DIFFUSION IN PACKED BEDS A T LOW FLOW RATES G E O R G E R O E M E R , J . S. D R A N O F F , A N D J . M. S M I T H Northwestern University, Evanston, Ill.
Diffusion rates were measured in a 2-inch I.D. column, packed with l / g - to '/Z-inch spherical particles and through which nitrogen flowed. Carbon dioxide was introduced into the center of the bed and its concentration measured 4.75 inches downstream. Measurements were made in the low range of modified Reynolds numbers, from 3 to 80,where data are scarce. Modified Peclet numbers were computed from the data by using a finite-source solution to the equation expressing the concentration as a function of position. For the range of conditions studied the simpler point source solution would give results about 10% low. The importance of axial diffusion was analyzed by computing the effect of the magnitude of the axial diffusivity on the radial diffusivity. For the low Reynolds numbers employed in the present study the final results were obtained b y assuming that the axial and radial diffusivities were equal.
mass transfer rates in packed beds have been studied a t high flow rates by Bernard and Wilhelm (7), Plautz and Johnstone ( 5 ) ,and Fahien and Smith ( 4 ) . However, there is little information in the region of Reynolds numbers less than 100. Because of the persistence of molecular diffusion as the flow rate approaches zero, the effective radial diffusivity will not become zero a t zero Reynolds number. Dorweiler and Fahien (2) investigated this region for a bed of '/a-inch spheres in a 4-inch column. T h e purpose of the present work is to study mass transfer rates for several particle sizes a t Reynolds numbers below 100 and to determine the importance of longitudinal diffusion upon the computed values of the radial diffusivitv. ADIAL
Data were obtained in a 2-inch I.D. glass column packed mith spherical Plexiglas spheres l, 8 to l ' 2 inch in diameter. Diffusivities Mere calculated from steady-state measurements of the concentration of carbon dioxide, which was introduced as a tracer gas at the entrance to the bed. The flow rate of nitrogen through the bed \vas varied from 14 to 85 lb./(hr.) (sq. ft.) corresponding to a modified Reynolds number range of 3.4 to 82.
Table 1. Nominal Particle Diameter, Inch
Properties of Beds Actual Diameter and Variations, Inch
Void Fraction
0.115 i 0 . 0 0 2 0.233 3 ~ 0 . 0 0 3 0.364i0.003 0.494i0.004
0.39 0.41 0.43 0.45
withdrawn from the top of the bed was determined in a GowMac Model JOC-015 thermal conductivity cell, using nitrogen as a reference gas. The cell was calibrated by preparing mixtures of carbon dioxide and nitrogen of known composition. The average concentration was known from the measured flow rates (meters F1 and F 2 in Figure 1) of the two gases. The beds were packed by adding the particles through a funnel to the column. The void fractions were measured by determining the particle densities and bulk densities as packed in the bed. These results along with the size range of the particles are given in Table I.
Apparatus
vs"-
The carbon dioxide was introduced into the center of the column 12 inches above the base of the packing (Figure 1) through a '/d-inch I.D. brass tube. The gas stream was sampled, at the center, through a '/lG-inch I.D. copper tube which was held in place with a steel ring. The sampling tube opening was placed about 3/8 inch above the top of the packed bed. This distance has been found by Fahien ( 3 )to be the optimum for this type of measurement. Preliminary experiments were made over a range of depths of packing between injection tube and top of bed. At high depths (13 inches) the radial concentration profile becomes almost flat, making accurate evaluation of the diffusivity difficult. Low bed depths ( l l / z inches) gave erratic results. The final data were taken with a bed depth of 43/4 inches. The operating conditions were approximately 1 atm. and 70" F. The concentration of carbon dioxide in the sample 1
284
Present address, University of California, Davis, Calif. I&EC FUNDAMENTALS
"I I
V3
F3 m
m
V4
Figure 1.
Schematic flow diagram
F 4
Results Table II. Particle Site, Inch
No difference was observed in the concentration measurements when runs a t the same operating conditions were repeated without repacking the bed. Definite differences were observed when the bed was repacked, and the variations increased with the particle size. T o eliminate these random changes as far as practical, three to five runs were made a t the same conditions with repacking of the bed after each run. The average results for each packing size and flow rate are shown as C / C A values in Tablc 11. Also included is the average of the percentage differences of the indivldual concentration measurements from these C / C A values. Finite Source and Paint Solutions. If it is supposed that a differential equation can be used to describe the concentrationposition relationship in a packed bed, the steady-state equation is
1/8
1/4
3/8
1/2
I n this expression angular symmetry in the cylindrical tube is assumed and radial velocity components are neglected. For a finite source of tracer gas of concentration C,, the boundary conditions are:
C(r, 0 ) = C F ; 0 C(r, O:I = 0; R
C(r, m )
=
< r < aR > r :> aR
J1 (A,) = 0. This equation may be solved for @ by applying it a t 5 = L / R and e = 0, the point where the concentration ratio, C/C, was measured. I t is interesting to compare Equation 7 with the solution used by Bernard and Wilhelm (7), which supposes that the tracer gas is fed into the main stream at a point on the axis of the tube. Their solution is
(4) (5)
CA
Concentration Data and Calculated Peclet Numbers Modz3ed ModiJed c/cA Reynolds Peclet Xumber, Pe Numbtr, Re Average Dzff., 7 0 3.56 3.4 1.56 5.9 2.49 3.5 5.94 7.8 7.42 11.9 3.06 2.5 8.94 16.2 3.65 8.1 9.25 20.5 3.78 6.3 6.90 6.8 1.52 2.6 9.57 2.04 2.8 15.5 10.1 2.11 5.7 23.9 11.1 32.4 2.32 9.3 12.4 41 . O 2.60 13 10.5 10.2 1.49 5.2 10.8 23.3 1.56 5.5 1.67 11 12.0 35.8 12.4 1.76 13 48.6 11.2 1.43 22 61.4 11 .o 13.6 1.26 7.4 13.6 31.1 1.43 11 14.0 47.8 1.49 10 18.0 1.92 21 64.8 16.5 81.9 1.71 29
(6)
where a is the radius of the injection tube as a fraction of the tube radius, R. The solution of Equations 1 to 6 when the axial diffusivity, D,, is taken equal to DTis : in which k , are the eigenvalues and 9 is a different Peclet number equal to uR/2D,. To illustrate the deviation between the two solutions, modified Peclet numbers, d,u/D,, obtained from the data for the l/8- and '/l-inch particles, are plotted in Figure 2. The curves suggest that the simplified, point-source equation yields Peclet numbers about 10% too low.
g=1+
This expression is in dimensionless form, where @ is a Peclet number equal to uR/LI,. The eigenvalues, A,, are roots of
II
7
"
a
n
5
6
IO
I4
18
22 Re
Figure 2.
26
30
34
38
dbG/M
Comparison of point and finite source solution VOL. 1
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LU
18
I
I
I
1
I
16 14 12
10 8 6 4
"
2
0
10
20
30
40
50
60
REYNOLDS NUMBER, dp Figure 3.
The solution for this equation with the same boundary conditions (Equations 2 to 6) is:
By applying this expression a t the exit and on the axis of the tube-that is, ( = L / R and 6 = 0, C / C A values can be evaluated in terms of the Peclet number, 4. This was done for n = 0, 1, and 5 and three curves were prepared for C / C A us. Peclet number, one for each value of n. These curves and the experimental values of C/C, were then used to obtain the Peclet number. Calculations were carried out for the 1/8-inch particles a t varying Reynolds numbers to evaluate the influence of flow rate and for different-size particles a t constant flow rate to determine the influence of d p . The results given in Table I11 show that the effect of accounting for axial diffusion is to increase the value of the radial diffusivity derived from the data and to reduce the radial Peclet number. Under all conditions the decrease in Pe is about 1OY6 in going from n = 0 to n = 1. Further increases in the value of the axial diffusivity have little effect on the radial Peclet number. I t appears then that a t low Reynolds numbers an error of I&EC FUNDAMENTALS
3/8"
A
1/21'
70
80
G/r
Peclet numbers in packed beds
Effect of Axial Diffusion on Radial Diffusivity. Studies have shown axial diffusivities to be several times greater than the corresponding radial values a t moderate and high Reynolds numbers. As the flow rate decreases, the molecular contribution to the diffusivity becomes larger, and a t very low flow rates the effective diffusivity, D , should be equal to the molecular value, when corrected for the void fraction of the bed. At these conditions it would be expected that D,/D, = 1. O n the other hand, Dorweiler and Fahien (2) neglected axial diffusion in treating their data a t low Reynolds numbers. Since the ratio D J D , will be greater than, or equal to unity, the justification for neglecting axial diffusion rests upon the magnitude of the axial concentration gradient. T o investigate this question more thoroughly Equation 1 was rewritten in terms of n, the ratio D,/D,, as follows:
286
A
10% would be made in the experimental results by neglecting axial diffusion. Final Results. In view of the figures given in Table 111, it was decided to compute the final values of D, and the modified Peclet number by taking n = 1-that is, using Equation 7. The results are given in the last column of Table I1 and in Figure 3. The decreasing curves a t low Reynolds numbers are due to the effect of streamline flow. Here the molecular diffusivity is the major contribution to D . This is independent of velocity, so that the modified Peclet number decreases in direct proportion to the Reynolds number. At higher Rcynolds numbers the turbulent contribution becomes important and this is a function of the fluid velocity. At still higher Reynolds numbers beyond the range of this study, the Peckt number becomes essentially constant, The maxima in the data points for the two largest particle sizes may be suspect, since these results were the least reproducible (see Table 11). O n the other hand Dorweiler and Fahien (2) observed the same flat maximum at a Reynolds number of about 40 with '/r-inch spherical particles. At low flow rates the effective diffusivity increases with particle size, as shown in Figure 3. Extracting D , from the modified Peclet number in Table I1 and plotting the results us. particle size give an approximately linear relationship at R e = 10 and R e = 20. T o reconcile the effect of particle size Fahien and Smith (4)suggested the empirical ordinate shown in Figure 4. The results for lI8-,l / d - , and 3/8-inch
Effect of Axial Diffusion on Radial Diffusivities Reynolds No., Re 3.4 7.8 12 16 20 10
10 10
n = O 3.9 6.7 8.2 9.8 10.1 7.8
9.0 10.8
Peclet X o . n = l 3.6 6.0 7.4 8.9 9.2 7.0 8.2 10.0
n = 5 3.3 5.8 7.2 8.6 9 .O 6.8 7.9 9.8
--I - - - T I 11----j7--r , I BERNARD 8 WILHELM I
I
DORWEILER -FAHIEN
o
2
Figure 4. number
4
8
15 30 R e = d,Clp
Nomenclature - ratio of radius of injector tube to radius of packed
a
D, D, d, d,
column
= concentration = = = = = =
G = JQ,J I = k, =
L n
=
Pe
=
-
120
200
Correlation of modified Peclet number with d,/d, and modified Reynolds
particles are presented on this plot as well as the results of previous studies. Thr: data reported in this investigation agree reasonably well with the curves of Bernard and Wilhelm and Fahien and Smith in the overlapping range of flow rates. The results of Dorweiler and Fahien are significantly higher. This deviation would he somewhat less if axial diffusion had been considered in the :Dorweiler and Fahien analysis.
c c, c,
60
OTHER AUTH ORS I/ 8" SPHERE 'HERE S
of carbon dioxide a t a point in the bed, lb. moles/(cu. ft.) average concentration determined from measured flow rates of nitrogen and carbon dioxide concentration of carbon dioxide in injector tube effective diffusivity in radial direction, sq. ft./sec. effective diffusivity in axial direction, sq. ft./sec. diameter of spherical packing, ft. diameter of packed column, ft. mass flow rate, lb./(hr.)(sq. ft.) Bessel functions of first kind, zero and first order eigenvalues in point source solution, Equation 8 length of pack.ed bed, 4.75 inches ratio of axial and radial effective diffusivities modified Peclet number, d,u/D,
r
R Re U f
= = = = =
radial distance from axis of tube, ft. radius of packed column, ft. modified Reynolds number, d,G/p average superficial velocity in packed column, ft./hr. axial distance from injection tube, measured in direction of flow, ft.
GREEKLETTERS = eigenvalues in finite source solution, Equation 7 Xi @ = a form of Peclet number, vR/2D, $ = a form of Peclet number, vR/D, = dimensionless radial distance, r / R 0 = dimensionless axial distance, z / R E ,u = viscosity, lb./(hr.)(ft.) literature Cited (1) Bernard, R. A., Wilhelm, R. H., Chem. Eng. Pro,or. 46, 233 (1950). ( 2 ) Dorweiler, U. P., Fahien, R. W.. AZChE Journal 5 , 139 (1959). (3) Fahien, R. W., Ph.D. thesis, Purdue University, Lafayette. Ind., January 1954. (4) Fahien, R. W., Smith, J. M., AIChE Journal 1, 28 (1955). (5) Plautz, D. A , , Johnstone, H. F., Ibid., 1, 193 (1955).
RECEIVED for review October 23, 1961 ACCEPTEDJune 26. 1962 Financial assistance provided by the Technological Institute. Northwestern University.
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