C. E. SCHWARTZI
AND
J. M . SMITH
PURDUE UNIVERSITY, LAFAYETTE, IND.
T h i s work was undertaken because recent heat transfer results a n d preliminary d a t a o n velocity had suggested t h a t , contrary t o common supposition, t h e velocity across t h e diameter of a packed bed m i g h t n o t be uniform. D a t a obtained in pipe sizes f r o m 2 t o 4 inches showed t h a t a peak velocity occurred approximately one pellet diameter away f r o m t h e pipe wall. For D l / D p less t h a n 30 t h i s peak velocity ranged f r o m 30 t o 100% greater t h a n t h e velocity a t t h e center of t h e tube. T h e results indicate t h a t unless Dt/Dp i s greater t h a n about 30, i m p o r t a n t velocity variations exist across a packed bed. Such variations m a y be i m p o r t a n t in analyzing t h e operation of catalytic reactors a n d packed-bed heat exchangers. m
I
N T H E design of gas-solid catalytic reactors and packed-bed
1
heat exchangers it is customary to assume that the distribution of flow is uniform across the diameter of the bed. Kear the center of a large, uniformly packed bed of solid particles this assumption appears a reasonable one. However, simple geometrical considerations indicate that near the wall of the containing pipe the flow channels are larger in area and smaller in resistance than near the center of the pipe. Also, in close proximity to the wall, skin friction would tend to reduce the flow to a low value. These two factors suggest that the assumption of a uniform velocity may be unsatisfactory under certain combinations of packing size, tube size, and flow rate. Two recent exploratory investigations bear out the necessity for considering radial velocity variations in packed beds. Arthur, Linnett, Raynor, and Sington ( 1 ) studied the flow distribution of air through a bed of charcoal granules (7- to 18-mesh size) in a glass tube 4.83 em. in inside diameter. The flow was separated into several parts a t the top of the bed by the insertion of thin rings concentric to the tube wall. By takins special precautions to ensure t h a t equal resistance was offered to the air stream in each section, the flow rate was measured. The results did not give point velocities but did indicate that the flow rate reached a maximum a short distance from the tube wall and decreased from that AIR OUT point in both directions, gradually toward the center, and sharply toward the wall. Morales, Spinn, and Smith ( I I ) , using circular, hot-wire anemometers, measured point velocities at five radii in a standard 2-inch pipe packed with cylindrical pellets l/8, =/4, and " 8 inch in size. These r e sults also showed a maximum velocity near the wall, with a sharply decreasing value as the wall was approached more closely. As the peak velocity was up to 100% greater than the value a t the center of the pipe, it appeared that the errors involved in the assumption of a uniform profile could be significant. Similar gradients were observed for all three packing sizes, the smallest of which corresponded to a D,/D, = 16. I n view of these preliminary S E C T I O N A-A studies, a more c o m p r e h e n s i v e STRAIGHTENING VANES 1
Present address, The Texas Co., New
York, N. Y.
Figure 1.
investigation of velocity profiles in packed beds was undertaken with the objectives of (1) determining under what conditions the uniform velocity assumption would be valid, and (2) correlating and explaining the magnitude of the observed velocity profiles. Data were obtained in 2-, 3-, and 4-inch pipes using l/8-, I/C, and 1/2-inch spherical and cylindrical pellets, corresponding to a range of Dt/D, from 5 to 32. The length of the cylindrical pellets was equal t o the diameter. In order to include the range of operating conditions encountered in commercial reactors and heat exchangers average velocities from 0.42 foot per second [G= 114 pounds/(hour) (sq. feet)] to 3.59 feet per second [G = 1110 pounds/(hour) (sq. feet)] were investigated. VELOCITY-MEASURING INSTRUMENTS
It was decided in the beginning that the precise measurement of
velocities within the packing of the bed would be virtually impossible. Therefore, as in previous investigations ( 1 , I 1 )> the flow rate was studied in the empty pipe just downstream from the packing. This meant t h a t the velocities so obtained mere superficial values, in that they were based upon the total void and nonvoid area. The objective was to determine the average velocity that would existdin the bed a t a given radial position based upon the total area rather than the actual void area. Such a measurement is a n average velocity because the void fraction varies with distance along the axis of the pipe (a periodic variation with a period about equal to the dimension of AIR IN the pellet). In order to attain this objective, the effect of distance of the measuring device from the packNEYOMETER ing must be examined. If this distance is too large, the observed velocity profiles may be more descriptive of empty-pipe conditions than those existing within the packed bed. If the distance is too small, velocity components perpendicular to the direction of flow may be significant. This problem was attacked experimentally and is discussed in a later section. Three experimental methods were considered: (1) the insertion of sleeves t o divide the flow into several parts each a t a different radial position, (2) a Pitot tube to measure point velocities, and (3) hot-wire anemometers placed a t Diagram of Apparatus different radial positions in the 1209
flow in a n empty pipe and also under turbulent flow conditions introduced by the insertion of screens upstream and downstream from the anemometer. I n the range of overlapping velocities the two methods of calibration gave excellent agreement. The circular anemometer also gives directly an integrated average velocity at a given radial position. The validity of this average velocity does not require an exactly linear calibration curve of voltage us. velocity: provided the distance between the top of the packing and the anemometer is sufficient to dampen out large variations in velocity with position on the anemometer circumference. Directly over a pellet the velocity v3ll be low, and between two pellets it will be high. However, as the gas leaves the top of the packing these variations will disappear as a result of transfer of momentum, and a uniform velocity mill be established. If the position of the anemometer is correct, this average velocity will be measured. E X P E R I M E N T A L APPARATUS
A schematic diagram of the apparatus used is shown in Figure 1. The pipe section containing the bed mas 4 feet long and mounted in a vertical position for all pipe sizes. Straightening vanes were inserted at the bottom of the pipe as indicated in the figure. The packing material was supported in most cases by an iron screen (20-gage wire, l/ls-inch openings). I n the 4-inch pipe tests were also run using a perforated plate with 1/16-inch holes on l/r-inch centers. The velocity measurements were found to be unaffected by the change in method of support. The packing bed depth could be varied from 3 to 23 inches. The average velocity for the entire bed was obtained from rotameters in the exit line. The temperature of the air was approximately 73" F. and the gage pressure at the exit of the bed a fev inches of water. Anemometer Assembly.
Constant-current type anemometers
of 0.003- and 0.004-inch platinum wire v~ereconstructed for each
Figure 2.
Anemometer Assembly for 4-Inch Pipe
packed bed. The first method was used by Arthur et al. ( 1 ) with apparent satisfaction. However, the results do not give velocities a t definite radial positions but integrated flow rates over small cross-sectional areas of the bed. The difficulties experienced by Arthur et al. in maint,aining the same resistance in the separate flow paths indicated t h a t the method mould not be particularly suitable for the large number of measurements necessary in a comprehensive investigation. Method 2 was examined by constructing a small-diameter ( l/rz-inch) Pitot tube and a micromanometer for pressure measurements. It was found that a manometer sensitive enough for precise measurement of kinetic heads corresponding to velocities up to a few feet per second was also sensitive to outside influences, and very sluggish in its approach to an equilibrium reading. Furthermore, the validity of the results obtained is questionable when a Pitot tube is Calibrated under conditions of turbulence that may be different than those existing when it is used as a velocity-measuring device. Finally, the Pitot tube gives a point velocity, so t h a t it would be necessary to make a number of measurements around a circumference in order to obtain a satisfactory average value for any one radial position. The hot-wire anemometer is in some \Tags an ideal device for precise measurement of low velocities, and a system of five circular anemometers (Figure 2 ) was chosen as the measuring device for this work. A particular advantage of the instrument is that the results are not influenced by differences in turbulence level, provided the wire used is not smaller than 2 to 3 mils. King ( 7 ) showed that wires of varying diameter (but greater than 2.5 mils) and under varying conditions of turbulence gave identical results for the velocity in the direction of flom, when properly calibrated. This result was verified in the present investigation by calibrating the anemometers under conditions of streamline
1210
pipe size in accordance with the procedures recommended by Ower ( I S ) and King ( 7 ) . The assembly for the 4-inch pipe is shown in the two photographs of Figure 2. The mount was machined to close tolerances from 3/9-inch Plexiglas. The platinum wire was supported and held in the proper radial position by passing it through small holes in mica strips which were cemented to the mount. Twenty-gage copper leads were spot-welded to the anemometer wire, and to minimize the disturbances to the flow, the leads had a short vertical run from the anemometers before passing radially to the mount (see Figure 2). The ends of the leads were soldered t o phone jacks screiJ-ed into the Plexiglas mount from the outside. The five anemometers were located a t radii determined by the equation
where
n T
R
= anemometer KO. (1, 2, = radius of anemometer = radius of pipe
3, 4, or 5 )
This arrangement permitted the average velocity to be estimated by a simple arithmetic average of the velocities a t each anemometer. The necessity for tedious graphical integrations to compute approximate average velocities was thus eliminated. I n order to minimize disturbances to the flow pattern, a separate mount was constructed for each of the five anemometers for the 2-inch pipe. For the 3-inch data, two mounts were made, one containing anemometers I, 3, and 5 and a second holding 2 and 4. The power supply for the anemometer circuit consisted of four automotive-type, 6-volt batteries connected in series. Voltages across the anemometers and standard resistance were measured by a Leeds and Northrup precision potentiometer with a range of 0 to 16 volts and a precision of 0.0001 volt.
Packing the Bed. In packing the bed the larger pellets were placed in the pipe virtually one a t a time to prevent bridging and the resultant uneven packing. The small pellets were fed into the pipe through a funnel which caused them to sprag out evenly across the pipe diameter. The pipe was not hammered t o obtain a denser packing because results so obtained were nonreproducible. Air was passed through the bed before the
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 6
anemometers were inserted, in order t o remove dust and small particles which might have damaged the anemometers. CALIBRATION OF ANEMOMETERS
For low velocities the anemometers were calibrated by placing them in vertical, galvanized steel pipes which had an undisturbed length of 18 feet. Nikuradze ( l a ) ,on the basis of extensive data1 set a limiting length of 40 pipe diameters for a fully developed velocity profile in an open pipe. As 18 feet corresponded to 108, 72, and 54 diameters for the 2-, 3-, and 4-inch pipes, it was assumed that the flow was parabolic for Reynolds numbers less than 2100. From the measured average velocity, the calibration of the anemometers was made using the equation u = 2 v [1
athere
2c
-
= velocity a t radius
(g]
is necessary if the results are t o be significant. Repeated checks of point velocities without repacking the bed indicated a maximum deviation of 2%. If, however, a point velocity was rechecked after the pipe was emptied and repacked, the variability in packing increased the deviations. I n obtaining the final data the bed was twice, so that three values were obtained for each set of conditions, The absolute deviation from the mean of the three replications was 9%, although maximum values
8LASlUS 1/7
POWER LAW
(2)
r
&"
c
In order to extend the calibration curves t o higher velocities, the method of using screens to obtain flat velocity profiles (15-17) was employed. Teago (17)found that flat profiles extending almost to the wall of the pipe could be obtained by placing screens upstream and downstream from the measuring instrument. Experimental measurements indicated that the linear velocity over the flat portion of the profile was equal to the average velocity divided by 0.94 for a wide range of Reynolds numbers above 2100, in agreement with the work of Teago. Substantiation of this method of calibration in turbulent flow is shown in Figure 3, where the velocity profile in the empty 4-inch pipe is shown for a Reynolds number of 7000 as determined from the screen calibration data. The velocity was measured a t the end of the 18-foot section corresponding to an undisturbed length of 54 diameters. 91so shown on the graph is the profile based upon the Blasius l / 7 power law. For Reynolds numbers this low, a power of '/B gives a better fit with the data of Nikuradze ( I d ) and Stanton and Panne11 (15),as pointed out by Nikuradze. Using an exponent of '/e gives almost exact agreement with the curve based upon the screen calibration. The data of Stanton and Pannell(16) showed that the ratio of the average velocity to maximum velocity was 0.79 for a Reynolds number of 7000. The ratio determined from the curve in Figure 3 is also 0.79. As an illustration, the calibration curves for the 4-inch pipe a t an air temperature of 73" F. are shown in Figure 4. I n the 2and %inch pipes the shorter lengths of the anemometer wires permitted operation a t high temperatures (dull red heat, approximately 1000" C., in still air). Hence the calibration curves were n& affected by changes in the air temperature Of f50 With the longer anemmeter wires used in the 4-inch pipe, the at wire temperature was lower, making it necessary to two air temperatures, 73" and 81" F. All of the data were obtained at temperatures within this 8" F. interval. I n the calibration procedure all anemometers were first annealed for 4 hours a t a dull red heat. The calibration curves were checked periodically to determine if vaporization or changes in wire structure had occurred. With proper annealing it was found that the calibration remained constant for at least a month of intensive use. As a l h a l check on the calibration, Profiles were measured in the test section (4feet long) before adding packing. Results for the 4-inch pipe are shown in Figure for three flow rates' though developed flow conditions were not Obtained in a Of show the 4-foot length, the streamline and turbulent flow in empty pipes. PRECISION OF DATA
Reproducibility. Because a packed bed is a random syskm and cannot be exactly duplicated, even with controlled methods of packing, a statistical analysis of the velocity measurements June 1953
0
01
0 2
03
0 4
RADIAL
Figure 3.
08
0.6
POSITIOY
0.1
- r/R
0.8
09
1.0
Velocity Profile for E m p t y 4-Inch Pipe 54 Diameters Long Average veloclty. 3.37 feet per second
as high as 25% were observed. In general the deviation increased as D,/Dt became larger, Although repacking the bed twice for each point tripled the total number of measurements, such a procedure was deemed necessary in view of the average reproducibility of 9%, Mass Balance. One measure of the accuracy of the results is the comparison of the average velocity as measured by the rotameters with the value obtained by summing the point readings obtained from the five anemometers and dividing by 5. The average deviation for all the data wa8 -2.5%$. The absolute value of the maximum deviation was less than lo%, Distance of Anemometers above Packing. If the anemometers were too ,,lose to the top of the bed, the indicated velocities were too high at large flow rates. This was attributed to large velocity components perpendicular to the dibe sensitive. rection of flow, to which the anemometers Figure 6 shows this effect for anemometer 2 ( r / R = 0.55) in the 2-inoh pipe. At the higher averagevelocities the curve rises sharply as the distance of the anemometer above the top of the bed decrease8 toward zero. At low average velocities the components perpendicular to the axial direction were not signzcant, as indicated by the horizontal curves. At anemometer position 2, the velocity in the bed is less than in the empty pipe. Therefore, as
INDUSTRIAL AND ENGINEERING CHEMISTRY
1211
Table l a .
Velocity Profile Data for 2-Inch Pipeand 23-Inch Bed Depth l/ac
u*/v uy/V us/V ua/V
UL/V
Ul/V
u2/V ua/V u4/V us/V ui/V
uz/V w3/V
ur/V us/V ui/V
u2/V us/V uJV us/V
1/40
3/ac
5/325
1/45
3s:s
Average F'elocity 0.97 Foot/Second 1.05 1.06 0.92 0.77 1.09 0.94 1.25 1.26 1.19 1.33 1.21 1.04 1.23 1.26 1.32 1.11 1.21 1.11 1.09 1.06 1.08 1.30 1.20 1.20 0.45 0.46 0.59 0.70 0.46 0.63 Average Velocity 1.51 Peet/Second 0.98 0.74 0.99 0.88 0.96 1.12 1.04 1.11 1.01 1.09 1.16 1.12 1.24 1.08 1.14 1.12 1.09 1.28 1.19 1.17 0.68 0.75 0.85 0.70 0.77 Average Velocity 2.20 Feet/Second 0.92 0.89 0.75 0.99 0.84 1.07 1.08 0.98 1.08 0.98 1.17 1.18 1.05 1.12 1.12 1.25 1.11 1.17 1.09 1.17 0.74 0.78 0.82 0.88 0.87 0.80 0.98 1.14 1.11 0.86
Average Velocity 0.86 0.71 0.95 0.92 1.11 1.08 1.17 1.11 0.80 0.89
RESULTS
1.02 1.19 1.13 1.03 0.59 0.91 1.15 1.16 0.99 0.63
2.68 Feet/Seoond 0.92 0.83 0.86 1.05 1.14 1.06 1.20 1.18 1.22 1.27 1.11 0 . 9 8 0 76 0.64 0.84
a Each value represents average of three runs. pellet; '/is = '/a-inch spherical pellet, etc.
l/gc
= '/s-inch cylindrical
Table lla. Velocity Profile Data for 3-Inch Pipeand 23-Inch Bed Depth '/SO
ui/V
u*/V u3/V ur/V us/V
0.83 0.85 0.98 1.12 0.99
3/8C '/IC 6/Q25 '/as Average Velocity 1.62 Feet/Second 0.62 0.68 0.80 0.90 0.60 0.80 0.86 0.99 0.86 0.89 1,15 1.21 1.04 1.08 1.07 1.20 1.00 1.16 1.24 1.13 0.99 0.87 0.91 0.97 1.00
'/io
ducible. There was no significant difference betm een orders, from which it may be concluded that the anemometers maintained their calibration with use.
3/8S
081 0.88 1.04 1.06 0.80
'/28
0.77 0.97 1.09 1.01 0.89
Average Velocity 2.64 Feet/Second 0.61 0.79 0.82 0.85 0.85 0.69 0.52 0.76 0.90 0.70 0.82 0.86 1 . 0 3 0.83 0.84 0.77 1.12 1.04 1.11 1.20 1.07 1.16 1.09 1.01 1.04 1.09 1.07 1.18 1.20 1.17 1.21 1.15 0.93 0.82 0.83 0.95 0.79 0.93 0.91 1.00 a Each value represents average of three runs. l/ac = l/s-inch cylindrical pellet; '/as = l/a-inch spherical pellet, etc.
Effect of Packing Depth. I n order to obtain results which would not be influenced by flow conditions upstream from the packing, the bulk of the data was taken a t a bed depth of 23 inches of packing. hIorales et al. ( 1 1 ) found that a t low bed dept,hs the velocity approached t h a t in an empty pipe. Experiments with '/?-inch spherical pellets in the 4-inch pipe showed no significant effect of bed depth from 3 to 23 inches. Effect of Total Flow Rate. Tables I, 11, and I11 present all the data as the ratio of the point velocity, u,to the average velocity, Ti. Analysis of t>heseresults indicat,es that ratio u / V is independent of the total flow rate. This is illustrated in Figure 7 for the case of 1/4-inchspherical pellets in the 4-inch pipe. The data plotted as u us. V give a straight line passing through the origin for each radial position. This direct proportionalit,y between the point velocity and the flow rate simrlifies the problem of correlating the profiles. Effect of Radial Position. To illustrat,e the type of velocity profile obtained: the data for l/s- and 1/4-inch cylinders (from Tables I. II,, and 111) are shown in Figure 8 for the 2-inch pipe, in Figure 9 for the 3-inch pipe, and in Figure 10 for the 4-inch pipe. Each experimental point is t'he average of the t'hree replications obtained by repacking t,hebed. Thedata for the spherical packings were similar. The solid curves were calculated using Equation 12. The interesting feat,ure of the profiles shown in the graphs is the maximum exhibited near the wall. The divergence from a flat profile increases as the packing size increases and the pipe size decreases. Thus, in the 2-inch pipe and 1i4-inch cylinders, where
ui/V uz/V uz/V ua/V u6/V
the distance betm-een the anemometer and the top of the bed increases, the velocity again increases, approaching the empty pipe value. The over-all result is a mininium in the velocity us. distance relationship illustrated by the three higher curvcs in Figure 6. From an analysis of curves similar to Figure 6 for all the operating conditions it was concluded that a distance of 2 inches between bed and anemometer would minimize the errors due to both the approach toward an empty-pipe profile obtained with large distances, and the sensitivity of the anemometer to velocity components perpendicular to the direction of flow. Hence all the data TT ere obtained with 2 inches separating the anemometers and the packing. Statistical Analysis of Data. I n order to determine the importance of such factors as repacking the bed and the distance of anemometer from top of packing, an analysis of variance (6) was carried out on a set of experiments which included five average velocities, and four distances between anemometer and bed packing, I / k , I , 2, and 3 inches. A statistical, factorial design of experiments JTas used in which the order of measurements was completely randomized. The runs were carried out with t r o different bed packings of 1/8-inch cylindrical pellets in the 2-inch pipe. The two-factor interaction betn een anemometers and height of anemometer above the bed indicated that there was a aignificant change in profile with distance of packing to anemometer. The very significant difference observed between anemometers showed that the profile was not flat. The difference between runs (repacking the bed) was not significant a t the 5 % confidence level, indicating that the results were reasonably well repro1212
4
6.0
3 3
e 0
0.4
0.0
1.2
2.0
1.6
A I R VELOCITY
Figure 4.
-
2.4
2.8
3.2
3.6
4.0
FT/SEC
Calibration Curves for 4-Inch
Pipe
7 3 O F.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 6
Table
Illa. l/ac
Velocity Profile D a t a f o r 4 - I n c h Pipeand 23-Inch Bed Depth '/ac 3/80 '/2C '/a28 '/a 3/8S '/a8 Average Velocity 1.01 Feet/Second
ui/VO.95 uz/VO.93 u3/V 0 . 8 7 u4/V 1.10 us/V 1 . 1 0
0.81 1.02 1.10 1.14 0.99
u3/V u4/V us/V
ui/VO.92 0.87 0.92 1.01 0.97
0.83 0.91 1.07 1.11 0.91
uv'V0.94 udV0.90 us/V 0 . 9 8 u4/V 1.01 u s p 0.98
0.84 0.89 1.06 1.10 0.94
uz/V
u1/ uz/
v v
0.91 0.90 0.97 1.02 0.97
u3/v
u4/v us/v Ul/V
us/
Average Velocity 2.64 0.84 0.66 0.75 0.91 0.73 0.89 1.04 0.92 0.96 1.07 1.18 1.05 0.91 1.20 0.96 Average Velocity 3.59
Feet/Second 0.91 0.69 0.93 0.75 0.98 1.02 1.08 1.18 0.96 1.07 Feet/Second 0.92 0.94 1.01 .1.09 0.98
v
uz/ ua/ V
u4/
0.66 0.75 0.99 0.69 0.74 0.99 0.83 0.92 0.99 0.97 1.00 0.99 1.41 1.10 1.11 1.38 1.19 1.35 1.05 0.99 Average Velocity 1.62 Feet/Second 0.61 0.71 0.89 0.71 0.71 0.71 0.89 0.93 0.93 0.99 1.00 0.95 1.26 1.09 1.11 1.31 1.25 0.99 0.98 1.10 Average Velocity 2.14 Feet/Seoond 0.70 0.64 0.76 0.91 0.67 0.89 0.94 0.75 0.96 1.00 1.03 1.00 1.22 1.06 1.10 1.34 1.22 0.98 0.98 1.10
v v
a Each value represents average of three runs. pellet; 1/68 = l/a-inch spherical pellet, etc.
0.67 0.77 0.95 1.26 1.09
0.69 0.98 1.09 1.25 1.14
0.70 0.89 1.06 1.27 1.08
0.67 0.83 1.05 1.20 0.97
0.62 0.81 1.07 1.24 1.03
0.64 0.85 1.05 1.16 0.97
0.63 0.83 1.09 1.21 1.02
0.64 0.85 1.05 1.15 0.98
0.65 0.83 1.09 1.20
0.64 0.84
0.66 0.84 1.07 1.25 1.02
1.06
1.18 0.95
occurred a t a distance of approximately one pellet diameter from the pipe wall, regardless of pipe and packing size. The deviation from a flat profile becomes more pronounced as the packing size increases. ' Effect of Pipe Size. As shown in Figures 8,9, and 10, the larger the pipe the more uniform the velocity. As the peak in velocity occurs at the same distance from the pipe wall (with a given pellet) regardless of pipe size, the flat portion of the profile extends further toward the wall the larger the pipe. This is illustrated by comparing the upper curves (for 1/8-inch cylinders) in Figures 8 and 10. THEORETICAL DEVELOPMENT
Concept of Flow in Packed Beds. An explanation of velocity variations radially in a packed bed must rest upon an analysis of the void space in the bed. Near the wall of the containing pipe the void fraction will be larger than near the center of the bed, because the packing elements must conform to the circular shape of the pipe wall. At a differential distance from the wall
1.00
*/a = I/s-inch cylindrical
bi c'
D,/D, = 8, the velocity gradients are large and the peak velocity is approximately 100% above the center value. I n contrast, in the 4-inch pipe packed with '/s-inch cylinders (Dt/D, = 32) the velocity is essentially constant out to a radial position of r / R = 0.7 and the peak velocity is but 20% greater than the center velocity. Effect of Packing Size. The maximum in the velocity profile
4
.AVE. VELOCITY 0 3.59 ~FT.../SEC
.
I
-
I a
w W
z
0
:: 4
>
8
P >
AVE. VEL.
ij
9
0197
> W
AVE.VEL.-O.42 0 .c
0 0
I
I
2
HEIQHT A B O V E B E D
Figure 6.
Figure 5. Velocity Profiles in 4 - I n c h Pipe Test Section w i t h o u t Packing
June 1953
-
3
4
INCHES
Effect of Height above Bed for Anemometer 2
the void fraction will approach unity, since the particles can make only a line or point contact with the pipe surface. Year the center of the pipe the bed should not be affected by the wall; hence in this central core the void fraction will have a constant minimum value. How close to the wall the void fraction will remain constant is a function of the size of packing. Flow through packed beds is, therefore, somewhat analogous to flow through a bundle of tubes, which consists of a central core, containing tubes of constant diameter, surrounded by tubes of progressively larger diameter. For a constant pressure drop, a discontinuous velocity profile would result, the larger flow rates occurring in the outer tubes of larger diameter. I n a packed bed, the flow paths are not parallel but interconnected. This results in momentum transfer, and the velocity becomes a more continuous function of the
INDUSTRIAL AND ENGINEERING CHEMISTRY
1213
radius. If only void space is considered, the profile would have a flat central section, viith the velocity increasing on either side as the pipe wall is approached. Actually the wall will exert a frictional force on the gas, E O that the velocity again decreases and approaches a zero value right a t the wall surface. This concept of flow in packed beds is considered to be the explanation for the maximum in the observed velocity profiles, as shown in Figures 8, 9, and 10, and is employed in the folloming section in developing quantitative expressions.
4.8 4.4
t
FIG.?
- V A R I A T I O N OF P R O F I L E MASS VELOCITY -
aP
=
D, 6 3
If it is assumed t h a t this expression, ~1hich was derived for the packed bed as a n hole, can be applied to point conditions, the pressure drop at any radial position in the bed can be evaluated. Considering just the central core, where the void fraction, 6, n ill be constant, Equation 4 indicates that if a velocity gradient exists in the bed, a pressure gradient also exists. -4ctually, as soon a6 a pressure gradient begins t o develop, the flow will redistribute itself radially in such a way that the pressure drop will be the same a t all radial positions. The driving force for this redistribution or transfer of momentum is the difference in pressure drop betn eera the center of the pipe, where no momentum transfer occurs, and
WITH
(114" S P H E R I C A L
PELLETS
packed bed. This can be accomplished by utilizing available information on the pressure drop through packed beds. Among the more important pressure drop studies are those of Carman ( 3 ) ,Kozeny ( 8 ) , Leva (9, I O ) , and Ergun ( 5 ) . D'Srcy (4), Of particular importance for present purposes is the effect of void space and velocity on the pressure drop. Leva's development may be simplified to the following form to show the effect of these two variables k i V 2 (1 - 6)
4"PIPE)
f
2.8
1.5
-
1.1
-
1.0
-
0.9
-
0.8
-
2.4 I-
5
FIG.8
-
I VELOCITY
I
I
I
I
I
PROFILES FOR 2" PIPE
-
2.0
0 P
U
1.6
v
1.4 0.8
0.4
0
0.4
0.8
1.2
1.6
AVERAGE
2.0 2.4 VELOCITY
-
2.8
3.2
3.6
3.6
FT/SEC
Prediction of Velocity Profiles. The development of a theory for the velocity distribution is particularly difficult because of the complexities introduced by the packing. The following treatment is based upon an expression for the shearing force in the gas introduced by Prandtl ( 1 6 ) and a number of simplifying assumptions. The results agree reasonably well with the experimental velocity and void-fraction data and the qualitative concepts presented above. However, this agreement cannot be considered a complete verification of the theory in view of the several assumptions made. Rather than use the Prandtl mixing length, the shearing force might be interpreted in terms of an eddy viscosity (kinematic) e, according to the expression T = E P ( d u l d r ) . By analogy to mass transfer in packed beds, where Baron ( 2 ) has shown that a linear relationship exists between eddy diffusivity and velocity a t high Reynolds numbers, the eddy viscosity might be taken proportional to the velocity. Continuation of this line of attack, however, did not lead to as satisfactory agreement with the experimental data as the mixing length concept. According to the Prandtl theory the shearing force, T, is related to the mixing length, L, by the expression (3)
I n packed beds the only part of the total shearing force that is related to the velocity gradient is that entirely in the gas phasei.e., turbulent shear stresses. The shear between the gas and the solid packing is not involved. Equation 3 can be integrated to obtain the velocity profile, provided a n expression can be obtained for the shearing force in terms of the fundamental variables of the
1214
Y a
0.5 0
0.1
0.2
0.3
0.4
RADIAL
0.5
0,6
POSITIOM
-
6.7
0.8
0.0
1.0
r/R
that a t any radial position. This pressure defect per unit length of bed, Pd, can be formulated in the following way from Equation 4: (5)
I n this equation the subscript zero denotes the center of the bed. Equation 4 gives the total pressure drop due to skin friction between the gas and solid particles and pipe wall and orifice losses as well as that due to turbulent shear in the gas alone. If it
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 6
1.4
- VELOCITY
F1G.Q
I
I
I
I
I
I
P R O F I L E S FOR 3" P I P
1.2
is assumed that the fraction of the total pressure drop due to turbulent shear in the gas is a constant, Equation 4 can still be used to determine the pressure defect by introducing a constant factor. This is designated as kz in Equation 5. The following force balance on a cylinder of radius T relates the pressure defect and the shear stress, T,
Equations 3, 5, and 6 can be used to determine the velocity profile. I n order to reduce the errors involved in using Leva's pressure drop correlation, the pressure drop was measured each time the velocity profile was determined. Then the parameter k l in Equation 4 was determined by the expression,
AP,Dp X
o EXPERIMENTAL POINTS
J.4
*EOUATION
k1 =
(19)
-
1.1
-
0.8
-
0.7
-
0.9
0
o
(7)
where A p e is the experimental pressure drop. The void space in Equation 7, 6,, is the average value for the entire bed and could be estimated with reasonable accuracy from Leva's experimental void fraction measurements (IO). Combining Equations 3,5, 6, and 7 leads to the result,
1.3
1.2
6a3
vz x ( 1 - 6a)
at
01
0.3
0.4
RADIAL
0.5
0.0
POSITION
0.7
-
0.8
0.9
1.0
V R
:::;
Within the limits of the assumptions employed, Equation 8 defines the velocity gradient in terms of the void fraction, pellet diameter, radius r, and mixing length L. The problem of integrating the expression to obtain the velocity profile varies in complexity with radial position. In the central core 6 may be taken as constant and equal to 60. Also in this region the mixing length, L (the radial distance that a small mass of fluid travels before losing its identity), will be assumed equal to D p / 2 . This is based upon the concept that flow approaching a pellet will divide, and a portion will pass on either side of the obstruction. In passing around the pellet each stream will mix with another stream bypassing the adjacent
1.2
FIG.
I-
11
-EXPERIMENTAL
V A L U E S OF K'
I
-1
as o
0.1
0.2
0.3
0.4
RADIAL
June 1953
0.8
0.0
POSITION
0.1
- r/R
0.a
as
LO v
INDUSTRIAL AND ENGINEERING CHEMISTRY
t
1215
.60
.50 .40
30
m ; n; .30
-
v)
-P
5 2
.30
2 c $
.40
>
0-
.60,
! h ! ! I II ! !
.so
~
2 .60 f
I ! ! ! I !
.so
.40
.4 0
In
30
o
0.1
az
0.3
0.4
RADIAL
Figure 12.
0.5
0.6
POSITION
-
0.7
0.8
0.9
1.0
o
o.e
9.1
r/R
Variation of Void Radial Position
0.3
0.4
RADIAL
Space w i t h
Figure 13.
as
0.6
0.7
Posiiion
- r/R
0.s as
0.1
Variation of Void Space w i t h Radial Position
2-inch pipe
3-inch pipe
age velocity, V , Thus, the numerator and denominator of the logarithmic term may be djvided by 1' to convert the equation to a form involving only the dimensionless ratio, u / V :
v-
0
0.1
0.Z
Figure 14.
0.3
0.4 0.5 0.6 RADIAL POSITION
0.7
- t/R
0.6
09
Equation 12 can be compared n i t h the experimental velocity profiles t o appraise the theory. For each pellet and pipe size i t was found that a constant value of k' TT ould give good agreement between experimental velocities and those determined from Equation 12 from the center to a distance of t n o pellet diameters from the pipe wall (see Figures 8. 9. and 10, which show experimental points and theoretical curve). Closer to the wall large deviations were encountered. indicating that the assumption of a constant void fraction was not valid in the wall region. The values of k' so obtained are listed in Table IV and plotted in Although the theory as shown Figure 11 as a function of D,/Dt. by Equation 10 does not suggest that D,/Dt is the sole variabl? affecting k'. a reasonably good correlation is obtained in Figure 11 for the range of pellets and pipe sizes studied in this investigation.
1.0
v a r i a t i o n of Void Space w i t h Radial Position
Table I V .
4-inch pipe
Pellet
pellet. With these simplifications Equation 8 may be integrated to give
1/80 1/40 5/8C
5 / m
(9)
1/41
3/68 1/50
I/&
where
5/8C
'/PC 5/32S 1/48
8/53 1/28 l/5C
and
1/40 3/8C
1/20 5/323
1/48
I n accord with the experimental data, Equation 9 postulates a velocity profile which is independent of the total flow rate or aver-
1216
5/88
112s
Pipe, Inches
2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4
4 4 4
Values of
k'
(Equation 12)
k' Exptl.
Calcd.
1.81 2.71 1.70 1.92 1.92 2.22 1.18 2.26 1.97 1.97 1.24 1.95 1.75 1.90 0.74 1.32 1.70 1 72 0.74 1.80 2.08 2.20
8.60 3 10 1.49 5.86 2.82 1,58 17.1 4.3 2.79 1.66 9.86 4.34 2.60 1.30
INDUSTRIAL AND ENGINEERING CHEMISTRY
14.2
4.39 2.40 1.43 7.95 3.29 2.08 1.20
ua. Foot/Sec. 0.84 0.60 0.90 0.85 0.85 0.90 0.78 0.60 0.65 0.77 0.82 0.60 0.76 0.73 0.02 0.80 0.62 0.70 0.92 0.92 0.60 0.60
Vol. 45, No. 6
It is also noted that Equation 10 predicts an increase in k' with a n increase in the ratio, Dt/D,, while the experimental data show the opposite trend. No significant difference is observed between spherical and cylindrical pellets in Figure 11. At low values of D,/Dt, k' approaches 0 and Equation 12 would predict a flat profile. Figure 11 along with Equation 12 can be used to predict velocities in a packed bed from the center to a distance of two pellet diameters from the pipe wall. Extrapolation of Figure 11 to higher values of D,/Dt or use for nonuniform packing materials is not justified. Equation 10 can be used to predict values of k ' , provided the pressure defect constant, kz, can be estimated. This was done by choosing JCZ so that the predicted velocity profile for the 3/8-inch spherical pellets in the 4-inch pipe agreed with the experimental measurements. Under these conditions kz = 0.0096 and the predicted k' values are also shown in Table IV. The comparison is fair for the large packing sizes but deviates greatly for the smaller two sizes, indicating that some of the assumptions involved in the theory are not valid for the small packing units. Of these, perhaps the IT eakest is the assumption of L = D,/2. Beyond the central core the void space varies, and it is not likely that Equation 8 can be integrated analytically even for a very simple functional relationship between 6 and radial position. Furthermore, there appear to be no data in the literature on the manner in which the void fraction increases near the pipe wall. Hence, in order to test the theory, the experimentalvelocity profiles near the wall were used to predict how the void space varies with radial position. To accomplish this the mixing length was again taken equal to D,/2 and Equation 8 was written in the form
velocity profile. These calculations lead to a series of values of void fraction and corresponding radial position for each combination of packing size and pipe size. The results are plotted us. radial position, r / R , in Figures 12, 13, and 14 for the three pipe sizes. The curves are drawn to a void fraction of 0.32 a t the center of the pipe, since this value corresponds to the maximum packing density. The shape of the curves is in agreement with theoretical considerations, in that the gradient is nearly zero up to approximately two pellet diameters from the wall and then increases progressively as the wall is approached. The wall effect extends into the pipe a greater distance for the larger packing
0*50 0.40
EXP., SHAFFER(14) CALGFROM EO. 13 I
1
I
1
-er.l,. O,A,O
I
1
I
1
1/4" SPHERES
0.30
P
'0.5
0
0.4
Ahumerical, stepwise integration procedure was then employed in Equation 13, starting the process a t a position two pellet diameters from the wall. Values of k' were available from the central core calculations and are the same as those given in Table IV. For each increment of radial position, the value of 8 was determined that would give exact agreement with the experimental
0.3
Figure 16.
Void Fraction vs. Radial Position for 3-Inch Pipe
0.50-EXPERIMENTALlSHAFFER(14)-0,A,
-
u
CALC. FROM EQ 13 - * ? A , =
0.48
I
\' I
EXPERIMENTAL, SHAFFER(14)
E!
CALCULATED FROM EO. 13
2
+
-016
-
1
1/4"SPHERES^ - ,
11-1,
e
1 I
-0.50
I
,A
5- 0.40
c
20 0.30 LL
'
0.55
0.50 0.40
0.30 0 Figure 15. June 1953
0.2 Void
0.4
'/ R
0.6
0.8
1.0
Fraction vs. Radial Position for 2 - I n c h Pipe
Figure 17.
Void Fraction vs. Radial Position for 4-Inch Pipe
INDUSTRIAL AND ENGINEERING CHEMISTRY
1217
units than the smaller ones, although this distance is approximately two pellet diameters in each case. The calculated void fractions are based upon the assumption of a constant mixing length. At the point of the peak velocity, approximately one pellet diameter from the wall, the influence of wall friction in reducing the velocity becomes important. I n empty pipe flow the mixing length has been found ( l a ) to vary directly as the distance from the wall for r / R greater than 0.93. Therefore, Equation 13 would not be expected to hold for r / R values much larger than those corresponding to the peak velocity. Hence the calculation of void fraction by Equation 13 was carried to a radial position just beyond the peak velocity. To carry the calculations closer to the wall it would be necessary to revise Equation 13 to take into account a variable mixing length. At the time Figures 12 to 14 Fere prepared there was no experimental information on the variation in void fraction with radial position. Since then Shaffer (14) has obtained such data for several spherical particles in 2-, 3-, and 4-inch pipes. The experimental technique consisted, briefly, in measuring thevolume of water necessary to attain various heights in a horizontal, packed bed. This procedure gave directly void fraction data as a function of chord length and this had to be interpreted mathematically in order to obtain 6 as a function of radial position. The results of Shaffer’s work are shown in Table V.
Table V.
larger radii the void fraction increases rapidly. This is verified by the direct void fraction measurements of Shaffer (14). ACKNOWLEDGMENT
This study was made possible by a grant-in-aid from the Research Corp. The authors n-ish t o acknowledge the financial a + sistance and interest of this organization in the project. NOMENCLATURE
diameter of pellet, feet diameter of pipe, feet mass velocity of gas (based upon empty pipe area), pounds/(hour)(sq. feet) Prandtl mixing length, feet pressure drop per foot of bed, pounds/(sq. foot)(feet) pressure defect, pounds/(sq. foot)(feet) radius of pipe = D J 2 , feet Reynolds number (modified Re based upon pellet diameter) average linear velocity, feet per second 32.2 feet/sec.2 constant in pressure drop Equation 4, pounds(sec. *)/ (feet4) constant in pressure defect eauation = 0.0096. dimensionless defined by Equation 9, dimensionless radius. feet point velocity a t radius r , feet per second point velocity a t center of pipe, feet per second point velocity a t r/R = 0.32, feet per second point velocity a t r/R = 0.55, feet per second point velocity a t r / R = 0.71, feet per second point velocity a t r/R = 0.84, feet per second point velocity a t r/R = 0.95, feet per second &a), dimensionless
Experimental Void Fraction D a t a
Radial position, 2-Inch Pipe
r/R
I/4s
a/es
0.24 0.62 0.72 0.80 0.85 0.93
0.32 0.40 0.41 0.38 0.44 0.45
0.32 0.38 0.46 0.41 0.49 0.63
3-Inch Pipe l/as 8/85 1/23 0 . 3 4 0.33 0 . 2 8 0.34 0.33 0.36 0 . 4 0 0.43 0 . 4 7 0 . 3 4 0.34 0 . 3 8 0.43 0.46 0 . 5 0 0.47 0.52 0.56
4-Inch Pipe >/as
a/es
1/zs
0.32 0.35 0.41 0.34 0.42 0.44
0.33 0.32 0.42 0.34 0.46 0.54
0.30 0.32 0.44 0.34 0.50 0.58
The information in Table T’ can be used to check the value of the theory (Equation 13) by comparing it with the computed void fractions shown in Figures 12 to 14. Such comparisons are presented in Figures 15 to 17, where the open circles represent Shaffer’s direct experimental measurements and the solid circles the computed 6 values from Equation 13. The general agreement between the two sets of measurements is good, and, it is believed, definitely establishes the hypothesis that the void fraction increases gradually as the pipe wall is approached and indicates the magnitude of that increase.
(F) (
void fraction a t radius r void fraction for central core of bed average void fraction for entire bed density of air, pounds per cubic foot shearing force per unit area, pounds per square foot
Subscripts = cylindrical pellet = spherical pellet LITERATURE CITED
CONCLUSIONS
The velocity profile for gases flowing through a packed bed is not flat, but has a maximum value approximately one pellet diameter from the pipe wall. The maximum or peak velocity ranges up to 100% higher than the center velocity as the ratio of pipe diameter to pellet diameter decreases. The divergence of the profile from the assumption of a uniform velocity is less than 20% for ratios of pipe diameter t o pellet diameter of more than 30. A tentative theory based upon the concept of momentum transfer and variation of void fraction with radial position agrees Tith the experimental velocity data. Application of the theory suggests that it is satisfactory to assume that the void fraction is constant a t its minimum value up to a distance of two pellet diameters from the pipe wall. At
1218
(14)
(15) (16) (17)
Arthur, J. R., Linnett, J. W.,Raynor, E. J., and Sington, E P. C., T r a n s . Faraday SOC.,45, 270(1950). Baron, T., Chem. Eng. Prog?., 48, 118 (1952). Carman, P. C., J . Sac. Chem. Ind., 57, 226 (1938); 58, B (1939). D’Arcy, H., “Les Fontaines Publiques de la Ville Dijon,” Paris, Victor Dalmont, 1956. Ergun, S., Chem. Eng. Progr., 48,89 (1952). Freeman, H. A., “Industrial Statistics,” New Pork, John Wiley & Sons, 1942. King, L. V., Phil. T r a n s . Rou. SOC.,214A, 373 (1914). Kozeny, J., Ber. Wien A k a d . , 135a,271 (1927). Leva, M., Chem. En#. Progr., 43,549 (1947). Leva, M., and Grummer, AI., Ibid., 43,713 (1947). Morales, AI., Spinn, C. W., and Smith, J. M., IND.ENG. CHEM.,43, 225 (1951). Nikuradze, J., Forschungschuft 355, Supplement Forsch. Gebiete Ingenieurw., 3 (September-October 1932). Ower, E., “The Measurement of Air Flow,” 3rd ed., London, Chapman & Hall. 1949. Shaffer, M. R., M.S. thesis, Purdue University, June 1953. Stanton, T. E., and Pannell, J. R., Phil. Trans. R o y . SOC., 214A. 199 (1914). Stoker,’R. L.’, IND’. ENG.CHEX.,38, 622 (1946). Teago, F. J., J . Inst. EEec. Engrs., 52,563 (1915).
RECXIVED for review July 12, 1952.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED March 6.
1953.
Vol. 45, No. 6
