MASS TRANSFER RATES IN PACKED BEDS

By s.
TARIQ .Al-ICHUDAIRI

ANAJBS'JIRACI.‘

submitted to the School for Advanced Graduate Studies of
Michigan State university of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

Doctor of Philosophy

Department of Chemical Engineering

1960

11

TARIQ AL-KHUDAYRI

ABSTRACT

mm chemical engineering processes involve mass transfer between a
packed bed and the gas or liquid stream flowing through the bed. The
driving force for mass transfer is the difference between the concentration
of the active component in the bulk of the fluid and at the solid surface.
This concentration difference may be caused, for example, by a chemical
reaction on the surface of a catalyst, by evaporation of a component
from the solid, or by adsorption.

A part of the resistance to mass transfer is diffusional resistance
in the fluid itself. This resistance is usually expressed in terms of its
reciprocal, the mass transfer coefficient, kg.

A correlation has been established in this thesis for predicting the
use transfer coefficient in packed beds. This correlation is plotted in
Figure 17. Both the use transfer coefficient and the superficial velocity
of the fluid occur on the plot in dimensionless groups, which also include
fluid properties, bed void space, and surface area . Experimental data of
previous investigators on liquid-solid systems show an average deviation
from this correlation of 30$. Data on gas -solid systems show much larger
deviations.

At low flow rates, Re/(l- € ) < 100, .the correlating curve is a
theoretical one, based on a modified form of Graetz heat transfer equation
for parabolic flow in circular ducts. The modification or the Graetz
curve was achieved by assuming that the void spaces in a packed bed would
have the same lass transfer characteristics as short cylindrical channels
when the overall dimensions of the solid space -- the void volume and. the

void surface area -- are the same as those of the channels .

iii
TARIQ.AL-KHUDAXRI

At high flow rates, Re/(l- €‘)‘>>100, the correlating curve was based
on experimental data. This part of the curve deviates from the theoretical
curve of Graetz in its slope of 0.5, which is the same slope as that of the
Graetz equation for rod-like flow. The slope of the Graetz equation for
parabolic flow at high flow rates is 0.33. This change may be a result
of turbulence.

The greater deviations of gas-solid systems are attributed to the
fact that gases have higher diffusivities than liquids. As a result,
resistances other than the gas film resistance become increasingly
important and may be controlling.

In the experimental work of this thesis, precautions were taken not
only to minimize these other resistances, but also to eliminate other
effects which might hurt the reliability of the data taken. Special
consideration was given to wall effect, end effect, back mixing, radial
diffusion, and saturation. Nevertheless, the gas phase data showed a
considerable deviation from the curves of Figure 17.

The effects of reaction kinetics, of stagnant areas in the bed,
and of diffusional resistance in the solid phase or condensed liquid
phase were analyzed as possible causes for this deviation. No one of
these factors could have been controlling, although a combination of these
could have been present. It is also possible that an interfacial resistance
caused by intermolecular forces may have been present.

It will be of great value to conduct a program that investigates the

nature of this extra resistance.

Q‘s/W

APPROVED: Majortirofessor

 

MASS TRANSFER RATES IN PACKED BEDS

/ f
TARIQ AL-ICHUDAYRI

A THESIS

submitted to the School for Advanced Graduate Studies of
Nfichigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

Doctor of Philosophy

Department of Chemical Engineering

1960

ACKNOWLEDGEMENT

The author wishes to express his sincere thanks to Dr. Carl M. Cooper
for his kind guidance and constant supervision during the entire course

of this investigation.

ABSTRACT . . . . . .
.ACKNOWLEDGEMENT . . .
LIST OF TABLES . . .
LIST OF FIGURES . . .
INTRODUCTION . . . .
HISTORICAL BACKGROUND

THEORETICAL ANALYSIS

Mass transfer in packed beds

Molecular diffusion

Dimensionless groups

Machanism of flow

EXPERIMENTAL WORK . .

Design of apparatus

0

O

0

OF CONTENTS

Design of apparatus for present experiment .

Experimental materials and equipment . . . .

Experimental procedure .

METHODS OF CALCULATION

RESULTS . . . . . . . .

Experimental results .

Theoretical data

Experimental results of other investigators

DISCUSSION

0 O

Analogy between mass and heat transfer . . .

Theoretical approach at low Re .

vi

ii

viii

16
16
17
19
20

31

33
31+
1+0
1&8

59
59
62
69
99

99
100

Solutions of the diffusion equation

Representative Graetz plots

Flow in packed beds

Application of Graetz equation to packed beds . .

Experimental data

Deviation of experimental data according to Schmidt number .

Possible reasons for low data on gases . . . . .

Interfacial resistance .

Liquid phase resistance

Air resistance . .

Reaction rate

Back pressure equilibrium

Mass transfer in solid phase .

Effect of back mixing

Effect of radial diffusion .
Effect of bed geometry .

Previous correlations for predicting

Suggested plot .

CONCLUSIONS .
RECOMMENDATIOI‘IS . . . . .
APPEBJDH O I O O I I 0

Sample calculations

Table of nomenclature

Bibliography . . . .

mass transfer .

vii

100
101
103
106
107
108
110
111+
11h
11k
115
116
117
118
118
119

121

128
129
131
131+

Table Number

FD.)

\OOD-Qmm

IO

12
13
1h
15
16
17
18
19
20

21

LIST OF TABLES

Empirical Equations . .

Theoretical Data for Different Mechanisms
Dissociation Pressure of Copper Ammines
Properties of the System in the Present Experiments.
Original Experimental Data for Runs A.and B
Original Experimental Data for Final Runs

Degree of Absorption for Runs A and B

Time Effect on Absorption Rate . . .

Degree of Absorption for Final Runs

0

Mass Transfer Data of the Present Experiment .

Theoretical Mass

Theoretical Mass

Theoretical Mass

Theoretical Mass

Transfer Data . . .
Transfer Data . . .
Transfer Data . . .

Transfer Data . . .

Experimental Data of Wilke and Hougen (37) . .

Experimental
Experimental
Experimental
Experimental
Experimental
Experimental

Experimental

Data

Data

Data

Data

Data

Data

Data

of Resnick and White (28)

of Hobson and Thodos (13)

of MCCune and Wilhelm (23) .

of Hebson and Thodos (IA)

of DeAcetis (3)

of Taecker and Hougen (3h) .

of Shulman and Margolis (31)

viii

Page
15
3O
35

39
A6

1+7
5h
55

57
6O

63
66
6?
68

70

72
7h

77

81

23
2h
25
26
27
28
29

Experimental Data of Hurt (15) . . . ........

Experimental Data of Gaffney and Drew (8)

Experimental Data of Thoenes and Kramers(35) . .

Experimental Mass Transfer Coefficients . . . .

Excess Resistance . . .

Effect of’Reaction Kinetics

0000000000

85

91

96

97
98

Figure Number

1

CD4 O\\J‘| F’U)

10

l2

13
1h

15
16
17

LIST OF FIGURES

Page

Forward Diffusion in Packed Beds . . . . . . . . . 29
Illustrative Plot for Different Mechanisms . . . . 29
Dissociation Pressure of Copper Ammines . . . . . . 36
Diagram of the Packed Bed . . . . . . . . . . . . . hl
Flow Diagram . . . . . . . . . . . . . . . . . . . #3
Plot of Time Effect on Absorption Rate . . . . . . 56
Plot of FloW'Rate Effect on Degree of Absorption . 58
Plot of Experimental.Nass Transfer Data for

Different Mechanisms . . . . . . . . . . . . . 61
Theoretical Plot Based on Graetz Equation for

Parabolic and.Rod-Like Flow in Circular Ducts. 102
Theoretical Plot Based on Graetz Equation for

Different Mass Transfer Coefficients in Circular

Ducts . . . . . . . . . . . . . . . . . . . . 10h

Idealized Flow Path in Packed Beds . . . . . . . . 105
Plot of Experimental and Theoretical Data for

Mass Transfer . . . . . . . . . . . . . . . . 109
Experimental Mass Transfer Coefficients . . . . . . 111
Plot of Excess Resistance Evaluated for the

Present Experiment . . . . . . . . . . . . . . 113
Diagram of the Stagnant Areas in Packed Beds . . . 120
Plot of Empirical Correlations for Mass Transfer . 122
Plot of Suggested Curves for Mass Transfer in

Packed Beds . . . . . o . . . . . . . . . . . 12h

INTRODUCTION

MowcrIon

Packed'beds are used commercially in a wide variety of applications.
Catalytic reactors, ion exchangers, adsorbers, and tray dryers are important
examples in which packed beds are actively involved in chemical engineering
processes. Absorption.and distillation towers filled with Raschig rings or
Berl saddles are also packed beds, but in these cases the packing is simply
an inert medium for dispersing the liquid.

The physical phenomena that occur are similar regardless of whether
the packing itself or the liquid flowing over the packing takes part in
the process. Similar phenomena occur also in spray towers, in liquid-
liquid extraction columns, and in other dispersed homogeneous systems
where there is no packing. In all of these there is transfer of some
component from one phase to another and there is a series of resistances
to this transfer. For example, there may be resistances to diffusion
through one phase, resistances to diffusion through the other phase, or
resistances due to slow chemical reactions involved in the formation or
reaction of the active component.

In many processes the controlling resistance is mass transfer through
the continuous phase. For example, in a catalytic reactor where the reaction
rate is fast and the gas flow rate slow, we would expect that the rate of
mass transfer from the gas stream to the solid surface would determine the
size of the equipment. For a slow reaction and a fast flow rate, however,
are sham expect the kinetics of the chemical reaction to control. In
either case we need reliable information on mass transfer rates before we

can design the reactor with assurance.

The film theory has been a useful concept in dealing with mass
transfer, but it is not completely and quantitatively correct. Film
theory assumes that most of the resistance to mass transfer is caused
by a thin layer of stagnant fluid next to the solid surface. Recent
studies show that considerable resistance can also exist in the turbulent
gas stream and in the buffer region between the main stream and the film.
Nevertheless, mass transfer rates are conveniently GXPressed in terms of the
mass transfer coefficient, kG, Just as heat transfer rates are expressed
in terms of h, the film coefficient of heat transfer.

To calculate mass transfer coefficients, a relationship must be known
between the coefficient and the variables that influence it. These
variables are the density and viscosity of the fluid, the diffusivity
of the active component through the fluid, the superficial flow rate
of the fluid, and the shape and size of the spaces between the particles
in the packed bed. With such an array of variables, correlations are
best made by using dimensionless groups.

The common dimensionless groups for mass transfer are the Sherwood
number, the Schmidt number, and the Reynolds number. The Sherwood number
contains the mass transfer coefficient and the diffusivity, and is some-
what analogous to the Russelt number of heat transfer. The Schmidt number
contains only the physical properties of the fluid and its active component,
and is therefore similar to the Prandtl number of heat transfer. The
Reynolds number is, of course, a measure of flow rate. In the case of
packed beds, the expressions for these groups should be modified to take
into account the difference between superficial velocity and interstitial
velocity, and between particle diameter and effective diameter of inter-

stiees.

The analogies between heat transfer and mass transfer are well known.
These arise from the fact that the mechanisms for transfer are the same --
namely, molecular diffusion and convective mixing. Heat transfer correlations
for flow through pipes can therefore be used quantitatively to calculate
mass transfer to and from pipe walls. In the case of packed beds, however,
the information on heat transfer to and from the particles is even less
than it is for mass transfer. Information on momentum transfer in packed
beds-~that is, pressure drop--is well established, but attempts to demon-
strate the analogy between mass and momentum transfer for packed beds have
not been successful.

A number of correlations for mass transfer in packed'beds have been
proposed. Most of these involve a single-line relationship between the
Reynolds number and the product of Sherwood number multiplied by some
power of the Schmidt number. Earlier correlations define the Sherwood
and Schmidt numbers arbitrarily; therefore the variables in bed geometry
are not properly accounted for. Recent correlations are much better in
this respect, as they give mass transfer rates that are in good agreement
with most of the reliable liquid-solid data reported thus far. For gas-
solid systems, however, they leave much to be desired.

In view of the wide discrepancies between mass transfer correlations
for low velocity flow of gases through packed beds, it was decided to
investigate this region. Factors involved in the study of such systems
present many prdblems, each of which merits special study. A study of
the gas-solid system presupposes a knowledge of all factors involved,
such as fluid properties, packing properties, and nature of the flow,

as well as the effect of each factor on the others.

In pursuit of this requirement, the present work is intended to
eliminate some of the problems by studying a gas-film controlled system
using gas and solid beds of known constant properties. Systematic
deviations in the final results due to bed heterogeneity, wall and end
effects, temperature effects, and bed geometry were controlled.

Experimental data were obtained in an attempt to decide which of
the results of previous investigators were most reliable, and theoretical

calculations were made in an attempt to re-evaluate previous calculations.

HISTORICAL BACKGROUND

The mechanism of mass transfer in packed'beds has been studied by
many investigators. Among the earliest work reported on the subject is
that of Colburn (2), in which the Reynolds analogy between heat transfer
and frictional resistance to fluid flow was extended to mass transfer.
Colburn suggested that heat transfer correlations for flow through tubes
and.across tube banks could be extended to similar mass transfer
arrangements.

The later work of Chilton and Colburn (1) followed the same analysis.
In.addition, they presented charts for turbulent flow inside tubes,
parallel to plane surfaces, and across tubes and tube banks. The abscissa
for these charts is the Reynolds number. The ordinate in each case is a
heat transfer factor or, alternatively, a mass transfer factor. The heat
transfer factor Jh and the mass transfer factor Ja were expressed as
follows:

(Ii/cps) (ex/3 (1)
(kc/U) (Sc)2/ 3 (2)
It was reported that the Ja equation does not allow for any liquid

Jh

Jd

film resistance at the gas-liquid interface on the tubes, and that the
velocity in the equation refers to the relative motion between two phases.
Also the effect of free convection at low Reynolds number was not considered
in the above analogy.

.Among the early experimental data supporting the Chilton-Colburn
analogy for packed beds were those reported by Gamson, Thodos, and Hougen

(9). Their experiment involved a through-circulation dryer operated

with or without recirculation of air. Il'hey reported that values of Jd
were in good agreement with values of .1}, (about 8% lower on the average).
They recs-ended the equations:

J'd = 0.989 (Re)'o'h1 for Be >350

Jd = 16.8 (Re)'1 for Re <1+0

Sherwood (29) pointed out that the agreement between .14 and .1},
reported by Gamson et al was inconclusive. Since they I“ used a humidity
chart to determine the surface temperature of the particles, they were in
effect assuming that this temperature is equal to the adiabatic saturation
temperature. This assumption in itself leads to the conclusion that

Jh/J'd == Pr2/3/ Sc2/3 a 1.08 (for water)

Sherwood also pointed out that, if the surface temperature is not
the adiabatic saturation temperature, the true value of Jh would be only
slightly different from the calculated value . Values of J'd, however,
could be considerably different .

The same procedure of Gamson et al was followed by Wilke and Hougen
(37) with some modifications of wetting the packing and controlling the
best conditions. Once again the surface temperature was taken as the
wet bulb temperature. The new data and the previous data were plotted
to give a new equation.

J’d = 1.82 (Re)'°'51 for Re (100

Hurt (15) measured the height of transfer unit for gas film controlled
systems in packed beds with different sizes and shapes of packing. No
spherical particles were used, but the cylindrical particle data showed
close agreement beWeen heat and less transfer factors as Chilton and
Colburn had suggested. No such agreement between heat and mas transfer

data was obtained for other packing shapes. Hurt did not report fraction

voids or surface area of the beds he used. Hewever, Resnick and'White (28)
and Ergun (7) have analyzed Burt's data using assumed properties for the
bed. Resnick and White's plots showed considerable disagreement between
Burt's data and the equations of Wilke and Hougen (37). Different curves
were obtained for different size particles. However, it was interesting
to note that Burt's data for the water-air system coincided with those

of Wilke and Hougen (37) at high Reynolds numbers. For systems in which
Se is higher than for water-air, agreement is poor even at high.Re.

The effect of gas and liquid rate, temperature, packing size, and
humidity of inlet gas on the overall coefficient of mass transfer was
studied'by Dwyer and Dodge (5). They concluded that the gas film co-
efficient in a packed absorber is somewhat affected by liquor rate. It
was reported by Gilliland and Sherwood (11) that the mass transfer
coefficient kG varies with the 0.56 power of the diffusivity in wetted
wall towers. It was also reported by Sherwood and Holloway (30) that kG
varies with the 0.17 power of diffusivity in packed columns. The variation
of the diffusivity effect on kG in wetted wall columnsand packed beds has
been explained'by Dwyer on the basis that DIn is a stronger factor in laminar
flow than it is in turbulent flow.

In their study of gas film mass transfer coefficients, Molstad and
coaworkers (25) measured the rate of absorption of ammonia by water from
ammonia-air mixtures in packed'beds with nine different industrial packings.
They found that kGa does not vary with a constant exponent of the gas
rate. They also found that the diffusivity exponents could vary (O.l7o0.67)
in packed'beds. They also reported a noticeable effect of temperature on
kga.

Later, Surosky and Dodge (33) vaporized three organic liquids and

water into an air stream in a modified packed tower where end effects

were controlled by stream distributors. They reported that kGa varies as
the 0.72 power of the gas rate and the 0.15 power of the diffusivity of
the gas.

using the same technique as that of Gamson (9) for liquid film studies,
Habson and Thodos (13) correlated mass transfer with Reynolds number.
They found that separate linear correlations for each system and particle
size were required when mass transfer was expressed as kG, but one correlation
was adequate for all their data when the mass transfer factor Jd was used.
Furthermore, this correlation was reported to be independent of Du. They
recommended an expression to cover both.

Log (Jd) = 0.7683 - 0.9175 (log Re) + 0.0817 (log Re)2

A different procedure was employed by Resnick and White (28) to
correlate the rate of mass transfer for'both fixed beds and fluidized
beds of naphthalene particles. The Chilton-Colburn Ja factor was employed
to present the results, which did not agree with those of Gamson (9) or
Hurt (15). The disagreement with Gamson's results was attributed by the
investigators to the fact that Gamson's particles were larger in size.
Rcsnick and White found that, with smaller particle sizes, Ja factors are
much lower than those reported by Gamson. Furthermore, the smaller the
size the greater the deviation. They suggested a resemblance of this
behavior to roughness in unpacked pipes. Inconsistency with Hurt's
results was attributed to the possibility that their naphthalene beds
were packed differently from those of Hurt.

The 2/3 power for the Schmidt number, suggested by Chilton and Colburn
(1) was shown to be valid by McCune and Wilhelm (23) in the range of So =
1200 to 1500. Using 2-Naphth01 and liquid water, their results agreed

with those of Gamson and others (9, 13, 37), even though Gamson et al

10

used systems with Sc = 0.62. The agreement was close especially for
Be (120. Where disagreement was observed, it was attributed mainly to
the effect of particle size.
Taecker and Hougen (31+) confirmed Hougen's previous data (9 , 37)
using a modified Reynolds number.
R; = G lip/- f .

For spherical and cylindrical packing the following equations were Obtained:
-0.hl

 

1.251 (Re’) for Re, > 620

2.41;. (Re’)'°'5l for Re’ < 620

Jd

Jd

Following the same procedure of McCune and Wilhelm (23), Gaffney

and Drew (8) found the Schmidt number power to be 0.58 instead of 2/3.
Mass transfer from packing to organic solvent in a single phase was

investigated and the data were plotted as

It was found that the data was higher than those of Hougen and others
(9, 13, 23, 37).

With spherical packing and a void volume of h7.5% at 1 atmosphere
pressure, Hobson and Thodos (1h) measured the rate of vaporization of
water and some organic compounds into air, nitrogen, carbon dioxide, and
ammonia by the bed weighing method that most of the previous workers
employed. This work confirmed the results that had been reported before
(9, 13, 23, 37). It was found that the surface temperature varied from
top to bottom during the measurements, which were conducted at steady
state conditions. These actual temperature measurements were used in the
calculations to evaluate the driving force. They also reported that the
variation of surface temperature in the direction of the flow was linear.
However, the Schmidt number was considered constant and its variation

with temperature was disregarded.

11

To determine the effect of fluid properties on mass transfer in the
gas phase, lynch and Wilke (20) vaporized water into different gases and
found that the mes transfer coefficient varies with the 0.5 power of Sc
- for f’fld perpendicular to a single cylinder. It was also established

that 0 -
HT oe Sc '9 when compared at equal Re

and 51‘ o: Sco'l‘7 when compared at equal 02
The effect of gas properties on the use transfer coefficient was
studied with the effect of temperature and pressure by Shulnn and hrgolis
(31). They concluded that
.14 =.- 1.195 (Re/1-€ ,
AlsotheyrepcrtedthatJdisindependentofthepressureandthathis‘

)-O.36

therefore inversely proportional to the pressure. These relationships
had been established before by other workers (32).

12°92: (7) node use ofthe Reynolds analogy relating heat and ms
transfer to pressure drop. 0n the basis of this analogy, he attempted to
apply to use transfer in packed beds a correlation he had previously
estoblished (6) for pram drop in packed beds. He used the Reynolds
analog in'the following form.

In order to integrate this equation, he assumed two possible patterns

for flow of fluid through the bed, complete mixing and piston flow.
Forcompletemixing g1; yn::o._____ 2 AP
3! 7 € 70115

Forpiltonflow dy_ «ln y* “YO..- 2
y ‘T—;-n 76 pue

12

Using'the data of license and Wilheln (23), Ergun found that (7’) was
constant when piston flow is assumed, and that 7 = p DE

Combining these results with his previously derived pressure drop equation

AP= 1-69 [1 L 1-6 L
7)? ””‘sL' T 7.9"” 73 ° ‘51»—

he arrived at a general equation for use transfer in packed beds.

_<1_P_ 6‘

J9 =3 L 0 TT- Tlém e f% = 150 1% . 1-66 + 1.75 (3)

From this equation it my be seen that a straight line is obtained for a

 

plot of
l yn " yo
T vs. -;——- if couplets mixing exists
y ' Yn
l y* ' 3’0
—G-— vs. 1n -;—— if piston flow exists
y " yn

Experimental film of any workers, who used liquid-solid systems,
were eeployed by Ergun to demonstrate these relationships. They showed
that piston flow generally prevailed. However, poor results were obtained
when data of gas-solid systems were employed. While most of the gas-solid
data did not give a clear indication of the nature of the flow, Burt's
data (15) indicated complete mixing, though it did not agree with the
equation for (J3). Ergun attributed this failure to the uncertainity of
the properties of Burt's system.

The other mJor contribution of Ergun was the use of the dimensions
of the space between particles in the calculations instead of the diameter
of the particles theuelves. This is more logical since mss transfer is
analogous to the frictional resistance to flow. In dealing with frictional
resistance to flow, the wetted surface and the conduit volume are considered,
and in lass transfer the same should be done. The importance of using

proper dimensions we demonstrated successfully by Ergun when he showed

13

that Hurt's data for different particles sizes all fall on the same line.
Hurt, using particle dimensions instead of conduit dimensions, had reported
separate lines.

In recent Mes considerable attention has been paid to mixing in
the direction of flow. Experimental results on axial diffusivity have
been obtained by many workers for gases and liquids (17, 19, 36).

By means of the Ficks law equation for diffusion, an axial Peclet
number was reported by McHenry and Wilhelm (2h) . They obtained a value
of approxintely 2.0 for Pa. The same results have been obtained by '
assuming perfect mixing on each layer for a high bed as had been illustrated
by Kramers (17).

The most recent work on the subject was that of Thoenes and Kramers(35).
In their work, they reviewed previous data and.presented a collective
graph of

Sh/Scl/ 3 vs. Re/l-€

They conducted their experiment on a single sphere, taking into consideration
bed inhomogeneity and.axial diffusion. Two equations were obtained to
relate Sherwood number, Schmidt number, and Reynolds number. For gases
and liquids, the following equation was derived with a mean deviation of
t 10%.

an = 1.26 (Re)1/3(Se)1/3 + 0.051; (Re)°-8(Se)°°’* + 0.8 (Re)0'2
where the terms on the right side of the equation refer to the laminar,
turbulent, and stagnant layers respectively.

The various studies. which have been mde, were conducted with different
procedures and assumptions. Even the basic properties of the fluids used
for the experiments were obtained from different sources or derived by
different methods. This by itself could explain partly the wide distribution

of the data for mas transfer.

lh

Table 1 includes a list of all the major equations presented by

different workers and the flow conditions of their applications.
empirical equations were based on the experimental results of the

corresponding investigations.

These

15

TABLE 1

Empirical Equations for the Mass Transfer Factor
as Predictedfby Different Investigators

 

 

 

ref.
Ja = 16.8 (Re)'l for Re <: ho (9)
J'd = 0.989 (Re)"°°’*3L for Re > 350 (9)
Ja = 1.82 (Re)'o'51 for Re <: 100 (37)
“Id -0 2
1 0.19 (Re) - 73 for Re < 25 (28)
-5
d1:
J'd = 2.12:. (Re, )-0-51 for ne’< 620 (31.)
J4 = 1.251 (Re’ )")'ill for Re’) 620 (3h)
Jd = 1.625 (Re)'0'507 for Re < 120 (23)
Jd = 0.687 (Re)’°'327 for Re > 120 (23)
.111”: 1.97 (Re/6‘)'°'6l3 for Re/€ < 200 (8)
J5“: 0.29 (Re/€)‘°°25'* for Re/€ >200 (8)
log .14 = 0.7683 - 0.9175 log Re + 0.0817 (log Re)2 (13)
Je = 150 (1- €)/Re + 1.75 (7)
sn e 1.26 (Re)1/3 (Sc)l/3 + 0.051 (Re)0 8(Sc)o “ + 0.8 (Re)0°2 (35)
0.725
Jd = 0.111 (3)
(Re) - 1.5
Jd = 1-195 (Re/1' 0-036 (31)

 

* dp is in mm.

**'Jd = kG/U (Sc)o'58

16

THEORETICAL ANALYSIS

Nhss transfer ingpacked'beds

whee transfer is the transfer of components through a phase from
a region of high concentration to one of low concentration. This transfer
may be accomplished by molecular diffusion of the active component through
the rest of the phase, or it may‘be done by eddy diffusion which is the
mixing of the phase due to turbulence. There may also be mass transfer
from one phase to another. The resistance to such transfer is generally
considered negligible, and the composition on one side of the interface
can be assumed to be in equilibrium with the composition on the other side.
Thus there is a steady drop in the concentration on either side of the
interphase with discontinuous change, which may‘be positive or negative,
at the interphase itself.

In packed beds, resistance occurs in a film of fluid that covers
much of the surface of the packing and separates it from the‘bulk of the
flowing fluid. In case of equilibrium, the thermodynamic potential of
the component in the bulk of the fluid is the same as its thermodynamic
potential on the surface of the solid.

Mass transfer in.packed beds has been measured'by calculating the
rate of mass transfer between phases. This rate per unit length at any
cross section of the bed is equal to: AU dyf/dL.

The driving force for mass transfer is represented.by the difference
in concentration of a component in the bulk stream and the concentration

at the interface, Just as temperature difference is the driving force f0r '

17

heat transfer. The transfer of mass will continue through the film until
equilibrium is reached. Hence, the driving force could be represented as:
(y - y*), where y is the concentration of the active component in the
stream and y* is the concentration of the active component at the inter-
face or, in other words, the component concentration in equilibrium with
the fluid or solid on the other side of the interface.

By evaluating the driving force and the rate of mass transfer between
the phases, the resistance to mass transfer or its reciprocal, the mass
transfer coefficient kG, can be predicated

AUdyf/dL=kGaA(y-y*)
or

a (y-y*) dL

k6 is comparable to the heat transfer coefficient h except that in the

 

case of heat transfer the driving force is the temperature difference

(T - T*), while for mass transfer it is (y - y*), the concentration
difference. Also while h is measured as Btu per unit time per unit area
per unit temperature, k6 is measured as the volumetric rate, or volume

per unit time, per unit area.

Melecular diffusion

 

A distinction should be made between mass transfer and diffusion.
JMass transfer involves not only diffusion but also the motion of larger
masses or eddies where turbulence exists. Diffusion is the part of mass
transfer due to migration of molecules of the constituents of a system of
miscible material toward a region of lower concentration. The phenomena
of diffusion is brought about by random motion of the molecules. It was

described mathematically by Fick's law as follows:

Q= -Dm (By/31:)

18

where x is the distance traveled by the molecules while diffusing from
a higher to a lower concentration region. The proportionality factor
Dm is called the diffusion coefficient or diffusivity.

Diffusivity, Dm, has always been determined experimentally by
measuring the rate of flow of molecules between two regions represented
by flat planes, in a stagnant fluid. The stagnant system has been used in
order that no turbulence shall exist to cause eddy transfer. Dm also can
be estimated closely from semi-theoretical equations such as that of

Gilliland (10):

Dm=0.00h3 T3/2 ‘ [ 1 + 1]1/2
P (VII/3 +V21/3)d

M1 ”2

or by the more exact equation of Wilke (18) which will be illustrated
later. These equations were derived by considering the molecular kinetic
theory for gases and also the theory of osmosis in fluids.

If a slight motion of the fluid is considered, the distance traveled
by the molecules during the process of diffusion x is considered to be
perpendicular to the direction of flow. The diffusivity, therefore, will
represent the volumetric rate of diffusion per unit length of the distance
between two regions of different concentrations. This is comparable to
the mass transfer coefficient that represents the volumetric rate of mass
transfer per unit area through which mass transfer is taking place.

Let us consider a solid plane placed parallel to the direction of
flow of a fluid. .And, let this fluid be conshtuted.of two components one
of which can be shew by the solid. Then mass transfer of the absorbed
component will take place between the bulk of the stream and the interphase
between the fluid.and the solid. Part of this mass transfer is due to
diffusion and the other part is due to turbulence. Therefore, if turbulence

is stopped, the mass transfer will be solely due to diffusion and thus

19

um: = kc:
The distance x is a function of the geometry of the system. Accordingly,
it could be represented by the space in which the fluid is contained. In
packed beds, x is a function of the equivalent diameter which is expressed
in terms of the bed's volume and the surface area of packing.
zoo d.p € /l-€
Therefore, the terms Dm and kc; (1P 6 /l-€ are comparable for a stagnant

fluid in packed beds.

Dimensionless groups

 

The total resistance to mass transfer which is a function of fluid
turbulence as a whole resides in the laminar film, the turbulent regions,
and in the buffer region between them. was transfer across the laminar
film is determined mainly by the molecular diffusivity. Nbss transfer
across the turbulent and buffer regions in packed beds is dependent on
the motion of the fluid and its properties. Therefore, the mass transfer
coefficient is a function of the molecular diffusivity, the turbulence of
the flow , and the properties of the fluid.

kG=a(DmJU1p: € . dp’ l1)
Rearranging these terms into dimensionless groups, kG dp € /Dm (l- 6),

Re and Sc, the following relationship is obtained:

Wad-pl? = alfiljia . (Sc))

The term kG dp-C /DIn (l- C ) is the Sherwood number and represents the
ratio of mass transfer coefficient to molecular diffusivity.

The turbulent nature of the flow is properly presented by Reynolds
number Re. Re is a function of the velocity, density, and viscosity of
the fluid, and of a linear dimension representing the space through which

the fluid is moving. Therefore, in packed beds, the diameter of the

20

packing does not represent the conduit. Using the void fraction, the packing
diameter could be used properly in an effective modified Reynolds number
that is equal to

911’ ‘11) u p)1Re

“IT r”- 7‘1")== (1% ‘11—"
where (I'f/l- d dp) represents the equivalent diameter.
The properties of the fluid are well represented in the Schmidt number.

Sc == [.1/ ,0 Dm

The Schmidt number is comparable to the Prandtl number which represents

the properties of the fluid that are involved in the transfer of heat

through the fluid.
Pr = 01, u /k

From the definition of use transfer coefficient, when U and a are constants

_.§_ ___E’_£_
kc _ fife-W)

he can be evaluated from experimental data. It is very important to obtain
accurate measurements of U, a, L, and, most essential, y. y represents the
initial and final concentrations of the fluid. Equally important is the
accurate computation of the value of y*. y* could be the vapor pressure
for an evaporation operation such as drying, since the vapor pressure
represents the concentration of the active component at the interface

in equilibrium with both phases. When absorption is considered, as in

this experiment, y* is equal to zero if continuous and effective removal

of the component from the surface is accomptnhed by the solid phase.

Mechanism of flow in packed beds
In order to calculate Inc, from experimental data, the mechanism of
the flow has to be determined or assumed so that fdyf/Ay can be evaluated.

Several mechanisms have been suggested and investigated. Each one of

21

these mechanisms leads to a different way in which y changes from inlet
to outlet of the bed. In the following discussion, these mechanisms will
be summarized and [dyf/Ay will be evaluated. Throughout the whole
discussion, Ay will represent the driving force and dyf will represent
the concentration change in the direction of flow, or the change of
concentration of the fluid that is flowing.

(a) Piston flow: ‘This mechanism suggests that the whole bed is
divided lengthwise into an infinite number of infinitesmal stages along
which the concentration changes. In other words, the concentration of
the active component in the stream varies continuously along the bed in

the direction of flow. Thus:
dy Y - y*
_]r lnl

Ay Yo ' 3*

 

When the equilibrium concentration is very small, y*-= O, as in this

experiment, then

dyf _ y1
-«[N — 1n yo (6)

 

 

(b) Complete mixing: In this mechanism, the outlet concentration
is equal to the concentration of the stream throughout the bed. It
resembles the case where a stirrer is employed to assimilate the
concentration of the inlet stream with that of the fluid'body in a ,
container or a bed. Therefore, by this mechanism y is the same through-

out the bed and is equal to the outlet concentration yl.
Jf dyf yo ' yl

 

 

*
Ay Y - Y1
and when y* = O,
-fdyf - y—-—-—°-y1 (7)
AV - «-Yl

22

(c) Complete mixing in each layer: the theory of this mechanism
suggests that each layer of packing acts as a perfect mixer. The
concentration y in each layer stays constant throughout the whole layer
until the fluid enters the following layer. Therefore, if the bed is
constituted of one layer, the mechanism is identical with complete maxing
discussed above. For a bed constituted of n layers, the concentration

change on the individual layers is considered. When y*-= 0, then

 

de _ yo ' yl yl ’ YE yn-l ' yn
-.[Av"(-Y1)+(-Y2)+ +(-yn)

But

U dyr
kG: aL Av

 

and if U, a, L, and RG are constants, then

 

 

 

 

 

 

Yo-Yl = 3’1 'Y2 ___ ______ yn-i -yn
YI YQ yn
and
Y0 = yl =: y2 = ___ = yn-l
yl Y2 Y3 Yn
also
Y0 = (YO )l/n
Y1 Yn
or
i i
_ dyr = Q’i-i ' y1) = Z ( yi-l _ 1)

substituting for Yi-l by ( yo )l/n
yi yn

HI [dbl/“~11

 

 

23

If a layer of packing is assumed to be one diameter deep, n = __lL.

dp

(8)

 

£3- vi;- WWW]

(d) Forward diffusion: This mechanism includes the effect of eddy
diffusion and molecular diffusion on mass transfer in the direction of
the flow. It presents a mechanism that produces incomplete mixing but
not piston flow. In this mechanism, the components of the stream will
diffuse in different directions, including the forward direction. If
absorption is considered in a process, the concentration of the active
component will decrease and the concentration of the inactive component
will increase in the direction of flow. Therefore, according to the
theory of molecular diffusion, diffusion in the forward direction is
expected for the active component and in the backward direction for
the inactive component due to the direction of their respective driving
forces. .Accordingly, the velocity of the active component may be seen to
be greater than the average velocity while that of the inactive component
'will be less than the average velocity. Because of this, the concentration
of the active component in the stream which flows past a point in the bed
will'be greater than the concentration of the same component at that
point in the bed if axial diffusion did not occur.

The mathematical consideration of this mechanism may be approached
by considering a stream of gas passing a point in a packed'bed where
absorption is taking place (Fig. 1). Let N represent the volumetric
rate of flow of the active component at a point in the bed. Then the
rate of change of this volumetric rate in the direction of flow will

equal the rate of absorption.

21+

%— = 159 a A (Av)

If the forward movement of the active component in the stream is
considered, the transfer process in the axial. direction will be noticed
as a combination of three operations:

1. Transport associated with molecular migration of the
component; namely the forward molecular diffusion of the
active component when there is a change in its concentration
due to absorption.

2. Transport due to the relative microscopic motion resulting
from turbulence. in the fluid .

3. Transport due to the gross motion of the fluid across the
assigned point.

Accordingly, the volumetric rate of the active component in the axial
direction can be stated in terms of its eddy diffusivity caused by back
mixing in the space of the bed at that point, its molecular diffusivity,

and. its velocity.

N= [uy-(Dn+E)_a$XL_]A€ (10)

The second term on the right includes a statement of Fick's law of
diffusion and eddy diffusion. E represents forward transfer due to
longitudinal mixing and DI, represents forward transfer due to molecular
diffusion. The minus sign indicates that diffusion takes place in the
direction of decreasing concentration. The first term on the right side
of the equation corrects for the fact that Fick's law is strictly valid
only in a gaseous mixture that is at rest, , whereas motion of the fluid
is considered here.

If Equation 10 is differentiated with respect to the height or
length of the bed, assuming substantitally constant velocity for the

25

whole stream, and also constant diffusivity, then

%= [u-%—-(Dm+E) :3] A€ (11)

 

Equating Equatiom 9 and ll,

kGacm-e [u %—-(n.+E)$—]e

For the present experiment it is safe to assume constant velocity because
low concentrations are employed, or y is very small. Due to this, the
diffusivity also my safely be considered to be constant. Considering
the absorption in this experiment, y* = 0 and Ay = -y. Then the above

equation can be rearranged as follows:

[(%+E)%-u_% +kG .5; m] =0

To solve this equation, it is convenient to introduce the notation

of differential operators in which

me—

Then

[—(Dm+E)D2+uD+kG _E_]y=0
This linear differential operator of order 2 can be solved as a
quadratic equation, if u is again assumed to be substantially constant.

The roots of this equation are

1+ lIkGug—(Dml'E)

11" swung)
u

1- \[1+E_I‘G_$_(Dm+s)

1’2“ 2m. E)

 

 

 

 

m

 

6+

26

Then y = Cl erlL + c2er2L

at L = 0,
Yo = c1 + c2
atL=°<= , y=0then:
clerlL + c2er2L = 0
Since r1 is positive, then cl = 0

Then the true solution of the equation includes

 

 

y=c2er2L
or fhkga
dy_1- 1+?1T(Dm+E)
T_ 2 (Dm+E) (y)
__E_

By integrating this equation between the limits

y=yo,y=ym L=0andL=L.

 

tha
yn 1- —\[1+ €112 (Dm+E)

Yo= 2 (Dm+E)
—T—

 

1n (12)

 

To simplify this equation, more correlations among the different
terms are required. Kramers and Alberda (17) showed that the eddy diffu-
sivity is related to the superficial velocity and the particle diameters
in the bed. This relationship is expressed in a special type of Peclet
number which includes both the molecular diffusivity and the eddy

diffusivity.

pe=_&_u__
Dm+E

Kramer: (17) reported that this special Peclet number is approximately
equal to 2. This value of 2 for Fe was validated also by experimental

work of other investigators (21+) .

27

Accordingly, Equation 12 is reduced to the following form:

2 kG a L
yn 1 - W+ —T (dP/L)

yo Udp/L)
Rearranging this equation,

 

 

 

1n

 

2 kG a L yn ) 2
mg— (dp/L) - [(€(dp/L)1n(yo) - 1 - 1
But from the definition of the mass transfer coefficient for an absorption

process,

 

kG a L .. Jf dyf

Then

 

- f—g—f— = 5"qu [(€(dp/L)ln(::)-1)2-1] (13)

These different mechanisms, as presented by Equation 6, 7, 8, and 13
give different values for kG. However, under certain conditions they
approach each other. The nature of these mechanisms is illustrated in
a diagram (Figure 2) showing the path which the concentration changes
assume along the height of a packed bed. It can be seen that complete
mixing is attained by the flow path of layer perfect mixing if only one
layer of packing is used in the bed. Also Kramersand Alberda (17)
illustrated the approach of layer perfect mixing to the forward diffusion
mechanism if high beds are considered. From the definition of piston

flow and the definition of layer perfect mixing, the limit of

 

 

L
y
_£L_. ( ° )dp - should approach In yn when an infinite
dp 3’1 3’0

number of layers is employed. To prove this let yO/yl = i and
tip/L = m. At infinite stages, mat-O. Then the right side of Equations

6and8wiube—1niand-Cim-D/m. Tofind lim _i"'-1,1etim=z
mweo m

28

 

Then
m ln i = 1n Z
ln 1 = _%i. _§§_. (by differentiating with respect to m)
dZ _ m
Tar-i 1“
Also dm “
‘537’— 1
Then lim _(im - l) =-im ln 1 = _ 1n i
m-so m

Referring to the work of Kramers(l7), we found that if complete
mixing is assumed at each layer, the results approach those obtained
when forward diffusion is assumed and the number of layers is increased.

Therefore, it can be stated that Equation 6, 8, and 13 should give
results approaching each other as the number of layers of packing is
increased. To demonstrate this statement, Table 2 was prepared to include
numerical values of the driving force for a given ratio of initial and
final concentrations. These values were calculated as y0 ' yl

dyr
“fly
for given yl/yo, using Equations 6, 8, and 13 for one layer and five

 

layers.

29

 

 

 

 

 

 

Distancegl "
NO cm3Zsec ,/// Rh cm3/sec
on ,. 4%??? yfn -e:

 

Area A -JI | |/// T
yo concentration y. yh

Figure 1. Forward Diffusion in a Packed Bed

 

 

 

 

 

 

 

 

 

37:0-
y=yf 1. piston flow
2. perfect mixing in each layer
3. forward diffusion
h. complete mixing
1
y
‘2
3
an _. h
\
I I
O L
L

Figure 2. The Effect of Different Machanisms on Absorption
in Packed Beds

 

 

30

TABLE 2
Theoretical data representing the driving force as calculated by different
methods for given absorption rate. These methods are selected for systems*

of one or five rows of packing, assuming different mechanisms .

 

 

 

 

 

forward forward complete complete
2‘: piston diffusion diffusion mixing mixing
yo flow 1 row 5 rows 1 row 5 rows
0.9 0.9h8 0.928 0.9% 0.9 0.939
0.5 0.722 0.63h 0.702 0.5 0.672
0.1 0.391 0.268 0.358 0.1 0.308
The driving force, yn ‘ 3’0
y* - V
For piston flow, _ f dy = * 1n Yn
y - y yo
For forward diffusion, f g L €- dp ln yn 2 J
_ = - . """""" - l "l
y* - y 2 E a; [ ( I'— yo )
For complete mixing _ 511 = _ L [ ( Yo )dp/L _ 1]
y - y ‘19 ’11

 

* A fractional void of 0.1+ was assumed.

yo was assumed = 1%.

EXPERIMENTAL WORK

Design of Apparatus
The usual method for calculating the mass transfer coefficient from
test measurements is by applying the mass transfer rate equation:
k6,: __H_.d/l:£HL____
81- y-T"
Use of this equation necessitates making a few assumptions in order to
validate its application to a particular set of test data. These
assumptions may be summarized as follows:

(a) Hall effect: A.uniform and constant velocity across the packing
has been assumed by past investigators, even when the particles of the
packing were randomly distributed in the column. The velocity across a
section through the bed will not be uniform unless packing distribution
is uniform. This problem‘becomes particulary troublesome at the wall of
the column where the larger spaces give higher velocities and flow rates.
One measure that has been considered occasionally in the past to reduce
the wall effect to a minimum was the use of very small particles as
compared to the column diameter. This reduces but does not eliminate
wall effect.

Thus an average flow velocity, as used in the mass transfer rate
equation, does not represent the effective velocity in the packed column.
Better representation would be expected if a uniformly arranged cross
section was employed.

(b) Surface area of packing: The use of a homogeneous packing
throughout the bed would be desirable, especially when an average effective

surface area per unit volume of the bed needs to be considered. The

32

surface area of the packing per unit volume of the bed can be measured
or calculated if the packing is constituted of particles of nearly the
same size and shape, oriented uniformly throughout the bed. In this way,
the flow space can easily‘be determined.

(c) Approach to saturation (end value of y - y*): The concentrations
usually measured in mass transfer experiments, (y), are the inlet and out-
let concentrations. In beds with many rows of packing, especially at low
Reynolds number, downstream concentration approaches saturation (y*) . This
makes it difficult to measure accurately the downstream driving force
(yh - y*) at the end of the bed. However, it should be recalled that the
height of the bed is a strong factor in bringing about the saturation
stage. This is due to the extension of area of contact between fluid and
packing regardless of the mechanism and its effect on mass transfer.

Some investigations attempted to correct this weakness by using a dilution
method for packing, i.e., by distributing active packing throughout a bed
of inert packing. This method results in a problem of radial diffusion
that must be considered.

(d) Surface temperature: When (yn) approaches (y*) it is very
important to know y* accurately. In almost all pioneering experiments
that involved gas film control systems, water was evaporated into a gas
stream, usually air. Surface temperature was assumed to be the wet bulb
temperature, but no valid proof was given as to whether these temperatures
are actually the same. Hence a slight error in the temperature could
reveal a larger error in the determined driving force. The same could
be true, although to a lesser extent, when naphthalene was evaporated

into a gas stream.

33

In a more recent work, Thoenes and Kramer$(35) employed a unique
techniqueto estimate the mass transfer coefficient in which they
eliminated.most of the sources of error. Their work consisted of
evaporation of'a liquid in a single packing embedded in a group of
particles of the same size. The test was carried out at different
orientations of packing in the bed. The surrounding particles formed
a short ~«bed; placed in a special way to eliminate wall effect. Hew-
ever, the existence of only one moistened particle in a large volume
of the bed could produce an effect of radial diffusion. Another problem
that could have arisen from the use of such system was over- or under-
saturation of the particle with liquid. In case of over-saturation,
for example, the liquid may overflow over the surface, and thus be carried
by the stream of gas. For this reason, the mass transfer rate mtghd:not

be determined accurately.

 

Design of Apparatus for Present Experiment

This experiment was planned to measure the mass transfer coefficient
for a gas-film-controlled.system in the laminar region. In the light of
the above discussion, a gas mixture of known or easily measured properties
was passed through a packing where one of the components was absorbed
partially. In order to facilitate measurement of the gas concentration,
ammonia was chosen and simple acidAbase titration was employed. The
inert gas was selected on the basis of its known properties. In order
to insure ammonia absorption on the packing, the packing was coated with
a layer of a metallic halide that forms an ammonia complex. After
studying the thermodynamics of solid salt-ammonia reactions (Table 3,

Figure 3): cuprous chloride was chosen as a coating for the packing.

3h

The packing was chosen on the basis of its porosity and surface roughness.
Spherical particles of approximately equal size were used. The rough
surface was a factor to insure good coating.

The bed itself consisted of one active layer between two inert
layers of packing. Each layer consisted of twelve spheres, oriented in
a uniform section. The inert layers were used to eliminate end effect
by providing a‘bed-type flow pattern.

Hall effect was eliminated by surrounding the bed with a wall of
paraffin.wax, in which about half of each particle in the outermost row
of each layer was embedded (Figure h). Accordingly, each layer of
packing contains four whole spheres in the center, surrounded by eight
spheres partially exposed to the stream.

The structure of these layers was organized in such a way that they
fitted tightly on top of each other. The inactive layers were removable
and could.be placed in the same pattern after the active layer had'been
regenerated. In this way, reproducibility of packing conditions was

secured.

Experimental materials and equipment

 

(a) Fluid: The fluid was a gas mixture prepared in a cylinder
fitted with a stainless steel valve and pressure gauge. After the ammonia
was introduced, helium was added and the mixture was left for 72 hours or
longer to secure effective diffusion and complete mixing before use.
Different concentrations were made during the program. The anhydrous
ammonia and helium were obtained from Mtheson Company, Inc.

(b) Packing: The packing was made of alundum balls (5 Tumble: S)
obtained from Nbrton Abrasive Company. The balls were screened and then

selected on the basis of sphericity. Four reading were taken for the

TABLE 3

Dissociation Pressure of Copper Ammines (16)

35

 

System Temperature °K Pressure Atm.
Cu C12 - NH3 = Cu C12 + NH3 381.8 0.00h1
h26.6 0.088
h87.8 1.089
Cu Cl - NH3 = Cu 01 + NH3 305.8 0.0000h7
336.0 0.00066
3h9.0 0.00186
Cu 01 . 1.5 NH3 = Cu 01 - NH3 + 0.5 NH3 305.8 0.033
336.0 0.2h1
3h9.0 0.532
Cu c1 - 3 N113 ;_ Cu 01 . 1.5 NH3 288.1 0.200
+ 1.5 NH3 301.6 0.u21
305.8 0.u50
307.6 0.568
315.3 0.822

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m.
m / m...
.1... / u
. / m
m / h
x m mo i // //
is //r /
- / m /,// 3
[Z
24
m /
3 / / mm
m. //11/////r .I/II/l/l 59
NW / 4/
//
7 m
0 1 2 3 i. a.)
1 1 .m .m w m m

monogamoapo enammonm

Temperature °K

37

diameter of each ball, using a micrometer. An average diameter for the
selected group was Obtained with an average deviation of 1.89%.

The active packing was prepared by washing the balls and soaking
them in a slurry of cuprous chloride at room temperature under a—elight
vacuum. Then the balls were rolled on filter paper to absorb part of
the moisture, and were placed in a desiccator to dry. For use in the
test, the balls were placed in a prepared ring of wax to form one complete
layer.

The inert packing layer was prepared by placing twelve balls in the
desired geometric formation. These balls were.aeafied to each other by
placing a resin plastic compound at the points of contact. When the
resin dried after a few hours, a solid geometric figure of balls resulted.
Their surface area was not affected by the resin. To prepare the wax
ring for these layer (as well as for the active layer), hard paraffin wax
USP of Fisher Laboratory Chemicals was melted and poured into a prepared
structure made of a glass ring with inside diameter the same as the column,
and a glass puncher was placed in the middle. This puncher was made of
twelve glass tubes arranged in the same geometric figure as the desired
pattern. The diameter of each tube was equal to the packing diameter.

The height of this structure was equal to the diameter of each ball.
After the wax had cooled a little, the puncher was pulled out carefully
and the prepared ball layer was inserted in. A sharp knife was used to
prepare the whole block for the final shape.

In the case of the active layer, the wax ring was prepared by the
same method except that after the wax was solidified completely, the

inert ball layer was removed and the active balls fitted in.

38

(c) Absorption column: The column.was made of a glass tube with
ground glass fittings at the ends to house the inlet and outlet tubes.
The column had a thermometer near its inlet to measure the temperature
of the stream, and an open end manometer at the outlet. The manometer
served as an indicator for the change of pressure drop while changing the
stream to and from the analytical train.

The packing could be placed very tightly in the column. A.glass
sieve at the inlet of the column helped in distributing flow across the
column. The bed was placed approximately four ball diameters below the
outlet end of the column. The volume of the column after packing, plus
the volume of the outlet line up to the analytical train, was 70 milliliters
(for the final set of runs). The column was fitted with a three-way
valve at its entrance, to direct the flow from the gas mixing cylinder
into the column or to a washing bottle filled with dilute hydrochloric
acid solution. This bottle insured the absence of ammonia from the
surrounding atmosphere of the analytical train while the flow rate was
adjusted.

(d) Analytical apparatus: The analytical train consisted of two
125am1 washing bottles placed in series with a stainless steel "Precision"
wet test meter. The first bottle, connected directly to the absorption
column outlet, served as a sample bottle. It was filled with 35 milliliters
of dilute hydrochloric acid and.an indicator. The second bottle was a
detecting bottle, and contained dilute hydrochloric acid and an indicator
to detect any escaped ammonia before the gas passed on to the wet test
meter. Both bottles were furnished with fritted glass inlets immersed

in the acid to insure complete absorption of ammonia by the acid.

39

TABLE A

Properties of the System for this Experiment

Packing diameter, dp = 0.726 cm.
Number of particles per layer = 8
Calculated void fraction = O.h07.
Experimental void fraction = O.h55.
Theoretical void fraction = O.h76.

Cross sectional area of the column = 3.73 cme.

.__..=1
dp

Average density of the fluid = 0.000162 g/cm3 at 27°C.
Average absolute viscosity of the fluid = 0.0001965 g/cm. sec. at 27°C.

Diffusivity of the fluid = 0.926 cm2/sec. at 27°C.

Effective volume of the absorber

275 cc for runs of sets A and B.

Effective volume of the absorber

70 cc for runs of set C.

Initial concentration of ammonia

0.913% for runs of sets A and B.

Initial concentration of ammonia 0.1875% for runs of set C.

to

(e) Analytical reagents: The hydrochloric acid as well as the sodium
hydroxide were prepared from stock solutions. The prepared auxihnn hydr-
oxide was titrated against a standard solution of sulfuric acid. The
normality of the hydrochloric acid was checked by titrating against the
freshly standardized sodium hydroxide. In all these titrations methyl
red was used as an indicator.

Due to its sharp change of color in the presence of ammonia, cochineal
indicator was added to the dilute hydrochloric acid of the sample bottle
prior to the addition of ammonia. The Cochineal indicator was prepared
from ground dry cochineal. About 10 grams of the grind were extracted
with 100 milliliters of ethyl alcohol for five days. Three hundred
'milliliters of distilled water was added and the solution was filtered
before use.

The end point of the acidebase titration was taken when the orange
color of the solution changed to purple red in the presence of cochineal.
This end point was checked with that of methyl red. The properties and
the specifications of all the materials and equipment used in this work

are listed in Table h.

:Egperimental Procedure

 

Prior to the start of a run, the inlet gas was analyzed to determine
its concentration. This was done by using the following procedure without
the use of the bed in the column.

The hydrochloric acid sample was measured and placed in the sample
'bottle :27flgféeh drops of cochineal indicator was-added and the total
volume was brought to about 35 ml. After the system was checked for
leakage, gas was allowed to pass from cylinder "5' to chamber 8. After

the flow rate was adJusted and kept steady, gas was introduced to the

kl

 

 

/

 

 

 

 

A

F—

e
I
-_-—-—_

f

7 _

 

 

 

 

1h cm

 

 

 

 

“~‘n—e---——- ——-

 

 

 

 

 

 

 

 

 

 

 

 

l

thermomemter outlet
gas inlet ]t

 

 

L I '/
.|."'..I..rv A— ,'
/

gas outlet
manometer outlet

wax layer
inert packing
active packing
inert packing
glass sieve

     

 
  

i@®'®7®’
, I
>._~ A A A.

Cross section of
the bed

Figure h. Diagram of the Packed Bed

 

 

h2

column l'by using valve 6, at the same moment the time was recorded.
Valve 7 had been adjusted to allow the gas? to flow to the test bottle 2
before the run.began. .After the desired quantity had been recorded on
the wet test meter h, valve 7 was turned to shift the flow into chamber 9,
and the time was recorded.

The gas volume was recorded on the wet test meter h by taking readings
before and after each run. The temperature was recorded on thermometer 10,
to be used in determining the gas properties, and on thermometer ll of the
meter to be used in determining the mass flow rate. manometer 12 was
checked for any severe change in the pressure inside the system, especially
when valve 7 was shifted from one direction to the other.

The sample bottle was then removed, and the content washed into a
250 ml beaker for analysis. The analytical procedure used was a standard
method of acidebase titration to determine the amount of HCl reacted with
ammonia.

The same procedure was followed when the bed was placed in the column.
During all these runs, the atmospheric pressure was recorded. The detecting
bottle 3 was checked periodically, and showed that all the ammonia had‘been
absorbed in the sample bottle after passing through the column.

In order to insure that no parts of the apparatus would absorb ammonia
except for the packing or the acid in the sample bottle, a blank was run
by passing gas through the apparatus with everything present except the
active layer of packing. This test showed that no absorption took place
in the column.

Three sets of runs were conducted. The above outlined procedure

refers to the final set.

#3

 

 

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(a) first set ofruns (A): Itwss noticeddurimgtheruns ofthe
first set that for identical successive runs, where gas was flowing
continuously through the packing between the runs, different values for
absorption rate (or yl/yo) were obtained at the same flow rate (Figure 7).
It was thought that this phenomena was due to the surface saturation of
solids where solid diffusion became controlling. The apparent mass
transfer coefficient he would then be a function of time at constant
Reynolds number.

Duringthis setofruns, thegsswsssllowedtoflowthroughthe
packing before actual run was started. The spproxinte volume of gas
sampled for each run was one liter with an initial concentration of 0.9135.
mumsrunofthis setwssndsonsfrcshpscking.

To check this assumption of gas film versus solid diffusion control,
the scoond set of runs were conducted in the same apparatus.

(b) Second set of runs (B): Two runs of this set were nde separately.
on freshly prepared packing. Short successive but fast samples were taken
for each run at constant flow rate. The first run was conducted at a flow
rate of 0.0069 liters per second where four samples totaling 1.315 liter
were drawn in 190.8 seconds. The second run was conducted at a flow rate
of 0.02“. liter per second where four samples totaling 1.061 liters were
drawn in h3.5 seconds. Table 8 includes the original results of these
runs.

The results of these runs, as they are presented in Figure 6 indicate
that the age of the packing is important “Wt“ absorption rate.
Also they indicate that renewal of the packing no be necessary before
achmm'mmmammnymminm
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The effect of time on surface saturation.dcde it necessary to decrease
the initial concentration. Also to avoid saturation, small sample of gas
is more desired.

The size of the sample should be at least equal to size of the absorption
apparatus. This condition is necessitated by the fact that the volume considered
should be that of the gas absorbed and not of the air that initially existed in
the apparatus. For this reason, the first sample of the second run of this
set is discarded due to the fact that its volume is 0.1h6 liter while the
absorption sections volume is 0.275 liter. This sample could have made only
a portion of the leaving gas.

(c) Third set of runs (0): According to the result of the second set
of runs, the following recommendations were followed for this set of runs:

1. Initial concentration was reduced to 0.1875%.

2. Absorber sectionh volume was reduced to 70 ml, (using a shorter
column that housesa wax block'besidesthe packed bed).

3. Samples of about 0.5 liter were used.

1+. First runs on freshly prepared packing were considered for
calculating mass transfer coefficient.

5. Other runs were considered for comparison.

#6

TABLE 5

Original Experimental Data of This Work (first and second sets of runs)

 

 

 

Barometric Outlet Gas (Hel__ Duration Analytical Solutions

Run Pressure Temperature Volume of’Run fiElgTVBl. NaOH Vol.
mm C° 1. seconds cc cc

All 735.5 27.0 1.003 58.0 10.0 39.30
A12 735.5 27.0 0.991 1h.0 10.0 3h.10
A13 736.0 25.5 1.072 51.0 10.0 3h.15
Alh 736.3 23.9 1.03h 21.5 6.0 1h.h5
A15 7h5.8 2h.h 0.903 13.2 6.0 15.90
A21 735.5 28.0 1.000 32.8 11.0 h1.30
A22 7h5.6 27.2 0.8h2 200.2 8.0 25.10
A23 7h5.6 27.5 2.038 --- 15.5 39.20
A2h 7h5.5 26.3 1.025 3h.h 8.5 25.10
A25 7h5.5 26.5 1.058 67.2 8.5 23.70
A31 737.5 26.3 0.96h 27.6 9.0 30.25
A32 737.5 26.h 1.027 16.0 6.8 17.60
A33 737.5 27.7 1.060 18.2 6.0 13.85
A3h 737.2 27.6 1.019 10.0 5.9 13.37
A35 737.2 27.6 0.998 7.6 6.0 1h.05
Ahi 737.3 27.6 1.015 132.9 6.0 19.h0
Ah2 737.1 27.3 0.992 21.2 6.0 15.05
Ah3 738.0 29.5 1.007 11.8 7.0 19.80
Ahh 738.5 28.2 0.980 13.2 6.0 1h.5h
A51 7h0.6 2h.7 1.000 52.2 6.5 19.80
A52 7L0.5 26.5 1.008 72.6 6.0 17.20
A53 7h6.0 26.2 1.020 19.2 8.0 21.50
Ash 7A6.0 26.75 1.061 38.h 8.0 21.h0
A55 7h5.1 25.1 1.06h 3h.0 8.0 21.50
A56 7h5.2 25.0 1.071 10.0 8.1 20.20
310 7h2.9 26.2 0.300 h5.0 6.0 25.20
311 732.9 26.2 0.3h0 --- 6.0 23.60
312 7h2.9 26.2 0.315 M5.0 6.0 22.50
313 7h2.9 26.2 0.3u5 50.u5 6.0 21.85
32 7h3.9 25.6 0.1u6 --- 6.1 25.65
320 7h3.9 25.6 0.363 13.8 6.0 23.h5
321 7h3.9 25.6 0.257 11.0 7.5 29.60
322 7A3. 25.6 0.295 12.5 6.0 22.30

1+7

TABLE 6

Original Experimental Data of This Work (final set of runs)

 

 

 

Barometric Outlet Gas (He) Duration Analytical Solutions
Run Pressure Temperature Volume of Run HCl Vol. NaOH Vol.
mm C° 1 seconds cc cc

011 7h2.3 21.0 0.510 no.2 2.0 7.60
012 7h2.3 21.0 0.h88 39.2 2.0 7.30
021 737.0 22.8 0.580 26.3 2.0 7.15
022 737.0 22.8 0.635 --- 2.0 6.75
023 737 0 23.2 0.h5h 10.0 2.0 7.35
02h 737.0 23.2 0.605 6.1 2.0 6.80
025 737.0 23.2 0.531 16.8 2.0 7.25
031 7h1.9 22.0 0.h00 --- 2.0 7.h5
032 7h1.9 22.0 0.h00 -—- 2.0 7.35
033 7h1.9 22.0 0.h96 9.3 2.0 7.15
03h 7h1.9 22.0 0.335 1A. 2.0 7.55
0A1 73h.3 21.8 0.5h5 19.0 2.0 7.60
Cue 735.3 21.9 0.355 12.5 2.0 7.85
051 735.h 22.1 0.hh5 1h.0 2.0 8.05
052 730.5 23.5 0.515 12.5 2.0 7.h0
053 730.5 23.5 0.u93 23.7 2.0 7.25~
061 735.1 20.8 0.h60 10.8 2.0 7.75
071 735.2 21.1 0.h8h 9.0 2.0 7.55
081 735.1 22.0 0.u76 6.0 2.0 7.60
091 735.1 22.0 0.h88 5.0 2.0 7.55

#8

METHODS OF CALCULATIONL

The calculations include the computation of concentration y, surface

area of mass transfer a, void fraction in the bed 5', the height of the

bed L, particle diameter dp, and the volumetric rate of gas passing

through the bed v. It also includes the calculation of such dimensionless

groups as Reynolds number Re, a mass transfer factor Sh (the Sherwood

number), and the Schmidt number Sc from the physical properties of the

bed and the fluid as well as the experimental measured data.

(a) Calculation of y: From the original data of the experiment, the

following information was obtained.

1.

2.

\0 CD4 O\U‘I FL»

10.

11.

Initial gas concentration of ammonia yo.

Volume of hydrochloric acid used in the sample bottle 28
milliliter.

HCl normality b8.

Volume of sodium hydroxide used for back titration zb milliliter.
NaOH normality bb.

Duration of run 6 seconds.

Flow rate of exit gas v liters per second.

Exit gas temperature in the wet test meter T1 °K.
Barometric pressure P in mm of mercury.

Water vapor pressure p in mm of mercury at temperature T1.

Gas temperature in the bed T °C.

Then, for the gas leaving the bed, the following calculations were applied.

Mbles of ammonia = (baza - bbzb) x 10'3

1&9

Moles of helium = 37:71. . 423.12..

when R is the ideal gas constant = 62.361 ms lit./°K mole
(bang-bbzb) x 10-3
(baza'bbz'b) x 10'3 + (P-p)( 60/33

 

y =

(b) Calculations of bed properties:

1 . Equivalent diamter for the bed. The equivalent diameter is
represented by a linear function based on the packing diameter and space/
solid ratio.

equivalent diameterc-c € dP/l-E
In this work, the spherical packing was oriented uniformly in cubic
arrangements . Therefore, the bed could be divided into cubes each of which
contain a solid sphere whose diameter is equal to the side of the cube.

Hence :

 

€ __ cube volume - sphere volume
- E " sphere volume

6‘ = “93 ' 7po36 ___, 0.h76
r7? ——T—l—,T.p/6 ‘67s?

Then, the equivalent diameter = 0 . 91 dp.

H

01‘

2. Actual space volume of the bed. The measured actual void is

0.165. This value is different from a theoretical value of 0.14.07 which is

based on the presence of four whole particles and eight half particles in

each layer. The difference my be due to the ineffective technique in wax
wall formation.

mscd on the actual value of € , the actual velocity is evaluated as

follows :

U
“=‘€—=—F—0-55

50

3. Effective surface area of the packing. The surface area per

unit volume of bed if obtained as
2
77'dp§&5

Using the actual void value,

a

6(0-5h5) = 3.27
dp
(c) Calculations of fluid properties: For a mixture that has less

3:

than 1% ammonia and more than 99% helium, the density as well as the
‘viscosity of the mixture could be considered as that of helium. However
the diffusivity was calculated using empirical equations. Two methods
were used to calculate the diffusivity for the sake of comparison.

1. Gilliland equation (10).
3/2 l 1
D = 0.00h3 T '\/____.+.___.
m P (VII/3+ v21/3)2 M1 M2

P = 1 atm.

 

 

v1 h/0.126 = 31.8 cc/g.mole (helium)

v2 10.3 + 3 (3.7) = 21.6 cc/g.mol (ammonia)

(V11/3 + Val/3)2= 35 25

When the temperature is taken as the average temperature of the
gas in the bed for all the runs, 27°C, then at atmospheric pressure,

_ 0.00h3 (300)3/2 1 1 1/2
D" ‘ 35.25 OTI“*'TT7'

Dm = 0.353 cm2/sec
2. Hirschfelder equation (12).

_ B T3/2'VRNh.+ M2)/M1M2
" P (r12)? W1”) (1 -A>

 

 

This equation was simplified by Wilke and Lee (18). Referring to that work,

the following constants were Obtained from their tables.

51

6‘12 = U6‘1 é‘2

 

 

€ .
—-—E$3—l=\/(315)(6.03) = h3.6
.5J3——— = 6.87 for T = 300°K
€12
w1(1) = 0.h1 for KT = 6.87 as obtained from Wilke's table (18).
- 2512
/ /
rig = r1 2 r2 = 2.62h2+ 2.7 = 2.662
(1 - A) = 1

(9.29 x io-h) (5200) (h + 17)1/2

D” (2.662)2 (O.hl) E x 17

 

Dm = 0.926 cm2/sec

By comparing experimental diffusivities for different systems with those
calculated by the above mentioned methods, it was found that the maximum
deviation for Gilliland's equation is h6.8¢ while maximum deviation for
Wilke's method is 16%. The average deviation of the latter method is 3.9%
as compared to 20% for the first method.

Accordingly, the diffusivity value for the present system was taken
as 0.926 cm2/sec.

(d) Calculations of the dimensionless groups:

1. Schmidt number.

 

 

Sc = ‘1
DD...
3 _ 0.1965 x 10-3
6 “ -3
O.l62.x 10 (0.926)
80 = 1.31

Density and viscosity values were the experimental values for helium (27).

2. Reynolds number.

Re __ Edp ,DU
1"; _ (1'€)#€

 

U = _%_.= _X§¥7%92_.= 268.3 v cm/sec

52

 

 

 

Re _ 0 1 d l3 268. 3 v
r?“ '9 P' 72— THIS—5"
= 0.91 (0.726) (0.162 x 10-3) , (268.3 v)
(0.1965 x 10'3) 0.355
II§%_=322V
3. Sherwood number.
d k
but k0 = U fay: 3268. v "Jf dyr

and.when L/dp= l and Dm— = O. 926 cm2/sec, then

Sh _(0. 91)(0. 726)(268. 3-v) ["Jf dyr ]
(3 27)(0 926) “ ‘

= 58.l+v [-jiyj]

Therefore Sh/Scl/3 = 53.3 v [_ J{-::f ]

 

 

 

-‘Jf—:;f is calculated by using Equations 6, 8, or 13,
according to the expected mechanism.
Sh/Scl/3 is a mass transfer factor used to represent the data of
this work as well as the experimental data of other investigations.
(e) Calculation of Sh/Scl/3 for other data when reported differently:
1. For data reported as Jd.
Sh/Sc1/3 = 6 Re Jd/|—€
2. For data reported as Je.
Sh/Sc1/3 = Je R€Vt'€'

6 (Sc)173

3. For data reported as HT.

6' SgéRe
'1??? 3 ' HTa

In all these cases, Re was considered as dP GAL( , based on superficial

Sh/Scl/3 =

 

 

53

velocity, or _
Re a U
(r) Calculations of sh/3c1/3 and Re/(l- e ) for the theoretical data:
Theoretical data were reported aslu vs. 6 where ( 6 = Re.Pr.d/L).
These data were considered as he d/n... vs. (Re. Sc. d/L) fqr conduit, where

d _ 1: volume of conduit
" We area

Inpackedbed,

c v 6 3 E -
see—:11: “L—LLW 7;,19 4‘ 8 =—?— «re—.5
4"16'"1—§'€dp

 

 

Then
' d
w. *6 ,_}.ng.,.%. if-.. .m.
Therefore Sh/Sc1/3 e' (1.5/Sc1/ 3) - 1‘0 <1
"T,

h 0Ac _(:d2 u d d
6=W4—%=+OB’OT
Re

Bk 6 2 up c_ 1L2 c E
6(T)TI—T?-dp] oTols‘ -(2—) Wes. -

Re is based on the average velocity in a parallel direction to the

 

wall. If a packed bed is considered, the actual velocity in the bed does
not represent the average velocity. The vector component of the actual
velocity would represent the average velocity more accurately .

For an idealized bed (Figure 11) the average velocity is (1.1!»lh u).

Therefore if the void fraction is h0%,

 

3
‘fTeg’='(‘i.6‘)"1.uluEP5c '6

=2.38—a%--—8C-Z—=2.38 $.91.—

TABLE?

Degree of Absorption for Runs A and B~
(first and second sets of runs)

Outlet Gas

 

 

Run Flow Rate (v) Concentration yl
liter/second Y1 mole o/o yo

A11 0.0173 0. 538 0. 589
A12 0.0708 0.853 0.93h3
A13 0.021 0.761: 0.8368
Alh 0.0h8 0.809 0.8861
A15 0.068h 0.8377 0.9175
A21 0.0305 0. 703 0. 77
A22 0.00h2 0. 7126 0. 781
A23 00:39 0.791 0. 867
A2h 0.0298 0.720 0. 789
A25 0.0157 0.785 0.860
A31 0.0319 0.763 0.8357
A32 0.06h2 0.816 0.9266
A33 0.0582 0.831 0.9102
A31: 0.1019 0.858 0.9313
A35 0.131 0.86». 0.963
Ahl 0 . 0076h 0. 553 0. 6057
Ah2 0.0h68 0.813 0.8905
Ah3 0.0860 0.81:0 0. 9200
AM 0.07112 0.89 0. 9299
A51 0.0192 0.6725 0.7366
A52 0.0139 0.6995 0.7662
A53 0.0531 0.816 0.8938
Ash 0.0276 0.793 0.8686
A55 0.031 0.787 0.8620
A56 0. 1071 0.86h 0.9h63
311 0.0069 0.363 0.3976
312 0.0070 0.627 0.6870
313 0.0068: 0.691: 0.760
321 0.02%. 0.57 0.6213
322 0.02% 0.7063 0.773

55

TABLE 8

Experimental Data of the Second Set of Runs to Illustrate the Time Effect
on Surface Absorption of the Packing at Constant Flow Rate

 

 

Run Flow Rate Gas Volume Total Time of yl Sh/Sc1/3
liter/sec Passed in liters Contact/seconds IQJBZ
310 '0.0069 0.300 h3.5 0.055 1.035
311 0.0069 0.6h0 92.8 0.363 0.3h0
312 0.0069 0.955 138.5 0.627 0.138
313 0.0069 1.315 190.8 0.69h 0.101
32 0.02hu 0.1u6 5.98 0.100 2.880
320 0.02hh 0.509 20.85 0.366 1.190
321 0.02hu 0.766 31.h 0.570 0.613
322 0.02hh 1.061 h3.5 0.706 0.332

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Degree of.Ahsorption for linal Runs

TAEEE 9

(Third set of runs)

57

 

 

Outlet Gas
Run Flow’Rate (v) Concentration yl
liter/second y1 mole o/o yo

011 0.0127 0.1078 0.575
012 0.0125 0.153 0.816
021 0.0205 0.1h82 0.790
022 0.02205 0.1759 0.938
023 0.0h5u 0.1601 0.85h
02h 0.0992 0.1811 0.966
025 0.0316 0.1h96 0.798
031 0.080 0.1628 0.868
032 0.080 0.179h 0-957
033 0.0533 0.171% 0.91h
03h 0.0229 0.1788 0.932
chl 0.0287 0.1175 0.627
0&2 0.028h 0.1369 0.730
051 0.03178 0.0686 0.366
052 _0,0h12. 0.1575 0.830
061 0.0h26 0.1153 0.615
071 0.0538 0.1h21 0.7579
081 0.0790 0.135h 0.7221
091 0.0976 0.1h51 0.7739

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59

RESULTS

Experimental results

 

a. Table 5 contains the original experimental data for the first and
second sets of runs (A and B runs) for the present experiment. Table 6
contains the original experimental data for the final set of runs (C runs)
of the present experiment. Original data includes temperature and pressure
readings of each run. Gas volume, and the duration of each run, and acid and
base volumes used for the analysis were also included in the original data.

Five new packings were used during the first set. Two new packings
were used for the second set and nine packings were used for the final set.
The number of runs conducted on each packing varies from 1 to 6.

b. Tables 7 and 9 contain flow rate, final concentration, and fraction
of ammonia absorbed in each run (see section on methods of calculations).
Figure 7 shows degree of absorption as a function of flow rate. The lower
curve refers to the first runs that were conducted on freshly prepared
packing. The upper curve refers to the rest of the runs.

0. Table 8 contains data of the second set of runs (set B) that was
designed to show the time effect on rate of absorption at constant flow rate.

Figure 6 expresses mass transfer rate as a function of time. Curve 1
of Figure 6 refers to data obtained at low flow rate and curve 2 refers to
data obtained at a somewhat higher flow rate. The shape of either curve is
not clear at low values of 6 because no reliable samples were obtainable

for short periods of time.

TABLEIO

Experimental mas Transfer Data of This Work Presented for Different
Mechanisms at Different Flow Rates

 

Re Sh/Sc1/3 Sh/Sc1/3 3h /Sc1/ 3
Run '17:— Piston flow Complete mixing Forward diffusion
All 5.57 0.h88 0.6M. 0.51:8
311 2.22 0.3% 0.558 0.1+O8
321 7.85 0.613 0.783 0.678
011 809 0.375 0.500 0.1t18
021 7.10 0.278 0.312 0.285
chi 9.25 0.716 0.911 0.792
061 13.73 1.108 1.h25 1.228
071 17.33 0.795 0.916 0.8h5
081 25.15 1.37h 1.623 1.16
091 31.15 1.31:0 1.520 1.115

m m a m m mOH

61

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masses oooaasoofl . o

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0]

CD \Ou‘ad' Mk.

0

62

d. Table 10 contains the final data expressed in dimensionless groups
(Re/l- E , and Sh/ 801/ 3) for different mechanism ; piston flow, forward
diffusion, and complete mixing. The data of Table 10 is plotted in.Figure 8.

e. Table 27 contains experimental values of k0: and the hypothetical
excess resistance that may account for the deviation of experimental data

from the theoretical curve. This excess resistance was obtained as follows:

Excess resistance = 1 1
E5 ks
when k is the mass transfer coefficient as obtained from the theoretical

8
curve. Figure 1% presents the excess resistance as a function of Re/l-Ef .

Table 27 also includes values for Sh/Sc1/3 calculated from experimental
data assuming that ammonia is to diffuse through air before reaching the

solid.

Theoretical data

 

a. Tables 11 and 12 contain heat transfer data for parabolic flow in
circular ducts. The data which.was originally obtained from numerical
tables (26) was reported as Nu vs. 6 .(6 =(h/7T) Gr). mas transfer data
wascbroanrom the original data and presented as follows:

Sh = 1.5 nu

Rc/l- 6 = 2.38 6 /80 when L/d? = l
The data of Table 11 was based on log-mean average driving force. For
Table 12, the arithmetic average of the driving force was used.

b. Table 13 contains heat transfer data for rodulike flow in circular
ducts. The data of this table was sealed off a curve that represent Nu vs.
Gr (22) using the logemean average of the driving force.

c. Table 1% contains heat transfer data for parabolic flow in circular
ducts. The data of this table was scaled off a curve (22) that represents

the local heat transfer coefficient as a function of Gr.

63

TABLE 11

Theoretical Mass Transfer Data Calculated from Graetz Equation (26 ) for
Parabolic Flow Using Log-Mean for Average Driving Force

 

(5 Pr Re / 1- € Nu Sh Sh/Scl/3
1 1 2.38 3.68 5.52 5.52
2 l h.76 3.76 5.6h 5.6h
h 1 9-52 3-86 5-79 5-79
7 1 16.66 h.01 6.01 6.01
7 10 1.67 h.01 6.01 2.80
10 1 23.85 n.16 6.2h 6.2h
10 10 2.38 h.16 6.2h 2.91
15 l 35.75 h.h1 6.62 6.62
15 10 3.57 h.h1 6.62 3.08
20 l h7.60 8.70 7.05 7.05
20 lo n.76 h.70 7.05 3.27
30 1 71.h0 5.22 7.83 7.83
30 10 7.1h 5.22 7.83 3.6h
60 1 132.80 6.h8 9.72 9.72
60 10 1h.28 6.h8 9.72 n.52
60 100 l.h3 6.h8 9.72 2.09
130 1 310.00 8.23 12.35 12.35
130 10 31.00 8.23 12.35 5.75

TABLE 11 (continued)

Theoretical Lass Transfer Data Calculated from Graetz Equation (26) for
Parabolic Flow Using Log-Mean for Average Driving Force

 

(5 Pr Re/l-€ Nu Sh Shchl/ 3
130 100 3.10 8.23 12.35 2.h9
200 1 u76.00 10.h0 15.60 15.60
200 10 37.60 10.h0 15.60 7.26
200 100 n.76 10.h0 15.60 3.1h
too 1 952.00 13.h0 20.10 20.10
hoo 10 95.20 13.h0 20.10 9.36
A00 100 9.52 13.h0 20.10 h.05
1000 1 2385.00 16.10 2h.15 2h.15
1000 10 238.50 16.10 2h.15 11.23
1000 100 23.85 16.10 2h.15 h.86
2000 l h760.00 20.30 30.h5 30.h5
2000 10 h76.00 20.30 30.h5 1h.17
2000 100 h7.60 20.30 30.h5 6.1h
hooo 1 9520.00 25.60 38.h0 38.h0
hooo 10 952.00 25.60 38.h0 17.86
#000 100 95.20 25.60 38.h0 7.73
#000 10“ 0.95 25.60 38.l+0 1.79

TABLE 11 (continued)

55

Theoretical Lass Transfer Data Calculated from Graetz Equation (26rfor
Parabolic Flow Using Log-Mean for Average Driving Force

 

__j§g Pr 3e/1-§;_ nu 3h sh/3c1/3
10000 10 2385.00 3h.80 52.20 2h.30
10000 100 238.50 3h.80 52.20 11.26
10000 101‘ 2.39 3h.80 52.20 2.h3
20000 10 h760.00 h3.80 65.70 30.60
20000 100 h76.00 h3.80 65.70 1h.20
20000 10" n.76 h3.80 65.70 , 3.06
h0000 100 952.00 50.20 75.30 16.25
3000 10h 9.52 50.20 75.30 3.52

*-

(5 is abscissa on Norris and Streid heat transfer plot.

Pr is Prandtl number for heat transfer or Schmidt number, So, for mass transfer.
Re/l-€‘ is 3.38 6/Pr, assuming L/d.p = 1.

Nu is ordinate on Norris and Streid plot.

Sh is 1.5 Nu as explained on page 53.

1131312

Theoretical Lass Transfer Data Calculated from Graetz Equation (26) for
Parabolic Flow Using Arithmtic Averages for the Average Driving Force

 

(5 Re/l- 6 Nu Sh
1 2.38 0.50 0.75
2 h.76 0.99 1.h9
h 9.52 1.92 2.88
7 16.66 2.86 h.28
10 23.85 3.hl 5.11
15 35-75 3-96 5.9%
20 h7.60 n.38 6.57
30 7l.h0 5.02 7.53
60 1h2.80 6.32 9.h7

67

TABLE 13

Theoretical Mass Transfer Data Calculated From the Graetz Equation (22) for
Rod-Like Flow, Using Log-Mean for the Average Driving Force

 

Gr Re/l-E' Nu Sh
l 3.01: 5.7 8.55

2 6.08 5.7 h 8-55

A 12.16 5.7 8.55
5 15.18 5.7 8.55
10 30.1w 5.8 8.70
30 . 91.20 8.5 12.75
100 30800 15.0 22.50
1:00 1216.00 25.5 38.20

1000 301LO.00 39.0 58. 50

68

TABLE 1h

Theoretical Data Calculated from Graetz Equation (22) for Local Mass
Transfer Coefficient where Parabolic Flow is.Assumed.

 

Gr Re/l-E‘ mu sn
1 3.0h 3.66 5.h8
10 30.h0 3.66 5.h8
20 60.80 3.75 5-72
3h 103.30 h.00 6.00
50 151.80 h.h0 6.60
70 212.50 n.75 6.97
200 608.00 6.30 9.u5
h00 1216.00 8.15 12.22
700 2125.00 10.00 15.00

1000 30h0.00 11.50 17.25

69

In Figure 9, the upper curve represents the heat transfer coefficient
(as Nu) for a rod-like flow in pnuhddfbpds. The lower curve represents
the heat transfer coefficient for a parabolic flow in circular ducts.

Figure 10 presents the local, the log-mean, and the arithmetic heat

transfer coefficient in circular ducts.

Experimental results of other investigations

 

a. Tables 15 to 25 contain the experimental data of other investigations.
All the data was obtained as reported originally by the authors. The data
was reported originally in terms of a mass transfer factor or height of
transfer units and Re or a modified Re. The data of Tables 22 and 25 were
scaled off the original curves due to the absence of tables.

Figure 12 presents the experimental data along with the theoretical
curves on a Sh/Scl/3 vs. Re/l-€ plot. The theoretical curve was based
on the Graetz solution to predict mass transfer coefficient for a parabolic
flow in circular conduits using a log-a. mean driving force.

b. Tables 26 and Figure 13 presents mass transfer coefficients as
obtained experimentally by different investigators for gas systems at low
flow rates.

0. Table 29 and Figure 16 presents mass transfer data as calculated by
using different empirical equations which were suggested by different
investigators. In using these equations, a Schmidt number of one and a void
fraction of O.h were assumed. Curve 9 of Figure 16 was based on the Graetz
solution for parabolic flow in circular conduits using a log-mean driving

force.

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Suggested plot

Figure 17 presents the suggested plot for predicting mass transfer
coefficient in packed beds. This curve fits most of the experimental data
covered in this work.

The coordinates of this curve are Sh/Sc1/3 and Re/l- 6 . At low values
of Re/l-E (Re/l-E < 100) the curve is theoretically based on the
solution of the Graetz equation for a parabolic flow in circular ducts
assuming a log—mean driving force. At high flow rates, the curve assumes

a slope ofone half.

Mass

TABLE 26

Transfer Coefficient 1‘6 as Obtained by Other Investigators

for Different Flow Rates

 

Author Watem 2Dm Re kG
cm /sec TE— cm/sec

Hobson a. Thodos Butanol-C02 0.07 65.6 0.595

Toluene-C02 0.162 53.0 1.67

68.0 1.517

Octane-air 0.0599 51.h 0.813

76.h 0.885

Butanol-Ng 0.082h 59.0 1.125

163.0 1.772

8h.0 1.392

237.0 2.560

no.6 0.927

390.5 2.220

67.0 0.89h

DeAcetis water-air 0.2h5 25.1 0.h88

30.7 0.600

52.1 0.959

81.1 1.056

30.9 0.5h2

h1.8 0.637

Hurt Naphthalene-air 0.06h 91.5 0.h32

518.0 2.hh5

2h.5 0.269

256.0 1.h86

20.5 0.368

TABLE 26 (continued)

95

Nhss Transfer Coefficient kG as Obtained by Other Investigators
for Different Flow Rates

 

Author System 022:0 %_ (fig/sec
Hurt (continued) 157.0 2.635
11.0 0.256
Taecker 8c Hougen Water-air 0.259 58.0 0.866
7h.0 0.858
no.3 2.0831;
100.0 2.896
178.0 2.500
350.0 23.120
733-0 8.180
Resnick & White Naphthalene-air 0.0611. 19.7 0.083
7.88 0.0h32
Naphthalene-H2 . ,. 2.96 0.0196
2.3 0.0172

11031327

Excess Resistance Calculated from the Present Experimental Data
and the Theoretical Data for bass Transfer Coefficients

 

 

 

Run Re rimental Theoretical Excess Sh/Scl/3
'32— };E? IIEG 1A3 Resistance for air*
on h.09 0.575 1.7% 0.126 1.61h 0.896
021 7.10 0526 2.3».0 0.119 sun 0.661;
chi 9.25 1.100 0.908 0.117 0.791 1.710
061 13.73 1.700 0.588 0.115 0573 2.650
071 17.33 1.225 0.815 0.112 0.701 1.882
081 2555 2.105 0571; 0.109 0.365 3.285
091 3155 2.055 0586 0.105 0.381 3.202
1311 2.22 0.521
321 7.85 0.910
A1], 5.57 0.7h8

 

* Diffusivity for air = 0.2218 cmZ/sec

TABLE 28

Experimental Data Presenting the Volume of Gas Reacted
or Absorbed per Second for a Given Flow Rate

 

l[:ec yl/yo yo-Y1 (yo-1127sic103* D;::::g*
0.01 0.h85 0.0965 0.965 0.1333,
0.02 0.580 0.0787 1.575 0.1hh6
0.03 0.626 0.0701 2.10 0.1h99
0.05 0.670 0.0619 3.09 0.15h5
0.07 0.71h 0.0536 3.75 0.1591
0.09 0.757 0.0u56 h.105 0.1639

 

* (yo-yl) v = absorption rate of ammonia.

** Driving force = (Yo "Yl) = ln—mean concentration.
-ln(y17§07

 

TABLE29a

Mass Transfer Data Obtained by Using Empirical Equatio Proposed
by Different Investigators, and Reported as Sh/Sc1 3*.

 

 

 

Re Gamson Wilke H0bson Taecker Gaffney DeAcetis
T? 19) (37) (13) (311) (8) (3)
1.7 11.20 1.21 1.16 1.12 0.75 -0.966
5.0 11.20 2.11 1.73 1.92 1.15 20.7
16.7 11.20 3.71 2.95 3.16 1.78 1.52
50.0 11.20 6.12 1.10 5.91 2.79 5.73
166.7 -—- 11.60 12.0 10.71 1.15 9.50
500.0 19.10 --- 29.61 18.10 10.85 16.37
1666.7 38.60 --- 121.0 38.60 26.60 31.20
5000.0 71.60 --- 521.0 71.60 60.60 57.80
TABLE 29b
Re Ergun Thoenes
T? (7) (35)
10 27.90 1.32
30 33.70 6.32
100 51.10 10.01
300 112.30 16.10
1000 316.00 28.93
3000 898.00 51.82
* Sc = 1 0

0.1

"3
u

99

DISCUSSION

Analogy between mass and heat transfer

 

In a stagnant fluid, diffusion is caused by the random movement of
the molecules which'by their mixing tend to equalize their concentrations.
By the same movement, temperature in the fluid is equalized too. This
analogy between molecular diffusion and.heat conduction can be extended
to similar processes in moving fluids. Any fluid motion which brings in or
carries away solute also brings in or carries away heat. In both processes,
a driving force and a resistance determine the rate of transfer. For mass
transfer the driving force is the concentration gradient. Fbr heat transfer
it is the temperature gradient. The resistance fer both mass and heat
transfer is directly proportional to the thickness of the stagnant fluid
and inversely proportional to the area and to the diffusivity or thermal
conductivity. With these similarities, a very sound analogy has been
established'between mass and heat transfer at low concentrations where a
linear relation between driving force and concentration may be assumed.

Accordingly, the effect of this analogy is as follows. The nature
of the flow in both cases is represented by the Reynolds number. The
Sherwood number for mass transfer and the Nusselt number for heat transfer
are equivalent since thegrepresent the ratio of the actual to the stagnant
transfer rates in both cases. The Sherwood number includes a linear
dimension of the system, the molecular diffusivity, and the mass transfer
coefficient. The Nusselt number, on the other hand, includes the same
linear dimension, the thermal conductivity, and the heat transfer coefficient.
In addition, the Schmidt number in mass transfer and the Prandtl number in

heat transfer are equivalent. The Schmidt number represent the fluid

100

properties for mass transfer, while the Prandtl number represents the

fluid properties for heat transfer.

Theoretical approach at low Re

 

At low Reynolds number (Re <<:2100 for circular pipes), the flow in a
conduit is usually viscous. Under these conditions, the velocity at any
point in the fluid may be calculated from the viscosity, the pressure drop,
and the conditions at the conduit wall. Likewise, the solute concentration
at any point in the fluid may be calculated from the diffusivity, the velocity
distribution, and the concentrations near the wall.

The required calculations involve the solution of partial differential
equations. The most general form of these partial differential equations
is the Fourier-Poisson equation, which was derived mainly for heat transfer.
This equation applies to any shape of conduit with steady or unsteady state
flow, and to any type of boundary conditions. For steady state flow, in
particular conduits with particular flow patterns, the differential equations
may be simplified and integrated to give the mass transfer coefficient at
the wall. These integrations are often complex, but solutions have been
obtained for many of the more important cases.

In turbulent flow the flow geometry is not clear; rather it is complicated
by back-mixing. In such a case, local velocities cannot be defined theoretically.
Therefore, it is difficult to predict an exact correlation between flow and

mass transfer rates.

Solutions of the diffusion equation

 

Mass transfer in viscous or creeping flow which occurs at low Reynolds
number can be determined theoretically by solving the differential equation
for molecular diffusion in a moving fluid. The most accepted solutions of

the equation are those of Graetz (h), which were derived initially for

101

heat transfer in circular ducts. The boundary conditions and necessary
assumptions for these solutions involve the following:

a. Concentration is constant at the duct wall. .

b. Fluid properties ( I), Dm,[1.) are constant.

0. Forward diffusion in the direction of flow is negligible.

d. Concentration conditions are in a steady state.

e. Flow pattern is either parabolic or rod-like.

A rod-like flow is a condition in which the fluid velocity is the
same at any point in the fluid. In practice, this may be expected to
occur only with systems such as flowing sand.where all the slip occurs
at the wall. Rod-like flow also may be expected to occur as a fluid
passes through an orifice. On the other hand, a parabolic flow is a
condition in which the fluid velocity has a parabolic relationship to
diameter. This flow pattern is normally expected with newtonian fluids
of constant viscosities.

For turbulent flow at high Reynolds number; Martinelli (21) worked
out a relation between Nu, Re, and Pr, based on an assumed flow pattern
for turbulent flow. The same relation could.be applied to Sh, Re, and Sc,
but the range of’Re and Sc in which the Nbrtinelli relations can be applied

are outside the range of the experimental data covered in this thesis.

Representative Graetz plots

 

Fer circular ducts, two plots were Obtained to represent heat (or mass)
transfer coefficients in the viscous region (see Figure 9). The upper
curve was Obtained when rodzlike flow existed (22) and the lower curve was
obtained when parabolic flow existed (26). In'both cases, a log-mean heat

(or mass) transfer coefficient was calculated.

102

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103

The mass transfer coefficient is identified as a local coefficient,
log-mean coefficient, or arithmetic coefficient. The differences among
these three coefficients are due to the driving force used to determine
each. In order to illustrate the differences among these coefficients,
consider a circular conduit in which a fluid is flowing in one direction.
If the fluid has an initial concentration which is different from the
interfacial concentration, then mass transfer between the fluid and the
wall takes place. When the interfacial concentration stays constant along
the conduit, the driving force can be calculated at any section of the

conduit by this relationship:

L
l
T211215— = T 1‘6 W”) “L

O

The local mass transfer coefficient is based on the driving force
that is calculated for thelenat conditions (or L = L).

If an average mass transfer coefficient for the entire length of the
conduit is desired, an average driving force must be determined. If this
average value for the driving force is the arithmetic average between the
entrance driving force (L = O) and the exit driving force (L = L), then the
mass transfer coefficient Obtained is the arithmetic mass transfer coefficient.
If the average value of the driving force between the entrance and the exit
of the conduit is the logarithmic mean, then the log mean mass transfer
coefficient is obtained.

Figure 10 shows that these coefficients (heat or mass transfer coefficients)

differ from each other at different flow rates.

Flow in packed'beds

 

A flow pattern for randomly packed'beds cannot be defined precisely
since the pattern depends on how the packing falls. A hand packed'bed has
a definite flow pattern which depends on how the packing is stacked. It

has been customary in the past (7, 35) to consider that two packed beds

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105

would have the same mass transfer characteristics if the overall dimensions,
the void volumes, and the void surface areas are the same for the two beds .
Thoenes' experiments (35) showed some variation with packing orientation,
but in his final correlation he neglected the effect of this variation on

the mass transfer «coefficient.

Mixing
regions _—c—*

Flow /
channels '

Figure 1.1..

  

 

 

Idealized Flow Path in a Hand Packed Bed

Figure 110. is a two-dimensional representation of a regularly packed
bed of spheres. Figure 11b is a two-dimensional representation of a hypo—
thetical bed in which voids are in the form of cylindrical channels
(indicated by arrows) which Join together at points (indicated by the
large dots) where complete mixing takes place . Both Figures 11a and 11b
would look somewhat different in three dimensions .

If the void volumes and areas are made the same for two beds and if
the same assumption made by previous authors concerning mass transfer
characteristics of packed beds is again used, then a theoretical analysis
of Figure llb should give nass transfer relations which can be used for
Figure 11a. Figure llb is relatively easy to analyze since the Graetz

equations for cylindrical conduits can be applied to it.

106

application of Graetz equation to_packed.beds

In applying the Graetz equation for cylindrical conduits to packed
beds, the preceding analysis of a hypothetical bed and its short cylindrical
channels is considered. Therefore, L/dP is equal to unity. Since these
hypothetical cylindrical channels are diagonally oriented in the bed
(Figure llb), the vertical component of the velocity vector in these
channels will be calculated to represent the average velocity in the
Graetz equation.

Parabolic flow takes place when Newtonian gases and liquids flow
through tubes of uniform cross section. Since the hypothetical bed
was assumed to consist of cylindrical channels, then the Graetz solution
for parabolic flow will be used. However, if the spaces between the
packings in a bed.behave like an orifice, rod-like flow would have been
considered since it is possible to Obtain a uniform velocity distribution
across an orifice.

This particular solution of the Graetz equation is employed to
predict the log-mean mass transfer coefficient. The use of the log-mean
coefficient in predicting mass transfer rate in packed'beds is more valid
than using the local or the arithmetic average coefficients. In order to
illustrate this point, let us consider each coefficient for the hypothetical
bed separately.

a. The local mass transfer coefficient is based on the driving force
at the end of each channel (where L = dp). Therefore, in order to represent
the mass transfer coefficient in the whole channel, an average driving force
should be used and not a local one.

b. If the arithmetic average mass transfer coefficient is considered,
an arithmetic average driving force should be employed As the flow rate

decreases, the mass transfer coefficient will approach zero since the

107

driving force does not approach zero when saturation is reached and mass

transfer ceases, i.e., y* = yl

Ada :ké‘ [W— 2

c. The logarithmic mean mass transfer coefficient is based on the log

dN yo + yl ]

mean of the driving force in the channel. The log-mean mass transfer

coefficient does not approach zero as the flow rate decreases.

 

dN = k1 (y* - n) - {y* - yo)
A d5 J M (fl - yl‘)
y* ‘ Yo

Emperimental data

 

Figure 12 indicates that experimental data fit the theoretical curve
fairly well from a value of Re/l-éf of 50 to 500. Large deviations are
found to exist at low Reynolds number (Re/l- € <50). In some cases
experimental data would have to be about 35 times as large to fit the
theoretical curve. These deviations have been a subject of discussion by
many investigators. Ergun (7), for instance, tried to explain the low
values obtained for gases on the basis of complete back mixing. Thoenes
and Kramer$(35) mentioned the possibility of bed geometry as a factor in
this deviation.

At high Reynolds numbers (Re/l- €>soo), experimental data apparently
rise above the theoretical curve with a slope which is higher than that
of the theoretical curve. This deviation can be attributed to either or
both of the following factors:

a. The effect of turbulence: The theoretical curve was based on
viscous flow. If turbulence exists, better mass transfer would be expected
due to the formation of eddies. The effect of the eddies would be reinforced

by the thinning of the film at the interface due to higher flow rate.

108

b. A change of flow mttcrn: It was assumed, when solving for the
theoretical curve, that the velocity distribution was parabolic as in a
cylindrical channel. However, since the actual channels are not cylindrical,
the velocity profflcny be more uniform than parabolic, if the packing
behaves like an orifice.

The latter factor seems reasonable when we consider that the new
slope of the experimental points is approxilataly one half as _ compared to
the theoreticalle of one third at high Reynolds number. Theoretically,
if a rod-like flow‘is assumed, a slope of 0.5 is obtained according to
Figure 9.

In general, it is found that ass transfer data on liquid systems
fit the theoretical curve better than use transfer data on gas systems.
This phenomenon is clear at low flow rates (Figure 12). An example of
gas data which deviate widely fro. the theoretical curve are the data of
Hurt (15), of Besnick (28), and of this thesis. Those deviations are

more noticeable at low Reynolds numbers .

Deviation or dagger-mm data according to Schmidt mam:

Fro- Figurc 12, it appears that experimental data for systems of various
Sc are in reverse of what they should be theoretically. Experimtal data
represent] systems with Sc of l to 10 falls below those represent-3 systems
with Sc of 10,000 on a Sh/3c1/3 vs. Re/(l- e ) plot.

Theory in this respect is not likely to be wrong, regardless of the
flow pattern chosen. In circular ducts, Sh is constant at low Be. This
is attributed to the fact that ass transfer continues even when fluid
motion is slowed or stopped, due to molecular diffusion. It can be stated
that at a certain low flow rate, the fluid velocity has no more effect on
mas transfer, and molecular diffusion will be the controlling factor.

Il'his is true if no other factors such as reaction, surface tension, etc.

89

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mtuvnzu<nn<z 600.232

.02. .>z<nSOU zoo.- xuoou

 

.IHJU>0 202.." H >I n ~U_IIF_K<UOJ

 

JUN—Q .Oz

110

exist besides velocity and molecular diffusion. Since Sh represents mass
transfer rate in a certain conduit, its limit will‘be determined solely‘by
the thickness of the film through.which molecular diffusion is taking
place and the nature of the flow.

Since Sh is constant at low flow rate, then Sh/Scl/3 must increase
with decreasing Sc and vice versa. The experimental data failed to follow

this theoretical approach.

Possible reasons for low data on gases

 

Nhss transfer rate for gas systems in packed'beds will‘be controlled
not only‘by the driving force but also‘by an overall resistance Be. This
resistance is made of a series of resistances such as gas or liquid film
resistance l/kg, resistance due to impurities l/ki, resistance due to
the reaction kinetics l/kr, etc...

1 l 1
= ____-+ +-_Er_.
RG kg k1 r

 

The overall resistance RG, can be calculated from the apparent mass

transfer coefficient kG that is usually obtained from experimental data.

Fer liquids, k is often in the range of lO“h to 10‘5 cm/sec at low

8
Re, and therefore it is the controlling resistance. waever, for gases,
k8 is large and often in the range of 0.1 to 1.0 cm/sec at low Re. There-
fore any other resistance, even a small one, can contribute to the overall
resistance. I

If mass transfer in gases is considered, where no other resistance
exists except the gas-film resistance l/kg, then kg = kg; or the apparent
mass transfer coefficient is equal to the theoretical mass transfer

coefficient in.a gas—film controlling system.

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112

Apparent mass transfer coefficients at low'Reynolds number, as obtained
experimentally for gases by different investigators, show a wide range of
values (Figure 13). Some of these coefficients, each with the diffusivity
of its system, give Sherwood numbers close to the theoretical. Other
coefficients failed to do so, hence deviation from the theoretical curve
is noted. In order for these values of k6 to give Sh numbers that fall
on the theoretical curve, kG would have to be much larger than any of the
figures shown (Table 26). This means that experimental resistance (R3)
equals theoretical resistance (l/kg) plus additional resistance, where
the theoretical resistance is calculated from the theoretical curve.

The highest values of kg measured at low Re (Re:>5) are those of the
experiments of this thesis, and even these are not high enough to bring
the overall mass transfer coefficient up to the theoretical curve.

Figure 1h shows the extra resistance calculated on the basis of what
is required to adJust the data of this work to the theoretical curve.

Extra resistance = overall resistance (l/kG) - theoretical gas-film
resistance (l/kg).

Some of the possible factors that may contribute to this extra
resistance are:

a. Interfacial resistance.

b. Liquid—phase resistance.

c. Resistance due to presence of air.

d. Effect of reaction rate.

e. Effect of back pressure equilibrium.

f. Effect of mass transfer in solid phase.

g. Effect of back mixing.

h. Effect of radial diffusion.

1. Effect of bed geometry.

113

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Interfacial resistance

The possibility of the presence of a resistance in the interface has
been mentioned by other authors. This resistance may be due to inter-
molecular forces at the interface. Also small amounts of impurities which
may gather at the liquid-gas interface or at the solid-gas interfaces, as
in this experiment, could create new resistance.

This interfacial resistance could be very large if the fluids contain
impurities. Qabéhnvapsniin the air, for instance, could increase this
resistance if the packing were exposed to the air and used for a run‘before

the air was evacuated.

Liquid-phase resistance

 

In the experiments of this thesis, no liquid phase was employed. How-
ever, if water vapor from the surroundings is condensed on the packing, a
liquid film will be formed. This liquid film will provide another resistance
to diffusion. This resistance is often controlling since diffusivity in
liquids is usually much lower than diffusivity in gases.

Other authors (3, 1h, 35) measured mass transfer rates for the
evaporation of pure liquids such as water, para-xylene, etc. Liquid
resistance in these cases is meaningless since there is no necessity for
liquid diffusion from the bulk of the'flqwdto the surface. Accordingly,

the only other resistance in these cases would be resistance to evaporation.

Air resistance

 

There is a possibility that some of the air which is in the bed prior
to the run remainsin the packing during the run. If this air is not
purged prior to measurement of the rate of mass transfer, part of it may
cling to the packing and form a sheath on the surface. This sheath presents

a resistance to diffusion between the bulk and the solid.

115

In order to avoid saturation of the bed, the packing was not purged
with testing gas in the final experimental runs of this thesis. Also the
packing was not purged with inert gas prior to the run, in order to avoid
diluting the testing gas during the run. Therefore there is a possibility
that air remained in the bed. However, the air could not account for all
of the apparent resistance.

Table 27 shows that if all the packing is covered with a layer of
air, Sh/Scl/3 is increased by a factor of 2.h. Even with this assumption,
the experimental data does not approach the theoretical curve. Therefore,
if trapped air is a factor, some other resistances would have to be present

too.

Reaction rate

 

In the experiments of this thesis, mass transfer is accompanied'by

a reaction between the active component of the gas and the solid.
CuCl-I-NH3 = CuCl - N113

In predicting the mass transfer coefficient, it was assumed that
mass transfer and not reaction kinetics was controlling.

The average concentration of active component in the fluid at the
surface of the solid must be equal to the average concentration of active
component in bulk of the fluid, provided that reaction kinetics is
controlling and gas phase resistance is negligible.

Table 28 shows the rate at which the reaction occurred versus its
average concentration in the bulk of gas. It may be seen that the rate
of absorption increases fourfold as compared to a small increase (of
1.5 times) in the concentration of NH3 as the flow rate is increased

from 0.01 to 0.09 liters per second.

116

If reaction rate is controlling, such an increase in absorption rate
may not correspond to a small increase in ammonia concentration. 0n the
other hand the increase in.ahsorption rate may result from a ninefold
increase in flow rate, if mass transfer is controlling.

Therefore, reaction kinetics may cause a small part of the resistance

to mass transfer but it may not be the controlling factor.

QEEEAEressure equilibrium

The dissociation pressure of the various ammonia complexes is shown
in Figure 3. Since the gas was flowing at 300 to 305'K and nearly atmo-
spheric pressure, the dissociation pressure of CuCl'NH3 is between 0.000031
to 0.0000h5 atmospheres. This vapor pressure is much less than the partial
pressure of ammonia during all runs, since the initial ammonia concentrations
were 0.913% and 0.1875$ for the first two sets of runs and last set of runs,
respectively. Moreover, the vapor pressure of CuClg-NH3, which could be
formed if part of the cuprous chloride was oxidized to cuprice chloride
prior to the runs, is about 0.00001 atmospheres at room temperatures.
Apparently equilibrium back pressure is not a factor that results in
deviation, as long as CuCl or CuClg is present on the surface of solid.

However, if all the CuCl is converted to CuCl’NH3 then the back
pressure of the new complexes, CuCl°l¥l/2 NH3 or CuCl°3NH3 would'be about
0.033 atmospheres and 0.0h5 atmospheres respectively (see Table 3). In
such cases back pressure would be a factor. But since during the experimental
runs of this thesis the intial partial pressures of the gas did not exceed
the vapor pressure of CuCl-l-l/2 N33 and CuCl-BNH3, such complexes may not
have been present at any time.

Therefore, back pressure could not have a significanteffect on the

resistances to mass transfer as it was measured in the experiments of

this thesis.

117

Mass transfer in solid phase

 

Figure 7 shows that the rate of absorption in runs conducted on freshly
prepared packing is much higher than the rate of absorption in runs conducted
on packing that was used before. This phenomenon is attributed to penetration
of ammonia into the solid phase. When gas first contacts freshly prepared
packing, gas-film diffusion is controlling. Once the surface layer became
saturated, solid diffusion of ammonia or it ‘a; complex through the surface
layer became controlling. Accordingly, the apparent mass transfer coefficient
will be a function of time at constant flow rate.

This was demonstrated by a series of very short runs conducted on the
same fresh packing at constant flow rate. Figure 6 shows the changes in
the mass transfer coefficient with time at constant flow rates. Due to the
lack of enough data, curves 1 and 2 may follow either a'or E} If either
§_or b_levels off when 59 approaches zero, then a steady state condition
exists until the surface is saturated. Accordingly, the mass transfer
coefficient can be predicted more accurately by scaling off its value
from the curve at that region.

To avoid surface saturation of the fresh bed during every run, the
initial concentration was reduced and a minimum volume of fluid was used
for the runs in the final set of experiments. This modification in the
experimental method was designed to reduce possible saturation of the
surface of the packing. The effect of this technique was demonstrated
by noticing that the excess resistance as calculated from the experimental
and theoretical data (Figure 1%) decreases as the flow rate increases.

Solid diffusion may be expected to be independent of flow rate and there-

fore it may not account for all the excess resistance.

118

Effect of back mixing

 

All the experimental data in Figure 12 were treated on the basis that
a piston flow mechanism exists in the bed. Ergun (7) suggested that more
or less complete back mixing can exist in the packing. Using his approach
(Equation 3), he employed Hurt's data (15) to show that back mixing exists.
Using a driving force that is based on this assumption, Hurt's data are
found to fit the theoretical curve. It is possible that in Hurt's particular
experiments, back mixing may have been present to this extent, due to some
unusual experimental conditions.

The low kG values of this thesis cannot be explained on the basis that
complete mixing exists, since all of the bed except one row of packing was
inert and therefore values of yl/yO were not large. Values of yl/yo ranged
from 0.35 to 0.95, and in this range values of kG are approximately the
same whether piston flow or complete mixing is assumed (Table 2).

Furthermore, when Kramersand Alberda (17) investigated forward
diffusion and complete mixing at each layer in the bed, they found that
both mechanisms yield approximately the same data for beds with many layers.
Using their analysis, the driving force was calculated for forward diffusion
and complete mixing. The calculated data are approximately the same as
those calculated for piston flow when one layer was assumed in the bed.

Therefore, back mixing may not have an effect on the apparent mass
transfer coefficient which was Obtained experimentally in this thesis

(see Figure 8).

Effect of radial diffusion

 

To avoid a situation in which the concentration of the fluid might fall
to a point where it would approach equilibrium with the bed, some experimenters

used inert packing with active packing particles dispersed among the inert

119

ones. Such systems were used by DeAcetis (3), Thoenes (35), and in the
experiments of this thesis. DeAcetis (3) employed a packed bed where

the active packings were dispersed randomly in the inert packing. Thoenes
(35) used a single active particle in his bed. The experiments of this
thesis involved a bed made of one complete layer of active packing placed
between inert layers.

The difference between the present experiments and those of others,
who employed this technique, is in the possibility of radial diffusion.
Radial diffusion could be a factor that affects the overall mass transfer
rate when a single sphere or an attenuated bed is used as in Thoenes' work
and DeAcetis' work, but it could not be a factor in the experiments of

this work.

Effect of bed geometry

 

In presenting mass transfer data in packed beds, earlier investigators
used the packing diameter to represent the effect of bed geometry. Ergun (7),
showed that the space volume and the surface area of the packing are more
proper to represent the effect of bed geometry on mass transfer rate. With
a proper use of the surface area and the void, Ergun's coordinates make
experimental data fit the theoretical curve more closely. Mereover, Ergun's
analysis of the bed geometry made it possible to apply the Graetz equation
for circular conduits to packed beds.

Thoenes (35) showed that even with the proper coordinates, up to two-
fold variation in mass transfer coefficient may be observed‘for p-xylene-
C02 system in cubic and body center cubic geometry form). However, Thoenes’
work may have been complicated by radial diffusion.

In any case, the geometry effect may be involved in void fraction
and in exposing active surface to the flow. Part of the void spaces in

a packed bed may contain stagnant fluid. This is most likely to occur

120

in the spaces between the packing of succesive rows where the packing
shields the space from the main flowing stream. A clearer picture is
Obtained if the flow through this space is compared to the flow throughan
orifice. It is clear that the surface area of the conduit that lies
directly behind the orifice may not be exposed to the flow.

A bed with spaces in a square arrangement such as the one used in
the present experiment is shown in Figure 15a. Very little flow is likely
to occur next to the spheres except in the open diamond shaped cross
section between longitudinal rows of packing as illustrated in Figure 15b.
Figure 15c shows a possible flow profile along the bed where only part

of the packing area is exposed.

Diamond shaped space Stagnant fluid
cross section

in the space

 

 

 

 

b
Longitudinal section A cross section Possible flow profile
of bed of the bed along the bed
Figure 15

A Packed Bed with Spheres on Square Centers

Accordingly, diffusion from the bulk of fluid to the unexposed surface
of the packing will be slower than diffusion to the surface in contact with
the flow. If a different fluid such as air is trapped in the stagnant space,
kG will be different in both cases. The difference in the value of kG will
not only be due to the different molecular diffusivities in the two media
but also due to the effect of the different surface area considered and the

actual velocities that will effect Re.

121

Previous correlations for predicing mass transfer

There have been a number of empirical equations suggested by different
investigators to correlate mass transfer rate and flow rate in packed beds.
Figure 16 shows that large differences exist among the curves of these
equations. It may be noticed also that some of them are far from the
theoretical curve of the Graetz equation. All the curves which approach
zero at low values of Re are theoretically incorrect since mass transfer
is appreciable even at zero velocity. The only curves that present a
limiting value for Sh are those of Gamson (9) and Ergun (7). But the
limiting values for both of these curves are far from the theoretical
value given by the Graetzicurve. At high flow rates (high Re), the curves
seem to fall in the same region with the exception of those of Ergun (7)
and of Hobson and Thodos (13).

The Ergun curve (7) was based on an assumed analogy between mass
transfer and pressure drop (momentum transfer). Apparently the analogy
was not correctly drawn since the limiting value of the curve is extremely
high.

Gamson's curve (9) at low flow rates was based on extrapolation of
the high flow rate data through a single point in the low flow rate
region. One of the authors of that investigation suggested (37) that
curve 6 of Figure 16 would be more logical.

The peculiar shape of the DeAcetis curve is due to extrapolations
of his empirical equation beyond the range in which his data were taken.
At high flow rates, his curve follows Graetz curve closely. But the use
of his empirical correlation should be limited to values of Rej>8
(or Re/l- 5‘ >13). This limitation will eliminate the possibility of
obtaining a negative mass transfer coefficient or a coefficient that

approaches infinity.

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123

It should be stated here that the Thoenes and Kramerscurve (37) fits
the experimental data'better than the other curves and that it is also the

closest to the theoretical curve of the Graetz equation at low values of Re.

Suggested plot

 

0n the basis of the preceding analysis, a plot (Figure 17) is suggested
for mass transfer rate in packed'beds which can be used for practical purposes
with minimum deviations. This curve fits most of the experimental data
fairly well.

The part of this curve at low flow rates (Re/l- {(100) is based on
the Graetz equation. At high flow rates, (Rafi-6 >100), the curve assumes
a slope of (0.5) which is higher than that of Graetz equation. Theoretically
this curve is reasonable since it reaches an asymptotic value when the flow
rate approaches zero. The asymptotic value of this curve is necessitated
by the contribution of molecular diffusion to mass transfer in packed beds.

At high flow rates, the experimental data rather than the Graetz curve
was taken as basis for this curve. This is Justified by the fact that the
experimental data may be more valid because the effect of turbulence was

not considered in the Graetz equation.

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125

CONCLUSIONS

The results and conclusions of the present work are summarized as
follows:

a. A correlation between flow rate and mass transfer coefficient
was Obtained to predict mass transfer rate in packed beds. The correlation
was presented as a plot that expresses the mass transfer coefficient as a
function of flow conditions and fluid properties. This curve fits most
of the experimental data covered in this work.

b. The part of the curve at low flow rates fits the Graetz equation
when this equation is applied to packed beds. At high flow rates, the
curve must assume a higher slope than that of the Graetz equation in order
to fit the experimental data. This may be due to turbulence that increases
actual mass transfer rate.

c. The void volume and void surface area of the bed are more valid
to use than packing diameter when expressing the mass transfer characteristics
of the bed. If the packing diameter is used, a correction needs to be
made to take care of variations in void fractions. Such corrections were
used to modify the Graetz equation for circular conduits to fit the
experimental data for mass transfer in packed beds.

d. Deviation of gas data from the theoretical curve can be explained
on the basis of an unknown resistance which contributed to the total
resistance more than the gas-film resistance. Liquid data does not
deviate as much as gas data since liquid-film resistance is so high that
it controls the overall resistance. The additional resistance can be
attributed to many factors such as interfacial resistance, liquid4film
resistance, diffusion through the other phase, reaction kinetics,

saturation, etc.

126

e. The deviation of the data of the present work is attributed misly
to a combination of factors or to an interfacial resistance of unknown
nature.

1‘. Some of the causes for deviation can be controlled by special
experimental techniques. In the course of the present experiment, any
factors such as back mixing, saturation, forward or radial diffusion, wall

effect, and end effect were controlled.

127

RECOMMENZAIIONS

In order to understand better the operation of mass transfer in
packed beds, the following points are suggested for further investigations.

a. Theoretical work to solve the partial differential equation that
represents mass transfer rate in packed beds.

b. Experimental work to determine the nature of the extra resistance
which appears to be present in gas-phase experiments at low flow rates.
The same work may be extended to high flow rate for comparison.

c. Improvement of the column and packing used in this work in such
a way as to permit regeneration of the packing while in the column.

d. Use of different gas-solid systems whose reaction rates are

better known, for the experimental work.

APPENDIX

129

SAMPLE CALCULATIONS

To illustrate methods of calculation outlined beginning on page MB of
this thesis, the fourth run of set C is used here as an example.
1. Calculation of yl/yo.
From experimental data, the following were Obtained:
ba ; 0.1097 (acid normality for all runs of set C)
bb = 0.02601 (base normality for all runs of set 0)

Also from Table 6, the following were Obtained.

za = 2.00 cc 0 v = 0.5h5 l.
zb = 7.60 cc P = 73h.3 mm 38°
T = 21.8°c v = 0.02872 l/sec

From water vapor pressure tables, p = 19.6 mm Hg.
effective volume of gas = 0.5h5 - 0.070 = 0.h75 1.

R = 62.361 mm l/°K mole

‘g-moles of“he11um = 73h°3 ‘ 19°5 . O'h75 = 0.018h5

 

29u.9 627361
g-moles of ammonia = (2 x 0.1097 - 7.6 x 0.02601)10-3 = 2.17 x 10‘5
then y1 = 0.1175%
also yo = 0.1875% (calculated for all runs of set C)

yl/yo = 0.627
(Effective volume of gas for runs Ch through 09 = 0 v - 0.07,
effective volume of gas for other runs = 0 v)

2. Calculation of - Jfayany

for piston flow, 'JLEXE' = _ in 0,627 = 0.h669
“AV

for complete mixing, _ dyf = 0.1875 - 0.1175 = 0 5957
“AV 0-1175 °

 

130

3. Calculation of dimensionless group.

Re/l- 6‘
Sh/Sc1/3

322 x .02872 = 9.25

53.3 x 0.02872 x 0.h669 = 0.716

These results are presented in Table 10.

1+. Calculation of mass transfer coefficient.
G _ - —
aL J ~Av

_ 268.3 x 0.02872 [_ ]
_ 3.27 (L/dgj 1n 0.627

kG = 1.10

 

 

5. Calculation of individual resistance from theoretical gas film
resistance and acutal resistance which is considered as total resistance.
Total resistance = 1/kG (experimental)

l 1
_. k

kG 3

At Re/1-€ = 9.25, experimental mass transfer coefficient is 1.1.

+ excess resistance

 

At the same Re/l- E , theoretical Sh/Scl/3 = 5.6 (from Graetz curve).

 

kg = camel/3) (Sci/3)[(1- eve] . mam/a,
_ (0.52h) (0-92§) _
— (5.6)(1.091) (0.h76) (0.726) _ 8.56
1/kG = 1/1.1 = 0.908
1/kg = 1/8.56 = 0.117

unknown resistance = 0.908 - 0.117 = 0.791 sec/l

131

TABEE OF NOMENCLATURE

The fundamental dimensions are represented by the following letters:
force F, length L, mass m, time t, temperature T. When special units are

used, these are specified in the text.

a Surface area per unit volume of bed 1/L
A. Cross sectional area of a conduit L2
AP Surface area of one packing particle L2

ba. Acid normality
bb Base normality

Constant (Hirschfelder equation) 9.2916 x 10'h

U1

c1,c2.. Constants

cp Heat capacity at constant pressure FL/mT
d (Circular) conduit diameter - L

d? Particle diameter L

D Differential operator

Dn Molecular diffusivity L2/t

E Eddy diffusivity L2/t

G was velocity m/L2t
Gr Graetz numbergGA cp/kL

HT Height of a transfer unit L

h Heat transfer coefficient F/LTt

Ja mass transfer factor defined by Equation 2
J5 iMass transfer factor defined'by Equation 3

Jh Heat transfer factor defined by Equation 1

132

k Thermal conductivity F/tT
kg Gas film mass transfer coefficient L/t
kg Apparent mass transfer coefficient L/t

K Boltzmann's constant (erg./degree)

L Bed length or channel length L
MiMé.. Molecular weights of gases 1, 2, ..

Nu Nusselt number h d/k

n Number of layers in'bed

N Volumetric flow rate in'bed L3/t
P Total pressure F/L2
p vapor pressure of water F/LZ

Pe Peclet number for forward diffusion d? U/(Dm + E)

Pr Prandtl number = cp LL/k

Q Rate of diffusion m/t L2
R Gas constant Fl/mT
RG Total resistance to mass transfer t/L‘
Re Reynolds number = 6'22 .'_£le_

[1.

6'
Re Modified.Reynolds number = g'G'-1 /- AP

r1, r2,... Roots of a differential equation

r1, ré,... Mblecular radii of gases 1, 2, .. L
r12 Collision radius (ri’ + ré )/2 L
Sc Schmidt number = LL

p Dm
Sh Sherwood number _ 6‘ fig kc

1- 3 Dn
T _Fluid temperature T
T1 Meter temperature T

(T-T*) Driving force for heat transfer T

133

u Actual velocity of flour L/t
U Superficial velocity of flow L/t
v Volumetric gas rate of leaving inert component 1.3 / t
V1,V2,... Molecular volule of components 1, 2,...
at their normal boiling points L3/m
W1C” Collision integral (Hirschfelder equation)
2 Distance in the direction of diffusion L
y Concentration of active component at any section in the bed
yr Concentration of the flowing stream
y* Component concentration in equilibrium with that on the other
side of the interface
yo Initial concentration
yn Concentration of the component leaving the nth layer
23 Volume of acid used for a run 1.3
sh Volume of base to neutralize excess acid L3
7’ Proportionslity factor
Correction factor (Hirschfelder equation)

A
6‘ Void fraction in the bed

6,162” minimum energy of attraction of components 1, 2, . . FL
5 Duration of a run t

u Viscosity of fluid I/Lt
,0 Density of fluid m/L3
6 ll Gr

 

1r

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10.

12.

13.
11+.
15.
16.

17.
18.

19.

13h

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