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ABSTRACT
THE RELATION OF MEAN PARTICLE SIZE AND
THE SIZE DISTRIBUTION TO DIFFUSION
OF ACETONE VAPOR THROUGH AIR IN
A PACKED COLUMN
By

Andrew N. Paquet

The subject of diffusion of one vapor through another
as occurring in a packed column or porous media is a very
important area in many fields of science and engineering.
One of these areas is in the field of soil science. In
the soil many physico—chemical mechanisms occur, but one of
special interest to this study is the rate of transfer of
gaseous components. How these gases transfer helps determine
the biological activity which will occur in the soil. This
in turn determines uses of the soil by man.

The purpose of this study was to determine diffusion
rates of gases through air in a porous media. More speci-
fically, it was of interest to study the affect of the size
distribution of particles comprising the porous matrix in
relation to the diffusion rate through the matrix.

The experiments were conducted in glass columns packed
with spherical glass beads of various sizes. Tests were

conducted to determine the mean size and standard deviation

Andrew N. Paquet

of the particle mixtures, the porosity, and the tortuousity
of the bed. The procedure was initiated by running tests

in empty columns so as to be sure that the literature
(accepted) value of a diffusion coefficient for a particular
vapor—air system could be obtained. The system under study
in this work was acetone vapor diffusing through air. Acetone
was metered into the glass columns and in a given period of
time the amount that had evaporated could be measured by use
of a calibrated pipet. This allowed for the calculation of
the flux of acetone vapor for a given diffusion length. By
using the ideal gas law, which approximately describes the
behavior of the vapor-air mixture, and correcting for experi-
mental temperature and pressure the diffusion coefficient
could be obtained. All experiments were conducted in an
environmental control chamber to facilitate constant experi-
mental conditions.

The results of this study show that the particle
distribution does affect the tortuousity of the porous matrix.
However, a clear affect on the porosity by the particle
distribution was not discerned. In regards to the above
findings it can be stated that the particle distribution
does affect the rate of diffusion of acetone in air through
the packed bed. That is, as the distribution of particle
sizes becomes narrower the path becomes less tortuous.

This, in turn, results in an increased diffusion rate (higher

diffusion coefficient).

THE RELATION OF MEAN PARTICLE SIZE AND
THE SIZE DISTRIBUTION TO DIFFUSION
OF ACETONE VAPOR THROUGH AIR IN

A PACKED COLUMN

By

Andrew N. Paquet

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

1974

To

ii

Jan

 

ACKNOWLEDGEMENTS

I would like to express my appreciation to all those
persons who showed interest and gave assistance to me in
my work when it was requested. Included among the people
are Dr. D. K. Anderson, Paul Husted, and Roger Hinson.

Special appreciation is extended to my graduate advisor
Dr. George Coulman whose assistance, interest, and friend-
ship made this work easier and enjoyable. I can not thank
him enough.

Lastly, I would like it to be recognized that financial
assistance from National Science Foundation Grant GI-2O

made this study possible.

111

TABLE OF CONTENTS

Page

DEDICATION . . . . . . . . . . . . . . . 11
ACKNOWLEDGEMENTS . . . . . . . . . . . . . iii
LIST OF TABLES. . . . . . . . . . . . . . Vi
LIST OF FIGURES . . . . . . . . . . . . . Vii
INTRODUCTION . . . . . . . . . . . . . . 1
BACKGROUND . 3
Tortuousity - 6
Porosity . 7
EXPERIMENTAL METHODS AND APPARATUS . 9
RESULTS AND DISCUSSION . . . . . . . . . . . 20
SUMMARY AND CONCLUSIONS. . . . . . . . . . . 30
RECOMMENDATIONS . . . . . . . . . . . . . 31
BIBLIOGRAPHY . . . . . . . . . . . . . . 32
APPENDIX A (General References) . . . . . . . . 33
APPENDIX B . . . . . . . . . . . . . . . 3U
APPENDIX C . . . . . . , . , . , . , , , 36

iv

LIST OF TABLES

Table

Page
1. Experimental Results for Diffusion of
Acetone in Air in an Empty Column . . . . 22
2. Experimental Results for Diffusion of
Acetone in Air in a Packed Column . . . . 23
3. Comparison of Tortuousity with Increasing
Size Distribution Range . . . . . . . 2A
A. Dimensions of the Glass Columns. . 3U

LIST OF FIGURES

Figure Page
1. Experimental Apparatus, . 18
2. Determination.afAcetone Level in Pipet . . l9
3. Illustrative Diagram of Read Size

Distributions for Four Bead Mixtures . . . 25

vi

INTRODUCTION

The subject of diffusion through a porous media plays
a fundamental role in many areas of science and engineering.
This subject has been investigated by many people in the
past. However, certain aspects of this area of study either
remain unexplored or, if information is available, obscure.

One particular application of the subject of diffusion
in a porous media is in the field of soil mechanics. In
view of the recent concern for environmental control and
pollution abatement, means for proper disposal and assimula-
tion of human and animal body wastes into natural systems
is of growing concern. One means of processing this waste
and simultaneously deriving economic benefit is by spray
irrigation. This involves applying the waste to a given
area ofland and taking advantage of natural processes, within
the soil for decomposition and assimilation of the waste.

To dothis optimally requires a thorough understanding of the
biological and physical mechanisms involved.

The research that was undertaken deals with a small
part of the physical mechanisms that occur in the soil. In
order to understand optimal use of the soil for waste disposal
and assimilation, it is necessary to have information about
gas exchange rates. This gas exchange occurs both within
the soil and at the soil-atmosphere interface. The gas

1

exchange rates involve the diffusion of the gas through a
media, and the physical structure of the media.

In light of this, the following study was undertaken
to investigate the relationship of the particle size and
the particle size distribution to diffusion of a vapor
through a packed column. Furthermore, it was desired to
provide useful information for use in a mechanistic mathe-
matical model to optimally describe spray irrigation procsses.

To obtain the desired information, the porosity and
tortuousity of the packing were needed. These were inde-
pendently obtained experimentally. Furthermore, the
tortuousity of the porous media (packing) was related to
the experimentally obtained diffusion coefficient (effective).
The relationship between tortuousity and the diffusion
coefficient was derived by use of the law of conservation of

mass. Bear (1) gives the following expression:

.. "I ‘ -
30A _ 3((DABTL1_’(apA/3fi(

I 3x.
4

 

Q)

where DAB is the diffusion coefficient for a binary gas
system through an empty column.

In the following pages background information for this
study is presented, followed by a description of the methods
used to obtain experimental data. The experimental results

are then presented and discussed.

BACKGROUND

In describing mass transfer processes in a porous media,
one naturally relies upon the equation of mass conservation
to describe, in mathematical terms, the physical process
that is occurring. When this is done, it is readily seen
that a term called the tortuousity is present. Bear (1)
developed the conservation of mass equation for a porous
medium. In his development, he assumed that the media was
isotropic and that the porosity was constant. The following
was his result:

-*- --
36A 3[(DL'*DABTL1 )apA/ax. 8(pAV£)]

= J L, _ ._m____
32 3x. 8x.
4 L

_ t -u
where DABTij EDAB is the "coefficient of molecular diffusion
t

in a porous medium," and fij is the medium's tortuousity.
Notice that for a fixed DAB the value of DAB will decrease

as the diffusion path becomes more tortuous. This requires
fij‘ ("tortuousity") to decrease as the path becomes more
tortuous. Furthermore, if there is no bulk mechanical motion,

the coefficient of mechanical dispersion, D.

Lf’ is equal to

zero. Also, under these conditions Pi is equal to zero.
Consequently, the equation describing diffusion in the porous

media reduces to:

- - I -
335 _ atthBTij )(apA/axj)1

at 3x.
L

 

The concept of tortuousity involves the concept of
a matrix whose components are the products of cosines of
angles between the directioncn‘the streamlines of flow. It
can therefore be expressed in terms of a coordinate system.
Furthermore, the tortuousity can be interpreted as a
characteristic path length formed by the connection of porous
spaces in which diffusion occurs.

In determining the tortuousity of a porous media, it is
helpful to express the transport phenomena occurring in this
process in terms of fluxes as defined by Fick. For the
problem at hand, concern iscflfld'with a single fluid saturat-
ing the porous media. It is assumed that transport does not
take place in the solid phase. Furthermore, there are no
interactions in the form of ion exchange, adsorption, etc.,
occurring between the constituents of the solid packing and
the saturating fluid. Therefore for a homogenous fluid
(5A = constant) at rest (9:0), taking the above conditions
into consideration, the diffusive mass flux with respect to

mass average velocity can be expressed as:
t
j = -Dd-gnad DA
or, in terms of per unit area of porous media,

+* it - _I
Q = n] = -nDdT - gnad pA = -nDd-gaad pA’

- 1
Therefore, nDd =nDd

_t
The quantity Dd is defined as the

2*
T
coefficient of molecular diffusion in a porous medium. If
Dd is known, which in general it is (or can be determined),

_¢
and Dd can be experimentally obtained, it is readily seen

it
how T , the medium's tortuousity, can be acquired.

The diffusion coefficient, D is the diffusion coefficient

d’
for a binary gas system obtained by diffusion of one gas
through a stagnant layer of another gas. This involves the
use of Fick's first law of diffusion through a stagnant gas
film. The theoretical discussion and mathematical expressions
are developed in numerous books and articles; the following

is from Bird, et al. (2). The most useful expression being

the following:

 

‘pDAB/RT’ P32
N _ Kn [ 1
AZ z=z ‘22 21’ 75—3—1

I

(vaB/RT)

P
(12-2,)(P

 

(PAI’ A2)

B’zn
This form, as can be seen, allows for the determination
of Dd=DAB, the diffusion coefficient, by measuring the flux,
the diffusion path length, and the particle pressures of the
binary components (ref. Lee and Wilke). The partial pressures
can be determined by temperature measurements. The diffusion
path length should remain constant as a theoretical consid-
eration. However, by assuming a quasi-steady state condition,

and using an average of the initial and final lengths, the

problem of a changing diffusion path length can be circum-
vented. The flux can be determined experimentally by
utilizing a vessel of constant dimensions and (determining
by measurement) the amount of substance that has evaporated

or diffused out of the column.

Tortuousigy

 

The following explanation may be helpful in understand-
ing how tortuousity is related to the distribution of particle
sizes. The tortuousity is an indication of the effective
length along which diffusion occurs. This effective length
is caused by the manner in which the individual beads pack
themselves, or in other words, the geometry of the packing.
Furthermore, it is felt by this author, that this geometry
is related to the mean size of the beads (also, ref. Brown
(5)). Between the beads, pores occur (through) which the
vapor diffuses. When the beads are of various sizes,
smaller beads can occupy the porous spaces between larger
beads. This in turn causes the vapor to take a more
"tortuous" path as it diffuses through a porous media. If
the distribution of particle sizes is broad, i.e. the
standard deviation is fairly large, there is a greater
chance of smaller beads occupying the space between larger
beads. This will act to form a more tortuous path. Also,
it may be that this situation results in a smaller porosity
for the packing. As a result of the definition of I‘, it

will decrease in value as the path becomes more tortuous.

Porosity
Porosity, as defined by Bear (1), Bird, et al. (2),

and many others, is the ratio of the volume of the void

space (Vv) to the bulk volume (VB) of a porous medium:

VU (I-VS)
€=VE=__VB___

where epsilon is the porosity and Us is the volume of the
solids contained in the bulk volume. As can be seen, the
porosity is a dimensionless quantity and is expressed as
a percentage. In a porous medium, the pores may be inter-
connected or not. In regards to flow through a porous
media, only interconnected pores are of interest. Therefore,
the measurement of an effective porosity is desirable
because the fraction of interconnected pore volume is mea-
sured. The porosity of unconsolidated materails depends on
thepacking of the particles, their shape, arrangement, and
the size distribution. The particle size distribution
affects the porosity in that smaller particles may occupy
the pores formed by the larger particles. This may act to
reduce the porosity. Porosity may vary with depth into a
column due to compaction from the weight of the material
above. For short distances, though, this effect is small.
The measurement of porosity can be done directly by
measuring thevolume displacement of water by the beads, or

by dividing the weight of the beads by their density.

In previous work on diffusion in porous media, many
investigators found various relationships between the
porosity of the medium and the diffusion coefficient.

Smith and Brown (1933), in working with a COZ—air-soil
system, according to Linvill report a linear relationship
between the porosity of the soil and the diffusion
coefficient of the gases diffusing in the soil. Linvill
also reports that Penman (19M0) also found a linear relation-
ship. For separate runs with C02, C82, and acetone where
the porosity of the medium ranges from 0.0-0.7 Penman
reports the expression D'/vd = 0.665.

Linvill (1969); however, found that the ratio bd/vd'
or that the diffusion coefficient, varied exponentially with
porosity. He worked with an OZ-air—soil system. The form
of his relationship is bd/Dd = aeb, where a and b are
experimentally obtained constants.

Actual values of porosities for various types of packing
materials with different geometrical structures are given in
Brown (5). For spheres (spherecity = 1.0), the porosities
range from 0.38—0.A7. Glass spheres (spherecity = 1.0) are
reported to have a porosity of about 0.41. This same
reference indicates that porosity may vary with the diameter

of the particles comprising the packing.

EXPERIMENTAL PROCEDURE AND APPARATUS

The experimental procedure and apparatus used in this
study was quite similar to that employed by other investigators
in obtaining vapor phase diffusion coefficients (ref. Lee
and Wilke; Stefan; McMurtie and Keyes). The experimental
work was separated into two phases. The first phase being
conducted in empty columns while the second phase employed
packed columns.

In the first phase of experimentation, the method con-
sisted of having a liquid contained in the bottom of a
vertical tube. This liquid was allowed to evaporate and
its vapor to pass up the length of the empty column. At
the tube opening, another gas (air) was passed by at a
sufficient rate to keep the partial pressure of the vapor
at a level corresponding to theinitial composition of the
gas. In the case where the gas is vapor free, this initial
composition is zero. The diffusion coefficient can then be
determined by the rate at which a given quantity of vapor
has passed out of the apparatus.

The variables to consider in this method were the
temperature at which the experiment was conducted, the
ambient pressure, the flow rate of air passing over the tube

Opening, and the diffusion path length (as related to the

10

air flow rate, and also the falling off of the liquid level
as a result of evaporation).

Temperature affects the rate of evaporation. If the
vapor behaves ideally, then there is a linear relationship
between the rate of evaporation and temperature. It is
desirable to maintain a constant temperature if a constant
evaporation rate (or flux) is desired.

Pressure also affects the rate of evaporation for a
well behaved vapor. A higher pressure will result in less
vapor, and vice versa. Consequently, it is desired to
maintain constant pressure during the experiment, or to
isolate it from other factors and measure evaporation with
changing pressure.

The flow rate of air passed by the tube opening facili-
tated the removal of vapor as it arrived there. It also
allowed for the maintanance of a concentration difference
from the liquid level at the bottom of the tube to the tube
opening. With no concentration difference there would be no
evaporation. However, the rate of evaporation can be affected
if the air flow rate is too great or too small. This affect
is manifested in the actual diffusion path length. If the
air flow rate is too great, turbulence occurs at the tube
opening with the result being a shortening of the actual
diffusion path length. The opposite affect occurs if the

air flow rate is too low (ref. Lee and Wilke).

11

A further problem with the diffusion path length vary-
ing was encountered due to the evaporation of the liquid.
As the liquid evaporated, the liquid level dropped. Since
it is desirable to maintain the liquid level at a constant
position, a means to do just that was developed. A reservoir
of liquid was connected to the column. Periodically, when
the liquid level had dropped a small amount, a small amount
of liquid could be added to the column. By this method the
diffusion path length could be determined by using the
arithmetic average of values at the beginning and at the
end of the diffusion process. This resulted in a small error
for the actual value of the diffusion coefficient experi-
mentally measured. In regards to this, quasi-steady state
conditions rather than steady state conditions were main-
tained.

In this study an environmental chamber was employed to
control and adjust the ambient temperature and the air flow
rate. Pressure was monitored by a barometer (calibrated)
housed within the chamber.

It was of course necessary to calibrate and adjust the
air flow rate within this chamber for reasons already
mentioned. All conditions were set and monitored so that
the affects of the air flow rate would be isolated. The
temperature was set at 30°C, and pressure was monitored.
Subsequently, it was found that an air flow rate of 30% of

maximum was sufficient for operation. These conditions

l2

(temperature and air flow rate) were established as the
experimental operating conditions.

After having established the operating conditions,
experiments to obtain diffusion coefficients employing the
following procedure were conducted (refer to Figure l):

1. Read level in pipet.

2. Uncork pipet, open valve to transfer acetone

to column.

3. Set level of acetone at a predetermined sight

line on column reservoir.

A. Close valve, cork up pipet.

5. Read and record new level in pipet.

6. Repeat procedure at various time intervals.

In the course of these experiments many problems were
encountered, incorporating changes in the apparatus, before
good consistent results were obtained. These problems
stemmed mainly from two sources, leaks in the system and
the air flow rate. The problem with the air flow rate was
due to not knowing the maximum flow rate (circulation rate)
in the environmental chamber. The circulation rate was
rheostatically controlled and the dial was calibrated in
percentage of the maximum flow rate. Consequently, an
acceptable flow rate of air was determined by trial and
error. The importance of the circulation rate has been
previously discussed. However, the problem with the air
flow rate was alleviated only after the leaks were stopped.

Only then could the affect of the flow rate be isolated.

13

In the original experimental setup, calibrated burets
instead of pipets were used as part of the acetone reservoir.
The burets had stopcocks so that acetone could be metered
into the column. It was found that acetone leaked out of
the burets at the stopcocks. Apparently the acetone dissolved
the stopcock grease after a period of time so that the seal
between the stopcock and the buret was destroyed. After
varying attempts at correcting this problem, it was decided
to change the experimental apparatus and use pipets and
valves (see Figure 1). This helped correct much of the
problem, but some leaks still remained. These leads came
from the polyethylene hose--glass nipple joinings both at
the pipet and the column. This problem was alleviated by
using a silicone sealant at these points (manufactured by
General Electric Company).

A third source of error was associated with the accuracy
with which the calibration of the pipet could be read. This
was overcome by using finely calibrated rulers placed next
to the markings on the pipet and taking a sighting by a
telescope (see Figure 2). This setup allowed readings
accurate to the thousandths of a milliliter.

It was very important to take thetime to obtain good
consistent results from this phase (the first) of the
experiment. Contingent upon these results and the subsequent
operation of the apparatus was the basis for acceptance of

the results for the second phase of the experiment. The

1“

first phase of the experiment was used to obtain a well
working experimental apparatus and to verify the "literature"
value of the binary diffusion coefficient of acetone vapor

through air (as reported by Perry's Chemical Engineer's

 

Handbook).

The result of correcting all the above mentioned
problems resulted in the attainment of the goals of the
first phase of experimentation.

Having established the correct procedure, as already
stated, it was possible to procede to the second phase of
the experimentation. In this phase the same operating
conditions, methods, and equipment as used in the first
phase were employed. However,instead of measuring diffusion
through an empty column, diffusion was measured through a
packed column.

The column was packed with glass beads of a certain
size distribution and mean diameter. Furthermore, these
beads were of a uniform spherical shape (approximately 90%
true spheres), and their specific gravities ranged between
2.H6 and 2.99 g/ml. These beads were manufactured by the
Microbeads Division of the Cataphote Company. The porosity
of the packing was measured by weighing a known volume of
beads and then dividing this value by the density (specific
gravity) of the beads. The value obtained was the volume
of tmabeads in the known (bulk) volume weighed. The former

volume was designated the bead volume (V3), and the latter

15

volume was called the bulk volume (Vb). The porosity was

then calculated by use of the following expression:

E = .YX. = (vb-vé)
Vb 0b

where Vv is the void volume. The porosity obtained in this
manner was checked by another procedure. If the known volume
of beads (Vb) was placed in a given volume of water, the
resoltant volume of beads plus water should be equal to the
sum of the bead volume (V6) and the given volume of water

(vw).

Apparatus

 

The apparatus used was similar to that which was employed
by other investigators (see Figure l). The diffusion
columns were constructed of glass. Plastic columns were
originally used, but they proved to be inadequate because
it was desirable to use acetone. Acetone is a rather good
solvent for the plasticizers in the plexiglass. The length
to diameter ratio for the five glass columns used was
approximately 12:1. The liquid reservoir was also constructed
of glass. The reservoir had sight lines placed every
sixteenth of an inch apart along its length. Furthermore,
the reservoir and the column were joined by a ground glass
joint.

A glass pipet with a volume of five milliliters,

calibrated into tenths of a milliliter, was connected to the

16

diffusion column by polyethylene tubing (OD=0.25 inches,
ID=0.l25 inches). Acetone was placed in the pipet and
transferred through the tubing into the column. The acetone
used was reagent grade (98.8% pure). Flow of the liquid

was regulated by use of values equipped with Swagelok
fittings.

The polyethylene tubing was secured to the pipet and
the column at their respective nipples by using a silicone
adhesive that set in 2H hours. The adhesive was manufactured
by the General Electric Company.

After the liquid was transferred from the pipet to the
column, the liquid level in the pipet was accurately deter-
mined. This was facilitated by use of a precisely calibrated
ruler and a sighting telescope (see Figure 2). The ruler
was placed behind the pipet, both being supported by a burst
clamp. The liquid level could then be accurately read by
magnifying the set-up with the telescope (Figure 2).

The pipet was cooked to prevent evaporation of acetone.
The loss of acetone vapor out of the pipet when it is
uncorked during the readings is small in volume therefore
introducing negligible error in volume readings.

The whole set-up was housed in a controlled environment
room. The use of this room allowed for the maintenance of
a constant temperature and air flow rate. A barometer
housed in this room allowed for the measurement of pressure

when ever a reading of the acetone level was done.

17

The values for the column dimensions are listed in

Table A in Appendix B.

18

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FIwREZ. DETERMINATION of' ACETONE

IN ’PIPET

RESULTS AND DISCUSSION

Some representative results for the first phase of the
experimentation are given in Table 1. It will be noticed
that the error percentage of some of the experimentally
obtained diffusion coefficients deviates greatly from the
value of the diffusion coefficient as reported by Perry's

Chemical Engineer's Handbook. These values are presented

 

for illustrative purposes in regards to the problems encount-
ered, as mentioned in the previous section. Obviously, the
value with the large positive error indicates a leaky system
as indeed was the case. The value with the significant
negative deviation was a result of the air circulation rate
being too low. Because the flow rate was slow, the acetone
vapor was not removed from fluetube opening at a sufficient
rate. This resulted in effectively increasing the diffusion
path length. Consequently, by using the tube length as the
diffusion path length in the computations a negative error
resulted. The value of the diffusion coefficient with the
small error is an example of the experimental results after
the problems with the set—up and the experimental method were
corrected. Consistently good results, with the error less
than 5%, were obtained when the apparatus and experimental

technique became established.

20

21

Having proved that good results could be consistently
obtained gave confidence in applying the technique to the
second phase of the experiment.

For the first phase of the experiment a gas phase
diffusion coefficient for acetone in air was established as
0.13A5 square cm per second, at 30°C and 7A0.0 mm Hg pressure.

For the second phase ofthe experiment with diffusion
of acetone through air in a column packed with glass beads,
representative results are given in Table 2. In Table 3
values of the tortuousity resulting from various packing
distributions are presented. The experimental conditions
were 30°C and various pressures. A complete set of experi-
mental data is given in Appendix C.

From Table 2 it can be seen that some of the bead distri-
butions are not specifically known. These distributions are
designated Mixes #l, #2, and #3. They are mixtures of
various narrower bead distributions prepared to obtain added
data on the affect of the size distribution with respect to
thetortuousity, and also get a tighter packing arrangement
(i.e. a smaller porosity). The bead mixtures were constructed
around the basic narrow distribution with a mean size of
0.019896t0.00297l centimeters. Mix #1 was a broad distribu-
tion of sizes, with Mix #2 having a little narrower range,
and Mix #3 still narrower than the other two. Figure 3 below
illustrates the size distributions and how the three mixtures

approach the basic narrow bead distribution (designated as

22

TABLE l.—-Experimental Results for Diffusion of Acetone in
an Empty Column (Representative Values).

_.—-— __._-_

 

DACEAEXP(cm2-sec-l) DACE,REAL(cm2 sec-1) Error(%)
0.1345 0.1301 + 3.A
0.1062 0.129“ -17.9

0.19u1 0.1327 +A6.3

 

23

 

 

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mmmma0.0 H>0N00.0 0505.0 mmoa.0 23.0
mwmm00.0 HO0H00.0 30am.0 QOHH.0 23.0
mwmw00.0 HO0H00.0 N02w.0 mHHH.0 32.0
“Eovcmoz AEoVwH>oQ .me AmpfimsoszOpve AHIomm.mEovmxm.mo<Q anamopom

 

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24

--Comparison of Tortuousity with Increasing Size
Distribution Range.

..____ ___ _
--=

Tortuousity Size Distribution(cm) No. of Samples

 

.62918 0.165860: 6.9% 5
.71122 0.019896i1u.9% 6
.77923 0.050788i17.9% A
.57960 0.0312u012o.9% 5
.82713 0.008289:23.7% 6

 

25

mm! #4“

A 1me #3
A mun.
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NIX #I

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0.?»qu cm

FIGURE 3- I HUSTRATIVE

DIAAAAm m: Bum SIEE DISTRIBUTIDIJ;
Fan. Faun. BEA]; mIIITIJILES

26

"Mix #4" for comparative purposes). From Table 2 it can be
seen that as the bead size distribution of these mixtures
approach the basic narrow distribution the tortuousity of

the mixtures also approach the tortuousity of the basic
mixture (Mix #A). This result indicates that tortuousity

is affected by the distribution of sizes of the beads. If

the size distribution is narrow, i.e. the beads are close in
size and in relatively similar quantities, then the tortuousity
(I’) is increased. If, however, the size distribution is
broad, the tortuousity is generally decreased. This is

shown in the results given in Tables 2 and 3. Table 3 shows
an exception for the case with an average tortuousity of
0.57960. This can partially be accounted for by the character
of the bead size distribution for this mixture. The distri-
bution curve for this mixture was not smooth and unimodal,

but had a "hump" at both ends as well as in the center of

the distribution curve. With this being the case it is
difficult to compare this set of data with the results from
the other size distributions.

It will be noticed that a relatively narrow range of
values were obtained for the porosities of the bead mixtures.
The beads were randomly packed into the columns, and after
each run the packing was changed. An investigation of the
experimental results yield no progression of porosity with
particle sizes. Unfortunately, though, the range of data

(on porosity) is not extensive enough to provide reasons to

27

the lack of progression in the data. As mentioned previously,
it was hoped that the mixtures #l, #2, and #3 would have had
lower porosities so as to broaden the range of data and

help elucidate the affect of porosity on gas phase diffusion.
As a result of these mixtures though, it is felt by this
author that a combination of bead sizes with the influence

of compaction of the bed will yield lower porosities.

It was mentioned that the columns were repacked after
each run. This was done so as to maintain the randomness of
the packing geometry which might possibly have been lost as
a result of settling. Settling could occur by disturbing the
columns as a result of activity in the environmental room
and with taking readings of the acetone level in the columns.

The experiments were conducted to obtain data that would
allow for the computation of the effective diffusion
coefficient, DACE' The tortuousity could then be computed
by using the value obtained for bACE and DACE’ the diffusion
coefficient of acetone through air from an empty column, in
themanner described in the Background Section.

In the analysis of the data two statistical techniques
were employed. A linear regression of the data was performed.
This data was of a temporal sequence, or continuous with time.
The linear regression of this data yielded information, via
the computed slope of the line, as to the variability of
results. It will be noticed in the results given in Appendix

C that the slopes were very low, hence proving consistent

28

results. If the slope of the line was very great, it would
indicate that diffusion was not constant with time which would
be false for the experimental conditions in this study.

The second statistical technique used was the least
squares test. The purpose of this test was to find out
again the consistency of the results. But, for this test,
each time a data reading was taken it was considered a
separate experiment. That is there was no consideration for
continuity with time. The result from this test would yield
an average value for the data and could be compared to the
intercept of the line obtained from the linear regression
technique for an indication of consistency of results.

In regards to the values shown on the computer outputs
in Appendix C it will be noticed on a few of these sheets
that some rather large exponents occur on some of the values.
This is due to a programming error on the part of the author
of not properly dimensioning the array in which these values
are stored.

It was mentioned in the Background Section that in this
exercise it was assumed that no Knudsen diffusion or that
no adsorption on the bead surfaces occurs. The former can
be verified mathematically. However, it is a little more
difficult to prove that there is no adsorption on the bead
surfaces. The beads are inert to the acetone vapor, and the

porosity of the packing is sufficient for the vapor to pass

29

through the column. Hence, it is generally accepted that
if adsorption is occurring its affect is negligible.

In comparing the results of this study with those of
studies given earlier in the paper, it will be noticed that
generally agreement was not obtained. It was stated that
Penman obtained an average tortuousity of 0.66. It is felt
by this author that Penman's work was too simplistic, and
it did not concern itself with the effect that different
packing distributions have on the tortuousity of the matrix.
The work that Linvill performed appears much more realistic,
and hence it is felt that a logarithmic relationship exists

for tortuousity and porosity.

SUMMARY AND CONCLUSIONS

For diffusion of acetone vapor in air through an empty
column it was found that the acceptable literature value
could be obtained within 5% error.

For diffusion of acetone vapor in air through a packed
column (a porous bed) it was found that the size distribution
of beads influenced the tortuousity and porosity of the
packing. This in turn determined the (effective) diffusion
coefficient for the system being studied.

Narrow ranges of porosity were obtained. It is felt
that this was partly due to improperly mixing the various
bead sizes and partly due to a lack of compaction of the
bead bed. The values of porosity that were obtained are
generally near the maximum attainable value. These porosities
represent packing geometries close to orthorhombic or cubic
geometries.

The average diffusion coefficient obtained for a packed
column system was 0.09660 square cm per second. This value
is associated with a tortuousity of 0.7182:0.0665. However,
the use of an average tortuousity may be rather limited due
to the fact that the tortuousity is affected by the size

distribution of the packing.

30

RECOMMENDATIONS

There are many areas of engineering where information
of the nature given in this work is desired. Much additional
work is needed to complete the study undertaken here.
Specifically additional information acquired at lower
porosities is desired, for it is these lower porosities that
occur in natural soil systems. Therefore it is recommended
that additional experimentation, if it should be undertaken,
focus on attaining packed columns with porosities around
0.30. Furthermore, a series of packed beds having a range
of porosities is desired. Information from a study of this
sort should yield a useful (empirical) mathematicalexpression
that could be used in soil studies, fluidized bed systems,
adsorption columns, and other important applications in the

field of engineering.

31

BIBLIOGRAPHY

BIBLIOGRAPHY

Bear, Jacob; Dynamics of Fluids in Porous Media;

 

Chap. 2,4; American Elsevier Publishing Co., Inc.;
New York (1972)-
Bird, R. Byron; Steward, Warren E.; Lightfoot,

Edwin N.; Transport Phenomena; Chap. 16,17; John

 

 

Wiley & Sons, Inc.; New York (1960).

Lee, C.Y. and Wilke, C.R.; "Measurements of Vapor
Diffusion Coefficient"; Industrial and Engineering
Chemistry 46(11), 2381 (1954).

Linvill, Dale Edward; MLS. Thesis; University of
Connecticut (1969).

Brown, George G.; Unit Operations; Chapter 7;

 

John Wiley and Sons, Inc.; New York (1950).

32

APPENDIX A

GENERAL REFERENCES

McMurtrie, R.L., and Keyes, F.G., J. Am. Chem. Soc. 19,
3755(1948).
Wilke, C.R., Chem. Eng. Progr. 46, 95(1950).

Carslaw, Introduction to the Mathematical Theorngf

 

 

the Conductance of Heat in Solids, New York, Dover

 

Publication, 1945.

Van Bavel, C.H.M.; "Gaseous Diffusion and Porosity

in Porous Media", Soil Sci. 13, 91-104(1952).

Shearer, R.C., and Millington, R.J., and Quirk, J.P.;
"Oxygen Diffusion Through Sands in Relation to
Capillary Hysteresis--Quasi Steady State Diffusion of
Oxygen Through Partly Saturated Sands", Soil Sci. 191,
432-436(1966).

Perry's Handbook for Chemical Engineer's, McGraw-Hill

 

Publishing Co., 4th Edition.

33

APPENDIX B

34

TABLE 4.--Dimensions of the Glass Columns.

 

 

Diameter (ID in inches) Length of Tube (inches)
0.5444 6.230
0.5800 6.341
0.5750 6.306
0.5758 6.211
0.5422 6.288

 

0’6

35

NOMENCLATURE

3)

average density of component A( gm'cm—

dispersion coefficient (cm2'sec-l

diffusion coefficient (cm2'sec-l)
average tortuousity tensor
distance coordinate of component 1 (cm)
diffusion coefficient vector (cm2'sec-l)
average velocity (cm'sec—l)

mass flux relative to moving coordinates
(gm-cm’z-seC'l)

mass flux relative to stationary coordinates
(gm-cm'Z-sec’l)

system pressure (mm Hg)

partial pressure of component 1 (mm Hg)
length at level 1 (cm)

-1°-1

gas constant (cal-gmole ~ K )
temperature (°K)

media's porosity

media's tortuousity

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