pL po pt
u
= liquid density, lb./cu. ft.
vapor density a t atmospheric pressure, lb./cu. ft. = vapor density, lb./cu. ft. = surface tension, Ib./ft. =
SUBSCRIPTS b g
L o s u LL’
(8) Jakob, M., “Heat Transfer,” Vol. 1, pp. 640-7, Wiley, New York, 1949. (9) Jakob, M., Linke, W., Physik 2. 36, 267-80 (1935). (10) Johnson, V. J., gen. ed., “Compendium of the Properties of Materials at Low Temperature (Phase I),” Wright Air Development Division, WADD Tech. Rept. 60-56, Part I (July 1960). (11) Levy, S., J . Heat Transfer 81, Ser. C, 37-42 (1959). 12 McNelly, M. J., J . Imp. Coll. Chem. Eng. SOC. 7, 18-34 (1953). 13 Malkov, M. P., Zeldovitch, A. G., Fradkov, A. B., Danilov, I. B., “Industrial Separation of Deuterium by Low-Temperature Distillation,” Proc. Second United Nations International Conference on Peaceful Uses of Atomic Energy 4, 491-8 (1958). (14) Miyauchi, T., Yagi, S., SOC.Chem. Engrs., Japan 25, No. 1, 18-30 (1961). (15) Mulford, R. N., Nigon, J. P., Dash, J. G., Keller, W. E., “Low Temperature Heat Transfer Studies,” U.S. At. Energy Comm., Natl. Sci. Found., Washington, D. C., LA-1416 (1952). (16) Murphy, G. M., et al., “Production of Heavy Water,” 1st ed., pp. 91-2, McGraw-Hill, New York, 1955. (17) Nishikawa, K., Mem. Fac. Engr., Kyushu Univ. 16, No. 1, 1-28 11956).
[i
boiling gas or vapor = liquid = standard pressure = saturation = vapor or gas = evaluate a t wall surface temperature
= =
literature Cited
(1) Cryder, D. S., Gilliland, E. R., Ind. Eng. Chem. 24, 1382-7 (19 32). (2) Drayer, D. E., “Experimental Investigation of the Heat Transfer Coefficients for Boiling and Condensing Hydrogen Films,” Ph.D. thesis, IJniv. of Colorado, Boulder, 1961. (3) Forster, H. K., Greif, R., J . Heat Transfer 81, Ser. C, 43-53 (1959). (4) Forster, H. K., Zuber, N., A.Z.Ch.E.J. 1, 531-5 (1955). (5) Gilmour, C. H., Chem. Eng. Progr. 54, No. 10, 77-9 (1958). (6) Hughmark, G. A., “Statistical Analysis of Nucleate Pool Boiling Data,” A.1.Ch.E. Natl. Meeting, Cleveland, Abstract 41, May 1961. (7) Insinger, T. H., Bliss, H., Trans. A m . Znst. Chcm. Engrs. 36, 491-516 (1940).
(21) Westwater, J. W., “Advances in Chemical Engineering,”
Vol. 1, Chap. 1, Academic Press, New York, 1956. (22) Westwater, J. W., PetrolChem. Engr. 33, No. 10, 53-60
(1961). RECEIVED for review February 17, 1964 ACCEPTED November 30, 1964
HEAT TRANSFER IN A ROTATING ANNULUS D. K. PETREE, W. L. DUNKLEY, AND J. M . S M I T H University of California, Davis, Calif, The effective thermal conductivity, k,, of water was measured in a vertical annulus in which the inner cylinder was heated and the outer cylinder and bottom plate were rotated. Qualitative analysis of the flow potterns due to natural convection and rotation indicates that the effect of rotation is to counteract the flow caused by natural convection. The experimental results verify this and establish the rotation speed, Bo*, at which the heat transfer rate becomes a minimum. At this speed k , approaches the true thermal conductivity of the fluid, and the heat transfer characteristics are poorer than with no rotation. Above Qo*,the effective thermal conductivity increases.
EAT
transfer to rotating bodies of fluid is important in such
H diverse areas as satellite operation (2, 3 ) , meteorological
predictions ( 9 , 70), and food canning. This paper is concerned with heat transfer across a vertical annulus filled with water. Further, the work is restricted to the conditions where the inner heated cylinder is stationary and the outer cylinder rotates. Under stationary conditions, heating of the inner cylinder may induce fluid motion due to density gradients. T h e induced circulation increases the heat transfer rate across the fluid annulus above that expected by radial conduction. This does not occur a t very low heat inputs (4)-that is, below a critical Rayleigh number (see Nomenclature). In the experimental work reported here, the heat input was always above this critical value. The chief objective was to study the effects of rotation on heat transfer by natural convection. The experimental method was to measure heat inputs and the radial temperature difTerence across the annulus. From thew data the effective thermal conductivity could be computed. T h e combined effects of a large radial gap between cylinders, high heat fluxes, and appreciable Coriolis forces (due to the rotating bottom disk) result in a complicated
system. Dynamically three velocity vectors must be‘ taken into account. Thermally heat is transferred by natural and forced convection and conduction. The velocity and temperature profiles are mutually dependent, making a rigorous mathematical description complex. However, the experimental arrangement is such that the effects of natural convection and rotation oppose each other. There have been numerous studies of fluid velocity between rotating cylinders, particularly on problems of dynamic stability (7, 72, 75). For isothermal flow it can be shown theoretically that there should be no limit to stable flow if only the outer cylinder is in motion, the flow remaining laminar at all rotational speeds. For many years experimental data did not agree with the theoretical predictions (73). However, by very careful experimentation Schultz-Grunow (74) has recently demonstrated the theory to be correct. H e observed that laminar flow occurred a t the highest possible speeds of his apparatus, corresponding to Reynolds numbers five to eight times the magnitude previously thought possible. T h e effect of natural convection on heat transfer in a stationary annulus of fluid has been correlated in terms of the Rayleigh number (product of Grashof and Prandtl numbers) VOL. 4
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171
(4, 8 ) . The critical value, mentioned earlier, is about lo3, but this increases as a result of rotation (7, 5, 6 ) , and the increase is a function of the Taylor number, ArTi,,, and also depends upon the boundary conditions : both cylinders rotating or one stationary, end surfaces of the fluid free or rigid, etc. Investigations of the influence of rotation on heat transfer, after natural convection has been established, are limited. Publications most closely related to this work are those of Dropkin and Globe ( 3 ) , Kuo (9, 70), and Lemlich ( 7 7 ) . In the first work, mercury in a rotating cylinder was heated from a bottom, horizontal plate and cooled a t the top plate. The results were correlated by plotting KNu against iVT, for lines of constant ( A r X u N R a ) . After a critical Taylor number was reached, the heat flux decreased with increasing rotation speed. Curiously enough, inexplicable, periodic temperature fluctations, superimposed upon normal random fluctuations, were observed. In Kuo’s work (9) it was supposed that the outer cylinder was heated, both were rotating, and air was contained in the annulus. Kuo considered, theoretically, the development of wave motions at high radial temperature differences and a t high rotation speeds. This was found to obscure, partially or totally, the unidirectional convective flow a t normal rotational speeds, when the thickness of the annulus was large. Experimental
The coaxial cylinders, containing water in the annular space, are shown schematically in Figure 1. The outside diameters of the stainless steel cylinders were 2 and 3 inches, providing an annular space 7/16 inch in width and 12l/2 inches high. A bottom plate was attached to the outer cylinder by screws set through a welded flange. A thin rubber gasket was inserted to assure a water tight enclosure. A vertical steel rod. protruding downward from the center of the bottom plate, holds a belt pulley and ends in a ball-bearing race. This allowed rotation of the outer cylinder with low frictional resistance. Another 1-inch ball bearing on the bottom of the inner cylinder fitted tightly into a cylindrical recession in the bottom plate. A larger 2-inch ball bearing was fitted onto the top of the inner cylinder and in an aluminum housing as shown in Figure 1. A heavy, removable steel collar, equipped with setscrews, gripped the inner cylinder above the top bearing in order to hold the cylinder rigid. Vertical alignment of the two cylinders was adjusted by lateral movement of the collar. A dial indicator independently mounted and calibrated in thousandths of an inch, served to check the alignment of the cylinders and to measure eccentricity and wobble of the outer cylinder. The belt drive permitted rotation of the outer cylinder up to 240 r.p.m. An electrical heater, constructed of a Glas-Col heating tape, wound on a 11/2-inch0.d. stainless steel cylinder, was inserted into the hollow center of the inner cylinder. Spacers held the heater in place and created a l/l,-inch air gap between heating tape and inner surface of the inner cylinder. The heat flux was measured directly with a Weston Model 310 wattmeter. Figure 2 shows the location of the four thermocouples (30gage copper-constantan) on the inner cylinder and the single junction on the outer cylinder. The junction heads were soldered into small grooves on the cylinder surface, and the lead wires were embedded for about 1 inch along the circumference, in grooves filled with epoxy resin. The cylindrical surfaces were then sanded to remove irregularities. T h e leads from thermocouples on the inner cylinder were passed through small holes (later waterproofed) to the axis and then up and away from the apparatus. Each lead wire from the couple on the rotating, outer cylinder was attached to a separate aluminum ring-one ring above, and the other below, the junction. The aluminum rings were held in place with setscrews pushing against Lucite blocks attached to the outer cylinder. This arrangement also insulated, electrically, the aluminum rings from the cylinder. For transmitting the thermocouple current a single, stationary. graphite brush was in contact with each ring. 172
l&EC
FUNDAMENTALS
b
C
*
Figure 1 .
Rotating heat transfer annulus
-Outer
C y l i n d e r 1
Figure 2. in annulus
Thermocouple locations
Scope and Precision of Data
Measurements were made with the apparatus in surroundings a t room temperature. T o reduce variations in heat transfer to the surroundings, cardboard shields were used to decrease air currents. Three levels of heat input, Q = 10, 25, and 40 watts, were employed and these caused temperatures on the cylindrical surfaces ranging from 90' to 170' F. and radial temperature differences ( T I - Tz)from 3' to 16' F. Even at the maximum radial temperature differences the change in thermal conductivity of water is insignificant. The rotation speed, 0 0 , was varied from 0 to 240 r.p.m. T h e data were obtained with the coivcentric axis as vertical as possible. Only one height-annulus geometry was used. The five thermocouples were checked for precision before and after installation and found to give identical readings within *1 gv., which corresponded to the accuracy of the potentiometer. The temperatures on the inner cylinder were stable a t steady-state conditions and are estimated to have an accuracy of 0.1" F. A check on the precision of these values and on radial symmetry was furnished by comparing Tl and T,. These locations were a t approximately the same height on the inner cylinder, but diametrically opposite. The average difference, T , - T I ,was 4-0.3' F. and the maximum deviation f0.8" F. These differences were always positive because the thermocouple corresponding to T,,, was a t a slightly higher point on the cylinder than that corresponding to T1. The thermocouple on the outer cylinder gave readings which were subject to random fluctuations with a superimposed periodic oscillation. Both types of fluctuations increased in amplitude with heat input and decreased as the rotation speed increased. T h e correct TZwas taken as the estimated mean temperature, about which the random fluctuations occurred, over several minutes of observation. As a result of this uncertainty in Tz, the data points a t low noand high Q (Figure 4) scatter. Errors in the measured heat flux were estimated to be less than 5% for all runs. Some loss occurred from evaporation of water, and by conduction through the top and bottom of the test section. T h e reproducibility of the data was checked by operating in overlapping ranges of rotation speed from one day to the next. After each day's runs the apparatus was dismounted and cleaned. This eliminated the film of oil which otherwise would be deposited on the cylindrical surface over a period of time. T h e smooth curves formed (Figure 4) thus attest to the reproducibility of the data. A slight but noticeable wobble was observed when the outer cylinder was rotated. This was due to slight imperfections in the vertical alignment of the welded base plate and pulley shaft, and lack of complete rigidity due to a minute lateral play in the ball bearings. To define this wobble, and thus associate the heat transfer data with a quantitative measure of the geometry of the system, the dial indicator was used to determine the magnitude of the lateral movement. It was found that the range of movement per revolution remained essentially constant during a run. For the entire data the range varied from 0.032 to 0.045 inch. Figure 3 illustrates the displacement as a function of circumferential distance for a specific run for which the range was 0.043 inch. Results
The measured data were used to calculate a n effective thermal conductivity, k,? (for the water in the annulus), from the equation :
Cy lin der
Dial Indicator
L 0
-d E
Run No. 40-18
0.630
.'.'*..'..
0 0,
CL
'-.
'..
b 0.610
t
.......
U
0.590 0 .-
Range o f Lateral Displacement 0.043In. I*
0
.-0
1
-. . . C.
a
I
The results are reported in Figure 4 where k, is plotted against for the three Q values. Numerous runs for each heat input are shown on the figure for the stationary case (00 = 0). The minimum values of k , read from the curves are given in Table I along with the corresponding rotation speed, a,*, and the thermal conductivity, k, of water. This was evaluated a t the arithmetic mean of the temperatures of the inner and other cylinders, a t mid-height. Other required properties were also determined in this manner. It is noted from Table I that (k,),,, closely approaches k a t the lowest heat input. Discussion
Heat Transfer with No Rotation (0, = 0). Above a critical Q, density gradients in the annulus will result in a movement of elements of fluid upward along the heated inner cylinder, and downward and inward from near the outer cylinder. When the convection motion has become established, the flow lines will be as described in Figure 5 with stagnant pockets of hot and cold fluid as indicated. As Q increases the fluid velocities rise and the energy transport by convection increases, aidemonstrated by the experimental data ' = 0 in Figure 4. at & Investigations of natural convection between parallel hot and cold plates, summarized by Eckert and Drake ( 4 ) ,indicate that the ratio k,/k can be correlated in terms of the product of the Grashof and Prandtl numbers. For values of (NG,,Vp,) from about lo3 to lo6 the data can be represented by the equation
Although the concentric cylinder arrangement for the data reported here is different from that for parallel plates, the results are very similar. Figure 6 shows that the data can be described by:
k , / k = 0.11 (.V~r,Vpr)O25 = 0.11 VOL. 4
NO. 2
25
(3)
MAY 1965
173
(S,,)O
*Q=lOw Qs25w
Il,,rprn
Figure 4.
Effect of rotation on effective conductivity
will remain stable and laminar a t all values of Q,,, In the event of small transverse disturbances, the centrifugal force acts to oppose them. Next consider a flat plate or disk rotating a t constant speed about an axis perpendicular to the center of the disk. Fluid will flow around and outward near the disk, and inward and down near the axis as shown in Figure 7. If the rotating disk is enclosed in a cylindrical housing and a stationary axis of finite diameter is added, the flow lines would be as depicted in Figure 8. Hence the isothermal effect of rotation in the apparatus of Figure 1 is to produce simultaneously tangential flow about the axis and flow downward along the inner cylinder and radially outward along the bottom plate. Addition of a heat flow out from the inner cylinder will tend to cause a rotating flow in the opposite direction. as illus-
Heat Transfer with Rotation. The fluid movement accompanying rotation in the experimental apparatus is due to motion of both the outer cylinder and the bottom plate. For an isothermal, long annulus there is no velocity component in either the radial or vertical directions. The fluid streamlines are concentric circles about the vertical axis, so that the flow is of the annular, tangential type. With the outer cylinder rotating and the inner one stationary, the flow
Inner Outer Cylinder Cylinder
Cold
,of
Sta nant Pocket j o t Fluid
Cold Table I.
Q,
Watts
tagnant Pocket
f Cold Fluid
10 25 40
Figure 5. Natural convection flow patterns in an annulus
Effect of Rotation on Minimum Effective Conductivity
01, B.t.u./(Hr.) (Ft.)("F.) 0,66 0.90 1.09
kAi20
(ke1rninj
B.t.u./(Hr.) (Ft.)(OF.) 0.37 0.47 0.45
'do*,
R.P.M. 46 67 88
6
Gr Pr Figure 6 . 174
l&EC FUNDAMENTALS
Natural convection with zero rotation
k, B.t.u./(Hr.) (Ft.)("F.) 0.36 0,37 0,38
Disk Figure 8. Flow near on enclosed, rotating annular disk
isk Figure disk
7. Flow near a rotating
trated in Figure 5. The net motion of the fluid will depend upon the magnitude of the two counterflows, and therefore upon no and Q . Since any motion will increase the heat flow due to convection, k , will be increased above the normal thermal conductivity of water for net flow in either direction. These concepts explain the nature of the curves in Figure 4. At low rotation speeds, the flow due to natural convection predominates and k , is high. Increasing no tends to reduce the net flow. The minimum k , corresponds to a net flow approaching zero and k , bvould be expected to be the same as the normal conductivity, k . This is seen in Table I to be almost exactly the case a t Q = 10 Watts. Of course end effects and wobble would tend to prevent the complete counterbalancing of currents due to convection and rotation. Hence some net flow may be expected. This could explain why the minimum k , is greater than k for higher heat inputs. The rotation speed, no*,a t which the minimum occurs appears to be linear in Q (Table I). After the minimum is reached, further increases in no lead to higher values of k,, showing that the flow current due to rotation (Figure 8) predominates. When rotation is combined with natural convection, k , / k should be a function 01' the Taylor number in addition to IV,,. The nature of this relationship can be seen by replotting the data of Figure 4 in terms of dimensionless quantities. The ratio k , / k can be considered a Nusselt number defined in terms of the width. a , of the annulus. ke - ha - - k k The abscissa of Figure 4 depends upon the rotation speed =
Dropkin and Globe ( 3 ) pointed out that (NNU,VCIRa) was a dimensionless heat flux and plotted their data, for mercury in an assembly heated from the bottom, in this way. At very low N T a the lines were horizontal because the rotation speed was below the critical value. Above this speed the curves dropped toward a value of NNu = 1 just as those in Figure 9, even though the system and geometry were different. Their data extended only to 00= 25 r.p.m. and the reverssl in magnitude of .Vnuwas not observed. The vertical temperature differences, T , - T,, on the inner cylinder also demonstrated the countereffect of rotation on natural convection. Thus At, = T t - To was positive at low values of QO and became negative a t high rotation speeds. is the point The rotation speed a t which Atz reversed-that, where At, = 0-increased with Q , but the values were higher than %*. The reason for this is not known. Possibly it i related to convection due to the centrifugal force. Conclusion
Rotation of the outer cylinder and bottom plate in a vertica heated annulus can reduce the heat transfer rate to tha
(4)
The heat flux is related to the product NNuNRa, as can be seen from the following equations
- _-
h(AT),
k
= (NNu) -
Am
a
(AT),
(6)
Combining Equations 6 and 7 gives
Equation 8 shows that ,VxulVRa is a dimensionless heat flux, and for a given fluid and apparatus a constant value of Q is equivalent to a constant jVSUA\IRB Hence the data obtained when k , is measured as a function of Qo, a t constant Q, should be plotted as >VYuus. .V-p* with lines of constant iVNuAVRa. The results in Figure 4 have been replotted in this way in Figure 9.
5 Figure 9. Experimental numbers
data
VOL. 4
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NO. 2
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MAY
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1965
Taylor
175
approaching the conduction contribution. T h e explanation appears to be due to a fluid motion which counteracts natural convection. Further increases in noresult in a n increased heat transfer due to the establishment of a net flow of fluid due to rotation.
GREEKSYMBOLS a = thermal diffusivity, k/pc, sq. ft./sec. = coefficient (volume) of thermal expansion, P P = density. lb./cu. ft. = fluid viscosity, lb./hr. ft. P = rate of rotation, r.p.m. 00 = rate of rotation a t (k,),,,, r.p.m. R0*
Acknowledgment
SUBSCRIPTS
T h e financial assistance of the Department of Food Science and Technology, University of California, Davis, is gratefully acknowledged.
1 2
Nomenclature Am
= log-mean area of cylinder, sq. ft.
a
= width of annulus, r2 - r l , ft. = specific heat, B.t.u./(lb.) (OF.)
C
acceleration of gravity, ft./(hr.)z
g
=
h
= heat transfer coefficient: B.t.u./(hr.) (sq. ft.)(OF.)
k k,
=
= = = =
L
Nxu Npr NRa NTB
=
= = = =
Q
r
T, =
Ti,
=
Tz Tb
=
Tt (AT), (AT),
= =
=
thermal conductivity, B.t.u./(hr.)(ft.) (OF.) effective thermal conductivity, B.t.u./(hr.) (ft.) (OF.) height of annulus, ft. Grashof number, g/3a3(AT),pz/p2 Nusselt number, k , / k = ha/k Prandtl number, c p / k Rayleigh number, g / 3 a 3 ( A T ) , / ( a p / p ) Taylor number, 40,+a4pZ/p2 heat input (radial heat flow rate), B.t.u./hr. radius of cylinder, ft. temperature at midpoint of inner cylinder, O F . (TI and T, are at points diametrically opposite to each other) temperature at midpoint of outer cylinder, OF. temperature near bottom of inner cylinder, O F . temperature near top of inner cylinder, OF. radial temperature difference, T I - Tz, OF. vertical temperature difference, T , - Tb, OF.
( O F . ) -l
= inner cylinder = outer cylinder
literature Cited
(1) Chandrasekhar, S., “Hydrodynamic and Hydromagnetic Stability.” Chap. 2, 3, 7, 8, Oxford University Press, London, 1961. (2) Dropkin, D., Gelb, G., “Heat Transfer by Natural Convection of Mercury in Enclosed Space when Heated from Below and Rotated,” National Heat Transfer Conference, Boston, Mass., ASME Paper 63-HT-19 (August 1963). (3) Dropkin, D., Globe, S., J . Appl. Phys. 30, 84 (1959). (4) E,c,kert, E. R. G., Drake, R. M., Jr., “Heat and Mass Transfer, pp. 328-31, McGraw-Hill, New York, 1959. (5) Fultz, D., Nakagawa, Y . , Proc. Roy. SOC.(London) Ser. A, 231, 211 (1955). (6) Fultz. D.. Nakagawa, Y., Frenzen, P., Phys. Rev. 94, 1471 (1954). ( 7 ) Kaye, J., Elgar, E. C., Trans. A.S.M.E. 30, 753 (1958). (8 Kralssold, H., Forsch. Gebtfte Ingenteurw. 5 , 186 (1934). (91 Kuo, H.-L.. J . Meteorol. 9, 260 (1952); 11, 399 (1954); 13,521 (1956). (10) Kuo, H. L., Tellus 5 , 475 (1953). (11) Lemlich, Robert, TND. ENG. CHEM.FUNDAMENTALS 2, 157 (1963). (12) Lewis, J. W., Proc. Roy. SOC.(London) Ser. A, 117, 388 (1927). (1 3) Schlicting, H., “Boundary Layer Theory,” McGraw-Hill, New York, 1960. (14) Schultz-Grunow, F., Z a m m 39, 101 (1959). (15) Taylor, G. I., Proc. Roy. Soc. (London) Ser. A, 157, 565 (1936).
RECEIVED for review April 16, 1964 ACCEPTED December 3, 1964
WALL EFFECT IN COUETTE FLOW OF NON-NEWTONIAN SUSPENSIONS STANTON R. M O R R I S O N AND JOHN C. HARPER
Agricultural Engineering Department, University of California, Davis, Calif. Suspensions of fibrous particles in Couette flow exhibited yield stresses, apparent wall slip, and time-dependent behavior. The behavior exhibited cannot be observed in a nar!ow-gap viscometer, and data from such instruments should b e used with caution.
that consist of a solid phase suspended in a liquid common in many industries. They may display flow characteristics not found in simple liquids, such as pseudoplasticity. time effects, yield stresses, aggregation of particles, and wall effects. I n the food industry, many food purees and concentrates are made u p of fibrous particles and exhibit these characteristics to a greater degree than d o simpler nonNewtonian suspensions. Apparent slippage at a surface is likely with any suspension which has a yield stress. T h e suspension is necessarily discontinuous at a surface, and if the frictional forces between the material and the surface are less than those which bind it together, there will be a slippage of the body of the material. T h e particles of the suspension, however, are surrounded by a liquid. a thin layer of which will form between the bulk of the LUIDS
F are
176
I&EC
FUNDAMENTALS
suspension and surface, and a velocity gradient will occur across this layer. This slippage phenomenon has long been recognized (7, 4, 6 , 7, 75, 77-20), but no investigators have reported direct measurements. Some investigators (2, 5, 74), in fact, have not considered the phenomenon even when it might be expected to be important. Wall slippage is often assumed from the fact that flow curves for a fluid vary with tube diameter when a capillary tube viscometer is used, or with gap width when a coaxial cylinder instrument is used (9, 72, 76, 27). A slip velocity can be calculated with these methods if certain assumptions are made. This paper reports results of direct visual measurement of slip velocity with a coaxial cylinder viscometer and suspensions of fibrous particles.
