I
RUXTON H. VILLET a n d RICHARD H. WILHELM Department of Chemical Engineering, Princeton University, Princeton, N. J.
Knudsen Flow-Diff usion in Porous Pellets This technique for measuring effective diffusion constants is valuable for determining deviations from kinetic behavior in catalytic reactions T H E R E ARE TWO CLASSICAL TECHNIQUES
for the direct measurement of gas transport in porous pellets: the flow of a single gas through a pellet under a total pressure gradient and the diffusion flux of multiple gases under partial pressure gradients. As is well known, these procedures are identical mechanistically for flux in the Knudsen region. Matters of convenience, precision, and sturdiness serve to distinguish between various experimental arrangements. In the first category of techniques, Gilliland, Baddour, and Russel ( 2 ) caused various gases to flow through a porous plug held in neoprene tubing. The steady-state pressure drop corresponded to the permeability of the plug. The very low flow rates were measured by a volumetric displacement technique involving intricate regulation of mercury flow into a gas buret. The pioneering study in the second category was that of Wicke and Kallenbach (9), who arranged for C O Z and nitrogen to diffuse in opposite directions through a tablet of porous carbon cemented into a glass tube. Diffusion constants were computed from gas flow rates and gas compositional analyses as determined by thermal conductivity. Hoogschagen ( 3 ) , by contrast, used a gravimetric technique. Oxygen diffused through a tablet mounted in rubber tubing and reacted with active copper. From change in weight of the latter, the diffusion constant was computed. Weisz (7) modified and improved Sreatly the convenience of the WickeKallenbach technique. A catalyst pellet forced into slightly undersized Tygon tubing was exposed to a measured flow of nitrogen gas on one face and hydrogen gas on the other. Concentration of the hydrogen which diffused through the pellet was measured with a thernial conductivity cell. A partial pressure difference of 1 atm. was maintained across the pellet. From these measurements and with pellet dimensions, the diffusion constant was computed. The procedure described here follows the pressure gradient technique and is characterized by mechanical and operational simplicity. A single gas is caused to flow, by means of a pressure gradient, through a number of pellets inserted into a corresponding number of
parallel-connected plastic tubes, the number in question being selected to give a convenient lapse of time to the experiment as well as a satisfactory precision measure for the diffusion constant. Manometers and constant volume flasks are arranged to permit a simultaneous measurement of both the pressure differential across the pellets and the total volume flow rate of the gas through the pellet assembly. The diffusion constant is computed from these quantities. The system is permitted to progress only a small extent toward pressure equilibrium. Although the technique involves measurement of a time-variant differential pressure across the ends of the pellets, it is shown that the system, effectively, is a t steady state. Thus quasi-stationary conditions exist, permitting the use of Fick’s first, rather than second, law in reducing the data. The technique is applied to new and to regenerated cracking catalyst pellets, and a statistical design of the variables investigated is presented. Agreement with expectations based on theoretical postulates is noted. Furthermore, results compare favorably with those obtained using the modified Wicke-Kallenbach technique on the same population of catalyst pellets.
Theory Knudsen diffusion conditions under a quasi-steady state situation were antic-
ipated, because the mean free path was estimated to be a t least 15 times greater than the pore radius. The rate equation describing transport across a given cross-section of pore of length Az for Knudsen flow is (5):
The assumption made in deducing this formula is that transport proceeds in random kinetic movements punctuated by collisions with and by diffuse reflections from the pore wall. Collisions between molecules are negligible. A “short-tube” modification of this equation by Kennard (4) does exist, and it could properly be employed when a large number of junction points exist. However, it was decided arbitrarily to restrict present work to the “long-tube” formula, since the internal pore geometry was unknown. Differences between formulas could incur a computed difference in the value of the diffusion coefficient of a factor of 4/3. When Equation 1 is modified by the gas law, P = ckT, the following version of Knudsen’s equation results:
A Knudsen diffusion constant D x , is defined thus: dn - = A D x -dC di dz
(3)
Knowledge o f the rates a t which molecules are transported within porous materials i s important. For instance, if the objective in catalytic work is to measure chemical parameters alone, pore diffusivity must be measured and its effect checked to ascertain that i t does not cause a n appreciable deviation from true kinetic behavior. Furthermore, in commercial chemical reactors, the amount o f catalyst required for a given conversion level depends upon the extent to which pore diffusion restricts the reaction rate. Any criterion for this limiting effect must involve an estimate o f diffusion coefficient (7). Investigations o f underground o i l a n d gas reservoirs a n d filtration studies are among a host o f applicafions.
VOL. 53,
NO. 10
OCTOBER 1961
837
0016
i
7
k
I718
I
l
I 716
O0I2 ""In
t
1714
-.
1.712
x
0
I
n-
1.710
u
w
s
170s
1.706
I704 FOR
H P a ~ N P ~ +
o
20
40
60 TIME,
80
120
100
140
160
MINUTES
I702
30
60 TIME,
Typical plots of log pressure function vs. time for hydrogen and nitrogen a t atmospheric pressure (right)
77 and 102
I20
90
I50
MINUTES
cm. of mercury (left) and sub-
Data points showed a linear relationship, and lines were drawn visually
Comparison of Equations 2 and 3 gives:
Equation 3 with the gas law. Integration yields :
:]/2.3ZV, from which DKiscalculated:
DK The constant thus is independent of absolute pressure, directly proportional to pore radius, and inversely related to the square root of molecular weight. Theory of the Experiment. There are two experimental arrangements, as depicted (below). Case A is an arranqe-
CASE A
CASE
B
ment in which constant pressure upstream to the pellets is maintained, and the pressure downstream varies with time in a constart volume, constant temperature vessel. Case B relates to a closed system in which the pellets are contained between two constant volume vessels with different pressures as initial conditions. I n present experiments, Case A was used for superatmospheric conditions and Case B for subatmospheric. CASE A. One or more pellets of length L and total cross-sectional area A are secured in parallel tubes. The upstream gas flow is regulated to be at a constant pressure, Po. O n the downstream side the pressure, Pt in a vessel of volume, V , is brought to a n initial value, Pi. Constant upstream pressure is maintained. The relation between pore diffusivity and the change in pressure, P, with time is based on the differential equation: dt
- AZV DE
( P - Po)
which arises from a combination of
838
Thus a plot of log ( P o - P , ) / ( f ' , - P)us. t is predicted to give a straight line of slope ADK/2.3ZV, and diffusivity is calculated from: DK =
2.3 (slope) ( 2 )( V ) CYA
(6)
where a is a geometric correction factor for spheroidal pellets. CASEB. I n this case the pellets are contained between two volume elements, VI and I/%, of a closed system. Pressure in vessel 1 is higher than in vessel 2. Diffusion through the pellets leads to a rise in pressure PZ and a decrease in P I . (Note that single gases are used; the subscripts 1 and 2 refer to the vessels on either side of the pellet.) The differential equation for this case is :
=
CYA
Steady-State Nature of Experiment. To make a statement about the validity of the steady-state assumption, accumulation within the pellet must be compared with flux across it. If accumulation is exceedingly small in relation to flux, then assumption of a quasistationary state holds good. The procedure (6) was to soive Fick's second law for the boundary conditions of Case B, givingp = f ( t , t ) . From this equation the RT D 3 y and the flux, accumulation, - A D dp F T&, were evaluated.
less ratio,
The dimension(Accumulation)(Pore Volume), Flux
was found to be 0.069. This seems to be negligibly small, justifying the use of Fick's first law. Experimental
System continuity gives:
A combination of these equations leads ultimately to :
where
Solution to Equation 7 is the following: 2.310g[Pi/C
- G/KC]
-Kt
(9)
Thus a plot of log [PI - G / K ] US. t should be a straight line of slope ADK
INDUSTRIAL AND ENGINEERING CHEMISTRY
Silica-alumina cracking catalyst pellets were examined in regard to internal pore diffusivity. A single gas was used for each measurement. Incremental pressure changes in the system bounding the pellets were measured with a mercury manometer and collameter. The pressure changes per unit of time were a measure of the gas flow through the pellets. The system was regulated at constant temperature. A factorial design was drawn up to examine the efTects of pressure, different gases, and different porous material. I t also served to compare results with diffusion measurements on pellets from the same population but with a modified Wicke-Kallenbach technique. Equipment. T o get a reasonable lapse of time for one experiment, four pellets were mounted in parallel for each run. Flasks on either side of the
KNUDSEN FLOW-DIFFUSION pellets were each approximately 1-liter capacity. M I , Mz, and M3 were mercury manometers. T h e system (right) was regulated a t room temperature by means of a water bath. Upstream pressure was kept constant with a ventingtype diaphragm regulator. Each pellet had been forced into l/a-inch-diameter Tygon tubes attached to 1.5-inch-diameter brass barrels by means of 0.25inch Polyflo fittings. Valves a, b, c, d, and e isolated sections of the equipment as required. Materials. T h e porous pellets tested were silica-alumina cracking catalyst, both fresh and regenerated. Both types were 5 to 6 mesh in size. The €allowing properties were measured by Cook (7) using the mercury porosimeter technique : Surface Mean Area, Sq. Pore Pore Meters/ Radius, Fraction Gram A.
Catalyst Regenerated Fresh
0.447 0.464
143 243
49.5 31.4
Fresh and regenerated catalyst were selected because on use in a chemical reactor the pore diameter changes and there is a difference in surface area. Diffusing gases were prepurified nitrogen and technical grade electrolytic hydrogen. Procedure. T h e equipment was purged thoroughly with the gas under test before each trial. Incremental changes in mercury level were measured to 0.005 cm. on a collameter a t intervals of 15, 20, or 30 minutes, depending upon the diffusion rate. Runs lasted from 1 to 2 hours. Net change in pressure differential across the pellets was about 2 cm. of mercury. Pellet diameter was measured using a microscope; the pellet was illuminated through the transparent tubing. Reduction a n d Aaalysis of Data. Effective diffusion constants were estimated from the data in the following manner: CASE A: PRESSURES ABOVE ATMOSPHERIC. As indicated by Equation 5, plots of log (Po-PJ/(Po-P) us. t gave straight lines. T h e data points were so
d
3 ri==i
II
it
TO __o
ATMOSPHERE
T O VACUUM __o
PUMP
HZ OR N
----+
ENTERIN
PRESSURE REGULATOR
POROUS P E L L E T S IN TYGON T U B E S
Two experimenlal arrangements were used, a one-flask system (Case A) and a two-flask system (Case B) close to a linear relationship that it was unnecessary to run regression analyses. Using visual judgment, the graphs were drawn. A typical plot is shown (p. 838, left). Slopes were substituted into Equation 6. Since pellets were spheroidal in shape, a geometric correction factor of 0.78 was used. This value was obtained by Weisz (7) after comparing diffusion measurement results for cylinders and spheres. CASE B: PRESSURES BELOW ATMOSPHERIC. Working from Equation 9, plots were made of log (Pl-G/K) us. t, and straight line graphs were drawn visually. Typical plots are shown (p. 838, right). Equation 10 was used to estimate diffusivities. I t was planned to answer a number of questions with as little experimentation as possible. T h e information desired was : 0 Effect of different gases on diffusivity 0 Effect of fresh and regenerated catalyst on diffusivity 0 Effect of pressure on diffusivity 0 Comparison of diffusion constants between alternate techniques (present and modified Wicke-Kallenbach methods).
Accordingly, an experimental design was prepared to yield the information sought. T h e table is a precise description
of this design: A 2 X 3 X 3 experiment, nested with respect to samples within catalyst but split-plot with regard to gases and pressures within catalyst. Data were taken in random order to eliminate extraneous effects.
Results Major variables investigated were: Variable
Range
Catalyst Gas Pressure upstream of pellets Pressure downstream of pellets Temperature
Fresh, regenerated
Hz,Nz
52-102 cm. Hg
absolute 1-4 cm. Hg absolute 25' j z '1 C.
Diffusion constants are presented in the table. Opportunity was afforded for comparison between experimental methods through the cooperation of the Socony Mobil Research Laboratory. There, internal pore diffusion coefficients for hydrogen were determined by the modified Wicke-Kallenbach technique on 30 pellets each of the fresh and regenerated catalyst. Results were 0.0070 + 0.00087 and 0.0100 f 0.0018 sq. cm. per second for fresh and regenerated catalyst samples, respectively.
Diffusion Constants Were Determined Using This Statistical Design
cz
c1
H
A Pi
Pz Ps
0.0079 0.0076 0.0073
B 0.0081 0.0078 0.0071
N B
H E
C A C D F 0.0023 0.0097 0.0109 0.0104 0.0024 0.0027 0.0081 0.0026 0.0097 0.0108 0.0107 0.0024 0.0030 0.0080 0.0022 0.0023 0.0095 0.0100 0.0102 0.0079 0.0022 CI = Fresh catalyst PI = 102 cm. Hg ahsolute CZ = Regenerated catalyst PZ = 77 cm. Hg absolute H = Hydrogen P 8 = 52 cm. Hg absolute N = Nitrogen -4, B, C; D, E, F are sets of four parallel pellets.
D 0.0039 0.0038 0,0026
VOL. 53, NO. 10
N E 0.0030 0.0030
0.0028
F 0.0033 0.0031 0.0030
OCTOBER 1961
839
Statistical Treatment of Data. An analysis of variances was performed. Effects of the variables investigated and interactions among these variables were estimated. From the statistical information, statements were made regarding the significance of the results [“t-test” ( 7 ) 1. Detailed calculations are presented elsewhere (6)
Effect O f Pressure. STATISTICAL EVIDENCE.Consider fresh catalyst diffusivities recorded a t pressure P I = 102 cm. of mercury and P2 = 77 cm. of mercury. From a “t-distribution” table, for degrees of freedom = 2 (since n’ = 31, t’ = 0.2 a t P’ = 0.86. This means that there is a n 867; probability of getting the value t‘ = 0.2 and a strong probability of getting values 0.00803 or 0.00780 by chance for the diffusion coefficient a t the pressures Discussion considered. I n other words, diffusivities Effect of Different Gases. STATIS- cannot bc distinguished at these two pressures. TICAL EVIDENCE.Theoretical diffusivity Now consider regenerated catalyst ratio for hydrogen and nitrogen is 3.74. diffusivities observed a t pressures PI = .4verage values of diffusivity Z!Z stand102 cm. of mercury and Pa = 52 cm. of ard deviation, recorded in this rvork, mercury. P’ = 0.36 a t t’ = 1.18 for do not include the value 3.74. Hence, 2 degrees of freedom. Thus There is a the difference appears significant, since good chance (36%) of getting either it is not due to experimental error but 0.01033 or 0.00990 when measuring possibly to some physical phenomenon. diffusivity at pressures PI and Ps. Hence, PHYSICALINFERENCE. Since the difdiffusion constants are again indisfusivity ratio is lower than that required tinguishable. by theory, it would appear that nitrogen PHYSICAL INFERGZICE. T h e independis diffusing faster than if the transport ence on pressure of diffusion coeffimechanism were merely diffuse Knudsencients obtained within the pressure range type reflection from pore walls. This examined ( 3 2 to 102 cm. of mercury) could arise if nitrogen forms an adis in accord with Knudsen theory. sorbed layer on the surface of the pore Comparison of Alternate Techwall. Thus two-dimensional surface EVIDENCE.Variniques. STATISTICAL diffusion may be operative. With a ance per four pellets and per pellet greater surface area the effect would be tested in the present n-ork is smaller more pronounced. Fresh catalyst has a than variance per individual pellet surface area of 243 square meters per tested by the modified Wicke-Kallenbach gram and gives a diffusivity ratio of technique. 3.154, whereas regenerated catalyst Consider both methods of measurehas a n area of 143 square meters per ment applied to fresh catalyst. With gram and its diffusivity ratio is 3.221 38 degrees of freedom, P‘ < 0.005 a t (slightly nearer the theoretical). t’ = 3.98. Thus, the probability of Effect of Fresh and Regenerated Catalyst. STATISTICALEVIDENCE. getting values 0.00776 or 0.00700 by chance for diffusion coefficient when Knudsen theory requires that diffusivity using alternate methods is less than ratio, fresh to regenerated cata!yst, be as 0.57,, which is very slender. I t is the ratio of pore diameters (ratio = reasonable to conclude, therefore, that 0.634). This value is not included in alternate methods yield results which are the range given by measured average significantly different when fresh catalyst diffusivity ratio i standard deviation. is under test. Thus the difference appears to be due For regenerated catalyst, however, to a factor other than experimental there is a 55% probability of getting error. either 0.01021 or 0.01000 by chance for PHYSICALINFERENCE. To explain diffusion coefficients. This means that this disparity, a hypothesis of catalyst regenerated catalyst diffusivities obaging is suggested. Twenty nine months tained from alternate experimental methelapsed since the pore radius measureods are essentially indistinguishabie. ments were made on the particular catPHYSICALINFERENCE. As run, this alyst samples. Over several years, work has a favorable result. If both silica-alumina could well undergo strucmethods used individual pellets, the tural changes through change in water present method would have had at least content. This would change the efcomparable precision, if not better. fective pore radius and, consequently, the diffusion coefficient would be difCatalyst aging is again cited as an ferent from that anticipated. Since explanation. There was a lapse of regenerated catalyst has undergone a several years betlveen application of the vigorous thermal treatment in a cattwo comparative techniques to the same alytic regenerator, this would probably samples. As mentioned before, greater confer a greater stability to change with stability is expected with the heattreated regenerated catalyst. T h e extime. I t is assumed, in proferring this cellent agreement between regenerated explanation, that the geometrical concatalyst results and the apparent disfigurations inside a catalyst pellet reagreement using fresh catalyst seems to main essentially the same on regenerabear out this hypothesis. tion-that is, tortuosity does not change.
840
INDUSTRIAL AND ENGlNEERlPlG CHEMISTRY
Acknowledgment
The authors thank J. C. Whitwell for his assistance in the statistical analysis, P. B. Weisz and the Socony Mobil Oil Research Laboratory for supplying the materials tested and the comparative data, Donald Jost for a helpful review of the work, and Walter McKee for advice about mechanical problems.
Nomenclature
A
maximum cross-sectional area, sq. cm. C = concentration, molecules or moles per cc. df = degrees of freedom D = effective diffusion constant, sq cm./second G = constant = ADicRTnJZVlVZ k = Boltzmann constant K = subscript, referring to Knudsen region m = molecule weight, grams n = number of molecules or moles n‘ = number of trials n, = total number of moles N ~ , I ,= ? number of determinations p = partial pressure, cm. H g absolute P = pressure, cm. H g absolute P’ = probability r = pellet radius, cm. R = gas constant S = sum of squares t = time, seconds t’ = statistical t-distribution factor T = absolute temperature V = volume of flask, cc. X = response 8 = mean value of responses 2 = distance, cm. Z = maximum length of peIIet in direction of flow, cm. a = geometric correction factor foi spheroidal pellets =
References (1) Cook; P. A. C . , Ph.D. thesis, Princeton Univ.. 1959. (2) Gilliland, E. R., Baddour, R. F. Russel, J. L., A.I.Ch.E. Journal 4, 90-6 (1 5 X). \ - - - -,
(3) Hoogschagen, J., IND. ENC. CHRM. 47, 906 (1955). (4) Kennard, E. H., “Kinetic Theory of Gases,” McGraw-Hill, New York, 1938. (5) ICnudsen, M., Ann. Physik 28,75 (1909) : 35. 389 (1911). (6) Villet,‘ K. H., M.S. thesis, Princeton Univ.. 1958. Wheeler, A , , “Advances in Catalysis,” 111, Academic Press, New Y c (9) Wicke, E., Kallenbach, R., Kolloid-Z. 97, 135 (1941). RECEIVED for review September 30, 1960 ACCEPTED March 31, 1961 Symposium on Mechanism of Interface Reactions, Division of Industrial and Engineering Chemistry, Baltimore, Md.. December 1959. Work supported by Consolidation Coal Co. fellowship to R. H , Villet.
