D A V I D G. T H O M A S
. I
Part ZZ.
TURBULENT TRANSPORT CHARACTERISTICS
he object of these studies was to relate the turbulent
Ttransport characteristics of non-Newtonian suspen-
.
sions prepared from symmetrically shaped particles of micron-sized metal oxides to the easily measured laminarflow and hindered-settling characteristics of the suspensions. In the first part of this two-part series (Indusfrial and Engincning Chernrsfry,November 1963), the Bingham plastic model was chosen to represent the laminar-flow data, largely on the basis of engineering convenience. The principal factors affecting the magnitude of the Bingham plastic coefficients of a given suspension were found to be the concentration and particle sue. Consideration of the physicochemical forces between particles indicated that the extrinsic physical property relationships (that is, those which are affected by suspension flmculation) could be expected to be quite general, applying to aqueous suspensions of metal oxides and solids which act as reversible gas electrodes. Experimental data for laminar flow and hindered settling of flocculated suspensions of such metallic oxides as titania, urania, kaolin, thoria, magnesia, and alumina were in accord with this prediction provided the particles were roughly symmetrically shaped. This part of the series summarizes the turbulent transport characteristics of the same suspensions used in the laminar-flow studies. The turbulent transport characteristics were shown to be unique functions of the intrinsic and extrinsic physical properties of the suspension independent of the composition of the solid phase. Although not proven conclusively, there is some evidence that particle shape affects turbulent flow data in a way which cannot be predicted from simple laminar-flow measurements. However, the deviations are sufficiently small that they are relatively unimportant for engineering purposes. A major question that must be resolved during the course of any study of the turbulent transport characteristics of nowNewtonian fluids is the appropriate viscosity
AUTHOR David G. Thomas is a DeueloprnenfEngineer in
fhb
Reactor Division of Oak Ridge Notional Laboratory, Oak Ridge, Tenn. This installafion is operated by Union Carbide Carp. for the Afomic Energy Cmnmission.
term for use in calculating parameters, such as the Reynolds number or the dimensionless velocity profile coordinates, of importance in analyzing the turbulent flow data. It is an axiom of rheology that even the simplest proposition must be verified experimentally. For instance, it was not until 1911 that sufficient information on friction loss of Newtonian fluids had been accumulated to permit Blasius (26A) to correlate the data empirically and prove that the turbulent friction factor curve for different liquids was a unique function of the Reynolds number when the Reynolds number was calculated using the laminar viscosity. Then in 1913 the National Physical Laboratory (29A) showed that the same empirical curve was valid for both Newtonian liquids and gases. It is not entirely clear that the question of the most meaningful viscosity term for use in turbulent nonNewtonian flow correlation has been resolved ( 2 4 354. In all probability the question can he resolved experimentally without resorting to ex cafhcdra decisions. One of the simplest criteria is that the Reynolds number be defined in such a way that the turbulent non-Newtonian friction factors would agree with the turbulent Newtonian friction factors ( 7 4 . This would probably be misleading since it implies that the scale and intensity of turbulence in a non-Newtonian system is substantially the same as in Newtonian systems provided there is Reynoldsnumber similarity. The general consensus seems to be that this is not so, and that the elements responsible for non-Newtonian behavior must damp the turbulence characteristics in some way. Another viscosity that has heen used ( 3 4 70A) is the apparent viscosity evaluated a t the wall shear stress. This is the natural viscosity for correlating laminar-flow data. However, this form of the viscosity has not been tested over a sufficient range of tube diameters to insure that the tube diameter &ects are absent in turbulent flow. Re-examination (4A) of previously published data (70A)showed that there was a slight diameter effect when using the apparent viscosity to correlate the data even though there was only a fourfold variation in tube diameter. Recently, the prior methods (34 7A, IOA, 27A) of calculating the apparent viscosity fir turbulent flow correlations has been questioned and a more rigorous procedure proposed (7 IA) ; V O L 5 6 NO. 1 2 D E C E M B E R 1 9 6 3
27
however, there are insufficient data available to check this proposal. A set of phenomenological criteria for a visksity that would be particularly useful in engineering is:
-The friction factor is a unique function of RGnalds number, independent of tube diameterfor any partimlar suspmsion. -The turbdent non-Newtonian friction factors are always less than the values f w Newtonian fluids, presumably due to suppression of turbulewe by dissolved or suspended non-Newtonian elements. -The turbulent non-Newtonian friction factors are always greater than the laminar ualues for Neutonianf[uids. Undoubtedly, other criteria can be developed based on velocity profile measurements and ultiniately on fundamental turbulence measurements on the non-Newtonian materials themselves. In the absence of these more fundamental criteria, the desirability of having an engineering correlation that fulfilled the first item above was given precedence in these studies.
Friction Loss Characteristics
Analysis of data for the turbulent friction loss of suspensions of titania, h o l m , and thorium oxide showed that the limiting viscosity a t high rates of shear, 9,fulfilled the three criteria given above for a suitable viscosity (30A. 32.4). This analysis also showed that the effective viscosity (Equation 25 in the November article) was not satisfactory since its use resulted in a definite diameter effect for tubes having a ninefold range in diameters. Typical data for two different suspension concentrations are shown in Figure 12. The coordinates of the diagram on the left permit the correlation of the laminar-flow data by a single line with the turbulent flow data for different tube diameters branching offon different lines. The good agreement of the laminar-flow data, independent of the tube diameter, ensured that the wall effects were negligible for the systems studied and, therefore, the coefficient of rigidity and yield stress were unique properties of the suspension. On the right side of Figure 12, the same data are plotted as f versus NE, = D V p / p coordinates which, with the proper selection of the viscosity, permit the correlation of the turbulent flow data for any given suspension by a single line. The good agreement of the turbulent flow data demonstrates that there was no detectable wall effect for tubes from to 1 inch in diameter and that friction factor similarity with high yield stress suspensions was obtained using the limiting viscosity a t high rates of shear, 9,in the calculation of the Reynolds number. Data Correlation. Turbulent friction facton for non-Newtonian suspensions were always below those for Newtonian fluids; however, two entirely different trends with increasing Reynolds number were observed depending on the value of the non-Newtonian properties (32A). For yield values less than 0.5 Ib.,/q. ft. the suspension friction facton tended to approach those for Newtonian fluids as the Reynolds number was increased. 28
INDUSTRIAL A N D ENGINEERING CHEMISTRY
W I L L M I 1 S1KBS.r. LB.dn.2
Figure 72. Pseudo-shcm dingram and Faming friction-factor plot for concentrotcd suspmrionr showing og8emmf of laminar and ttububnt data, respcc6ively, as tubs dzamatrr IUOS wried. AN ruspenrionr WETCof thoria. For thcjrst three rc63 of points listed in the key, surpcnrion profimfics wcm as follow^: 0 = 0.70, = 0.69 lb.,/ft.z, 7 = 5.7 cp. For the sccond three, 0 = 0.76,r. = 1.25 1b.,/fLP, and '1 = ?O.l cp. Thc CpUnliOn of fhc two lines dcsignoted ( A ) ir 4.0 lag ( N E J d f i - 0.40
?/dy =
i-
t
.
VIELO SIKESS, LO.i/SP.
.
R
Figure 13. E&t of yield dress m BIosiur cm&mts ReyM[drnlrmbnandth Fmulingjricrimrfactor
rdati.;! ihr .:
..
..
,
lElwDLDS
awua Ir
....
-7 OYP
Figure 9 from Part I. Fricrion fuctof-Reynoldr numbn d i ~ p fmm lamimrpour of Binghamplustiz motnidx in roundpipes. ( A largm chart 4ppCms on P q c 27 of tht Novmbn mticle.) Thc light diagonal lincr hnve thc equmkm:
I n contrast to this, the friction factors for suspensions with yield values greater than 0.5 lb.,/sq. ft. tended to diverge from the Newtonian values as the Reynolds number was increased. The extent of the change in behavior is illustrated in Figure 13, which shows data for the Blasius coefficient, B, and exponent, b, plotted,\as a function of yield stress. (The use of the yield stress in this plot is only a convenient way of indicating the degree of non-Newtonian behavior and does not imply that it is the only variable or importance.) The values of B and b were determined by fitting the friction factor line for each suspension with the expression :
f
= BlvRe-b
Dimensional analysis shows that, aside from the Reynolds number, two different combinations of suspension physical properties are possible, V / M and gCpr,x2/pz. These two parameters must be combined additively in order to represent the change of slope shown in Figure 13. The expression for the coefficient and exponent of the Blasius equation (Equation 32) for the entire range of yield values becomes :
+ W) (& + *)
B
=
0.079 (0.
(33)
b
=
0.25
(34)
where 0 = p / v and @ = gcprvx2/pL2.When the yield stress is zero and 7 and /L are equal, Equations 33 and 34 reduce to the commonly accepted values for Newtonian fluids. The constants and exponents of these equations were determined using a nonlinear least squares procedure ; the results were (32A) : 0.48 d = e = 2
a =
c = 0.15 x
=6
x
lO-+ft.
Friction loss data are also available for non-Newtonian suspensions of flexible ( 5 4 9A) and rigid (704 13A) needle-like particles. I n general, the behavior of suspensions of such particles is more complicated than that of suspensions of equiaxial particles. For instance, laminarflow studies show that suspensions of rigid needle-like particles display time-dependent shear-thinning or shearthickening behavior (13A). With flexible fibers, there is the possibility of entanglement resulting in viscoelasticity and/or shear thinning behavior with the added complication of flow with a n annular region of suspending medium next to the wall (5A). Turbulent friction loss data for rigid needle-like particles are difficult to compare with data for equiaxial particles because in both cases there are no fundamental turbulence data and even on a macroscopic scale there is the problem of defining a unique viscosity. One way of making this comparison (35A) is on the basis of the magnitude of the friction factors predicted for a particular set of data by the proposed correlations for equiaxial (32A) and for needleshaped particles (1OA). I n two particular cases, extensive data were available for a suspension of equiaxial particles. The physical property data required for use in the two proposed correlations were evaluated objectively from the laminar-flow shear-diagrams taking especial care that there were good laminar-flow data from
a small tube viscometer corresponding to the wall shear stress of the turbulent flow points obtained with tubes of large diameter. (The existence of laminar flow for the data taken with the small tube was proven by two facts: visual observation of the suspension as it jetted from the viscometer tube and the good agreement of the laminarflow data with the Hedstrom line on a plot o f f versus D V p / ? ; that is, Figure 9 [see page 281.) T h e results of such a comparison (35A) showed that in general the correlation for equiaxial particles (Equations 32 to 34) was in good agreement with the experimental data for suspensions of equiaxial particles, whereas the correlation for turbulent friction loss of needle-like particles predicted friction factors 22 to 36Oj, larger than the experimental values for equiaxial particles when using the laminar-flow properties for equiaxial particles evaluated using the procedure recommended by Dodge and Metzner (70A). Thus particle shape apparently affects turbulent flow data in a way which cannot be predicted from simple laminar-flow measurements. However, the deviations are sufficiently small that they are relatively unimportant for engineering purposes. Design Procedure. For values of the yield stress less than 0.3 lb.,/sq. ft., the @ term contributes less than 5% to the final value of the Blasius coefficient and exponent of Equations 33 and 34. This means a considerable simplification in design procedure for suspensions with a moderate yield value because a single plot of friction factor versus Reynolds number can be prepared with v / p as the non-Newtonian parameter. For larger values of the yield stress, the complete expression for B and b must be used. I n either case, the Hedstrom number should be evaluated and checked against Figure 9 to ensure that flow is not laminar for the given value of the Reynolds number, D Vp/v. Aqueous suspensions of symmetrically shaped particles of titanium oxide and magnesium metal have been shown to possess friction loss characteristics when flowing through fittings (valves, ells, etc.) which are virtually the same as those for Newtonian fluids of the same density (38A). Phenomenological Analysis. The value for x in Equations 33 and 34 is of the same order of magnitude as the ratio of Y / U * for the suspending medium (for example, for a viscosity of 0.9 cp. and a density of 1.0 g./ml., the value of Y / U * is 6.0 x 10-B ft. when r w = 7.5 lb.,/sq. ft. and the suspension density is 1.5 g./ml,), Since u* is experimentally proportional to u', the fluctuating component of the mean velocity (25A), then x also is the order of the scale of eddies for which viscous forces are important (75A); that is, x is the distance over which the yield stress might be expected to exert an influence on the structure of turbulent flow. By application of Prandtl's simplified equations for the friction loss in round pipes to the suspension data (26A, 32A), it was possible to infer additional information about the phenomena which occur during turbulent flow of non-Newtonian suspensions. The relation between the dimensionless wall thickness layer, N = uw* 8/11, and the suspension characteristics was found to be : N = 11.4 ( V / ~ ) O . ' ~ (35) VOL. 5 5
NO, 1 2
DECEMBER 1 9 6 3
29
This equation reduces to the commonly accepted value for Newtonian fluids when ?I = p. Such a thickrning of the wall layer has been indicated experimentally in several different studies (3A, 7A, 27A) ; however, no prior attempts have been made to evaluate the thickening quantitatively. One possible explanation of the gradual thickening may be obtained by comparison with the results of detailed studies of Newtonian fluids. These studies showed that the production of turbulence was at a maximum very close to the wall, at a distance y + -11.4 = llr-i.e., the value given by Equation 35. However, the energy of turbulence was not completely dissipated where it was produced, but there was a displacement toward the axis due to diffusion in the direction of decreasing turbulent intensity (25A). Now, flocculated solids in suspension would be expected to markedly suppress the turbulent intensity, particularly in the vicinity of the tube axis where the shear stress is less than the apparent yield stress. Following the szme arguments advanced for Newtonian fluids, damping of the turbulent fluctuations on the tube axis due to the flocculated solids results in an additional displacement of the maximum in the turbulence dissipation and production away from the wall, and consequently a thicker wall layer, as given by Equation 35. This explanation is also consistent with the observation that friction fzctors of low yield stress suspensions converge toward Newtonian friction factors as the Reynolds number is increased. That is, the increase in Reynolds number is accompanied by an increase in the , wall shear stress, hence by decreasing values of rV ’ T ~ thus weakening the strength of the sink on the centerline and consequently causing less deviation of the wall layer thickness from the values expected for Neivtonian fluids, and in general a behavior more characteristic of Newtonian fluids. I n contrast to the behavior of low yield stress suspensions, the friction factors of high yield stress suspensions diverged from the Newtonian fluid line. This suggests that turbulent fluctuations were damped throughout the tube cross section by the suspended solids, and that the flow was becoming more laminar in nature. This is supported by the effect of yield stress on the von Karman coefficient, K , calculated from the friction factor values using Prandtl’s equations (32A). The value of K was substantially constant up to a yield stress of 0.5 lb.,,’sq. ft. and decreased thereafter. Since K is commonly considered to be a measure of the average intensity of turbulent fluctuation ( Q A )the decrease in K means that the turbulent intensity was damped for suspensions with yield values greater than 0.5 lb.,/sq. ft. Similar deductions have been reported by Vanoni (37A) in studies of Newtonian suspensions; in addition, he observed that the effect increased with increasing concentration and decreasing particle size. By replacing rY by Equation 15 and x by v/u*, the expression of @ in Equations 33 and 34 becomes : @ = #‘[(A1/O,*)j(PU*Z/ge)] (36) where A1 is the force constant in Equation 15 which characterizes the attraction between particles, D, is the 30
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
10
2
I
10-3
10-2
DIA IN m33lS
HT TR 378
VI ‘;33
175
AP
635
1 6 x
34
196 4 8 x IO5
f
I 05
04 REYNOLOS NUMBER
I
C6
9 v
Fzgure 74. Heat transfer a n d j u i d j 7 o u . characteristics of thorza Jlurries using the lamttang vascosity at high rates of shmr t o calcvlute the Iieynolds and Prandtl numbers. For these data, 7 = 8.2 cp. = 0.0055 Ib./ft.-spc., r y= 0.46 lb./sq.ft.. and p = 7 3 3 lb./cu. f t .
+
particle size, is the volume fraction solids, and u* is the friction velocity. O n this basis, is proportional to the volume fraction solids cubed and the ratio of the attractive forces between particles to the disruptive forces due to turbulent fluctuations. Hence, the relative importance of the turbulent fluctuations at any given flow rate is diminished by the addition of more solids or by the increase of the attractive force term by a reduction in particle size. This is in agreement with the observations of Vanoni, cited above. Heat Transfer to Flocculafed Suspensions
Turbulent heat-transfer measurements were made on non-Newtonian suspensions in the same equipment used for the friction loss tests (30A). A comparison of the results of the two different kinds of measurement allowed the general features of non-Nebvtonian suspension heat transfer to be readily identified. As with the friction loss data. the limiting viscosity at high rates of shear was shown to be a suitable viscosity for correlating the heattransfer data. Laminar and turbulent data for a suspension having a yield stress of 0.5 lb.,/sq. ft. taken with tubes of 0.318 and 1.030 inches in diameter are shown in Figure 14. These data show the characteristic displacement of the laminar data with tube diameter correspondin5 to the different values of the Hedstrom number, gcpr,D2/q2. The heattransfer and fluid-flow data having the same Hedstrom number show that the departure from laminar flow occurs at identical Reynolds number for both the heattransfer and flow data. The heat-transfer data for fully developed turbulent flow for both tube diameters are correlated by the conventional Newtonian line : h - - - 0.027 N E e - 0 , 2 N p 7 - 2 ’ 3 ( ~ , / q ~ ) ’ . ’ ~ (37) CPG when lVpr = c p v / k , provided that the Reynolds number is 3 to 5 times the critical value for the onset of non-
laminar flow. This range of Reynolds numbers for the transition region corresponds very closely to the heattransfer transition range observed with Newtonian fluids, and in fact, except for the displacement in critical Reynolds numbers due to the non-Newtonian laminar characteristics of the suspension, the non-Newtonian heattransfer data are very similar in appearance to Newtonian heat-transfer data. Design Procedure. The suspension heat-transfer data shown in Figure 14 show a dip region at Reynolds numbers extending from the critical value for the transition to about 4 ( N R J C . A similar dip regionwas first identified for Newtonian fluids by Colburn (8A). Since the value of the critical velocity for the transition (3 to 8 ft./sec. for slurries with values of the yield stress from 0.075 to 0.5 lb./sq. ft.) is already approaching the range of velocities commonly used in heat exchanger design, a graphical procedure similar to the one originally proposed by Colburn (BA) appears to be the most suitable method for avoiding the ambiguities associated with design for this dip region. The design procedure recommended is : 1. Calculate the value of the Hedstrom number and identify its location on a Fanning friction factorReynolds number (DVp/v)plot containing the Hedstrom number grid as a parameter (Figure 9 of the November article, repeated on page 28). 2. Locate the turbulent-flow friction-factor line on the same plot by using Equations 32, 33, and 34. 3. Calculate the laminar-flow j-factor from Equation 31 by using the value of the critical Reynolds number, (ArRe)c,determined from the intersection of the laminarand turbulent-flow lines obtained in steps 1 and 2 above. Plot this value on a Newtonianj-factor-Reynolds number plot, and locate the laminar heat-transfer line from this point with a slope of - 2 / 3 . 4. Connect the laminarj-factor point at ( N R J cto the turbulent-flow Newtonian j-factor curve at N,, = 4 with a smooth curve characteristic of the j-factor curve in the dip region. O r if desired, a n analytical
procedure given by Petersen and Christiansen (24A) may be used to determine the transition region curve. Heat- and Momentum-Transfer Analogies
Newtonian heat- and momentum-transfer analogies (76A) developed by Colburn, Prandtl, von Karman, Martinelli, and Metzner and Friend (ZOA) were in good agreement with the experimental data for non-Newtonian suspensions. This was to be expected since the suspension Prandtl number was about 10. Although the best fit was observed with the Martinelli analogy ( 1 6 4 ) (which was developed following von Karman's assumption of a buffer layer between the laminar sublayer and the turbulent core), the scatter of the data probably does not warrant the use of such a complex expression. Instead, the von Karman analogy (Equation 38) may be recommended. (f/2)NP:'a = 1 54j7i{(1vP, - 1) In [I 5 / a ( N ~-, 1 > 1 j (38)
+
+
+
If the Martinelli analogy is used, a considerable simplification is afforded by using the tabulated values of parameters given by Knudsen and Katz (16A). With values from these tables. the Martinelli equation reduces to
4fT= 2.120 log N,,Z/f + 17.48
for a Prandtl number of 8 and to
dfx=
1.715 log N R e 4 f
+
19.07
(39) (40)
for a Prandtl number of 11. The experimental frictionfactor and the j-factor data (30A) for Reynolds numbers greater than four times the critical number are plotted versus N ~ ~ d The j . Martinelli lines in Figure 15 as for the two different Prandtl numbers, Equations 39 and 40, are included, together with appropriate lines for the other analogies. All the turbulent-flow suspension heattransfer and pressure-drop data are included on a single plot and show reasonably good agreement with both the Martinelli and von Karman analogies.
4f3
Figure 75. Comparison of heat- and momentum-transfer analogies with data f o r non-Newtonian suspensions. For the j r s t j v e sets of points listed on thejgure, physical properties of the slurries were: r y = 0.075 lb./sq. f t . , 17 = 2.9 cp. = 0.0079 lb./ft.-sec., Npr = 8.2. For the bottom two sets, these were: r y = 0.46 lb./sq. ft., 17 = 8.2 cp. = 0.0055 lb./ft.-sec., NpT= 7 7 . 2
*0318 A 0 318
'378
0318
1030
I26 175
E 1030
a7
196
0 318 8 1030
378
635
I75
196
A
I
($
252
635 635 635
----
2 FRIEND AND METZNER
I
196
VOL. 5 5
NO. 1 2
DECEMBER 1963
31
Suspension Nucleate Boiling
Nucleate boiling heat-transfer measurements have been made with aqueous T h o z suspensions containing up to 0.105 volume fraction solids (344). Boiling took place from the surface of 1/16- and l/g-inch diameter platinum tubes submerged in slurry. The results showed that the heat flux was proportional to the nth power of temperature difference between the tube surface and the bulk fluid :
q/A = ( A T ) " (411 For the slurries studied, the heat flux at a A T of IO" F. was about lo4 B.t.u./hr. s q . ft. regardless of slurry concentration. However, the value of the exponent, n, decreased as the volume fraction solids was increased. The value of n was 3.3 with no thorium oxide present and approached unity a t a volume fraction solids of 0.10. Typical results are shown in Figure 16. The maximum heat flux attainable under nucleateboiling conditions (often called the critical heat flux or burnout heat flux) at slurry concentrations of 200 g. of thorium per kg. of water was about the same as for water. However, at a concentration of 1000 g. of thorium per kg. of water, the burnout heat flux was 210,000 B.t.u./hr. sq. ft. compared with a value of 490,000 B.t.u./hr. sq. ft. for water under corresponding conditions. At constant heat flux, the temperature difference between the heated tube surface and the fluid saturation temperature increased 5 to 6" F. per hour. This result might be explained by a "soft" film that surrounded the
'i6
I
V O L U M E FRACTION OF SOLIDS
TUBE O I A ,
INCHES 1/16
i a I I8
1 ;A i
8
I'
6
5 4
7
_-
1 ^?
1
2
3 r 5 C 9 , 3
I c::
AT 'F,
Fzgure 76. Effect of thoria suspenszon concentration on nucleate boiling from a platinum tube. Forced conuectionjow rate about one g.p.m. zn tank containing about 3gallons of slurry 32
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
heated metal surface. This film was apparently less than 1/32 inch thick and was never distinguishable as an adhering film after the tube was removed from the slurry system. No hard cakes were observed on the surface from which boiling took place during any of the tests. The nucleate-boiling tests were made with aqueous thorium oxide slurries which had non-Newtonian laminarflow characteristics and which were almost Newtonian under turbulent-flow conditions. No phenomena were observed which could be attributed to the effect of the solid particles on the gross physical properties of the slurry. For example, the non-Newtonian laminar-flow characteristics of the slurry had no discernible effect on the nucleate-boiling heat transfer ; instead, both the decrease in burnout heat flux and the effects shown in Figure 16 were attributed to the deposition of a film of solids onthesurface caused by vaporization of liquid at the surface (20A). I n studies of subcooled boiling burnout with nonKewtonian suspensions, a decrease in burnout heat flux for the suspensions was observed when compared with water flowing at the same velocity and degree of subcooling (72A). However, it was postulated that this decrease was due to a decrease in the effectiveness of the bubble convection loop caused by the suspension yield stress, rather than a film of solids on the surface. As yet, there is insufficient data to prove whether the observed effect is due to one mechanism or the other or to a combination of both. Flocculated Suspension Transporl
The minimum transport velocity is defined a s the mean stream velocity required to prevent the accumulation of a layer of stationary or sliding particles on the bottom of a horizontal conduit. I n studies with flocculated suspensions, two flow regimes were observed depending on the concentration of the suspension (31A). In the first, the suspension was sufficiently concentrated to be in the compaction zone and hence had an extremely low settling rate. The second regime was observed with more dilute suspensions which were in thc hinderedsettling zone and settled ten to one hundred times faqtar than suspensions which were in compaction. Concentrated Suspension Transport. The difference between the dilute and concentrated flow regimes are clearly evident when the experimental data shown in Figure 17 are examined. For dilute suspensions (characterized by a low yield stress in accord with Equation 15), the minimum transport velocity was substantially a constant for any given suspension; this velocity was appreciably greater than the velocity required for turbulent flow. As the concentration was increased, the minimum transport velocity also increased, but not as rapidly as tha transition velocity, so that eventually the two curves coincided. Thus for sufficiently large concentrations, the suspensions were in compaction and the minimum transport velocity \vas essentially the velocity at transition to turbulent flow. ID reality the minimum transport Reynolds number was 10% to 60y0 greater than that calculated from the intersection of the appropriate Hedstrom line (Figure 9) and the
........................
j
?
KAOLIN-HI
0
t-
.......................... IUBI OIAMEIER [INCHES) 1.045
A 2.056 0
3.97
.
0
l I - l . . . l .. I...li. I"-4
2
3
4
6
810-3
YdLO SIRISS ( 3 lBi./SO.
F1.I
rigwe 77. Effect of yield stress on minimum trnnspmt uel0cily. Velocity for transition from lmninm to furbulctlrflow ipuolr minimum honrport velocity at Imge vulues of yidd stress
turbulent flow l i e calculated from Equations 32. 33, and 34. In all probability the true minimum transport velocity corresponds to the end of the transition region rather than the beginning. T o a first approximationthis velocity is given by Equation 26. The critical concentration dividing the dilute and concentrated flow regions was determined experimentally to be the same as that observed in hindered-settling measurements as the floc structure bridged across the tube. In the absence of experimental data similar to that shown in Figure 8, the critical concentration dividing the two regimes may be estimated from Equation 22 by assuming that depth, H, equals diameter, D . Dilute %uspensionTransport. In classifying dilute suspension transport phenomena, it is useful to 'consider the case of infinite dilution. Under these circumstances, the minimum transport problem for dilute suspensions can be divided into two major flow regimes (36A) (Figure 18). For flow regime I, a particle in contact with the channel wall would be immersed in an essentially laminar sublayer when (D,u.*lv)5. In addition, the diagonal limes on Figure 18 for wnstant values of the product (D,uo*/v)(U,/u,*) = D,Li,/v gives the Reynolds number (and hence the drag coefficient) for particles settling in a quiescent fluid. For dilute suspensions in the hindered-settling range, theflocs are not in contact. Hence, energy must be supplied in excess of that required to initiate turbulence in order to overcome the tendency of the floes to settle. Thus the criterion given for concentrated suspensions would no longer be sufficient ;instead the minimum transport condition for dilute suspensions was found to be a function of both the floc settling velocity and the intensity of turbulent fluctuations (374. The settling
rate provides a measure of the tendency of the: particles to settle out, whereas the turbulent fluctuations provide a driving force to maintain the particles in suspension. The measurement of the turbulent fluctuations in flocculated non-Newtonian suspensions was beyond the scope of the suspension transport studies. However, Laufer (774 has shown that the turbulent fluctuations of Newtonian fluids at any given radial position are proConportional to the friction velocity, u* = sequently, the friction velocity was used as a measure of the turbulent fluctuations. In fact many investiga-
dgx.
V' =
& I03
I
IO-
Figurc 18. Flour regime closrifdion for minimum transport correla- , tim VOL 55
NO. 1 2
DECEMBER 1 9 6 3
33
d
tions have shown that the ratio of particle terminal settling velocity to friction velocity is the principal factor affecting the distribution of solids in a flowing fluid ( 6 4 78A). The experimental data for dilute suspension transport showed that the friction velocity at the minimum transport condition was substantially independent of pipe size in the range 1- to 4-inch diameter and also independent of concentrationupto the limit given by Equation 22. I n addition, all of the data (37A) for dilute flocculated suspensions of thoria and kaolin were in flow regime I as defined in Figure 18. Additional regime I data were available for nonflocculated suspensions of glass beads in water (21A, 33A) and fine coal dust in air (22A). These data are shown in Figure 1 9 using the coordinates suggested by Figure 18. The particle diameter for flocculated suspensions was taken to be the floc diameter, calculated in the manner described above in the section on hindered settling of flocculated suspensions. A least-squares analysis gave (33,4):
The coefficient and exponent are in rather good agreement with the predictions of a phenomenological analysis (0.0104 and 3, respectively) which assumed that particle transport occurred when lift, due to Bernoulli forces resulting from the velocity gradient across a small particle resting on the pipe wall, was sufficiently large to overcome the gravitational forces on the particle. (Parenthetically, it may be noted that the transport process for large size, flow regime 11, particles is quite complicated as might be guessed by examination of Figure 19. Since these suspensions are not flocculated, they are outside the scope of the present paper but additional information may be found in reference 33A.) Epilogue
This discussion of the transport characteristics of nonNewtonian suspensions must be considered as more in the nature of a progress report than a final report for so little has been done and so much remains to be done. Certainly, these studies represent the first extensive investigation of laminar and turbulent heat, momentum, and mass transfer characteristics of the same suspensions under comparable conditions and all the data displayed remarkable internal consistency. But only integral measurements (for example, over-all pressure drop, mean velocity, or over-all heat-transfer coefficients) were made. h‘ecessarily, the analysis of these results was based more on plausibility arguments than on the facts one could obtain from differential measurements such as velocity and temperature profiles or concentration gradients. Despite these limitations, the investigations fulfilled one engineering requirement-that of producing general relations suitable for design purposes. Application of the principles of colloid chemistry showed that the laminar-flow physical property relations could be expected to be of widespread usefulness, applying to all aqueous suspensions of metal oxides or solids which act as a reversible gas electrode provided the particles 34
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are symmetrically shaped. SVith the same restrictions on particle shape, the turbulent flow correlations were also shown to be general in that they were functions only of the intrinsic properties of the two phases and the laminarflow properties of the suspensions. In addition to providing design relations, the results furnish some insight into processes that may be occurring as well as establish limits into which more rigorous treatments must fit. The question remains, what can be done to improve our fundamental understanding of the transport characteristics of suspensions? Under laminar flow conditions, analysis shows that the properties of dilute suspensions of solid spherical particles are simple functions of the fluid properties, the volume fraction solids, and in some cases the particle density and diameter. Yet this represents almost the limit of our ability to derive rigorous theories. There has been some success in treating dilute suspensions of nonspherical solid particles or dilute suspensions in which the particles are liquid drops. But attempts to extend theory to concentrated suspensions, where particleparticle collisions leading to doublet and triplet formation must be accounted for, leave something to be desired. The further complications accompanying consideration of irregularly shaped particles or the reduction of particle size to near colloidal dimensions has hardly been touched theoretically. Factors that must be considered in the latter case include the nature of the attractive force between particles, the effect of electrolytes on the magnitude of this force, surface roughness of the particles, the geometrical arrangement of particles in a floc under static conditions, the effect of shear on the rate of formation and destruction of flocs, and on the arrangement of particles within the floc, and so forth. O n the experimental side, work of Mason and Bartok (79A) on the behavior of suspended particles in
laminar shear is a n outstanding example of the kind of research that is required to provide a rigorous test of theory. Mention might also be made of the recent work by Giesekus (14A) on the motion of particles suspended in constant velocity gradient flow field. As for turbulence studies with flocculated suspensions, the outlook is not favorable. The primary reason for this, of course, is the great difficulties that have been encountered in the development of turbulent flow theory for Newtonian fluids. One area of turbulent flocculated suspension research that is important and also which is within the capability of existing experimental techniques is the determination of accurate velocity profiles for non-Newtonian suspensions in pipes and channels ( 3 A ) . Some progress has been achieved on theoretical and experimental studies of the statistical properties of flowing gas-solid suspensions (23A, 28A) but no results are available for non-Newtonian suspensions. NOMENCLATURE A A1
= area, sq. ft. = force constant, lb.,
6 , c, d, e = exponents, dimensionless B = coefficient, dimensionless = specific heat, B.t.u./lb. O F. c, D = diameter, ft. du/dr = velocity gradient, sec. = Fanning friction factor, ( D A j / 4 L )/ ( p V 2 / 2 g , ) dimensionless f g, = conversion factor, (lb.,/lb.,)(ft./sec.2) gL = gravitational constant, ft./secZ G = mass flow rate, lb.,/hr. sq. ft. h = heat-transfer coefficient, B.t.u./hr. sq. ft. O F. j = ( h / ~ , p V ’ ) ( N p , ) ~dimensionless ’~, k = thermal conductivity, B.t.u./hr. sq. ft. F./ft. L = length, ft. N = wall layer thickness, dimensionless N x e = Hedstrom number, g,pruD2/+ Np, = Prandtl number, c,v/ke N R ~= Reynolds number, Dvp/q or DVp/pe q = heat flux, B.t.u./hr. T = temperature, O F. = particle settling rate, ft./se_c. Ut u* = friction velocity, d g c r w / p ,ft./sec. = minimum transport friction velocity at infinite dilution, uo* ft. /sec. V = mean stream velocity, ft./sec. = transition velocity, ft./sec. V, x = length, O Y / U * , ft. = dimensionless distance from wall, y u * / v y+ a,
Greek Letters CY = volume immobilized water/volume solid, dimensionless = wall layer thickness, ft. 6 = von Karman coefficient, dimensionless K = coefficient of rigidity, lb.,/ft. sec. 7 6 = p / ~ dimensionless , = viscosity of suspending medium, lb.,/ft. sec. p = effective viscostiy, lb.,/ft. sec. pLe = kinematic viscosity, sq. ft./sec. v p = density, lb.,/cu. ft. = shear stress, Ib.,/sq. ft. r = wall shear stress, D A f / 4 L, lb.,/sq. ft. rw = yield stress, lb.,/sq. ft. ru rP = g,pryx2/p2,dimensionless = volume fraction solids, dimensionless @
Subscripts b = bulk f = floc I = suspending medium p = particle s = suspension w = wall
LITERATURE CITED (1A) Alves, G. E., Boucher, D. F., Pigford, R. L., Chem. Eng. Progr. 48, 385 (1952) (2A) Bird, R. Byron, A.1.Ch.E. J . 2, 428 (1956). (3A) Bogue, D. C., Metzner, A. B., IND.ENC. CHEM.FUNDAMENTALS 2,143 (1963). (4A) Bowen, R. L., Jr., Chrm. Eng. 68, No. 15, 143 (July 24, 1961). (5A) Bugliarello, George, Daily, J. W., Tapfii 44, 881 (1961). (6A) Chien, Ning, “The Present Status of Research on Sediment Transport,” Trans. Am. Soc. CiuilEng. 121, 833 (1956). (7A) Clapp, R. M., “Turbulent Heat Transfer in Pseudoplastic Non-Newtonian Fluids,” p. 652 in “International Developments in Heat Transfer, Part 3,” Am. SOC.Mech. Eng., 1961. (8A) Colburn, A. P., Tram. A.l.Ch.E. 29, 174 (1933). (9A) Daily, J. W., Bugliarello, George, Tappi 44, 497 (1961). (10A) Dodge, D. W., Metzner, A. B., A.l.Ch.E. J. 5 , 189 (1959). (1 1A) Eissenberg, D. M., “Measurement and Correlation of Turbulent Friction Factors of Thoria Suspensions a t Elevated Temperatures,” A.1.Ch.E. J . , in press. (12A) Eissenberg, D. M., “Boilin Burnout Heat Flux Measurements in a NonNewtonian Suspension,” 51st dational Meeting, A.I.Ch.E., San Juan, Puerto Rico, Sept. 29, 1963. (13A) Gabrysh, A. F., Eyring, H., Cutler, I., J . Am. Ceram. Soc. 45, 334 (1962). (14A) Giesekus, Hanswalter, Rhcologica Acta 2, No. 2, 112 (1962). (15A) Hughes, R. R., IND.END.CHEM.49, 947 (1957). (16A) Knudsen, J. G., Katz, D. L.,“Fluid Dynamics and Heat Transfer,” McGrawHill, New York, 1958. (17A) Laufer, John, “The Structure of Turbulence in Fully Developed Pipe Flow,” National Advisorv Committee for Aeronautics Reo. 1174 (1 954). (18A) Marris, A. W., Can. J . Technol. 33, 470 (1955). (19A) Mason, S . G., Bartok, W., “The Behavior of Suspended Particles in Laminar Flow,” p. 16 in “Rheology of Disperse Systems,” Pergamon Press, New York, 1959. (20A) Metzner, A. B., Friend, P. S., IND. ENG.CHEM.51,879 (1959). (21A) Mur hy, Glen, Young, D . F., Burian, R. J., “Progress Report on Friction Loss of &wries in Straight Tubes,” Ames Laboratory Rept. ISC-474 (April 1, 1954). (22A) . , Pattenon. R . C.. J. Enr. Power (ASME1, 81.43 (19591 , ~ ~ (23A) Peskin, R. L., “Some Effects of Particle-Particle and Particle Fluid Interactions in Two-Phase Flow S stems,” p. 192 in Proc. 1960 Heat Transfer and Fluid Mechanics Inst., Stanford bniv. Press, Stanford, Calif. (24A) Petersen, A. W., Christiansen, E. B., t o be published. (25A) Rouse, H., ed., “Advanced Mechanics of Fluids,” p. 302, Wiley, New York, 1959. (26A) Schlicting, Herman, “Boundary Layer Theory,” 4th ed., p. 503, McGrawHill. New York. 1960. (27A) Shaver, R. G., Merrill, E. W., A.I.Ch.E. J . 5 , 181 (1959). 1, 33 (1962). (28A) Soo, S. L., TND. ENC. CHEM.FUNDAMENTALS (29A) Stanton, T. E., Pannell, J. R., Phil. Trans. RoyalSoc. 214, 199 (1914). (30A) Thomas, D. G., A.I.Ch.E. J . 6, 631 (1960). (31A) Thomas, D . G., A.l.Ch.E. J . 7, 423 (1961). (32A) Thomas, D . G . , A.1.Ch.E. J . 8 , 266 (1962). (33A) Thomas. D. G.. A.1.Ch.E. J . 8. 373 (1962). ( 3 4 4 Thomas, D. G., Chem. Eng. Prog. Symp. Ser. 57, No. 32, 182 (1960). (35A) Thomas, D. G., “Significant Aspects of Non-Newtonian Technology,” p. 669 in “Progress in International Research on Thermodynamic and Transport Properties,” ed. by J. F. Masi and D. H . Tsai, Academic Press, New York, 1962. (36A) Thomas, D. G., “Transport Characteristics of Suspensions: Part IX. Representation of Periodic Phenomena on a Flow Regime Diagram for Dilute Suspension Transport,” A.1.Ch.E. J.,in press. (37A) Vanoni, V. A,, Trans. Am. Soc. Civil Eng. 111, 67 (1946). (38A) Weltman, R. A., Keller, T. A,, “Pressure Losses of Titania and Magnesium Slurries in Pipes and Pipeline Transitions,” Nat. Adv. Comm. for Aeronautics TN-3889, January 1957. .
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