AXIAL DISPERSION OF LIQUID

1 N PACKED BEDS

McGILL UNIVERSITY

AXIAL DISPERSION OF LIQUID

1 N PACKED BEDS

by

RAM TIRTH KHANNA, B.Sc., B.Tech. (Hons.)

Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfi 1 lment of the requirements for

the degree of Master of Engineering

MONTREAL, QUEBEC

April, 1966

Chemical Engineering

AXIAL DISPERSION OF LJQUJD 1 N PACKED BEDS

Ram Tirth Khanna

ABSTRACT

M. Eng.

The axial dispersion of liquid in a fixed bed of dry

randomly packed Raschig rings was investigated in a 1-foot

diameter column for three sizes of packing: 0.5, 1.0 and 1.5-inch.

The primary variables investigated were particle size, bed height,

liquid and gas flow rates. The transient response technique using

a step function input was used.

Diffusion and random walk models have been used to

obtain axial Peclet Number, Pe = ude/DL· The results are pres­

ented as relationships between the axial Peclet Number, the

Reynolds Number of the dispersed liquid phase, Re= udejJ~,

Galle lei Number, Ga = d~gp 2 ! /' 2, and gas phase Reynolds Number,

ReG = deGIJl .

•

i i

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation

to ali those persans who ren~ered help ln this investigation and

in particular to the following:

To Dr. W. J. M. Douglas of the Chemical Engineering

Department for his counsel, guidance and encouragement.

To the Pulp and Paper Research lnstitute of Canada for

assistance in the form of equipment grants and for the extensive

use of their library, shops and other faci lities.

To Mr. R. A. Lindsay of the P.P.R.I .C. for his assist­

ance in the design, construction and maintenance of the electronic

instrumention.

To McGil 1 University for financial assistance in the

form of University Graduate Fel Jowships.

To members of the Computing Centre of McGi 11 University

for thetr cooperation.

To the Staff and Graduate Students of the Chemica1

Engineering Department for their help.

i i i

TABLE OF CONTENTS Page

ACKNOWLEDGMENTS

LIST OF ILLUSTRATIONS

LIST OF TABLES

1. INTRODUCTION

Il . L 1 TERATURE SUR VEY

i i

v

vi i

9

INTRODUCTION 9

HOLDUP JO

RESIDENCE TIME DISTRIBUTION AND LONGITUDINAL DISPERSION 16

Axial Dispersion in Pipe Flow 16 Axial Dispersion in Single Phase Flow

Through Packed Beds 19 Axial Dispersion in Two Phase Flow

Through Packed Beds 27

111. RESIDENCE TIME DISTRIBUTION AND TECHNIQUES OF EVALUATING EFFECTIVE LONGITUDINAL DISPERSION COEFFICIENT

1 n t r od uc t i on Transient Response Comparison of Input Signais Mixing Models Mixing-Cel 1 Model Diffusion Madel Statistical Random Walk Madel

IV. EXPERIMENTAL APPARATUS AND OPERATING PROCEDURE

v.

EQUIPMENT Flow System Instrumentation Conductivity Probe Electronics

EXPERIMENTAL PROCEDURE

CORRELATION AND DISCUSSION OF EXPERIMENTAL RESULTS HOLDUP

33 34

38

39 41 45 51

55 61 64 66 71

84

VI.

PECLET NUMBERS

SUMMARY

NOMENCLATURE

BI BLIOGRAPHY

APPENDICES

87

106

108

Ill

1. Experimental Data and Calculated Results 118

11. Electrical Circuits and Operating Instructions 134

iv

v

LIST OF ILLUSTRATIONS

Figure Page

2

3

4

5

6

7

8

9

JO

1 1

1 2

13

14

15

16

17

18

Material Balance Over a Section of Packed Bed 4

Liquid Phase Dispersion Data for Single Phase Flow

Response Curves for an Impulse Input Function

Response Curves for a Step Input Function

Response Curves for a Sinusoidal Input Function

Mixing Cel 1 Mode)

Material Balance Components of the Diffusion Model

Theoretical Breakthrough Curves for Various Column Peclet Numbers

Flow Diagram of the Experimental Set-up

Photograph of Experimental Set-up

Photograph of Water Distributor

ln let Flow Profiles for Various Liquid Mass Velocities

Photograph of Air Distributor

Photograph of Conductivity Probe Assembly

Basic Circuit Diagram for Conductivity Measurement

Photographs of Six-Channel Amplifier

Vfsicorder Response vs. Solute Concentration Plot, showing the Linearity of the Measuring System

Comparison of Purging and Feeding Step Inputs

23

35

36

37

42

47

51

56

57

59

60

62

65

67

69

70

72

Figure

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

Comparison of Liquid Holdup in Packed Beds

Effect of Liquid Flow Rate with No Gas Flow on Typical Response Curves

Effect of Particle Diameter on Typical Response Curves

Effect of Gas Flow Rate on TypicaJ Response Curves

Effect of Packed Bed Height on Typical Respon se Curves

End Effect Correction vs. Liquid Flow Rate

Operating Liquid Holdup in Packed Beds

Effect of Liquid Phase Reynolds Number on Peclet Number Without Gas Flow

Dispersion for Liquid Flow Through a Packed Bed Without Gas Flow: This Study

Dispersion for Liquid Flow Through a Packed Bed Without Gas Flow: Various Studies

Effect of Mean Liquid Velocity on Effective Axial Dispersion Coefficient for Raschig Rings

Effect of Mean Liquid VeJocity on Effective Axial Dispersion Coefficient for Spheres

Effec t of Gas Ve 1 oc i ty on the Di spe rsed . Liquid Phase Peclet Number

General Correlation of Dispersion Data for Counter-Current Air-Water Flow Through Packed Bed

Basic Circuitry for Amplifier-Demodu1ator­Adder Units

74

76

78

79

81

82

86

88

94

96

100

10 1

103

105

137

vi

Table

Il

Ill

IV

LIST OF TABLES

Nomenclature for Figure

Packing Characteristics

Experimental Data and Calculated Results for Liquid Flow

Experimental Data and Calculated Results for Simultaneous Counter-current Gas­Li qui d F 1 ow

vii

5

63

119

127

1. INTRODUCTION

Many segments of the chemical and process industries,

including the petroleum, pulp and paper, and metal lurgical

industries have processes which involve the contacting of a gas

phase and liquid phase, with and without a solid phase also being

present. Frequently these contactors take the form of packed or

fluidized beds. The contacting may be for the purpose of removing

entrained solids from the gases, for carrying out a chemical reac­

tion, for heating or cooling by direct contact of the phases, for

mass transfer such as absorption or dehumidification, or for any

combinations of these operations. ln view of the key position

occupied by gas-liquid contacting operations in the process indus­

tries, it follows that attention should be focussed on the funda­

mental aspects of heat, mass and momentum transfer in these sys­

tems. ln a recently completed study, Chen ( 15) studied the liquid

phase mixing characteristics of a novel gas-liquid contactor, in

which the contacting occurs in the presence of a low density solid

phase which is in a state of random motion due to the combined

action of the gas and liquid flow over the solid. ln correlating

his results for the mixing characteristics of the liquid phase,

Chen found that the even simpler case of mixing in a packed bed

with a static solid phase was reported in an incomplete and con­

flicting way. Since a packed bed provides a natural point of

reference for desired further studies of gas-liquid contacting in

a mobi Je bed, and since packed beds are of basic interest due to

their widespread use in the process industries, it was apparent

that a specifie study of longitudinal dispersion or axial mixing

2

in a packed bed was required.

Axial dispersion has been neglected in the calculation

of mass transfer coefficients in packed beds. Mass transfer data

obtained from wetted wal 1 or dise columns are not directly ap~li­

cable to the prediction of performance of packed absorbers. This

is because little is known about the flow of fluids over packing

surfaces. True values of mass transfer coefficients based on

the actual existing concentrations of materials can be evaluated

when the extent of this dispersion is known. The classical

design approach has been to assume a uniform velocity across the

bed, with 'piston-flow' through the bed. However, residence time

distribution measurements have shawn that this is far from being

an accurate description of the flow (8, JO, 20, 26, 58, 70, 83).

ln most real cases neither of the two limiting cases of 1plug

flow• or 'perfect mixing' apply. The Jack of established quan­

titative generalisations do not permit the inclusion of the axial

dispersion factor into rational design methods.

Axial mixing tends to reduce the concentration driving

force for mass transfer or chemical reaction from that Which

would exist for plug flow. A high degree of mixing may sometimes

be an advantage, for exampJe when consecutive reactions occur

( 107). For any counter-current contacter, the concentration

gradients in beth phases are Jowered by such mixing and the

driving force for mass transfer may be decreased considerably

(72). For a given column under conditions of axial mixing the

height of transfer unit, reported on the basis of Colburn•s equa­

tions, are higher than that calculated from actual mass transfer

coefficients. lt follows, that in order to be a~Je to predict

accurately the performance of any contacting deviee, precise

knowledge of the extent of axial mixing in both phases must be

known.

Consider a material balance over a thin cross-section

3

of a packed absorption tower as shown in Figure 1. There are

three modes of material transfer in and out of the section. With

the fol Jowing simplifying assumptions, one can arrive at basic

differentiai equations relating concentration variation along the

packed bed:

1. Uniform constant velocity in the bed.

11. No radial concentration gradients.

1 Il. The mixing is characterized by Fick's Law, where

effective dispersion coefficient i s used instead

of molecular diffusivity.

IV. 1 nterphase trans fer rate i s given by;

Rate = Kla(.cl - mcG)

whence for liquid phase:

2 -DL€L del+ L del+ KLa(cl - mcG) = 0

dx2 PL dx

and for gas phase:

( 1 • 1 )

( 1 . 2)

( 1 • 3)

The case for DL and DG equal to zero reduces equation ( 1.2) and

( 1.3) to the expressions which are currently used in mass transfer

FIGURE

MATERIAL BALANCE OVER A SECTION

OF PACKED BED

4

TABLE 1. NOMENCLATURE FOR FIG. 1

Symbol Meaning

Variables

G

L

A

p

ê

D

c

x

m

Subscri pts

Mass velocity of gas phase, lb./hr.-sq.ft.

Mass velocity of liquid phase, 1b./hr.-sq.ft.

Cross sectional area of tower, sq.ft.

Density of fluid, lb./cu.ft.

Fraction of bed occupied by fluid, cu. ft./cu. ft.

Effective diffusivity, sq.ft./hr.

Concentration, lb. mole/cu.ft.

Position in bed, ft.

Over-all Jiquid phase mass transfer coef­ficient, (lb. mole/cu.ft.)/hr.-(lb. mole/cu.ft.)

* Equi librium constant, cL= m cG where c~ is the concentration of a liquid in equi librium with a gas of concentration, CG.

L Liquid phase

G Gas phase

5

e e

D

(L,~cl) ( LPAcl) x LIQUID L x+ ÂX ... LIQUI

PHASE

(-DL A •t ::L ) ( del) -DLAcL r -dx . x *

X+ÂX

-- ~-----~----- -- - .... ---- ........ - -- -- --- - • ----- - ·-·-- INTERFACE

(G,:cG) - cAcG) 1\; x+âx x GAS -GAS --

( D A• dcG) PHASE ( DoA•G de~ -G G ëiX x dx + âx

x x+âx

*FLUX ACROSS INTERFACE = KLa Aâx (cL- mcG)

6

design. The resulting equations are fi rst order differentiai

equations and are easïer to apply than the second order equations

( 1,2) and ( 1.3). The fact that DL and DG are never zero in any

real situation indicates the importance of equations ( 1 .2) and

( 1 .3) in predicting the performance of continuous flow mass trans­

fer systems.

The phenomenon of axial dispersion is indicated by the

spread of residence times of the individual elements of a fluid

stream passing through any flow vesse!. Although the distribution

of residence times describes the steady state behaviour of a sys­

tem, i t is most readï ly obtai ned by the unsteady state method

known as the transient response technique. !n this technique a

particular input function may be imposed on the system so as to

yield the distribution directly, ln the present study~ a step

change of a salt tracer concentration has been employed as the

input signal. The transient response was obtained by subsequent

measurement of the outlet concentration as a function of time.

The transient response, in the usual situation~ is

inadequate to make a unique prediction of system performance.

The fact that the outlet residence time distribution does not

resolve the case history of each element of reactant fluid in the

system has already been weil documented (22, 110, Ill). However,

when a mathematical formulation which simulates the dispersion

process within the system is also known 9 it is possible to char-.

acterize the dispersion process with the correlating parameter of

the mode!. A bread review of models whlch have been used to des­

cri be the dispersion process in the flow systems is presented.

7

The present investigation may in many ways be considered

as a continuation of industrial and theoretical studies of gas-

liquid contacting involving 11 floating beds 11,

11 turbulent bed con-

tactors11, or in more general terms, "mobile beds" ( 15~ 29, 53,

137). ln order to establish a more reliable base for studying the

behaviour of mobile bed contactors~ however, it was found neces­

sary to conduct a caref4l study on a fixed bed system first. This

thesis is concerned with the study of the liquid phase response to

a step function in tracer concentration in the inlet liquid stream.

The primary objective of this study has been to evaluate the mixing

occurring in the axial direction in the dispersed liquid phase for

dry random packed beds of ceramic rings, with and without simul­

taneous counter-current gas flow. Most of the data aval Jable in

the literature regarding single-phase and two-phase flow relate to

laboratory experiments. The need for additional data on single

and two-phase flow in fui 1 scale equipment has been widely

stressed (59). The present investigation has been undertaken to

furnish badly needed data on axial dispersion of liquid phase, the

measurements being made on equipment which is intermediate in

size, i.e. larger than many laboratory investigations of the past,

but smal 1er than industrial equipment.

As a logical prelude to further studies, complicating

effects such as absorption or chemical reaction between the liquid

and gas stream have been avoided. An inert air-water system has

been selected for the purpose. The basic parameters studied are

liquid flow rate, gas flow rate, and packing diameter. ln view

of the need for experimental data in near industrial size equip

8

ment, it was decided to conduct studies in 1-ft. diameter column,

using industrial packing of 0.5, 1.0 and 1 .5-inch Raschig rings.

9

1 1. LITERATURE SURVEY

1 NTRODUCT ION

Nearly ali of the research in the area of packed beds

has dealt with gross properties such as pressure drop, holdup and

overall transfer coefficients. This is not surprising, since a

packed bed forms such a complicated fluid dynamic environment

that investigators have to describe the performance on the basis

of measured gross properties. For example, experimenters on

packed columns have made extensive reference to holdup (36, 39,

79, 85' 1 15' 1 1 6) .

Though residence time distribution for a continuous

flow system has been described as early as 1934 by MacMulin and

Weber ( 1 12), the application of this knowledge to packed beds is

of relatively recent origin. The methods of treating nonideal

flow behaviour in actual process equipment either have only

recently been developed or are yet to be developed. ln this

section the significant literature pertinent to the present

residence time distribution and axial mixing study wi 11 be

reviewed. This wi 11 include a review of holdup studies, axial

mixing studies in continuous flow systems, and comparison and

description of different flow models which have been put forth

by different investigators.

ln chemically reactive systems 9 the performance is

dependent upon the residence time distribution of the reactants

and the kinetics of the system. The performance of the system is

predictable for a change which is a linear function of an inten­

sive fluid property. However, for non-1 inear systems, such as

10

is the case for any reaction which is not first order, the know­

Jedge of residence-times and the kinetics of the system furnish

necessary but not sufficient conditions for predicting the per­

formance. For such cases a knowledge of actual experimental flow

pattern in the system is required in order to be able to predict

the performance. Since, in the present study a non-reactive sys­

tem has been chosen, the pertinent literature on the effect of

mixing on chemical reaction rate has been excluded from this

review. Further in two phase flow, only the literature directly

concerned with counter-current flow through packed beds has been

reviewed.

1. HOLDUP:

The volume fraction occupied by the Jiquid in an absorp­

tion column at any time during operation is frequently referred

to as holdup. Under normal operating conditions, the gas in a

bed occupies a Jarger portion of the voids than the liquid phase.

lt wi Il be seen that the experimental techniques for evaluating

the holdup are relatively simple but cannot be applied to the gas

phase. ln most of the previous works the fraction of total void

volume occupied by gas has therefore been determined from the

knowledge of liquid holdup.

ln the analysis of the performance of packed co1umns

such as those used for absorption, distillation, humidification

etc., estimation of liquid holdup in the voids of the bed would

Jead to better understanding of the mechanism of mass transfer.

Importance of holdup in mass transfer processes is shown by the

number of attempts made to correlate and explain data on the

Il

basis of the Jiquid holdup in such columns (36, 39, 85, 114, 115,

1 1 6).

Payne and Dodge (79) were the pioneers of holdup

studies. They reported holdup data on 10 mm glass rings in a

2.84-inch diameter column. They determined the amount of liquid

necessary to wet the walls and packing, which we now cali "static

holdup", by introducing a known amount of liquid into the column

and measuring the drained excess. The dynamic holdup was also

measured, starting from the condition of a steady-state flow of

Jiquid through the tower. Al 1 experiments were carried out

without gas flow through the column. After steady state was

attained, the liquid flow was stopped and simultaneously the

drainage was col Jected and measured. The sum of the drainage and

the amount necessary to wet the system was taken to be the holdup.

Their primary interest was to apply holdup data to mass transfer

absorption data, and no attempt was made to correlate their data.

lt remained for Fenske, Tongberg and Quiggle {36) to

give names to the individual components of the total holdup~ HT·

The static holdup, H5 , is defined as the liquid in the packing

which did not drain from the packing when the liquid supply to

the column is discontinued. lt is thus the amount necessary to 1

wet the tower packing. The amount of liquid that drained from

the packing after the liquid flow was suddenly eut off from

steady flow, is cal led the operating holdup, Hop· The total

liquid holdup, HT, is defined as the total liquid in the packing

under operating conditions. The relation between the three

holdups is given by:

12

( 2. 1 )

Simmons and Osborn (86) working with beds of spheres

and coke, concluded that the operating holdup varies linearly

with the mass flow rate of the liquid phase:

H0 p = bL ( 2. 2)

The constant, b, in equation (2.2) is independent of the packing

type but depends on the nature of the liquid. The liquids reported

in their work were water and kerosene. At about the same tïme,

Uchida and Fugita {99) were conducting experiments on beds of rings

and broken solids. Furnas and Bellinger (39) determined the

operating and static holdups for Raschig rings and Berl saddles.

They also concluded that the gas flow rate has negligible effect

on the liquid holdup below the loading point. The same conclusion

was arrived at by Elgin and Weiss (33) who studied the effect of

gas flow rate on the holdup in beds packed with Berl saddles,

rings and bal ls. Cooper, Christi and Perry ( 1 13) employed holdup

data to interpret the effect of gas velocity and high liquid rates

on Height of Transfer Unit (HTU) values measured in the desorption

of carbon dioxide in packed towers. Jesser and Elgin (34) in

their comprehensive study of liquid holdup in beds of glass

spheres, Berl saddles and carbon rings gave the fol Jowing relation

for operating holdup:

H = bls op ( 2. 3)

The exponent, s, depends on the type of packing used but not on

the size of the particles. The coefficient, b, is proportional to

the surface area per unit volume of packed bed. Static holdup was

13

found to be constant for given packing size and shape.

Shulman, UJ !rich and Wei ls (85) measured holdups for

various packings by weighing the entire JO-inch diameter, 36-inch

packed bed with a highly sensitive suspension system. They gave

fol lowing empi ri cal equation for total holdup:

H - bls T --

o2 (2.4)

where D is the diameter of a sphere possessing the same surface

area as the piece of packing. The coefficients, b and s, are

functions of the geometry, size and material of packing. The

static holdup, which is a measure of the accumulated semi-stagnant

liquid, was related by equation (2.5):

( 2. 5)

The coefficient a, depends on the material and shape of the pack­

ing whi le rn is a function of shape of the packing only.

Most of the correlati·ons proposed for prediction of

holdup may be observed to be dimensionally inconsistent. By

making use of the fact that holdup is virtually independent of

the gas flow rate below the loading point and using dimensional

analysis, a generalised quantitative correlation was presented by

Otake and Okada (78) for a wide variety of combinations of packing

geometry, size and physical properties of the liquid. Combining

their data with those of previous studies {33, 34, 86, 99), they

reported the following correlation for operating holdup for

Raschig rings, Berl saddles and spheres of sizes from 1/4-inch to

1- inch.

14

= 1. 295 (~Lr 671 d!~plO. 44 adp ( 2. 6)

Davidson ( 1 14), based on theoretica1 considerations,

obtained a simi Jar correlation as given below:

(2.7)

The correlation (2.8), proposed by Varrier and Rao ( l 17) appears

to be a ·modification ·of Otake 1 s correlation in which d3 has been p

rep laced by 1 /N.

= 7 .. 1 2 ( L ) 0 . 6 7 ( :Jfi_ )-0 . 44

1 /3 NJ1.2

N }J.

( 2. 8)

where N is number of pieces of packing per cubic foot of packed

volume.

Recently, Mohunta ahd Laddha (73) gave a generalized

correlation (2.9) for predicting the liquid phase holdup in terms

of packing density, equivalent spherical diameter (defined as

diameter of a sphere having the same volume as a piece of packing),

for operating holdup for Raschig rings, Lessing rings and spherical

packings.

( 2. 9)

Direct measurement method was employed ln al 1 earlier

works. Otake and Kunugita (76) were the first to apply tracer

techniques in the study of holdup. Residence time of the liquid

in the bed was used to calculate total Jiquid holdup. They found

that total liquid holdup was proportional to the interstitial

15

ve1ocity. They were successful in determining static ho1dup by

extrapolating to zero ve1ocity on a plot of HT versus interstitia1

velocity. Their results for static and operating holdup for

Raschig rings are given by equations (2. 10) and (2. l 1).

H5

= 0.038 d

Hop= 1.75 x J03(~~(d3

;;;2

) - 1

where d is diameter of rings in centimeters.

(2.10)

(2.11)

De Maria and White (24) employing transient response

technique to gas phase related the fraction of the total voidage

occupied by gas by equation (2. 12).

4 -6 ( )-2.31 = 0.90 x 10-3. 3 x JO d/dt Rel (2.12)

and since

(2.13)

total holdup was given by equation (2. 14)

HT = f 1 - 0 . 9 0 x 10-3. 4 3 x 10 d 1 dt ReL ( 2. 1 4) [

-6( ,-2.31 ]

This resu1t is independent of the gas flow rate 9 as indicated by

previous investigators. As indicated earlier that fG is much

higher than HT, the above relationship should be used with caution

because the determination of total holdup involves subtracting a

large number from another large number to geta smal 1 number by

difference. The usefulness of equation (2.1 1) is also restricted

16

by the fact that it involves interstitial velocity, which depends

on the liquid holdup in the system. The fact that this equation

does correlate experimental data qui te satisfactori ly more than

offsets this criticism, especial ly so when it is possible to

determine actual residence times of the liquid phase in the system.

Therefore this is by far the most general correlation avai Jable

for the determination of holdup. Transient response technique has

also been successfully used for the measurement of liquid holdup

by Schiesser and Lapidus (83) in their studies of flow distribu­

tion in a tric~e-bed column.

2. RESIDENCE TIME DISTRIBUTION AND LONGITUDINAL DISPERSION

Unti 1 recently the transport of mass in the direction

of flow by a 11 di ffusion mechanism11 has been neglected i·n the study

of rate processes. A number of theoretical studies ( 1, 2, 4, 5,

8, 9, 20, 25, 43, 58, 63, 65, 93, 94, 101, 126, 127, 128) and

experimental investigations (10, 11, 13, 15, 24, 26, 30, 31, 41,

42, 46, 48, sa, 68, 70, 72, 76, 82, 87, 96) have been made to

determine the nature and magnitude of the axial dispersion mecha­

nism. Various investigations regarding the axial dispersion that

have been reported in the Jast decade for non-fluidised continuous

flow systems can be categorised under three headings~

1. Single-phase flow through empty pipes,

1 1. Single-phase flow through packed beds,

111. Two-phase flow through packed beds.

1. AXIAL DISPERSION IN P·IPE FLOW: Taylor (93), in 1953, con­

sidered Poiseul 1 le flow where axial dispersion is due to parabolic

17

veloclty profile and transverse molecular diffusion. He described

the system as equivalent to one having a flat velocity profile

equal to the mean velocity V, and an effective axial dispersion

coefficient, DL. This mode! impl icitly defines the axial disper­

sion coefficient~ which Taylor found to be

where Dv is the molecular diffusivity of the fluid species. The

effective Peclet number in Poiseul Ile flow, according to Taylor

(93) and van Deemter et al, ( 1 19)$ becomes:

(2.16)

This analysis is valid only if the time constant for transverse

molecular diffusion ( ~ d~/Dv) is not much larger than the average

residence time, i.e.

(2. 17)

which yields,

Pe x L/Dt>J92 (2.18)

where Lis the Jength of tube.

Since the values of Pe for Jiquids as reported by

Taylor (93) are of the order of Jo-2 9 this leads to excessively

high values of L/dt for liquids.

Aris (120, 121) extended this work. By using the method

of moments he showed that the axial dispersion effect is additive

and thus modified equation (2. 15) to glve:

(2.19)

18

He also generalised the entire treatment to include ali types of

velocity distribution with any vesse! geometry. He showed that

the coefficient given as 1/48 by Taylor is real ly a function of

tube shape and velocity profile. Taylor (94) later extended the

treatment for turbulent flow in tubes and correlated the effective

axial dispersion coefficient wîth fanning friction factor.

Tichacek et al. (96) using experimental velocity

profiles over a wide range of the turbulent region (Re = 2,500 -

2.19 x 106) calculated axial Peclet number which show limited

agreement with the experimental results of Fowler and Brown (38)

and Keys (52). He also found that effective longitudinal disper-

sion coefficient was quite sensitive to variations in velocity

profile. Sjenitzer (88) correlated a large number of measure­

ments, sorne of which were performed with commercial pipe !ines,

and proposed the empirical relationship,

( 2. 20)

.For turbulent flow in pipes at Re.>10 4, the Peclet number for

longitudinal dispersion is much higher than for laminar flow and

near plug flow behaviour results. This is because of transverse

transport of mass and momentum due to turbulence which flattens

velocity profile and greatly increases radial mixing.

Experimental results reported by a number of investi­

gators for dispersion in tubes are in rather good agreement with

each other. Taylor•s approximate solutions for reasonably large

values of residence time predict nearly the same results. How­

ever, for slow flows the effect of Peclet number has not been

established satisfactori ly and the limits of applicabi lity of

19

the Taylor-Aris solution are in doubt.

Very recently, Ananthakrishnan, Gi 11 and Barduhn ( 1),

using modified Peaceman and Rachford ( 122) numerical method,

have obtained a complete numerical solution to the equation

describing laminar flow in tubes with both axial and radial mol­

ecular diffusion. Their numerical results for both small and

large values of Peclet number corroborate the Taylor-Aris asymp­

totic solution for large values of residence time. Crookewit et

al. (18) have given results for flow in an annular region.

The spread of residence times in laminar flow in pipes

is a consequence of poor radial transport. The dispersion for a

viscous flow is shown to be considerably reduced by spira11ing

the tube. Secondary flow is produced in any curved conduit by

the action of centrifugai force which acts most strongly on the

fluid near the centre of the curved path. ln tubes, this has

shown to produce two symmetric circulation patterns (57).

Kramers (59) suggests that the secondary flow in a curved pipe

results in considerable decrease in the spread of residence

times. Koutsky and Adler (57) improved the transverse dispersion

by winding the tube into a helix. As a consequence, the apparent

longitudinal dispersion èoefflcient for.:taminat flow in curved

tubes is surprisingly sma11.

11. AXIAL DISPERSION IN SINGLE PHASE FLOW THROUGH PACKED BEDS:

When a single homogeneous fluid flows through the interstices of

packed bed, the elements of the fluid undergo random changes of

velocity. These variations in the local velocity cause a disper­

sion in the direction of flow causing a self-mixing of the flowing

20

fluid. One of the practical interests in the effects of longi­

tudinal mixing has been in continuous flow through process equip­

ment for the purpose of mass or heat transfer between the fluid

and the solid phase or of a chemical reaction. One of the

examples of recent interest is the process of chromatographie

separation, where axial mixing is undesi rable, si nee it leads to

a broadening of the peaks and a Joss of selectivity (100).

The most common approach to describing fluid dispersion

has been the assumption of a single eddy dispersion mechanism in

the bed. Whi le this may not be physical Jy correct ( 123), the

mathematical description of fluid dispersion assuming a single or

average eddy dispersion coefficient for the bed often results in

a successful correlation of the data for single phase flow.

Study of axial dispersion in packed systems appears to

have started at Princeton University where Deisler and Wi Jhelm

(26) in 1953 studied dispersion in beds of porous solids using a

frequency response technique. They were able to relate the effec­

tive dispersion coefficient to amplitude attenuation and phase lag

of a sinusoidal input. McHenry and Wilhelm (79), employing the

same technique but with improved precision, concluded- that the

axial Peclet number for spheres based on particle diameter was

not detectably different from 2(actual range, 1.6 to 2.3} over

the· range of Reynolds number studied ( 10-400). The significance

of the value 2 derives from the case of perfect mixing in every

void of the bed, which leads theoretically to a Peclet number of

2. The published gas residence time distribution.results of

De Maria and White (24} show serious asymmetry about the mean,

21

indicating more uneven distribution of gas (channeling). For a

range of gas phase Reynolds numbers from 5.5 to 200, they found

that the Peclet number, ud/DL, is independent of Reynolds number

and equal to 1 .94.

Sine lair (87) using Mercury vapour in an air stream and

ultra-violet absorption principle, measured longitudinal disper­

sion by frequency-response technique. The longitudinal Peclet

numbers for Reynolds number of 4.1 and greater, agree with those

of other workers in that values of the Peclet number of about 2

are obtained. Carberry and Bretton ( 13) have briefly reported a

few experimental results of longitudinal dispersion in packed

chromatographie columns, using helium as a tracer in air. These

measurements at low Reynolds numbers (Jess than 1) yielded a value

of the Peclet number consistent with the molecular-diffusion coef­

ficient of the helium-air system. This substantiates the princi­

ple that at low flow rates the effective dispersion coefficient

should reduce to the molecular diffusivity.

lt has been easier to measure concentrations of tracer

material in liquids than in gases, a fact that has resulted in

more information being now avai Jable for liquids than for gases.

Oanckwerts 1 (20) widely quoted 1953 paper was the first of its

kind and provided the fundamental basis for axial mixing studies

using residence time distribution of the fluid flowing through a

continuous system. He concluded that for any incrementai volume

of fluid entering a packed bed, there would be an infini te number

of possible residence times dependent entirely on the path taken

by the incrementai volume through the bed. He also introduced

22

the concept of 11 hold-back11 and 11 segregation 11, which were later

modified to explain real systems. He applied his analysis to a

single experiment in which a step input of tracer was introduced

into a bed packed with 3/8-inch Raschig rings. The Reynolds

number, dl, of the flowing water was 22 and the analysis of break-JI through curve gave a Peclet number, ud, of 0.55 which is shown in

DL Figure 2. The experimental residence time curve was found to be

in good agreement with that predicted from 11 diffusion-model 11•

Kramers and Alberda (58) used the frequency response

technique to determine axial dispersion. Their analysis showed

that frequency response diagram for a system with perfect piston

flow superimposed on longitudinal diffusion is identical with

that of n equal volume perfect mixed stages with the same total

residence time, and obtained:

= 2n (2.21)

where n is the number of equal volume, perfectly mixed cel ls in

series. The validity of equation (2.21) is restricted to very

high values of n, but has been used to obtain rough estimate of

the longitudinal dispersion in a fluid flowing through a packed

bed. Their experiment at two water flow rates corresponding to

Reynolds number of 75 and 150 gave the Peclet number of 0.965 for

both the runs and are also shown in Figure 2.

Aris and Amudson (4) worked out the case for spheres

packed in a rhombohedral arrangement and found that a perfect

mixing cel 1 corresponds to a length equal to 0.816 times the

particle diameter or:

FIGURE 2.

LIQUID PHASE DISPERSION DATA FOR SINGLE

PHASE FLOW THROUGH PACKED BED

23

e

6 J..

4 ~

2 J..

- 1.0 Q)

a. 6

4 c

2 l 0.1

1 1 ' • • 1 1 J 1 .... 1- 1

A - CARBERRY a BRETTON ( 13)

x - DANCKWERTS (20)

B - EBACH a WHITE (31)

+ - KRAMERS a ALBERDA (58)

C - LI LES a GEANKOPLIS ( 68)

D - STRANG a GEANKOPLIS ( 92)

D + c

D

c A

1 1 • 1 1 • • • l 1 1

2 4 6 810 2

ReL 4 6 100 2

-·

e

• --

-

---•

-1

4

24

(2.22)

However, for random packed beds they obtained the same equation

(2.21).

Ebach and White (31) used dye as a tracer and employed

sinusoïdal and impulse input in a bed packed with glass spheres,

Raschig rings, Berl saddles and lntalox saddles. The values

obtained for Peclet number scattered in the range of 0.5 to 1.0,

over a Reynolds number range of 0.01 to 200. Variations in liquid

viscosity from 1 to 26 centipoises were found to have no detect­

able effect on DL· They used the equivalent diameter, de, (defined

as the diameter of a sphere with the same volume as the particle)

and correlated their data by

(2.23)

for deuP Jess than 100.

J1 Carberry and Bretton ( 13) in their study of axial dis­

persion of a continuous water phase, flowing in 1.0 and 1.5-inch

diameter column pointed out that for particle Reynolds number,

d u p !p. , between 5 and 100, the Pee let number, ud/DL i s approxi­

mately constant with extreme values lying between 0.2 and 1.3

Their results representa significant departure from those of

Ebach and White (31). There is sorne uncertainty with respect to

the results of Carberry and Bretton due to the presence of long

tai ls in their experimental residence time curves, which they

referred to as 11 bed capaci tance". They offered cri ti ci sm for the

25

work of Kramers and Alberda on the grounds that the use of phase

angle data rather than the amplitude attenuation in the original

calculations, yields axial dispersion coefficients 50 to lOO%

different from those reported.

Jacques and Vermeulen (124) employed an electrical con­

ductivity cell placed di rectly in the bed to measure the response

to a step input of salt solution. They varied the porosity of the

bed by packing 3/4-inch spheres in four geometrie patterns. The

use of 1/4 and 3/4-inch Raschig rings, 1-inch lntalox saddles and

1/4-inch pellets along with the spheres gave a good range of

particle geometries. Using equivalent spherical diameter (as

defined for equation 2.23) they obtained good correlation of

their data. For a Reynolds number Jess than 30, the Peclet number

is approximately O. 18 and 0.67 above 200.

Strang and Geankoplis (92) used di lute solutions of 2-

naphthol as a tracer and obtained concentrations by measuring the

transparency of the liquid in a void gap to ultra-violet light.

They also employed frequency response technique. For Reynolds

number, dl, from 10 to 50, they found the Peclet number to be

0.88 for~lass beads and 0.56 for Raschig rings. They pointed

out that short columns gave incorrect results owing to end

effects. ln a study directed exclusively to determine the effect

of packed length and particle diameter on the Peclet number,

Li les and Geankoplis (68) reported that DL is not a function of

length. The Peclet numbers of 3 and 6 mm beads were approximately

the same, whi le those for 0.47 mm beads were much lower. This was

attributed to a possible change in flow characteristics for fine

26

particles of this dimension. The effect of interstitial velocity

on DL for spheres was given by fol lowing empirical equation:

(2.24)

Cairns and Prausnitz ( 10 9 11) used an electrical conduc­

tivity cel! to measure the response to a step input of salt solu­

tion. Three sizes of spheres were investigated and the results

given as a plot of udh/DL versus dh, where dh is the hydraulic

diameter for a packed bed and includes the effect of bed diameter.

For spherical packing, the hydraulic diameter is given by:

€ dt dh = ------=-----~---

3 dt ( 1 -f) + 1 2 d

(2.25)

Over a range of udh from 0.04 to 20 sq.cm./sec., they found that

the Pee let number varies from about 0.3 to 1 .0.

Gottschlich {43) tried to explain the discrepancy

between the axial Peclet numbers for gas-flow and liquid-flow

experiments by including the effect of a stagnant fi lm, This fi lm

has been compared with the stagnant fi lm calculated from mass

transfer experiments and calculated that the two kinds of fi lm

appear to be the same. Hiby (48), Cairns and Prausnitz (JO) have

indicated that the presence of wall effects gives higher disper­

sion. Radially integrated results are shown to be higher due to

a non-flat velocity profi Je, especially when the ratio of diameter

of bed to the diameter of particle, dt/dp, is Jess than 15.

27

111. AXIAL DISPERSION IN TWO-PHASE FLOW THROUGH PACKED BEDS:

The problem of evaluating the Peclet numbers of each

phase of a two phase system presents more challenging difficul­

ties than for single-phase systems. This accounts for the meagre

data avai Jable for these systems. The main problem centers around

the determination of concentration of the two phases since the

presence of one phase interferes in the determination of concen­

tration of the other.

Kramers and Alberda (58) in their study of the frequency

response of 0.7 m high packing of 10 mm Raschig rings found that

the residence time distribution was as that predicted by JO to 20

perfect mixers in cascade. They reported increase in longitudinal

dispersion with decreasing liquid Joad but found that the gas

velocity had little influence below the loading point. At zero

gas flow rates, the Peclet number, ud/DL varies from 0.3 at a

superficial liquid flow rate of 3200 lb./(hr.-sq.ft.) to 0.5 at

a liquid rate of 6900Jb./(hr.-sq.ft.).

ln a particularly significiant study reported in 1958,

Otake and Kunugita (76) determined the Peclet number of the dis­

persed water flow in a bed of 0.785 and 1.55 cm Raschig rings

for Reynolds number range of 70-100 and gas flow of zero or

13 lb./(hr.-sq.ft.). Their results for laboratory size equip­

ment were correlated by: 0 5 0.333

Ot) = 0.527(714. (~fl L L L

( 2. 26)

28

or ( ) 0 5 ( 3 2) -0 . 3 3 3

~~~t=l.897 dï/ L. d;/ L (2.27)

As shown by the correlation, the Peclet number is independent of

the gas flow rate. Since their flow rates were alwqys below the

loading conditions this result is to be expected from holdup find­

ings. The values of Peclet numbe·r in the case of irrigated packed

towers were found to be very high as compared with those in other

types of flow reactors. They concluded that the liquid flow in a

packed bed tower closely resembles the piston flow, the more so

by increasing the length of the bed and the mean velocity of

liquid and decreasing the diameter of packing. However, equation

(2.26) fai led to correlate the data of Hofmann (49), who used a

column of industrial size. ln 1965, Chen ( 15) reported experi­

ments with fixed beds of0.5, 1.0 and 1.5-inch spheres using

transient response technique. When Chen related his experimental

data for the Peclet number of the dispersed water phase by equa­

tion (2.28), he found significant departure from that obtained by

0 ta ke et a 1 . ( 7 6) .

( 2. 28)

Jacques et aJ. (50) conducted longitudinal dispersion

experiments in a 2-ft. diameter gas absorption column. Using

Einstein•s statistical mode! (32) their reported Peclet numbers

for both phases were found to be much lower~ perhaps the lowest

of those in previous studies. Liquid phase Peclet numbers were

found to increase with liquid flow rates and no quantitative

effect of gas rate was observed.

29

Stemerding ( 129) for a 10-ft. high column packed with

10 mm. rings found the axial dispersion coefficient for the con­

tinuous water phase to be essentially constant, i.e. to be inde­

pendent of water flow rate over the range 0.2 to 1.0 cm./sec.,

and dependent only on the air flow rate, ln a simi Jar study for

the case of countercurrent flow of liquid and gas over a completely

submerged packing of Raschig rings conducted by Ot~ke et al. (77),

results were found which contradicted those obtained by Stemerding

( 129), but were correlated weil by the equation (2.29):

Pe = 1.425(Ga)-0.333 (Rel)0.777 ( 2. 29)

ln a trickle-bed reactor, the flow pattern of water was

investigated by Lapidus (61) and Schiesser et al. (83). The extent

of axial mixing occurring was found to be so small asto justify

the assumption of plug flow behaviour.

Recently, Sater (82) used radioactive tracer technique

which had the advantage of measuring the concentration of a stream

without disturbing the flow patterns. The technique developed by

Aris (2) for measuring the response at two positions, thus doing

away with the necessity of having a precise knowledge of the tracer

input, was applied. Sater claimed to have obtained reliable

results for both liquid and gas under two-phase flow conditions.

Working from the lines of Otake and Kunugita (76) he could cor­

relate his data for liquid phase Peclet numbers under two-phase

flow conditions by the equation (2.30)~

[ ~~ ) = L

19.4[ 1/L r-747 (d3g~2)-0,69 (ad}l.97

r L fl L

( 2. 30)

30

However, Sater 1 s correlation, (2.31), for the gas phase Peclet

number under two phase flow conditions predicts values of the

Peclet number to be one-sixth of the values reported by De Maria

and White ( 24).

( ~~ l = o.o585 (adlz.5a ( ïl r.66a x 10-o.ooz59(JJIL (2.31)

De Maria and White (24) found that gas phase Peclet

number decreased with increasing liquid and gas Reynolds numbers.

Their calculated liquid phase Peclet number for Raschig rings were

satisfactori ly correlated by the equation:

( 2. 32)

Since the pioneer work of Danckwerts (20) in 1953 and

Taylor (93, 94) in 1954, there has been considerable interest in

the field of axial dispersion of fluid in continuous flow systems.

Unsteady-state stimulus-response tracer technique has been applied

without exception. Although there has been considerable progress

in the development of theory for these systems, there is an acute

need for new experimenta] techniques and research tools to explore

the micro structure of these flow systems. Brenner (9) and

Miyauchi (72) have presented a mathematical solution to diffusion

madel which does not requi re the assumption of an infinite packed

bed. Mickley et al. (71), in an attempt to elucidate the mecha­

nism of transport phenomena in packed beds by making basic fluid

flow measurements of velocity profiles and turbulence parameters,

distinguished between commonly confused (8) eddy diffusivity and

dispersion coefficients. The eddy diffusivity is the proportion-

3 1

ality coefficient between the local rate of momentum or mass

transfer and the time-mean velocity in a turbulent flow field,

whi Je the dispersion coefficient is a constant appearing in dif­

fusion models of a packed bed and is a gross property of the bed

which depends on the flow patterns. They compared calculated

values of eddy diffusivity with the dispersion coefficient

obtained from the Jiterature, and showed that eddy diffusivities

are much smaller than either radial or axial dispersion coef­

ficients.

The effect of axial dispersion in heat and mass transfer

·studies has been shown by Ogburn (75), Sleicher (89), Epstein (35),

Carberry ( 12), Douglas (28), Levenspiel (63), Stemerding (91) and

Miyauchi (72). From thei r studies it can safely be concluded that

plug flow is a poor approximation in many practical design prob­

Jems.

lt is evident from this review that there is Jack of

agreement in the magnitude of the Jiquid phase Peclet number for

a given liquid Reynolds number. ln fact sorne workers (JO) have

questioned the justification of using Reynolds number as the cor­

relating parameter on the ground that the fluid viscosity has very

little effect on the dispersion (31). Others {49~ 76) have felt

the necessity of introducing other parameters like packing diame­

ter, tube diameter, void fraction, etc. Also, the analyses have

been based on experimental studies for which it has been necessary

to assume a perfect input signal, which of course can never quite

be realized. Peclet numbers were evaluated from the concentration­

time response on the assumption that the concentrations were

32

measured directly in the packing at the end of the test section

(5). Needless to say, there does not exist any simple method to

do this, and experimenters have to use measurements taken just

outside the end of packed bed. The errors caused by these dif­

ferences between the mathematical madel and practical application

are difficult to estimate. Further, most of the work reported to

date has been with laboratory size equipment and packing. Since

scale up procedure for these flow systems is so uncertain,

Kramers (59) emphasized the need for experimental work in near

industrial size equipment, bath for single as wei 1 as two-phase

flow.

Thus, even though there is much reported data for

single phase flow systems, it is obvious that further experimental

work is needed in which al 1 possible experimental errors are mini­

mized. Two-phase flow is inherently more complicated than the

single phase flow and there are relatively much Jess data avai J­

able. Further~ the reported results of De Maria (24) and Sater

(82) for the gas phase Peclet number show wide disagreement,

whi le the liquid phase Peclet numbers reported by Otake (76) and

Sater (82) show much scatter with the published correlations.

Even more important, the results of Chen ( 15), Otake (76),

Lapidus (61) and Schiesser (83) do not lead to any generalised

correlation. Abnormally low values of Peclet number reported by

Dunn et al. (30) makes one realise the difficulties to be encoun­

tered in two-phase system due to possible flow irregularities in

their system.

1 1 1 •

RESIDENCE TIME DISTRIBUTION AND TECHNIQUES OF EVALUATING EFFECTIVE LONGITUDINAL

DISPERSION COEFFICIENT

1 NTRODUCT ION

33

The purpose of contacting fluids in process equipment

is to modify them one way or the other. The performance of con-. . . .'. tact1ng equ1pment is determined by the flow structure in the

equipment. 1 t has not yet been possible to explore the inter­

stices of the bed, because the presence of probes interferes with

the flow patterns and true information cannat be obtained. Also,

fluid flow in a packed bed is difficult to describe mathemat­

ical ly, because of the complex flow patterns involved (33, 34,

39). However, experimental methods which have been in use for

some time now in process dynamics and control studies have found

increasing use in the last decade in exploring the behaviour of

process equipment. The stimulus-response technique, though it

fai ls to give the exact flow path in the bed, does however give

partial information about the gross flow behaviour in continuous­

flow systems. Specifical ly, the transient response of such a

system gives an indication of how long different elements of

fluid remain in the vesse]. This is not sufficient to give a com­

plete description of the flow behaviour, but often is relatively

easy to obtain and interpret. Coupling this information with the

theoretical models of the system, one can evaluate the parameters

of the model which simulates the real situation.

34

The behaviour of ali continuous physical systems can

be approximated by linear differentiai equations. No real system

is exactly linear. lt will be seen later in this chapter that

longitudinal dispersion in packed bed can be simulated by a linear

second-order differentiai equation with constant coefficients.

TRANSIENT RESPONSE

The input signal is simply a tracer introduced in a

definite manner into the fluid stream entering the system. The

transient response methods involve the use of an input signal and

subsequent meas.,urement of the output as a function of time. The

time-variant output signal is then mathematical Jy given by the

Convolution Integral of the system. Step, impulse, sinusoïdal

(cyclic) and random signais have been described in the literature.

Figures 3, 4 and 5 show various input functions and their response

curves. Sine waves (2, 26, 31, 44, 68, 70, 77, 87, 90, 92),

impulse ( 13, 21, 31, 130) and step functions ( 10, Il, 15, 20, 24,

30, 61, 83, 124, 128) have been used to obtain information about

axial dispersion phenomenon.

For flow systems a signal which modifies an intensive

property of the fluid being processed must be used. Of these,

the most readi ly measurable properties are temperature and con­

centration. Concentration measurement is straightforward and

easiest of the two and thus has been adopted as a source of signal

in this study.

FIGURE 3

RES PONSE CURVES FOR AN

IMPULSE INPUT FUNCTION

35

w 1.0 Ct>

a-----r-----t--+ PWG FLOW -----1

0 1.0

INTERMEDIATE CASE

DIMENSIONLESS TIME, t 18

FIGURE 4

RESPONSE CURVES FOR A

STEP INPUT FUNCTION

36

e

0

~ (.)

Il

LL

1.0

0

STEP INPUT

INTERMEDIATE

CASE

PLUG FLOW

1 r 1-----

0 1.0

DIMENSIONLESS TIME, t/8

e

2.0

FIGURE 5

RESPONSE CURVES FOR A

SINUSOIDAL INPUT FUNCTION

37

z 0

W!;;:{ 0::> =>z 1-w __..._ a.. :Et( <t

w CJ)

- z .. 0 LIJ a. :e ~ ~

a::

b z w => 0 w a:: LL.

~ 'NOil~~lN3~NO~ ~3~\1~1

38

COMPARISON OF INPUT SIGNALS

Impulse signal requires instantaneous injection of

sufficient amount of tracer into the ingoing stream during a

period of time which must be very short compared to mean resi­

dence time, an operation which is very difficult to achieve in

practice. The use of an impulse can however be justified in

cases where average residence time is large compared to the time

of injection. Practically it is difficult to satisfy this con­

dition for cases in which short residence times and high flow

rates are encountered. The impulse input is especially useful

when radio-active detection techniques are used. The step input

on the other hand is simpler and more practical to obtain. This

input signal is produced by an abrupt change in tracer concentra­

tion. A continuous analysis of the effluent fluid gives an

5-shaped plot of tracer concentration versus time, which, when

expressed in dimensionless form, is generally referred as an

F-diagram { 20). 1 t has been much simpler and accurate in prac-

tice to use step input, e.g. by the use of solenoid or toggle

valve in a tracer injection Jine.

The spread in C-curve and the shape of F-diagram give

an indication of the dispersion as wi Il be shown later. The area

under the response-curve for step input di rectly gives residence­

time which can be related to Jiquid holdup of the system. The

mean residence time and liquid holdup is given by:

e = { 3. 1 )

39

= (3.2)

Apaft from step and impulse inputs, the sinusoïdal input

has found extensive application. The use of the sinusoïdal func­

tion derives from the fact that although step and impulse inputs

are conceptually very straightforward and easy to use, they tend

to be i naccurate under ei ther of two ci rcumstances:

(i) the mean residence time in the system of interest is

very short,

(ii) the change in concentration in the outlet occurs during

a very small interval of time (i.e. there is a close

approximation to plug flow).

To meet this difficulty Wilhelm (26) and Kramers (58) developed an

alternative method of using sinusoïdal inputs. No direct informa-

tion, like mean residence time, can be extracted from the response.

The mathematical analysis is greatly simplified because of the

continuous nature of sine function. With the use of the diffusion

model, the effective axial dispersion coefficient can be evaluated

from equation (3.3):

(3.3)

Ml X 1 NG MO DELS

lt has already been pointed out that the transient

response in itself is insufficient to give any detai led informa­

tion about the flow pattern. Also, it has been impractical to

obtain and lnterpret actual experimental flow patterns. The

rational . design of fixed bed systems requires a mathematical

formulation (or madel) to describe the physical behavid~r of

40

the system. If such a mode! approximates the real flow behaviour,

the response curves predlcted from the madel wi 11 match the

experimental :transient response. Even ln the absence of chemical

reaction, the deve 1 opment of a sui tab 1 e mode 1 wh i ch adequate 1 y

includes the residence time distribution of elements flqwing

through the bed and the various gradients of heat and mass is a

difficult problem. However, even though a mathematical madel may

not include many of the complications of the real system, it may

be satisfactory for sorne particular purposes. Although a poor

madel may contain too many parameters, the number of parameters

used does, nonetheless, give an indication of how closely the

madel predicts the real behaviour. However, whi Je a multi­

parameter madel may closely approximate actual performance, the

associated mathematical campi ications wi li 1 imi t i ts usefulness.

For packed beds and tubular reactors it is usually

adequate to assume one-parameter models, since frequently these

are satisfactory for predicting. system performance. Because of

the essentially discrete physical nature of a packed bed, it is

not surprising that attempts have been made to formulate a madel

different from that used to describe the mixing characterfstics

in continuum flow. Sorne of the most common mixing models for

particulate flow systems are described next.

41

MIXING-CELL MOOEL

The mixing-cell model (2, 13, 55, 58, 63, 100, 134)

assumes that the packed bed can be characterised by a cascade of

perfectly mixed cel Js. ln packed columns, the voids are con­

sidered to constitute the unit cells for such mixing. At high

Reynolds number flow (turbulent-flow regime), the individual voids

may each approach perfect mixing. Fig. 6 i 1 lustrates this analogy.

ln each ce11, perfect mixing is assumed to occur, such that the

effluent from the cell has the same composition as the fluid at

al 1 points within the cell. ln practice, even if the mixing is

not complete in each void, a series of voids may be represented

theoretically by a m.fxing cel 1 ( 135).

A mass balance on a single component wi 11 result in the

fol1owing differentiai equation:

Vn rn = ven - 1 (3.4)

where Vn = volume of the nth ce11

= rate of production of component

v = volumetrie flow rate.

The volume of each cell times the number of cel Js equal

to the void volume;

( 3. 5)

so that

{ 3. 6) v nv n nu

FIGURE 6

MIXING CELL MODEL

42

C)...l Z...l -LIJ ~ 0 - c:\1 l") • :e

where V = total bed volume

€ = porosity of the bed

h = length of the bed

u = average flow velocity

Substituting equation (3.6) in (3.4), one gets:

h den nu dt

h nu

43

( 3. 7)

The mixing in the bed is therefore characterised by only one par­

ameter, n, the number of perfectly mixed cel Js in the bed.

For non-reactive systems which are subjected to an

impulse input, the solution to equation (3.4) by Laplace Transforms

gives the response function, referred to as the C-curve. For a

series of n equal volume, perfectly mixed cells the solution in

dimensionless form is given by:

n Tn-1 -nT c .. n = e ( 3. 8)

(n - 1 ) !

where T = dimensionless ti me, t(j

t = ti me variable

.(} = tota 1 average res i denee ti me

wi th mean f = ( 3. 9)

and variance about mean cr2 = 1/n (3.10)

Further, equation (3.4) for non-reactive systems~ rn = 0, can be

wr i tten as:

den v -- + (3.11) dt vn

44

den n n __ + Cn = en 1 e e -dt

(3.12)

as n vn = (i.v . (3.13)

Equation (3. 12) is a linear differentiai equation, whose solution

(59) for a step change, wi th initial conditions, en = 0 for

t = o(~) and al 1 n, and c0 = fort =0(+), is given by

F( t) = 1 -nT x [ 1 + nT + (nT)~ . - e . . . ..

2J

+ 1 (nT)"- 1] (~.14) {n - 1 ) !

lt is interesting to note that the slope of F-diagram from equation

(3. 14) is in fact the C-curve, given by equation (3.8). For large

values of n, Sti rling•s approximation for the factorial leads to:

(3.15)

(3.16)

This model does not give a di reet indication of the

mixing occuring in the system, but by matching experimental res­

ponse curves with those predicted by equation (3.14), it is pos­

sible to evaluate n, the number of equal volume perfectly mixed

cells in cascade .. Once n is determined, it isstraight-forward to

make predictions of system performance. The mode! has been

extended to include the case of unequal cells in series ( 132) and

to a three-dimensional array of mixing cells {25). lt will be

shown that the number of equal volume perfect mixers in series,

n, corresponds to one•half of the column Peclet number, which is

the mixing parameter of the diffusion mode!.

DIFFUSiON MODEL,

Mixing in a packed bed is the result of splitting,

acceleration, deceleration, and trapping of elements of fluid

45

as they pass through the bed. if these individual modes of mixing

are repeated a large number of t1mes 1 and if the length over which

a single mixing effect acts is smal 1~ the resulting overal 1 mixing

process can be described by the diffusion mode]. in this mode! it

is assumed that the actual dispersion process in a packed bed can

be described by the same mathematfcal mode! as is used for molecu­

lar diffusion, with the additional superposition of a convective

piston flow. Thus, the dispersion is described by a modified form

of Fick 1 s Law, where an effective dispersion coefficient is used

in place of mo1ecular diffusivity.

ln the absence of chemical reaction and interphase

transfer, the general continuity equation describing the concen­

tration vartation with time, at any point ln the bed, for a con­

stant density fluid, is given by:

utk_ + o (~ + l eSc)= l:k b; r 6~2 r {)r ~ (3.17)

This is second order partial differentiai equation and

it has not been possible to find its solution with the boundary

conditions of the system and the input signal. !n view of the

mathematical limitations, the fol Jowing assumptions are made to

reduce equation (3.17) to a form which can be integrated:

1. Radial concentration gradients are negl igible~ i.e. Dc!O r =O.

2. Radial velocity gradients are negligible so that no radial

mass transport occurs.

3. The quantity of solute transferred by axial mixing is

directly proportional to axial concentration gradient.

4. Effective axial dispersion coefficient is constant ali

along the length of the bed.

46

5. The system is linear, i.e. system response is independent

of amplitude of signal. This also involves the assumption

that no Joss of tracer occurs by adsorption or chemical

reaction with the packing material.

Whi le real systems of industrial scale may deviate

appreciably from the first two assumptions, it has been common

practice in laboratory investigations to distribute the liquid

uniformly over the entire cross-section so that quite a flat

velocity profile is obtained. This also enables one to introduce

a radially uniform input signal. ln fact, previous work (10, Il,

13) has indicated that the radial gradients are smal 1. Whatever

contribution these radial gradients make to the axial dispersion

is included in effective dispersion coefficient, DL, and hence

is a source of discrepancy between results obtained with different

columns.

A simplified form of equation (3.17) has been obtained

as follows. Considera material balance over a thin section of a

packed bed as shown in Fig. 7.

Input - Output =Accumulation - Production

-0

L ~: x + uc L - [-0 L tx L + 6x ] ,. uc lx

+6x

= ~ 6x- r6x (3.18)

FIGURE 7

MATERIAL BALANCE COMPONENTS OF THE DIFFUSION MODEL

47

ACCUMULATION • l-f â x

R ATE OF PRODUCTION • r â ~ JFLOW

-0: ~ uc 1

VOLUME x OCCUPIED

BY __ l_l_~ PACKING - âx --

j_ AND ------- t l l OTHER

PHASE -oll uc bX

l~ 1 _~ .. _E~_....:j

whence, be

r - b t

48-

(3.19)

which for r = 0, i.e. in the absence of chemical reaction, reduces

to the form:

( 3. 20)

This simplified form of equation (3. 17) and its solution has been

widely discussed in the literature (4, 9, 20, 46, 59, 63, 84, 108,

131, 133) for a variety of boundary conditions. Whi Je Kramers

{59) gave the solution for a sinusoïdal input, the solutions given

by Danckwerts (20) and others refer to a step input function.

Danckwerts gave solution for an infinite bed whi le the solution

given by Aris (4) is for semi-infinite bed. However, the most

general finite-boundary solution for a finite bed, with approp­

riate boundary conditions at the inlet, x= 0, for a step change

in concentration from c 0 to 0, was published in 1962 by Brenner

( 9).

Consider the mixing process occuring at the inlet to

the bed. With steady state flow conditions (i.e. no accumulation)

at the inlet (x= 0), the rate of removal of tracer from just

inside the bed inlet by diffusion and convective plug flow is

equal to the rate of arrivai of tracer from the outside by con­

vective plug flow, so that:

uc(O-) = uc(O+)- DL t:<o+l, x= o, t>o (3.21)

49

Simi lar materlal balance at the outlet, x= h, gives:

uc(h+) = uc(h-) - DL ~~ (h-) (3.22)

With a purging step, regardless of the mechanism of material trans-

port, one can never have a downstream concentration lower than an

upstream value (20, 108). Thus the solution of equation (3.20) is

requi~ed for the case of only positive values of ~, i.e.

c(h+) :> c(h-) for a feeding step and c(h+) <: c(h-) for a

purging step. This requires occurrence of an extremum (maxima or

minima) of concentration somewhere in the bed. The only phys­

ically acceptable condition is that at x= h,

D ~ c = 0 ~ for a 1 1 t )0 ( 3 . 2 3) L bx

and since DL ~ 0,

be bx = 0, x = h, t>O (3.24)

and hence c(h-) = c(h+)

The initial conditions for a step input of tracer from c 0 to 0

are:

c = 0'

c = c 0 ,

x< 0,

x > 0,

t = 0

t = 0

The solution to equation (3.20) as given by Brenner (9) is:

( 3. 25)

( 3. 26)

50

F(t) = c/c0 = 0.5 erfc[ 0.5(N/T)0 ·5 ( 1 - T)]- (NT/ff)0.5 x

x [3 + 0.5N (1 + T)J-exp [- 0.25N (1- T)2/T] +

+ [0.5 + 0.5N(3 + 4T) + 0.25N2 ( 1 + T) 2) x

x exp (N).erfc [o.5(N/T)0 ·5 (1 + T)] (3.27)

Figure 8 shows this solution for different values of columnPeclet

number, N. By compa~ing these plots with the experimental tran­

sient response, one can evaluate Peclet number, N, for the system.

Brenner•s finite boundary solution, which takes into account the

diffusional ~aterial transport both at the inlet and exit of the

bed, is part,cularly useful for a system which approaches almost

perfectly mixed stage (N--o). For column Peclet number, N,

greater than 20, the unbounded solution (infinite length) of

Danckwerts {20) gives practically the same resultas equation

(3.27). Further, for the unbounded solution, the variance is

given by;

( 3. 28)

which by comparison with equation (3, 10) gives,

N = 2n (3.29)

i.e., the column Peclet number is twice the number of equal vol-

ureperfect mixers in cascade.

STATISTICAL RANDOM WALK MODEL

The mode! proposeq by Einstein (32), and which has been

discussed by Cairns and Prausnitz (11), Dunn et al. (30), and

Jacques and Vermeulen (50) 9 considers the motion of corpuscles

FIGURE 8

THEORETICAL BREAKTHROUGH CURVES FOR VARIOUS COLUMN PECLET NUMBERS

51

z 0 -ti a: 1-z L&J 0 z 0 0

L&J 1-~ ..J 0 (1)

1--x I.LI

·8

·6

·4

·3

·2

·10

·08

·06

·04

P•œ 10

0 ·5 1·0 1·5 2·0 2·5 NUMBER OF DISPLACEMENTS

52

of tracer material in the main stream fluid. The mixing process

i s descri bed as a sequence of 11 motion 11 and 11 rest 11 phases. The

motion phase is considered to require much Jess time than the

rest phase. For a packed bed the motion phase may be taken as .'

the period when the fluid element is passing at high velocity

through the narrow constriction between the particles, and the

rest phase as the period when it is in a void space.

The mode! is constructed by considering the probabi lity

that a given tracer corpuscle wi 11 be at a given longitudinal

position, x, from the entrance to the bed, after a certain period

of time, t, since introduction. This probabi lity is expressed

mathematically by the following equation:

P(x,t) dx dt= e-x-t dx dt ( 3. 30)

If equation (3.30) is applied to a consecutJve set of n motion

phases and n rest phases, then the probabi 1 i ty of fi ndi ng a

packet of fluid at a relative position x= N mixing lengths away

from the inlet in its random walk, and at relative time t = t 1,

is given by:

p(N,t 1 ) = Y:00

[exp( -N-t1)] N~ . t1n

n = 0 n. n! . (3.31)

Equation (3.31) can be converted to a continuous function of the

form

( ) - N - T 1 .... ,.--;-;:;:-; p N ,T 1 = e 1 0 ( 2 v NT 1 )

which is the impulse response or C-curve for the system.

response to a step input at time t 1 = 0 is then given by:

(3.32)

The

FN(T) - Il' e-N-t' 10

(2-..[Ntô) dt' - 0

53

(3.33)

where 10 is a Bessel function of zero order and first kind of an i

imaginary argument. A good approximation to the integral equation

(3.33} has been obtained ( 136) in the form:

FN(T) =o.~ [1 + erf <VT" -"{N- 0,125/;f'f' -0.125t'V'Ni] ( 3. 34)

From a comparison of equation (3.34) with the solution of diffu­

sion mode! at large x and t, it is found that the mean mixing

length, 1, is defined by equation (3.35):

so tha t

N = xl 1 = xu/DL

r• = ut/1 = u2t/DL

T = T 1 /N = ut/x

( 3. 35)

(3.36)

( 3. 37)

(3.38)

A comparison of equation (3.36) and (3.29) would show that the

size of n equal volume perfect mixers in cascade corresponds to

twice the mixing length.

The fami liar breakthrough curve for a step function

input is obtained by plotting FN(T} versusTfor a given value of

N. By numericaJ approximation-{50) for sufficiently large N,

the column Peclet number, N, is found to be related to the mid­

point slope, s, by

N = 4T(s2 - 0.8

where s = (dF/dt)F=o.s x t 0 .5

(3.39)

(3.40)

54

Equation (3.39) is the final equation which has been used in

this study for the purpose of computing Peclet number. ln fact,

the Peclet number obtained by this madel is essential ly the same

as given by Diffusion mode!, especially when there is not much

mixing in the system, i.e. for systems which are closer to the

plug flow limit than they are to the completely back-mixed limit.

Further, equation (3.39) indicates that this method of determina­

tion of the Peclet number is very convenient as it requires con­

siderably Jess calculations than other methods (4, 13, 26, 31,

48, 92, 104, 123). The finite-boundary diffusion mode! and the

random-walk have very simi Jar midpoint-slope values ( 135). A Iso,

breakthrough curves given by the two models have qui te simi Jar

shapes over thei r entire range. For practical purposes, there­

fore, the two models may be said to yield identical results.

55

IV. EXPERIMENTAL APPARATUS AND OPERATING PROCEDURE

EQUIPMENT

Flow System

A schematic drawing of the packed tower and accessories

is given in Fig. 9.

The 2' long test sections were made of 12-in. I.D.

flanged pipes of plexiglass and aluminium. The packing support

grid was Jocated between the gasket and flanges connecting the

gas distribution system to the column test section. The packing

support consi sted of a screen placed on a 2" x 3" rectangular

grid of 1/8-in. iron strips. Five conductivity probes were

placed just below the screen, and were mounted on travelling

supports which enabled probes to be placed at any radial posi­

tion. The probe tips were always 1.0 cm. below the supporting

screen.

A cylindrical tee, 12" x 12" x 18", connected to a

hol low cylindrical section placed above the packed section provided

ample de-entrainment space. The liquid distributor was connected

through this tee so that the outlet of the liquid distributor was

just above the 2' section of packed bed. The liquid supply system

permitted the use of water flow rates from 1500 to 10,000 lb./

(hr.-sq.ft.). A spider type liquid distributor allowed ample free

space for the passage of exit gas. The distributor, as shown in

Figure 11, consisted of 52 1/8-in. O.D. copper tubes connected to

a 1 -1 /2" di ame ter by 1" deep brass box. The sma 1 1 si ze of brass

56

FIGURE 9

FLOW DIAGRAM OF THE EXPERIMENTAL SET-UP

FIGURE 9. FLOW DIAGRAM

A Air Fi 1 ter

B 0 rif i ce P 1 a te

C lnclined Manometer

D Blower

E Control Damper

F Air Distributor

G Calming Section

H Conductivity Probe

Packed Section

J Loading Port

K Water Distributor

L Tracer Solenoid Valve

M Quick Opening Valve

N Tracer by-pass Solenoid Valve

P Pressure Tap

Q Water Rotameter

R Water by-pass Solenoid Valve

S Tracer Rotameter

T The rmome ter

U Water Strainer

V Tracer Storage Tank

W Nitrogen Cylinder

X Leve 1 Gage

Y Leve 1 Gage

t­l&J .J z -

57

FIGURE 10

PHOTOGRAPH OF EXPERIMENTAL SET-UP

box helped to obtain a sharp step input. The small diameter

tubes were chosen so that they remained fui 1 even when not in

use. The tubes of the distributor extended within 1/4-in. of

58

the column wall. The flow rate was essential ly zero at the

column wal 1. The flanged cover on the cylindrical tee was used

to center the distributor in the column. Figure 12 shows the

percentage deviation of the flow rate in a given section of the

distributor from the average flow rate corresponding to a uniform

flow profile. 1 t i s apparent that a good uni form di stri but ion

was obtained with this system.

A 24-in. high converging cone above a 24-in. diameter

cylindrical section helped to obtain a flat gas velocity profile.

lt was realised that this conical section alone would not be

sufficient to obtain uniform gas distribution. Thus 1 the hori­

zonta! gas flow from the blower was directed upwards by a bundle

of 32, 2-in. diameter aluminium tubes. Further the diameters of

these tubes were adjusted to obtain an even gas distribution

across the cross-section just below the packed bed. Flatness

of velocity profiles were tested using a constant current hot­

wire anemometer in two perpendicular directions. Air flow rates

of up to 1000 lb./(hr.-sq.ft.), i.e. 160 s.c.f.m., could be

obtained with this system under normal operating conditions.

Figure 13 shows the photograph of air distributor. The air flow

rate was metered by an orifice meter with radius taps installed

in the suction line of the blower.

The tracer used was O. IN solution of potassium chloride.

Pressurised nitrogen was used to force the solution through the

59

FIGURE Il

PHOTOGRAPH OF WATER DISTRIBUTOR

FIGURE 12

INLET FLOW PROFILES FOR VARIOUS LIQUID MASS VELOCITIES

60

WALL

+10

t. WATER FLOW RATE, lb./ hr.

5 4: 3 2 2 3 4 5 6 7 . 8060

WALL

5740

3600

1395

SECTION NUMBER

61

rotameters to the instant-action Hoke two-way solenoid valve.

The solenoid valve was attached just above the brass mixing tee.

The injection nozzle was 0.0611 in diameter and was placed along

the centre line facing the incoming flow of water so that the

tracer line remained fui 1 of the tracer solution between experi­

ments. An automatic timer and switch provided a remote control

for the solenoid as weil as a simultaneous record of the exact

time it was switched on or off.

The packing characteristics are shawn in Table 11.

1 nstrumen tati on

Electrical conductivity method was used to measure

the concentration of tracer. This choice was guided by the

requirement of accurate, rapid and continuous monitoring of

tracer concentration in the water stream. The signais from the

concentration cells were recorded on a 14-channel 11 Visicorder 11,

an instrument which uses highly sensitive, electromagnetical ly

damped galvanometers. The signais that enter the Visicorder

are converted into moving beams of light by moving mirror

galvanometers and are focussed on to the photo-sensitive paper.

The paper speed could be adjusted at any one of the four speeds

of 25, 5, 1 and 0.2 in./sec. This along with an averaging

amplifier provided a high speed response recording system and

was found very satisfactory for the purpose. Cairns and

Prausni tz {JO, 11), Hennico et al. ( 135) and others have used

simi Jar concentration measurement techniques in their respective

s tud i es.

62

FIGURE 13

PHOTOGRAPH OF A 1 R D 1 STR 1 BUTOR

63

TABLE 11. PACKING CHARACTERISTICS

1 . Nominal di ame ter, in. 0.5 1.0 1.5

2. Equivalent diameter, ft. 0.01975 0.02613 0.04111

3. Porosity, dry packed. 0.541 0.726 0.731

4. Pieces per cu. ft.' N t 1200 1250 352

5. Surface a rea, a, ft. 2/ft.3 139.3 63.0 39.35

6. Outsi de di ame ter, cm. 1. 303 2.680 3.972

7. 1 n s i de d i ame te r, cm. 0. 705 1 .850 2.873

8. Hei ght, h, cm. 1. 235 2.460 3.746

9. A rea pe r pi ece, sq. cm. Il .56 46.85 10 3. 95

10. Volume per piece, cu. cm. 1 . 16 7.33 21.62

Il. S tati c Ho 1 dup, cu. ft.

Hs, cu. ft./ 0.0326* 0 .o 149* 0.00893*

*Data of Shulman et. al. (85)

64

Conductivity Probe

The conductivity cel 1 consisted of two 1-mm. diameter

platinum electrodes embedded in a I-mm. slotted cylindrical body

of 1-cm. diameter x 1.3-cm. long. Each probe was constructed by

immersing a U-bend of platinum wire of 1-mm. diameter x 1.2-cm.

long, into liquid Epoxy resin in a cylindrical mould. After the

resin had set the solid body was machined to the desired size

and shape, and was slotted at the U-bend, cutting a I-mm. gap in

the platinum wi re perpendicular to i ts length. Because of the

smal 1 size of the probe, it was possible to measure the concen­

tration of as smal 1 as 1-cu.mm. of fluid almost instantaneously.

For operation in counter-current air-water flow system,

it was necessary to keep a continuous flow of liquid through the

electrodes, a requirement which presented sorne difficulty at high

gas flow rates. A liquid col Jecting deviee as shown in Figure 14

was used for this purpose. A flexible vinyl tube at the lower

end of the collecter provided a siphoning action. An adjustable

pinch cock was used to keep the tube and electrodes fui 1 with a

continuously flowing water stream. ln order to avoid electrical

interference ( 11cross-talking 11) due to the presence of ether

probes in the water stream, it became necessary to put a thin

brass sleeve over the solid Epoxy probes. When the same voltage

which was tmpressed on the electrodes was also applied to this

sleeve, a considerable improvement was obtained in cell perform­

ance over that of unshielded cel ls, presumably because of the

elimination of stray currents which would otherwise exist between

unshielded cells.

65

FIGURE 14

PHOTOGRAPH OF CONDUCT 1 V 1 TV PRO BE ASSEMBLY

66

Electronics

The basic electronic circuit used to measure the con­

ductivity is shown in Figure 15. Slmi Jar net work has been

successfully employed by Cairns and Prausnitz (11), Dunn et al.

(30) and ethers (58, 135). A constant voltage JO kc osci llator,

used as the audio range a.c. voltage source, was connected across

the conductivity çeJJ and the current detecter resistor, r. If

ris chosen to be small compared to the resistance of the solu­

tion, the voltage across r is proportional to the current through

it, which in turn is proportional to the conductance of the solu­

tion in the cel 1, 1/R. For the present study, r =51 ohm was

chosen so that it was only 2% of the smallest value of R encoun­

tered. The voltage Vr, was amplified and demodulated, resulting

in a voltage which was directly proportional to the conductance

of the solution. The use of an adding circuit allowed the signal

from a reference cel 1 placed in the water stream entering the

test section to be subtracted from the signais of the cel ls

measuring conductance at the bed outlet. The net resulting d.c.

voltage signal was used to drive a moving mirror-type galvanometer

in Honeywell's Hei land type 906C Visicorder. The galvanometers

had a high sensitivity of 1.19 mv./inch, and a very low time

constant which depended on the resistance of the external circuit.

Because of these special features of high sensitivity and very

low time constant, it was possible to obtain accurate records of

concentration of the emerging solution streams.

Since a single probe in a 1-ft. diameter column can

hardly be expected to give the true picture of the actual response

FIGURE 15

BASIC CIRCUIT DIAGRAM FOR CONOUCTIVITY MEASUREMENT

67

e

IOKC.

OSCILLATOR

--

CONDUCTIVITY CELL

r

AMPLIFIER­

DEMODULATOR

A ODER

REFERENCE REF. REF. CELL

r --

e

14-ch. VISICORDER

of the system, it was decided to use as many as five probes

located at random at five locations at the end of the bed to

obtain a radially averaged concentration. For this purpose,

68

the amplifier-demodulator network was designed to accommodate

six simultaneous signais from the probes, and a separate averag­

ing circuit was used to obtain the arithmetic mean of from one

to five individual signais. These individual signais as wei 1 as

averaged signais could be recorded on the Visicorder simultane­

ously. Also, an automatic timer and remote control solenoid

switch were incorporated into the system. Thus, a sharp pip

was obtained on the recording paper each time the solenoid valve

was switched on or off, thereby indicating the exact instant of

imposing a step function input on the system. Figure 16 shows

photographs of the six-channel ampliffer-demodulator with JO kc

osci 1 Jator and associated power supplies. The detai led circuitry

appears in the Appendix 11. The equipment required about 45

minutes to get stabi lized. The operating procedure as indicated

in Appendix 11 had to be followed strictly for an accurate nul 1

adjustment before making any experimental runs.

Figure 17 shows the Jinearity of Visicorder response

for different potassium chloride concentrations. The !east

square line gives the net Visicorder deflection for a given salt

concentration. The high speed response characteristic of the

system as a whole is indicated by the fact that 62.3% of the

final steady state value was attained for any step in a time

interval of 0.041 seconds or Jess.

FIGURE 16

PHOTOGRAPHS OF SIX-CHANNEL AMPLIFIER DEMODULATOR

69

-

FIGURE 17

VISICORDER RESPONSE VS. SOLUTE CONCENTRATION PLOT, SHOWING THE

LINEARITY OF THE MEASURING SYSTEM

70

.d -~ :::; ~ .d -1 ., 0 -z 0 -~ v 0: t­z LLI u z 0 u

N 0: LLI u <[

' a: \ - t-\ \ \ \

NOISIAIO - 9NIOV3Y Y30~0~1SIA

7 1

EXPERIMENTAL PROCEDURE

ln order to obtain consistent and reproducible data,

the experimental procedure was standardized so that the runs

were made under identical conditions throughout the investigation.

The step change in tracer concentration was used rather

than a pulse input since it was impractical to injecta pulse of

sufficient amount in an interval short enough relative to the

mean residence time. Further, a sudden pulse of tracer would have

disturbed the flow pattern in the packed bed. The mean residence

time varied from 2 to 15 seconds depending upon flow rates of

liquid and gas and packing diameter. Both feeding-in and purging­

out step changes of concentration were used in the evaluation of

Peclet Number from the resulting F-diagrams. Figure 18 shows the

response to purging and feeding step input. lt can be seen that

the deviation between the two response curves is practical ly

negligible. The mean residence time from both types of input

were found to be nearly the same. For the purpose of i 1 lustra­

tion, Figure 18 has been drawn using dimensionless coordinates.

lt was also found that the amount of entrainment is

negligible so that the water flow rate could safely be assumed

to remain constant.

The tower was dry packed by dumping packing from the

top with intermittent tapping. The voidage of the packed section

was measured by fi 1 ling the column with a known volume of water.

The volume of water in the voids was then the difference of

actual volume of water used and the empty volume of the packing

FIGURE 18

COMPARISON OF PURGING AND FEEDING STEP INPUTS

d = 0. 5 inch p

d = 1 0 inch p .

dp = 0.5 inch

72

e e

z 1.0 0 -~ 1-- .8 z LLJ (.)

~ .6 (.) L, tb. /(hr. sq. ft.) : 7895

en f3 _. .4 z 0

G, lb./(hr.sq.ft.): 0

'"*" • FEEDING BED HEIGHT, ft. ' 2 ....

0 PURGING -dp, ln. 0•5

-Cl)

~ .2 :E -ca ..

0 (.J

' (.J 0 .4 .6 .8 1.0 1.2 .2 1.4 1.6 1.8

t/8, DIMENSIONLESS TIME

e

z 0

~ a: ~ z

~ 0 (.)

en cn LLI _J z 0 ën z

1.0

.8

.6

.4

LLI :E .2 -0

... 0

(.)

' (.)

- - -1

~

1

1

1

0

e

-r. . .

L, lb./( hr. sq. ft.) = 3950

'1. G, lb./lhr.sq.ft.)' 0·

BED HEIGHT, ft: 2

dp t

. 1n. --\

0 FEEDING

\ • PURGING

.4 .8 1.2 1.6 2.0 2.4

t/8, DIMENSIONLESS TIME

e e

z 1.0 0 - L, lb. 1 (hr. sq. ft.) 7895 tt :

0::: G, lb. /( hr. sq. ft.) : 0 ..... .8 BED HEIGHT, ft. 2 z :

LLJ dp, in. : t· 5 (.)

~ .6 ï ~~ SYMBOL (.)

(J) 1 - 0 FEEDING (J)

. 4 L ~ • PURGING LLJ _J z 0 ëi)

.2 z LLJ ~ ë5

0 (.)

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 ' c.J

t 18, DIMENSIONLESS TIME

73

g.rid. The mean values of porosity thus measured and other packing

characteristics are shown in Table 11 .(p. 63).

A further check on the applicabi lity of transient res­

pense technique for investigating mixing characteristics in packed

beds was provided by indirect liquid holdup measurement. The

mean residence time for each flow rate was obtained by integrating

the F-diagram obtained using the step change in concentration of

KCl tracer. This was then corrected for end effects which are

described later in this chapter. The total liquid holdup obtained

from equation (3.2) compares very well with that reported by

Shulman et al. (85) and Sater (82) and is as shown in Figure 19.

The fact that this indirect determination of total liquid holdup

agrees quite closely with that of direct measurements by liquid

collection method further ensures the suitabi lity of this tech­

nique for the present study.

To inject tracer solution, the tracer reservoir was

pressurized to about 5 psi higher than in the water line. After

the water was turned on and steady state attained, the conductiv­

ity cel ls were adjusted to proper positions. The flow rates,

through the liquid collecter where the conductivity cells were

located, were adjusted so that the cells always ran full.

As described earlier, one hour was al lowed for the

amplifier-demodulator unit to become stabi lized. Zeroing was

then done for no signal input to the unit. The range adjustments

on the Visicorder were done by imposing a feeding step,input of

pota$ium chloride tracer concentration and adjusting the volume

FIGURE 19

COMPARISON OF LIQUID HOLDUP

1 N PACKED BEDS

74

"'tJ CD

..c • .... .,._ •

:::J CJ

' • .... .,._ ::; CJ

... a. :::>

1

c _. 0 J:

c -:::> 0 -_.

..

•

J

2 ~ BED HEIGHT, ft. 1 2

dp = 0•5 in. • e dp = 1.0 in. ...

1.~ t dp • 1·5 ln. • c;>

6 1- 0

4 1-

-o

2 1-

Î 1 1

DATA OF SHULMAN

0 ET AL.

~ie~ -o o- e o&

0 0 0~ :t!-o ., ... ~.cre ~ 9.._ J o-

-o ~ -o ~ o-Y

<ro-G, lb. /(hr. sq. ft.) : 0

WATER TEMP. '70 °F.

~ 0.01 1 1 1 1 1 1 1 1 1 1

2 4 6 81000 2 .4 6 810000

L, WATER MASS VELOCITY, lb. /(hr. sq. ft.)

e

-

-

--·

-

75

control of each of the five galvanometers so that each ~t~flected

to the same extent. When the tracer input was discontinued, ali

the 1ight spots from the galvanometers should return to the same

position as before the injection. ln the case where they fai 1

to do so, the potentiometer contro11ing the amplitude of the

signal from the reference cell to each of the five channels was

adjusted to bring al 1 or at !east one to the initial zero posi­

tion. The feed-back potentiometers of the remaining channels

were then adjusted to bring back ail the light spots to the zero

position. The procedure was repeated unti 1 the deflection of

each of the five channels became exactly identical.

For taking any run, the water flow rate through the

column was started at a predetermined constant value and the

system allowed to come to a steady state for at !east 15 minutes.

A water level indicator at the bottom of the column provided a

further indication of when steady state h9s been attained. The

step input of tracer was started by throwing the solenoid valve

open. The tracer was al lowed to flow into the column unti 1 the

exit concentration at al 1 the sample points reached a steady

constant value. At this stage the valve was de-energied to

obtain purging-out tracer breakthrough curve. ln most of the

runs a paper speed of !-inch per second was found to be quite

adequate.

For the first part of the study, the experiments were

performed without any counter-current air flow. Figure 20

illustrates a typical set of experimental response curves for

this case. The purpose of these experiments was to provide

FIGURE 20

EFFECT OF LIQUID FLOW RATE WITH NO GAS FLOW ON TYPI CAL RESPONSE CURVES

dp = 0.5 inch

dp = 1 • 0 inch

dp = 1 .5 inch

76

e e

~ 1.0 -t:f G , lb. 1 ( hr. sq. ft. ) : 0 ~ BED HEIGHT, ft. 1 1 z .8 LIJ d P, in. . 0•5 . (.) z 0

.6 ~ A\ B\ C\ \ L, lb./( hr. sq. ft.) (.)

en A 9865 en B 5920 LIJ ..J

.4 ~ \ \ \ ""- c 3950 z 0 D 1975 -en z IJ.J .2 ~ -0

... 0

u

' 0 2 4 6 8 10 12 14 16 18 u

TIME, SECONDS

e e

z Q 1.0 !ct G, 1 b./( hr. sq. ft.) : 0 c::

BED HE IGHT, ft. 2 1- ' z .8 d p' in. w

(.) z

.6 f ~~ \ \ L, lb. /(hr. sq. ft.)

0 (.) A 9865

Cl) 8 5920 Cl)

1 \ \ \ \. c 3950 w _J .4 t" \ \ \ \. D 1975 z 0 -Cl) z w .2 ~ -Cl

0 <J

' <J 0 2 4 6 8 10 12 14 16

TIME, SECONDS

e e

z Q 1.0 !::(

G, lb./( hr. sq. ft.) a: 1 0 1- BED HEIGHT, ft. 1 2 ~ .8 d p • in. • 1•5 (.) z L, lb. 1 ( hr. sq. ft.) 0 (.) .6 A 9865

- - - B 7895 Cl)

~ .4 1 't~~ '\ c 5920

D 3950

Q Cl) z ~ .2 ë5 ..

0 0

' 0 0 2 4 6 8 10 12 TIME, SECONDS

dispersion data for water flow through an absorption column.

Further, since the effect of gas flow rate was to be studied,

this provided a reference point for comparison of later runs.

Experimental data and calculated results for runs without gas

flow are listed in Table 111 of Appendix 1. Figure 21 shows

the dependence of experimental response curves on particle

di ame ter.

77

For the second part of the study, experiments were

conducted with a large number of combinations of gas and liquid

flow rates. The time al lowed for the flow conditions to attain

a steady state was increased to about 20 minutes for each run.

Experimental data and calculated results for the counter-current

flow of air and water are listed in Table IV of Appendix 1.

Figure 22 shows the dependence of experimental response curves on

gas flow rate.

Further, to investigate the effect of packed bed height

on axial dispersion, the runs were carried out at packed bed

heights of 1 and 2-ft. Figure 23 i llustrates the typical res­

pense curves for the two packed bed hei ghts.

Average residence time for each run was obtained from

equation {3. 1} by numerical integration of the digitised analog

records from the Visicorder. The integrated values of average

residence time were corrected for the end effects due to mixing

in the water distributor and delay due to the empty section.

The end effects were determined by finding the resi­

dence times in beds of decreasing height at a number of water

mass velocities. The difference of residence time in bed of

FIGURE 21

EFFECT OF PARTICLE DIAMETER ON TYPiCAL RES PONS E CURVES

d = 0.5 inch p

d = 1.0 inch p

d = 1 . 5 inch p

78

z 0 ij a:: 1-z LLJ (.) z 0 (.)

(/) Cf) LLI ...J z 0 00

e

1.0

.8

.6

.4

z ~ .2 -c

... 0

()

' ()

1 ~·

1 l.l ' - -

1- "" L ""\ \

0 2 4

e

~ . . . .

L, lb. /( hr. sq. ft.) : 5920

w G , lb. /( hr. sq. ft.) 1 0

BED HEIGHT, ft. : 2

SYMBOLS dp, in. h Q 0•5

0 1.0

\._ 0 1.5

6 8 10 12 14 16 18

TIME, SECONDS

FIGURE 22

EFFECT OF GAS FLOW RATE ON TYPICAL RESPONSE CURVES

dp = 1.0 inch

d = l . 5 inch p

79

e e

~ 1.0

~ L , lb. /(hr. sq. ft) -: 1975 0::: BED HEIGHT, ft. : 2 ...... 8 z dp , in. w (..) SYMBOL G , lb./(hr. sq. ft.)

~ ~~ ~ ~~ C> 704 (..) .6 ~ <D 553

(/) 1 0~ ~ ~" Q 390 (/)

~ .4 r """ . '\.... m_~_ • 197 z

1 Q •- ~ um~ o 0 0 ëi5 ~ .2 ::?! a

.. 0 0

~ 2 4 6 8 10 12 14 16 18 20 (.)

TIME, SECONDS

-..: ... 0

. C\1 Il) c::r C\1 C\1 . .,

en - ..: -Il) .1:

~ . 0 (J) Il) .0 0 .. .. .. - -. .. ~ CX) 0 - .. C) . ... -... ... .. crt-., :a: ..J C)

. (/) ..: c 0

.1: - ·- m a LLI • 0 () 0) - :a: .. :lE z .......

.d a. >- 0 0 , (/) - (.) .,LLJ

..J m L&J (.0 (/)

.. L&J ::E -v 1-

C\1

0 (X) (.0 . C\1 v

• . .

NOil~~lN3~NO~ SS31NOISN3~1G ' 0~1~

80

zero height {obtained by extrapolating the above data) and the

timé of free fal 1 was used to correct the integral residence

time. lt has been shown (15) that the mixing occuring in the

external s~ction is negligible and as such no attempts have been

made to account for it. Figure 24 shows the end corrections

applied as a function of 1 iquid mass velocity. Further, it has

been assumed that the presence of gas phase does not affect the

above end correction, which has been determined at zero gas flow.

The total Jiquid holdup was then determined by equation

{3.2). Statistical Random-walk mode] has been used to evaluate,

from the mid-point s !opes of the breakthrough curves, the mi xi ng

parameter, N, also commonly cal led column Peclet number. Also,

a number of experimental curves were transferred to a dimension­

Jess semi-logarithmic plot and the resulting curves were compared

with the computer-plots of the solution of finite-boundary dif­

fusion mode! to evaluate N. 1 t was found that the Peclet number,

N, thus determined was very much the same as that obtained by

using mid-point slopes of breakthrough curves in equation (3.29).

lt was, therefore, decided to use mid-point slopes for the evalu­

ation of column Peclet number.

The packing Peclet number which takes account of the

size of packing is computed from

( 4. 1 )

For any particular run, the effective axial dispersion coefficient,

DL, was calculated from the knowledge of other variables, viz. u

and dp.

FIGURE 23

EFFECT OF PACKED BED HEIGHT ON TYPICAL RESPONSE CURVES

h = 12 inch

h = 24 inch

81

~ U') 0 U')

1-<.0 . (X) 0 ::J: en <!) -LLJ

::J: . C\J -.. .. .... "' -0 - LLJ - al ,..: . -.... .... .

0" . . en 0" c:: _j 0 en ·-..: 0 -.c ...: .. al - .c a. ~ • 0 (/)

........ -........ ., >- 0

.ci . Cl) z - .c

"' .. a> 0 ..J (!) (.)

w (/)

.. <.0 w

:E -1-

v

0

0 . (X) . <.0 . C\J . v . -NOil'i~lN3~NO~ SS31NOISN3V\IIO ' 0 010

e e

z 0 1.0 -ti L, lb./( hr. sq. ft.) : 3950 a:: 1- .8 z G , 1 b . 1 ( hr. sq. ft. ) : 0 w dP.' in. : 1•5 (.) z 0 .6 (.) r ~ \ SYMBOL BED HEIGHT Cl) ft. Cl) w _J .4 L '"< ' • 2 z 1 Q '- 0 Q Cl) z w .2 ~ ë5 ..

0 (.)

0 1 2 3 4 5 6 7 8 9 ' (.)

TIME, SECONDS

FIGURE 24

END EFFECT CORRECTION VS. LIQUID MASS VELOCITY

82

.,..._ •

8 .... ....

• ..J g. 0"' 0 Cl)

m - • .... :eoe .J: >-

' Cl) . ..0

r:! c.o -·- lt) .. • lt) .. a.O~ >-"0 1::: u

v 9 w > Cl) Cl) <(

(\J ::E

0:: w !ëi 3=

0 .. 0 ...J 0

lO 0 (X) v ro . - . . 0

SON0~3S '3~11

83

Equipment for the facile reduction of experimental data

were not avai Jable and considerable difficulty was encountered in

data reduction. The primary problem in the data reduction stemmed

from digitizing the data manual ly. lt is the feeling of the author

that with automatic analog-to-digrtal conversion capabi lities, the

data reduction could be accompli shed relatively easi ly.

V. CORRELATION AND DISCUSSION OF EXPERIMENTAL RESULTS

The information to be obtained from the expèriments

carried out in this work can be placed under two categories:

1. Determination of liquid holdup by transient-response

technique and to compare the results with directly

measured published values for confirmation of the

validity of the experimental technique.

1 1 Correlation of the effect of operating factors on the

axial dispersion in packed beds.

HOLDUP

84

As a comparison with results avai Jable in the literature,

the total liquid holdup, HT• is plotted versus the liquid mass

flow rate, L, on Figure 19, along with the data of Shulman,

Ul Jrich and Wells (85). The total liquid holdup values in the

present investigation have been obtained from average residence

time with the use of equation (3.2), whi Je Shulman et al. deter­

mined the total holdup by weighing the entire packed tower. An

inspection of Figure 19 shows good agreement with the values

reported in the literature.

Referring to equation (2.6), Otake and Okada (78) found

that the operating holdup is given by:

Hop= 1.295 (~)0.676 (d~;:~2)-0.44 (adpl

= 1.295 (Rel)0 · 676 (Gal)-0 · 44 (adp) (2.6)

85

ln order to calculate operating holdup from total holdup with

the equation;

= Hl - Hs ( 2. J )

a knowledge of static holdup, H5 , is essential. The values of

static holdup were therefore taken from the data of Shulman et

al. (85) who made direct measurement of static holdup. The values

used are:

1/211 Raschig rings, Hs = 0.0326

, .. Raschi g rings, Hs = 0.0149

1-1/2 11 Rasch i g ri n gs, Hs = 0.00893

Otake and Okada (78) were able to correlate their ope rat i ng holdup

along with the data of ether investigators (33, 34, 86, 99) to

within a range of~ 15% using equation (2.6). Experimental data,

given in Table 111 of Appendix 1, were therefore converted to the

form given by equation (2.6) and are shown in Figure 25. The

solid line represents Otake 1 s correlation. Also plotted are the

results of Sater (82) and Shulman et al. (85). Since the results

of the present study along with those of Sater and Shulman do not

contradict Otake's correlation, it can be inferred that equation

(2.6) gives reliable estimates of the operating holdup. lt

further demonstrates the suitabi lity of the transient response

technique for the present investigation.

86

FIGURE 25

OPERATING LIQUID HOLDUP IN PACKED BEDS

0 0

. _j ...J <( <( <[ a

<( ~ 1- 1-0 w LLI

ca z w ~ a: ~ ..J w <( ::> ti 1- ::1: 0 (f) (f)

1 <[ m

Q) <D ~

.. w N ëi)

..... ::1: (!) -w ::1:

a w m

C\J

(\J

-.t:l <(

m

Il') 0 Il') 0 . ....;

fi • 0

(t • .. 6 9 9

000 w -( dpo) 1( z-rl )do 1- ~t·o ad 5 ç(JP H

<D

C\1

0 ..J 0 Q) - et:. Q)

<D

C\1

0 -

87

PECLET NUMBERS

The· values of Peclet number for different operating

conditions are shawn in Table 111 of Appendix 1. The results

for single phase flow through packed bed reported in the litera­

ture are usually based on nominal diameter a-nd superficial veloc­

ity. For the purpose of comparison the results, for water flow

through the bed in the absence of gas flow, are shawn in Figure

26.

ln trying to correlate any data based on packed beds,

it is essential to describe the geometry of the packing. There

has been considerable confusion about the proper choice of the

characteristic length in Reynolds and Peclet numbers. The data

reported in the literature make use of almost every conceivable

defini ti on of diameter for packings. Wilhelm ( 137) and Cairns

(JO) defined it in a manner simi Jar to the hydraulic diameter

as used for friction factor in pipes:

_:4....!>.( ..;..f..;..r.;:;.e.;:;.e _v..;..o;:;..:..l u::.:m.:.:.:e::;......:o::;.;f:..____;f:....l:...:u:...:i-=d:....c..) dh = ...... . wetted area

which for spherical packings and single phase flow, where total

liquid holdup equals the void volume of the bed, becomes

( 5. 1 )

€: dt = --~------~------

1 dt ( 1 - €: ) 2 dp

(5.2) + 1

FIGURE 26

EFFECT OF LIQUID PHASE REYNOLDS NUMBER

ON PECLET NUMBER WITHOUT GAS FLOW

88

0 ...J Q) C\J (i) 9 • 0 a: -- 0 <t

(X)

...: <D

-... . . . - c c c ..... ·- ·- -:cao 0 10 (!). • • wo - -:c

Il Il Il Q(i) 0 a. a. g. w "a "a "a CD

0 C\.1

C\J Q Q) <D v. - 0

,8d

Sater (82) defined equivalent hydraulic diameter

for packing particles in two phase flow by,

d = Hr h 4/dt + a

89

{5.3)

lt is interesting to note that since holdup varies with the

liquid flow rate, dh depends on flow rate as well as packing

size. lt is thus not a true geometrie factor. The use of such

a characteristic length dimension would certainly complicate any

correlation in which it was used.

Otake and Okada (78) used the nominal packing size, dp,

as the characteristic length dimension, together with the surface

area per unit volume, a, to describe the geometry of the packing

in their correlation for holdup. Since the product adp charac­

terizes the shape of the packing, it requires inclusion of one

more dimensionless number to already too many dimensionless

numbers affecting the axial dispersion process. ln order to be

able to evaluate ali the constants involved in a general correla-

tion involving al 1 the relevant dimensionless numbers, the experi-

mental data required would have to be prohibitively large. Such

a programme would prove formidable, especially when the reduction

of analog data to a digitized form had to be done manual ly.

However, the approach appears feasible if an automatic analog-to­

digital conversion unit were used.

Ebach {31) used equivalent spherical diameter defined

as a diameter of sphere having the same volume as the particle

for correlating his results of axial mixing of liquid flowing

through a bed of Raschig rings. Though his results are correlated

fai rly weil for single phase flow, Dunn et al. (30) have used

equivalent diameter, de, defined as the diameter of a sphere

90

with the same surface-ta-volume ratio as a packing particle, for

correlating his results for two-phase flow. With this definition

of equivalent diameter, the diameter for spherical packing is in

fact the actual diameter. ln the absence of any other suitable

definition for characteristic length dimension, the results of the

present investigation are correlated using this definition of

characteristic Jength for use in Reynolds and Peclet number.

Characteristic velocity of flow has been defined by:

u = h (5.4) e lt has been qui te common { 15, 77, 82) to use this definition of

characteristic veloci ty in computing effective axial dispersion

coefficient from Peclet number. Whi le this definition of charac­

teristic velocity has been retained for Peclet number, Sater (82)

used liquid mass velocity for computing Reynolds number of the

flow. Since characteristic veloci ty defined by the above equation

can be expected to be a better representation of the actual flow

velocity through the bed than the superficial velocity, it would

be used in ali correlations reported in this study.

lt is a wei 1 established fact that gas flow does not

appreciably affect liquid holdup in packed bed and that the latter

is related to average residence time of liquid phase. lt would

be assumed that stationary gas phase did not have any significant

influence on the mixing characteristics of the liquid phase flow­

ing through the bed (30, 58, 82). A dimensional analysis of the

91

following variables,

for a fixed bed without gas flow would lead to a maximum of four

dimension Jess groups. These are, the Pee let, Gal le lei and Reynolds

numbers and the aspect ratio.

With respect to the effect of the aspect ratio, it is

interesting to recall the study of De Maria (24) which indicates

that the variation of particle diameter to bed diameter ratio

from 0.0625 to O. 1250 does not appreciably affect the axial mixing.

ln a simi Jar study of Otake and Kunugita (76) aimed particularly

to investigate the effect of the geometrie factor, d/dt and h/dt

on the axial dispersion of liquid flowing over Raschig rings in a

packed column, the generalized correlation has the form:

Pe = 1 . 895 (Re )0 · 5 (Ga) -0 · 333 (5.5)

They established the absence of an effect of d/dt over the diameter

ratio range of O. 145 and 0.231. The diameter of the column was

not varied in the present case but three sizes of 0.5, 1.0 and

1.5 inch Raschig rings provided three values of aspect ratio,

d/dt, i.e. 0 .042, 0.083 and 0. 125. 1 n the 1 i ght of the above

information, it was decided to exclude diameter ratio from the

cor re 1 at ion.

lt is a wei 1 established fact that Reynolds number is

the best correlating group for the fluid flow whi le Pee let number

accounts for the dispersion phenomenon. The dimensional analysis

suggests inclusion of a dimensionless group involving accelera­

tion due ta gravity. lt is interesting to note that in almost

92

ail generalized correlations, for liquid holdup in packed beds,

given by equations (2.6), (2.7), (2.8), (2.9) and (2.11), the

gravity group, commonly cal led Gal lelei number, enters as an

important correlating parameter.

As an extension to the work of Sherwood and Holloway

for the effect of operating variables on the liquid phase gas­

absorption coefficient, Van Krevelen and Hoftizer (138) proposed

a dimensional correlation which included an effect of gravity:

(5.6)

The effect of gravity was deduced from the theoretical relation

for the thickness of a 1aminar fal1ing liquid fi lm and the use

of this fi lm thickness as the characteristic length dimension in

the Sherwood number. This assumption had no rigorous justifica­

tion. Nevertheless, their work did cali attention to the import­

ance of the acceleration due to gravity in determining the nature

of the liquid fi lm over the packing and to the fact that a

dimensionless correlation of liquid-phase mass transfer coef­

ficients in packed columns must inc1ude the effect of gravity.

The studies of Vivi an and Peaceman ( 139) in 1955, Davidson ( 140)

in 1957, Onda, Sada and Murase (115) in 1959 and Vivian, Brian

and Krukonis ( 103) late in 1965, ail indicate that the Gal Je lei

number is an important correlating parameter for mass transfer

studies in packed beds. Though there is sti 11 no agreement

about the exponent on this group in the dimensionless correla­

tion for liquid phase mass transfer coefficient, it can safely

93

be concluded that a complete representation of simultaneous gas­

liquid contacting system must involve the gravity group in sorne

form or the ether. Whi le rigorous justification for the inclusion

of this group in correlating their resu1ts for liquid dispersion

in packed beds is lacking, the studies of Chen ( 15), Otake and

Kunugita (76) and Sater (82) point out that gravity group is an

essentia1 correlating parameter.

ln a study of axial dispersion of spheres fluidized

with liquids, Kennedy and Bretton ( 141) also included this group

for correlating their data. Further, the inclusion of Gal lelei

number as one of the correlating parameters is attractive espe-

cially because it contains ali the pertinent physical properties

of both packing and fluid.

The dimensional analysis indicates a relationship of

the type,

lude) = A(deuP)8 (d~gp2 )C

D L f1 L J12 L ( 5. 7)

ln order to evaluate the two exponents B and C, at least two of

the variables, one in each dimensionless group, must be varied.

The two variables, Land de, were varied in the present work.

The variation in viscosity was found to be negligibly small

owing to the smal 1 variation in temperature of water. The func­

tional relationship between Peclet, Reynolds and Gal le lei number

which best represents al 1 the data of the present investigation

was found to be:

Pe = 0.789 (Re)0.383 (Ga)-0.21 (5.8)

FIGURE 27

DISPERSION FOR LIQUID FLOW THROUGH A PACKED

BED WITHOUT GAS FLOW: THIS STUDY

94

1 1 • • 1 . ~ . v

• rr>

- ~ . C\1

<kD

'Q 0 0 . 0 -~

~ . (X) ~ \Q Q) Q . U) a:: -

~ r- ~ . v

- • Q <D 0 C\1

·à· - 0 () -. C\1 - 1-

:t: = = --(!) U') 0 U') - . LLI 0 . . :t:

" Il Il

0 0 Q. Q. Q. . 0 r- LLI -c -c -c m

...1 _j l .1 ...1 1

rn C\J 0 (X) <.0 v -1 ~.0 ( DE)) ( 9d)

This equation along with ali the experimental data on which it

is based is shawn in Figure 27.

95

For comparing the results of present study with other

investigators, the data of Sater (82) for 0.5 in. Raschig rings

were converted to the form given by equation and are shown in

Figure 28, along with the experimental data of Chen (15) for

three sizes of spheres.

Otake and Kunugita (76) have also reported a simi Jar

study with 1.5 cm. and 0.75 cm. Raschig rings, but thei r data

could not be transformed to the desired form because the data

for mean residence time and dry bed porosity were not avai Jable.

A comparison of the data of Sater {82) with that of the present

study indicates a considerable discrepancy with respect to the

effect of Reynolds number. lt is rather unusual to observe such

a large depenpence of Peclet number on Reynolds number of the

dispersed phase. The correlations proposed both by Otake (76)

and Sater (82) take the form:

3 2 c [ ~a -A [ fJ J: ( d ;P l (5.9)

The values of coefficients B and C as reported are:

Sa ter: For a 1 1 Rel, B = 0.747, c = -0.693

0 take: low ReL, B = 0.5, c = -0.333

High Rel, B = 0. 16' c ::;:: -0 . 1 1

lt may be noted that in both these studies only a few packing

FIGURE 28

DISPERSION FOR LIQUID FLOW THROUGH A PACKED

BED WITHOUT GAS FLOW: VARIOUS STUDIES

96

-t\1 • 0 .........

c (!) ._..... ......... Q)

0.. ....,.,;~

100

6

4

2

10

6·

4

2

1.0

6

4

2

0.1

oo

0 Q]

golf) Do

gf!p 0 CHEN ( 15)

cB D

100 2

0 SATER (82)

- THIS WORK

4 6 1000 2

Re

4 6 10000

97

sizes have been used. Whi le Sater used only one size of Raschig

rings, 0.5-inch, Otake used 1.5 cm. and 0.75 cm. Raschig rings.

For evaluating the exponent on Gal lelei number, C, Sater used

the variation of viscosity of water due to change in temperature.

Since the viscosity varied from J. JO cp to 1.35 cp only, it is

not surprising to find that the reported value of C had wide

confidence limits, viz.

C = -0.693 ~ 1.095 ]95% confidence limit

ln Otake's study, the only variable in the Gal Je lei number was

d. Since it entered as d3, it is quite understandable that the

exponent on Gal le lei number cannat be determined accurately.

However, in the present study three sizes of rings were used and

the results were found to correlate reasonably wei 1 with equation

(5~8).

Further, Figure 28 shows that the results reported by

Chen are much higher than those obtained in the present study.

Since he used spheres whi le rings have been used for this study,

the difference in the two results can be ascribed to the effect

of particle shape. Also, it points out the necessity of intro­

ducing an additional parameter to al low for the effect of packing

shape. ln other words, the Gal Jelei number alone is insufficient

to correlate the data for any packing geometry.

The fact that Peclet number increases with liquid flow

rate indicates that plug flow approaches at high flow rates.

This is expected because of higher mixing in each void of the

bed due to increased turbulence at high flow rates. The counter-

98

current gas flow would sti Il further increase turbulence and

thus the mixing in each packing void. lt can be expected that

in the presence of counter-current gas flow, the liquid phase

Peclet numbers should be higher than for no gas flow condition,

especial ly when there is no channel ing.

The values of Peclet number reported by Sater are

lower than the present study. The reason for this difference

becomes apparent if one critical ly observes his experimental

setup and technique. He used a 5-ft. bed in a column of 4-inch

diameter. The inlet water discharged directly from the inlet

pipe at the centre of the column since no inlet liquid distribu­

tor was used. Bonifaz (7) has indicated that the values of

Peclet number are considerably lower for the case of maldistribu­

tion of flow over the cross-section of the bed. This is espe­

cially true for the wal 1 flow which invariably occurs in a packed

bed. Sorne recent work on the mechanism of liquid spread in

packed columns has shown that, in smal 1 diameter columns, a large

proportion of liquid flows down the wall, even when the ratio of

column diameter and packing size is greater than the recommended

Jimits of eight or twelve to one. lt is worthwhile to recall a

recent study of wall flow in packed columns by Templeman and

Porter (95) which indicates that wall flow attains an asymptotic

value for a 6-inch (or Jess) diameter bed of height 5 feet.

Specifical ly, wal 1 flow in a 4-inch column for a constant total

water flow of 2000 lb./hr.ft: could approach a value as high as

54% for half inch Raschig rings. Sater could not avoid the wall

flow either by increasing the diameter, reducing the packing size

99

or decreasing the bed height. Further, since he measured con­

centration externally using a radioactive tracer technique, his

measurements would be expected to yield low values of Peclet

number because of the large wal 1 effect which would exist in a

5-ft. long, 4-inch diameter column.

The value of effective dispersion coefficient has been

calculated from the Peclet number. The dependence of effective

dispersion coefficient on characteristic water velocity, u, is

shown in Figure 29. For comparison purposes, the results of

Chen ( 15) with spherical packings are shown in Figure 30. Chen

correlated his data by:

DL= 0.169 (u)0.417 (d)0.66 (5.10)

Since no definite trend is noticeable for data on Raschig rings,

no empirical relation has been attempted. However, the value of

effective dispersion coefficient could be obtained using equation

(5.8). lt can be seen from Figure 29 and Figure 30, that effec­

tive dispersion coefficient increases faster with characteristic

water velocity for rings than for spheres. The greater degree

of axial mixing in case of rings results, probably, from the

increased by-passing, trapping and short-circuiting of the liquid

due to sharp corners in case of rings as compared to the case for

smooth spheres.

The results of the axial dispersion study with simul­

taneous countercurrent gas-liquid flow are shown in Figure 31.

Liquid phase Peclet numbers under these conditions were found to

be only slightly dependent on gas flow rate below the loading

FIGURE 29

EFFECT OF MEAN LIQUID VELOCITY ON EFFECTIVE AXIAL DISPERSION COEFFICIENT

FOR RASCHIG RINGS

100

• 0 Q)

~ (\1

.. ..... z w -(.) -I..L I..L w 0 (.)

z 0 ëïj 0:: w a. (/) -0

6

4

3

2

... O·OI ..J

0

8

6

BED HEIGHT, ft. a

dp = 0·5 in.

dp = 1.0 in.

dp = 1·5 in.

2 -

1 2

t> 0 <D

Q e

3 4

U, LIQUID VELOCITY, ft./sec.

5 0·6

FIGURE 30

EFFECT OF MEAN LIQUID VELOCITY ON EFFECTIVE AXIAL DISPERSION COEFFICIENT FOR SPHERES (AFTER CHEN ( 15)}

101

0 dp = 1·5 in.

. u •

e dp = 1·0 in . 0 dp = 0·5 in.

• oi': .... ·03 ....

1 1-z LLI -0 -LI. ·02 LI. I.LI 0 0

z 0 -Cl) 0:: ~· (/) -0 ·01

1 .... 0

·008

·1 ·2 ·3 ·4 ·5 ·6

ü- LIQUIO VELOCITY- ft./sec.

102

point. Since holdup is also only slightly affected by gas flow,

the experimental results show agreement with that expected from

the results of holdup studies. This is further substantiated by

the fact that under the condition of no gas flow, the holdup and

axial dispersion are both dependent on the same variables, as

i llustrated by equation (5.8) and (2.6). Although the dependence

of axial dispersion on gas flow is slight, it is more noticeable

for !-inch Raschig rings than for 1.5-inch Raschig rings. Also,

the fact that the Peclet numbers with gas flow are higher than

for no gas flow indicates that countercurrent gas flow helps in

creating more mixing in each void of the bed.

lt has already been observed that the values of Peclet

number would be lower in the presence of channeling than when it

is absent. Thus the results shown in Figure 31 may be viewed as

the sum of two opposing effects, viz. increasing Peclet number

because of more turbulence and hence better mixing in packing

voids, vs. decreasing Peclet number because of any channeling or

wal 1 flow. ln view of the use of a 52-jet water distributor and

relatively short beds, one has every reason to believe that the

wal 1 flow and channeling effect, if at ali present, were suf­

ficiently small to cause 1 ittle effect on the results obtained.

Further, since holdup studies point out that the gas flow would

have only a slight influence on Peclet number, and this has indeed

been found to be so, the channeling and wal 1 flow appear to be

unimportant in the present study.

The Peclet number correlations for simultaneous -gas

liquid flow could be extrapolated to zero gas flow and have been

10 3

FIGURE 31

EFFECT OF GAS VELOCI TY ON THE

01 S PERSEO LI QU 10 PHASE PECLET NUMBER

dp = 1.0-inch

d = p 1 .• 5- i ne h

e

7 6 5

L , lb. /(hr. sq. ft) = 1975

SYMBOL : e A

PACKING SIZE, in . ' 1

e

3950 5920 7895 8880

• Q) f) 0

B

CD l ~ 8_ A a.. 4 _CDo • 1> •-e= -e~

~ e -.,= 1 éRa e_.__e-... 3 !A -e-

0.2

0 100 200 300 400 500 600

. G, GAS MASS VELOCITY, lb. /( hr. sq. ft. )

700

e e

1.0 .... 1 -···-· -8 PACKING SIZE, ln : 1• 5 -

L, lb. /(hr. - sq. ft.) \ 3950 1975 -6 BED HEIGHT, ft. : 1 2 2 -

r SYMBOL () 0 o- -

tf 4 0 ()

c-t> -o-8: 0

3 1- 0 -

0.2 _l 1 1 1 1 1

0 200 400 600 800 1000 1200 G, GAS ·MASS VELOClTY, lb. /(hr. sq. ft.)

e e

1

1.0 ~ BED HEIGHT , ft. l 1 2 -

SYMBOL . C> 0 . 8 r L, lb./(hr. sq. ft.) . 5920 . -

PACKING SIZE ln. • 1. 5

6 L -cf

4 ~ 0 0 C> 1>- ... .,. 1) 0 0

0.2 1 1 1 1 1

0 200 400 600 800 1000 G, GAS MASS VELOCITY, lb./{hr. sq. ft.)

e e

1 1 1 1 1

1.0 -BED HEIGHT, ft. : 1 2 -

8 SYMBOL : t> 0 -L, lb./( hr. sq. ft.) : 7895 -

6 PACKING SIZE, in.: 1• 5 -

:x -Q)

~ Q_

6 ~ -0 0

3 L -

0.2 1 1 1 1 1

0 200 400 600 800 1000 G, GAS MASS VELO CITY, lb. /( hr. sq. ft.}

104

found to yield essentially the same values as those obtained

experimental ly. The results for non-reactive, simultaneous,

countercurrent, gas-liquid flow have been correlated by equations

(5. 11) and (5. 12) and are shown in Figure 32.

For 1-inch Raschig Rings:

Pe = 0. 789 (Re)0.382 (Ga)-0.21 x 10 36.9 x 10-5 ReG

( 5. 1 1 )

For 1 . 5 - i ne h Ra sc h i g Ri n gs :

Pe = 0. 789 (Re)0.383 (Ga)-0.21 x 106.33 x 10-5 ReG

(5.12)

These correlations, though stiJl not entirely general,

have the advantage that they reduce to equation (5.8) when gas

flow rate reduces to zero. 1 t i s apparent that considerable work

would be required to obtain a generalized correlation for ali sizes

with a11 packing geometries.

FIGURE 32

CORRELATION OF DISPERSION DATA FOR SIMULTANEOUS COUNTERCURRENT AIR-WATER

FLOW THROUGH PACKED BED

dp = 1.0-inch

dp = 1.5-inch

105

\ <D

' 0

• lO

\ rt>

0 CD 0 <D CD \ CD 0

0

--r rt> U') Ol Q CD

\ 1"-0 If)

0 C\1 C\1 e ~ Ol U')

l 0 .(!)

0 0 Q)

.U') • C\1 0:: Ol rt)

U') 0 0 1"- <D .:..:. lO Ol

.. .. - • ..,.; . 5 0 .... Q . ... cr w en

...J N -..: ~

(/) .s::. -...... ~ (!) . z .a Cl) -- ~ .. 0 ...J f

0

C\J q 1'-(.0LO ~ 0

ra.o ( 0~) ~e~-ol etJ > ad-

e

2 1- L, lb. /(hr. sq.ft) : 1975 3950 5920 7895 8880

SYMBOL : e 0 (]) 0- () PACKING SIZE, : 1. 5 in. -C\1 .

0 ......... c ~ 1.0 o- ([J.

(]) 0 0 ,

(X) , . 0

1 ......... Q)

0::: ......... Q) a..

6 5

4

0.3 0

<D-0--<D-Il.: QlD O-<D a- o- 6 Q •

200 400 600

ReG 800 1000

e

106

SUMMARY

1. The results of this investigation indicate that the transient

response technique may be used with a satisfactory degree

of precision to determine axial dispersion coefficients

for liquid flowing through a packed bed, with or without a

counter-current flow of gas. Feeding and purging step

function inputs are found to give identical results.

2. The voidage of the packed bed with simultaneous counter­

current gas-Jiquid flow can be calculated from a knowledge

of the mean residence time of the liquid and the corre~

sponding volumetrie flow rate.

3. The values of the total and operating holdup of the liquid

phase, as calculated from the transient response of the

system, agree wei 1 with values reported in literature.

4. The axial mixing through the bed as expressed by Peclet

number or effective axial dispersion coefficient is depend­

ent on liquid and gas flow rates, particle size and geometry,

but is independent of packed bed height. The extent of

mixing in the liquid phase decreases with increasîng flow

rates of either the gas or liquid, the liquid flow rate

having more effect than the gas flow rate.

5. A 11 gravity group'' has been included in the correlation. The

results in the absence of gas flow have been correlated by

the equation

Pe = 0.789 (Re)0 ·383 (Ga)-0 · 21 ( 5. 8)

107

and for simultaneous counter-current gas liquid flow by

the equation

for 1-inch Raschig rings,

Pe = 0.789 (Re)0.383 (Ga)-0.21 x 1036.9 x 10-5 ReG

for 1 .5-inch Raschig rings,

Pe = 0.789 (Re)0.383 (Ga)-0.21 x 106.33 x 10-5 ReG

6. The Peclet number of the dispersed liquid phase is only

si ightly affected by gas flow rate. This finding is further

substantiated by the fact that liquid holdup also is only

slightly affected by gas flow.

.e

A

Ao

Al

a

B

b

c

co

Cn

c

d, dp

de

dh

dt

DL

Dv

Dr

F

f

g

G

Ga

NOMENCLATURE

Unless otherwise specified, the symbols used in this thesis have the following meaning.

- Cross-sectional area of packed bed, sq.ft.

- lnlet amplitude concentration, mass/unit volume

- Outlet amplitude concentration, mass/unit volume

- Surface area of packing per unit volume of packed bed

- Constant, in holdup relations

-Tracer concentration at time t, mass/unit volume

-Maximum tracer concentration, mass/unit volume

-Tracer concentration in nth cell

- c/c 0 , dimensionless concentration

-Nominal packing diameter, inch

- Equivalent diameter, inch (page 90 for definition)

- Hydraulic diameter, as defined by equation (2.27)

- Bed diameter, inch

- Effective axial dispersion coefficient, sq.ft./sec.

- Molecular diffusivity, sq.ft./sec.

- Radial dispersion coefficient, sq.ft./sec.

- Step response function, dimensionless

- Fanning friction factor 2 -Acceleration due to gravity, 32.2 ft./sec.

- Gas mas s ve 1 oc i ty, 1 b. 1 ( h r . - sq . ft. )

- Gallelei number, d3gp2 , dimensionless

f2

108

h - Packed bed height, inch

- Operating liquid holdup, cu.ft./cu.ft.

- Static liquid holdup, cu.ft./cu.ft.

-Total liquid holdup, cu.ft./cu.ft.

- Liquid phase mass transfer coefficient, ft./hr.

- Overal 1 gas phase mass transfer coefficient, lb.mole/(hr.-cu.ft.-atm.)

- Overall 1 iquid phase mass transfer coefficient, lb.mole/(hr.-cu.ft.-( lb.mole/cu.ft. ))

- Mixing Jength

L - Liquid mass velocity, lb./(hr.sq.ft.), also Jength variable

m - a constant

N - Column Peclet number, uh/DL, dimensionless

Nsc - Schmidt number, fl!}JDv n Number of ideally mixed cel ls in series,

p

Pe

Pe'

Q

r

R

Re

dimensionless

- Probabi lity symbol

- Peclet number based on equivalent diameter of packing, u de/DL

- Peclet number based on nominal diameter of packi ng, u dp/DL

- Quantity of tracer material injected, moles

- Rate of production of tracer

- Radius of the tube, ft.

- Liquid phase Reynolds number based on superficial liquid velocity and nominal diameter, dplhJ, dimension 1 es s r

- Liquid phase Reynolds number based on mean liquid velocity and equivalent diameter, u de}JAI , dimension 1 ess r

- Gas phase Reynolds number based on superficial gas velocity and equivalent diameter, deGh/ , dimensionless r

109

/

110

s - Laplace Transformation variable, dimensionl.ess

t

T

u

v

v

y

z

- Time variable, sec.

- t@ , dimensionless time

Average axial 1 iquid veloci ty in packed bed, ft./sec. = lnterstitial velocity for single phase flow = Characteristic velocity for two phase flow

- Superficial velocity, ft./sec.

- Volumetrie flow rate, cu.ft./sec.

-Total volume of bed, also velocity

- Volume of nth cel 1

- Length variable, inch

- Ordinate in figures

- Reduced length, x/h, dimensionless

-Total void fraction, cu.ft./cu.ft. bed volume

- Fraction of total void fraction occupied by gas, cu.ft./cu.ft. bed volume

-Mean residence time, sec.

- Viscosity of gas, lb./(ft.sec.)

- Viscosity of liquid, lb./(ft.sec.)

- Density of liquid, lb./cu.ft.

1 •

2.

3.

4.

5.

6.

7.

8.

9.

JO.

1 1 •

1 2 •

13.

14.

15 .

1 6.

1 7 •

18.

19.

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MacMulin and Weber, Trans. Am. lnst. Chem. Engrs., ll, 409 ( 19 34-35).

Cooper, C. M., R. J. Christi and J. C. Perry, Ibid., fl, 979 ( 1941).

1,

4,

D9vidson, J.F., Trans. lnstn. Chem. Engrs., fl, 131 (1959).

Onda, K., E. Sada, and Y. Murase, A.I.Ch.E.J., ~, 235 (1959).

Shu1man, H. L., C. G. Savini, and R. V. Edwin, A.I.Ch.E.J., .9_, 479 ( 1963).

Varrier, C.B. S., and K. R. Rao, Trans. lndian lnst. Chem. Engrs., 13, 29 {1960-61).

Denbigh, K. G., J. Appl. Chem., l, 227 (1951).

Deemter, J. J. van4

J. J. Broeder and H. A, Lauwerier, App1. Sc i • Res. , ~~ 3 7 ( 195 6).

Aris, R., Proc. Roy. Soc. (London}, 235A, 67 ( 1956).

1 2 1 . A r i s , R . , 1 b i d . , 2 5 2A , 5 3 8 ( J 9 59 ) .

122. Peaceman, D. W., and H. H. Rachford, J. Soc. lnd. Appl. Math., 1, 28 ( 1955).

123. Klinkenberg, A., H. Krajenbrink, and H. J. Lauwerier, lnd. Eng .. Chem. ~~ 1202 ( 1953).

117

124. Jacques, G. L. and T. Vermeulen, University of California Radiation Laboratory, UCRL Report 8029.

1

125. Cairns, E. J. and J. M. Prausnitz, lnd. Eng. Chem., .2.,1., 1 44 1 { 1 9 59 ).

126.

127.

128.

129.

130.

131.

132.

133.

1 34.

135.

136.

137.

138.

139.

140.

141 .

Turner, G. A., Chem. Eng. Sc i . , 1, 156 ( 1958).

Turner, G. A. , Chem. Eng. Sc i . , l.Q., 14 ( 1959) .

Levens)iel, 0. ' and W. K. Sm i th, ( 195 6 .

Chem. Eng. Sc i . , 6, 227

Sterne rd i ng, s. ' Chem. Eng. Sc i . , ~~ 209 (1961).

Hoogendoorn, Amsterdam.

c. J . ' and J. Lips, S he 1 1 Labo ra tory,

Wehner, J. F., and R. H. Wilhelm, Chem. Eng. Sei., 6, 89 { 1959).

Mason, D. R., andE. L. Piret, lnd. Eng. Chem., 42, 817 ( 1950) .

Lapidus\ L., and N. R. Amundsen, J. Phys. Chem., 984 (1952J.

Carberry, J. J., A.I.Ch.E.J., 4, 13M (1958).

Hennico, A., G. Jacques, and T. Vermeu1en, University of California, Berkeley, Lawrence Radiation Laboratory, UCRL Report 10696, March 1963.

KI inkenberg, A., lnd. Eng. Chem., 46, 2285 ( 1954).

Wilhelm, R. H., Chem. Eng. Progr., !±2., 150 (1953).

Van Krevelen, D. W., and P. J. Hoftizer, Rec. Trav. Chim. 66' 49 ( 19 4 7 ) .

Vivian, J. E., and D. W. Peaceman, A. 1 .Ch.E.J., 1, 437 ( 195 6).

Davidson, J. F., Trans. lnst. Chem. Engrs. (London), li, 51 (1957).

Kennedy\ S. C., and R. H. Bretton, A.l .Ch.E.J., l, 24, ( 1966}.

1 18

APPENDIX

EXPERIMENTAL DATA AND

CALCULATED RESULTS

• e

TABLE 1 1 1 . EXPERIMENTAL DATA AND CALCULATED RESULTS

·-No GAS' RUNS

Packing Diameter: 0.5 inch Ga= (d~gp2 >tfL 2 = 2.01 x to 7

Liquid Li qui d Parti c le Tota 1 Bed Flow Reynolds Pee let Li qui d Y* Height Rate Number Number Holdup

L Rel Pe• HT h lb./hr./ - - cu. ft./ cu. ft./ inch. sq. ft. cu. ft. cu. ft.

1975 34 0.587 0.0745 11.9 24

3950 68 0. 757 0. 1085 21.55 24

5920 102 0.714 0. 1290 27.37 24

6905 J 19 0.710 0. 1486 33.00 24

7895 136 0.795 0. 16165 36.65 24 .. 8880 153 0.878 0.1607 36.4 24

9865 170 0.872 0. 1710 39.3 24

*Y = H {Ga)0 · 44(ad )-J op L p

\.0

• e

TABLE Ill. EXPERIMENTAL DATA AND CALCULATED RESULTS:

NO GAS RUNS (cont.)

Packing Diameter: 0.5 inch Ga = ( d~g p2>!p.2 = 2. Il x 106

Liquid Average Li qui d Partie le Axial Bed Flow Li qui d Reynolds Pee Jet Dispersion Y* Height Rate Ve 1 oc i ty Number Number Coefficient

L U. Re Pe 2DL h lb./hr./ ft./sec. - - ft./ sec. - inch. sq.ft.

1975 0. 1346 256 0.2781 0.010 5.93 24

3950 0. 1694 282 0.359 0 .0 1 17 7. 65 24

5920 0.213 405 0.338 0.0129 7.2 24

6905 0. 2154 410 0.337 0.01325 7.2 24

7895 0.2261 430 0.377 0.01229 8.04 24

8880 0. 25 61 487 0.416 0 .o 131 8.88 24

9865 0.267 508 0. 413 0.0133 8.81 24

*Y = Pe GaO · 2 l

N 0

• e

TABLE Il 1 • EXPERIMENTAL DATA AND CALCULATED RESULTS

NO GAS RUNS ( cont.)

Packing Diameter: 0.5 inch. Ga = ( d ~ g p·2 ) 1 p 2 = 2 . 0 1 x JO 7

Li qui d Li quid Parti c 1 e Tota 1 Bed Flow Reynolds Pee let Li qui d Y* He i ght Rate Number Number Holdup

L Rel Pe• HT h lb./hr./ - - cu. ft./ inch. sq. ft. cu.ft.

1975 34 0.666 0.08121 13.8 12

3950 68 0.696 0.1022 19.80 12

5920 102 0.836 0. 1205 24.85 12

6905 119 0. 630 0. 1445 31.8 12

7895 136 0.798 0. 1558 35.0 12

9865 170 0. 91 0.1755 qo .5 12

'*Y = H (Ga )0 · 44( ad ) -l op L p

N

e

TA BLE J Il . EXPERIMENTAL DATA AND CALCULATED RESULTS:

NO GAS RUNS (cont.)

Packing Di-ameter: 0.5 inch Ga = (d~g p2 >1 fl 2 = 2. Il x 106

Li qui d Average Li qui d Parti c le Axial Bed Flow Liquid Reynolds Pee let Dispersion Y* Height Rate VelocJty Number Number Coefficient

L u Re Pe DL h lb./hr./ ft /sec. - - ft~/sec. - inch sq.ft.

1975 0. 1 125 206 0.3158 0,007 6. 725 12

3950 0. 1799 330 0.33 0.0108 7.04 12

5920 0.2286 419 0.396 0.01235 8. 45 12

6905 0.222 407 0.299XX 0.0147 6.38 12

7895 0. 235 431 0.378 0.01345 8.05 12

9865 0. 265 477.5 0.431 0.0119 9.2 12

*Y = PeGa O · 2 J xx Not shown in figures.

e

N N

e e

TABLE Ill, EXPERIMENTAL DATA AND CALCULATED RESULTS

NO GAS RUNS (cont.)

Packing Diameter: 1 . 0 i ne h. Ga = ( d ~ g p2 ) 1 f1 2 = 1 6 . 1 x 10 7

·uquid Li qui d Partie le Total Bed Flow Reynolds Pee Jet Liquid Y* Height Rate Number Number Holdup

L Rel Pe' HT h lb./hr./ - - cu. ft./ inch. sq. ft. cu. ft.

1975 68 1.02 0.04423 23. 1 24

2960 102 0.975 0.05373 30.6 24

3950 136 0.91 0. 0591 1 34.8 24

4935 170 1.07 0.06550 39.85 24

5920 204 1 . 0 1 0.0723 45. 15 24

6905 238 1.02 0.07728 49. 1 24

7895 272 1. 34 0.0855 55.6 24

*Y = 0.44 -1 H0 P(Ga)L (adp) N w

• e

TABLE 1 Il . EXPERIMENTAL DATA AND CALCULATED RESULTS:

NO GAS RUNS (cont.)

Packing Diameter: 1.0 inch Ga = (d~gp 2 >!J.L 2 = 4.9 x 10 6

Liquid Average Liquid Parti cl e Axial Bed Flow Li qui d Reynolds Pee 1 et Dispersion Y* Height Rate Ve 1 oc i ty Number Number Coefficient

L u Re Pe DL h lb./hr./ ft./sec. - - ft?/sec. - inch. sq. ft.

1975 0. 20 70 502 0.32 0 .o 1704 9. 45 24

2960 0.2551 620 0. 305 0.0219 9.0 24

3950 0. 3093 750 0. 285 0.02906 8.4 24

4935 0.349 847 0.334 0.029 9.85 24

5920 0. 379 920 0.316 0.0325 9.31 24

6905 0. 4139 1004 0.3184 0.0 345 J 9. 25 24

7895 0. 428 1029 0. 42 0.03222 12.4 24

*Y = PeGa0 · 21

N .s=-

• e

TABLE 1 1 1 . EXPERIMENTAL DATA AND CALCULATED RESULTS

NO GAS RUNS {cont.)

Packing Diameter: 1 . 5 inch. Ga = ( d ~ g p2 ) 1 p. 2 = 54. 3 5 x JO 7

Li qui d Liquid Parti c 1 e Teta 1 Bed Flow Reynolds Pee 1 et Li qui d Y* Height Rate Number Number Holdup

L ReL Pe • HT h lb./hr./ - - cu. ft./ inch. sq. ft. cu. ft.

1975 102 1.1 0.02932 29. 1 24

3950 204 1 .065 0 .04778 55.5 24

5920 306 l. 42 0.06107 74.5 24

7895 408 1 . 215 0.07639xx 96.4xx 24

9865 510 1 .091 xx 0.07366?<X 92 .5xx 24

3950 204 1 .095 0 .04092 45.75 1 2

5920 306 1.2 0.0536 64.00 12

7895 408 1 . 2 1 0.0838xx 107 .oxx 12

*Y = H0p(Ga)~· 44(adp)-l xxNot shawn in figures. N V"1

e e

TABLE Ill. EXPERIMENTAL DATA AND CALCULATED RESULTS:

NO GAS RUNS (cont.)

Packing Diameter: 1.5 inch Ga = { d; g p 2 ) J tf = 1 . 9 2 x JO 7

Liquid Average Liquid Parti c 1 e Axial Bed Flow Liquid Reynolds Pee 1 et Dispersion Y* Height Rate Velocity Number Number Coefficient

L u Re Pe DL h lb./hr./ ft./sec. - - ft?/sec. inch. sq.ft.

1975 0. 4260 1650 0.361 0.04382 12.29 24

3950 0. 3825 1460 0.3505 0.04486 1 1 . 9 24

5920 0. 45 13 1846 0. 4091 0.0400 13.9 24

7895 0. 4785 1826 0.400 0 .04946 13.6 24

9865 0. 4879 1862 0.3595xx 0.05582 1 2. 3 24

3950 0.4467 1431 0.36 0.03966 12.23 l 2

5920 0.431 1645 0. 3944 0 .04506 13.42 12

7895 0.5167 1845 0.398 0.06225 13.5 2 12

*Y = PeGa0 · 2 1 xx Not shown in figures.

N <l'

• e

TABLE IV. EXPERIMENTAL DATA AND CALCULATED RESULTS:

Packing Diameter: 1.0 inch

Li qui d Gas Flow Flow Rate Rate

L G lb./hr./ lb./hr./ sq. ft. sq.ft.

1975 196.5

1975 389.5

1975 468.0

1975 553.0

1975 638.0

1975 704.0

1975 0

3950 305.0

3950 400.0

x from Fig. 26

xx from Fig. 27

Li qui d Reynolds

Numbe r Re -

490

484

466

465

480

349

540XX

705

656

Parti c 1 e Pee let Number

Pe -

0. 3225

0. 355 3

0.3738

0. 3825

0. 3904

0. 40

0.30JX

0. 3775

0. 4089

Ga = (d~g p~! jJ2 = 4.9 x 106

Gas Reynolds y

Number ReG

-

J 1 3 0.886

224.5 0.985

270 1 .045

319 1 .071

368 1.08

405 1. 255

0 0.8

176 0.9

230.5 1 .007

Y = Pe(Re)-0.383(Ga)0.21 L

Bed Height

h inch

24

24

24

24

24

24

24

24

24

N ""-~

e e

TABLE IV. EXPERIMENTAL DATA AND CALCULATED RESULTS:

(cont.)

Pack i ng Di ame ter: 1.0 inch Ga = (d~g pjl p2 = 4.9 x 10

6

Liquid Gas Li qui d Parti c 1 e Gas Bed Flow Flow Reynolds Peclet Reynolds y Height Rate Rate Number Number Number

L G Re Pe ReG h lb./hr./ lb./hr./ - - - inch sq. ft. sq.ft.

3950 502 567 0.3882 289.5 1 .004 24

3950 571 351 0. 2894xxx 329.0 0.908 24

3950 602.5 503 0. 399 347.5 1 . 09 24

3950 0 620xx 0.3135x 0 0.788 24

5920 305 550 Q • 42 ]XXX 176 1 • JO 3 24

5920 400 471 0. 3654 230.5 1 . 0 21 24

5920 502.5 382 0. 45 30xxx 290 1 . 37 3 24

5920 55 1. 5 310 0. 4543xxx 318 1. 49 24

5920 0 75QXX 0. 329x 0 0.77 24

x from Fig. 2 6

xx from Fig. 27 Y = Pe(Re)-0.383(Ga)~· 21 N 00

xxx Rej ec ted

e

TABLE IV. EXPERIMENTAL DATA AND CALCULATED RESULTS:

Packing Diameter: 1.0 inch

Liquid Gas Flow Flow Rate Rate

L G lb./hr./ lb./hr./ sq.ft. sq.ft.

7895 202.5

7895 305.0

7895 402.5

7895 502.5

7895 0

8880 305.0

8880 402.5

8880 0

x f rom F i g . 26 xx f rom F i g . 27

Liquid Reynolds

Number Re -

792

71 l

794

525

930XX

860

682

l050XX

(con t. )

Parti c 1 e Pee 1 et Number

Pe -

0.3766

0.3818

0.4334

0. 4071

0.36X

0. 4240

0. 4325

0.376X

Ga = d~g p21 f'2 = 4.9 x 10 6

Gas Reynolds y Number

ReG -

116. 6 0.862

176 0.910

235 0.988

290 1.09

0 0.776

176 0.936

235 1 .048

0 0.77

Y = Pe(Re)-0.383(Ga)0.21 L

---

Bed Height

h inch

24

24

24

24

24

24

24

24

e

N \.0

e e

TABLE IV. EXPERIMENTAL DATA AND CALCULATED RESULTS:

{con t.)

Packing Diameter: 1.5 inch Ga = (dJg p21 p-2)= 1.92 x JQ7

Li qui d Gas Li qui d Parti c 1 e Gas Bed Flow Flow Reynolds Pee 1 et Reynolds y He i ght Rate Rate Number Number Number

L G Re Pe ReG h lb./hr./ lb./hr./ - - inch sq.ft. sq.ft.

1975 1068 1220 0.3888 967 0. 865 24

1975 0 IOOOXX 0. 3455x 0 0.83 24

3950 300 1370 0 . 3 1 1 272 0.666 24

3950 512 1366 0. 3565 465 0.767 24

3950 710 1200 0. 3418 645 0.77 24

3950 919.5 1150 0. 4288 835 0.955 24

3950 0 1300xx 0.3505x 0 0.77 24

5920 222.7 1573 0. 4026 202 0.82 24

5920 408.5 1654 0.4144 371 0.8288 24

x from Fig. 26 y = Pe(Re)-0.383(Ga)~·21

xx f rom F i g . 2 7 w 0

- e

TABLE IV. EXPERIMENTAL DATA AND CALCULATED RESULTS:

(con t.)

Packing Diameter: 1.5 inch Ga = ( dé g f 21 p2) = 1. 9 2 x l 0 7

Li qui d Gas Li qui d Parti c le Gas Bed Flow Flow Reynolds Peel et Reynolds y He i ght Rate Rate Number Number Number

L G Re Pe ReG h lb./hr./ lb./hr./ - - - inch sq.ft. sq. ft.

5920 637.5 1356 0.3789 579 0.816 24

5920 815.0 1187 0. 3865 740 0.877 24

5920 0 J4QOXX 0.3618X 0 0.78 24

7895 222.7 1911 0. 3116XXX 202 0.585 24

7895 300 1824 0.3738 272 0.714 24

7895 406 1729 0. 3584 369 0.7 24

7895 512 1687 0.3958 465 0.785 24

7895 61 1 1480 0. 3847 555 0. 80 24

7895 710 1222 0.4434 645 0.989 24

x from Fi g. 26 y = Pe(Re)-0.383(Ga)~·21 w

xx from Fig. 27 xxx Rej ec ted

e e

TABLE 1 V. EXPERIMENTAL DATA AND CALCULATED RESULTS:

(cont.)

Packing Diameter: 1 . 5 inch Ga = ( d~g flt f 2) = J. 92 x 101

Liquid Gas Li qui d Parti c le Gas Bed Flow Flow Reynolds Pee 1 et Reynolds y Height Rate Rate Number Number Number

L G Re Pe ReG h lb./hr./ lb./hr./ .. - inch sq.ft. sq.ft.

7895 0 J750XX 0. 3945X 0 0.77 24

9865 301 1862 0. 3484 273 0.663 24

9865 513.5 1652 0. 4066 466 0.8132 24

9865 713.3 1243 0. 3407 647 0.755 24

9865 D 2000XX 0.4195x 0 0. 775 24

3950 500 1060 0. 3602 454 0.845 12

3950 710.5 1180 0. 3854 645 0.875 12

3950 1010.5 1005 0. 4387 917 1.06 12

3950 0 J300XX 0. 3505X 0 0.778 12

-x from Fig. 26

Y = Pe(Re)-0.383(Ga)~· 21

xx from Fi g. 27 w N

• TABLE 1 V. EXPERIMENTAL DATA AND CALCULATED RESULTS:

Packing Diameter:

Li qui d Gas Flow Flow Rate Rate

L G 1b./hr./ lb./hr./ sq. ft. sq.ft.

5920 505.5

5920 710.5

5920 905.5

5920 0

7895 303

7895 506.5

7895 710.5

7895 0

x from Fig. 26

xx from Fig. 27

l . 5 inch

Li qui d Reynolds

Number Re -

1334

1068

691

J4QOXX

1401

1262

1060

1750xx

(cont.)

Ga = ( d~ g f'l 1 p. 2) = 1 . 9 2 x 1 0 7

Partie le Gas Bed Peel et RÂynolds y Height Number umber

Pe ReG h - inch

0.3889 460 0.839 12

0. 41 1 645 0.965 12

0. 4158 822 1 . 1 1 12

0.36J8X 0 0.78 12

0. 3964 275 0.842 12

0.4161 460 0.916 12

0.4116 645 0.97 12

0. 3945x 0 0.77 1 2

Y = Pe(Re)-0.383(Ga)0.21

e

w w

APPENDIX Il

ELECTRICAL CIRCUITS AND OPERATING INSTRUCTIONS

134

GALVANOMETER AMPLI F 1ER

0 pe rat i n g 1 n s truc ti ons

Preliminary

Zero

Switch on negative 44 volts power supply. Switch on fi lament supply. Switch on 105 and 250 volt supplies. Allow to warm and stabilize for 40 minutes.

Switch al 1 function switches to GND. Swi teh a Il avera ger swi tches to OFF. Switch al 1 galvanometer switches to OFF. Rotate al 1 amplitude controls fully CCW (min.)

1 - Reference Channel

(a) Galvanometer switch to ON (b) Push zero button on back and adjust coarse zero

control for nul! on recorder (c) Release zero button and adjust fine zero control

for null on recorder (d) Galvanometer switch to OFF

2 - Chan ne 1 1

3 -

4 -

5 -

6 -

(a) ( b)

( c)

( d)

Galvanometer switch to ON Push zero button on back and adjust coarse zero

control for nul 1 on recorder Release zero button and adjust fine zero control

for nul 1 on recorder Galvanometer switch to OFF

Channel 2

Sa me as Channel 1

Channel ~ Sa me as Channel 1

Channe 1 4 Sa me as Channe 1 1

Channel 2 Sa me as Channel 1

NOTE: lt is important that the Reference Channel be adjusted fi rst.

135

Ca 1 i br at ion

(a) Al 1 function switches to CALIBRATE.

(b) Al 1 galvanometer switches to OFF.

(c} Adjust ali galvanometers fo,r a mechanical zero.

(d) Rotate the reference channel amplitude control fui ly CCW (min.)

(e) Turn galvanometer switches for channels 1 to 5 to ON. (Ref. to OFF).

(f) Adjust amplitude controls for channels 1 to 5 such that each galvanometer wi 11 deflect equal ly ( i . e. fi ve uni t s) .

(g} Without altering these five settings, turn the reference channel amplitude control CW unti 1 one or more galvanometers return to a zero position.

(h) Adjust the potentiometers under the chassis to return the remaining galvanometers to zero.

136

FIGURE 33

BASIC CIRCUITRY FOR AMPLIFIER­DEMODULATOR-ADDER UNITS

137

e

~LIFIER CONVERTER-CRA.DEL 1

r-------~-~--~--~---~-~-----~---~--, ·~~ 1 1

6AU6 1 Il l l l l 6AL5 .il2AXZ l 1 220K 820K 150K 1

-45V

AMPLIFIER CONVERTER-CHA.Jm'EL 6

1 1 1 1~ 1

1 1 10 1 1 1 1 1 1

"' 250V

r----~------------~--~--------------, 1 (Reference) 1 1 1

~ Identical tc channel 1 except that the connections tc the diode are reversed.

1 - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

L---~-----------------~----------~--~· . .

e

lDDER 1

·----~------------~-, t TO VISI-'

1 ~ CORDER 6U8

105V

ON

All resistors are i Watts unless otherwise epecified.

Channels 1 to 5 are identical.

~. 2M, and j.9M resistors are 1~.

e

TO ADDERS

1 zero

--

6U8 250V

-45V

3.9M

AVERAGING AMPLIFIER

lOS V

TO on

e

39K ' • 1)

i 12

ALL RESISTORS ARE 1/2 W UNLESS OTHER­WISE SPECIFIED

2M a 3.9M RESISTORS ARE 1 -y.