T H E PLASTIC FLOW BEHAVIOR OF FLOCCULATED KAOLIN SUSPENSIONS ALAN S.
M I C H A E L S AND J U S T I N C. BOLGER'
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Data on shear stress vs. shear rate were obtained for aqueous, flocculated kaolin suspensions, using concentric cylinder viscometers. Equations which predict the effects of kaolin concentration, shear rate, and chemical content of the ambient fluid were derived, based on a structural model which assumes that in a flocculated suspension, the basic flow units are small clusters of particles (plus enclosed water) called flocs. At low shear rates, the flocs group into clusters, or aggregates, which may form networks extending to the walls of the container and giving the suspension a finite yield stress. At high shear rates, the aggregates are broken down into individual flocs, and the model predicts a straight-line relationship between shear stress and shear rate. HE kinetics of sedimentation of flocculated aqueous kaolin 'slurries, and the compressibility and density of the resulting sediments, were satisfaci orily correlated in a preceding paper (77), and interpreted in terms of a structural model postulating the existence, in the!ie suspensions, of essentially indestructible particle clusters, or flocs. These flocs could associate randomly to form weakly bonded aggregates, whose size and configuration are governed by the intensity of the local shear field. At low shear rates, the aggregates form tenuous networks which give the suspension its plastic and structural properties. I n view of the success of this analysis in dealing with the settling behavior of flocculated suspensions, it was considered appropriate to attempt 10. use this same approach to study the relationships between shear stress and shear rate of such suspensions in Couette (linear shear-gradient) flow. Most previous studies of the flow behavior of suspensions have dealt with dispersrd, nonattracring particles. For these systems, the usual starting point is the Einstein equation (3):
where p is the Newtonian viscosity of a dilute suspension containing a volume fraction, @, of nonattracting spherical particles. Einstein derived this equation by calculating the viscous energy losses due .o the presence of the particles. For Couette flow, the total energy dissipation rate per unit volume is E,,, = TG. Therefore: E,,,, = TG = poG2(1 $- 2 . 5 9 )
(2)
Later workers have artempted to extend the Einstein equation to cover nonspherical particles (8) or to account for the effect of particle size (27) or size distribution (2, 79). Vand (25)and Mason and Bartok ( 7 4 , among others, have attempted to extend the range of the Einstein equation to higher particle concentrations by considering the additional energy dissipated because of particle collisions. The flow behavior of flocculated suspensions, wherein the particles attract each other to form flocs and floc networks, is much more complex t h m for nonattracting particles. Floc1 Present address, Avco Research and Advanced Development Division, Wilmington, Mass.
culated suspensions are generally non-Newtonian, and the "apparent viscosity," v? defined as d G , is a nonlinear function of the shear rate. McDowell and Usher (77), in an early paper in this field, showed that their flocculated cadmium sulfide and copper ferrocyanide sols had finite yield stresses and very high apparent viscosities which decreased as shear rate increased. The authors proved, by measuring electrical conductivity, that the rigidity a t rest was due to a continuous particle network! and that when sheared, the network conductivity decreases. T h e fall in viscosity with shear rate was explained by assuming that the loosely packed particle aggregates are broken u p by shear forces, thereby releasing trapped water and reducing the effective solid volume concentration. Mason (73) and Reich and Vold (20) have studied the effects of agitation upon floc size, and have proposed that a dynamic equilibrium is established between floc growth and destruction at any shear rate. High shear rates shift the equilibrium in the direction of higher dispersion. Several workers, including Bingham and Green ( 7 ) and Harper and Riseman ( 7 ) ,have regarded plastic suspensions as plastic solids a t rest and at low shear rates, and as fluids a t high shear rates. Roscoe (22),in a summary article, gives a general discussion of the relaxation flow of false-bodied SUSpensions at low shear rates. Roscoe cites the results of several relaxation tests in which a solid is first stressed above its yield value to produce a permanent deformation and then held in fixed position (constant strain). The stress needed to maintain this strain decreases as the material creeps or relaxes. This relaxation is due to the local breakdown and reformation of links in the network structure. Roscoe then postulates, by analogy, that once the yield stress of a suspension is exceeded, there is a range of stresses in which d r ;dG is equal to the sum of two viscosity terms: one related to the modulus of rigidity and one which accounts for viscous drag. Roscoe's summary article was intended to provide a qualitative understanding of relaxation flow rather than a quantitative basis for correlating rheological data. I t predicts, for example, that 7 should be linearly related to G at low shear rates, whereas actual shear plots usually curve sharply at low shear rates, reflecting the physical fact that the suspension structure, and therefore the values of the parameters which characterize the structure, varies with shear rate. I n addition, Roscoe VOL. 1
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Procedure
T h e kaolin used in this study \vas a well crystallized acidbleached product of rather narrow particle size distribution [90 weight 7 0 betlveen 0.2- and 2.0-micron equivalent diameter (41. T h e properties of this clay, and the procedures employed to prepare suspensions of varying degrees of flocculation, have been described (77).
?
E $60 \
W w
Z
> n I1
Flocculated Slurries Studied. SLURRYA. Untreated kaolin, p H 4. This was PD-10 grade kaolinite, mixed with distilled water in a Waring Blendor. At a kaolin volume fraction, @K > 0.03, the p H of slurry .4 was about 4.0. SLURRYB. Untreated kaolin, pH 6. This was identical to slurry A, except that the p H was raised to 6.0 by the additior of N a O H . SALT-TVASHED SLL-RRIESC A N D D. Kaolin samples \ v e x extracted with 1-I' S a C l at p H 3, according to the procedure described by Martin (72). Slurry C was salt-washed twice and then adjusted to pH 4.5 and a chloride ion concentration of 0.001S. Slurry D was identical to slurry C, except that the p H was adjusted to 6.5 lvith S a O H . CARD-PACKSLURRYE. I n contrast to the card-house flocced slurries A! B, C , and D, slurry E had a platelet face to platelet face card-pack structure. Slurry E was prepared by raising the p H of slurry A to 9.0 and then adding enough NaCl to bring the C1- concentration in the supernatant to 0.06.V.
W 3 W
40 X
" 0
5 W
LL
w
2 20 2 W
3
0 LL 0
+ 0 0
1 2 0 10 S H E A R R A T E , R.P.M. ( X 2.10 = sec.-I)
Figure 1 .
20
Kaolin volume concentrations, aK, were determined by measuring the suspension density, ps, lvith a 25-ml. pycnometer and then calculating from the material balance equation:
Merrill-Brookfield viscometer shear plots Slurry A
PS
predicts that the continuous structure is completely disrupted a t high shear rates, making the suspension behave as if it contained nonattracting particles. This is not generally true for flocculated suspensions, wherein the interparticle attractive forces are as strong a t high as a t low shear rates. Shear forces may change the particle orientation and may preclude any large scale agglomeration, but this ivork will show that, even ar high shear rates. the strength of the attractive force between flocs is the most important factor influencing the relationship between shear stress and shear rate.
0
10
Figure 2.
20 T I M E , MINUTES
I&EC FUNDAMENTALS
30
Shear stress vs. time at constant shear rate Slurry A.
154
- PIL. =
@K ( P S
- P72)
(3)
,4t 25' C., p K = 2.58 and pW = 0.997 gram per cc. Floc volume concentrations, Q F , were obtained from the calculated values of @ K by- the use of the appropriate curve in Figure 13 of the previous study (77). These plots of @F L I S . aK Ivere obtained from measurements of the final sediment volumes of settled beds. These sediment volume measurements also showed that the aggregate volume coefficient. C,,, \vas, equal to about 3.8 for all five slurries:
$K
= 0,053
40
60
80
1600
i0 0
e
800
ln W
2
>.
0
ln ln W
c
r
k
o
ln
Lz
a
W
5
-- 800
0
Figure 3.
2000 4000 G. SHEAR RATE, SEC - 1
6000
Recorder tracings measured with Hagan viscometer Slurry A
Shear stresses were measured using t\vo concentric cylinder Couette-type viscometers. Data a t 101%.shear rates. 0 to 42 sec. -l: Irere taken \rith a viscometer using standard eightspeed LI'T and four-speed R\'X Brookfield drive heads coupled to a concentric rotor and stator unit designed by hferrill (7.5). Seyei,al :?pica1 shear plots obtained with the hlerrill-Brookfield viscometc:r arc sholvn in Figure 1. This instrument was also used to measurt: yield stresses by a procedure illustrated in Figure 2. After the Viscometer had r u n a t a lo\v steady speed! the cirive motclr \vas turned off. For a plastic suspension, the torque scale reading did not fall to zero. but leveled off a t some steady shear stress value ivhich \vas then recorded as the yield stress?T ~ . Data a t high shear rates, 0 to 10.000 set.-', were taken Trith a rotary viscomtrter [Hagan Corp., Pittsburgh, Pa. ( S ) ] . T h e Hagan viscomerer is equipped with a continuous automatic recording system {vhich provided the shear plots reproduced in Figure 3. T h e geometric parameters of both viscometers a r e given in Table I. Results and Discussion
Ideally, a complete scslution to the problem of the Couette flo\v of a flocculated suspension would entail deriving a n equation Fvhich \t-odd predict shear stress as a function of each of the pertinent independent variables-i.e., shear rate, viscosity of the suspending medium, solids concentration, temperature, time, viscometer dimensions, and suspension structural properties. For the partial solution desired in this \vork. however, thelfollowing simplifications have been made.
Sufficient time has been alloIved for structural equilibrium to be attained a t any shear rate. T h e d a t a points are taken from the steady-state shear stress values. T h e flow in the viscometer approaches Couette flow between t\vo infinite parallel plates. This is a good assumption for the viscometers used in this work, Lvherein the gap widths are small, the rotor diameters are large, and the size of the flow units is generally small relative to the gap widths. According to the structural model, a flocculated suspension a t rest is composed of rigid flocs which tend to cluster into large but comparatively weak aggregates. IVhen such a SLISpension is sheared. it is reasonable to assume that these aggregates break down into smaller fragments. as shown in Figure 4. At the low shear rates covered by the Merrill-Brookfield viscometer, the aggregates are deformed and reduced in size as shear rate increases, but the assumption is made that all the flocs are still contained in the aggregates. T h e aggregates form nehrorks over this lo\v shear rate range, although the strength of the aggregate net\vork a t any shear rate will probably be different from the strength of the structure a t rest. .4t high shear rates (Figure 4,right) the aggregates are assumed to be broken doivn completely into individual flocs. There is no continuous structural network. T h e flocs still attract each other as strongly as ever, but the shear forces pull the floc clusters apart as quickly as they form. A s a result there are many temporary doublets, triplets, etc.: which result from floc-floc collisions, but there is assumed to be no gross aggregation. To apply this model to the prediction of shear stresses. consider the deformation of a plastic solid in \vhich. for stresses
All d a t a are taken a t constant temperature of 25" C. ..\I1 suspensions are aqueous. ptr (25" C . ) = 0.89 cp.
Table I.
Viscometer Dimensions MerrillBrookfield Shear rate range, sec. --L 0-42 Shear stress range, dvnes/sq. cm. 0-6 1 Gap width, microns 1500 Rotor diameter, cm. 6 25 IVetted rotor area, sq. cm. 372
Hagan 0-1 0,000 0- 2,000 4'0 3 72 46 7
Figure 4. teff.
Structural model at low and high shear rates 0 to 4 2 sec.-l VOL. 1
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AUGUST 1962
155
less than the yield stress, the strain is elastic and recoverable, and the strain rate is zero. For T - T~ > 0, the solid creeps and acquires a permanent strain. The problem then reduces to determining the force necessary to produce deformation a t a given rate. Roscoe (22) proposed one relationship between shear stress and strain rate by assuming that the creep resistance was due to the sum of a structural viscosity, which was a measure of the difficulty with which network bonds are broken, plus a purely viscous resistance to flow. This statement can be written as: 7
- T.V
= 7CR
+
(5)
‘I’
Equation 5 departs from Roscoe’s assumption in that the stress terms are redefined in this work to allow for changes in ~ defined as suspension structure as shear rate is varied. T . is the yield strength of the aggregate network a t any shear rate. At G = 0, T~~ equals r y , the conventional yield stress. At very high shear rates, the model requires that T‘%,= 0. Lim 7.v =
~y
C-0
Lim G-
TV =
0
(7)
m
T~~ has the same meaning as in Roscoe’s work, being the shear force required to break the structural bonds a t a rate corresponding to the creep rate, G. This treatment, however, does not require Roscoe’s assumption that the creep viscosity is equal to the modulus of rigidity times the relaxation time. is that portion of the shear stress used to overcome purely viscous drag forces. This resistance arises because the deforming units must move relative to the viscous suspending liquid. The force balance expressed in Equation 5 can be converted to an energy balance, which states that the total energy dissipation rate, E,,,, in ergs per cubic centimeter per second, is:
SG
=
7.y
Et,, =
G
+
Ex
TCR
Gf
rY G
+ ECRf E”
(8a)
Network Energy Term. T o explain the meaning of E,, the network term in Equation 8, consider a spring, fastened at one end while the other end is stretched by a cord. Stretching the spring requires work, but this work is completely recoverable, provided the unloading is carried out reversibly. Suppose, however, that the cord breaks during the stretching operation, allowing the spring to snap back and dissipate the stored energy as noise, internal friction, etc. T h e situation is similar for a flocculated suspension. As the suspension is sheared, the network bonds are stretched and deformed in the shear direction. Eventually the yield stress of each network link is exceeded and the broken network fragments snap back to momentarily unstrained positions. Deforming the bonds requires work, but because of the irreversible unloading process. all of this network deformation energy is dissipated in local viscous osci 11ations. A number of authors have shown that the yield strength of a flocculated suspension is proportional to the cube of the solids concentration (78, 24) and inversely proportional to the square of the particle diameter (24). Kaiser ( 9 ) in a study of the mechanical strength of compacts of colloidal zinc oxide particles found that the crushing strength of such a compact was proportional to the parameter H a 3 / d 2 , where H is the interaggregate bonding force, @ is the volume fraction of aggregates in the compact, and d is the mean aggregate diameter. If the kaolin aggregate network can be regarded as a similar aggregate l&EC FUNDAMENTALS
a
HA @A3 da2
(9)
For large spherical aggregates, the number of floc-floc bonds which can be formed at the contact point between two aggregates should be proportional to dA2,/dF2. Then :
Combining Equations 4, 9, and 10:
The previous settling rate study (77) denionstrated that there exists a concentration below which the aggregates settled individually and the network compressive stress was zero. Equation 11 should therefore be modified to allow for the possi~ go to zero a t some floc concentration larger bility that T , may than zero :
The parameter aF0is the minimum floc concentration which is required to form a continuous aggregate network. Equation 12, however, lacks a t least one additional parameter. Inasmuch as T . ~arises from some structural configuration in which the rate of formation equals the rate of destruction of ~ depend upon the geometric orientanetwork linkages, T . must tion of the aggregates-i.e., whether these are arranged randomly, as chains, or in some other particular orientation. ~ also depend upon a n orientation function, Therefore, T , must F ( G ) , which can be a function of shear rate but must be dimensionless and independent of H F ,d,, and @.,
(8b)
Equation 8 states that the total energy input to the viscometer rotor is used to do work against the resistance of the continuous structural network, to break structural bonds, and to overcome viscous drag.
156
compact, then a t low shear rates the network strength should vary as :
Equation 13 has been obtained by dimensional reasoning, rather than by a rigorous derivation, so that F(G) should be regarded mathematically as an arbitrary parameter which is defined by Equation 13. F ( G ) would be expected to depend upon the dimensions of the particular viscometer used to measure shear stress. The utility of Equation 13 rests, therefore, upon its ability to correlate a particular set of experimental data in which all independent variables are held constant except shear rate, solids concentration, and the magnitude of the interparticle attractive force. Creep Energy Term, EcR. T h e second term of Equation 8 accounts for the energy used to break network bonds. The only breakable bonds in an aggregate network are, by the structural model, the bonds between flocs. I t takes energy to pull two flocs apart. Yet when flocs recombine by collision, the separation energy is not recovered, but rather is dissipated in local disturbances around the contact points. As a result, the total bond energy dissipation rate is: =
No. of bonds broken cc. sec.
(
) (W)
(14)
M/ is the average work, in ergs, required to move two flocs from close contact to complete separation. According to Hamaker ( 6 ) , the attractive force between two spherical flocs in close contact is:
Since da is assumed to be much larger than dF, the last two terms of Equation 23 may be neglected, so that:
Therefore :
No. of bonds formed collision If sufficient time is allowed for structural equilibrium to be established a t each shear rate, the rate a t which bonds are broken should be equail to the rate a t which bonds are reformed by collision, so that: No. of collisions E(n
=
No. of bonds formed
)
( 7 7 (collision
) (W)
(17)
5K
cJ
CAP
- 1) (2)’
(24)
Then, by analogy with Equation 19 :
No. of aggregate collisions cc. Sec.
(;
G@A)($:~)
(25)
Substituting Equations 16, 24, and 25 into Equation 17 gives the creep energy dissipation term a t low shear rates:
If steady state has not been reached, and the rates of formation and of breaking bonds are not equal, then the suspension structure and the shear stress must change with time. This The constant “attraction” coefficients of Equations 20 and is the qualitative explanation of the shape of the “relaxation” 26 are related by the equation: curves in the time plot of Figure 2. Equation 17 can be solved for two limiting cases: a t IOM. shear rates where all the flocs are assumed to be clustered into aggregates, and a t high shear rates, where the aggregates are Viscous Energy Term. T h e third term of Equation 8 assumed to have broken down completely into flocs. accounts for exactly the same type of viscous energy losses At high shear rates the flocs are assumed to be spheres of as does the Einstein equation. At low shear rates, the total uniform diameter. The collision probability for a single effective solids concentration is aA,while a t high shear rates sphere in a laminar Couette flow is given by Mason and Bartok the concentration is aP. Therefore:
(74):
+ 2.5Ca~a~) At high shear rates E, = pwG2(1+ 2 . 5 @ ~ ) At low shear rates E, = fi,G2(1
2 G@F Collision probability == x
No. of floc ~ _ collisions _ =-
cc. sec.
(5
~@ R ) ( % ) G
(18)
(19)
The number of bonds formed per collision is 1. Therefore, combining Equations 15, 17. and 19, the creep energy term at high shear rates is:
T h e solution for E,, a t lo\r shear rates requires a n estimate of how many bonds are formed when two spherical aggregates collide. For this estimate, assume that the aggregates are uniform spheres of diameter dA, and that when two aggregates collide they coalesce completely into one larger sphere, of diameter dp. If the flcics, within each aggregate, are assumed to be uniform spheres t h r n each floc is in contact with K.,. other flocs. The volume fraction of flocs, in the aggregates, is 1 ’CAP. Applying Rumpf’s (23) formula for the average value of K , for random spheres a t volume fractions of about 0.25 gives:
If the aggregates are large, compared to the flocs, then each floc at the outer aggregate surface is in contact with about half as many other flocs as each floc in the interior. ‘The number of flocs at the surface, per aggregate, is:
(28)
(29)
Predicted Equations. T h e complete solution a t high shear rates is given by substituting Equations 7, 26, and 29 into Equation 8 and simplifying: T
= A&F’
+ p&(l + 2 . 5 @ ~ )
(30)
For the solution a t low shear rates, Equations 6, 26, and 28 are substituted into Equation 8. After clearing terms: 7
-
TY
=
Ar.
* da
*F’
+ pLILG(l+ 2.5 C A ~ F )
(31)
Experimental Results
Shear plots of the types shown in Figures 1 and 3 were obtained for various concentrations of each of the five slurry systems studied. Transient structural effects produce the “hysteresis loops” shown in Figure 3. These transient effects are also illustrated by plots of shear stress as a function of time after step changes in the viscometer speed a t low shear rates, as in Figure 2. The resulting “relaxation” curves show that the steady-state readings are reproducible, and do not change with time even when the viscometer is r u n for long periods. Figure 5 is a schematic representation of the steady-state , Bingham yield value, shear stress values. T h e symbol T ~ the
The number of flocs at the surface is decreased after the two aggregates coalesce. The difference can be computed from Equation 22 by taking the diameter after coalescence to be
-
2 times the original diameter. The total number of flocfloc bonds in the system is K , ’ 2 times the number of flocs in the interior of aggregates, plus K , / 4 times the number of flocs a t the outer aggregate surfaces. From these assumptions, it follows that : No. of bonds formed collision
~
T
4(Cm [2.5
-
1)
X
0 G, SHEAR RATE, SEC-1
(2)’
-8.9
($) + 8 1
(23)
Steady-state shear plot kaolin suspensions VOL 1
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AUGUST
1962
157
is defined as the extrapolated intercept of the linear portion of the shear plot. vs, the Bingham viscosity, is defined as the slope of this straight-line segment. High Shear Rates. T h e terms AI: @F, and p m in Equation 30 are all independent of shear rate. Therefore, Equation 30 predicts that a plot of T os. G should be a straight line a t high shear rates. T h e slope of this line, le, should be: 75 = 4
1
+ 2.5
(32)
~ J F )
T o check Equation 32. 12 concentrations of slurry X were tested in the Hagan viscometer. a t shear rates from 0 to 10,000 set.-' I n every case, the straight line illustrated in Figure 3 was observed a t shear rates above 1000 set.-' T h e slopes of these straight lines are presented in Figure 6. A s this figure shows, a complete verification of Equation 32 is not possible because the Hagan viscometer is not sufficiently sensitive to measure viscosities below 2.5 cp.. and so cannot be used for floc concentrations beloiv about 30 volume Yc. Extrapolation of the data to zero concentration, however, indicates that the Bingham viscosities should fall some\vhere betxveen the Einstein line and the locus predicted by the \-and equation for dispersed, uniform spheres. Figure 6 also sho\vs that Eilers' d a t a ( 2 ) for dispersed spheres. of diverse sizes, agree fairly \vel1 with the .'individual, spherical floc" correlation. These facts support the assumption that the aggregates are broken down into separate flocs a t high shear rates? and that the effect of the flocs upon the viscous energy dissipation rate can be predicted by the Einstein equation in terms of aF. One might argue that Figure 6 does not prove the exiswnce of flocs a t high shear rates. because the data could also be correlated by using the primary particle concentration. a K ; together \vith a hisher value of the Einstein constant. If: however, the data of Figure 6 \vere plotted against B K rather
than Q F 3 the slope a t the origin would imply a n Einstein constant of about 28. This is much too high, even for platelets of axial ratio 10 to 1 . Viscosity data for dispersed kaolin suspensions indicate that the best value for the Einstein constant is about 10. A second objection to interpreting the high shear behavior in terms of primary particles is that there is no physical reason why the viscosity should increase so rapidly a t a primary particle concentration of 16 to 18%. which it would do if the data \vere plotted against aK5whereas there is an excellent reason \vhy vB should increase rapidly as Q F approaches 0.62. This concentration, 0.62. was the value assumed in the previous sedimentation study for the packing fraction of the flocs in their random close-packed configuration. Bingham Yield Value, T ~ ..At low floc concentrations, the position of the line asymptotic to the low shear rate data points could be estimated. T h e slope of this asymptote line \vas taken to agree ivith the Einstein viscosity of Figure 6 and the intercept at G = 0 \vas recorded as T ~ .Bingham yield values measured by- this method for the five slurries studied are shoivn in Figure 7. These data confirm Equation 30. Ivhich predicted that T~ should be proportional to @;I. (33)
From Figure 7 ; the constant A i in Equation 33 is found to be about 100 dynes per sq. cm. for all five slurries. T h e floc diameter, d F >is estimated to be of the order of 5 to 10 microns. A not unreasonable assumption for the magnitude of a0 is 5 -4,;lvhich allows a monomolecular water layer on each contacting floc surface. The corresponding estimate of A0 from Equation 33 is about 10-'0 to lO-"erg. This is larger than the van der ilraals constant of lo-" erg usually assigned to aqueous colloidal systems (70), suggesting that the attractive forces
I
1
0
I
Figure 7. Correlation of Bingham yield values by Equation 33 158
l&EC FUNDAMENTALS
I
0 X
2
I Card Pack Slurry E q=603
I
t
(DF.
,025)'
J
I - All
1
Card House Slurries
+ Slurry
B
0 slurry c Slurry D
0
0.1
0
$F,
Figure 8.
0.2 FLOC VOLUME CONCENTRATION
Dependence of yield stress on floc concentration
are due to hJ-drogen bonding and electrostatic attractions in addition to van der Waals forces. 'The reasonable values calculated for A0 lend. it is believed, considerable support to the floc model and to the (admittedly oversimplified) geometric representation of floc interactions. Equation 33 also helps explain the rather surprising result that the attraction coefficient. -1I. has approximately the same value for all five slurries. 'rhis requires that chemical treatment must change both - 4 0 and d,' in about the same ratio. This is not unreasonable. .-ls changes in the chemical composition of the arnbienr fluid cause the kaolin to become more strongly flocculated: t'oth the external attractive forces between flocs-i.e.. .4[,-and the cohesive forces Ivithin the flocs should increase. As shown hy Mason ( 7 3 ) . Reich and \.old (70): and the previous Lvork on sedimentation (17). the equilibrium floc size in an)- shear environment should also increase as internal cohesive forces increase. Yield Stress. Equarions 6 and 13 predict that the yield . be: stress. T ~ should rY
I
=
>44@,. -
@'roj3
0.4
0.3
(34)
For a given slurry. .qS is a constant which can be related to .&I1and -42 through Equations 1 3 and 27.
predicted by Equation 34. 'l'he magnitude of Q F O ,ho\vever, is smaller when measured in a narroLv gap viscometer than when measured in large settling tubes in which floc concentrations from 0.05 to 0.07 (77) were required to form continuous aggregate networks. Results at Low Shear Rates. To verify Equation 31, shear stresses \\'ere measured for various concentrations of each of the five slurry systems. a t each of the eight viscometer shear rates of 0.21. 0.42, 1.0. 2.1. 4.2. 8.4. 21. and 42 s e ~ . - ~Kaolin concentrations, aKjwere measured by pycnometer (Equation 3 ) and converted to floc concentrations. Q F . via Figure 13 of the previous Lvork (77). Of the remaining quantities in Equation 31. p K = 0.00893 poise and C,,. was found (77) to be about 3.8 for all five slurries. The coefficient can be calculated from Equation 2':
T h e value of A I , from Figure 7.was found to be about 120 dynes per sq. cm. for slurries A. C: D: and E and 100 for slurry B. l'his leaves only the network yield strength, T . ~ .and the diameter ratio. d3, d,, unknown in Equation 31. T o solve for these: Equations 31 and 34 combine to give: T
=
p u . G ( l f 2.5 C.4F @.F) f
dp
- @P2 d.4
(35) T~ \vas measured a's illustrated in Figure 2. by reading the final value to Jvhich ihe shear stress falls after the viscometer has been shut off. 'The results for all five slurry systems are presented in Figure 8 : Lvhich sho\vs that Equation 34 correctly represents all of the data a t floc concentrations beloit- about 0.25. which corresponds to the aggregate concentration range (pA4< 1) in ivhich tk'e model of Figure 4. left. applies. For the four card-house slurries, .43 is about 200 dynes per 34. cm., while for the card-pack slurry. -43 = 600 dynes per sq. cm. Regarding the h r n i of the data plots. T~ does not go to zero a t the origin, but rather at the finite floc concentration
2.12
f .A,
T V
- (@F
-
@FO)~
TI.
(36)
From Figure 8: aF0= 0.025 for all slurries. and A 3 = 200 dynes per sq. cm. for the four card-house slurries and 600 dynes per sq. cm. for slurry E. Therefore: Equation 36 can be written in the foirn. 1. = -1I.Y
+ I3
Lvhere
VOL.
1
NO. 3
AUGUST
1962
."
M = A3 rwG
B G - Az
(40)
TY
dF
(41 1
rwG da
0.1 0.2 X, CONCENTRATION FUNCTION
0
-411 terms in X and Yare known. I- contains all the viscosity data and is a function of shear rate, concentration, and chemical treatment. X depends only on floc concentration. The unknowns M and B depend upon shear rate and chemical treatment but are independent of concentration. Therefore a plot of Y us. X should give a straight line for each shear rate. From the slopes and intercepts of these lines, network strength and aggregate size can be calculated as functions of the shear and chemical treatment. Figures 9, 10. and 11 show the data plotted in this manner for slurries -4 and E. At each shear rate, the predicted straight line is obtained. Similar straight lines were also obtained for the X-Y p!ots at each shear rate for slurries B, C, and D. The points show more scatter a t the very lowest shear rate-e.g., Figure 10-than at shear rates over 2 set.? because of the difficulty in obtaining accurate shear stresses near the origin, where the shear plot is nearly vertical. T h e points in Figures 9 and 11 diverge upward: above the predicted line, a t values of X greater than 0.2. This is because X = 0.2 corresponds to aF = 0.26. If C,, = 3.8. than @a goes to unity at aF = 0 . 2 6 , so that there is no free interaggregate water and the model of Figure 4, left. no longer applies. This agrees with the results of the previous sedimentation study, in which the slurry behaved as if it were one large aggregate at aF> 0.26.
0.3
Figure 9. Correlation of low shear rate data Equation 37
by
Slurry A
0
40,000
4000
30,000
3000
I
2
El Iu
z
-c
0
2
z
0 2
z
>.
t
g 2000
20,000
-
Y>
v)
0
z>
$
$
i
q--+
1000
10,000
G = 4 2 SeCT'
0
Figure 10.
0
0.1 0.2 X, CONCENTRATRATION FUNCTION
Correlation of low shear rate data
37 Slurry A
160
j
0
0
l&EC FUNDAMENTALS
by Equation
0.2 0.1 X , CONCENTRATION FUNCTION
Figure 1 1. Correlation of low shear rate data by Equation 3 7 Slurry E
Effect of Shear Rate on Aggregate Diameter. Figure 12 gives the values of dA,/dF,calculated via Equations 27a and 41 from the Y intercepts of the data plots. As the model predicts. d, decreases rapidly as shear rate increases. Initially, at G close to zero, Figure 12 gives a value of 80 to 100 for d,/d,, implying that dA = 400 to 1000 microns (assuming 5 to 10 microns). Eventually, at high values of G, dF d,/d, should decrease to unity as the aggregates break down completely into flocs. T h e aggregate sizes of Figure 12 agree reasonably well with the previously obtained settling data (77). For example, for slurry A, the Stokes settling velocity was found to be 300 cm. per hour for 200-micron spherical aggregates. ,4t this size and velocity, the maximum shear rate at the aggregate surface is about 10 set.? From Figure 12 at G = 10 set.-', d, 'dF is about 30 for slurry A. Assuming that d, = 5 to 10 microns, this gives d, = 150 to 300 microns, Jvhich agrees \vith the settling diameter. Network Strength. Figure 13 shows the values of T , , , / T ~ calculated from the slopes of the X - Y plots by Equation 40. According to Equations 6 and 7,
60
40 C
t
u u
2C
'.v should equal unity at 7 1'
G = 0, but should approach zero at high shear rates.
For the four card-house slurries, however, the network strength increases a t very low shear rates, passing through a maximum value of 2 to 3 timer; 7 y before starting to decrease. This initial increase in equilibrium network strength probably explains the unusual fact that the descending branches of the hysteresis loops of Figure 3 lie above the ascending branches. I n most previous works (22):the ascending branches ivere found to be uppermost. T h e only parameter in Equation 13 which is a function of shear rate is the orientation function, F(G). Therefore:
If a n assemblage of sticky spheres is sheared slowly in a narrow channel, it is reasonable to suppose that the spheres rend to form chains which are oriented principally in the shear direction, and which thereby form a stronger network than Jvould a purely random configuration. At a given volume concentration and shear rate, small spheres should attain this preferred orientation more readily than large spheres, because of their larger number and greater mobility. Card-pack flocs of kaolin particles are considerably smaller (76) than card-house flocs, which, together with the values of dA!dF shown in Figure 12, indicates that the aggregates formed from card-pack flocs are very much smaller than aggregates of card-house flocs. As C + 0, the card-house floc aggregates become too large to maintain the favorable orientation maintained by the cardpack floc aggregates, which could account for the threefold lower value of A 3 for ]:he card-house slurries in Figure 8. As shear rate increases, however, the card-house floc aggregates decrease in size and attain the orientation which leads to the ~ shown in Figure 13. The threefold increased values of T . 'T* initial difference in the initial value of T~~ for the card-pack slurry ultimately brings the values of T~~ for all five slurries into general agreement a t shear rates greater than about 5 set.-' T h e effect of shear rate upon suspension structure is given by the two dimensionless groups, M and B, in Equations 40 and 41. Since A2 and '43 are proportional to HF (Equations 15, 27, and 35), it follows that both M and B involve the dimensionless group ,uir,GdF2/HF, which expresses the ratio of the magnitude of the shear forces which tend to pull the flocs apart to the magnitude of the attractive forces which hold the flocs together. This suggests that a more extensive study of the
0 0
40
29
G, SHEAR RATE, SEC.-'
Figure 12. Ratio of aggregate diameter to floc diameter vs. shear rate
0
Figure 13. rate
20 G, SHEAR RATE, S E C . 7
40
Aggregate network strength vs. shear
VOL 1
NO. 3
AUGUST 1962
161
parameters T~~ and dA and their shear rate dependency might lead to a shear stress-shear rate relationship in which the flow behavior of a flocculated suspension would be completely characterized by the geometry and interparticle force constant of a single floc. T h e fact that such a large quantity of data, in three independent variables (five slurry systems. a t floc volume concentrations from 0 to 0.25 and a t shear rates from 0.2 to 42 set.-? can be correlated by Equation 31 in terms of just tlvo empirically determined functions [ F ( G ) and
&]
is very en-
dF
couraging, although admittedly it is no proof of the correctness of the derivations. What makes the treatment credible is that the values of these txvo functions are reasonable in terms of the model, and agree with the previously obtained settling data (77). Furthermore, Equation 31 is general enough to apply to suspensions having other types of shear plots. If the aggregate structure does not change hvith shear rate-i.e., 7\- and dA are constant-then Equation 31 predicts the Bingham plastic straight line. If the solids net\vork cannot support a )-ield stress (r.,. = O j , Equation 31 predicts a pseudo-plastic shear plot. And finall)-. if the attractive forces bet\veen flocs did not exist. then T . ~ . .40. -41.rll! and -13 lvould all be zero. and Equations 39 and 31 reduce to the NeLvtonian Einstein q u a tion. Conclusions
This study supports the concept, developed in the previous settling rate study, that the floc, rather than the primary particle, is the basic flow unit in low shear processes such as laminar Couette flow and gravity sedimentation. These flocs can be regarded as rigid spheres Ivhich tend to cluster together into weak aggregates. These aggregates grow by collision, break down by shear forces, and are able to form the extended networks which give the suspension its structural properties. Floc and aggregate structural parameters deduced from the settling rate correlations may be substituted directly into the shear stress equations. T h e most significant consequence of this investigation. it is believed, has been the successful unification of suspension sedimentation and rheological behavior, in terms of just t\vo structural parameters: floc volume coefficient ( C F K )and interfloc bond strength, The former quantity is related to primary particle size and shape and the magnitude of the interparticle bonding forces, while the interfloc forces appear to be controlled by interparticle forces alone. .4pplicability of this analysis to other flocculated suspension systems remains to be established. Nomenclature a0
= minimum distance between two floc surfaces in close
A0
=
AI
=
A2
=
-43
=
B
=
CAP = = C F ~
d,
=
dF
=
ECR =
162
contact, cm. attractive force constant between two flocs, ergs constant coefficient defined by Equation 20, dynes,’ sq. cm. constant coefficient defined by Equation 27, dynes,” sq. cm. constant coefficient defined by Equation 35: dynes’ sq. cm. dimensionless group defined by Equation 41 @*“aF= ratio of aggregate volume concentration to volume concentration of flocs within aggregates @ F : @ ~ = ratio of floc to kaolin volume concentration average aggregate diameter, cm. average floc diameter, cm. creep energy dissipation term, Equation 8b, ergswcc. sec.
l&EC
FUNDAMENTALS
= dimensionless orientation function defined by Equa-
tion 13 shear rate, set.-' interfloc adhesive force, dynes dimensionless group defined by Equation 40 number of floc-floc bonds between two aggregates in contact = work required to separate two flocs? ergs = dimensionless concentration function, Equation 39 = dimensionless viscosity function, Equation 38 = = = =
x Y
SUBSCRIPTS u!
K F A
= = = =
water kaolin flocs aggregates
GREEK T G, apparent viscosity, poise Bingham viscosity, defined by Figure 5, poise Piu. = viscosity of water, 0.00893 poise a t 25’ C. P S = suspension density, grams cc. 7 = shear stress measured in Couette viscometer, dynes ’ sq. cm. r g = Bingham yield value, defined b) Figure 5,dynes sq. cm. r.v = yield strength of aggregate netuork. Equation 5, dynes sq cm. T Y = yield stress, Equation 6. dhnes sq cm. @F = floc volume fraction = aggregate volume fraction @FO = a constant floc concentration beloxv which T I = 6
1 7 =
17B =
Literature Cited (1) Bingham. E. C., Green, H., Am. SOG.Testing Mater. Proc. 19, 11, 640 (1919). (2) Eilers. H.. Kollozd Z . 97. 313 (1941). (3j Einstdin, i., Zbid.:27, 137 ( l j 2 0 ) . ’ (4) Georgia Kaolin Co., Elizabeth, N. J., “Georgia Kaolin Handbook.” Bull. TSBH-10 (1956). (51 Hagan Corp., Pittsburqh. Pa., Bull. MSP-125A (1956). (6) Hahaker, H.’C., PhysZa.4, 1058 (1937). ( 7 ) Harper, R. C., Jr.: Riseman, J. J., Colloid Sci. 9, 81 (1954). (8) Jeffery, G. B., Proc. Roy. Soc. London A102, 161 (1922). (9) Kaiser. R . L., Sc.D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1961. (10) Koelmans, H . , Overbook, J. Th. G.: Discussions Faraday Soc. 18, 52 (1954). (11) McDowell, C. M., Usher, F. L.: Proc. Roy. SOC.A131, 40927, 564-76 (1931). (12) Martin, R . T . , Proc. 5th Conf. on Clays and Clay Material, Natl. Xcad. Sci.-Natl. Research Council, Publ. 5 6 6 , 23 (1958). (13) Mason, S. G., Pulp Paper .Wa,o. Can. 49, No. 13, 9 9 (1948). (14) Mason, S. G., Bartok, \V.: “Rheology of Disperse Systems,” C. C. Mill, ed.: Chap. 2. Pergamon Press, London. 1956. (15) Merrill, E. I V , , unpublished drawings on file in Department of Chemical Engineering. Massachusetts Institute of Technology. (16) Michaels, A. S., “Ceramic Fabrication Processes,” TV. D. Kingery? ed., Chap. 2, \Viley, New York, 1958. (171 Michaels. A . S.. Bolner. - . J. C.. IND. ENG. CHEM.FUNDA‘ ~ ; E N T A L S 1, 24 (1962). (18) Norton, F. H., Johnson, A. L., Lawrence, IV. G., J . ‘4m. Ceram. Soc. 27, 149 (1944). (19) Orr, C., Blocker, H . G., J . ColloidSci. 10, 24 (1955). (20) Reich, I . , Vold, R. D., J . Phys. Chem. 63, 1497 (1959). (21) Robinson, J. V., J . Phys. Colloid Chem. 55, 455 (1951). (22) Roscoe. R . , “Flow Properties of Disperse Systems,” Chap. 1 , J. J. Hermans, ed., Interscience, New York, 1953. (23) Rumpf, H.. Chem.-In,q. Tech. 30, 144 (1958). (24) Thomas, D. G.. ”Progress in International Research on Thermodynamic and Transport Properties,“ Chap. 61, ASME, Academic Press. New York, 1962. (25) Vand, V., J . Phys. Chem. 52, 277, 300 (1948).
RECEIVED for review December 29, 1961 ACCEPTED May 21, 1962 Division of Industrial and Engineering Chemistry, ACS, Symposium on Dynamics of Multiphase Systems, University of Delaware! Newark, Del., December 1961.
