MIXING IN A FLUID FLOWING
THROUGH A PACKED BED
by
B,
A,
P.-V. Danckwerts
A.
Oxford University
1939
B.
Sc.
C. Sugden
London University
1943
Submitted in Partial Fulfillment
of the
Requirements for the Degree of Master of
Science in Chemical Engineering Practice
from the
-Massachusetts Institute of Technology
1947
A
Signature of Authors
ri
w
-
-
/1
Signature of Professor in
Charge of Research
Signature of Head of Department
(
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May 26, 1947
Professor J. S. Newell
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
Dear Sir:
In accordance with the regulations of the
Faculty, we hereby submit a thesis entitled "Mixing
in a fluid Flowing Through a Packed Bed," in partial
fulfillment of the requirements for the degree of
Master of Science in Chemical Engineering Practice.
Very sincerely yours,
A. C. Sugden
P. V. Danckwerts
/pr
291792
&a
K NOW LE D
NEKE X'7.
The authora wish to take this opportunity to express
their thanks; to Professors E.R.Gilliland, A..K.Smith,
and ELS.Mickley for their many helpful critiisms and
auggestions.
Thanks are also due to J.Polack for his auggestions
and encouragement in- all.
stages of the research.
TABLE OF CONTENTS
I StMMA~cr.
......
..
........
* ..
. 2.
II INTRODUCTION.......................3
III APPARATUS AND PROCEDURE ......... 10
IV RESULTS...........................I4
V DISCUSSION OF RESULTS..............15
VI CONCLUSIONS.....................*.23
VII RECCOMIENDATIONS FOR FUTURE
WORK.o*****........25
VIII APPENDICES
A .GEOMETRY OF SPHERICAL PACKINGS,
AND DERIVATION OF.A GEOMATRICAL
MIXING COEFFICIENT.............28
B-.
FLUID FLOW THROUGH PACKED
TOWERS........................39
C. DERIVATION OF "PROBABILITY"
EQUATION, AND COMPARISON WITH
WILSONEQUATION..............
50
D. CALIBRATION OF FLOW MAN,
OMETER AND CO.RIMITRIC ...
STANDARDS.*.**..........*....58
E. ADDITIONAL NOTES ON
EXPERIMENTAL TECHNIQUE. ......
59
F. SAMPLE CALCULATION............63
H. NOMENCIATURE, DEFINITIONS
AND SYMBOLS...................66
I. BIBLIOGRAPH..*..**..........68
TABULATED RESULTS
GRAPHS
FIGURES
IS
U MMA R Y
I
I
S UMMARY.
The mixing processes which occur in a liquid
flowing through a packed tower have been studied and measured.
Water was passed up a packed tower, and an indicator solution of methylene blue injected at the axis.
The rate at which mixing occurred was determined by taking
samples from various points and measuring their concentrations.
The process of mixing was found to conform to
the normal diffusion law, or to the law derived from
"random walk" theory. (the two give very nearly the
same
expression),
and an "apparent diffusion coefficient" D
could be calculated from the distribution.
For spherical particles (with which most of the
work was done):
00
/-
(where (Re) is the modified Reynolds' number), when (Re)
was in the range 10 - 100.
In this range it is considered that flow is
turbulent in some parts of the packing,
others.
The mixing process is
and streamline in
thought to be caused by:
2
local turbulence (leading to "eddy diffusion) : splitting
and scattering of streams by channeling (caused by
fluctuations in the specific resistance to flow of various
parts of the packing) : and possibly by displacement of
elements of the liquid when they impinge on the particles.
II IN-T RO D U-C T I ON.
INTRODUCTION
Utilitarian aspect of the problem.
At the moment fixed-bed catalytic
reactors are designed largely on the "hit-or-miss" or "build-'and-try"
principle, simply because it is not possible to calculate the dimensions
and other characteristics of a reactor from small-scale experiments.
Since the rate of a gaseous reaction at any point in the catalyst bed
depends on the temperature and composition of the gas, it is clear that
the rationalisation of reactor design must depend on a knowledge of the
coefficients of heat-9nd--mass--transfer through the bed.
latter, it
is
As regards the
known that the molecules of the gas do not in
general travel
in straight lines parallel to the apparent axis of flow, but follow a
devious path, so that mixing takes place transversely and longitudinally.
The temperature and composition of the gas at any point will, therefore,
depend, among other things, on the rate at which this mixing takes place.
Object of Present Work.
The experimental work described in this
thesis was designed to throw some light on the mechanism and quantitative
characteristics of the mixing which occurs when a fluid flows through a
bed of particles of uniform size and shape.
handle than gases,
it
Since liquids are easier to
was decided to do the preliminary work using water
as the moving fluid, on the supposition that there would be a general
similarity to the case of a gas-stream, particularly at c6mparable
Reynold's numbers.
Although it was desirable to cover the range of
Reynolds' numbers which might be expected in a catalytic reactor, it was
also of interest to gain a picture of the behavior of the system over
as wide a range as possible.
In practice, the upper limit of flow rate
/f
was fixed by the limitations of the apparatus, and the lower by limitations in the technique of measurement employed; the higher Reynolds'
numbers were of the order of magnitude of those which might be encountered
in a reactor.
Nature of the Phenomenon.
In a gas-stream, mixing may be expected
to occur in several ways during passage through a packing:
sion, caused by thermal motion of the molecules:
normal diffu-
mixing due to turbulence:
and possibly scattering caused by the diversion of various parts of the
stream as they impinge on solid-surfaces.
In a liquid it was assumed
(and with justification, as the experiments showed) that the normal diffusion coefficient would be negligibly small in comparison with the other
effects at the rates of flow employed.
We are thus left with turbulent
mixing (or eddy-diffusion) and with the possibility of another distinguishable mechanism, which will be called displacement mixing.
Although no previous work on packed tubes had been published, a good
deal has been done on eddy-diffusion in open tubes.
fluid in
Mixing occurs in a
turbulent flow, because any element of fluid may have a backward,
forward or sideways component of velocity superimposed on its
velocity.
mean flow
The size of the "element" undergoing these fluctuations, the mag-
nitude of the random velocities attained and the mean distance travelled
between changes of direction are indeterminate; but the effective result
is a mixing process, analogous to molecular diffusion, and susceptible to
similar mathematical treatment.
In the case of molecular diffusion, the coefficient of diffusion D
is defined by the expression
A.bY
'
CI
where t
is the rate at which molecules of the diffusing species cross
area A of a plane at right-angles to the X-coordinate, c being the concentration of the diffusing species in molecules per unit volume.
This
is equivalent to the expression
The foregoing expressions are true for .the case where the concentration
varies in one dimension only.
For the more general case of variation
in 3 dimensions:
The general validity of these diffusion laws can be demonstrated
by considering the effects of random, independent movements of the
diffusing molecules, such as occur in a gas.
In a stagnant mass of gas,
for instance, containing two or more species of molecule, any macroscopic
variations in composition will in time disappear, because purely random
movement will cause a net transfer of molecules of each species from regions where their concentration is high into regions where it is low.
It is only necessary to assume that an individual molecule is equally
likely to move in any direction at any moment, and that the mean length
of individual displacements in all directions is equal to show that the
diffusion laws given above will follow. (see Kenard's "Kinetic Theory of
Gase s" ).--
Sob jee-
ti tr li
l Wias PoeetiOled
lae
In turbulent motion the elements undergoing random displacement are
of macroscopic dimensions, and not individual molecules; the velocity of
displacement will be comparable to the flow-velocity, and hence much slower
than the thermal velocity in a gas; and the "mean free path" will no
doubt be much greater than for molecular movements.
It would be expected,
however, that the net effect, in a system large enought to dwarf the
fluctuations due to turbulence, would be analogous to the molecular diffusion.
The theory of eddy diffusion, which will not be discussed in detail
here has been developed largely by G. I. Taylor (10); the work of v. Karman
(11), Prandtl (20), Dryden (12), Sherwood and Woertz (13),
Goldstein (15) may also be consulted.
Murphree (14),
In place of the "mean free path" of
kinetic theory the concept of "mixing length" is employed-the mixing
length being the mean distance which an eddy travels before breaking up
and losing its
identity; and the root mean square velocity is
replaced
by the mean instantaneous deviating velocity in a direction perpendicular
to the main flow.
Neither of these quantities can readily be derived
from the parametus of the system; in fact the present state of the theory
does not admit of a priori predictions of the eddy diffusivity (or eddy
diffusion coefficient) such as can be made for the molecular diffusivity
D from kinetic theory.
However, experimental work has shown that the turbulent mixing., process
can be described by the normal mathematical diffusion laws, substituting
an eddy-diffusivity E in place of D
-
see Sherwood and Woertz (13),
Sherwood and Towle (5), Weddell (17), Hawthorne (18), all of whom were
investigating turbulence in open tubes, containing no packing.
A similar
conclusion was reached by Kalinske and Pien (19), who investigated mixing
in open troughs.
When a fluid flows through a bed of solid particles at a sufficient
rate to give rise to turbulence, therefore, it is to be expected that a
7
diffusion-like mixing process will be caused by eddy-diffusion, but it
is not possible to make quantitativepredictions.
At lower rates of
flow, where turbulence is non-existent or negligible, it would appear
doubtful at first sight whether mixing would occur at all; it might be
expected that any two particles which entered the system close together
would remain close together.
This certainly is the case in a fluid in
streamline flow past a single sphere (see Fig. IV(i).
However, considera-
tion of conditions in an assembly of spheres suggest that two particles
which are separated by the splitting of the stream by a sphere may never
regain their former relative positions, owning to the diversion of streams
of fluid by other spheres in the assembly into the "gap" between the
particles (see Fig. IV (ii)).
Thus it is possible, without postulating
discontinuities in the velocity distribution such as occur in turbulent
motion, to conceive of purely streamline or viscous flow leading to permanent separation of particles which were originally close together.
This is in effect a mixing process, and an attempt has been made to derive
an expression for the corresponding mixing or diffusion coefficient,
making certain simplifying assumptions (see Appendix "A").
This coefficient
is referred to as the "geometrical" mixing coefficient, and the process as
displacement mixing.
However, considerations advanced in Appendix B suggest
that the situation is complicated by the existence of channels in the-packing,
which lead to gross variations in the mass-velocity from region to region,
so that the path of a particle is probably not determined soldy or even
predominantly by the configuration of the solids in its immediate neighbourhood.
The only experimental investigations of fluid mixing
in a packed tower have been carried out by Rabinowitz and
Roessel (4)
, who worked out the apparatus and technique
which have, with certain modifications, been used in the
present work.
It is thought, however, that their quantit-
ative results were probably in error, owing to the excessive
rate at which the indicator solution was introduced (see
Appendix E ).
Diffusion and random scattering;. Rabinowitz and Roessel
used an equation known as the Wilson equation (6) to calculate
the mixing or apparent diffusion coefficient kmxwx from
measurements of concentration at various points in the tower.
The Wilson equation is a solution of the general diffusion
equation given earlier in this section, after the appropriate conditions have been inserted, and making the assumption that diffusion parallel to the axis of flow (i.e.
longitudinal mixing) may be ignored.
The present authors
derived an equation similar to that used by Kalinske and
Pien (19), by considering the distribution attained by
particles undergoing random scattering;
the derivation
and final form of this "probability equation" is given in
Appendix 0.
There is a discrepancy between the Wilson equation
and the probability equation, although ynder the conditions
of the experiments described here the divergence was within
the experimental error.
The reaon for the discrepancy
between the two equations is discussed in Appendix C.
III
APPARATUS
AND
P ROP 0E D U R E
/0
111.
Apparatus and Procedure
(See Appendix E forfurther details)
Apparatus. This consisted of a constant head tank,
providing a flow of water which was metered by an orifice
and manometer.
The water flowed up through a container filled
with coarse packing, and then through a 4-inch glass tower
which contained the packing under investigation (supported
on wire gauze), and finally overflowed at the top of the
tower. The indicabor solution was introduced at the bottom
of the tower by a capillary tube set on the axis.
Samples
could be withdrawn from various points on th e packing by means
of five 1/64 inch steel capillary tubes inserted into the
packing from above;
these were rigidly mounted on a metal
frame which held them vertically with their open ends in
one horizontal plane and situated along a diameter of the
tower.
The samples were collected simultaneously in five
test-tubes by applying a vacuum to the air-spaces of the latter.
The tips of the sampling tubes were usually about an inch
below the upper surface of the packing, and the total depth
of packing in the tower was varied as convenient.
Procedure. Water was admitted at thedesired constant
rate, and then the indicator solution ( methylene blue, 25
gms./litre) was admitted thrugh the injection tube.
The rate
of injection of indicator was adjusted
to give a convenient
depth of colour for
determinations (see
Appendix E).
the concentration
When steady state hed been reached a vacuum was
applied to the collecting-tubes, and samples withdrawn from
the tower.
The first portions of liquid collected were thrown
away, as they served to wash out the collecting-tubes.
samples properv were then collected.
The
It appeared desirable
to do this as slowly as possible, to avoid drawing into the
tubes: liquid franm remote,parts of the system. The time required
to, withdraw liquid through the: capillary tubes at the same
linear' velonity- as that
of the streami through the packing would
have been prohibitive,, so the lowest convenient rate was used
-- 10-20 minutes for a sample: of about 25cc.
from each tube--I.e.
2-20 times the linear- flow velocity of the stream.. The effect of
this was to- increase- the effective area of cross sezt:Eom from
which,the sample was withdrawn;; it
is
not believed too have dis-
-torted the results. It had been determined (see Appendix C)that
the: interpretation of the results did not depend on a constant
rate of injection of indicator, as long as the mean concentration
of the effluent was used for C* in the computation.
Since the
injection rate was not positively controlled, samples of
effluent were taken at regular intervals during the sampling
period, and mixed.
There was a tendency for air to come out
of solution and form bubbles on the packing, and it was feared
Preheating of the water
that this might affect the results.
wa' tried and found to be useless.
The effect was mitigated by
stirring the packing before each run, thus dislodging the
bubbles, which did not reac4.a size comparable with the packing
particles during the course of a run.
Colorimetric analysis of methylene blue solutions. Methylene
blue was used as an indicator because it can be estimated
accurately and easily by its colour at great dilutions.
Owing
to the limitations on the rate of injection of indicator (discussed in Appendix E), the concentrations to be measured were
as low as 1/4000 that of the indicator solution injected.
The
solution used, since it contained only 25 gms. of methylene
blue to a litre, was considered unlikely to differ
greatly in
density or viscosity
from pure water.
A Duboscq type colorimeter was used.
analysis was a 1000
The standard solution used for
dilution of the indicator solution.
structed experimentally,
A curve was con-
showing color-density as a function of concentra-
tion (departure from Beerts law is
great).
A number of sub-standards of
To determine the concentration of
various concentrations were also used.
a sample, its color-density was determined by comparison in the colorimeter
with one of the standards (the nearest one, because color as well as density appeared to alter with concentration) and its concentration read off
the curve.
Pressure-gradient Measurements.
In order to measure the pressure-
gradient through the packing at various velocities, pressure-taps were
inserted below the gauze supporting the packing and above the top of the
This arrangement gave very erratic
p:.cking, and connected to a manometer.
results, probably because air bubbles collected under the gauze and much
Since the greatest conceivable Pitot effect
increased its resistance.
was estimated to be negligible, it was decided to insert tubes directly
into the packing.
A "pressure-probe" of the type shown in Fig. VIII was
used, the ends of the tubes'being turned upwards to prevent any bubbles
entering or blocking them.
Although results were inclined to be erratic,
they were very much more reproducible than with the original arrangement.
Calculation of Diffusion Coeffiqients.
Several sets of samples having
been taken with the flow velocity constant, a plot was made of
C
e
D
y
(cx
(*)being
averaged for all samples with the same v and X) and the best
straight line consistent with the "probability"' equation was drawn through
the points.~ Either the slope or intercept of the line gave the value of
FD.
Details of the calculation are given in Appendix F.
IV
RESULT S.
IV
RESUL TS
The following is a table of the results obtained in this investigation
Packing type
and size
6 m glass
Free
volume
0.36
Superficial
Velocity CMS/sec
Reynolds Number
Friction Factor
volue
Pressure gradient
Diffusion
Dynes/sq. C
Coefficient
2
Dyes/s.
CMCmOms
/sec.
0.25
11.99
61
0.0129
0.0103
0.314
15.05
52.0
0.0174
0.0165
0.32
15.31
51.5
0.0179
0.0200
1.07
51.3
24.5
0.0956
0.0834
1.66
79.5
19.5
0.183
0.196
1.85
88.7
17.5
0.204
0.225
15.5
0.279
0.389
75.5
0.0301
0.0105
beads
2.30
0.36
4 mm glass
0.28
110.
8.93
beads
P/8" catalyst
0.40
spheres
h"
Berl Saddles
4'3/8-inch
0.59
0.69
21.9
41.0
0.0995
0.0334
1.22
38.9
29.0
0.220
0.0812
1.75
55.8
23.3
0.363
0.133
2.26
72.0
20.5
0.534
0.172
2.36
75.3
20.0
0.568
0.224
1.06
92.2
11.4
0.0240
0.0201
1.06
92.2
11.4
0.0240
0.228
1.91
97 (nominal)
32.8*
0.0766
0.354
see appendix B
was manufacturer's nominal diameter. Displacement measurement of volume gave mean diameter as 11 mm.
V
D
U S S IO N
OF
R E S U L T S
/,
V. Discussion of Results
Accuracy of Experimental Work.
The chief source of inaccuracy
was the fact that the concentration distribution was not continuous;
particularly at low rates of flow, the concentration varied sharply
from point to point.
Since the samples were withdrawn in effect from
a point source, it was necessary to take a considerable number of individual samples and average their concentrations in order to achieve
reliable results.
The situation is analogous to that of plotting the
density of shots on a target; if samples consist of small, constant
areas of the target, the deviation between individual samples taken in
the same region will be least at the centre, and will become larger as
one goes outwards to regions of lower density.
In the same way, the
variation between individual samples taken in corresponding positions
became greater as one movea( further from the axis of the tower.
An experiment was performed to show the point-to-point variations
in concentration.
The five sampling tubes were arranged so that they
formed two compact groups, the members of each group being very close
together.
The following results were obtained:
4.
d
(cms):0.97
1.16
1.40
(a) ::10.3
18.4
8.3
(b)
25.3
14.2
9.0
..............
2.87
3.10
..............0.33
0
0.37
0
...............
The samples (b) were taken immediately after (a), wit hout disturbing
the apparatus.
It can be seen that the relative concentrations do not
decrease uniformly as -, increases, that large variations occur within
short distances, and that the configuration changes somewhat with time.
In the above experiment (Runs 67 and 68) the value of
. was 20 cms.
and of the Reynolds' number 16.
It is felt that the error in the measurements of concentration
was negligible, compared to those arising from the above effect, since
they were reproducible to within about 3%.
A possibility of error lay in the fact that the linear rate at which
samples were withdrawn through the sampling tubes was greater than the
linear velocity through the tower (see Section III).
The effect of this
was to draw in liquid from regions remote from the actual sampling-point.
Here, again, however, it is felt that since the velocity of withdrawal
was at most 30 times the rate of flow, and hence the effective sampling
area was only about 5 times that of the tube in diameter, the error introduced was small compared to that due to fluctuations in the concentration.
Indeed, by averaging the concentration over a wider area than the
cross-section of the sampling-tube, the sample-to-sample variation may
have been reduced.
Other sources of error which have been considered, and which are
felt to be probably unimportant are: the excess of the speed of injection
of indicator over the flow velocity, and the influence of air .bubbles on
the effective shape and free volume of the packing (see section III.)
The ne*t error in the diffusion coefficient is difficult to estimate.
From the consistency of the graphical correlation plots, it is considered
that the maximum error is about 30%, and the average error between 5 and
10%.
The error in the measurement of the pressure-drop through the packing was of the order of
ing.
--
- ,ms.e
of water for 20 cms. of pack-
The reason for this error (when using the "probe" described in
Appendix E) is not known; it obviously amount to a considerable relative
error at low velocities and in the case of packings of low specific
17
resistance, such as 3/8-inch spheres and Berl saddles.
In fdture it
might be desirable to use a considerably greater depth of packing to
measure pressure-gradients.
Agreement with distribution equations.
An inspection of graphs
9 - 18 will show that considering the discontinuities in the system,
the distribution of concentration as found by averaging several measurements agreed remarkably well with either the Wilson or the probability
From an engineering point of view it
equation.
(if
is immaterial which law
either) is followed exactly; however, the probability expression is
more convenient,
since it
is
easily integrated over a circular section;
it is thus possible to calculate, for instance, the length of a tower of
given diameter which is required, under given conditions, tolliron-out"
or homogenise a fluid stream of non-uniform concentration (such a calculation is made, for example, on pages 21, 22 of Vol. I of the original
lab. notes).
Under the conditions obtaining in a catalytic reactor it
is thought that abrupt discontinuities in concentration are unlikely to
be encountered.
In the first place, the Raynolds' number is likely to
be such that considerable turbulent mixing will take place; secondly,
the gaseous molecular diffusion co-efficient may be of the same order of
magnitude as the coefficient of mixing due to flow, so that interdiffusion will occur between neighboring regions at all rates of flow.
Correlation of Results.
Assuming that the magnitude of D depends
only on the superficial velocity WY the fractional free volume, F, the
density
and viscosity/of the fluid, and the diameter of the pack-
ing particles d, and that the tower diameter does not influence it (see
Appendix B), dimensional analysis shows:
is a function of
or
is a function of
For the spherical packings, all but D, v and d are constant.
Graph V
shows a plot of D against
Since the friction factor
(where
A
f, defined for spherical particles as
p is the pressure drop in absolute units through a depth L of
packing) is a unique function of the Reynolds' number (see Appendix B),
D was also plotted against f (graph IV).
Within the range of conditions
covered by the experiments, these plots show that, approximately:?
003
.
(F was donstant at 0.36 for all the experiments except those involving
11 mm.beads and Berl saddles; hence its part in the correlation cannot be
determined.)
It is clear that whatever the exact form of these two func-
tions, d can be eliminated between them; this would be true even for non-
spherical particles, using arbitrary values of d.
shape of particle, it
)
AP
Thus for any given
must be possible to express D as a function of
and of some dimensionless factor to account for varia-
tions in the fractional free volume.
If it is desired to correlate data
in which the shape of the particles and the fractional free volume vary
independently of each other and of the other variables, then an empirical
factor EM
it
have to be introduced to account for each of these.
However,
appeared to the authors that shape and free volume might not appear
in the correlation expression as independent variables, since their effects might be absorbed into
A
applied on the assumption that
j
only, and it
.
is
Hence dimensional analysis was
a function of
uA
was found that
is a function of
Since/I- and f were constant throughout, D was plotted against
/IAf
(Graph VII)
For the spherical particles,
2
and the single value of
by this expression.
('C:)-
.O.Z(C)
0-2
for Berl saddles was within 15% of that given
Although not conclusive, this is a promising result,
since Berl saddles are far removed from the sphere in shape, and their
fractional free volume was 1.6 times that for the spherical packings.
If
further experimental work on non-spherical packings confirms the
validity of this relationship, it
would enable values
should be extremely useful, since it
to be predicted for such packings from
pressure-drop measurements alone, and these are relatively easy to carry
out or (in
some cases) to find in
the literature, while any expression
involving the friction factor or Reynolds number suffers from the indeterminacy of these numbers for irregular shapes.
It is interesting to note that if the Reynold's number for 1/4-inch
Berl saddles is calculated using this nominal diameter as d, the single
value of D obtained for this packing falls close to the same line as that
relating D to Re for spheres.
However, it is not felt that this is more
than a coincidence in view of the considerations put forward in Appendix B6
In particular, the free volume for the saddles is much larger than for the
spheres (0.59 and 0.36 respectively), and since there is every reason to
believe that D depends on the free volume, it
tion.
should enter into the correla-
In addition, there is some reason for thinking .that the "nominal" Reynolds'.
number for Berl saddles does not correspond numerically with the Reynolds'
number for spheres-
i.e. that the change from turbulent to streamline flow
does not occur over the same numerical range in the two cases. (See Appendix B).
Interpretation of Results.
It does not appear feasible to provide a de-
tailed and quantitative picture of the phenomena which lead to the type of
mixing observed.
There is good reason to believe that in all cases the
flow through the packing was turbulent in some spots and streamline in
others; hence one would not expect the diffusion co-efficients to bear any
more simple a relationship to the Reynolds number in this region than does
the friction-factor.
It is suspected that at least two mechanisms are
,iixixng
effective in this range-
turbulent/or eddy-diffusivity, and "displacement"
mixing; since the relative effects of the two mechanisms are probably varying in importance, a quantittative analysis would be difficult, even were
it possible to predict the effect of one mechanism alone, which it is not.
The fact that the simplified "geometrical" displacement co-efficient derived in Appendix B gives an approximation to the observed results at
2/
low rates of flow may or may not be significant.
(Graph VI shows a plot
of the values of D predicted by this expression for varying values of the
Reynolds number; taking
ship has the form
f=/
F= o31
D = 0,00065 (Re)
).
-o,
,
the relation-
Further experiments at still
lower rates of flow would be required to show whether the apparent agreement
is coincidental.
As far as could be observed in qualitative experiments,
channelling, caused by fluctuations in the specific resistance to flow from
point to point in the packing, is responsible for some of the mixing which
took place.
No quantitative theory of this effect is offered; it would be
expected to lead to a "random-walk" type of distribution, but the value
of the coefficient is difficult to estimate.
arily
It would not appear necess-
to lead to a coefficient of the same magnitude as that calculated
for the "geometrical" coefficient.
The experimental results indicate that there is probably a mixing-effeet even when the flow-velocity is
5) that flow is
so low (i.e.
completely streamline.
Reynolds'
number less than
In such conditions point-concentra-
tions would be expected to vary abruptly between zero and the original indicator concentration; thelmixing" would consist in the scattering of filaments
of the indicator across the tower.
In a regularly-arranged bed of isometric
particles channelling could presumably not occur.
It would be interesting
to investigate such a system at low Reynolds numbers.
Comparisonwith Heat-Transfer Data.
Since experiments on the transfer of
heat in a packed tower through which a fluid is
out at the Institute at the moment,
of the present work on the subject.
flowing are being carried
some remarks may be made on the bearing
The overall heat-transfer coefficient
or apparent conductivity in a stream of gas passing through a packed tower
is
presumably due to three different types of mechanism -transfer
due to
the mass-transfer caused by the mechanism investigated in the present work;
2
nprmal thermal conductivity of the gas; and transfer of heat
from gas to solid, conduction through the solid, and re-transfer
The following expression includes a term for
to the gas beyond.
each mechanism:
R'
S/
where k' is the overall apparent cogiductivity, D is the apparent
the
diffusion coefficient as measured in the present work,
density of the gas, and S its specific heat.
F is the fractional
free volume - also equal to the fraction of any plane free from
solids and hence available for gaseous conductivity;
/ (
74)
is a function of the conductivity of the material of which the
packing is composed and of the velocity and physical properties
of the gas.
Stadig (2/) has shown that for air at a Reynolds' number in
the neighbourhood of 100, the value of k' is of the order of
0.2 B.t.u
/
saddles.
Using our value of D for Berl saddles at a comparable
hr.-ft.- F., when the packing used was '-in.
Berl
Reynolds' number, it is found that the first term is about 1/10
of k', while Fk
is about 1/40 of k' .
Hence most of the trans-
fer apparently takes place by the third mechanism - i.e. by the
path gas-solid-gas.
VI
0 03N 0 L U S I O! N S.
VI
CONCLUSIONS.
(a) Mixing, in the sense of scattering of elements of fluid
through the packing,
Reynolds'
occurs over the entire range studiqd -
numbers from 10 to 110.
(b) - The general form of the concentration-distributin in the
tower at steady state was consistent with either the Wilson
diffusion equation or with the probability equation derived
from the"random walk"theory.
expressions,
These lead to slightly different
but under the conditions of the experiments the
difference between the two was less than the experimental error.
(c) The apparent diffusion-coefficient so determined increases with flew-velocity and with the diameter of the particles.
For spheres:
over the range studied.
(d) The expression:
3
which does not refer to particle diameter, correlates the single
value obtained for Berl saddles reasonably well with those for
spheres.
(e)
The nature of the flow through the packing is considered
toAviscous in some parts of the packing and turbulent in others
over this range of Reynolds' numbers.
Mixing is thought to be
caused partly by local turbulence; partly by the divergences of
flow caused by channeling ( which is in turn caused by local
fluctuations in the specific resistance to flow ); and also,
perhaps, partly by divergences caused
when the stream divides
to flow past packing-particles.
(f) The distribution of concentration of the indicator in
these experiments appeared to be discontinuous, particularly
at the lower rates of flow, when it varied sharply fromi point
to point; at higher rates turbulence seemed partially to even out
these discontinuities.
In all cases it was necessary to make
and average many individual point measurements to obtain reliable results.
There appeared to be quite a marked change in
the appearance of the stream leaving the packing for relatively
small changes in Reynolds'
(g)
number, when the latter was about 40.
As long as the diameter of the tower is more than five
or six times that of the particles, it is concluded that none
of the characteristics of flow (e.g., pressure-gradient,
apparent diffusion-coefficient) are affected by the ratio of the
two.
V
X E C 0M E- N D A T 1--0IO N S
FO R
FU-. T U R E
W O R K.
VII
Recommendations for Future Work
heads
Recommendations for future work will be made under the following
(i)
Variation of fluid properties and velocity.
(ii)
Variation of packing.
(iii)
-
Pressure-gradient measurements.
(iv) Elimination of air-bubbles from water.
(v)
Qualitative experiments bearing on mechanism.
(vi)
Improved methods of sampling.
(vii) Other modifications to apparatus.
(i)
All the present work was carried out in the transition region
between streamline and turbulent flow.
It would be of interest to
extend the measurements to the purely streamline ard purely turbulent
regions.
The former requires an improved sampling technique, the lat-
ter various changes in
the apparatus.
The viscosity of the water might
be altered considerably by adding small quantities of cellulose ethers.
Finally the investigation should be extended to gases.
(ii) Further work on spheres of different sizes, on Berl saddles and on
irregular particles is required, particularly to see whether the mixing
can be correlated with the pressure gradient.
It would be of interest
to investigate the effect of changes in free volume in the case of a pack-
ing (such as Berl saddles) where this can be varied independently of other
factors.
iii) Correlation with pressure gradierit requires an improved technique for
measuring the latter, with greater accuracy than the method used by the
present authors.
It may well be advisable to measure pressure-drops through
considerably greater depths of packing.
(iv)
The formation of air bubbles on the packing may prove to be serious
in some circumstances-e.g. when using a very fine packing.
The trouble
could be eliminated, it is thought, by giving the water from the taps no
chance to warm up while on its way to, or in passage through, the tower.
Thus the constant head tank and all pipes might be lagged, and the -tower
itself
waterjacketed.
The addition of a wetting agent to the water might
prevent bubbles clinging to surfaces in the tower.
(v)
The use of a two-dimensional transparent model might give some in-
sight into the mechanism of the mixing process -
e.g. cylindrical blocks
between two sheets of glass, or standing upright in a shallow tray through
which water flows.
If cryolite crystals, which have the same refractive
index as water, were used as the packing particles, it should be possible
to see a good deal more of what is
(vi)
going on than when glass beads are used.
As has been mentioned, point sampling is unsatisfactory, particularly
at low Reynolds numbers, because of the point to point fluctuations in concentration.
This means that a large number of samples must be taken and
their concentrations averaged.
It is suggested that area sampling be used.
It has been shown that a standard probability distribution is followed; the
integration of this expression over a circular area concentric with the injection-axis is simple.
All that is required is a method of finding the
mean concentration of indicator in the fluid flowing through a given circular
cross-section.
A suggested experimental arrangement is shown in Fig. XI.
The diameter of the sampling-tube should be carefully chosen with regard
to conditions so that approximately half the indicator flows through it,
otherwise results will not be accurate.
(vii)
If packing-particles of larger diameter than those used by the pres-
ent authors are used, it
diameter.
would be advisable to construct a tower of greater
To obtain reasonable results samples must be taken a good many
packing-diameters from the injection point; on the other hand, samples must
be taken at cross-sections where the concentration at the wall is small
compared to that at the centre.
These two requirements will be incom-
patible if the packing diameter is more than about 1/10 that of the tower.
It may also be advisable, particularly when using packings of low specific
resistance, to use a larger calming section at the bottom of the tower to
ensure that flow is
symmetrical.
The container used for the purpose was
felt to be too small to act efficiently at high rates of flow.
Vi1. A P P E N D L0 E S.
APPENDIX A
The Geometry of Spherical Packings and Derivation of a
Geometrical mixing Coefficient.
If a mass of spheres of uniform size is placed
in a glass vessel, it can be seen that the arrangement of
the spheres is not regular.
Tapping the vessel or
running a rod through it increases the regularity of the
packing on the lower parts of the wall and decreases
the volume of the aggregate; however, it can be seen
that the arrangement of the spheres in contact with
the upper parts of the walls, and of the spheres at
the surface, is far from regular.
It appears that the
system as a whole departs widely from any geometrically
regular system of packing; experiments designed to find
the free volumeF indicated that the closeness of packing
varies considerably with variations in the way in which
the system is treated (e.g., tapped, stirred, rammed).
The sphere-diameter, d, and the fraction of
the total volume not occupied by the substance of the
spheres, F, are the important numerical characteriestics
of a spherical packing.
and constant.
d, of course, is quite definite
F, which may be expected to have an
important influence on fluid flow through the packing
is, as already remarked, dependent on various factors.
It is felt however that under the conditions of the
experiment its
value was essentially constant, since
the packing was thoroughly stirred at ,
under water
between each run, so that the spheres may be assumed
to have settled into their equilibrium position under
much the same conditions each time.
Neglecting for the moment the effect of the
wall of the confining vessel, we may regard the packing
to be isotropic and uniform, in the mass.
If the
fraction of free volume is F, and the number of spheres
in unit volume is N, we have
Nird3
F = 16
Consider a plane drgwn through the packing;
the fraction of the area of this plane lying outside
the spheres is A.
Hence in a lamina of thickness dh,
the free volume will be Adh.
Thus in a finite volume
composed of a very large number of such laminae, in
each of which the free volumeis Adh, the total free
volume will beJAdh = Ah.
Since the total volume will
be h, we have
F
=A
h
i.e., the fractional free area in any plane is equal
numerically to the fractional free volume of the mass.
SO
By a similar argument it can be shown that if
a straight line is drawn through the packing, the fraction of its length which lies outside the spheres is
equal to F.
Such a straight line will cut every sphere
whose centre lies in a cylinder centered on the
and of diameter d (the sphere diameter).
line
The volume of
such a cylinder will be 1- units per unit length, and
4
since there are N spheres per unit volume, and hence N
sphere-centres per unit volume, the line will cut
NTd 2
or
3(1_-_F)
---
4
spheres per unit length.
2d
Similarly, unit area of any plane cuts all the
spheres lying in a volume of 2d, which is .l
-
TTd
F) spaheres.
2
Furnas (1) argues that the effective free area
for the flow of a fluid is not F.
He concedes that the
fractional free area of a plane drawn through the
packing would be F, but argues that this is not the
cross-section available for fluid flow, since " 'free
area' is by no means all effective in allowing the passage
of (fluid) as the bulk of the area is blocked by overlying
pieces of solid material".
He mentions, but does not
reproduce, geometrical considerations which show-the
effective free area for spherical packings to be a fraction of F.
The authors are unable to agree with his
argument, so far as they understand it, and consider that
3/
at any rate for smooth, non-re-entrant shapes such as
spheres the free area must be taken as F.
Furnas devotes a good deal of attention to the
eff ect of the wall of the confining vessel on the
pressure-gradient--flow-rate relationship of the packing.
There is
a prima facie case for supposing that the
effect of the wrall is
to increase the free volume in
the layer of spheres immediately adjacent to it, and
thus to offer a path of lower resistance than through
the main body of the packing.
Consider a layer of spheres
resting on a plane; there is a considerable amount of
free volume between the plane and the spheres.
If now
the surface of the plane were embossed with hemispherical humps, these would partially fill the spaces between
the spheres and the plane, and reduce the free volume.
This latter arrangement may be supposed to approximate
more closely to the conditions in the body of the
packing.
However, as hasa.already been mentioned, the
spheres in contact with the walls tend to be more
regularly arranged, as far as can be seen, than those
in the body of the
packing;
under the conditions of
our 4experiqgents, quite- a-close approach to hexagonal
packing was attained over the major part of the towerwall.
This would tend to reduce the free volume in the
outer layer.
A calculation has been made to find the
free volume in a layer of thickness d/2 adjacent to a
plane on which spheres of diameter d are lying in hexa-
gonal (i.e., closest possible) packing (see Fig.I);
The value of F in this layer is about 0.4, and since
the value of F in the body of the packing is about 0.36,
the difference is not great.
More will be said in
Appendix B on this subject in relation to flow.
A
number of experiments were carried out to find the
6h an-io os 5XAapfa5 a49Wy.
overall free volume in packing confined in essel/in
%
wall area
which the ratio ------------- varied widely, with a
total volume
view to extrapolating to zero value of this ratio to
find the value of F in the body of the packing.
The
earlier experiments appeared to show a trend of the
type which would be expected--i.e., F decreased as the
proportion of wall area to volume of the vessel decreased.
At this time, however, the effect of the
treatment of the packing on F was not fully realized.
A later set of experiments, conducted in cylindrical
vessels, in which the packing was consolidated under
water to a constant volume by a standardized method of
ramming before F was determined, showed that the value
of F was substantially constant at 0.36 - 0.37 provided
that the diameter of the containing vessel was more
than say five times greater than the diameter of the
packing.
It is considered that the increase of free
volume above a certain value of the diameter ratio is
Table
Free Volumes for Glass Spheres in
Cylindrieal Towers
Particle diameter
Tower diameter
0*42
0.43
0.37
0.36
0.36
0.36
.247
.225
.150
.130
.064
.043
due to an effect on the mode of packing in the interior
of the vessel, rather than to an excess of voids in the
immediate vicinity of the wall.
The 11 mm.
spheres (nominal diameter 3/8 ins.)
gave a value of F of 0.4; ratio of particle to tower
diameter was 012.
But the particles were not entirely
uniform in -size and shape, which may account for this
high value.
The free volume was determined in one of two
ways:
(a)
Some water was poured into a cylindrical
vessel, some packing added and consolidated, and the
apparent volume, V,, of the packed region and total volume
V 2 , of packing and water measured.
More packing was then
added and consolidated as described; the apparent (V3 )
and total volumes were noted again,,
em -
V 4 - V2
V3 meV3.
(b) Packing was consolidated under water in
Si
in the tower of the apparatus.
Water was run off
from a tap at the bottom of the tower until the surface stood at a certain height in.the tower; a measured
volume of water was then run off,and the fall in height
of the water surface was observed.
Both these methods eliminate the "end-effects"
which might be expected at the bottom and top of the
vessel.
Derivation of a "Geometrical" Mixing Co-efficient.
Preliminary consideration of the mixing of
fluids in a bed of spheres led to the idea of basing
a derivation of a "mixing-coefficient" on premises which,
while probably unrealistic, might yet give an idea of
the order of magnitude of the actual co-efficient under
limiting conditions.
The derivation is based on the
idea of the splitting of streams in streamline motion
through the bed--discussed in the Introduction and illustrated in Figure IV (11).
Consider an inelastic particle of negligible
size falling through a bed of spheres whose diameter is
d.
The particle will move in a vertical straight line
until it strikes a sphere; it will then "skirt" the
sphere in a great circle
path (taking the shortest route
to the edge) until it rolls off and falls in a straight
line to the next sphere.
It has been shown earlier in this appendix that
a straight line drawn through a bed of spheres cuts
-
-(1--
spheres per unit length, where F is the
d
2
fractional free volume.
It was also shown that the
fraction of the length lying outside the spheres is F.
2Fd
Hence the mean free path between spheres is -------3(1 - F)
consideration will show that the path
A little
of the particle considered can out no new sphere while
it
is
skirting a sphere, whereas a straight line drawn
through the packing can cut no new sphere while it is
"inside" a sphere.
ErggIgg
Hence
of straigtine
=D
total length-i ength inide~ggherps
total length
total length
-
length inside spheres
total length
path
dist. dropped in skirtingf
Fre legthof
artclepat
Total vertical dist. dropped
=Total
vertical dist.-vertica-4-l.
Total vertical distance
Vertical distance droped in skirtin
Total vertical distance
pt
54
As Figure II is intended to convey, if a
vertical straight line strikes a sphere at a certain
spot, the length of the line lying within the sphere
will be twice the vertical distance dropped by the
particle, if it strikes the same spot, while it is
skirting the sphere.
3
--
(1-
2
. ------
d
F)
Since a straight line cuts
spheres per unit length, and (1 - F) of
of its length lies inside the spheres, the mean length
of its path through a sphere is -.
2d
Hence the mean
3
vertical distance dropped by the particle while skirting
a sphere is half of this, or
d
Since the number of
spheres struck by the particle per unit length of frre,
vertical path is the same as the number cut by the
straight line per unit length of free path, it can
easily be shown that the number of spheres struck by the
particle in unit length of total vertical path is
-. F)
d(l + F)
The projection of a sphere onto a horizontal
plane is a circle of diameter d, and the particle
fall anywhere within this circle.
may
The probability that
it
falls at a distance r from the centre of the circle
is
-
Tr.dr
or -O
2
Trr
V
hence the average value of the
,
distances from the centre for all possible points of
impact is
d
8
d2
r2dr3
3
Hence -is
the mean horizontal displacement undergone
by the particle per impact.
Now suppose that the
particle moves at a constant vertical velocity u.
unit time it will undergo
ments of mean value
-
3
.
9U
In
/1-F
) horizontal displace{e/* F)
These displacements will be
completely random in direction, in that the direction
of any one will be unrelated to that preceding it.
Under these circumstances we can apply the "random walk"
theory to calculate the mean net horizontal displacement
(The theory is dis-
of the particle in a given time.
cussed in Kennard's "Kinetic Theory of Gases" f )The
expression for displacements occurring in three
dimen-
sions (as in a gas) is
where X is the mean square of the net displacement, Iis the mean length of one displacement, t the time, and
In a gas, an individual
Z the frequency of displacements.
displacement may be in any direction in space, and
is the mean of the projections of the individual displacements onto a single straight line.
The displacements
which we are considering take place horizontally only,
so that the mean of the projections of individual
displacements onto a given line will be
the value already calculated, C
under consideration,
2.27
.
,I
having
Hence for the particle
Since Z had been shown to be
3I4
If
a large number of particles move through the bed in
the way described, the result will be that a process
analogous to diffusion will take place horizontally.
It can be shown (see Kennard's "Kinetic Theory of Gases")(2..-)
that the diffusion co-efficient D will be
hence for the system considered
for horizontal diffusion.
For particles of other than spherical shape, a
similar (but more difficult) treatment would also give
an expression of the form
where A
is a constant.
The free volume of the 4-mm. and 6-mm. spheres
used in this work, F= 0.37.
given above
u-= Fv
and
Substituting in the expression
(Re)
dvg,/ ,
we have
Substituting F= 0.37, we have D= 0.00065 (Re).
This ex-
pression is plotted on Graph VI, where it is compared
with the line obtained by extrapolating the experimental
plot of D vs. (Re)
APPENDIX B
Fluid Flow through Packed Towers.
The treatment of fluid flow through beds of
solids is much more complicated than for that through
geometrically simple conduits, and the correlation of
experimental data has been at the best semi-empirical,
with the theoretical background of the expressions open
to some doubt.
Three papers by Furnas (1), Chilton
and Colburn (2) and Meyer and Work (3) present most of
what is known on the subject.
A summary is given in
Perry's "Chemical Engineer's Handbook",
The basis of the correlation equations is a
modified Reynold's Number (hereafter referred to simply
as the Reynold's number):
and the Fanning Equation
where d = particle diameter (actual or fictitious), v
superficial fluid velocity--i.e., linear rate of flow if
packing were removed from containing tower.
Consider a fluid flowing through a bed of
spheres at a velocity so low that turbulence is ruled
out; and follow a single particle in the fluid through
the bedl
At any instant it can be pictured as flowing
through a conduit of irregular and changing cross-section.
If
it is near the center of such a conduit it will be
travelling faster than if it is near the wall; a section
through the bed would show a number of such "conduits",
where flow is brisk, surrounded by)ne6
of decreasing
velocity, with stagnancy at the surfaces of the spheres.
However, the maximum velocity through all these conduits
is not equal; fluctuations in the arrangement of the
spheres provide "channels", which are conduits whose
cross-sections are by chance larger than the average
for a considerable distance.
These channels are not
continuous throughout the depth of the bed (unless flowrates are high enough to lift the spheres), but peter
out and are replaced by others in other parts of the
cross-section.
In any cross-section, therefore, there
are regions in which the maximum velocity is much higher
than the average, and others ih which it is much less.
As the location of these channels varies from section
to section, the particle will wander with respect to its
original axis of flow as it passes through the packing.
Moreover, two particles which are originally close to-to
gether may become widely separated in this fashion.
This
type of behavior is in marked contrast to that of a fluid
flowing past an isolated sphere in streamline motiona
The flow-paths of particles of fluid in this
h4t
case are shown in Figure IV (i); it can be seen that
any particle has the same situation with respect to the
axis of the sphere before and after the encounter, so
that no displacement or splitting of the stream can
occur.
Mixing by channeling does not exclude mixing
by the mechanism illustrated in Figure IV (i1) and discussed in the Introduction.,-
ie,
As a further- consequence of channeling, it may
occur that a stream in a conduit reaches a relatively
high velocity and then debouches into a region where
the velocity is low.
As a result, turbulence may be
set up locally even when flow in most regions is streamline.
This picture of the nature of the flow through
the packing is supported by some qualitative experiments
carried out with 6-mm. glass spheres in an 8-cm. glass
tube.
A layer of the spheres 4 or 5 diameters thick was
supported on a wire gauze, and water was run through the
tube at such a velocity that flow in the open part of
the tube was steamline.
This was confirmed by injecting
a solution of methylene blue through a capillary tube
beneath the layer of packing; the linear rate of injection was less than that of flow, so that the dye was
drawn out into a smooth filament.
The following phen-
omena were observed:
(a)
On several occasions the filament, after
broadening to a ribbon a few millimetres wide, split
into two distinct parts (with apparently no col/d
in the
intervening region), which then diverged and entered the
packing at points some 6-8 frd diameters apart, well
off the axis of the tube.
(b)
Filaments which entered the packing emerged
from the top either distorted into broad and irregular
sheets, or split into several streams; the edges were
sharp, and remained so up the tube.
The extreme dis-
tance apart of any two parts of the visible eCWk'4
was
not more than 3-4 sphere diameters.
(c)
The cola
streams emerging from the
top of the packing did not emerge in a direction parallel
to the axis of the -tube, but inclined at small and
various angles to it.
At steady state an emergent
filament appeared as a stationary line of a curved or
spiral shape, tending to straighten out higher up the
tube.
(d)
When the colO*M-/t4
filament first
entered
the packing, it could be seen to accelerate, and the
emergent streams could be seen to be moving considerable
faster than the average velocity of the water in the
open tube.
Under some conditions, turbulence could be
seen in the shape of very fine, uniform ripples at the
boundary between an emergent filament and the relatively
stagnant fluid surrounding it.
Figure V shows a typical situation.
When the rate of flow was increased to the point
at which the flow in the open tube was turbulent, the
filament leaving the injector tube was considerably
shredded and contorted before reaching the packing.
Instead of passing steadily through one point in the
bed, it oscillated rapidly over the whole surface.
In
the course of routine experiments it was observed that
at higher rates of flow some kind of oscillation or
fluttering of the coloured fluid was going on in the
packing near the injection-point.
change in appearance of the
fluid
There was a marked
aving the top of the
packing as the velocity was increased.
At low velocities
the colour emerged in streaks or filaments --
i.e., there
were discontinuities in the concentration; at high velocities the colour distribution from point to point was
continuous.
The change occurred at a Reynold t s number
(calculated for the packing) in the neighborhood of 40.
When using 11-mm. white alumina spheres, observations could be made on filaments of colour which
channeling brought to the wall of the tube.
It could
be seen that providing Re was not too high, an almost
steady state was reached, the position of a filament
hardly changing, except for local turbulence in spots.
The path of such a filament was. very devious.
It ap-
peared to be governed by channeling effects, rather then
by purely geometrical displacements, such as those considered in Appendix A.
Whatever the exact nature of the mechanism of
flow through pacing, it is generally admitted that there
is no abrupt transition from streamline to turbulent
motion.
For spheres the flow is entirely streamline
below Reynolds Number = 5 (3); the pressure-gradient
through the packing is then proportional to the velocity-i.e., the friction factor f in the Fanning equation is
inversely proportional to'-the Reynolds Number,
At suf-
ficiently high Reynolds numbers (the lower limit is not
accurately known), flow appears to become entirely turbulent, and f is proportional to the (-0.15) power of
Reynolds Number.
There is a transition region where the
exponential relationship between f and Re varies continuously (see Graph I and Perry (9))
All the experiments conducted by us were probably in the transtion region;--values of Re (Reynolds
Number) lay between 10 and 110.
It is suggested that the behaviour of the
coloured filament immediately before and after entering
the packing in the experiments described shows in an
exaggerated form the processes occurring within the
packing--splitting by channeling, and localised turbulence which may lead to eddy-mixing.
It cannot be said
that this picture of the flow gives any reason to suppose
that the geometrical mixing co-efficient derived in
Appendix A will correspond to the facts; however, experiments show this to be actually a fair approximation for
the value of the co-efficient observed at low rates of
flow (see Graph VI).
As regards the influence of the wall of the
confining vessel on the distribution of the elements
of the packing, and hence on fluid flow, it has already
been stated in Appendix A that, for the case of spheres,
both geometrical considerations and actual measurements
of F indicate that the free volume in the part of the
packing adjacent to the wall is equal or very nearly
equal to that in the body of the packing.
Furnas (1)
derived a semi-empirical expression for a co-efficient
A
to be used in a modified form of the Fanning equation:
(
is supposed to be a factor which is the same
where
for all sizes of particle at a given Reynolds Number,
while
Aj
is a function of the ratio of the particle
diameter to the tube diameter, which corrects for the
influence of the tube wall on the free volume of the
packing (Perry (9)).
'/e ag/off experiments on free volume, already referred
to, indicated that the overall free volume is constant
as long as the tube diameter is at least 5 times as great
as the particle diameter.
Moreover, both White (7) and
Uchida and Fujita (8) found that the pressure-drop was
uninfluenced by the ratio of particle diameter to tube
diameter as long as this was less than 1/6.
er
4
dried
periment on this question,
to perform a critical exThe tower of the apparatus
was packed normally with 6-mm. glass spheres, water was
run through at an ascertained velocity, and the pressuregradient through the packing observed.
A thin sheet
of metal was then rolled into a rough cylinder or
spiral scroll, and thrust co-axially into the packing
in the tower.
The area of the metal surface (counting
both sides) was about equal to the area of the tower
wall in contact with the packing; if, therefore, the
layer of packing immediately adjacent to a flat surface
offers a path of lower specific resistance to the flow
of liauid than the body of the packing, one would have
expected the pressure-gradient to be lower, at the same
velocity, than before.
Unfortunately, owing to the un-
certainty involved in the pressure-gradient measurements
(see Appendix E), all that can be said is that if there
was an effect it was about of the same order of magnitude as the experimental uncertainty.
One would have
expected, following Furnas(l), that the pressure through
the packing would drop by some 0.1
but unfortunately the
to a
-
0.2 inches of water,
measurements were not reproducible
greater accuracy than this.
Chilton and Colburn (2) used the factor Af given
by Furnas in correlating their experimental results and
those of previous workers.
As far as their own data are
concerned, we have re-plotted them without reference to'
A
--i.e., plotting
,
as calculated from the normal
Fanning equation, against the Reynolds Number. Jd
that for those experiments in which the ratio of
particle diameter to tube diameter was less than 1/6,
the correlation obtained was as good as, or better than,
that obtained when using Af- ; when this ratio was
greater than 1/5, the friction factor was markedly low.
Finally,
have measured pressure-
gradients and calculated from them friction-factors
(without using A ) for 4-mm. and 6-mm. beads in a
tower 9 cms. diameter.
The friction-factors for given
Reynolds' numbers for the two sizes are very close together (see Graph I); it is true that the values of f
for the smaller size appear to be some 5% higher than
for the larger size, but in view of the errors in the
pressure-gradient measurements) "Wd 4
whether this
is significant.
To sum up, our own free-volume measurements,
Chilton and Colburn's data, and the opinions of other
investigators appear to lead to the conclusion that as
long as
the diameter of the confining tube is 5 or 6
times as great as the diameter of the packing particles,
variations in the former will not affect the flow through
the packing.
This implies that in4(e& experiments the linear
rate of flow was substantially constant, in the mean,
in all parts of the cross-section of the tower.
The
matter has been stressed because it is of some importance
in correlating any measurements concerned with flow
through packings.
In particular, the equations #Z imM
used to calculate the diffusion co-efficients from
concentration measurements were derived O1the assumption that the mean linear rate of flow is uniform
across the tower.
Were this not so, a correction would
have to be applied in order to calculate the true mean
linear velocity from the volumetric rate of flow.
The treatment of non-spherical particles is
more difficult, and it is necessary to define their
diameter arbitrarily.
White (7) has correlated pressure-drop data for
Raschig Rings and Berl saddles in the following way:
for a given shape of packing, an empirical relationship
is drawn up between the nominal diameter dP of the
particles and a corrected diameter d
.
The nominal
diameter is used to calculate the Reynold's number, and
then the friction factor corresponding to this is read
off the Chilton-Colburn curve.
can then be calculated by using
The pressure-gradient
the Fanning equation,
with this value of the friction factor, and O
place of the diameter.
in
In other words, the pressure-
gradient for an irregular shape is equal to that for
spheres at the same Re (using nominal diameter of particles),
multiplied by the factor (A); or we may say that the
friction factor for an irregular shape is that for spheres
at the same Re, divided by ( I)
particle diameter throughout.
using the nominal
The data for Berl saddles
are scanty, but for rings the procedure is justified in
that the
vs. Re curve "breaks" at the same value of
Re as for spheres.
This may not be so for Berl saddles;
White's data are all in the turbulent region and do not
cover the "break".
(See also
saddles, plotted on Graph II.)
1/,edata on 1/4-inch Berl
Formulae for
for rings
and saddles are given in Perry (9).
It would be more satisfactory, when dealing
with non-spherical particles, to use a correlating expression containing no reference to particle diameter,
and an attempt
has been made to derive one (see "Cor-
relation of Results").
If the free volume F can vary independently of
the particle shape, then F must also be introduced into
the correlating expressions.
by Meyer and Work (3);
TktI subject was studied
see also Perry (9).
lem, however, did not occur in 1
work.
This prob-e
Appendix 0
Derivation of "Probability" Equation for computation of Diffusion
Coefficient from Concentration Measurements.
Comparison with Wilson
Diffusion Equation
In the course. of flow through a packing, mixing may be expected to
to occur in several ways--e.g. molecular diffusion, eddy diffusion,
and mixing caused by the geometrical characteristics of the packing.
These are all alike in that they are caused by displacements of elements of the fluid which are large in number,
random in direction,
and small compared to the mean total displacements to be measured.
The effectiveness of these various processes in causing mixing depends
on the frequency and average length of these random displacements;
in
the case of a liquid, the mean free path of a molecule in thermal motion is
so small that prolecular diffusion plays a negligible part
under under the conditions of experiment.
The distribution of a number of particles which,
Common
aantre,
(random,
tion".
starting from a
have undergone displacements of the kind described
numerous,
small) ;i,
given by the "normal probability func-
This ginction expresses the probability of a particle which
was originally at the origin being now in a plane which cuts the
A - axis perpendicularly at x:-
where K
is the mean square of the x- coordinates of all
Similarly:
2
IT1
the particles.
for displacement along the y- coordinate.
the X & y coordinates only is
This if displacement along
considered, & if
we take 0
equal to
at any time (i.e. diffusion isotropic) we have
N
where
and PXy is
the Probability of the X & y coordinates having simultanedusly
the values (Xy).
Consider a very thin stream of an indicator fed into the axis of
a stream of moving fluid, & suppose for the moment that diffusion
takes place only in directions at right-angles to the direction of
Then the above expression gives the probability of a given mo-
flow.
lecule of the indicator being at a specified point at a distance
r
from the axis, when 3?1. tefers to all the indicator molecules entering the stream at the same instant.
It
can be shown (see Kennard()) that the co-efficient fof dif-
fusion, D, will be given by the expression
-ZL
where 4t is the time which has elapsed since the indicator entered the
stream.
(The validity of this expression is discussed later in this
appendix).
//kre.
eL
for the probability of a particle being at a specified point -distant
rfrom the axis, in a plane distant ut from the point of entry, where
u = linear flow velocity (see fig. X)
sampling
point.
Since all
,Fig.
the indicator molecules entering the stream at a given ins-
tant reach this plane at the same time,
the expression for Pr also
gives the fraction of the total number of indicator molecules entering the stream which pass through the differential area dX.
if N molecules enter in unit time,
unit time.
Thus
then NPr pass through this area in
For the purposes of measurement,
but finite cross-section a is
dy.
used* it is
a sampling tube of small
assumed that Pr is
effectively
constant over this area.
In any plane drawn through the packing perpendicular to the direction of flow the free area unoccupied by the subtance of the packing
is
a fraction F of the total.
The average linear velocity across the
plane will then be v/u, where v = V/A, V being the volumetric rate of
flow, & A the total area.
Thus in unit time a volume av/F of fluid
flows through an element a of free area.
Now Pr expresses the pro-
bability of a particle being in a given overall area of the crosssection; since some of this area is occupied by the substance of the
packing, & the particle cannot be in this part, the probability of it
being in in an equal element of free area is Pr/F.
Thus the total
number of indicator molecules passing through a small element a of
free area in unit time is NaPr , and thttk concentration
Fdx.dy
is therefore
53
NV Pr
-/
If the sampling - point is situated in a plane distant X from the point
at which the indicator enters,
also
N = C * V, where 0* is the average concentration of indicator in
fluid. stream.
the entire ..
Hence
C
C
4irPD
-r
I-r 4F b
It can be seen from the derivation of this expression that if
samp-
ling is continued at a constant rate for a finite time, & if 0 is the
mean concentration of the sample so collected, whi4e 0* is the mean concentration of the efuent over the same period; then providing all the
other factors remain constant,
variations in
the rate of injection of
indicator are inmaterial.
This expression is only valid so long as diffusion in directions
parallel to the axis of flow can be neglected without error.
Otherwise,
similar reasoning leads to the expression:
a
172
However,
the simpler expression seems justified by the experimen-
tal results.
Previous workers (4,
evaluation of D.
5) have employed,
the Wilson Squation for the
This equation Was originally4derived (6)
for the con-
duction of heat from a point source through an isotropic stream of moving
fluid, but owning to the mathematical identity between the fundamental
equations for heat transfer & for diffusion, can be applied with equal
validity to the latter.
Lile the probability equation, it neglects the
effect of diffusion in the direction of flow.
For an open tube,
the Wilson Equation has the form:
Co)Wecting for the space effect of the packing & transforming by multi-*
plying top & bottom of the exponent by
(V+X
t
,
we have
N
F_ JUL)
()=
This form show that the expression is
Equation when r = Q, & that it
identical with the Probability
will approximate it
Since under our experimental conditions 7
of x 2 (& usually much less)
~
closely when Yl<
was never more than.1/10
was never more thanX
9F
,
1.5 times
X.
be 5%, in
the expOaents 2.5%, as between the two expression3.
or
Hence the maximum variation in the co-efficients would
The ex-
perimental error was greater than this, so that no comparison can be
made between the two.
The Probability Equation was used because its derivation was more
fully understood, because it
appears to correspond to the mechanism of
the mixing process, & because it
than the Wilson Equation (e.g. it
is
mathematically somewhat simpler
can be easily integrated).
Assum-
ing that both the Wilson Equation & the Probability Equation have
been derived from their starting assumptions without error,
the dis-
crepancy can only be due to the non-validity of the expression:
55
where 5E2 is the mean square displacement undergone by an element
of fluid in time t .
It was by using this expression to sub-
stitute for 22 in the"random-walk" distribution that D was
introduced into the probability equation.
The derivation of
this expression is given in Kennard (22).
In the course of the
derivation it is tacitly assumed that the concentration gradient
in any direction is constant; since, in fact, in the case under
consideration the gradient varies in all three dimensions,a
discrepancy will arise from this assumption, and D in the diffusion equation is
not, in fact,
equivalent to 2 2 /2t
in
the
random walk theory in such a case.
Since the probability equation has been used throughout,
D is
defined as
z2 /2t
; however,
a numerically equal value
would have been obtained had the Wilson equation been used, and
D defined accordingly.
More accurate experiments would be re-
quired to determine which expression more nearly corresponds to
the facts; it is clear that neither can be more than an approximation, since both lead to finite values of the concentration
at all distances from the axis, while practically the concentration falls to zero at a finite distance.
Moreover, both are con-
tinuous functions, while actual diffusion of any kind involves
the movement of discrete elements of matter.
of the
A general survey
validity of the diffusion and-proability treatments is
given in the following notes.
(a) The diffusion equation is an arbitrary or empirical
expression with no theoretical basis.
The expression which it
gives for the rate of transfer at a point refers only to the concentration-gradients at that point, and not to conditions in
other parts of the system.
(b) The probability equation is derived from the random
walk theory, and will be strictly
true when transfer is due
to random displacements of the type specified in that theory.
(c) For the special case when the concentration-gradient
on any axis in the system I.s constant along that axis (extrapolated to infinity), it has been shown by Einstein thatthe probability and diffusion-equations are identical in form, and that
D in the former is then equivalent to 32 /2t in the latter.
(d) However, for the general case the two are not
it is clear that they cannot be, since the diffusion
equivalent;
equation refers only to conditions at a point, while to express
in terms of the random walk theory the quantity of material
crossing a given plane requires a knowledge of the concentration
of the substance at all distances away from the plane in both
directions.
(The fact that : x:n the Wilson equation, for
instance, the difference is not great is because in fact it
is those parts of the system which are close to the plane in
question which contribute most to the transfer across it; the
error in assuming the concentration-gradient constant over a short
distance will not be great, while at greater distances the contribution to the total will be so small that large errors in
assessing this contribution are of little importance).
(e)
Thus
1ie diffusion equation is only strictly applicable
in cases where the physical conditions are such that the con-
centration-gradient is
constant :along any given axis.
It
is
clear that this can never be the case when a solution containing
a finite concentration of the diffusing substance is
injected into
the pure solvent.
If
the arrangement used in the present work were replaced,
-for ibstance, by that shown above, indicator solution being
injected from a linear source, and the direction of diffusion
being limited to the x-axis, it is
clear that the concentration-
gradient on the latter would vary with x .The diffusion and
probability distribution functions would still
forIns; in
other words,
the discrebancy is
be of different
not clue to the
geometrical conditions imposed by the point-injection, radial
diffusion arrangenment used in the present work.
Appendix D.
Calibration of Flow Manometer and Colorimetric Standards.
Water was run through the apparatus and through the
Flow Manometer.
manometer until it was certain that no air was left in the tubes.
constant-head tank was allowed to fill
The
until overflow was constant; the
flow-rate was then adjusted to a chosen value by the control-valve.
A
reading of the pressure-drop across the orifice was then taken, and the
flow-rate obtained by timing the collection of a measured volume (about
10,000 cc) of water as it over flowed from the tower.
A second reading
of the manometer was then taken; if it differed from the first, the two
were averaged.
When the volumetric rate of flow was plotted against the
manometer head on Jg-g
is
shown in
paper, the slope was exactly 1/2.
The plot wasa-
Graph 22.
Colorimetric Standards.
A portion of the original indicator solution
made up (24.80 gms/litre) was diluted 1,000 times, and used as a reference
standard (referred to in the lab. notes as s/1000).
Methylene blue solu-
tions of various other known concentrations were then compared.with the
in the Duboscq colorimeter.
S/1000
The color-densities of the various solution$
were plotted against their concentrations, thus giving aburve from which
the concentration of any solution of known color-density could be determined.
It was noticed that it was advisable to compare solutions which
were as close as possible in concentration, as the color appeared to vary
with the concentration.
Tap-water was used as the diluent throughout.
The color-density/concentration curves are given in graphs 20 and 21.
Appendir .EAddittional Notes on Experimental Technique.
Packings.
Most of the work was done with 4-mm. and 6-mm. glass heads.
These were cleaned with chromic acid before starting the experiments
to discourage air-bubbles from clinging to them.
It is necessary to treat the packing in a standardised way in
order to keep the free: volume constant.
In this work, the glass
beads were stirred under water with a thin rod after every run,
and measurements of free volume and pressure-gradient were made
with the packing in a similaryses
1of consolidation.
The 3/8-in. (nominal-diameter) spheres and --in. Berl saddles
could not be stirred in this way.
when it was
ftll
They were dumped into the tower
of water and lightly tamped down.
The alumina spheres as received were not uniform in shape or
size., so they were sorted by hand to remove the more obviously
deformed or outsize ones.
The volume of a known number of the
sorted spheres was then measured by displacement, and the mean
diameter calculated.
It was found to be 11 mm.
(0.43 in) which
is not very close to the nominal diameter of 3/8 (0.375) inch.
Methylene Blue Solutions. It was found advisable to warm the solution
when making it up, to make certain that all the dye dissolved.
It dissolves slowly, and it is difficult to see whether any solid
is left.
The exact concentration of .the indicator solution need-
not be known, -since it does not enter into the calculation. A
solution containing 25 - 30 gms/litre was used; this approaching
saturation, and as will be explained later, the highest practicable
concentration was required.
A concentration of about 1/1000 of this was found to be the easies t
to estimate accurately and quickly.
When less than
of this concent.'
ration, the accuracy of the determination is poor, and when more than
2 or 3 times this, the sample must be diuted to give accurate results.
For a superficial flow velocity V= 1 cm/sec, using a packing of
spheres, the true mean Eineatr velocity through the packing is about
3 aub'sec. - /, /tk4z
r vol" e- v -_k- ren of /feo
70 cc/sec. Since the diameter of the injector tube was approximately
.08 cm.,
injection of indicator solution at the time linear flow, velo-
city ( as was originally considered desirable) would give an effluent
solution with a concentration equal to
76
indicator solution.
Since the concentration of the effluent gives an
idea of the range of concentration of the samples which wifJl have to
be analyzed, and since the concentration of the methylene blue solution can
aot
be greatly increased,
it
is necessary to inject the in-
dicator solution at a higher velocity (say 2-5 times as *att)
of the main stream if
to be obtained.
than that
samples of a suitable range of concentration are
No significant variation in
the observable results
was noticed when the injection rate was varied, so that the procedure was probably permissible.
Apparatus.
Fig. VII.
The nanometers used to'meter the flow are shown in
There was a tendency for air to collect in the tubes lead-
ing from the metering orifice to the namometer, so the latter were so
constructed :w to allow water to be run through them and flush out air
6/
before readings were taken.
In the case of- the water manometer, used
for flow rates up to about 150 cc/se.,
water was allowed to run out of
the right-hand limb, the lower stop-cock was then closed, and the puffer
used to force the water levels over into the two left-hand limbs (slugs
of air which formed in
the left-hand limbs in
the process could be re-
moved by raising and lowering the levels by alternately puffing in
and allowing it
to escape).
prevent air being sucked in
air
The upper stop-cock was then closed, to
through the puffer by the negative pressure
developed beneath the metering orifice when the water valve was wide open.
The mercury manometer,
used only at the highest attainable flow-rates,
was so designed that when the mercury reservoir was lowered, water could
be run through both limbs and out of the lower stop-cock.
The steel capillary tubes used to collect the samples were very
flexible, and in order to fix the position of their open ends with certainty it was necessary to attach them to a rigid frame, which is shown in
Fig. IX.
It was considered desirable for the liquid leaving the top of
the tower to overflow symmetrically, at any rate at low flow-velocities,
in order not to distort the flow in the tower.
To ensure this, a 6-inch
high metal extension was attached to the top of the glass tower, and 6
equally-spaced
lAO
inch holes were drilled in it, in a horizontal plane
some 4 inches from the top.
While at higher rates of flow the water
overflowed at the top of this extension-piece, at lower rates it flowed
exclusively through the six holes; the flow through each was equal and
practically unaffected by any small tilt of the tower, because this would
produce only a small fractional change in the head above any one hole.
The connections between the metal container at the bottom of the
tower, the tower itself, and the extension piece at the iDp were secured
and made leak-proof by the use of rim cement, and by a plastic waterproofing material which was forced tightly into the joint after assembly.
SAMPLE CALCULATION
APPENDIX F.
Analysis of Runs 76 - 83 inclusive. All concentrations of the samples are expressed,
as was the general case throughout, with reference to a standard solution of
1
1000
The concentrations
the concentration of the original indicator solution injected.
of the effluent and the samples were found as outlined in Section III (Procedure).
The general statement of the probability expression used throughout these analyses
is of the form
Cx
V
C
where
C
Concentration of sample collected
C*
Concentration of effluent liquid
X-
Axial distance between the injector capillary and sampling plane
V
Volume of water flow
F
Free volume of the packing
D
Diffusion coefficient Cms 2/sec.
r
Radial sampling distance from axis of the tower
v
Superficial flow velocity of water.
c.c./sec.
The general approach is to plot Log,,
line is given by
~~
i
V
Cms/sec.
4
and the intercept by Logi
when the slope of the
V
.
The best line is
drawn through the points obtained, consistent with satisfying the equation above.
Upon averaging the individual results of Runs 76-83, Table A is obtained. Table B
shows the results for the individual runs.
TABLE A
Run
Numbers
Tube
C L.
C*
CX.
CX
r2
)
Volumetric rate
of floW
Superficial
flow velocitir
Number
(average)
76-83
Log
(average)
(cic./sec)
1
288
2.460
0
2
224
2.350
0.011:
3
143
2.156
0.064
4
60.2
1.780
0.129
5
10.8
1.034
0.225
21.7
CMS/sec0.0314
A plot is made of Log
It
CX.
S(average)
vs.
2
r
x..
is seen that the slope of the line on the plot - --
'7'
9.2 FD
or substituting F D :-f- 0.314
= 0.00594
9.2 x 5.75
But the intercept on the graph at 12
0 is given by log10
V
477 FD
Intercept = 2.464= log1
V
10 477 FD
where V is the volume of flow
/sec.cc.
Then 291= 21.7
47 FD
,
FD
0.00594
i.e. the values of F D as calculated from the slope and the intercept
on the plot, give the same value for F D .
A'eee'lD
.00594
F.
=
.00594
0.36
0.0165
cm 2
sec
Centre-Line Sampling Method._ In some cases samples were taken on
the centre-line of the tower only.
thus,
for a given value of V, it
is
Thens
Cx
only necessary to average
the values of CX for all the samples to find the best value of FD.
C*
Table
Run
No
76
Tube
No:
78
r 2C
x
C
r
x
0
364
0
0.860
0.230
394
105
0.266
1.51
0.110
50.5 3.05
0.129
5
0.020
9.17 5.30
0.225
1
4.38
1
7.90
1
2
3
4
77
0
B
2
3
4
5
2
3
4
5
0.795
5.15
3.70
1.45
0.44
5.10
3.20
23.5
23.5
0.051
0.245
418
0
492 0.266
354 1.51
138 3.05
42
5.30
23.5
0.675
275
178
112
0
0.266
1.51
91.6 3.05
6.52 5.30
2.63
0.187
23.5
0.46
110
98
0
0.266
0.0113
0.064
0
0.0113
0.064
0.129
0.225
0
0.0113
0.064
0.129
0.225
0
0.0113
1
2
2.15
1.93
3.
4
5
1.68
0.63
0.105
1
2
5.30
2.40
3
4
5
1.15
0.51
0.035
8l
1
2
3
4
5.07
2.50
1.62
0.24
0.015
23.5
0.255 470
231
150
22.2
1.39
0
0.266
1.51
3.05
5.30
0
0.0113
0.064
0.129
0.225
82
1
2
1.072
0.63
23.5
0.205 124
73
0
0.266
0
0.0113
3
0.060
4
0.020
1
2
1.17
0.995
3
1.07
79
80
83
4
5
0.545
0.09
86
1.51
32.2 3.05
5.37 5.30
23.5
0.368 340
154
0
6.266
1.51
74
32.7 3.05
2.24 5.30
23.5
0.064
0.124
0.225
0
0.0113
0.064
0.129
0.225
6.95 1.51
0.064
2.34 3.05
0.129
0.135 216
176
189
0
0.266
1.51
96.3 3.05
15.9 5.30
0
0.0113.
0.064
0.129
0.225
Appendix H.
A
Af
Nomenclature,
definitions and symbols.
cross-sectional area of tower - sq. cms.
factor to be used in
faaing equation to correct for variations in
ratio of particle diameter to tower diameter.
C
root mean square velocity of particle in random motion (in 3 dimensions).
C
concentration of sample taken at point (X , 4. ) -- 1000 x(cc.
injected indicator solution per cc. of samplek,.e. -4/7i
/-
of
sde
C*
mean concentration of total stream passing through tower-same units
as C.
d
particle diameter -- cms.
dp
nominal particle diameter (cms) for irregular shapes -eg.
D
apparent diffusion co-efficient (sq. ems per sec.) as defined in Appendix C.
D'
co-efficient of molecular diffusion
E
Berl saddles.
eddy diffusion coefficient.
f
tIanning friction factor, as defined in Appendix B.
F
fraction of overall volume of packing which is not occupied by the
substance of the packing particles.
L.
depth (ems) (measured parallel to axis of tower) of packing through
which pressure drop isA P dynes per sq.cm. when superficial velocity
is V ems. per sec.
n
number of molecules (defined by context.)
N
number of spheres ih unit overall volume of packing.
AP
pressure-drop (dynes per sq.cm., unless otherwise stated) through
depth L cms. of packing at superficial velocity v cms. per sec.
r
distance of sampling-points from axis of tower (cms).
t
time (secs.)
u
mean linear velocity ( in direction parallel to tower axis) of fluid
through interstices of packing - cms per sec.
v
v6lumetric rate of flow divided by cross-sectional area of tower,
A
or superficial velocity of flow (cms. per sec.)
V
volumetric rate of flow, or volume of fluid flowing through tower in
unit time -cc. per sec.
x
distance of cross-sectional plane in which sample is taken from that
of injection-point-cms. (See Appendix C).
Z
collision frequency, or frequency with which particle in
changes direction.
daK6LorCY o
fL4oL
-
random motion
APPENDIX I
*
BIBLIOGRAPHY
1.
Furnas, C. C.
2.
Chilton, T. H. & Colburn, A.P. Trans. A.I.Ch.E., XXVI pp. 178-196 (1931)
3.
Meyer, W. G. and Work, L. T., Trans. A. I. Ch. E, XXXII, 13-33, (1937).
4.
Rabinowitz, B. and Roessel, T. B. Mixing Rate in Liquid Model Towers,
S.B. Thesis, Chem. Eng. M.I.T. 1946.
5.
Sherwood, T. K. and Towle W. L. Ind. and Eng. Chem. 31, 457, (1939).
6.
Wilson, H. A. Proc. Cambridge Phil. Soc. 12, 406, 1904.
7.
White, M. Trans. A. I. Ch. E. Vol. XXXI, (1936) P. 390.
8.
Uchida, S. and Fujita, S. (i) J. Soc. Chem. Ind., Japan, 724B, 791B (1934).
9.
Perry, J. H. Chemical Engineer Handbook 829, (1941)
U. S. Bureau of Mines Bulletin 307
(1929)
10.
Taylor G. I. (1) Proc. Roy.Soc. London A 159.1 496 (1937)
(11)Proc. Roy.Soc. bondon A 151, 421, 1935'.
11. Karman, Th. von J. Aernaut. Sci I, 1, 1934.
12. Dryden, H. L. Ind. and Eng. Chem. 31, 416, 1939.
Eng.
13. Sherwood, T. K. and Woertz, B. B. Ind. and/Chein. 31, No. 8, August 1939.
14.
Murphree, E. E. Ind. and Eng. Chem. 24, 727, 1932.
15.
Goldstein, S. "Modern Developments in Fluid Dynamics", Vol. II,Oxford 1938.
16. Woertz, B. B. "Mass transfer by Eddy Diffusion" Sc. D. Thesis Chem. Eng.
M.I.T. 1939.
17. Weddell, D. "Turbulent Mixing on Gas Flames and its Reproduction on Liquid
Models" Sc. D. Thesis, Chem. Eng. M.I.T. 1941.
18.
Hawthorne, W. R. "The Mixing of Gas and Air in Flames", Sc.D. Thesis Chem.
Eng. M.I.T. 1939.
19.
Kalinske A.A. and Pien, C. L.
Ind. and Eng. Chem. 36, 220-223,1944.
20. Prandtl, L. in W. F. Durand's Aerodynamic Theory Vol. III J. Springer 1935.
21.
Stadig, J. "Heat transfer in a packed bed", S.&
A'e ahvA- E
/<-~
0,a
bgo
m
u)vg
Thesis Chem. Eng.M.I.T.1947.
-//rp
TIABUIATioI
GEAPT$..
FIGURJFES..
oF REsuLATs.
_________
RUN NUMBER&
INCLUSIVE
PACKING TYPE
AND DIAMETER
DIFFUSION
COEFFICIENT
2,
FREE
FLOW
AXIAL DISTANCE
VELOCITY
FROM INJECTOR
'O SAMPLING
PT,ANE (MS
VOLUME F.
IDMS_'/seeIm/sv.C
6 mm
2-8
0.082
0.36
I
I
I.
cx
TUBE I TUBE II.
.TUBE
TTT
Average
TUBE
ITVT
centralI
TUBE
v
value of'
TUBE Il TUBE I TUBE
ii
ITTT
.-
I
c.x
TUBE
IV
TUBE
v
i
CMS
Radial Sampling distance
T BE
TUBE I|TUBE II TUBE
TTT
v
__________
0
1.07
t
TUBE
v
1.03
2.06
3.14
4.3
1.03
2.06
3.14
4.3
glass beads
2-
107
10.9
68
20.2
130
49.5
6.3
10.25
175
32.3
3.58
10.25
128
188
2 (a)
10.25
160
2 (b)
10.25
186.;
3 (a)
10.25
3 (b)
4
.81149.9
4
5
20.5
156
131
47.5
6
20.5
130
135
75.72
7
20.5
188
166
37.4
12.1
8
20.5
326
233
152.0
50.0
15
6 mm
0.196
0.36
89
22.4
200
158
78
9.9
16.6 I--0
1.66
glass beads
10.6
102
62.5
15.9
0.63
10.6
122
66.0
19.6
0.44
11
10.6
164
55.2
12
10.6
98
43.7
9
6.15
10.1
121
9 - 12
13
15.15
162
113
30.4
0.57
14
15.15
129
140
34.2
0.95
15
15.15
131
90.5
13.1
0.79
13 - 15
1---
4.87
2 -8
9
157
141
55.6
81.1
12.9 10.53
125.9
0.77
I
I
RUN NUMBERS
INCLUSIVE
DIFFUSION
PACKING TYPE
FLOW
VELOCITY
6m
2
CMS/SEC
CMS /sec___
0.2,5
0.36
AXIAL DISTANCEC_
PLANE
CMS
Average Value of
0*C*
FROM INJECTOR
TO SAMPLING
AND DIAMETE COEFFICIENT VOLUME F,
_-D
16 - 23
FREE
TUBE I TUBE I]
centra~
TUBE
III
TU3E
TUBE
TUBE I
V
'I
TUBE
II
distance CMS
CR
TUBE
III
TUBE
TUBE
IV
V
1.85
TUBE
I
0
E
II
II
1.03
2.06
0.535
1.08
TUBE
IV
3.14
T
UBE
V
4.3
glass beads
76.6
62.5
17
57.0
12.11
18
99.5
59.0
17.3
0.555
19
167
152
19.8
1.27
16
12.9
15.9
3.39
3.64
0.645
16 - 19
163
71.9
10.0
1.03
21
103
42.6
16.6
0.545
22
96.9
120
5.2
0.73
23
131
70.5
6.8
2.1
20
9.75
20 - 23
24 - 31
6 m
0.0103
0.36
1.01
100
71.-4
126
76
214.1
1.55
9.6
1.11
0
0.25
glass beads
352
244
46.2
25
216
72.2
22.9
LO.6
26
297.5 |124
38.9
1.4
27
141.5
79.0
2.66
12.3
24
124
0.79
24 - 27
28
411
291
55.7
29
377
230
129
30
393
215
40.0
9.3
31
361
244
222
21.0
17.15
32
6 m
glass beads
0.020
0.36
0.32
318
160
25.9
33
332
151
tOO.0
34
254
258
95
38.7
35
241
183
99.0
49.5
36
172
485
106
35
17.2
200
40.1
387
246
112
3.92
4.3
24 - 31
32 - 40
249
8.75
1.66
2.26
RUN NUMBERS PACKING TYPE
INCLUSIVE
AND DIAMETER
IDIFFUSION
COEFFICIENT
D CMS
2
/sec
,CL__CX
FREE
FLOW
AXIAL DISTANCES
C*
FROM INJECTOR
VOLUME F
VELOCITY
CMS/SEC
TO SAMPLING
PLANE
CMS
Average value
TUBa
TUBE I
central
TUBE
TUBE
TUBE
II
III
IV
TUBE
V
32 - 36
37
13.65
404
268
67.3
38
13.65
240
112
67.7
39
13.65
87
112
116
37.4
40
13.65
93
156
105
2.0
38 - 40
41 - 49
4 mm
0.0105
0.36
Radia l Sampling distance CMS
of C*
TUBE
TUBE
TUBE
TUBE
41.5
II
266
158
85
206
162
79
19.7
42
238
48.0
5.8
360
218
105.4
6.3
43
338
510
300
39.9
44
423
260
130
3.25
45
424
384
72
45
41 - 45
352
46
21.9
221
294
199
33.2
47
21.9
405
306
79.5
10.3
48
21.9
242
127
63.5
14.9
49
21.9
536
333
120
68.0
376
46 - 49
I
TUBB
TUBE
Iy
y
4.2
0
225
II
16.8
0.28
17.35
TU8
__L_
glass beads
41
TUBE
V UBk5
314
265
131
113
20.0
31.6
0.535
1.08
1.66
2.26
RUN NUMBERS
PACKING TYPE
FREE
DIFFUSION
VELOCITY
COEFFICIENT
INCLUSIVE
50
57
-
AND DIAMETER
4 mr,-.
2
D CMS
FLOW
FROM INJECTOR
TUBE
II
TUE
ITE
IIIIII
TUBE
375
0.69
0.36
Radial Sampling distance CMS
Average value of
CX
C*__
_
TO SArING PLANE EI T
central
VOLUME F
/seeCS/SEC
0.033
AXIAL DISTANCE
TUBE
282
TUBE
II
TUBE
TUBE
TUB
TUBE
II
153
37.5
13.4
0
0.535
TUBE
TTT
1.08
TUBE
TV
1.66
TUBE
V
2.26
glass beads
59
-
50
18.2
383
450
220
51
18.2
262
116
13.5
52
18.2
330
234
214
66
11.6
53
18.2
364
405
290
35.2
9.26
54
18-2
277
196
60.0
29
5.5
18.2
528
107
64.5
21.2
1.9
56
18.2
396
211
120
35.3
6.47
57
18.2
480
521
217.2
71.0
66
4 mm
0.133
0.36
43.3
73.0
0
1.75
1.11
2.14
3.34
glass beads
59
16.05
133
152
35
60
16.05
302
123
8.025
61
16.05
16.05
201
212
66.4
51.2
4.69
44.1
62
59
63
-
-
3.52
212
62
63
22.35
202
89
24.4
1.69
64
22.35
214
140
30.1
6.65
65
22.35
176
104
30.5
66
22.35
160
45.7
42.1
98
15.0
94.5
31.8
4.1
188
66
3.16
67
Experiment+ L
runs
For trial
purposes
only.
68
69
-
75
4 mm
0.0812
0.36
1.22
:EN1iTjR ITNE SAMPLING
glass beads
69
4.4
139
70
6.7
155
71
8.95
188
72
16.30
259
73
18.90
248
74
21.6
273
75
18.1
210
CENTE LINE j3AMFLING
4.27
RUN NUMBERS
INCLUSIVE
PACKING TYPE
AND DIAMETER
DIFFUSION
FRE
D0EFFICIeNT
D CMS
VOLUME
se/
FLOW
AXIAL DISTANCE
VELOCITY
FROM
CMS/sEC
TO SAMPLING
PLANE CMS
F
Average value of
Rladial Sampling Distance
C*C*9
INJECTOR
TUBE'l
{entral
TUBE
II
TUBE
III
(cm].)
TUBE
IV
TUBE
V
TUBE
I
TUBE
II
TUBE
III
TUBE
IV
V
92 - 103 (cont)
101
22.60
72.5
102
22.55
79.8
103
22.40
101.5
104 - 115
4 mm glass
beads
0.224
0.36
2.36
CENT
104
22.0
175
105
22.1
183
106
21.9
197
107
22.1
179
108
22.2
147
109
22.1
167
110
22.0
162
111
22.0
130
112
22.0
215
113
21.9
111
114
22.0
129
115
22.2
123
116 - 121
1/4 inch
Berl saddles
0.354
0.59
1.91
LINE SAMPLII
13.3
54.3
117
13.5
34.0
118
13.5
55.3
119
13.2
71.2
120
12.7
38.8
121
13.1
47.8
CE] NTER I INE SAMPLING
CEMTER IINE SAMILING
CENTREI LINE S4MPLINGI
116
160
50.2
RUN NUMBERS
INCLUSIVE
PACKING
DIFFUSION
TYPE AND
COEFFICIENT
DIAMETER
2/
sec
CMS
______
FREE
VOLUME F
_
FLOW
VELOCITY
C
_SE
Average value of
AXIAL DISTANCE.CE.
FROM INJECTOR
TO SAMPLING
PLANE CM3
TUBE I
TUBE
centralIi
TUBE
II
IV
_
__I_
72 - 75
76 - 83
6 mm
0.0165
0.36
0.314
288
23.5
76
364-
394
105
50.5
9.17
77
418
492
354.
138
42
78
275
178
178
92
6.5
79
110
98
86
32
5.4
80
340
154
74
33
2.24
81
470
231
150
2.2
1.39
82
124
73
83
216
176
91
-
-
II
TUBE
TUBE
III
TUBE
IV
V
43.0
60.2
10.8
TUBE
TUBE
4 mm
glass beads
0.172
0.36
2.26
7
2.34
189
96.3
15.9
184
182
93
85
130
77
86
296
124
87
138
148.6
4.16
88
135
70.5
6P30
89
114
116
5.40
90
184
171
16.25
91
173
208
25.0
15.9
0.515
0
0
169
20.6
224
126
1.08
1.22
2.2
-
103
6 mm
0.389
0.36
2.30
C4[NTRE LI*
glass beads
92
22.5
93.5
93
22.5
76.5
94
22.5
101
95
22.7
87.5
96
22.75
91.0
97
22.75
93.0
98
22.45
94.5
99
22.50
145
22.70
85
100
MS
TUBE
Iv---
(coitinued)
1.74
3.35
13.08 11.3
1.13
7.22
24.7
7.47
CENTdE LINE
92
TUBE
JJL
V
246
lass beads
84
Ra ial S apling Distance
C
_*
SAMPLINIO 90
I-
AMPLING
2.29
4.11
RUN NUMBERS PACKING TYPE
INCLUSIVE
AND DIAMETER
DIFFUSION
FREE
COEFFICIENT
D CS2/sec
VOLUME F.
FLOW
VELOCITY
FROM INJECTOR
TO SAMPLING PLANE
I
___/____CMS
122 - 130
122
3/ t A1
20 3
0.200
0.40
1.057
catalyst
spheres
CX.
0_
AXIAL DISTANCE
65.3
123
13.8
64.8
124
14.1
36.8
125
13.5
73.5
126
13.2
.00
127
14.0
43.4
128
14.1
139
129
14.0
58.0
130
13.5
54.1
131 -142
131
3 8" A1 0
2 3
catalyst
spheres
0.227
0.40
TUBE
TUBE
TUBE
TUBE
II
III
IV
V
CENTR4 LINE StMPLING
13.5
Average value of
TUBE
I
72.8
TUBE
TUBE
.LTT
TTT
x
O~
Radial Sampling Distance (cms)
TUBE
T
TUBE
V
CENTRE LINE: AMPLING
CENTt RE LINE SAMPLINO
1.057
CENTR
13.5
37.8
132
14.2
57.8
133
13.9
76.7
134
13.9
62.2
135
14.0
57.8
136
14.2
50.2
137
14.7
67.4
138
13.75
94.5
139
13.75
102
140
13.50
55.4
141
13.50
20.2
142
13.50
50.0
LINE d AMPLINGI
63.5
Ii-
~fil
D
60--
F
N
c
-
C-:T
f-
S4'T
200
/0
-
F o\-
-
F/-
6-
6
,-
v
1
-..--.
---
10a
F
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I-.-
3
o
.FSU9F*
20
11
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-
F
3
IO20
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FI3: I
I
21
1'IG: II
Skirting and cutting paths.
N
f
/
/
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FI
LI
Mean lateral displacement on
Freat-ercle path a/6.
ac
&tr
FIG: IV (II)
Streamline flow in
pack In.
Streamline flow
past sphere.
'e
FIG: V.
Distprbaxice of flow by
RATE
0f
frjIX W i-'J-
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Me, A j le tit B1 VL
5U
111
TL 5
W~.te r
~
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/k
Control Valve
141
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till'
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11
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FIG: VII.
er cury manometer.
anometer.
Apparatus for measuring
-
.
pressure-gradient throu6h
-pakin6-
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n
.
injection
noint
point
FIG; IX
FIG:X
Holder for samplins tubes.
zle,
u::ked out here
while stirring with tube
FodrIGr.Area sampling
device.
I-6
,