I
A. W . SISKO Research and Development Department, Standard Oil Co. (Indiana), Whiting, Ind.
a
The Flow of Lubricating Greases A relatively simple flow equation is used to fit experimental flow data, and can be applied to other materials as well as greases
G R E A S E S are non-Newtonian fluids with high viscosities at low shear rates and low viscosities a t high shear rates. Thus, they do not flow away from the outside of gears and bearings where shear rates are low, and they lubricate effectively at the bearing or gear surfaces where shear rates are high. The dependence of viscosity on shear rate must be known if power losses in bearings or pressures required for flow through pipes are to be determined. Flow equations previously described (9, 72, 73) cover narrow ranges of shear rates-two (9, 73) conclude that grease flows substantially as a Bingham body; the third (72) describes flow in terms of the theory of rate processes ( 5 ) . Data over a wide range of shear rates are needed to test the adequacy of any flow equation ( 2 ) . Accordingly, flow data obtained over a wide range of shear iates have been used for testing flow equations in the literature. A simple flow equation has been derived, which accurately describes the flow of the greases studied over a shear-rate range from 0.04 to 22,000 set.+ This equation may well find application to other disperse systems. .
Experimental An ASTM viscometer (7) was modified by adding extra capillaries. A second pump was added to obtain three flow rates, which with the standard set of capillaries having length-to-diameter ratios of 40 to 1 provided shear rates from 10 to 22,000 set.-' A set of 12 to 1 capillaries, having the same bore as those having a ratio of 40 to 1, corrected pressures read with the longer capillaries for entrance effects and piston drag. The range of this instrument was extended into that of the pipe viscometer by two pairs of capillaries-one pair having a diameter of 0.657 and the other
of 0.912 cm. Lengths of the first pair were 25.4 and 78.6 cm. and of the other, 41.6 and 147.2 cm. For adequate pressure response, these capillaries are longer than those having ratios of 40 to 1 and 12 to 1. They extend the range of the instrument down to 1 sec.-l The pipe viscometer consisted of a galvanized iron pipe 22 feet long and 1.5 inches in diameter, attached to a 5-gallon grease kettle. The first 2 feet of the pipe served as an orienting zone, and pressuredrop measurements were made over the remaining length. The use of an orienting zone and the high length-todiameter ratio of the pipe (160 to 1) made entrance corrections unnecessary. Pressure was measured with a 60-p.s.i. Bourdon gage fitted with an adapter, which was threaded into a T on the pipe. The adapter contained a thin plastic membrane positioned at the inside pipe wall. The gage and adapter were filled with light mineral oil. In operation, the kettle is capped and slowly stirred while air pressure 'forces the grease through the pipe. At the discharge end, samples ark collected by sharply cutting off the moving column of grease. T h e greases, commercial products made from petroleum oils and commercial thickening agents, were observed a t a temperature of 25' C. They had the following compositions :
The measured flow rates and pressure drops are converted to apparent viscosity, q., nominal shear rate, .E, and shearing stress a t the wall, Fw, defined as
where R is the tube radius, L is the tube length, P is the pressure drop across the tube, and Q is the volumetric flow rate. One (7) of the several techniques (7, 7 7) available was used for obtaining true viscosity, 11, and shear rate, $. Test of Flow Equations The literature on the rheology of greases describes grease flow in terms of the Bingham body concept and of the theory of rate processes. As a Bingham body, grease would possess a yield point and its flow in a capillary would be given by the Buckingham (4) equation. T h e theory of rate processes ( 5 ) has been used ( 7 7) to develop a general equation for the flow of non-Newtonian fluids. All of the greases studied showed linear plots of log q a vs. log d and of log q us. log $ at low shear rates, with slopes ranging from -0.776 to -0.946. Buckingham Equation. T h e flow of a Bingham body is expressed in terms of the plastic viscosity, vpl, by the equation VpI
Oil Vin.-
Thickener, Grease A B
C D
E
Thickener Calcium fatty acid Lithiumhydroxy stearate Sodiumtallow Calcium fatty acid Hydrophobic silica
70
cosity, SUB at 1OOOF.
7.2
260
6.5
813
8.9
500
8.5
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13.0
813
I
F - f
= --
Y
where F is the shearing stress and f is the limiting shearing stress below which no flow occurs. This equation was integrated by Buckingham for flow in a tube, where no flow occurs below a yield pressure drop, P,.
The constants are evaluated easily by a technique based on dimensionless numbers ( 8 ) ; a nomograph constructed by VOL. 50, NQ. 12
DECEMBER 1958
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Figure 1. Buckingham equation does not fit flow data of grease A Slopes of calculated and observed points disagree at low shear rates
Hedstram (6) is convenient for this purpose. A trial-and-error process is applied to the data from two flow determinations, and the constants vp, and P, are found. Typical results for the application oi the Buckingham equation to grease flow are compared with the data for grease A in Figure 1. The calculated curves, on plots of log 7, us. log s, have a slope of -1.000 at low shear rates. This disagrees with slopes found for the greases; consequently they are not Bingham bodies.
Ree-Eyring Equation. A general formula for viscosity (72) has been derived from the theory of rate processes. Viscosity is
where xnj Pn,and a, are parameters of the theory. The flow of lime-base grease ( 3 ) is described in terms of two flow units, Newtonian and non-Newtonian. For this situation, the equation is rewritten
The simplest means for finding the three parameters A, B, and pz is to use the equation in the form
which applies for large P z y , where sinh-' In 2fl2+. At small values of y , the contribution of A is negligible and a plot of log q us. log is linear. Selecting two data points in this shear-rate region
pa+
+
Figure 2. Threeparameter Ree-Eyring equation does not fit flow data of grease C
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INDUSTRIAL AND ENGINEERING CHEMISTRY
FLOW OF LUBRICATING GREASES 100 ow
I
I
IO 000
I
I
I
,i
I
m
0 1,000
$ P
PIPE VlSCOLlE v)
0 ASTH VISCOMETER
1.000
W
0. 100
6
10
100
10
I
001
0.1
1
1000 10,000
10 100 9 , SEC-1
100,000
Figure 3. Curves calculated from new equation give good agreement with flow data of all greases studied
and solving simultaneous equations in F and log .j give the values of K and D. Constant A is the best value. at high Y K1og.j 0' shear rates, of q - -- - Figure Y .j' 2 shows the equation fitted to the data of grease C. 7 =
6.00
1
10
y,
+ 1160-ilog i. +-2760 Y
Viscosities calculated from the equation rise above the observed values a t low shear rates and fall below a t high shear rates. Plots of the log of the viscosity of the non-Newtonian flow unit us. log y become linear when P 2 . j becomes very large. However, when P 2 . j becomes very large the slopes on these plots become equal to -1. The slope is given by
Ree-Eyring viscosity equation, which describes flow in terms of one Newtonian and two non-Newtonian flow units. However, testing this equation would require the evaluation of five parameters -A, B, Pz, C, and 03: sinh-1 B2.i s=A+B-------
P2.i
sinh-1 p3+
+
c---
P3-i
The difficulty of determining these five parameters makes it more practical to seek a new equation for evaluation. -(1---)
1 sinh-1 Pzi.
The disagreement of these slopes with the observed slopes leads to the conclusion that the three-parameter equation (72) lacks the flexibility needed to fit the flow data. The data might be fitted by the
.
N e w Equation for Grease Flow
As in the three-parameter equati , the grease is considered as being c posed of two flow units, one Newtonian and one non-Newtonian. The total shearing stress required to produce the
-
SEC.-'
same shear rate for both units is the sum of the shear stresses required for each. For the non-Newtonian unit, the observed power dependence on shear rate is used. F = a 4 C b-?
where a, b, and n are constants. Dividing by gives the viscosity as a function of shear rate-i.e., The parameters for this equation are easily determined. At low shear rates, the contribution of to the total viscosity is much larger than that of a. The equation may then be written as log q = ( n - 1) lcg .j log b. A plot of log q us. log .j is linear with a slope of n - 1, and b is the value of q when
+
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DECEMBER 1958
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, shearing stress at the wall, 2110 dynes per sq. crn., was substituted into the flow equation:
I 0 OBSERVED I - CALCULATED
F = 2.70.j
+ 1940+"~'3g
and a shear rate of 1.78 set.-' found. Substituting this shear rate into the integrated efflux equation gives a calculated flow rate of 4.61 cc. per second. The observed value is 4.66 cc. per second. Velocity Profiles in Flowing Grease. In photographic study of grease velocity profiles in a pipe ( g ) , velocity distributions resembled plug flow and were fitted to the equation for the velocity profile in a Bingham body. In the region where plug flow is predicted by the equation, the observed and calculated values of velocity differed markedly. The shear rates covered involve principally the effect of the non-Newtonian flow unit. The power dependence of shear rate on shearing stress, for this unit, leads to a relation between velocity u and radius r in a tube of radius R (70) given by u = B (R*- 7') where
.cj
w
tn
and
2
I
0
I
2
R A D I U S , CM. Figure 4.
Radial distribution of grease velocities in pipe flow Pipe radius 1.27 cm. ( 9 )
+=
1. Constant a is found from the best value of ( 7 - b+"-').
As shown in Figure 3, calculated and observed data agree well. The constants for each grease are: Grease
a
b
A
2.70 5.34 6.00 22.0 22.6
1940 2300 2380 2000 4650
B C D E
n 0.139 0.196 0.224 0.175 0.054
Calculating Pressure Drops in Pipe Flow. The new equation provides an accurate method for calculating pressure drops in pipe flow. With data from a capillary viscometer, the more convenient nominal shear rate, s, can be used. The pressure required to produce a desired flow rate in a pipe of length L and radius R is given by p = ""( a i
+ ci")
where a, c. and n are constants. Constants a and n are the same whether nominal shear rate, S, or true shear rate. +, is used and c is a constant for nominal shear rate. Pressure drops in pipe flow can be calculated from data obtained with
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rotational viscometers. Solution of the efflux integral is required,
Q
= $m2
(- 2 ) d r
=
$Tr2+dr
The variable of integration is changed from r to +, using integration by parts.
Substituting the expression for r , integrating, and applying the boundary = 0 give an condition that Q = 0 a t equation relating pressure drop to flow rate.
+
when n = 1 the fluid is Newtonian, F = (a b)?, and the equation reduces
+
As an example of such a calculation the flow rate of grease A in the 20-foot, 1.5-inch-diameter pipe ( L = 605.9 cm., R = 2.03 cm.) was calculated for a pressure drop of 18.25 p s i . The
INDUSTRIAL AND ENGINEERING CHEMISTRY
Substituting the data from Figure 11 ( 9 ) gives an equation for the velocity distribution in this grease: u = 0.450 (1.27g.63- r9.53). Figure 4 shows a plot of this equation against the points taken from the reference. The calculated and observed values agree, especially in the region 0.5 < Y < 1.0 cm., where the Bingham equation predicts absence of flow. literature Cited
(1) Am. SOC.Testing Materials, "ASTM Standards on Petroleum Products and Lubricants," 1955, D 1092-55, Philadelphia. (2) Arveson, M. H., IND.EIG. CHEY.24, 71 (1932). (3) Biott, J. F. T., Samuel, D. L., Zhid., 32, 68 (1940). (4) Buckingham, E., Am. Soc. Testing Materials, Proc. 21, 1154 (1921). (5) Glasstone, S., Laidler, K. J., Eyring, H., "Theory of Rate Processes," McGraw-Hill, New York, 1941. (6) Hedstrom, B. 0. A,, IND.ENG.CHEY. 44, 651 (1952). (7) Krieger, I. M., Maron, S.H., J . A$@. Phys. 25, 72 (1954). (8) McMillen, E. L., C h m . Eng. Progr. 44. 537 11948). (9y,-Mahncke, h.E., Tabor, W., Luhricatzon Eng. 11, 22 (1955). (IO) Metzner, .4.B., Adoances in Chem. Eng. 1, 77-153 (1956). (11) Mooney, M., J . Rheol. 2, 210 (1931). (12) Ree, T., Eyring, H., J . Appl. Phys. 26, 793, 800 (1955). (13) Singleterry, C. R., Stone, E. E., J . ColloidSci. 6, 171 (1951). RECEIVED for review October 26, 1957 ACCEPTED August 18, 1958
