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Ind. Eng. Chern. Process Des. Dev., Vol. 18, No. 4, 1979
Measurement of the Rheological Characteristics of Slowly Settling Flocculated Suspensions Guillermo Sarmienlo, Phillip G. Crabbe, David V. Boger,' and Peter H. T. Uhlherr Department of Chemical Engineering, Monash University, Clayton, 3 168, Victoria, Australia
The rheological properties of a lightly flocculated red mud suspension at its natural pH of 10.4 were investigated using a simple capillary rheometer specifically designed for slowly settling suspensions. A simplification of Bagley's end correction method is successfully applied to the capillary measurements to correct for entrance and exit effects. The capillary flow curves obtained with the new rheometer and corrected for end effects and for tube diameter effects are compared with independent data taken with a Weissenberg rheogoniometer and with a pilot-scale flow-loop. Excellent agreement is obtained and it is suggested that the capillary rheometer can be a most useful instrument in the minerals processing industry.
Introduction In the Bayer alumina process large quantities of the waste slurry red mud are produced. This mud is usually pumped to pondage for sedimentation and recovery of the supernatant liquor. The satisfactory pipelining of red mud is an important aspect of the process. Hence, the rheological properties of red mud need to be known, and it is convenient for industrial applications to have available a simple and reliable method of obtaining such data. A suitably designed capillary rheometer provides a convenient method for measuring the rheological properties of slowly settling suspensions. As well as being a simple instrument, a capillary rheometer provides data which can be used for direct scale-up in laminar flow and which can be used as a basis for design in turbulent flow. Capillary Rheometer For fully developed flow in a tube the wall shear stress is given by
S, = DAp/4L
(1)
where A p / L is the pressure gradient necessary to maintain the flow in a tube of diameter D. In a vertical capillary rheometer the pressure drop is usually measured from an upstream fluid reservoir to the atmospheric exit of the tube rather than between two points within the tube itself. Corrections must therefore be made to the pressure drop measurement to allow for the head of fluid above the tube exit, for the losses a t the tube entrance and exit, and for the kinetic energy of the stream issuing from the capillary tube. The first of these corrections is easily made, using the equipment dimensions and the fluid density, but the other corrections are more difficult.
End Corrections The well known procedure due to Bagley (1957) is used to correct for both entrance and exit effects. The Bagley method involves the addition of a fictitious length to the actual length of the capillary tube to account for entry and exit losses. The shear stress is calculated from S , = D A p m / 4 ( L + eD) (2) where A p , is the pressure drop measured in the instrument, with a tube of length L. This pressure drop corresponds to the fully developed pressure drop for a tube of length ( L + eD). If now the volumetric flow rate Q, is measured in a series of tubes of the same diameter but of different lengths, then, for constant Q,, a plot of S p , 0019-7882/79/1118-0746$01.00/0
against L I D is linear (Jastrzebski, 1967; Skelland, 1967). The intercept on the L I D axis gives the value of -e. The end correction e can thus be determined as a function of flow rate Q, and eq 2 can be used to calculate the wall shear stress as a function of the flow rate. The measured pressure drop across the system, Ap,, can alternatively be represented as the sum of the pressure drop for fully developed flow in the tube of length L and a term Ape representing the excess pressure loss at the entrance, at the tube exit, and corresponding to the exit kinetic energy. Thus APm = AP + APe
(3)
As the length of the capillary tube is increased, the relative importance of Ape in eq 3 decreases and it will become negligible compared with A p where L approaches infinity. Thus the true shear stress at the wall is given as limL,, = D 4 p m / 4 L . This limit is conveniently obtained from a plot of DApm/4L against D I L for constant values of D and Q,. The true value of S , is obtained by extrapolation of the curves to the D A p m / 4 L axis. Hence t h e true wall shear stress can be determined directly without first determining t h e end correction e. Diameter Effects In certain suspensions the velocity gradient in the vicinity of the tube wall may induce particle migration away from the wall region, leaving only the dispersion medium adjacent to the wall surface. A consequence of this phenomenon is the reduction of the apparent viscosity near the wall and it manifests itself as a diameter dependence of the pressure drop, which can be modelled as apparent slip between the fluid and the solid surface (Higginbotham et al., 1958; Maude, 1959; Segre and Silberberg, 1963; Seshadri and Sutera, 1970). The phenomenon is not yet fully understood. In order to accommodate apparent slip at the tube wall, the general Rabinowitsch-Mooney equation must be modified. This equation defines the wall shear rate for fully developed, steady, laminar flow of a time-independent fluid through a circular tube. For nonzero velocity at the wall the equation becomes (Jastrzebski, 1967; Gorislavets and Dunets, 1975) 32Q,/xD3 = 8blS,/D
+ ( 4 / S , 3 ) s 0s w S2f(S)d S
(4)
Here, f(S) is a function of the fluid constitutive equation and bl is a slip coefficient defined by 1979 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
bl =
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(5)
vw/sw
where V, is the effective velocity at the wall and characterizes the anomalous behavior of the fluid at the wall. Recent work (Jastrzebski, 1967; Gorislavets and Dunets, 1975) shows that bl also depends on the diameter of the tube bl = (2/DIab
(6)
where b is a corrected slip coefficient and a is a constant for a given system such that 0.5 I a I 1.0. Equation 4 may thus be rewritten as 32Q,/rD3 = 2a+3bS,/Da+'
ssw
+ (4/Sw3) 0
S2f(S)d S (7)
A plot of 32Q,/rD3 against l/Da+' for constant S, therefore yields a straight line. The slope of this line allows the corrected coefficient b to be calculated. The parameter a is determined by trial and error so that the plot of 32Q,/rD3 vs. l/Da+' is linear. A numerical value of 1 is common for suspensions. Once b is known, the flow rate Q in the absence of a diameter effect is given by (Jastrzebski, 1967)
Q = Q, - rD2S,b1/4
(8)
32Q/rD3 = 32Q,/rD3 - a/Da+'
(9)
or alternatively where = 2a+3bsW
(10)
Thus a is the slope of the lines discussed above.
The Flow Curve Capillary rheometer measurements yield corresponding values of Ap, and Q, for fixed L and D. The flow curve required is the relationship between true shear stress and true shear rate. In order to calculate the latter from the former, a plot of DAp,/4L vs. 32Q,/rD3 is first constructed for each tube diameter. A family of lines results, corresponding to different tube lengths. The variation of Ap, and L may be obtained from these lines for constant values of apparent shear rate 32Q,/rD3. A correction for L is then made either by the Bagley method or by the alternative method outlined above, to give the true wall shear stress, S,, as DAp/4L. A family of semi-corrected flow curves can now be constructed, DAp/4L vs. 32Q,/rD3 for constant values of D. The variation of 32Q,/rD3 with D can be obtained from this family of curves for constant values of DAp/4L and the slip correction can be made as described above. The flow curve S, vs. 32Q/rD3 (= 8VID) can now be constructed and this curve may be used directly for scale-up to larger pipes provided that the following conditions in the rheometer and in the large scale line are met: equal temperatures both in laminar flow, the fluid properties are time-independent, and the wall shear stress in the large pipe falls in the range covered in the rheometer experiments. The quantity 32Q/rD3 or 8V/D represents the apparent shear rate and must be modified by yet another parameter before the true shear rate 9, at the tube wall can be found (Skelland, 1967)
4,
= [(3n'+ 1)/4n1(8V/D)
(11)
The value of n'can be found from the slope of a log-log plot of apparent shear rate, 8V/D, against true shear stress,
SW.
SOLIDS C O U C I N T R A T I O U
SOLIDS C O N C € N l R A l l O U
Figure 1. Concentration profiles in a settling column of suspension at different times. Suitable positions of the capillary entrance and of the air tube are shown.
Experimental Equipment and Procedure Capillary Rheometer for Slowly Settling Suspensions. For suspensions which settle relatively slowly, it is possible to design a capillary tube system that does not require continuous agitation of the suspension. To achieve this, use is made of the different regimes of sedimentation shown by most suspensions. Thus, suspensions such as metallurgical pulps (Scott, 1968),calcium carbonate (Comings, 1940), Kaolin (Michaels and Bolger, 1962), thoria (Kearsey, 1963), and red mud (Bujdoso and Orban, 1973) show two types of settling behavior determined by the slurry concentration. Sedimenting suspensions of low and intermediate initial concentration exhibit three zones whose extent changes slowly with time. This behavior is shown in Figure 1 (Bujdoso and Orban, 1973). A clear layer (zone 1) develops a t the top of the column and beneath this is a zone of constant concentration (zone 2). The concentration in zone 2 is the same as the initial concentration of the suspension, provided that all particles are sufficiently small (say
