SFI: A Simple Rheological Parameter for Estimating Viscosity

A. V. Shenoy Department of Energy and Mechanical Engineering, Shizuoka University, Hamamatsu, Japan

• A simple rheological parameter, namely, a solution flow index (SFI) for estimating the viscosity of Newtonian and non-Newtonian inelastic solu- tions, is introduced. SFI is defined as the weight of the solution that flows out of a capillary tube under gravity in 10 min. Test solutions are character- ized on a standard Brookfield-type cone-and-plate viscometer, and the viscosity versus shear rate curves generated are coalesced into master curves that are concentration- and temperature-independent using SFI as a normalizing parameter. The master curves developed in this manner are useful in generating rheograms of uncharacterized samples merely from their SFI values. A modified Arrhenius-type equation is proposed for predicting the SFI of solutions at different temperatures. Further, a model based on the altered free volume state concept is proposed to predict the SFI values of solutions at various solute concentrations. The predictions of the model are compared with experimental data and found to give good agreement.

Keywords: solution flow index, master curve, viscosity measurement, non-Newtonian fluid, inelastic fluid

INTRODUCTION

Rheometry is the measuring arm of rheology, and its basic function has been to quantify the rheological parameters of importance. The rheological techniques available to this date can be found in various books on rheometry and rheology such as those of Waiters [1], Nielsen [2], Whorlow [3], and Cheremisinoff [4] and hence will not be discussed here. However, one point worth noting is that rheometry has been used at various levels of sophistica- tion depending on the requirement of accuracy. The in- dustrial approach has always been to look for convenient and easy methods of measurement that have a reasonable degree of accuracy. On the other hand, the academic bias has been to search for greater accuracy in measurement through an increased level of sophistication, especially when dealing with non-Newtonian fluids. New types of viscometers [6-10] are being sought even now with a view to alleviate the deficiencies in the existing measurement techniques. At the same time, there are efforts such as those of Shenoy et al [11-16] to elevate single-point measurements so that reasonably accurate rheological data can be quickly obtained without the use of very expensive sophisticated equipment and trained personnel.

The present work represents a search for a simple yet reasonably accurate rheological parameter that could be used for characterizing Newtonian and non-Newtonian inelastic fluids. A number of test solutions are systemati- cally characterized for their rheological behavior using a standard commercial cone-and-plate viscometer. At the

same time, a solution flow index (SFI) is determined that is based on the time required for a fixed volume of liquid to flow through a capillary tube. The SFI is then used to normalize the viscosity versus shear rate curves obtained from the viscometer in order to generate a master curve, which is shown to be concentration- and temperature- independent. The master curve is then used for regenerat- ing rheograms of uncharacterized samples simply from the knowledge of SFI determined at the concentration and temperature of interest.

A modified form of the Arrhenius equation is proposed in order to estimate the SFI value at different tempera- tures without resorting to experimentation. Further, a model based on the altered free volume state concept is proposed to predict the SFI values of solutions on the basis of solute concentration. The predictions of the model are compared with the experimental data and found to give good agreement. Thus the SFI value at different concentrations can be estimated without having to deter- mine it experimentally.

THE SFI CONCEPT

Definition

The solution flow index (SFI) is defined as the weight of solution that flows under gravity in 10 min at constant- temperature conditions through a capillary of fixed diame- ter and length.

Address correspondence to Dr. A. V. Shenoy, Department of Energy and Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432 Japan.

Experimental Thermal and Fluid Science 1993; 6:324-332 © 1993 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010 0894-1777/93/$6.00

324

This definition is rather an arbitrary one but has been so chosen to be consistent with the well-known rheological parameter used in polymer melt rheology, the melt flow index (MFI). In the case of MFI [14], there are standard prescribed conditions for measurement based on ISO Rl133, ASTM 1238D, DIN 53735, and JIS K7210 that have evolved gradually through years of persistent re- search activities. There is also commercially available equipment known as melt flow indexer for measurement of MFI. The situation is not the same for SFI as it is a parameter that is being introduced for the first time. There is no standard equipment to measure it, and hence the choice is entirely personal and arbitrary.

The aim of this presentation is to prove that the SFI concept works effectively and that it is quick and easy to determine. A simple buret such as the one shown in Fig. 1, which is commercially available at the reasonably low price of ¥ 8820 (June 1991 price), or about $65 U.S., is chosen as the trial solution flow indexer for the measure- ment of SFI. High-temperature SFI measurements cannot be done in the buret shown in Fig. 1. Hence, a little modification is done without much financial investment as shown in Fig. 2 so that there is added facility to heat the solutions. The buret is jacketed with a glass tube through which constant-temperature water is allowed to flow from a constant-temperature bath. The modified buret then represents a type of capillary viscometer with a reservoir on the top that can be heated to the temperature of interest.

Method of Measurement

The buret is filled with the solution whose SFI is to be determined. The filled buret is left as is for some time until all bubbles in the liquid sample have vanished. The stopcock is then opened, and the solution is allowed to flow out until the solution level in the buret reaches a fixed final mark, after which the stopcock is closed. In the present case, the initial mark is arbitrarily set to the buret mark of 70, which gives h i = 0.4707 m. A clean preweighed beaker is placed below the capillary of the buret, and the stopcock is opened to allow the solution to flow into the beaker under gravity. At the same time the stopwatch is started for the measurement of flow time. When the solution level in the buret reaches the arbitrarily set final buret mark of 50, that is, hf = 0.367 m, the stopcock is closed and the stopwatch is simultaneously stopped. The beaker filled with the solution is then weighed to deter- mine the weight W (in g) of solution that flowed out of the capillary in the time t (in seconds) recorded by the stop- watch. Note that if the density of the solution is predeter- mined, then the solution flowing out need not be weighed, as it would be known from the fixed volume contained between the h i and h marks. In such cases, the time of flow between the two harks would suffice.

Data Analysis

The solution flow index (in g /10 min) is calculated as

W SFI = - - × 600 (1)

t

A Simple Rheological Parameter for Viscosity 325

_ ....... Iii

Figure 1. Schematic diagram of solution flow indexer.

The solution flow through the capillary of the buret is laminar, and hence the volumetric flow rate of the fluid would be given by the well-known Hagen-Poiseuille law

Q = "rrD 4 AP/128rlaL (2)

However, data generated in any capillary tube viscometer are known to require corrections due to four contributions to frictional losses. Hence the Hagen-Poiseuille law needs

f

®

t

J

O

Figure 2. Schematic diagram of solution flow indexer with high-temperature measurement facility. 1, Buret; 2, beaker; 3, jacket; 4, temperature control; 5, water bath; 6, pump; 7, autobalance; 8, CMC solution; 9, thermometer.

326 A.V. Shenoy

to be corrected for (1) kinetic energy effects, (2) the head of the fluid above the tube exit, (3) end effects, and (4) effective wall slip.

Correction 1 When the solution moving slowly in the buret reservoir suddenly encounters the capillary, it begins to move quickly, resulting in a fall in pressure. This effect of kinetic energy change can be taken into account in the Hagen-Poiseuille equation by replacing A P with A p - m p v 2, where v is the average velocity given by v 4 Q / z r D 2. The numerical factor m is a correction factor evaluated by a number of workers for Newtonian fluids as shown in Table 1, which is taken from Merrington [17]. The commonly used value for m is 1.125. Yamasaki and Irvine [9] have shown that for power-law fluids this value changes with power-law index as shown in Table 2. Note that m in the present paper is equivalent to C / 2 in Yamasaki and Irvine [9]. However, Metzner [18] proposed that the correction for non-Newtonian fluids could be taken to be the same as that for Newtonian fluids without much loss in accuracy. Metzner 's [18] proposal received the support of other workers such as Ng et al [19] and Tung et al [20] through their experimental efforts. Using the correction as discussed above, Eq. (1) can be rewritten in the corrected form as

r r D 4 [ A p - (16mpQ2/ zc2O4)] (3)

*/a = 128QL

where A p = 0 .5pg(h i + h f ) (4)

It is obvious that A P varies with time throughout the experiment because the fluid level keeps decreasing as the fluid moves from the h i mark to the h f mark. Hence, the average value is chosen in Eq. (4).

Correction 2 In the present work, the measurements are done consistently using the same fluid head for all solutions having relatively close densities. Hence it is assumed that the second correction is not significant.

Correction 3 The end effects, though significant, would cancel each other as SFI will be used as a normalizing factor for all solutions. Hence, inclusion of this third correction is assumed to be not significant.

Correction 4 The effective wall slip is neglected in the present case as a first approximation. If the theoretical

Table 1. Values of Correction Factor m for Newtonian Fluids as Determined by Different Workers

Researcher m

Reynolds Hagenbach Couette Wilberforce Boussinesq Riemann Swindells Knobbs Jacobson

0.50 0.79 1.00 1.00 1.12 1.124 1.12-1.17 1.14 1.25-1.55

Source." Taken from Merrington [17].

Table 2. Values of Correction Factor m for Power-Law Fluids

Power-Law Index n rn

1.0 1.125 0.9 1.090 0.8 1.053 0.7 1.013 0.6 0.968 0.5 0.918

Source: Taken from Yamasaki and Irvine [9].

prediction does not match the experimental result, this correction could always be introduced.

Based on the definition of SFI, the relationship between Q and SFI can be written as

Q = SFI x 10 -3 /600p (5)

Substituting Eq. (5) into Eq. (3) gives

75,000p*rD 4 AP [1 - (m × lO-6SFIZ/22,500pTr2D 4 A p ) ]

~Ta = 16 SFI L (6)

o r

r/a SFI = K ' ( 7 )

(1 - CCSFI 2)

where K ' is a constant determined by the test conditions and the solution density and is given as

K ' = 75,000p2zrD4g(hi + h f ) / 3 2 L (8)

and CC is a constant correction factor given by

m × 10 - 6

CC = 11,250pZlr2D4g(h i + h f ) (9)

In order to evaluate CC, the values of m, hi, hf, and D must be known. The value of m can be taken from Table 1 or 2, or the popular value for Newtonian fluids of 1.125 can be used as a first approximation, h i and hf are easily measured using an ordinary length scale, h i = 0.4707 m and hf = 0.367 m were determined as an average of five readings. The diameter D of the capillary cannot be measured easily with accuracy. Hence, an indirect method is employed to obtain the value of CC. Two test solutions are chosen as standard, namely, water with a low viscosity and honey with a high viscosity. Using the apparent vis- cosities r/w = 0.0009 P a . s and 7/h = 2.15 P a - s and the SFI values of these two solutions, SFI W = 655 g / 1 0 min and SFI h = 5.76 g / 1 0 min in Eq. (7), CC is evaluated by solving the equations simultaneously. The value of CC thus determined is 2.22 × 10-6. Now D is evaluated from Eq. (9) using the values CC = 2.22 × 10 -6, m = 1.125,

3 2 p = 998 k g / m , g = 9.807 m / s , h i = 0.4707 m, and h r = 0.367 m. The value of D thus determined is 0.00086m. Using this value of D along with the values of viscosity r/w = 0.0009 P a . s and SFI~ = 655 g / 1 0 min for water and the value CC = 2.22 × 10 6, K ' is evaluated as 12.39 from Eq. (7). Using this value of K ' in Eq. (8), the value of A P / L is determined as 1.54 × 106 P a / m . It is worth

A Simple Rheological Parameter for Viscosity 327

noting in passing that SFI is measured under nearly con- stant shear stress conditions given by

r = O AP(1 - CC SFI2 ) /4L (10)

The explicit value of shear stress cannot be determined because SFI remains unknown. However, for values of SFI < 100, the bracketed term can be taken as identically equal to 1, and thus the shear stress in the SFI apparatus can be approximated using

r = D A P / 4 L = 331 Pa (11)

with D ffi 0.00086 m and A P / L ~- 1.54 x 106 P a / m . The apparent shear rate ~/, is given by

Ya = g n ( 4 Q / r r D 3 ) (12)

o r

~/a/SFI = K" (13)

where K n is a function of the power-law index for struc- turally complex non-Newtonian power-law-type solutions as given by Skelland [21] and takes a value of 8 for Newtonian fluids as follows:

Kn = 8(3n + 1 ) / 4 n (14)

and K" is a constant defined as

K" = K n / 1 5 0 0 w D 3 (15)

Using 0 ~- 998 k g / m 3 and D = 0.00086 m, the value of K" = 26,570 is obtained for Kn = 8. Equations (7) and (13) are of a form similar to those obtainedby Shenoy et al [11] when relating melt flow index to rheogram. There- fore, it should be possible to coalesce 7/a versus ~/a curves of polymer solutions of different SFIs (ie, solutions of the same polymer at different concentrations or different tem- peratures) by plotting % . S F I / ( 1 - CC. SFI 2) versus 4/,/SFI. Once a master curve is formed, it should be possible to get the viscosity versus shear rate curve at any intermediate concentration or temperature simply by de- termining the SFI of the sample under those conditions.

SFI Determination

The SFIs of the solutions are determined by the method outlined above. For Newtonian fluids the values obtained at room temperature are given in Table 3. In the case of water, whose SFI value is high, an average of five readings is taken. The maximum deviation from the average is _ 0.75%. For all other solutions with lower SFI values, the error bound is even lower, and hence not more than two readings are required to determine the value.

For CMC solutions, the SFI values are given in Table 4 for room-temperature measurements and in Table 5 for high-temperature measurements.

Rheograms

The viscosity versus shear rate curves are generated using a Brookfield-type viscometer, type Visconic EHD manufactured by Tokimec Inc. It is among the simplest cone-and-plate viscometers commercially available but does carry quite a high purchase price, on the order of ¥ 494,400 (June 1991 price), or $3644.

The viscosity data obtained for the four Newtonian samples are shown in Fig. 3. The various curves obtained for six different concentrations of CMC solutions at room temperature are shown in Fig. 4, and those for the 2.5%

Table 3. Newtonian Solutions and Their SFI Values

Sample SFI Avg. SFI Solution Material (g / l O min) (g / l O rain)

1 Honey 5.76 2 Glycerol 19.14 3 Maple syrup 37.40 4 Water 659.7

657.5 654.3 655.0 653.2 649.7

DEVELOPING CONCENTRATION- AND TEMPERATURE. INDEPENDENT

MASTER CURVES

Test Solutions

Two categories of test solutions are used. Newtonian fluids. Water (from tap), glycerol (Koso

Chemical Co. Ltd.), maple syrup (Morinaga Seika), and honey (Co-op) are chosen as typical Newtonian fluids to cover a wide range of viscosities.

Non-Newtonian inelastic fluids. Samples made by dissolv- ing in water various amountg of sodium carboxymethyl cellulose (CMC) polymer (Wako Pure Chemical Industries Ltd.) are used as typical non-Newtonian inelastic fluids.

Six different concentrations are used ranging from 1.5 to 3.0%. The concentration range is chosen on the higher side because this particular quality of CMC does not deviate much from Newtonian behavior at lower concen- trations of 0.1-1%. The six specified concentrations are used for room-temperature measurements.

A concentration of 2.5% CMC is chosen for high-tem- perature measurements at eight different temperatures between 18 and 50°C.

Table 4. SFI Values for CMC at Different Concentrations

Sample Vol. Fraction qb SFI (g / 10 rain)

A 0.0300 4.9 B 0.0275 8.4 C 0.0250 10.5 D 0.0225 10.5 E 0.0200 32.3 F 0.0150 94.5

Table 5. SFI Values for 2.5% CMC at Different Temperatures

Sample Temp. (°C) SFI (g / 10 rain)

G 18 10.5 H 20 13.6 I 25 15.9 J 30 18.7 K 35 22.6 L 40 26.4 M 45 32.4 N 50 35.6

328 A.V. Shenoy

i l l CMC solution at eight different temperatures are shown in Fig. 5.

-1 i 0

- 2 10

- 3

-4 10

1 l - 1 0 ~ i O s

Figure 3. Viscosity curves for four Newtonian solutions at room temperature obtained with the commercial viscometer.

Master Curves

The rheograms in Figs. 3 -5 are coalesced using the corre- sponding SFI values from Tables 3-5, respectively, by plotting r/a • SFI / (1 - CC- SFI 2) versus 4/a/SFI to obtain master curves as shown in Figs. 6-8.

It is to be noted that the master curve in Fig. 6 for Newtonian fluid is a straight line defined by Eq. (7) with K ' = 12.39 as determined earlier.

Figures 7 and 8 are the master curves for the inelastic non-Newtonian CMC solutions that are concentration- independent and temperature-independent , respectively. Superimposing the master curve in Fig. 8 on Fig. 7 gives the unique master curve shown in Fig. 9, which is concen- tration- and temperature-independent .

Deve lop ing a Modified Arrhenius Equation for Determinat ion of SFI at Different Temperatures

The well-known Arrhenius equation written in terms of zero-shear viscosity is given as follows:

1)1 ~o, r2 exp (16) ~0. r , T2 /'1

Based on the inverse relationship between viscosity and SFI shown in Eq. (7), the modified form of the Arrhenius equation can be written as

S F I T 2 ( I - C C S F I 2 1 ) [ E ( 1 1 )] (17)

SFITI( 1 _ CCSFI22) = exp ~ 7"2 T1

The SFI values from Table 5 are plotted in Fig. 10 with respect to the reciprocal temperature. The slope of the line in Fig. 10 multiplied by the gas constant R = 8.3232

~a

± u ! : ~ : :~ : l i I I I I i i r l l l l I ! . . . . . . . . i i i i ,

I ] I I I l l : : I I I I I t I i l l l

, ! l ' i l l I I I ~--- ~ ~ ,

' - ~ , ~,~,..I' ~

I !/1[I1 '~ 1 . . . .

d - - 4 - - 4 - - 4 4 -

vs~cMc I i i i! - I C 2. 5O~CMC : : : : :

O 2. 25~.CMC ~--~ S 2. OO~CMC , I I F ~. ~o~cMc, i

1 1 0

i

1 0 2

I I IIIIII [ r I I I I I

I I I I I I l l

l l 1 i : : i :

: : : : t : : :

i i i l i l l I I I I t I I

X I I I I I 1 1 [ [ I [ [ 1 1

" - , ~ ' - ~ " ~ I I [ [ [ I

---.2"-~ 1111[I

. . _ . _ . x ~ ! - ~ ! ! ! ! ! !

: : : : : : : :

: : : : : : : :

10 ~

Figure 4. Viscosity versus shear rate curves for six CMC solutions of different concentrations at room temperature obtained with the commercial viscometer.

T ? a

i 0

-2 10

1 1 £~ 1 0 ~ 10 3 fo

Figure 5. Viscosity versus shear rate curve for 2.5% CMC solution at eight different temperatures from 18 to 50°C obtained with the commercial viscometer.

A Simple Rheological Parameter for Viscosity 329

2

i0

10

-1 10

i0 -' 1 10 10 2 f , ~ / S F I

Figure 6. Master curve obtained for the four Newtonian solutions of Fig. 3 using the corresponding SFI values from Table 3.

J / ( m o l • K) is the activation energy, which in the present case is calculated to be equal to 27.8 k J / ( m o l . K). It is worth noting that Eq. (17) is of a form similar to that proposed by Saini and Shenoy [12] for MFI.

D e v e l o p i n g a Mode l Based o n the Altered Free V o l u m e State C o n c e p t for C oncentra t ion D e p e n d e n c e

The viscosity of the polymer solution is basically the resistance of the solution to flow and hence is a function

2 /o! ! :

I I I lll

© 10 ,,+,

" I r, ! i i i . . . .

t I I I I I I I

I I I ]1111 I I ] ! I ] ] 1 [ 1 I I

IIllll ] I I i i Ill[l! ~ I

2 ! ! l a ~ . + .

n,r~L l i l l l l l l I I I I I '~?~1 f I I I I I I

I I I I I I [ l~t--I I I I I I I !rlll l ] ! 1 '

, 1tll U) I [ I ,jjj ;4 20:0-' L 4 0 ~ ~ j , ~,,]~, ~_ I I I I I ] 1 [ 2 5 : C M 4 5 : 0 I I I 1 I I I ] 1

I I I I l l l J 3 o o N 5 0 0 I I ] [ ! I ] I I I I L l [ , I i l l

I lO 10 ~

a / S F I

Figure 8. Master curve obtained for the 2.5% CMC solutions at the eight different temperatures of Fig. 5 using the corre- sponding SFI values from Table 5.

~2 [.t 03

b i

r, O3

of concentration. Increasing concentration decreases the mobility of the polymer chains within the solution. This mobility is in turn related to the free volume available for motion. Hence, the relationship between viscosity of the polymer solution and its free volume can be written in terms of Doolittle's equation as

In r/a(T, ~b) = In A + B / f ( T , cb) (18)

where "0a(T, ~b) is the apparent shear viscosity of the solution at temperature T and with volume fraction qb of

a2_ r, [D

(D ©

\

[u 03

2 2 i 0 l f ]

I 1 1 1 I I I r n l t l : : : : : I I I I I I . . . . . I l l ,],]], [ I l I I Z : : : : : I l l l l I I I I I I I iiiii IIrElI IIE]Jl ~,2

IIIlII I1[111 i i i i i

I I I I ' ¢.Jll, i i i [ [ [ ~ + . . . . it ', ', ', " ' . . . . . . I ] I I I l l l l I ' ~ W , ~ I I I I I I I I

I l l l l l ...... IIII1[- i iiii .7\

B 2. 7 5 N C M C 1 C 2 5 0 % C M C . . . . . . b-. D 2 : 2 5 ~ C M C ~ j , , ] ] I ',',',',',', U}

2 , O O X C M C i l ; ] i ] ;;;;;; E F 1. 50%CMC /lllll IIIIII

IIIIII lit I1[111 I I [ ,~, l l l l lllli

r l ! l l iL i

1 10 10 2 7~n/S F I

I I I I I I I I I I I I l l I I I I I

i iiiiii i i i i i i i i i i i i

: I I II I Ill A 3. 00% G 18"O i

~ J t][ B 2. 75~ H 20"C ~['[" c 22 50x [ 25~ i [ [ I III : 25g J 30"(3 I I I I I I ~, 2 o a x 3 5 " c I I I I l i t : 50% L K 40"0 ,'

45"C M 50"0 il [ [ J LLI [ I I J !

1 10

a / S F I

Figure 7. Master curve obtained for the six CMC solutions of Figure 9. Unique master curve obtained by superimposing Fig. 4 using the corresponding SFI values from Table 4. the master curve from Fig. 8 on the master curve from Fig. 7.

330

2_

O

!

v

2_ u)

A. V. Shenoy

2

10

S

G ', 8 "C H 2 0 " C I 2 5 " C J 3 0 "C K 35"C

4 5 " C N 50"C

i , I 1

0

i

i

i

± / 1 ~ (

-3 10

Figure 10. Variation of SFI with reciprocal temperature in kelvins.

1

log aSF !

where

the polymer; A is a constant dependent on the nature of the continuous phase, that is, water; B is a constant equal to 1 based on the arguments of Fujita and Kishimoto [22]; and f (T , 4~) is the free volume of the solution at tempera- ture T and with volume fraction ~b of the polymer.

Based on the inverse relationship between viscosity and SFI shown in Eq. (7), Eq. (18) can be rewritten as

[ SFI(T'~b) ] A, 1 In = In (19)

1 - CCSFI(T , ~b) 2 f ( T , oh)

The above equation would hold even for the continuous phase (ie, water), and hence

[ SFI (T '0 ) ] A, 1 In = In (20)

1 - CCSFI(T,O) 2 f (T ,O)

Now the altered free volume state resulting from the addition of the polymer to the water can be considered to reduce the free volume of the water in a manner that is a linear function of 4~ as given by Fujita and Kishimoto [22]. Thus,

f ( T , eb) = f ( T , 0 ) - /3(T)~b (21)

where /3(T) is the difference between the free volume of the water and that of the polymer solution. Combining Eqs. (19)-(21) and rearranging the terms gives

2"303f2(T' 0 ) ( - ~ ) ( 2 2 ) - 2.303f(T, 0) + /3(T)

SFI(T, 0) / [1 - CC SFI(T, 0) 2]

asFI = SFI(T, ~b)/[1 - CCSFI(T, 4:}) 2] (23)

It is worth noting that Eq. (22) is of a form similar to that proposed by Shenoy and Saini [23] for MFI. The above

L

©

o

\ \

L~ • <J'

0 . 4

0 . 3

0 . 2

. . . . . . T ! i

i • 4L

30

j / i

/ i / /< A 3 . oo oMo

B 2 . 7 5 N C M C C 2 . 5 0 % 0 M C

O0~CMC F t ~50%CMC

L i

, 1 ]

4O 60 TO

Figure 11. Variation of SFI with volume fraction of the CMC polymer at room temperature.

equation predicts that a plot of 1/(log aSF I) versus 1/4, should be linear. To check this out, the samples given in Table 4 are considered and Fig. 11 is formed using the SFI value of water, which is taken as 655 g /10 min from Table 3. It can be seen that, as expected, the points fall rather well on a straight line.

RESULTS AND DISCUSSION

The unique concentration- and temperature-independent master curve shown in Fig. 9 is very useful for generating viscosity versus shear rate data of unknown samples with- out resorting to expensive equipment. To simplify the procedure, the master curve is now fitted with the follow- ing equation based on the modified Carreau model.

.s. 0sFi[ ( tl ( 1 - CCSFI 2) 1 - C C S F I 2 1 + (ASFI) 2 - ~ i

(24)

In the present case, the model constants arc found to be as follows:

r? 0 SFI 1 - CCSFI 2 12.39; ASFI = 0.95; N = 0.18

(25)

Using these model constants, Eq. (24) can be rewritten as 0 . 1 8 ,,o SF, [ t l

1 2.22 × 10 .6 SFI 2 = 12.39 + 0.903[ - ~ SFI ] J

(26)

Equation (26) can now be used for generating the rheogram of any unknown concentration in the intermedi- ate region. A concentration of 1.75% CMC is now chosen as Sample X. The SFI of this solution is found to be 50.8. Using this value of SFI in Eq. (26), the viscosity versus shear rate curve is generated as shown in Fig. 12 by the

A Simple Rheological Parameter for Viscosity 331

1

[+ir, . . . . . . . . . . .

IT[II I II111 ~++-)a IIlll I III[I 11111 I ll[ll

+ _ ILLII I lllll

i . . . . . . . . . Illl; l : lit', I I I I I I I I I t l l l l

l fill flirt t +lift ~2~I " fill Jill[ l l [ [ l Ill[ IIlll I Illll

Illl Illll [ II[ll N EXPERIMENT THEORY

.... IIIII 1 ..... Illl iJlll I IIIJl Ill Illll I Illll Illll 1 Illll

I IIllJ I lllll 1

10 10 ~ 10 a

Figure 12. Viscosity versus shear rate curve for 1.75% CMC solution at room temperature predicted from the master curve of Fig. 9. The points X show the values obtained with the commercial viscometer for the same solution•

solid line. The solution is now characterized on the vis- cometer, and the values obtained are plotted as points X on the curve. It can be seen that the match is rather good. In fact, the error bound between the predictions from the SFI method and the actual viscometer measurements lies between + 5%.

C O N C L U D I N G REMARKS

A new simple rheological parameter, the solution flow index (SFI), has been proposed. It has been shown~that SFI can be effectively used for coalescing viscosity versus shear rate curves at any temperatures for polymer solu- tions with different concentrations. The master, curve ob- tained is independent of temperature and concentration. The proposed concept gives a simple, fast method for obtaining the rheogram without the use of expensive sophisticated equipment if the master curve for the partic- ular polymer solution is known. A modified form of the Arrhenius equation has been suggested for estimating the SFI values of solutions at higher temperatures. An altered free volume state model has been adapted to estimate SFI at different polymer concentrations.

The present method is very useful for estimating the viscosity of Newtonian and non-Newtonian inelastic solu- tions. However, it cannot be used for solutions that un- dergo structural changes during shear, such as suspen- sions, foams, or emulsions. Further, it would not be appli- cable to highly elastic liquids because the entry flow from the buret to the capillary and the exit flow would both be affected by the extensional properties of the elastic liquids.

I wish to express my gratitude to Mr. Takayuki Shimizu and Mr. Masahiko Inoue for carrying out the experiments under my guidance as part of their research for graduation and for drawing the figures.

B

CC

D E £

g

hf

hi

K,,

K '

K"

L

m

MFI n

N

Ap Q

R SFI

SFIrl SFIT2

t

NOMENCLATURE

asm ratio of SFI with different free volumes defined in Eq. (23), dimensionless

A constant dependent on the nature of the continu- ous phase (water), dimensionless

A' constant dependent on the nature of the continu- ous phase (water) and other parameters relating to SFI measurement, dimensionless constant equal to 1 based on the arguments of Fujita and Kishimoto [23], dimensionless constant correction factor given by Eq. (9), (Pa. s • m ) - 2

capillary diameter of the solution flow indexer, m activation energy, kJ / (mol • K) free volume of the solution, a function of temper- ature and volume fraction of solute, m 3 acceleration due to gravity, 9.807 m / s 2 final height of liquid in the solution flow indexer, m initial height of liquid in the solution flow indexer, m function of power-law index for inelastic non- Newtonian fluids given by Eq. (14), dimensionless constant given by Eq. (8), kg. Pa constant defined by Eq. (15), kg-1 capillary length of the solution flow indexer, m correction factor whose values are given in Tables 1 and 2, dimensionless melt flow index, g /10 min power-law index for an inelastic non-Newtonian fluid, dimensionless power index in the modified Carreau model pro- posed by Eq. (24), dimensionless pressure head loss given by Eq. (4), Pa volumetric flow rate through the solution flow indexer, m3/s gas constant, 8.3232 J / (mol • K) solution flow index, g /10 min solution flow index at temperature Tt, g /10 min solution flow index at temperature T 2, g /10 min time for flow of liquid through solution flow indexer from h i to h f , s

v average velocity of flow through solution flow indexer, m / s

W weight of liquid flowing in time t through solution flow indexer from h i to hp g

Greek Symbols

difference between free volume of water and that of the polymer solution, m 3

~/a apparent shear rate, s-1 r/a apparent shear viscosity, Pa . s 7/0 zero-shear viscosity, Pa- s

~70rt zero-shear viscosity at temperature Tt, Pa . s T/0r2 zero-shear viscosity at temperature T 2, Pa. s

A fluid relaxation time, s p density of fluid, k g / m 3 ~" shear stress, Pa 4' volume fraction of polymer, dimensionless

332 A . V . Shenoy

REFERENCES

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1977. 3. Whorlow, R. W., Rheological Techniques, Halstead (Wiley), New

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5. Park, N. A., and Irvine, T. F., Jr., The Falling Needle Viscometer: A New Technique for Viscosity Measurements, Warme- Stoffu- bertrag., 18, 201-206, 1984.

6. Park, N. A., and Irvine, T. F., Jr., Measurement of Rheological Fluid Properties with the Falling Needle Viscometer, Rev. Sci. lnstrum., 59, 2051-2058, 1988.

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8. Butcher, T. A., and Irvine, T. F., Jr., Use of the Falling Ball Viscometer to Obtain Flow Curves for Inelastic, Non-Newtonian Fluids, J. Non-Newtonian Fluid Mech., 36, 51-70, 1990.

9. Yamasaki, T., and Irvine, T. F., Jr., A Comparative Capillary Tube Viscometer to Measure the Viscous Properties of Newto- nian and Power-Law Fluids, Exp. Thermal Fluid Sci., 3, 458-462, 1990.

10. Abdel-Wahab, M., Giesekus, H., and Zidan, M., A New Eccen- tric Cylinder Rheometer, Rheol. Acta, 29, 16-22, 1990.

11. Shenoy, A. V., Chattopadhyay, S., and Nadkami, V. M., From Melt Flow Index to Rheogram, Rheol. Acta, 22, 90-101, 1983.

12. Saini, D. R., and Shenoy, A. V., A New Method for the Determi- nation of Flow Activation Energy of Polymer Melts, J. Macromol. Sci., !122, 437-449, 1983.

13. Shenoy, A. V., and Saini, D. R., Upgrading the Melt Flow Index to Rheogram Approach in the Low Shear Rate Region, J. Appl. Polym. Sci., 29, 1581-1593, 1984.

14. Shenoy, A. V., and Saini, D. R., Melt Flow Index: More than Just a Quality Control Rheological Parameter. Part I, Adc. Po(vm. TechnoL, 6, 1-58, 1986.

15. Shenoy, A. V., and Saini, D. R., Melt Flow Index: More than Just a Quality Control Rheological Parameter. Part II, Ado. Polym. Technol., 6, 125-145, 1986.

16. Shenoy, A. V., Practical Applications of Rheology to Polymer Processing, in Encyclopedia of Fluid Mechanics, Gulf Publishing Co., Houston, Tx., 1st Edition, Vol. 7, pp. 961-989, 1988.

17. Merrington, A. C., Viscometry, Arnold, London, 1949. 18. Metzner, A. B., in Handbook of Fluid Dynamics, V. L. Streeter,

Ed., McGraw-Hill, New York, 1961. 19. Ng, K. S., Cho, Y. I., and Hartnett, J. P., Heat Transfer Perfor-

mance of Concentrated Polyethylene Oxide and Polyacrylamide Solutions, A1ChE Syrup. Ser., 199, 76, 250, 1980.

20. Tung, T. T., Ng, K. S., and Hartnett, J. P., Influence of Rheologi- cal Property Changes on Friction and Convective Heat Transfer in a Viscoelastic Polyacrylamide Solution, Heat Transfer, 6th Int. Heat Transfer Conf., 5, 329, 1978.

21. Skelland, A. H. P., Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967.

22. Fujita, H., and Kishimoto, J., Interpretation of Viscosity Data for Concentrated Polymer Solutions, J. Chem. Phys., 34, 393-398, 1961.

23. Shenoy, A. V., and Saini, D. R., Interpretation of Flow Data for Multicomponent Polymeric Systems, Colloid Polym. Sci., 261, 846-854, 1983.

Received April 16, 1992; revised November 25, 1992