Part I
PHYSICAL PROPERTtES AND LAMINAR TRANSPORT CHARACTERISTICS DAVID 0. THOMAS
on-Newtonian fluid mechanics is a curious blend N of theory and empiricism interwoven with approximations and hypotheses. The thmny has not reached the same stage of development as had mechanics at the time of Newton for as yet there is no commonly accepted rheological equation of state which satisfactorily describes the relation between the shear s t r e ~ sand the rate of shearing strain, let alone the composition, the thermodynamic variables, and time. Furthermore the necessaiy and sufficient measurements required to characterize a given material cannot be speci6ed as yet for any except the simplest boundary conditions. Despite these deficiencies non-Newtonian fluids and suspensions are being handled on a very large scale in many industries and many proposed uses involving such fluids have been studied through the pilot plant stage. Typical of those products or processes in which the non-Newtonian characteristics of suspensions are important are the paint (63, solid propellant (79) and the petroleum industries. I n the latter case over 180 million dollars per year are spent for drilling muds and chemicals for use in well drilling operations (26). Among the proposed applications are the addition of magnesium or boron as fine particles to jet fuels to enhance either the range or the thrust (4, the propulsion of interplanetary rockets by elecnostatic acceleration of electrically charged particles of colloidal dimensions (50), and the synthesis of hydrazine (75) or ethylene glycol (33) in chemonuclear reactors using a suspension fuel. The information described in the present paper was evolved as a portion of the development program for aqueous homogeneous breeder reactors. One possible conception (34) of such a plant would generate 330 megawatts of electrical power while operating at a thermal power of 1140 megawatts. This power could be generated in a two-region machine in which a slurry containing 200 g. of thorium and 14 g. of uranium per liter is circulated 18
INDUSTRIAL AND ENGINEERING CHEMISTRY
Flow of a non-Nmtw‘m supnuion in a showing the distinctive uclm’typrofi[c of a B i n g h plastic m a t d . Nem the wall, shearing f m m arb highest, andp”rticle agglmaks me b r o h up 01 rolled into d l , relatiwdy dense masses. I n the center of theflpe, shear u bdow
through a 7 foot diameter core at a rate of 90,000 g.p.m., while a slurry containiig 1000 g. of thorium per liter is circulated through a 2 foot thick blanket region at a rate of 30,000g.p.m. The purpose of this paper is to illustrate how theory, simplified models, and empirical correlations can be combined to give the quantitative description-required for the design of large scale systems--of the laminar and turbulent transport characteristics of suspensions. The approach followed in the research program was to relate the easily measured hindered-settling and laminar-flow propertis to the more important turbulent-flow characteristics, such as heat and momentum transfer and the minimum transport velocity for solids. When necessary, experiments were camed out to determine the relative importance of the particle characteristics, such as size and shape, on the laminar-flow physical properties. I n general, the present data apply to all aqueous suspensions of regularly or irregularly shaped, equiaxial, solid particles in the 0.1- to 20-micron size range which are composed of metal oxides or hydroxides or of materials which may act as reversible gas electrodes. Particles having the shape of needles or plates are definitely excluded, although it is believed that once the plates or needles are broken up into roughly equiaxial shape the proposed relations will provide a satisfactory engineering approximation. The properties of homogeneous suspensions of solids in liquids are, in some cases, simply functions of the properties of the pure materials and their concentration, and consequently will be referred to as intrinsic physical properties. Examples are the density, specific heat, and to a certain extent, thumal conductivity. These particular properties of a suspension can be calculated for any concentration and temperature if data for the pure materials are available. However, other properties, V O L 5 5 NO. 1 1
NOVEMBER 1 9 6 3
19
F z p e 7 . EFFECT OF ELECTROLYTE CONCENTRATION
O N THE
SLIP11116 P I M E
b fl
2
h I
double layer
DISTANCE
Normal distribution of potential in the
Reversal of potential in the double layer
such as settlimg rate and viscosity, are functions of particle size and shape as well as concentration. Furthermore, as the solids particle size is reduced toward colloidal dimensions, physicochemical forces become significant. Under these circumstances, not only does the presence or absence of, and the concentration of electrolytes become important but also the rate of shear (or velocity gradient) may exert a profound influence on such physical properties as the viscosity. These properties will be referred to as extrinsic physical properties.
The treatment is identical for all potential problems, hence the results are equally valid for electrical or thermal conductivity, electrical permittivity, etc. However, recent experimental studies (28,29) show that Equation 1 gives results that may be 5 per cent low at a volume fraction solids, +, of 0.1 and as much as 30 per cent low at values of = 0.3. Bruggeman (4) has extended Equation 1 by accounting for the ratio of the major to minor axes of the particles and for the ratio k,jk,, the result was:
+
llminiic Phyrlcal RoperHer
Intrinsic suspension properties are those properties (such as density) which are functions of the properties and concentrations of the pure materials alone. In all Cases these calculated properties are free from restrictions due to the presence of electrolyte, rate of shear, and so forth; and some, such as the viscosity, represent minimum values below which even the non-Newtonian property values can never be reduced. Since both the Newtonian viscosity and hindered-settling relations for suspensions are important factors in the evaluation of floc properties, they will be discussed briefly in the following paragraphs together with procedures for calculating the suspension density, heat capacity, and thermal conductivity, properties which are required for the heat transfer and fluid flow correlations. Suspension Density and Heat Capacity. The density of a suspension is the sum of the products of pure component density and the volume fraction of that component. The heat capacity of a suspension is the sum of the product of the pure component heat capacities on a weight basis and the weight fraction of that component all evaluated a t tlie bulk-mean suspension temperature. Suspension Thermal Conductivity. Maxwell derived the following expression for the electrical conductivity of discontinuous media (36):
20
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Equation 2 gave a satisfactory fit to the data for a variety
of different metal and metal oxide suspensions when
x
had values between 2.0 and 3.1, depending on the shape and thermal conductivity of the powders (29). Suspension Viscosity. The viscosity of suspensions of spherical noninteracting particles is a function only of the volume fraction solids. Einstein (70) has shown that, for sufficiently dilute suspensions of spherical particles, the viscosity is given by: PS
M I (1
+ 2S+)
(3)
He assumed that the system was incompressible, that there was no slip between the particles and the liquid, no inertial effects, and that the macroscopic hydrodynamic equations held in the immediate neighborhood of the particles. Einstein’s theory has been extended ( 5 8 , 1 I , 20, 22,59, 63,64) to greater concentrations by considering the hydrodynamic interaction between par-
En&u in the Reactm Diuison of Oak Ridge National Lobmatmy, Oak Ridge, Tmn. This instaiiation is operated by Union Carbide Cmj.fm the Atomic Enmgy Commission
AUTHOR David G. Thomas is a Dmelopmmt
ticles. Although no one equation adequately represents the data for all systems, many of the equations can be written in terms of a power series: P. =
+ + Ai@ + AaO' + . . . I
P Z [ ~ Ai9
(4)
The above references quote values for the coefficients in the following ranges:
2.5 < A i < 5.5 2.5 < A i < 14.1 2.5 < A I < 36.3
(7) Representative expressions (3,23, 35, 44,47, 52) for f(9) show a maximum deviation of about 40 per cent for any given value of 9; an average expression valid for the range 1 < U,/Uo < 0.08 is (58)
(8)
Although values for the higher coefficients have not been evaluated, attempts to fit the data empirically with a power series having positive coefficients were unsuccessful for volume fraction solids greater than 0.3. One alternative expression which fitted over the entire range was (59): P* = 1 + 2.59 + l0.05# + 0.062 exp [I Pi
ciently large to minimize colloidal forces. This can be expressed as:
is%+]
(5) Theoretical and empirical treatments have also included such nonspherical particles as ellipsoids (n), dumbells (57),and rods (5, 48). Since these particles may sweep out a variable volume of fluid as they rotate in a velocity gradient, the suspension viscosity is increased by an additional factor which is primarily a function of the axial ratio of the particles. Suspension Settling Bate. Single spherical particles settle with a velocity given by Stokes's law:
provided the particle Reynolds number is less than one, (D,Uopl/p, < 1). The presence of additional particles decreases the settling rate due to hydrodynamic interference among the particles as well as to the displacement of fluid as the particles settle. To the 6rst order, the decrease in settling rate is a function only of the volume fraction solids provided the particle size is suffi-
Exhinsic Physical ProperHes
Reduction of one or more of the particle dimensions to less than 5 to 10 microns results in increased relative importance of the physicochemical forces responsible for the distinctive behavior of colloidal materials (32). Although not truly colloidal, suspensions in which the particle sue is 0.1 to 20 microns are often observed to be flocculated. I n a flocculated suspension the particles stick together in the form of loose irregular clusters or flocs in which the original particles can s t i l l be recognized. Under these circumstances, small additions of electrolyte as well as the rate of deformation may markedly affect those physical properties, such as viscosity and settling rate, which are dependent on floc structure. As yet, there is no relation which gives a general description of the effect of electrolytes on the suspension physical properties. However, fundamental studies (9, 53) of the surface chemistry of aqueous suspensions suggest qualitative explanations of the observed hindredsettling and flow behavior in terms of the amount and nature of ions adsorbed on the surface of the particles. The adsorption of ions at the solid-liquid interface is often interpreted in t e r m s of a double-layer structure which has a characteristic electrokinetic potential (Figure 1 ) . The potential differencewhich exists in the neighborhood of the interface arises because of the unequal tendencies of ions to be distributed between the solid and liquid phases. The relative magnitude of the repulsive force due to this potential difference and the attractive force due to Van der Waals' forces determine whether the VOL 55
NO. 1 1
NOVEMSER 1 9 6 3
21
particles experience a net attraction or repulsion. Since the attractive forces originate within the solid, little can be done to modify them. However, the repulsive forces are dependent on the amount and nature of the electrolyte in the suspension, and thus changes in these variables are responsible for the drastic changes in physical properties often observed with suspensions of small particle size material. As illustrated in Figure 1, the potential due to adsorbed ions may be either positive or negative, depending on electrolyte concentration. Flocculated suspensions are often observed (7) to have potentials between 30 mv. and - 30 mv. Thus a stable (deflocculated) suspension may flocculate upon the addition of electrolyte as the potential is reduced, then become deflocculated upon the further addition of electrolyte as the charge is reversed, and finally flocculate again as the concentration of electrolyte is increased sufficiently to compress the thickness of the adsorbed layer of ions. T h e remainder of this paper will be concerned primarily with the characteristics of suspensions in the flocculated state, because it is while in this state that they display their non-Newtonian characteristics. Application of the principles of colloid chemistry (32) to the present problem shows that the potential determining ions for oxides, hydroxides, and solids which can act as reversible gas electrodes are hydrogen and hydroxyl ions. This means that suspensions prepared with materials from any of these categories should exhibit the same general behavior as the p H is varied. Since the flocculated suspensions under consideration always have a small surface charge, the attractive force between particles is largely determined by the London-Van der Waals' force. For metal oxides, the magnitude of this force is largely determined by the oxygen atoms (32). Consequently, the laminar flow properties of aqueous suspensions of metallic oxide particles of similar size, shape, and concentration may be expected to be quite similar. Rheological Characteristics of Flocculated Suspensions. EXPERIMENTAL TECHNIQUES. The rheological properties of suspensions may be determined using either rotational or capillary tube viscometers (62). I n the present studies the capillary tube viscometer was chosen because the data obtained with it can be compared directly with data from tubes having diameters of engineering interest (55). Regardless of which type is chosen, care must be used in selecting instrument dimensions in order to minimize the magnitude of corrections to the data. Although procedures for calculating the value of the corrections for non-h-ewtonian suspensions are generally unavailable. some of the interferring phenomena are well-known qualitatively and include :
+
-entrance and exit losses, -wall effects dependent on the ratio of particles to tube diameter, -viscous heating effects, -nonuniformity of particle distribution. Uncertainties in the magnitude of the entrance and exit losses may be minimized by choosing tubes with large 22
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
length to diameter ratio ( 7 ) . il'all effects (which always result in a reduction of the apparent viscosity from its true value) may be minimized by choosing a large tube diameter relative to the particle size; however, this must be balanced against the advantage of high shear rates which can be achieved using small tube diameters ( 7 , 60). Viscous heating at high shear rates also produces a reduction in the apparent viscosity; the evaluation of the magnitude of this effect is complicated by the fact that not only does the heating distort the velocity profile (18), but the effect of temperature on the shear diagram is often not well understood (60). The compromise capillary tube dimensions chosen for these studies (55) was a tube with a diameter of 0.124 inches and an L / D ratio of 1000. SVith this tube diameter, wall effects were minimized as shown by good agreement of the data for tubes from 0.124 to 1.024 inches in diameter. Limited information on entrance and exit corrections indicate that the maximum correction was 3.5 per cent with an uncertainty somewhat less than this. Although the maximum mean temperature rise of the fluid due to viscous heating was always less than 2.5' C., no attempt was made to correct for the maximum local temperature rise which would be expected to occur in the high shear rate region adjacent to the wall. Settling of particles during the viscometer measurement was Figure 2. suspension
Typical laminar-Jozu shear diagram f o r a thorium oxide
unimportant because, as shown later, the flocculated suspensions were in the compaction regime (that is, substantially zero settling rate) for all except very dilute suspensions when using the '/*-inch diameter viscometer tube. PHYSICAL PROPERTIES.Aqueous suspensions of lyophobic particles often have non-Newtonian laminar flow characteristics distinguished by a nonconstant relation between the rate of shearing strain, du/dr, and the shear stress, T . Typical data are represented as a true shear diagram by the curve ABCDEF in Figure 2. At high rates of shear, the suspension curve (region AB) approaches a limiting value but never crosses the curve for the suspending medium. At lower rates of shear, the slurry curve (region BCD) deviates markedly from the curve for the suspending medium. At very low rates of shear, the slurry data are observed in the region DEF with the data usually being closer to curve DE than to curve D F. The central problem in the field of rheology is the derivation of general equations of state relating the rate of shear strain, the shear stress, the thermodynamic variables, the composition, and time. The derivation of such relationships has long intrigued the theoretician and recent review articles (37, 49, 60, 67) contain extensive bibliographies referring to their efforts. No further dis-
cussion of the generalized theories will be given in the present paper except to note that none of the available expressions are really satisfactory for use in engineering correlations which require values for the apparent viscosity or the limiting viscosity a t high rates of shear. Despite the lack of generalized equations of state, progress has been achieved in problems of practical interest using simplified flow models. Three commonly used models are (60):
1. POWERLAW:
- (du/dr)"
=
(g,/K)T
(9)
Equation 9 can be integrated to give the mean flow only if n and K are truly constants. I n order to circumvent this difficulty, Metzner (37) has proposed that the power law be defined in terms of the mean flow: (8VD)"' = ( g c / K ' ) r ,
(10)
2 . BINGHAM PLASTIC:
8v/g,D
- r,) when r 2 du/dr = 0 when r < r y = ( r d d [1 - (4r,/3rw) 4-
(du/dr) = (gC/T)(T
(11)
rY
(12)
I
(ry4/37-tu4)
(13 )
Figure 3. Application of three rheological models to data f o r a shear-thinning non-Newtonian susjbension (kaolin in water, 0.25 volume fraction solids). The Bingham plastic model was chosen to refiresent the data
VOL. 5 5
NO. 1 1
NOVEMBER 1963
23
3.
EYRING:
I n order to illustrate how well the different models fit experimental data, typical results taken with a kaolin suspension flowing in tubes from 0.124- to 0.944-inch inside diameter are shown in Figure 3 as the rate of shearing strain versus the wall shear stress (56). The experimental data were converted to these coordinates using the Mooney-Rabinowitsch (47)expression :
_ -du- dr
(i 4 [ 1
-t
dln (8V/D) d l n (DAp/4L)
I)
8V -
D
(15)
The coefficients appearing in Equations 9, 11, and 14 were determined to give the best fit to the data in the shear rate range l o 3 to l o 4 sec.-l. The values of the coefficients are shown in Table I ; these values were used to calculate the curves shown in Figure 3. All the TABLE I.
RHEOLOGICAL COEFFICIENTS ~-
Power taw
Bingham Plastic
Eyring 5
lb.,,J(
I
-
r2/= A1$1+3/Dp2
(16 )
where $1
= exp 0.7[(S/S0) - 11
(17)
The constant A1 had a value of 210 lb., micron2/ft.* or 2.27 X Ib., when the particle size was given in microns or feet, respectively. The coefficient of rigidity was shown (Figures 6 and 7) to be given by:
i ~
0.86
10.0451 0.0075
i
8.7 8.3
1
1
8.3
ft.)(sec.)1.578
models fit the data reasonably well in the shear rate range l o 3 to lo4. However, the fit outside this range is frequently not as satisfactory. Of these three expressions, the Bingham plastic model was chosen to represent the suspension data. The reasons for this were: -It is a simple, two parameter flow model. -It fits the data sufficiently well a t high shear rates that accurate apparent viscosities can be calculated from the experimentally determined flow parameters. -It can be extrapolated to give a limiting value of the viscosity at high rates of shear that is always greater than the viscosity of the suspending medium. Although many solutions of the equations of motion for different boundary conditions have been obtained using the Bingham plastic model (74, 42,43), there have been few experimental studies which attempt to compare the theoretical predictions with the actual behavior of real materials and, in fact, there is reasonable doubt as to the existence of a true yield stress. Most laminar flow studies have used the Bingham plastic model to calculate apparent viscosities in the high shear rate regions where the question of the existence of a true yield stress is relatively unimportant. Since this is the way that the Bingham plastic equations will be used in the subsequent portions of this paper, all physical property values were evaluated in the high shear rate regions of l o 3 to IO4 sec-l. Recently, it has been suggested (72)that a true limiting viscosity may he obtained from data taken at shear rates greater than 104 to 106 sec.-l or alternatively 24
by measuring the Newtonian viscosity of the particular suspension after deflocculation with a suitable electrol) te The usefulness of these suggestions has not as yet been assessed because of the difficulties, pointed out above, of obtaining data at high shear rates free from errors due to wall effects and viscous heating. The utility of the Bingham plastic model as an equation of state can be increased by determining the factors responsible for the magnitude of r and r Y . The yield stress of aqueous suspensions of thorium oxide, kaolin, titanium oxide, aluminum oxide, graphite, magnesium oxide, and uranium dioxide having particle sizes in the range 0.35 to 13 microns was shown (55) to be proportional to the cube of the volume fraction solids (Figure 4) and inversely proportional to the square of the particle diameter (Figure 5) :
INDUSTRIAL A N D ENGINEERING CHEMISTRY
when the particle diameter was given in microns, The shape factors and $ 2 were introduced to account for the asymmetric particle shape, which was observed for all particles greater than 2 microns in diameter as shown by the breaks in the curves in Figures 5 and 7. The particular form of Equation 18 was chosen to agree with the theoretical results (Equation 3) in the limit for large particle size suspensions. The limit for small particle sizes (of the order of colloidal dimensions) has not been established. Care must be used in applying Equation 16 through 19 to suspensions having concentrations greater than 0.2 to 0.3 volume fraction solids of micron sized particles, because at some concentration in this range the flow behavior changes from shear-thinning to shear-thickening (dilatancy) (38). It is not yet possible to predict either the exact value of the concentration at which this phenomena may be expected or the magnitude of the effect. The relation for the yield stress (Equation 16) has been derived from the suspension characteristics using a simple model for the floc assuming symmetrically shaped particles (58). Similarly, the rheological properties of suspensions of platelike kaolin particles have also been derived from simplified floc models (39, 40). The rheological properties of suspensions of needle shaped particles have recently been shown to be very complicated (76, 77). As might be expected, the suspensions are shear-rate thinning but in addition they have shown time dependent thinning and thickening at constant values of rate of shearing strain. Additional details may be found in the references cited above.
Figura 4. E p c f of uolumc fraction of solids on the yicld stress of the s u p m i o n . Particlc rharocfzristicr are: titanium dioxide, D, is 0.40 mimom and 0 is 7.6; graphite, 2.35 mimom nnd 7.6; thorium oxide (upper curua) 0.74 mimom and 2.2; rhoriwn oxide (lower curve) 7.35 mimom and 7.4: kaolin, 2.85 rnicrom and 4.0
Figure 7. Effect of pm6iclc d m & m the coe@imt of rigichry of the SUIpension. The chmgc in SI@ ar particle si= inrrnurs is explained a( o shapt-factcta @€st
Figwe 6. Effect of solids mmtr&m OR the m&imt of rigid+ of the surpmrion. Pm6icle charoclnirrics arc the s m c as those in Figwe 4
V O L 5 5 NO. 1 1
NOVEMBER 1 9 6 3
25
T h e effect of electrolyte concentration (57) on the yield stress and coefficient of rigidity of aqueous suspensions often shows little variation with p H in ranges of practical interest. In one particular case, the p H of a suspension in the absence of additives was 8.1 over a range of concentrations from 0.069 to 0.1 54 volume fraction solids. Shear diagrams were obtained a t 0.090 volume fraction solids as a function of pH. Although they varied in a regular manner, the physical property values determined in the p H range from 4 to 12 were within a band of h 2 0 per cent. The weak dependency of the rheological properties on the p H was characteristic of essentially all the suspensions studied regardless of the physical or geometrical properties of the particles. This means that a scatter of less than A20 per cent was introduced into the correlations for yield stress and coefficient of rigidity due to p H variations since the suspensions usually had p H values from 6 to 10. Settling-Rate Characteristics of Flocculated Suspensions. Dilute suspensions of flocculated particles also follow Equation 7 except that the volume fraction solids must be increased to account for the floc structure. T h e increase in volume fraction solids may be accounted for by one additional parameter, a, the ratio of immobilized water volume to solid volume (52, 58). T h e volume fraction solids and density of the floc are then given by: dl =
PI =
PP
(1
+44
(20)
+aP
The floc diameter may be calculated from Stokes’s law (Equation 6 and Equation 21) using a settling rate determined by extrapolation of the hindered-settling data to zero concentration. Typical data for hindered settling of a flocculated suspension are shown in Figure 8 as a function of suspension concentration and container diameter. I n the dilute regions, settling rates were independent of tube diameter. However, as the concentration was increased, the settling rate decreased markedly a t certain critical concentrations which were dependent on the tube diameter. This phenomenon is believed to be due to the bridginy of flocs across the container with consequent reduction in the settling rate. From the limited data available, the concentration, +c, a t which bridging will occur may be estimated from the empirical relation (58):
with a mean deviation of 5 1 2 per cent. This differs from the relation proposed by Michaels and Bolger (39) in the value of the numerical constant and in the term which accounts for the support given the floc structure by the container bottom. The decrease of the critical concentration for compaction with decreasing container diameter given by Equation 17 is the primary reason horizontal tube viscometers are believed to give data representative of homogeneous suspensions. This is because suspensions in compaction settle a negligible 26
INDUSTRIAL AND ENGINEERING CHEMISTRY
amount during the length of time the material is in the viscometer tube. I n the absence of viscometer data, rheological properties may be estimated from a-values determined from dilute suspension hindered-settling rates using the empirical expressions (58):
where and
k l = 1.35 X l o p 2 lb.//ft.?
In v/,u = 2.5 4
+a
The only precaution that must be used in evaluating a is that the suspension must be quite dilute as, for instance, the region to the left of 0.025 volume fraction solids in Figure 8. The value of (1 a ) is simply the ratio of the effective volume fraction solids calculated for a given reduction in settling rate from Equation 8 to the true volume fraction solids responsible for the reduction.
+
l a m i n a r Transport Characteristics
In principle all the theoretical transport relations for Kewtonian laminar flow may be taken over into nonNewtonian flow theory by simply accounting for the different shape of the velocity profiles provided the flow is rectilinear-that is, steady shearing motion such that the surfaces of constant speed are either planes or portions of right circular cylinders ( 1 3 ) . I n practice difficulties may be experienced due to complicated stressstrain relations or due to nonconstant values of parameters in the simpler flow models. I n the present studies the Bingham plastic model fitted the data sufficiently well and little difficulty was encountered from this source. Figure 8. Hindered rettling rates for Jiocculated Thoz suspensions. A t higher concentrations, container diameter becomes a factor
Friction Loss. There are two equally satisfactory methods for presenting laminar friction loss data. I n the first the effective viscosity for Bingham plastics (6) is defined as:
I - -
~
-
l I -2
3rw ' 3
1
results when it is assumed (6) that the transition for Bingham plastic materials occurs at a Reynolds number of 2100. (Actually, data indicate that the non-Newtonian character of the material causes transition to occur over a range of Reynolds numbers from 1900 to about 6000 [57].) Then, using the approximate form for the effective viscosity (Equation 25) :
\rwl
When the Reynolds number is calculated using Equation 25 for the viscosity, all laminar flow friction loss data should, by definition, fall on the usual Newtonian curve, f = 16/NR,. A second possibility was suggested by Hedstrom (24) as a result of a dimensional analysis which showed that two parameters could be developed from the Bingham plastic equation (Equation 13). These parameters were a Reynolds number, DVp/q, and g,p7,D2/qz, a number which has since been called the Hedstrom number although, as a matter of fact, Oldroyd was the first to point out its significance (42). The relation among the Fanning friction factor, the Reynolds number (DVplq), and the Hedstrom number is shown in Figure 9. Additional discussion of laminar flow data will be given in the section on turbulent flow. Laminar-Turbulent Transition. An appreciation of the technological significance of different numerical values for the parameters in the Bingham plastic equation can be gained from Figure 10, which shows the effect of yield stress, coefficient of rigidity, and tube diameter on the critical velocity for the onset of turbulence. I n reality the transition process in non-Newtonian suspensions must be quite complicated, but Figure 10 shows clearly that the value of the yield stress is the primary factor governing the transition. A similar conclusion
and recognizing that for large diameter tubes (diameters >1 inch), the term on the right in the brackets is large compared with unity, the critical velocity becomes:
The significance of Figure 10 and Equation 27 is that for yield stresses greater than 0.1 to 0.5 lb.,/ft.2, the critical velocities already exceed those usually considered economic for large-scale engineering equipment. A recent analysis (27) of the laminar-turbulent transition for fluids with a yield stress showed good agreement with the data a t low values of the Hedstrom number. At larger Reynolds numbers, the predicted values were somewhat less than the data. This was attributed to the inadequacy of the Bingham plastic model in describing the velocity profile data. However, another possibility for the discrepancy may be that the mechanism proposed by the author (21) is a t considerable variance with other detailed experimental and theoretical results which show that transition in Newtonian fluids is due to three-dimensional disturbances in the immediate vicinity of the wall (30, 37).
Figure 9. Friction factor-Reynolds number diagram for laminarflaw of Bingham plastic materials in round pipes
VOL 5 5
NO. 1 1 N O V E M B E R 1 9 6 3
27
, 1 l
Laminar Heat Transfer. For steady shearing motion such that surfaces of constant speed are either planes or portions of right circular cylinders, rectilinear flow is observed whether the fluid is Newtonian or nonNewtonian. The significance of this is that all analytical results for rectilinear, Newtonian flow are immediately available for rectilinear non-Newtonian flow provided the non-Newtonian velocity profile is accounted for (73). F'igford followed this approach in deriving corrections to Leveque's solution for the laminar heat-eansfer problem which account for deviations from the parabolic velocity profile due to non-Newtonian properties (46). The modified equation is: hD/k = 1.61(&Da/8kL)"'
(28)
For isothermal non-Newtonian flow, B = 8yV/D. The value of y may be evaluated by differentiating the pseudoshear diagram for rectilinear flow and using this exnreueion: 3
r=,+
d ( I n 9
(29)
and 30-further modified to include a Seider-Tate correction for nonisothermal conditions-have been used to correlate (54)data taken with nowNewtonian suspensions (Figure 11). When using the Seider-Tate form of the equation, the correct value of the numerical constant in Equation 28 is 1.86. The resulting equation is:
The system geometry included tubes of 3/8- and 1-inch nominal diameter with heated lengths of from 87 to 378 L/D ratio. Suspension characteristics included a sixfold variation in yield stress from 0.075 to 0.46 lb.,/fta. The mean value for the data w a s in excellent agreement with the theoretical line and 95 per cent of the data were within a band of *15 per cent. The validity of using the t a m (l),Jq)o.lkas the Seider-Tate correction has not received sufficientexperimental verification; however, the good agreement of the data with Equation 31 warrants its use as a first approximation. Recapitulation
I
The exact differential may be determined in the Bingham plastic case:
Subsequent analytical studies (25) showed that Equations 28 and 30 were accurate for Nusselt numbers greater than 8.3; graphical procedures were presented for smaller values of the Nus& number. Equations 28 28 l
INDUSTRIAL AND ENGINEERING CHEMISTRY
The object of these studies was twofold: to define a particular set of experimental techniques for evaluating the intrinsic and extrinsic physical properties of flocculated suspensions; and to relate these properties to the laminar and turbulent transport characteristics of the suspensions using simplified flow models. All the data displayed remarkable internal consistency and, in general, apply to all aqueous suspensions of regularly or imgularly shaped, equiaxial solid particles in the
0.1- to 20-micron size range which are composed of metal oxides, hydroxides, or materials which may act as reversible gas electrodes. Using a simplified flow model, the laminar flow physical properties for a given concentration may be estimated either from particle size data or from a series of hindered-settling measurements. More accurate data may be determined directly from a laminar flow shear diagram obtained with a capillary tube viscometer. These data were shown to provide a satisfactory basis for calculating the laminar heat transfer and friction loss characteristics of a given suspension. I n the second part of this paper, the turbulent transport characteristics of suspensions will be described and correlations for the turbulent transport data will be presented which use the extrinsic and intrinsic physical properties given above. NOM ENCLATU RE A Ai A2,3
B
c CP
D
d du/dr f
sc H
h k
K, K L n, n’ “e N Pr
NR e AP
s/so
vc x CY
= relaxation coefficient, Equation 14, set.-' =
force constant, 1b.f
= numerical constants, dimensionless
coefficient, Equation 14, ft.2/lb., coefficient, Equation 14, lb.,/ft. sec. specific heat, B.t.u./lb. OF. diameter, ft. = differential operator = velocity gradient, sec. = Fanning friction factor, (DAp/4L)/(pV2/2gc). dimensionless = gravitational constant, ft./sec2 = conversion factor (hm/1b.,) (ft./sec.2) = depth, ft. = heat-transfer coefficient, B.t.u./hr. ft.a O F . = thermal conductivity, B.t.u./hr. ft.2 “F./ft. = dimensional coefficient (Equations 9 and 10) = length, ft. = exponents (Equations 9 and 10) = Hedstrom number, gCpryD2/v2 = Prandtl number, c,v/ks = Reynolds number, Dvp/7 or D V p / p e = pressure drop, lb.,/ft.2 = platelike particle surface area/equiaxial particle surface area, dimensionless = Stokes’s law settling rate, ft./sec. = suspension settling rate, ft./sec. = mean stream velocity, ft./sec. = transition velocity, ft./sec. = constant in Equation 2, dimensionless = volume immobilized water/volume solid, dimensionless = 8 r V / D , set.-' rY/da,lb.//ft.2 = coefficient of rigidity in lb.,/ft. sec. = viscosity of suspending medium, lb.,/ft. sec. = effective viscosity, lb.,/ft. sec. = density, lb.,/ft.3 = shear stress, lb.f/ft.2 = wall shear stress, DAp/4L, lb.,/ft.2 = yield stress, lb f/ft.2 = volume fraction solids, dimensionless = shape factors, dimensionless = log mean standard deviation of particle size = = = =
SUBSCR I PTS b c
f 1
P 5 W
= = = = = = =
bulk critical floc suspending medium particle suspension wall
LITERATURE CITED (1) Bingham, E. C., “Fluidity and Plasticity,” McGraw Hill, New York, 1922. (2) Boutaric, A., Vuillaume, M., J . Chcm. Phys. 21, 247 (1924). (3) Brinkman, H. C., App/. Sci. Res., Sect. A . Al, 27 (1949). (4) Bruggeman, D. A. G., Ann. Phys. (Leiprig) Ser. V 24, 636 (1935). (5, Bur ers I J , M .]“,O n the Motion of Small Particles of Elongated Form Suspended in a iscous Liquid ” in “Second Report on Viscosity and Plasticity,” North Holland Pub. Co., Amsterdam, 1938. ( 6 ) Caldwell, D. H., Babitt, H. E., IND.END.CHEM.33, 249 (1941). (7) Davies, J. T., Rideal, E. K., “Interfacial Phenomena,” p. 392, Academic, New York, 1961. (8) de Bruyn, H., “The Viscosity of Concentrated Solutions of Polymers,” in Proceedings of the Institute of Rheology, Vol. 11, p. 95, Amsterdam, 1949. (9) Douglas, H . W., Burden, J., Trans. Faraday Sac. 5 5 , 350, 356 (1959). (10) Einstein, A,, Koiloid-Z. 247, 137 (1920). (11) Eirich, F. R., Margaretha, H., Bunzl, M., 75, 20 (1936). (12) Eissenberg, D . M., p. 277 in Deuelopments in Theoretical and Applied Mechanics, Vol. I, Plenum Press, New York, 1963. (13) Erickson, J .L., Quart. Appl. Math. 14, 318 (1956). (14) Ericksen, J. L., Arch. Rational Mech. and Analysis 8, 1 (1961). (15) Fritsch, C. P. Jr., Gustafson, M. R., Cusack, J. H., Miller, R . I., Chem. Ens. Prog. 57, 3, 37, (1961). (16) Gabrysh, A. F., Eyring, H., Cutler, I., J . Amer. CeramicSoc. 45,334 (1962). (17) Gabrysh, A. F., Eyring, H., Shimizu, M., Asay J., J . Appl. Phys. 34, 261 (1963). (18) Gerrard, J. E., Appeldoorn, J. K., Philippoff, W., Nature 194, 1067 (1962). (19) Greek, B. F., Dougherty, C. F., Mundy, W. J., Ind. Ens. Chem. 52, 974 (I 960). (20) Guth, E., Simha, R., Kolloid-Z. 74, 266 (1936). (21) Hanks, R. W., A.I.Ch.E.J. 9, 306 (1963). (22) Happel, John, J . Appl. Phys. 28, 1288 (1947). (23) Hawksley, P. G. W., “Some Aspects of Fluid Flow,” Arnold Press, New York, 1950. (24) Hedstrom, B. 0. A,, IND. ENa. CHEM.44, 561 (1952). (25) Hirai, E., A.I.Ch.E.J. 5 , 130 (1959). (26) Hyde, R. B., Jr. Can. Min. Met. Bull. 52, 170 (1959). (27) Jeffery, G. B., Proc. Roy. Soc. (London), A102, 161 (1922-23). (28) Jefferson, T. B., Witzel, 0. W., Sibbitt, W. L., IND.ENO. CHEM.50, 1589 (1958). (29) Johnson F. A. “The Thermal Conductivity ofA ueous Thoria Suspensions ” United KiLgdom Atomic Energy Authority Report A I R E R / R 2578 (June 1958j. (30) Klebanoff, P. S., Tidstrom, K. D., Sargent, L. M., J . Fluid Mech. 12, 1 (1962). (31) Kovasznay, L. S. G. “A New Look a t Transition,” in “Aeronautics and Astronautics,” Pergarno; Press, New York, 1960. (32) Kruyt, H. R., “Colloid Science,” Vol. I, Elsevier Pub. Co., Amsterdam, 1952. (33) Landsman, D. A,, Nucl. Ens. 7 (2), 50 (1962). (34) Lane, J. A,, MacPherson, H. G. Maslan, F., “Fluid Fuel Reactors,” Addison Wesley Press, Reading, Mass., i958. (35) Loeffler, A. L., Jr., Ruth, B. F., A.1.Ch.E.J. 5, 310 (1959). (36) Maxwell, J. C., “Electricity and Magnetism,” 3rd ed., p. 440, Dover Pub. Co., 1954. (37) Metzner A. B. “Flow of Non-Newtonian Fluids ” Section 7 in “Handbook of Fluid Dymmics,”’V. L. Streeter, ed., McGraw-Hili, New York, 1961. (38) Metzner, A. B., Whitlock, M., Trans. Soc. Rheol. 2, 239-45 (1958). (39) Michaels, A. S., Bolger, J. C., Ind. Eng. Chem. Fundamentals 1, 24 (1962). (40) Ibid., p. 153. (41) Mooney, M., J . Rheo/o,qy 2, 210 (1931); Rabinowitsch, B., 2. Physik. Chem. (Leiprig) A145, 2-26 (1929). (42) Oldroyd, J. G., PYOC. Cambridge Phil. Soc., 43, 100 (1947). (43) Ibid., p. 396. (44) Oliver, D. R., Chcm. Eng. Sci. 15, 230 (1961). (45) Olson, W. T., Breitwieser, R. “NACA Research on Slurry Fuels Through 1954,” NACA Rept. R M E55 B14 (April 21, 1955). (46) Pigford, R. L., Chem. Eng. Prog. Symp. Ser. 151, 17, 79 (1955). (47) Richardson, J. F., Zaki, W. N., Chem. Eng. Sci. 3, 65 (1954). (48) Riseman, J., Ullan, R., J . Chem. Phys. 19, 578 (1951). (49) Schowlater, W. R., “Non-Newtonian Viscosities and the Equation of Motion,” p. 687 in “Progress in International Research on Thermodynamics and Transport Properties,” J. F. Masi, D. H. Tsai, eds., Academic, New York, 1962. (50) Schultz R. D., Wiech, E., Jr., “Electrical Propulsion with Colloidal Materials,:’ in “Advanced Propulsion Techniques,” R. S. Penner, ed., Pergamon, New York, 1961. (51) Simha, R., J . ColloidSci. 5 , 386 (1950). (52) Steinour, H. H., IND.ENC.CHEM.36, 618, 840-901 (1944). (53) Street, N., Australian J . Chem., 9, 467 (1956). (54) Thomas, D, G., A.I.Ch.E.J., 6 , 631 (1960). (55) . , Ibid.., 7. 43T (1961). .~ ,(56’ Thomas, D. G., “Trawnorr Characterictics of Suipensioni: Par. \I.’’ p. 704 in “Progrejs in International Research on Thermodynamic and Transport Properrics,” J. F. hfasi, D. H. Tsai, eds., Academic, Ncw I’ork, 1962. (57) Thomas, D. G., A.ICh.E.J., 8, 373 (1962). (58) Ibid., 9, 310 (1963). (59) Thomas, D. G . , unpublished data. (60) Thomas, D. G., “Significant Aspects of Non-Newtonian Technology,” p. 669 in “Progress in International Research on Thermodynamic and Transport Properties,” J. F. Masi, D. H. Tsai, eds , Academic, New York, 1962. (61) Truesdell, C., J. Rational Mechanzcsund Analysis 1, 125 (1952). (62) Va? Wazer, J. R., L ons, J. W., Kim, K. Y., “Viscosity and Flow Measurement, Wiley, New Yori, 1963. (63) Vand, V., J . Phys. C3 Colloid Chcm. 52, 277 (1948). (64) Ibid., p. 300. (65) Weltman, R. N., “The Rheology of Pastes and Paints,” in “Rheology,” Vol. 111, F. R. Eirich, ed., Academic, New York, 1960.
&.
.
~~
~
VOL. 5 5
NO. 1 1
NOVEMBER 1 9 6 3
29
