Capillary viscometer for non-Newtonian liquids

JOUIRNAL OF COLLOID SCIENCE 15, 89-96

(1960)

CAPILLARY VISCOMETER FOR NON-NEWTONIAN LIQUIDS A. W. Sisko Research and Development Department, Standard Oil Company (Indiana), Whiting, Indiana Received August I, 1959 ABSTRACT A capillary viscometer having a wide range of shear rates (0.02 to 40,000 sec.-1) has been developed, and factors contributing to the response of the instrument have been analyzed. The analysis showed that time and sample can be conserved by the proper selection of pressure gauges and by the exclusion of air. The instrument has been used to study the flow behavior of lubricating greases over a range of shear rates wide enough that both pipe and capillary viscometers are usually needed to cover it. INTRODUCTION

Capillary viscometers are widely used for measuring the flow characteristics of liquids. In these instruments, the viscosity of Newtonian liquids is determined from the measurement of a single pressure drop and a single flow rate. Non-Newtonian liquids, however, require observations that extend over a range of flow rates. Usually the flow rate is measured at various pressures, or the pressure at various flow rates. Viscometers of the latter type are particularly convenient to use and have found wide acceptance in grease research (1-4). A desirable feature of viscometers for non-Newtonian liquids is that they allow the determination of viscosity over a wide range of shear rates. D a t a over such a range provide means for severely testing theories of flow. Suitable data have been obtained by the use of capillary viscometers in conjunction with pipe viscometers (5, 6). The inconvenience of operating two viscometers and the saving in time and material that would be realized prompted the development of a single instrument of wide shear-rate range. DESCRIPTION

Viscometers in which pressures are measured at fixed flow rates require nonpulsating volumetric pumps, a wide range of flow rates, and rapid pressure response. The new instrument features a piston pump with a 224-fold range of flow rates and strain-gauge pressure transducers. I t also includes a sample cylinder with floating piston and stainless-steel capillaries. The shear-rate range is 0.02 to 40,000 sec. -1 89

90

818KO STRAIN GAUGE /

LIMIT swrrc.

/ \ RELIEFVALVE

P,STON\ GEAR BOX~,

I

~)

ol ~

\

CYLINDER, t..~-~l

"

RESERVO,R

/E Y

1 ~MOTOR

11"

S A M P L ECYLINDER

THERMOCOUPLE ~

WELL

FIG. 1. Diagram of viscometer.

The arrangement of components is shown in Fig. 1. A 500-c.c. piston pump (Ruska Instrument Co., Houston, Texas), driven by a synchronous motor, forces a hydraulic fluid (SAE-40 oil) into the sample cylinder of the instrument. Delivery is controlled by a gear box that provides 28 piston speeds to give flow rates from 1.56 X 10-3 to 350 X 10-3 c.c. per second. Without leaking, the pump dispenses up to its maximum operating pressure of 4000 p.s.i. It is equipped with a travel-limit switch. Pressure in the hydraulic fluid is measured by strain-gauge transducers. Three gauges with ranges of 30, 500, and 5000 p.s.i, are used. The 30- and 500-p.s.i. gauges are protected by relief valves and can be isolated from the system by manual valves; the 5000-p.s.i. gauge and the pump are protected by a 4000-p.s.i. rupture disk. (For simplicity, Fig. 1 shows only one of the strain gauges and relief valves.) The gauges are operated from a dry-cell power supply equipped for independent zeroing of each gauge. A low-resistance microammeter measures the gauge output. The sample cylinder is a 660-c.c. pressure vessel with smooth internal walls. The threaded caps of the cylinder have O-ring seals that hold full pressure when tightened by hand. The selected capillary and a thermocouple well are threaded into the cap at the sample end. Three pairs of stainless-steel capillaries are used. The members of each pair have the same radius but differ in length. The length difference permits correction for friction of the floating piston and for capillary-entrance effects. OPERATION

The sample cylinder is filled by setting it upright, removing the cap at the sample end, and pouring or packing the sample into the cylinder. The end cap is replaced, the cylinder is inverted, and the other end is filled with hydraulic oil. The cylinder is then attached to the hydraulic-oil outlet, and the desired capillary is threaded into place. Care must be taken to avoid introducing air when filling and assembling the sample cylinder,

CAPILLARY VISCOMETER FOR NON-NEWTONIAN LIQUIDS

9~-

as air slows the response of the instrument. The valve below the oil reservoir is opened, and the pump cylinder is filled by retracting the piston. The power supply is turned on and adjusted to 12 volts. After a 15-minute warm-up period, the strain gauges are individually zeroed against a reference head of oil in the reservoir, the reservoir valve is closed, and the viscometer is ready to operate. The pump is started and a gear setting is selected. As sample is discharged, pressure in the instrument rises and levels off at a value determined by the flow rate and the viscosity. Pressures are measured at the same flow rates in each member of a capillary pair. CALCULATIONS In capillary viscometry, flow rate Q and pressure drop P are measured and used in Poiseuille's equation to obtain viscosity v, shearing stress at the wall F, and rate of shear #. These quantities are defined by -

~R4P 8LQ ;

F

RP 2L '

~

4Q ~-R~'

where R and L are the radius and length of the capillary in centimeters, P is measured in dynes/cm. 2, and v is measured in poises. The rate of shear in a capillary varies from zero at the center to a maximum at the wall. For non-Newtonian fluids, viscosity varies with shear rate, and use of these relationships results in the calculation of an apparent viscosity and a nominal shear rate at the wall. When data from viscometers of different geometries are to be compared, these quantities must be converted to viscosity and shear rate. The standardized treatment of viscometric data (7) is convenient for doing so. In this viscometer, flow rates in cubic centimeters per second are obtained directly from the gear-box setting by multiplying by a proportionality factor. The factor is calculated from the speed of the motor, the gear reduction, and the area of the piston. Pressm'e in the hydraulic fluid is directly related to strain-gauge output in microamperes, t~a. The constant for each gauge was determined by calibration at 12 volts against a dead-weight gauge. Constants for the three gauges used are: 30-p.s.i. 500-p.s.i. 5000-p.s.i.

0.4454 p.s.i./pa. 5.780 p.s.i./~a. 59.95 p.s.i./ga.

Capillary dimensions were determined with a traveling microscope and checked by the use of viscosity standards. The dimensions, in centimeters, of the three pairs of capillaries are:

92

SISKO

Radius

L

0.4415 0.09299 0.02330

88.74 19.05 4.541

S

17.78 3.91 0.976

where L and S are the lengths of the long and short capillaries. Corrections for piston friction and capillary-entrance effects are made from pressure drops measured at the same flow rate in the long and short members of a capillary pair. The true pressure drop, P, is then given by L P = ( P ~ - - PS) L ~ , where PL and Ps are the pressure drops in the long and short capillaries. RESPONSE

Time and material are saved if the pressure drop P rapidly comes to equilibrium at any pumping rate. Delay in reaching equilibrium is caused by extrusion of the sample, expansion of the sensing element of the pressure gauge, and compression of any entrapped air. The time required can be calculated from an over-all rate equation: Q=Q~+Q~+Qa

summing capillary flow, gauge filling, and air compression. The term Qc is represented by Poiseuille's equation. An expression for Qg is obtained by assuming a linear dependence of gauge volume on pressure, P - P0 = b(Vo - Vo), where P0 is atmospheric pressure, Vg is the gauge volume, V0 is the gauge volume at atmospheric pressure, and b is a compressibility constant. The flow rate into the gauge is: Qo-

dV 1 dP dt - b dt

An expression for Qa is derived from the ideal-gas law, ( P + Po)Va = nk, where Va and n are the volume and moles of air and k is a constant. The rate of compression of the air is: Q~

dV nl~ dP dt - (P + Po)2 dt "

The over-all rate equation then becomes lrR 4 1 dP Q = ~PWbd[+(PWPo)

nk

dP ~ dt"

Integrating and applying the condition that the pressure drop is zero when t = 0 gives the response equation:

CAPILLARY

VISCOMETER

FOR

NON-NEWTONIAN

LIQUIDS

93

Q

t

lnQ _ AP

n£ I P A In Q(P -f- Po) 7 "F APo + Q _Po(P -F Po) + APo -F Q Po(Q - A-fi)J' where A = ~'R4/8L~. In the absence of entrapped air, the response equation simplifies to: t

1 ~lnQ

O _ AP"

Substituting the expression for A and solving for P gives the change in pressure drop with time

p_

8LQn ~R 4

I 1 - e ~r~Rib "1 t|. J

When the final pressure drop for the particular flow rate has been reached, no more liquid flows into the gauge, Qc = Q, and P = 8LQv/rR 4. Sample and time are conserved if the factor R4b/Lv is large. With a fixed set of capillaries, response is speeded by increasing the value of the gauge constant b. This means that the volume of the sensing element must change little with pressure. For this reason, manometers are unsuitable. More desirable

60i

L

i

,

,

WITHOUT AIR

40

uS tD

~ e~

2o

o

i 0

i

i

I0

20

I

TIME, SEC.

FIG. 2. Calculated effect of air on response.

30

94

S~SKO

is a gauge in which the deflection of the sensing element is mechanically or electrically amplified. Strain-gauge pressure transducers have high values of b and are thus particularly suitable. The response equation permits calculation of the time required to reach 99 % of the final pressure drop with or without entrapped air present. Figure 2 shows the calculated response curves for these conditions. The calculations are for the flow of a liquid of 100 poise viscosity in a capillary 10 cm. in length and 0.1 cm. in radius. At a pumping rate of 0.135 c.c. per second, the final pressure is 50.0 p.s.i. The gauge constant b was assumed equal to 100 p.s.i./c.c. In the absence of entrapped air, 17.2 sec. are required to reach 99 % o f full pressure. T o determine the effect of 1 c.c. of entrapped air, a reasonable value for the gas constant was assumed, nk -16.2 p.s.i.c.c., and P0 was taken as 14.7 p.s.i. The time required for 99 % of full pressure with 1 c.c. of air present is 30.8 sec. I:~ESULTS The viscosities of several Newtonian and non-Newtonian liquids were determined with the viscometer. For a Newtonian fluid, F = ~. A plot of log ~ against log F is a straight line of slope I, and log v = log F when log ~; = 0. Figure 3 shows such a plot for a commercial lubricating lOS

........

I

........

I

........

I

. . . . . . .

i

d b,bJ

10 4

-

10 3

_

.<

ILl I

-0-

10 2 10 2

r

q

ITII

I

10 3

SHEAR

I

CAPILLARY

PAIR

C

CAPILLARY

PAIR

B

1 II,llfl

~

I

I

STRESS,

I

I tlll~

10 4

10 5

DYNES/CM

FIG. 3. Flow of SAE-40 oil at 25°C.

2

I

n u

I

10 6

CAPILLARY

]05 =

VISCOMETER

i i lllh

FOR

I , J;l!ill

I

NON-NEWTONIAN

i I !ll{JJ

I E I]lFll

LIQUIDS

I J illlJI

, i llil~

o CAPILLARY VISCOMETER • CONE-AND-PLATE VISCOMETER

~ 0 4 -%' -%

~-

O U ~,~ >

95

z

-~

-

,I

_

~02

~, ---

10

1

I

i I Ililll

i~!!i

10 -~

1

I ! IIIIh 10

10 2

] I IIIIII

I i Illlil

10 a -I

10 4

i ;iiliil IO S

SHEAR RATE, SEC.

FIG. 4. Flow of lithium-soap grease. oil that shows Newtonian behavior and has a viscosity of 4.08 poises at 25°C. The data were obtained with two of the capillary pairs. The flow behavior of a non-Newtonian fluid, a lithium-soap grease, was determined in the viscometer and also in a cone-and-plate viscometer (8). Viscosity and shear rate can be calculated directly from data obtained with the cone-and-plate viseometer because the sample is uniformly sheared in this instrument. Apparent viscosities and nominal shear rates from the capillary viscometer were converted to viscosities and true shear rates, Figure 4 shows the effect of shear rate in these viscometers on the viscosity of the grease. The data from both instruments fall on a single smooth curve. An equation that has been proposed for the flow of lubricating greases (6): = 5.00 + 3,300 ~-0.812 fits the data well. CONCLUSION

The design of the new viscometer gives a response as rapid as is practically possible. With it the effect of the concentration and geometry of dispersed particles, as well as that of the viscosity of the continuous phase on the flow of non-Newtonian systems, can be studied. With suitable temperature control, the dependence of viscosity on temperature can be determined.

96

SISKO ACKNOWLEDGMENTS

The author thanks Professors I. M. Krieger and S. H. Maron of Case Institute 05 Technology for their contributions to the project. REFERENCES 1. American Society for Testing Materials, "ASTM Standards on Petroleum Products and Lubricants," Philadelphia, 1955, D1092-55. 2. ARVESON,M. H., Ind. Eng. Chem. 24, 71 (1932). 3. BRUNSTRUM,L. C., AND STEINBRUCH, l~., Institute Spokesman 13, No. 8, 10 (1949). 4. MARUSOV,N., Institute Spokesman 15, No. 5, 8 (1951). 5. SUMMERS-SMITH, D., in "Proceedings of Conference on Lubrication and Wear~" The Institution of Mechanical Engineers, London, 1957. 6. SISKO, A. W., Ind. Eng. Chem. 50, 1789 (1958). 7. KRIEGER, I. M., AND MARON, S. H., J. Appl. Phys. 25, 72 (1954). 8. McKENNELL, R., in V. G. W. Harrison, ed., "Proceedings of the Second International Congress on Rheology," p. 350. Academic Press, London, 1953.