Chin, D. T.. Litt, M., J. Eiectrochem. SOC.,119, 1338 (1972). Frank-Kamenetskii, D. A., "Diffusion and Heat Exchange in Chemical Kinetics," 2nd ed, J. P. Appleton, Ed., Plenum Press, New York, N. Y., 1969. Heymach, G. J., Ph.D. thesis, University of Pennsylvania, Philadelphia, Pa., 1969. Johnson, D. L., Saito, H . , Polejes, J. D., Hougen, 0. A,, A.I.Ch.E. J.. 3, 411 (1957). Joksimovic-Tjapkin, S. M., Delic, D., Ind. Eng. Chern.. Fundam., 12, 33 (1973). Klug, H. P., Alexander, L. E., "X-Ray Diffraction Procedures." Wiley, New York, N. Y., 1954. Lehmkuhl, G. D., Hudson, J. L.. Chem. Eng. Sci.. 26, 1601 (1971). Levich, V. G.. "Physicochemical Hydrodynamics." Prentice-Hall, Englewood Cliffs, N. J., 1962. Litt. M., Serad, G., Chem. Eng. Sci.. 19, 867 (1964). MacDonald, D. D., Wright, G. A,, Can. J. Chem., 48, 2847 (1970). Mann. R. S.. Khulbe. K. C., Can. J. Chem., 47, 215 (1969). Mann, R. S.,Khulbe, K. C., J . Catai.. 17, 46 (1970). Mattox, D. M., Electrochem. Jechnol., 2, 295 (1964). Mattox, D. M., Surface Physics and Chemistry Division, Sandia Laboratories, Albuquerque, N. M., private communication, 1970. Olander, D. R.. lnd. Eng. Chem., Fundam.. 6, 178, 188 (1967) Riddiford, A. C.. in "Advances in Electrochemistry and Electrochemical
Engineering,"Vol. 4 , P. Delahey, Ed., Wiley, New York, N. Y., 1966. Satterfield, C. N.. Ma, Y . H., Sherwood, T. K., Inst. Chem. Eng. (London) Syrnp. Ser., No. 28, 22 (1968). Satterfield, C. N., Pelossof, A . A , , Sherwood, T. K., A . i.Ch.E J . , 15, 226 11969i. Schlichting, H., "Boundary Layer Theory." 6th ed, translated by J. Kestin, McGraw-Hill, New York, N . Y., 1968. Schuit, G. C. A , , van Reijen, L. L., Advan. Catal.. 10, 242 (1958) Serad, G., Ph.D. Thesis, University of Pennsylvania, Philadelphia, Pa., 1964. Sherwood, T. K.. Farkas, E. J . . Chem. Eng. Sci.. 21, 573 (1966). Sinfelt, J. H., in "Applied Kinetics and Chemical Reaction Engineering," American Chemical Society, Washington, D. C., 1967. Spaeth. E. E., Friedlander, S.K., Biophys. J . . 7, 827 (1967). Turkevich. J.. Kim, G., Science. 169, 873 (1970). Weller, S., A.l.Ch.E. J., 2, 59 (1956). White, D. E., Ph.D. Thesis, University of Pennsylvania, Philadelphia, Pa.. 1972.
Receioedfor reoiew October 5 , 1973 Accepted February 14, 1974 This paper was presented a t the J o i n t AIChE/CSChE M e e t i n g in Vancouver, B.C., Sept 1973.
EXPERIMENTAL TECHNIQUE
A Simplified Continuous Viscometer for Non-Newtonian Fluids John L. Scheve,' William H. Abraham,* and Earl B. Lancasjer2 Engineering Research institute and Department of Chemical Engineering, Iowa State University, Arnes, lowa 50070
A s t u d y was made to evaluate a simple pipeline- viscometer capable of continuous measurement of true viscosity of non-Newtonian fluids. The prototype model consists of two pipes of slightly different diameters connected in series, with prbvision for measurement of axial pressure gradients and flow rate. Mathematical analysis establishes a fundamental relationship among the three measured quantities and t h e corresponding shear stresses and rates of strain at t h e fluid-pipe wall interfaces. Experimental verification was obtained using glycerine, carboxymethyl cellulose solutions, and starch pastes a s tests fluids. Results agree with shear curves obtained by application of classical capillary analysis.
Introduction There is an established demand for instruments capable of making rapid, accurate viscosity measurements in the field of industrial control viscometry. Viscosity is an important control parameter in the manufacture of many materials such as polymer melts, emulsions, greases, and suspensions because their usefulness depends on unique and predictable flow behavior. It is possible to measure the true viscosities of these materials with on-line instruments, although these devices are typically quite limited in applicability or else complicated and expensive. For example, capillary viscometers are used to monitor the viscosity of Newtonian fluids on a continuous basis and are desirably simple and relatively inexpensive for such use. However, application of the capillary viscometer to online measurement of non-Newtonian viscosities is not
' E. I. DuPont de Nemours 8 Co., Inc., Victoria, Tex. 77901
USDA Northern Regional Research Laboratory, Peoria, Ill. 61604
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practical, since measurements are required at several different flow rates in order to establish the rate of change of pressure gradient with respect to change of flow rate. If the capillary viscometer were directly applicable to online measurement of true viscosity of non-Newtonian fluids, the capillary instrument might be more attractive than complex devices now used, because of simplicity and low cost. The goal of this investigation was to test experimentally a proposal for a simple pipeline viscometer capable of measuring true viscosities without the complications of multiple flow rates and data reduction which are inherent in the classical capillary device. The design concept, in effect, involves replacing the multiple flow rate operation and data reduction of the capillary approach by simultaneous measurement of two different pressure drops and a sinele flow rate. A prototwe - _ developed from this concept consists of two pipes of slightly different diameter connected in series, with provision for measurement of the
axial pressure gradients in the two pipes and the single volumetric flow rate. The three continuously measured values may be used in a direct calculation of true viscosity. Theoretical analysis shows that the instrument is applicable to arbitrary non-Newtonian viscous fluids. Theoretical Development The design concept for the proposed pipeline viscometer is based on a theoretical development quite similar to the derivation of the Rabinowitsch (1929) equation which applies to the traditional capillary viscometer. The essential difference between the two derivations is the treatment of the measurable parameters, radius, R, volumetric flow rate, Q, and pressure gradient, AP/L. The Rabinowitsch derivation is based on the interdependence of flow rate and pressure gradient in a single capillary of known radius. Conversely, the pipeline viscometer is based on the interdependence of pressure gradient and pipe radius for a single, constant flow rate. The mathematical procedure used to develop the serial-pipe viscometer principle is therefore very much the same as that used for the Rabinowitsch equation. We suppose an incompressible fluid in axial flow under steady-state laminar conditions in a long pipe of circular cross section. Under these conditions, even for a fluid which exhibits elastic as well as viscous effects, the shear stress in the fluid is related to velocity gradient as
Under the same conditions, a standard momentum balance yields the well-known relation between shear stress and pressure gradient
Likewise, an integrated form of the equation of continuity is
Q = SR2arV,dr
(3)
Equation 3 is integrated by parts to give
(4) On the usual assumption of no wall slip, the leading term on the right-hand side of eq 4 is zero. The remaining term of the right-hand side may be modified by a change of variables. Equation 2 establishes that RrrZ/rR may be substituted for r, which leads to the expression
(5) This equation is a general relationship among flow rate, shear stress, and rate of strain for the system. For the derivation of the Rabinowitsch equation, eq 5 is differentiated with respect to T R holding R constant using the Leibnitz formula (Bird, et al., 1960). After simplification, the following relationship between shear stress and rate of strain results
This expression may then be used to calculate rate of strain by graphically differentiating the curve of 4Q/nR3 us.
TR.
For the serial pipe concept being developed, eq 5 is differentiated with respect to R holding Q constant to obtain
This expression may be readily simplified to an expression for the rate of strain at the wall 3Q
d In R
The final derivative term of eq 8 may be evaluated directly from experimental measurements of axial pressure gradients in two consecutive pipes of slightly different radius, by approximating the derivative with a difference expression to obtain
where the term R refers to either pipe radius and the subscripts 1 and 2 identify the two different pipes. For a power law fluid, both eq 8 and 9 reduce to (%)R
= -%[1+&]
(10)
where N is the exponent of the constitutive expression for a power law fluid
Thus for a power law fluid (including Newtonian fluids as a special case) the working eq 9 for rate of strain is exact. For other non-Newtonian fluids, eq 9 is approximate. In the limit of small differences between pipe radii, the approximate expression for rate of strain approaches the exact value given by eq 8. The viscometer concept proposed in this study depends upon the use of eq 9 to calculate rate of strain at the interface between fluid and pipe wall. The corresponding shear stress is calculated in the usual way, using eq 2 at the radius R of the interface. Since the experimental variables of flow rate and pressure gradients are continuously measured, the proposed viscometer is capable of providing a continuous indication of true viscosity. Experimental Procedure The test materials used in this study were glycerine, Dow Chemical Co., sodium carboxymethylcellulose (CMC), “Carbose” D, Wyandotte Chemical Co., and Pearl cornstarch, Penick and Ford. The glycerine was used as received, and the CMC was dispersed into solution using accepted techniques. Starch pastes were prepared from the cornstarch by cooking a uniform starch slurry a t 90 f 2°C for 1 hr. All of the materials were tested a t room temperature of approximately 25°C. The prototype viscometer is basically a simple instrument; its fundamental components are shown schematically in Figure 1. The jacketed reservoir has provisions for being heated with low-pressure steam or cooled with water. The Moyno pump is a positive displacement screwtype pump. Volumetric flow rates of fluid from the pump were determined to be proportional to pump speed, with a slight correction for changes in discharge pressure. Four interchangeable two-pipe test sections with static pressure taps were provided. These test sections were fabricated from Yz-in., 3/4-in., 1-in., and ll/z-in. standard steel pipe. The first segment was a schedule 40 pipe and the second was a schedule 80 pipe of the same nominal diameter. This design gave a difference of 5-16’70between the inside Ind. Eng. Chern., Fundam., Vol. 13, No. 2, 1974
151
STATIC TAPS
(NOMINAL D I M E T E R )
PUNA
M O Y N O PUMP
n 4
Figure 1. Schematic representation of the prototype pipeline viscometer
radii of the two pipes. Also, a distribution manifold and return section fabricated from lyz-in. pipe were provided. In addition to these components, two Statham differential pressure transducers and a Honeywell millivolt recorder were used to measure and record pressure drops in the test sections. With these components, all of the information required to calculate the rate of strain from eq 9 is readily available. That is, the pressure drops produced by a fluid flowing through two calibrated pipes in series and the constant volumetric flow rate are known quantities. A number of precautions were taken to assure that pressure drops did not involve any entrance effects. An entry length of 3-5 f t was provided prior to the first of four pressure taps in each half of the two-pipe test sections, and a calming length of at least 2 f t was provided after the fourth tap. This design was conservatively based on previous study of non-Newtonian entry effects (Dodge and Metzner, 1959) and was intended to ensure fully developed velocity profiles in the test sections. Good agreement between measurements with the four taps indicated no entrance effects were present. In addition, a separate study (Scheve, 1971) of entrance effects was made with a 5% starch paste test fluid and additional pressure taps. This study indicated that end effects were not detectable beyond the initial five pipe diameters. Results and Discussion Newtonian Fluid. The measurements with glycerine were used to check whether the experimental viscometer worked properly in the simple case of a Newtonian fluid. As expected, the ratio of pressure drops in successive pipe sections was experimentally independent of flow rate, temperature, or even contamination of the glycerine with water. The measured viscosity was within 2.5% of literature values for a water-glycerine mixture of 99.5% purity, which was the approximate composition of the glycerine as received. Small amounts of water contamination did lower the viscosity markedly in later runs, but the experimental viscosity could be calculated by either the usual Hagen-Poiseuille equation or by the two-pipe viscometer formula with an agreement within 1% between the two calculation procedures; exceptions resulted from the nominal l1/z-in. diameter test sections, which consistently yielded results different from those for test sections of K-in., 3/4-in., and 1-in. nominal diameter. By repeating the calculations with the radius of the schedule 80 ll,&in. pipe changed from 0.751 to 0.757 in. the viscosity values became internally consistent among the four sizes of test sections. A check of the schedule 80 ll/z-in. pipe indicated a possible reason for the problem. The pipe was slightly elliptical with a variation of 0.01 in. in the outside diameter measured with a micrometer. Based on the glycerine measurements, the value of 0.757-in. inside radius was used in all calculations for the pipe. The pipe radii in all other cases were based on measurement of outside diameter and calculation of average wall 152
Ind. Eng. C h e m . , F u n d a m . , Vol. 1.3,No. 2, 1974
RUN B 0 TWO-PIPE 0
CAPILLARY
dV
RATE OF STRAIW,
2
YF-'
Figure 2. Comparison of two-pipe and capillary results for a 4% CMC solution
thickness. The average wall thickness was determined by measuring the pipe steel density and weighing a measured length of pipe. An alternative procedure of determining inside pipe radii was attempted, which was a water displacement method. Results were more scattered and were not used. It should be noted that the two-pipe calculations are rather sensitive to small errors in the pipe radii, as consideration of the logarithm terms of eq 9 will make clear. The glycerine results show that the two-pipe viscometer behaves according to theoretical predictions in the simple case of a Newtonian fluid. The results were also used to correct the measured radius of the schedule 80 lyz-in. pipe from 0.751 to 0.757 in. It is suggested that the simplest procedure for determining the effective geometry of a twopipe viscometer would be by calibration with a Newtonian fluid of known viscosity. Power Law Fluid. Sodium carboxymethylcellulose (CMC) was chosen for this investigation because of its known pseudoplastic behavior; the power law model is applicable to CMC solutions over the shear stress range investigated. As stated earlier, the two-pipe viscometer principle is exact for this case. To check the two-pipe viscometer results, rates of strain were calculated for the individual test section segments using the Rabinowitsch capillary equation (6). The derivative term of eq 6 was evaluated graphically. The results of the comparison of two-pipe and capillary techniques are illustrated in Figure 2. These results for a 4% CMC solution are representative of similar runs made with 2% and 6% CMC solution. The results produced by the two different procedures show very good agreement with one another. We suspect that the experimental scatter observed in some runs at higher rates of strain is mostly attributable to errors of pressure measurement. Temperature variations of the order of 1°C were probably less important. The technique used to determine line pressures in the CMC series of runs involved balancing a diaphragm at an interface between CMC and water (Scheve, 1971). Failure to satisfactorily balance the diaphragm could have resulted in occasional pressure errors of considerable magnitude. Overall, the CMC results support the contention that the two-pipe and capillary procedures are theoretically equivalent for a power law fluid. Starch Pastes. The starch paste runs were made to evaluate performance for a non-Newtonian, nonpower law fluid. The two-pipe principle is not as exact in this case as it was in the power law fluid case. Thus, the starch paste results would indicate whether the two-pipe concept could be applied successfully for such a nonexact case.
1603
1400 RUk A
SUN B
n
0
TWO-PIPI CAPILLARY
- 0 0
300
-- -
0
L--
I 400
500
700
600
800
a"
RATE Or STRAIh
d2
ec-
Figure 3. Comparison of two-pipe and capillary results for a 2% starch paste
RUNA 4
RUN B 0 TWO-PIP[
A
:[
CAPILLARY
IO0
0
,
,
50
100
,
,
150
203
RATE OF STRAIN.
2
,
I
200
,
,
250
3M
350
(ec-'
Figure 5. Comparison of two-pipe and capillary results for a 7% starch paste RUNA
RLN B
D
C
A
TWO-PIPE CAPILLARY
100-
1
-
I
0 O
Irn
?(a
3w
5w
433
m
7M
am
dV
- >
U ' E OF 5TRAIh.
f
xc-'
Figure
4. Comparison of two-pipe and capillary results for a 5% starch paste
Figures 3, 4, and 5 show the basic shear curves for starch pastes containing 2, 5 , and 7% starch by weight. The results for the starch pastes are comparable to the CMC measurements, except that the 7% starch paste results are more scattered. The principal cause of the scatter is believed to be experimental errors of measuring differential pressure. Sensitivity of the two-pipe viscometer to errors of pressure measurement could be greatly reduced by redesign. The scatter of the 7% starch paste results is not caused by the approximation introduced into working eq 9 for the two-pipe viscometer. To establish this assertion, calculations were carried out to predict the performance of the two-pipe viscometer in the absence of experimental errors of pressure measurement. The constitutive equation of the 7% starch paste is adequately described by the HerschellBulkley law (Brodkey, 1967)
Coefficients in this case are T O = 400 g/cm sec2, K = 49.12, and N = 0.59. The velocity gradient is expressed in reciprocal seconds. Equation 12 was combined with the relations developed earlier to obtain a computer solution for the expected wall stress and rate of strain for such a fluid at a known flow rate in a pipe of known radius. Values so obtained will be referred to as exact values. In turn, eq 9 was used to calculate rate of strain values which one would expect to obtain with a two-pipe viscometer. Such rate of strain values will be referred to as approximate. The pipe radii and volumetric flow rates used correspond to actual experimental
values for the 7% starch paste. A comparison of exact and approximate rate of strain values provides a direct measure of the error caused by the approximation used in developing the working eq 9 for rate of strain. The ratio of approximate and exact rates of strain is ideally 1.0. The departure from 1.0 is a measure of the approximation introduced by the working eq 9 for rate of strain measured in a two-pipe viscometer. The ratio was calculated for the larger and smaller of the two pipes of each of four two-pipe test sections at five different flow rates of 7% starch paste. The 20 values for larger pipes show that the approximate rate of strain is 0.98-0.99 of the exact value, and the 20 values for smaller pipes show that the approximate rate of strain is 1.01-1.02 times the exact value. Thus, the fundamental error associated with a derivative approximation (eq 9) is of the order of 1-2'70 for the prototype apparatus. For control purposes the theoretical errors of a two-pipe viscometer would probably be considered negligible, particularly after suitable calibration. The practical limits of the device would be set by the need for close temperature control and accurate pressure measurements. The scatter of the 7% starch paste results in the present study is believed caused by errors of pressure measurement, rather than any theoretical limitation of the twopipe viscometer. A different technique was used for differential pressure measurements with starch pastes than with CMC solutions and this technique may have been less accurate. Instead of a balanced diaphragm interface, a direct water-starch paste interface was maintained in each pressure tap near the pipe wall. This was accomplished by bleeding approximately 2 cm3 of water/hr into the pressure tap. The reason for resorting to a different technique with starch pastes was to overcome the tendency of these pastes to gel in the pressure taps (Scheve, 1971). The specific reason for the larger experimental scatter of results with the 7% starch paste as compared to 2% or 5% pastes is believed to involve increased sensitivity of the two-pipe viscometer calculations to pressure measurement errors for the case of the 7% paste. A critical calcuInd. Eng. Chem., Fundam., Vol. 13, NO. 2,1974 153
lation is the rate of strain defined by eq 9 rewritten in somewhat different form below
L
The most demanding case is a highly pseudoplastic fluid with a small difference of pipe radii in the two parts of the test section. Thus, for the series of runs with 7% starch paste in nominal 1-in. pipe, the ratio of pipe radii is 1.05/1 and the expected ratio of pressure gradients approximately 1/1.10. Under these conditions, a 1% error of measurement of each differential pressure could cause a 20% error in the calculated rate of strain. The problem is much less severe for dilatant fluids and could be considerably reduced even for the 7% starch pastes by redesign of the viscometer. If the ratio of radii were changed from 1.05 to 1.15, for example, the effect of pressure measurement error would be reduced by a factor of 3 in the case cited. Overall the starch paste results support the contention that two-pipe and capillary procedures are closely equivalent for any arbitrary non-Newtonian fluid. Conclusions The results of this investigation have shown that a workable device for continuous measurement of true viscosity of non-Newtonian fluids can be developed from the two-pipe concept. Furthermore, such a viscometer can duplicate capillary results quite closely. Accuracy limits depend primarily on errors of differential pressure measurement and selection of optimum geometry. In addition, the two-pipe concept could be adapted to a process monitoring or process control instrument. Such an instrument would have the simplicity of a capillary viscometer together with the capability of continuous determination of true viscosity of a non-Newtonian fluid.
Acknowledgments This document is a report of work done under a contract from the U. s. Department of Agriculture and authorized by the Research and Marketing Act of 1946. The grant was supervised by the Northern Regional Research Laboratory and the Engineering Research Institute, Iowa State University, Ames, Iowa. A fellowship was provided John L. Scheve by the Procter and Gamble Company. Nomenclature
K = power law constant L = length,cm N = power law exponent P = pressure A P = differential pressure, g/cm sec2 Q = flowrate, cm3/sec R = radius, cm r = radial distance, cm V, = velocity (axial), cm/sec Greek Letters shear stress at pipe wall, g/cm secZ TR 7 r z = shear stress, g/cm sec2 T O = yield shear, g/cm sec2 Subscripts 1 identifies smaller of two pipes 2 identifies larger of two pipes Literature Cited Bird, R . B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena," p 732, Wiley. New York, N . Y., 1960. Brodkey, R. S., "The Phenomena of Fluid Motions," p 389, Addison-Wesley, Reading, Mass., 1967. Dodge, D. W., Metzner, A. B.,A.I.Ch.E. J . 5, 189 (1959). Rabinowitsch, B., Z. Phys. Chem. A145, 1 (1929). [Cited in Bird, R. B., Stewart, W . E., Lightfoot, E. N.. "Transport Phenomena," p 67, Wiiey, New York, N. Y., 1960.1 Scheve, J. L., M.S. Thesis, Iowa State University, Ames, Iowa. 1971,
Received for review M a r c h 15, 1973 Accepted October 23, 1973 Presented a t t h e AIChE M e e t i n g , Dallas, Texas, F e b 23, 1972.
COMMUNICATIONS
Use of Infinite-Dilution Activity Coefficients for Predicting Azeotrope Formation at Constant Temperature and Partial Miscibility in Binary Liquid Mixtures T h e conditions for azeotrope formation at constant temperature are directly determined in t e r m s of t h e infinite dilution activity coefficients, whereas those for partial miscibility are expressed in terms of t h e constants of t h e van Laar equation, which can b e evaluated from infinite-dilution activity coefficients. T h e relations found are tested on experimental data.
Jaques and Lee (1966) have proposed a method for predicting azeotrope formation a t constant temperature and partial miscibility in binary liquid mixtures. Their method is limited by the necessity of having experimental liquid-vapor equilibrium data in order to calculate the con154
Ind. Eng. Chem., Fundarn., Vol. 13, No. 2, 1974
stants of the Redlich-Kister empirical equation of third order. It is our purpose here to determine the conditions in terms of infinite-dilution activity coefficients, since a number of empirical or semiempirical techniques have
