FLOW THROUGH POROUS MEDIA
(Viscous Flow in Multiparticle Systems)
Cubical Assemblages of Uniform Spheres J O H N HAPPEL
AND
N O R M A N EPSTEINl
DEPARTMENT OF CHEMICAL ENQINEERINQ. NEW Y O R K UNIVERSITY, NEW Y O R K . N . Y .
T h i s study employs a cubic assemblage of fixed immobilized spheres as a model for experid mental study of the relationship between pressure drop and fractional void volume. Results. are compared with data on fluidization and sedimentation f r o m t h e literature. It i s shown) that, for t h e most part, this model predicts t h e pressure drop satisfactorily. Therefore, it m a y be inferred t h a t hydrodynamic resistance is the major factor contributing t o pressure drop as compared with other possible variables, such as solid-solid friction. It also appears t h a t a cubic model can serve as the basis for further theoretical treatment which should have practical significance.
P
RBSSURE drop in fluidization and sedimentation velocity
.
in hindered settling are aff ctcted primarily by the fluid-solid friction encountered in such systems. There are, however, a host of other factors that may contribute to the magnitude of these variables. Among these are such things as mutual collisions of the particles, interparticle bridging, and particle agitation and rotation. The investigation described in this paper was an attempt to isolate these other factors by making measurements on mechanically suspended and immobilized assemblages of equal diameter smooth spheres, in the fractional void volume range corresponding to fluidization and hindered settling. By comparing these data to fluidization and sedimentation data a t the same void volume and particle Reynolds number, it was hoped to ascertain whether or not these complicating factors contribute measurably to the pressure drop or sedimentation velocity. The configuration employed was principally simple cubic. The results of Martin et al. ( 2 3 ) fofuniform size smooth spheres stacked in simple cubic arrangement thereby represent the lower porosity limit of the present study. The case of an isolated sphere represents the upper limit. Approximate solutions of the Navier-Stokes equations for flow through swarms of spheres have been limited to viscous flow. For this reason and because of the small Reynolds numbers encountered in fluidized and sedimenting systems, the particle Reynolds number range investigated was kept relatively low. The experimental results were used to test these approximate solutions, as well as more empirical ones. EQUIPMENT AND PROCEDURE
The equipment for making flow measurements on suspended assemblages of spheres consisted of a series of bead assemblages and an aqueous glycerol circulating system. The circulating system started with a rotary displacement pump, driven by a 1.5-hp. motor, feeding into a 25-gallon-perminute capacity Fischer and Porter Ultra-Stable-Vis rotameter. The pum drew liquid from an open 20-gallon tank, to which the liqui% wa8 ultimately returned for recirculation. Between the rotameter and the tank was a vertical 4inch-diameter glass tube 12 inches long, flanged with cork gasket at both ends to sections of 4-inch standard iron pipe. This 4inch diameter conduit, illustrated schematically in Figure 1, housed the bead assemblages. The other flow lines were all 1-inch standard brass pipe and were screwed directly and a t right angles to the Cinch 1
iron sections. Liquid from the rotameter passed into the lower iron section, through the glass tube, and from the upper iron section back to the recirculsting tank. Leads from a differential manometer were connected to pressure taps immediately below the lower iron-to-glass flange and immediately above the upper flange. The manometer was an 8-mm. inverted glass U-tube, inclined a t various angles to the horizontal. Thermometers were inserted a t the exit from the rotameter, a t the inlet and outlet of the &inch conduit, in the upper connecting lead to the differential manometer, and adjacent to the glass manometer tubing. The circulating liquid was a transparent solution of approximately 91% glycerol in water. The spheres for the assemblages were 5-mm. cellulose acetate beads; the particle to tube diameter ratio thus was kept to 1 to 20. The beads, with approximately 0.043-inch holes, were strung on round brass rods of similar diameter, 3 feet long, and glued into position by means of Du Pont clear windshield sealer. The ends of the rods were threaded, inserted into appropriately drilled brass disks temporarily mounted outside the flow system, positioned manually, and then locked by nuts a t both ends. The entire assemblage was then inserted into the 4-inch conduit, as illustrated in Figure 1. The bottom brass disk was held in position by three l/a-inch setscrews in the lower iron pipe, while tension was exerted on all the rods by tightening the wing nut above the upper disk. This was made possible by a l/d-inch stove bolt connecting the center of the top disk to the wing nut, via a hole in a crossbar resting on the walls of the Cinch pipe. The beads were mounted so that they appeared in the center glass section, where they could be observed visually. The entire apparatus is shown in Figure 2. Relevant apparatus dimensions are given in Table I. The bead diameter was determined by taking micrometer readings on 50 randomly selected beads and averaging the result. The mean and maximum deviations from the average were 0.7 and l.S%, respectively. Rod diameters showed negligible deviations from the value cited. The diameter of the glass tube, which
Table
I. Dimensions of Apparatus and Assemblages
Bead diam. 4.90 mm. Rod diam. '1.10 mm. (0.0433inch) Av. tube diam., 10.0 om. (3.94inches) Distance between pressure taps, 46 cm. Cubic Assemblage No. 1 2 3 4 Tetragonal Side of unit cell, mm. 10.0 5& 4fi 6 ~ 5 ... Void fraction, based on unit cell 0.9383 0.8255 0.6595 0.8999 0.9383 No.of rods 76 145 225 112 76 No. of beadshod 24 14 10 16 24 0.9405 0.8395 0.6885 0,8968 Void fraction, over-all 0.9405
Present address, The University of British Columbia, Vancouver, B. C.
June 1954
INDUSTRIAL AND ENGINEERING CHEMISTRY
1187
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT was not perfectly unifolm because of imperfections in the wall, was detei mined b), measuring the volume of liquid displaced over a giwn height.
-NUTS
LOCKING RODS
HERMOMETER
RECIRCULATION TANK
TO DIFFERENTIAL 5-MY. CELLULOSE ACETATE BEADS
GLASS TUBE
( SET( 3)%REV-
-
NUTS -..-J' LOCKING RODS
4"-
1-
REDUCIMG COUPLING
structed jimply by raising every other rod in cubic assemblage Yo. 1, unit cell side of 10 mm., through a distance of 5.0 mm., so that in any horizontal layer no two beads were closer than 10 d2 mm. from each other. The number of horizontal layers of beads was thus doubled, while the number of beads pcr layer was halved. The void fraction, howwer, remained t h e same a8 for cubic assemblage S o . 1. Streamline Flow. Jnk injections at, a number of points along the diameter of the conduit immediately below the glass tube indirated the actual prevalence of streamline flow in the operating Reynolds number range. In each case, an ink jet introduced at a given distance from the conduit n d l s passed through the entire height of the assemblage without dissipating into the body of the fluid, and maintained the same position with respect to the walls on leaving t'he t'ube as it did on entering. Apparent>lyt'he rods, rrhich extended ~ w l beyond l the liquid inlet to the outlet from the 4-inch conduit, acted as straightening vanes to smooth out turbulent entrance and exit effects in the conduits, particularly in the middle glass section occupied by the bead assemblages. Those jete introduced near a rod described closely, though smoothly, the alternate contours of rod and beads. This provided justification for treating exposed rod and beads as hydraulic resist>ancesiii series.
-LIQUID FROM THERMOMETER ROTAMETER
I' STD PIPE
"-3g
, -
DIAMETER,$
THICK.
DRILLED BRASS DISK I' NIPPLE I' GLOBE VALVE FOR
FLUSHING CWOUIT
Figure 1.
Diagram of 4-Inch Conduit
Cubic Assemblages. The properties of the four cubic assemblages which were investigated, are recorded in Table I. Discrepancies between the theoretical void fraction, based on a unit cell calculation, and the actual over-all void fraction in an assemblage are due to the fact. that the regular lattice arrangement was interrupted by the walls, which caused the porosity near the walls ta be different from the porosity in the core of the assemblage. I n the case of each assemblage, runs were performed on the rods alone as well as on the rods plus beads. The empty tube was also calibrated. Flow rate JYab controlled and varied by means of a gate valve preceding the rotameter. Variation of kinematic viscosity was obtained by change in the temperature of the glycerol solution from about 27' to 43 O C., brought about by the gradual heating of the cilculating pump. The range of pressure drop measurable by the manometer was rendered versatile by using monometer liquids of different densities-dyecolored benzene and nitrobenzene-and by varying the inclination of the U-tube. When steady state prevailed for a given setting of the control valve, as indicat,ed by the attainment of a maximum reading on the different,ial manometer, readings of the various thermometers, the rotamet,er, and the manometer were simultaneously made. Where the flow rate was too small t o be accurately determined on the rotameter, it was measured directly by temporarily connecting the exit line from the 4-inch conduit to a weigh tank and clocking the time interval for a given weight of liquid to be collected. For a group of runs performed on a given day, a sample of the circulating aqueous glycerol was taken and kinematic viscosity measurements made with an Ostwald viscosimeter. I n addition to the simple cubic assemblages, one body-centered tetragonal assemblage was also investigated. This was con-
1188
Figure 2.
Experimental Apparatus
CORRELATION OF D A T A
The experimental measurements were used t o evaluate the variables; p, p , V , APw, APIVK!and PPWRS:and the correspond~, ing dimenBionless groups, APm,lpV2, A P w R / ~ V APwRsipV', and d V p ! p . Figure 3 is a log-log plot of each of these three VP AP/PVZgroups as ordinate versus das abscissa, based on the P
data obtained for cubic assemblage KO. 1. The raw data obt,ained for the other three atsemblages are not presented. Figure 4 compares the data for cubic assemblage No. 1 wit,h data for the tetragonal assemblage. Except for the fourth cubic assemblage, which viae run primarily for t)he purpose of checking the
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Vol. 46, No. 6
FLOW THROUGH POROUS MEDIA consistency of results on the other three in viscous flow, the particle Reynolds number range was about 0.4 to 11, corresponding t o Reynolds numbers of 8 to 220, based on column diameter. As expected a t such low Reynolds numbers, the log-log plots of APw/pV2 and APrn~/pSr~ versus dVp/p yielded straight lines with a slope of minus 1. These data were fitted numerically with equations of the form
and
The equivalence between the drag coefficient, C, and the expression, 4APsd/3L( 1- e)V2p is predicated on the assumption that the residual pressure drop attributable to the spheres, APs, is identical with the pressure drop which would be caused by the presence of the spheres alone, were the fluid unbounded, APs'. This amounts to assuming that the influence of the bounding wall on the residual pressure drop, APs, is negligible compared to interactions between neighboring spheres. The pressure drop is then obtained by a summation of the force exerted on each sphere. .4s pointed out by Happel and Byrne in an accompanying paper ( I d ) , this assumption is probably invalid a t low solids concentrations, as e approaches 1.0. However, this method of plotting is convenient for comparison with fluidization and sedimentation data.
where F ( E )is a different conPtant for each of the four assemblages, by virtue of the different rod arrangements. For each assemblage, however, the plots of A P W R S / P Vversus ~ dVp/p yielded a slight curvature toward decreasing negative slope which was due t o the onset of inertial effects. By using the method of least mean squares, the numerical data for each cubical bead assemblage were fitted with an equation of the form
The constants, obtained for each assemblage and appearing in equation I, 2, and 3, are tabulated in Table 11.
Table I I.
Experimental Results on Cubic Assemblages 1 0.9405 24.00
Void fraction Height ( L ) ,om. Dimensionless
___
ronstanhin - _-
Equation 1 F ( e ) , Equation 2 a, Equation 3 21, Equation 3 A.Eauation7 B ; Eduation 7
14.08 97.2 185.2 4.410 48.4 2.02
Cubic Assemblage No. 2 3 4 0.8395 0.6885 0,8968 9.898 5,656 13.57 14.08 184.1 346.0 7.782 74.4 3.20
14.08 295.9 980.0 3.900 231 1.44
14.08 169.6 269.4 34.42 56.8
...
Martin et al. (83) 0,4764
... ... ... ... ...
01
02
05
LO
2
5
IO
20
50
Ix)
&L
P
Figure 3.
Flow through Cubic Assemblage No. 1
1020
...
By assuming that the tube and rods represented hydraulic resistances in series, the residual pressure drop attributable to the rods, APR,was calculated for each assemblage from the expression
a t equal values of dVp/p. By employing the same assumption, the residual pressure drop attributable to the spheres (beads), APs, was calculated for each assemblage from the relation
where the residual pressure drop due to the rods was corrected for the fact that with the beads mounted in place, the entire length of the rod was no longer exposed. I n this manner, each assemblage gave rise t o an equation of the form
01
e
5
1.0
2
5
+-
IO
20
50
100
Figure 4. Comparison of Simple Cubic and Tetragonal Assemblages
or, alternatively
3L(1
- e)V2p
which can be abbreviated t o
A
- +B
(d;p)
(7)
Values of the various constants are recorded in Table 11, which includes also the experimental results of Martin et al. (%), for a simple cubic assemblage a t its minimum possible fractional void volume of 0.4764. If Equation 8 is changed to the form (9)
June 1954
INDUSTRIAL AND ENGINEERING CHEMISTRY
1189
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT as N R , ~CNR,-+A; , that is, where inertial forces are negligible relative to viscous forces,
term, the ratio, N R ~ / A ' R ~at~ ,a constant value of CXh, represents the ratio, VIVO for the given conditions. T'alues of V/VO for the various cubic assemblages, corresponding to several values of both N R and ~ ~I \ r ~ e were , thus calculated and are recorded in Table 111.
or 4APsd = A L L 3L(1 - e)B2p dVp
Table I I I .
Calculated Velocity Ratios T'/ VQ
Stokes' law for an isolated sphere in viscous flow is
Cubic Assemblage No. 2 3
1
Martin et aZ.
4
(23)
.VEXo
0 . 3 (Equation 14) l , O ( C S & = 26.5) 1.3(CN& = 35.7) 2 . 3 ( C N i e = 69) 10 (CAT& = 410) 11 (CAI'$, = 468) 16 (CiV$, = 761)
Comparison of equations 11and 13, e m p l o y i n g the assumption that APs = AP;, indicates t h a t for viscous flow
h h 0.007 (Equation 14)
0.496 0.496
10
0.675
100
1000
8000
-I
Substitution of this relation in Equation 11 gives A = 48, and thus VIVO = 1/2. This indicates that values of VIVO as computed from pressure drop measurements tend to approach a limiting value of 1/2 as €-+I, in contrast to sedimentation measurements where the value of V/Vo + 1 as e -+ 1. For values of I V R2~ 0.3, Equation 15 probably does not hold. Equation 9 can be transformed to the form CN;, = A N R ,
+ BW;,
(16)
which gives a relation between C X & and N Rfor ~ each of the various assemblages. An analogous graphical relationship for an isolated sphere was obtained by replotting the usual C versus I V R ~curve ~ as CjVie versus N R ~using ~ , tabulated values ( 2 7 ) . Since the dimensionless group, CNh, does not contain a velocity
1190
0.01 0.1 1.0
T h e group, VIVO, r e p r esents the ratio of the velocity a t which a given fluid approaches an assemblage in viscous flow to Stokes' ve002 locity for an iso04 05 06 07 08 09 ID lated sphere of POROSITY - 6 equal diameter Figure 5. Experimental Approach in t h e s a m e Velocities for Cubic Assemblages fluid medium, Relative t o Approach Velocities for under the same Isolated Sphere frictional drag. It also represents the ratio of settling velocities of assemblage and isolated sphere of equal diameter and density in the same viscous fluid medium, where wall effects can be neglected-Le., e
/
'
0.6
0.7
-
/
005
a 0.3
0.4
0.5
POROSITY
-
0.8
0.9
1.0
E
Figure 6. Comparison w i t h Sedimentation and Fluidization Data for Viscous Flow
The fact that the data on stationary assemblages are based on pressure drop measurements, whereas the data of Steinour and Mertes are based on actual velocity measurements, explains this discrepancy a t high values of e. Thus, caution must be exercised in the interpretation of velocity and pressure drop data applied interchangeably in dilute phase fluidization and hindered settling operations. This treatment, in which wall effects are neglected, has been applied in a great many previous investigations. A number of possibile explanations exist for the slight discrepancy a t values of e below 0.8. These are the presence of channeling, solid-solid friction between particles, and differences in configuration between sedimenting and fluidized beds as compared with a cubic assemblage. Friction between particles is not believed to be important in viscous flow systems (21, 26) and, if present, would tend to make the fluidization line fall below the corresponding data for flow through assemblages shown in Figure 6, rather than above. The importance of configuration is not too well defined, but it is believed to exert a relatively small effect. The data plotted in Figure 4 indicate no difference in results for simple cubic and body-centered tetragonal assemblages a t the same voidage. There also appears t o be no question that a t the lower limit of voidage possible for a cubic packed assemblage, e = 0.48, the data on pressure drop for random and cubic packing are in agreement (Table IV). On the other hand, the theory of Burgers ( 428) indicates that in sedimentation a very dilute random arrangement should show a substantially higher VIVOthan an ordered one of the same fractional void volume. The effect of channeling of liquid through a system of fluidized particles appears to offer the best explanation for the presence of higher values of VIVOthan are observed in a corresponding fixed assemblage. Several investigators have concluded that, a t least in the small units employed aa fluidized bed reactors in the laboratory, there is a general flow pattern for the solid which
I N D U S T R I A L A N D E N G I N E E R I N G C H E MISST R Y
1191
vciy close to the experimental data on the cubic assemblages A plot of the equation devel-
I
I
1 ~
FLOW THROUGH POROUS MEDIA
,
__
I
IO
I SEDIHENTbTION OF GLbSS SPHERES IN WATER (N&'12
I
X PLYlDlZATlON OF C U S S
B'
= dimensionless constant
B
= dimensionless constant
C
= d r a g coefficient,
d F( E )
= particle for spheres diameter = dimensionless constant
in Equation 6
in Equation 7
SPHERES
1
4 x AP,'d/3L( 1 - € ) V 2 p ,
in Equation 2; represents APWRdlpT.' in viscous flow L = length of bed iVBe = multiparticle Reynolds go number, d V p / p 9 N R e o = singleparticleReynold: number, d V o p / p A P ~= observed pressure drop 00 for fluid flowing in I I I empty column due to presence of ;containing walls A P W R = observed pressure drop due to fluid 0 0 03 04 05 06 07 08 09 IO 03 04 05 0.6 0.1 0.8 0.9 1.0 flowing in column POROSITY- L POROSITY - G containing rods A P ~ R , s= observed pressure Figure I O . Comparison with Data of Figure 9. Comparison with Sedimentadrop due to fluid Lewis, Gilliland, and Bauer (22) for tion and Fluidization Data for N R =~ 1.3 ~ flowing in column N R = ~ 2.3 ~ containing both r o d s a n d beaids (spheres) APR = residual pressure drop due to presence of rods; d e A possible explanation for the better agreement a t the higher fined by Equation 4 Reynolds number is that solid-solid friction, small or negligible AP5 = residual pressure drop due to presence of spheres in viscous flow, becomes increasingly important as the Reynolds (beads); defined by Equation 5 AI'; = theoretical pressure drop which would be caused by number is increased. This is consistent with the conclusion presence of spheres (beads) alone in an unbounded reached by Morse (65) that energy dissipation by particle colmedium lisions in gas-solid fluidization becomes significant in the lower transitional Reynolds number region. This effect is greater for gas-solid than for liquid solid fluidization (65, 53). CONCLUSIONS
Data are presented for pressure drop through cubic assemblages of stationary spheres, from low values of Reynolds number (below N R = ~ 0.3) to values of N R =~ 16. ~ These data are replotted in a form convenient for comparison with data on fluidization and sedimentation. On the basis of these data and a theoretical analysis of the relationship between pressure drop and velocity required to suspend spheres in a dilute assemblage, it appears that wall effects should be considered in correlations of data of fractional voidages when E 2 0.9, especially when viscous flow, N R
