Rheol. Acta 14, 404~-09 (1975)

Department of Chemical andBiochemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania (USA)

A general model for the effective viscosity of pseudoplastic and dilatant fluids

St. W. Churchi l land R. U. Churchi l l

With 8 figures (Received April 10, 1973)

Nomenclature

b

x

y(x) yo(X) y~(x)

z

~A qB ~0 ~o(C)

~(c)

% c

CA

~0

Cl/2

arbitrary constant in Sisko model (eq. [5]) arbitrary exponent in eq. [1] independent variable dependent variable limiting behavior of dependent variable as x ~ 0 limiting behavior of dependent variable as x~oo original dependent variable arbitrary constant in Sisko model (eq. [5]) and Bird-Sisko model (eq. [6]) arbitrary exponent in eqs. [2] and [8] effective viscosity = shear stress/rate of shear effective viscosity at c = CA empirical constant in eqs~ [2] and [8] limiting value of effective viscosity as c --+ 0 limiting behavior of effective viscosity as c--, 0 limiting value of effective viscosity as z ~ oo limiting behavior of effective viscosity as c ~ oo rate of shear arbitrary constant in Bird-Sisko model (eq. [6]) shear stress arbitrary constant in eqs. [2] and [8] shear stress at ~) ~ 0 in Bingharn model shear stress at t/= (q0 + r/oo)/2

Churchill and Usagi (1) have demonstrated that data for many phenomena can be success- fully correlated in terms of the general expression

y*(x) = y(x) + y%(x) [1]

where yo(x) and y~(x) are the asymptotic solutions or limiting correlations for small and large values of x, respectively. If these limiting functions are known the best value of the arbitrary exponent n can be determined from one or more intermediate values of y and x. If the dependent variable is an increasing power of x, n is positive. If the dependent variable is a decreasing power of x, n is negative. In this latter case negative values of n can be avoided, if desired, by defining the dependent variable y as 1/z where z is the original dependent variable. 156

Construction of general model for pseudoplastic fluids

Eq. [1] appears to have particular value for correlation of the effective viscosity of pseudo- plastic and dilatant fluids as a function of shear stress. For a pseudoplastic fluid with the behavior indicated in fig. la only the limiting values of ~o and ~oo rather than the limiting functional dependences are ordinarily known.

tog 7?

~o ~~, / /~m( ~" ) (b)

,d, Lf4~o / log -c

Fig. 1. Asymptotic behavior of pseudoplastic and dilatant fluids

This is insufficient information to apply eq. [1] directly. What is needed is either t/o(~ ) or ~~(z), as indicated in fig. lb. If for example, asymptotic behavior for large shear stress of the general form

Churchill and Churchill, A general model for the efJective viscosity of pseudoplastic and dilatant fluids 405

tl~(T) = tloo + tl @- '

is postulated where # > 0, and if y is defined as 1/(tl - tloo), eq. [1] takes the form

(tl - tloo) "= (t lo - tloo)" + - -~A

which can be rearranged as

Simplifieation of model [2]

Eq. [4] can be simplified somewhat before evaluating the constants by arbitrarily choosing tl = tlo - ~~. This was not done originally in eqs. [2] and [3] only in order to permit direct reduction to the Frederickson-Sisko and Bingham

[3] models. Eq. [4] then becomes

( ; I ~~~ t lo - t loo 1 + - - . [7] tl - - tloo %'A (.,o ~~;+((,,o ,,~) ~ 7 ~4~

t l - tl oo tl CA I /

Reduction of general model to prior models

Eq. [4] includes as special cases many of the models previously proposed to represent vis- cosity data for pseudoplastic fluids:

1) For n = 1 eq. [4] has the form of the equation proposed by Meter (2).

2) For n = 1 and tl~ = 0 the form of the Ellis model is obtained.

3) For n = 1 and # = 2 the form of the Reiner- Phillippoff model is obtained.

4) For n = 1 and with Y in place of z the form of the equation proposed by Cross (3) is obtained.

5) The model proposed by Sisko (5) is equivalent to

tl = , l~ + be [5 ]

which is a special case of eq. [4] for n = 1 and

[6 ]

case of eq. [13] in terms of

tlo ~ oo in terms of ) rather than z. 5) For n = 1 and in the limit as tlo ~ oo the form

called the Sisko model by Frederickson (4) is obtained. Bird (6) refers to the Sisko model as

tlo

which is a special rather than t.)

7) Forn=l ,#= 1, t lo=0, r l=t l0oandTA=% eq. [4] reduces to the Bingham model for I~1 > to .

8) For n = 1 and tl o~ : 0 eq. [4] approaches the Ostwald-deWaele (power-law) model as t increases.

In all of the above special cases n is taken as unity. In each of these cases the remaining constants can readily be identified with those in the conventional form of the particular model.

Extension of general model for dilatant fluids

The behavior of a dilatant fluid is indicated diagrammatical ly in fig. l c. The functional behavior in the two limits is indicated in fig. 1 d. This time the general form

~~(t) = t l~ - ( t l~ - tlo) [8 ]

can be postulated, again with fl > 0, for the asymptotic behavior for large shear stress. Choosing y as 1/(tloo - tl), eq. [1] takes the form

tloo - tlo = 1 + [9] tloo q

which is equivalent to eq. [7]. Thus the same general model is applicable to both pseudo- plastic and dilatant fluids.

Evaluation of constants in model

The five constants in eq. [7] can be evaluated systematically as follows:

1) A plot of log tl vs.log t such as in fig. 1 yields the limiting values tlo and tloo as indicated.

2) The asymptote at large t of a plot of log (t 1 - tloo) vs. log t as in fig. 2 defines # and rA.

3) The exponent n can be evaluated by orte

I \ \ . . . . . . . . . . . . ~ : - \ \

~,~ -"q~ . . . . . . . . . . . ~" - i "~\" .~,-sto pe

l.og ( "q- '%)

vA log T

Fig. 2. Determination of/9, ~A and ~4 27

406 Rheologica Acta, Vol. 14, No. 5 (1975)

of the several procedures proposed by Churchill and Usagi (1). If t /= ~/A is known for t = "c A then

n= In 2/ln (-~-A0- ~/q ) . [io3

I I I I I I I I I

v~i,ol_

l 2 10 -'t 1 10

Fig. 3. Logarithmic chart for determination of n

Alternatively values of (t/ - t/~)/(qo - q~) can be plotted vs. l t / rA I ~ on logarithmic coordinates. n is then chosen by comparison with the family of curves corresponding to discrete values of n as illustfated in fig. 3. A more critical evaluation can be made by plotting values of ( t / - rl~)/ (r/o - r/~) vs. I'c/zAI a from 0 to 1 and values of

(~ ~~)1~ i ~1 ~ - - vs. from 1 to 0 on r/o - q~ ta arithmetric coordinates, n is chosen by com- parison'of the plotted values with curves rep- resenting discrete values of n as illustrated in fig. 4 which is actually the inverse of the plot proposed by Churchill and Usagi. Fig. 4 is an expanded plot of the variance from the limiting solutions. Hence any scatter in the data will be

exaggerated. In this application the scatter may also be magnified by a poor prior choice of the other four constants. Conversely the final corre- lation is relatively insensitive to the value of n and a round number or simple fraction can generally be chosen without serious error.

4) Values of ~ are offen not available at sufficiently large -c to define t/o ~ in a plot such as fig. la. A better value may be obtained iteratively by recalculating r/~ from eq. [7] for large t after tA and/~ are determined, i. e.,

"c

t/o - t/ tA

as z --. oQ . [11] z IP ~ 7oo

1 I TA In cases where even this procedure does not yield a satisfactory value for zoo an approximate value or zero may be chosen arbitrarily; the correlation will still be reasonably valid for "c less than the maximum experimental values.

It is very desirable to use experimental values of t/ and "c to evaluate the constants. Values of the constants determined for the various simpler models will not necessarily carry over to the more general model.

Al ternat ive fo rm for cor re la t ion

Instead of postulating eq. [2] for large q asymptotic behavior of the general form

,1o(~) = ,7o - (,1o - ,7oo) ~ B23

can be postulated for small 'c for a pseudoplastic fluid as suggested by fig. lb, and y chosen to be 1/(~/o - t/) leading to

.~_2-~___~. = 1 + - - . [13] \ t /o - t / t

1.0

0,8

~"q~ o.6

"r]O ~ho o.4

0.2

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0,4 0,2

L0

0.8

-qo-7/~-~ r a t ON.

02

o

Fig. 4. Arithmetic chart for precise determination of n

Churchill and Churchill, A general model for the effective viscosity of pseudoplastic and dilatant fluids 407

Similarly for a dilatant fluid the expression

! ~ [14] ~o(~) = ~o + (,Io - ~oo) z~ ,

can be postulated for small v instead of eq. [8] for large ~. The choice of y as 1/(t/ - r/0) then leads to eq. [13] in this case as well.

Eq. [13] is identical to eq. [-7] only for n = 1. One or the other expression may be slightly more successful for correlating a particular set of data. The eonstants in eq. [12] can be eval- uated by a procedure equivalent to that outlined above for eq. [7].

Sign i f icance of constants

Two of the five constants in eq. [7] and [13], ~o and ~1~o, can be interpreted in terms of the limiting behavior; two constants, ~a and fl, can be interpreted in terms of the intermediate power-law-type behavior; and n can be inter- preted as the appropriate order of the mean of the two limiting solutions. White and Churchill (7) have asserted that very precise data are needed to define so many constants uniquely. If the data do not have this degree of precision they can be fitted almost equally well over a range of values of some of the constants. In this event the correlation may be successful but physical interpretation of these constants cannot be justified. This is almost certainly the case with typical rheological data.

The use of n = 1 in the models mentioned above is arbitrary and has no theoretical justification. Reasonable success in correlation has apparently been attained because of the insensitivity of eq. [7] to the value of n and

because of the flexibility afforded by the other four constants.

Bird (6) noted for a number of fluids that the value of cA obtained for the Ellis model (eq. [7] with n = 1 and ~o~ = 0) from a log-log plot of t/0/~ rs. ~ was not equal to the value of z at t/o/~ = 2. This discrepancy can be minimized by using eq. [7] instead of the Ellis model and noting that rA equals ~ at (t/o -t/oo)/ (t/ - r/oo) = 21/".

Example

The experimental data of Ashare (8) for a 5~ wt solution of monodisperse polystyrene with a molecular weight of 1.8 10 in Aroclor 1248, a chlorinated diphenyl, can be used to illustrate the use of eq. [7] for correlation. These data were chosen because they are available in tabular form and extend to a sufficiently high rate of shear to indicate an asymptotic value for the effective viscosity.

Fig. 5 indicates that these data do not define values of t/o and t/~o very precisely. A value of 160 poise was derived for t/oo by the iterative procedure described above. The value of 7600 poise derived for t/o by Ashare was accepted to expedite comparison with his correlation. The asymptote drawn in fig. 6 yields a slope fi ~ 8/3 and an intersection with t / - r/~ = t/0 - t/~ = 7440 poise at rA ~ 1920 dyne/cm 2. The corresponding value of t /A- t/oo ~ 3000 poise. Hence n = ln2/ln (7440/3000) = 0.763. Fig. 7 is a plot of the data in the form of fig. 4 with the indicated values of t/o, t/~, zA and fi. Curves representing n = 0.70, 0.75 and 0.80 are included. The scatter in the data at large and

10 a - -

G 0

poise

103

n : ,~ :12

Eq7 n 3/4 ~/1920 dyne/cm a ~'~" . . . . . . . Eq.13, n = 5 /3 , )3 = 8 /5 , r a = 2150 dyne/cm 2

~o i i ; t I r I p F [ i I T i I i , ~0 10 2 10 3 10 4

T-dynes /cm 2

,r]

r r I 10 5

Fig. 5. Experimental data and final correlations for 5% wt, 1800000 MW poty- styrene in Aroclor 1248

27*

408 Rheologica Acta, Vol. 14, No. 5 (1975)

io 4 ~~~~ 7~ i I I i ~ i , J %-% ~ . . . . ,4~--

('~-'q~)/2 "r/A- ~ ~10 3 ~ ~ : }~ . . . . .

b r / -T /~

poise %

102 ~,

~,~~ . ~( slpe = -12/5

slope = -8/3 ~'~~/ V,

1o , , , , I , , , L I V',, I , I I 10 2 10 3 ,i0 4 10 5

T- dynes /cm 2

Fig. 6. Determination of fl, rA, Zl/2 and /~A for 5% wt, 1800000 MW polystyrene in Aroclor 1248

small shear stresses, and hence the uncertainty in t/o and t/~o, is even more evident in this form. Considering the scatter of these data and the insensitivity of the correlation to the value of n a round value of 3/4 can be chosen.

Fig. 6 was also used to construct a correlation with n arbitrarily chosen as unity as in the Meter equation. The intersection of q = (No - q~)/2 with the eurve sketched through the data yields zl/z = 1600 dyne/cm 2. The asymptote for large shear stress through z = "LI/2 and t / - 0oo = qo - qoo has a slope of 12/5.

The alternative form of eq. [13] was also tried as illustrated in fig. 8. The uncertainty in the location of the asymptote for decreasing shear stress is considerably greater than for increasing shear stress with this set of data. The indicated asymptote, which was necessarily located rather arbitrarily, has a slope of 8/5 and yields a value of ZA = 2150 dynes/cm 2. The corresponding value ofn = ln2/ln(7440/4900) = 1.66 ~_ 5/3.

10 rt/z "r A j I I l I I ! IJ J I l I I I

~o~~ . . . . . . . . . . . . . . . . . L . i - / .~~~o oooo %-~A - - - ~~/ I

~72~ o /

poise 1 3 / i #

o / / / / /

/ /

slope = 8/5 ~ / / . slope = I2/5

# Y / /

/ / / /

// i / 10 /I i t T J I I I T 1 I 10 2 10 3 10 `+

T-dynes /cm 2

Fig. 8. Alternative determination of #, zl/ ~A and t/A for 5% wt, 1800000 MW polystyrene in Aroclor 1248

As noted above the same values of # and ~1/2 are necessarily obtained from eqs. [7] and-[13] if n is chosen as 1.0. In the absence of this information an asymptote with a considerably lesser slope would have probably been projected from t/o - r/ = qo - r/~o and z = tl~2.

The curve representing eq. [7] with 0o = 7600 poise, t/o~ = 160 poise, fl = 8/3, rA = 1920 dynes/ Cm 2 and n = 3/4 is seen in fig. 5 to represent the data over the entire range of as closely as justified by their scatter. The curve representing eq. [13] with fl = 8/5, rA = 2150 dynes/cm 2 and n = 5/3 is seen to differ only slightly on the higher side. This discrepancy could be eliminated by a slight adjustment of the constants. The curve representing eqs. [7] and [13] with B= 12/5, zA= 1600 dynes/cm 2 and n= 1.0 also appears to provide an acceptable correlation. It is evident that plots in the form of figs. 6 and 8 provide a far more critical test of the consistency of the data and the success of the correlating

~/-16o 74z+0

L4 / I J I J J I I I

!

0.6 n = 0.8

O.q

0.21 I I I I I 0.2 0.8 1.0 0.2

I I I 0.4 0.6 0.8 0.6 0.4

(T/1920) 8/3 (1920/r) s/3

i.#

'1.0

7,+o J~lgzoJ

0.6

Q4

o.z Fig. 7. Determination of n for 5% wt, 1800000 MW 0 polystyrene in Aroclor 1248

Churchill and Churchill, A 9eneral model for the effective viscosity of pseudoplastic and dilatant fluids 409

equations than fig. 5. The scatter and deviations are effectively hidden by the logarithmic scale and choice of ordinate in fig. 5. Fig. 7 provides a still more critical test.

All three representations are considerably better than achieved by Ashare (8) using the Bird-Carreau (9) and Spri9gs (10) equations. These latter expre~sions have however the advantage of incorporating empirical constants which are theoretically related to other rheo- logical properties.

Other applieations

A number of other sets of data for the effective viscosity have been examined and successfully correlated by this procedure. The best value of n has differed significantly from unity in all of the cases which have been examined. In many of these cases it has been possible to construct a satisfactory, if poorer correlation, with n arbi- trarily taken as unity as in the previously proposed models. Eqs. [73 and [-13] can also be used to construct correlations for the effective viscosity as an explicit function of the rate of shear rather than of the shear stress and for the dynamic viscosity as a function of the frequency of oscillation.

Conelusions

The model and alternative model proposed herein are more general and less constrained than previous models which have been proposed or used for correlation of the effective viscosity of pseudoplastic and dilatant fluids with the shear stress.

These models are also adaptable for corre- lation of the effective viscosity directly with the rate of shear and for correlation of the dynamic viscosity with the frequency of oscillation.

The procedure proposed for evaluation of the constants is straightforward and should yield the previously proposed models if the data so indicate. However, this degeneration has not occurred in any of the sets of data which have been examined.

Plots such as figs. 6, 7 and 8 provide a far more critical test of the data and of the success of the correlating equation than plots such as fig. 5.

Insofar as eq. [-7] is a successful model, figs. 3

and 4 are general plots for all pseudoplastic and dilatant fluids.

Acknowledgement

The assistance of Profs. Drs. L. VMclntire and W.E. Forsman is greatly appreciated.

Summary

A new and very general expression is proposed for correlation of data for the effective viscosity of pseudo- plastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expres- sion. A straightforward procedure is outlined for evaluation of the arbitrary constants.

Zusammetfassung

Eine neue und sehr allgemeine Formel wird fr die" Korrelation der Werte der effektiven Viskositt von strukturviskosen und dilatanten Flssigkeiten in Ab- hngigkeit von der Schubspannung vorgeschlagen. Die meisten schon frher vorgeschlagenen Methoden werden hier als Spezialflle dieser Gleichung gezeigt. Ein einfaches Verfahren fr die Auswertung der willkrlichen Konstanten wird beschrieben.

Referencs 1) Churchill, S. W. and R. Usagi, Amer. Inst. Chem.

Eng. J. 18, 1121-1128 (1972). 2) Meter, D. M., Ph. D. Thesis, Univ. of Wisconsin,

Madison, Wisconsin (1963). 3) Cross, M. M., J. Appl. Polymer Sci. 13, 765 774

(1969). 4) Frederickson, A. G., Principles and Applications

of Rheology, Prentice-Hall (Englewood Cliffs, N.J. 1964).

5) Sisko, A. W., Ind. Eng. Chem. 50, 1789 1792 (1958).

6) Bird, R. B., Canad. J. Chem. Eng. 44, 161 168 (1965).

7) White, R. R. and S. W. Churehill, Amer. Inst. Chem. Eng. J. 5, 354 360 (1959).

8) Ashare, E., Ph. D. Thesis, University of Wisconsin, Madison, Wisconsin (1970).

9) Bird, R. B. and P. J. Carreau, Chem. Eng. Sci. 23, 427-434 {1968).

10) Spriggs, T. W., Chem. Eng. Sci. 20, 931-940 (1965).

Authors' addresses:

Prof. Dr. Stuart W. Churchill, Depar{ment of Chemical and Biochemical Engineering University of Pennsylvania Philadelphia, Pennsylvania 19174 (USA)

Renate U. Churchill International Chemical Engineering P. O. Box 627 Media, Pennsylvania, 19063 (USA)