FLUID MECHANICS IN CHEMiCAL ENGINEERING
I
M. PRAUSNITZ' and R. H. WILHELM Princeton University, Princeton,
N. J.
Turbulent Concentration Fluctuations in a Packe Quantitative measurements on scale and intensity of turbulence in packed beds explain theoretical and experimental relationships between concentration fluctuations and turbulence parameters T
1 HE IMPORTANCE of packed beds in chemical technology has encouraged the development of rational design methods for packed bed reactors. These methods require data for turbulent heat and mass transfer rates in the interstices of the packing material. Such data have been reported (7, 2, 4, 6) and the results have been expressed in terms of eddy diffusion coefficients. Further advances in rational design methods require deeper insight into the mechanism of turbulent transport and quantitative information on the turbulence parameters that determine eddy diffusivity. The objective of the present investigation was a detailed view of the process of turbulent mass transfer. I n a turbulent fluid true steady state is never reached, as the velocity at any point is not constant with time but fluctuates about a constant time-average velocity. When a time-average concentration gradient exists in a turbulent fluid, the concentration a t a fixed point within the fluid is not constant with time but fluctuates in a manner determined by the turbulence. This work gives theoretical and experimental relationships between concentration fluctuations and turbulence parameters in a packed bed.
an irregular manner, colliding and displacing each other in space. An observer focusing attention on a fixed point within the turbulent fluid notices that the fluid mass at that point is constantly changing; a t any given instant the aggregates of molecules occupying a fixed small volume in the turbulent fluid are not the same as those which occupy the same volume a moment later. If the molecules observed in the first instant are of a different species than those replacing them at a later instant, a concentration fluctuation occurs. These elementary concepts are applied here to a turbulent fluid flowing in a packed bed. Fluctuation Mechanism in Packed Bed. Consider a fluid flowing upward in a vertical tube filled with small solid particles. At a point on the axis of this tube a tracer material is injected at a linear velocity equal to that of the main fluid. Neglecting wall effects, it has been shown (4) that the concentration of tracer material at a point downstream from and not near the point of injection is given by
where Ca= time-average concentration of tracer material at coordinate portions x , y, and z x , y = distances measured from the
Theory In the present rudimentary state of knowledge about the motion of fluids in packed beds the use of a modification of Prandtl's simple theory of turbulence is considered appropriate in preference to Taylor's statistical theory. According to Prandtl's mixing length theory, the motion of small masses within the turbulent fluid is analogous to the motion of molecules as described in the kinetic theory of gases. According to this concept, fluid masses move about in Present address, Department of Chemical Engineering, University of California,
Berkeley, Calif.
978
P
a
C, Pe
U dp E
axis in plane perpendicular to the mean flow direction = axial distance, measured from point of injection = tube radius = concentration after complete mixing-Le., at z = m = Peclet number = Ud,/E = fluid linear velocity within bed = particle diameter = eddy diffusivity
and where the prime (') is an operator denoting division by dp. Latinen (4) and Bernard and Wil-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
helm ( 7 ) have shown that the area average Peclet number is constant for Reynolds numbers exceeding 200, where the Reynolds number is based on particle diameter and superficial velocity. We fix our attention on a certain horizontal plane within the tube a t some height z where 2 is not very small. It is then convenient to rewrite Equation 1
where , Pe a ' z c, = C___ 4z '
-
= 4t' -
Pe
Equation 2 defines a set of circular time-average concentration contour lines whose centers are at the axis of the tube, As the fluid masses move about from one radial position to another, concentration fluctuations result. The basis of the proposed mechanism is that concentration fluctuations are caused by velocity fluctuations. However, the velocity fluctuations in question here are not the temporarily rapidly fluctuating velocities of conventional turbulence. Rather, they are velocity fluctuations of relatively low frequency, associated with a more or less random flow such as occurs in packed beds. While fluid displacements occur in any turbulent system, they alone are not sufficient to effect concentration fluctuations. It is necessary to have a concentration gradient. For a packed bed in a region not near the point source, the concentration gradient in the radial direction is much larger than that in the axial direction, except in the immediate vicinity of the center. At all points other than those near the center, therefore, the radial displacements of fluid elements are responsible for concentration fluctuations; although axial displacements no doubt occur in the turbulent fluid, they do not in this case con-
tribute significantly to concentration fluctuations. Formulation of Concentration Fluctuation Theory in Two, Dimensions. The instantaneous concentration is designated by C. The instantaneous concentration fluctuation therefore, is given by C - C,. The following assumptions are made :
1. Concentration fluctuations are caused by displacement of fluid elements : C ( X , Y ,t ) = Ca ( X
+ 4)
+P,Y
where p and q are stationary random position variables. 2. Variables p and q are statistically independent. Their joint probability distribution is normal:
P(P, 4 )
= a;
1
+ 92) --(4*
u2
e
where u is a scale of turbulence. Assumption 1 is based on the mechanism of turbulence, which is inherent in the mixing length theory of Prandtl. I t states that at a given instant the element of fluid whose concentration is observed at some fixed point, is, in fact, a fluid element which a moment ago was a t a different point, p units away in the x direction and q units away in they direction. The concentration of this fluid element is assumed to be equal to the time-average concentration associated with the point ( x P,Y Assumption 2 is based on the fact that a packed bed with a point source on its axis is axially symmetric, and that small displacements are more probable than large ones. Assumptions 1 and 2 taken together are, in effect, a definition of a scale of turbulence. The mean-square value and the meanabsolute value of the concentration fluctuation are given by
Equations 5 and 6 are valid only a t points not in the immediate region of the axis. They show that the relative concentration fluctuation, the ratio of the concentration fluctuation to the timeaverage concentration, is proportional to the radial distance from the axis, to the scale of turbulence, and to the Peclet number; and is inversely proportional to the axial distance from the point source. These equations also show that for a given point in a packed bed the relative concentration fluctuation depends only on the scale of turbulence and not on the turbulent intensity. Because the Peclet number is independent of the Reynolds number in the turbulent region, the analysis predicts, therefore, that at a given position in a packed bed of constant particle diameter the amplitude of the relative concentration fluctuation is independent of the Reynolds number. The dimensionless scale of turbulence designated by u' can be considered the length of an average fluid displacement in the plane perpendicular to the direction of flow. Equations 5 and 6 show that when compared a t the same r' and z' the relative concentration fluctuations are independent of the particle size. Equations 3 and 4 can also be integrated for the special case where r = 0. For that case we obtain
IY
Figure A. Mechanism of concentration fluctuations Concentric circles are concentration contour lines Displacement of fluid elements according to mixing length concept Element 1 displaces 2 Element 2 displaces 3 Element 3 displaces 4, etc.
We now expand this in a Taylor series, using cylindrical coordinates with axial symmetry:
+
where r2 = x 2 y 2 as before and 1 is a scale factor. If all derivatives higher than the first are neglected, root-meansquare values taken, and all linear terms made dimensionless by division by d,, Equation 11 becomes
4-3
+ + SI.
=
d+-\ - bC
(12)
p(7).]
The derivations of Equations 7 and 8 are shown in (7). According to Equation 1, the radial time-average concentration gradient vanishes a t r' = 0. Equations 7 and 8 show that although this concentration gradient vanishes a t the center, the concentration fluctuation does not vanish, even if the axial gradient is not
[It can be shown that = Equation 12 states that the concentration fluctuation is proportional to the absolute value of the time-average concentration gradient. This gradient can be found by differentiating Equation 1 with respect to r'
or
W A L L - ~ A $ T ~ % E ORING W I T H S E T SCREWS OVERFLOW P O R T S UNIVERSAL COMPOUND VISE
The integrals in Equations 3 and 4 are unwieldly ; the detailed integrations are shown by Prausnitz (7). However, for the following conditions approximations have been obtained : For r2 = xa
+ ya >>us, z >>d,
and -7 -rau2< 1 2 m4
Equations 3 and 4 now become d(C
- C,)z ca
-
considered. These equations, however, are probably of little use, as the axial contribution to the fluctuations is probably the important one a t the center of the bed. Generalization of Concentration Fluctuation Theory in Two Dimensions. The instantaneous concentration fluctuation, AC, a t point x , y is given by the expression AC = C ( X ,Y )
1.414 u'r' m2
(5)
- Ca(x, Y)
P Y R E X CAPILLARY AClO F E E 0 TUBE
TO FLOWMETER AND CONSTANT-HEAD
T O FLOWMETER AND ACID EGG
(91
On introducing the first assumption, the instantaneous fluctuation becomes AC = C,(X
P. Y 4- 4 ) -
C,(X, Y )
(10)
Figure B.
Elevation of test column
VOL. 49, NO. 6
JUNE 1957
979
Comparing Equation 13 with Equations
5 and 6 and substituting, we have
Equations 14 and 15 are valid for any time-average concentration distribution. dc
arovided that
For the special case of a point source experiment in which the tubular container wall reflections are important it has been shown (7) that the time average concentration is given by:
that their joint probability distribution is normal :
where c is the scale used before. This probability distribution assumes that turbulence in a packed bed is isotropic. On the basis of these assumptions the mean-absolute value of the relative concentration fluctuations for the special case Y = 0 has been computed to a first approximation :
The derivation of Equation 19 is in (7). The root mean square value of the concentration fluctuation was not computed,
where 1c/, a‘ are the roots of Jl(lc/%o’)= 0 Substitution of Equation 16 into 14 and 15 yields
Equations 17 and 18 predict the absence of fluctuations a t the wall where the time-average concentration gradient vanishes. This prediction is probably not correct. The hypothesis that concentration fluctuations are proportional to the concentration gradient is only a first approximation. I t would probably be better to say that the fluctuations depend on the first as well as the higher concentration derivatives. These higher derivatives do not vanish at the wall. The restrictions on Equations 17 and 18 disappear at a distance from the wall when
1 I >> 1. dc
~
dc
;IE
The equations may
then be used in reducing concentration fluctuation data. Extension to Fluctuations in Three Dimensions. The two-dimensional treatment may be extended to include the contribution of the axial displacements to the concentration fluctuations. The same assumptions and procedures are used. The first assumption now is that the concentration fluctuation, AC, is given by AC
=
Ca(* + P , Y
+ P,Z +
S)
- ca(x,~,z)
The second assumption is that the stationary random position variables p , q, and s are statistically independent and
980
but by analogy to Equation 7 we write
The relative concentration fluctuation is proportional to the scale of turbulence in Equations 19 and 20 but proportional to the square of the scale in Equations 7 and 8. If the Peclet number is set equal to about 10 and g ’ is set equal to about 1/5, Equations 7 and 8 give approximately the same results as Equations 1 9 and 20. If the scale is smaller, the relative concentration fluctuations, neglecting the axial component, give results lower than those obtained when the axial component is considered. When the scale is larger, the reverse is true. The predictions of Equations 1 9 and 20 are based on the assumption of isotropy. Recent data (3, 5 ) indicate that this assumption is probably not correct. However, as a rough first approximation, these equations may be used with appropriate fluctuation data to compute the ratio of the axial to radial scale.
ured in a packed tube with an acid tracer introduced into the main water stream. Spherical lead packing was contained in a 2-inch tube. The tracer was a mixture of hydrochloric acid and methanol, adjusted to the density of water. Time-average concentrations and concentration fluctuations were measured directly above the packing by a small, calibrated, movable electrical conductivity cell which essentially consists of two platinized platinum wire ends approximately 1 mm. apart (Figure 1). Suitable electronic apparatus with frequency response u p to about 2000 cps. was developed. This equipment and method for investigating turbulence in liquids have been described ( 8 ) . Experimental Arrangement. Water was pumped in succession from a constant-head tank through a flowmeter, a brass fluid distributor, a calming section, the borosilicate glass pipe test section. and finally to overflow ports in the exhaust section. The tracer hydrochloric acid-methanol mixture was pumped by a borosilicate glass acid egg using nitrogen. The mixture flowed through ball-and-joint connected borosilicate glass tubing into a thick-walled borosilicate glass capillary tube having an internal diameter of 1.4 mm. The capillary tube, which was the tracer source, was tapered and ended at adjustable positions on the axis of the test section. A Lucite pipe exhaust section 2 inches in inside diameter and 18 inches long was joined to the top of the borosilicate glass pipe by flanges. The top of the Lucite pipe was supported laterally by setscrews from a wall-fastened ring. The conductivity probe was held firmly by a brass collar, mounted in a universal compound vise. This vise permitted lateral readings to 0.001 inch. Particles were of lead shot manufactured by the Stoeger Arms Corp. The two sizes were 0.147 i 0.004 and 0.250 i 0.006 inch. Size was measured with
r
BINDING P O S T S
Experimental
T o provide an experimental basis for judging the previously developed theory, concentration fluctuations were meas-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
FRONT VIEW
Figure 1
SIDE V I E W
Conductivity probe
o
FLUID M E C H A N I C S a micrometer on 20 randomly selected particles. Procedure. All time-average and fluctuating acid concentrations were measured by the calibrated conductivity probe. T a p water was used and the very small contribution of the conductivity of this fluid was corrected for in the calibration procedure. After proper vertical alignment of the equipment, experiments were started by setting the acid and water rates. The probe was positioned with the compound vise a t the point showing the largest acid concentration. This point, called the dynamic center, was close to but not always identical with the geometric center. A radial traverse was then made for four quadrants, using the dynamic center as the reference point; the two double traverses were perpendicular to each other. In all the experimental runs the orientation of the probe was such th& the line joining the two electrodes was tangent to a circular concentratibn contour line having its center a t the axis. T o determine whether or not the probe orientation affected the magnitude of the concentration fluctuations, two runs were made with the probe first in its normal position and then rotated 90'. I n making these runs the probe was randomly varied between clockwise and counterclockwise rotation. The data (7) indicate that probe orientation does not significantly affect results. Time-average concentrations, the mixed-average effluent concentration, the mean-absolute value, and the meansquare value of the relative concentration fluctuations were measured. The mixed-average acid conductivity was measured in large effluent samples taken from the system. Terminal Conditions a t Bed Exit. As the analytical conductivity cell was
----- - 1 4 2
20
not placed within the bed of particles but rather a t the upstream terminus of the bed, it became important to discover whether bed properties were in fact being measured by this procedure. I n one set of experiments the relative concentration fluctuations were measured as the probe was situated at different vertical positions above the bed, radial position being maintained constant. Figure 2 which presents the results, leads to the conclusion that the concentration fluctuations are substantially constant for several millimeters above the bed, within which distance the probe normally is placed. On the basis of these results it is presumed that bed properties are measured by such a closely placed probe. The curve in Figure 2 (right) which rises after the initial plateau and is followed finally by a decrease in the relative fluctuations may, perhaps, be explained by the following reasoning. The fluctuations increase rapidly, probably because the scale of turbulence tends increasingly to be determined by the containing tube rather than by the particles. At very large distances above the bed mixing is almost complete; the concentration gradient therefore is diminished and the fluctuations fall. T o reach large Reynolds numbers without fluidizing the bed, a retaining screen is necessary. T o establish whether the presence of a screen a t the bed exit affects concentration fluctuations, fluctuation measurements were obtained directly above a bed of 0.147-inch particles. In one experiment a '/lO-inch stainless steel wire mesh screen was positioned directly a t the top of the bed. I n another experiment the screen was absent. The results (Figure 3) show that the screen dampens fluctuations substantially. Retaining screens, therefore, were not used in the present work.
T o achieve the maximum possible upper velocity limit in an unconfined bed, lead particles were chosen because of .their high density. Results
Time-average concentration and concentration fluctuation measurements were made for two particle sizes, four bed heights, and Reynolds numbers ranging from 216 to 566, Reynolds number being based on particle diameter and superficial velocity. Time-average concentration data were used to calculate values of the Peclet group which appears as a parameter in the previously developed theory of concentration fluctuations. Fluctuation data were restricted to the above limited range of variables.
1. The velocity should be large enough to have reached the fully turbulent region but not so large that fluidization sets in. Previous studies (7, 4) have shown that fully developed turbulence sets in a t a Reynolds number of about 200. 2. If particle size is too small, fluidization may occur; if too large, t4e ratio of tube diameter to particle diameter may become so small that the assembly of particles can no longer be considered homogeneous. I n this work most of the experiments were done with a tube-to-particle diameter ratio of 13.6 and some were done with a ratio of 8. 3. If bed height is too short, the time-average concentration distribution may be severely asymmetric about the axis; if too high, time-average concentration gradient may be too low. When this gradient is small, the concentration fluctuations are also small and difficult to measure. Original experimental data have been recorded (7). Time-Average Concentration Distribution. Values of the Peclet groups
dp = 0.147 inch :13.0 om. r' 3 2.72
Height
Re =-460
dp :0 25 i n c h Height = 16.5 cm r ' = 16 Re = 428
o
10
I5
20
25
DISTANCE ABOVE BED, MILLIMETERS
Figure 2.
DISTANCE ABOVE BED. MILLIMETERS
Variation of concentration fluctuation with distance above bed VOL. 49, NO. 6
JUNE 1957
981
-
12
/ dp
:0.147
H e i q h t = 6.6
/
z
I
I
I
I
/
IC
inch
dp
crn.
J
= 0.147 i n c h cm.
H e i g h t = 18.3
r ' = I 36
c 20
I
R e = 216
Re = 3 3 0
c
8
z W u n w
a
6
#Jo 4
2
Individual D a t a Points A r e Averages O f D a t a Far Four Q u a d r a n t s
I I I I 1 5 10 15 20 25 DISTANCE ABOVE B E D , M I L L I M E T E R S
Figure 3. Variation of concentration fluctuation with distance above bed, with and without retaining screen
Figure 4. Effect of radial distance on relative concentration fluctuation
calculated from the time-average concentration measurements (Table I) are in good agreement with those of Latinen ( 4 ) and Bernard and lVilhelm ( I ) . who used a dye-tracer technique rather than the present acid-tracer method. The Peclet group calculated from the dye tracer studies has an average value of 10.5 for Reynolds numbers larger than about 200, tube-toparticle diameter ratios larger than about 10, and bed heights larger than about 40. The Peclet group value falls slightly as the bed height becomes less than about 30 particle diameters and is somewhat larger when the ratio of tube diameter to particle diameter is below about 10 (Table I). Asymmetry of the time-average concentration distribution with respect to the axis was not serious for most of the experimental runs. When the bed height was a t least 30 particle di-
1.
Peclet Group for Spheres in 2-Inch Tube
Table Bed Height,
Lead
Particle Reynolds Size, Peclet Cm. Number Inch Group 0.147 10.7 18.3 216 0.147 10.2 250 0.147 10.8 332 0.147 10.3 465 0.147 10.6 13.0 216 0.147 10.0 332 0.147 10.7 465 0.147 9.0 10.0 216 0.147 8.8 332 0.147 8.5 6.6" 332 0.147 8.4b 332 0.250 13.5 16.5 566 a Bed tapped for close packing. With '/ls-inch stainless steel retaining screen.
982
0
I
I
I
I
I
I
2
3
4
5
ameters, the concentration maximum was within 1 or 2 mm. of the geometric center. At bed heights in the neighborhood of about 20 particle diameters the asymmetry of the bed was considerable; when the packing was poured into the tube, the displacement of the dynamic center from the geometric center was sometimes as much as 1 cm. However, asymmetry could be much reduced by repeated tapping of the bed with a mallet to effect close packing. This was done for the two runs at bed heights of 6.6 cm. Concentration Fluctuations. For the restrictive conditions of 7 r2 d, and - - u2 < 1, 2 m4 Equations 5 and 6 predict relative r.m.s. and relative mean-absolute value concentration fluctuations directly proportional to the dimensionless radial distance, r'. Typical data (Figure 4) show that this proportionality is achieved experimentally a t all radial points except the center, where the first two restrictive conditions are violated. Equations 5 and 6 predict, furthermore, that the ratio of slopes for r.m.s. and meanabsolute value fluctuation lines should have a value of 1.414/1.128 = 1.254. In a series of 10 such experiments covering a range of particle size, bed height, and Reynolds number conditions, the average experimental value of the ratio of these slopes is 1.248 i 0.031. Slopes of the lines of Figure 4 are according to Equations 5 and 6 proportional to u', the dimensionless scale of turbulence as defined by assumption 2. Values of B' for different experimental conditions are shown in Table 11. Experiments with a particle size of 0.147 inch were performed a t several bed heights; those for the larger size, 0.25
r2
INDUSTRIAL AND ENGINEERING CHEMISTRY
>>
u2, z
>>>
6
RADIAL DISTANCE, PARTICLE DIAMETERS
inch, were at a single bed height. I t may be concluded that the dimensionless scale is substantially constant and independent of particle diameter. Stated in another way, the scale size in the plane perpendicular to the direction of mean flow i s about one quarter of a particle diameter. Furthermore, the dimensionless scale is independent of the Reynolds number and the same numerical value of the scale is obtained by mean-absolute and r.m.s. values of the concentration fluctuations. The system is independent of Reynolds number (Figure 5), there being no systematic variation of the relative fluctuation with Reynolds number a t any radial position.
Table I!.
Scale of
Turbulence
(Dimensionless) Fluctuations Bed Measured by S c e Height, Reynolds Absolute R.ni.s. Cm. Number value value d, = 0.147 inch. d T / d , = 13.6 18.3 216 0.256 0.258 250 0.249 0.246 332 0.262 0.259 465 0.274 0.267 13.0 216 0.252 0.254 465 0.271 0.253 6.6a 332 0.259 0.259 332 0.1051 0.100' Av. (excluding run with screen) 0.258 d, = 0.25 inch.
16.5
dl/d, = 8
0.248 0.247 0.224 Av. 0.237 a Bed tapped for close packing. With 1/1rinch stainless steel retaining screen.
*
371 428 566
0.239 0.248 0.217
FLUID MECHANICS
Figure 5. Wect of radial distance on relative concentration fluctuation, ' h w i n g independence of Reynolds number
Effect of Wall ReRe*ions. The generalized theory of concentration fluctuations, as &pressed in Equations 14 and 15, states that when
I
,
#
,
centration kuchration is proportional to the absolute value of the timeaverage radial concentration gradient. This theoretical prediction is well substantiated by the expuimental results in Figure 6, which shows the variation of the relative r.m.s. concentration fluctuation with radial position, including a close approach to the wall. The timeaverage radial concentration gradient of the tracer must needy vanish a t the wall. This condition, however, doea not assume physical importance until
-_ll___l
Figure 6. Hiect of wall reflections on relative concentration fluduotion
a measurable amount of tracer material reaches the wall. In order to explain Flgure 6, where wall reflections are important, consider first the time-average radial concenuation profiles given in Figure 7. ThaK curves represent the same experimental conditions as t h o r in Figure 6. B (Figure 7) correctly reprexnts the data when diffusion extends in substantial amount to the wall. This curve is exp d by Equation 16. A, on the other hand, neglects wall reflections and is expressed by the simpler Equation I, whic!,serves well for small radial distances. To this point emphasis has been placed dn results that can properly be described in terms of timeaverage fadial concentration profdea e*prwec*
l
,
by Equation 1. In evaluating the results in the present section, however, the timeaverage radial concentration gradient cannot be computed from the derivative of the concentration distribution, Equation 1. Rather the derivative of the more precise Equation 16 must be used. The continuous line in Figure 6 was computed using the generalized thwry and the concentration gradient as computed by differentiation of Equation 16. The dotted line was computed on the basis of Equation 1, which negkc*i the effect of the wall. E x p i mental results show dearly that wncentration fluctuations are propodonal to the absolute value of the concentration gradient. The previous results, prcxntcd in Figrm 4 and 5, c o m p o n d to the experi-
ments on the left lobe of the curve in Figure 6. Generalized Concentration Fluctuations. The generalized theory of concentration fluctuations may be used to correlate fluctuation data under a variety of conditions, except for measurements taken at the tube center or at the tube wall where dC,/dr = 0. Figure 8 presents a composite of experimental results covering different particle sizes, Reynolds numbers, and radial and axial positions. The experimental points shown in Figure 6 are included in Figure 8. A single straight line is obtained, whose slope, as indicated by Equation 14, is proportional to the dimensionless scale of turbulence. The value of the dimensionless scale by this method of calculation is 0.251. Fluctuations a t T u b e Center. The magnitude of concentration fluctuations at the tube center may be estimated by the use of Equations 7 and 8 and 19 and 20 with the experimental value of u’ of Table 11. Equations 7 and 8 consider only radial fluctuations; Equations 19 and 20, however, consider radial and axial fluctuations with the assumption that the turbulence is isotropic. Table I11 compares the experimental concentration fluctuations a t the center with those calculated. The experimental results did not appear to vary significantly with the Reynolds number; values in Table 111, therefore, are the averages of several Reynolds numbers at any particular bed height. In all cases the observed concentration fluctuations are larger than those calculated. The high experimental results suggest that axial fluctuations are not negligible and that turbulence in a packed bed is not isotropic. A possible explanation for the high results is that dispIacements parallel to the mean flow are larger than those perpendicular to the mean flow. The scale u‘ appearing in Equations 19 and 20 i s probably larger than that found in experiments where only radial fluctuations are detectable. The above results may be used in a first-order approximation of the extent of anisotropy in packed beds. The experimental and calculated values shown in Table 11 are used to estimate the ratio of the average displacement in the z direction to that in the x or y
Table 111. Bed Height, Cm. 18.3 13.0 6.6 16.5 a
With
984
Particle
Diameter,
Inch 0.147 0.147 0.147 0.250
direction. Using Equations 19 and 20 with the experimental results, we calculate an over-all experimental scale gexp’. The radial dimensionless scale, u’, shown in Table 11, is the average displacement in the xy plane; we call this radial scale u ‘ ~ . The average displacement along the x (or y) coordinate, ut,> is therefore given by c r ‘ ~ ‘ ~ ‘ 2 . We let the average displacement along the z axis be u’$. We can then estimate the ratio U ‘ J U ‘ , by writing
+ + 52)
2 uzeXp.= 1/3(uZ2
gy2
(21)
Equation 21 can be rewritten to give
If the experimental values of Table u’‘/cr’* is about 7.0. Scale and Intensity of Turbulence in Packed Bed. The theory of eddy diffusion shows that the eddy diffusivity is proportional to the product of the r.m.s. value of the velocity fluctuation and the scale of turbulence (9). Assuming that the proportionality factor is unity, the Peclet group can be written
I1 are used, ratio
p e s -Ud, E -
u d ~ dz I,
or
1 = Il’, Pe
(23)
where u, = velocity fluctuation along a coordinate perpendicular to the mean flow I, = scale of turbulence along a coordinate perpendicular to the mean flow l’, = dimensionless scale of turbulence = L/dn I = turbulent intensity = d z / U The relationship between l J 2 and u’ is not known with certainty. However, Q ’ is the average jump distance in plane xy perpendicular to the mean flow; the component of this average jump distance in either the x or y direction is given by a‘/&. A reasonable assumption, therefore, is that I, = a‘/-&. The concentration fluctuation studies show that the dimensionless scale, u’, is equal to about 0.25. The Peclet group has a value of about 11. If these values are substituted in Equation 22, the turbulent intensity, I, is equal to about 50%. Acknowledgmenf
The authors wish to acknowledge
Concentration Fluctuations a t Center
0.354 0.50 0.835 0.861
Nomenciature = tube radius C = instantaneous concentration C, = time-average concentration C, = concentration, C a Pe a I 2 a
Rel. Concentration Fluctuation, 70 Absolute Value R.M.S. Value Exptl. Eq. 8 O Eq. lQa Exptl. Eq. 7‘ Eq. 20‘ 0.784 1.09 2.54 1.65
with thanks the valuable contribution of Louis Howard, Princeton University, Department of Mathematics, in the evaluation of essential integrals in this work. The assistance of Walter McKer and Daniel Hogan in the construction of apparatus is appreciated. Support of this study through a Celanese Corp. of America Fellowship in Chemical Engineering and a grant from the Shell Development Co. is gratefully acknowledged.
0.211 0.298 0.581 0.366
c’ = 0.25.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1.33
1.70 3.39 2.18
0.501 0.707 1.18 1.22
0.298 0.420 0.825 0.516
4z d, = particle diameter dt = tubediameter E = eddy diffusivity I = turbulent intensity, d/uz”/U J = Bessel function of first kind 1 = distance increment in r coordinate I, = scale of turbulence along coordinate ( x or y ) perpendicular to mean flow m2 = 4z’/Pe p = distance increment in x coordinate Pe = Peclet number, Udp/E q = distance increment in y coordinate Re = Reynolds number, dPUQ r s = distance increment in z coordinatc U = mean point velocity Lro = superficial velocity u, = velocity fluctuation along coordinate ( x or y) perpendicular to mean flow x, y j 2 = Cartesian coordinates u = scale of turbulence $% = defined by JI(+%U’) = 0 7 = kinematic viscosity = division of a length term by d,
= d w
Literature Cited ( I ) Bernard, R. A., Wilhelm, R. H.. Chem. Eng. Progr. 46, 233 (1950). (.2 .) Fahien, R. W., Smith, J. M., A.I.CI1.E Jouriar I, 2s (1955j . (3) Kramers, H., Alberda, G., Chem. Eng, Sci. 2, 173 (1953). (4) Latinen, G. A., Ph.D. thesis, Princeton Universitv. 1951. McHenry, K. W., Wilhelm, R. H., A.I.Ch.E. Journal 3, 83 (1957). Plautz, D. A , , Johnstone, H. F., Ibid., 1,193 (1955). Prausnitz, J. M., Ph.D. thesis, Department of Chemical Engineering, Princeton, 1955; microfilm from Universitv Microfilms, Ann Arbor Mich. (8) Prausnitz, J. M., Wilhelm, R. H., Rev. Sei. Instr. 27. 941 (1956). ( 9 ) Sherwood, T. K:, Pigford,’ R. I,.) “Absorption and Extraction,” Chap. 11, McGraw-Hill, New York, 1952.
RECEIVED for review January 2, 1957 ACCEPTEDApril 15, 1957 Division of Industrial and Engineering Chemistry, ACS, Symposium on Fluid Mechanics in Chemical Engineering, Lafayette, Ind., December 1956.
