PHYSICAL MODEL FOR MASS TRANSFER INA PACKED ”
BED '

Thesis for the Degree of Ph. D.
MiGHiGAN STATE UNWERSHY
RAYMOS‘D LEON PORTER

1973 V

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PHYSICAL MODEL
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presented by

RAYMOND LEON PORTER

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ABSTRACT

PHYSICAL MODEL
FOR MASS TRANSFER
IN A PACKED BED

By

Raymond Leon Porter

A new method for calculating mass, heat and momentum transfer
between particles of a fixed bed and the fluid flowing through it is
shown. Overall mass and heat transfer coefficients and pressure loss
per unit length of bed are computed from fluid preperties--viscosity,
heat capacity, superficial velocity, thermal conductivity, density,
diffusion coefficient of active component through the fluid; and the
bed characteristics--porosity, particle size, specific surface per
unit volume and an index defining the distribution of passage cross
sections within the bed. Values calculated for gases in the Reynolds
number range from 5 to 33,000 show an average deviation of 3% from
literature correlations [5, 8, 15, 31, 40, S4, 58]. Values for
liquids in the Reynolds number range from 0.003 to 33,000 and for
Schmidt numbers up to 70,600 deviate an average of 5% from literature
results [15, 27, S4, 59, 60]. These figures are for fixed beds with
voids fractions ranging from 0.38 to 0.70.

It is believed that the values calculated in ranges not corrob-
orated by experimental investigators are of equivalent accuracy.

This is because the method deve10ped in this thesis is not a simple

Raymond Leon Porter

correlation of experimental data, but is based on a theoretical
treatment of a reasonable physical model for a packed bed using
principles of fluid dynamics and transport phenomena.

The physical model consists of a network of passages arranged
in parallel and series with complete mixing assumed at the passage
junctions. The passages are assumed to have a distribution of
cross sections as described by the index mentioned above. This
distribution of cross sections has an effect on coefficients
computed for the complete Reynolds number range. Its effect is
greatest at extremely low Reynolds numbers where it gives Nusselt
and Sherwood numbers which are considerably lower for the bed than
for the limiting values of the individual passages. .

In the region of fully developed velocity profiles through the
passages, treatment of the passages as cylinders with lengths equal
to packing size proved to be satisfactory and convenient. In the
region of developing boundary layers the length was taken to be half
the packing size to allow for boundary layer separation over surfaces
curved in the direction of flow. Typically it occurs at about 90
degrees around the curve for surfaces such as cylinders or spheres.

The method presented here is in the form of a computer program
due to the complexity of handling different cross sections in parallel_

with different flow patterns in the various cross sections.

PHYSICAL MODEL
FOR MASS TRANSFER
IN A PACKED BED

BY

Raymond Leon Porter

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Chemical Engineering

1973

ACKNOWLEDGMENTS

The author would like to thank Dr. Carl M. Cooper for his
expert guidance during the writing of this thesis. The patience
and understanding of Dr. Cooper, the other members of the Michigan
State University Chemical Engineering Department faculty and my

family are appreciated very much.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENT . . . . . . . . . .
LIST OF TABLES . . . . . . . . .
LIST OF FIGURES . . . . . . . . . .
INTRODUCTION ... . . . . . . . . . .
LITERATURE SURVEY
PACKED BEDS . . . . . . . . . .
FLOW PHENOMENA . . . . . . .
THEORETICAL ANALYSIS
MODEL OF A PACKED BED . . . . .
'MODEL EQUATIONS . . . . . . . .
OPERATION OF MODEL. . . . . . .
RESULTS . . . . . . . . . . . . .
DISCUSSION OF RESULTS . . . . . . .
SUMMARY . . . . . . . . . . . . . .
CONCLUSIONS . . . . . . . . . . . .
RECOMMENDATIONS FOR FUTURE WORK . .
APPENDIX
A. DERIVATION OF MODEL EQUATIONS
B. DISTRIBUTION INDEX . . . . . .

c. (VY2)m VARIATION WITH REYNOLDS

iii

NUMBER

21

. 24

. 38

89

. 102

. 104

. 107
. 114

. 119

Page

D. SUMMARY OF MODEL EQUATIONS . . . . . . . . . . . . . . . . 122
E. COMPUTER PROGRAMS . . . . . . . . . . . . . . . . . . . . . 125
F. SHERWOOD NUMBERS, UNIFORM AND NON-UNIFORM PASSAGES . . . . 132
TABLE OF NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . 134
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . 139

iv

13.

14.
15.
l6.
l7.
18.
19.

20.

Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison
Comparison

Comparison

[59] . .

Comparison

of Model
of Model
of Model
of Model
of Model
of Model
of Model
of Model
of Model
of Model
of Model
of Model

of Model

[59] . . . . . .

Comparison
Comparison
Comparison
Comparison
Comparison
Comparison

Comparison

of Model
of Model
of Model
of Model
of Model
of Model

of Model

LIST OF TABLES

with Chu, Kalil and Wetteroth [8] .

with Chu, Kalil and Wetteroth [8] .

with Chu, Kalil and Wetteroth [8] . . .

with Thoenes
with Thoenes
with Thoenes
with Thoenes
with Thoenes

with Bennett

and Kramers [54] .
and Kramers [54] .....
and Kramers [S4] . . . . .
and Kramers [S4] . . . . .
and Kramers [S4] .

and Bradshaw [3] . . . . .

with Kasik and Happel [31]. . . . . . .

with Kusik and Happel [31]. . . . . . .

with Williamson, Bazaire and Geankoplis

with Williamson, Bazaire and Geankoplis

with Wilson
with Wilson
with Wilson
with Wilson
with Wilson
with Wilson

with Wilson

0 O O O O O I O O O O O O O

and Geankoplis [60] . . . .
and Geankoplis [60] . . .
and Ceankoplis [60] . . . .
and Geankoplis [60] . . . .
and Geankoplis [60] . . . .
and Geankoplis [60] . . . .

and Geankoplis [60] . . . .

,so

51

$2

53

.54

S6

S7

58

60

Table Page

 

21. Comparison of Model with Wilson and Geankoplis [60] . . . . 69
22. Comparison of Model with Galloway and Sage, Spheres [15]. . 72
23. Comparison of Model with Galloway and Sage, Spheres [15]. . 73
24. Comparison of Model with Galloway End Sage, Spheres [15]. . 74
25. Comparison of Model with Galloway and Sage, Spheres [15]. . 75

26. Comparison of Model with Galloway and Sage, Commercial
Packing [15] . . . . . . . . . . . . . . . . . . . . . . 76

27. Comparison of Model with Galloway and Sage, Commercial
Packing [15] . . . . . . . . . . . . . . . . . . . . . . 77

28. Comparison of Model with Galloway and Sage, Commercial _
Peeking [15] O I O O O O O O O O O O O O I O O O I O O O 78

29. Comparison of Model with Galloway and Sage, Commercial
PaCking [15] O O O O O I O O O O O O I O O O O O O O O O 79

30. Comparison of Model with Galloway and Sage, Commercial
PaCking [15] O O O O O O O O O O O O O O I O O O O O 0 O 80

31. Comparison of Model with Petrovic and Thodos [40] . . . . . 82
32. Comparison of Model with Jolls and Hanratty [27]. . . . . . 83
33. Comparison of Model with Wilkins and Thodos [58]. . . . . . 84
34. Summary of Results for Cases. . . . . . .......... 85
35. Summary of Results for Liquids. . . . . . . . . . . . . . . 86
36. Example Using Model . . . . . . . . . ........... 88
37. Constants for Equation 44 . . . . . . . . . . . . . . . . . 120

38. (W2 Variation with Reynolds Number ........... 121

)m
39. Computer Program Listing for Table 2 . . . . ....... 126
40. Computer Program Listing for Table 36 . . . . . . . . . . . 128

41. Sherwood Numbers, Uniform and Non-Uniform Passages ..... 133

vi

10.
11.
12.

13.

14.

15.

LIST OF FIGURES

Random Packed Bed of Spheres . . . . . . . . . . . . .
Model of a Random Packed Bed . . . . . . . . . .
Distribution Index . . . . .

Flow in a Passage . . . . . . . . . .

Nusselt Number in Tubes . . . . . . . . . . .

Effect of Schmidt Number on Mass Transfer at Low Reynolds
Nmnbers O O O O O O O O O O O O O O O O O O O O

Sherwood Numbers for Uniform and Non-Uniform Passages

Comparison of Model with Wilson and Geankoplis [60], Low
Reynolds Numbers . . . . . . . . . . . . . . . . . . .

Comparison of Model with Wilson and GeankOplis [60], High
Reynolds Numbers . . . . . . . . . . . . . . . . . . .

Comparison of Model with Literature Correlations . . . .
Comparison of Model with Literature Correlations . . .
Comparison of Model with Literature Correlations . . .

Comparison of Model with Data of Wilson and
Geank0plis [60] . . . . . . .

Simple Cubic Array of Spheres

Rhombohedral Array of Spheres . . . . . . . . . . . . .

vii

Page

24

25

30

32

36

43

44

92

93

95

96

116

117

118

INTRODUCTION

A large number of chemical processes, especially catalytic
processes, are carried out in packed bed reactors. With improvements
in catalytic reactors, ion exchange columns, leaching beds, chromat-
ographic columns and gas adsorbers, it has become increasingly
important to predict accurately the performance of packed beds.

Proper design involves a knowledge of heat, mass and momentum
transport between fluid and solid surfaces.

Mass transfer between the packing and the flowing fluid can
occur in either direction depending upon the process. In either case
the physical phenomena is similar. There is a transfer of some chemi-
cal specie to and from the phase boundary through a series of resist-
ances. Resistances may be due to diffusion in either phase, laminar
or turbulent convection in the fluid, or due to slow chemical formation
or reaction of the active chemical substance.

In some processes the controlling resistance is in the flowing
fluid, such as in a catalytic reactor where there is a slow flow rate
and a fast chemical reaction rate. In such a case the size of the
equipment is determined by the mass transfer rate between the fluid
and solid particles. When the fluid flow rate is rapid and the
reaction rate is slow, chemical kinetics dictates the design.
Knowledge of the mass transfer rate for a particular process is
essential for design and it is dependent upon the type of heterogeneous

system.

2

Mechanism of mass transfer has been explained by several theories.
The film concept assumes that most of the resistance to mass transfer
occurs in a stagnant layer next to the solid surface. Mass transfer
through the film is by diffusion. Boundary layer studies show that
there is also considerable resistance in the bulk flow stream and in
the buffer region next to the film adhering to the solid surface.
Another completely different explanation called the 'penetration
theory' holds that the flowing fluid is a mass of eddies which
continually expose fresh surfaces of fluid to the solid. No matter
which theory is followed mass transfer rates are generally expressed
in terms of a mass transfer coefficient, kc, just as heat transfer
rates are given in terms of a heat transfer film coefficient, h.

Variables influencing mass transfer coefficients in packed beds
are size and shape of the voids, viscosity and density of the flowing
fluid, and the diffusivity of the active substance in the fluid.
As with many other chemical engineering processes, correlations are
effected using dimensionless groups. The common ones for mass
transfer are Reynolds number, Schmidt number and Sherwood number.
Reynolds number is a measure of fluid flow rate. Schmidt number
contains only the physical prOperties of the fluid and its active
component which makes it similar to the Prandtl number of heat
transfer. Sherwood number contains the mass transfer coefficient
and the diffusivity and is analogous to the Nusselt number of heat
transfer. For packed beds particle diameter is commonly used in
place of effective diameter of voids and fluid velocity based on
empty cross sectional area is used in place of interstitial velocity.

For flow through pipes the analogy between heat and mass transfer

exists because they both occur due to molecular diffusion and

3

convective mixing. Thus heat transfer correlations can be used to
calculate numerical values for mass transfer rates to or from pipe
walls. There is, however, less literature data concerning heat
transfer in packed beds than there is for mass transfer.

Momentum transfer, in terms of pressure drop, in a packed bed
can easily be determined from existing equations. However, attempts
to show the analogy between momentum and mass transfer have not been
successful.

Most of the correlations for mass transfer in packed beds are
given in the literature by a relationship between Reynolds number
and Sherwood number divided by Schmidt number to the one-third
power. Recent correlations give mass transfer rates which are in
reasonable agreement with reliable reported data. The best corre-
lations are for liquids flowing at high Reynolds numbers. For gases
at lower Reynolds number flow rates, correlations are more difficult
because diffusion coefficients are so much higher for gases than for
liquids. Equations necessarily have to be more complicated.

Keeping in mind all of the complexities of packed beds it was
decided to formulate a physical model for a packed bed which would
take into account factors such as fluid properties, packing arrange-
ments, nature of flow and the inter-relationships among these factors.
Due to the ready availability of digital computers it was thought
that a fairly sophisticated model could be devised which could be
readily solved by the computer for desired results.

For simplicity reasons it was decided to derive the model on
the basis of heat transfer and then make the necessary analogies for

application to mass transfer. It was also thought that some

4
correlation could be obtained between mass and momentum transfer in
packed beds, namely an equation in which mass transfer coefficient

is a function of pressure loss per unit length of bed.

LITERATURE SURVEY

PACKED BEDS

 

The scientific means by which mass transfer occurs in packed
beds has been investigated by many persons. One of the first
investigations was by Colburn [9] in 1933 who wrote an analogy
between frictional resistance to fluid flow, heat transfer and mass
transfer which was based on flow through tubes and across tube

banks. He reported the equation:
Nu = 0.33 Re'6 pr1/3

for heat transfer across tube banks. Using the same analysis
Chilton anlColburn [7] later suggested as a basis for correlation

of heat transfer data the following equation:

2/3

h/CG Pr = J = f(Rep)

h

and for mass transfer a similar equation:
2/3 . .

~Graphs of J versus Reynolds number were presented for turbulent flow
inside tubes, across tube banks and parallel to flat plates. They
reported that the mass transfer equation disregarded free convection
at low Reynolds numbers and any liquid film resistance at the gas-
liquid interface on the tubes. Mass velocity was for the relative

motion between the two phases.

By using water evaporation data from a through circulation
dryer experiment Gamson, Thodos and Hougen [19] reported values
of Jd averaged about 8% lower than Jh values. They assumed in
the calculations that the surface temperature of the particles was
equal to the adiabatic saturation temperature. One of their

recommended equations was:

Jd = 16.8 (Rep)'1 for Rep< 40

Sherwood [50] pointed out that if the surface temperature were
not at the adiabatic saturation temperature the Jd values could vary

widely whereas the J values would vary little.

h
Wilke and Hougen [57] used the same type of experiment as Gamson
et. al. and by controlling heating conditions and changing the method

of wetting the packing arrived at a different equation.
J . 1.82 (Re )'-$1 for Re < 100
d p p

They also assumed the surface temperature to be equal to the adiabatic
saturation temperature.

Hurt [25] used different sizes and shapes of packing and measured
the height of a transfer unit for gas controlled systems. He showed
good agreement between heat and mass transfer factors when employing
cylindrical particles. The relationship between height of a transfer

unit and Jd is:

Jd - Sc2/3/(Ht)a

where a is the specific surface area per unit volume. The agreement
was poor for other packing shapes. Hurt did not, however, report the

surface area or the voids fraction of his packed beds.

Resnick and White [45] ran experiments with fixed and fluidized
beds of naphthalene particles. Results showed Jd values lower than
those of Gamson et. al. which was attributed to the use of smaller
particles.

McCune and Wilhelm [36] obtained data for the mass transfer in
both fixed and fluidized bed between flakes of P-naphthol and
flowing water. Gamson [18] collected data for water evaporation
from porous particles into a flowing air stream. Hobson and Thodos
[24] observed data during mass transfer to water or methyl ethyl

ketone adsorbed on fixed bed particles. BrStz [6] analyzed these

authors' data and came up with the equations:

-.41

Jd I 1.46 (Rep)

(1 - c)’61 for Rep/(l - c)> 100

_ -l _ 1.2 _
Jd 17 (Rep) (1 e) for Rep/(1 c)< 100

using an equivalent diameter for particle diameter.
By changing temperature and pressure, the effect of gas
properties on the mass transfer coefficient was studied by Shulman

and Margolis [51]. They reported that J was independent of pressure

d
in their equation:

Jd - 1.195 [Rep/(l - an"36

Robson and Thodos [24] measured evaporation rates of water and
organic compounds from spherical packing into air, carbon dioxide,
ammonia and nitrogen. It was found that the temperature of the bed
decreased linearly in the direction of flow but the Schmidt number
remained practically constant, so the temperature effect was

disregarded.

Ergun [12] correlated a mass of experimental data and arrived
at an equation for pressure drop in packed beds in terms of

dimensionless groups.

(-AP)chpe3/[pu2L(l - 5)] 150(1 - e)/Rep + 1.75

At low Reynolds numbers the 1.75 is negligible and at high Reynolds
numbers it is dominant.

Chu, Kalil and Wetteroth [8] correlated data on heat and mass
transfer in liquid-solid and gas-solid systems and arrived at the

equation:
Jd - 1.77 [(Rep/(l - e)]'-44 for 30 < Rep/(l - e) < 5000

. Epatein [ll] determined an axial mixing factor to correct heat
and mass transfer coefficients to account for non-plug flow in packed
beds. The fixed bed is treated as a series of perfect mixers in his
mathematical treatment.

‘ Thoenes and Kramers [54] determined mass transfer coefficients
for fluids flowing around single active Spheres surrounded by similar
inactive spheres using eight different packing arrangements. The
acive Spheres were either soluble in the flowing fluid or were
porous and soaked with liquid “thich evaporated into a gas stream.
:Graphs of [Shp/Sc1/3][e/(1 - 2)] (which is equivalent to
eJdRep/[l - 2]) versus Rep/(l - S) were presented. A review of 438
mass transfer measurements was expressed by the equation:

Shp[e/(l - e)] . 1.0 Sc1/3[Rep/(1 - an”2

which was said to be good for a Rep /(1 - e) range between 40 and 4000,

a voids fraction range between 0.25 and 0.50 and a Schmidt number
range between 1 and 4000. It was said to have a mean deviation of

t 10%. An even better correlation was obtained by assuming that the
total mass transfer was due to three contributions: laminar
convective transfer, turbulent convective transfer and one for
diffusion in stagnant areas. The latter is important for gas flows
at Reynolds numbers less than 500. For gases the stagnant regions
near the contact points of adjacent spheres are important because
diffusion coefficients for gases are so much larger than for liquids.

The following equation was correlated:
Shple/(I - c)] . 1.26 Sc1/3[Rep/(1 - 5)]1/3 + 0,054 Sc-4[ReP,(1 _ 6)].8
* 0.8 [Rep/(1 - e)].2

in which the first term is for laminar convection, the second for
turbulent convection and the last for diffusion.

Al-Khudayri [1] made a correlation for predicting the mass trans-
fer coefficient in packed beds. The correlation is a plot of
[Shp/Sc1/3][e/(l f 5)] versus Rep/(l - e). For liquid-solid systems
the deviation of experimental data of other investigators was 30% and
was higher for gas-solid systems. His experimental work consisted
of the absorption of ammonia fvom a helium-ammonia flow stream onto
-the surface of 0.726 cm. diameter alundum spheres coated with
copper II chloride. Laminar flowsware used amd mass transfer
coefficients were calculated from the data. His results checked
closely with those of other investigators for gases at low flow rates.
Al Khudayri pointed out that void volume and void surface area are

more valid to use than packing diameter when expressing the mass

lO

transfer characteristics of the bed. If packing diameter is used,
a correction needs to be made to compensate for variations in voids
fraction.

DeAcetis and Thodos [10] made careful temperature measurements
of air and packing surface during the evaporation of water from the
surface of porous ceramic spheres into an air stream. They found
that contrary to usual assumptions, temperature of the packing sur-
face is not the same as the wet-bulb temperature of the entering
air, unless high air flow rates are used. They summarized their
data and that of other investigators up to 1960 in graphs of Jh and
Jd versus Rep. The ratio of Jh to Jd reported was 1.51 compared to
the value of 1.08 given by Gamson, Thodos and Hougen which was
obtained on the assumption that the temperature of the evaporating
surface was that of the wet-bulb temperature of the air entering
the bed.

Bradshaw and Bennett [5] measured mass transfer coefficients

for air flowing through short beds of naphthalene Spheres and

cylinders. They reported the equation:

1/3

J . 2.0/Re Sc + 1.97/Re 1’2
d p p

which was said to cover the Reynolds number range from 40 to 10,000.
Sen Gupta and Thodos [22] analyzed the data of other workers
-and found Jd to be inversely proportional to voids fraction for mass
transfer to flowing gases in packed beds.
Kusik and Happel [31] made a theoretical study of gas diffusion

rates in packed beds using a free-surface model (spherical particle

surrounded by a spherical envelope) with boundary layer theory.

11

The Reynolds number (Rep/e) range covered was from 100 to 1000 with
voids fractions of 0.3 to 1.0. Simplified forms of momentum and
diffusion equations were solved to predict dissolution rates in a
particle bed and to analyze the effects of molar velocity perpendicular
to the spherical catalytic surface. Boundary layer equations were
solved by using Pohlhausen's method of introducing polynomials
describing velocity and density distributions. One equation arrived

at was :
Sh/Sc1/3=O.93[e - 0.75(1 - e)(e - 0.2)]‘1/2Rep1/2

The authors confirm that the change in boundary layer thickness due
to normal convective velocity was the same for boundary layers on
spheres as for flat plate geometry. The results Of the study
showed that the assumption in film theory that the film thickness
does not change with mass transfer rate is correct. This means that
film theory can be used for the bulk of chemical engineering problems
involving heterogeneous catalysis.

Williamson, Bazaire and Geankoplis [59] obtained liquid phase
mass transfer coefficients for packed beds of benzoic acid spheres

with water passing through. The recommended equations were:
St SCo53 = 2.40 (Rep/;)'-66
for Rep]: from 0.08 to 125 and:
St Sc-58 - 0.442(Rep/E)'°31

for Rep/e from 125 to 5000.

12

Galloway and Sage [16] developed analytical expressions for
heat and mass transport from Spheres to a turbulent air stream
taking into account Reynolds number, sphere diameter, and level of
turbulence. Sphere size ranged from draplets to 1 ft. in diameter
and superficial Reynolds number values ranged from 2 to 1.33 x 106.
By analyzing the works of other investigators it was shown by graphi-
cal means that the assumption of Nusselt number to be a single valued
function of the square root of Reynolds number leads to an average
deviation of 60%. Their analysis begins with the Frossling [14]
equation for macroscOpic transfer from spheres with zero turbulence:

Nu . 2.00 + 0.552 Re 1/2Pr.1/3
P“
Relationships derived for predicting convective thermal and material

transport were:

[(Nu - 2.00)]/Re l/Zpr.1/3 - 0.538 + 0.1807 01/2 + 0.323
a (a + 0.0405)Re 1/2
t t p

[(Sh - 2.00)1/Repa1/25c.1/3 . 0.439 + 0.1807 01/2 + 0.234

a (a + 0.0500)Re 1/2
t t p

where d is sphere diameter and at is the longitudinal level of
turbulence.

I Mickley, Smith, Korchak [37] measured velocity profiles and
turbulence parameters in the voids of a 1 ft. square bed of
rhombohedrally arranged 1.5 in. diameter table tennis balls. A

hot wire anemometer was used and the superficial Reynolds number

13

range covered was from 4780 to 7010. Turbulence energy spectra
showed that eddy shedding behind the particles did not occur in
the voids between spheres. Since high heat transfer coefficients
are known to be caused by eddy shedding and high turbulence level,
the high local heat transfer coefficients in the voids characteristic
of rhombohedral packing must be explained by a high level of turbu-
lence intensity. The mean void velocity showed a maximum within
1.5 particle diameters from the wall. There was a 10% difference
in the mean velocity at the center of the bed compared with the
maximum region.

Rhodes and Peebles [46] determined local mass transfer rates
at room temperature by measuring radius changes in 1.5 in. diameter
benzoic acid spheres by passing water around them at various flow
rates using simple cubic and rhombohedral packing arrays. The
Reynolds number range covered was from 166 to 3410 based on superficial
velocity. Mass transfer tests were carried out by placing the test
sphere in an assembled array of inert, insoluble spheres. Analysis
of results for the cubic array (voids fractionw-0.4764), in terms
of Sherwood number versus degrees from the front stagnation point,
suggested that the flow pattern around the test sphere had the
following characteristics: (a) The region around the forward contact
point (0 to 10 deg.) showed the minimum mass transfer rate. (b) The
over-all maximum mass transfer rate occurs between 50 to 80 degrees
forward from the front stagnation point. It was suggested that this
is the ring of attachment of the boundary layer of the sphere above.

In this region mass transfer rates are 2.2 to 3 times the over-all

14

average. A streamline arriving at this location splits into two
streamlines: one circling upward forming the principle eddy of the
wake of the preceding sphere and the other streamline attaching
itself as a boundary layer that follows along the sphere surfaee
until it reachasa ring of separation between 103 and 122 degrees
depending upon the Reynolds number. (c) A region of essentially
zero mass transfer occurs where there is a point of contact between
spheres. (d) The region to the rear of the separation ring is a
wake region where the local mass transfer rates are less than the
average over the entire sphere. In the rhombohedral array (voids
fraction m 0.2595) the orientation of packing was such that each
sphere was entirely behind another sphere in the flowing stream,
thus giving the limiting case for investigating extremes of local
mass transfer rates. Considerably higher maximum Sherwood numbers
were reported between 30 and 50 degrees from the front stagnation
point than for the cubic array.

Wilson and Geankoplis [60] reported studies of mass transfer
and reviewed earlier works. They used a bed of randomly packed
benzoic acid spheres with an average diameter of 0.251 in. Water or
propylene glycol were allowed to flow down through the bed. For
superficial Reynolds numbers between 55 and 1500 and voids fractions
between 0.35 and 0.75 they recommend:

0.31

eJ

d I 0.250/Rep

Between superficial Reynolds numbers of 0.0016 and 55 the equation

given was:

_ 2/3
eJd 1.09/Rep

15

This eqUation was shown to correlate data over a Schmidt number
range of 165 to 70,600.

Petrovic and Thodos [40] determined mass transfer factors in
a packed bed by vaporizing water and heavy hydrocarbons from the
surface of 0.0721 to 0.370 in. diameter random packed spheres into
air. Using this data and recalculating various other studies by
Thodos and coworkers to correct the data for axial mixing, their

recommended equation is:

I I

The results of this study covered the superficial Reynolds number
range between 3 and 230 for voids fractions between 0.416 and 0.778
and were said to hold for solid-gas systems subjected to either
upward or downward flow.

Satterfield [48] compared the equations of Wilson and Geankoplis
and Petrovic and Thodos and found that they only differ 15% or less
over a range of superficial Reynolds numbers between 55 and 1500.

Gillespie, Crandall, and Carberry [20] measured local and
overall heat transfer coefficients in two random packed beds of
l in. diameter brass Spheres. Air was passed through the packing
at flows corresponding to a Reynolds number range of 120 to 1700,
based on superficial velocity and Sphere diameter. Local heat
.transfer coefficients were measured in the first, second and nine-
teenth layers of packing. Average heat transfer coefficients were
determined at 25 places in the bed. By examining the local heat
transfer distribution the existence of a laminar boundary layer was

verified. Highest values of heat transfer coefficient were obtained

16

for the surface perpendicular to the bulk flow in the bed. It was
also observed from heat transfer coefficient profiles that at high
Reynolds numbers the flow may rejoin the sphere and begin to build
another boundary layer which subsequently separates. The effect of
repacking of the bed was to change the range of local heat transfer
coefficients, but the variation within the range was about the same.
The entrance effect of the bed has been shown to result in a lower
heat transfer coefficient in the tOp layer than in the bulk of the
bed. This has been attributed to a lower incident flow rate and
turbulence intensity. The effect of lateral position on average
heat transfer coefficient showed higher coefficients near the wall
than at the center of the bed.

Wilkins and Thodos [58] studied the evaporation of n-decane in-
to air from the surface of 0.1 in. diameter celite spheres in both
a random packed bed and a fluidized bed. Using their results and

those of other investigators they obtained the relationship:

cJ - 0.589/Rep°°427

d

Jolls and Hanratty [27] used electrochemical techniques and studied
details of flow around an instrumented l in. diameter nickel plated
brass-bronze ball located 7 to 8 inches from the top in a dumped
bed (voids fraction - 0.41) of l in. diameter glass spheres.
.Reynolds number (based on empty cross—section) ranged from 5 to 1100
and the Schmidt number of the flowing fluid was 1700. A transition
from laminar to turbulent flow was found to occur in this system
over a Reynolds number range from 110 to 150. The electrochemical
reaction consisted of the reduction of the ferricyanide ion on the

nickel cathode (test Sphere) and oxidation of the ferrocyanide ion

17

on a nickel pipe anode located outside the column. Electrode lead
wires were placed at various points arOund the sphere. With the
exception of the very rearward portion of the Sphere the effect of
Reynolds number on the local mass transfer rate was the same as that
predicted by boundary layer theory. At the rear of the sphere local
mass transfer measurements indicated a larger variation with Reynolds
number, apparently due to separation. The effect of Reynolds number
on the overall mass transfer to a sphere in either a bed of inert or
active spheres indicates a slightly higher power on the Reynolds
number dependency than that predicted by boundary theory. This
research showed that the flow pattern varied from sphere to sphere.
In order to make meaningful results it was necessary to average
measurements from a large number of experiments. Reasonable
correlation was obtained by assuming the Sherwood number varied with
Re 0°57 and Sco'ss.

P Galloway and Sage [17] used an instrumented 1.5 in. diameter
copper sphere in a 12 in. diameter, 20 in. long column containing
a rhombohedral array of 1.5 in. uniform diameter spheres in order
to make local heat transfer measurements. Inert spheres were made
of celluloid and partial spheres were used on the inside surface
of the cylindrical wall to fill out the array and reduce wall effects.
The study concerned determination of local heat transfer coefficients
.as a result of steady flow of air and covered a range of superficial
Reynolds numbers from 875 to 3618 with the effect of turbulence
being noted. Local air velocities in flow passages were measured
directly. Analyzing available literature data their model provided

an analytical expression which was found to represent transport

18

from Single cylinders and spheres, and arrays of cylindrical,
spherical, and commercial packing. Overall deviation was 9.8%.

The model also predicted the height of a gas phase transfer unit

in commercial packed columns being irrigated with liquid within 12%
for twelve cases involving absorption and vaporization. .The basis
of their boundary layer model was that the local Frassling number

l”Sci/3]) was independent of surface

(Fs, equivalent to [Sh-2]][Re
shape and configuration of packing. Consequently such a model should
apply equally well to any packing material. Their general equation

was 3

u/kc - (Ht)a - ReSc/Sh - Rel/28c2/3/[Fs + 2/(Rel/28c1/311

Galloway [15] presented results of earlier analyses of data using
uniform Spheres, cylinders and commercial packing. The expression

given for mass transfer in beds of spheres was:

A 1 3 2
sap . 2/[1 - (1 - cp) / 1. °°55[(°pG/")(€p _ 6b)11/ Sc1/3

1/3
+ 0.30 Zt(Zt + 0.05)(DpG/uep)5c

in whick SP is the voids fraction of the packing, ch is the voids
fraction of relatively stagnant regions in the bed, and 2t is the
turbulence level.

Haring and Greenkorn [23] dev010ped a statistical model of a
porous medium with non-uniform pores which matched experimental
capillary pressure, permeability and dispersion data. The model
was constructed with two parameter distribution functions for pore

radius and pore length. Orientation of pores was considered random

in all directions. Various properties of a porous medium were found

19

by integrating over joint distributions resulting from the model.
Dimensionless quantities were used for pore length and radius so they
varied from O to 1. To make the model non-uniform the dimensionless
terms were each assumed to be diStributed according to the beta
function. The permeability of the model was found by relating the
average velocity in an individual pore to the average velocity of
all pores. The permeability-porosity ratio, which causes dissipation
due to entrance-exit effects, was found to be a function of the
average pore radius squared and the pore radius distribution. For
most flow situations of interest to engineers, the residence time
of the fluid in the individual pore is much smaller than the time
needed for appreciable mixing due to molecular diffusion within that
pore. Neglecting molecular diffusion, expressions for dispersion
coefficient were found by determining the probability distribution
of the position of a marked particle after a random walk of inde-
pendent steps through the model. The diSpersion coefficient was
found to be dependent on both pore radius and length distributions.

Wegner, Karabelas, Hanratty [56] made studies of the motion of
dye streamers in a rhombohedral array (voids fraction -«O.26) of
3 in. diameter Plexiglas spheres. The test sphere containing the
dye taps was located in the tenth layer of a fifteen layer bed.
Similar flow patterns were observed at superficial Reynolds numbers
.0f 82 and 200. Nine distinct regions of reverse flow were noted.
Flow was described as steady at the lower flow rate and unsteady at
the higher one. 1

Van Der Merwe and Gauvin [55] investigated flow deveIOpment in

packed beds by setting up an experimental apparatus using a regular

20

arrangement of ten banks of seven centimeter diameter spheres. A
skewed arrangement was also tested where the spheres were arranged
on 0.375 in. rods at an angle where the mean flow direction made
equal angles with the three principal axes of the packing. The
pressure drag coefficients on the central Sphere of each bank were
determined for air having Reynolds numbers of 27,000 and 10,000.
It was found from the distribution of local pressure measurements,
which allowed determinations of the separation and reattachment
points of the boundary layer on the central sphere of each bank,
that the boundary layer behavior on a sphere in a packing is similar
to that of a Single Sphere. The skewed arrangement showed a lower
pressure drag coefficient than did the regular arrangement at the
same Reynolds number.

Karabelas, Wegner and Hanratty [28] studied the effect of Grashof,
Reynolds and Schmidt numbers on mass transfer rates to liquids
(Sc . 1600) from cubic arrays of spheres. For Reynolds below

Rep - 110, the correlation equation was:

Shp a 0.46 (Gr Sc)-25

They also give a summary of other authors' correlations for heat

and mass transfer data.

21

FLOW PHENOMENA

Graetz [21] in 1883 made the first analysis for the development
of the temperature profile in a round tube. He assumed the velocity
profile was fully developed at the tube entrance for the two cases
of uniform and parabolic velocity. Nusselt substantiated Graetzls
solutions independently in 1910. Pohlhausen [42] solved the problem
of heat transfer to a fluid in laminar flow parallel to a flat plate.
The velocities and temperatures are approximated by polynomials in
y having coefficients that are functions of x. The coefficients are
determined by satisfying the boundary conditions at the plate and at
the edge of the boundary layer using integral forms of the equations
of continuity, motion and energy for the boundary layer.

Leveque [33] modified the problem of heat transfer to a fluid in
laminar flow in a pipe with constant temperature walls. He assumed the
parabolic velocity profile to be completely developed before the fluid
enters the section of pipe where the heating begins. A thermal
boundary layer is then assumed to develOp, superimposing itself on
the already developed velocity profile. The following equation was

-formulated:
Nu . 1.077 [0 Re Pr/L]1/3

This equation gives the same values for Nusselt number in the region

[0 Re Pr/L] greater than 100 as the more complicated Graetz equation.

22

The Leveque equation is not valid beyond the length where the thermal
boundary layer reaches the center of the pipe.

Norris and Streid [38] analyzed the problem for laminar flow in
flat rectangular ducts and suggested that entrance region Nusselt num-
bers for simultaneously develOping velocity and temperature profiles
might be obtained using results for heat transfer from a flat plate.

Langhaar [32] postulated that the pressure gradient in the
transition length of a tube is higher than in a region of laminar
flow because of increased frictional loss and increased kinetic energy
if fluid as it passes downstream. He used linear approximation methods
to solve the Navier-Stokes motion equations involving frictional flow
for the case of steady flow in the transition length of a straight
tube. A family of velocity profiles was determined which were
defined by means of Bessel functions. The pressure function was then
derived from the computed velocity field by means of the general
energy equation.

Sparrow [53] studied the simultaneous develOpment of temperature
and velocity profiles in flat rectangular ducts. Laminar flow and
constant wall temperature were assumed. Thermal and velocity
boundary layer calculations were made using the Pohlhausen method.
Nusselt numbers were reported for the Prandtl range from 0.01 to 50.
By plotting D Re Pr/L (Graetz.number, 02) versus Nusselt number it was
found that there is a separate curve for each Prandtl number in the
entrance region. In contrast when a parabolic profile is assumed at
the entrance, there is a single curve which satisfies all Prandtl
numbers. In order to compare his results with those of Norris and

Streid, the Pohlhausen solution gave the equation:

23
Nu = [0.664/¢][D Re Pr/L11/2

For Prandtl numbers equal to or greater than one, many investigators
have found 0 - PSI/6. For Pr less than one Sparrow has a plot of
e vs. Pr.

Kays [30] studied the problem where the thermal and hydrodynamic
boundary layers develop at the same time by combining Langhaar's
results for the developing velocity profile with a numerical solution
of the differential energy balance. His solutions were limited to
fluids with a Prandtl number of 0.7. Kays pointed out that for high
Prandtl number fluids, such as oils, the assumption of a fully
developed velocity profile at the tube entrance does not affect the
heat transfer mechanism because the velocity profile is established
much more rapidly than the temperature profile at the place where
heating begins. However, for fluids with Prandtl numbers near unity,
such as gaSes, the velocity and temperature profiles develop at
nearly the same rate along the tube. As a result experimental data
Showed considerably higher Nusselt numbers than predicted by the
assumption of a parabolic profile throughout the tube. The Graetz
parabolic velocity solution provided lower limit Nusselt numbers,
while the Graetz slug flow solution gave upper limiting values.

Kays showed that the Pohlhausen flat plate solution using Langhaar
‘velocity profiles gave intermediate Nusselt numbers to those with
parabolic and slug flows. As D/L approaches zero, Nusselt number
approaches a minimum value of 5.75 for slug flow and 3.656 for
parabolic flow. Kays postulates that the Pohlhausen solution should

approximate actual performance near the tube entrance.

THEORETICAL ANALYSIS

MLQAEAQKEDEED

Packed beds are commonly made by packing tubes or cylindrical
vessels with solid particles such as cylindrical catalyst pellets,
Raschig rings, spheres, etc. These beds are generally used to effect
mass transport between the bulk of a fluid flowing through the bed
and the fluid-solid interface (or the interface with a second fluid
which wets the solid). Usually the solid particles which make up
the bed distribute themselves in a random fashion, but sometimes,
especially in research on the characteristics of packed beds, the
packing is placed in the bed in a regular pattern. Figure 14 and
15 in Appendix B Show the spatial arrangement for two types of

regular fixed beds.

 

 

 

Figure 1. Random Packed Bed of Spheres

The mass, momentum, and heat transport characteristics of packed
beds have been investigated by others in a vast number of experiments

resulting (after correction for wall and and effects) in a large

24

25

number of correlating equations for transport in the bulk of the
packed bed. These equations are generally expressed in terms of a
packed bed Reynolds number together with a Sherwood and Schmidt
number (or alternatively a Nusselt and Prandtl number). The tacit
assumption is commonly made that these correlating equations may be
used to design beds with a different packing material and a different
random packing arrangement as long as the dimensionless variables are
in the same range as the experiments.

While such extension of the correlating equations could lead to
erroneous results (for example, when cylindrical packing happens to
arrange itself in a manner which blocks the fluid) they have been used
in this manner with some success. As might be expected, the corre-
lating equations are extended more successtlly when the dimensionless
variables are defined in terms of the average interstitial velocity
u]: rather than superficial velocity u, and interstitial hydraulic
radius e/a, or c Dp/6(l - e), rather than the spherical packing
diameter DP' Thus a packed bed may be viewed more appropriately as
a network of channels of varying Shapes and sizes rather than as an
aggregate of solids. The active part of the bed is the voids. It is

the solids which are inert.

 

 

 

 

 

Figure 2. Model of a Random Packed Bed

26

This thesis grows out of the concept that the transport
characteristics of a random packed bed can be computed from a
physical model consisting of a simplified network of channels.
Al-Khudayri [I] assumed a network of uniform cylindrical channels
with mass transport in the individual channels governed by the
Graetz [21] equation. McCabe and Smith [35] use a similar model to
derive the Ergun [12] equation for pressure drOp in packed beds.

The model used here is more SOphisticated than these. It includes
channels of varying diameters, thus simulating the stagnant and
active flow regions which occur in real packed beds. And it provides
for mass transfer in the regimes of boundary layer formation and
separation, and incipient turbulence. Specifically, this model, or
combination of models, may be described as follows:

1. For computing velocity distribution in the bed and for

computing mass transfer at low velocities, the physical
model used is a network of cylindrical channels all of a
length equal to the diameter of equivalent spherical pack-
ing. On the average these channels are at an angle of 45°
with the axis of the bed, and the distribution of diameters
is described by a parameter XS. The void volume per unit
bed volume and the surface per unit bed volume are the same
as in the real bed.

2. The distribution of velocities in the bed is computed

assuming that all channels have the same pressure drop and
that that pressure drOp may be computed from Langhaar's

analysis [32] of the entrance of a circular pipe. This

27

'gives very low velocities in low diameter channels, and

high velocities in large diameter channels.

Mass transfer in the channels is computed in accordance

with the type of flow occurring. At the lowest velocities
with fully developed velocity and concentration profiles

the asymptotic Sherwood number for cylindrical tubes is

used. At somewhat higher velocities with developed veloc-
ities and developing concentrations the Leveque equation is
used to compute the Sherwood number. Both of these equations
derive from rigorous application of basic fluid dynamic and
transport principles to the flow regimes described. I
At higher velocities and diameters both the velocity profile
and the concentration profiles are developing. The treatment
developed by Blasius and Pohlhausen [42] for flow over a
surface parallel to the direction of flow is applicable here
except that the real surface formed by spheres and cylindrical
packing curves in the direction of flow. Since boundary
layer separation occurs at about half way around a sphere or
cylinder, the length of the boundary layer in_the Pohlhausen
equation is taken as half the length of the channel.

At still higher velocities a somewhat different physical
model is used to simulate mass transport in a packed bed.
Instead of regarding the fluid as flowing through a network
of cylindrical passages, it is regarded as flowing normal to
a bank of cylinders, again with the same void fraction and
surface as the real bed. This model gives incipient turbu-

lence at much lower velocities than cylindrical passages

28

and in this respect behaves more like a real packed bed.
The Colburn [9] equation developed for heat transfer in
fluids flowing across banks of tubes is applicable to this
model.

6. In this thesis a single equation is used to compute the
Sherwood number in all the flow regimes described above and
in the transition regions between them. This equation
states that the Sherwood number in any case is equal to the
feurth root of the sum of the fourth powers of Sherwood
numbers computed by all the equations described above. This
is a somewhat arbitrary combination of these equations, but
it does give values which are in pretty good agreement with
the transition between developing concentrations and developed
concentrations as derived rigorously by Graetz [21].

7. Overall mass transfer in the bed is then computed on the
basis that all the concentrations leaving a given layer of
channels mix to an average concentration before entering the
next layer of channels.

Obviously what is described above is not a rigorous derivation of
transport in a randOm packed bed from the equations of continuity,
motion, energy, and mass transfer. It is, however, a combination of
_ rigorous analysis and reasonable approximation to the transport
behavior of a fluid flowing through a physical model designed to
simulate many of the phenomena which occur in packed beds. Mass
transfer coefficients computed from this model are therefore a
priori predictions as to how random packed beds Should behave over

a wide range of operating conditions. This is much different from

29

correlating equations which represent a posteriori fits to limited
range data taken on a particular packed bed.

The model equations are derived on the basis of heat transfer
for simplicity reasons and then converted to mass transfer by

substituting the appropriate dimensionless variables.

DERIVATION Q2 MODEL EQUATIONS
Primary units for quantities used in the derivations are:
heat H, mass M, length L, time t, force F, and temperature T.
Consider that the flow cross-section is distributed among

the various diameters so that:

S/sIll - (D/Dm)s (l)

where: S total cross-sectional area of passages

having diameters less than 0
sm - total cross-sectional area of all passages
D - diameter of a given passage
D - maximum passage diameter present

5 - exponent which depends upon the distribution
of passages

The average passage diameter Dav is determined from equation 1
by multiplying 4 times the average hydraulic radius. Average
hydraulic radius is calculated by dividing SIn by the total perimeter

of all passages. Since the perimeter of a given circular cross-section

30

is ND = 4S/D:

Dav ' F.1— (2)
D

After integration (See Appendix A):

Dav 3 Lil on (3)
Let: XS = l/S
Dav = (1 - x3) 0In (4)

When X5 = 0, S = c, all passages have the same diameter. When
XS 3 1, Dav . 0. This requires that substantially all of the

surface to be located in passages of infinitesimal diameter.

Dh‘yxs. 0.01

 

 
 

 

 

 

O S 5m

Figure 3. Distribution Index

Ordinary packed bed parameters are:
u a superficial fluid velocity, based on empty cross section (L/t)
e a voids fraction, voids volume/total bed volume (L3/L3)

a a packing surface area per unit bed volume (L2/L3)

31

Dp - particle diameter (L)

p a fluid viscosity (M/Lt)

p a fluid density (M/L3)

k a thermal conductivity of fluid (H/LtT)

C a heat capacity of fluid (H/MT)

J- diffusion coefficient of active component in fluid (L2/t)
Volumetric hydraulic radius in terms of fixed bed parameters

is the volume of voids divided by the packing surface area. Therefore

the average equivalent diameter of a cylindrical passage is:

Dav = 45/8 (5)

Consider the fluid to be perfectly mixed before entering a
given layer of passages so that the entering temperature T1 is the
same for all passages. Consider the wall temperature Tw constant
throughout the layer so that (T1 - T") 8 ATI is also uniform. However,
since different temperatures are reached at the end of different

passages, T2 and (T2 - T") 2 AT are not uniform. Assume the length

2
of a passage L to be equal to a particle diameter D , and the average
angle between the flow direction in the passages and the axis of the
bed to be 0.

In order to determine the ATZ/ATl ratio for a given passage a

heat energy balance per unit of time is made:
m C dTp a - h (TP - Th) e D dL (6)
where: m . mass flow rate of fluid in a passage (M/t)

T a temperature of fluid at any point in a
passage (T)

32

h a fluid film heat transfer coefficient (H/thT)

 

Figure 4. Flow in a passage

Rearranging equation (6) and integrating over the length of the

passage:
L

1;.
111C (Iii-IIWDIOL
r-T
T. ' V) O

In [BIZ/1T1] - - 5.3.1:.

The mass velocity of the fluid is G = m/(nD2/4).

1 AT AT s-flL
'11 2/ 1] 600

The dimensionless groups--Reynolds number, Re a DG/u; Nusselt

(7)

(8)

(9)

number, Nu s hD/k; and Prandtl number, Pr = Cu/k--are then substituted

into equation (9).

._ 4NuL

(10)

33

It is then convenient to introduce Y = D Re/L, a parameter given in

Langhaar's article [32].

‘41'u

 

ln [ATZIATI] =- - Y? (11)
V
Solving for ATz/ATI:
Mm
AT /AT- 2 e V (12)
2 1

In terms of dimensionless groups the average Reynolds number of

the fluid, based on the superficial velocity direction, is defined as:

1
g)! 035 .S
Reav I D... I. R. ‘0‘. D J 5:- (13)

The average temperature change ratio is calculated by integrating

equation (12) over the distribution of passages.

‘Nm
‘ . 'F D s
f a Re case "' <13;-
(ATZ/AT1)av = 3* ._________

(14)
R. case 23 135;

 

e
A relationship for Y is determined from the mechanical energy

balance of a fluid streamline at the entrance of a tube, the pressure

loss equation for fully developed laminar flow, and a correction

to account for the pressure loss in the transition length.
Disregarding elevation effects and assuming a fluid of con-

stant density, the Bernoulli equation [35] for potential steady-state

34

flow along a streamline is:

-‘{-+-'£wdw=0 (15)

where: P a static pressure of fluid (F/Lz)
9 . density of fluid (M/L3)

w a velocity component perpendicular to the
cross section of channel (L/t)

gc - gravitational constant (LM/th)
Since the average velocity in the tubes is u/e and w in the mixing

sections between layers is negligible:

(16)

 

I.
By dividing both sides of equation (16) by 2l’:,:,,‘!..and defining V (the

number of 'Velocity heads') :- muff—:13; , then

at the tube entrance.
Beyond the transition length where the laminar flow pattern is
fully developed the Hagen-Poiseuille equation [35] for pressure loss

in a round tube applies.

- AP = flit“ (13)

$

t>IAQ

Since Y 3 D RE/L 3 W

V = 64/Y (19)

35

Therefore the equation V I l + 64/Y or its equivalent,

 

Y I \IVYZ + 1024 - 32 (See Appendix A), satisfy the limiting conditions
at high Y and at low Y, but in the intermediate region (transition
length) a correction is needed. This region is important to the model
because of the distribution of passages. Langhaar [32] made a theoret-I
ical study of the pressure losses in the flow develOping region of a
tube and his results are used in this thesis to make the needed
correction.

Langhaar's analysis begins with the Navier-Stokes differential
equation of motion for flow perpendicular to the channel cross section.
He solves the differential equation using the equation of continuity and
valid approximations. The solution is a family of velocity profiles
defined by Bessel functions. The pressure function is then determined
from the computed velocity field by means of the general energy equation.
From these equations then Langhaar calculates a table of values for 4/Y

versus V. For purposes of this thesis the table is converted into the

 

equation:
2 B
a 4 - - ——_——.—_
Y (‘Ivv +102 32)(1 ".N") (20)
where: B,A I constants
RT = (W2)°25 (21)

By analyzing Langhaar's data the best fit seems to be when: B = 5.8,
A I 175.

The Nusselt number in a given passage is determined by combining
the limiting value and three other equations using the fourth power

averaging method.

36

Nu - ((3.656)4 + (1.615)4(Y Pr)4/3 + (0.664(2 nuzprl”)4 e (22)

1/3 4 .25

(0.33 Re-6 Pr ) )

Equation 22 is a continuous equation and represents a weighted
average of the limiting Nusselt number for fully developed laminar
flow in tubes [30], the Leveque equation for developed velocity and
developing temperature laminar flow profiles [33], the Pohlhausen
equation for developing laminar velocity and temperature profiles
[42], and the Colburn equation for heat transfer in turbulent flow
across tube banks [9]. The factor 2 in the Pohlhausen equation
compensates for the formation of two boundary layers in one length

of channel as previously described.

 

log-tog P393?

 

51/
3*
N“ W
c Q 6V0?
Jaga!$:1%’
vgyove FWiII
3.55 - L ID I

 

 

The

Figure 5. Nusselt Numbers in Tubes

As is seen in equation 22 the Nusselt number for boundary
layer formation, developed laminar flow and turbulent flow is
proportional to the one-third power of the Prandtl number times

Reynolds number to a power which depends upon flow conditions.

37

For the analogy between heat and mass transfer the following

terms are defined:

Nu I average Nusselt number, Dth/k

8V

kc

Shav I average Sherwood number, Davkc/‘V

I mass transfer coefficient based on superficial velocity (L/t)

Shp I Sherwood number based on particle diameter, Dpkc/J?

. . f‘
Sc I Schmdt number of fluid, Q5

Rep I Reynolds number based on particle diameter, DquVr

Relationships derived in Appendix A are:

Rep - 1.5(1 - e)Reav

 

Shp - 1.5(1 - c)Shav/c

e 6Kc‘
Shp'7:1: *' auar

 

(23)

(24)

(251

(26)

For mass transfer Sherwood number and Schmidt number are

similar, respectively, to Nusselt number and Prandtl number of heat

transfer. Therefore:
1/3 ... 1/3
Shav/Sc -—-Nuav/Pr

In terms of mass transfer then:

1/3 x

Shav/ Sc (1: Rem,

with the value of x depending upon the type of flow.

follows that:

(27)

(28)

It then

(29)

38

where C1 is a proportionality constant.

Equation 29 expresses the mass transfer characteristics Of a
packed bed in terms of voids volume and voids surface area and many
of the literature correlations use varying forms Of this equation.

For a bed of spherical particles the following relationship

is derived in Appendix A.

L - Op - 6(l - c)/a (30)

OPERATION 9f Lg MODEL

 

The general procedure used to mathematically solve for

.891 and ‘3ng Tg—z from the model is:

(A) Bed porosity (e) and Pr (Sc) are set at desired values.
The angle 9 is assumed to be 45’, so case- I 0.707.

(B) XS is assigned a value of 0.3 (See Appendix B).

(C) A value for (W2)m is assumed, the magnitude of which depends
upon the voids fraction and the desired value for Rep/(1 - a)
(See Appendix C).

(D) Reav and (ATz/ATl)av are evaluated from equations 13 and 14 by
integration. Dav/0m is calculated from equation 4. For each
value of S/Sm, the following sequence of equations is used:
(a) DID. from equation 1

(b) W2 from (VY2)n(D/Dm)4 (31)

39

(c) RT from equation 21

(d) Y from equation 20 XS
(e) From Appendix A: Re I 1.5 Y 1'7! (ii—'9) (I-XS) (32)
(f) Nu from equation 22

(E) Rep/(1 - c) is then determined from equation 23.

(F) Nuav is calculated from (See Appendix A):

é PrReAv 1 (AI:
3" ' - 5(:-£)cose :1 AT, M (33’

(6) Finally by combining equations 25 and 27:

Se"; .- 5 “0’3
Model equations are summarized in Appendix D.
The mass transfer coefficient can then be determined from
equation 26.
a 551;.“ ShF G
1‘c" —_"“ [Sch Te]

The pressure loss per unit length of bed can be calculated from

the model and the Ergun equation (See Appendix A).

q {730 - 010- X5)+(VY‘)- (35)

-AP AL I
/ IZG 3e e‘ 2 D.

A correlation between mass transfer coefficient and pressure
loss per unit length of bed can be made by combining equations 28

and 35:

40

1: '-AP [64.61 93: 50,1!» [Eh—'1‘] 36
c' TC zvar‘u-e 0"0- X5)7"(VY") 56° '-‘ I )

 

The heat transfer coefficient can be determined by combining

the definition of Nuav and equation 5.
h - Nuavka/«t e (37)

Computer programs showing the Operational steps of the model

are given in Appendix E

RESULTS

The principal advantages Of the computerized model of this
thesis compared to previous correlations are its flexibility and
its coverage of larger ranges Of Reynolds and Schmidt numbers and
bed perosities. Literature correlations are generally for data
Obtained from specially constructed laboratory beds. Graphs of
data are usually in the form of Colburn 'J' factors versus Reynolds
number which are easily compared with results of this model. Authors'
equations containing Colburn 'Jd' factors are changed to equations

containing Sherwood number by:

J . kc/(u.Sc2/3)

d

Si ce: R I D
n ep p u p/u
SC = u/(plfl
Sh IDk
P P c”r
Then: Shp/(Rep Sc) I kc/u

1/3

I Shp/(Rep Sc )

Correlation equations Often contain specially defined axial
mixing or turbulence correction factors and apply only for limited

ranges of packed bed parameters. Data have been Obtained by

evaporating various liquids from porous solid particles into gas

41

42

streams, dissolving pellets of slightly soluble solids into flowing
liquids or extracting liquids from porous solids into flowing water.

Correlations for gases at relatively low Reynolds numbers
(below 250) are most difficult because mixing in the axial direction
becomes increasingly significant and it is hard to avoid essentially
equilibrium conditions at the exit even in a short packed bed.

In this work boundary layer theory is considered by using the
Pohlhausen equation in the model to account for the development of
temperature, concentration and velocity profiles in the entrance
region of a conduit. The model also contains the Leveque equation
for developing temperature and developed velocity profiles. These
are important for gases because the temperature, concentration and
velocity profiles develop simultaneously whereas for viscous liquids
the velocity profile develops first. For gases, therefore, heat and
mass transfer occur at a much greater rate in this region than
downstream where profiles are fully developed.

{Figure 6 shows the effect of a low Schmidt number (gases) on
the model at low Reynolds numbers. It can be seen that mass transfer
is much greater than for a liquid with a high Schmidt number.

Figure 7 shows the effect of distributed cross-sections on the
rate of heat and mass transfer. A distribution index (XS) of 0.3
.gives Nusselt and Sherwood numbers at extremely low Reynolds numbers
which are only about one-third of the amount they would be if all
passages were of the same diameter. Data for the graph are given in
Table 41 in Appendix F.

An equation to account for turbulence is also incorporated into

the model. It is based on the Colburn equation for turbulent flow

43

Voids Fraction I .4
X8 = .3

 

0.001 0.01 0.1 1.0

 

322 g 6'49.
l"‘ an”

Figure 6. Effect of Schmidt Number on Mass Transfer
at Low Reynolds Numbers

44

newsman.— Eomgcaéez 13 30.3.5 new whose—.2 100335 .5 omen:

l4
vs?

.ooo.o~ .OOOA .ooH .o— o.~ ~.c Ao.o moo.o

n >4Nk

 

~n~.~

2 52.5.:

o.m v.mv~

45

heat transfer across tube banks. Jolls and Hanratty [27] and
Karabelas, Wagner and Hanratty [28], using electrochemical techniques
report that in a dumped bed of l in. spheres having a voids fraction
of 0.41 that a transition from laminar to turbulent flow occurred
over the Reynolds number range of 110 to 150 (Rep/(l - e) I 186 - 255).

Table 1 shows the results of the model not using the turbulence
equation. The model equations used were identical with those of
Table 2 with the exception of the omission of the turbulence equation.
Comparison of the two tables shows that turbulence affects the results
above a Reynolds number of 260.

Tablesl to 33 compare Shp/Sc1/3Ic/(1 - e)] - 315233.17,- - ORE/ADS
(computer print-out) values from the model with those obtained by
using various authors' equations and graphs. The Reynolds numbers
given are Rep/ (1 - e) I 6 u Q/af I OUR/AZ (computer print-out).

Table 39 is an example computer program used to compute Table 2 and
is found in Appendix E. (W2)In values were selected from Appendix C
to produce the Reynolds number range desired at the voids fraction of
the bed. The equations listed in the headings are those of the
authors and the Reynolds number ranges given are in terms of

Rep/ (1 - 6).

Tables 2 and 3 compare the model results with those of Chu,

Kalil and Wetteroth [8]. Their correlation equation is for mass
transfer in packed and fluidized beds to a gas, Schmidt number of

2.57, covering a Rep/(l - c) range from 30 to 5000 and bed porosities

of 0.38 and 0.64.

Jd - 1.77[Rep/(l - an"44

46

TABLE 1. COMPARISON OF MODEL WITH CHU, KALIL AND WETTEROTH [8]

CHU I KALIL + HETTEROTH (19531
EQUATION J 8 1.77/REEII.44
SCHMIDT NUMBER 8 2.57 (GASES)
VOIOS FRACTION = 0.38

REYNOLDS NUMBER RANGE = 30 - 5000
XS 3 0.3

REYNOLDS
I6UR/AZ)

29.7577
40.4643
54.3287

72.0920.

94.6633
123.1697
159.0187
203.9735
260.2463
330.6098
418.5302
528.3253
665.3528
836.2363

1049.1404
1314.1057
1643.4611
2052.3297
2559.2521
3186.9540
3963.2883
4922.3948

MODEL CHU DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
5.2150 4.4975 0.1375
5.9654 5.3421 0.1044
6.7986 6.3004 0.0732
7.7186 7.3819 0.0436
8.7323 8.5984* 0.0153
9e8490 909641 ‘000116
11.0793 11.4965 '0.0376
12.4354 13.2165 -0.0628
13.9321 15.1485 -0.0873
1505866 1703210 ”0.1112
17.4186 19.7662 -0.1347
19.4506 22.5206 -0.1578
2107076 2506251 -001804
24.2168 29.1246 -0.2026
27.0080 33.0692 “0.2244
30.1131 37.5136 -0.2457
3305668 4205189 “002666
3704066 4801521 -002872
41.6733 5404879 -003075
4604115 6106094 -0032?“
51.6702 69.6094 -0.3471
57.5034 78.5915 -0.3667

47

TABLE 2. COMPARISON OF MODEL WITH CHU, KALIL AND WETTEROTH [8]

CHU + KALIL + HETTEROTH (1953)
EQUATION J 8 1.77/REEIO.44
SCHMIDT NUMBER 8 2.57 (GASES)
VOIDS FRACTION 8 0.38

REYNOLDS NUMBER RANGE 3 30 - 5000
XS 8 0.3

REYNOLDS
I6UR/AZ)

29.7577
40.4643
54.3287
72.0920
94.6633
123.1697
159.0187
203.9735
260.2463
330.6098
418.5302
528.3253
665.3528
836.2363
1049.1404
1314.1057
1643.4611
2052.3297
2559.2521
3186.9540
3963.2883
4922.3948

MODEL CHU DEVIATION
(6KE/ADS) (ORE/ADS) FRACTION
5.3352 4.4975 0.1570
6.1447 5.3421 0.1306
7.0588 6.3004 0.1074
8.0872 7.3819 0.0872
9.2432 8.5984 0.0697
10.5422 9.9641 0.0548
12.0020 11.4965 0.0421
13.6425 13.2165 0.0312
15.4874 15.1485 0.0218
17.5643 17.3210 0.0138
19.9058 19.7662 0.0070
22.5495 22.5206 0.0012
25.5381 25.6251 -0.0034
28.9198 29.1246 ‘0.0070
32.7488 33.0692 -0.0097
41.9954 42.5189 -0.0124
47.5545 48.1521 -0.0125
53.8453 54.4879 -0.0119
69.0052 69.6094 -0.0087
78.0956 78.5915 *0.0063

48

TABLE 3. COMPARISON 0? MODEL WITH CHU, KALIL AND NETTEROTH [8]

REYNOLDS
IbUR/AZ)

CHU + KALIL + HETTEROTH (1953)
EQUATION J 8 1.77/REE*!.44
SCHMIDT NUMBER 8 2.57 (GASES)
VOIDS FRACTION 8 0.64 r
REYNOLDS NUMBER RANGE 8 30 - 5000

30.6966
40.0189
51.7523

66047391

84.9081
107.9628
136.7743
172.7587
217.6751
273.6998
343.5172
430.4298
538.4942
672.6872
839.1108

1045.2445
1300.2556
1615.3801
2004.3909
2484.1720
3075.4214
3803.5146
4699.5597

XS = 0.3
MODEL CHU DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
8.5324 7.7076 0.0966
9.6402 8.9417 0.0724
10.8636 10.3265 0.0494
12.2154 11.8805 0.0274
13.7110 13.6258 0.0062
1503684 1505878 ’0001‘2
1702084 1707957 “000341
19.2550 20.2824 -0.0533
21.5353 23.0848 -0.0719
2400789 2602438 ‘000899
2609188 2908046 “001072
30.0909 33.8172 -0.1238
33.6344 38.3366 -0.1398
37.5924 43.4238 -0.1551
42.0125 49.1464 -0.1698
46.9470 55.5795 “0.1838
52.4542 62.8071 -0.1973
58.5991 70.9229 -0.2103
65.4544 80.0319 -0.2227
73.1013 90.2516 -0.2346
8106313 10107137 -002460
91.1471 114.5663 -0.2569
101.7643 128.9749 -0.2673

49

This equation converts to:
Shp/Sc1/3[c/(l - c)]I 1.77 e[Rep/(l - c)]'S6

Table 4 shows a comparison of one equation of Thoenes and Kramers
[54] and the model. They measured the rate of mass transfer between
a flowing fluid and the surface of one active Sphere in the middle
of a regular bed of spheres. Eight different geometric configurations
of spherical packing were used. They present graphs interpreting
their data, but do not list the data in tabular form. One equation

given is:

sup e/(l - c) . 1.26[Rep/(1 - 5)]1/3 Sal/3 + o.os4[aep/(1 - c)]°8 Sc°4

+ 0.8[Rep/(l - .)]'z

The first term is said to be for laminar convective transfer, the
second for turbulent convective transfer and the third for diffusion
in the stagnant regions near contact points of adjacent spheres. The
last.term is said to account for a large part of mass transfer in
gases at Reynolds numbers less than 500. Another equation listed in

this same article is:

Shp C/(l - E) a l'olkep/(l _ 6)]l/2 SCI/3

' which they say checks within 1 10% for all of their 438 mass transfer

measurements. The ranges for this equation are given as: voids

fraction, 0.25 to 0.50; Schmidt number, 1 to 4000; Reynolds number,

40 to 4000. Tables 5 through 8 compare this equation with the model.
Table 9 shows the equation of Bradshaw and Bennett [5] who

measured mass transfer coefficients for air passing through various

TABLE 4. COMPARISON OF MODEL WITH THOENES AND KRAMERS [54]

THDENES C KRAMERS
EQUATION

SCHMIDT NUMBER
VOIDS FRACTION
REYNOLDS NUMBER R
XS = 0.3

REYNOLDS
(6UR/AZ)

39.9038
54.1952
72.6815

96.3457

126.3956
164.3293
212.0186
271.8090
346.6431
440.2082
557.1130
703.0961
885.2755
1112.4486
1395.4568

1747.6302~

2185.3349

'2728.6450

3402.1703
4236.0747

50

(19581

SH/SCIII/3 = 1.26 REE**l/3 + .054 REE**.B
SC*0.067 + .8 REE**.2/SC**1/3

= 1.0 (GASES)
= 0.32
ANGE = 40 - 4000

MODEL THOENES
IbKE/ADS) (6KE/ADS)

5.4498 7.0087

6.2858 7.8628

7.2529 8.8089

8.3596 9.8574

9.6171 11.0220
11.0419 12.3193
12.6556 13.7704
14.4823 15.4002
16.5496 17.2393
18.8897 19.3235
21.5400 21.6954
24.5441 24.4050
27.9519 27.5106
31.8200 31.0800
36.2119 35.1921
41.1983 39.9385
46.8582 45.4261
53.2794 51.7792
60.5602 59.1428
68.8104 67.6867

DEVIATION
FRACTION

-0 0 2860
-0.2508
-002145
-001791
“001460
-0.1156
-0.0880
-000633
-0.0416
-0.0229
-000072
0.0056
0.0157
0.0232
0.0281
0.0305
0.0305
0.0281
0.0234
0.0163

TABLE 5. COMPARISON OF MODEL WITH THOENES AND KRAMERS [54]

THOENES 8 KRAMERS (1958)

EQUATION SH 3 1.0 RE**1/2 SCIIII3
SCHMIDT NUMBER 3 1.0 (GASES)

VOIDS FRACTION * 0.40

REYNOLDS NUMBER RANGE = 40 - 4000
XS 8 0.3

REYNOLDS
(6UR/AZ)

40.6615
54.3969
71.9472
94.2029
122.2702
157.5324
201.7237
257.0198
326.1454
412.5041
520.3302
654.8720
822.6120
1031.5341
1291.4520
1614.4116
'2015.1856
2511.8839
3126.7029
3886.8476

MODEL THOENES DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
602981 603766 ’000124
7.2164 7.3754 -0.0220
802533 804821 '000277
9.4165 9.7058 -0.0307
10.7198 11.0575 -0.0315
12.1816 12.5511 -0.0303
13.8236 14.2029 -0.0274
15.6701 16.0318 -0.0230
17.7488 18.0594 -0.0175
20.0921 20.3101 -0.0108
22.7369 22.8107 -0.0032
25.7253 25.5904 0.0052
29.1049 28.6812 0.0145
32.9285 32.1175 0.0246
37.2553 35.9367 0.0353
42.1509 40.1797 0.0467
47.6885 44.8908 0.0586
53.9497 50.1186 0.0710
61.0260 55.9169 0.0837
69.0199 62.3445 0.0967

52

TABLE 6. COMPARISON OF MODEL WITH THOENES AND KRAMERS [54]

REYNOLDS

THOENES 8 KRAMERS (1958)

EQUATION SH 3 1.0 RE001IZ SC**1/3
SCHMIDT NUMBER 8 4000. (LIQUIDS)

VDIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 40 - 4000

XS 3 0.3

(6UR/AZ)

40.6615
54.3969

71.9472‘

94.2029
122.2702
157.5324
201.7237
257.0198
326.1454
412.5041
520.3302
654.8720
822.6120

1031.5341
1291.4520
1614.4116
2015.1856
2511.8839
3126.7029
3886.8476

MODEL THOENES DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
6.5896 6.3766 0.0323
7.5172 7.3754 0.0188
8.5594 8.4821 0.0090
9.7288 9.7058 0.0023
11.0404 11.0575 “0.0015
12.5116 12.5511 “0.0031
14.1628 14.2029 -0.0028
16.0181 16.0318 “0.0008
18.1055 18.0594 0.0025
20.4576 20.3101 0.0072
23.1118 22.8107 0.0130
26.1103 25.5904 0.0199
29.5005 28.6812 0.0277
33.3357 32.1175 0.0365
37.6747 35.9367 0.0461
42.5833 40.1797 0.0564
48.1347 44.8908 0.0673
54.4106 50.1186 0.0788
61.5023 55.9169 0.0908
69.5127 62.3445 0.1031

53

TABLE 7. COMPARISON OF MODEL um: THOENES AND was [54]

REYNOLDS

THOENES 8 KRAMERS (1958)

EQUATION SH 8 1.0 REOIIIZ SCI81/3
SCHMIDT NUMBER 8 1.0 (GASES)

VOIDS FRACTION 8 0.50

REYNOLDS NUMBER RANGE 8 4O - 4000
XS 8 0.3

(6UR/AZ)

39.6532
52.2723
68.2434

88.3577'

113.6052
145.2278
184.7829
234.2199
295.9704
373.0563
469.2186
589.0736
738.3027
923.8861
1154.3883
1440.3116
1794.5291
2232.8186
2774.5188
3443.3339
4268.3221

MODEL THOENES DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
7.3841 6.2970 0.1472
8.3994 7.2299 0.1392
9.5265 8.2609 0.1328
10.7772 9.3998 0.1278
12.1676 10.6585 0.1240
13.7167 12.0510 0.1214
15.4457 13.5934 0.1199
17.3790 15.3042 0.1193
19.5443 17.2037 0.1197
21.9731 19.3146 0.1209
24.7011 21.6614 0.1230
27.7678 24.2708 0.1259
31.2173 27.1717 0.1295
35.0983 30.3954 0.1339
39.4648 33.9762 0.1390
44.3770 37.9514 0.1447
49.9018 42.3618 0.1510
56.1140 47.2527 0.1579
63.0978 52.6737 0.1652
70.9478 58.6799 0.1729
79.7704 65.3323 0.1809

54

TABLE 8. COMPARISON OF MODEL WITH THOENES AND KRAMERS [S4]

REYNOLDS

THOENES 5 KRAMERS (1958) .
EQUATION SH 8 1.0 RE981/2 SC'81/3
SCHMIDT NUMBER 8 4000. (LIQUIDS)

VOIOS FRACTION 8 0.50

REYNOLDS NUMBER RANGE 3 4O “ 4000

X5 3 003

(6URIAZ)

39.6532
52.2723
68.2434

88.3577'

113.6052
145.2278
184.7829
234.2199
295.9704
373.0563
469.2186
589.0736
738.3027
923.8861
1154.3883
1440.3116
1794.5291
2232.8186
2774.5188
3443.3339
4268.3221

MODEL THOENES DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
7.6487 6.2970 0.1767
8.6622 7.2299 0.1653
9.7893 8.2609 0.1561
11.0424 9.3998 0.1487
12.4365 10.6585 0.1429
13.9894 12.0510 0.1385
15.7218 13.5934 0.1353
17.6582 15.3042 0.1333
19.8265 17.2037 0.1322
22.2586 19.3146 0.1322
24.9899 21.6614 0.1331
28.0603 24.2708 0.1350
31.5139 27.1717 0.1377
35.3994 30.3954 0.1413
39.7710 33.9762 0.1457
44.6887 37.9514 0.1507
50.2196 42.3618 0.1564
56.4388 47.2527 0.1627
63.4301 52.6737 0.1695
71.2884 58.6799 0.1768
80.1198 65.3323 0.1845

55

size naphthalene spheres and cylinders. Beds were randomly packed,

4 in. diameter and 5 to 10 in. high. .Their correlation equation is:

1/3 1/2

Jd I 2.0/Rep Sc . 1.97/Rep

which is said to cover the Rep range from 400 to 10,000. In terms

Of this work the equation is:
Shp/Sc1/3te/(1 - efl' 2.0 E/(l _ e)Sc1/3 + 1.97 6/(1 - e)1/2
[Rep/(1 - c)]'5

with a Rep/(l - 2) range from 667 to 16,667.

A theoretical study of gaseous diffusion rates in packed beds
using a free surface model (spherical particle surrounded by a
spherical envelope of fluid) and boundary layer theory was made by

Kusik and Happel [31]. They give the equation:
Shp/Sc1/3 Repl/Z a 0.93/(e _ 0.75(1 _ €)(e - .2))0.5

which they say is applicable for Rep]: range of from 100 to 1000

and a voids fraction range from 0.3 to 1.0. The Reynolds number
range converts to Rep/(l - e) between 67 and 667 at a porosity of 0.4
and between 233 and 2330 at a porosity of 0.7. Tables 10 and 11

are for voids fractions of 0.4 and 0.7.

Liquid mass transfer coefficients fOr randomly packed beds Of
benzoic acid spheres and water were measured by Williamson, Bazaire
and GeankOplis [59]. Two equations are reported, each covering a
different Reynolds number range.

The equation:

58

s: Sc' - 2.4(Rep/e)"66

56

TABLE 9. COMPARISON OF MODEL WITH BENNETT AND BRADSHAW [3]

BRADSHAW 8 BENNETT (1961)

EQUATION J 8 2.0/(RE SCIIII3) + 1.97/RE'*.5
SCHMIDT NUMBER 8 2.57 (GASES)

VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 667 - 16667

XS 8 0.3
REYNOLDS MODEL BRADSHAW DEVIATION
(6URIAZ) (6KE/ADS) (6KE/ADS) FRACTION
673.5189 26.2911 27.3747 “0.0412
845.8481’ 29.7285 30.5601 “0.0279
1060.4576 33.6169 34.1015 -0.0144
1327.4116 38.0163 38.0375 “0.0005
1659.0617 42.9933 42.4097 0.0135
2070.5549 48.6220 47.2641 0.0279
2580.4587 54.9854 52.6506 0.0424
3211.5306 62.1762 58.6243 0.0571
3991.6634 70.2983 65.2462 0.0718
4955.0464 79.4691 72.5835 0.0866
6143.5903 89.8206 80.7107 0.1014
7608.6738 101.5020 89.7104 0.1161
9413.2845 114.6816 99.6743 0.1308
11634.6421 129.5498 110.7038 0.1454
14367.4106 146.3218 122.9116 0.1599
17727.6347 165.2408 136.4224 0.1744

57

TABLE 10. COMPARISON OF MODEL WITH KUSIK AND HAPPEL [31]

KUSIK a HAPPEL (1962)

EQUATION SH = 0.93/(E-O.75(l-E)(E-0.2))II.5
REII.5 SCIIl/B

SCHMIDT NUMBER I 1.0 (GASES)

VOIDS FRACTION 0.40
REYNOLDS NUMBER RANGE 8 67 - 667

XS = 003

REYNOLDS MODEL KUSIK DEVIATION
(6UR/AZ) IbKE/ADS) (6KE/ADS) FRACTION

66.1489 7.9246 7.0153 0.1147

76.94301 8.5270 7.5660 0.1127

89.2338 9.1685 8.1480 0.1113
103.2058 9.8513 8.7627 0.1105
119.0673 10.5783 9.4120 0.1102
137.0542 11.3526 10.0979 0.1105
157.4337 12.1777 10.8226 0.1112
180.5085 13.0574 11.5887 0.1124
206.6214 13.9954 12.3986 0.1140
236.1608 14.9961 13.2553 0.1160
269.5664 16.0640 14.1618 0.1184
307.3356 17.2039 15.1214 0.1210
350.0301 18.4214 16.1376 0.1239.
398.2840 19.7220 17.2140 0.1271
452.8119 21.1121 18.3546 0.1306
‘514.4185 22.5984 19.5634 0.1343
584.0083 24.1880 20.8447 0.1382
662.5978 25.8885 22.2029 0.1423

TABLE 11. COMPARISON OF MODEL WITH KUSIK AND HAPPEL [31]

REYNOLDS
(6UR/AZ)

232.1225
262.9783
297.7868
337.0403
381.2897
431.1519
487.3167
550.5560
621.7330
701.8132
791.8767
893.1315
1006.9285
1134.7789
1278.3733
1439.6023
1620.5811
1823.6756
2051.5325
2307.1125

KUSIK 8 HAPPEL
EQUATION

(1962)
SH = 0.93/(E-0.75(1-E1(E-0.2)1".5

RE*|.5 SCii1/3

SCHMIDT NUMBER =
VDIDS FRACTION =
REYNOLDS NUMBER RANGE

MODEL

(6KE/ADS)

25.0388
26.6128
28.2842
30.0589
31.9433
33.9440
36.0679
38.3225
40.7154
43.2549
45.9495
48.8034
51.8412
55.0582
58.4704
62.0892
65.9269
69.9967
74.3124
78.8888

(GASES)

233 - 2330

KUSIK

(6KE/ADS)

23.6251
25.1464
26.7589
28.4680
30.2791
32.1982
34.2311
36.3845
38.6650
41.0796
43.6360
46.3419
49.2057
52.2362
55.4428
58.8352
62.4240
66.2201
70.2353
74.4819

DEVIATION
FRACTION

0.0564
0.0551
0.0539
0.0529
0.0520
0.0514
0.0509
0.0505
0.0503
0.0502
0.0503
0.0505
0.0508
0.0512
0.0517
0.0524
0.0531
0.0539
0.0548
0.0558

59

is said to cover a Rep/e range from 0.08 to 125 for a bed porosity of

0.4 and a Schmidt number of 1000. This converts to:

.09

Shp/Sc1/3fc/(l - ci . 2.4 Sc 81°66/(l - e)'66 [Rep/(1 _ eH.354

for Rep/(l - e) from 0.053 to 83. Table 12 analyzes this equation.

The other equation listed is:

58 1

St Sc‘ 3 0.442(I?.ep/e).'3

covering a Rep/c range from 125 to 5000. This equation converts to:

1/3 .09 61.31

[e/(l - 5)]: 0.442 Sc /(1 - e)°31 [Rep/(l - e)]°69

Shp/Sc

at a Rep/(l - 2) range between 83 and 3333 for a voids fraction of 0.4.
Table 13 compares the equation with the model.

Two equations are reported by Wilson and Geank0plis [60] for
mass transfer from randomly packed beds of benzoic acid spheres to

water and prepylene glycol solutions. They report the equation:

6 Jd - 1.09 Rep’2/3

for the Rep range from 0.0016 to 55, Schmidt numbers varying from
950 to 70,600 and bed porosities between 0.35 and 0.75. This converts

to the equation:

1/3 1/3

Shp/Sc [e/(l - e)]= 1.09/(1 - 6) Rep

for a Rep/(l - 6) range from 0.0027 to 92 at a bed porosity of 0.4 and
0.0053 to 183 at a bed porosity of 0.7. Tables 14 through 18 compare

the model results with this equation. The other equation:

a .1d - 0.25 Rep’°31

60

TABLE 12. COMPARISON OF MODEL WITH WILLIAMSON,
BAZAIRB AND GEANKOPLIS [59]

NILLIAMSON 5 BAZAIRE 8 GEANKOPLIS (1963)
EQUATION ST SC'*.58 8 2.4 (RE/E)**-.66
SCHMIDT NUMBER 8 1000 (LIQUIDS)

VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 0.053 - 83

XS 3 003

REYNOLDS MODEL WILLIAMSON DEVIATION
16UR/AZ) (6KE/AOSI (6KE/AOS1 FRACTION
0.0528 0.5484 0.5034 0.0820
0.0789 0.6308 0.5769 0.0855
0.1177 0.7268 0.6609 0.0906
0.1755 0.8376 0.7570 0.0961
0.2614 0.9647 0.8669 0.1014
0.3889 1.1105 0.9922 0.1065
0.5778 1.2779 1.1352 0.1116
0.8569 1.4702 1.2979 0.1172
1.2675 1.6912 1.4827 0.1233
1.8690 1.9455 1.6920 0.1303
2.7443 2.2380 1.9280 0.1385
4.0072 2.5745 2.1929 0.1482
5.8102 2.9613 2.4881 0.1597
8.3509 3.4053 2.8147 0.1734
11.8776 3.9137 3.1729 0.1892
16.6937 4.4938 3.5622 0.2073
23.1624 5.1532 3.9817 0.2273
31.7142 5.8997 4.4307 0.2489
42.8610 6.7420 4.9085 0.2719
57.2182 7.6897 5.4151 0.2957
75.5354 8.7540 5.9514 0.3201

61

TABLE 13. COMPARISON OF MODEL WITH WILLIAMSON,

REYNOLDS

BAZAIRE AND GEANKOPLIS [59]

WILLIAMSON 8 BAZAIRE 8 GEANKDPLIS (19631
EQUATION ST SC'*.58 8 0.442 (RE/E1**-.31
SCHMIDT NUMBER 3 1000 (LIQUIDS)

VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE I 83 - 3333

XS = 0.3

I6UR/AZI

83.6600
108.9902
140.8616
180.8422
230.8992
293.4984
371.7243
469.4210
591.3615
743.4491
932.9637

1168.8604
1462.1372
1826.2865
2277.8505
2837.1057
3528.9034

MODEL WILLIAMSON
(6KE/ADS) I6KE/ADSI

9.1886 6.1578
10.4353 7.3908
11.8335 8.8218
13.4021 10.4816
15.1635 12.4066
17.1440 14.6400
19.3742 17.2324
21.8891 20.2428
24.7289 23.7395
27.9387 27.8008
31.5691 32.5162
35.6763 37.9883
40.3230 44.3338
45.5790 51.6864
51.5220 60.1985
58.2389 70.0451
65.8272 81.4264

DEVIATION
FRACTION

0.3298
0.2917
0.2545
0.2179
0.1818
0.1460
0.1105
0.0752
0.0400
0.0049
-0.0300
-0006‘08
“000994
-001339
-00 168‘.
”002027
’002369

62

TABLE 14. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

WILSON 8 GEANKOPLIS (I966)
EQUATION E J = 1.09/RE*'2/3
SCHMIDT NUMBER = 950 (LIQUIDS)
VOIDS FRACTION = 0.40

REYNOLDS NUMBER RANGE = 0.0027 - 92

XS = 0.3
REYNOLDS MODEL WILSON DEVIATION
(6UR/AZ) (6KE/ADSI I6KE/ADS) FRACTION
0.0027 0.2416‘ 0.2134 0.1166
0.0053 0.2785 0.2686 0.0357
0.0107 0.3314 0.3379 -0.0195
0.0424 0.5082 0.5343 -0.0511
0.1665 0.8214 0.8431 -0.0262
0.3290 1.0459 1.0578 ~0.0113
0.6473 1.3298 1.3255 0.0032
1.2662 1.6898 1.6576 0.0190
2.4532 2.1472 2.0665 0.0376
4.6819 2.7284 2.5632 0.0605
8.7345 3.4650 3.1555 0.0893
15.7971 4.3926 3.8445 0.1247
27.5323 5.5482 4.6266 0.1661
46.1803 6.9723 5.4971 0.2115
74.7931 8.7131 6.4556 0.2590

63

TABLE 15. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

WILSON 8 GEANKOPLIS (1966)
EQUATION E J = 1.09/RE'i2/3
SCHMIDT NUMBER = 70600 (LIQUIDS)
VOIDS FRACTION = 0.40

REYNOLDS NUMBER RANGE = 0.0027 - 92

XS = 003

REYNOLDS MODEL WILSON DEVIATION
(6UR/AZ) (6KE/ADS) (6KE/ADS) FRACTION
0.0027 0.2070 0.2134 -0.0309
0.0053 70.2633 0.2686 -o.ozoo
000107 003342 003379 “000111
0.0213 0.4234 0.4250 -0.0037
0.0424 0.5355 0.5343 0.0023
0.0841 0.6764 0.6713 0.0075
0.1665 0.8537 0.8431 0.0124
0.3290 1.0770 1.0578 0.0178
0.6473 1.3591 1.3255 0.0247
1.2662 1.7163 1.6576 0.0341
2.4532 2.1701 2.0665 0.0477
4.6819 2.7475 2.5632 0.0670
8.7345 3.4805 3.1555 0.0933
15.7971 4.4049 3.8445 0.1272
27.5323 5.5579 4.6266 0.1675
46.1803 6.9799 5.4971 0.2124
74.7931 8.7192 6.4556 0.2596

64

TABLE 16. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

WILSON 8 GEANKOPLIS (1966)
EQUATION E J = 1.09/RE**2/3
SCHMIDT NUMBER = 950 (LIQUIDS)
VOIDS FRACTION = 0.70

REYNOLDS NUMBER RANGE = 0.0053 - 183
XS = 0.3

REYNOLDS MODEL WILSON DEVIATION
(6UR/AZ) (6KE/ADS) (6KE/ADS) FRACTION
0.0050 0.3839 0.4183 ~0.0896
0.0101 0.4783 0.5260 -0.0996
0.0200 0.6051 0.6610 '0.0924
0.0397 0.7706 0.8302 -0.0774
0.0786 0.9814 1.0420 -0.0616
0.1548 1.2478 1.3062 -0.0467
0.3034 1.5850 1.6345 -0.0312
0.5896 2.0120 2.0396 -0.0137
1.1306 2.5520 2.5339 0.0071
2.1237 3.2320 3.1264 0.0326
3.8752 4.0802 3.8204 0.0636
6.8225 5.1235 4.6130 0.0996
11.5576 6.3869 5.4992 0.1389
18.8821 7.8977 6.4767 0.1799
29.9160 9.6911 7.5505 0.2208
46.2762 11.8153 8.7322 0.2609
70.3346 14.3362 10.0399 0.2996
105.5814 17.3412 11.4957 0.3370
157.1198 20.9410 13.1245 0.3732

65

TABLE 17. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

WILSON 8 GEANKOPLIS (1966)

EQUATION E J = 1.09/RE'52/3
SCHMIDT NUMBER = 70600 (LIQUIDS)
VOIDS FRACTION = 0.70

REYNOLDS NUMBER RANGE = 0.0053 - 183

XS = 0.3

REYNOLDS MODEL WILSON DEVIATION
(6UR/AZ) (6KE/AOS) (6KE/ADS) FRACTION
0.0050 0.3979 0.4183 -0.0513
0.0101' 0.5035 0.5260 -0.0446
0.0200 0.6362 0.6610 -0.0390
0.0397 0.8030 0.8302 -0.0339
0.0786 1.0129 1.0420 -0.0286
0.1548 1.2777 1.3062 -0.0223
0.3034 1.6122 1.6345 -0.0137
0.5896 2.0358 2.0396 *0.0018
1.1306 2.5721 2.5339 0.0148
2.1237 3.2483 3.1264 0.0375
3.8752 4.0932 3.8204 0.0666
6.8225 5.1335 4.6130 0.1013
11.5576 6.3946 5.4992 0.1400
18.8821 7.9037 6.4767 0.1805
29.9160 9.6958 7.5505 0.2212
46.2762 11.8191 8.7322 0.2611
70.3346 14.3394 10.0399 0.2998
105.5814 17.3441 11.4957 0.3371
157.1198 20.9436 13.1245 0.3733

TABLE 18. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

REYNOLDS
(6UR/AZ)

92.6051
115.0941
142.2887
175.1088
214.6648
262.2967
319.6189
388.5730
471.4868
571.1448
690.8685
834.6116

1007.0720
1213.8246
1461.4782
1757.8623
2112.2491

2535.6173'

WILSON 8 GEANKOPLIS
EQUATION E J
SCHMIDT NUMBER
VOIDS FRACTION 3 0.40

REYNOLDS NUMBER RANGE 8 92 - 2500
XS = 0.3

MODEL

(6KE/ADS)

9.6456
10.7156
11.8924
13.1871
14.6124
16.1827
17.9146
19.8266
21.9396
24.2767
26.8637
29.7285
32.9020
36.4179
40.3129
44.6272
49.4048
54.6939

(1966)
0.25/RE'&.31
(LIQUIDS)

WILSON

(6KE/ADS)

6.6633

7.7417

8.9619
10.3417
11.9022
13.6672
15.6642
17.9245
20.4836
23.3812
26.6621
30.3764
34.5799
39.3351
44.7115
50.7870
57.6484
65.3929

DEVIATION
FRACTION

0.3091
0.2775
0.2464
0.2157
0.1854
0.1554
0.1256
0.0959
0.0663
0.0368
0.0075
-0.0217
-0.0509
'0.0801
'0.1091
-0.1380
-0.1668
-001956

TABLE 19. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

REYNOLDS
(6UR/AZ)

92.6051
115.0941
142.2887
175.1088
214.6648
262.2967
319.6189
388.5730
471.4868
571.1448
690.8685
834.6116

1007.0720
1213.8246
1461.4782
1757.8623
2112.2491
2535.6173

WILSON 8 GEANKOPLIS
EQUATION E J
SCHMIDT
VOIDS FRACTION = 0.40
REYNOLDS NUMBER RANGE
XS = 0.3

NUMBER

MODEL

(6KE/ADS)

9.6513
10.7209
11.8973
13.1917
14.6169
16.1870
17.9188
19.8305
21.9439
24.2807
26.8679
29.7328
32.9061
36.4223
40.3172
44.6318
49.4093
54.6977

(1966)
0.25/RE**.31
(LIQUIDS)

92 * 2500

WILSON

(6KE/ADS)

6.6633

7.7417

8.9619
10.3417
11.9022
13.6672
15.6642
17.9245
20.4836
23.3812
26.6621
30.3764
34.5799
39.3351
44.7115
50.7870
57.6484
65.3929

DEVIATION
FRACTION

0.3095
0.2778
0.2467
0.2160
0.1857
0.1556
0.1258
0.0961
0.0665
0.0370
0.0076
’000216
-0.0508
-000799
‘001089
”001379
”001667
-0.1955

68

TABLE 20. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

REYNOLDS
(6UR/AZ)

181.3178
227.8180
285.7430
357.8186
447.3915
558.5660
696.3739
866.9824
1077.9505
1338.5433
1660.1180
2056.5974
2545.0501
3146.4014
3886.3040
.4796.2047

WILSON 8 GEANKOPLIS (1966)
EQUATION E J = 0.25/RE09.31
SCHMIDT NUMBER 950 (LIQUIDS)
VOIDS FRACTION 0.70

REYNOLDS NUMBER RANGE = 183 - 5000
XS = 0.3

MODEL WILSON DE
(6KE/ADS) (6KE/ADS) FR
22.4266 13.1327
25.0308 15.3733
27.9346 17.9744
31.1730 20.9922
34.7842 24.4910
38.8100 28.5439
43.2961 33.2348
48.2931 38.6597
53.8570 44.9288
60.0503 52.1685
66.9420 60.5240
74.6098 70.1625
83.1405 81.2761
92.6312 94.0858
103.1907 108.8460
114.9411 125.8496

VIATION
ACTION

0.4144
0.3858
0.3565
0.3265
0.2959
0.2645
0.2323
0.1994
0.1657
0.1312
0.0958
0.0596
0.0224
-0.0157
”0.0548
’0009’99

TABLE 21. COMPARISON OF MODEL WITH WILSON AND GEANKOPLIS [60]

REYNOLDS
(6URIAZ)

181.3178
227.8180
285.7430
357.8186
447.3915
558.5660
696.3739
866.9824
1077.9505
1338.5433
1660.1180
2056.5974
2545.0501
3146.4014
3886.3040
~4796.2047

WILSON 8 GEANKOPLIS
EQUATION E J
SCHMIDT NUMBER 8 70600
VOIDS FRACTION
REYNOLDS NUMBER RANGE
0.3

MODEL

(6KE/ADS)

22.4291
25.0332
27.9367
31.1755
34.7864
38.8115
43.2987
48.2949
53.8578
60.0525
66.9437
74.6112
83.1416
92.6330

103.1971
114.9396

(1966)
0.25/RE'*.31
(LIQUIDS)

183 - 5000

WILSON

(6KE/ADS)

13.1327
15.3733
17.9744
20.9922
24.4910
28.5439
33.2348
38.6597
44.9288
52.1685
60.5240
70.1625
81.2761
94.0858

108.8460
125.8496

DEVIATION
FRACTION

0.4144
0.3858
0.3566
0.3266
0.2959
0.2645
0.2324
0.1995
0.1657
0.1312
0.0958
0.0596
0.0224
-000156
“0.0547
-000949

70

is for a Rep ranging from 55 to 1500 with Schmidt numbers and voids
fractions being the same as for the other equation. This equation

converts to:
sup/5c1/3fe/(1 - £)1- 0.25/(1 - 2) Rep'69

for a Rep/(1 - 5) range from 92 to 2500 at a porosity of 0.4 and 183
to 5000 at a bed porosity of 0.7. Tables 18 through 21 show com-
parison of results of the model with the equation.

In his thesis Galloway [15] reports equations containing
turbulence factors and graphs of c Jd versus ReP for beds of spheres,
cylinders and commercial packing. Tables 22 through 30 show results
obtained by estimating equations from his graphs and using these
equations for comparison with the model. Estimated equations for

spheres are:

e Jd - 0.85 Rep"5°

for a Rep range between 3 and 10,000 and a Schmidt number of 1000.

This converts to:
Shp/Scl/3[e/(l - c)]- 0.850 - e)"5[aep/(1 - e)]'5

for a Rep/(l - a) range between 5 and 16,700 for a bed porosity of 0.4'

and between 10 and 33,333 for a voids fraction of 0.7.
c Jd - 0.95 Rep"51

for a Rep range between 10 and 10,000 and a Schmidt number of 1. This

converts to:

Shp/Sc1/3[€/(l - 6)]. 0.95(1 _ €)'051[Rep/(1 _ 5)].49

71

for a Rep/(l - c) range between 17 and 16,700 for a bed porosity of 0.4
and between 33 and 33,333 for a voids fraction of 0.7. Tables 22 to
25 analyze these equations.

Estimated equations for commercial packing are:

e Jd - 0.7 Rep“48

for a Rep range between 35 and 2000 and a Schmidt number of 1. This
converts to:

5 /S<=1/3 E/(l - 611- 0.7(1 - e)’°48[ge /(1 - c)]'52
hp P

for a Rep/(1 - c) range between $8 and 3333 for a bed porosity of 0.4

and between 117 and 6667 for a voids fraction of 0.7.

-e32
c Jd I 0.23 Rep

for 3 Rep range between 2000 and 10,000 and a Schmidt number of
1000. This converts to:

sup/SCUSIC/(l - c)] I 0.230 - £)-'32[Rep/(1 - “1.68

for a Rep/(1 - a) range between 3333 and 16,667 for a bed porosity of

0.4.

c ad - 0.50 aep"4l

for a Rep range between 35 and 2000 and a Schmidt number of 1000.

This converts to:

sup/scum“: - .)]- 0.500 - .y-“mepm , 01-59

72

TABLE 22. COMPARISON OF MODEL WITH GALLOWAY AND SAGE, SPHERES [15]

REYNOLDS
(6UR/AZ)

GALLOWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR BEDS OF
SPHERES E J 8 0.95/REOO.51
SCHMIDT NUMBER 8 1 (GASES)

VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 17 - 16700

16.0226'

27.8987
46.7518
75.6586
118.9646
182.9820
276.9902
414.6284
615.7833
909.1413
1335.7102
1953.7778
2845.9635
4129.2699
5969.3868
8600.9973
12356.5448
17706.9452

XS 8 0.3
MODEL GALLOWAY DEVIATION
(6KE/ADS) (6KEIADS) FRACTION
‘02321 ‘07993 -0013‘0
5.3202 6.2980 -0.1837
6.7193 8.1108 ‘0e2070
8.4574 10.2685 -0.2141
10.5737 12.8180 -0.2122
13.1487 15.8287 -0.2038
16.2933 19.3942 ‘0.1903
20.1469 23.6330 '0.1730~
2‘08866 28.6870 '001527
3007331 3‘07212 ‘001297
37.9542 41.9242 'OeIO‘b
46.8719 50.5120 -0.0776
57.8739 60073‘9 ’000‘9‘
71.4299 72.8860 “0.0203
88.1137 87.3116 0.0091
108.6299 104.4228 0.0387
133.8460 124.7084 0.0682
164.8315 148.7499 0.0975

73

TABLE 23. COMPARISON OF MODEL WITH GALLOWAY AND SAGE, SPHERES [15]

REYNOLDS
(6UR/AZ)

GALLDWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR BEDS OF
SPHERES E J 8 0.95/RE88.51
SCHMIDT NUMBER 2 1 (GASES)

VOIDS FRACTION 8 0.70

REYNOLDS NUMBER RANGE 8 33 - 33333

34.3932
52.8744
80.0105
119.7385
177.7973
262.4626
385.5627
563.9079
821.3252
1191.5570
1722.3868
2481.4956
3564.7579
5107.9823
7303.5199
10423.7565
14854.3412
21141.1899
30056.9690

XS 8 0.3
MODEL GALLDWAY DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
10.0932 9.9371 0.0154
12.3347 12.2682 0.0053
14.9978 15.0291 -0.0020
18.1788 18.3116 '0.0073
21.9951 22.2256 -0.0104
26.5873 26.8989 -0.0117
32.1195 32.4771 ’0.0111
38.7822 39.1276 “0.0089
46.7982 47.0439 -0.0052
67.9957 67.6232 0.0054
81.8735 80.8725 0.0122
98.5255 96.5798 0.0197
118.5122 115.1950 0.0279
142.5150 137.2532 0.0369
171.3632 163.3893 0.0465
206.0669 194.3568 0.0568
247.8578 231.0495 0.0678
298.2387 274.5268 0.0795

74

TABLE 24. COMPARISON OF MODEL WITH GALLOWAY AND SAGE, SPHERES [15]

REYNOLDS
(6UR/AZ)

7.2776’

13.2947
23.4360
39.7536
65.0221
103.1153
159.6190
242.7263
364.4930
542.5527
802.4290
1180.7022
1729.4439
2522.5093
3664.5050
5303.5616
7649.4773
10999.4387
15774.4283

MODEL

(6KE/ADS)

3.2283
4.0952
5.1789
6.5186
8.1595
10.1587
12.5910
15.5542
19.1772
23.6257
29.1067
35.8728
44.2276
54.5361
67.2397
82.8762
102.1049
125.7371
154.7732

GALLDWAY 8 SAGE
ESTIMATED EQUATION FOR BEDS OF
SPHERES E J 3 0.85/RE89.50
SCHMIDT NUMBER 3 1000
VOIDS FRACTION 3 0.40
REYNOLDS NUMBER RANGE 3 5 - 16700
XS 3 003

(1967)

(LIQUIDS)

GALLDWAY

(6KE/ADS)

2.9603
4.0011
5.3123
6.9188
8.8485
11.1430
13.8639
17.0962
20.9501
25.5602
31.0846
37.7062
45.6348
55.1137
66.4279
79.9147
95.9752
115.0876
137.8224

DEVIATION
FRACTION

0.0830
0.0229
-0.0257
80.0613
"0e0844
'0.0968
-0.1010
-0.0991
-000924
-000818
-0e0679
-000511
-0.0318
‘0. 0105
0.0120
0.0357
0.0600
0.0846
0.1095

TABLE 25. COMPARISON OF MODEL WITH GALLDWAY AND SAGE, SPHERES [15]

REYNOLDS
(6UR/AZ)

13.3576
21.6167
33.9899
52.2805
79.1400
118.4652
175.9381
259.7546
381.6315
558.2222
813.1324
1179.7914
1705.5391
2457.4278
3530.4414
5059.1272
7234.0508
10325.0707
14714.2601
20942.4748
29775.2226

SPHERES
SCHMIDT
VOIDS FRACTION 3 0.70

REYNOLDS NUMBER RANGE 8 10 - 33333
XS 8 003

GALLDWAY 8 SAGE
ESTIMATED EQUATION FOR BEDS OF
E J = 0.85/RE8*.50
NUMBER

MODEL

(6KE/ADS)

6.7951
8.3831
10.2660
12.4967
15.1469
18.3106
22.1053
26.6711
32.1721
38.7982
46.7716
56.3548
67.8620
81.6717
98.2436
118.1347
142.0238
170.7337
205.2717
246.8614
296.9975

(1967)

(LIQUIDS)

GALLDWAY
(6KE/ADS)

5.6718
7.2152
9.0475
11.2209
13.8056
16.8909
20.5844
25.0115
30.3165
36.6658
44.2526
53.3041
64.0898
76.9305
92.2088
110.3814
131.9924
157.6902
188.2467
224.5806
267.7847

DEVIATION
FRACTION

0.1653
0.1393
0.1186
0.1020
0.0885
0.0775
0.0688
0.0622
0.0576
0.0549
0.0538
0.0541
0.0555
0.0580
0.0614
0.0656
0.0706
0.0763
0.0829
0.0902
0.0983

76

TABLE 26. COMPARISON OF MODEL WITH GALLDWAY AND SAGE,

COMMERCIAL PACKING [15]

GALLDWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR COMMERCIAL
PACKING E J 3 Oe7/RE..e#8
SCHMIDT NUMBER 3 1 (GASES)

VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE ‘ 58 - 3333
X5 8 0.3

REYNOLDS
(6URIAZ)

59.4799
78.4085
102.3652
132.5366
170.4078
217.8417
277.1744
351.3298
443.9556
559.5844
703.8279
883.6104
1107.4541
1385.8291
~1731.5826
2160.4675
2691.7937
3349.2280

MODEL GALLDWAY
(6KE/ADS) (6KEIADS)
7.5302 7.4861
8.6058 8.6427
9.8115 9.9280
11.1626 11.3553
12.6788 12.9407
14.3824 14.7034
16.2989 16.6654
18.4574 18.8519
20.8915 21.2912
23.6398 24.0146
26.7462 27.0562
30.2597 30.4537
34.2353 34.2479
38.7340 38.4834
43.8238 43.2091
49.5802 48.4786
56.0881 54.3509
63.4420 60.8915

DEVIATION
FRACTION

0.0058
-000042
-000118
-000172
-0.0206
'0.0223
-00022‘
-0.0213
-0.0191
-0.0158
-0001 15
-00006‘
-0.0003

0.0064

0.0140

0.0222

0.0309

0.0402

77

TABLE 27. COMPARISON OF MODEL WITH GALLDWAY AND SAGE,

COMMERCIAL PACKING [15]

GALLOWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR COMMERCIAL
PACKING E J 8 0.7/RE58.48

SCHMIDT NUMBER 8 1 (GASES)

VOIDS FRACTION 8 0.70

REYNOLDS NUMBER RANGE 8 117 - 6667
XS 8 0.3

REYNOLDS
(6UR/AZ)

120.1870
151.5724
190.7316
239.5497
300.3476
375.9781
469.9422
586.5336
731.0155
909.8388
1130.9102
1403.9211
1740.7517
2155.9658
2667.4182
3296.9977
4071.5381
5023.9338
6194.5064

MODEL GALLDWAY
(6KE/AOS) (6KEIAOS)
18.2114 15.0524
20.3618 16.9825
22.7569 19.1381
25.4265 21.5459
28.4033 24.2350
31.7229 27.2373
35.4241 30.5874
39.5495 34.3235
44.1459 38.4876
49.2648 43.1262
54.9635 48.2905
61.3058 54.0378
68.3626 60.4314
76.2135 67.5419
84.9474 75.4479
94.6639 84.2367
105.4746 94.0057
117.5047 104.8630
130.8944 116.9292

DEVIATION
FRACTION

0.1734
0.1659
0.1590
0.1526
0.1467
0.1413
0.1365
0.1321
0.1281
0.1246
0.1214
0.1185
0.1160
0.1137
0.1118
0.1101
0.1087
0.1075
0.1066

78

TABLE 28. COMPARISON OF MODEL WITH GALLDWAY AND SAGE,

COMMERCIAL PACKING [IS]

GALLDWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR COMMERCIAL
PACKING E J 8 0.50/RE88.41
SCHMIDT NUMBER 8 1000 (LIQUIDS)
VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 58 - 3333
XS 8 0.3

REYNOLDS
(6URIAZ)

59.4799
78.4085
102.3652
132.5366
170.4078
217.8417
277.1744
351.3298
443.9556
559.5844
703.8279
883.6104
1107.4541
1385.8291
1731.5826
2160.4675
2691.7937
3349.2280

MODEL GALLDWAY
(6KE/ADS) (6KE/ADS)

7.8287 6.8676

8.9100 8.0835
10.1229 9.4605
11.4831 11.0180
13.0089 12.7792
14.7218 14.7717
16.6471 17.0275
18.8143 19.5839
21.2574 22.4832
24.0153 25.7732
27.1319 29.5075
30.6564 33.7459
34.6437 38.5549
39.1548 44.0085
44.2578 50.1891
50.0283 57.1888
56.5509 65.1107
63.9206 74.0705

DEVIATION
FRACTION

0.1227

0.0927

0.0654

0.0404

0.0176
-0.0033
-000228
-0.0409
80.0576
“Go 0731
-0.0875
-0.1007
-0.1128
-0.1239
”0.1340
-00 1431
-0.1513
-0.1587

79

TABLE 29. COMPARISON OF MODEL WITH GALLDWAY AND SAGE,

COMMERCIAL PACKING [15]

GALLDWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR COMMERCIAL
PACKING E J 8 0.23/RE8*.32

SCHMIDT NUMBER 8 1000 (LIQUIDS)
VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 3333 - 16667
XS 8 0.3

REYNOLDS
(6UR/AZ)

3419.3315
3770.5474
4156.7638
4581.3815
5048.1279
5561.0876
6124.7370
6743.9821
7424.1993
8171.2809
8991.6844
9892.4867
10881.4438
11967.0551
13158.6356
14466.3931
15901.5144
17476.2585

MODEL GALLDWAY
(6KE/ADS) (6KE/ADS)
64.6694 68.5171
68.3292 73.2275
72.1945 78.2479
76.2764 83.5982
80.5869 89.2993
85.1385 95.3735
89.9444 101.8448
95.0186 108.7383
100.3758 116.0810
106.0317 123.9015
112.0026 132.2304
118.3061 141.1000
124.9604 150.5449
131.9852 160.6019
139.4007 171.3100
147.2289 182.7108
155.4928 194.8487
164.2157 207.7707

DEVIATION
FRACTION

”000594
'0.0716
-0.0838
-0.0959
“0.1081
-001202
“001323
'001443
“001564
-001685
-0.1806
-001926
-0.2047
“0.2168
-0.2289
-0e2409
°0.2531
-0.2652

80

TABLE 30. COMPARISON OF MODEL WITH GALLOWAY AND SAGE,

COMMERCIAL PACKING [15]

GALLDWAY 8 SAGE (1967)

ESTIMATED EQUATION FOR COMMERCIAL
PACKING E J 8 0.50/RE88.41
SCHMIDT NUMBER 8 1000 (LIQUIDS)
VOIDS FRACTION 8 0.70

REYNOLDS NUMBER RANGE 8 117 - 6667
X5 8 0.3

REYNOLDS
(6UR/AZ)

120.1870.

151.5724
190.7316
239.5497
300.3476
375.9781
469.9422
586.5336
731.0155
909.8388
1130.9102
1403.9211
1740.7517
2155.9658
'266704182
3296.9977
4071.5381
5023.9338
6194.5064

MODEL GALLDWAY
(6KE/ADS) (6KE/ADS)
18.4361 13.8186
20.5849 15.8458
22.9782 18.1467
25.6457 20.7583
28.6203 23.7218
31.9377 27.0829
35.6369 30.8927
39.7602 35.2081
44.3547 40.0928
49.4720 45.6183
55.1695 51.8648
61.5108 58.9226
68.5671 66.8936
76.4176 75.8925
85.1519 86.0486
94.8687 97.5080
105.6806 110.4352
117.7119 125.0161
131.1033 141.4601

DEVIATION
FRACTION

0.2504
0.2302
0.2102
0.1905
0.1711
0.1520
0.1331
0.1144
0.0960
0.0778
0.0599
0.0420
0.0244
0.0068
-000105
-0.0273
'0e0‘49
~0.0620
-000789

81

for a Rep/(1 - c) range between 58 and 3333 for a bed porosity of 0.4
and between 117 and 6667 for a voids fraction of 0.7. Tables 26 to
30 show these equations.

Data for mass transfer in randomly packed beds of spheres to
gases at low Reynolds numbers is given by Petrovic and Thodos [40].
The values given are corrected for axial mixing. Their recommended

equation is:

e Jd - 0.357/Rep'359

and is recommended for a Rep between 3 and 230. This converts to

the equation:
Shp/Scl/sfc/(l - c)]- 0.357 Rep’641/(1 - c)

with a Rep/(1 - e) range from S to 390. Comparison of this equation
with the model is in Table 31.

Table 32 compares the model with the equation of Jolls and
Hanratty [27]. They used electrochemical techniques to study mass
transfer rates to an active sphere in a dumped bed. They report

the equation:

1/3 _ .58
Shp/Sc 1.44 Rep

_ to be good for a Schmidt number of 1700, voids fraction of 0.41 and
a Re range between 35 and 140. This converts to:

P

5hp/5c1/3[e/(1 - e)]- 1.44 c/(l - 6) Rep'58

at a Rep/(1 - 0) range between 59 and 237.

82

TABLE 31. COMPARISON OF MODEL WITH PETROVIC AND THODOS [40]

PETROVIC 8 THODOS (1968)
EQUATION E J 8 0.357/RE8*.359
SCHMIDT NUMBER 8 3 (GASES)
VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 5 - 390

XS 8 0.3

REYNOLDS MODEL PETROVIC DEVIATION
(6URIAZ) (6KE/ADS) (6KE/ADS) FRACTION
4.9733 2.6797 1.1991 0.5525
7.1760 3.0551 1.5168 0.5035
10.2537 3.5051 1.9067 0.4560
14.4859 4.0366 2.3794 0.4105
20.2101 4.6556 2.9456 0.3672
27.8271 5.3662 3.6159 0.3261
37.8124 6.1719 4.4012 0.2868
50.7344 7.0783 5.3138 0.2492
67.2819 8.0954 6.3678 0.2134
88.3002 9.2360 7.5800 0.1792
114.8381 10.5147 8.9707 0.1468
148.2052 11.9482 10.5641 0.1158
190.0427 13.5558 12.3895 0.0860
242.4094 15.3604 14.4813 0.0572
307.8858 17.3888 16.8798 0.0292
' 389.6970 19.6724 19.6321 0.0020

83

TABLE 32. COMPARISON OF MODEL WITH JOLLS AND HANRATTY [27]

REYNOLDS
(SUR/Al)

58.8424
66.6772
75.3981
85.0932
95.8594

JOLLS 8 HANRATTY (1969)

EQUATION SH/SC881/3 8 1.44 RE88.58
SCHMIDT NUMBER 8 1700 (LIQUIDS)
VOIDS FRACTION 80.41

REYNOLDS NUMBER RANGE 8 59 - 237

XS 8 0.3

107.8039
121.0455
135.7153
151.9587
169.9367
189.8276
211.8288
236.1587

MODEL JOLLS DEVIATION
(6KE/ADS) (6KE/ADS) FRACTION
7.9148 7.8308 0.0106
8.8847 9.0418 -0.0176
9.4082 9.6990 '0.0309
9.9591 10.3928 '0.0435
10.5389 11.1254 -0.0556
11.1491 11.8986 -0.0672
11.7914 12.7149 80.0783
13.9287 15.4467 -0.1089
15.5500 17.5326 -0.1274

84

TABLE 33. COMPARISON OF MODEL WITH WILKINS AND THODOS [58]

REYNOLDS
(6UR/AZ)

30.5370
41.3359
55.2024
73.0405
95.5950
124.0220
159.7301
204.4754
200.4010
330.4450
417.0749
527.0341
003.2337
033.0319
1044.5050
1307.5790
1034.4309
‘2040.0195
2542.0420
3104.7524

WILKINS 8 THODOS (1969)

EQUATION E J 3 0.589/RE88.427
SCHMIDT NUMBER 8 3 (GASES)

VOIDS FRACTION 8 0.40

REYNOLDS NUMBER RANGE 8 33 8 3333
X5 3 003

MODEL WILKINS
(6KE/ADS) (6KE/ADS)
5.5967 5.1958
6.4317 6.1801
7.3700 7.2988
8.4226 8.5643
9.6028 9.9915
10.9258 11.5989
12.4092 13.4086
14.0731 15.4469
15.9415 17.7445
18.0426 20.3371
20.4093 23.2650
23.0791 26.5740
26.0949 30.3151
29.5045 34.5448
33.3614 39.3260
37.7251 44.7281
42.6617 50.8281
48.2448 57.7118
54.5568 65.4746
61.6895 74.2231

DEVIATION
FRACTION

0.0716

0.0391

0.0096
80.0168
-000404
-000616
-000805
“000976
80.1131
80.1271
-001399
-001514
80.1617
80.1708
-001787
80.1856
’001914
80.1962
80.2001
-002031

85

TABLE 34. SUMMARY OF RESULTS FOR GASES

REYNOLDS DEVIATION AVERAGE
TABLE VOIDS NUMBER FRACTION DEVIATION
NUMBER FRACTION RANGE RANGE FRACTION

31 0.40 5-390 .55 -.- 0 +0.15
22 0.40 17-16,700 -.13-+- -.21-.- .10 -0.02
2 0.30 30-5000 .16 -.- -.01 +0.03
3 0.64 30-5000 .10 +-.27 -0.15
33 0.40 33-3333 .07 + -.20 -0.10
23 0.70 33-33,333 0 +.07 +0.02
4 0.40 40-4000 -.29 ->- .02 -0.04
5 0.40 40-4000 -.03 -e- .10 +0.04
7 0.50 40-4000 .14 —>- .12—>.10 +0.14
26 0.40 50-3333 0+ -.02 4.04 +0.01
10 0.40 67-667 .11 ->- .14 +0.12
27 0.70 117-6667 .17 + .11 +0.13
11 0.70 233-2330 .05 +.06 +0.05
9 0.40 667-16,667 -.04 +.17 +0.00

AVERAGE PERCENTAGE DEVIATION 8 +34

86

TABLE 35. SUMMARY OF RESULTS FOR LIQUIDS

REYNOLDS DEVIATION AVERAGE
TABLE VOIDS NUMBER FRACTION DEVIATION
NUMBER FRACTION RANGE RANGE FRACTION
14 0.40 0.0027-92 .12+-.05 —+-.26 +0.06
Sc=950
15 0.40 0.0027-92 -.03 —>- .26 +0.06
56-70600
16 0.70 0.0053-103 -.09 + .37 +0.05
Sc-9SO -
17 0.70 0.0053-183 -.05 -—+-.37 +0.06
Sc-70600
12 0.40 0.053-03 .00 —0- .32 +0.15
24 0040 5-16.700 008+ -010 + 011 70002
25 0.70 10-33,333 .16 -+- .05 -+ .10 +0.00
6 . 0.40 40-4000 .03—0- 0 -—.- .10 +0.04
0 0.50 40-4000 .10-0- .13 ->- .10 +0.15
20 0.40 50-3333 .12—> -.16 - - -0.04
32 0.41 59-237 . .01....- -.13 -0.03
13 0.40 83-3333 .33 +-.24 +0.03
10 0.40 92-2500 .30 -e--.20 +0.05
30 0.70 117-6667 ‘ L25 + -.00 +0.05
20 0.70 103-5000 ‘ .41->- -.10 +0.14
29 0.40 3333-16,667 -.O6 +-.27 +0.15

AVERAGE PERCENTAGE DEVIATION 8 +54

87

In order to demonstrate the usefulness of the computer model
the results of an example problem, taken from Satterfield [48],
are shown in Table 36. Table 40 in Appendix E is the program
listing for Table 36. This program calculates h, kc and pressure
loss per unit length of bed by reading in the standard packed bed
parameters and fluid properties. Actual Reynolds number (REI) is
the Rep/(l - a) calculated from bed parameters and fluid properties.
Calculated Reynolds number (REE) is the Rep/(1 - 6) calculated from
(VY2)m. The (W2)m used is calculated from the actual REI. Since
the relationship between (VYZ)m and the Reynolds number changes
with the porosity of the bed, the constants A1 through A9 have to be
changed accordingly. The values for these constants are given in

Table 37, Appendix C.

88

TABLE 36. EXAMPLE USING MODEL
GIVEN FLUID PROPERTIES

VISCOSITY (V15) 8 0.092 LB/(FT.HR)

HEAT CAPACITY (CP) 8 0.90 BTU/(LBoDEG F)
SUPERFICIAL VELOCITY (VEL) 8 1320 FT/HR

THERMAL CONDUCTIVITY (AK) 8 0.131 BTU/(FToHRcDEG F)
DENSITY (RHO) 8 1.05 L8/(CU FT)

DIFFUSION COEFFICIENT (DIF) 8 0.0296 SQ FT/HR

GIVEN BED CHARACTERISTICS
BED POROSITY (EP) 8 0.40

SPECIFIC SURFACE (ASP) 8 311 SQ FT/(CU FT)
PARTICLE DIAMETER (DPA) 8 0.01285 FT

COMPUTER RESULTS

ACTUAL CALCULATED
SCHMIDT REYNOLDS REYNOLDS
VYSM NUMBER NUMBER NUMBER
165593.6539 2.9601 322.6467 322.1269
HEAT MASS
TRANSFER TRANSFER DP/DL
COEFFICIENT COEFFICIENT PSI/FT

433.9091 98.0404 0.0397

DISCUSSION OF RESULTS

The model equations of this thesis simulate heat and mass
transfer rates in a randomly packed bed better over a wide range
parameters than any of the empirical equations we found in the
literature. This is because the equations are derived from basic
fluid dynamics and transport phenomena principles. Other authors'
equations are obtained by drawing arbitrary straight lines through
scattered data points and in some cases using special mixing or
turbulence factors to fit their data. Some of the earlier authors
did not recognize that Jd was inversely proportional to voids
fraction. ch, and not Jd, is shown to be a function of Reynolds
number by Thoenes 0 Kramers [54], Gupta 8 Thodos [22], Wilson 8
Geankoplis (60], and others. The model equations also cover the
entire range of Reynolds numbers, Schmidt or Prandtl numbers, and ,
voids fractions whereas literature equations are for limited ranges.

Comparisons between the model and empirical equations are
given in Tables 2 to 33 and summarized in Tables 34 and 35.

The equation of Chu, Kalil and Wetteroth [8] is analyzed in
‘Tables 2 and 3. Their equation is said to apply to both packed and
fluidized beds and shows no dependency of Jd on voids fraction. In
the article they show data for fixed beds with voids fractions of
about 0.4 and expanded beds of higher porosities. Table 2, which is
for a voids fraction of 0.38, shows a much better correlation than

Table 3, which is for a porosity of 0.64. The equation of Chu,

89

90

et. a1., gives mass transfer rates that are proportional to voids
fraction compared to the model equations which indicate less
dependency. For example, at an interstitial Reynolds number of

3900, the increase in mass transfer by the Chu equation is 0.64/0.38
whereas the model equations show an increase of only the square

root of this ratio. In addition mass transfer rates are necessarily
higher in fluidized beds due to the increased surface contact between
solid and fluid which is the case in Table 2..

Thoenes and Kramers [54) present the equation shown in Table 4,
which contains three additive terms. One term is for mass transfer
in laminar flow, one for turbulent flow and one for stagnant areas.
The packing was arranged in a body-centered cubic configuration.
Analysis of results indicates that mass transfer was better in the
bed at low Reynolds numbers than the model shows. This could be
accOuntable to regular packing. In a regular packed bed there are
bottlenecks in which the fluid flows at a much higher rate than the
average velocity. Consequently mass transfer is greater in these
areas. The effect on overall transfer rate would reasonably be
greatest at lower Reynolds numbers.

Tables 5 to 8 compare a simplified formula presented by Thoenes
and Kramers in the same article which they say has a mean deviation
of t 10%. Agreement is reasonably goOd for beds with porosities of
-0.4 but model resUlts average about 15% higher than the given
formula for a voids fraction of 0.5. No tabular data is listed, but
lines on graphs presented fOr packed beds with porosities of 0.48
generally show higher rates of mass transfer than for the lower

porosities.

91

The equation of Bradshaw and Bennett [5] shown in Table 9 is
in terms of Jd instead of EJd. Also the data from which the equa-
tion was derived shows a 25% standard deviation.

Kusik and Happel [31] use a free surface model to derive their
equation which is compared to the model in Tables 10 and 11. They
used boundary layer theory in the derivation. As the Tables
indicate, correlation is better at a voids fraction 0.7 than for 0.4.
This would seem reasonable for a free surface model which is described
as a Sphere surrounded by a spherical enveIOpe of fluid.

Williamson, Bazaire and Geankoplis [59] present two equations,
one for low and one for high Reynolds numbers. These comparisons
are shown in Tables 12 and 13. Agreement is not too good, especially
in the Reynolds number region where the two equations coincide.

There is considerable scattering of the data and these two equations
seemed to be the best fit.

Wilson and Geankoplis [60] used the data of the previous
article by the senior author and new data to present two new equa-
tions which are analyzed in Tables 14 to 21. The first four Tables
are for void fractions of 0.4 and the others for 0.7. It can be
seen that changing the Schmidt number from 950 to 70,600 affects
the results only at low Reynolds numbers. Figures 8 and 9, which
follow, show graphically the answers in Tables 14 and 18. The model
results follow closely the authors' equations, except in the inter-
secting region. Again in order to divide the data into two correlating
equations, it was necessary to have larger deviations at intermediate

Reynolds numbers. Similar graphs would result by plotting Tables 15

and 19, etc.

92

—--- Model

-- Table 14

 

Figure 8. Comparison of M0001 with Wilson and G0ankoplis [60]
Low R0ynolds Numbers

93

100

10

1.0

---- Model

Tables 14 8 18

 

Reg ‘ 61.42
I-e hf

Figure 9. Comparison of Model with Wilson and Geankoplis [60]
High Reynolds Numbers

94

Galloway [15] presents graphs in his thesis that are correlations
of his data and other authors. The equations presented contain
turbulence intensity factors and are difficult to compare with the

model. Two of the graphs given are plots of Sh/Scl/S

versus
Reynolds number; so equations were estimated from them that are
comparable to the model. Tables 22 to 25 are for beds of spheres
and the results compare reasonably well with the model. Tables 26
to 30 are for commercial packing. Again the results compare
favorably.

Petrovic and Thodos [40] give an equation for mass transfer to
gases. Table 31 shows poor correlation at low Reynolds numbers
but the data presented in the article shows considerable scattering
especially at low Reynolds numbers. He also uses axial mixing
factors of Epstein [11] in his analyses.

Wilkins and Thodos [58] use the previous data of the senior
author and others and give a new equation for mass transfer to
gases which varies considerably from the previous equation. These
results are given in Table 33. This equation gives higher mass
transfer rates at corresponding Reynolds numbers.

Jolls and Hanratty [27] give an equation for mass transfer for
an isolated sphere in a bed of inert spheres. Table 32 shows that
- mass transfer is slightly better than for the model. This would
seem logical since the model is for a randomly packed bed of active
spheres.

Figures 10, 11, and 12 compare the model results with literature
equations for gases and liquids at voids fractions of 0.4 and 0.7.

These Figures show that the model equations agree with the various

95

Schmidt Number = 1 (Cases)
Voids Fraction = 0.4

---- Model

Table
Table
Table
Table 10
Table 22
Table 31
Table 33

 

10 100 1000 10,000

 

Re : ‘uQ
41-2 ar

Figure 10. Comparison of Model with Literature Correlations

96

Schmidt Number I 1000 (Liquids
Voids Fraction 8 0.4

---— Model

Table 6 -
Table 13 -
Table 18 -
Table 24 -
Table 28 -
Table 29 -
Table 32 -

 

R -6
7—35“Tur‘

Figur0 11. Comparison of Model with Literature Correlations

97

Schmidt Number 8 1000 (Liquids)
= 7

Voids

-—-- Model

1 Table 20 - Re£.f601
2 Table 25 - Ref.[|51
3 Table 30 - Ref.C'5J

 

Figure 12. Comparison of Model with Literature Correlations

98

authors' correlations better than the correlations do with each other.
For this reason the deviation fractions in Tables 2 to 33 are based
on results of the model equations. As is indicated in Tables 34
and 35 the average deviation for gases and liquids are +34 and +51,
respectively. Root mean square deviation, which is a measure of
data scattering, is not applicable. Root mean square deviations
were determined to be 13.0% for gases and 14.2% for liquids.
Satterfield [48] uses the equation of Petrovic and Thodos (40]
and calculate. heat and mass transfer coefficients for a hydrode-
sulfurization reactor packed with cylindrical catalyst pellets.
Table 36 shows results using the model of this thesis. Units for
the answers are: heat transfer coefficient-+8tu/ft2hroF; mass
transfer coefficient-~ft/hr; pressure drop per unit length of
Dede-pounds force per square inch per foot. Agreement between the
computer answers and those of Satterfield is about 10% due to the

use of the Petrovic equation.

SUMMARY

Through the years a very large number of articles have appeared
in the literature, representing a huge expenditure of research time
and effort in the study of heat, mass and momentum transfer in
packed beds. Many of the authors have presented correlation equations
for mass transfer coefficients covering varying ranges of fluid flow
rates, physical properties and bed characteristics. The method of
computing mass transfer coefficients develOped here differs from
most of these correlations, since it is based on a physical model
and does not employ arbitrary empirical constants to fit a specific
set of data.

If we compare the values predicted by the literature correlations
with those computed by this new model, we find that the root-mean-
square deviation for the literature correlations studied is about
13.5%, whereas the average deviation between the physical model
results of this thesis and these same correlations is about 4%. In
other words the mass transfer results from the model agree better
with authors' results than a comparison of authors' results with
'one another.

The physical model is derived from basic principles of fluid
flow and transport phenomena. The bed is considered to be randomly
packed with spheres. The channels between the spheres are treated

at low'loymolds mulbers as parallel cylindrical tubes with different

99

100

cross sections. The distribution of cross sections is described by a
distribution index, XS. The bed is assumed to be divided into layers
of these parallel passages with the length of each passage equal to
the diameter of the spheres used. The fluid from all of the tubes in
each layer mix before entering the next layer. Flow in the passages
is treated as laminar and the pressure drop across each layer is the
same through each of the parallel conduits.
Mhss transfer coefficients computed using this model are the
ones for equimolal counter-diffusion, for low concentrations, or for
other cases where J is equal to N. These coefficients are therefore
entirely analogous to heat transfer and the latter may be computed by
substituting Pr for Sc and Nu for Sh. Also in deriving the model
most of the basic transport phenomena equations used-~Leveque,
Pohlhausen, Colburn--were originally derived for heat transfer. For
simplicity, therefore, the model was derived on the basis of heat
transfer and converted to mass transfer by substitution of the
apprOpriate dimensionless variables.
Starting equations are heat and mechanical energy balances

across a passage with constant temperature walls. A correction is
added to account fer the higher pressure gradient in the transition
length. Nusselt number is calculated (a) from a weighted average of

the limiting value fer fully developed laminar flow, (b) from the
Leveque equation for developed velocity and developing temperature
profiles, to) from the Pohlhausen equation for developing velocity
and temperature profiles and (d) from the Colburn equation for heat
transfer acress tube banks. These are all combined into a continuous

equation which smooths out the transition ranges between the regimes

101

described by the individual equations. Average Nusselt and Reynolds
numbers are then determined by integrating over the distribution

of the cross sections in the layer. Overall Nusselt and Prandtl
numbers of heat transfer are converted to Sherwood and Schmidt numbers
of mass transfer. Since most of the literature correlations are in
terms of Sherwood number divided by Schmidt number to the one-third
power, they are easily compared with the model.

Due to the complexity of doing the mathematics of the model
equations, they are solved by means of a computer program.

The model equations cover a much broader range of Reynolds and
Schmidt numbers and bed porosities than do any of the literature
correlations. The Colburn equation is used to account for turbulent
heat and mass transfer at high Reynolds numbers. At low Reynolds
numbers the distribution of cross sections is particularly important
since uniform passages would give higher Sherwood numbers than
experimental results show. This should be particularly important
for gas chromatography where mass transfer occurs at extremely low

flow rates in packed beds containing finely divided particles.

CONCLUSIONS

The results and conclusions of this research are summarized

as follows:

Overall mass and heat transfer coefficients and pressure
loss per unit length of bed can be predicted with reasonable
accuracy using the physical model of this thesis. Fluid
properties that need to be specified are: viscosity,

heat capacity, superficial velocity, thermal conductivity,
density, diffusion coefficient of active component through
the fluid. Bed characteristics which have to be known are:
porosity, particle size, specific surface per unit volume
and an index defining the distribution of passage cross-
sections within the bed.

The model equations cover wider Reynolds and Schmidt

number ranges than do any of the literature correlations.
Mass transfer results using the model equations show
deviations from literature correlations of 3% for gases

and 5% for liquids in the Reynolds and Schmidt number

ranges reported.

Mass transfer values calculated in.Reynolds and Schmidt
number ranges not corroborated by experimental investigators
are believed to be reasonably accurate because basic
principles of fluid dynamics and transport phenomena are used

in developing the model.

102

103

The distribution of cross-sections introduced into the
model has an effect upon coefficients computed over the
entire Reynolds number range. The effect is greatest,
however, at extremely low Reynolds numbers where it
gives Sherwood and Nusselt numbers which are much lower
for the bed than for the limiting values for individual
passages.

Treatment Of the passages between the spheres as layers

of parallel tubes with mixing between layers proved to be

satisfactory and convenient.

The Pohlhausen and Leveque equations adequately describe

transfer in the flow develOping regions and the Colburn

equation simulates the turbulence flow results. For

simplicity reasons the model was derived on the basis of

heat transfer.

RECOMMENDATIONS FOR FUTURE WORK

After analyzing the results of this thesis the following

suggestions are made for future investigations:

a.

Design carefully controlled experiments to cover a wide
range of Schmidt, Reynolds numbers and bed porosities to
further verify the results of the model. Investigate
Reynolds number regions not previously explored. With

more controlled experiments we would be justified in making
a more sophisticated model.

Using turbulent boundary layer theory or some other
theoretical method, investigate the turbulent region in

more detail to obtain a better theoretical model.

Determine the effect of distributed cross sections on

results using cylinders or commercial packing, such as
Raschig rings, instead of spheres.

Refine the fourth power method of evaluating the transition
regions when calculating Nusselt number.

Investigate the effect of particle shape on the distribution
coefficient, XS, at low Reynolds numbers. Design experiments
for gas flow mass transfer in beds of finely divided particles.
Assume venturi shaped cross sections or passages with flat

walls to see if a better model can be formulated.

104

105

g. A theoretical model could be attempted assuming passages
with non-isothermal walls. The partial differential
equations involved, however, would be more difficult to

solve.

APPENDICES

106

APPENDIX A

107

APPENDIX A

DERIVATION 0F MODEL EQUATIONS

DERIVATION 95 D

 

.3!
45m
890
4
”,1 Dds
Total perimeter from: 2170 855.39.- d8
0

Since: 8 n.3— D5
D S
m

s
d8 -._2. s 05-1 d0
D5
I)
4 sm

”v ' 5.5713119...) . 0“»

s
”111 /s

l)avflg 1571—"

 

Ds 2 d0

s
”111 /s
D 8
av D 5'1
III

 

(s-l)

D v = [(5 - l)/S] Dm

108

(1)

(3)

109

DERIVATION 95,;

V 8 1 + 64/Y

VY2 8 Y2 + 64 Y

72 + 64 v + 1024 - w2 + 1024

2

(Y + 32)2 - vv + 1024

v . (w2 + 1024)”2 - 32

MODEL EQUATIONS ASSUMING PARTICLES ARE SPHERES

The parallel cylindrical passage model is related to spherical

particles using the subscript p to represent such particles.

Volume of sphere . i 11' D: l- l

 

SCI-Race area of sphere f p" T 0

Dp 8 6(1 - c)/a (30)
Since: a . 4 s/Dav (5)

0p - 1.5 Dav(l - c)/c I (30)

Rep .. 831.393 (24)

Since: Rem, - Dav 11 Q]: [I

Dav 8 4 e/a (S)

414!
v

R°av . (39)

 

110

Rep 8 1.5(1 - c)Reav

Since: Shav = Dav kc/JV

D :- 6.:
3" 1571-01

Shp . Dp kc/e
_ .51
5"“ fin.
Since: Dp 8 6(1 - c)/a
G 6310
5"? vs T 0+5

Re a Y L/D

Since: L . Dp . 1.5 Dave—31+)

-=--.—-'-°~ (44

Since: Dav - (1 - XS)DIn

(23)

(38)

(25) .

(30)

(26)

(38)

(4)

111

Re I 1.5 Y(1 - XS)[(1 -e)lc](Dm/D)
Since: D/Dm = (S/sm)XS (1)
XS '
Re = 1.5 Y(1 - XS)[(1 - c)/c](Sm/S) (32)
The average Nusselt number is calculated from an energy

balance over a layer of passages. For one passage at the angle

6 it has been previously shown that:

Nu . - [(0 Re Pr)/(4 L)][ln(AT2/AT1)] (10)

A similar equation can be written for the average Nusselt number

of heat flow perpendicular to the superficial velocity direction.

Nuav I . [(DavReavPr)/(4 L cos 0)][ln(AT2/8Tl)av] (40)
Since: L = 1.5 Dav [(1 - E)/c] (38)
Nuav = - {[ReavPr c]/[6 cos 6(1 - c)]}[1n(AT2/ATl)av] (33)

The average Stanton number 15 defined as: Stav I Nuav/(Reavpr)+

Stav I - {8/[6 cos 6(1 - e)]}[ln(AT2/AT1)av] (41)

112

MOMENTUM'EQUATION

 

The pressure loss per unit length in terms of the parameters of
the model is determined by the fellowing procedure from the Ergun [12]

equation.

ZEEELS‘ 0’ ‘3 ——— an ."'(L") .0 138‘
{Leeseu‘fl-B R0,

 

Ergun equation:

. -AP 10 0' 6)
be" By ' Q 1. cos 0 1541-4)

-(;9‘5—1¢‘=L)° ‘ ,, L)

(VIZ). - szm/D)‘ (31)
4-
M2) . -AP 21.29:
I r“

BIY‘BmII' 1.19.5

H z. 2" of en 0 a‘h-e)

Since: L I 01) I 6(1 - c)/a (30)
BY Iatvv‘). 2"?qu
g
Q 0. u a to: 0

Since: Dav/DI I (1 - XS) (4)

113

4-‘5
:g m‘). c‘n-mo-xss r
H t D” «so a" :0 Q‘

Since: Dav ' 4 5/8 (S)

1 (VP)... (Iv ¢)(c-xs)‘s‘r"

EY- ._
11.0 1 use uq

 

The Ergun Equation in terms of the model is then:

 

O o
£19 0' c’ - “"3" ("‘x"”qr—) 43’! . '"7Mt‘ +1.1: (42)

QLe-south-é me ”sou

Solving for pressure loss per unit length of bed:

, , u‘r‘ “-60.4346,”Ill
“W W, + (35)

 

APPENDIX B

‘ 114

APPENDIX B
DISTRIBUTION INDEX

XS was assigned a value of 0.3 by comparing model results with
other authors' results from random packed beds. Figure 13 compares
the equation of Wilson and Geankoplis with the model for X8 values
of 0, 0.25 and 0.50. After studying this graph and other similar
plots from other authors it was decided to use XS - 0.3 in all cal-
culations. Dm/DP ratios for simple cubic (Figure 14) and rhom-
bohedral (Figure 15) arrays or any other regular arrangement of

Spherical packing can be approximated by the following method.
Dav - (1 - ‘xsmm (4)
XS 3 l - Dav/0m

Since: Dp - 1.5 Dav(1' e)/e (33)

"S’l'fiifin‘a

Since in regular packing the passage cross sections would be uniformly

distributed (X880):

l.
D.’% = m ‘4”

115

116

Voids Fraction - .4

 

0.001 0.01 0.1 1
Reg .3 bug
l-G &

Figure 13. Comparison of Model with Data of
Wilson and Geankoplis [60]

117

 

fi
‘4

 

 

Volume of Cube - Dp

no ‘
Solid Volume in Cube - “7;:-
Voids Fraction (e) - :45 s 0.41s

DI: ,, cheat

9' " ISM-0.415) z °'“°"

 

Figure 14. Simple Cubic Array of Spheres

 

118

 

 

 

 

Volume of Pyramid - 'é' (Area of Base) (Altitude)

'I
D ‘
a . ‘- °'
- flaw-Wm (5%;- 1
- 0.118 Ops
| .2:
Solid Volume in Pyramid - 4 (a) t". )
- 0.0373 01,3
- . M1) =
Vaids Fraction (c) m 0'". 0.260
D_n z 0.2; ~
, 1.50-0.24.) " 0'2.“

 

Figure 15. Rhomhohedral Array of Spheres

 

 

APPENDIX C

119

APPENDIX C
(VYZ)In VARIATION WITH REYNOLDS NUMBER

As is shown in Table 38 the (VYZ)m value to use to produce a
desired Reynolds number depends upon the porosity of the bed.

Equation 44, which is in the form:
In (W2) - Al + A2 (ln REI) + A3 (ln REUZ + -----+ A9(ln mans

can be used to predict the correct (VY2)n. Using the data from
Table 38 the average constants--Al, A2, A3, °'-, A9--were determined

by a least squares technique using the Gauss-Jordan elimination method.

TABLE 37

 

CONSTANTS FOR EQUATION 44

EP I 0.3 EP 8 0.4 EP 3 0.5 EP 3 0.6 EP I 0.7
A1 .4124SE+01 .45844E+01 .500618+01 .SSllGE+01 .58816E+01
A2 .104068001 .10358E+01 .10360Efi01 .123085+01 .1058SE+01
A3 -.8546BE-02 -.61421E-02 .104385-02 -.7829ZE-02 .27823E-01
A4 -.30444E-04 .40949E-02 .726988—02 -.90$07E-02 .111835-01
AS .260675-02 .236346-02 .188415-02 .322178-02 .49379E-03
A6 -.S6269£-04 -.l8728E-03 -.2606SE-03 .891635-04 -.30353E-03
A7 -.62718£-04 -.43577E-04 -.2726SE-O4 -.5966SE-04 .63015E-06
A8 .698415-05 .618365-05 .530918-05 .392528-05 .354458-05
A9 -.22253£-06 -.21780£-06 -.2074l£-06 -.76254E-07 -.188535-06

120

VYSH

0.5005
0.1005
0.2005
0.4005
0.8005
0.1605
0.3205
0.6405
0.1285
0.2565
0.5125
0.1025
0.2045
0.4095
0.8195
0.1635
0.3275
0.6555
0.1315
0.2625
0.5245
0.1045
0.2095
0.4195
0.8385
0.1675
0.3355
0.6715
0.1345
0.2685
0.5365
0.1075
0.2145
0.4295
0.8585
0.1715
0.3435
0.6875
0.1375
0.2745

00
01
01
01
01
02
02
02
03
03
03
04
04
04
04
05
05
05
06
06
O6
07
07
07
07
08
08
08
09
09
09
10
10
10
10
11
11
11
12
12

TABLE 38. (VYZ)In VARIATION WITH REYNOLDS NUMBER

5P=0.3

0.8725-02
0.1735-01
0e34SE‘01
006866-01
0.1365 00
0.2695 00
0.5325 00
0.1045 01
0.2045 01
0.3965 01
0.7565 01
0.1405 02
0.2545 02
0.4415 02
0.7395 02
0.1195 03
0.1875 03
0.2885 03
0.4365 03
0.6535 03
0.9705 03
0.1435 04
0.2105 04
0.3075 04
0.4485 04
0.6505 04
0.9405 04
0.1355 05
0.1945 05
0.2785 05
0.3985 05
0.5695 05
0.8105 05
0.1155 06
0.1645 06
0.2335 06
0.3305 06
0.4695 06
0.6655 06
0.9425 06

121

Ep30e4

0.5615-02
0.1115-01
0.2225-01
0e441E'01
OeBTSE'Ol
0.1735 00
0.3425 00
0.6735 00
0.1315 01
0.2545 01
0.4865 01
0.9055 01
0.1635 02
0.2845 02
0.4755 02
0.7685 02
0.1205 03
0.1855 03
0.2805 03
0.4205 03
0.6245 03
0.9215 03
0.1355 04
0.1975 04
0.2885 04
0.4185 04
0.6045 04
0.8705 04
0.1255 05
0.1795 05
0.2565 05
0.3655 05
0.5215 05
0.7415 05
0.1055 06
0.1495 06
0.2125 06
0.3015 06
0.4275 06
0.6065 06

5.0.6.0.?

EP'OeS

0e37‘E'02
0e74‘E'02
0.1485-01
0.2945-01
OeSBBE'OI
0.1155 00
0.2285 00
0.4495 00
0.8775 00
0.1695 01
0.3245 01
0.6035 01
0.1085 02
0.1895 02
0.3175 02
0.5125 02
0.8055 02
0.1235 03
0.1875 03
0.2805 03
0.4165 03
0.6145 03
0.9025 03
0.1315 04
0.1925 04
0.2785 04
0.4025 04
0.5805 04
0.8345 04
0.1195 05
0.1705 05
0.2435 05
0.3475 05
0.4945 05
0.7035 05
0.9985 05
0.1415 06
0.2015 06
0.2855 06
0.4045 06

VOID FRACTIONS (5P) OF
0.3.0.4.0.

EP'Oeb

0e249E‘02
0e‘966-02
0.9875-02
0.1965-01
0.3895-01
0.7705-01
0.1525 00
0.2995 00
0.5855 00
0.1135 01
0.2165 01
0.4025 01
0.7265 01
0.1265 02
0.2115 02
0.3415 02
0.5365 02
0.8255 02
0.1245 03
0.1865 03
0.2775 03
0.4095 03
0.6015 03
0.8795 03
0.1285 04
0.1855 04
0.2685 04
0.3875 04
0.5565 04
0.7965 04
0.1135 05
0.1625 05
0.2315 05
0.3295 05
0.4685 05
0.6655 05
0.9455 05
0.1345 06
0.1905 06
0.2695 06

EP'0e7

0.1605-02
0.3195-02
0.6345-02
0.1265-01
0.2505-01
0.4955-01
0.9785-01
0.1925 00
0.3765
0.7285
0.1385
0.2585
0.4665
0.8115
0.1355
0.2195
0.3455
0.5305
0.8025
0.1205
0.1785
0.2635
0.3865
0.5655
0.8235
0.1195
0.1725
0.2485
0.3575
0.5125
0.7325
0.1045
0.1485
0.2115
0.3015
0.4285
0.6075
0.8615
0.1225
0.1735

The values in the 5 columns to the right are Reynolds numbers,

Rep/(l - a), calculated from the model using (VYZ). values in the

left column.

 

APPENDIX D

122

 

APPENDIX D
SUWARY OF IDDEL EQUATIONS

Given Data: Voids Fraction (6)
Schmidt number (Pr)
cos 0 a 0.707
XS I 0.3

Assume a value for (W2)In depending upon the Rep/(l - e) desired.

By graphical integration solve for:

a.” - (1 - XS) cosI f ‘4 (-‘-£-)”4(‘3:)

d‘ir -:—:: R, (...ng m

LR g. Tx‘déi'»

The sequence used for the graphical integration is:

 

(AT2/AT1)‘V '

For SIS“ values between 0 and 1, determine:
vvz - (wz). (seams)

Y I ( VY + 1024 - 32)(l - 5.8/(RT + 175IRT))

Re - 1.5 Y(l - x3)(.'L.!.) ‘53.)”

123

(l3)

(14)

(31)

(21)

(20)

(32)

 

124

1/2 1/3 4

4 . 1.6154(v pr)"3 + (.664 (2 Y) Pr )

Nu I ((3.656

.6Pr1/3 4 .25

+ (.33 Re ) )

After Reav and (ATZ/ATl)aV have been determined:

Rep/(l - c) I 1.5 Reav

1e Fk'lhe £5129)
nuav. - fill-0‘98. 5'1 AT. MI
8 e 1.: Na
'35 1-0 ' PC"

k-"‘°"’ Ea: +3]

c .’ 6»!

1 a? r";. - (1"(1- xs)4( v v),

- L .
Ap/A ‘26 ‘6 a? D?

h I Nuav k a/(4 e)

(22)

(23)

(33)

(34)

(26)

(35)

(37)

 

APPENDIX E

125

 

.3

C) (1 rs (5C! (3 (1

OMWCI (1

F5 (5F! r1 (5 (5

APPENDIX E

TABLE 39. COMPUTER PROGRAM LISTING FOR TABLE 2

HRITEISoII

FORMATIIHI)

MRITEISeZI

FORMATII4I/121X'CHU + KALIL + HETTEROTH (19531'./21X
I'EQUATION J 3 1.77/REEII.44'./21X
Z'SCHMIDT NUMBER 8 2.57 (GASESI'o/ZIX
3'VOIDS FRACTION I 0.38'./21X
4'R5YNOLDS NUMBER RANGE I 30 - 5000'0/21X
5'XS I 0.3'/////)

HRIT51593I

FORMATI9X

1'R5YNOLDS MODEL CHU DEVIATION
2'I9X

3'16UR/AZ) (6K5/ADS) (6KE/ADSI FRACTION'
4el/I

FORMATI4FI7.4)

COSINE THETA

CT‘eTO.’

CONSTANT IN LANGHAAR CORRECTION

B’Sea

SCHMIDT NUMBER EQUALS VISCOSITY DIVIDED BY
DENSITY AND OIFFUSIVITY

PR32e57

BED VOIDS FRACTION

EPI.38

EPRIII.-5P)/EP

DISTRIBUTION INDEX

XS'.3

V TIMES Y SQUARED MAXIMUM

VYSM82600.

DO 12 NIlg22

VYSMIVYSMII.5

S DIVIDED BY S SUBSCRIPT M

SR 3 1e0625

START GRAPHICAL INTEGRATION TO FIND
AVERAGE REYNOLDS NUMBER AND

AVE TEMPERATURE DIFFERENCE RATIO

00 10 I 3 1116

SRISR-.0625

D DIVIDED BY 0 SUBSCRIPT M

OR I SRIIXS

V TIMES Y SOUARED

VYSI VYSM I ORG-4

VARIABLE IN LANGHAAR CORRECTION

RT 3 VYS..e25

Y EQUALS DIAMETER TIMES REYNOLDS NUMBER
DIVIDED BY LENGTH

Y I ((1024.+VYSIII.5 - 32.1'11. - B/IRT + 175./RTI)
REYNOLDS NUMBER IN A PASSAGE
REIl.SIYIEPR/DRI(l.-XS)

126

 

IS

10

12

127

TABLE 39 (cont'd,)

NUSSELT NUMBER IN A PASSAGE

AA=3.656**4+1.615**4*(Y*PR)**1.33333

BB=(.664*(2.*Y)**.S*PR**.33333)**4

CC=(.33*RE**.6*PR**.33333)**4

GNU=(AA+BB+CC)**.25

IF(I-1)15,15,9

LOGARITHM OF MAXIMUM TEMPERATURE DIFFERENCE RATIO, SR - 1

ALXM=-4.*GNU/PR/Y

STARTING VALUE FOR INTEGRAL TO FIND AVERAGE REYNOLDS NUMBER

SUMRI-RE/Z .

STARTING VALUE FOR INTEGRAL To FIND AVERAGE TEMPERATURE RATIO

SXOMRI-RE/Z. ;
SUM OF REYNOLDS NUMBERS ‘
SUMR-SUMR+RE/UR I
TEMPERATURE DIFFERENCE RATIO DIVIDED BY MAXIMUM RATIO
XOXMIEXP(-4.*GNU/PR/Y-ALXM)

SUM OF TEMPERATURE DIFFERENCE RATIOS DIVIDED BY MAXIMUM RATIO
SXOMR=SXOMR+XOXM*RE/DR

CORRECTION FOR THE INITIAL VALUE

SUMR=SUMR+RE/DR/2.

CORRECTION FOR THE INITIAL VALUE

SXOMR=SXOMR+RE/DR/2.

AVERAGE REYNOLDS NUMBER FOR A GIVEN VYSM
REAI(l.-XS)*CT*SUMR/l6.

AVERAGE TEMPERATURE DIFFERENCE RATIO DIVIDED BY MAXIMUM RATIO
DTAOMISXOMR/SUMR

AVERAGE TEMPERATURE DIFFERENCE RATIO FOR A GIVEN VYSM
XXI-(ALOG(DTAOM)+ALXM)

AVERAGE STANTON NUMBER

STAIXX/6./EPR/CT

PARTICLE REYNOLDS NUMBER DIVIDED BY (1 - VOIDS FRACTION) - 6UR/AZ
REE=1.S*REA

AVERAGE NUSSELT NUMBER

GNUAISTA*REA*PR

6KE/ADS FROM MODEL EQUATIONS

SSTIl.S*GNUA/PR**.33333

DRE/ADS FROM THE EQUATION 0F CHU, KALIL AND NETTEROTH
CHU=1.77*EP*REE**.S6

DEVIATION FRACTION BASED ON MODEL 6KE/ADS

DEVG=(SST-GRU)/SST

WRITE(S,4)REE,SST,CHU,DEVC

CONTINUE

CALL EXIT

END

 

 

 

 

128

TABLE 40. COMPUTER PROGRAM LISTING FOR.TABLE 36

HRITEI59II
FORMATIIHIYIZI/II
WRITEISgSI
FORMATIZTX'GIVEN FLUID PROPERTIES'e/IIZX
'VISCOSITY IVISI = 0.092 LB/IFToHRI'p/IZX
'HEAT CAPACITY (CF) 3 0.90 BTU/ILBTDEG FI'v/IZX

(1

(T O n n

on

N

2
3 .

4 'SUPERFICIAL VELOCITY (VEL) : 1320 FT/HR'./12X
5

'THERMAL CONDUCTIVITY (AK) 8 0.131
HRITEISob)

BTU/(FT.HR.D5G FI'I

FORMATIIZXI'DENSITY IRHOI 8 1.05 LB/(CU FTI'a/IZX

I 'DIFFUSION COEFFICIENT IOIFI 3 0.0296 SQ FT/HR'o/l/ZSX

2 'CIVEN BED CHARACTERISTICS'QFIIZX

3 'BED POROSITY (EP) = 0.40'0/12X

4 'SPECIFIC SURFACE (ASP) 3 311 SQ FT/ICU FTI'o/IZX

5 'PARTICLE DIAMETER IDPAI 3 0.01285 FT'I

HRITEISeTI

FORMATI/////3OX'COMPUTER RESULTS.)

HRITEISQZI

FORMATII/IIX

I. ACTUAL CALCULATED'
Zc/IIX

3' SCHMIDT REYNOLDS REYNOLDS.
Re/IOX ‘

5'VYSM NUMBER NUMBER NUMBER.)
COSINE THETA

CT‘oTOT

CONSTANT IN LANGHAAR CORRECTION
835e8
DISTRIBUTION INDEX

XS=.3

VISCOSITY OF FLUID
VIS=.O92

HEAT CAPACITY OF FLUID
CP=.9

SUPERFICIAL VELOCITY OF FLUID
VEL=I3ZO.

THERMOCONDUCTIVITY OF FLUID
AK=.13I

DENSITY OF FLUID

RHO=1.05

DIFFUSIVITY OF ACTIVE COMPONENT
0153.0296

BED VOIDS FRACTION

EP=.4

SPECIFIC SURFACE OF PACKING
ASP=3II.

DIAMETER OF PARTICLE

 

 

0 00

0

00 D

mm o O

129

TABLE 40 (cont‘d.)

DPA=.01285

GRAVITATIONAL CONSTANT

GC=4.17E+08

EPR=(l.-EPl/EP

SCHMIDT NUMBER EQUALS VISCOSITY DIVIDED BY
DENSITY AND DIFFUSIVITY
SC=VISIDIFIRHO

PRISC

ACTUAL PARTICLE REYNOLDS NUMBER
REO=DPAIVELIRHOIVIS

ACTUAL REYNOLDS NUMBER = OUR/Al
REI=REO/(I.-EPI

CONSTANTS FROM TABLE 31
Al=.4584425+01

AZ=.1035827E+01

A3=-.6142123E-02

A4=.40949ISE-02

A5=.23b3402E-02

Abs-.187284E-03

A7=-.4357685E-04

A8=.6183554E-05

A9I-.21780075-06

EQUATION TO FIND V TIMES Y SQUARED MAX'
FROM ACTUAL REYNOLDS NUMBER
A=ALOGTREII

AZIAI+AZIA+A3IAII2
AZIAZ+A4IAI53+ASIAII4+A6IAIIS
AZIAZ+A7IAII6+A8IAII7+A9IAII8

V TIMES Y SQUARED MAX

VYSM=EXPTAZT

S DIVIDED BY S SUBSCRIPT M

SR 2 1.0625

START GRAPHICAL INTEGRATION TO FIND
AVERAGE REYNOLDS NUMBER AND

AVE TEMPERATURE DIFFERENCE RATIO

DO 10 I 3 ler

SRISR-.0625

D DIVIDED BY 0 SUBSCRIPT M

OR = SRIIXS

V TIMES Y SQUARED

VYS= VYSM . DRII4

VARIABLE IN LANGHAAR CORRECTION

RT : VYS'.025

Y EQUALS DIAMETER TIMES REYNOLDS NUMBER
DIVIDED BY LENGTH

Y = ((1024.+VYS)II.5 - 32.1!(1. - B/(RT + 175./RT!)
REYNOLDS NUMBER IN A PASSAGE
RE=I.SIYIEPRIDRI(l.-XS)

NUSSELT NUMBER IN A PASSAGE

 

n on on a On an on

000000

(7 (7f? (7

(700

O

H
H

.-
‘0

130

TABLE 40 (cont'd.)

AA=3.656**4+1.615**4*(Y*PR)**1.33333
BB=(.664*(2.*Y)**.S*PR**.33333)**4
CC=(.33*RE**.6*PR**.33333)**4
GNU=(AA+BB+CC)**.25

IF(I-1)Is,15,9

LOGARITHM OF MAXIMUM TEMPERATURE
DIFFERENCE RATIO, SR = 1
ALXM=-4.*GNU/PR/Y

STARTING VALUE FOR INTEGRAL

TO FIND AVERAGE REYNOLDS NUMBER
SUMRa-RE/Z.

STARTING VALUE FOR INTEGRAL TO FIND
AVERAGE TEMPERATURE DIFFERENCE RATIO
SXOMRs-RE/Z.

SUM OB REYNOLDS NUMBERS
SUMR=SUMR+RE/DR

TEMPERATURE DIFFERENCE RATIO

DIVIDED BY MAXIMUM RATIO
XOXM=ExP(-4.*GNU/PR/Y-ALXM)

SUM OF TEMPERATURE DIFFERENCE RATIOS
DIVIDED BY MAXIMUM RATIO
SXOMR=SXOMR+XOXM*RE/DR

CORRECTION FOR INITIAL VALUE
SXOMR=SXOMR+RE/DR/2.

CORRECTION FOR INITIAL VALUE
SUMR=SUMR+RE/DR/2.

AVERAGE REYNOLDS NUMBER FOR A GIVEN VYSM
REA-(1.-XS)*CT*SUMR/16.

AVERAGE TEMPERATURE DIFFERENCE RATIO
DIVIDED BY MAXIMUM RATIO
DTAOM=SXOMR/SUMR

AVERAGE TEMPERATURE DIFFERENCE RATIO
FOR A GIVEN VYSM
XX--(ALOG(DTAOM)+ALXM)

AVERAGE STANTON NUMBER
STA=XX/6./EPR/CT

PARTICLE REYNOLDS NUMBER DIVIDED

BY (1. - VOIDS FRACTION) a 6UR/AZ
REE=1.S*REA '
AVERAGE NUSSELT NUMBER
GNUA-STA*REA*PR

6KE/ADS FROM MODEL EQUATIONS
SST=1.S*GNUA/PR**.33333

HEAT TRANSFER COEFFICIENT
AH=GNUA*AK*ASP/4./EP

MASS TRANSFER COEFFICIENT
AKC=ASP*DIF*SC**.33333/6./EP*SST
PRESSURE LOSS PER UNIT LENGTH OF BED
DPDL-9.*ASP**2*VIS**2*(I.-EP)**2
DPDLaDPDL*(1.-XS)**4*VYSM/128.

 

a};

131

TABLE 40 (cont'd.)

DPDL=DPDLlGC/EP**4/RHO/DPA/144.
HRITEIS.3)VYSM,PR.REI.REE
FORMATI/4F17.4)

HRITEISI4I

FORMAT(/////6X

W

I

1' HEAT MASS
TRANSFER
COEFFICIENT

2' TRANSFER

3' COEFFICIENT
FORMATI/9X3Fl7.4)
HRITEI5.8)AH9AKCVDPDL
CALL EXIT
END

0)

F URES SUPPORTED

1F
T E NDED PRECISIJN
(I

A
x
0 s
R

E
E
I
:9 E REQUIREMENTS FOR
“OHMDN O VARIABLES 172 PROGRAM
EVD OF COMPILATION

[I XEQ

'o/BX

1312

DP/DL'9/7X
PSI/FT')

 

I
I
n
.

APPENDIX F

132

 

APPENDIX F

TABLE 41. SHERWOOD NUMBERS, UNIFORM AND NON-UNIFORM PASSAGES

REYNOLDS
(4UR/AZ)

0.0002
0.0004
0.0009
0.0018
0.0037
0.0074
0.0147
0.0293
0.0582
0.1154
'0.2279
0.4483
0.8785
1.8971
3.2358
8.0289
10.8788
18.9190
31.8884
51.1938
80.4313
123.8393
187.0805
279.9594
415.8919
813.8238
901.4041
1318.3273
1920.0885
2785.5598
4028.4345
5800.9157
8333.1087
1 1940.4843
1‘I072.5998

SCHMIDT NUMBER 8 1
VOIDS FRACTION 8 .4

SHERWOOD
(XS303’

1.1300
1.1302
1.1310
1.1320
1.1337
1.1364
1.1410
1.1493
1.1646
1.1937
1.2459
1.3294
1.4542
1.6377
1.9065
2.2956
2.8416
3.5732
4.5129
5.6792
7.0988
8.8263
10.9363
13.5222
16.7032
20.6271
25.4737
31.4590
38.8428
47.9403
59.1366
72.9045
89.8261

110.6193
136.1690

133

REYNOLDS
I4UR/AZI

0.0007
0.0015
0.0030
0.0060
0.0120
0.0239
0.0476
0.0945
0.1872
0.3698
0.7278
1.4240
2.7603
5.2708
9.8294
17.7197
30.6258
50.6487
80.5665
124.4274
188.4019
281.8330
418.5266
618.4262
909.9348
1333.2542
1945.2162
2826.2189
4090.1040
5898.1543
8478.8974
12156.1121
17388.4383
24825.4019
35386.6635

SHERHDOD
(X5801

3.6560
3.6560
3.6560
3.6560
3.6560
3.6561
3.6563
3.6568
3.6580
3.6611
3.6690
3.6890
3.7392
3.8611
4.1339
4.6595
5.5049
6.6785
8.1774
10.0291
12.3023
15.1029
18.5683
22.8646
28.1877
34.7702
42.8924
52.8964
65.2029
80.3305
98.9195
121.7597
149.8247
184.3144
226.7064

 

{3

.‘Il..- _.

TABLE OF NOMENCLATURE

134

 

‘4,

TABLE OF NOMENCLATURE

Primary Quantities

 

§ZEB£1. Dimension Name
F Force
H Heat
L Length
M Mass
T Temperature
t Time

Secondary Quantities

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Symbol Name Dimensions
r—- "“‘
H L M T t
8 Packing area per unit volume of -1
bed
C Heat capacity of flowing fluid 1 -l -1
D Diameter of a given passage 1
Dav Average diameter of a passage 1
Dn Maximum diameter of any passage 1
Dp Particle diameter ,1
Diffusivity of solute in flowing 2 -l
‘5 fluid
G Mass velocity of fluid in a -2 l -1
given passage
Gav Average mass velocity of fluid -2 I -1
LL perpendicular to cross-section

 

135

 

 

 

136

TABLE OF NOMENCLATURE (cont'd.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Symbol Name Dimensions
' F II L M T t
gc Gravitational constant -1 l l -2
h Average Fluid film heat transfer I -2 —l -1
coefficient parallel to the bed
axis
k Thermal conductivity of fluid 1 -1 -l -1
k Average mass transfer coefficient 1 -1
L Length of parallel passages l
m Mass flow rate in a given passage 1 -l
P Pressure exerted by fluid 1 -2
S Total cross-sectional area of all 2
passages having diameters less
than D -
Sm Total cross-sectional area of 2
all‘passages
T1 Temperature of fluid entering l
a passage
T2 Temperature of fluid leaving 1
a passage
T Temperature of fluid at any 1
p point in a passage
Tw Temperature of the passage wall 1
u Average linear velocity of fluid, 1 -1
based on empty cross-section,
perpendicular to the cross-
section of the bed
w Velocity component perpendicular l -1
to the cross-section
p Fluid density -3 l
u Fluid viscosity -1 1 -l

 

 

 

 

 

 

 

 

 

SSZbeol

l\,J3

(21.x

LIKES

I!
‘eav

II
ep

137

TABLE OF NOMENCLATURE (cont'd.)

chsrl‘sns

Name

 

Voids fraction (voids volume/total bed volume)

Average angle between passages and average flow

direction in bed (degrees)

Constants in Langhaar [32] correction factor for Y

Constant and exponent in: ShPIScl/3 [c/(l - 8)]

3 C1[Rep/(l - 6)]x

Exponent which depends upon the distribution of

passages

l/s

Dimensinnlessfima

Name

 

Graetz number

Colburn mass transfer factor
Colburn heat transfer factor
Nusselt number in a passage

Average Nusselt number in all
passages

Prandtl number of fluid
Reynolds number in a passage

Average Reynolds number in
all passages

Reynolds number based on
particle diameter

Basic Formula

Cu/k
0 Ch:

Dav Gav,"

Dp uQ/u

 

 

EESZELOI
IQFT

138

TABLE OF NOMENCLATURE (cont'd.)

Dimensignlgss 919335

Name

 

Variable in Langhaar [32]
correction factor

Schmidt number
Average Sherwood number

Sherwood number based on
particle diameter

Average Stanton number
Velocity head
Maximum VY2 factor

Parameter of Langhaar [32]

Basic Formula

(vyz).25

.11.
u!
Dav kc/D

DP kc/D

h/C Gav

- P
Q Iiélu’dd
VY2(Dm/D)4

D Re/L

BIBLIOGRAPHY

139

 

10.
11.
12.

1&3.

11%.

15.

11L

17h

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"7'1? TA'LTAT'TT'WTTRETT