T h e viscosity level of the oil is also important. At equivalent shear stresses, low-viscosity oils will run a t higher velocities than high-viscosity oils, and the residence time in the capillary is shorter. There is, therefore, less time for heat transfer. However low-viscosity oils have flatter viscosity-temperature curves, so the heat does not have so great a n effect. T h e over-all effect of these factors is that they also tend to counteract each other. As a broad generalization, then, all oils will tend to behave similarly-that is, they will become limited by heating a t roughly the same shear stress. Computer solutions using oils of different viscosities and viscosity-pressure-temperature characteristics show that the limiting shear stress does not vary by much more than a factor of 3. Generally low-viscosity oils appear to show somew hat greater apparent viscosity loss. Difficulty in Obtaining the Isothermal Wall. Our own difficulties in obtaining a truly isothermal wall have been described. A check of two other cases shows that other investigators have also been unable to achieve this condition. Fenske et al. (2) reported data on API oil 101, a paraffinic mineral oil having a viscosity of 54 cp./77' F. (155 S.S.U.! 100' F.; 103 V.I.). They also used a capillary 10 cm. long but with a smaller radius, 0.017 cm. The viscosity-temperature and viscosity-pressure characteristics are known, inasmuch as this oil was one of those measured in the ASME PressureVicosity Report ( 7 ) . Using these data, computer cases were run for both the adiabatic and isothermal-wall cases. Figure 9 shows that the experimental points fall midway between these two cases. This fact is not pertinent to the authors' interpretation of their data, for they were aware that heating was occurring and they made suitable corrections for it. But the measurements were run in a constant temperature bath, and the capillary walls were clearly not a t this temperature. A similar examination was made of data obtained by Stringer (9). These data werle capillary measurements of a nonwaxy mineral oil (REO-153) a t 5' F. (-15' C.). At that temperature the oil had a viscosity of 24.5 poises, and very steep viscosity temperature and viscosity-pressure slopes. T h e viscositytemperature coefficient was determined from experimental data; the viscosity-pressure coefficient was estimated from correlations with other oils anq from extrapolation of similar oils
given in the ASME Pressure-Viscosity Report. T h e capillary was 8.0 cm. in length, 0.034 cm. in radius. Figure 10 compares the experimental points and the computed curves for the two cases. This capillary ran closer to the adiabatic case than to the isothermal-wall case. I t can therefore be concluded that it is very difficult to obtain a truly isothermal wall in a n experimental apparatus. Semiadiabatic conditions seem to be much more prevalent. Conclusions
Significant heating occurs in a capillary even if the wall is isothermal. Compared to the adiabatic case, the isothermal wall allows a t best a fivefold increase in shear stress before heating becomes limiting. Oil viscosity and viscosity-temperaturepressure characteristics are relatively unimportant, but short thin capillaries will greatly reduce heating effects. I t appears to be very difficult to obtain a n isothermal wall experimentally. Nomenclature
L
= = Q = Y = R = T = pa = T =
P
capillary length, cm. pressure drop, dynes/sq. cm. volume flow rate, cc./sec. radial distance, cm. capillary radius, cm. temperature, ' F. apparent viscosity = nPR4/8 QL, poises shear stress a t capillary wall = PR/2L, dynes/sq. cm.
literature Cited (1) Am. SOC. Mech Engrs., ASME Pressure-Viscosity Report,
1953.
( 2 ) Fenske, M. R., Klaus, E. E., Dannenbrink, R. W., ASTM, Philadelphia, Pa., ASTM Spec. Tech. Publ. 111 (1951). (3) Cerrard, J. E., Steidler, F. E., Appeldoorn, J. K., IND. ENG.
CHEM.FUNDAMENTALS 4, 332 (1965). (4) Gruntfest, I. J., Trans. Sac. Rheol. 7, 195 (1963). (5) Hersey, M . D., Physzcs 7, 403 (1936). (6) Hersey, M. D., Zimmer, J. C., J . Appl. Phyr. 8, 359 (1937). ( 7 ) Horowitz, H. H., Znd. Eng. Chem. 50, 1089 (1958). ( 8 ) Kearsley, E. A., Trans. Sac. Rheol. 6,253 (1962). (9) Stringer, J. R., Rohm & Haas Co., unpublished results. RECEIVED for review June 1, 1965 ACCEPTED November 12, 1965 Division of Petroleum Chemistry, 150th Meeting, ACS, Atlantic City, N. J., September 1965.
HELICAL FLOW OF A NON-NEWTONIAN PO LY I SO BUT Y LEN E SO L UT IO N ALBERT C. DIERCKES, J R . , ' A N D
W . R . SCHOWALTER
Department of Chemical Engineering, Princeton University, Princeton, N . J .
the past decade theoretical work of No11 and others (4, 72, 74) has provided a means for comparing results obtained with non-Newtonian fluids in different types of viscometers. It has also clearly indicated pitfalls that might exist when one attempts to utilize results obtained in one geometry and flow condition for predicting behavior of a given fluid in a different geometry or under different flow conditions. Of particular importance are predictions which apply to a n incompressible simple fluid, defined as one for which the stress DURING
Present address, Procter and Gamble Co., Cincinnati, Ohio.
on a n element of fluid is determined by the history of the rate of deformation of the fluid element. One of the chief conclusions arising from simple fluid theory is the fact that results obtained from one type of flow can in certain special cases be used to predict flows in certain other geometries. I n particular, rheological measurements in a capillary tube should be sufficient to predict velocity profiles in the (concentric) helical flow system shown schematically in Figure 1. Helical flow is generated by rotating one or both of two concentric cylinders while simultaneously imposing a n axial pressure gradient upon the fluid in the annular space. VOL 5
NO. 2
MAY 1944
263
Measurements of pressure drop as a function of flow rate have been made for helical flow of a solution of 3% (by weight) polyisobutylene (Eniay Vistanex L-100) dissolved in Decalin. A helical flow pattern was generated by pumping the solution through an annular space formed with two concentric cylinders while rotating the outer cylinder. Shear rate in the annular space ranged from 27 to 21 0 set.-' Experimental results were compared with predictions based upon rheological measurements performed on the fluid in a tubular viscometer and upon the theory of simple fluids. Close agreement between measurements and computations was obtained, indicating consistency with the postulate that the polyisobutylene solution acted as a simple fluid.
to the velocity field exhibited by the fluid. Suppose that the position of a fluid element a t time r is given by coordinates t ( ~=) [E'(.), E2(r),E3(r)]with respect to some orthonormal system of base vectors. We denote the contravariant coordinates [ ' ( r ) a t some reference time t by E i ( t ) = x i . Then, for a viscometric flow, one must be able to write
I n this and subsequent equations the summation convention is applied to repeated indices unless a statement to the contrary is made. I t is convenient to take the present time, t , as reference time. Then s = t - 7, and positives represents time measured into the past. The g k l are components of the metric tensor for time 7. Thus the square of the element of length
A ( i j f j and
A(zjtj are, respectively, components of the first and second Rivlin-Ericksen tensors (75) written with respect to a n orthonormal system of base vectors. T h e Rivlin-Ericksen tensors in steady flow are defined by
Figure 1.
Representation of helical flow
One naturally wonders whether any actual fluids exist which behave in accord with the predictions for simple fluids. T h e work presented here is a n attempt to establish a partial answer to that question. Flow rate-pressure drop data obtained for helical flow of a 3% solution of polyisobutylene dissolved in Decalin are found to be consistent with predictions of simple fluid theory. Previous Work
Theoretical. Theory pertinent to the work described in this paper rests on the concept of a simple fluid undergoing a viscometric flow. These terms have been amply discussed in recent rheology literature (2-4) ; consequently, the presentation here treats only some of the essential ideas. [The reader wishing to dispense with a mathematical description of viscometric flows can immediately proceed to Equation 5, after accepting viscometric flows as the class of flows existing in most viscometers. ] A viscometric flow is a steady flow characterized by the form of a n equation relating the strain a t any point in the fluid 264
l&EC FUNDAMENTALS
and a comma indicates covariant differentiation. Furthermore, if the flow is viscometric, the orthogonal coordinate system in which Equation 1 is written must be one for which the Rivlin-Ericksen tensors can be written in the form
where K is a scalar which may vary with position. I t is often called the shear rate. Flows which exist in the usual types of viscometers such as capillary tubes or concentric cylinders obey the requirements for a viscometric flow. The rheological behavior of a n incompressible simple fluid in a viscometric flow is completely described by three material functions (3): tlz
=
tzz
- t33 - t33
t13
=
til
(5)
T(K)
t23
=
Cl(K)
(6)
=
Uz(K)
(7)
0
(8)
T h e variable K is obtained from Equations 3 and 4. T h e ti, are components of the (symmetric) extra stress tensor, T.
T is related to total stress, S, by S=T-lp
(9)
We are concerned only with the case where the inner cylinder is stationary and the outer cylinder rotates with angular velocity 0,. Boundary conditions for the flow are
where trS = -3p and trT = 0. A viscometric flow of a n incompressible simple fluid is then described by Equations 1 thraugh 9 along with the momentum equation
w(R,) = O w ( R 0 ) = 00
u(R,)= u(R,) = 0 Dv Dt
=
pg
4- V . S
= pg
- Vp
+ V*T
(10)
I-
(18)
For helical flow of a simple fluid Coleman and No11 (5) have shown that the nonzero components of T are
and the continuity equation V.V
= 0
(1 1)
TUBEFLOW. I n a horizontal capillary tube one readily
7( K )
tTZ = u’
finds
ti
1
- --ar+2
b
r
iI
where
c ( r ) = axial component of velocity.
Letter subscripts on t ,which refer to a specific coordinate system, such as t,,, imply that physical components are intended. Assuming that the function ti) is invertible, one can write for a tube of radius R
tZL
=
-
1 3
Ul(ti)
+ 2(u’)2 3-
ti2
+
(rw,)2
(19)
~
UZ(4
1
where ti = [ ( u t ) ’ (rul)Z]l’Z A significant result is that the velocity profile is not dependent upon the normal stress functions, u1 and g ~ :
T h e volumetric flow rate, Q , is found from
Q =
SR
2~rv(r)dr
0
Integrating by parts
T h e equations given above are in a convenient form for computing U ( Y ) and Q when the function T ( K ) is known. However, for present purposes one wishes to determine T ( K ) from measurements of pressure drop and flow rate. This can be done by differentiating Equation 15 Ivith respect to a to obtain a form of the Rabinowitsch equation (4, 7 7 ) . T h e result is
I n these equations several new quantities have been introduced which have sufficient physical meaning to warrant further explanation : M is the torque per unit height exerted on a cylindrical surface of fluid. It is not a function of Y , indicating that M is the torque transmitted to the walls of the apparatus. Constant a is the driving force per unit height and unit cross section which is exerted in the axial direction (23)
+
where 7 [ J indicates the functional dependence of T . Thus from experimental measurement of Q as a function of a, one may determine ti). HELICAL FLOW. T h e type of helical flow described above is also a viscometric flow ( 2 ) . However, the coordinate system in which Equations 1 and 4 are satisfied is not convenient, since it changes orientation with position a t a given flow rate, and also when flow rate is varied. Consequently, we describe the flow in a cylindrical system Y , 0, z ( z being taken as positive in a n upward vertical (direction), and postulate
where P = p pgz Constant b is related to the radius a t which u’ = 0. It can be found by evaluating the integral of Equation 20 between Rt and Ro.
can be found by combining Equations 19A and 19B to 7 (ti)
form
VOL. 5
NO. 2 M A Y 1 9 6 6
265
A number of other approaches to solution of the equation of motion for helical flow exist (8, 74, 77). However, the development sketched above is most pertinent to the present work. Experimental. Tanner (76, 18) has performed the only previous helical flow experiments known to the authors. Actually, his apparatus contained a small annular gap to provide a special case of helical flow-a combination of channd flow and simple rectilinear shear flow-to conform to the simplifications made in Tanner’s theoretical analysis. His experiments were performed with various solutions of poly(methy1 methacrylate) and polyisobutylene. Results were compared with predictions made using a constitutive equation of the Oldroyd type (73). Tanner found agreement with the Oldroyd model when shear rate in the apparatus varied over only a narrow range, and when the viscosity ratio of the fluid (apparent viscosity a t zero shear rate/apparent viscosity a t very large shear rate) was Less than 6 . Flow Equations for Present Work
Since the chief purpose of this work is to compare actual
Figure 2.
dP
- --
bz
Helical flow apparatus
+ 1 a-d, (”&.) ~
=0
I
results in
[
(v’)2
d’
+
+ n(rw‘)2 + f y n - 1)w’w” &’)* -t (m’)’
_._.. _.-
Combin; working equations (271
1;
0’
-t
~ --” a~d 32 results in the final pair olF
+ 1 ) ( ~ ’ )+~ (n + 2) (rw’)z
-
nr
and
These equations are to be solved in accord with the boundary conditions given in Equation 18. Volumetric flow rate is computed from Equation 22.
CALMING CHAMBER
TOP HEADER
TOP OUTER CYLINDER
l - l
GROOVE FOR TOP OUTER CYLINDER EXTENDER
-
DlSTRl BUTER
-SOCKET
OUTER CYLINDER
BOTTOM OUTER CY L I NDER
1
7 L:O :::
OUTER
CYLINDER BOTTOM
EXTENDER
Figure
Figure 3. Exploded schematic diagram of helical flow test section
Figure 4. Cutaway drawing of inner cylinder showing one pressure talp
BY-PAS S " 2
VALVES FOR
Q
MEASUREMENT
5.
Schematic drawing of flow system
Description of Fluid and Apparatus
A detailed description is provided by Dierckes ( 8 ) . Fluid, The fluid chosen for this study was a 3% (by weight) solution of polyisobutylene (PIB) dissolved in Decalin. The PIB was \'istanex L-100 provided by the Enjay Co. ; Decalin, a practical grade mixture of cis and trans isomers, was obtained from the hlatheson Co. Several factors prompted this choice of fluid. It is relatively inexpensive and easy to handle, exhibits appreciable non-Xewtonian behavior without excessive zero shear viscosity, is transparent (an important requirement for subsequent work with tracer particles), is stable over long periods of time, and has frequently been used for careful rheological studies. Tube Viscometer. T h e viscometer consisted of a 25-quart pressure sterilizer acting as a reservoir, to which were fitted t\vo brass tubes with inside diameters of 0.390 + 0.002 and 0.152 i 0.002 inch. Two tubes were selected to provide a large range of shear rate without excessively large or small flow rates or pressure drops. Fluid was driven through either tube by applying nitrogen pressure to the reservoir. During a run the pressure in the reservoir was maintained essentially constant. Flow rates were controlled by the magnitude of nitrogen pressure in the reservoir and by globe valves located a t the downstream end of each tube. Pressure drop in the tubes was measured between two taps located 50 diameters apart on each tube. The upstream tap on the large tube was located 23 diameters downstream from the reservoir and 8 diameters downstream from a n on-off valve. For the small tube, the first pressure tap \vas located 46 diameters from the reservoir and 20 diameters from an onoff valve. Each pressure tap consisted of a hollow annular ring brazed to the tube. This annular ring provided a chamber of fluid around the tube and was connected to the inside of the tube by three '/s?-inch diameter holes spaced on the circumference of the tube. ,4port from the annular ring was connected to one leg of a manometer. Reservoir and tubes were immersed in a water bath which was controlled to maintain constant temperature to within = t 0 . l o C. The tubes were supported with carefully placed spacers which ensured a horizontal tube axis. Pressure drop was measured with a three-fluid manometer. Working fluid in the manometer was one of two Meriam manometer fluids (density = l .75 or 2.95 grams per cc.) or mercury. Since the Meriam oils were miscible with the fluid being tested, a layer of water was used to prevent mixing. Height of fluid in the manometer legs could be read to =t1 mm. Volumetric flow rate was measured by collecting tube efflux over a known time, weighing the sample, and determining its density with a pycnometer. VOL. 5
NO.
2
MAY 1966
267
Helical Flow System. A close-up view of the rotating outer cylinder and some auxiliary equipment is shown in Figure 2. An exploded schematic diagram of the cylinders and adjacent equipment is presented in Figure 3.
30-
TUBE 0 0
a
ISNER CYLISDER.T h e inner cylinder was made from a straight piece of commercial brass tubing, 3-inch O . D . , '/g-inch wall, and 18 inches long. The cylinder \vas turned on a lathe to a n outside diameter of 2.980 =t 0.001 inches. For about 11/8 inches on each end the cylinder was reduced slightly in diameter to match sockets in the headers into which the cylinder was placed. Eleven '/2-inch holes were drilled through the cylinder wall, spaced every l l / y inches along the length and equally spaced around the circumference. These holes, which served as pressure taps, formed a helix on the cylinder. Pressure was transmitted from the holes to the manometer through a series of brass tees which were brazed to the inner cylinder before its outer surface was machined smooth. One of these pressure transmission tubes is sketched in Figure 4. OUTER CYLINDER AKD EXTENDERS. The outer cylinder was fabricated from commercial brass tubing with '/*-inch wall. A piece was chosen which had an inside diameter of 3.987 0.002 - 0.003 inches. The tube was carefully flanged to an over-all length of 18 inches and bolted to the extenders shown in Figure 3. Each extender was 4.000 inches in I.D. and 3'/4 inches long. The apparatus was sealed with Teflon 0rings. The bottom extender contained a 6-inch brass gear (not shown on Figure 3) with which torque was transmitted for rotation of the outer cylinder, Each extender was carefully aligned to the outer cylinder with a Lucite jig which ensured concentricity to within 10.001 inch. HEADERS.These pieces, shown in Figure 3, served as entry and exit ports for the fluid, and also provided a bearing surface for the rotating outer cylinder. The bottom header contained an SKF ball thrust bearing which supported the outer cylinder. Assembly of inner and outer cylinders to the headers was accomplished with an eccentricity of the two cylinders of less than 0.005 inch. Headers were mounted in a holding assembly shown in Figure 2, which was designed to provide vertical alignment of the cylinders. The calming sections of the headers were joined to the fluid piping system by distributors, one of which is shown a t the upper right of Figure 2. The plastic multitube connection to headers reduced transmission of pump vibration to the test section and helped to distribute fluid evenly at the entry and exit of the annular space. Thermocouples in top and bottom headers permitted measurement of fluid temperature. MOTORA N D TRASSMISSION. The outer cylinder was driven by a \Vestinghouse 1-hp. induction motor. Torque from the motor was transmitted through a Vickers variable speed reversible hydraulic transmission. Output from the transmission was delivered to the gear on the lower extender through a series of gears and a flexible coupling, which was included to isolate the flow system from vibration of the motor or transmission. The outer cylinder could be rotated a t a speed in excess of 200 r.p.m. FLUIDCIRCULATION. A schematic drawing of the circulation system is presented in Figure 5. A Moyno pump was used to pump fluid through the system. The pump was driven at a speed which provided a flow rate of approximately 20 gallons per minute. The drum shown in Figure 5 had a capacity of 30 gallons. Piping was 2-inch copper tubing soldered to cast brass fittings. Of special importance are the "valves for Q measurement" in Figure 5 . These were quick-opening ball valves coupled together to act simultaneously, enabling diversion of fluid into a weigh tank. The common valve handle activated microswitches, and these switches automatically started and stopped a timer which recorded collection time of a sample in the weigh tank. Though not shown in Figure 5, test fluid passed through a concentric pipe heat exchanger within which cooling water could be circulated. Operation of the heat exchanger was not required, however, and all runs were performed a t 25' i 2" C. Measurements. Rotational speed was measured with a
+
268
I&EC FUNDAMENTALS
E
2
LARGE LARGE LARGE SMALL SMALL
MANOMETER FLUID MERIAM ( B L U E 1 F L U I D MERIAM *3 FLUID MERCURY MERIAM #3 FLUID MERCURY
oc
;I / 0.5
LOG
7,
(DYNESICM')
Figure 6. Viscometer data for 3% Decalin at 25" C.
polyisobutylene in
General Radio Co. Strobotac. In addition rough settings were made with the aid of a tachometer mounted on the apparatus. Pressure drop was measured with the manometer described earlier. By means of a manifold system pressure difference was readily measured between the tap nearest the entry of the helical flow system and any other pressure tap along the inner cylinder. The method for measuring flow rate has been described. Experimental Procedure
Data were taken with fluid circulating a t a rate of about 20 gallons per minute and the outer cylinder rotating a t 100 r.p.m. Manometer response time was approximately 15 minutes. Readings were not recorded until they remained constant for a t least 2 minutes. Volumetric flow rates were measured by collecting five samples over periods ranging from 5 to 25 seconds. Since the pump was of the positive displacement type, initial experiments verified that simultaneous pressure drop and volumetric flow rate measurements were not necessary. Results
T u b e Viscometer. a logarithmic plot of
These data are presented in Figure 6 as
(
D D
=
2 3 )
against drag a t the wall
[ i R= - t r r ( r = R ) ]for 3% PIB dissolved in Decalin a t 25" C. The straight line which the data approach a t low shear rate represents Newtonian behavior. Over the range 120 < T~ < 630 dynes per sq. cm. the data are represented extremely well by a power law (Equation 26). Power law constants, extracted from the data with the aid of Equation 27 are
K
=
9.4 dyne sec.O.ll/sq. cm.
n
=
0.77
An expanded portion of Figure 6, showing the degree of a least squares fit to the power law, is presented in Figure 7. Because of its possible utility to others, the curve of Figure G has been replotted in Figure 8 to illustrate the dependence of
2.4
t
i
''I 1.4
20
22
26
2.4
20
LOG TR (DYNES / C M 2 )
,
Figure 7. Viscometer data for 3y0polyisobutylene in Decalin at 25" C. 1.4 < log D < 2.35
I
I
0.5
1.0
I
I
I
2.5
3.0
3.5
I
1.5 2.0 LOG y ( S E C - ' )
Figure 8. Shear stress as a function of shear rate for in Decalin a t 25' C.
3% polyisobutylene
shear stress on shear rate. T h e power law constants given above are valid over the range of shear rates between 27 and 208 set.-'
Helical Flow. EXPERIMENTS. Results reported here were obtained with the outer cylinder rotating at 100 r.p.m. and a driving force - aP/az := 819 dynes/sq. cm./cm. In Figure 9 are shown the differences hl, in manometer fluid levels between the leg connected to the lowest or reference tap (tap 1) and that connected to tap j. T h e taps were spaced a t axial intervals of 3.81 cm. \701umetric flow rate was measured by determining the slope of a least squares fit of volume of fluid collected over a certain time. Data from five samples are presented in Figure 10. T h e slight displacement of the line from the origin is associated lvith the short time required to open and close the valves. The slope of the line indicates a flow rate Q = 20.9 gallons per minute. C o h w v T A T I o N s . A test of the extent to which the PIB followed predictions of simple fluid theory is afforded by comparison of results given above with those predicted on the basis of data collectecl from the tube viscometer. LTsing
aP
= 819 dynes/sq. cm./cm. and the values of K and n
a2
found from the tube viscometer, computation of Q from Equations 22, 33, and 34 yielded Q = 20.7 gallons per minute. The comparison is summarized in Table I.
T A P NUMBER ( j l Figure
9.
Pressure drop measurements in helical
flow for 370 polyisobutylene in Decalin a t 25" C. Table 1.
Comparison
Ri R,
- 9'az fl,
Fluid
Volumetric flow rate, gal./min.
of
Experimental and Theoretical Results
3.785 cm. 5.065 cm. = 819 dynes/sq. cm./cm.
= =
=
100 r.p.m.
Experiment
Theory
37, (by weight) polyisobutylene in Decalin
Power-law fluid with K = 9.4 dyne sec.O,77/sq. cm., n = 0.77
20.9
20.7
Discussion
T h e chief result of this work follows from the excellent agreement between the experimentally determined volumetric flow rate, Q = 20.9 gallons per minute, and the computed value, Q = 20.7 gallons per minute. Though data were obtained a t only one flow rate, the shear rate within the helical flow system ranged from 27 to 210 sec. T h e combination of axial and angular flow precludes any radius of zero stress in the fluid such as exists in pure annular flow. Hence, though the power law is used for T ( K ) , its failure to describe fluid behavior a t low rates of shear is not important. Agreement between experiment and computation is in VOL. 5
NO. 2
MAY 1966
269
TIME t ( S E C )
Figure 10. Volumetric flow rate measurements inhelical flow for 3% polyisobutylene in Decalin at
25" C.
accord with simple fluid theory and lends confidence to the hope that some real fluids behave in a manner consistent with predictions based upon a simple fluid. The discrepancy between measured and predicted values of Q is well within experimental error. This was determined by computing volumetric flow rate for deviations of 3 ~ 1 %in n, K , and
bP ---. aZ
These three quantities were believed to be subject to errors near 1%. Other variables, such as Ri, R,, and no, were subject to much smaller errors. T h e effect on Q of 1% de-
bP
viations, taken one a t a time, in n, K , and - -, is shown in az Table 11. These results show variations in Q greater than that between the experimental value and that computed on the basis of tube viscometer results.
Table II. Brackets on Calculated Q Caused b y tions in K, n, and
Rs R, n,
bP
* 1%
Devia-
-bZ
3.785 cm. 5.065 cm. = 100 r.p.m. hP __ = =
-_
a,?'
n 0.778 0.762 0.770 0.770 0.770 0.770
270
K , Dyne DyneslSq. Cm./ Sec.n/Sq. Cm. Cm
.
9.40 9.40 9.49 9.31 9.40 9.40
I&EC FUNDAMENTALS
819 81 9 81 9 81 9 827.2 810.8
Q,
Gal. /Min. 19.7 21.7 20.4 20.9 20.9 20.4
This work also has produced a number of additional pieces of information which should be useful to future experimenters. One is, of course, a carefully determined rheogram (Figures 6 and 8) for 3% PIB solution in Decalin a t 25" C . over a range of shear rates between 4 and 1500 sec-'. Another interesting result concerns the question of entry length. This point caused the authors considerable concern in design of both the viscometer tubes and the helical flow apparatus. I n determining location of the pressure taps on the viscometer tubes, the work of Collins and Schowalter ( 6 ) was used as a guide. However, that work does not account for any fluid elasticity, and it is known that PIB solutions are viscoelastic. Lengths of 8 and 20 diameters existed between a valve and the upstream tap of the large and small viscometer tubes. respectively. The excellent consistency found between rheological measurements in different tubes (Figure 6) indicates that at least as far as pressure measurements are concerned, the flow was fully developed a t the first pressure tap. Similarly, in the helical flow apparatus there was little prior work on which to base the required size of the entry region. Some crude approximations were made from combining the predictions of Collins and Schowalter for tube flow with the computations of Bird and Curtiss (7). They considered time required for the profile of a Newtonian fluid in circular Couette flow to develop when the fluid was accelerated from a rest position by sudden acceleration of one of the cylinders. The straight line connecting pressure a t successive taps along the inner cylinder (Figure 9) indicates a fully devrloped wall pressure in helical flow over the axial region of interest. Finally, we return to the primary result of this paper-viz., agreement of observed volumetric flow rate in helical flow with predictions based upon simple fluid theory. Though Poiseuille tube flow and helical flow are both viscometric flows, they have some nontrivial differences. Of particular importance is the difference in the way in which adjacent elements of fluid are sheared in the two geometries. In Poiseuille flow, as well as in circular or plane Couette flow, all streamlines in the flow field have the same orientation with respect to the apparatus boundaries. This is not the case in helical flow. Thus the change in orientation of adjacent fluid filaments could have a noticeable effect on rheological behavior, especially in the case of polymer melts or solutions where one envisions considerable interaction between adjacent molecular chains. No such effect was observed in the experiments reported here. The present results, though consistent with simple fluid theory, are insufficient to establish that 3% PIB in Decalin behaves as a simple fluid in viscometric flow. [Denn has noted (7) that experiments with viscometric flows are insufficient to distinguish between a simple fluid and certain types of anisotropic fluids.] T h e only material function studied here was T ( K ) ; normal stress behavior, characterized by C I ( K ) and u ? ( K ) was , not studied. Some measurements of normal stresses have been reported by Markovitz (70). Lastly, measurement of volumetric flow rate is not sensitive to the details of the velocity profile but only to integration of the profile over the annular space according to Equation 22. Since simple fluid theory permits prediction of the velocity field. work is in progress to measure, by a n optical technique, velocity profiles in helical flow. Conclusions
Dependence of shear stress on the rate of shear is shown in Figure 8 for 3% polyisobutylene (Vistanex L-100) in Decalin a t 25" C. Over the range of shear rates between 27 and
208 sec-'. the effective viscosity is described by a power law cm. and (Equation 26) with constants K = 9.4 dyne ~ec.O.~?/sq. n = 0.77. T h e relationship among volumetric flow rate, pressure drop, and relative rotational speed of the cylinders for helical flow of the polyisobutylene solution can be predicted from the flow equations. Specifically, this can be done when the effective viscosity is adequately represented by a power-law model over the range of shear rate existing in the helical flow field. T h e polyisobutylene used in this study is known to exhibit effects which result from the elastic nature of the solution. However, these effects did not interfere with prediction of volumetric flow rate from knowledge only of the effective viscosity of the fluid. Thus, to the extent of the measurements made, the polyisobutylene behaved as a simple fluid. Nomenclature = =
- dp/& in tube flow - dpP/dz - pg in helical
= = = = = =
tensor component defined in Equation 4 constant 4Q/aR3 acceleration due to gravity component of the metric tensor pressure difference between first pressure tap and j t h pressur: tap in helical flow system, cm. of manometer fluid with density of 2.95 grams/cc. unit tensor constant in Equation 24 torque per unit height in helical flow system power law constant in Equation 24
= = = = = = = = = =
p
flow
+ Pgt
mean pressure defined by - l / 3 tr S volumetric flow rate tube radius outer radius of inner cylinder in helical flow system inner radius of outer cylinder in helical flow system = cylindrical coordinate system = stress tensor = t - - 7
= S+Ip = present time = component of T = velocity vector
Vi
= covariant ith component of v
= x l , 2,x3 = = -i = 6ij
U(Y)
K P 01
02 T T(K)
axial component of velocity in tube or helical flow coordinates in a n orthogonal coordinate system shear rate, equivalent to K Kronecker delta, equal to 1 for i = j and 0 for
izj
= rate of shear, defined by Equations 3 and 4 = density = t l l - t33 = t22 - t33 = time = t12
R)
7R
= drag on tube wall = -trr(r =
Qo
= rotational speed of outer cylinder, r.p.m.
W(T)
= rotational speed of fluid in helical flow system
€ s-
= dummy variable = dummy variable
SUPERSCRIPT I = d/dr Literature Cited (1) Bird, R. B., Curtiss, C. F., Chem. Eng. Sci. 11, 108 (1959). (2) Coleman, B. D., Arch. Rational Mech. Anal. 9, 273 (1962). (3) Coleman, B. D., Noll, Walter, Ann. N.Y. Acad. Sci. 89, 672
(1961). (4) Coleman, B. D., Noll, Walter, Arch. Rational Mech. Anal. 3, 289 (1959). (5) Coleman, B. D., Noll, Walter, J. Appl. Phys. 30, 1508 (1959). (6) Collins, Morton, Schowalter, W. R., A.I.Ch.E. J . 9, 804 (1963). (7) Denn, Morton, personal communication, 1965. (8) Dierckes, A. C., Jr., Ph.D. thesis, Princeton University, 1965. ( 9 ) Fredrickson, A. G., Chem. Eng. Sci. 11, 252 (1960). (10) Markovitz, Hershel, Phys. Fluids 8, 200 (1965). (11) Metzner, A. B., Adoan. Chem. Eng. 1 , 98 (1956). (12) Noll, Walter, Arch. Rational Mech. Anal. 2, 197 (1958). (13) Oldroyd, J. G., Proc. Roy. Soc. (London) A245,278 (1958). (14) Rivlin, R. S . , J. Rational Mech. Anal. 5, 179 (1956). (15) Rivlin, R. S., Ericksen, J. L., Ibid., 4, 323 (1955). (16) Tanner, R. I., Chem. Eng. Sci. 19, 349 (1964). (17) Tanner, R. I., Rheol. Acta 3, 21 (1963). (18) Ibid., 26. RECEIVED for review August 30, 1965 ACCEPTED December 1, 1965 American Institute of Chemical Engineers, Philadelphia, Pa., December 1965. Work supported by the National Science Foundation through grant G-14442, through a fellowship, and through support of computer facilitiesunder grant GP 579. Partial support was provided by the Research Corp., the Ford Foundation, and the National Aeronautics and Space Administration through grant NsG-665.
A N A L Y S I S OF BREAKAGE IN D I S P E R S E D PHASE S Y S T E M S KENNIETH J. V A L E N T A S , l O L E G H B I L O U S , * AND NEAL
R. A M U N D S O N
University cf Minnesota, Minneapolis, Minn.
phase reactors are of common occurrence in many industrially important processes such as liquidliquid extraction, gas absorption, and emulsion polymerization. I n the past the design of dispersed phase reactors has been based largely on empirical methods and only recently has active interest been shown in developing a sound theoretical basis for design. Rational design requires a knowledge DISPERSED
2
Present address, Sinclaii: Research, Inc., Harvey, Ill. Present address, Melbourne, Australia.
of the distribution of particle sizes in the reactor. In the general case the poorly understood phenomena of breakage and coalescence have a strong influence on the characteristics of the dispersed phase. The problem can be simplified and meaningful results obtained by first analyzing only the breakage process, since coalescence is often reduced to a negligible level in many systems. This is particularly true in dilute dispersed phase systems or when surface-active agents are present to increase the resistance to coalescence. When a solute is present, mass VOL. 5
NO.
2
MAY 1966
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