I
P. R. CROWLEY' and A. S. KITZES Oak Ridge National Laboratory, Oak Ridge, Tenn.
Rheological Behavior of Thorium Oxide Slurries in Laminar Flow These data are an integral part of information needed in designing the blanket system of a thorium oxide slurry breeder reactor
FLUIDS
are classified rheologically as Xewtonian or non-Newtonian. The velocity gradient, dV/dr, of Newtonian fluids in laminar motion is directly proportional to the shear stress acting on the fluid, r, or
Equation 1 is known as Newton's law and p is termed the coefficient of viscosity. A plot of the velocity gradient us. shear stress is known as a shear diagram. Figure l , A, illustrates a shear diagram of a Newtonian fluid; the reciprocal of the slope of this curve represents the viscosity. Non-Newtonian fluids are classified and compared by means of shear diagrams, illustrated in Figure l , B to E. Figure 1, B, is a general shear diagram for the plastic class of non-Newtonians. Using the definition of viscosity expressed in Equation 1, it may be seen that the viscosity of plastic fluids remains infinitely large until a definite value of shear stress is reached. Further increase in shear stress is then accompanied by a decrease in viscosity. The shear stress at the point where the viscosity becomes finite is designated as the yield stress, 7,. Bingham plastics (Figure 1, C) may be considered special cases of the plastic class, in which the shear diagram
is a straight line above the yield stress. The reciprocal of the slope of this line is termed the coefficient of rigidity, q, expressed in units of viscosity. The fluid illustrated in Figure 1, D, represents the pseudoplastic class of nonNewtonians, in which the viscosity is a finite value, PO, at rest and decreases as the shear stress is increased. Figure 1, E, represents the dilatant class, which differs from the pseudoplastic class only in that viscosity increases as shear stress increases. An application of the viscosity concept is the prediction of pressure drop caused by the laminar flow of fluids in lengths of circular conduit. Mooney (7) has derived a general expression, Equation 2, relating the coordinates of shear diagrams with the average velocity in a pipe, V,, the diameter of the pipe, D , and the pressure drop, AP, across a length, L, of the pipe.
(4)
A plot of 8Va/g,D us. rD is termrd a pseudo shear diagram. Pseudo shear diagrams for all types of non-Newtonian fluids may be obtained from shear diagrams by graphical integration of Equation 3. In some cases, however, the relationship between shear stress and velocity gradient may be expressed mathematically and Equation 3 integrated directly. For Bingham piastics the true shear diagram is described by Equation 5 and integration of Equation 3 yields Equation 6.
(5)
where T D equals DAP/4L. form of Equation 2 is
Present address, General Mills, Minneapolis, Minn.
88 8
When Equation 1 is introduced into Equation 3, the integration yields for Newtonian fluids
INDUSTRIAL AND ENGINEERING CHEMISTRY
An integral
Farrow and Lowe (5) have suggested that some pseudoplastic fluids are described by the relationship of Equation 7 and hence the pseudo shear diagrams of these fluids have the form of Equation 8, where n is a constant greater than unity and typical of a particular suspension.
(7)
paints a t infinite shear can be correlated with concentration by the equation pLm = PLlog-1
Other workers have proposed analytical expressions describing pseudoplastic materials. Powell and Eyring (9) have suggested the form of Equation
- g-1c
[rz)
1 + Bsinh-l
1 dV
x)] = (9 )
where A, B, and C are constants of a particular fluid, Christiansen, Ryan, and Stevens (4)have derived pipe-flow equations for the Powell-Eyring relation and presented their results in graphical form. Reiner and Riwlin (70) have derived an expression for the form of pseudo shear diagrams for pseudoplastics based on a model suggested by Hatschek (6) : TnF
where po is the rest viscosity, p m is the viscosity a t infinite shear, and F is a constant. Hatschek’s model assumed that the particles in a suspension carry with them fluid envelopes, sometimes referred to as lyospheres, which have a maximum volume when the fluid is resting and decrease toward a minimum in volume with increasing velocity gradient. Reiner and Riwlin also assumed in their derivation of Equation 10 that the effective volume fraction of solids decreases with shear exponentially, and that Einstein’s equation relating the viscosity of a Newtonian suspension to the volume fraction of solids applies. Zettlemoyer and Lower (72) have applied Hatschek’s concept of immobilized fluid envelopes to calcium carbonate-polybutene oil suspensions and found that the viscosity of these suspensions can be expressed by a modified form of Einstein’s equation. Their expression, shown in Equation 11, includes a correction factor for the increase in the effective volume of the solids :
Ps
- PL PL
=
K* (1
+ KP A ) S
(11)
where K1 is constant for a system of given constitution, K Pis another constant, A is the surface area of the solids so that K A is the volume of the “adsorbed” layer of vehicle, and S is the volume fraction of the pigment. Asbeck, Scherer, and Van Loo (7) have recently found that the viscosity of
ukS
1-s
(12)
where p L is the viscosity of the suspending fluid, S is the volume fraction of pure solids, and u and k are constants for a particular system. The equation was extended to express the viscosity of paint suspensions as a function of concentration and “shearing velocity,” E, for “shearing velocities” greater than 300 set.-' Their final expression is
where a! and 6 represent two additional constants of the system. Efforts have also been made to correlate the yield stresses of non-Newtonians with concentration. Bingham (3) reports that the yield stress of a certain clay increased in proportion to the increase in concentration after a minimum value of concentration was reached. Babbitt and Caldwell (2) found a similar phenomenon in clay, although the increase in yield stress was not strictly linear. Norton, Johnson, and Lawrence (8) report that the yield stresses of certain kaolinite slurries increase with the third power of concentration.
Model and Analysis The model used in this paper for the analysis of non-Newtonian flow is similar to the one proposed by Hatschek. I t is postulated that the dispersed particles in a non-Newtonian suspension are solvated and the amount of liquid carried in the solvated envelope of bound water varies with shear stress. I n order to simplify the analysis, it is further postulated that the solid particles of the suspension are spherical in shape and of uniform radius, r, and that at any given shear stress, the envelope is of uniform thickness, t . If it is postulated that the viscosity of the suspension is a function only of the volume fraction of solids present, as in the case of Newtonian suspensions, only two functions remain unknown in order to complete the analysis-the expression for the viscosity of a suspension in terms of the volume fraction of effective solids and the relationship between shear stress and envelope thickness, t . A relationship between suspension viscosity and volume fraction of solids has been proposed by Robinson (7 7) as
B
1 C
I
D
E Figure 1. Shear diagrams of Newtonian and non-Newtonian classes of fluids A. 6. C. D.
E.
Newtonian fluid Plastic fluid Bingham plastic fluid Pseudo plastic fluid Dilatant fluid
VOL.49, NO. 5
MAY 1957
889
where x is termed a frictional coefficient and y is the relative sediment volume or volume ratio of sediment to solids. Selecting Einstein’s value for x and the porosity of the settled bed as 0.40, as in the case of randomly packed spheres, x becomes 2.5 and y is 1.667, and Equation 14 may then be written as PA9 - P L -
2.5s
1
PL
-
1.667s
(15)
In the case of solvated particles, the volume fraction of effective solids, ST, may be substituted directly for S in Equation 15 and the viscosity of a suspension of uniform solvated spheres becomes
Figure 2. Pseudo shear diagram as a function of solidity, water slurry at 25.0’ C.
With the postulate that the dispersed particles are spherical and the envelope thickness uniform, the envelope thickness may be related to particle radius by the volume ratio of liquid in the envelopes to solid particles, v, as
SI for thorium oxide-
t =
Y
+
[(v
1)”3
-
I]
(17)
Because the volume fraction of effective solids, ST,is the sum of the volume fractions of pure solid, S, and fraction of liquid immobilized in the envelope, ST and S are related by the equation ST =
s(1
+
(18)
Y)
Combining Equations 17 and 18, the volume fraction of effective solids may be written in terms of volume fraction of pure solids and the ratio t/r as ST = S(t/r
t or
(DAP/4Lll/b.
+ 1)a
(19)
Combining Equations 16 and 19 yields an expression for the viscosity of a nonKewtonian fluid in terms of viscosity of the suspending liquid, volume fraction of pure solids, and the shear stress-dependent ratio, t / r , or
SeC.)
Figure 3. Shear diagrams as a function of solidity, S, for thorium oxide-water slurry including lines of constant viscosity, a through m
Table 1. Values of S, pSl and T from Figure 3 and Computed Values of S = 0.0739 S = 0.0903 S = 0.0552 S = 0.0350 Curve a b c d e J’ Q
h i j
k
I m
890
Ps, Lb.M/
7
v
1b.F/ sq. ft.
6.13 6.73 7.23 7.68 7.95 8.30 8.55
0.230 0.155 0.125 0.105 0.095 0.086 0.078
9.87
0.060
Ib.F/
Ft.-Sec.
ST
Y
sq. ft.
0.00145 0.00185 0.00238 0.00290 0.00350 0.00425 0.00490 0.00605 0.00728 0.00947 0.0126 0.01975 0.0516
0.287 0.346 0.393 0.427 0.454 0.479 0.494 0.513 0.527 0.543 0.557 0.572 .o.589
7.20 8.88 10.24 11.21 11.96
0.116 0.067 0.045 0.039 0.033
m
0.600
16.15
0.020
INDUSTRIAL AND ENGINEERING CHEMISTRY
s
v
Ib.F/ sq. bt.
v
0.0091 lb.F/ sq. it.
0.400 0.325 0.286 0.242 0.212 0.190 0.173 0.154 0.140
4.31 4.47 4.68 4.83 5.01 5.16 5.33 5.53
0.700 0.636 0.563 0.506 0.446 0.393 0.340 0.290
3.98 4.18 4.32 4.48 4.62 4.77 4.94
0.130
5.64
0.270
5.05
v
r lb.F/ sq. f t .
5.13 5.48 5.68 5.94 6.13 6.34 6.53 6.74 6.97 7.12
7
ST and v
7
=
S = 0.1138 7 lb.F/ Y
sq. ft.
0.830 0.755 0.695 0.616 0.540 0.455 0.375
3.63 3.77 3.88 4.02 4.17
1.196 1.073 0.950 0.805 0.665
0.340
4.27
0.590
T H O R I U M OXIDE SLURRIES 17
I
I
I
-.
S
3
1
I
S
.-
1
I
0 0.0350 0 0552 A 0 0739
L
I 0.8
0.4
(T) Shear
20
L
Stress (lb./sq. ft.)
T
Determination of Function t/r
The relationship between shear stress and envelope thickness is not known quantitatively and, therefore, must be determined experimentally. A suspension of thorium oxide particles in water containing approximately 1500 p.p.m. of sulfate added as thorium sulfate was conditioned by circulation for over 100 hours a t 300’ C. in a pressurized pumploop system. Shear diagrams for various concentrations of thorium oxide were prepared from data obtained by using a pipeline-type viscometer. A tube, 67.5 inches long with an inside diameter of 0.125 inch, was employed as the test section. The ends of the tube were machined square and corrections for the kinetic effects were made using the relationship (D$)=(!g) measured
inpD Pa L 4gc
’
and viscosity (the intercepts of the constant viscosity lines and shear diagrams). Shown in Table I are values of volume fraction of total effective solids, ST, obtained from Equation 16 for each viscosity. From Equation 18, the ratio of envelope volume to particle volume, v, was evaluated at each shear stress. These values are tabulated for each concentration and shear stress in Table I. Plots of v us. T are shown in Figure 4. Sample Calculation of Quantity t/r. The viscosity, p , of any point on line a of Figure 3 may be computed from Equation l as
. - /0.00145 \ ST =
lSZ
(0.000597) -
. = 0.287
0.000597
The shear diagram for the suspension having a concentration, S, of 0.0350 intersects line a at T = 0.116 1b.F per sq. ft. (Figure 3). The ratio of envelope volume to solids volume for the suspension having a concentration, S, of 0.0350 a t the shear stress, 7,of 0.116 l b a Rper sq. ft. may be computed from Equation 17 as
s(1 f
ST =
V)
o r v = ST -- 1
S
or
p =
7
--
or
(211
where m was assumed equal to 1.12 as recommended by Bingham (3) for Newtonian fluids. The data are shown as pseudo shear diagrams in Figure 2, where SVa/gcD, the average rate of shearing strain, is , shear stress a t the plotted against T ~ the wall. True shear diagrams, shown in Figure 3, were constructed from the pseudo shear diagrams by using the Mooney Equation 2. Lines of constant viscosity ( a through rn) were constructed on the true shear diagrams covering the complete range of viscosity. Values of shear stress were evaluated as function of concentration
(IbJsq. It)
Figure 5. Relation of t / r ratio to shear stress for thorium oxidewater slurry (log plot)
Figure 4. Bound water as a function of shear stress for thorium oxide-water slurry
p =
0.290 lb 200 - 0 . 0 0 1ft.-sec. 452
Ratio t / r for the suspension having a concentration, S , of 0.0350 at the shear stress: 7, of 0.116 1bsF/sq. ft. may be computed from Equation 1 6 as t- = [(v
The volume fraction of effective solids represented by line a of Figure 3 may be calculated from Equation 13 as Ps = P L [1;22-+2;T]
or
ST
= 1.2
(”-”= I ) PL
24, + 1 PL
or using pL = 0.000597 Ib.M/ft.-sec. at 25’ C.
+ Iy’a - 11
= l(7.20
+
1)1/8
- 11 = 1.016
To determine the functional relationship between envelope thickness and shear stress, ratio t/r was plotted us. T on logarithmic grids (Figure 5). The values plot as a straight line and the slope has a value of approximately -0.2. Similar plots for other thorium oxide slurries have been made; the lines are essentially parallel. With this in mind, it is possible to VOL. 49, NO. 5
4
MAY 1957
891
express the relationship between envelope thickness and shear stress quantitatively as
Combining Equations 20 and 22, the final expression for the viscosity of a nonNewtonian slurry is
Equation 23 may be combined with Newton’s relationship; the form of the shear diagrams of these materials becomes
Then, from Mooney’s equation in the integral form, the expression for the pseudo shear diagram may be written
The lower limit of the integral of Equation 25, T ~ may , be evaluated from Equation 24 when dV/dr is zero or
Unfortunately, Equation 25 lvould be excessively laborious to evaluate numerically, because of the form of the function of r in the integral. Therefore, the integral of Equation 25 can be evaluated using a computer and the results plotted as the function (8V,/D)(pL/4g,) us. rv in families of S for various values of c.
Conclusions
T h e actual existence or importance of the lyosphere envelope, which is an integral part of the analytical model, has been questioned by many authors. Although this work does not constitute a verification of the existence of lyospheres, the results demonstrate the utility of the lyosphere concept. The quantitative relationship among suspension viscosity, liquid viscosity, solidity, and shear stress, expressed in Equation 23, makes it possible to calculate the viscosity of a non-Newtonian having the flow properties exhibited by thorium oxide slurries as a function of shear stress and concentration, when only one experimentally determinable constant, c, is known. Using Equation 23, constant c may be determined from
892
anv one measured viscosity or yield stress at a known concentration. From a more practical standpoint, the analysis makes possible the calculation of the pressure drop in any pipe for any concentration of slurry by using Equation 25 and the measurement of one pressure drop in a given pipe a t any one concentration. A valuable corollary of this conclusion is that by knowing the yield stress at one concentration, the yield stress at any slurry concentration may be predicted from Equation 26. A conclusion of secondary importance is that non-Kewtonian fluids satisfying the relationship of Equation 23, such as thorium oxide slurries, are neither pseudoplastic nor strictly Bingham plastic in behavior. In order to satisfy Equation 23, a fluid must necessarily have a yield stress a1 all concentrations of solids above infinite dilution. From Equation 26 it may be seen that rplequals zero only when S equals zero, and hence the fluid cannot be a pseudoplastic. That the fluid cannot satisfy Equation 23 and be Bingham plastic may be seen from a comparison of Equation 5 for a Bingham plastic with Equation 24. However, a material can comply with Equation 23 and still appear to be a Bingham plastic, as demonstrated by the shear diagrams for low solidity in Figure 3. Several limitations of the proposed analysis are evident. First, the data thus far obtained have been limited to Iow temperatures. Therefore, the applicability of the analysis to high temperatures-Le., 100’ to 300’ C. where the rheological properties of thorium oxide slurries are most pertinent for use in reactor systems-has not been proved. Secondly, the data verifying the analysis have been obtained only with systems of thorium oxide and water, thereby limiting its generality. A third limitation of the analysis results from the relatively small range of the variables of concentration and shear stress explored. I t is possible that the relationship between t / r and r deviates from that shown in Figure 5 at values of shear stress less than 0.02 pound per square foot and greater than 1.1 pounds per square foot. However, a similar plot for a different thorium oxide system indicated the validity for shear stress u p to 4.5 pounds per square foot and solidities up to 0.155.
Nomenclature
A , B, C = constants in Powell-Eyring equation D = diameter of circular conduit, feet E = “shearing velocity” in Asbeck equation, sec.-l F = constant in Reiner-Riwlin equation K1, Kz = constants in Zettlemoyer-Lowe equation
INDUSTRIAL AND ENGINEERING CHEMISTRY
L AP
length or circular conduit, feet pressure drop across length L of circular conduit, lb.,/sq. foot S = volume fraction pure solids or solidity, cu. feet per cu. foot S, = volume fraction effective solids or effective solidity, cu. feet per cu. foot dV/d7= rate of shearing strain or velocity gradient, set.+ V, = average velocity in circular conduit, fect per second. c = constant 32.2 ft.-lb.,v gc = conversion factor, sec.2-lb.n k , u, a, 6 = constants in Asbeck equations m = coefficient n = exponent in Fdrrow-Lowe equation r = radius of particles, feet t --. thickness of liquid envelope, feet x = frictional coefficient in Robinson equation y = relative sediment volume, cu. feet per cu. foot 7 = coefficient of rigidity, lb.,w/ft.-sec. p = viscosity of Newtonian fluid, lb.“/ ft.-sec. = viscosity of suspending liquid, Ib.,/ft.-sec. po = viscosity of pseudoplastic at no shear stress, lb.,/ft.-sec. ps = viscosity of suspension, lb.M/ft.sec. ,urn= viscosity of pseudoplastic at infinite shear stress, lb..w/ft.-sec. Y = volume ratio of envelope to pure solids, cu. feet per cu. foot p = density of fluid, lb.M/cu. foot T = shear stress, Ib.F/sq. foot TD = shear stress at diameter D in circular conduit, lb.p/sq. foot r y = yield stress, lb.F/sq. foot = =
literature Cited (1) Asbeck, W. K., Scherer, G. A., Van Loo, M., IND.ENG.CHEM.47, 1472
(1955). (2) Babbitt, H. E., Caldwell, D. H., “Laminar Flow of Sludges in Pipes,” Univ. of Ill. Engineering Experiment Station, Bull. 319 (Nov. 14, 1939). (3) Bingham, E. C., “Fluidity and Plasticity,” McGraw-Hill, New York, 1922. (4) Christiansen, E. B., Ryan, N. W., Stevens, W. E., A.Z.Ch.E. Journal 1, 544 (1955). (5) Farrow, F. D., Lowe, G. M., J . T e x tile Znst. 14, 414 (1923). (6) Hatschek, E., “Viscosity of Liquids,” Van Nostrand, London, 1928. (7) Mooney, M., J . Rheol. 2, 210 (1931). (8) Norton, F. H., Johnson, A. L., Lawrence, W. G., J . Am. Ceram. SOG. 27, 149 (1944). (9) Powell, R. E., Eyring, H., Nature 154, 427 (1 944). (10) Reiner, M., Kiwlin, R., Kolloid Z . 44, 9 (1928). (11) Robinson, J. V., J. Phys. and Colloid Chem. 53, 1042 (1949). (12) Zettlemoyer, A. C., Lower, G. W., J. Colloid Sci. 10, 1 (1955). RECEIVED for review June 29, 1956 ACCEPTED November 14, 1956
