to the operating liquid loadings of existing natural draft cooling towers. This would indicate that economic considerations result in operation of the tower close to the maximum liquid loading. This is very similar to the case of a gas stripping unit, where economic considerations result in the operating liquidgas ratio close to the maximum liquid-gas ratio. If a tower was designed to operate at a fraction, f, of the maximum liquid loading, then
L (operating) = fL(,,,)
g
G
(23)
Equation 23 is then a very simple design equation, since f, b, and S are known and the operating residual tower number can be obtained. For a circular tower of diameter D with a total liquid load of W , from Equation 6, (24) The determination of either H or D from this equation will depend on the ratio of D to H which was used to obtain factor f.
=
fraction
= acceleration due to gravity
= gas flow rate
H = height of tower HTU = height of a transfer unit
- enthalpy i L = liquid flow rate N = resistance N'TU = number of transfer units s = specification number - water temperature t T = tower number or air temperature depth of packing P = air density A = change a = Merkel's cooling factor
z
Tdoperating) = f T R ( m n x ) TR(operating) = 0.38 fzb-1/2So.808
f
=
SUBSCRIPTS DB IW max
ow
P
R W
WB
= = = = = = = =
dry bulb inlet water maximum outlet water packing residual water wet bulb
Acknowledgment
The author thanks the Mathematics Laboratory, University of Surrey, for time on the Sirius computer.
Furzer, I. A., IND. ENG. CHEWPROCESS DESIGNDEVELOP. 7, 555 (1968).
Nomenclature
b D
= =
Literature Cited
defined by Equation 8 diameter
RECEIVED for review December 21, 1567 ACCEPTED June 10, 1968
MODIFIED JET PUMPING OF SOLID SPHERES H E R B E R T SUSSKIND, R O B E R T ODETTE,' Brookhavan National Laboratory, Upton, N . Y .
AND WALTER
BECKER
17973
The removal of individual layers of balls from the surface of an ordered packed bed without disturbing the remaining packing was investigated. The modified jet pumping procedure could be used in the countercurrent fuel movement in the ordered bed fast reactor, as well as for the selective unpacking of beds of spheres in other industrial applications. Liquid jets entering two opposite sides of a square cross-section column through rectangular nozzles above the surface of the packed bed produced drag forces on the balls at a critical flow rote that were strong enough to entrain them and carry them out with the outgoing liquid. The distance of bed penetration increased with this critical flow rate. Such variables as liquid flow, ball density and diameter, nozzle velocity and inclination, and column size were investigated, and a mechanism 0.755 In X. for this model was postulated. The data could be correlated by the expression R = 2.63
+
TUDIES
were carried out at Brookhaven National Labora-
S tory to develop a sodium-cooled, fast breeder reactor concept in which packed beds of c '/b-inch-diameter spherical fuel particles were used in 12 X 12 X 48 inch steel containers (Susskind et al., 1 9 6 6 ~ ) . These studies culminated in a unique system in which perfectly ordered beds of spheres packed in a rhombohedral array were obtained consistently, even though the balls were dropped randomly into the containers. This method of packing provides the basis for the ordered bed fast reactor concept (Epel et al., 1966; Susskind et al., 1965). More generally, it applies to the packing of beds in any industrial application in which precisely known, high densities of 1 Present address, Massachusetts Institute of Technology, Cambridge, M a s . 02139.
solids and large heat and mass transfer surfaces are desirable. Countercurrent fuel movement (Epel et al., 1966) involves periodic rearrangement of the fuel balls, accomplished by mechanically or hydraulically handling them, either directly inside the reactor vessel or in an external hot cell. The purpose of the work described was to investigate the selective removal of individual layers of balls by hydraulic means from the top of a packed bed, ideally one layer a t a time, without disturbing the remaining packing. This was accomplished by the interaction between these balls and liquid jets entering the column just above them, representing a modification of the usual jet pump, which uses a high-velocity liquid to entrain a low-velocity or stagnant fluid. In this instance, solid spheres packed in the beds are effectively entrained. VOL. 7
NO. 4
OCTOBER 1 9 6 8
565
The following variables were studied in a series of 60 test runs leading to the development of a general correlation useful in the design of systems for removing balls from any packed bed: water flow rate, 0 to 75 gallons per minute (corresponding to velocities of 0 to 5 and 0 to 1.7 feet per second, respectively, in the two test columns); ball density, 1.15 to 7.8 grams per cc.; ball diameter, s',1 to '/z inch; nozzle width, ' / I 6 to 1/4 inch; nozzle velocity, 0 to 71 and 0 to 13 feet per second in the two test columns; upward nozzle inclination, 30' to 90' from the vertical; and ratio of column width to ball diameter, 3.6 to 30. The correlation previously shown by Susskind et al. (1966b) has been improved, and a model was postulated for this complex flow system (Susskind et al., 1966a). The testing program was not completed because of the curtailment of work on advanced reactor concepts by the U. s. Atomic Energy Commission. The principal results of the program obtained prior to the curtailment are reported here. Further details are given by Susskind et al. (1966b).
n 4SIDE DIMENSIONS OF COLUMNS:
, 7 9 5 " ~1.795" 41'-1/8" WIDE (15"EI.) 4I0-1/16"WIDE* (12 3/4"EI.)
4lo-1/I6" WIDE (12 5/8"EI.)
90°-I/8"WIDE (IO 1/4" El.) 30- I / ~ " W I D E
90°-1/8" WIDE (97/8"El.) 30'-1/8"WlDE (938"EI.) 70'-I/8"WIDE ( 6 5/8"EI.)
(91/2"EI.)
700- I W " W I D E ( 6 1/4"EI.)
Previous Work
The removal of solid balls from a packed bed by means of jetted liquid bears a strong resemblance to the action of a jet pump. Good reviews of the available information, including performance characteristics of jet pumps, are available in the literature (Cunningham, 1957 ; Folsom, 1948 ; Gosline and O'Brien, 1934; Reemsnyder, 1964). The analysis and design of pumps using the same liquid for both the driver and entrained fluids were placed on a firm basis by Gosline and O'Brien (1934). O n the other hand, no information described a system in which one of the components is a solid. The jet pump has, however, been used to pump a mixture of solid particles in suspension in the liquid as long as the average velocity of the mixture was above the critical velocity required to maintain the solid particles in suspension. O'Brien and Folsom (1937) have shown that the flow characteristics of the mixture are similar to those for the pure liquid possessing the density of the mixture. The critical velocity depends upon the concentration of solids and the settling velocity of the particles in the liquid. However, even this information is mainly applicable only to that part of the system studied here in which the balls have already been entrained and are flowing parallel to the axis of the column-Le., some distance above the nozzles. Furthermore, the jetted liquid in this system does not enter axially, as was assumed in the previous work. Comparatively little work has been done on energy dissipation in ducted jets which occurs in this system. Although considerable data exist on free jet mixing, it is difficult to deduce general laws for ducted jet systems. The difficulty increases as the position of the nozzle is changed from the axis of the duct to the walls, or if the cross section is changed from a circular to a rectangular one. Curtet and coworkers (Barchilon and Curtet, 1964; Craya and Curtet, 1955) and Becker and coworkers (Becker et al., 1963; Williams and Becker, 1963) were the first to recognize the general criteria for the similarity of ducted jets. Williams and Becker (1963) found that the flow pattern in a turbulent ducted jet is dependent on a single parameter, which they called the throttling number, Th, defined as the momentum ratio of the mixed fluids in the duct to that of the total inducted and jetted fluids. The throttling number varies between 0 and 1, and the regime of recirculation exists when T h is small. They found experimentally that recirculation stopped at as low a throttling number as 0.43. Craya and Curtet (1955) derived the similarity parameter, rn, which is a unique function of the kinematic and dynamic velocities of streams entering a confined-jet sys566
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
I
r
!
i
U
U
COLUMN 1
COLUMN 2
I
LATER CHANGED TO V4" WIDTH NOTE: ELEVATIONS MEASURED FROM TOP OF GRID PLATE
Figure 1.
Lucite test columns
tem. Becker et al. (1963) modified this parameter and were able to relate it to the maximum recirculation rate as a function of the total flow. This was verified experimentally (Barchilon and Curtet, 1964). Dealy (1964) also discussed the essential features of turbulent jets. Considerable information exists on the drag and lift forces exerted by a moving fluid on single spheres as well as spheres in the presence of others (Rowe, 1961; Rowe and Henwood, 1961; Struve et al., 1958; Young, 1960; Zenz and Weil, 1958). Rowe and Henwood (1961) found that the maximum lift observed on a sphere in a packed array of others was ~ 0 . 6of the drag on the same isolated sphere. The fluid velocity required to support a particle is extremely sensitive to the separation between the spheres. Increases of the drag coefficient of almost two orders of magnitude were obtained when the sphere formed part of a regular packed array, but this is reduced when a neighbor leaves, the lift increasing at the same time. The net result is that neighbors tend to move toward the path of the departing spheres. As more and more spheres are removed, the average drag per sphere falls rapidly. Rowe and Henwood also showed that surfaces formed by particle assemblies normal to the direction of flow are stable and possess a property analogous to surface tension. Before the minimum fluidization velocity can be reached, the topmost layer of particles will be subjected to the greatest drag and will, therefore, be the first to lift and become entrained. Fluidization will start from the top and develop downward as each layer lifts and allows the one below it to follow suit. Experimental Work
Materials and Equipment. The experiments described here were conducted in two square Lucite columns, 1.795 and 3.725 inches on a side, respectively (Figure 1). A series of paired rectangular nozzles was machined horizontally along two opposite sides of the columns. The nozzles were '/16, and '/4 inch wide, respectively, spanned the entire column
wall, and were inclined upward at angles varying between 30' and 90" from the vertical. Each pair of nozzles was located at a different height in the columns and those not in use were sealed with brass plates. Each bed was packed in an ordered array on a flat Lucite plate provided with suitably spaced countersunk depressions. These plates, which sealed the bottom of the columns, were provided with holes for drainage of liquid after the experiments were completed. The beds were packed with 0.125, 0.250-, and 0.500-inchdiameter stainless steel, aluminum, and nylon halls. Careful measurements of the balls showed that they varied in size hy less than +0.001 inch in diameter. Their surfaces were smooth. Water was recirculated through a surge-supply tank by means of a 3-hp. centrifugal pump. The flaw rate was controlled with a manually operated throttling valve and pump bypass circuit and measured with a previously calibrated rotameter. The water was circulated through l'/r inch copper tubing connected to the various column nozzles
the two nozzles (Figure 2) until the first balls became fluidized, and then increased in 2-gallon per minute increments until all the balls had been removed from the column or the maximum flow rate was reached. The height of the fluidized zone in the columns was measured from the start of water flow until the first hall carry-over occurred. Each flaw rate was then maintained for 1 minute, which was usually sufficient time to remove all the balls corresponding to that particular flow rate. The pump was turned off at the end of every flow increment, and the number of balls removed in that increment was determined from the height of bed remaining in the column. The number of halls fluidized in each flow increment was also measured and, with those carried out of the column, provided a measure of interlayer mixing. Care was taken to ensure that the volumetric flow through each nozzle was approximately equal. Flow patterns in the column were studied with the help of red dye injections into the system 2nd high-speed movies (1000 frames per second) of the action. Results and Discussion
Figure 2. Test column 1 packed with %-inch-diameter stainless steel balls Removal of top bed layer with walertlowing through l w o %-inch.wide rectangular nozzle$ inclined 70' from yerticd
with heavy-duty flexible rubber hose. The entrained balls were collected in an overhead Lucite classifier, from whence they could be returned to the columns again through a ball valve. Experimental Procedures. A column was packed in the ordered geometry by pouring in the balls until the bed height corresponded to the level of the pair of nozzles to be used. The packed column was then connected to the flow system. Water was allowed to flow into the column very slowly so as not to disturb the bed, while the air was bled off through a vent on the top. Water circulation was then started through
At each water flow rate, measurements were obtained of the expansion of the fluidized ball zone prior to initial ball carryover from the column, the number of ball layers set in motion by the flowing liquid, and the number of ball layers completely removed from the column by the flowing liquid. These data were then plotted for each run, and while the curves for the different runs varied in magnitude depending upon the exact experimental conditions, the following similarities in shape were noted. The fluidized ball zone expanded as the water flow through the nozzles located above the bed was increased. This height approached infinity as initial carry-over of the balls began in the column. When these critical flows were expressed as superficial fluid velocities, they varied between 20 and 80% of the terminal velocities of single spheres. This compares with the 67 to 70% reported in the literature for normal fluidization (Stepanoff, 1966). Except for the lowest liquid flows (shortest distances between the bed surface and the nozzles), the curves for the number of ball layers in motion inside the columns with flow rate paralleled those for the number of ball layers actually carried out of the columns. The number of ball layers in motion was 2 to 3 layers greater than that removed a t the same flow, indicating slight interlayer mixing. I n practice, the initial mixing could be minimized by locating the nozzles a t a greater height above the initial packed bed and starting ball removal with a greater liquid flow. High-speed movies, taken for selected experiments in conjunction with the introduction or removal of a red dye from the liquid, very clearly showed the formation of a series of paired vortices projecting downward into the column. Direct measurements of the film strips indicated that the length of the vortices was approximately equal to one column width. Greater bed penetration occurred in the areas of strong vortex flaw around the potential core of the vortex, while relatively little activity occurred in the law-velocity areas, resulting in the stepwise, rather than continuous, ball removal with flow that was found. Ball Density. As would be expected, the density of the halls had a very marked effect on their removal from the columns. Substantially more nylon than aluminum balls were removed, and more of both than stainless steel, a t identical liquid flow rates (Figure 3). The well-defined steps for the stainless steel and aluminum curves may also be seen. The changes in slope were not so pronounced for nylon, probably because of its low density, which is very near that of water (1.15 and 1.00 grams per cc., respectively). The ball carry-over from column 1 was plotted directly against ball density at constant water flow in Figure 4; the VOL. 7 NO. 4
OCTOBER 1 9 6 8
567
I
I
I
1
1
I
COLUMN A R E A - 1.795"~1.795" N O Z Z L E WIDTH 1/4" I N C L I N A T I O N - 4 l 0 F R O M VERTICAL B A L L SIZE 1/4"
1
I
I
1
COLUMN A R E A - 1.795"X1.795" N O Z Z L E WIDTH- I / B " B A L L SIZE l/4"
-
9""r
-
-
,3
8
5
NYLON
3
/'/
_I
/'
50
U
0
w
>
2
ALUMINUM
40
a v)
a W
3 30 _I _I
m a
%
20
[L
W
m
E
n -
3
0
50
10 20 30 40 TOTAL WATER FLOW RATE, g a l l r n i n
IO
60
Figure 3. Effect of ball density on ball removal from column 1 for different packings I00
1
I
I
I
I I I I I
1
1
I
1
0
0
IO
20 30 40 TOTAL WATER FLOW RATE, gal/min
50
1 1 1 1
Figure 5. Effect of nozzle inclination on ball removal from column 1 for different packings
\
1 60
-
z
COLUMN AREA 1.795"~1.795'' N O Z Z L E INCLINATION- 41° FROM VERTICAL B A L L S I Z E - 1/4"
E
2 J
,410
50
30'
E
NOZZLE WIDTHS
U
COLUMN AREA- 1 . 7 9 5 " ~1.795" NOZZLE WIDTH O,A,o=l/8"
0
a
= 1/4"
'
0 W16" BALL SIZE - 1/4" TOTAL WATER FLOW RATE45 gal/rnin
I
IO
I
I
I
I I I I I
IO0 B A L L DENSITY, I b / f t 3
1
I
I
I
40
70°
I
i
1 I l l
IO00
Figure 4. Effect of ball density on ball removal from column 1 at constant flow
-
5a v)
a 30
W
5 -1 -I
20 LL
0
a
k!
5z
IO
n "
curves showed a definite dependence on the nozzle inclination. While not shown, ball removal wa.s also affected by the column geometry. The curves for nozzles inclined at 41" and with widths of 1/16 and 1/4 inch were also superimposed in Figure 4. Their slopes were identical to that of the 30' curves, and, as would be expected, their position straddled the one for 30' (and 1/8-inch-widenozzles). Nozzle Inclination. The nozzle inclination determines the vertical component of velocity of the water and the size and shape of the high-velocity, noncirculating region in the column above the nozzles. Thus smaller inclinations will not only provide a greater effective vertical component to the balls, but will also be able to accelerate them in the region of the jets for a greater distance than is possible at larger inclinations (Figure 5). I n addition, the balls will suffer fewer collisions and tend to recirculate less above, as well as below, the jets. Not only is the area between jets through which the 568
1/8"
0
I&EC PROCESS DESIGN A N D DEVELOPMENT
0
IO
20
30
40
50
60
T O T A L WATER FLOW RATE, g a l / m i n
Figure 6. Effect of nozzle width on ball removal from column 1 for different packings
balls must depart larger for small nozzle inclinations, but the vortices above the jets are also smaller and weaker, which lessens the chance of ball recapture or mixing. Greater bed penetration was observed to occur with smaller nozzle inclinations because of an increase in the effective column cross section seen by the jets, thus lowering the throttling number and increasing recirculation. Similarly, more ball layers were removed in the larger cross-section column. Energy dissipation due to wall effects was also reduced. As shown in Figure 4, the ball density contributed a supplementary effect to the extent of ball carry-over from the column. Nozzle Velocity. Water jets from the smaller nozzles
1
z
I G50
5 -
K
I
I
I
1
1 .
I
COLUMN A R E A - 1.795"~1.795" NOZZLE INCLINATION 41' FROM VERTICAL B A L L S I Z E 1/4 TOTAL WATER FLOW RATE 4 5 gal/rnin
-
-
-
LL
1
i
COLUMN A R E A - 1.795"~1.795" N O Z Z L E WIDTH 1/8" T O T A L W A T E R FLOW R A T E 45 g a l l m i n
-
I-
36 W
I
n w
m
u 5
9 v)
W
I 0
2 4 2d
I
l
5
3 0
v 3
5
z
U
n
g 2
0
I
I
I
I
I
I
10 20 30 40 50 N O Z Z L E VELOCITY, ft/sec IN EACH NOZZLE
0
I
60
Figure 7. Effect of nozzle velocity on ball removal from column 1 for different packings
I ,
70
I
-
I
I
I
COLUMN AREA 1.795"~1.795" N O Z Z L E WIDTH- 1/8" INCLINATION-30' FROM VERTICAL
3 a v)
i ' a' m
n "
t
0
0.12.5 0.250 B A L L DIAMETER, I N C H E S
0.375
Figure 9. Effect of ball diameter on ball removal from column 1 at clonstant flow 1/8" ALUMINUM
balls were removed from the column than larger ones for all the materials studied. The effect of ball diameter is also shown in Figure 9 at constant water flow. Identical values were obtained for the slopes of the aluminum and stainless steel as well as aluminum and nylon curves at each nozzle inclination, but the values varied with nozzle inclination and column geometry.
$ 60 1
0 V
H K 0 LI
0
Y 40 K
Postulated Mechanism for Ball Removal Process
v)
K W >
4 -J 1
d b
20
n w m 2 3
z
0
0
IO
20 30 40 TOTAL WATER FLOW RATE, gol/min
50
Figure 8. Effect of ball diameter on ball removal from column 1 for different packings
possessed a much higher energy and therefore produced stronger vortices and deeper bed penetration. The jets will also impart added momentum to the balls to enhance their removal from the column. I n every case, therefore, the liquid flowing through the l/,,-inch-wide nozzles, which had the greater velocity, removed more balls than that flowing through the '/l-inch-wide ones (Figure 6). A cross plot of ball carry-over against nozzle velocity at constant water flow is shown in Figure 7. The curves all had the same slope. At very low velocities the curves drop to zero, since a minimum flow is required to start ball removal. Ball Diameter. T h e effect of ball diameter on ball carryover is shown in Figure 8. Many more layers of the smaller
The mechanism for the selective entrainment of individual ball layers from the top of an ordered packed bed without disturbing the remaining packing and for their subsequent transport out of the column is very complex. The depth of bed penetration increases with higher flow rates of the liquid entering two opposite sides of the square column through rectangular nozzles above the bed surface. The authors postulated that ducted turbulent jets are formed by this liquid entering the stagnant fluid in the column, which then forms recirculation eddies, or vortices, below the jets. The boundary conditions imposed by the container walls are such that the growth of the jets by continual entrainment of the surrounding liquid is stopped when it reaches the duct wall (Figure 10). If the amount of fluid is too small to satisfy the entrainment capability of the jets based on momentum considerations, compensation will occur through the generation of vortices. The initial pair of vortices then will generate a chain of vortex pairs down into the column, their energy decreasing with distance from the nozzles. These chains may be considered as energy transmission linkages whose length depends primarily upon the energy imparted by the fluid jets to the first vortex pair, the extent of the viscous damping losses in the relatively stagnant fluid, and wall friction losses. The lowest vortex in the chain sweeps the surface of the bed to entrain the balls which are then transported upward through the series of progressively more energetic vortices. They receive a final addition of energy when moving past the fluid jets, until they have gained enough momentum to leave the column overhead with the outgoing liquid (Figure 11). VOL. 7
NO. 4 O C T O B E R 1 9 6 8
569
MAIN FLOW AREAS
VELOCITY PROFILES AS FUNCTION OF Z
AREA OF UNIFORM VELOCITY
I\
& (ENTRAINS MBIENT FLUID)
RECIRCULATING
Generalized Correlation
On the basis of the postulated model, an empirical correlation was developed with the help of a digital computer at M I T to describe this complex system. The average strength of each vortex, S, was assumed to be a function of some, due to the complexity of this system as yet unknown, power of the average velocity of the first vortex, u g ,
S
It",
a
(11
vOa
Then the magnitude of penetration of the packed bed by the entraining liquid is determined by the strength, So, of the first vortex and its subsequent transmission to the vortices below, The authors assumed that the vortex strength decays exponentially as the distance from the nozzles, z, increases because of viscous damping in the fluid and wall friction. Thus
Figure 10. Schematic representation of flow pattern and velocity profiles in a ducted turbulent jet
s = Soe-Z'Z
(2)
The total bed penetration is determined by the point at which the velocity of the fluid is just sufficient to suspend the balls. The minimum vortex strength, S,, is proportional to the terminal velocity, u t , of a sphere in an infinite volume of fluid, or
where
The vertical component of the nozzle inclination may be expressed by cos~(p/2), half the angle being selected to avoid a discontinuity at 90'. A dimensionless strength parameter X,which is a measure of the useful jet work and includes the angular nozzle correction, may be defined as
x = (S/S,) cosb(q/2)
(5)
The average velocity of fluid in the first vortex can be related to the volumetric fluid recirculation rate, q,, by UP
Figure 11. Typical representation of the flow pattern in test column 1 Two lets and train of vortices below them, as well as vortices above jets and subsequent axial flow at top
The high-speed movies showed that it was practically impossible to achieve dynamic balance of flows through the two nozzles, and that consequently the point of contact of the two jets would oscillate in a regular pattern about a central point. This imbalance caused enough movement of the vortices below to permit them to sweep back and forth along the bed surface, which resulted in a smooth surface rather than in concave digging, as might have been expected. The balls were generally ejected at the angle of the nozzles and, even in the small column, underwent but few collisions with the column walls. The flow pattern of the flowing liquid, as well as that of the entrained balls, paralleled the axis of the column after a few column widths. The vortices formed below the nozzles were fairly stagnant, while those above moved to some degree with the jetted liquid. 570
l & E C PROCESS D E S I G N A N D DEVELOPMENT
a
q7/Ac
(6)
and a recirculation constant, R,, can be defined as the fraction of the total flow rate, Q1,which is recirculated, or
R,'
= qr/Qt
(7)
On the basis of the work of Barchilon and Curtet (1964), R, can be related to the system variables through the similarity parameter, m, by the expression
R, = Amd
+B
(8)
After proper substitutions are carried out, Equation 5 is reduced to
X
=
(Amd
+ B)'(Qt/Acvl)acosb ( 9 1 2 )
(9)
The similarity parameter, m, may be expressed in terms of the throttling number, Th, suggested by Williams and Becker (1963) as m = d(1
-
Thz)/2Th2
(10)
where
Th = ['/z
ACP,VR~/(I~ '/zA CP,U%)
10.6
(11)
Table I. Best Values of Exponents and Coefficients Determined from Computer and Analytical Computations
Parameter
e f
2 25 1 0.5 10 1"
2.0 2.0 0.6
A B
0.4 0.6
D
...
0.1 0.2 2.63 0.755
a
t
C
d
1 .o 1 .o
...
(14)
which is related to the distance between the nozzles and the bed surface. I t neglects the fine flow patterns observed during ball removal. The final form of the correlation is
1 .o
R
= Cf
D In X
A computer program was written to select the best values (Table I) of exponents a to f and constants A and B, and then to determine constants C and D from the experimental data plotted in Figure 12. The program was designed to maximize the correlation coefficient and minimize the standard deviation, resulting in the expression
f0.46 0.9
Standard deviation in R Correlation coefficient Correlation coefficient of population at 90% confidence level Max. value Min. value Error at mean extpl. values, % ' Average interlayer mixing a
R = Noe(Q/Dc)'
Best Values Empirical computer analysis
Analytical basis
Based on the observed vortex geometry, ball removal from the column is expressed as a dimensionless parameter, R, or
0.97 0.88 *20 4 . 1 5 ball layers
R
=
2.63
+ 0.755 In X
(16)
where
X = (O,ld(A,/Aj)
Included experimental observations.
- 0.5 - 0.2)1.2(QE/A,
cos2 ( p / 2 )
and
R
and vk = ut
+ A,v,/A,
(momentum of fluid in column)
ut = velocity of inducted fluid
I,
=
A,P,v,~ (momentum of jet)
For the system studied here, Equation 11 may be modified to the approximation
Th'
= 1/A,/2AC
(12)
and since there are two impinging jets, these are inclined at angles greater than O o , and the liquid in the column is stagnant-Le., ut is zero. Substituting Equation 12 in Equation 10 gives
- 0.5
m = Z/(A,/Aj)
=
NoQ/Dc
(18)
A correlation coefficient of 0.9 and a standard deviation of h0.46 in R were obtained. The error at the mean of the experimental values was =t20%. The empirical values obtained
(13)
from the computer were in good agreement with the analytically derived values obtained from Barchilon and Curtet (1964) and by assuming exponential energy damping. T o compare exponents c and d, only their product should be considered. Repeated experimental runs to measure ball removal under similar conditions indicated a standard deviation in the measured ball removal (and simple ball fluidization) of about 3~0.35. However, not enough values were obtained to provide good statistics. The measurement of flow rates and geometries was much more accurate than that of the number of balls removed. Other sources of experimental difficulties included the fact that in some of the runs the balls were completely
EXPERIMENTAL ERROR IN
R.2.63t0.755
0.0I
In 4
10.0
0.10 1.00 STRENGTH PARAMETER, X
Figure 12. Correlation of liquid flow through nozzles in a packed column with ball removal from ordered packed bed VOL. 7
NO. 4
OCTOBER 1 9 6 8
571
removed from the column before maximum flow rates were reached. I n several cases there appeared to be a shock effect when the flow was initiated, which aided bed penetration, while in other cases ball removal took place in spurts. The correlation does not account for the effect of viscosity on vortex formation and propagation. However, the results should be applicable to fluids whose viscosities are close to that of water. A large source of error in the correlation was the failure to account for the effect of mixing between ball layers and the steplike nature of the bed penetration by the liquid, since a bmooth exponential decay had been assumed. This experimental system is only one of several approaches that might be optimized to give the best results. For example, interlayer mixing might be minimized and ball removal increased by introducing a small axial flow of fluid through the bed. This should not be large enough, however, to fluidize the bed. Increased ball removal might also be obtained by the installation of several pairs of nozzles along the column walls to be employed in succession. Different column and nozzle geometries might also prove beneficial. Acknowledgment
The authors gratefully acknowledge the contributions of James Sauls and George Arnold in conducting the experi,merits; W. E. Winsche, C. J. Raseman, and E. A. Mason in their helpful suggestions and advice; and R. J. Walton in taking high-speed movies. Nomenclature
A = coefficient used in Equation 8 A, = column area A . = area of one nozzle B = coefficient used in Equation 8
c = coefficient
used in Equation 15 drag coefficient D = coefficient used in Equation 15 D, = column width D, = ball diameter gc = conversion factor, 32.2(1b.,) (ft.)/(lb,J (sec.)z I = relaxation length m = Craya-Curtet similarity parameter N o = number of ball layers removed from column q1 = volumetric fluid recirculation rate in column Q = height per layer of bed Qt = total volumetric flow rate of fluid in column R = ball removal from column, NoQ/D, R, = fraction of total flow rate which is recirculated in column s = average strength of any vortex so = average strength of first vortex s, = minimum vortex strength required to suspend balls T h = throttling number ut = terminal velocity of a single particle in an infinite liquid medium CD =
572
I & E C PROCESS D E S I G N A N D DEVELOPMENT
vu = average velocity of fluid in first vortex X = dimensionless strength parameter defined by Equation 17 z = distance below nozzles = nozzle inclination from the vertical ’p pf = fluid density p r = ball density
SUPERSCRIPTS = exponent used b = exponent used = exponent used c d = exponent used e = exponent used f = exponent used a
in in in in in in
Equation Equation Equation Equation Equation Equation
1 5
7 8 14 14
literature Cited
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Foisom1.R. G., Chem. Eng. Progr. 44, 765 (1948). Gosline. J. E., O’Brien, M. P., Uniu. Calif. (Berkeley) Publ. Enz. 3, 167 (1934). O’Brien, M. P., Folsom, R. G., Uniu. Calif. (Berkeley) Publ. Ens. 3, 343 (1937). Reemsnyder, D. C., Symposium on Fully Separated Flow, A. G., Hansen, ed., p. 81, American Society of Mechanical Engineers, New York, 1964. Rowe, P. N., Trans. Inst. Chem. Engrs. (London) 39, 175 (1961). Rowe, P. N., Henwood, G. A., Trans. Inst. Chern. Engrs. (London) 39. 43 (1961). StepAnog A. ’ J., “Pumps and Blowers-Two-Phase Flow,” p. 171, Wiley, New York, 1966. Struve, D. L., Lapidus, L., Elgin, J. C., Can. J . Chem. Eng. 36, 141 (1958). Susskind, Herbert, Maresca, Michael, Becker, Walter, U. S. Patent 3,294,645 (1966a). Susskind, Herbert, Maresca, Michael, Becker, Walter, Odette, Robert, U. S. Atomic Energy Commission, Brookhaven National Laboratory, Upton, N. Y.,Rept. BNL 50024 (T-442) (June 1966b). Susskind, Herbert, Winsche, W. E., Becker, Walter, Nucl. Appl. 1, 405 (1965). Susskind, Herbert, Winsche, W. E., Becker, Walter, U. S. Atomic Energy Commission, Brookhaven National Laboratory, Upton, N. Y., Rept. BNL 50022 (T-441)(June 1966~). Williams, G. C., Becker, H. A., Tappi 46 ( 8 ) , 153A (1963). Young, D. F., J . Hydraulics Diu. Am. SOC.Civil Engrs. 86, 47 (1960). Zenz, F. A., Weil, N. A,, A.I.Ch.E. J . 4, 472 (1958). RECEIVED for review December 22, 1967 ACCEPTEDMarch 13, 1968 Work performed under the auspices of the U. S. Atomic Energy Commission.
