hfat transfer in two phase flows in packed beds yu …

HfAT TRANSFER IN TWO PHASE FLOWS IN PACKED BEDS

by

YU PENG WAN ,, B. S.

A THESIS

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of lexas lech University in

Partial Fulfillment of the Requirements for

the Degree of

BASTER OF SCIERCE

IN

CflEMICAI ENGINEERING

Approved

Accepted

A u g u s t , 1984

^;^^>V^ CONTENTS

CHAPTER

I. INTRODDCTION 1

II. LITERATURE REVIEW 4

Flow Regimes 4 Trickle Flow Regime 5 Pulse Flow Regime 5 Spray Flow Regime 6 Flow Maps 6 Interaction Regimes 8 Importance of Flow Regime in Two-Phase

Flews 8 Axial Dispersion of Two-Phase Flows in Packed

Eeds 12 Definition of Dispersion 12 Dispersion Models 13 Dispersion Coefficient Correlations 14

Heat Transfer ?lechanisms in Packed Eeds 18 Wall Heat Transfer in Two-Phase Flow in Packed

Beds 18 Radial Heat Transfer in Two-Phase Flows in

Packed Eeds 22 Experiments and Models 23

Hatheitatical Models for Two-Phase Flows 25 Mathematical Models for Single-Phase Heat

Transfer in Packed Beds 26 Cortinuous-Sclid Model ( CS model) 27 The Schumann Model 28 Dispersion-Concentric Model (DC model) 30

Techniques for Measuring Packed Bed Heat Transfer Coefficients 32

Steady State Measurements of Heat Transfer 33 Unsteady State Heat Transfer Measurement 34 frequency Response "lethod 35 Imperfect Pulse Input Method 36 Step Input Method 36 Tailor's Model for '^wo-Phase Flow in backed

Eeds 37

I I I . EXPEFI^FNTAI SET-DP 38

T^xper i Inen ta l A p p a r a t u s and P r o c e d u r e s 38

i i

Packed Bed and Temperature Measuring Devices 38

Physical Constants Qsed in the Experiment 42 Heating System and Water and Air Supply

Systems 45 Operating Procedures 49

IV. MATHEMATICAL MODEI AND SOIUTICN 52

Laplace Transform Solution 58 Time Domain Fitting 62 Expanding a Known Transfer Function in the

Time Domain 63 The Optimization of the Error Function 66 Numerical Method Osed in Non-Linear Optimum

Seeking Method 68 Parameter Estimation by Imperfect Pulse Theory 72

Weighted Moment Method 73 The Sensitivity of the First Moment 74

V. PREPARATION OF EXPERIMENTAL DATA AND ESTIMATION OF PHYSICAL PARAMETEES 77

Data Treatment Ey Using the Smoothing Spline Method 77

Estimation of Physical Properties and Bed Parameters 81

Calculation of Dynamic Liquid Hold-Up 8 1 Calculation of the Homogeneous Sensible

Heat 82 Calculaton of Vh 82

Calculation of Vh By using Weighted Moment Method 83

Comparison Between Vh Calculated from the Two Different Methods 85

VI. ESTIMATING HEAT TRANSFER AND THERMAL DISPERSION COEFFICIENTS 89

Parametric Sensitivity 89 Sensitivity Study for Heat Transfer Coeffi

cients 89 Sensitivity Study for the Dispersion Coeffi

cient 94 Calculated Peclet Number Results 97 Correlation of the Asymptotic Peclet Num

bers 107 Magnitudes of Individual Heat Transfer

^iechanism 109 Comparison of Individual Heat Transfer

Mechanisms at Different Flow Regimes 112 Heat Transfer Mechanisms ^or Step Type

Inputs 115 Transition of Heat Transfer Mechanism 121 Discussion 121

• • •

111

Heat Transfer Coefficient Calculated by Neglecting Dispersion

Heat Transfer Coefficient in Two-Phase Pulsing Flow in Packed Bed

Heat Transfer Coefficient Results Comparison of the Heat Transfer Coeffi

cients with Others

VII. NUMMARY

Recomnendations

123

125

130

130

138

140

B I B L I O G I A P H Y 141

A P P E N D I )

A. I I S T OF THE EXPERIMENTAL DATA

E . ICTATICK

COMPUTEE PROGRAM

148

174

177

IV

LIST OF FIGURES

1 . Flow P a t t o r n Diagram f o r Non ^earning and Foaming S y s t e m s ( C h a r p e n t i e r and F a v i e r ( 1 9 7 8 ) ) 9

2- Two P h a s e Flow Map Us ing I n t e r a c t i o n Reginies ( S n e c c h i a ( 1 9 7 9 ) ) 10

3. Schematic Fepresentation of Individual Heat Transfer Mechanisms in Two Phase Flows in Packed Eed 19

4. Schematic Diagram of the Distributor and Sprayer Used in the Experiment 40

5. Schematic Ciagram of Radial Temperature Distribution Within the Packed Bed 41

6. Schematic Eiagram of the Experimental Apparatus for

the Two-Phase Flew Heat Transfer Experiments 43

7. Schematic Eiagram of the Packed Eed nn

8. Schematic Ciagram of Cpg at Different Temperatures 48

9- Calculated Response of the System to Unit Pulse Input 61

10. The Effect of Period on the Calculated System

Response to a Unit Pulse Input 67

11. Flow Chart of the Optimum Seeking Method 70

12. Time Domain Fitting of Output Signal 71

13. Optimum Laplace Operator for Step Input Signal. Curve A Is for Zeroth Moment, Curve B Is for First Foment 75

14. Comparison of Measured Input and Output Temperatures with the Calculated, Smoothed Curves , 79

15. Comparison of the Temperature Response in the System for Different Flow Regimes 80

16. Calculation of Vh by the Weighted Moment Method Using Iguation 5-8 85

17. Experimental Values of Vh Calculated Using the Weighted Moment Pethod 86

18. Ezaiple of a Parametric Error Hap for Heat Transfer ID Single Phase, Gas Flow (Wakao,1979) . 90

19. Effect of Changes in Peclet Nuib€rs on the System Outpjut Temperature, T2(t), Responding to a Perfect Pulse Input in the Trickle Flow Regime (Pe(OFt)=0. 155 at ReG=55.6, ReL=1.4) 95

20. Effect of Changes in Peclet Numbers on the System Output Teaperatulre, T2(t), for an Experimental Step Input in the Trickle Ilcw Regime (Pe (OFt)=0.155 at ReG=55.6, Fel=1.4) 96

21. Effect of Changes in Peclet numbers on the System Output Teiperature, T2(t), Responding to a Perfect Pulse Input in the Pulse Flov Regime (P€<opt)=0.374 at HeG=652.2, ReL=1.68) 98

22. Effect of Changes in Peclet Numbers on the System Output Temperature, T2(t), for an Experimental Step Input in the Pulse Flew Regime (Pe(opt)=0.374 at ReG=652.2, Rel=1.68) 99

23. Effect of Gas Phase Velocity on Keasured Peclet Numbers (ReL=1.4) 101

24^ Effect of Gas Phase Velocity OJ Keasured Peclet Numbers (ReL=1.6S) 102

25 • Effect of Gas Phase Velocity on Beasured Peclet Numbers (ReL=1.9€) 103

26o Effjsct of Gas Phase Velocity on Measured Peclet Numbers (R€L=2.24) 10'!

27- Ei;f€Ct of Gas Phase Velocity on Measured Peclet Numbers <R€L=2.52) 105

28. The Asymptotic Peclet Numbers at Different Rel 108

29o Com{arisen of Contrilutions Made by Convection Dispersion and Fluid-to-Particle Heat Transfer to the Overall Heat Transfer Rate in Response to a Perfect Pulse Input in Pulsing Flow Regime. 113

30. Com {arisen of the Contributions Kade by Convection Dispersion and Fluid-to-Particle Heat Transfer to the Overall Heat Transfer in Response to an Experimental Input in Trickle and Pulsing Flow Regime- i15

31. Effect of Changes in Ei Numbers on the Contriluticns of the Heat Transfer Mechanisms

vi

in Irici'le Flow with an Experimental Step Change in Temperature. (PeG=5S.6, ReI=2-4) 117

32. Comparison of Mechanisms Contribution to Overall Heat Transfer in Trickle and Pulse Flow Regimes (ReG=55.6, ReL=1.4, Bi (opt) = 0. 576, Pe (opt) = C-155 and ReG = 326.e, ReL=1.4, Pi (opt) = 1.594, Pe (opt)=0-288) 119

33. Comparison of Mechanisms Contributions to Overall Heat Transfer in Pulse Flow Pegime- (ReG=571.4, ReL=1.q) 120

34. Comparison of Mechanisms Contributions to Overall Heat Transfer in Pulse and Trickle Flow Regime (ReG=55.6, 328, 652.2, with ReL=1.96) 122

35. Error Function Versus Bi Number Calculated for Run #1 128

36. Error Function Versus Bi Number Calculated for Run #10 129

37. Effect of Gas Velocity on Measured Bi Number (ReL=1.4) 131



40- Effect of Gas Velocity on Measured Bi Number (ReL=2.2i4) 134


42. The Asymptotic Bi Numbers at Different ReL 137

Vll

IIST OF TABLES

1 . P h y s i c a l Cons tant s i n the System 45

2 . Haximuffl and Hinimun of Operating Parameters 46

3 . Power of Heaters 46

4 . D i f f e r e n c e Between Vh's C a l c u l a t e d from D i f f e r e n t Hethod 85

5. Error of Curve Fitting by Using different Vh 88

6. Error Functions Evaluated for ReG=112.8 , R€l.= 1-96 91

7. Error Functions Evaluated for EeG=408, R€L=1.96 92

8. Error Function Calculated for Trickle and Pulsing Regimes 93

9- Error Function Calculated for Different Heat Transfer Coefficient of Pun «1 126

10. Error Function Calculated for Different Heat

Transfer Coefficient of Bun #10 127

11. Experimental Cata for Run # 1, Run # 2 149

12. Experimental Data for Run # 3, Run # 4 150

13. Experimental Data for Run # 5, Run # 6 151

14. Experimental Cata for Run # 7, Run # 8 152

15. Experimental Cata for Run #9, Run ii 10 153

16. Experimental Result for Run # 11, Run #12 154

17. Experimental Result for Run # 13, Run # 14 155

18. Experimental Result for Run n 15, Run # 16 156

19. Experimental Result for Run « 17, Run # 18 157



22. Experimental Result for Hun « 23, Run « 24 160 viii

23. Experimental Fesult for Run

24. Experimental Result for Run


26. Experimental Fesult for Run










« 25,

« 27,

^ 29,

« 31,

* 33,

« 35,

f 37,

« 39,

« 41,

t 43,

f 45,

i 47,

t 49

Run

Run

Run

Run i

Run i

Run 1

Run (

Run i

Run <

Run <

Run i

Run '

1 26

1 28

i 30

1 32

1 34

1 36

\ 38

» 40

» 42

I 44

\ 46

I 48

154

155

156

157

158

159

160

161

162

163

164

165

166

IX

CFIAPTER I

INTRODUCTION

Theoretical and experimental studies of two phase

flows in packed beds are becoming increasingly important

because of the widespread application of packed bed sys

tems in industry. The application of two phase packed bed

flow research to problems of reactor design in the petro

chemical industries is clear. However, in the transporta

tion and extraction of oil, in coal gasification, nuclear

energy and solar energy, new applications of two phase

flow packed bed technology are becoming equally important.

The gas-liquid cocurrent flow mode allows high flow

rates of both liquid and gas, without the problem of

flooding encountered in countercurrent situations. Cocur-

rent flow rates are mainly limited by the pressure head

available. Cocurrent operation reguires that the gas-li

quid transfer resistance not be rate-limiting in the pro

cess, however, because cocurrent devices only provide one

eguilibrium contacting stage.

The commercial design of two phase packed bed reac

tors is often based on data obtained from existing plants.

This procedure can be time-consuming and costly. Research

based on theoretical principles of two phase flows in

packed beds has been limited by insufficient hydrodynamic

knowledge of the system, especially in the liquid phase. 1

*!ost of the gas- l iquid cocurrent packed beds are used

in the petrochemical and chemical industries as contacting

devices for c a t a l y t i c react ions , in which the gas md l i

quid are passed in paral le l flow through a bed of so l id

cata lys t p a r t i c l e s . In many gas- l iguid ca ta ly t i c reactions

there are large heat e f f e c t s due to exothermicity or en-

dothernic i ty . There i s a need to remove or replenish heat

in the bed in order to maintain a suitable reaction temp

erature and to suppress s ide react ions . An understanding »

of the heat transfer properties i s necessary for the design of a gas- l iquid cocurrent packed bed.

In describing heat transfer within packed beds, sev

eral typical parameters are used: heat transfer c o e f f i

c i ents to account for par t i c l e - to - f lu id heat transfer and

heat transfer at the wall for large reactors; dispersion

c o e f f i c i e n t s to describe hydrodynamic mixing of f l u i d s ;

and the f lu id residence time distribution in the bed.

These parameters are not d irect ly measurable and instead

must be found by parameter estimation techniques based on

experimental data and a mathematical model of the system.

Tailor (1982) developed a two-parameter equation to

describe f lu id - t o - p a r t i c l e heat transfer in two-phase

flow in packed beds. Although a parametric s e n s i t i v i t y

study indicated th i s approach should be capable of

determining heat transfer parameters with reasonable

accuracy, the experimental problems with generating a

sinusoidal thermal input prevented i t s use .

A different approach was followed in this study. A

numerical method based on Laplace transforms and time do

main fitting was developed for estimating packed bed heat

transfer parameters based on experiments with an imperfect

step input signal. The changes in sensitivity were exa

mined both experimentally and mathematically.

The specific tasks in developing this technigue in

clude:

1. Derive a set of partial differential eguations de

scribing heat transfer between a homogeneous, two-

phase fluid and the solid bed, solving this system

of eguations in the Laplace domain and inverting

the solution in the time domain by a new numerical

method which is specific for this system.

2. An experimentally measured temperature response

curve was used to find the heat transfer parameters

as functions of flow parameters. The numerical

method decribed in Task 1 was used to recover the

heat transfer pararaeters-

3. The relative importance of individual mechanisms

responsible for heat transfer was examined by com

paring their magnitude under different flow condi

tions using both the numerical and experimental

results.

CHAPTER II

LITERATDRE REVIEW

In solving a heat transfer problem with convection

present there is always a conjugate problem in fluid me

chanics. A unique feature of two-phase flows, the flow-re

gime, is of particular importance in this study.

Basically, there are four flow regimes in two-phase

packed bed flows:

1. The gas continuous flow at low gas and liquid rates

(trickle flow).

2. Dispersed-bubble flow at higher liquid rates (11-

guid continuous).

3. Pulsing flow at increasing gas rates (gas and 11-

guid separated in slugs).

4. Spray flow at higher gas rates (liguid dispersed).

The boundaries of the flow patterns were determined

by visual observations (Talmor, 1977), sharp increases in

pressure fluctuations (Chou et al-, 1977; Sicardi et al.,

1979), and variations in the apparent electrical conduc

tivity (Matsunra et al., 1979).

In several studies (Fukushima and Kusaka, 1977;

Matsuura et al., 1979) the air-water system was

investigated. However, Chou et al. (1977) and Sicardi et 4

5

a l . , (1977) have indicated that the e f f e c t of phys ica l

propert ies of the f l u i d s on the t r a n s i t i o n from the g a s -

continuous to the puls ing- f low regime i s very s i g n i f i c a n t .

Other e f f e c t s such as p a r t i c l e shape, surface roughness

and s i z e , or the reactor t o p a r t i c l e diameter r a t i o can be

q u i t e s i g n i f i c a n t in e s t a b l i s h i n g flow p a t t e r n s .

Tr ickle Flow Regime

In the t r i c k l e flow regime the gas phase i s cont inu

ous and l i q u i d t r i c k l e s over the packing in the form of a

f i l m , r i v u l e t s and drops. Osually the lowest flow ra te s in

two phase flow w i l l c rea te a bubble regime where the gas

phase i s d ispersed through a continuous l i gu id phase.

This region may have two kinds of flow: laminar g a s - l i q u i d

flow and laminar l i q u i d - t u r b u l e n t gas f low.

Pulse Flow Regime

As the gas flow r a t e i n c r e a s e s , the drag force be t

ween gas and l i g u i d w i l l a l s o i n c r e a s e . At some point the

drag force w i l l become s u f f i c i e n t l y large to cause turbu

lence in the l i g u i d phase. Some l i q u i d may be separated

from the l i q u i d film and create s lugs or drops. When the

gas flow r a t e i s further increased , the drops become large

enough to block the vo idages . This causes the v e l o c i t y of

both phases to increase through increas ing the pressure

head necessary for f low. This kind of pulse wave w i l l

then tend to propagate along the whole bed. Pulsing always

begins at the bottom of the bed, with the top of the co

lumn rippling. Velocities are greater at the bottom of the

column due to the loss of pressure head through the column

and expansion in the gas phase. The expansion in the gas

increases the volumetric flow rate of gas phase.

Spray Flow Regime

Further increasing the gas flow rate will tend to

squeeze the pulsing slugs together and the flow pattern

will switch to spray flow. In spray flow, the gas is con

tinuous, with liguid being carried as mist along the co

lumn. Packings are wetted by a liguid layer. This layer

becomes thinner as gas flow rate is further increased.

Flow Haps

Weekman and nyers(1964), Turpin and Huntington (1967) ,

Charpentier and Favier (1975) , Talmor (1977), Han (1982),

Shah (1979), Halfacre (1978), Gianetto et al. (1978), Hof-

mann (1978), aoid Satterfield (1975) have given different

flow maps for nonfoaming and foaming systems. These flow

maps differed only by the parameters used as coordinates

in indicating the flow patterns.

Weekman and Myers (1964) used gas and liguid rate as

the coordinates to present the flow regimes for the air

and water system. Turpin and Huntington (1967) modified

7

this flow map by plotting GL/GG vs. GG for the same

system. These maps are good only for the system they stu

died (air-water) .

Charpentier and Favier (1975) developed maps for the

combination of organic and inorganic systems. They took

into consideration the influence of physical properties

such as viscosity, density and surface tension, on the

system. Flow regimes were presented for foaming as well

as nonfoaming systems.

Talmor (1<;77) proposed a flow map using the gas-to-

liquid ratio and driving force-to-resistance force ratio

as the two correlating parameters. The resistance force

in the system is inertia and the three driving forces are

viscous forces, surface tension and gravity (for downward

operation). The ratio of the three forces versus inertia

was used as a single operating parameter to be plotted

against GG/GL.

Halfacre (1S78) studied the effect of surface tension

on the flow pattern by using isopropanol and water solu

tions. He made a modification on Talmor»s map at Talmor's

boundary at the pulsing range.

Han (1982) examined the two most important parame

ters, flow regime and pressure drop, by using liquid

mixtures instead of pure liquids to study the behavior of

foaming and nonfoaming systems. The liguid mixtures used

were isopropanol-glycerol-ethylene glycol mixtures and

8

ispropanol-water mixtures. The r e s u l t s , when compared with

Talmor»s, showed that Talmor's flow map can not f i t the

foaming system and only p a r t i a l l y f i t s the nonfoaming s y s

tem. Figure 1 i l l u s t r a t e s a t y p i c a l flow map for foaming

and nonfoaming systems adapted from Gianetto and S icardi

(1978) . In t h i s f i g u r e , G and L stand for the mass flow

rat« and a , p, ^ p, stand for the surface tens ion , v i s

c o s i t y and the dens i ty of the mater ia l , r e s p e c t i v e l y . The

subscr ip t s L and G stand for the l iquid and gas .

In terac t ion Regimes

Another way to des ignate flow patterns was proposed

by Baldi and Specchia (1977) . They defined two d i s t i n c t

flow regimes, the poor - in terac t ion and the h i g h - i n t e r a c

t ion regimes. They considered t r i c k l e flow or gas -cont inu

ous flow to be a poor i n t e r a c t i o n regime and spray and

puls ing f lows to be h igh- in terac t ion regimes. Based on

t h i s c r i t e r i o n , S icardi e t a l . (1979) have prepared a modi

f i ed flow map for non-foaming sytems. This map agrees wel l

with the work of Charpentier and Favier (1975). Figure 2

shows the Specchia and Baldi map.

Importance of Flow Regime in Two-Phase Flows

There are s e v e r a l flow models proposed to expla in

transport phenomena in two-phase flows in packed beds .

10 r2 3 6 10 -1 3 6 10 3 6 10

G/A£ fkg/m2sl Source of the lines-:

a: Gianetto (1970)

b: Sato (1973)

c : Charpentier and Favier (1975)

d: Chou (1976)

e: Specchia and Baldi (1977)

Figure 1: Flow Pattern Diagram for Non Foaming and Foaming Systems (Charpentier and Favier (1978))

10

^ r -^ ^^ r r r* VO 0 \ 0 \ ON r ^ I—I i-H 1—« o\ >^>-^^^ ^^ to tn sz

•H 'H o X jc jcz tn D o o r-o c* U^ *--

re ^ x: u 04 u

u/ O Ui

CN

M 3

Il4

fM r^

(/I

E

11

Some of them use as many as six parameters to correlate

the experimental data for mass or heat transfer. The

large number of parameters is due to the following facts:

1. liguid distribution within the bed is very impor

tant in the interpretation of data. In two-phase

flow systems, flow regimes are divided into stag-

rant and dynamic regions and mass transfer occurs

letween these two regions.

2. In different flow regimes, various equations must

le set up to describe hydrodynamic conditions in

the packed beds. Some of the parameters can be ne

glected in certain regimes to simplify the solu

tion.

3. Is described in the following sections, there are

sometimes sudden changes in the physical properties

cf the system as the flow regimes change. This is

because as the flow regime changes, the liguid dis

tribution and pressure drop will have an abrupt

change. An example of this is from Han (1982) who

indicated that there is a sudden change in pressure

drop when the flow regime changes from trickle to

fulse flow in a foaming system. However, in non

foaming systems the pressure drop is a very smooth

function across regime boundaries.

12

Axial Dispersion of Two-Phase Flows in Packed Beds

Definition of Dispersion

Since not all the fluid in the bed is flowing at the

same velocity (which is the assumption of ideal piston

like flow) , some parts of the liguid will be retained in

the system longer than the others (lower renewal rate) .

This results in micro or macroraixing. Dispersion is the

mechanisB used in describing the mixing and residence time

distribution data for various flow.

nany factors way affect raacromixing of fluids in a

transport processes. Various models are summarized in the

reviews of Gianetto et al. (1978) and Shah (1979) . They

showed that a comparison of the dispersion parameters ob

tained from any given model depends heavily on the method

of analysis used with the model. This is to say that even

when the same model is used to describe the system, the

resulting parameter, though model dependent, will also de

pend on the method of numerical analysis. Sicardi (1980b)

confirmed this conclusion. For this reason both solution

of a descriptive model and the method of numerical emaly-

sis employed is usually important for flow through a

packed bed.

13

Dispersion Models

The mathematical models used for interpretation of

liquid residence time distribution data can be categorized

according to the number of parameters used in the model:

1. Two-parameter model (PD or piston dispersion model)

used by Michell (1972). This model assumes plug

flow in the system. Dispersion and total liguid

hold-up were used as the two parameters. This mo

del was not successful because the physical pheno

menon of the liquid trickling on the packing is too

complex to be explained by dispersion only.

2. Three-parameter model (cross-flow model). In this

model, the liguid distribution is divided into two

zones: the stagnant zone and the dynamic zone. Li

quid hold-up, the mass transfer coefficient and the

dynamic hold-up are the three parameters used in

the cross flow model. Hochman (1969) and Hoogend-

oorn (1965) used this model to calculate mass

transfer coefficients in the air-water system. Pe

clet numbers for the individual flowing phases were

calculated and correlated by the moment method. In

their work both authors found it is very difficult

to achieve a satisfactory reproducibility of the

experimental results. This was attributed to the

somewhat random character of the dispersion

coefficient. A more rigorous model was recommended.

3. Four Parameter Hodel (PDE or piston dispersion

exchange model) This model was derived from the

three parameter model by assuming that axial dis

persion is superimposed on plug flow. The result

ing model has the advantage of having enough param

eters to describe the system and yet still remain

simple. Some of the five-parameter-models can be

simplified to this model.

Due to various kinds of input signals and

different numerical methods, Hatsuura (1976) deter

mined parameters by using the least mean square er

ror, while Bennet (1970) used a graphical method)

to determine these parameters, all the parameters

except total liguid hold-up are subject to doubt in

their true significance. Therefore, the dispersion

coefficients are not always reliable for comparison

between values obtained from different methods and

models (Sicardi, 1980). This explains why there is

guite a scattering in the Peclet numbers which have

been ccllectd from various authors. (Sicardi,

1980) -

Dispersion Coefficient Correlations

Host dispersion coefficient data in the literature

have been obtained using mass tranfer experiments (Hochman

15

and Effron (1969), Lerou (1980), Nakamura (1983), Van

Swaaij (1969)). The results from these works show that

dispersion has a significant effect on residence time dis

tribution data in gas-liquid trickle bed reactors. When

the Peclet number for the gas phase was correlated, it was

seen to be a function of liguid and gas flow rate, as well

as the size and nature of the packings. The liguid Peclet

number is a function of the liquid flow rate and the size

and nature of the packing only.

The Peclet numbers for the trickle flow regime are

lower than for single-phase flow through packed beds at

low liguid flow rates. Peclet numbers for both of the op

eration methods mentioned above will achieve an asymptotic

value( which is not a definite number and will depend on

the method of analysis) for high liquid flow rates. Matsu

ura (1976) gave 1.5 as the limiting value for the disper

sion coefficient.

Hochman and Effron (1969) presented their results for

gas phase dispersion coefficient as:

Pe - 1.8 Re-^-7 10-^-^5 ReL ^ ^ ^

Pe -D

U D (2-2)

a

16

where Da is the dispersion coefficient and D is the

diameter of the particle. U is the interstitial velocity

of the fluid in the packed bed. The Reynolds numbers are

based upon the packing diameters and the void fraction of

the bed.

ReG - G(, D/ (1 - e) p^ (2-3)

ReL ^ D/ (1 - G) y (2-U)

It must be mentioned that most of the systems report

ed in the literature used air and water as the two fluids-

Therefore, use of these correlations should be restricted

to the air-water system also.

The above results are for the trickle flow regime

only. Very few works have been published for the pulsing

flow regime. Lerou (1981) and Hatsuura (1976) used diffe

rent models to explain axial dispersion in different flow

regimes. Lercu (1981) described flow in the pulsing re

gime as alternate layers of gas-rich and liquid-rich slugs

passing through the packings. Due to the differences in

velocity and to rapid mixing between the two layers, slug

flow may be described by a model which regards the liguid

phase as consisting of two continuous phases flowing

cocurrently at different velocities. There is a continuous

exchange between the two phases. The results show that at

17

sufficiently high gas flow rates, dispersion can be

neglected due to the large Pe calculated. Liguid disper

sion decreases with increasing liguid flow rate and is

smaller than that found in the trickle flow regime-

There are interactions between the distribution of

liquids and the dispersion. Hatsuura (1976) assumed that

the liquid hold-up consists of two parts,one stagnant and

the other dynairic. The liguid flows in an axially dis

persed plug flow and transport occnrs between the two li

guid parts. Eguations were derived for both the stagnant

and dynamic zones. The result was a four parameter model

which contained axial dispersion, dynamic hold-up, the

mass transfer coefficient and the residence time distribu

tion to describe the system. Hatsuura's results show that

Pe is a constant at low Reynolds numbers and rises gradu

ally in the range 150 < Re < 400 and then is again cons

tant at higher Re. He also indicated that axial mixing was

caused mainly ty hydrodynamic mixing.

Dispersion is not a controlling factor in the high

interaction regimes. Other effects such as convective

transport are more important. This is due to limitations

on hydrodynamic mixing. In the low interaction regimes,

the liguid and the packings are at thermal equilibrium and

hydrodynamic effects can be more inportant.

18

Heat Transfer Mechanisms in Packed

Beds

Based on research with packed bed catalytic reactors

in single phase flows, the heat transfer mechanisms may be

divided into six different categories:

1. Radial or axial mixing in the fluid.

2. Radial and axial conduction in the solids-

3. Fluid to particle heat transfer.

U. Fluid to wall heat transfer.

5. Solid to wall heat transfer.

6. Intraparticle heat transfer-

Hears (1971), Gunn (1974), Wakao(1981) found that the

relative importance of heat transfer mechanisms follows

the order: 1. Radial heat transfer.

2. Fluid to particle heat transfer.

3. Intraparticle heat transfer.

Figure 3 depicts these mechanisms.

Wall Heat Transfer in Two-Phase Flow in Packed Eeds

The wall heat transfer coefficients have been found

to depend on the hydrodynamic regimes present in two phase

flow in a packed bed. Huroyama (1977) and Hatsuura

(1979c) developed a model which includes five

contributions to the wall heat transfer coefficient, hw:

19

(A)

(B)

(G)

A: Radial fluid heat t ransfer B: Axial fluid heat transfer C: Fluid-to-Wall heat t ransfer D: In t rapar t i c l e heat t ransfer E: Solid-to-wall heat t ransfer F: F lu id- to-par t ic le heat t ransfer G: Direction of f luid flow

Figure 3: schematic Representation of Indiv idual Heat Transfer Hechanisms in Two Phase Flows in Packed Bed

20

1. The apparent wall heat transfer coefficient, hwO.

For a stagnant situation, hwO is a function of par

ticle size.

2. The apparent wall heat transfer coefficient, hws,

which represents the heat transfer between the

fluid and the wall near the contact surface between

the particles and the wall. This was correlated as

a function of liguid hold-up.

3. The true heat transfer coefficient, hw, correlated

as a function of the gas-phase Reynolds and Prandtl

numbers for the lew-interaction regime, and liquid-

phase Reynolds number for the high interaction re

gime.

U. The apparent heat transfer coefficients, hwg and

hwl, accounting for heat transfer due to radial gas

and radial liquid mixing, respectively.

Haroyama obtained wall heat transfer coefficients us

ing water and agueous glycerol solutions as the liquid

phases. He found hw is a constant in the high interaction

regime. This coefficient is about one order of magnitude

higher than for single-phase flow. He reported the a cor

relation in terms of Nusselt number for the wall heat

transfer coefficient in high interaction regime as:

h D %L^ 1/3 D G. 0.8 ^ - 0.092 ( — ) ( i ^ ) J2-5)

\ \ ^ ^ \

21

where kL is the thermal conductivity of the liguid and CpL

is the heat capacity of the liquid.

Specchia (1979) obtained correlations for low and

high interaction regimes separately. In the low interac

tion regimes hw increases when VG increases. This may be

explained by the effect cf an increase in velocity of the

liguid film over the wall and the wetted wall area. Spec-

hia's correlation for the low interaction regime was:

— = 0.057 ( PveL)°- ^ (Pr)^/^ (2-6)

where:

«^- GL V^t '^"•'*»

Pr = ^ PL (2-7B)

Specchia showed that hv is constant in the high in

teraction regime. At certain VG and VL values the wall

becomes entirely covered by a liguid film. Increasing the

liguid velocity can be counterbalanced by the effect of

liguid distribution in the radial direction, with result

that hw is constant. This result is the same as found by

Huroyama (1976). The asymptotic value, when expressed in

terms of Nusselt number, has the average value 2. 1

22

KJ/m sK. Huroyama»s correlation is recommended for

estimating hw in trickle beds. This correlation was ob

tained using the small particles normally found in cata

lyst systems. The complex result proposed by Hatsuura

(1976)may not apply to other fluids and different operat

ing conditions. The Specchia and Baldi (1979) correlation

is probably not applicable for small packings.

Radial Heat Transfer oJ Two-Phase Flows in Packed Beds

Heat transfer in packed beds becomes very important

when there is a reaction taking place in the packed bed,

since the heat involved will usually increase proportion

ally with concentration and rate of reaction. When there

is a large heat effect, temperature gradients within the

bed may be very important. Radial heat transfer can not be

neglected because when the reactor is of finite length,

the thermal gradients in the radial direction are no lon

ger negligible. The solids within the bed will also con

tribute to the heat transfer from the solids to the vail.

A review of radial heat transfer behavior and modeling of

fers insight into axial heat transfer mechanisms as well

as description of a significant part of the total heat

transport process.

23

Experiments and flodels

W€€kman and Hyers (1965) developed a packed bed heat

t r a n s f e r model which neglected the heat transfer at the

w a l l . I l l r e s i s t a n c e to heat t rans fer was included in an

e f f e c t i v e r a d i a l thermal c o n d u c t i v i t y , ker . This ker was

found t c be 2-*l times larger in two-phase flow than for a

l iqu id f i l l e d , s i n g l e - f l o w reactor at the same flow rate

as the air-water system.

Hashimoto (1976) measured r a d i a l temperature p r o f i l e s

for a packed ted in two-phase f low, including the wall

temperature, at various depths in the bed. The data were

analyzed using a two parameter, steady s t a t e model. A

c o r r e l a t i o n for the e f f e c t i v e radia l thermal c o n d u c t i v i t y ,

ker, which covered both low and high in terac t ion regimes

was derived. The kel term, which i s the e f f e c t i v e l i g u i d

rad ia l l i x ing contr ibut ion , was dominant and at c e r t a i n

l i gu id l a t e s was larger than the value for l i g u i d s i n g l e -

phase flow.

The Hashimoto e t a l . c o r r e l a t i o n , which was obtained

with s irel l p a r t i c l e s (0.26x10-^ cm spheres ) , g ives a con

serva t ive es t imate of ker. It i s recommended for e s t i m a t

ing the e f f e c t i v e rad ia l thermal conduct iv i ty (Smith,

(1983)) for a ted in two-phase flow.

Specchia and Ealdi (1979) used the assunption of Yagi

and KuEii (1S57) that the e f f e c t i v e radia l thermal

conduct iv i ty can te expressed as the sum of two

24

c o n t r i b u t i o n s : a stagnant e f f e c t i v e thermal conduct iv i ty

ke, which i s independent of hydrodynamics (conduction con

t r i b u t i o n for molecular conduct iv i ty) and a contribut ion

a s s o c i a t e d with the convect ive radia l flow of gas , keg,

and l i q u i d , k e l . The e f f e c t i v e rad ia l thermal conduct iv i

ty i s given by:

ker=ke • keg + kel (2-8)

Dif ferent c o r r e l a t i o n s were derived for individual flow

regimes.

ke l for the low i n t e r a c t i o n regime and VG=0 was cor

r e l a t e d with the l i q u i d Peclet number,

PeL= 0.041 ( ReL )°-®^ (2-9)

In the high i n t e r a c t i o n regime: FeL i s a function of

hold-up and packing geometry.

PeL - 338 (ReL)°'675 g 0.29 ^ ^ ^,^ y2.7 ,2-10)

The data obtained by Speechia and Baldi (1979) were

correlated and cover a wide range of operating conditions,

packing shapes and sizes, solid particle thermal conduc

tivities and reactor diameters. The term ke is a function

of bed void fraction and the thermal conductivity of the

solid and the gas. Both ke and keg were derived from a

study of gas flow through packed teds. These two terms

25

are smaller than kel. The Specchia and the Hashimoto

correlations give Peclet numbers in good agreement for ra

dial heat transfer in the liguid phase.

Hathematical Hodels for Two-Phase

Flows

Two phase flows are very d i f f i cu l t to simulate. In

part t h i s i s the result of problems in describing a very

complex gas- l iguid in ter face . The result ing partial dif

f erent ia l eguations are d i f f i c u l t or impossible to so lve .

Several methods for simplifying the description of two

phase gas- l iguid flows have been described by Hestroni

(1982). Simplifying approaches include use of local i n

stantaneous eguations, use of instantaneous space-averaged

equations, use of time-averaged equations, and use of com

posite-averaged eguations.

In some cases a multiphase mixture may be considered

as a s i n g l e , f i c t i o n s f l u i d . This leads to one-fluid mo

d e l s , where the equations are almost ident ica l to s i n g l e -

phase eguations. The c r i t i c a l problem in th is approach i s

the determination of mixing rules for the physical proper

t i e s . A number of mixing rule approaches are poss ib le .

However, one-f luid models are not extremely powerful s ince

they do not account for l iquid distribution variations

with flow regime.

26

Hultifluid models consider each phase as a distinct

fluid, but are not necessarily sensitive to flow regime

changes. A complete description of a multiphase flow sys

tem also needs topological laws, to describe the flow

structure as well. H ultifluid models treat each fluid as

an independent phase, with the link between phases speci

fied by the transfer law. The models used by Kan (1983)

and Hatsuura (1976) are examples of multifluid models.

They used the concept of dynamic and stagnant zones for

the application of topological and transfer laws.

Hathematical Hodels for Single-Phase Heat Transfer in Packed Beds

The mathematical models used for single-phase flows

in packed beds can be divided into two categories: pseudo-

homogeneous and hetrogeneous. Pseudohomogeneous models

assume there is no temperature gradient between phases in

the two phase system (solid and fluid). This kind of mo

del will be correct when the convective term is the domi

nant heat transmission term. Convection dominates when the

flowing velocity is high.

The heterogeneous models postulate that local temper

ature differences are affected by solid-fluid heat trans

fer. The rate of solid-fluid heat transfer is influenced

by the fluid velocity. These models are closer to real

conditions and are more frequently used. Several different

27

assumptions may be used in developing a heterogeneous

model, each leading to a somewhat different final result.

The most important of these appproaches will be developed

below.

Continuous-Solid Hodel ( CS model)

In this model, the fluid is assumed to be in dis

persed plug flow. Axial heat conduction is considered only

in the solid phase. The equations to describe this model

are the following:

'''--"'^ - A^(T-V (2-11A) at 3x ^ f pf ^

^ s h a (2-1 IB)

pS 8

where h i s the p a r t i c l e - t o - f l u i d heat transfer c o e f f i

c i ent .

Equation 2-11A i s the d i f f erent ia l eguation for the

f lu id phase and Equation 2-11B i s for the so l id phase.

The f l u i d - t o - p a r t i c l e heat transfer coe f f i c i en t s are ob

tained ind irec t ly from measurements of axial qas tempera

ture p r o f i l e s . These prof i l e s are f i r s t used to evaluate

the e f f e c t i v e thermal conduct iv i t i e s of the bed ty

assuming that the gas and part i c l e s are at the same

28

temperature at any point within the bed. This single-phase

model demands an exponential gas temperature profile

(which the measured profiles satisfy) dependent only on

the effective thermal conductivities of the bed. There is

an algebraic relation between the effective thermal con

ductivity and the heat transfer coefficients and the mea

sured exponential gas temperature profile. The heat trans

fer coefficients are calculated from this algebraic

relationship.

Turner (1967), Cybulski (1975), Kunii (1961) used the

continuous solid model, but their results, when used with

steady state measurements, show that the fluid temperature

calculated with low heat transfer coefficients from the CS

model agree with temperatures computed using a single-

phase model. The solid temperature evaluated on the CS

model failed to match the temperatures on the single-phase

model.

The Schumann Hodel

This model was proposed by Schumann (1929) using the

assumptions:

1. The fluid is in plug flow and there is no disper

sion.

2. No temperature gradients exist within the

particles.

29

The eguations derived based cn those assumptions

were:

» 'ef »'T .. 3T ha r n C T ' " - n r (T - T ) (2-12A)

' ^ 3

a c

k s

* s ps

2

9 T

a X

h a

P c ^s ps

( 1 - e ) ^ - S— -^r-^ + " ' • ( T - T ) (2-12B)

Eguation 2-12A is the equation for the fluid phase and

Equation 2-12E is for the solid phase.

Furnas (1S30), Saunders (19^0), lof (1948), Cop-

page(1956), and Handely(1968) used this model to estimate

heat transfer parameters by graphical methods for the sin

gle-phase system. Dayton (1952), Heek (1961), and Shear

er ( 1962) used sinusoidal input techniques based on this

model to find Nusselt numbers for gas-solid systems.

Their results were accurate at high Reynolds numbers when

compared to previous work, but were not acceptable at low

Reynolds numbers. As pointed by Liftman (1966), at low Pe

conduction in the solids and dispersion in the fluid phase

affect the frequency response to such an extent that this

simple model will not fit.

30

Dispersion-Concentric Hodel (DC model)

This model has been recently developed and several

works have been published by Gunn (1970), Bradshaw (1970),

and Wakao et.al. (1976). The basic assumptions of the or

iginal DC model were:

1. The fluid is in dispersed plug flow.

2. The solid particle temperature profile exhibits ra

dial symmetry.

The assumption for the solid temperature profile ap

parently is not true at low flow rates because this model

will predict erroneous Nusselt numbers at low flow rates.

Wakao (1976) proposed that the dispersion coefficient must

be considered as well. The new dispersion coefficient is

defined as:

s ps

instead of :

k "ax = 1 — (2-14)

e p C s ps

where R is the radius of the particle and 0 is the in

terstitial velocity of the fluid in the packed bed.

The basic equation for the fluid phase in the

modified DC model was:

a T a T „ aT h a r T - T I )

at ^ a X* °" ^ pf V ^"""^

31

The b a s i c egua t ion d e s c r i b i n g the t empera tu re p r o f i l e

within the s o l i d s was:

3T^ k 3 X 2 31 . ^ S . « _ l . . r . . ^ s ^ s ^ J2-16)

3t PgCpg 3 r r 3r

Only two r e s e a r c h e r s have publ ished t h e i r r e s u l t s fo r

hea t t r a n s f e r in a packed bed us ing the DC model. They a r e

Gunn (1S70, 1971, 1974, 1977) and Hakao (1977a, 1977b,

1978, 1?79, 1S61, 19€2a, 1982b). Gunn (1970) was t h e f i r s t

to publ i sh the s o l u t i o n of t h i s model for d i f f e r e n t boun

dary c o E d i t i o n s as well a s d i f f e r e n t k inds of i n p u t s i g -

isals t o the packed bed sys tem. Gunn a l s o suggested a num

ber of t echn iques for e s t i m a t i n g heat t r a n s f e r parameters

using the Laplace t ransform method.

Gum (1974) v e r i f i e d t h e a p p l i c a b i l i t y of t he modi

f ied DC model ty conduct ing a hea t t r a n s f e r experiment in

s packet bed with s i n u s c i d a l l y varying heat i n p u t . He

measured the ampl i tude r a t i o and phase angle d i f f e r e n c e

between the i n p u t / o u t p u t s i g n a l s . Air was used as the

f l u i d ard the tec! was packed with m e t a l l i c or g l a s s beads .

Reynolds numbers ranged from 1 to 300 for the gas phase .

Heat t r a n s f e r c o e f f i c i e n t s and d i s p e r s i o n c o e f f i c i e n t s

were c a J c u l a t e d using time domain f i t t i n g . As shown by

Cunn (1S77), the omission of therieal d i spe r s ion i s as

impor tan t a t high Reynolds numbers a s i t i s a t low

Reynolds numbers. The conf idence l e v e l for the parameters

32

Reynolds numbers. The confidence level for the parameters

is also higher at high Reynolds numbers than at low Rey

nolds numbers.

The same conclusions were also reached by Wakao

(1979). Wakao used the modified DC model to correlate

heat transfer data from a number of previous works. The

final correlation was:

Nu - 2.0 + 1.1 (Re)°- (Pr)^^^ (2-17)

Wakao showed that d i spers ion p lays an important r o l e in

f l u i d - t o - p a r t i c l e heat transfer at low Reynolds numbers,

owing to the thermal equil ibrium between the s o l i d s and

the f l u i d . At high Reynolds numbers, the macromixing i s

more important than the molecular conduct iv i ty mechanism

and dispers ion i s mainly due to the turbulent mixing.

Techniques for Measuring Packed Bed Heat Transfer Coefficients

There are two major categories of methods for deter

mining the fluid-to-particle heat transfer coefficients,

steady state measurements and unsteady state mesurements.

The unsteady state experimental methods have gained much

attention in chemical engineering research because the

time distribution better characterizes the system and may

also provide a similarity criterion for the scale-up of

equipment.

33

Steady State Measurements of Heat Transfer

Work of t h i s type has been reported for r e l a t i v e l y

simple process systems such as s ing le -phase flow in pipes

and in packed beds. Steady s t a t e heat transfer measure

ments were used by Ramson, Thodos and Hougan (1943) for

the evaporation of water from porous c a t a l y s t carr i er s in

an a i r stream during the constant drying period. The meth

od f a i l e d at low Reynolds numbers because the temperature

d i f f erence between the gas and p a r t i c l e at the o u t l e t of

the bed was so small that the mean temperature d i f ference

between the gas and p a r t i c l e s in the bed could not be d e t

ermined accurate ly . The only parameter in t h i s kind of ex

periment was the f l u i d - t o - p a r t i c l e heat transfer c o e f f i

c i e n t s .

Glaser and Thodos (1958) maintained steady s t a t e con

d i t i o n s in a packed bed by pass ing an e l e c t r i c a l current

through a bed of m e t a l l i c p a r t i c l e s and then measuring the

f l u i d and s o l i d temperature d i f f erences to obtain the heat

transfer c o e f f i c i e n t s .

S a t t e r f i e l d and Resnick (1954) studied the decomposi

t ion of hydrogen-peroxide on polished c a t a l y t i c metal.

P a r t i c l e temperature was measured to c a l c u l a t e heat t r a n s

f er c o e f f i c i e n t s .

The steady s t a t e measurement method has been

c r i t i c i z e d by Wakao (1977,1979) and Gunn (1974). The

34

reason is that at steady state, the temperature of the

particle and the fluid is the same. Erroneous heat trans

fer coefficients may be obtained because of unreliable

measurement of the low temperature difference. The coef

ficients thus obtained have no physical significance. The

heat transfer coefficients vere low possibly due to the

errors in measuring the final temperatures of solid and

liquid at the bed exit.

Unsteady State Heat Transfer Heasurement

The fundamental principle of unsteady state measure

ment is to impose a time varying input signal on a given

system and measure the output signal (response) of the

system. The input signal and output signal are linked

through an equation that contains the parameters of inter

est. By manipulating an appropriate mathematical model of

the system, these paramters can be identified-

There are several types of input signal that can be

imposed on a system, a step input, an imperfect pulse in

put and a sinusoidal input signal. The step input is usu

ally used in heat transfer measurements because it is ea

sier to generate this kind of signal. An imperfect pulse

signal is used in mass transfer measurements. Due to the

conservation of mass, the imperfect mass pulse method has

a higher accuracy when compared to heat transfer

35

measurements. An imperfect thermal pulse input usually

does not contain enough energy for accurate measurements,

because the temperature variation is not large.

Sinusoidal inputs are very difficult to generate- The

frequency can not be too high due to limitations on the

heating unit. However, sinusoidal inputs have been used

specifically in heat transfer studies under the name of

the frequency response method.

Frequency Response Hethod

This method uses a sinusoidal signal as the input.

After a sufficiently long time, the response of the system

will differ from the input signal in amplitude and phase

angle. The amplitude ratio of the input and output signals

together with the phase angle allow calculation of the

parameters of the system.

Littman (1968), Lindauer (1967), Wakao (1980), Gunn

(1974) have used the freguency-response method in single-

phase gas flow through packed beds. Gunn (1974) studied

the dispersion effect within a packed bed using the fre

quency response method, with special attention to low Rey

nolds number regimes. He used the DC model to simulate

the packed bed system. By measuring the temperature at

the exit of the bed and analyzing the amplitude ratio and

phase angle difference between the input and output

temperature waves, it was possible to obtain the Nusselt

mt^,.

36

number and Peclet number for the system . He found that

these numbers are much larger than those measured in the

corresponding mass transfer systems.

Imperfect Pulse Input Hethod

Pulse testing is widely used in chromotography and

many works related to mass transfer, but it is seldom used

in heat transfer. As indicated in the previous section,

the pulse testing method has the advantage of being suita

ble for application both of the moment method and curve

fitting in the time domain as a means of parameter estima

tion. This provides an indirect but very efficient way to

determine heat and mass transfer parameters, curve fitting

of the response to an imperfect pulse will be much more

accurate than when using step input signals.

Step Input Method

Step inputs are seldom used in heat transfer experi

ments although it is relatively easy to generate this kind

of signal. Bradshaw (1970) obtained heat transfer coeffi

cients for packed beds by using a step input with packed

beds of alumina and steel balls. The Laplace transform

method was applied to calculate the parameters. The

response curve is not very sensitive toward changes in the

heat transfer parameters with step inputs compared to the

sensitivity to a pulse type input signal.

37

Tailor's Hodel for Two-Phase Flow in Packed Beds

Tailor(1982) derived an eguation for heat transfer in

two-phase flows in packed beds. The equation is based on a

one-fluid DC model with fictious physical constants calcu

lated from existing mixing rules. However, this equation

did not involve any information about the liquid distribu

tion within the system. Tailor used this model to deter

mine the parametric sensitivity of the system toward the

material used in the packed bed and the frequecy used in a

sinusoidal input- He used a similar method to Wakao

(1979) to find the best fit parameters to the model. This

work indicated that steel spheres have the highest accura

cies in measuring the heat transfer coefficients and axial

dispersion coefficients. He proposed to use a sinusoidal

wave input to the system and found that the higher the

freguency, the higher the accuracy in measuring the heat

transfer coefficients and the axial dispersion coeffi

cients.

In this work, an experiment is designed to verify the

Tailor model in low and high interaction regimes. The re

sults are presented in the following chapters.

CHAPTER III

EXPERIMENTAL SET-OP

Experimental Apparatus and Procedures

Packed Bed and Temperature Measuring Devices

The experimental apparatus consisted of a 3.3 ft

(100.6 cm) long by 3 inch (7.6 cm) inside diameter tranpa-

rent acrylic resin column. The bed was made up of 3mm

diameter, spherical glass beads. The bed had an average

voidage of 0.36. The voidage was measured by taking the

weight difference of a beaker, which had a diameter near

that of the column, packed with beads and the same beaker

with water added to the void volume.

The total length of the testing section was 33.03

inches (83.9 cm). There were 3.75 inches (9.5 cm) of

packing above the top of testing section and 1.75 inches

(4.5 cm) below it. This was intended to minimize the ent

rance effects and exit effects. Pressure gauges made by

the Marsh Instrument Company with a measuring range from

0-15 psig were used. The gauges were attached at the ends

of the testing section.

The support for the packed bed was made of stainless

steel mesh. The disengaging zone was made of the same

material as the packed bed. Three holes 3/4 inch (1.9cm)

38

39

in diameter were drilled at the bottom of the column for

the exits for fluids from the column.

At the entrance to the bed there was an orifice plate

made of plastic with holes drilled on an even spacing.

This plate served as a distributor. Glass tubes(chimney

type) of 1/4 inch (0.635 cm) ID and 1/8 inch (0.317 cm) CD

were implanted on the holes. The tubes were separated from

each other by a guarter inch. The height was the same for

all tubes implantd. This was to make sure that the initial

distribution of the fluid was made as even as possible.

A spray located 1.5 inch (3.8 cm) above the distribu

tor plate was used to spray the water evenly on the dis

tributor. This spray was made of 1/2 inch O.D. (1.27 cm)

copper tubing. Holes of 1/16 inch (0.16 cm) diameter were

drilled with 1/4 inch (0.635 cm) spacing. The eveness of

flow of the water across the packed bed was assumed based

on having the same temperature across the cross sectional

area. A sketch of the distributor and spray is provided

in Figure 4 and sample radial temperature profiles are

shown in Figure 5.

Thermocouples were inserted 33.03 inches (83.32 cm)

apart(which is the distance between entrance and exit of

the bed) and were set at different immersion depths ( one

at the center of the column and the other 3/4 (1.9 cm)

inch and 1/4 inch (0.635 cm) from the wall for measurement

of the temperature variations across the column ). These

Sprayer Detail

r i o 0 0 0 0 0 0 0 0 0 i- 10 0 0 0 0 0 0 0 0 0

<>0

:i o

0 00 0 0000 0 0

r 2. 4

I I

Holes in tube wall were 0.0625 inch diameter •

Distributor Gross Sectional View

ncn n n n nnnn 3 inches

C:(spacing between tubes) 1/4 inch width of tube 1/4 inch height of tube 1/4 inch

Distributor Top View

o o o o o o O o o o

O O O Q O 0 0 0

O O O G o O O O O O O O O O O 0 0 0 o o , ^ o o o o O Q

3 inches

Figure 4: Schematic Diagram of the Distributor and SD'raver nsed in the Experiment. Spray

41

15.2-

--15.(1-o

b 4->

2 14.^

l l 4 .

14.2.

0

ReG=285.7, ReL=1.4 (plusing flow)

i 1 0.5 1.0

Distance Frcjm Wall

i 1.5

— » (inch)

18.2

--18w0 o

•wl7.9

^17 .6 u Ii7.4

17.2

17.0

ReG =58, ReL=1.4 (trickle flow)

•

<

0 0.5 1.0

Distance From Wall

1.5

(inch)

Each point i s the average temperature of the thermocouple

in thirty seconds.

Figure 5: Schematic Diagram of Padia l Temperature D i s t r i b u t i o n Within the Packed Bed

42

thermoccuplGs were 1/16 (0.16 cn) inch, grounded,

stainless steel, type K. The thermocouples were guarded by

a steel wire gau2e. The gauze shield was made so that it

is not larger than 1/2 x 1/4 inch. A 1/8 (0.3175 cm) inch

hole was drilled near tip of the gauze to allow the fluid

to pass through the gauze. The gauze was used to avoid

contact between the thermocouple tip and the solid pack

ings to ensure measurement of the fluid, rather than the

solid temperature. The thermocouples were linked to a di

gital recorder(Esterline Angus PE-2064) to record the

temperatures within the column. The recorder could be pro

grammed to reac at different time intervals for different

channels. A schematic of the whole apparatus is shown in

Figure t and a detailed drawing of the bed, showing ther

mocouple locations is shown in Figure 7.

Physical Constants Used in the Experiment

The average physical properties important to the heat

transfer experiments are summarized in Table 1. Table 2

summariaes the range of operating parameters use in the

study-

43

A Packed bed

B Gas rotameter

C Water rotameter

D Air heater

E Water heater

F Saturator

F i g u r e 6

G Regulator

Schematic Diagram of the Experimental Apparatus for the Two-Phase Flow Heat Transfer Experiments

Air i n l e t 44

Water i n l e t

(B)

A: Thermocounles B: Pressure gauges C: Bed ex i t D: Sprayer

E: D i s t r i bu to r F : Screen t o hold packings

- - : Center l ine of the bed. GH: Distance of the thermocouple

from the w a l l , 0.25 inch.

Figure 7: Schematic Diagram of the Packed Bed

45

TABLE 1

P h y s i c a l C o n s t a n t s i n t h e S y s t e

A i r ( 80a)F ) S a f e r ( 80aF ) G l a s s

C.0735 62.4 Dens j t y (A) ( l b / f t ^ 5 V i s c o s i t y x105 1.24 ( I b / f t - s e c ) Condcctivity (E) (Etu/ft-hr-aF) Heat Capacity (E) (Btu/lb-5lF) source of the table: (A)ncmentum, Heat and Mass Transfer. Welty, Wicks (E)Ctemical Engineering Handbook, Perry.

170

0.0152

1.85

57.8

0.353

1-0

0.45

0.2

Heating System and Water and Air Supply Systens

The heating system was designed in a parallel flow

pattern. The fluids were heated by heaters from the Linde-

burg Co. of different power levels. Variacs controlled

each heeter so that the guantity of heat input could be

varied over a vide range. Table 3 shows the combinations

of the beaters for different power outputs. The heating

pipes were galvanized iron of one inch (2.54 cm) OD. The

heating lengths for the water and air sides were 67 and 4

inches, respectively.

The water flow rate was measured using a calibrated

rotameter from the Dwyer Co., with aaximum flow rate of 1

46

TABLE

Maximum and Minimun of C

Flow rate of Gas(SCFM)

Flow rate of Hater (GPH)

Temperature of water (F)

Temperature of air (F)

Pressure of the column at top. (psig)

Pressure of the column at bottom, (psig)

Density of air(Ib/ft3>CA>

2

ip. erat ing

M a x .

7.5

0.9

100.

100.

15.

3.

0.0757

Paramete rs

0.

• " 1

Min. 1

1.0 1

0.5 f

70. 1

70. 1

0. 1

0. !

0728 1

TABLE 3

Power of Heaters

Power(Watt) Number of Heaters

1200(230 V) 3 750(115 V) 4 500(115 V) 4 250(115 V) 2 150(115 V for air) 2

' r*''! of w a t e r . T'hp r c t a t n e t e r was c a l i h r . ^ t p > l hy l i r p c t f low

mea.^iiremon t s . The a i r f low r a t e was ii.^a.sure.i hy i F i s h e r

( F P - 3 / 4 - 2 1 - G - 2 lO/"^"^) r o t a m e t p r w i t h a maxin-.um flow r a t e of

a i r of 10 SCF?1- T h i s r o t a m e t e r war. c i l i b r a t e d u s i n g t h e

p r e s s u r e b ^ l l in t h e u n i t o p e r a t i o n s l a b o r a t c r y .

The r a n g e o^ t e m p e r a t u r e c h a n g e f o r a l l e x p e r i m e n t s

was s e t a t 18 -20 ( F ) . T h i s magni tu^ .e was c h o s e n b e c a u s e :

1. The gas phase modified heat c a p a c i t y , CPG, can he

t r ea t e r l as a c o n s t a n t over the t e n p e r a t u r e range

1R-20 (F) (see F igure 3 ) ,

2 . The h e a t i n g system could not t o l e r a t e a l a rge h e a t

ing load .

The a i r was s u p p l i e d through the r r e s s u r i z e d pipe

l i n e in the old power p l a n t . The a i r was in t roduced to

the bottom of a v e r t i c a l s a t u r a t o r of t o t a l l ength 34

(S6. 36 cr\) i n c h e s . The s a t u r a t o r con ta ined water so t h a t

when the a i r e x i t e d , i t was s a t u r a t e d . S a t u r a t i o n was con

firmed by measuring the wet and dry bulb t empera ture of

t h e e x i t i n g a i r of t h e s a t u r a t o r . A wire gauze was placed

before the e x i t of t he s a t u r a t o r t o prevent water from en

t e r i n g the a i r h e a t i n g s e c t i o n . Before a i r was in t roduced

i n t o the column, i t was heated to the des i re '^ t e m p e r a t u r e ,

which was measured a t t he bed e n t r a n c e .

'i'/ater was in t roduced t o the sys ton by pass ing the

water through the c a l i b r a t e d rotam'^tor anl th roujh a

h e a t i n g s e c t i c n . Before the mixture of f l u i d s f lowei

48

o E

ro O

40 60 80 100 TEMPERATURE, F

120 140

F igure 8 Schematic Diagram of Cpg at D i f f e r e n t Temperatures

49

through the column, they were first mixed at the

distributor.

In the initial stages of the experiments, the air was

saturated in a saturator which was heated by a constant

water hath. The bath was to keep the exiting air tempera

ture from the saturator close to the temperature of the

water before heating. The air was heated by passing it

through the heater and then mixing it with the water which

is also heated to the same temperature as the air.

The imperfect step input was based on a change from a

higher to a lower operating fluid teirperature. The reason

for using a negative step is that switching the stream of

gas and liguid from hot to cold is much easier than from

cold to hot in this system. The cold temperature of air

and water was very stable and with continuous flow the

hotter water and air temperatures could also be stabi

lized. Ball valves were used for guick hot-cold switch

ing.

Operating Procedures

1. Check the offset of the digital recorder (PD-2064,

by Esterline-Angus Co.). The offset was always in

the range of 0.2-0.3 C. Check the temperatures at

the top and bottom of the column see if they agree

within the range of instrument error.

50

2. Turn on the air and water inlet line and determine

the inlet air and water temperature. Measure the

air temperature after the saturator. If the satu

rated air temperature does not match the water

temperature before heating, adjust the water bath

temperature to make the temperatures more nearly

egual.

3. Adjust the air line feed temperature by adjusting

the variacs to obtain a column temperature 10 C (20

F) higher than system inlet.

4. Adjust the water supply in the same way.

5. Run the column at steady state for 6-10 minutes.

The steady state was determined by the temperature

read-outs from the recorder. If the temperature of

each channel did not have a variance larger than

the offset of the instrument for 6-10 minutes, this

was considered a steady state.

6. Register time zero on the recorder by extending the

time to a preset time. When the time is up, switch

the hot streams for cold streams. The process of

switching reguired less than 3 seconds, which is

the interval for successive recordings of tempera

ture.

7. Check the temperature shown at the recorder see if

the temperature at the bottom and top of the column

has reached a new steady state reading.

51

8, Rerun the whole process for a new set of flow

rates.

•Vi

CHAPTER IV

MATHEMATICAL MODEL AND SOLOTION

The b a s i s for data a n a l y s i s i s a descr ipt ion of heat

t rans fer in two-phase flows in a packed bed in the form of

a mathematical model. This model dep ic t s the system in the

form of lumped parameters. The parameters must be able to

account for the fo l lowing heat transfer mechanisms:

1. P a r t i c l e - t o - f l u i d heat t r a n s f e r .

2 . Axial d i spers ion heat t rans fer .

3 . Convective heat t ransfer within the packed bed.

When t r a n s i e n t methods were used, mechanisms and parame

t e r s can not be re la ted without using a model. Direct

measurement of the d i spers ion c o e f f i c i e n t and p a r t i c l e - t o -

f l u i d heat t rans fer c o e f f i c i e n t i s d i f f i c u l t , but use of a

model can provide an accurate , i n d i r e c t way of measuring

the paramters.

A one-dimensional model s imi lar to the one-phase D-C

model developed by T a i l o r (1981) i s used with the assump

t i o n s :

1. There are no r a d i a l temperature gradients in the

bed.

2 . Thermal and thermodynamic eguil ibrium between the

gas and l i g u i d i s a t t a i n e d .

3 . The s p e c i f i c heat c a p a c i t y , Cpg, defined as 52

53

PG = ^y^ (4-1)

where } \ , the enthalpy of saturated air, is cons-s

tant. The studies by Tailor (1981) and by Weekman

(1965) suggest that if the column is operated over

a small temperature range ( AT below 20 F) this as

sumption is true. Tailor also found that in the

range of 70-100 (F), Cpg is approximately a cons-

tan t.

4. The particle surface area is completely wetted. Ac

cording to Charpentier (1976), for liguid flow

rates greater than 5 Kg/mz sec, the particles are

completely wetted. In this work, the liguid flow

rate was at least 6.88 kg/m^ sec.

5. The solid packing has concentric temperature pro

files. Actually, there is conduction between the

solids. Also the effect of convection is such that

the temperature profile will not be exactly concen

tric, but of interest is the mean heat flux over

the surface of the particle. The local heat flux on

the surface of the particle is affected by varia

tions away from a concentric profile, but the moan

is not.

54

Based on these assumptions, the energy balance egua

tion for the fluid phase is:

c[ ( 1 - 6)P(, C^ D^^ + B P L C P L [ > 3 L ] - ^ 3 z

(4-2)

d z

- c( (1 - 6)Pc Cp + Bp CpL) - I f

where

a= s p e c i f i c surface area of the par t i c l e s , f t ^ / f t '

A = cros s - s ec t iona l area of the tube, ft^

Cp_= s p e c i f i c heat of saturated a i r ,

BTu/lb-5F

C = s p e c i f i c heat of l iquid , BTO/lb-aF

D^= axia l gas dispersion c o e f f i c i e n t , f t^/s aCj

D - = axial l iguid dispersion c o e f f i c i e n t , f t^ / s

G = superf ic ia l gas mass v e l o c i t y , Ib / f t^-s G

h = partical-to-liguid heat transfer coefficient,

BTa/ft2-s-aF

R = diameter of the particle, ft-

T = temperature of the gas and liguid, iF.

Ts= solid temperature, flF.

t = time, sec Z = axial coordinate, ft

G = bed void fraction, dimensionless.

6 = liguid hold-up, dimensionless.

55

t>L- gas density, lb/ft'

pQ= liquid density, It/ft^

ReG* gas phase Reynolds number, defined as Re_ = G^ D/i, e G G •' G

ReL= liquid phase Reynolds number, defined as Re.= (1 D/lir '

Equation 4-2 may be simplified by using mixing rules

to create pseudo one-fluid physical constants in order to

solve the model. Eased on the work cf Tailor (1981), this

eguation can be sinplified to:

- V |3L + D ill +-!i-2- ( T I -T)'Î- l'*- )

where:

K„=(l - 0)Cpp. PG -^ P L ^ (4-4)

H= ' ^'"PG "G PL

i =(VPG ^ VPL>/S (4-5)

â= — âL+ — âG i^'^)

Eguation ^-3 can be used to describe the systen ty

using the hoBogenecus properties defined in Eguations 4-4

to 4-6. The whole system, though composed of two phases,

then acts like a single-phase system.

56

One boundary condition for the fluid phase is:

B.C. 1: at 2=0 , 1 = T1 (t) (4-7)

Here T1(t) is an arbitrary, but known, function of time.

This boundary condition is necessary because in the exper

iments the temperature disturbance at the entrance could

not be regarded as a perfect step function but was a func

tion of time.

The second boundary condition is

B.C. 2: z—>oo , T—>T0, (4-8)

where TO is the initial temperature of the liquid and gas

phase. This assumption is eguivalent to assuming a semi-

infinite length for the reactor. The assumption can be

used when the l/I ratio ( where 1 is the distance away from

the bed exit where the signal was recorded and L is the

distance between two measuring ports) , is larger than

0.01- ((Wakao (1982)). In this work the 1/L ratio was

0-088". The initial condition for the packed bed model is:

I.e. 1: t = 0 , T = TO. (4-9)

The e g u a t i o n f o r the s o l i d phase i s the same as in

the DC model:

2 3T k„ 3 T 2 3T„

^ - ^ ( —r-^ + 1 ) ^t PgCp 3 r r 3r (4-10)

57

The boundary conditions are:

B.C. 1: at r=0 ,Ts is finite (4-11)

3 T s B.C- 2: at r=R ,k | = h ( T - T L R)(4-12)

^ 9 r r=R ^ ^^

where T is the temperature of fluid phase, a function of

time and Z. The initial condition is:

I.e. 1: at t=0 ,T=TO (4-13)

Eguations 4-10 and 4-3 form a system of partial dif

ferential equations coupled together by the boundary con

ditions describing heat transport between the solid phase

and the fluid phase. An analytical solution to this prob

lem has been presented by Rasmuson (1982), but this solu

tion does not serve for heat transfer parameter estima

tion. This is because the boundairy condition used by

Rasmuson is a boundary condition of the first kind rather

than of the third kind. Highly oscillating terras (sines

and cosines of high freguency) will prevent the nonlinear

optimum seeking method from converging because of round

off errors in calculating oscillatory terms in the

analytical solution-

A Laplace transform solution provides an alternative

method to solve the eguations. The Laplace solution can be

58

used foi parameter estimation via the moment method and

time dciain fitting as described in later sections.

laplacg Transform Solution

The solution in the Laplace domain can be obtained in

a closec form. Equation 4-10 may be solved together with

Equatioc 4-11 and 4-12 by applying the Laplace transforma

tion on both sides of Equation 4-10. Rote that in Equa

tion 4-12, the Laplace transformation of T(2,t) is treated

as a constant. The Laplace transform solution gives:

•Is(s)/T(s) = G1(s, h. Da) (4-14)

where ATIR^H k

G1= s w coth(w)-l (4-15)

K„ s RH - 1 + w coth(w) H

S

"s -

k s (4-15B))

P c s ps

T(s) anc Ts (s) are the Laplace transforms of the solid and

fluid phase temperatures, respectively. The function,

Gl(s, h. Da), is the link between these two phases. This

relaticr can le used when solving Eguation 4-3. Eguation

59

4-14 states that if the variation in temperature of the

fluid phase is known at any point along the axis of the

packed led, the corresponding temperature of solid can be

calculated via Equation 4-14-

Taking the Laplace transform cf Equation 4-3, togeth

er with Equations 4-4 to 4-9, gives:

d? T(s) d T(s) D^ -5 - y„ -s( 1 + aGl) - 0 (4-16)

a z d z

To o b t a i n Equation 4 - 1 6 , Ts ( s ) was rep laced by T(s)

and G1 (s) froir Eguation 4 - 1 4 . The r e s u l t i s an ordinary

d i f f e r e n t i a l eguat ion with Z , h. Da and s a s parameters

and/or v a r i a b l e s . The d i f f e r e n t i a l eguat ion can be so lved

e a s i l y to g e t :

• I 2 { s ) / T l ( s ) = G ( s , h , ,Da) ( 4 -18 )

where

G = exp(—^^ (1 - 'n ) ) ( ' • - IS) 2 \)

a

«=i ^ - - T ^ d ^ M ^ ) G 1 ) j , . ^ g , j

I 4> =

^ T T R ' •J

60

Or»

T2(s) = ; e'^^ ( T2(t) - TO) dt (4-20) 0

oo

-St Tl(s) = / e ^^( Tl(t) - TO) dt (4-2 1)

Tl(s) and T2(s) are the Laplace transforms of the

temperature measured at the bed entrance and exit respec

tively. The function G(s,h,Da) is the transfer function

between T1 and T2.

Eguation 4-18 relates the input signal at the bed

entrance and the response at the bed exit. Given a known

input profile for the bed entrance, the response at the

bed exit can te predicted through the transfer function G.

When the input signal is in the form of perfect im

pulse, the Laplace transform of T1 is 1. From Equation

4-18,

T2 (s)= G (S, h. Da) (4-22)

When the correct parameter values cf h and Da are known, G

can be inverted into the time domain so that the response

signal subject to a perfect pulse input can be obtained.

Figure 9 is a typical response to a perfect pulse input.

This figure is also the time donain value of G when G was

inverted into time domain.

61

ID

<

cr 0.03-

0.02-

0.01 -

0.00-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 I 1 1 1 1 I I ' ' ' " ' ' ' I ' ' * ' ' ' ' ' ' I ' '

200 UOO 600 TIME (SEC)

800 1000

Figure 9: Calculated Response of the System to Unit Pulse Input

62

Time Domain Fitting

In the method of time domain fitting, the measured

response signals are compared with signals predicted in

the time domain. If the two response curves agreed well,

the final parameter values used in the searching process

may be regarded as the optimum values. In time domain

fitting it is important to examine the effect of changes

in parameter value on the shape of computed response curve

(sensitivity analysis).

Equation 4-18 is considered to be the transfer func

tion between the solid phase and the fluid phase. Since

the transfer function contains unknown parameters. Da and

h, the major task in this work is searching for the best

parameters to fit the system.

The transfer function in this system is so complicat

ed that the usual Laplace inversion techniques can not be

applied. A numerical approximation of the transfer func

tion may be used to overcome this difficulty. The analyt

ical form of the function is not available, but certain

functions can te expanded as a Fourier series. This re

guires that the transfer function be null after a certain

time. The algorithm is presented in the following section.

63

Expanding a Known Transfer Function in the lime Domain

Before expanding a function of time as a Fourier ser

ies, the functional values at different tiies and the per

iod of the function must be deternined in order to calcu

late the Fourier coefficients by integration over the

desired interval. Osing the Laplace transform avoids

these difficulties by means of the following procedures. A

priori knowledge of the period of the function and the

function value at discrete points corresponding to diffe

rent times is rot cecessarj.

T2 (t) can be calculated via the theorem of convolu

tion through Eguation 4-18:

c T2(c)- / Tl( C) f(t -C )<1C (4-28)

0

From Equation 4-28, it is known that f (t)=0 after t >

t2 - t1, where t2 and t1 are the times when the tail of

the output and input signals vanish for an iaperfect pulse

input. If the period, 21, is not known exactly, the func

tion f(t) may be approximated by a fourier series as long

as the period is taken larger than t2- If the Laplace

transform, G, of a given function of time, f(t), has the

property that when t > 2T, f(t)=0, the Laplace transform

can be writtec as:

60

C ( s ) » re-^^ £(t)dc ( , _ 2 „

- { * ^^"^^^ (4-29A)

S u b s t i t u t i n g s* imr/T' i n t o Equation 4-29A g i v e s :

G ( L 0) ) « / ^ e'^^^ f (c )dt (a_30)

2T 2T « / coswt f ( t )d t - i / sin'i)t f ( t )d t (4-30A)

0 0

0) = —^ — (4-30B)

Hence the transfer function can be expanded in the form;

f (t)=L-i{ c } (4-31)

o r , expanded in Four ier form

f ( t )» E A cos(n7Tc/T) + B sin(mrt/T)

over the range 0 < t < 2T (4-32)

where:

65 1 2T

n * ^ f(t)cos(mrc/T)dt (*«-33) T 0

i 2T ^n' ~T—^ f(t:)sin(nTTc/T)dt (4-34)

0

2T

AQ = 2 T " ^ ^ >'' (4-35) 0

The Fourier series coefficients were obtained frcn

the Laplace transform in the manner described above. The

transfer function G does have the property that after an

extended time 2T, it will be a null function. In this

work 2T is chosen as the time elapsed for the system to

reach a new steady state in the experiment. Such property

can be seen from Figure 6, together with the fact that the

two phase flow system is a stable system. This means the

system will not diverge from one stable state to another.

This can te judged from the second law of thermody

namics. For a pulse input to the system, the system will

eventually be at the original state after some time has

elapsed. This is because the the total input to the system

is a finite amount and there are no generation terras in

Equation 4-3. The energy input at t=0, would be dispersed,

convected and transferred to the particles through

different heat transfer mechanisms for this specific

system- The time domain approximation of f (t) can te

66

accomplished by substituting these Fourier coefficients

into Eguation 4-32.

The next step was to use this approximated function

and the property of convolution to compute the output sig

nal via Equation 4-18. The integral was computed by using

Simpson's rule. Figure 10 shows the computed f(t) using

the method described in this section. In the figure 500

terms were used to approximate the output function. The

figure is plotted for T and 2T (T is the period) . This

figure shows that the choice of the period has no influ

ence on the computed result. The integration process and

the calculation of sines and cosines, as can be seen from

Eguations 4-32 and 4-28, are all very time-consuming in

this work. The CPU time required to compute each set of

parameters was the order of 1.5 hr on a VAX-11/780 system

due to the large number of iterations required to con

verge.

The Optimization of the Error Function

The purpose of time domain fitting is to find the

best parameter values so that the calculated temperature

vs. time curve matches the experimental curve well. A cri

terion must be set up in order to measure the guality of

fit. The objective function used was minimization of the

sum of squares of the difference between the experimental

67

^ • ^

0.04

0.03

Ui cr <

UJ CL

UJ

Q.02

0^01

Legend:

0« 4ci(l^

• Period i s 21

Period i s T

0 10 30

TIME (SEC)

- * — * —

50 70

Figure 10: The Effect of Period on the Calculated System Response tc a Unit Pulse Input

68

and theoretical response signals. The error function is

thus defined as:

2T / ( T2 - T2 ) dt 0 cal exp ,.. -_,

ERROR - / ( (4-36)

2T

. { ^2^oal^^

Since T2 is a function of t, z, h. Da, it is obvious that

the error is a function of h and Da also. By systematical

ly varying the values of these two parameters, an optimum

error (minimum) and the corresponding h and Da which are

the best fit values for each run can be determined.

The error was typically in the range of 0.03 to

0.0D8. The parameters which fit the criterion that Error

< 0.03 were accepted as the best fit parameters.

numerical Hethod Used in Non-Linear Optimum Seeking Method

The error function defined in Equation 4-36 is bimo-

dal in nature. There may exist more than one set of sub-

minimum or submaximum solutions. To find the absolute mi

nimum of the function, a direct search method (combination

of the Davies, Swann, and Campey (DSC) and Powell's

method) was used. Finding optimum values of the regression

variables can be accomplished by adjusting all the

69

variables simultaneously or by adjusting each variable one

at a time. The first method involves the calculation or

estimation of partial derivatives and suffers from roun

doff error.

The single variable approach was used in this work.

This resulted in a unidimensional search in one variable,

while the other was held constant. The DSC method was

first applied to find the approximate location of the op

timum value of the variable. The DSC search was used be

cause it "brackets" the optimum value very rapidly. The

Powell search was then employed because the method rapidly

converges to the optimum, once the optimum has been brack

eted. The flow chart shown in Figure 11 shows the struc

ture of the optimum seeking algorithm.

The optimum seeking method is cyclic in nature- The

heat transfer coefficient is optimized first. The disper

sion coefficient was then optimized using the newly op-

timzed heat transfer coefficient. Then the heat transfer

coefficient was reoptimized using the newly optimized dis-

perion coefficient. The cycle was repeated until the de

sired value for ERROB defined by Equation 4-36 was

reached.

Figure 12 shows how the direct search method worked.

Line A is the input temperature profile at bed entrance

and the set of line B's are the calculated output

temperature profiles at successive iteration cycles. The

70

h OPTIMIZA-TION. DSC SEARCH

POWELL SEARCH

a OPTIMIZA-TION. DSC SEARCH

I

POWELL SEARCH

Nomenclature:

ACCUR = Termination Criteria; Minmun

Allowable Difference of Regres

sion Variables of Subsequent

Cycles

IC = Cyole Number

Figure 11: Flow Chart of the Optimum Seeking Method

71

a> Q.

•H CO

- • 4) > S o

c o •H - P 10 U 0) 4->

m r

p-« m •

oo

r^ <M • •

o m CO

•» CD •;

CO

• i - l <0

•H •-! •P •H C

m ir\ rsi

• o

VO •

rr\

S •» rr\ CQ

ro

r>. o <M

• o

• O P H »H

VO CO

•»

lA CQ

< ! •

. H

"r~

VO 0 \

• I - l II

«J

00 VO t -H 11

O a>

Of

lA I

O I

lA .—t

I

( J.) 01 -:i

72

square are the experimental output temperature value. The

vertical axis is plotted as the reduced temperature, which

is the recorded temperature subtracted from the initial

temperature of each run. In the direct search method, the

smallest step size allowed in the search was 10-^ to

10-6-

Parameter Estimation bjr Imperfect Pulse Theory

Most of the published studies of axial dispersion

have been based on the assumption that the imposed changes

on the system are actually imperfect steps or pulse in

puts- Aris (1959) showed that a perfect pulse function is

unnecessary if the transient tracer concentration is mea

sured at any two points in the system-

The imperfect pulse method is characterized by a num

ber of important advantages. The principal advantage is,

as mentioned, that a mathematically perfect tracer is un

necessary. In addition, Bischoff (1963) has emphasized

that the method can be used for determination of parame

ters for any model that can be represented by a transfer

function. This is important in this work because we can

not have a perfect step input and the imperfect pulse

theory must be applied.

71

Weighted Moraent " lethod

The moment method, which i s a method to r e l a t e t h e

n-th moment of the f u n c t i o n t o c e r t a i n parameters , s u f f e r s

from m a g n i f i c a t i o n of e r r o r s i n the t a i l of the curve in

t h i s s t u d y . I t was not p o s s i b l e t o recover the heat t r a n s

f e r parameters through the moment method, as the parame

t e r s c a l c u l a t e d by t h i s method sometimes became n e g a t i v e -

An a l t e r n a t i v e method which weights every part of t h e

r e s p o n s e curve d i f f e r e n t l y was proposed by Cstergaard and

n i c h e l s e n ( 1 9 6 9 ) . The working eguat ion for t h i s method i s

t h e f o l l o w i n g :

oo

/ e"^''T2(c)dt

^"^ =— ( 1 - / 1 " ) ) (^-37) / e-^^l(c)dc 2 D 0

The definition of B in the square root is the same as in

Fguation 4-19. Eguation 4-37 becomes an eguation in h and

Da- Solution of Fquation 4-37 together with the s given

and the first derivative of 4-37 with respect to s, gives

the parameters h and Da. By using the weighting factor

g-st for the frontal portion of the curve, the error will

be minimised due to the exponential decrease in the

weighting factor.

74

The S e n s i t i v i t y of the F i r s t Momen t

Not all the Laplace operators, s, when substituted

into Eguation 4-37 would give the best estimation of par

ameters. Different types of input functions have diffe

rent types of optimum Laplace operators for use in the

weighted moment method, as indicated by Anderssen (1971).

The relative variation (R.V.) is an indication of the er

ror involved when estimating the k-th weighted moment- The

equation given by Anderssen is the following:

R.V.

a

2' -2ksc 2 2k 1/2 0

/ e ^ f (t) t* dt 0

(4-38)

where sigma is the standard deviation of the noise of the

signal, T is the total time span, and f(t) is the the in

put signal. By plotting P.V./sigma vs. s, the range of s

which is best for parameter estimation may be obtained.

Figure 13 illustrates this fact for a step type signal.

Tjnfortunately, for the first moment estimation there

is no optimum s for parameter estimation, since the error

increases drastically for a step type input. This plot

also reveals there is an optimum s for the zeroth moment,

when s > 0. The optimum s for the zeroth order moment

is not as restricted as for the first moment. The first

moment is very sensitive to the changes in the Laplace

Tl 0.000

75

- 0 . 0 2 S -

-O.OSO-

-G.07S-

> UJ o O

SE o z < »— </> UJ o • y

< 1—t

< > UJ > »-t >-<

1 UJ a:

-0.100

-0.12S

-0.150

-0.175

- 0 . 2 0 0 -

- 0 . 2 2 5 -

- 0 . 2 5 0 -

Curve A : z e ro th monent

Curve B : 1s t moment

B

'WT "f T • v'^r—* •« » - T T » » * » ^ * T | * f « » * l » T T | 9 W'W'^ "W ^ T ' T ' V m W T '

0.00 0.02 0.04 0.06 0.08 0.10

LAPLACE OPERATOR, S

F i g u r e 13: Optiicuir L a f l a c e C p e r a t c c f o r S t e p I n p u t S i g n a l . Curve A I s f o r Z e r o t h Mcirent, Curve E I s f o r F i r s t Moment

76

operator. The insonsitivity of the first moment prevents

use of this method to determine the heat transfer parame

ters, since two simultaneous equations are needed to det

ermine h and Da from the experimental data. This reguires

use of the first moment. As will be seen in the next

chapter, the zeroth moment is useful as a means of calcu

lating the hoiogeneous flow velocity-

CHAPTER V

FEIPARATICN OF IXPEEIHEMTAL DATA AND ESIIEATION OF FHTfSICAl

PABAHETEES

Data Ire^tment Bjf Dsinq the Smoothing Spline Bethod ~

If a set of data is regarded as a set of samples from

a contiiuous function, the spline method is a good approx

imation to interpolate data between these samples. The ap

proximate function which is obtained by the spline method

is the smoothest interpolation relation possible- The

smoothirg spline method may be modified to incorporate the

minimized sguare-error criterion with the smoothing re-

guiremect.

In the experiments, the temperature versus time data

had inherent fluctuations due to the measuring eguipment-

In ordei to smooth the input data, a program was developed

to minisize the sguared error of the experimental data vs.

the smoothed data representation.

In this approach, the objective signal was treated as

a space function s(x) and the sguared error was defined

as:

SQ(s) - l"™ ( r. - s.)2 (5-1) j-1 J J

Here r. was the original functional value and s. was the J ^ J

approxiHated functional value. The guantity that measured

the smoothness of the approximated value was defined as: 77

78

^ = ^ \ s f ^ l ( x ) ) 2 d x (5-2)

fheres*- -' (x)i£ the L-th crdeE deriiiative cf £(x)

The criterion used here was to minimize:

J(s)=b Sfi(£)*(1-b) snis) (5-3)

where t was a constant that was adjusted to either empha

size minimizing the sguare error as b — > 1 or maximizing

smoothness as t—>C- The b value in this work was always

in the neighborhood of 0-99 which was computed in the pro

gram.

If 1=2 ir Eguation 5-2, then the s(x)is a cubic

spline. Since the routine could interpolate the function

al values tetueei) discrete experisental points with high

confidence levels, this was used tc interpolate the exper

imental data.

Figure 14 shows how this routine works- The dotted

points are the raw data. As can te seen from the figure,

the variations in the measuring device can be smoothed,

especially at the front and tail of the experimental

points. Fijure 1? shews the sirocthed experimental data for

different flow re-jiires plotted against the time elapsed

for the heat transfer process. In this fijuce, it can he

79

(0

> u o

• a

x: + j o o E (0

J C 4 J -P C

0) (4 «0

CO Qi c • H

1—1

O Q.

f H «0 4-> C 0) E

TJ f.4 s.

> U 3 u

4) 3 CO O O 4-) Q. 3 CO 0> <D C

01 1—«

9 ' CO LU

(0 ^

o rg

o »H (M

O 0 0 »H

130

o *-» *VJ o • ^ V

CO

g • H h-

O 0 \

o NO

o •

TJ 0)

U3 4J • * 0 0 a o V e 0 to o

% fCO c «

10

S S CUO ar-^ M <0

u •c Q) 0) M . « ; S» 4J

« 4J SC i H

* CM

o in

n u O 3 0} 4^ •H cc M M n <d « w Q 4 p l > B e u O 0) 9 U H u

•> ^

(D M 9 C7<

•H Ct4

CO VO rsl O ^4

CO VO <M

( J,) *01 -1

80

• • o o AO VO ^ \d-lA l/>

II

'i CC

• k

<*• •

»H II

^

II o

r> II

h". II

O

• k

<*• •

' H II

^

oe «

<t •

f - i II

V - 1 0) Q£

QC

& ^

•p (0

per

E

for

o ^ •p (0 (4

a E V

« 1 -t -

put

c

4*

tpu

3

• h

J t •

ri II

. J «

vac oe &

V

u 3 •P (0

per

E

for

4)

^ •P *0 u

mpe

4> fl> 1 -

h-

put

c

• p

tpu

3 M O HH O

^ i-t

CD < CD •-• c j - (NJ

o o

o oo

o NO

o UJ in UJ

o

9 M

O CM

O Ol tf] (Q

ffi B •H

M 9 9 m 4J

h o V f H

•

o;

^ «

«M»H O Q

ti u o o (QtM

•H U B 10 V U*^ B m O P^ U CO

0)

u 9

•H

lA

I

O •—I

I I

O

( J^) *01 - 1

81

seen that t h e curve drops much f a s t e r for the p u l s i n g

regime ( curve A) than f o r the t r i c k l i n g f low regime ( curve

B) . Th i s may he due t o the l a r g e r gas flow ra te which

caused t u r b u l e n c e in t h e s y s t e m . The exper imenta l curves

a t d i f f e r e n t f low reg imes reguired a comparable l e n g t h of

t ime f o r t h e p r o c e s s t o s h i f t from one s teady temperature

t o a n o t h e r .

Es t imat ion of P h y s i c a l P r o p e r t i e s and Bed Parameters

C a l c u l a t i o n of Dynamic Liquid Hold-Op

The l i q u i d h o l d - u p was c a l c u l a t e d using the Mi-

doux(1976) correlation:

1 + 0.66 x'*'*

V - ^ fi <5-aA)

where 5 6 are the pres sure drops for the l i g u i d and gas L * 0

phases when l i g u i d and gas were f lowing c o c u r r e n t l y i n the

packed bed. Both p r e s s u r e drops were c a l c u l a t e d using t h e

Ergun e q u a t i o n . Equation 5-U was developed for the a i r - w a

t e r system and i s d i r e c t l y a p p l i c a b l e to t h i s work.

82

Calculation of the Homogeneous Sensible Heat

In Tailor's work there was a need to calculate both

the homogeneous heat capacity and density. Based on the

results shown in Equation a-3, it is seen that calculation

of both properties was not necessary. The product of these

two quantities was calculated instead. In this work, the

homoqeneous sensible heat was defined as:

Kh= (I - 6)C_- p^ -6 p, C (5-5) PG 'C " L PL

This mixing rule was based on the assumption that the

fluid is composed of a homogeneous mixture of gas and li

guid.

Calculaton of ^h

Vh, when multiplied by Kh gives a measure of the

ability of the combined fluid phases to transport energy.

The equation for the bulk flow of energy can be written

as:

GG Cpg + GI Cpl=G(l - 6)p^Cp^ + cfiV Cp p

= CV„K„ (5-6)

83

where VG, VI are the interstitial gas and liquid

velocities, respectively. Vh can te calculated from rear

rangement of Eguation 5-6 to:

This was suggested ty Tailor(1981). He tr€at€d Vh as a

known parameter which can be calculated from Equation 5-7.

Calculation of Vh Ey using Weighted Moment Method

The weighted moment method is not useful in estimat

ing parameters for certain types of input signals, but as

can he seen from Figure 9, the error for the zeroth moment

is always the same no matter what the value of s is. This

specific property for the zeroth moment of the step input

signal can be used to calculate some important physical

parameter from the zeroth moment.

In Equation a-8, autiply both sides by 1/s and let

s—> 0 and the R.H.S (right hand side) of the eguation be

comes

/ e"®42(c)dc „ ^ I 0 V z (5-8)

-'-lev-- ^ (1 - / l ^ ) ) / s / c Tl(c)dt a 0

84

After calculating the L.H.S. (left hand side) numerically,

the plot of the t.H.S. vs. s will be a straight line with

K V H e s) (5-g) H H

as the intercept and

- z ^'' ^ AC D 2

<P i-^^—)^-^ (K . 1-l^K ) (5-10) " " s -•o H H

as the slope (see Figure 16) .

Vh may be calculated from Equation 5-10 based on the

Kh calculated previously. Figure 17 is the plot for Vh

calculated by this method. Vh calculated via this new

method showed that this is a monotonically increasing

function of gas phase Reynolds number. The magnitude of Vh

is between the gas phase velocity and liquid velocity

based on the void cross sectional area of the packed bed.

Table 4 shows the Vh's calculated by different methods for

a constant water flow rate and varying gas flow

rate (increasing from the top of the table tc the bottom).

This table also shows that the gas flow rate has a

significant influence on Vh. As the gas flow rate

increased for a fixed water flow rate, the calculated Vh

also increased. It is also noted from Table 4 that the two

85

-17,42

-17.44 -

-17.46

CM h -17.48

-17.50

-17.52 -

-17.54 Q 8 10

S X 10"^ (sec"^)

Figure 16: C a l c u l a t i o n of Vh by the Weighted Moment Method Osing Eguation 5-8

86

^l ReL=lA B: ReL=1.68 C: ReL=1.96 D: ReL=2.2^ E: ReL=2.52

0 .W-

O.SS-

0.S0-

O.HS-

^ o,wA

0.3S-

0 . ^ -

0.25-

0.20-

0.1S-

0.10-" ' I ' ' I "•! I I' I I ' " I ' W I I •!" ' I .1 T 2S SO 7S 100 12S ISO 17S 200 22S. 2S0 27S 300 32S 3S0 * 37S

*CC X 0.5

Figure 17: Experimental Values of Vh Calculated Osing the Weighted Moment Method

87

Vh's calculated from the different methods have the same

order of magnitude in the trickling flow regime.

TABLE a

D i f f e r e n c e Between Vh»s C a l c u l a t e d from D i f f e r e n t Method

Vh ( f t / s e c . A) Vh ( f t / s e c , E) Flow Regime

0 . 1 2 3 3 0 .1527 t r i c k l e 0-1670 0 .1976 t r i c k l e 0 .1904 0 .2375 pu l se 0-2368 0 .2759 pu l se 0 .2579 0 .3127 pulse 0 .2784 0-3493 pu l se 0 .3422 0 .4210 pul se

A: Vh c a l c u l a t e d by weighted moment method. B: Vh c a l c u l a t e d by T a i l o r ' s method.

Comparison Between Vh Ca lcu la ted from t h e Two D i f f e r e n t ?Iethods

From Equation 4 - 3 , t h e r e are three p h y s i c a l parame

t e r s , the homogeneous s e n s i b l e h e a t , the homoqeneous v e l

o c i t y and l i q u i d h o l d - u p , which can be determined by the

methods decr ibed above- The heat t r a n s f e r parameters , h

and Da, can be c a l c u l a t e d from the error funct ion de f ined

by Equation 4-36 v ia an optimum seek ing method as soon as

t h e p h y s i c a l parameters are determined. Table 5 shows the

comparison of the v a l u e s of the error funct ion us ing

88

d i f f e r e n t Vh's c a l c u l a t e d by Eguation 5-7 and Eguation

5 - 9 . The homogeneous s e n s i b l e heat c a l c u l a t e d from Egua

t i o n 5-5 was used in a l l the c a l c u l a t i o n s .

TABLE 5

Error of Curve F i t t i n g by Osing d i f f e r e n t Vh

Error/Vh by T a i l o r Error/Vh t h i s work Flow Regime

0-1026 0-02199 pulse 0 .09598 0 .03759 pulse 0-09791 0 .03120 pulse 0-13279 0 .02777 pulse 0 .12283 0 .02493 pul se 0-11002 0 .02294 pulse

Comparing the f i r s t and the second column of Table 5 ,

i t i s found tha t the error f u n c t i o n c a l c u l a t e d by us ing

Vh's from Eguat ion 5-7 i s f i v e t o s i x t imes l a r g e r than

us ing Vh's from Eguation 5 - 9 - When the Vh's from Eguation

5-7 were used as a known parameter to c a l c u l a t e h and Da,

the va lue of the error f u n c t i o n i s l arger than 0 . 1 , more

than t h a t found us ing Vh's from Eguation 5 - 9 . This i n d i

c a t e s tha t t h e Vh's from Eguation 5-9 can f i t the model

b e t t e r than Vh's from Eguation 5 - 7 . I t i s important to

note t h a t Equation 5-7 i s a ra ther a r b i t r a r y mixing r u l e

e s t i m a t e for Vh, w h i l e Eguation 5-9 u s e s the exper imenta l

measurements as a b a s i s for c a l c u l a t i n g the parameters .

CH^PTE^ VI

ESTIMATING HEAT TRAKSFFIR AND TtERIAL DISPERSION CnEFFICIENTS

P a r a m e t r i c S e n s i t i v i t y

S e n s i t i v i t y S t u d y f o r Heat T r a n s f e r C o e f f i c i e n t s

The p a r a m e t r i c i n s e n s i t i v i t y of an o b j e c t i v e f u n c t i o n

t o a p a r a m e t e r i s a l w a y s a p r o b l e m i n t h e use of m u l t i d i

m e n s i o n a l opt imum s e e k i n g m e t h o d s . Wakao (1979) used e r

r o r maps t o i n d i c a t e t h e s e n s i t i v i t y of t h e o b j e c t i v e

f u n c t i o n t o w a r d c h a n g e s i n t h e model p a r a m e t e r s from t h e

d i s p e r s i o n - c o n c e n t r i c mode l . F i g u r e 19, a d a p t e d from Wak

ao ( 1 9 7 9 ) , shows t h a t when t h e R e y n o l d s number i s low

( l e s s t h a n 2 5 0 ) , t h e h e a t t r a n s f e r c o e f f i c i e n t can n o t be

d e t e r m i n e d from t h e opt imum s e e k i n g method he d e v e l o p e d

f o r t h e s i n g l e - p h a s e ( a i r ) s y s t e m b e c a u s e t h e o b j e c t i v e

f u n c t i o n i s n o t s e n s i t i v e t o t h e h e a t t r a n s f e r c o e f f i

c i e n t .

I n t h i s work t h e o b j e c t i v e f u n c t i o n i s t h e e r r o r

f u n c t i o n d e f i n e d a s :

2T / ( T2 - T2 )^dc 0 ca l exp

ERROR - / ( ( 6 -1 ) 2T / T2^^ . dt 0 f ^ ' i l

e<5

90

12

10

8

I CO

s

0 0.1

Re = 5.1 , , k = 0.2 Wm"TC" s

Nu

A 0.06 B 0.03

10 100

Figure 18: Example of a Parametr ic Error Hap for Heat Transfer i n S i n g l e Phase , Gas Flow (Wakao,1979) .

The error f u n c t i o n c a l c u l a t e d i s a f u n c t i o n of T2(t )

c a l c u l a t e d from the f o l l o w i n g e g u a t i o n :

T2(t)= / Tl( ^) f ( t -^ )dC 0

(4-28)

Ihi^.

91

From E q u a t i o n 6-1 and 4 - 2 8 , i t i s known t h a t t h e e r r o r

f u n c t i o n i s an i m p l i c i t f u n c t i o n of T 2 ( t ) . From Equa t ion

4 - 2 8 , T 2 ( t ) i s a l s o a f u n c t i o n of f ( t , h . Da) and Tl (t) .

S i n c e Tl ( t ) i s a known f u n c t i o n of t ime o n l y , t h e c a l c u

l a t e d T2( t ) i s a l s o a f u n c t i o n of h e a t t r a n s f e r C o e f f i

c i e n t and a x i a l d i s p e r s i o n c o e f f i c i e n t . T h i s o b j e c t i v e

f u n c t i o n i s t o he o p t i m i z e d ove r two p a r a m e t e r s , h and Da-

The e f f e c t o f chanq inq p a r a m e t e r s on t h e c a l c u l a t e d v a l u e

of T 2 ( t ) i s examined t h r o u g h t h e f o l l o w i n g t a b l e s f o r

d i f f e r e n t flow r e g i m e s .

The e r r o r f u n c t i o n d e f i n e d i n Egua t ion 6-1 i s q u i t e

s e n s i t i v e t o c h a n g e s i n Da, but i s no t s e n s i t i v e t o c h a n g

es i n t h e h e a t t r a n s f e r c o e f f i c i e n t s . Tab le 6 g i v e s an

example of t h i s phenomena f o r run #22 i n t h e t r i c k l e flow

r e g i m e .

TABLE 6

E r r o r F u n c t i o n s E v a l u a t e d f o r ReG=112.8 , ReL=1.96

E r r o r 0 .04509 0-C3989 0 .03596 0 .03575 0 .03574 0 .03574 0 .03573 0 .03573

Pe 0 - 4 1 3 0 0 . 2 4 1 5 0 - 2 2 2 1 0 . 2 1 0 3 0-2094 0 -2085 0 .2084 0 .2084

Bi 21 .90 37 .85 50 .79 63 .74 7 6 . 6 9 89 .64 102.59 115 .53

5ik.>

92

The table shows at this low air flow rate, Pe converges to

an asymptotic value rapidly. But Ei did not converge even

though the error function showed that error was decreasing

for increasing Bi. The rate of decrease in error was very

slow for very large changes in Bi. This same pattern was

found in all of the trickle flow regime data- Hence, it

is evident that Bi could not be determined in the trickle

flow regime-

TABLE 7

Error Functions Evaluated for ReG=408, ReL=1.96

Error Pe Bi

0.05588 0.2032 26.90 0.05525 0-3243 39.85 0.05405 0-3417 52.79 0.05196 0.3423 65.74 0.04801 0.3384 78-69 0.04709 0.3346 91.64 0.04709 0-3345 104-59

In Table 7, air flow rate was increased and the flow

regime was pulsing flow- However, the sensitivity of the

error function to the parameters is similar to that seen

in Table 6.

The first column in Table 8 shows the error function w

calculated for the trickling flow regime and the third co-

93

TABLE 8

Error Funct ion C a l c u l a t e d f o r T r i c k l e and Puls ing Regimes

1 PeG=117.8

1 E r r o r 1 0 . 2 0 9 9 1 0 . 0 3 0 0 1 0 . 0 2 7 4 5 1 0 . 0 2 6 7 8 5 1 0 . 0 2 6 4 9 1 0 . 0 2 6 3 4 I 0 . 0 2 6 2 4 1 0 . 0 2 6 1 8 1 0 .C2613 1 0 . 0 2 6 0 9 1 0 . 0 2 6 0 7 1 0 . 0 2 6 0 5 1 0 . 0 2 6 0 3 1 0 . 0 2 6 0 1 1 O.C2600 1 0 . 0 2 5 9 9 I 0 . 0 2 5 9 8 1 0 . 0 2 5 9 7 1 0 .C2597 1 0 . 0 2 5 9 6 1 0 . 0 2 5 9 5 1 0 . 0 2 5 9 5 1 0 . 0 2 5 9 4 1 0 . 0 2 5 9 4 1 0 .02594 1 0 . 0 2 5 9 3 f 0 . 0 2 5 9 3 1 0 . 0 2 5 9 3 \ 0 . 0 2 5 9 3 1 0 . 0 2 5 9 3

EeL=1 .96

B i 0 . 1

5 . 1 1 0 . 1 1 5 . 1 2 0 . 1 2 5 . 1 3 0 . 1 3 5 - 1 4 0 - 1 4 5 . 1 5 0 . 1 5 5 . 1 6 0 . 1 6 5 . 1 7 0 . 1 7 5 . 1 8 0 . 1 8 5 . 1 9 0 . 1 9 5 . 1

100 .1 105 .1 110 .1 115 .1 1 2 0 . 1 125 .1 130-1 135 .1 140 .1 1 4 5 . 1

BeG=326.8

E r r o r 0 .3891 0 .0407 0 .0386 0 .0380 0 .0377 0 .03756 0 .03745 0 .03737 0 .03731 0 .03727 0-03723 0 .03721 0-03718 0-03716 0 .03715 0-03713 0 .03712 0 .03711 0 .03710 0 .03709 0 .03708 0 .03707 0 .03707 0 .03706 0 .03706 0 .03706 0 .03706 0 .03705 0 .03705 0 .03704

•

lumn gives the error function calculated for the pulsing

flow regime. Both data sets were calculated for their op-

94

timum P e c l e t numbers. Although the heat t r a n s f e r c o e f f i

c i e n t changes from 5 .1 t o 1 4 5 - 1 , the e r r o r func t ion only

changes 0.5''?- I t i s concluded tha t the o b j e c t i v e f u n c t i o n

i s not s e n s i t i v e to p a r t i c l e - t o - f l u i d heat t r a n s f e r c o e f

f i c i e n t . The heat t r a n s f e r c o e f f i c i e n t can not be d e t e r

mined from the optimum s e e k i n g method by the error func

t i o n d e f i n e d in Equation 4 -28 - I t i s ev ident from the

t a b l e tha t t h e r e i s an a s y m p t o t i c va lue f o r the error as a

f u n c t i o n of B i -

S e n s i t i v i t y Study for the D i s p e r s i o n C o e f f i c i e n t

Fiqure 19 shows the p l o t of f (t) v s . time for Pe (op

timum) c a l c u l a t e d u s i n g the t ime domain f i t t i n g method,

a long with c u r v e s f o r d e v i a t i o n s from Pe (opt) of -40% and

-801? r e s p e c t i v e l y - This corresponds t o 66% and 400%

changes in t h e d i s p e r s i o n c o e f f i c i e n t . Da- Figure 20 i s

the p l o t of t h e corresponding T2(t) ' s c a l c u l a t e d by us ing

the same f ( t ) ' s as i n Figure 19. I t i s e v i d e n t on compar

i n g t h e f i g u r e s t h a t a -40% d e v i a t i o n i n Pe can r e s u l t a

very s i g n i f i c a n t v a r i a t i o n i n t h e b a s i c shape of f ( t ) - The

f ( t ) curves in t h i s p l o t a l l d e v i a t e from each other i n

peak h e i g h t , breakthrough t ime and the the t ime for the

peak maximum.

Small changes in t h e P e c l e t number cause much l a r g e r

changes in the shape of T2( t ) than the same r e l a t i v e

w

95 0.3 •#w«

0.25

A: Pe<ont)

B: 4 : f^eviation

C: 80% deviation

0;2

UJ

e UJ

0.15

0.1

0.05

0

20 40 60

TIME (SEC)

80 100

Figure 19: Effect of Changes in Peclet Numbers on the System Output Temperature, T2 (t), Responding to a Perfect Pulse Input in the Trickle Flow Regime (Pe(opt)=0.155 at ReG=55.6, ReL=1.4)

96

0

— Pe (opt) *AQ% deviation +80% deviation

-5

-10

o I

-15

-20 j_i 0 20 40 60

TIME (SEC)

80 100 120

Figure 20: E f f e c t of Changes i n P e c l e t Haabers on the System Output Temperature, T2 ( t ) , for an Exper iaenta l Step Input in the Trickle Flow Regime (Pe (opt )=0 .155 at ReG = 5 5 . 6 , BeL=1.4)

97

changes in Blot numbers. The stronger effect of changing

Pe on the calculated T2(t) results a greater sensitivity

than in the case of changing Bi.

This effect is more evident when the three curves in

Figure 21 compared to those in Figure 19- Figure 21 is a

plot of f (t) for the pulsing regime and Figure 19 is for

the trickling regime. In Figure 19, the deviation of the

40% and 80% curve with respect to the base curve is larger

than in Figure 21. This is because Da in the pulse regime

is larger than in the trickling regime- The 40% or 80% de

viation of Pe in the pulse regime will cause more change

in f (t) than it will cause in the pulse flow regime. Fig

ures 20 and 22 are the plots of T2(t) 's using the same

heat transfer coefficient and dispersion coefficient as in

Figures 19 and 21. Figure 22 is a plot of T2(t) using

f (t) from Figure 21. This plot shows that T2(t) is also

sensitive to changes in Pe-

Calculated Peclet Number Results

Based on the results described above, it is evident

that the objective function is sensitive to changes in the

dispersion coefficient. Hence, it is reasonable to use the

parameter estimation approach developed here to calculate

experimental values for the dispersion coefficients.

However, as indicated by Hochman (1969), there is usually

considerable scattering in dispersion coefficient data.

98

0.05

0.04 Pe(opt)

*•** 40% deviation

80% deviation

0.03

^0.02

<

UJ Q.

0.01

0.

1 r

40 * 60 80

TIME (SEC)

100

Figure 21: E f f e c t o f Changes i n P e c l e t Numbers on the System Output Temperature, T2 ( t ) , Responding t o a P e r f e c t P u l s e Input i n t h e Pu l s e Flow Regime (Pe (opt) = 0 . 3 7 4 a t ReG=652.2, ReL=1.68)

99

a

1 — r T — T

UJ

s tc ii! -5 - 5

Pe(opt)

40% deviation

(The second count

from left)

• * 80% deviation

-18

-15 -

e 18 28 38 « TIME (SEC)

SB 68 78

Figure 22: E f f e c t of Changes i n Pec l e t Numbers on the System Output Temperature, T2 (t) , for an Exper iaenta l Step Input in the Pulse Flow Regime (Pe (opt)=0-374 at ReG=652.2, ReL=l. 68)

100

particularly at low flow rates. Eochman found there are

significant and apparently random variations in dispersion

coefficient from point tc point within the bed, probably

because of shifting liquid streams and instability inhe

rent to two phase flows.

Figures 25 through 27 show the mean values of from

four sets of Fe calculated, based upon the experimental

T2 (t) values. The standard variation and the flow regime

for eack data point were also indicated in the figures-

These figures shon there is a shift of Pe from one cons

tant value to another. The aymptotic value for Pe in the

trickle flow regime ranged from 0. 12 to 0.2 as the gas

phase Fcynolds number ranged from 50 to 200.

There is transition between the trickle flow regime

and the pulse flow regime. The asymptotic value of Fe for

the pulse flow regime was 0.3 to 0.4 for gas phase Rey

nolds numbers ranging from 400 tc 700. This shift was

also observed ty Hatsura (1976). Dispersion increases as

the mixing within the system is increased. However, when

a certain limit is reached, the dispersion attains a cons

tant le\el.

The order of magnitude of Fe values for two-phase

flows aie in the same range (Pe ranges from 0-1 to 1 and

Re ranges from 1 to 130) as those in single phase flows

reported by Gunn (1975,1574). Gunn obtained the Peclet

numbers by using sinusoidal temperature inputs to a

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106

g a s - s o l i d s i n g l e - p h a s e system and measuring the gas

temperature response . He found that the ax ia l Peclet num

bers for m e t a l l i c and nonmetal l ic b a l l s ranged from 0.01

to 1.3 when the p a r t i c l e ' s Reynolds numbers varied from

0.1 to 100.

In two-phase f lows much of the data come from mass

t rans fer s t u d i e s . Lerou (1980) showed the Pe i s the order

of 400 when the bed length i s used ins tead of p a r t i c l e

diameter to c a l c u l a t e Pe. The r a t i o of diameter of g l a s s

beads in t h i s work to the length of reactor in Lerou's

work i s 590. Therefore the range of Pe in h i s work i s in

the order of 0.7 when the diameter of the g lass beads i s

used t o c a l c u l a t e the Pe. The gas s u p e r f i c i a l v e l o c i t y

ranged from 0 to 3 f t / s e c .

Hochman (1969) obtained experimental data for a ir -wa

ter system. In h i s mass t rans fer experiment, KSCN was

used as the t r a c e r t o measure the res idence time d i s t r i b u

t ion of the system. The data were correlated for the two

phases i n d i v i d u a l l y . The PeL was found to vary from 0.1

to 0-6 when the l i g u i d Reynolds numbers varied from 2 t o

80. The gas phase Pec l e t number ranged from 0.02 to 2 .0

when the gas phase Reynolds number varied from 2 to 1000.

natsuura (1976) a l s o measured a x i a l dispersion in

mass t rans fer for two-phase down flow in a packed bed. He

used a model that s p l i t s the flow in to dynamic and

stagnant r e g i o n s . The a x i a l d i s p e r s i o n , dynamic hold-up

107

and mass transfer coefficient between the two phases were

used as parameters. The Pe values found ranged from 0.43

to 1.7 when Red (Reynolds number in dynamic region)ranged

from 10 to 1000- For Red egual to 150, Pe rose gradually

until Red was near 400 and reached an asymptote value of

1.7.

Correlation of the Asymptotic Peclet Numbers

The asymptotic Peclet numbers were correlated with the li

guid phase Reynolds number. The Peclet numbers over the

gas phase Reynolds number range from 400 to 700 were aver

aged for each liguid phase Reynolds number. This average

was used as the asymptotic Peclet number. The data are

shown in Figure 28. The data may be represented by the

empirical relation:

Pe (asymptote) = 1.63 (ReL) * (6-2)

which comes from a least sguare curve fit.

108

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109

Ragnitudes of Indiv idual Heat Transfer Mechanism

There are three i n d i v i d a a l nechanisms involved in the

model used in t h i s work: p a r t i c l e - t c - f l u i d heat t r a n s f e r ,

d i spers ion heat transfer and convect ive heat t rans fer . The

•agni tnde of heat t rans ferred by each of these Bechanis is

can be c a l c u l a t e d v ia T2(t ,2) and the p a r t i a l d e r i v a t i v e s

of T2{t ,2) with respec t to z . Since there i s no a n a l y t i

c a l s o l u t i o n for T2(t ,2) when c a l c u l a t i n g the p a r t i a l der

i v a t i v e s , a numerical method must be used. T2(t ,2) can be

ca l cu la ted from the deconvolution of T1( t , 2=0) and f ( t , 2 )

from Equation 4-18, i . e . :

T2 (t) = / Tl ( X) f (t-X . z) d X ^ " ^ 0

When the first derivative of T2(t,2) with respect to z is

calculated, the convective term can be shown to be:

v„ JJ^ . v„ / n ( X ) ( — f(c,z) U . t- X > « --' " 3 Z ° 9z

Since Tl(t,2=0) is a fixed function of t at z=0. The

partial derivative with respect to i can he brought into

the integral. In Eguation 6-4, the problem was to

determine the first derivative of f (t,z) vith respect to

110

2. It is reguired that the first derivative of f(t,z) have

the same value as f (t)=0 as time approaches infinity-

From the basic Eguation 6-3, it is seen that for a

pulse type input, individual terms such as accumulation,

convective heat transfer...etc., will all go to zero after

an extended period of time. This means that the first

derivative of f (t,z) with respect to z does have the same

limit of zero as that of f (t) - From Eguation 4-28:

t T2 (C) - / Tl ( X) f (t-X . z) d X (4-28)

The first derivative of 72(t,z) with respect to z can be

calculated through the convolution intergral of Tl (t) and

the Laplace inverse cf the first derivative of G(3,2) with

respect to z (which is the Laplace transform of f(t) with

respect to 7.) to give:

/^T1(0 L"^ { G )\^^^ r dC (6-5) 3 z 3 z 0

= / T1(0 — L"M G }| .d (6-5A) 0 3 z t: t ^

t -1 ^^ (6-5B)

° 3z ' ^ "

/ T1(X) L"^ { - ^ ( 1 - B^/^) G }| ^ j._ ^ dX (6-5C)

a

Ill

In the sane way, the dispersion term can be calculated

fron the second derivative according to the expression:

* V 3 T t -1 ^ 1/2

D 2 - D / Tl ( X) L { ( 1 - B '' )

a (6-6) V z

X exp( ( 1 - B /2 jjj I ^ ? D

a

The p a r t i c l e - t o - f l u i d heat t r a n s f e r tern can be ca lcu la ted

from Eguation 4-3- The f i r s t p a r t i a l d e r i v a t i v e of T2(t)

with r e s p e c t to t i n e i s egu iva lent to the Laplace inverse

of sT2 ( t ) . Therefore:

Fron Equation 4-3:

. V |TL. +D i ^ + - ^ - ( T I „ - T ) - | f - (U-3) • H32 a^,^ ep„Cp„^ s'r-R 3t

Since a l l the t e r n s , convec t ion , accumulation, d i spers ion ,

are to be ca lcu la ted fron Eguations 6-1 through 6 -7 , the

c a l c u l a t i o n of the p a r t i c a l - t o - f l u i d heat transfer can be

ca lcu la ted a s :

^ ^ ( T2(t) - T (C) ) (6-8) s h

112

Osing Ecuations 6-1 to 6-8 it is possible to conpare the

nagnitudes of the individual heat transfer nechanisn con

tribution at any time during the transient process, as

¥ell as at any location within the packed bed.

^^lJ£l§2£ of Individual Heat l£^5^££ Jgchanisms at Different

Flow Regiines ""

The response of the system to a perfect pulse input

is presented in Figure 29 to show the individual nagin-

tudes of the h€at transfer mechanisBS- In the figures to

cone, the vertical axis is the contribution by each indi

vidual teat transfer mechanism in units of BT0/(ft2 sec) x

(ft3 P/ITU). The units in the first bracket stand for the

heat transfer rate and the units in the second bracket

stand for the honogeneous sensible heat. These units are

used because e^ery tern in Eguation ^-3 was divided by the

homogeneous sensible heat for ease of calculation. The

horizontal axis is the elapsed time for the heat transfer

process in units of seconds.

Ficure 2S is a plot of the individual heat transfer

mechanism contribution of the system subject to a perfect

pulse fcr ReL=^C6.8, ReG=1.26 (pulse flow regime). Ihis

plot sbows a conservative estimation of the relative

113

o m

4->

u.i5

^ 0.1

* Convective heat t ransfer Pa r t i c l e - t o - f l u id heat t ransfer Dispersion heat t ransfer

ReG =^408, ReL =2.52 Bi(opt)=5.6 Pe(opt)=0.4335

o 0) V)

CM

« » -

ZD

UJ

!5 CC

CC UJ

u. CO < CC

< UJ X

0.05

0.

-0.05

-0 .1

-0 .2 60

TIME(SEC)

Figure 29: Comparison of C o n t r i b u t i o n s Hade by C o n v e c t i o n , E i s p e r s i o n and F l u i d - t o - P a r t i c l e Heat T r a n s f e r t o t h e O v e r a l l Heat Transfer Eate in Pesponse t o a P e r f e c t P u l s e Input in P u l s i n g Flow F e g i n e .

114

magnitude of the heat t rans fer mechanisms s ince the heat

t rans fer c o e f f i c i e n t was chosen as the lowest of the va

l u e s found in the f i r s t sec t ion of t h i s chapter. This i s

because i f a larger heat transfer c o e f f i c i e n t was used to

c a l c u l a t e the contr ibut ions of the individual heat t rans

f e r mechanisms, the p a r t i c l e - t o - f l u i d heat transfer would

have a more s i g n i f i c a n t part on the t o t a l heat transferred

in the p r o c e s s .

In the pulse flow regime, the peak appears a l i t t l e

e a r l i e r than in the t r i c k l e flow regime. This i s because

the heat t r a n s f e r rate i s higher in the pulse flow regime.

However, when an experimental s t ep input i s used as Tl ( t ) ,

the r e s u l t s are d i f f e r e n t . In Figure 30 the same parameter

values as in Figure 29 are used, but the experimental i n

put was used as the input function T l ( t ) . In the same f i g

ure, the heat transfer r a t e for the t r i c k l e flow regime

was added for comparison - Figure 3 0 i l l u s t r a t e s the heat

transfer ra te d i f f e r e n c e between the pulse and t r i c k l e

flow regimes. Figures 29 and 30 show there are large d i f

ferences between pulse and s tep inputs , as the peak be

comes more ev ident in the s t e p input c a s e . The plot shows

that t h e r a t e of heat t rans f er for a s t ep input w i l l f i r s t

pass through a raaximun and then fade t o 2ero as the system

returns to egu i l ibr ium.

In Figure 30, there i s an i n t e r e s t i n g phenomenon. The

t o t a l time reguired for the system to return to

115

ID

CO

o tf)

CM +J «*-

I— CO

UJ I— < CC

a < or

<

32

1

0

-1 0 10 20 30 40 50 60 70

TIME(SEC)

Figure 30: Conparison of the Contr ibut ions Hade by Convection Di spers ion and F l u i d - t o - P a r t i c l e Heat Transfer to t h e Overall Heat Transfer in Hesponse t o an Sxperimental Input in Tr ick le and Puls ing Flow Begime.

116

eguilibrium for the the trickle flow regime is longer than

for the pulse flow regime. This indicates if the packed

bed is in the pulse regime, it will take less tine for the

heat transfer process to be completed. Since the curve

for the pulse flow regime has a long tail and most of the

heat is transferred in the front portion of the processes,

this can also be utili2ed to cut the tine for certain pro

cess to complete the heat transfer.

Figure 31 shows additional evidence of the insensi

tivity of the system to changes in heat transfer coeffi

cient to error functions- In this plot there are three

curves, with values of Ei of 1.078, 20 and 40 respec

tively. From this figure it is observed that no natter

how h was increased, the curves for Bi of 20 and 40

still cone close to each other.

Heat Transfer Mechanisms for Step Type Inputs

As determined in the previous section, dispersion

plays a minor part in the total amount of heat transferred

between the fluid and solid for perfect pulse inputs.

This section examines the relative importance of the indi

vidual heat transfer nechanisns by using the tenperature

input at the entrance of the packed t€d as Tl(t). Run f 1

and Bun # 5 were taken as exanples. Run # 1 was in the

trickle flow regine and Run # 5 was in the pulsing flow

regine.

117

2.0

S 1.5 CD

U_

<M

O

Vi

1 \

1.0

UJ >

<

cr cr UJ

I 0.5 < cr < UJ X

0.0

-0.5

• t—I-

1 I I . Bi(opt)

Bi= 20 and 40 nes pectively

Curve A: convective heat transfer

ChjTve B: particle-to-fluid heat transfer

Curve C:dispersion

i I i 0 10 20 33 AQ 50 60 70

TIME (SEC)

Figure 31: Ef fec t of Changes in Bi Kuabers on the Contr ibut ions of the Heat Transfer nechanisns in T r i c k l e Flow with an Experimental Step Change in Tenperature. (ReG=55-6, ReL-2.4)

118

Ficure 32 conpares the magnitudes of the individual

heat transfer nechanisns for different flow regines- Fig

ure 32 shows that the dispersion will not increase signi

ficantly with an increase in the gas flow rate at a cons

tant licuid flon rate. However, the particle-to-fluid

heat transfer rate increases sharply with a change fron

trickling to pulsing flow. In the pulsing regine the peak

of the curve appears earlier than in trickle flow due to a

larger beat transfer coefficient resulting fron a larger

convective effect-

Figure 33 is a plot of heat transfer rate by indivi

dual h€€t transfer nechanisns for the operating conditions

of R€L=1-4, R€G=571-4, which is in the pulsing region-

Again, the dispersion contribution is almost negligible

compared to the total heat transferred. As the liguid flow

rate is further increased, the asount of dispersion heat

transfer will te even snaller-

Frcn the plots, it can be seen that the peaks for

each heet transfer mechanism occur at nearly the same time

and all fade out to zero for an extended period of tine.

The particle-tc-fluid heat transfer for the trickle flow

region is nearly six times larger than the dispersion heat

transfei. For larger liguid and/or air flow rates, the

percentage of dispersion heat transfer in the total heat

transferred within the system will be even less. If the

dispersion tern were to te neglected, the laximun error in

119

2.0

5 1.5 3

CD

O 0) (0

<M. •P 4.

CD

<

cr UJ

1.0

^ 0.5 <

<

0.0

-0.5 0 20 40 60 80 100

TIME (SEC)

Figure 32: Coipacison of Mechanisas Contribution to Overall Heat Transfer in Trickle and Pulse Flow Eegines (HeG=55-6, PeL^I-U, Bi |opt)=C.576, Pe(opt) =0.155 and ReG = 326.8, R€l=1.4, Bi (cpt)=1. 594, Ee (opt) =0. 288)

120

= 3 I— CD

U -

fVJ

CD

UJ

!<

cr cr UJ

z < cr

< UJ X

i.e

8.8

8.6

8.4

8.2

I ^ ^ ^

, \ Legend:

J ^

• • • . . 8.8 »»»*»»»»'

•8.2

* * Par t i d e - t o -f lu id heat trans

Conveotive heat t ransfer .

8 28

+ Dispersion

heat transfer

lee 129 TIM': (SEC)

Figure 33: Conparison o f Hechanisms C o n t r i b u t i o n s t o O v e r a l l Heat Trans fer in P u l s e Flow Regine. (R€G = 5 7 1 - 4 , BeL=1.4)

121

calculating the total heat transport could be less than

than O-S'' for the pulse flow regime.

Transition of Heat Transfer nechanisn

When the flow regines transition from one to another,

there are sometimes transitions in the phase di.stribution.

Figure 34 examines whether there is a change in the heat

transfer nechanisns for the transition fron the trickle

flow to the pulse flow regime. Run #31, Run #35, and Run

•39 are examined-

Figure 34 shows that the heat transport curves for

the pulse flow regime have a sharper peak than those for

the trickle flow regine. The heat transport curves for

the trickle flow regime have a longer break-through time

than those in the pulse flow regime. This indicates that

the heat transfer process in the trickle flow regine needs

more time to complete than in the pulse flow regime. The

heat transport curves in the pulse flow regime has the

same break through time for different operating condi

tions.

Ciscussion

It appears that the dispersion term can be neglected

in the pulsing flow regime. This same conclusion was also

reached by Lerou et. al.,(198 1) in their mass transfer

122

C 3 3

CQ

4J

CVJ <0

UJ

< CC

QC UJ

^ 1 < CC

<

0

Legend:

Run #39 (pulsing) Run ilf35 (pulsing)

•Run #31 (trickle)

A: Par t io le - to - f lu id heat transfer

B: Convective heat transfer Dispersion heat transfer

-1

« ^ ^ ^ ^ ^ ^ ^ ^ ^% ^% ^ ^ ^ ^

I I 0 10 20 30

TIME (SEC)

40 50 60

Figure 34: Comparison of Mechanisms Contributions to Overall Heat Transfer in Pulse and Trickle Flow Regime (ReG=55.6, 328, 652 .2 , with Rel=1-g6)

123

work. Lerou worked with an a i r - w a t e r system and showed

t h a t t h e d i s p e r s i o n e f f e c t i s s m a l l e r in the pulse l a g l n e

conpared t o d i s p e r s i o n i n the t r i c k l e f low reg ine - This

d i f f e r e n c e i s because the p u l s e s cause d i f f e r e n t l i g u i d

s t r e a n s t o mix , r e s u l t i n g i n a decrease in the t o t a l d i s

p e r s i o n .

Heat Transfer C o e f f i c i e n t C a l c u l a t e d by N e g l e c t i n g

D i spers ion

Based on the d i c u s s i o n s in the previous s e c t i o n , the

d i s p e r s i o n can be n e g l e c t e d when the system i s in the

p u l s i n g r e g i n e . This makes the two-parameter D-C model be

come a s i n g l e parane ter n o d e l , where only the p a r t i c l e - t o -

f l u i d heat t r a n s f e r need t o be c o n s i d e r e d . Eguation 4 - 3 ,

with t h e d i s p e r s i o n n e g l e c t e d , becomes:

- V H _ + ^ ^ ( r \ - T) = -^^ (6-9)


B.C. 1: at 2 = 0 , 1 = H (t).

B.C. 2: at z >« , T = 10.

The i n i t i a l c o n d i t i o n i s :

I . e . 1: at t = 0 , T=TO

(6-9A)

(6-9B)

(6-9C)

124

The heat transfer eguation for the solid phase is :

3T^ D% . 9T

3 r r D r


B.C. 1: at r = C , T is finite. (6-10A)

dT

9r

The in i t i a l condition i s :

B-C. 2: at r = S , k s ^ ^^ ^ _ (6-10B)

I-C- 1: at t = 0 r T = TO- (6-10C)

All the boundary conditions for the fluid and solid phases

are the sane as in Chapter IV. The eguations were solved

by the sane Laplace transforn nethod and nunerical nethod

to invert the Laplace transform into time domain as in

Chapter TV. The physical constants, liguid hold-up, homo

geneous sensible heat, and the homogeneous velocity were

the same as in Chapter V. The heat transfer coefficient

was found by using the sane optimum seeking method as

stated in Chapter IV-

125

Heat Transfer Coefficient in Two-Fbas€ Pulsing flow in Packed Eed

The new model was developed specif ic-il ly for search

ing fcr the optimum heat transfer coefficients in the

pulse flow regimes. However, the experinental data in the

trickle flow regime were also used to calculate the opti

mum heat transfer coefficient. It was found that the sen

sitivity of the error function to the heat transfer coef

ficient was lew, as expected. Table 9 is the error

function calulated for Bun #1, which is in the trickle

flow regime.

There was little change in the calculated error as

the Bi shifted from the optimum value to a higher value-

The slope of (changes in error)/ (changes in Bi) is almost

zero- Ihe optimun Bi calculated will not have much physi

cal meaning.

As the flow regime shifts from trickling to pulsing,

the sensitivity cf the error function to the heat transfer

coefficient improves. lafcle 10 shows the error function

calculated for Bun #10- In this table, it is found that

the sensitivity of the error function to Fi inpcoves, com

pared to Eun n . The Di calculated will h.ive more physical

significance than that in Run n . The error function as

functions of heat transfer coefficient shewn in Figures 35

and 36 illustrate this.

126

TABLE 9

Error Function Calculated for Different Heat Transfer Coefficient of Run #1

Error Calculated

0.79E-0 0 .29E-0 0 .14E-0 0 . 14B-0 0. 14E-0 0 . 14E-0 0 .14E-0 0. 14E-0 0. 14E-0 0 .14E-0 C.14E-0 0 .14E-0 0 , U B - 0 0 . 14B-0 0 .14E-0 0 . 14E-0 0 .14E-0 0 .14B-0 0 .15E-0 0 .15E-0 0- 15E-0 0 .15E-0 0-15E-0

T r i c k l e Flow

Heat Transfer C o e f f i c i e n t

Bi Number

0 . 126E-01 0.176B-01 0.226E-01 0.276E-01 0.326E-01 0.376E-01 0.426E-01 0.476E-01 0.526E-01 0.576E-01 0.626E-01 0.676E-01 0.726E-01 0.776E-01 0.B26E-01 0.876E-01 0-926E-01 0.976E-01 0.102E+00 0.107E*00 0.112E+00 0. 117E+00 0.122E+00

0.50E+00 0.69E+00 0.89E4-00 0.109E401 0.129B401 0-14BE*01 0.168E+01 0-188E*01 0.208E+01 0.227E+01 0-247E+01 0-266E+01 0.287E+01 0.306E+01 0.326E+01 0.346E+01 0.366B+01 O-SBeE^OI 0.405E+01 0.425E+01 0.445E+01 0-465E+01 0.484E+01

(Opt)

This c o n c l u s i o n was a l s o reached by Wakao(1979). He

used a DC nodel t o s i n u l a t e heat t r a n s f e r i n a

s i n g l e - p h a s e s y s t e n . Although the model had the heat

t r a n s f e r c o e f f i c i e n t and the d i s p e r s i o n c o e f f i c i e n t as the

127

TABLE 10

Error Function Calculated fcr Different Heat Transfer Coef f i c i ent of Run #10

Error Calculated

0.S92E-01 0 .314E-01 0.2I I0E-01 0 .229E-01 0 .242E-01 0 .261B-01 0.28 I E - 0 1 0.30 I E - 0 1 0.318E^01 0 .334E-01 0 .348E-01 0 .361E-01 0 .373E-01 0 .383E-01 0.392B-01 0-40 I E - 0 1 0 .408E-01 0 .416E-01 0 .422E-01 0 .428E-01 0.H34E-01 0.439E-01 0.a44E-01

Pulse Flow

Heat Transfer C o e f f i c i e n t

0.101E-01 0 . 151B-01 0.201B-01 0.25 IE-01 0.301E-01 0,351E-01 0.401E-01 0-451E-01 0.501E-01 0.55 IE-01 0-601E-01 0.651E-01 0.701E-01 0.751E-01 0.801E-01 0.851E-01 0.901E-01 0-951E-01 0.lOOE^OO 0. 105E4-00 0. IIOE^OO 0.115E+00 0. 120E + 00

Bi Munber

0.40E^00 0.59E+00 0.79E+00 0.993E+00(Opt) 0.119E*01 0- 138E+01 0.158E*01 0.178E+01 0.198E+01 0.217E*01 0.237E+01 0.257E+01 0.277E+01 0.296E+01 0.316E*01 0-336E+01 0.356E*01 0.376E+01 0-395E'»-01 0.415B*01 0.435E+01 0.455E+01 0.474E+01

two parameters, he f ixed dispers ion c o e f f i c i e n t before

f inding the optinun Husselt number, Nu. The Nusselt

numbers ca l cu la ted had a higher confidence l eve l when the

gas (air) flow rate was higher.

128

SO

70

60

50

o 40

X

1 30

20

10

\ •

1 1 1 i 1 \

20 40 60 80 100 120 h X 10"^ (Btu/seo-ft -F)

Piqure 35: Error Function Versus Bi Nunber Ca lcu la ted for

129

I ;'"

80 _

70

60 .

50 -

'o ^0

g cr g 30

20 -

10 ,

20 40 60 80 100 h X 10"- (Btu/seo-ft -F)

Figure 36: Error Function Versus Bi Nunber Calculated for Run «10

130

Heat Transfer Coefficient Results

Figures 37 to 41 show the calculated heat transfer

coefficients plotted as a function of the gas phase Rey

nolds nunber. However, there are certain errors involved

in the paranters, especially when the gas flow rate is

low. The dashed line in the figures for the trickle flow

regine neans the number is not reliable, based on the sen

sitivity analysis given in previous section.

Bi tends to be constant in the pulse flow regine for

fixed ReL and increasing ReG- As the liguid flow rate is

increased, the heat transfer coefficient is larger for the

sane gas flow rate. The heat transfer coefficient is a

weak function of gas flow rate for a fixed liguid flow

rate-

Conparison of the Heat Transfer Coefficients with Others

Wakao (1979) collected data for particle-to-fluid

heat transfer coefficient from other works for single

phase flows and correlated those data by using the disper

sion-concentric model- The result was:

0« 0 33 '^""' Nu = 2.0 + 1.1 ( Re )"•* ( Pr ) ' "

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Th€ asymptotic values of Ei were correlated as

functicB of Be l . The data are shown in Figure 42- The

corre la t ion for the data shown in Figure 41 i s :

Bi = 0.402 (ReL) 1.75 (6-12)

This eguation shows that the asymptote Ei i s a strong

function of l i g u i d phase Reynolds number. A comparison of

the heat t rans fer c o e f f i c i e n t s ca lculated from Eguation

6-11 and 6-12 showed that the heat transfer c o e f f i c i e n t s

in two phase flows are larger than those in the s ing le

phase gas flows with the same Reynolds number by as much

as a factor of 4. There are no l i t e r a t u r e data for

f l u i d - t c - p a r t i c l e heat transfer in two-phase flows ava i la

ble for comparison.

137

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CHAPTER VII

SUMMARY

The reason that dispersion plays a much more impor

tant role in the trickle flow or low flow rate region is

that the fluid flows slowly and thermal eguilibrium is

readily established between the fluid and solid- The dis

persion coefficient is a function of molecular conductivi

ty for the packing material and the degree of turbulent

mixing in the system-

The system is turbulent in the pulse flow regime- The

turbulent mixing dominates the molecular conduction pro

cess, but the degree of turbulent mixing does not increase

significantly for higher gas or liguid flow rates. The

dispersion heat transfer mechanism was neglected in the

pulse flow regime in this work.

It was learned that the error function is insensi

tive to chages in Bi. This may be due to the effect of

the input function. The step type inpnt function was used

in this work. The small change in f (t) caused by changing

heat transfer coefficient was masked by integration in the

convolution process. Tailor (1982) used the sinusoidal

wave as the input function and higher sensitivity of f(t)

to changes in Ei were obtained.

The particle-to-flaid heat transfer and convective

heat transfer mechanism dominates the the system in the

138

139

pulse flow regime. The packing icaterial may affect the

particle-to-fluid heat transfer rate, since the heat

transfer coefficient is a function of material hut was not

studied here- It may be inferred from the model that the

packing material can also affect the sensitivity of the

model to the heat transfer coefficient. Tailor's (1982)

study indicated steel packings gave a more sensitive error

function.

The particle-to-fluid heat transfer coefficient is a

strong function of liguid flow rate in pulsing flow. The

larger the liguid flow rate, the larger the heat transfer

coefficient. This was because the liguid had a larger heat

capacity than cid the gas-

In the two-phase heat transfer process, the gas ef

fected the pressure head of the system and the bulk veloc

ity of the systett. This caused the convective contribu

tions to be dominant- But there was a limit on the

contribution of the convective heat transfer to the total

heat transfer- this can be seen from Figure 41. The Bi

nnumbers tend to be constant after the gas Reynolds num

bers reached a certain limit because the convective heat

transfer is liiited to certain extent-

140

^£££i5>6ndations

1. mother model which accounts for l i gu id d i s t r i b u

tion and exchange between dynamic and stagnant

2ones should be proposed to so lve the heat transfer

problem in the two-phase flow as some other recent

»ork in mass transfer as by Hatsurra (1976).

2 . I higher l i g u i d flow rate should be used to improve

the s e n s i t i v i t y of the model to the p a r t i c l e - t o -

f luid heat transfer c o e f f i c i e n t .

3. r i f f e r e n t packing materia ls should be used to study

the s e n s i t i v i t y of the model to the p a r t i c l e - t o -

t luid heat t ransfer c o e f f i c i e n t s .

BIBLICGRAPHK

Anderssen , A. S . , H h i t e , E- T . , "parameter Est imat ion by t h e Transfer Funct ion Method" Chem. Eng- S c i - , 25 , 1 0 1 5 - 1 0 2 1 , 11970) . " '^

Anderssen, A- S - , S h i t e , E. T . , "Parameter Est imation by the Weighted Moment Method" Chem. Eng. S c i . , 26 , 1203 -1219 , ( 1 9 7 1 ) .

Beimescfc, H. E - , K e s s l e r , D- P . , " l i g u i d - G a s D i s t r i b u t i o n Measurements in the P u l s i n g Regime of Two-Phase Cocurrent Flow i n Packed Beds" AIChE. J - , J 7 , 1160, (1971) -

Bennet , A. , Gcodr idge , F . , "Hydrodynamic and Mass Transfer S t u d i e s in Packed Absorption Columns-Part I : Axia l D i s p e r s i o n " T r a n s , I n s t - Chem- Inqrs- 4 8 , T232, ( 1 9 7 0 ) -

Bradshav, A . V . , Johnson, A . , Mc l a c h l a n , N. B . , "Heat Transfer between Air and Nitrogen and Packed Beds of Non-Eeacting S o l i d s " Trans. I n s t n . Chem. F n g r s . , 48^ T77-184 , (1S70) -

Buffham, B- A . , Rathor , M- N. , "The I n f l u e n c e of V i s c o s i t y on A i i a l Mixing in T r i c k l e Flow in Packed Beds" Trans. I n s t . Chem. J n ^ r s - , 5 6 , 266 , (1S78) .

B u n e l l , D. G-, I r v i n , H- G-, Os l son , R. W-, Smith, J . H., " E f f e c t i v e Thermal C o n d u c t i v i t i e s in Gas-So l id Systems" I J C , 4 1 , 1 9 7 7 - 1 9 8 1 , ( 1 9 4 8 ) .

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