Rheologica Acta P, heol. Acta 22, 417-419 (1983)

SHORT COMMUNICATIONS

The kineüc energy correction in viscometry

M. Keentok

Department of Mechanical Engineering, The University of Sydney

Abstract. When calibrating U-tube capillary viscometers a kinetic energy correction is required for liquids in a certain viscosity range and in the past an unsatisfactory method has been used for its determination. This correction may be obtained with reasonable accuracy by least squares fitting flow time data to the corrected Poiseuille's law. The flow times of totuene (20 ° C and 25 ° C), benzene (25 ° C) and water (25 ° C), together with literature values of viscosity were used to determine the calibration constant C and the kinetic energy correction B of a size A U-tube viscometer. The agreement with the manufacturer's C and previously reported B is very good. A modified kinetic energy correction of Cannon et al. does not appear to give better accuracy.

Key words: Viscometry, U-tube viscometer, kinetic energy correction, Poiseuille's law, calibration of viscometers

1. Introduetion

When measurements of viscosity are made with U-tube capil lary viscometers, a kinetic energy correction is required when the efflux time (or flow time) t is less than some critical time te, which depends on the viscometer dimensions [1-4]. In practice this means that a kinetic energy correction must be calculated for liquids having viscosities less than ~ 2cSt and for liquids with larger viscosities if the correct size viscometer is not available. The kinematic viscosity v is calculated from t using the corrected Poiseuille 's law

v = c t - ~ / t (1 )

where C is a viscometer constant and B/t is the kinetic energy correction. The Couette correct ion is incorpora ted in C. Fo r the viscometer used here (BS/UsizeA) t« = 530 sec and for liquids with v ~ 0.7 cSt corresponding to t ~ 250 sec the kinetic energy correction is ~ 1 ~ and thus it is desirable to det.ermine B to an accuraey of 10 or better. 887

2. Determinat ion o f the kinet ic energy eorrecüon

The currently recommended procedure for cal ibrat ion of a U-tube capil lary viscometer requires [1, 2]:

1) determinat ion of C by using a l iquid of accurately known viscosity having a flow time t > to in the viscometer under calibration,

2) determinat ion o f B by measuring the flow times t~ and t2 of two liquids having accurately known viscosities v~ and 1) 2 and then calculating

t l t 2 /~ = ~ ( v 2 t 1 - vl t2).

1 2 - - t 1 (2)

While procedure (1) may be satisfactory, procedure (2) is inaccurate because of compounding of errors arising in eq. (2). The author obtained B values which were approximate ly 2-2.5 times too large by this method.

In this report the coefficients C and B will be determined by fitting eq. (1) to da ta for three liquids with

418 Rheologica Acta, Vol. 22, No. 4 (1983)

viscosities v~ and flow times t i using a least squares method. Eq. (1) may be written

v~ t~ = C t ~ - B (3)

and by setting yi = v~ ti, x i = t~ we obtain

Yi = Cx~ - B. (4)

It is assumed hefe that B, as well as C, is a constant. An alternate form of the kinetic energy correction has

been proposed by Cannon et al. [5] who investigated the Reynolds number dependence of the kinetic energy correction. F r o m the corrected Poiseuille 's law

C = n r 4 g h 8 V L ' (5)

m V B = s ~ ~ ' (6)

where Vis the volume of t iming bulb, h the mean head of liquid, r the capillary radius and L the capil lary length (wirb Couette correction).

The coefficient m varies with Reynolds ' number R = 2 V / T r r v t , [6,7] and by using specially designed viscometers Cannon et al. showed that m = 0.037R 1/2,

which leads to

E vi = Ct i t~ (7)

1.66 V 3/2 where E - L ( C D ) I / 2 . The advantage ofeq. (7) is that E i s

a constant whereas B varies with R. A least squares fit was also done to eq. (7) using the

substitutions Yl = vl t~ and xi = t~. The flow times were measured for a size A U-tube

viscometer ( B S 188; C = 0.002792); the reproducibil i ty was better than 0.1 ~ . The liquids were all analytical grade. The flow times are listed in table 1 together with the smallest and largest of the l i terature kinematic viscosities vi. The accuracy of C, B and E were est imated by performing the least squares fit using the smallest l i terature value of v i (first line ofeach entry in table 1) and the largest (2nd line). Hence the following results are obtained for eqs. (1) and (7) respectively:

C = 0.002794___ 0.000011, B = 1.87_+ 0.85, C = 0.002786 +__ 0.000007, E = 346 +_ 157.

These coefficients were used in eqs. (1) and (7) to calculate the viscosities listed in table 2, which are compared with

Table 1. Data used for least squares fit to eqs. (1) and (7). For each liquid at a given temperature the first line is based on the srnallest literature value of v i and the second on the largest

Liquid qi vi ti Eq. (1) Eq. (7)

(cp) (cSt) (sec) vi ti t 2 t i t~

toluene 20°C 0.5848 0.6754 245.4 165.74 60221 40673 0.5900 0.6815 245.4 167.23 60221 41041

25°C 0.5500 0.6387 231.7 147.99 53685 34289 0.5520 0.6410 231.7 148.53 53685 34412

benzene 25 ° C 0.6010 0.6880 249.5 171.66 62250 42828 0.6018 0.6890 249.5 171.89 62250 42890

watet 25 ° C 0.8904 0.8931 321.7 287.31 103491 92428 0.8904 0.8931 321.7 287.31 103491 92428

14778000 14778000 12439000 12439000 15531000 15531000 33293000 33293000

Table 2. Kinematic viscosities cal¢ulated from eqs. (1) and (7). The literature value quoted is the mean of those given in table 1

Liquid v (oSt) v (oSt) 6v (%)

Eq. (1) Eq. (7) Literature Eq. (1) Eq. (7)

toluene 20°C 0.6780 0.6779 0.6785 -0.1 -0.1 25°C 0.6393 0.6391 0.6399 -0 .1 -0.1

benzene 25 ° C 0.6896 0.6896 0.6885 +0.2 +0.2 watet 25 ° C 0.8930 0.8929 0.8931 - 0.0 - 0.0

Keentok, The kinetic energy correction in viscometry 419

the mean of the l i terature values. The maximum difference is ~ 0 . 2 % which is approximate ly the est imated accuracy o f the l i terature values. The computed C in both cases agrees with the nominal C to 0.2 % or better. The accuracy of the coefficient B will be considered in the next section. I t is clear from table 2 that there is no advantage in using eq. (7) for this viscometer over the Reynolds numbers encountered here.

Table3. Experimental and theoretical values of m and calculations of B for a size A U-tube viscometer

m B (calculated) Source (cSt. sec)

Experimental 0.97 1.55 0.29 0.46 1.12 1.78 1.13 a) 1.80 a)

Theoretical 1.12 1.78 0.5-1.12 0.8-1.78

Bond [6] Cannon et al, [5] Swindells et al. [7] This work

Boussinesq [8] Dinsdale, Moore [4]

a) B is measured, m is calculated from eq. (6).

3. Comparison of theory and experiment

We will concentrate here on eqs. (1), (5) and (6). Table 3 lists experimental and theoretical values of m and values

V 4.8 of B calculated from eq. (6) using

8~L 8n x 12 = 1.59 cSt. sec. F o r the work repor ted here 40 < R < 82 and so we may take R ~ 60. Clearly, except for Cannon et al. the experimental values are consistent when allowances are made for experimental accuracy. Boussinesq [8] est imated the kinetic energy correction theoretically, assuming an unsteady, accelerating flow in the capillary entrance region and arr ived at m = 1.12 which is in excellent agreement with experiment. The

other theoretical values cited by Dinsdale and Moore [4] are smaller; however Dinsdale and Moore indicate that most experimental values lie close to 1.12.

4. Condusion

The kinetic energy correction can be a significant one in the cal ibrat ion o f U-tube viscometers and a more satisfactory method for obtaining this correction is by performing a least squares fit o f (vl, ti) data to eq. (1). The v i da ta should be either measurements made in a master viscometer or reliable l i terature values. The work repor ted here suggests that there is no advantage in using eq. (7) over eq. (1) for the part icular viscometer used.

Acknowledgement

The author gratefully acknowledges the assistance ofR. Gear in making the viscosity measurements.

Referenees

1. British Standard Methods for the Determination of the Viscosity of Liquids in C.G.S. Units BS188 (1957).

2. Merrington, A.C., Viscometry, Edward Arnold&Co. (London 1949).

3. Barr, G., A Monograph of Viscometry, Oxford Univ. Press (London 1931)

4. Dinsdale, A., Moore, F., Viscosity and its Measurement, Chapman and Hall (London 1962).

5. Cannon, M.R., Manning, R. E., Bell, J. D., Anal. Chem. 32, 355 (1960).

6. Bond, W.N., Proc. Phys. Soc. 34, 139 (1922). 7. Swindells, J. F., Hardy, R. C., Cottington, R. L., J. Res. NBS

52, 105 (1954). 8. Boussinesq, J., Comptes Rendus 113, 49 (1891).

(Received September 21, 1982)

Author's address:

M. Keentok Department of Mechanical Engineering The University of Sydney Sydney, N.S.W. 2006 (Australia)