R. F. DYE1 and J. M. DALLAVALLE2 Georgia Institute of Technology, Atlanta, Ga.
Diffusion of Gases in Porous Media The diffusional characteristics of porous media are of practical as well as theoretical interest. Information and data on diffusion rates in porous materials are important, particularly in catalysis and adsorptidn, where over-all kinetics or adsomtion rates may be entirely controlled by the diffusional characteristics of the system
IN
A study of gas diffusion in porous media samples possessing a number of porosities, experimental measurements were made a t each porosity with samples of different thicknesses to check uniformity of packing, and the quantities necessary for evaluating the exact solution of the flow pattern equation were carefully measured. The diffusion process considered here can be represented by a partial differential equation (Fick's second law of diffusion) :
I t is not immediately apparent whether diffusive flow in a porous medium can adjust itself sufficiently rapidly to changing concentration, so as to resemble time-independent flow a t all times. Therefore, an exact mathematical treatment for unsteady conditions is necessary for rigorous results. Unsteady flow of gases through porous media is directly analogous to the unsteady conduction of heat,, which has been investigated a t great length ( 3 ) . By using proper boundary conditions and interfacial concentration proportionalities corresponding to experimental details, relationships readily handled by techniques common to heat conduction problems are developed and diffusion coefficients are quickly determined for media specimens of known characteristics. The analysis presented here is free from some of the restrictive conditions which characterize earlier work (2, 6); in addition to its theoretical interest, it commands attention because of the very general employment of the heat conduction solution.
Equipment and Experimental Procedure Experimental observations were made for unsteady - state, one - dimensional, countercurrent gas diffusion in porous media samples using carbon dioxide and nitrogen gases. Diffusion runs were made for porous specimens compressed from powdered potassium perchlorate. This medium was chosen Present address, Process Development Division, Phillips Petroleum Go., Bartlesville, Okla. Deceased.
because it possesses a wide compressibility range, and porosity (used synonymously with void volume) can be varied with negligible effect on particle size and shape. The material used had mean particle and mass diameters of 20 and 60 microns, respectively. The particles were essentially impermeable to the gases in question. Test specimens were cylindrical plugs l1/2 inches in diameter, in lengths of 1, 2, 4, and 8 inches, pressed dry from the perchlorate powde? into cylindrical holders. Desired porosities were achieved by pressing the calculated weight of material (based on the absolute density) into the holders. Test specimens had porosity values ranging from 0.20 to 0.40. Media samples with values larger than 0.40 were fragile and difficult to handle and those with values smaller than 0.20 required excessively high pressures in preparation. Nitrogen and carbon dioxide gases were of high purity, purchased in steel cylinders from comniercial supply houses, Equipment. Figure 1 shows a schematic diagram of the diffusion equipment. Pure gases were introduced to the system from cylinders at A and B and on entering the system passed through an electrically heated preheater which consisted of two 20-foot coils of 1/4-inch copper tubing, mounted with a doublepass arrangement in an insulated tank of water. Preheater constant temperature control was obtained by means of a thermoregulator. The preheater was placed in the system just before the constant temperature bath (containing the diffusion cell), in which the gases were heated to approximately 2' F. above the bath temperature. Fifteen-foot coils of l/4-inch copper tubing, placed in the lines between the preheater and the diffusion cell proper within the bath, brought the gas temperature down to that of the bath. Construction details of a diffusion cell half are shown in Figure 2. As used in the experimental work, two cell sections were clamped together, with the porous plug mounted in a flanged cylindrical holder separating them. This construction provided a definite cell geometry, in that the volumes on both sides of the porous plug enclosed between the cell half and
porous medium were equal and always remained the same irrespective of plug length. During a n experimental run the gas in each half of the diffusion cell separated by the plug was analyzed by means of a thermal conductivity gas analyzer. This instrument measures changes in the thermal conductivity of a gas and thereby changes in composition, as a function of the change in resistance of a heated platinum filament. The analyzer sensing system was two Gow-Mac Model 3 0 4 thermal conductivity units. I n operation, a stream of gas was circulated through each conductivity cell by means of a small bellows pump, which also maintained homogeneity of gas concentrations in the gas chambers. The analyzer circuit was that of a Wheatstone bridge with operation based on constant current conditions and degree of bridge unbalance. The conductivity cells were empirically calibrated with gases of known composition preliminary to diffusion operation. The constant temperature bath housed the diffusion cell and the thermal conductivity cells and was maintained at 104' F. (40' C.) for all runs. This tem-
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I
H
I-
r
J
I
ID! I
A
I
B
Figure 1. Diffusion equipment is a symmetrical arrangement A$.
G a s entry points Preheater Cooling coils F,G. Corron-Cerveny pumps H,I. Diffusion cell chambers J. Plug holder K . Vacuum pump L,M. Thermal conductivity units
c.
D,E.
VOL. 50, NO. 8
AUGUST 1958
1 1 95
contact with gases of concentrations &(t) and lC(t), may be had by Duhamel's theorem ( 3 ) . The solution of Equation 1: which satisfies boundary conditions i, ii, and iii, is simply C =
3"
O L E S (EACH
c1
+
CZ.
Making use of the proportionalities expressed by Equation iv, the problem may be solved in the form
Figure 2.
Construction details of diffusion cell
Half volumes on both sides of the porous plug remained the same, whatever the plug length
perature, chosen because no provisions were made for cooling the bath, was maintained by a thermoregulator system to within f0.2" F. Thermal conductivity units are somewhat temperaturedependent, but under Conditions of operation in this study, a 1 F. temperature change resulted in a negligible change in the gas analysis reading. Temperature observations at various points in the equipment were made by means of a Type K 2 Leeds & Northrup potentiometer used in conjunction with standardized iron-constantan thermocouples. Procedure. Before a diffusion run was started, a systematic series of leak tests was conducted-an absolute necessity for satisfactory operation. Runs were not attempted until leaks were minimized to a point where the diffusion cell could be evacuated to approximately 2 mm. of mercury. h'ormally, two runs were made for each porous plug, one with the plug initially saturated with one of the two gases used, and another with the plug saturated with the second gas. This saturation step was necessary to define the concentration distribution in the plug at the beginning or diffusion (solution of the problem is greatly simplified when one of the components is uniformly distributed-Le., when Co(.x) = Cn, at t = 0). T o achieve this preliminary saturation, one of the cell chambers was isolated by closing a thin sliding gate fitted to one end of the plug holder. This gate was opened at the start of diffusion runs, and during diffusion runs changes of composition us. time were recorded for each cell chamber as determined by the thermal conductivity gas analyzer. Data and Results Solution of Partial Differential Equation. The problem considered is that of countercurrent diffusion in a porous plug separating two gas volumes, 1 and 2. It is assumed that the plug has been given a preliminary treatment, to justify statement about the initial distribution of gas concentration-i.e., Co(.x)-in it. Diffusion is allowed to proceed for a
1 196
definite time and the gas volumes are kept homogeneous by mixing or stirring. The concentrations in 1 and 2 change from their original values, IC, and 2CC, to different values, 1C(t) and & ( t ) , in time t , and from a knobyledge of these values the diffusion constant of one of the gases is to be determined as it diffuses into the other. Assume the plug is of unit cross section and is bounded by planes x = 0 and x = 1. Homogeneous gaseous mistures of volumes g and h, and initial concentrations cC,, and IC,>,respectively, are in contact with the two sides of the plug. The first extends from x = 0 to x = -g; the second mixture, in contact rvith boundary I: = I , extends to x = 1 h. The initial distribution of concentration across the medium of "effective" volume, I , is C,(x). After time t the concentrations in the two gas volumes will be denoted bi- &(t) and 1C(t) and the distribution in the medium by C(n,t). Then the boundary conditions for which Equation 1 may be solved are:
+
C = C,(x)
at t
C(0, t ) = & ( t ) ;
0 for 0
