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Rheology of concentrated slurries of particles of natural size distribution produced by grinding
Powder
Technology,
56 (1988)
293
293 - 299
Rheology of Concentrated Slurries of Particles of Natural Size Distribution Produced by Grinding C. TANGSATHITKULCHAI
and L. G. AUSTIN
College
Sciences,
of Earth
and Mineral
PA 16802
(U.S.A.)
(Received
March 14, 1988; in revised
Department
of Mineral
108 Steidle
Building,
University
Park,
form July 27,1988)
SUMMARY
The rheological properties of concentrated slurries of natural size distributions of ground coal and quartz were determined in a cup-andbob viscometer in the laminar flow region, using a ridged bob and cup. Organic liquids of the same density as the solids were used to prevent settling of the solid. It was found that slurries had a true yield stress, followed by a transition to a Bingham plastic character (with a higher extrapolated Bingham yield stress) for shear rates above 40 s I. Empirical equations were developed to describe the results as a function of the Rosin-Rammler characteristic size and distribution modulii of the particle size distributions, for solid concentration of 30 to 60 vol.%. The two most significant parameters were volume per cent solids and size modulus, with the distribution modulus m having a minor effect between m = 0.4 to m = 0.9. Results from both coal and quartz were fitted by the same equations for plastic viscosity and yield stresses when these were expressed relative to the viscosity of the carrier liquid. Existing theories of slurry rheology do not appear to be adequate to explain the results.
INTRODUCTION
There is substantial literature on the theoretical prediction of rheological properties of dense slurries [ 1 - 71. However, many of the treatments contain no allowance for the shape or fineness of the size distribution of the solid dispersed in the slurry since they use the volume fraction of solid as the only variable. In addition, most of the experimental investigations of slurry rheology are in water systems where the results can be 0032-5910/88/$3.50
Engineering,
strongly influenced by electrical double layers and minor dissolved components which affect the degree of dispersion or flocculation of the finer particles. As part of an investigation of the influence of rheology on the power required to drive tumbling ball mills, it was necessary to have a systematic method of predicting slurry rheological properties for the size distributions of solid produced naturally by the grinding process. In the lack of suitable predictive equations, either theoretical or empirical, it was decided to develop an empirical description specifically for such natural size distributions.
EXPERIMENTAL
The rheological properties of the slurries were determined in the laminar region using a Haake Rotovisco RV-3 cup-and-bob viscometer, with the MK50 and MK500 heads which have maximum torques of 0.49 and 4.9 N.cm, respectively. The stators and cups (MVIIP and SVIIP models, 2.6 and 1.5 mm gaps, respectively) have vertically grooved surfaces, with grooves 1 mm deep by 1 mm wide spaced 2 mm apart, to prevent slip. An air bubble was retained in the recessed bottom to reduce the end effect. All experiments were carried out at a constant temperature by using a heating-refrigerating bath and circulator system to maintain the temperature at 25 + 0.5 “c. Two materials were studied, a crystalline white quartz and a coal (low volatile bituminous, Hardgrove Grindability Index = 90,16 wt.% ash content). Controlled grinding of these materials in a laboratory ball mill (200 mm diameter with 25 mm steel balls) for various times gave size distributions which could be fitted by the Rosin-Rammler @ Elsevier Sequoia/Printed
in The Netherlands
294
I 10 PARTICLE
I wlo
loo SIZE k+ pm
Fig. 1. Rosin-Rammler size distribution plots of quartz batch ground in a ball mill at 40 vol.% solids in water.
equation over most of the size range (see Fig. 1):
The liquids used to prepare the slurries were mixtures of perchlorethylene and aliphatic napthas (American Minechem Co., PA) to give specific gravities of 2.64 (for use with quartz) and 1.36 (for use with coal). Use of these non-polar liquids had three major advantages: the slurries produced were essentially non-settling; the liquids are excellent wetting and dispersing liquids for the solids; the chemical effects of pH, etc., present with water are avoided. The liquids had Newtonian viscosities of (2.8)(10e3) and (1.2)(10e3) Pa.s, respectively. The maximum operating speed was chosen to lie within the laminar flow region using the definition of Reynolds number for Newtonian couette flow, Re = ~~NPR,~CS - 1)
m
P(3C)=l-exp
[
01 i
(1)
where P(X) is the cumulative fraction less than sieve size X, lz is the fineness modulus and m is the distribution modulus. Additional samples were also prepared by blending the required amounts of ,/2 sieve fractions and various fl sub-sieve fractions. The sub-sieve fractions were prepared in a Donaldson A-12 air classifier (Donaldson Co., MN), with the finest fraction being a minus 2 pm fraction (see Table 1). TABLE
1
Particle size distribution of test samples for rheological determination as represented by the distribution modulus m and size modulus k of the RosinRammler equation Material
Distribution modulus
Size modulus
Preparation
k
(pm)
Quartz (40 vol.%)
where N is revolutions per second; p is the liquid density; R,, is the bob radius; S is the cup/bob diameter ratio; p is the coefficient of viscosity, and [B, 91 (3)
Recrit =
It was assumed that this relation would apply to the non-Newtonian slurry with /,Ltaken as the apparent coefficient of viscosity at the highest shear rate and p is the total slurry density. Figure 2 gives a typical result of torque uersus rotor speed, showing that the rheological regime was pseudo-plastic with yield stress followed by Bingham plastic at higher shear rates. Therefore the shear rate at the bob surface was calculated from the Kreiger and Maron [lo] equation, -_
m
(2)
cc
dv dr
1.22
=
q,
[1+k&
-1)
b
S2
Coal (40 vol.%)
Ball milling in water
Quartz (54 vol.%) Coal (54 vol.%)
0.78
Coal
0.50
14,40
Coal
0.40
lo,25
Blending I
+
k,(-$ - 1)2]
k,=
G(l+ f
k,=
:
Q2 -
6s’
1
L 1nS
1nS)
(4)
295
:
60
so-
I
0
0
20
40
60 ROTOR
60 SPEED,
100
120
140
r~rn
Fig. 2. Typical rheological data of torque uersus rotor speed for coal slurries at various concentrations (Rosin-Rammler size modulus, 50 pm, distribution modulus, 0.78; liquid s.g., 1.36; liquid viscosity, 0.0012 Pa.s).
20
Fig. 3. Shear as a function distribution modulus, 10 0.0012 Pa.s).
40 SHEAR
60
60 s-’
RATE.
100
120
stress versus shear rate for coal slurries of solid concentration (Rosin-Rammler modulus, 0.40; Rosin-Rammler size pm; liquid s.g., 1.36; liquid viscosity,
50,
I
I
I
0.5
0.6
,
d(ln T) n”Z _ d(ln N) where T is the bob torque at the wall calculated from 2nRb2hrb, h being the immersed bob depth and rb the shear stress. The yield value was determined by running the bob at 1 rev./min for 2 to 3 min and noting the torque after the rotation was stopped.
EXPERIMENTAL
0.1 SOLID
0.2 0.3 VOLUME
0.4 FRACTION
Fig. 4. Effect of solid concentration and fineness of size distribution on the true yield stress for quartz slurries (Rosin-Rammler distribution modulus, 0.92; liquid s.g., 2.63; liquid viscosity, 0.0028 Pa.s).
RESULTS
The results contained the features shown in Fig. 3, that is, a true yield value r,, a Bingham yield value ray and a ppL, and a transition region from zero to about 20 to 40 s-l shear rate. Typical variations of these values with slurry concentration C (volume fraction of solid in the slurry) and size modulus 12for a given distribution modulus m are shown in Figs. 4, 5 and 6. Figure 7 gives a typical result expressed as relative apparent viscosity (defined by the apparent viscosity at a shear rate divided by the Newtonian viscosity of the suspending liquid) versus shear rate, as a function of volume fraction of solids. It is clear that all three variables C, k and m affect the rheological behavior for the denser slurries. The pattern of rheological behavior of “pseudo-plastic with yield stress followed by Bingham plastic” has been reported by other
._g@~ 0-s 0
_I 0.1 SOLID
2
:
0.2 0.3 0.4 VOLUME FRACTION
0.5
0.6
Fig. 5. Effect of solid concentration and fineness of size distribution on the plastic viscosity of quartz slurries (see Fig. 4).
workers [ 11 - 131 studying coal slurries in water, oil, methanol and coal-derived liquids. The behavior does not agree with the results
296
0 0.1
0
SOLID
I 0.2 0.3 0.4 VOLUME FRACTION
0.5
0.6
Fig. 6. Effect of solid concentration and fineness of size distribution on the Bingham yield stress of quartz slurries (see Fig. 4). 1
10
100
PARTICLE
SIZE.
1000
pm
(bl
Fig. 8. Rosin-Rammler distribution plots of coals samples prepared by blending various fractions of sieve-size and subsieve-size particles. (a), Distribution modulus = 0.5; (b), distribution modulus = 0.4.
reasonable accuracy over the bulk of the mass, so possibly pseudo-plastic behavior is associated with multi-modal size distributions.
TREATMENT
0
20
40 SHEAR
60
80 RATE,
100
120
s-l
Fig. I. Variation of relative apparent viscosity with shear rate as a function of particle size and solid concentration for quartz slurries having a RosinRammler distribution modulus of 1.22.
reported by Klimpel [14] for ground ores and coals slurried with water; they report a “pseudo-plastic with yield stress” behavior with constant values of Flow Index n in the expression (5) for a particular C and particle size distribution. Toda et al. [ 151 report “pseudo-plastic behavior without yield stress” for mixtures of two fi size fractions, one much finer than the other. Figure 8 shows that the blends prepared for the work reported here fitted a Rosin-Rammler size distribution with
OF RESULTS
To avoid presenting the data purely in the form of a large number of figures, empirical equations were developed by curve-fitting to the data. The basic equation used was chosen to be 7 _=_
TY
P
/J
+
(6)
where Q is a function to be determined which allows for pseudo-plastic behavior at low shear rates, and 7y, Tgy, ppL are functions of C, k and m. Q becomes 1 at higher shear rates and eqn. (6) reduces to the Bingham plastic equation. Since other studies [2] indicate that the apparent viscosity under given conditions is directly proportional to the Newtonian viscosity of the suspending liquid, /Jsay, the values were expressed as a ratio to this viscosity by dividing by 1-1as shown. Figure 9 shows the values of Q as a function of shear rate. It was felt sufficient to approximate the ranges of Q with a single empirical function
a
5
10
I5 SHEAR
20 RATE.
25
30
35
s-’
Fig. 9. Variation of correction function Q for pseudoplastic behavior at low shear rate (<40 s-r).
Q =1
-exp[0.32(~~‘2]
(7)
The yield stresses and plastic viscosity were fitted to power functions, as illustrated in Figs. 10 and 11. The constants in the equations were correlated with C and k by trialand-error choice of fitting functions. Once the functions had been chosen the constants in the functions were determined with a nonlinear regression routine, to give 0.3 < C < 0.6
(8)
0.3 < c < 0.55
(9)
at least for 0.4
Substituting in eqns. (8) and (9) would reintroduce m into the equations. For example, for \L= 0.8, so that k* is the 80%passing size, k = k*/(1.609)““, and (6500)(1.609)"' k*
1
4-48c”3s
(84
ROSIN-RAMMLER
SIZE
MODULUS,
k, pm
Fig. 10. Correlation of measured yield stress with particle size distribution parameters and solid concentration.
I SOLID VOL. FRACTION
L
-
EPUATION
9
1022L 1
H) ROSIN-RAMMLER
SIZE
100 MODULUS,
k. pm
Fig. 11. Correlation of Bingham yield stress with particle size distribution parameters and solid concentration.
For a given value of k* and C, a smaller value of m would give a higher yield stress, as expected. This reasoning shows that a higher yield stress is associated with a smaller m provided that the characteristic size k* is taken as greater than the 63.21%passing size. The important conclusion is reached that the yield stresses are not dominated principally by the amount of ultra fine material but depend also on the total size distribution, so that (at least for natural Rosin-Rammler distributions in
298
non-interactive liquids) the 63.21%passing size is an index of yield stress irrespective of the distribution modulus m. In a similar manner, the variation of plastic viscosity with the characteristics of the particle size distribution were fitted by the empirical equation
c
1.1 +exP(-14.2m2.6)
1
x-
i
1-c
k-o.21
0.L IO
Wa) It can be seen that for any given concentration of solid C and a fixed value of k, there is a minimum in plastic viscosity at a particular value of m. For example, the function f(C!, m) has a minimum value of 1.6 at about m = 0.45 for C = 0.3, but the value of m giving the minimum moves to higher values as
I
I1,lll
I
SIZE
(11)
Again, higher values of C and smaller values of k give increased plastic viscosity, as illustrated in Fig. 12. The effect of m is more complex. The first term on the right-hand side of the equation has a minimum at m = 0.7. On the other hand, the effect of m in the exponent of the second term depends on whether C is greater or less than 0.5. Table 2 shows values of the function f(C, m) defined by
I
I
/
100 MODULUS, lrn
,,l,l
1000
Fig. 12. Variation of relative plastic viscosity with Rosin-Rammler size modulus as a function of per cent solids in the suspension of quartz (constant Rosin-Rammler distribution modulus, 0.92; liquid s.g., 2.63; liquid viscosity, 0.0028 Pa.s).
concentration increases, being about 0.8 for C = 0.6. The accuracy of the equations in predicting the apparent viscosities for the 100 tests at 5 or more shear rates per test was obtained by determination of the standard deviation of relative difference between a computed and measured result, giving a value of standard deviation of error of about 10%.
DISCUSSION OF RESULTS AND CONCLUSIONS
Figure 13 illustrates a major conclusion: the effective laminar viscosity is of course
TABLE 2 Values of f(C, m) in eqn. (lla) Distribution modulus m
1.0 0.9 0.8 0.7 0.6 0.5 0.4
Function f( C, m) Volume fraction of solid C = 0.3
0.35
0.4
0.45
0.5
0.55
0.6
2.017 1.728 1.677 1.672 1.644 1.593 1.607
2.600 2.223 2.155 2.152 2.127 2.093 2.196
3.284 2.810 2.727 2.723 2.700 2.700 2.940
4.114 3.519 3.415 3.413 3.401 3.453 3.898
5.130 4.389 4.260 4.260 4.260 4.389 5.130
6.398 5.477 5.322 5.342 5.580 5.580 7.146
8.000 6.857 6.653 6.663 6.716 7.146 8.940
299
theory of the rheology of concentrated slurries which does not include a strong dependence of viscosity on particle size cannot be correct. However, the weak dependence of viscosity on m, for a given value of characteristic size k, suggests that the viscosity is not dominated by interactive forces between very small particles (less than 2 pm), since there is a big difference in the relative amount of these very small particles, and the specific surface areas, between a particle size distribution of m = 0.8and one with m = 0.5.
ACKNOWLEDGEMENTS
I
I
I11111
10 ROSIN- RAMMLER
I,,,,,
100 SIZE
1000 MODULUS. pm
Fig. 13. Computed variation of effective relative viscosity at 100 s-‘[,Q/p = (~n,./lOO~ + c(p~/p)J based on the fitting equations, for a Rosin-Rammler size distribution modulus of m = 0.6.
strongly dependent on the volume fraction of solid in the slurry but it is also strongly affected by the characteristic size of the particle size distribution. Table 2 illustrates another important conclusion: within the range of Rosin-Rammler size distribution modulus 0.4 < m < 0.9,the effective laminar viscosity is not strongly dependent on the value of m. In addition, the results for coal in a liquid of Newtonian viscosity of (1.2)( 10p3) Pa.s and those for quartz in a liquid of (2.8) (10p3) Pa.s both fitted the same set of equations in reduced viscosity and yield stresses. The surface chemical properties of coal and quartz are quite different, and the liquids are not expected to give electrochemical double layers on the surfaces. These results suggest that the strong influence of characteristic particle size is not due to packing, since packing is a function of m rather than k, nor due to absorbed layers on the particles. Any
This work was partly funded by the Cooperative Program on Coal Research of the Pennsylvania State University and partly by NSF Grant CBT-8414214.
REFERENCES
8 9 10 11
12 13 14 15
D. G. Thomas, J. Colloid Sci., 20 (1965) 267. I. R. Rutgers, RheoZogicoZActa, 2 (1962) 305. H. Kambe, Znt. Chem. Eng., 9 (1969) 164. V. V. Jinescu, Znt. Chem. Eng., 14 (1974) 397. M. Mooney, J. Coil. Sci., 6 (1951) 162. H. Eilers, Kolloid Z., 97 (1941) 313. I. M. Kreiger and T. J. Dougherty, Tmns. Sot. Rheol., 3 (1959) 137. G. I. Taylor, Phil. Trans. Royal Sot. London, A223 (1953) 289. R. B. Bird, W. E. Stewart and E. W. Lightfoot, Transport Phenomena, Wiley, New York, 1960. I. H. Krieger and S. H. Maron, J. App. Phys., 25 (1954) 72. G. Papachristodoulou, H. Boghossian and 0. Trass, Proc. 4th Znt. Symp. on Coal Slurry Combustion, Pittsburgh Energy Technology Center, 4 (1982) 1. C. Moreland, Canadian J. of Chem. Eng., 41 (1963) 24. R. Darby and B. A. Rogers, AZChE J., 26 (1980) 310. R. R. Klimpel, Powder Technol., 32 (1982) 267. M. Toda, M. Kuriyama, H. Konno and T. Honma, Powder Technor., 55 (1988) 241.
Technology,
56 (1988)
293
293 - 299
Rheology of Concentrated Slurries of Particles of Natural Size Distribution Produced by Grinding C. TANGSATHITKULCHAI
and L. G. AUSTIN
College
Sciences,
of Earth
and Mineral
PA 16802
(U.S.A.)
(Received
March 14, 1988; in revised
Department
of Mineral
108 Steidle
Building,
University
Park,
form July 27,1988)
SUMMARY
The rheological properties of concentrated slurries of natural size distributions of ground coal and quartz were determined in a cup-andbob viscometer in the laminar flow region, using a ridged bob and cup. Organic liquids of the same density as the solids were used to prevent settling of the solid. It was found that slurries had a true yield stress, followed by a transition to a Bingham plastic character (with a higher extrapolated Bingham yield stress) for shear rates above 40 s I. Empirical equations were developed to describe the results as a function of the Rosin-Rammler characteristic size and distribution modulii of the particle size distributions, for solid concentration of 30 to 60 vol.%. The two most significant parameters were volume per cent solids and size modulus, with the distribution modulus m having a minor effect between m = 0.4 to m = 0.9. Results from both coal and quartz were fitted by the same equations for plastic viscosity and yield stresses when these were expressed relative to the viscosity of the carrier liquid. Existing theories of slurry rheology do not appear to be adequate to explain the results.
INTRODUCTION
There is substantial literature on the theoretical prediction of rheological properties of dense slurries [ 1 - 71. However, many of the treatments contain no allowance for the shape or fineness of the size distribution of the solid dispersed in the slurry since they use the volume fraction of solid as the only variable. In addition, most of the experimental investigations of slurry rheology are in water systems where the results can be 0032-5910/88/$3.50
Engineering,
strongly influenced by electrical double layers and minor dissolved components which affect the degree of dispersion or flocculation of the finer particles. As part of an investigation of the influence of rheology on the power required to drive tumbling ball mills, it was necessary to have a systematic method of predicting slurry rheological properties for the size distributions of solid produced naturally by the grinding process. In the lack of suitable predictive equations, either theoretical or empirical, it was decided to develop an empirical description specifically for such natural size distributions.
EXPERIMENTAL
The rheological properties of the slurries were determined in the laminar region using a Haake Rotovisco RV-3 cup-and-bob viscometer, with the MK50 and MK500 heads which have maximum torques of 0.49 and 4.9 N.cm, respectively. The stators and cups (MVIIP and SVIIP models, 2.6 and 1.5 mm gaps, respectively) have vertically grooved surfaces, with grooves 1 mm deep by 1 mm wide spaced 2 mm apart, to prevent slip. An air bubble was retained in the recessed bottom to reduce the end effect. All experiments were carried out at a constant temperature by using a heating-refrigerating bath and circulator system to maintain the temperature at 25 + 0.5 “c. Two materials were studied, a crystalline white quartz and a coal (low volatile bituminous, Hardgrove Grindability Index = 90,16 wt.% ash content). Controlled grinding of these materials in a laboratory ball mill (200 mm diameter with 25 mm steel balls) for various times gave size distributions which could be fitted by the Rosin-Rammler @ Elsevier Sequoia/Printed
in The Netherlands
294
I 10 PARTICLE
I wlo
loo SIZE k+ pm
Fig. 1. Rosin-Rammler size distribution plots of quartz batch ground in a ball mill at 40 vol.% solids in water.
equation over most of the size range (see Fig. 1):
The liquids used to prepare the slurries were mixtures of perchlorethylene and aliphatic napthas (American Minechem Co., PA) to give specific gravities of 2.64 (for use with quartz) and 1.36 (for use with coal). Use of these non-polar liquids had three major advantages: the slurries produced were essentially non-settling; the liquids are excellent wetting and dispersing liquids for the solids; the chemical effects of pH, etc., present with water are avoided. The liquids had Newtonian viscosities of (2.8)(10e3) and (1.2)(10e3) Pa.s, respectively. The maximum operating speed was chosen to lie within the laminar flow region using the definition of Reynolds number for Newtonian couette flow, Re = ~~NPR,~CS - 1)
m
P(3C)=l-exp
[
01 i
(1)
where P(X) is the cumulative fraction less than sieve size X, lz is the fineness modulus and m is the distribution modulus. Additional samples were also prepared by blending the required amounts of ,/2 sieve fractions and various fl sub-sieve fractions. The sub-sieve fractions were prepared in a Donaldson A-12 air classifier (Donaldson Co., MN), with the finest fraction being a minus 2 pm fraction (see Table 1). TABLE
1
Particle size distribution of test samples for rheological determination as represented by the distribution modulus m and size modulus k of the RosinRammler equation Material
Distribution modulus
Size modulus
Preparation
k
(pm)
Quartz (40 vol.%)
where N is revolutions per second; p is the liquid density; R,, is the bob radius; S is the cup/bob diameter ratio; p is the coefficient of viscosity, and [B, 91 (3)
Recrit =
It was assumed that this relation would apply to the non-Newtonian slurry with /,Ltaken as the apparent coefficient of viscosity at the highest shear rate and p is the total slurry density. Figure 2 gives a typical result of torque uersus rotor speed, showing that the rheological regime was pseudo-plastic with yield stress followed by Bingham plastic at higher shear rates. Therefore the shear rate at the bob surface was calculated from the Kreiger and Maron [lo] equation, -_
m
(2)
cc
dv dr
1.22
=
q,
[1+k&
-1)
b
S2
Coal (40 vol.%)
Ball milling in water
Quartz (54 vol.%) Coal (54 vol.%)
0.78
Coal
0.50
14,40
Coal
0.40
lo,25
Blending I
+
k,(-$ - 1)2]
k,=
G(l+ f
k,=
:
Q2 -
6s’
1
L 1nS
1nS)
(4)
295
:
60
so-
I
0
0
20
40
60 ROTOR
60 SPEED,
100
120
140
r~rn
Fig. 2. Typical rheological data of torque uersus rotor speed for coal slurries at various concentrations (Rosin-Rammler size modulus, 50 pm, distribution modulus, 0.78; liquid s.g., 1.36; liquid viscosity, 0.0012 Pa.s).
20
Fig. 3. Shear as a function distribution modulus, 10 0.0012 Pa.s).
40 SHEAR
60
60 s-’
RATE.
100
120
stress versus shear rate for coal slurries of solid concentration (Rosin-Rammler modulus, 0.40; Rosin-Rammler size pm; liquid s.g., 1.36; liquid viscosity,
50,
I
I
I
0.5
0.6
,
d(ln T) n”Z _ d(ln N) where T is the bob torque at the wall calculated from 2nRb2hrb, h being the immersed bob depth and rb the shear stress. The yield value was determined by running the bob at 1 rev./min for 2 to 3 min and noting the torque after the rotation was stopped.
EXPERIMENTAL
0.1 SOLID
0.2 0.3 VOLUME
0.4 FRACTION
Fig. 4. Effect of solid concentration and fineness of size distribution on the true yield stress for quartz slurries (Rosin-Rammler distribution modulus, 0.92; liquid s.g., 2.63; liquid viscosity, 0.0028 Pa.s).
RESULTS
The results contained the features shown in Fig. 3, that is, a true yield value r,, a Bingham yield value ray and a ppL, and a transition region from zero to about 20 to 40 s-l shear rate. Typical variations of these values with slurry concentration C (volume fraction of solid in the slurry) and size modulus 12for a given distribution modulus m are shown in Figs. 4, 5 and 6. Figure 7 gives a typical result expressed as relative apparent viscosity (defined by the apparent viscosity at a shear rate divided by the Newtonian viscosity of the suspending liquid) versus shear rate, as a function of volume fraction of solids. It is clear that all three variables C, k and m affect the rheological behavior for the denser slurries. The pattern of rheological behavior of “pseudo-plastic with yield stress followed by Bingham plastic” has been reported by other
._g@~ 0-s 0
_I 0.1 SOLID
2
:
0.2 0.3 0.4 VOLUME FRACTION
0.5
0.6
Fig. 5. Effect of solid concentration and fineness of size distribution on the plastic viscosity of quartz slurries (see Fig. 4).
workers [ 11 - 131 studying coal slurries in water, oil, methanol and coal-derived liquids. The behavior does not agree with the results
296
0 0.1
0
SOLID
I 0.2 0.3 0.4 VOLUME FRACTION
0.5
0.6
Fig. 6. Effect of solid concentration and fineness of size distribution on the Bingham yield stress of quartz slurries (see Fig. 4). 1
10
100
PARTICLE
SIZE.
1000
pm
(bl
Fig. 8. Rosin-Rammler distribution plots of coals samples prepared by blending various fractions of sieve-size and subsieve-size particles. (a), Distribution modulus = 0.5; (b), distribution modulus = 0.4.
reasonable accuracy over the bulk of the mass, so possibly pseudo-plastic behavior is associated with multi-modal size distributions.
TREATMENT
0
20
40 SHEAR
60
80 RATE,
100
120
s-l
Fig. I. Variation of relative apparent viscosity with shear rate as a function of particle size and solid concentration for quartz slurries having a RosinRammler distribution modulus of 1.22.
reported by Klimpel [14] for ground ores and coals slurried with water; they report a “pseudo-plastic with yield stress” behavior with constant values of Flow Index n in the expression (5) for a particular C and particle size distribution. Toda et al. [ 151 report “pseudo-plastic behavior without yield stress” for mixtures of two fi size fractions, one much finer than the other. Figure 8 shows that the blends prepared for the work reported here fitted a Rosin-Rammler size distribution with
OF RESULTS
To avoid presenting the data purely in the form of a large number of figures, empirical equations were developed by curve-fitting to the data. The basic equation used was chosen to be 7 _=_
TY
P
/J
+
(6)
where Q is a function to be determined which allows for pseudo-plastic behavior at low shear rates, and 7y, Tgy, ppL are functions of C, k and m. Q becomes 1 at higher shear rates and eqn. (6) reduces to the Bingham plastic equation. Since other studies [2] indicate that the apparent viscosity under given conditions is directly proportional to the Newtonian viscosity of the suspending liquid, /Jsay, the values were expressed as a ratio to this viscosity by dividing by 1-1as shown. Figure 9 shows the values of Q as a function of shear rate. It was felt sufficient to approximate the ranges of Q with a single empirical function
a
5
10
I5 SHEAR
20 RATE.
25
30
35
s-’
Fig. 9. Variation of correction function Q for pseudoplastic behavior at low shear rate (<40 s-r).
Q =1
-exp[0.32(~~‘2]
(7)
The yield stresses and plastic viscosity were fitted to power functions, as illustrated in Figs. 10 and 11. The constants in the equations were correlated with C and k by trialand-error choice of fitting functions. Once the functions had been chosen the constants in the functions were determined with a nonlinear regression routine, to give 0.3 < C < 0.6
(8)
0.3 < c < 0.55
(9)
at least for 0.4
Substituting in eqns. (8) and (9) would reintroduce m into the equations. For example, for \L= 0.8, so that k* is the 80%passing size, k = k*/(1.609)““, and (6500)(1.609)"' k*
1
4-48c”3s
(84
ROSIN-RAMMLER
SIZE
MODULUS,
k, pm
Fig. 10. Correlation of measured yield stress with particle size distribution parameters and solid concentration.
I SOLID VOL. FRACTION
L
-
EPUATION
9
1022L 1
H) ROSIN-RAMMLER
SIZE
100 MODULUS,
k. pm
Fig. 11. Correlation of Bingham yield stress with particle size distribution parameters and solid concentration.
For a given value of k* and C, a smaller value of m would give a higher yield stress, as expected. This reasoning shows that a higher yield stress is associated with a smaller m provided that the characteristic size k* is taken as greater than the 63.21%passing size. The important conclusion is reached that the yield stresses are not dominated principally by the amount of ultra fine material but depend also on the total size distribution, so that (at least for natural Rosin-Rammler distributions in
298
non-interactive liquids) the 63.21%passing size is an index of yield stress irrespective of the distribution modulus m. In a similar manner, the variation of plastic viscosity with the characteristics of the particle size distribution were fitted by the empirical equation
c
1.1 +exP(-14.2m2.6)
1
x-
i
1-c
k-o.21
0.L IO
Wa) It can be seen that for any given concentration of solid C and a fixed value of k, there is a minimum in plastic viscosity at a particular value of m. For example, the function f(C!, m) has a minimum value of 1.6 at about m = 0.45 for C = 0.3, but the value of m giving the minimum moves to higher values as
I
I1,lll
I
SIZE
(11)
Again, higher values of C and smaller values of k give increased plastic viscosity, as illustrated in Fig. 12. The effect of m is more complex. The first term on the right-hand side of the equation has a minimum at m = 0.7. On the other hand, the effect of m in the exponent of the second term depends on whether C is greater or less than 0.5. Table 2 shows values of the function f(C, m) defined by
I
I
/
100 MODULUS, lrn
,,l,l
1000
Fig. 12. Variation of relative plastic viscosity with Rosin-Rammler size modulus as a function of per cent solids in the suspension of quartz (constant Rosin-Rammler distribution modulus, 0.92; liquid s.g., 2.63; liquid viscosity, 0.0028 Pa.s).
concentration increases, being about 0.8 for C = 0.6. The accuracy of the equations in predicting the apparent viscosities for the 100 tests at 5 or more shear rates per test was obtained by determination of the standard deviation of relative difference between a computed and measured result, giving a value of standard deviation of error of about 10%.
DISCUSSION OF RESULTS AND CONCLUSIONS
Figure 13 illustrates a major conclusion: the effective laminar viscosity is of course
TABLE 2 Values of f(C, m) in eqn. (lla) Distribution modulus m
1.0 0.9 0.8 0.7 0.6 0.5 0.4
Function f( C, m) Volume fraction of solid C = 0.3
0.35
0.4
0.45
0.5
0.55
0.6
2.017 1.728 1.677 1.672 1.644 1.593 1.607
2.600 2.223 2.155 2.152 2.127 2.093 2.196
3.284 2.810 2.727 2.723 2.700 2.700 2.940
4.114 3.519 3.415 3.413 3.401 3.453 3.898
5.130 4.389 4.260 4.260 4.260 4.389 5.130
6.398 5.477 5.322 5.342 5.580 5.580 7.146
8.000 6.857 6.653 6.663 6.716 7.146 8.940
299
theory of the rheology of concentrated slurries which does not include a strong dependence of viscosity on particle size cannot be correct. However, the weak dependence of viscosity on m, for a given value of characteristic size k, suggests that the viscosity is not dominated by interactive forces between very small particles (less than 2 pm), since there is a big difference in the relative amount of these very small particles, and the specific surface areas, between a particle size distribution of m = 0.8and one with m = 0.5.
ACKNOWLEDGEMENTS
I
I
I11111
10 ROSIN- RAMMLER
I,,,,,
100 SIZE
1000 MODULUS. pm
Fig. 13. Computed variation of effective relative viscosity at 100 s-‘[,Q/p = (~n,./lOO~ + c(p~/p)J based on the fitting equations, for a Rosin-Rammler size distribution modulus of m = 0.6.
strongly dependent on the volume fraction of solid in the slurry but it is also strongly affected by the characteristic size of the particle size distribution. Table 2 illustrates another important conclusion: within the range of Rosin-Rammler size distribution modulus 0.4 < m < 0.9,the effective laminar viscosity is not strongly dependent on the value of m. In addition, the results for coal in a liquid of Newtonian viscosity of (1.2)( 10p3) Pa.s and those for quartz in a liquid of (2.8) (10p3) Pa.s both fitted the same set of equations in reduced viscosity and yield stresses. The surface chemical properties of coal and quartz are quite different, and the liquids are not expected to give electrochemical double layers on the surfaces. These results suggest that the strong influence of characteristic particle size is not due to packing, since packing is a function of m rather than k, nor due to absorbed layers on the particles. Any
This work was partly funded by the Cooperative Program on Coal Research of the Pennsylvania State University and partly by NSF Grant CBT-8414214.
REFERENCES
8 9 10 11
12 13 14 15
D. G. Thomas, J. Colloid Sci., 20 (1965) 267. I. R. Rutgers, RheoZogicoZActa, 2 (1962) 305. H. Kambe, Znt. Chem. Eng., 9 (1969) 164. V. V. Jinescu, Znt. Chem. Eng., 14 (1974) 397. M. Mooney, J. Coil. Sci., 6 (1951) 162. H. Eilers, Kolloid Z., 97 (1941) 313. I. M. Kreiger and T. J. Dougherty, Tmns. Sot. Rheol., 3 (1959) 137. G. I. Taylor, Phil. Trans. Royal Sot. London, A223 (1953) 289. R. B. Bird, W. E. Stewart and E. W. Lightfoot, Transport Phenomena, Wiley, New York, 1960. I. H. Krieger and S. H. Maron, J. App. Phys., 25 (1954) 72. G. Papachristodoulou, H. Boghossian and 0. Trass, Proc. 4th Znt. Symp. on Coal Slurry Combustion, Pittsburgh Energy Technology Center, 4 (1982) 1. C. Moreland, Canadian J. of Chem. Eng., 41 (1963) 24. R. Darby and B. A. Rogers, AZChE J., 26 (1980) 310. R. R. Klimpel, Powder Technol., 32 (1982) 267. M. Toda, M. Kuriyama, H. Konno and T. Honma, Powder Technor., 55 (1988) 241.