New method to determine characteristic time of viscoelastic fluids

INT. C0~4. ~ A T ~ S S TRANSFER 0735-1933/83/010077-06503.00/0 Vol. i0, pp. 77-82, 1983 @Pergamon Press Ltd. Printed in the United States

TO DETERMINE CHARACTERISTIC TIME OF V I S C O E L A S T I C F L U I D S

NEW

METHOD

E.Y. K w a c k and J.P. H a r t n e t t Energy Resources Center U n i v e r s i t y of I l l i n o i s at C h i c a g o Box 4348, Chicago, I l l i n o i s 60680

(Cc~municated by J.P. Hartnett and W.J. Minko~ycz)

ABSTRACT An e x t r a d i m e n s i o n l e s s p a r a m e t e r i n v o l v i n g the r a t i o of the e l a s t i c and v i s c o u s forces, the W e i s s e n b e r g number, is n e c e s s a r y to c o r r e l a t e the f r i c t i o n and h e a t t r a n s f e r b e h a v i o r of v i s c o e l a s t i c fluids. The e v a l u a t i o n of the W e i s s e n b e r g n u m b e r r e q u i r e s a m e a s u r e m e n t of the first n o r m a l force d i f f e r e n c e or, a l t e r n a t i v e l y , the d e t e r m i n a tion of the c h a r a c t e r i s t i c time of the fluid. We s u g g e s t that the c h a r a c t e r i s t i c time of v i s c o e l a s t i c fluids can be d e t e r m i n e d by a simple p r e s s u r e d r o p m e a s u r e m e n t . The basis for the p r o p o s e d m e t h o d is p r e s e n t e d . More detailed m e a s u r m e n t s are r e q u i r e d to v a l i d a t e the p r o c e d u r e .

Introduction It is w e l l k n o w n tain h i g h m o l e c u l a r viscoelastic

fluid.

such v i s c o e l a s t i c flow are r e d u c e d solvent.

that

weight The

The t u r b u l e n t

of v i s c o e l a s t i c [1-2],

the

of the

solvent

ty of these

polymers

friction

fluids below

the a d d i t i o n

amounts

of cer-

to a s o l v e n t

results

in a

factor

and h e a t t r a n s f e r

in fully e s t a b l i s h e d

the v a l u e s

associated

heat transfer

fluids is a f f e c t e d

level of m e c h a n i c a l and solute

fluids.

of small

with

by the p o l y m e r

[5-7], w h i c h the

77

[3-4]

behavior

concentration and the c h e m i s t r y

are r e l a t e d

friction

channel

the pure

and h y d r o d y n a m i c

degradation

In general,

turbulent

of

to the e l a s t i c i

factor

and the h e a t

78

E.Y. Kwack and J.P. Hartnett

transfer

decrease

tration,

while

they

To a n a l y z e elastic

(up to a point) increase with

the f r i c t i o n

fluids,

at least

as the first n o r m a l

force d i f f e r e n c e trations with

gests

one a d d i t i o n a l

fluids.

the c h a r a c t e r i s t i c

time,

that one n e w d i m e n s i o n l e s s

is n e e d e d

to c o r r e l a t e

viscoelastic

fluids.

the

dimensional

in a d d i t i o n

parameter

approach

is to deal measure

transfer

number,

Ws,

of

analysis

the W e i s s e n b e r g

and heat

The W e i s s e n b e r g

high c o n c e n -

Dimensional

parameter,

time of

the first n o r m a l

and a c c e s s i b l e

fluid.

such

to those n o r m a l l y

for r e l a t i v e l y

the usual

friction

of v i s c o -

or the c h a r a c t e r i s t i c

a useful

of a v i s c o e l a s t i c

concen-

of shear.

behavior

Unfortunately,

Accordingly,

polymer

hours

transfer

can o n l y be m e a s u r e d

of polymer.

the e l a s t i c i t y

and heat

into a c c o u n t

in N e w t o n i a n

increasing

increasing

force d i f f e r e n c e

the fluid m u s t be taken encountered

with

Vol. 10, No. 1

sug-

number,

behavior

is d e f i n e d

of

as:

Ws = ~V/d

where

(1)

~ is a c h a r a c t e r i s t i c

tic shear rate, Recently, Weissenberg and h e a t

time of the

V is the m e a n experimental

number

transfer

fluid v e l o c i t y

studies

[8-10]

is a key p a r a m e t e r behavior

fluid,

V/d

is a c h a r a c t e r i s

and d is the d i a m e t e r .

have

confirmed

in c o r r e l a t i n g

of v i s c o e l a t i c

fluids

the

that

the

friction

in t u r b u l e n t

pipe

flow. Previous

Methods

Since material

to D e t e r m i n e

the c h a r a c t e r i s t i c

property

constitutive

equation.

through molecular constitutive

of the

time,

solution,

l, of a p o l y m e r

it s h o u l d be o b t a i n a b l e

Constitutive

theories

solution

or t h r o u g h

equations

from a

are a v a i l a b l e

phenomenological

is a

either

nonlinear

equations.

Mizushina

et al.

[2] u s e d the R o u s e m o d e l

[ii]

based

on the

Vol. i0, No. 1

molecular

CHARACTERISTIC TIME OF VISCOEIASTIC FLUIDS

theory

polyethylene molecular

oxide

et al.

concentrations of c o m p l e x

using

data

such as the

measured

The above these

stress

over

generalized

the c h a r a c t e r i s t i c pointed models

New M e t h o d

used

lated

coefficient

i0 of Ref.

steady

time

a set

experi-

the dynamic

and

viscosi-

storage frequencies.

the a p p l i c a b i l i t y simple m o d e l s used

[15]

from

Such

considerable

tested

steady

is c l e a r l y

of the

data,

these

as

simple

shear v i s c o s i t y

superior

to have

of

in d e t e r m i n i n g

shear v i s c o s i t y

and Shaw

seemed

[9]).

[17] w h i c h

a very

the

friction

of aqueous

solutions factor

to the

slight

edge.

1 shows

to E y r i n g

model

time of the dilute

[8-10].

Based

on these

and the d i m e n s i o n l e s s

polyacrylamide

of the W e i s s e n b e r g Figure

is similar

the c h a r a c t e r i s t i c

polyacrylamide

times

as functions

model

to c a l c u l a t e

degraded

characteristic transfer

[16]

involves

of

to D e t e r m i n e

recently

and h i g h l y

Elbirli

model

The P o w e l l - E y r i n g was

only

time of

equations.

rates

limit

unknown.

a range

and the dynamic

several

that no one model

but the Eyring

require

can be e f f e c t i v e l y

the c h a r a c t e r i s t i c

data and c o n c l u d e d other,

[14].

constitutive

seriously

time using

out by Bird to o b t a i n

model

over

[13] w h i c h

of shear

Alternatively,

Newtonian

B

shear viscosity,

range

considerations

two approaches.

model

difference

a wide

are p o l y d i s -

are u s u a l l y

solutions

equations

steady

polymer

the c h a r a c t e r i s t i c

AP-273)

nonlinear

constitutive

in p r a c t i c e

distributions

the C a r r e a u

the first normal

modulus

(Separan

Unfortunately,

for m o n o d i s p e r s e

used

determined

phenomenological

phenomenological mental

[12]

polyacrylamide

developed

weight

time of dilute

solutions.

solutions

and their m o l e c u l a r

aqueous

ty,

were m a i n l y

but all p o l y m e r

Argumendo

the c h a r a c t e r i s t i c

(Polyox WSR-301)

theories

solutions, perse

to d e t e r m i n e

79

solutions

and R e y n o l d s

a representative

were

numbers

heat corre-

(see Fig.

f-Ws curve

for a

E.Y. Kwack and J.P. Hartnett

80

Vol. i0, No. 1

\ rneasured

Ws FIG. 1 F a n n i n g f r i c t i o n factor as a f u n c t i o n of W e i s s e n b e r g n u m b e r for a fixed R e y n o l d s number.

fixed R e y n o l d s istic

number.

If these

time can be d e t e r m i n e d

in a pipe or c a p i l l a r y erably

longer

assumed

number

value,

100 to e n s u r e

is k n o w n

then

value

greater

with

friction to t h a t

number

it is n e c e s s a r y

factor shown

of the

than the c r i t i c a l

a slightly

larger above

in Fig.

the W e i s s e n b e r g

the

friction

friction

it is i m p o s s i b l e

of the W e i s s e n b e r g

since

It is

factor

and the

number

of the e x p e r i -

factor

is e q u a l

number

the test

which will yield value.

The c h a r a c t e r i s t i c

to the

uniquely

the

to any

for friction.

in a n o t h e r a value

tube

of the

A figure comparable

1 can then be u s e d to d e t e r m i n e

number.

consid-

flow.

to d e t e r m i n e

Weissenberg

the a s y m p t o t i c

ratio

it m a y c o r r e s p o n d

to r e p e a t

diameter

drop measurement

fully e s t a b l i s h e d

between

the c h a r a c t e r -

to d i a m e t e r

for the R e y n o l d s

value

value

In this case,

pressure

tube w i t h a l e n g t h

If the m e a s u r e d

asymptotic

are valid,

by a simple

that the r e l a t i o n s h i p

Weissenberg ment.

than

results

the v a l u e

time m a y n o w be

of

Vol. I0, No. 1

determined

CHARACTERISTIC TIME OF VISCOELASTIC FLUIDS

81

from

I = Ws d/V

Since

(2)

I is a fluid p r o p e r t y ,

the v a l u e m e a s u r e d

capillary

tube can be used

for the

under

flow c o n d i t o n s .

This

any

valuable

for d i l u t e

the a l t e r n a t i v e

aqueous

procedures

same a q u e o u s

technique

polymer

in the pipe or polymer

solution

should be p a r t i c u l a r l y

solutions

in w h i c h

fail to y i e l d v a l u e s

case all of

of the c h a r a c t e r i s -

tic time. Nomenclature d

tube d i a m e t e r

f

Fanning

Re a

Reynolds

V

average

Ws

Weissenberg

Greek

Symbols apparent

p T

density w

wall

friction number

factor,

f = Tw/(pV2/2)

b a s e d on the v i s c o s i t y

velocity

at wall,

pVd/~

of fluid

number,

viscosity

lV/d

evaluated

at w a l l

of fluid

shear

stress

characteristic

time of fluid References

I.

G. M a r r u c c i and G. A s t a r i t a , T u r b u l e n t h e a t t r a n s f e r in v i s c o e l a s t i c liquids, Ind. Eng. Chem. Fundam. 6, 70 (1967).

2.

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3.

D.H. F i s h e r and F. R o d r i q u e z , D e g r a d a t i o n of d r a g - r e d u c i n g p o l y m e r s , J. Appl. P o l y m e r Sci., 15, 2975 (1971).

4.

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82

E.Y. Kwack and J.P. Hartnett

Vol. 10, No. 1

5.

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6.

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7.

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8.

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9.

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i0.

E.Y. Kwack and J.P. Hartnett, critical W e i s s e n b e r g numbers, 1445 (1982).

E f f e c t of solvent Int. J. Heat Mass

111.

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12.

A. Argumendo, T.T° Tung and K.I. Chang, R h e o l o g i c a l p r o p e r t y m e a s u r e m e n t s of d r a g - r e d u c i n g p o l y a c r y l a m i d e solutions, Trans. Soc. Reol., 22, 449 (1978).

13.

P.J. Carreau, R h e o l o g i c a l e q u a t i o n s from m o l e c u l a r theories, Trans. Soc. Rheol., 16, 99 (1972).

14.

R.B. Bird, E x p e r i m e n t a l tests of g e n e r a l i z e d N e w t o n i a n c o n t a i n i n g a z e r o - s h e a r v i s c o s i t y and a c h a r a c t e r i s t i c Can. J. Chem. Eng., 43, 161 (1965).

15.

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16.

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17.

R.E. Powell and H. Eyring, M e c h a n i s m viscosity, Nature, 154, 427 (1944).

from

chemistry Transfer,

on 25,

network

models time,

shear v i s c o s i t y

for r e l a x a t i o n

theory

of