R
= radius of tube in calculations for reaction along cylin-
R*
=
drical channel. cm dimensionless reaction rate,
{%
jR&(plate) ; (cylinder) Raw = average reaction rate over total reactive surface Re = Reynolds number, d v p / g t = seconds z i = average velocity, centimeter/second 4 1 ) = dimensionless velocity profile, for laminar flow, v ( 0 ) = (1 - s2) x = distance in axial direction, cm Y = distance in transverse direction, cm zi = dimensionless distance along x coordinate xD,/2ziR2 (cylinder)
6,
= stoichiometric coefficient
ei
=
dimensionless position along z coordinate,
= =
plate) dimensionless position along y coordinate, y/h density, grams/cm3
1 p
22Di (flat
3h20
literature Cited
Acrivos, A,, ChambrB, P. L., I n d . Eng. Chem.49,1025 (1957). Barron, A. N.. Hendrickson. A. R.. Wieland. D. R.. Trans. AIME. 225,409 (1962). Chamberlain, L. C., Boyer, R. F., Ind. Eng.Chem. 31, 400 (1939). Hoelscher, H. E., Cowhead, c., IND.ENG.CHEM.FCND.4M. 4, 150-4 (1965). Hendrickson, A. R., Rosene, R. B., Wieland, D. R., Division of Petroleum Chemistry. 137th Meeting- ACS, Cleveland, Ohio. April 1960. Powell, J. J. D., Computer J . 7, 155 (1964). Schechter, R. S.,Gidley, J. L., A.I.Ch.E. J . 15,339 (1969). Solbrig, c. W., GidasPow, D.7 Can. J . Chem. Eng. 45935 (1967). van Poollen, H. K., Jargon, J. R., Oil Gas J . 66,84 (1968). RECEIVED for review Xovernber 12, 1969 ACCEPTEDJuly 15, 1970 I .
GREEKLETTERS = apparent reaction order for weak acids = coefficients in series approximation t o boundary layer
a
Pn
= r(n) =
y
Droblem dimensionless radial position, r/R gamma function of variable n, tabulated function
Flow of Single-phase Fluids through Fibrous Beds Chwan P. Kyan, Darshanlal 1. Wasan,’ and Robert C. Kintner Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616
A pore model for the flow of a single-phase fluid through a bed of random fibers is proposed. An effective pore number, Ne,accounts for the influence of dead space on flow; deflection number, N6,characterizes the effect of fiber deflection on pressure drop. Experimental data were obtained with glass, nylon, and Dacron fibers of 8- to 28-micron diameter and with fluids of viscosity ranging from 1 to 22 cp. A generalized friction factor-Reynolds number equation i s presented. The effects of dead space in a fibrous bed on flow and of fiber deflection on pressure drop have no parallels in a granular bed.
THE flow of fluids through porous media has been a subject of investigation for many years. A considerable amount of research has been done on the flow through granular beds and many useful results have been obtained (Brownell and Katz, 1947, 1956; Ergun, 1952; Ergun and Orning, 1949). A lesser number of investigations have been done on the phenomena of flow of fluids through fibrous media, mostly in connection with aerosol filtration. General approaches pursued by most workers on the flow of fluids through fibrous beds involved the development of theoretical pressure drop equations from either a “channel model” or a “drag model.” The former was the more extensively used. N o s t workers using the channel model started with the Kozeny-Carman equation,
which in the friction factor form becomes
To whom correspondence should be sent. 596
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
I n this equation, t’he fact that k depends on fiber orientation and porosity had been observed and discussed by Sullivan and Hertel (Sullivan, 1941; Sullivan and Hertel, 1940) based on t’heir experimental work. Thus Equation 1 was inadequate for pressure drop correlations. Various workers using the channel model have elaborated upon Equat,ion 1 with modification for shape and orient,ation (Davies, 1952; Fowler and Hertel, 1940; Langmuir, 1942; Sullivan and Hertel, 1941). Most workers (Chen, 1955; Iberall, 1950; Wong et al., 1956) using t’he drag model rejected the applicability of the channel model because of the high porosity of a fibrous bed and derived a pressure drop equation by considering the drag forces due to fluid flow on the fibers. Wong et al. (1956) employed an effective drag coefficient, CD,, to account for the fiber orientation, interference of neighboring fibers, fiber ends, and nonuniformit’y of fiber distribution in the bed. They concluded that the fiber volume fraction, y, has a marked effect on Cn,. The higher the value of y, the higher is t’he neighboring fiber interference which leads to a higher C D , . They also not’iced the leveling off of the effective drag coefficient-Reynolds number plot a t Reynolds numbers greater than 6. Gunn and Aitken (1961) in their study of the mechanism
of the flow of air and water through packed glass fibers found that preswre drop data depended 011 the hi-tory of previous gas and liquid flow rates throiigh the bed. The effect of interference by neighlioriiig fibers ha.; been recentlj- explored Spielnian aiid Goren (1968). They considered :i hotly damping force to be proportional to the local velocity. S o satisfactory general pressure drop (or friction factor) correlation which takes into account the nature of the fibers and the wide range of poroqity for the flow of a single-phase fluid through f i b r o u he& has beeii olitaiiietl hitherto. Development of the Model
The most peculiar thing in the case of the flow of fluid through R fibrous bed is the fact that it gives an unexpectedly high pressure drop in spite of the high porosity of the bed. The causes of this high pressure drop are postulated as follows:
Front view of model
Figure 1 .
STAGNANT REGION
\-
{FLOW REGION
Only a fraction of the free space as calculated from the bulk density of the hed is available for fluid flow, the rest being occupied by stagnant fluid. Some energy is absorbed by deflection of individual fibers, causing an additional pressure drop other than those of a fluid dynamic nature. I n accordance with the foregoing postulatioiis, a model is proposed. Models such as “fibers parallel to the direction of flow,” “fibers normal to the direction of flow,” and “gridn-ork” iiiodels are not used because they do not Sire stagnant regions and caiiiiot be convenientlj. arranged to give a “normal high porosity.” The present niodel consists of inclined fibers intercrossed with those that are transverse to the direction of flow locked between them to make up a stable arrangement. The angle of inclination between two adjacent inclined fibers is cy, with spacing between parallel fibers charact’erized by a model spacing number, n. The model is diagrammatically sketched in Figures 1 and 2 . The fluid f l o w only through the n-ider spaces A B C D as shown in Figure 2 , the rest of the space of the eleiiientarunit of the niodel being occupied bj- the noiiflowiiig fluid. The fibers in the elementary unit of Figure 2 are being bent by the drag force of t’he fluid, leiiding to a dissipation of energy due to fiber deflection. Mathematical Relations Derived from Model. RELATION OF POROSITY ASD SP.4CIKG XUNBEE, n. Based on the shaded elementary volume of the bed, as shown in Figure 3, we have Total volume
=
(n Of) ( n D l ) (n D f sin
Total cell fiber volume 2
[ (n (1
n 3 D f 3sin
4 (n D f )
I)
(h D l z -
T
1---
x
n2 sin
Volume of = ( H ) ( X Y )= f l o region ~
CY
(4)
or ?r
=
1‘0 -
6)
sin a
(5)
EFFECTITE POROHTP OF BED, € 8 . Referring to Figures 1 aiid 2 for the elementary unit shown as shaded,
CY
1 :
n D f 3 ( n - 2.5)2 sin
CY
(6)
Since the value of n is usually about 8 and seldom goes below 5 (corresponding to E = 0.75), the error introduced by the above simplification is tolerable. From Equations 3 and 6, we have ee =
n
n D f 8 (n - 2 ) ( n - 3) sin
Of3
(3) fiber volume €=I---= total volume
1
Figure 3. Elementary volume of model
=
= n
x
Plan view of model
CY
=
I) + [
01) - Of2
=
cy)
6 Figure 2.
volume of flow region total volume
- (n - 2.5)2 n2
(7)
Combining Equations 5 and 7 and defining an effective pore number,
X,
=
d
(1 -
e)
x
(8)
s i n a - 2.5
Ind. Eng. Chem. Fundam., Vol. 9,
No. 4, 1970
597
Therefore,
we have €8
=
Ne2
(1 -
sin
CY
-
E)
A
(9)
The stagnant space and the volume occupied by the fibers in the bed are characterized by the number 2.5 in the above relationship. The value of CY in Equations 8 and 9 can be readily obtained b y applying to Equation 9 the limiting condition, Lim e, = 0, C+O yielding CY
=
30.17'
E
30'
(10)
form drag
2 HXY
drag
where the projected area,
A!
=
4n Of2
For a fibrous bed, it can be observed (TVong et al., 1956) that
Combining Equations 19,20, and 21 Kith 6 and 13, we have
EQUIVALENT DIAMETER, D e . Considering the elementary shaded unit in Figures 1 and 2, and defining, De
=
1 ($)form
4 cross-sectional area of flow
The term ( A P ) d e f l e o t i a n represents the possibility that the deflection of a fiber will absorb some energy of flow; it is accounted for as follows: Referring to the elementary shaded unit in Figures 1 and 2, maximum deflection is
wetted perimeter
we have
ACTUAL lrCLOCITY
THROUGH
where F is the drag force acting on the fiber, I f is the length of the fiber in the elementary unit, E is the modulus of elasticity of the fiber, and I is the volume moment of inertia of the fiber.
BED.
Equations 9 and 12 give [j* =
Work done
1.9895 ?r U Xe2(1 - E )
(13)
PRESSURE DROPEQCATION. Total pressure drop = pressure drop due to viscous flow losses pressure drop caused by pressure drop due to deflection of fibers-Le., form drag
=
F
Ymax
cz~2
if3
= ___
El
Work done per unit mass of fluid flowing through the elementary unit is
+
+
AP =
+
APt~ow
A P f o r m drag
+
APdeflaotion
(14)
The term (AP),,,, can be accounted for using the approach of Ergun and Orning (Ergun, 1952; Ergun and Orning, 1949),
dP dL
=
I.(
c*
Substituting Equations 4, 6, 13, 17, 20, and 21 in Equation 24 gives
Therefore, combining Equations 16, 22, and 25, the total pressure drop equation becomes
C17 De
The term on the right-hand side of Equation 15 accounts for the viscous losses only. The kinetic-loss term in the original relationship of Ergun and Orning (Ergun, 1952; Ergun and Orning, 1949) can be ignored for a fibrous bed, since i t is insignificant in the normal operation. Replacing and De in Equation 15, using Equations 11 and 13, yields
r*
FRICTIOS FACTOR EQUATIOS.Denoting N R , = ___ D f cpl I.(U
The term ( A P ) d r a g can be accounted for by a drag equation. Referring to the elementary unit as shown shaded in Figures 1 and 2, Drag force =
Cn
Aj
p
2
(R)drsg
=
(F)
formdrag
=
C D P U*2 AIH 2 (xy)H P
Eng. Chem. Fundam., Vol. 9, No. 4, 1970
(18)
- 4'
Reynolds number
(27)
deflection number and fik =
Work done b y drag force through H per unit mass of fluid is
598 Ind.
(22)
drag
AP Dy - Ne4 (1 L pU2
-
Equation 26 becomes
kinetic friction factor for a fibrous bed (29)
Several points are worth mentioning in connection with Equation 30. The term in parentheses which represents the coefficient
of -!is essentially a "permeability function,'' dependent NRe
upon porosity, E . The deflection number, N d , the ratio of the viscous drag of the fluid to t h e elastic force of the fiber, characterizes the effect of fiber deflection on pressure drop. The term ]Yez (1 - E ) accounts for the effective porosity instead of the apparent porosity, E , in the fibrous bed. Another form of Equation 30 can be obtained after dividing 1 i t throughout by the coefficient of the term -, as follows:
Figure 4.
NRe
Schematic diagram A. B.
C. D.
E.
where
F.
AP .-Dj Ne4 (1
L put f f f =k1
-
k2
-'r NeZ(1
-
G. H. J.
E)'
K. I. M.
E)
N. P. Q. 5.
and
of
equipment
Plexiglas test section Fibrous bed Air taps Three-way cocks Adjustable flanges U-tube manometer Manometer fluid reservoir Manometer fluid trap Iron pipe Rotameters Needle valve Centrifugal pump Recycle fllter 55-gallon storage tank Pressure gage Air seal
F
F
E
E Equation 31 is now simple enough for experimental verification,
D
Experimental Apparatus and Procedures
T h e equipment for the present investigat'ion (Figures 4 and 5) consisted essentially of a 55-gallon stainless steel recycle tank, centrifugal pump driven b y a 3/4-hp motor, two rotameters, a maiiometer, and an adjustable flanged connect'ion t o hold the fibrous bed test' section. The setup also included a needle valve for adjusting flow rate through the smaller rotameter, a manometer fluid reservoir, a recycle filter, and air-seal arrangement in the downst,ream of t h e test section. The fibrous bed test section (Figure 5 ) mas coniposed of a 13;4-inch-i.d. 14-inch-long Plexiglas section, two coarse supporting screens, two supporting aluminum rings, and a 'Ys-inch-thick fibrous pad. The functions of the pad were to damp out,any sudden change in pressure to avoid sudden cornpact'ing of the fibrous bed, to normalize flow, and to act' as an additional filt'er. Four 1/32-iiichpressure taps !!*ere located 3 inches apart, along the tube. Two l/s-inch air taps were also provided. Preweighed fibers were carefully dispersed in water and the fiber suspension was filtered portion b y portion into the test section under partial vacuum t o make a randomly packed fibrous bed, which was secured in the desired position b y supporting screens and aluminum rings. It was essential t o make a bed sufficiently compact' to stag rigid and st'able during experiment'al runs. The relative posit'ions of t,he fibrous bed and pressure taps are shown in Figures 4 and 5. The t'aps were used to measure the pressure drop through a n actual 3-inch length of bed, excluding the entrance and exit losses. During each run the rotamet'er reading, manometer reading, and temperature of fluid were not,ed. The flou- rate was gradually increased to maximum flow and then gradually decreased t o check the consist'ency of readings. It was important to make certain in each run that the fibrous bed was rigid and stable enough and t h a t there were no air bubbles in the line or manometer tubing. In the present investigation, viscosities of glycerol solutions were measured wit'h a n Oswald-Cannon-Fenske viscometer. Diameters of fibers were determined with the aid of a microscope.
Figure A. 6. C. D.
E.
F. G.
5. Fibrous bed test section
Supporting rings made from aluminum tube Coarse wire screen Fibrous bed Upstream fibrous p o d for flltering, flow normalizing, and damping Plexiglas pipe Airtaps Pressure taps
Experimental data were obtained with glass, Dacron, and nylon fibers of 8- t o 28-micron diameters and with water and aqueous glycerol solution of a viscosity range of 1 to 22 cp. Results and Discussion
Proportionality constants kl, k2, and k S in Equation 30 were evaluated by using two sets of experimental data plotted 1 as f,ic us. on a rectangular plot. Constants kl and k2 ~
NR,
were calculated from the slopes of the best straight line fitted to the experimental data of runs 1 and 5. kl and k 2 were found to be 62.3 and 107.4, respectively. Since the values for modulus of elasticity, E , for the type of fibers used were not available, the values of (ks/Egc) were found from the intercepts of the best fits as 1.74 and 25.4 secz-ft/lb mass for glass and nylon fibers, respectively. The group (k3/Egc)for Dacron fiber was estimated by using one data point in Equation 34 and found to be 29.2 sec2-ft/lb mass. Ind. Eng. Chem. Fundam., Vol. 9, No.
4, 1970 599
f"
01
I
IO
NRe Figure 6. Friction factor fibrous beds Run
1 2 3 4 5
6 7 8
9 10 11
0
0 H
A
A
-+
Fiber
DI
e
Glass Glass Dacron Nylon Nylon
8 8
0.919 0.868 0.829 0.765 0.682 0.892 0.901 0.894 0.896 0.895 0.820
18 20 28
Glass
8
Glass
8
X)Glass -0- Glass
8
0
CD
Gloss Dacron
8 8 13
Re
plot for
randomly packed Fluid
Water Water Water Water Woter
73% glycerol 63y0 glycerol 58% glycerol 5070 glycerol Water Water
w CP 0.87-0.9 0.9-0.945 0.9 0.91-0.96 0.9 2 1 .82 9.96 7.13 4.75 0.85-0.88 0.945
-Equation 34 calculated for indicated run - _ - -Best tit to d a t a
Thus, €or a randomly packed fibrous bed, Equation 30 becomes
(34)
Figure 6, a comparison of the present experimental data and the results predicted from Equation 29, shows our equivalent of a Kozeny-Carman plot. It is not identical, in that our Izl term contains a shape correction for fibers and their orientation. The solid lines represent the results calculated from the equation and the dashed lines represent the best fits to the two chosen sets of data. The proposed equation fits the data well. X pronounced dependence of the friction , porosity is indicated a t a constant value of factor, f f ~ on the Reynolds number. Figure 7 compares the present experimental data with the results predicted from the normalized friction factor Equation 31. This figure utilizes the k z term to give a much more satisfying correlation of our data, thus showing the necessity of the term. The figure clearly indicates that j n is independent of both the porosity and the deflection number a t low Reynolds numbers. However, a t high Reynolds numbers the curves level off, depending upon the magnitude of the deflection losses. These curves begin to deviate from a slope of -1 at different values of Reynolds numbers for different fibers and 600
Ind. Eng. Chem. Fundam., Vol. 9, No. 4,
1970
Figure 7. Normalized friction factor plot for randomly packed fibrous beds Run
2
0
3 4 6
A
Fiber
DJ
e
Glass
8 18 20 8
0.868 0.829 0.765 0.892
7 -0 Glass
8
0.901
Glass Dacron Nylon
Fluid Water Water Water
73%
w
CP
0.9-0.945 0.9 0.91-0.96 21.82
fd
0.0259 0.1060 0.1363 11.72
glycerol
63%
9.96
2.32
7.13
1.238
glycerol
8
Glass
8
0.894
58% alvcerol
9 -0- Glass
4.75 0.540 glycerol 10 0 Glass 8 0.895 W a t e r 0.85-0.88 0.0175 11 Dacron 13 0.820 W a t e r 0.945 0.2150 A = Equation 31 with f d = 0 C = Equation 31 with f d = 0.1363 B = Equation 31 with f d = 0.0175 D = Equation 31 with f d = 0.2150
8
0.896
50%
0
porosities. Moreover, the turbulent losses were estimated to be insignificant over the whole range of the Reynolds number encount,ered in the present experiments (Kyan, 1969). Thus, this behavior could not be attributed to the turbulent losses. Comparison of Proposed Correlation with t h e Literature Data. Pressure drop data for the flow of a single-phase fluid through a fibrous bed are relatively scarce in the literature. Furthermore, the data were reported on short beds where the upstream and downstream disturbances and the entrance scaling effect due to the deposition of suspended foreign particles on the froiit face of the bed could be allpreciable. In spite of these and other shortcomings, attempts were made to compare the proposed Equation 3 1 with the literature da.ta. Figure 8 shows a comparison of Equation 31 with data reported by Sheroiiy (1969)) for the flow of water through glass and nylon beads. The agreement is generally acceptable. The discrepancy between the calculated and measured values is att,ributed to the variations of AP/L with bed length. Figure 9 compares Equation 31 with the data of Spielman (1967) and Gunn and Aitken (1961), for water and air, respectively. The data generally lie above the proposed correlation, This discrepancy between the calculated and experimental values may be attributed to the collapsing of the bed a t these high porosities. If a fibrous bed is packed to a porosi1,y higher than that a t which the bed is both stable
toot'
I
I1
I
I
I 1 I It11
I
I I I l l
f" f"
I
0. I
IO
Figure 8. Comparison of data Equation 31 Run
0 V 0
A
0 A
m
Fiber Glass Glass Glass Nylon Nylon Nylon Nylon
Df
8 8 8 20 20 20 20
of Sherony (1969) with
e
0.895 0.894 0.894 0.851 0.850 0.787 0.850
I, Inches 2.0
Q.5 1 .o 0.5 2.0 1 .o 1 .o
0. I
fd
0.0242 0.0244 0.0244 0.0778 0.078 0.1278 0.078
IO
I
NRe Figure 10. plot
Effect of collapse of bed on friction factor
Porosity reduction of 1.5% in d a t a of Gunn and Aitken (1 Spielman (1967). Symbols and d a t a same as for Figure 9
961) and
and tight, it is liable to collapse under the action of fluid flow. Therefore, the true porosity will be lower than the apparent value. For example, if only a 1.5% reduction in porosity is assumed for data reported by Spielman and Gunn et al., and friction factor values are recalculated, the data lie below the proposed correlation as shown in Figure 10. Most of the other investigators (Brown, 1950; Lord, 1955; Wiggins et aE., 1939) have reported data on the flow of singlephase fluids through fibrous beds in the form of the Kozeny constant, k , as a function of bed porosity. F o r the sake of coniparison of the present correlation with their data, Equations l and 34 are combined and rearranged to yield an expression for the Kozeny constant. The Kozeny-Carman Equation 1 written for a bed of cylindrical particles takes the form
f"
A comparison of Equations 34 with 35 yields
N R. .-e Figure 9 . Comparison of data of Gunn and Aitken, and Spielman with Equation 3 1 Run
Df
e
0
6 3.5 3.5 6 12 9.73
0.942 0.944 0.945 0.946 0.924
m
0 A
A
0.95
1, Inches
0.30 0.13 0.33 0.13 0.23 0.81
Fluid Water Water Water Water Water Air
fd
0.0326 0.0941 0.0933 0,0317 0.0091 0.0029
Source Spielman Spielman Spielman Spielman Spielman Gunn and Aitken
k =
[62.3 N e 2 (1 - e) -k 107.41 e 3 [I 16 Ne6 (1 -
e)4
fd
NRel
g)2
(1 4-
Equation 36 shows that k depends on For a fibrous bed, since L I
+
E,
LVd, ~ V Rand ~,
(36)
D/
-. .tf
>> D f ,Equation 36 simplifies to
+
107.41 e 3 (1 i- fd LVR~) k = [62.3 -Ve2 (1 - e) __ 16Ne6 (1 - e ) 4
(37)
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970 601
Conclusions
-
INVESTIGATORS 0 WIGGINS ET.AL.
(1939)' W LORD BROWN(1955) (1950)
A
A fJ
a 20
LORD (1955) LORD ( 1 9 5 5 )
FIBERS GLASS; COPPER WIRE S GLASS ILK
FLUID ORIENTATION WATER EENZE~ERANDOM
CUPRAMMONIUM AIR VISCOSE RAYYON AIR
1 N
PREDOMINANTLY I r TRANSVERSE N D O M BUT
L
L-
An effective pore model is proposed for flow of a fluid through a randomly packed fibrous bed. Two dimensionless parameters, N d and N e , were obtained as a result of a theoretical development based on the proposed model. N d is a characteristic physical property group which is a measure of the effect of fiber deflection on pressure drop and N e accounts for the effect of stagnant space in a fibrous bed on flow. The effects of these parameters have no parallels in a granular bed. Friction-factor Equations 31 and 34 were developed for the flow of a single-phase fluid through a fibrous bed. The f,k or f n vs. N R e curve correlates the data satisfactorily. The effect of h T d or j d on pressure drop was found to be significant. An expression (Equation 37) for the Kozeny constant, k , was obtained. It shows that k is strongly dependent on Xd, NRe,and B and hence the usual one-term Kozeny-Carman equation is not applicable for flow of single-phase fluids through fibrous beds.
POROSITY, e Figure 1 1.
Kozeny constant Nomenclature
Comparison of data with Equation 37
= projected area of fiber in elementary unit =
ci, i =
1, 2
C D CDe
D De D/ E F
= = = = = = = = =
I
I
4b
I
2
I
=
1
4 DI~ UIO Re p(1-e)
N
Figure 12. number
I
6
=
i
Dependency of Kozeny constant on Reynolds
=
0
0
A
A
rn
0
-.
L
su2
(1 -
€)
normalized friction factor due to deflection as defined b y Equation 33 kinetic friction factor for fibrous bed as defined b y Equation 29 normalized friction factor for randomDacked fibrous bed as defined b v Eauation 32 conversion constant, (lbm/lbf)(in/sec2) height of elementary unit of model, 2nDf sin 1 5 O , in volume moment inertia of fiber = I
Malhotra (1 969). Water a t 68'F Symbols
4 nDf2, in2 numerical constants drag coefficient effective drag coefficient diameter of bed, in equivalent diameter of flow, in diameter of fiber, in or microns modulus of elasticity of fiber, lb/in (sec2) drag- force defined in Equation 23 AP 1 3 friction factor, - __ ___
Fibers
D/ Microns
Glass 8 Glass 8 Nylon 20 Nylon 20 Nylon 20 Nylon 20 Equation 37
E
0.895 0.894
I, Inches
2.0 1.0 1.0
0.850 0.851 0.804
0.5 1.0
0.850
2.0
= =
fd
0.0242 0.0244 0.0780 0.0778 0.1123 0.0780
=
(in) = Kozenv constant
k,, i
Nd
Ne
=
I/
602 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
1, 2, 3
numerical constant length of bed, in = length of fiber, in = length in general, in = length of fiber in elementary unit of the model, nDf, in = model spacing number = deflection number,
=
L L/ 1
Equation 37 is plotted for f d = 0 and for f d ' N R e = 0.1' and compared with the published k values in Figure 11. A good agreement is obtained between the predicted and experimental values for the randomly oriented fibers. Equation 37 also shows that for a constant value of N d and E , k increases linearly with X R ~This . behavior is depicted in Figure 12 and compared with the water data reported by Nalhotra (1969). The general trend appears to be correct. The agreement between the proposed correlations as given by Equations 31 and 34 and the present and literature data appears to be satisfactory. However, i t is very difficult to pack different beds to the same degree of randomness and porosity distribution. Therefore, the pressure drop is somewhat sensitive to variations in bed length. This effect was minimized in the present investigation by using a 3-inch length of bed, so that any nonuniformity was averaged out.
_
n
-L EDPP
N R ~ N R ~ P AP P' I
s
U U* W
= =
effective pore number as defined in Equation 8 = Reynolds number, D f U p/p(l - E ) pS(1 - 4 = Reynolds number defined as PU = pressure, lbf/in2 = pressure drop through bed length L , lbf/in2 = pressure drop through bed length H , lbf/in2 = f t 2 of packing surface per ft3 of packed volume, ft-' = superficial velocity through bed, volumetric flow rate in,sec cross-sectional area of bed' = actual velocity through bed, in/sec = work, (lbf) (in)
X
= =
Y
=
work done per unit mass of fluid, lbr-in/lb, width of flow area of model = (n - 3) D , cos 1 5 O , in length of flow area of model = (n - 2) D,, in I
.
GREEKSYMBOLS a Y c €s
cc P
angle of inclination between two fibers of model = 30.17' = fiber volume fraction = porosity = effective porosity = absolute viscosity of fluid, lb,/in sec or cp = density of fluid, lb,/ina =
literature Cited
Brown, J. C., Tappi 33, 130 (1950). Brownell, L. E., Katz, D. L., Chem. Eng. Progr. 43, 549-54, 601-12 119471. Chen, C. Y . , ?hem. Rev. 55, 595 (1955). Davies, C. N., Proc. Insf. Xech. Eng. (London) B1, 185 (1952). Ergun, S., Chem. Enq. Proor. 48 (21. 89-94 (19521. Ergun, S., Orning, A-A., Ind. Eng. Chem. 41, 1179 (1949). Fowler, J. L., Hertel, K. L., A p p l . Phys. 11, 496 (1940).
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Spielman, L., Goren, L., Environ. Sci. Technol. 2 (4), 279-87 (1968). Sullivan, R. R., J . A p p l . Phys. 12,503-8 (June 1941). Sullivan, R. R., Hertel, K. L., J . A p p l . Phys. 11, 761-5 (December 1940). Wiggins, E'. J., Campbell, W. B., blaass, O., Can. J . Res. 178, 318 (1939). Wong, J. B., Ranz, W. E., Johnstone, H. F., J . A p p l . Phys. 27, 161 (1956). RECEIVED for review September 29, 1969 ACCEPTEDJuly 22, 1970 Work supported through research grants WP 1452-01 and 12050 DRC, Federal Water Pollution Control Administration.
Experimental Observations of Sphere Migration in Couette Systems J. S. Halow and G. B. Wills Department of Chemical Engineering, Virginia Polytechnic Institute, Blacksburg, Va. 24061
Experimental measurements of the radial migration of spheres in Couette systems are reported. This migration carries the sphere to an equilibrium position near the midpoint of the annular region. The observed migrations varied from extremely rapid motions with overshooting of the midpoint to extremely slow migrations requiring many revolutions of the sphere about the Couette axis to reach equilibrium. With a correlation of the experimental data based on theoretical considerations, migration effects in Couette systems can be easily estimated.
R a d i a l migrat'ion of particles in sheared liquids has been a t,opic of considerable interest. X o s t studies of migration phenomena have considered particles suspended in Poiseuille systems. The first of these Poiseuille studies was made b y Segr6 and Silberberg (1961) with a suspension of neutrally buoyant' spheres ill a Sewtonian liquid. Their observations indicated that radial forces acting on rigid spheres carried t'hem to an equilibrium position at about 0.6 of the tube radius from the tube axis. The radially directed force created an annular zone of high particle concent'ration. The most complete study of traject'ories followed by single spheres in a tube system was made by Jeffrey and Pearson (1964). Observations with nonrotating spheres (Oliver, 1962) and spheres whose density differed from that of the liquid (Denson, 1965) have also been made with tube systems. Several other authors have report'ed on various aspect's of this phenomenon (Brenner, 1966; Theodore, 1964). Several authors have developed expressions for the lat'eral force acting on a sphere under viscous flow conditions.
Rubinow and Keller (1961) considered a rotating sphere translating through a stationary n'ewtonian liquid. By applying the method of matched asymptotic expansions, they derived a lateral force proportional to the velocity of translation and the sphere's rate of rotation. Saffinan (1956, 1968) considered a rotating sphere translating through a simply sheared fluid, perpendicular to the free stream velocity gradient. His calculation gave a laterally directed force proportional to the sphere's translational velocity and the free stream velocity gradient. I n Saffman's first-order approximation, the force was found to be independent of the sphere's rotation. Recently, Harper and Chang (1970) developed the complete lift tensor for the problem considered b y Saffman. Their results give the lateral force due to three-dimensional translation. Khile Poiseuille flow systems have received considerable attention, a basically simpler system, the Couette system, has been essentially ignored. The Couette system can give a nearly constant velocity gradient, and because theoretically derived expressions for migration-inducing force are best applied to Ind. Eng. Chem. Fundom., Vol. 9, No. 4, 1970
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