direct determination of the flow curves of no

Direct Determination of the Flow Curves of NonNewtonian Fluids. II.Shearing Rate in the Concentric Cylinder ViscometerIrvin M. Krieger and Harold Elrod

Citation: J. Appl. Phys. 24, 134 (1953); doi: 10.1063/1.1721226 View online: http://dx.doi.org/10.1063/1.1721226 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v24/i2 Published by the American Institute of Physics.

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JOURNAL OF APPLIED PHYSICS VOLUME 24. NUMBER 2 FEBRUARY. 1953

Direct Determination of the Flow Curves of Non-Newtonian Fluids. ll. Shearing Rate in the Concentric Cylinder Viscometer

IRVIN M. KRIEGER, Department of Chemistry and Chemical Engineering, Case Institute of Technology, Cle'Deland, Ohio AND

HAROLD ELROD, Department of Mechanical Engineering, Case Institute of Technology, Cle'Deland, Ohio (Received July 21, 1952)

Another method has been developed for obtaining the rate of shear '/IS shearing stress curves of non-Newtonian fluids from concentric cylinder viscometer data. The mathematical expression developed is a rapidly converging power series in Ins, where s is the cup to bob radius ratio. An estimate of error shows that under favorable conditions only two terms of the series are significant, and that terms past the third will hardly ever be needed.

INTRODUCTION

T HE first paper in this seriesl reviewed the theory of the concentric cylinder viscometer. There it was shown that an exact solution in closed form for the rate of shear can be obtained by treating the radius ratio of the cylinders as a variable. To use the method which was developed in this manner, one must obtain torque-angular velocity data with two or more bobs of different radii.

It was also shown that an expression for the difference between the rates of shear at the cup and at the bOb wall can be recovered from data taken with only one bob. In the present paper, a solution for this difference equation is obtained as a rapidly converging series.

THEORY OF THE CONCENTRIC CYLINDER VISCOMETER

The present treatment presupposes the existence of a functional rel3;tionship between the rate of shear dv/dyand the shearing stress F, namely,

dv/dy=g(F). (1) The first objective of viscometry is the recovery of this relation, called the flow equation, from experimental data. The flow equation may be expressed in analytic, tabular, or graphical form.

As in the previous paper, we consider a concentric cylinder viscometer having an inner cylinder or bob of length L and radius R l , and an outer cylinder or cup of radius R 2= sRI. The cup rotates with angular ve-locity n, while an external torque M holds the bob stationary. When the fluid in the annular space be-tween the cup and the bob is in laminar flow, the rate of shear at a distance r from the axis, where the fluid rotates with angular velocity w, is

dw dw g(F)=r-=-,

dr d Inr

while the shearing stress is F=M/27r'f2L.

(2)

(3)

Since M is constant under steady flow conditions, d InF= -2d lnr

and g(F) = -2(dw/d InF).

(4)

(5) At the bob surface, F=F l and w=O, while at the cup, F=F2 and w=n. Integrating (5) between these limits, one obtains

IfF! 1 iFI g(F) n= -- g(F)d InF= -- ~F.

2 rl 2 .'F. F (6)

In the previous paper, g(F) was obtained by differ-entiation with respect to s at constant F2, giving

g(FI) = s(an/as)F2 (7) From this equation an experimental method was de-veloped for determining the rate of shear from data obtained using two bobs.

When (6) is differentiated with respect to Fl , a dif-ference equation is obtained:l - a

dn 1 -=-[g(Fl)-g(F2)]. dFl 2Fl

(8)

The method described here for obtaining g(F) is based on a solution of this difference equation.

SOLUTION OF THE DIFFERENCE EQUATION Let us define a function heFt) by the relation

From Eq. (8),

Also, h(r2FI) = g(s-2F I) - g(s-4FI) ,

h(s~FI) = g(s-4FI)- g(s-6F I),

(9)

(10)

(11)

1 I. M. Krieger and S. H. Maron, J. Appl. Phys. 23, 147 (1952). 134

1M. D. Hersey, J. Rheol. 3, 196 (1932). M. Mooney, J. Rheol. 2, 210 (1931).


FLOW CURVES OF NON-NEWTONIAN FLUIDS. II 135

etc. Since s> 1 and g(O) = 0, 00

L h(s-2nF 1) = g(F 1). (12)

The sum is a slowly convergent one, which may be asymptotically evaluated using the Euler-MacLaurin sum formula,4

:. fen) = Lmf(X)dx+tU(O)+ f(m)] ..-0 0

r B2k + L --U(2k-l)(m)-f(2k-l)(0)]

k-l (2k) ! m-lfHl

+ E . P2r+l(X-i)J(x)dx. (13)

Here Bi is the ith Bernouilli number and Pi the Ber-nouilli polynomial. Applying this formula to the sum-mation in Eq. (12),

f. h(s-2nF1) = L"" h(r2xF1)dx+ t[h(F1) + h(O)] _0 0

1 [dh(S-2nF1)]"" 1 [d3h(S-2nF1)]"" +- -- " + .... (14) 12 dn _0 720 dn3 0

The integral is readily evaluated by making the sub-stitutions

y=s-2xF1 and

dx= -dy/2y Ins. Therefore, in view of the definition of h,

(15)

(16)

I "" -1 f.0 n h(S-2xF1)dx=- h(y)d Iny=-. (17) o 2 Ins 1"1 Ins The derivatives appearing in Eq. (14) are

d2n -4 Ins ,

d(lnr2nF 1)2 (18)

etc.

Since h(F1) and its derivatives are zero when Fl=O, the final asymptotic expression for the rateof shear may be written

n [ . d Inn (Ins)2 d2n g(Fl) = - 1+Ins--+--_-

Ins d InF 1 3n d(InF 1)2 (lns)4 d4n . ]

--. + .... (20) 45n d(lnF 1)4

4 J. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, Cambridge, 1946), p. 255.

A particularly attractive feature of the above result is the fact that the bracketed term is a power series in Ins. The better concentric cylinder viscometers are constructed with small clearances, so that Ins rarely exceeds 0.2 and is frequently 0.05 or less. The term d lnn/d InFl is unity for Newtonian fluids, but may take on higher values for non-Newtonian fluids. (Values as high as five have been observed in this laboratory.) The second term is therefore significant.

To estimate the error involved in terminating the series at any given point, it is convenient to consider the special case of a fluid whose flow equation is

g(F) = aFN, (21) where a and N are constants characteristic of the fluid.6-7 Integration of Eq. (6) with this value of g(F) gives

hence

etc. Thus

d lnn/d InF1=N, d3n/ d(lnF 1)3= N3n,

(22)

(23)

n [ (N Ins)! (N Ins)4 ] g(F1) = - l+N Ins+ +.... (24)

Ins 3 45

Hence the error in terminating the series at the second term is of order leN Ins)2, while carrying the third term reduces the error to 1/45(N Ins)4.

In this connection, it is evident that an instrument for which s is small is desirable. Comparison of two com-mon concentric cylinder viscometers brings out this point. For the Precision-Interchemica:l viscometer,S R2 = 1.500 cm, Rl = 1.300 cm, making s= 1.154. Here N must be less than 1.4 in order that the series may be terminated at two terms without exceeding 1 percent error, while N values up to 7.0 may be used if the third term is carried. For the Mooney-Ewart coni-cylindrical viscometer,9 for which Rl = 2.00, R 2= 2.10, and s= 1.05, the corresponding N values are 4.1 for two terms and 20.5 for three. The computational problem is thus much simpler in the latter case, since a second differ-entiation will rarely be required.

USE OF THE METHOD

In practice, M 'VS n data are obtained, and F 1 calcu-lated by multiplying M by the factor 1/211'R12L. A plot of logn 'Vs 10gFl is constructed. If this plot is linear, the exponential flow equation applies, and g(F) may be obtained by well-established techniques.7 If this graph is curved, graphical differentiation at points along the

6 A. W. Porter and P. A. M. Rao, Trans. Faraday Soc. 23, 311 (1927).

6 Farrow, Lowe, and Neale, J. Textile Inst. 19, T 18 (1928). 7 I. M. Krieger and S. H. Maron, J. Colloid Sci. 6, 528 (1952). 8 H. Green, Industrial Rheology (John Wiley and Sons, Inc.,

New York, 1949), p. 100. 9 M. Mooney and R. H. Ewart, J. Appl. Phys. 5, 530 (1934).


136 I. M. KRIEGER AND HAROLD ELROD

curve yields the slope m, where m=d InOld InFl (25)

For those cases where m Ins is less than 0.2, the third and higher terms may be neglected, and the rate of shear calculated with an error of less than 1 percent by

g(F1)=O/lns(1+m Ins). (26) When m Ins is greater than 0.2 but less than 1, an addi-tional graphical differentiation is required to keep the accuracy within 1 percent. Since

1 d20 dm - m2+--, o d(lnF 1)2 d InF 1

(27)

the third term may be obtained from slopes of a graph of m vs InF 1. The rate of shear is then

o [ (InS)2 dm ] g(F1)=- l+mlns+tcmIns)2+---- .

Ins 3 dIn~ (28)

In the rare case when m Ins is greater than 1, the fourth term may be required, although this has never been necessary for the systems studied in this laboratory. For most fluids, the first two terms should suffice.

CONCLUSIONS

An alternate method has been devised for the re-covery of the rate of shear of a non-Newtonian fluid from data obtained in concentric cylinder viscometers, thus permitting the direct determination of the flow curves of such fluids without prior assumption of a flow equation. The mathematical expression for the rate of shear is a power series in the logarithm of the radius ratio; the coefficients are derivatives of the angular velocity with respect to shearing stress. Under favorable conditions, terms beyond the second are negligible, while inclusion of the third term is almost always adequate. Instruments designed with radius ratios near unity are more satisfactory in this respect, since only one differential need be evaluated.

A paper is in preparation in which the flow curves of several fluids, obtained by the various direct methods, are intercompared. The work discussed herein was per-formed as part of a research project sponsored by the Reconstruction Finance Corporation, Office of Syn-thetic Rubber, in connection with the Government Syn-thetic Rubber program. The advice and encouragement of Dr. Samuel H. Maron are acknowledged.

JOURNAL OF APPLIED PHYSICS VOLUME 24, NUMBER 2 FEBRUARY, 1953

The Nature of the Coefficient of Friction J. T. BURWELL * AND E. RABINOWICZ

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts (Received August 18, 1952)

The modem theory of the friction between dry metal surfaces ascribes it to local minute welds or adhesions between the surfaces and suggests that for a given pair of surfaces the friction force is uniquely defined by the normal load alone. Herein it is demonstrated that this cannot in general be true and that some further con-dition of operation must also be defined. Experiments are reported indicating that one such possible condi-tion is the sliding speed so that the friction force is actually a function of the normal load and the sliding speed. It is pointed out that the speed can influence the friction force in two ways-one, by the resulting shear strain rate in the vicinity of the welded junction, and the other by the length of time taken for a junction of full strength to form.

T HE current theory of the mechanism of dry frictionl -3 pictures the force of friction as arising from the shear strength of minute welds formed be-tween two bodies in contact and distributed more or less at random over their apparent contact area. This can be expressed by the relation for the friction force

F=sA, (1) where s is the average shear strength of these welds and A the sum of their individual areas. Since it is these areas that actually carry the normal load W between

* Now with Horizons, Inc., Cleveland, Ohio. 1 R. Holm, Electric Contacts (Almquist and WikseUs, Uppsala,

1946). H. Ernst and M. E. Merchant, Proc. M.I.T. Sum. Conf. on

Surface Finish, p. 76, (June 1940). a F. P. Bowden and D. Tabor, The Friction and Lubrication of

Solids (Oxford University Press, New York, 1950).

the two bodies, we also have W=pm A , (2)

where Pm is defined as the flow pressure of the softer material in the vicinity of these local true contact areas. Eliminating A between these two equations leads to the familiar expression for the friction coefficient

(3) Bowden and Tabor emphasize the point that this is the ratio of two plastic properties of the weld and adjacent material. It has been customarily assumed that these two quantities are each constant for a given pair of contacting surfaces so that p. is also a constant.

Recently, however, McFarlane and Tabor' have 4 J. S. MacFarlane and D. Tabor, Proc. Roy. Soc. (London)

A202, 244 (1950).


direct determination of the flow curves of no

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