A CAPILLARY VISCOMETER FOR NON-NEWTONIAN LIQUIDS A R l E RAM ABRAHAM T A M l R
Easily built in the workshop, this multipurpose apparatus for
liquids hauing time-independent
viscosities has a working range that is almost unlimited here problems in handliig and transporting liquids W are encountered, the most important parameter is Viscosity. I t is used widely in correlations dealing with fluid mechanics, heat transfer, mixing, as well as polymer characterization in production and research. Where flow of water is concerned, the problem is relatively simple because any reliable viscometer is adequate. Viscosity is a constant ratio between shear stress and rate of shear: r = 7 (du/dr)
(1)
Also, several reliable tables are available which give Viscosity of water as well as of other simple liquids at various temperatures (70, 27). But trouble begins for nonsimple systems such as dispersions, colloidal solutions, and lubricating oils, as well as polymer melts and solutions. These liquids, ddined as non-Newtonian, and increasingly dealt with in research and industry, have been negatively defined as non-Newtonian for historical reasons. Also, they represent a group of substances which show different and sometimes contrasting behavior. Indeed, the only common point is that no simple law applicable to all exists. However, frequently the curve showing the relationship between shear streso and rate of shear, called flow curve, has some defined features. The work reported in this article deals with a major group of non-Newtonians, pseudoplastics, which shows a continuous decrease in slope of the graph for shear stress DS. rate of shear. This corresponds to a continuous decrease of the apparent viscosity when rate of shear rises. However, the complete picture is different. Careful investigators (78, 20) have found that these flow curves consist of several clear-cut sections (Figure 1): a Newtonian region at low rates of shear and another at high rates of shear. Then at medium rates of shear, a pseudoplastic structure region appears. When attempting to extend the flow curve by increasing velocity, a sharp increase of the
Figwe 1. Cartfur invlstigntors hum found thut complete ,¶ow curm far pseu&pIustics consist of clcor-ad rectionr-o Nzwtmiun section at both low and high rufts of shear, and a pseudoplustic region at medium rates of shear
apparent viscosity appears which indicates turbulence. However, apparent viscosity is restricted by definition to the ratio of shear stress to rate of shear in the laminar region only. Viscosity cannot be considered as a single valued parameter, and a simple Viscometer does not provide the desired data. A wide range flow curve can only be determined by a multispeed or multipressure viscometer. Of many attempts to simplify the mathematical relationships and linearize the flow curves, the “power law” is the mmt useful:
r = K(du/dr)’ (Continued on mx: fig#) V O L 5 6 NO. 2 F E B R U A R Y 1 9 6 4
(2)
47
Figure 2. The viscome6er.
Temperatwe and p e s w e
musl be rigidly conlrollcd o m 4 Wide range, but oper06ion is simple, rapid, ond&ibIe
which gives a straight line on log-log paper. The major advantage of this law is that only two parameters are involved (76). Unfortunately this law is not based on rheological fundamentals and is a purely empirical concept. In addition, the exponent n may change with rate of shear, as confirmed by some investigators (75,26). Thus, when viscometric data are needed for nonNewtonian liquids, a complete flow curve should be shown which includes at least the range of interest, although for future purposes, data beyond that range are preferable.
Equipmeml
Many instruments considered as multispeed or multipressure viscometers are available commercially, but most are rotating: concentric cylinders, rotating disks, or cone-and-plate. They are usually restricted in designed working range which cannot be altered easily. In addition, a good rotating viscometer capable of giving viscosity data over a wide range of shear of appropriate accuracy is usually expensive. However, many provide accurate information. Capillary viscometers for non-Newtonian liquids, described by several authors (2-5) can be built in regular workshops and, if design details are properly controlled, their range of operation, excluding low rates of shear, is almost unlimited. The viscometer built in our laboratories (Figure 2) consists of a 1-liter vessel designed for a maximum pressure of 1500 p.s.i.a. A water and an oil bath provide continuous circulation of temperature-controlliig liquids through a mantle. After filling, the vessel is closed and 48
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
connected to the nitrogen pressure pipeline. At its bottom, stainless steel capillaries are connected, having radii of 0.1 to 0.45 mm. and a ratio of length to radius of about 100 to 700. Fpessure is regulated by a manometer in the rangeof 0 to 30 p.s.i., 0 to 100 psi., 0 to 300 p.s.i., and 0 to 150 aim. Temperature is kept within *0.lo C. by using a thermometer inserted in a thermowd which is bored into the pressure vessel. Flow is regulated by valves. Various capillaries of different lengths and diameters are used. Operation is simple, rapid, and flexible. Temperature and pressure must be rigidly controlled over a wide range. I t should be emphasized that selecting the range of shear is easily done in this viscometer because of its extraordinary flexibility.
Theory of Nom-Newlomiam Viscosity
Apparent viscosity vs,, is defined as the point ratio of shear stress to rate of shear. As previously mentioned, this value varies with rate of shear for non-Newtonian liquids, except at very low or very high rates. The exact limits for those ranges differ for every system. In the authors' opinion, the apparent viscosity should be the natural substitute for the viscosity in all engineering calculations. Beyond the laminar region, the last value of qSP before onset of turbulence, is recommended. I n many cases this value equals that of the upper Newtonian viscosity. Now, it can be shown that the properly designed cone-and-plate viscometer usually gives the exact value of vaD. The concentric cylinder viscometer gives fairly exact values, provided the gap between the cylinders is very small compared to the radius of either
cylinder. Otherwise corrections are unavoidable and average values of shear rate are used. This introduces a complication because the flow profile or equation must a priori be known in order to calculate the average value. As to the capillary viscometer, some investigators have overlooked its simplicity and flexibility for nonNewtonian fluids characterization. It has been argued that since the conditions of shear vary from the center of the tube to the wall, an average value is needed. This is almost impossible to find without previous knowledge of the flow curve for the system. Our view is that there is no need for any average, but only for the value of shear stress and rate of shear at a single point in the tube. The obvious preferred point is a t the wall. A precise mathematical treatment done by Rabinowitsch (22) ends up with a n exact value of the shear rate at the capillary wall without assuming any known relationship. Good agreement has been obtained between flow curves for non-Newtonian liquids, measured by both rotational and capillary instruments ( 7 , 79). For capillary viscometers, the Rabinowitsch treatment was
applied, and the results provide the best proof for reliability of capillary viscometers for characterizing nonNewtonian liquids. Calculations are as follows: Equation 1 is used with qapsubstituted for q. Values of r and du/dr are taken a t the wall-Le., when r = R. For Newtonian liquids the following equations result : rw = ( A P . R ) / 2 L
(3)
(4) For non-Newtonian liquids, Equation 3 is still valid, but Equation 4 is modified:
where :
.o.xL/R
ap. (POISE)
1
1000 I
I
Ap=-
6.9 x 104
Tap. -1 D,
R
A
i
P (P.S.1.)
1500 1000
500
400
300
200 I50
100
:
50
0.07 0.06
0.04
IO
200
INSTRUCTIONS
1
0.02
0.01- 100
Figure 3 A .
Nomograph f o r shear rates covering several capillary geometries as a function of density and apparent viscosity VOL. 5 6
NO. 2
FEBRUARY 1964
49
,)I
(SEC -')
9
ap. (POISE)
1 O63-O.0l
-
--
__ ---? 0.02
2-0.06 ~"0.07 zrO.08 0.09 : 0.1 -:
't
a I
I
10-10
09 --
08--09
07--08 06 --07 05 --06 1-0 5
-~
104
LO 2
0
3
-
10
-
-
-0 2 1
-
INSTRUCTIONS
-
--Oh - 0.7 -70.8
- 0.9
-0 1
Figure 3B. Nomograph for shear rate at the wall, selected abparent viscosity, and capillary radius as related to Re by assuming several values of the slope defined by Equation 6 7
ap. (POISE)
R (MM.)
I Re t R 7 ap. m
p
07
(.=19-':)
04
c c.
02 04
INSTRUCTIONS
I
0 03
Re t
R e - R A 1 1v a p . /I 2 2-t /I v
Figuie 3C. hTomographf m telating Re to proper volume (for a constant assumed density) and time of Joiv 50
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
002
Equation 5, called the Mooney-Rabinowitsch equation, is an important contribution to the development of capillary viscometry. T o find the value of n’ (which is not necessarily a constant), only pressure and flow rate need be measured. It should be stressed that slope n’ is not, by definition, identical with the exponent n of the power law (Equation 2 ) . I t equals the latter only for a constant slope. The variables handled hitherto are pressure drop AP; flow rate, volume I’ and time t; capillary radius R; and length L. In addition, temperature T must be kept constant a t the desired value. Because weight-discharge measurements are more accurate, densities are involved too. T o design a suitable capillary viscometer, limits of rate of shear and shear stress must be determined. Rough estimates of working conditions may provide initial values, but it is wise to extend the working range as widely as possible. Thus, AP, Q(V, t ) , R, and L are required to get a flow curve describing viscometric data. The above variables are actually identical to those needed for Newtonian liquids. However, the straight line of Newtonian liquids can be determined with only two points, reducing to one when the origin (0, 0) is taken as one of the points. Based on these considerations working design nomographs were calculated (Figure 3 , A , B, and C ) . With these nomographs, limits of the above variables can be found easily for each actual case. Additional information has been included in the nomographs--e.g., the Reynolds number. This is important to ensure that it falls in the laminar region. Normally, the critical Reynolds number, Re,, is taken as 2100. However, for many polymer solutions this is not true (24). Whatever the critical Reynolds number is, it is essential to choose tEe variables so that Re < Re,. Reynolds number is related to the other variables as follows :
2 Qp 2 Vp Re = 2RDp ___ - _ _ - ___ Tap TRTap ~Rt~ap
(7)
The range of variables covered in our nomograph is limited to t = l/2-5 minutes, and V = 10-1000 cc. However, with the linear nomograph it is possible to extend the range by extrapolation. Re may be related to the shear rate D , by using Equation 5 and it is found that: Re =
D,” . R2p 2
Tap
= D,
(-)3n’4n’+ 1
2
(8) Tap
This equation completes the interrelation of all the variables. The nomograph may now be used to design a new capillary viscometer or to select operation condi-
AUTHORS Arie Ram and Abraham Tamir are in the Depart-
ment of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
tions with an existing one. Figure 3,A gives values of shear rate for several capillary geometries (represented by L/R), as a function of AP and qap. In Figure 3,B, the resultant D, and the chosen qap and R are related to the Reynolds number by assuming several values of n’. In Figure 3,C, the resultant Re is related to the proper volume (for a constant assumed density) and time of flow. It is thus possible to control the number of runs that may be accomplished with a given volume of liquid in the pressure vessel. Measuring Viscosify
Performance of the capillary viscometer is relatively simple. Once temperature is well under control, the only readings taken are pressure and weight of liquid issued at a known period of time. If the capillary length and radius are accurately known, then Equations 3 and 5 can be used to obtain the shear stress and rate of shear, both at the wall. From this, a flow curve is plotted and viscosity at any point in this range is known. If the radius is not accurately known it may be measured by filling the capillary with mercury or by direct optical enlargement. Alternatively, it may be eliminated by using a calibration method. In any case it is a well accepted rule to calibrate the whole system prior to use by reliable calibration liquids. The latter consist of stable Newtonian liquids, the viscosity of which has been carefully checked (at some defined temperature) by an authority such as Bureau of Standards. This calibration step is very important and should not be underestimated. Some workers prefer to prepare their own calibration liquids. Handbooks provide viscosity tables at various temperatures for water, glycerol (at various concentrations), and some other simple liquids (70, 27). Glycerol solutions are frequently used for calibration because of their wide range of viscosity (about 1 to 1500 cp. at 20’ C.). Water is still the simplest liquid, but its range of viscosity is so limited that corrections such as the kinetic energy correction as well as appearance of turbulence are unavoidable. I n our experience there are strong objections to the blind use of glycerol solutions. The first one is that glycerol is not a pure liquid, never reaching concentration of 100%. Moreover, the water content of the liquid is difficult to control because of the hygroscopy of concentrated glycerol. We have tried to prepare a glycerol solution by carefully weighing a sample from a fresh bottle of pharmaceutical glycerol, and then diluting with a measured volume of distilled water. In addition we have analyzed the water content of the pure glycerol by the method of Karl Fischer. The density of the diluted glycerol solutions was accurately checked at selected temperatures by using pycnometers. Severe discrepancies were found when comparing the values of measured viscosities (given by an accurate commercial viscometer at low shear rate) with the calculated ones based on tables relating specific gravity of glycerol solutions to concentration and viscosity (70). VOL. 5 6
NO. 2
FEBRUARY 1964
51
Unrecoverable drop in viscosity should be checked for
For example, water content of “pure” glycerol was found to be 4.1% by weight. Two solutions were prepared (Table I). TABLE 1.
PROPERTIES OF GLYCEROL SOLUTIONS
Temperature,
C
1
Solution 1 25
I
34.2
1’
q1, viscosity, cp.,
measured viscosity, cp., calcd. from p
q2,
(VI
-
70
772)/?l,
p ~ sp. , gr., calcd. from q (P2
cl,
-
Pl)/Pl,
% yo
21.99 30 1.1989
5.5
30
Solution 2 25
I
30
difference in the viscometric behavior of various materials is in the extent of the Newtonian range. The theoretical treatment of Eyring (7) actually supports this idea that every material is Newtonian at low shear rates only. I t should also be noted that other so-called Kewtonian liquids such as oils, were found to be non-Xewtonian at high shear rates (8, 77, 28-30). Corrections and Side Effacls
5: 26.9
98.1 47.6 1.2385
71.9 45.3
5.4
concn., wt.
To get accurate results, some well defined corrections are sometimes needed. Of these, the most important is for pressure dissipated to build up kinetic energy at the exit. Pressure Dissipated. This is given by AP
91
91.1
91 . o
86.4
The value of calculated viscosity checks well with the directly measured one, only when the water content found by an analysis is accounted for. However, when attempting to employ data of specific gravity in order to use the viscosity tables, the errors in the value of calculated viscosity may range from 27 to 48%) the higher value corresponding to the higher concentration. Figure 4 shows plots of Ap/(plo0 - p70) and Aq/ (q100- 770) against concentration of glycerol in the range of 70 to 1OOyoby weight. The subscripts 70 and 100 refer to the limits of concentration, while Ap and Aq refer to the cumulative change of density and viscosity, respectively. Although A p is a constant (equal to 0.00265 approximately, in cg. at 25’ C.) Aq varies from 1.57 cp. at 70% up to 170 cp. at 99%. This is the reason why a small error of about 57, in the reading of density leads to an error of 30 to 48ye in the value of viscosity. I t may therefore be concluded that the best way to calibrate the variable capillary viscometer is by Newtonian oils of known or carefully checked viscosity. I t is not surprising to find that Newtonian liquids behave like non-Newtonian when very high shear rates are met. Such a deviation for glycerol has been found for solutions in a concentric-cylinder viscometer ( 2 3 ) . We have verified this behavior in our capillary viscometer too. A similar behavior of glycerol solutions was mentioned by other investigators (6, 9 ) . This conforms to the general concept that the borderline between Newtonian and non-Newtonian is not rigid at all. Every liquid may become non-Newtonian at very high rates of shear, provided the laminar region is not interrupted by turbulence or breakdown of the structure. The only 52
INDUSTRIAL AND ENGINEERING CHEMISTRY
=
AP“ - mp@
(9)
where AP” is the measured pressure drop, and AP corresponds to the corrected value. This correction should be the first step before T is calculated. Equation 9 is usually called the Hagenbach correction. Most investigators use a value of rn = 1.12 which also accounts for the contraction entrance effect from the large diameter of the vessel into the capillary. In addition, the term of equivalent length L’ is commonly used instead of L in Equation 3 to take care ol end effects (boundary layer establishment) so that
L’
=
L f AL = L $- N R
(1 0)
AL is usually called the Couette correction. The value of N is often quite small and by using reasonable values of L / R in the capillaries (in our case L / R = 100-600) this correction becomes insignificant and
-g
60
‘=
6
s 2
40
23
20
0 GLYCEROL, WT. %
Figure 4. Change of density and viscosity for glycerol. T h e viscometer should be calibrated by Newtonian oils of known or carefully checked viscosity
frequently with a conventional low-shear viscometer
is hence L’ E L The correction for kinetic energy -. important only for very high velocities. Dissipation of Energy into Heat. Part of the pressure energy which has to overcome the force of friction is dissipated as heat according to the equation
M
= 17
(du/dr)*
(11)
where M is in terms of energy per unit time per unit volume. It is clear that a t high shear rates the value of M increases. I t may affect the readings and its extreme values should be estimated
(dT/dtj
P
c p
p
Q.C,AT
VapD2
(12 )
qapDZV
(1 3)
=
=
where V is volume of capillary.
( 7 7, 27). I t is therefore necessary to check at frequent intervals for unrecoverable drop of viscosity by using a conventional low-shear viscometer. Viscoelasticity. Elastic terms have not been discussed, and are beyond the scope of this article. NOMENCLATURE C, D D* L m
M n
n‘ N AP Q r
R t
T
Equation 7 can be used and Q can be replaced by another relationship including Re :
u
U V 7
vap p
AT,,,
2 D2RL C, Re
= ___
The mean value of shear rate should be taken. Therefore, if we design D , for shear rate at the wall, a n approximate value of D,/2 is preferred. So, the maximum rise of temperature resulting from dissipation of mechanical energy is
AT,,,
Om2RL 2 C, Re
= ___
A good check for the insignificance of the dissipation value is sometimes provided by a Newtonian straight line for the calibration oil in the appropriate range. Wall Effects. Wall effects may show up, once different values for viscosity of the same solution are obtained, by changing radii of the capillaries. The best way is to try different capillaries and to get an appropriate value for the minimum R needed. Turbulence. Turbulence should always be avoided. Unfortunately, with some polymer solutions there are many indications of a n “early” turbulence a t low Re’s. This problem will be treated separately in another article (24). Degradation. Polymer solutions of high molecular weight species can be “degraded” by chain-scission of the molecules under high rate of shear. The result is an irreversible decrease of viscosity. Though there is still disagreement if chain-cutting occurs in the turbulent region only, it is nevertheless an experimental fact
r
= specific heat = rate of shear
= rate of shear for Newtonian liquids = length of capillary = Hagenbach coefficient = energy dissipated into heat = exponent of power law = slope defined by Eq. 6 = Couette coefficient = pressure drop = volumetric flow rate = radial coordinate = radius of capillary = time = temperature = velocity, local = mean velocity = volume = viscosity, Newtonian = apparent viscosity = density, specific gravity = shear stress
Subscripts c
w
critical = wall =
REFERENCES (1) Brodnyan, J. G., Gaskins, F. H., Philippoff, W., Symp. on Non-Newtonian Viscometers, ASTM Special Tech. Publ., 299,1962. (2) Brodnyan, J. G., Gaskins, F. H., Philippoff, W., T r a m . Soc. Rheology I, 109 (1957). (3) Brodnyan, J. G., Kelley, E. L., Tram. Soc. Rheology V, 205 (1961). (4) Chinai, S. N., Schneider, W. C., J.AppL PolymcrSn‘. 7, 909 (1963). (5) Claesson, S., Lohmander, U., Die Makrs. Chcm., p. 461 (1961). (6) Dumanskii, I. A., Khaiienk, L. V., Koiloidn. Zh. 22,277 (1960). (7) Eyring, H., J.Cham. Phys. 4,283 (1936). (8) Grunberg, L., Nissan, A. H., Nature, p. 241 (Aug. 25, 1945). (9) Hagerty, W. W., J.Appl. Mcch. 17,54 (1950). 10) Handbook of Chemistry and Physics, Chemical Rubber Publishing Co. 11) Merrill, E. W., Mickley, H. S., Ram, A., J.PolymcrSa‘. 62, SI09 (1762). (12) Merrill, E. W., Mickley, H. S., Ram, A,, Perkinson, G., Trans. SOC.Rhcolopy V, 237 (1961). (13) Merrill, E. W., Mickley, H. S., Ram, A,, Stockmayer, W. H., J. Polymer Sa’. A-1, 1201 (1963). (14) Merrill, E. W., Mickley, H. S., Ram, A., Stockmayer, W. H., Trans. Soc. Rheology VI, 119 (1962). (15) Metzger, A. P., Brodkey, R. S., J. Appl. Sci. 7, 339 (1963). (16) Metzner, A. B., Reed, J. C., A.Z.Ch.E. J . 1, 434 (1955). (17) Neale, S. M., Chem. Ind., p. 140 (February 1937). (18) Ostwald, W., Kolloidni. Z . 36, 99 (1925). (19) Philippoff, W., Lubrication Sci. Tcchnol., p. 68-81 (1958). (20) Philippoff, W., Hess, K., 2. Physik. Chcm. B31, 237 (1936). (21) Porter, R. S., Johnson, J. F., J. Phys. Chcm. 63, 202 (1959). (22) Rabinowitsch, B., 2. Physik. Chcm. A145 1 (1929). (23) Ram, A., Sc.D. thesis, Mass. Inst. Technol., 1961. (24) Ram, A., Tamir, J . Appl. Poly. Sci., in press. (25) Reiner, Markus, “Deformation, Strain and Flow,” H. K. Lewis and Co. Ltd., London, 1960. (26) Shaver, R. G., Merrill, E. W., A.1.Ch.E. J. 5,181 (1959). (27) Swindells, J. F., Coe, J. R.,Godfrey, T. B., J . Res. Not. Bur. Sfd.48, No. 1 (1952). (28) Tollenaar, D., Bolthof, H., IND. ENQ.CHEM.38,851 (1946) (29) Weltmann, R. N., Znd. Eng. Chcm. (Anal.) 15,424 (1943). (30) Zbid., 40, 272 (1948).
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