I
JAMES W. DAILY and GEORGE BUGLIARELLO
with suspensions of differin the range Of papermaking concentration between 0.1 and 1.0% by weight have been conducted a t the M I T Hydrodynamics Laboratory, as part of a program sponsored by the Technical Association of the Pulp and Paper Industry (7, 3 ) . Flow features were observed, and friction losses measEXPERIMENTS
Massachusetts Institute of Technology, Cambridge, Mass.
A Particular Nan-Newtonian Flow Dilute Fiber Suspensions
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E VELOCITY PROFILES I Newtbnian Laminar 2- For the "Fiber" 3 . For the Suspension ~
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ANNULUS FORMATION HYPOTHESS
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FIGURE i A
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LONG LAC 17
f vs
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10% RUNS 25% 13 5 0 % m 75% C: 0
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,L-26
L-30, L-31
100%
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L-22 L-25 L- 25A L- 29
2" PIPE
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Figure 1 .
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Fraction factor vs. conventional Reynolds number based on water viscosity Annulus thickness may be computed from these plots
VOL. 51,
NO. 7
JULY 1959
887
ured in transparent vertical pipes of two different sizes. Flowing dilute fiber suspensions do not behave as ordinary Newtonian fluids. Attempts to use the more common rheological models-e.g., Bingham plastic and pseudoplastic-as the basis for correlation of experimental data and theoretical prediction of flow behavior have been generally unsuccessful, because the models are not in agreement with physical reality. The suspensions are in homogeneous two-phase systems, in which segregation of the suspended phase can take place. When their laminar flow through pipes is observed visually, it is seen to consist of two regions: a central one, where all the fibers gather in an entangled elastic structure (“plug”), across which the velocity is nearly constant (and practically equal to the bulk flow velocity, V ) , and a very thin peripheral clear-water annulus, across which the velocity drops from this constant value to zero approximately linearly. For suspensions of one of the fiber types studied at the Massachusetts Institute of Technology, a typical plot of friction factor f us. a conventional Reynolds number based on water viscosity is shown in Figure 1. Assuming the linear velocity gradient, the annulus thickness, d, may be computed from J plots by noting rW = pV/d = fpV2/’8. The results in Figure 2 agree with the visual evidence, d increasing with flow rate and, for a given flow rate, becoming smaller as concentration increases. Comparative data for fibers of different length show that, at the same flow rate, f is larger and d smaller for shorter, stiffer fibers. The two dashed curves in Figure 2 correspond to critical annulus Reynolds numbers of 200 and 310, beyond which the annulus becomes turbulent, and the f lines in Figure 1 change slope; 200 is a theoretical value for a two-dimensional Couette flow, and 310 an experimental critical value reported by Forgacs, Robertson, and Mason (2). The critical value may be expected to vary with type of fiber, concentration, etc., because the Couette flow is an oversimplification of the true conditions in the annulus, the edge of the plug being irregular, not well defined, porous, and elastic. A possible mechanism for the annulus formation, incorporating and amplifying published results ( Z ) , is suggested in Figure 1, A . First consider, for simplicity, a single flexible fiber oriented normally to the pipe axis and extending from wall to wall. Its presence alters the original velocity distribution of the flow (the Newtonian laminar flow parabola), which in turn interacts with the fiber and deforms it. The less stiff the fiber and the higher the flow rate-Le., the larger the drag forces acting on the “ fiber-the larger the gap between fiber
888
LONG L A C 17
0
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V (ft/sec
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Figure 2. Annulus thickness increases with flow rate and decreases with increased concentration
tips and wall. A network can now be imagined, as being formed by a large number of fibers. Depending upon concentration, it will possess different degrees of elasticity. For low concentrations the network should affect the velocity profile in a way similar to the idealized single fiber. With increasing concentration the plug stiffens and becomes less elastic, and the fiber deformation should be restricted mainly to the loose ends in the peripheral region of the plug. By this hypothesis, for a given drag on the fibers of a particular network (and hence given wall shear r W ) , the amount of deflection and extent of the annulus should be the same, irrespective of pipe size. In Figure 2, the computed thicknesses agree to within 10 to 20y0 in the two pipes, showing approximate agreement with this view (for this and the other wood pulp fibers studied). A universal correlation of suspension friction factors would be useful. However, in two annuli at the same temperature, the same d at the same V means the same T~ and same annulus Reynolds number, Vd/v = 8 / f . Hence, a successful correlation procedure must include the fact that the relationship between velocity gradient at the wall, V / d , and the ratio, V / D , depends on pipe size as well as suspension properties (for different pipe sizes, V and d are nearly constant, while D varies). Recently proposed procedures-eg., (#)-obtained by forcing a fit between experimental and Newtonian expressions for f to derive a generalized Reynolds number, do not meet this condition and
INDUSTRIAL AND ENGINEERING CHEMISTRY
do not allow prediction of flow resistance us. pipe sine. Conclusion
If the annulus formation mechanism described above is used, as being more accurately descriptive of the actual physical process, the prediction of the laminar flow resistance for all types of fiber suspensions will require a generalization of the information in such curves as in Figure 2. Thus it should be possible, through systematic experiments, to relate the deformation characteristics of the plug to the fiber characteristics, effect of concentration, and flow conditions. It should be recognized that there is a lower limit of concentration, below which the interlocking of fibers in an extensive network spanning the cross section, and hence the forming of a welldefined annulus, cannot take place. Literature Cited (1) Daily, J. W., Bugliarello, G., “Effects of Fibers on Velocity Distribution, Turbulence and Flow Resistance of Dilute Suspensions,” M.I.T. Hydrodynamics Lab., Tech. Rept. 30 (October 1958). (2) Forgacs, 0. L., Robertson, A. A., Mason, S. G . , “Hydrodynamic Behavior of Paper-Making Fibers,” Annual Meeting, Technical Section, Canadian Pulp and Paper Association, Montreal, January 1958. (3) Ippen, A. T., Daily, J. W., Bugliarello, G., TAPPI 40, 478-85 (June 1957). (4) Metzner, A. B., IND.ENG. CHEM.50, 1577 (1958). RECEIVED for review January 29, 1959 ACCEPTED April 27, 1959
