Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
AHR= ~ xl,kNRAHR,krkj= enthalpy change of the reactions on the plate j , where k is the index of a reaction and NR is the number of reactions 7, = Murphree efficiency in vapor phase on plate j T = contact time Literature Cited Auschuitz and Reitter, "Der Distilhtion unter vermicderen Druck in Laboratorium", 2nd ed, Cohen, Bonn, 1955. Beilsteins "Handbuch der Organischen Chemie", 2nd ed. Springer-Verlag, Berlin, 1948. Bromley, L. A. AIChE J . , 19, 313 (1973). Fredenslund, A., Jones, R. L., Prausnitz, J. M., AIChE J . , 21, 1086 (1975). Gerster, J. A., Chem. Eng. Prog., 59, 35 (1963). Gibson, Chem. I n d . , 971 (1931). Gordon, J. E. "The Organic Chemistry of Electrolytes Solutions", Wiley, New York, N.Y., 1975.
433
Horsiey, L. H., Adv. Chem. Ser., No. 2, (1962). Kahlbaum, "Siedetemperatur und Druck", Leipzig, 1945. Kahlbaum, Ber., 17, 1245 (1954). Meissner, H. P.. Kusik, C. L., Ind. Eng. Chem. Process Des. Dev., 12, 112 (1973). Meissner, H. P., Kusik, C. L., Ind. Eng. Chem. Process Des. Dev., 12, 205 (1973). Meissner, H. P., Tester, J. W., Ind. Eng. Chem. Process Des. Dev., 11, 128 (1972). Prausnitz, J. M., "Molecular Thermodynamics of Fluid-Phase Equilibria", Prentice-Hall, Englewood Cliffs, N.J., 1969. Reid, R. C.. Sherwood, T. K. "The Properties of Gases and Liquids", McGraw-Hill, New York, N.Y., 1958. Stull, D. R., Ind. Eng. Chem., 39, 517 (1947).
Received for review February 2 1 , 1978 Accepted December 27, 1978
Drag Coefficients for Irregularly Shaped Particles John
D. Hottovy and Nicholas D. Sylvester*
Resources Engineering Division, University of Tulsa, Tulsa, Oklahoma 74 104
This study was conducted to measure the terminal velocity of roundish but irregularly shaped particles, and to compare these data with generally accepted correlations. A group of twelve screen fractions of irregular1 shaped particles with a density of 0.88 g/cm3 were dropped through a column of liquid with a density of 0.50 g/cm at three different temperatures. Drag coefficients were calculated using an average diameter for each group of particles. Reynolds numbers (Re) ranged from 7 to 3000. The drag coefficients agreed with t h e standard drag curve for Re < 100. For Re ranging from 100 to 3000, the drag coefficients obtained were higher than predicted and were as much as 100% higher at Re of 3000.
x
Introduction The settling of solid particles in either gases or liquids is important in many industrial processes. The design of sludge settling, slurry pipe lines and fluidized beds all require knowledge of how quickly solid particles settle. This country's push for utilization of its large coal deposits via technologically improved processes requires knowledge as to how quickly coal particles settle in liquids and gases. The liquid velocity in a coal slurry pipe line must be high enough to suspend the coal particles but not so high that pumping costs are excessive. Also, large particles would require a higher liquid velocity and so higher pumping cost, but they would be easier to separate from the transport medium. Most data available in the literature involves smooth spherical particles, or regularly shaped particles like disks or cylinders. Particles encountered in industry usually are not smooth spheres but are irregularly shaped and do not have smooth surfaces. Some of the irregularly shaped particles wobble and tumble so that their interaction with the surrounding fluid is different than a sphere's. In several studies, it has been found that this different interaction changes the drag force exerted on the irregular body as compared to that of a sphere and thus decreases the settling velocity. An experimental study of the settling of irregularly shaped particles for Reynolds numbers ranging from 7 to 3000 is presented in this note. Background and Literature Review The movement of spherical objects through liquids and gases has been studied for some time. Many scientists 0019-7882/79/1118-0433$01.00/0
(Torobin and Gauvin, 1959a-c, 1960a,b) have taken data on various objects and in various media. After drag coefficients were computed and compared at various Reynolds numbers ( R e ) , a more or less consistent trend of lower drag coefficient (C,) with higher Re was observed (for Re < 5000). The plot of drag coefficient vs. Re became known as the "standard drag curve" (Perry and Chilton, 1973). However, some measurements of particle drag do not agree with the standard curve. Bailey (1973) notes that measurements of drag coefficients can differ due to particle sphericity, oscillations, free stream turbulence, and surface roughness. At low Re (CO.11, flow around a sphere is essentially symmetric fore and aft, and laminar in nature (Zarin, 1969). The drag force is predominantly due to viscous forces acting on the surface of the sphere. As Re increases to about 24, inertial effects increase in importance and the flow field starts to change. Taneda (1956) reports that at Re = 24, a vortex pattern is formed at the rear of the sphere. The boundary layer separates from the sphere where the flow meets the reversed flow of the vortex. Separation is due to the inability of the fluid, adjacent to the surface, to flow to the rear stagnation point against the adverse pressure gradient and surface friction, which resist flow (Zarin, 1969). As Re increases further, the vortex patterns grown in size and the inertial component of the drag force increases relative to the viscous force. The shedding of the vortices produces transverse forces on the particle, which cause freely falling or rising particles to follow zig-zag or helical paths (Margavey and Bishop, 1961). 0 1979 American Chemical Society
434
Ind. Eng. Chern. Process Des. Dev., Vol. 18, No. 3, 1979
For higher Re, the vortices are shed more frequently and this becomes nearly a continuous process for Re > 10000 (Zarin, 1969). The drag coefficient continues to decrease as Re increases up to 10000. As Re increase further, Cd remains relatively constant until Re equals 200000 where the boundary layer becomes turbulent. The drag coefficient for Re < 0.1 can be obtained accurately hy neglecting the inertial terms in the NavierStokes equations and equals 24/Re. Due to the significance of the inertial forces (or form drag) as a component of drag force, equations at higher Re numhers are only empirical approximations. Torohin and Gauvin (1959a), Greenkorn and Kessler (19721, and Perry and Chilton (1973) present many of the available approximations. Bailey (1973) discussed Cd data for spheres obtained by many authors. He noted several instances where sphere oscillations occurred for Re > 300 including Roos and Willmarth (1971), Zarin (1969), and Sivier (1967). The drag observed was higher than that of the standard drag curve. It was suggested that for Re, where particle oscillations take place, a nonlinear interaction takes place between wake vortex shedding and the drag on the particle. Apparently the particle oscillations are caused by the onset of vortex shedding which exerts a lateral force on the sphere. This lateral force is large enough to cause the spectacular effects seen, for example, when a “knucklehall” is pitched in baseball with a Re of about 6000 (Greenkorn and Kessler, 1972). A major problem associated with the study and correlation of drag on nonspherical bodies is finding a characteristic diameter. The quantifying of the size and shape of the body is not straightforward when progressing from spheres to geometric shapes to bodies with irregular shapes. Becker (1959) used coefficients called surface sphericity and form sphericity to correlate the Cd of regularly shaped geometric objects. However, obtaining average volume, surface area and projected surface area for irregularly and uniquely shaped particles is much more difficult since these numbers cannot he calculated. Nevertheless, Becker (1959) and Torobin and Gauvin’s (1960a) observations may he useful in anticipating what type of behavior might be shown by bodies less regularly shaped than spheroids, cylinders, rectangular parallelepipeds, and isometric polyhedrons. Becker found that up to Re of 5.5, Cd did not differ greatly from that of spheres. As Re increased, a transition region was observed for 5.5 C Re C 200, where depending on a particle’s lack of sphericity, the Cd values were slightly higher than those of a sphere. The less sphere-like and the higher the Re, the greater the deviation. As Re increased to 200-500, Becker noted that the particles tended to show more unpredictable motions, wobhling, rotating, etc. Also in this region and continuing up to Re = 3000, Cd deviated much more from sphere Cd data. From this discussion it is apparent that many things can affect C , data. For a given Re the most important factor is the particle shape. Experimental Section The particles used were irregularly shaped solid particles with a density of 0.88 g/cm3. The solids were classified using U.S. Standard Screens. Particles passing through a screen, hut retained on the next size smaller screen, were assumed to have an average particle diameter equal to the average of the opening width of the two screens. The particles of each class were screened at least twice to ensure that the smaller particles had passed through the screen. Table I shows the screen opening for each mesh size used and the diameters. Figure 1is a photo of the 0.000537-ft diameter particles magnified 50 times and clearly shows
Table I. Particle Size vs. Screen Size U.S. standard screen size
passed retained on throueh I
screen openinas. .. ft
av particle diameter, ft
~
10 12 16 20 30 35 50 60 80 100 140 200
12 16 20 30 35 50 60 80 100 140 200 230
0.006662 0.005508 0.003910 0.002758 0.001950 0.001642 0.000975 0.000817 0.000583 0.000492 0.000342 0.000242
0.005508 0.003910 0.002758 n.ooi95o o.ooi642 0.000975 0.000817 0.000583 0.000492 0.000342 0.000242 0.000208
0.00604 0.00471 0.00333 0.00235 0.00185 0.00 131 0.000896 0.000700 0.000537 0.000417 0.000292 0.000225
Figure 1. Photograph of 0.000537 ft diameter particles magnified 50 times.
the highly irregular shape of the particles studied. Photos of the other particle sizes are available (Hottovy, 1978) and show similar irregularities. The apparatus used is shown schematically in Figure 2 and the individual components are described below. 1. A large barrel (2 in. diameter) Jerguson sight glass was used as the test section. The Jerguson has two 9 in. long glass sections with 2 in. separating the sections. The velocity of the particles was measured over both the top 9 in. section and the 20 in. total length. 2. A 9 in. section of ’/& pipe was used above the Jerguson to allow acceleration of the particles before reaching the Jerguson. The largest particles needed about 0.5 in. to reach terminal velocity. 3. A Tufline plug valve on top of the 9 in. riser was used to put the particles into the Jerguson. By rotating the plug, solids were dumped into the liquid-full Jergusou. For each velocity measurement, a small pinch of solids was placed in the Tufline valve. This ranged from three
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 435
Table 11. Particle Velocity vs. Size and Temperature (Temperature, 80 95% confidence limits on velocity av av diameter, velocity, lower upper ft ft/s std dev limit limit O F )
0.00604 0.0047 1 0.00333 0.00235 0.001 80 0.00131 0.000896 0.000700 0.000537 0.0004 17 0.000292 0.000225
0.337 0.291 0.269 0.247 0.210 0.173 0.134 0.109 0.0929 0.0743 0.0459 0.0318
8
0.00604 0.0047 1 0.00333 0.00235 0.00185 0.00131 0.000896 0.000700 0.000537 0.000417 0.000292 0.000225
0.392 0.341 0.292 0.261 0.234 0.193 0.148 0.116 0.100 0.0762 0.0533 0.0365
Figure 3. Terminal velocity vs. particle diameter.
0.00604 0.00471 0.00333 0.00235 0.00180 0.0013 1 0.000896 0.000700 0.000537 0.000417 0.000292 0.000225
0.405 0.378 0.310 0.284 0.239 0.198 0.169 0.138 0.103 0.0827 0.0646 0.0427
5 DRAIN
TEMPERATUREINDICATOR
0.0311 0.0437 0.0131 0.00753 0.0162 0.0118 0.00611 0.00858 0.00300 0.00188 0.00574 0.00584
0.309 0.252 0.263 0.240 0.195 0.162 0.129 0.101 0.0906 0.0726 0.0421 0.0240
0.365 0.330 0.276 0.253 0.224 0.183 0.140 0.117 0.0951 0.0760 0.0497 0.0346
Temperature, 100 "F TO FLARE
2
Figure 2. Schematic diagram of apparatus. I
1
0.8
0.5 " 0.4 p 0.3 >r 0 , z 0.6
r c
I
1
I
l
I
l
I
TEMPERATURE 0 8O'F X IOO'F 0 12O'F
I
I
-1
I
t 0 0
d
0.374 0.3 24 0.282 0.246 0.223 0.185 0.144 0.112 0.0948 0.0728 0.0509 0.0331
0.410 0.358 0.301 0.276 0.244 0.202 0.153 0.120 0.1056 0.0795 0.0557 0.0399
Temperature, 120 "F 0.1 0.08
a 0.06 5 0.05 -
=.
0.0319 0.0306 0.0189 0.0228 0.0157 0.0122 0.00568 0.00421 0.00606 0.00443 0.00330 0.00784
_I
0.04
-
0.03-
t
0.02 0.01 1
0.I
I
02
1
1
0.3 0.4
l
1
I
0.6 0.8 I
2
3
I
l
PARTICLE DIAMETER
I
I
1
4
5
6
I
x 104f t
or four of the largest particles to many of the fine particles. In all cases the particle concentration was low enough to simulate the settling of a dilute suspension of particles. Next, the temperature was checked and the valve was turned. When about half of the particles had passed the top edge of the glass, two stopwatches were started manually. It was attempted to focus on a specific particle that was neither the fastest or slowest particle of the group. When the particles(s) reached the section divider, one stopwatch was stopped. The other stopwatch was stopped when the same, or a similar, particle reached the bottom of the second glass section. The experimental equipment and procedures are described in detail by Hottovy (1978). Results a n d Discussion The final results are an average of many runs. Many runs were made because a range of particle sizes was used and some of the irregular shapes were more streamlined than others. Also, timing was accurate to only *0.05 s, which gives an expected error of f2.7% for the large particles and the short distance. For the long distance and small particles, timing error was only *0.08%. No consistent bias was seen between the short (9 in.) and long (20 in.) distance measurements. Table I1 summarizes the averaged results. A plot of settling velocity vs. average particle diameter for the three temperatures studied is shown in Figure 3. The large particles fell faster than smaller particles and
0.0256 0.0194 0.00756 0.0121 0.00698 0.00872 0.00615 0.00511 0.00236 0.00442 0.00309 0.00354
0.391 0.366 0.306 0.276 0.235 0.194 0.165 0.135 0.101 0.0798 0.0625 0.0409
0.419 0.389 0.316 0.292 0.243 0.203 0.173 0.141 0.104 0.0857 0.0667 0.0445
Table 111, Solid and Liauid Densitv vs. TemDerature
temp, "F
viscosity, lb,/ft s
solid density, g/cm3
liquid density, g/cm3
density diff, g/cm3
120 100 80
0.000044 0.000049 0.000054
0.860 0.870 0.880
0.463 0.485 0.506
0.40 0.39 0.37
the rate increased with temperature. The large particles fell faster because of their larger weight relative to surface area. The higher temperature reduced the viscosity and increased the density difference between the liquid and solid, so the particles fell faster. Table I11 shows the viscosities, densities, and density differences at the three temperatures studied. Drag coefficients (Cd) and Reynolds numbers were calculated for the data in Table I1 and are shown in Table IV. The C d data defined a single curve when plotted against Re, as seen in Figure 4. The data agree very well with data presented by Bailey (1973), Perry and Chilton (1973),and Roos and Willmarth (19711, for Re less than 100. However, for Re greater than 100, the drag coefficients were higher than those of the standard drag curve. Higher C d values were also found by Becker (1959) and Roos and Willmarth (1971). They noted that oscillations of the particles increased C d above those of the standard drag curve. Deviations as high as 2570, for Re equal to 3000 were reported by Hill and Zukoski (1972).
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
436 Table
IV. Drag
diameter, ft
temp, " F
0.00604 0.0047 1 0.00333 0.00235 0.00180 0.00131 0.000896 0.000700 0.0005 37 0.000417 0.000292 0.000225 0.00604 0.00471 0.00333 0.00235 0.001 85 0.00131 0.000896 0.000700 0.000537 0.000417 0.000292 0.000225 0.00604 0.0047 1 0.003 3 3 0.00235 0.00180 0.0013 1 0.000896 0.000700 0.000 53 7 0.000417 0.000292 0.000225 6~
+z
wU
u.
5
tP
4
-
Coefficient
80 80 80 80 80 80 80 80 80 80 80 80
100 100 100 100 100 100 100 100 100 100 100 100 120 120 120 120 120 120 120 120 120 120 120 120 I
I
I
I
\
Re
cd
0.337 0.291 0.269 0.247 0.210 0.173 0.134 0.109 0.0929 0.0743 0.0459 0.0318 0.392 0.341 0.292 0.261 0.234 0.193 0.148 0.116 0.100 0.0762 0.0533 0.0365 0.405 0.378 0.310 0.284 0.239 0.198 0.169 0.138 0.103 0.0827 0.0646 0.0427
2056 1384 905 5 86 382 229 121 77.1 50.4 31.3 13.5 7.2 2594 1759 1065 672 474 277 145 89.0 58.8 34.8 17.1 9.0 2954 2150 1247 806 519 313 183 117 66.8 41.6 22.8 11.6
1.26 1.32 1.09 0.91 0.97 1.04 1.19 1.40 1.48 1.79 3.29 5.28 1.00 1.03 0.99 0.88 0.86 0.89 1.04 1.32 1.36 1.82 2.61 4.28 1.02 0.91 0.96 0.80 0.87 0.92 0.87
1
I
I
I
I
1.01 1.40 1.68 1.93 3.40 I
I
1
i
"\
W
0
3
E D
l
Acknowledgment Special thanks go to Phillips Petroleum Company for providing financial assistance to John D. Hottovy and for the experimental facilities. Thanks are also due to Mr. F. E. Axsom for assistance in setting up the equipment and Mr. D. D. Norwood for guidance and many useful comments.
Nomenclature
Literature Cited
-
08-
06 -
05 0.4 6
on a sphere of comparable size. This supports the assumption made on the roundish particles, diameters. The assumed diameter was calculated by averaging the width of the opening that the particle was retained on and the next smaller opening that the particle passed through. The finding that the shape of the particle did not change the drag coefficient for Re less than 100 agrees with Becker's (1959) observations on regularly shaped geometrical objects. Apparently, making an assumption that real world particles are spheres is valid for the calculation of drag force or settling velocity when the particle is roundish. For 100 < Re < 3000, the drag force that acts on an irregularly shaped particle is higher than that which acts on a sphere of comparable size. This was also noted by Becker (1959). Evidently the shape of a particle has an effect on the way vortices are formed and shed. The altering of the wake increases the inertial component of the drag coefficient. This change of C d is not solely due to particle oscillation. The maximum increase of C d observed by Hill and Zukoski (1972) on oscillating spheres was 25% when Re = 3000. At the same Re, C d was 100% higher for the irregularly shaped particles of this study. The observations of this study suggest that assuming a particle is a sphere for drag calculations is quite acceptable for Re < 100. However, for Re > 100,higher drag and thus lower settling velocities can be expected for irregularly shaped particles.
Cd = drag coefficient D = particle diameter Re = p V D / p = Reynolds number V = particle terminal velocity p = fluid density p = fluid Viscosity
TEMPERATURE 0 8O'F x IOO'F 0 IZO'F
3 -
P -
vs. R e y n o l d s Number
velocity, ft/s
I IO
1
20
1
1
I
40 60
1
100
300
400
IO00
3000
REYNOLDS NUMBER
Figure 4. Drag coefficient vs. Reynolds number.
Particle oscillations were noted for diameters larger than 0.0007 and Re near 100. This corresponds with the Re at which C d values started to deviate from those of the standard curve. The data show that the irregularly shaped particles behave similarly to solid spheres. Deviations from the standard curve occur when oscillation of the particles takes place. For Re less than 100, the drag force acting on the irregularly shaped particles is similar to the drag force acting
Bailey, A. 8.. J . FluidMech., 65, 401-410 (1973). Becker, H. A,. Can. J . Chem. f n g . , 37, 85-91 (1959). Greenkofn, R. A,, Kessler, D.P., "Transfer Operations", pp 206-213, McGraw-Hill, New York, N.Y., 1972. Hill, M. K., Zukoski, E. E., California Institute of Technology, Aerospace Research Laboratory Report, ARL-72-0017 (1972). Hottovy. J. D., M. S. Thesis, The University of Tulsa, Tulsa, Okla., 1978. Margarvey, R. H., Bishop, R. L., Can. J . Phys., 39, 1418-1422 (1961). Perry, R. H., Chikon. C. H., "Chemical Engineer's Handbook", 5th ed,McGraw-Hill, New York, N.Y., 1973. Roos, F. W., Willmarth, W. W., A . I . A . A . J . , 9, 285-291 (1971). Sivier, K. R., PhD. Thesis, The University of Michigan, Ann Arbor, Mich., 1967. Taneda, S., Repf. Res. Insf. Appl. Mech. Jpn., 4, 99-105 (1956). Torobin, L. B., Gauvin, W. H., Can. J . Chem. Eng.. 37, 129-141 (1959a). Torobin, L. B., Gauvin, W. H., Can. J . Chem. f n g . , 37, 167-176 (1959b). Torobin, L. B., Gauvin, W. H., Can. J . Chem. Eng., 37, 224-236 (1959~). Torobin, L. B., Gauvin, W. H., Can. J . Chem. Eng., 38, 142-153 (1960a). Torobin, L. B., Gauvin, W. H., Can. J . Chem. Eng., 38, 189-200 (1960b). Zarin, N. A,, Ph.D. Thesis, The University of Michigan, Ann Arbor, Mich., 1969.
Received for review March 7, 1978 Accepted February 14, 1979
