Nonlinear Analysis and Prediction of Time Series in Multiphase Reactors

49 Pages·2014·2.104 MB·English

by  Mingyan Liu, Zongding Hu (auth.)

SPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY  MULTIPHASE FLOW
Mingyan Liu
Zongding Hu
Nonlinear
Analysis and
Prediction of
Time Series
in Multiphase
Reactors
SpringerBriefs in Applied Sciences
and Technology
Multiphase Flow
Series editors
Lixin Cheng, Portsmouth, UK
Dieter Mewes, Hannover, Germany
For furthervolumes:
http://www.springer.com/series/11897
Mingyan Liu Zongding Hu
•
Nonlinear Analysis
and Prediction of Time
Series in Multiphase
Reactors
123
MingyanLiu Zongding Hu
School ofChemical Engineering School ofChemical Engineering
and Technology and Technology
TianjinUniversity TianjinUniversity
Tianjin Tianjin
People’s Republic ofChina People’s Republic ofChina
and
State KeyLaboratory ofChemical
Engineering
Tianjin
People’s Republic ofChina
ISSN 2191-530X ISSN 2191-5318 (electronic)
ISBN 978-3-319-04192-6 ISBN 978-3-319-04193-3 (eBook)
DOI 10.1007/978-3-319-04193-3
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013956739
(cid:2)TheAuthor(s)2014
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Acknowledgments
TheauthorsaregratefultotheNaturalScienceFoundationofChina(contractNos.
20106012 and 20576091), the Open Foundation of State Key Laboratory of Mul-
tiphase Complex System, Institute of Process Engineering, Chinese Academy of
Sciences and the Cheung Kong Scholar Program for Innovative Teams of the
MinistryofEducation(contactNo.IRT0641)forfinancialsupportofthework.The
authorsalsowishtothankDr.KlarenD.G.,Profs.KwaukM.,LiJ.,andLiuC.for
their helpful comments and suggestions. We want to thank the members of our
researchgroup team for their great contributionsfor this work.
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Experimental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Apparatus and Set-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Analysis Tools of Time Series Data. . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Attractor Reconstruction and Phase Plot. . . . . . . . . . . . . . . . . . 13
3.4 Correlation Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Kolmogorov Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.6 The Largest Lyapunov Exponent. . . . . . . . . . . . . . . . . . . . . . . 15
3.7 Local Non-Linear Short-Term Prediction . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Single-Orifice Bubbling Mechanism . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Time-Domain and Power Spectrum. . . . . . . . . . . . . . . . 19
4.1.2 Phase Plane Plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1.3 Kolmogorov Entropy and Correlation Dimension . . . . . . 22
4.1.4 Local Non-Linear Short-Term Prediction. . . . . . . . . . . . 27
4.2 Multi-Orifice Bubbling Behavior. . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Traditional Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Chaos Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.3 Flow Regime Identification Method . . . . . . . . . . . . . . . 35
4.3 Gas–Liquid–Solid Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . 37
4.3.1 Ordinary Gas–Liquid–Solid Fluidized Beds . . . . . . . . . . 37
4.3.2 Magnetized Gas–Liquid–Solid Fluidized Beds . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
vii
Abstract
This chapter reports our recent work on the nonlinear aspects or deterministic
chaosissuesinthesystemsofmultiphasereactors.Thereactorsincludegas–liquid
bubblecolumns,gas–liquid–solidfluidizedbeds,andgas–liquid–solidmagnetized
fluidized beds.Pressurefluctuations inthemultiphasesystemswere takenastime
series for nonlinear analysis, modeling, and forecasting. The qualitative and
quantitativenonlinearanalysistoolsincludeattractorphaseplaneplot,correlation
dimension,KolmogoroventropyandlargestLyapunovexponentcalculations,and
local nonlinear short-term prediction. First, chaotic bubbling dynamics and
transition mechanism in gas–liquid bubble columns with single orifice were
investigated.Second,thechaoticcharacteristicsingas–liquidbubblecolumnswith
multiorificedistributorwerepresented.Finally,nonlinearbehaviorsinothertypes
of multiphase reactors were explored briefly. The main results are as follows.
Bubbling process with single orifice exhibited deterministic chaotic behavior in a
certain range of gas flow rate. With the increase of gas flow rate, the sequence of
periodic bubbling, primary chaotic bubbling, advanced chaotic bubbling, and
jettingwassuccessivelyfound.Thesuddenincreaseoflocalnonlinearforecasting
errorofpressuresignalwithtimeprovidedanevidenceofthepresenceofchaotic
bubbling.Multivaluephenomenaofchaoticinvariantsresultedfromthemultiscale
behaviorexistingintheheterogeneousflowregimeinmultiorificebubblecolumns.
Flowregimesandtheirtransitionscouldbecharacterizedeffectivelybythechaotic
invariantsforthesemultiphasereactors.Forgas–liquid–solidfluidizedbeds,some
new flow regimes were identified by chaotic analyses. For gas–liquid–solid
magnetized fluidized bed, the values of chaotic invariants reduced with the
increase of magnetic field strength at given superficial gas velocity and liquid
velocity.
Keywords Gas–liquid bubble column (cid:2) Gas–liquid–solid fluidized bed (cid:2) Time
series (cid:2) Chaotic analysis (cid:2) Pressure fluctuation (cid:2) Prediction
ix
Notation
B Magnetic induction, T
C Correlation integral
ðrÞ
d Embedding dimension
D Correlation dimension
2
f Frequency, Hz
g Functional relationship between the current state
and the future state
G Predictor which approximates function g
HðxÞ Heaviside function
K Kolmogorov entropy, bit/s
K Order-2 Renyi entropy, bit/s
2;m
Lðt Þ Replacement length at evolution time t of k step,
k k
dimensionless
m Embedding dimension
M Total known number of data points
n Total number of replacement steps
N Length of restructured vector X
i
N Number of predicted points
p
P Pressure, p
a
P Experimentally measured pressure, p
e a
P Predicted pressure, p
p a
P Power spectrum
ðfÞ
Q Gas flow rate, ml/min
r Radius of hyper-sphere, dimensionless
X Restructured vector
i
x Component of restructured vector
t Time, s
U Superficial gas velocity, m/s
g
Greek Letters
d Standard deviation or size of the attractor
e Holdup, dimensionless
g
c Correlation coefficient
k Largest Lyapunov exponent
1
s Time delay, s
Dt Sample time interval, s
xi
Chapter 1
Introduction
Understanding the hydrodynamics in the gas–liquid bubble columns with single-
ormulti-orificeorinthegas–liquid–solidfluidizedbedsincludinggas–liquid–solid
magnetizedfluidizedbedsisofgreatimportanceforprocessindustriesbutitisstill
not well known due to the limitations of traditional analysis methods and the
complexityofmultiphaseflowsystemsthemselves[1,2].Itisinterestingtostudy
the bubbling hydrodynamics from the point of view of modern non-linear
dynamics [3–10].
Regardingasthebubblinghydrodynamicswithsingleorifice,TrittonandEgdell
[11] carried out a set of experiments in a beaker on bubbling from an upwardly
pointing vertical tube of an internal diameter of 1.2 mm. A period-doubling
sequence leading to chaotic behavior in the inter-pulse return maps was found by
analyzingthesignalfromahot-filmanemometerprobeplacedintheliquidnearthe
bubbleformationregion.Mittonietal.[12]performedasetofexperimentsovera
slightly broader set of conditions than that of Tritton and Egdell [11]. Similar
sequence was observed in the inter-pulse return maps by analyzing the gas inlet
pressure signal in the gas chamber. Nguyen et al. [13] investigated the spatio-
temporaldynamicsinatrainofrisingbubblesandfoundthattheanalogybetween
the rising bubbles in fluids and an inverted dripping faucet is weak. Li et al. [14]
studiedthein-linebubblecoalescenceinnon-Newtonianfluidsinabubblecolumn.
Bubble generation was through an orifice of varying diameter, submerged in the
non-Newtonianliquidatthecenterofthebottomsectionofthetank.Thesignalsof
frequency of bubble passage at different heights in the tank were measured by
optical probes of photodiodes placed at the external wall of the square duct. The
nonlinearanalysesrevealedthatthecoalescencebetweenbubbles obeys achaotic
and deterministic mechanism. Ruzicka et al. [15] studied the chaotic features of
bubbling from a submerged orifice above a gas chamber using the simultaneous
measurements of gas chamber pressure and liquid velocity fluctuations. Besides
return maps analysis, mean and variance, intermittency factor, distribution of
thelengthofjettingbursts,Kolmogoroventropy,K,correlationdimension,D ,the
2
Mann-Whitney statistic, power spectrum of signal were calculated to reveal the
underlying structure of the data. Although the typical period doubling sequence
leading to chaotic behavior was not found, the sequence of periodic bubbling,
M.LiuandZ.Hu,NonlinearAnalysisandPredictionofTimeSeries 1
inMultiphaseReactors,SpringerBriefsonMultiphaseFlow,
DOI:10.1007/978-3-319-04193-3_1,(cid:2)TheAuthor(s)2014
2 1 Introduction
chaotic bubbling and random bubbling was observed. Luewisutthichat et al. [16]
investigatedthepowerspectrum,KandD byanalyzingtherisevelocityandshape
2
fluctuations of single bubbles bubbling through an orifice in a two-dimensional
bubble column and found that the signals of bubble velocity and shape exhibit
chaos. Tufaile and Sartorelli [17] studied the dynamics of the formation of air
bubblesinasubmergedorificeinawater/glycerinsolutioninsideacylindricaltube.
Thedelaytimebetweensuccessivebubbleswasmeasuredwithalaser-photodiode
system. The sequence of period-1, period-2, period-3, period-4, period-2 and
chaotic bubbling was observed by analyzing the constructed bifurcation diagrams
andfirstreturnmaps.TufaileandSartorelli[18]studiedtheformationofairbubbles
in a submerged nozzle in a water/glycerol solution inside a cylindrical tube sub-
mitted to a sound wave perturbation. They observed a route to chaos via period
doublingasafunction ofthesound waveamplitude andestablishedrelationstoa
Henon-likedynamicswiththeconstructionofsymbolicplanes.Theysuggestedthat
the bubble formation be seen as an oscillator driven by a sound wave. They [19]
also found aroute tochaos via quasi-periodicity.
Mosdorf and Shoji [20] described nonlinear features and analytical results for
the chaotic bubbling from a submerged orifice of 2 mm in diameter and micro-
convection induced by the bubble generation was recorded using hot-probe ane-
mometerlocatedclosetotheorifice.Theyperformedthenonlinearanalysisforthe
time series data of hot-probe anemometer including the calculation of the largest
Lyapunov exponent and proposed a simple model to simulate the process of
interactionbetween theelastic bubble walland liquid. Theyfound thatone ofthe
reasons for chaos appearance is the nonlinear character of interaction between an
elastic bubble wall and the liquid stream.
Sarnobat et al. [21] studied the effect of an applied electric potential on the
dynamicsofgasbubbleformationfromasinglenozzleinglycerolexperimentally.
Theyfoundthatatconstantgasflow,bubble-formationexhibitedaclassicperiod-
doubling route to chaos with increasing potential. Although electric potential and
gas flow appeartohave similar effects on theperiod-doublingbifurcation process
forthissystem,therelativeimpactofelectrostaticforcesissmaller.However,the
relative impact of electrostatic forces for the case of insulating liquid and con-
ducting gas phases is comparable to flow forces.
Li et al. [22] studied the dynamics of a chain of bubbles rising in polymeric
solutions by stimulus-response simulation, birefringence and particle image ve-
locimetry. They identified two aspects as central to interactions and coalescence:
stress creation by bubble passages and their relaxation. The competition displays
complex nonlinear dynamics and can be described by a simple but explicit
mathematical model. Recently, Frank and Li [23] investigated the chaotic
dynamicsofabubblechainrisinginapolymericfluidbycognitivemodelingand
analytical analysis. Good agreement of the cognitive simulation results with
experimental data was obtained and this reveals that the period doubling bifur-
cation sequence is the route to chaotic behavior.
CieslinskiandMosdorf[24]discussedthenonlinearfeaturesoftheairbubbling
fromasubmergedglassnozzlessubmergedindistilledwaterinacylindricaltank.