reported in literature.

In addi ti.on to the force exerted normal to the screen by the

build up of the pressure drop in the. ink bank, an opposite force is

also exerted which tends to lift the squeegee from the screen. A's

shown in Figure 4-4(a) and (b), this upward lifting force is very

large when the squeegee angle is small but decreases rapidly as the

angle is increased. [81 An ·increase in the velocity of the screen

will also produce an increase lifting force on the squeegee.(8]

4.4 The flow rate through the screen

The ink flow rate through a single hole in the screen is cal­

culated using Equation 3-9 as a function of systematic changes in

the independent variables R, n, 90

, r and U, as ·Shown in Figure

4-5 a, b, o, d and e, respectively.

As shown in Figure 4-5(a), an increase in the radius of the

holes in the screen will cause .a logarithmic type increase in the

flow rate of ink t~rough the screen. This behavior is .logical, and

50

(I)

"Cl a,

,-1

.tJ

c:-(J\J

e p. C)

0 ........ s... C:

"Cl >, "Cl

9 ...-----·- -· - .... ' ...

7

--·-·------------- ---

U=75cm/s

r=1 .Ocm

m=300

n"'0,3

f ._ 5 ::s rn rn (I) s...

p..

(I) 'Cl a,

9

'.ri,,..... 6 ~ o e

C) p. ........ 0 C: r,.. >,

'Cl 'Cl

0

3-r------1----+-----l------J 2 10 18

9 (degree) 0

26

(a) Pressure drop as a function of squeegee

0

9 =5° 0

r=1 0cm

m=300

n=o.3

.~

------------

75 U ( cm/sec )

54

angle, 80

•

I I

150

(b) Pressure drop as a function of screen velocity, U.

Figure 4-4: a and b,

The pressur~ drop exertinc a lifting force on the squeegee as

a function of 80

and U.

51

10-12 ";!!--·-- --·-~ ..... --· .. --· ..... _ --·---

U•75cm/s

r•1 .Ocm

9 =5° 0

n=0.3

I

I 10-l:,,...-"--:~-----i-------+------+,-----J

0

0

0~004 0.008 0.012 0.016 R {cm)

Flow rate as a function of hole size, R.

0.4 0.8 n

U=75cm/s

r=1 .Ocm Q =50·

0

R=0.004cm

1.2

(b) Flow rate as a function of n.

Figure 4-5: a, b, c, d and e,

l I

I I

I i

i I I

!

1 • 6

The flow rate ·throubh a. sinGle hole in the screen as a function

of R, n, 80

, rand U.

52

12

... 0 ... >< 8 -() G) 0) ......

I"\ E! ()

0

·-·-· --· -- ·-··- -·-···-.. ---·--·-------........ -·--1 U•75cm/s

-r=1 .Ocm

R=0.004cm

n=0,3

I I ! I

I ! '

2 10 18 26 34 9

0 (degree)

(c) Flow rate a~ a function of squeegee angle, 99

. 10

,q-.....

0 ->< -() G) rn ......

I"\ E! ()

.........

G) ... ct! ~

) 0

r-i rz.

6.7

3.3

0 0 0.3 0.6

r(cm)

U=75cm/s

R=0,004cm

e =s0 0

n=0.3

0.9

(d) Flow rate as a function of distance, r.

Figure 4-5 c and d, continued.

I I I

I I I

1.2

10 .. ---·--··---· R•0.004cm

I.O r=1 .Ocm / .... 0 8 =5° I .... 0 ;

>< ,.

6.7 n=0.'3 . ,-.. . _: f t) : .... (I) . - . .. II)

......... I"\

E! t) .._,,

QJ 3.3 +> ~ ~

:a 0

,-l

rs..

0+------+------+------4------1 0 75 150

U (cm/sec)

(e) Flow.rate as a function of screen velocity, U.

Figure 4-5 e, continue.

54

a fast flow of inlc would be expected through a "loose" screen with

large openings because it offers little resistance to fluid . .flow.

An increase in the n value which characterizing the degree of

non-Hewtonian behavior of the ink, results in a large increase in

ink flow, as shown in Figure 4-5(b). Conversely, an increase in e . 0

or r would tend to dec~ease the ink flow rate through the screen ac­

cording to Figure 4-5(c) and (d), respectively. Finally, an in­

crease in the screen velocity will produce a corresponding linear

increase in the ink flow rate as shown in Figure l~-5(e).

4.5 Volume of ink delivered

To the printer, the most irnportant advantage of the mathemati­

cal model would be its ability to predict how the volume of ink

delivered to the substrate would be effected· by changing the ink

and/or the press conditions! Equation 3-14, was used to calculate

the volume of ink delivered per hole as a function or· the independ­

ent variables R, n and 80

, as shown in Figure 4-6a, band c, ~espec­

tively.

Figure 4-6( a) and (b) show· that an increase in R or n results

in a logarithir.ic trpe increase in tho volu1.1e- of ink delivered which

is consistent with the results obtained for the flow rate, Q, varil.!s

in the same manner. The prediction that the volu1.1t, of ink deliv·ered

would decreases as the squeegee angle, 00

, is incr~ased, as shown in

55

,-...

-1~ 10

t"\ -15 e 10 0

(I)

8 ;j

,-t

0 >

10-17

I

.... ···---------------~.

n=0,3

8 12 16

R (cm) x103

(a) Volur.1e of inl: dclive:red as a function of hollJ zize, R.

Figure 4-6: a, band c,

Voluwo of ink delivered as a function of H, n, and 80

•

56

-t"'I e ()

10-18

1 o-20-r------ir------+------+-------J 0 0.4 0.8 1.2 1.6

n

(b) Vol~me of ink delivered as a function of n.

Figure 4-6(b) continued.

57

10

8

U'\

0 6 R=0.004cm ->< ,..... n=0.3

t<"\ e C) ..._,,

Q) 4 e ::, ~ 0 >

2

0 2 10 18 26 34

80

(degree)

(c) Volume of ink delivered as a function of squeegee angle.

Figure 4-6(c) continued.

Figure 4-6(c), agrees with the trend reported by B. F. Dowds and by

N. Hiro.[ 15 ,19] Appendix B pres~nts the numerical values for the

volume of ink delivered over a wide range of independent- variable

values.

The theoretical model also predicts that the volume of ink

delivered is independent of the screen velocity as is evident from

inspection of Equation 3-14. The initial results of rotary screen

printing trial experiments, which were carried out by the

M. Lowenstein Corporation in the printing speed range of 7 to 45

yards per minute, confirm the ·theoretical predictions by showing

that the volume of ink delivered was independent of printing

speed.[20]

The limits of intergration along the·. screen, i.e. the upper c3:nd

lower limits of the r values, for the theoretical predi tions

presented in figure 4;;.6 (a), (b), and (c), were arbitrarily set at

10-6 to 1 cm. The lower limit of intergration is especially impor­

tant because the flow rate, normal pressure, and rate of shear all

increase dramatically as r becomes smaller and smaller. It is not

possible, furthermore, to set the lower limit of integration at r

equals to zero because all three quantites approach infinit~ as r

approaches zero, and hence they are not mathematically defined at

the singular point of r equals zero. The rationale for establishing

1 o-6 cm as the lower limit is that this size represents the smallest

59

pigment particle which can be present in the bank.

Calculations. of ink volume delivered were carried out in Figure

4-7 and 4-8 as a function of the variable upper and lower limits of

integration, respectively .Figure 4-7, which for all practical pur­

poses shows that the effect of increasing the size of the .ink bank,

shows that the vol~e of ink delivered increases rapidly up to about

one cm followed by a gradual increase with increasing r. Figure

4-8 shows the effect of volume of ink delivered when the bank s'ize

is fixed at one cm and the lower limit of integration is varied for

smaller and smaller va.iues. It is obvious from ~hese calculated

results that the bulk of the ink flow ·occurs at small values of r.

60 ·

10-16 ------·-----··

9 8

9 =5° 7 0

6 R=0,004cm

5 n=0,3

4 ,-...

r,"\ e () .......

3 a, s ::,

9 =50 ,-f 0 0 >

2 R=0,004cm

n=0,3

10-17 -t-----t----+-------l------l 0

Figure 4-7:

0,4 0,8 1.2 1.6

RU (cm)

Volume of ink delivered as a function of upper limit

of integration, RU.

61

10-16 ------·- --·-·· -- ·----------·---·---~-----·-----,

-17 10

R=0.004cm

n==(). J ,..._ t('\

e 0

10-18 ',,J

(1)

e ::, rl 0 >

10-19

10-20

10-5 10-3 1 Q-.1 10 O

RL ( cm)

Figure 4-8: Volume of ink delivered as a function of louer lir.1it

of integration, nL.

62

CHAPTER V

Conclusions and future work

5 .1 conclutions

The limitions of the mathematical model must be reconized for a

proper interpretation of theoretical results and conclusions based

on those resuJ_ts. The mathematical model considers two regions of a

rotary screen printing process, where region one considers the

forces corperative in the bank. of ink above the screen and region

two considers the flow of ink in the hole of the screen. The sub­

strate, i.e. the textile, is not taken into consider~tion ·with the

result that there is no pos~tive or negative pressures developed as

ink passes through_ the hole. The hydrostatic head of ink in the

bank is also not taken into consideration. Al though all of these

factors are considered to be potentially significa·nt, their inclu­

sion in the theoretical model. are proposed for future work and ex­

pansion of the theoretical model as presented in this ·report.

Based on computer calculation using the mathematical model

describing the hydrodynamic flow in region I and II on the press,

the following conclusions can be drawn:

1. The volume of ink .transferred through the screen is inde­

pendent o( the screen velocity.

63

,.

2, The ink flow rate and the volU111e of ink transfered

through the screen increases with increasing val ue_s of n.,

which characterize the non-newtonian nature of the ink,

but is independent of the m value, which is th~ apparent

viscosity at a rate of shear of 1 sec·1 ..

3, The volume -of ink transferred through the screen in­

creases with decreasing squeegee angle.

4. Although, the volume of ink delivered is increased by in­

creasing the size of the ink bank, the majority of ink

flow takes place at small values of r which is a measure-. . . .

ment of the distance from the point of contact of the

squeegee blade with the screen.

5, 2 F.uture research

The mathematical model must -be extended to include region III

and the effects of the substrate on irµc transfer during screen

process printing, The pressure normal to the screen which is in­

duced by the hydrostatic head of the bank must also· be added to the

present model. Both of these factors will increase ~he degree of

sophistication of the model presented in this report and can have

the effect of both increasing the magnitude of flow through the

screen and can alter the rela tfonship which have been predicted by

the present model,

references

1. A.Kosloff, Screen Process Printing, Times Pub. co., Ohio,{1950)2.

2. B.Zahn, Screen Process Methods of Reproduction, Drake, Ill., { i950) 13.

3. B.Miller and D.B.Clark, Textile Research, 3(1978)150.

4. B.Miller and D.B.Clark, Textile Research, 5(1978)256.

5. L.S.Penn, C.K.Nitta and L.Rebenfeld, Textile Research, 12(1981)324.

6. A.M.Schwartz and C.A.Rader, Chem., Phys. and Appl. Surf. Active Substances, Vol.12, edited by J._Th.G.Overback, N.Y.(1967)383.

7. R.L.Derry and R.S.Higginbotham, J. of the Society of Dyers and Colorists, ·59( 1953)569.

8. G.Boyacigiller, A Study of The Flow of Printing Pastes in Screen Pr~nting, Ph.D. Thesis,· (197.0)Victoria U. of Manchester.

9. H.Schlichting, Boundary Layer Theory, 7th ed., McGraw-Hill, N • Y , ( 1 97 9 )7 3 •

10. R.B.Bird, W.E.Stewart and E.N.Lightfoot, Transport Phenomena, John Wiley & Sons, N.Y.(1960)1-30.

11. H.Schlichting, Boundary Layer Theory, 7th ed~, McGraw-~ill, N.Y.(1979)112.

12. R.B.Bird, R.C.Armstrong and O.Hassager, ·Dynamics of Polymeric Liquids, John Wiley & Sons, N.Y.(1977)7.

13, R.B.Bird, W.E.Stewart and E.N~Lightfoot, Transport Phenomena, John Wiley & Sons, N.Y.{1960)102.

14. S.Middleman, Fun~~entals of polymer Processing, McGraw-HU!, N. Y, { 1977) 87,

15. B.F.Dowds, J, of '.!'he Society of Dyers and Coloists, 12(1970)512.

16. R.H.Perry and C.H.Chilton, Ch. E. Handbook, 5th ed·. McGraw-Hill, N.Y.{1973)5-33,

17. R.B.Bird, R.C.Armstrong and O.Hassager, Dynamics of Polymeric Liquids, John Wiley & Sons, N.Y.(1977)208.

18. L. W. C.Miles, Tex~ile Printing, Merrow, England( 1971) 25,

19. K.Teraji and N.Hiro, Sen-I-Gakkaishi (Japanese), 33, Vol. 2(.1977) 102.

20. Private communication with W. J. Kennedy at M. Lowenstein Corporation.

66

I

Appendex A

Solution for streat:i function,

From the Havier-Stokes equation 2-1, the equation for the

stream function, , is reduced to equation 2-4:

where ljJ is defined as

V = r

1 ~ r ae

And boundary conditions arc:

V (r,8=0) r

= - u

v8

(r,8=0) = 0

v (r,O=O ) r o

= 0

67

A-1

A-2

A-3

The reduced ?Javier-Stokes equation A-1 and the boundary c_on­

di tions n:ieet the criteria for homogeneous constant coefficient

fourth-order partial differential equation.

Since "for all values of r, at 9 = O; equation A-2 becoraes:

V = r

- .!.£!_ r ae - - u

So, by solv_ing e_quati_on A-4, 1/1 beco:iles:

1/1 = - UrG(8) + F(r)

A-4

A-5

where G( 9 ) is defined as a function of 9 only; t( r) is defined

as a function of r only.

Substituting in .equation A-3 the function for i.µ as described

in equation A-5 yields:

V = ~ = 0 e ar = -UG(8) + dF(r)

dr

Since g( 9) is function of only 9 ~nd F( r ) is function of only r 1

then for all values of r at 9 = 0:

68

dF(r)

ar - 0

indicating that

F(r) = C 0

where C0

is a constant.

Substitutin~ equation A~B in A-5 yields:

$ = - UrG(B) + C = rH(B) + C 0 0

where H( e ) ~ - U G( e)

Substituting equation A-9 in A-1 produces:

4 2 .,

~ H(B) + 2 ~ H(B) + H(.B) = 0

dB dB

Solvin~ equation A-10 for B( 9) yields:

69

A-7

A-8

A-9

A-10

H ( 8) = c1cos(8) + c2sin(8) + C 8cos(8) + C 8sin(8) . 3 4 A-11

Substituting for H( 9 ) in equation A-9, the strear:i function be-

co1:ies:

Ey substituting equation A-12 into A-2 and A-3, the constant C0

will disappear and the constants c1, c2, c3 and c4 can be solved by

using the boundary conditions.

Appendex·B

Numerical results of volume of ink delivered.

90 n R V

degree cm cm3

5.000 .200 .002 .239002E-22 5.000 .200 .004 .611846E-20 5 .ooo. .200 .006 .156809E-18 5;000 .200 .008 • 1 56 6 3 3E-17 5.000 .200 .010 .933603E-17 5.000 .300 .002 .434793E-16 '5. 000 .300 • 0014 ,350595E-14 5.000 .JOO .006 .457142E-13 5.000 .300 .008 .282702E-12 5.000 .300 .010 . 11 6 1 7 0 E- 11 5.000 .400 .• 002 .586440E-13 5.000 .400 .004 .265392E-11 5 . "'' • "j' '

t, ·,I/", . 21Hi82.S:-'.-10 ,·.

.. oo ·; .: co .008 .12010::(-09 5. }~/ . '(0 ;010 .409787E-09 5.000 .700 .002 .620058E-09 5.000 • 700 .004 .133526E-07 5.000 • 700 .006 .804263E-07 5.000 .100 .008 .287540E-06 5.000 .100 .010 • 772451E-06 5.000 1.000 .002 .252416E-07 5.·000 1. 000 .004 .403865E-06 5.000 1.000 .006 .20lJ457E-05 5.000 1.000 .008 .646184E-05 5.000 1. 000 .010 • 157760E-04

10.000 .200 .002 .11870~E-22 10.000 .200 .004 • 303880E-20 10 •. 000 .200 .006 .778811E-19 10.000 .200 .008 .777933E-18 10.000 .200 .010 • 46 3684E-17 10.000 .300 .002 .216868E-16 10.000 ,300 .004 .174871E-14 10·.000 ,300 .006 .228015E-13 10.000 .300 .008 .141007E-12 10.000 .300 .010 .579436E-12

71

10.000 .400 .002 .293131E-13 10.000 .400 .004 • 132656E-11 10.000 .400 .006 .123376E-10 lO. 000 .400 .008 .600333E-10 10.000 .400 .010 .204832E-09 10.000 .700 .002 .310786E-09 10.000 ·.100 .004 .669260E-08 10 .ooo .700 .006 .403114E-07 10.000 .700 .008 .144121E-06 10.000 .700 .010 .387169E-06 10.000 1.000 .002 .126655E-07 10.000 1.000 .004 .202648E-06 10.000 1.000 .006 • 102590E-05 10.000 1.000 .008 .324236E-05 10.000 1.000 .010 .791593E-05 15.000 .200 .002 .787989E-23 15.000 .200 .004 .201725E-20 15.000 .200 .006 .516999E-19 15.000 .200 .0.08 .516416E-18 15.000 .200 .0·10 .307808E-17 15.000 .300 .002 .144169E-16 1-5. 000 .300 .004 .116250E-14 15.000 .300 .006· .151579E-13 15.000 .300 .008 .937384E-13 15.000 .300 .010 ,385195E-12 i5.000 .400 .002 • 195005E-13 15.000 .400 .004 .882493E-12 15.000 .400. .006 .820755E-11 15.000 .400 .008 . 399371 E-10 15.000 .400 .010 • 136264E-09 15.000 .700 .002 .206940E-09 15.000 .700 .0"04 .445632E-08 15.000 .700 .006 .268416E-07 15.000 •. 700 .008 ,959642E-07 15.000 .700 .010 .257800£-06 15.000 1.000 .002 .843650E-08 15.000 1.000 .004 • 134 984E-06 15 .·ooo 1.000 .006 .683357E-06 15.000 .1 • 000 .008 .215974E-05 15.000 1.000 .010 .527281E-05 20.000 .200 .002 .588376E-23 20.000 .200 .004 .150624E-20 20 •. 000 .200 .006 .386033E-19 20.000 .200 .ooa .385598E-18 20.000 .200 .010 .229834E-17

72

20.000 .300 .002 .107725E-16 20.000 .300 .004 .868643E-15 20.000 .300 .006 .113262E-13 20.000 .300 .008 .700430E-13 20.000 .300 .010 .287825E-12 20.000 .400 .002 .145764E-13 20.000 .400 .004 .659651E-12 20.000 .400 .006 .613502E-11 20.000 .400 .008 .298524E-10 20.000 .400 .010 .101855E-09 20.000 .700 .002 .154756E-09 20.000 .700 .004 .333258E-08 20.000 .700 .006 • 2007 30E-07 20.000 .700 .008 .717650E-07 20.000 .700 .010 .192791E-06 20.000 1.000 .002 .631025E-08 20.000 1.000 .004 .100964E-06 20.000 1.000 .006 .511130E-06 20.000 1.000 .008 .161542E-05 20.000 1.000 .010 .394390E-05 25.000 .200 .002 .468292E-23 25.000 .200 .004 • 119883E-20 25.000 .200 .-006 .307246E-19 25.000 .200 .008 • 306900E-18 25.000 .200 .010 .162926E-17 25.000 .300 .002 .857768E-17 25.000 .300 .004. .691661E-15 25.000 .300 .006 .901857E-14 25.000 .300 .008 .557720E-13 25.000 .300 .010 •· 229182E-12 25.000 .400 .002 .116091E-13 25.000 .400 .004 .525366E-12 25.000 .400 .006 .488611E-11 25.000 .400 .008 .237753E-10 25.000 .400 .010 .811207E-10 25.000 .100 .002 • 123'287E-09 25.000 .100 ;oo4 .265491E-08 25.000 .700 .006 .159913E-07 25.000 .100 .008 • 571719E-07 25.000 ,700 .010 • 153587E-06 25.000 1.000 .002 .502765E-08 25.000 1.000 .004 .804424E-07 25.000 1.000 .006 .407240E-06 25.000 1.000 .008 • 128708E-05 25.000 1.000 .010 ,314228E-05

73