Determination of diffusion coefficient

Disclosed is a method of determining a diffusion coefficient for a diffusant in a particulate. An inert gas is passed through a particulate containing the diffusant. A parameter that is proportional to the concentration of the diffusant in the inert gas is measured over a period of time and the slope of the linear portion of that relationship is determined and is multiplied by a constant.

BACKGROUND OF THE INVENTION 
This invention relates to a method of determining the diffusion coefficient 
of a diffusant in a solid particulate. In particular, it relates to 
passing an inert gas through a bed of particulates containing a diffusant, 
measuring over time a parameter proportional to the diffusant 
concentration in the effluent gas, determining the slope of the linear 
portion of that relationship, and multiplying that slope by a constant. 
A diffusion coefficient indicates the rate at which a diffusant moves 
through a medium under a concentration gradient at a particular 
temperature and pressure. The diffusion of a chemical through a solid 
particulate is encountered in numerous industrial processes. Diffusion 
coefficients often must be known to properly design and operate these 
processes. For example, in a polyethylene manufacturing process, 
polymerization occurs in a flammable hydrocarbon solvent such as hexane. 
After the polymerization, the hexane solvent must be separated and 
recovered from the polymer to provide a clean resin product. The resin, 
usually in a form of powder, must be dried to a very low level to minimize 
the emission of hexane to the environment and the risk of explosion due to 
hexane build-up in storage vessels. When the hexane in the polyethylene is 
below 5%, the drying process becomes essentially a process of hexane 
diffusion in the polymer. The diffusion coefficient therefore is needed to 
properly design and optimize the process. As another example, crude 
poly(vinyl chloride) resins usually contain vinyl chloride monomer, a 
carcinogen. Its diffusion coefficient is needed to determine the 
conditions required to reduce the toxic vinyl chloride monomer 
concentration to a safe level. 
Most conventional methods of measuring the diffusion coefficient in a 
plastic material are based on a film permeability method similar to ASTM D 
1434. In such a method, a film made of the plastic material is placed 
between two chambers, one of which holds a constant concentration of the 
diffusant gas. The diffusant permeates through the film into the other 
chamber and, by measuring the diffusant concentration in the second 
chamber, one can obtain the diffusion coefficient. Although this method 
can produce good precision and accuracy for many practical applications, 
it has serious shortcomings if casting a film changes the morphology and 
physicochemical characteristics (such as crystallinity) of the material 
and if data for unaltered particulates are desired. In addition, at high 
temperatures and pressures, the mechanical integrity of the film may 
become a problem. And, for some materials, making a mechanically 
sustainable film simply is not possible. 
Diffusion coefficients for large solids can be determined by measuring the 
gain or loss in weight of the solid over time as the diffusant enters or 
leaves the solid. The accuracy of this method becomes questionable when 
the diffusant concentration is low. Although the method can also be used 
to measure the diffusion coefficients for fine particulates, its accuracy 
falls due to interparticle contacts and interactions, and poor ventilation 
at those points. 
Diffusion coefficients of polymer particulates can also be determined by, 
for example, packing a column with polymer-coated inert beads free of the 
diffusant, and chromatographing the diffusant by means of partitioning it 
between the polymer-coated beads and an inert carrier gas. The accuracy of 
that method also suffers because it requires coating inert particles, 
which changes the morphology of the polymer and may significantly affect 
diffusivity. 
SUMMARY OF THE INVENTION 
We have discovered a method of measuring the diffusion coefficient for 
particles without altering them. The method is simple yet much more 
accurate than prior methods, especially for small particles. Our method 
allows accurate measurements of intrinsic diffusion coefficients for 
particulates without forming the particulates into films. 
Moreover, in the method of this invention, it is not necessary to know the 
actual diffusant concentration in the solid, nor in the gas phase, making 
the method easier and simpler to carry out. That is a major advantage 
because it is generally very difficult and time-consuming to precisely 
measure the actual concentration of a fast diffusant in fine particles, 
particularly at elevated temperatures. For a fast diffusant, careful 
handling of the solids is required to avoid any loss of the diffusant 
prior to measurements. However, in the method of this invention, one needs 
to measure only a parameter that is proportional to the diffusant 
concentration in an inert gas that passes over the particles, something 
that is much more easily accomplished. 
The method of this invention can also be used to duplicate actual 
industrial gas/solid flow conditions, and determine the apparent diffusion 
coefficients under those conditions. These apparent or effective diffusion 
coefficients can then be used to determine the actual process efficiently 
under the industrial conditions.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
In FIG. 1, a temperature controlled oven 1 contains a fluidized bed 2 which 
consists of a glass cylinder 3 and a gas pervious glass frit diffuser 
plate 4, with a seal 5 at the top. Nitrogen from line 6 enters the oven 
and is split into line 7, which goes to fluidized bed 2, and bypass 8. A 
thermometer 9 measures the temperature of the powder 10 in fluidized bed 
2. The nitrogen passes out of fluidized bed 2 through line 11, pass filter 
12, and can be vented at valve 13. It can enter 6-port sampling valve 14 
where helium in line 15 takes the diffusant into gas chromatograph (GC) 
16. The results from the gas chromatograph are shown at 17, a plot of time 
versus a parameter proportional to the concentration of the diffusant in 
the nitrogen, in this case the GC area counts. 
The method of this invention can be used to determine the diffusion 
coefficient of a diffusant in a particulate. A practical range of sizes 
for the particulate is about 10 to about 5000 microns. While larger or 
smaller sized particles can be used, that range of particle sizes is 
considered to be practical because if the particle size is too small the 
time during which the measurements must be taken may be too short for 
on-line measurements. If the particle size is too large, precision is not 
impaired, but a very long time may be required to conduct the 
measurements. The invention is particularly useful with particulates 
having a diameter of about 10 to about 500 microns as it is difficult to 
determine the diffusion coefficient by other methods for particles within 
that size range. To obtain the most accurate results, the distribution of 
particle sizes in the samples tested should be as narrow as possible. A 
very narrow particle size distribution can be obtained, for example, by 
sieving a larger sample, taking a fraction, and re-sieving that fraction. 
Preferably, the particle size distribution of the particulate should be 
within 10% of the mean particle size. 
While the method of this invention can be applied to particles of almost 
any shape, it is most usefully applied to spherical particles or particles 
that are approximately spherical, such as cylinders (where 
height.apprxeq.diameter) or cubes. The particles may be of any solid 
material that is capable of absorbing a diffusant, such as wood, plastics, 
metals, and inorganic materials. Polymeric particulates, such as 
polyethylene, poly(vinylchloride), polyurethane, polystyrene, polyesters, 
and polymethylmethacrylate, are preferred as the invention is most 
applicable to that type of material. 
The diffusant can be a gas or a liquid, but it must leave the particle in a 
gaseous state. Thus, if the diffusant is not very volatile, it may be 
necessary to heat the particles. The initial concentration distribution of 
the diffusant throughout the particle must be uniform for accurate 
measurements. This can be accomplished by letting the particles sit in a 
closed atmosphere containing a constant diffusant concentration until 
equilibrium has been reached. Preferably, the particles are not saturated 
with diffusant because excessive adsorption of diffusant may swell the 
particles or otherwise alter their geometry and other physicochemical 
characteristics. A diffusant concentration of about 5 to about 40 wt % of 
saturation is preferred. Organic compounds, such as the C.sub.2 to 
C.sub.10 alkanes and alkenes, are commonly encountered diffusants. 
Any inert gas that does not react with or interact with the diffusant or 
the particles can be used in the process of this invention. Examples of 
such gases may include nitrogen, argon, helium, carbon dioxide, and air. 
The diffusion coefficient depends upon the temperature of the particles. 
The particles and the diffusion cell can be heated in an oven and the 
inert gas should preferably be heated to the same temperature as the 
particles. Generally, any temperature at which the diffusant does not 
condense into a liquid and which does not melt or soften the particles can 
be used. However, lower temperature is desirable if the particles are very 
small or the diffusant diffuses rapidly since a low temperature allows 
more time for the measurements. On the other hand, if the particles are 
large or the diffusion rate is low, a higher temperature can be used to 
shorten the experimental time. Most often, the temperature chosen is the 
temperature for the process to be studied. 
The analytical device which measures a parameter proportional to the 
concentration of the diffusant in the inert gas (where the parameter can 
be the concentration itself) can be of any type including, for example, 
gas chromatography (GC), photometric measurement (where the parameter is 
intensity of light absorbed or scattered), or chemical detector (where the 
parameter is the concentration of diffusant detected). Since at least two 
measurements must be taken and the interval between the measurements must 
be short if the particles are small, it is preferable to use a gas 
chromatograph as the analytical device. A gas chromatograph can be set up 
to automatically measure the parameter (GC area counts) three times per 
minute or more, thereby permitting a precise measurement of the diffusion 
coefficient of very small particles. A gas chromatograph is preferred 
because the capillary column contained in it focuses the gaseous diffusant 
injected into a discrete peak, which can be readily quantified as the area 
under the peak. 
The two or more measurements give the parameter proportional to the 
concentration of the diffusant in the inert gas for an interval of time. 
After a small initial time, the change in the natural log of the parameter 
per unit time (i.e., the slope of the plot of the natural log of GC area 
points versus time) will become constant. It is this constant or linear 
portion of the slope that is used to calculate the diffusion coefficient. 
The calculation is very simply made by multiplying the slope by a 
constant. For a spherical or near-spherical particle that constant is 
-R.sup.2 /.pi..sup.2 where R is the average radius of a particle. If the 
particles are in the form of flakes the constant is -41.sup.2 /.pi..sup.2, 
where 1 is flake thickness, and if the particles are in the form of 
needles the constant is -R.sup.2 /2.4, where R is the average radius of 
the needles. Other geometries can be approximated by an equivalent radius 
using the constant for a spherical particle. 
The particles, having the diffusant uniformly dispersed throughout, are 
placed in a fluidized bed. A sufficient amount of particles should be used 
so that the measured parameter is easily detectable. After reaching 
equilibrium at the desired temperature, the inert gas is passed through 
the particles. If the inherent diffusion coefficient is to be determined, 
the inert gas should fluidize the particles so that the entire surface of 
each particle is in thorough contact with the flowing gas. We have found 
that inherent diffusion coefficients obtained by the method of this 
invention correspond closely to inherent diffusion coefficients obtained 
for the same solids and diffusants by other methods. However, since in 
actual industrial operations the particles may be treated under conditions 
in which they are not completely fluidized, those conditions can be 
duplicated in the method of this invention by using the inert gas at a 
lower velocity so that the particles are either stationary or only partly 
fluidized. In this way, an apparent or effective diffusion coefficient is 
obtained which will enable one to make accurate calculations of the 
corresponding industrial set up. 
The following examples further illustrate this invention. 
EXAMPLE 1 
As shown in FIG. 1, a sample cell 9 cm.times.2.7 cm.times.2.5 cm having 
porous glass frit at the bottom and a glass fiber filter at its outlet was 
filled with either a 0.25 gm or a 1 gm sample of polyethylene powder 
containing hexane and placed in a temperature controlled oven. (In some 
experiments a copper tube with a steel mesh screen at the bottom was used 
instead of the glass cell.) Nitrogen gas was preheated to the desired 
temperature using a temperature controlled oil bath prior to entering the 
oven. Hexane concentration in the exiting nitrogen stream was monitored by 
a gas chromatograph equipped with a flame ionization detector. The GC was 
modified by installing a valve compartment which automated the control of 
a 6-port Valco valve equipped with a 500 .mu.l sample loop for sampling 
purposes. The sample loop was continuously flushed by the process stream 
with the aid of a Gorman-Rupp chemical feed bellows pump. A 15 m dimethyl 
silicone fused silica capillary column was used for analysis. The 
injection port was operated in a splitless mode. Helium was used as a 
carrier gas at 4 ml/min. The column temperature was 60.degree. C. 
isothermal. Injection port and detector temperatures were 225.degree. C. 
and 250.degree. C., respectively. The sample valve was programmed for 
three automatic injections per minute. The concentration decay trace was 
recorded on an integrator which measured the area of the hexane response 
peak. The sensitivity of the GC system was about 250 ppb hexane in 
nitrogen. The sample line was purged with dry nitrogen prior to an 
experimental run and was injected as a blank to assure cleanliness. 
Calibration standards were prepared by diluting a 4990 ppm (mole %) hexane 
standard in nitrogen obtained from Scott Gases. 
To insure that the powder sample had a narrow particle size distribution, 
it was sieved into various fractions. One of the fractions was chosen and 
re-sieved. The desired fraction from the second sieving was then analyzed 
by an optical technique. Usually, two sievings were sufficient to obtain 
the particle size distribution within 10% deviation of the mean. 
A known amount of reagent grade n-hexane (99+%) was spiked into a 
hexane-free polyethylene powder sample. The hexane concentration of the 
powder was in the range of 3500 to 5000 ppm as determined by headspace 
analysis or calculation. The oven was set to the desired temperature, 
which typically took 10 to 15 minutes. Once that temperature had been 
obtained, the cell was held for an additional two to five hours at that 
temperature. The nitrogen flow was set using a rotameter and made to flow 
through the bypass to check that its temperature was the same as the cell 
temperature. The nitrogen was then made to flow through the cell and the 
gas chromatogram was begun. It was experimentally determined that the 
optimum nitrogen flow was about 2000 cc/min for complete fluidization of 
the bed. The following table gives the results at the various 
temperatures, nitrogen flow rates, and particle sizes. 
______________________________________ 
N.sub.2 Flow 
(cc/min) Observed D (cm.sup.2 /s) 
______________________________________ 
RESULTS AT 60.degree. C. 
500 0.65E-8, 0.64E-8, 0.65E-8 (mean = 0.65E-8) 
1000 1.9E-8, 1.93E-8, 1.83E-8 (mean = 1.89E-8) 
1500 2.8E-8, 3.1E-8, 2.8E-8 (mean = 2.9E-8) 
2000 3.1E-8, 3.0E-8, 3.3E-8 (mean = 3.13E-8) 
2500 3.2E-8, 3.0E-8, 3.1E-8 (mean = 3.1E-8) 
3000 3.2E-8, 3.0E-8, 3.2E-8 (mean = 3.1E-8) 
2000** 3.1E-8 
RESULTS AT 70.degree. C. 
500 0.74E-8, 0.73E-8 
1000 2.2E-8 
1500 3.7E-8 
2000 6.0E-8 
2500 5.9E-8, 5.5E-8, 5.9E-8 (mean = 5.8E-8) 
3000 5.8E-8, 5.9E-8, 5.7E-8 (mean = 5.8E-8) 
2000** 5.5E-8 
______________________________________ 
N.sub.2 Flow Observed D 
(cc/min) PE Amount (g) 
(cm.sup.2 /s) 
______________________________________ 
RESULTS AT 80.degree. C. 
300 1.0 1.3E-8 
2000 1.0 4.7E-8 
3500 0.25 1.1E-7 
RESULTS AT 93.degree. C. 
300 1.0 1.9E-8 
1300 1.0 3.6E-8 
2000 1.0 6.7E-8 
______________________________________ 
**All runs used 127 .mu.m powder except that this run used 74 .mu.m 
powder. 
In the table "Observed D" means the diffusion coefficient as calculated by 
observing the GC area points, determining the slope of the natural log of 
the area points versus time, and multiplying that slope by -R.sup.2 
/.pi..sup.2, where R is particle diameter in centimeters. 
The above tables show observed D values for n-hexane in polyethylene in the 
60.degree.-93.degree. C. temperature range as a function of the flow 
rates. 
Referring to FIG. 2, the ordinate is the natural logarithm of gas 
chromatography (GC) area points, a parameter proportional to the 
concentration of the n-hexane in the nitrogen gas, and the abscissa is 
time in minutes. The slope of the linear portion of the Curve in FIG. 2 
was determined to be -0.0142 seconds.sup.-1. Since the particle size was 
127 .mu.m, R was 63.5 .mu.m (or 0.00635 cm), -R.sup.2 /.pi..sup.2 was 
-4.0855.times.10.sup.-6 cm.sup.2), and D (the intrinsic diffusion 
coefficient) was the slope times -R.sup.2 /.pi..sup.2 or 
5.8.times.10.sup.-86 cm.sup.2 /sec. 
The intrinsic diffusion coefficient for n-hexane/polyethylene was 
independently determined by M. Markelov and B. Kogarko of ACS Labs by the 
film permeation or time-lag method in "Service Report to OxyChem: 
Diffusion, Permeability and Solubility of n-hexane in Polyethylene", May 
1995. The value determined was 5.7.times.10.sup.-8 cm.sup.2 /sec. at 
70.degree. C.