I
SEYMOUR CALVERTl and ROBERT
H. MILLER
Engineering Research Institute, The University of Michigan, Ann Arbor, Mich.
Flow of Dense Pastes in Vertical Tubes Here is the initial approach to the possible use of paste fuel elements in nuclear reactors use of solid fuel elements in nuclear reactors requires high-cost fabrication and replacement techniques and scheduling of replacements on the basis of dimensional changes due to radiation. I t is possible that these problems could be avoided by use of a paste fuel composed of small particles of fissionable material plus a liquid such as sodium. This article covers the initial stages of a study of high-density paste flow, related to nuclear reactor application. Conduit sizes and flow velocities studied were small. However, upward flow may be generalized, and the results applied to systems involving larger sizes and flow rates. The hairpin system was selected for study (Figure 1) and three questions had to be answered :
T H E
What is the maximum density (or minimum porosity) at which paste will flow in a hairpin system and what influences the flowing density? How great is particle friction against the tube wall and what factors determine it? What is the driving force for flow and 1
Present address, Case Institute of
Technology, Cleveland, Ohio.
how is it related to solid and liquid flow rates and other variables?
Experimental Procedure
Literature answers the third question and the first partially. The driving force for flow results from weight of particles and the drag on them from liquid flowing past. For the paste systems studied, flow of fluid through the particle bed is laminar and the relationship among pressure drop, flow rate, particle size; and other variables is satisfactorily represented by the KozenyKarman equation ( 7). The general correlation of Mertes and Rhodes (2) for two-phase (fluid-particle) flow, based on flow around particle rather than through a porous medium, can sometimes provide the same information as the Leva correlation, but it does not tell the bed density nor wall friction. It does not take particle-toparticle forces into account and is, therefore, not suitable for describing high density paste flow when these forces are appreciable. Although Mertes and Rhodes did not carry concurrent upward flow runs beyond a solids fraction of 0.184, with their apparatus exit effects would have caused erratic results at high densities (high solids fractions).
40 to 300 microns (325 to 40 mesh) and paste flow velocity was 5 or 10 inches per
Fluid
Fluid *,
.-0
Particle sizes used ranged from about hour. Solid materials were glass beads, Ottawa sand, crushed quartz, and copper shot; the liquid was water. Paste can be made to flow vertically either up or down if the driving force results from particle weight plus drag exerted by the fluid moving through the column of particles. At the low flow rates investigated, the flowing density, which approximates settled density, is 55 to GO volume % of solids for rounded and spherical particles and 45 to 50% for angular particles. If solids led to the system are not restricted, uniformly dense flow can be obtained with the restricted exit tube. Exit restriction need not be a decrease in conduit size. The same effect can be obtained by adding a small upward flow section or changing the system so that a higher liquid-solid ratio is required to move solid particles away from the tube exit. Insufficient data are available to define quantitative design for any but
4
7"
Figure 1. The hairpin system was chosen for study
L
5
v)
d
sservoir
raduated ylinder Meter
HAIRPIN TUBE
WC
Paste from R9ervoir
b
ity High Density Paste-/ HORIZONTAL
Valve
DOWN WARD FLOW
FLOW (STRATIFIED)
Figure 2. The hairpin flow apparatus was used to evaluate flow rates with pressure drop and tube and particle parameters
,
VOL. 50, NO. 12
Pressure Tap To Manometer
DECEMBER 1958
1793
Figure 3
14
14
12
IO w i4
a
8
3 0 4 LL
2
6
4
Figure 4
b
4
k-
2 n 0 (crn Hg under H 2 0 / f t )
PASTE
FLOW RATE
Figure 5 hairpin geometry. A dense paste having particle-to-particle contact without substantial electrostatic interactions cannot be considered a fluid, either Newtonian or non-Newtonian. Two continuous phases are present, each capable of transmitting force. Upward Flow. T o relate flow rates with pressure drop and tube and particle parameters-i.e., to evaluate and define the frictional properties of pastes in terms of system variables-two experimental techniques were used: Flow rates and pressure drop were measured in hairpin tube systems, and frictional properties were determined in a rotating cylinder (Brookfield) viscometer. The hairpin flow appzratus (Figure 2) consisted of a paste' reservoir feeding downward into the hairpin, a waterfeeding and metering system discharging into the top of the reservoir, and a
1794
- cm3 / m i n
I 2 3 4 PRESSURE GRADIENT AP/L ( c r n H g under H, O / f t )
5 6 7 IN DOWNWARD L E G
.o
Figures 3 to 10. The technique required experimental determination of relation between conductance and porosity and cell constant for each segment of tubing between electrodes
D R Y SOLIDS FLOW (cm3/nin-cm"
Figure 6
hairpin exit which delivered discharge paste to a graduated cylinder. Pressure taps were located at several positions along the hairpin. Solid and liquid flow rates were determined from the weight and volume of samples collected in measured time intervals. The system was operated batchwise; solids were charged into the reservoir only at the start of a run. Exit Types. Hairpin tube exits have a great effect on flow. The principal' variations in exit design were: 180' and
INDUSTRIAL A N D ENOINEERING CHEMISTRY
90' bends of the same tube diameter as the hairpin, overflow from the upper end of the hairpin, a washed overflow provided by directing a low-volume stream of water down at the top of the overflow exit, and an overflow with a triangular notch cut in the upper end of the tube. The porosity of the moving column of solids was determined for some runs by electrical conductivity measurement. A dilute solution of sodium acetax was used as the electrolyte; platinum bead electrodes were fused into the glass walls
,PASTE FLOW 0.59
0.58 0-
0.57
a-
Decreasing rate for 17.8 mm tube Increasing rate for 8 mm tube
0.56
0.55
I
I
I
*45*0 0.530
4
8 DRY
Figure
7
-
SOLIDS
-
20
16
I
I - 28
24
32
FLOW RATE (cm3/min-cm'
SOLIDS
Figure 8
E .-A40
R A T E (cm3/min-cm2 )
12
I
50
N
DRY
I
I
Predicted for round sand
I
I
I
12
14
E m ' E
2 30 a
K
3
20
0
-1 &
-I
I
10
2 0
4
+
Figure 9 Figure 10
Table I.
Tube Figure 3
Dia., Mm. 10.3
0
b
,
,
4 5
10.3 10.0
6
10.3
7
10.3 17.8 17.8 11.0
0.530 t o 0.588
"0 1 2 3 4 5 DRY SOLIDS RATE (cm3//min-cm2 ) of the hairpin, and conductivity was
measured with an alternating current bridge. The technique required experimental determination of the relationship between conductance and porosity and of the cell constant for each segment of tubing between the two electrodes (Table'I, Figures 3 to IO). Viscometer Measurements. To determine the "viscosity" of sedimented
819 10 a
0 = overflow;
I
I
Relations between Pressure Drop and Other Variables
Tube Length, Ft. 2
Particles Size Type
Upward Leg Rowb
Exita
16/20 G . B. 32/35 G . B.
SO/lOO G. B.
o-(I-X) varyin9 from
1
2 4 6 8 0 DRY SOLIDS RATE (cm3/min-crn2 )
0 0 0 0 0 0
2.0-2.5 2.2-2.8 1.6-2.5 2.0-4.0 1.8-4.0 1.3-8.5 1.0-4.0 1.3-1.6 1-4-2.2 1.9-2.4 2.1-2.3 2.7-3.8 2.7-3.8 1.0 1.0 1.0 1.0 1.0 1.0
100/120 G. B. 170/200 G. B. 270/325 G.B. S 2 100/120 G. B. 2 100/140 0.S. 180' B 180' B 150/200 Cu 180' B -200 0.s. 180' B 4 100/140 0.S. 8 180' B 150/200 0.S. 1800 B 150/200 C. Q. 2 OC 140/200 Cu OC 200/325 Cu 2 S 100/120 G. B. W 2 100/120 G. B. W 2 100/140 0.S. W 2 40/60 C. Q. S = several; B = bend; W = washed. Approximate.
pastes a rotating cylinder viscometer was used. It was capable of turning the spindle a t 1, 2, 5 , and 10 r.p.m., and was fitted with a flat spiral spring which
*
DP
!
Microns 912 456 163 137 81 49 137 127 89 74 127 89 89 89 59 137 137 127 335
Notched.
connected the drive shaft with the spindle shaft and functioned as the sensing element for a torque pickup. Three cylindrical spindles were used: VOL. 50, NO. 12
0
DECEMBER 1958
1795
2 inches long from the bottom to the gage mark, and l / 2 , 3/4, and 1 inch in diameter. The range of torque pickup was from 0 to 500 units, each representing 0.1025 gram cm. Attempts to obtain meaningful data by running the spindle in a beaker of settled paste failed because of large and erratic variations in the normal force due to the solid particles bearing on the cylinder. The spindle could be stopped by pressing a finger on top of the solid bed 1 or 2 inches away from the spindle. The weight of particles above a given point on the spindle contributed to the normal force at that point to varying degrees, depending on arrangement of the particle bed. T o avoid this effect the system shown in Figure 11, whereby water could be forced up through the bed, was used. As the water rate is increased, frictional pressure drop becomes equal to the head due to the buoyant weight of the solids and there should be no variation in normal force acting along the length of the cylinder. Further increase in water rate results in an expansion of the bed and still provides the characteristic of weightless particles. This is the only way known to the authors for applying this instrument to dense paste systems with clear-cut significance. The data for 100/140-mesh Ottawa sand are presented in Figure 12. Because of the “stick-slip” phenomenon, intensified by the torque-spring coupling in the drive shaft, readings fluctuate widely and are shown as bars rather than points. The maximum scale reading is
VISCOMETER CAN RUN A T 1,2,5,EIIO R P M TORQUE PICKUP RANGE IS 0 TO 51.25 gm-cm
PASTE
RITTED
BED
DISK
uu
t
WATER
Figure 1 1 . To avoid variations this system was used
1 796
500, and this controlled the highest bed density which could be run in the apparatus. Where data stop below 500, it indicates that further increase in density (decrease in water rate) would cause the torque gage to go off scale. Data for 100/140-mesh copper shot and 100/120-mesh glass beads (Figure 13) are subject to uncertainty due to both the stick-slip effect and channeling at high water-flow rates.
100
Discussion and Results The frictional effect in an upward flow system can be estimated from viscometer data. Shear stress (force per unit area of cylinder surface) is approximately constant with the cylinder diameter at lower bed densities-i.e., torque is proportional to the square of the cylinder diameter. Assuming that this will hold for paste flow through tubes, frictional force exerted against the tube wall can be estimated. By extrapolation, the torque reading for the ’/e-inch diameter spindle in 100/140-me;h sand a t 55% solids is estimated to be 1.5. Thus, the torque is 0.154 gram cm., and the shear stress is 0.0116 gram per sq. cm. Wall-to-particle friction is conveniently expressed in terms of a dimensionless ratio in which the fluid frictional pressure drop is related to the buoyant weight of solids per unit length of tube. This “overweight ratio” is defined as:
where Row= overweight ratio (dimensionless) AP = frictional pressure drop (positive if it produces an upward force) r = tube radius P b = buoyant bed density of particles-particle density minus fluid density times volume fraction solids = ( P P - P I ) (1 - X ) L = bed length g = acceleration due to gravity ge = 32.17 Ib., ft./lb., sec2 If the overweight ratio is 1.O,force due to fluid friction is just equal and opposite to force due to particle weight and flow can occur if there is no wall friction. The ratio of wall frictional force to force due to buoyant weight of solids a t (1 X) = 0.55 for sand is then estimated from viscometer data to be (0.051/0), where D is tube diameter in centimeters. For upward flow in a 1-cm. diameter tube wall friction would add a resistive force equal to about 5% of the force due to particle weight. For a viscometer reading of 40 for copper shot at 55% solids the ratio of wall friction to bed weight is estimated as (0.14/0). This is not bad confirmation of the experimental observation that frictional effects are negligible for upward flow with a flushed exit. A closer check could be made by
INDUSTRIAL A N D ENGINEERING CHEMISTRY
IO
I
055
057
VOLUME
059
061
FRACTION
063
065
067
SOLIDS ( I - X )
Figure 12. For Ottawa sand rotational speed had a negligible effect
more elaborate techniques and a viscometer fitted with a zero-displacement torque pickup. However, this technique is valid only for upward flow and direct experimental observation of pressure drop is simpler. Hairpin System. The effect of exit design on hairpin flow is shown clearly in Table I and Figures 3 to 10. Where particles are washed away from the exit (no restraining force present) upward flow occurs when the frictional pressure drop is equal to the buoyant weight of the bed (overweight ratio = 1.0). I n some cases the 180’ return bends and plain overflow exit caused flow a t an overweight ratio of 1.0, but in most they caused flow at higher pressure drops in the upward leg. Also (Figure 5) overweight ratio was dependent on tube length for 180’ return bend exits. Tube lehgth does not affect overweight ratio when a washed exit is used. Thus the only reproducible case is that of the washed overflow exit. Some runs with the washed exit indicated a decrease in flowing density with flow rate. Density in the upward leg remained constant at the loosely settled density at all flow rates, once a high enough flow rate was attained; flow at higher than the loosely settled density is an unstable situation which can be eliminated by “breaking-in” the system a t a high flow rate or letting it run for a long period (Figure 8). Flow at less than the loosely settled density will not occur in a hairpin of uniform diameter. Several runs were made with hairpins on which the return bend was made of flexible (Tygon) tubing, so that it could be restricted by a screw clamp. The cross-sectional area of the bend could be reduced to a small
PASTE FLOW This relationship can be evaluated a t the point where the pressure drop per unit length is equivalent to the buoyant density of the solid particles in the bed. After doing this and changing some of the variables to more convenient units, we have
1000 I
l
0 500
500
1 ~
-
1 100
io0
f a 4
y
50
U
where C1 = a constant depending on shape and fluid properties, f and equal to(
2*g4510-4)
50
a
for water a t room temperature p s = density of dry solid, grams per cc. p = density of liquid, grams per cc. D, = particle diameter, microns
W !-
Y
s 52 I
'
l5
0
F
3
F
s
E
H
0-3/4" dla. spindle
1
1
0.55
0.57
I
VOLUME
1 0.59
I 0.61
1
1
I
1
0 - 314" dla. spindle Note One scale unit * 0.1025 gm.cm.
1
1
1
1
1
0.61 o..a 0.6s VOLUME FRACTION SOLIDS (I-X)
055
0.63 0.65
FRACTION SOLIDS (I-X)
0.57
0.59
Figure 13. Data for copper shot (A) and glass beads ( B ) are affected b y the stick-slip effect and channeling at high water rates
fraction (perhaps l/10) of its original value without impeding the solids flow rate sufficiently to decrease paste density in the upyard leg.
where
Correlation of Data Once the nature of the upward flow system was known, it became possible to describe its behavior on the basis of the following facts: The washed-exit system is the basic upward flow situation with no restraining force on particles a t exit. Friction between flowing paste and tube wall is negligible in upward flow. Paste flows a t approximately its loosely settled density in the upward leg and a t a somewhat higher density in the downward leg of a hairpin with washed exit. This description applies only to systems in which the solids input rate is not restricted; this covers the case of minimum ratio of liquid rate to solid rate. This is to differentiate the high-density flow regime from the low-density regime. in which for any solid rate there can be a range of liquid rates. The only problem remaining is to find a relationship which will indicate the liquid flow rate required to exert drag on the particles equal to their weight. The equation for the laminar flow of fluids through porous media is satisfactory. The method of correlation is
Vw
free liquid velocity, cc./ min.-sq. cm. (based on total cross-sectional area of tube) VT = total flow velocity, cc./ min.-sq. cm. V8 = dry-solids velocity, cc./ min.-sq. cm. X = volume fraction voids in flowing paste =
Equation 3 is represented in graphical form in Figure 14 and as a nomograph in Figure 15. The value of (1 - X) = 0.55 for average flowing densitv seems to be a fairlyvgood apiroxima;ion for rounded and spherical particles. Spherical particles such as glass beads exhibited flowing and loosely settled densities of about 57% solids; rounded sand ran about 55%. The loosely settled densities were determined in tubes of diameters from 8 to about 30 mm. and were not appreciably affected by tube diameter. Because effect of bed density is slight in' this range, this is not a crucial point.
Prediction of Flow Rates Application of the above correlation
Thus V p is the liquid velocity relative to the moving paste based on the total tube cross section. I t is this velocity which is related to the frictional pressure drop. Leva's correlation for the laminar flow of fluids through porous media is
for
Re
=
(y)
