Delivered by Ingenta to: Sophie Itumeleng Segwale IP : 168.172.0.254 Sat, 12 Feb 2011 17:33:31 Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 8, 8–16, 2011 Control of Nanoparticle Growth in High Temperature Reactor: Application of Reduced Population Balance Model II Oluranti Sadiku ∗ and Andrei Kolesnikov Department of Chemical and Metallurgical Engineering, Faculty of Engineering and the Built Environment, Tshwane University of Technology, Private Bag X680, Pretoria 0001, Republic of South Africa Aerosol processes are often model using the population balance equation (PBE). This article presents a study on the simulation of particle size distribution during nanoparticle growth with simultaneous chemical reaction, nucleation, condensation and coagulation. The method used to reduce the population balance model is the method of moments. Under the assumption of lognor- mal aerosol size distribution, the method of moments was employed to reduce the original model into a set of first-order ODE’s (ordinary differential equations) that accurately reproduce important dynamics of aerosol process. The objective of this study is to investigate if we can use the reduced population balance model for the control of nanoparticle size distribution and to investigate the pro- cess model sensitivity to the influence of disturbance. And subsequently use the model to control particle size distribution. The numerical result shows there is a dependence of the average particle diameter on the wall temperatures and disturbance has great influence on process model. The pro- cess model was used as a basis to synthesize a feedback controller where manipulated variable is the wall temperature of the reactor and the control variable the aerosol size distribution at the outlet of the reactor. Keywords: Population Balance Equation, Method of Moments, Particle Size Distribution, Disturbance and Average Particle Diameter. 1. INTRODUCTION In recent years, along with advancement in materials syn- thesis, it has been possible to embed nanoparticles with control size in bulk materials Zabarjadi et al. 1 Nano- particles always have novel properties that can be used for the development of new improved processing, making the control synthesis of nanoparticles vital to fields like chemistry, materials science and engineering, and the envi- ronmental sciences. Again, nanosized (i.e., smaller than tens of nanometres) particles often behave differently, both physically and chemically, than their larger counterparts. Thus, the synthesis of model materials for the fundamen- tal mechanisms of nanoparticle transformations, nucleation and growth as well as the physical and chemical prop- erties of nanoparticle are critical to understanding their behaviour in natural engineered systems. Nanoparticles are of great scientific interest as they are effectively a bridge ∗ Author to whom correspondence should be addressed. between bulk materials and atomic or molecular structures. A bulk material should have constant physical properties regardless of its size, but at the nano-scale this is not often the case. Size-dependant properties are observed such as quantum confinement in semiconductor particles, surface Plasmon resonance in some metal particles and superpara- magnetism in magnetic materials. The properties of mate- rials change as their size approaches the nanoscale and as the percentage of atoms at the surface of a material becomes significant. Aerosol processes is one of major interest for the gen- eration of nanopowders of nano- and micro-sized particles with well-defined product properties depending on parti- cle characteristics such as primary particle size, specific surface area and agglomerate structure. These processes have largely replaced other processes which involve mul- tiple steps of wet chemistry, due to the direct gas phase reaction of precursor vapour and the ease of separation of the particulate products from the gas. Strongly cou- pled aerosol systems exist in a number of applications 8 J. Comput. Theor. Nanosci. 2011, Vol. 8, No. 1 1546-1955/2011/8/008/009 doi:10.1166/jctn.2011.1650
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Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Copyright copy 2011 American Scientific PublishersAll rights reservedPrinted in the United States of America
Journal ofComputational and Theoretical Nanoscience
Vol 8 8ndash16 2011
Control of Nanoparticle Growth in HighTemperature Reactor Application of Reduced
Population Balance Model II
Oluranti Sadikulowast and Andrei KolesnikovDepartment of Chemical and Metallurgical Engineering Faculty of Engineering and the Built Environment
Tshwane University of Technology Private Bag X680 Pretoria 0001 Republic of South Africa
Aerosol processes are often model using the population balance equation (PBE) This articlepresents a study on the simulation of particle size distribution during nanoparticle growth withsimultaneous chemical reaction nucleation condensation and coagulation The method used toreduce the population balance model is the method of moments Under the assumption of lognor-mal aerosol size distribution the method of moments was employed to reduce the original modelinto a set of first-order ODErsquos (ordinary differential equations) that accurately reproduce importantdynamics of aerosol process The objective of this study is to investigate if we can use the reducedpopulation balance model for the control of nanoparticle size distribution and to investigate the pro-cess model sensitivity to the influence of disturbance And subsequently use the model to controlparticle size distribution The numerical result shows there is a dependence of the average particlediameter on the wall temperatures and disturbance has great influence on process model The pro-cess model was used as a basis to synthesize a feedback controller where manipulated variable isthe wall temperature of the reactor and the control variable the aerosol size distribution at the outletof the reactor
Keywords Population Balance Equation Method of Moments Particle Size DistributionDisturbance and Average Particle Diameter
1 INTRODUCTION
In recent years along with advancement in materials syn-thesis it has been possible to embed nanoparticles withcontrol size in bulk materials Zabarjadi et al1 Nano-particles always have novel properties that can be usedfor the development of new improved processing makingthe control synthesis of nanoparticles vital to fields likechemistry materials science and engineering and the envi-ronmental sciences Again nanosized (ie smaller thantens of nanometres) particles often behave differently bothphysically and chemically than their larger counterpartsThus the synthesis of model materials for the fundamen-tal mechanisms of nanoparticle transformations nucleationand growth as well as the physical and chemical prop-erties of nanoparticle are critical to understanding theirbehaviour in natural engineered systems Nanoparticles areof great scientific interest as they are effectively a bridge
lowastAuthor to whom correspondence should be addressed
between bulk materials and atomic or molecular structuresA bulk material should have constant physical propertiesregardless of its size but at the nano-scale this is not oftenthe case Size-dependant properties are observed such asquantum confinement in semiconductor particles surfacePlasmon resonance in some metal particles and superpara-magnetism in magnetic materials The properties of mate-rials change as their size approaches the nanoscale andas the percentage of atoms at the surface of a materialbecomes significantAerosol processes is one of major interest for the gen-
eration of nanopowders of nano- and micro-sized particleswith well-defined product properties depending on parti-cle characteristics such as primary particle size specificsurface area and agglomerate structure These processeshave largely replaced other processes which involve mul-tiple steps of wet chemistry due to the direct gas phasereaction of precursor vapour and the ease of separationof the particulate products from the gas Strongly cou-pled aerosol systems exist in a number of applications
8 J Comput Theor Nanosci 2011 Vol 8 No 1 1546-195520118008009 doi101166jctn20111650
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RESEARCH
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
including such diverse areas as materials synthesis druginhalation nuclear reactor safety multiphase combustionprocesses pollution formation flue gas cleaning and heatexchanger fouling and slagging Aerosol product suchas TiO2 B4C find widespread use as pigments rein-forcing agents ceramic powders optical fibres carbonblacks and semiconductor materials Numerous experi-mental studies have suggested that aerosol growth occursin stages beginning with the gas phase chemical reac-tion of the reactants to produce monomers or moleculesof the condensable species Panagiotis2 The monomersform unstable clusters which grow further by monomercondensation Beyond a critical cluster size nucleation ofstable aerosol particles occurs These particles grow fur-ther by coagulations (condensation and surface reaction ofsome other growth mechanisms Kalani and Panagiotis3)In aerosol processes the growth or coagulation rate whichis affected additively by Brownian and turbulent shearforces influences strongly on particle size size distributionand morphology And therefore it is of fundamental inter-est in the synthesis of nanoparticles by aerosol processesNumerical simulation provides a means of understandingand ultimately controlling the various dynamics trans-port and mass and energy coupling processes in aerosolsystemsA high temperature reactor for producing spherical
nanoparticles from vapor phase is a multi-compositionsystem varying appreciably in the flow and temperaturefields Nanoparticle synthesized in high temperature reac-tor often carries certain charges which can be further uti-lized in the electrostatic manipulation measurement andcontrol of such particles The nanoparticles produced athigh temperature often undergo rapid coalescence withcomplex rate laws Michael and Markus4 There is a needfor detailed understanding of nanoparticle formation espe-cially the material synthesis and combustion in high tem-perature processesThe dynamic model of aerosol processes are typically
obtained from the application of population material andenergy balances and consist of partial integro-differentialequation systems (where the independent variables aretime space and one or more internal particle coordinatessuch as particle volume and shape) Nonlinearity usu-ally arises from complex reaction nucleation condensa-tion and coagulation rates and their nonlinear dependenceon temperature The complex nature of aerosol processmodels has motivated an extensive research activity on thedevelopment of numerical methods for the accurate com-putation of their solution Numerical models can be viewedas mathematical frameworks that permit the interaction ofcomplex physical processes to be simulated The first stepin developing a numerical aerosol model is to assembleexpressions for the relevant physical processes such aschemical reactions nucleation condensation coagulation
etc The next step is to approximate the particle size distri-bution with a mathematical size distribution function Thefollowing are examples of solution methodsSectional methods (SM) are widely used to solve pop-
ulation equations In these methods the size spectrumis divided into a set of size classes In so doing dis-tinguishes between zero-order and higher-order methodsHigher-order methods use low order polynomials to repre-sent the particles within each section and can be regardedas a simple form of finite elements methods and will bediscussed in the next section They can suffer from stabil-ity problems and artificial dispersion whereas zero-ordermethods are more robust Using sectional methods com-putational domain is divided into rather small intervalsin which the solution is approximated by step functionsFor each interval one obtained an ordinary differentialequation which is coupled to neighbours depending on thediscretization scheme used Debra and Sonia5 described asingle moment sectional model to simulate the evolution ofan aerosol distribution that contains more than one chem-ical component The proposed method is based on divid-ing the particle domain into X sections with time variantsections boundaries To alleviate the problem with numeri-cal diffusion in the presence of the surface growth Kumarand Ramkrishna6ndash8 introduced a pivot technique combinedwith a moving grid and also the method of characteristicsAn additional method set of equation is solved to ensurethat a chosen set of properties is conserved Vanni9 cou-pled a sectional model to a detailed gas phase mechanismto calculate the soot particle size distribution in a contin-uous stirred tank reactor (CSTR)An alternative to sectional methods are the more
sophisticated finite element methods In the finite ele-ment approach the solution of the population balance isexpanded in series of polynomials For the coefficients ofthis expansion a set of equations has to be solved and thisis obtained by inserting the expansion into the populationbalance equation Various methods can be derived by dif-ferent nodes functions and time stepping schemes Themathematical discipline of functional analysis provides thetheoretical framework with which errors can be estimatedThis is of course a very attractive feature of finite elementmethodsAnother alternative to sectional methods for solving
the population balance equation are Monte Carlo (MC)methods They are easy to implement can account for fluc-tuations and can easily incorporate several internal coor-dinates In the case of nanoparticle modelling the numberof particles is so large that the fluctuations in particle num-bers can be neglectedOne of the applications of Monte Carlo methods is the
stochastic Monte Carlo (MC) method which is based onthe principle that the dynamic evolution of an extremelylarge population of particles Np(t) can be followed bytracking down the relevant particle events (ie growth
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
aggregation nucleation) Meimaroglou et al10 used thismethodJames and Panagiotis11 proposed a finite-dimensional
approximation and control of nonlinear parabolic partialdifferential equation (PDE) systems by combiningGalerkinrsquos method with the concept of approximate initialmanifolds known as non-linear Galerkinrsquos method Aleckand Costas12 used orthogonal collection of finite elementmethods to solve a continuous form of general popula-tion balance equation (PBE) Hailian et al13 developeda predictive controller for parabolic convectionndashdiffusionndashreaction systems operating in convection-dominatedregimes using the combination of finite differencesapproximation of the diffusion term and the method ofcharacteristics Dan et al14 designed a control algorithmon the basis of finite dimensional models to capture thedominant dynamics of particulate processes The leastsquares method (LSM) is a well-established technique forsolving a wide range of mathematical problems The basicidea in the LSM is to minimize the integral of the squareof the residual over the computational domain in thecase when the exact solutions are sufficiently smooth theconvergence rate is exponential Daora and Jakobosen15
applied the least square method to solve the populationbalance equationThe method of moments (MOM) is computationally the
most efficient approach to obtain a numerical approxi-mation to the moments of population balance For thisreason this method is often used when simulating prob-lems where transport of particles in a flow with com-plex geometry is essential In the area of nanoparticlemodelling two techniques have been used so far Oneof the techniques is the quadrature method of moments(QMOM) which is a more recent technique Dorao andJakobson16 derived (QMOM) in two ways ie the stan-dard quadrature method of moments which is a numer-ical closure for method of moments (MOM) and themethod of moments in the method of weighted residuals(MOM-MWR)The second technique is the method of moment
with internal closure (MOMIC) Diemer and Ehrman17
developed a design for the comparison of reconstructeddistributions from moment with direct calculation via sec-tional method The design was used to probe sensitiv-ity of distribution reconstruction and problem solution tomodel size for both the MOMIC and QMOM approachesBarret and Webb18 compared some approximation meth-ods for solving the aerosol dynamic equation The methodscompared are quadrature method of moments the finiteelement method (FEM) Luguerre quadrature and Associ-ated Laguerre quadrature Suddah and Mark19 also usedquadrature method of moments to solve the problem ofobtaining closure of the moment equations for the coag-ulation growth diffusion and thermophoretic terms to beexpressed in their original forms
In the first instance MOMIC has been developed todescribe the formation and oxidation of soot particlesIn its early form the method is based on univariatedescription of spherical soot particles in the free molec-ular regime An alternative approach for obtaining themoments of the PSD is the quadrature method of moments(QMOM) In this method the moments are calculatedassuming the PSD can be represented as weighted multi-dimensional Dirac delta function The weights and thenodes are then chosen to satisfy the transport equationsfor the moments of the PSD The advantage of thisapproach is that due to the choice of delta functions thereexists no closure problem Gerber and Mousavi20 appliedthe quadrature method of moments to the polydisperseddroplet spectrum in transonic steam flows with primaryand secondary nucleationThe method of moments (MOM) is computationally the
most efficient approach to obtain a numerical approxima-tion to the moments of population balance For this rea-son this method is often used when simulating problemswhere transport of particles in a flow with complex geome-try is essential The dominant dynamic behaviour of manyaerosol processes can be accurately captured by a modelthat describes the evolution of the three leading momentsof aerosol size distribution21 The aim of this work isto investigate the sensitivity of the process model to theinfluence of disturbances and to use the reduced popu-lation balance model for the control of nanoparticle sizedistribution The measurement of particle size distributionis a distinguishing feature in production of nanoparticlesbecause the particle size provides the critical link betweenthe product quality indices and the operating variables2
Thus the ability to effectively control the shape of thePSD is essential for regulating the end product quality ofthe process Many applications require a close control ofthis distributed particle length scale in order to achieve thehighest performance
2 PROCESS DESCRIPTION
The premixed preheated reactants (titanium tetrachlorideand oxygen gas) are injected into the reactor where fol-lowing exothermic reaction takes place The products ofthese reactions are titania monomers and chlorine gas
TiCl4g+O2grarr TiO2s+2Cl2g (1)
The size of a single TiO2 molecule (monomer) is largerthan the thermodynamic critical cluster size As a resultthe difference between chemical reaction and nucleationcannot be noticed thereby implying that the rapid chemi-cal reaction leads to nucleation burst The coagulation ofTiO2 monomers leads to an increase in the average particlesize and a decrease in particle concentration
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
21 Population Balance Model
The population balance equation consists of the followingnonlinear partial integro-differential equation22
n
t+ Gv z xn
v+ cz
n
zminus Ivlowastvminusvlowast
= 12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (2)
The first term on the left hand side of Eq (2) describethe change in the number concentration of particle vol-ume interval v v+dv and in the spatial interval z z+dznv z t denotes the particle size distribution function vis particle volume t is time z isin 0L is the spatial coor-dinate L is the length of the process The second termon the left hand side gives the loss or gain of particles bycondensational growth the third term on the left hand sidewhich is cznz corresponds to the convective trans-port of aerosol particles at fluid velocity cz and the fourthterm on the left hand side accounts for the formation ofnew particles of critical volume vlowast by nucleation rate I Ivlowastvminusvlowast also accounts for gain and loss of particlesby condensation Gv z x Ivlowast and v vminus v x arethe nonlinear scalar functions and is the standard Diracfunction The mass and energy balance model which pre-dicts the spatio-temporal evolution of the concentrations ofspecies and temperature of the gas phase given by Ashishand Panagiotis23 has the following form
dx
dt= A
dx
dz+ f x+ gxbzut
+ Aint
0a v xdv (3)
where xz t is an n-dimensional vector of state variablesthat depends on space and time A A are constant matri-ces f x gx a v x are nonlinear vector functionsut is the axially distributed manipulated input (eg walltemperatures Tw1 and Tw2 and bz is a known functionwhich determines how control action ut is distributedin space The last term on the right hand side of Eq (3)accounts for mass and heat transfer from the continuousphase to all the particles in population The gain and lossof particles by Brownian coagulation is described by thefirst and second term on the right hand side of Eq (2)respectively
12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (4)
Gv z x and are the condensational growth andcollision frequency function respectively for which two
different expressions are used for free molecule size andcontinuum size regimes14 The free molecule size regimetakes the following form
GFMx v z= B1v13Sminus1 where
B1 = 3613v1nskBT 2m112
FMx v z= B2
(1v+ 1
v
)12
v13+ v132
B2 = 34166kBTv1m112 (5)
And the continuum size regime takes the followingform
GCx v z= B3v13Sminus1 where
B3 = 48213Dv1ns D = 8kBT m1123
C = B4
(Cv
v13+ Cv
v13
)v13+ v13 B4 =
2kBT3
(6)
In Eqs (5) and (6) T is the temperature S is the sat-uration ratioD is the condensable vapour diffusivity isthe mean free path of the gas is the fluid viscosity ns
is the monomer concentration at saturation (ns = PskBT where Ps is the saturation pressure) m1 is the monomermass v1 is the monomer volume r is the particle radiusCv= 1+B5r is the Cunningham correction factor andB5 = 113257 Lastly the nucleation rate Ivlowast is assumedto follow the classical Becker-Doring theory given by theexpression below (Pratisinis)24
I = n2s s1kBT 2m1
12S22913
times12sum
expminusklowastInS2 (7)
where s1 is the monomer surface area and klowast is the numberof monomer in critical nucleus and is given by
klowast =
6
(4sum
InS
)3
where
sum = v231 kBT and is the surface tension (8)
3 LOGNORMAL AEROSOL SIZEDISTRIBUTION
The population balance model in Eq (2) is highly complexand does not allow the direct use for numerical com-putation of the size distribution in real-time To over-come this problem and to accelerate the computationsmethod of moments was employed to reduce the popu-lation balance model to a set of ODEs for the momentsof the size distribution In order to describe the spatio-temporal evolution of the three leading moments of thevolume distribution (which describes the exact evolutionof the lognormal aerosol size) a lognormal function wasemployed in moment model which was applied to popu-lation balance model
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
31 Moment Model
We assumed that the aerosol size distribution can beadequately represented by lognormal function which isdescribed as
nv z t= 13v
1radic2In
exp[minus In2vvg
18In2
](9)
where vg is the geometric average particle volume and is the standard deviation The kth moment of the distribu-tion is defined as
Mkt=int
0vknv z tdv (10)
According to the properties of a lognormal functionany moment can be written in terms of M0 vg and asfollows
Mk =M0vkg exp
(92k2In2
)(11)
If Eq (11) is written for k= 01 and 2 then vg and canexactly be expressed in terms of the first three momentsof the distribution according to the following relations
In2 = 19In
(M0M2
M21
)and vg =
M21
M320 M
122
(12)
In this subsection the ODEs describing the temporal evo-lution of the three leading moments of the size distributionfor the free molecule size and continuum size regime arepresented
311 Free Molecule Size Regime
The ODE system that describes the spatio-temporal sizedistribution of the kth moment of the aerosol size distri-bution is computed by substituting Eq (5) into Eq (2)multiplying by vk and integrating over all particle sizesThat gives the temporal evolution of the zeroth momentwhich is affected by nucleation and coagulation
dM0
dt= minuscz
dMo
dz+I
minusB2b0M0M16+2M13Mminus16+M23Mminus12 (13)
where the coefficient b0 is used for the relationship
(1v+ 1
v
)12
= b0
(1
v12+ 1
v12
)(14)
and it was computed by the expression b0 = 013633+0130922minus0130223 in publication of Pratisinis24 The evo-lution of M1 (aerosol volume) which is affected by con-densation is given by
dM1
dt=minuscz
dM1
dz+ Ivlowast +B1Sminus1M23 (15)
And the second moment M2 depends on nucleation andcoagulation according to the formula
dM2
dt= minuscz
dM2
dz+ Ivlowast2+2B1Sminus1M53
+2b2B2M76M1+2M43M56+M12M53 (16)
where b2 is used as b0 but for coagulation kernel of thesecond moment b2 is computed by the expression b2 =01339+0135 minus0132142+0130293 (Pratisinis)24
312 Continuum Size Regime
The spatio-temporal evolution of the kth moment of theaerosol size distribution in the continuum regime is com-puted by substituting Eq (6) into Eq (2) multiplying byvk and integrating over all particle sizes gives the temporalevolution of the zeroth moment M0 M1 and M2
dM0
dt
=minusczdM0
dz+ I minusB4
[M2
0 +M13Mminus13+B5
(43
)13
middot M0Mminus13+M13Mminus23
](17)
dM1
dt=minuscz
dM1
dz+ Ivlowast +B3Sminus1M13 (18)
dM2
dt=minuscz
dM2
dz+ Ivlowast2+2B3Sminus1M43
+2B4
[M2
1M43M23+B5
(43
)13
middot M1M23+M13M43
](19)
4 ANALYSIS OF RESULTS ANDDISCUSSION
This section describes the application of moment modelof the aerosol flow reactor for the purpose of nonlinearcontrol of the reactor Under the assumption of lognor-mal aerosol size distribution the mathematical model thatdescribes the evolution of the first three moments of dis-tribution together with the monomer and reactant concen-tration and temperature takes the following form
dN
d=minusczl
dN
dz+ I prime minusN 2
dV
d=minusczl
dV
dz+ I primeklowast +Sminus1N
dV2
d=minusczl
dV2
dz+ I primeklowast2+2Sminus1V +2V 2
dS
d=minusczl
dS
dz+CC1C2minus I primeklowast minusSminus1N
12 J Comput Theor Nanosci 8 8ndash16 2011
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
dC1
d=minusczl
dC1
dminusA1C1C2
dC2
d=minusczl
dC2
dminusA2C1C2
dT
d=minusvzl
dT
dz+BC1C2T +ET Tw minus T (20)
= FMCFM +C
C = K
[1+ expIn2+B5Kn1
r primeg
times exp(12In2
)1+ exp2In2
]
FM = rprime12g b0
[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)](21)
= FMC
FM +C
FM = vprime23g exp2In2 and
C = 4Kn1
3vprime13g exp
(12In2
)(22)
= FMC
FM +C
C = 4Kn1
3vprime13g exp
(72In2
)and
FM = vprime23g exp8In2 (23)
= FMCFM + C
FM = r prime12g b2 exp(32In2
)
times[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)]
C = K
[1+ expIn2+B5Kn1
r primegtimes exp(minus12In2
)
times(1+ expminus2In2
)](24)
where C1 and C2 are the dimensionless concentrations ofthe reactants Tw T are the dimensionless reactor and walltemperature respectively A1 A2 B C E are the dimen-sionless quantities Table II gives the process parametersused in the simulationDimensionless quantities for the model of Eq (20)
according to Ashish and Panagiotis23
A1 = kP0y20RT0 A2 = kP0y10RT0
B = P0kHRy10y20RT20 CP Ci = yiy10T
C = Navky10y20P0RT02ns0 E = 4URT0DCpP0
Table I Dimensionless variables by Ashish and Panagioti
N =M0ns V =M1nsv1 Aerosol concentration and volumeV2 =M2nsv
21 Second aerosol moment
= 2m1kBT 12nss1 Characteristic time for particle growth
K = 2kBT 3ns Coagulation coefficient and Nucleation rateI prime = Ins
Kn1= r1 Knudsen number
vprimeg = vgv1 Dimensionless geometric volumer primeg = rgr1 Dimensionless geometric radiusz= zL Dimensionless distanceczl = czL = t Dimensionless velocity and time = t Dimensionless time
Source Reprinted with permission from [23] K Ashish and D C PanagiotisChem Engng Sci 54 2669 (1999) copy 1999 Elsevier
T = T T0 and Tw = TwT0
In this chapter the process model of the equations inEq (20) was numerically solved in by using simulinka part of (Matlab) soft ware environment it is an inter-active computing package for simulating and analysingdifferential equations mathematical models and dynamicsystems The computation was done using a multi stepsolver known as ode 15s (stiffNDF) solver ode 15s isa variable-order solver based on the numerical differenti-ation formulas (NDFs) Optionally it uses the backwarddifferential formulas BDFs Sergey25 Table II shows theprocess variable parameters used for solving the processmodel of Eq (20)Since the objective of this study is to control nanopar-
ticle growth with desired particle distribution in high tem-perature reactor we studied the effect of wall temperature
Table II Process model parameters for the simulation study
L= 1135 m Reactor lengthD = 01301 m Reactor diameterP0 = 101000 pa Process pressureT0 = 2000 K Inlet temperaturey10 = 0134 Inlet mole fractions of O2
y20 = 0136 Inlet mole fractions of TiCl4U = 160 J mminus2 sminus1 Kminus1 Overall coefficient of heat transferHR = 88000 J molminus1 Heat of reactionCP = 16151325 Jmolminus1Kminus1 Heat capacity of process fluidMWg = 14130times10minus3 kg molminus1 Mol wt of process fluidK = 11134 m3 molminus1sminus1 Reaction rate constant= 6137times10minus5 kg mminus1 sminus1 Viscosity of process fluidlogPsmmHg PVT relation
=minus4644T +013906 logTminus01300162T +913004
= 01308 N mminus1 Surface tensionv1 = 51333times10minus29 m3 Monomer volumeNav = 613023times1023molminus1 Avogadrorsquos constantR= 813314 J molminus1 Kminus1 Universal gas constantkB = 11338times10minus23 JKminus1 Boltzmannrsquos constant = 0135 Sigmarg = 1135e06 m Geometric radiusr1 = 0135e06 m Monomer radiusvg = dp36 Geometric volumes0 = 1131587Eminus19 m Monomer surface areaTw = 800 K 500 K Wall temperatureRe = 2000 Reynolds number
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
including such diverse areas as materials synthesis druginhalation nuclear reactor safety multiphase combustionprocesses pollution formation flue gas cleaning and heatexchanger fouling and slagging Aerosol product suchas TiO2 B4C find widespread use as pigments rein-forcing agents ceramic powders optical fibres carbonblacks and semiconductor materials Numerous experi-mental studies have suggested that aerosol growth occursin stages beginning with the gas phase chemical reac-tion of the reactants to produce monomers or moleculesof the condensable species Panagiotis2 The monomersform unstable clusters which grow further by monomercondensation Beyond a critical cluster size nucleation ofstable aerosol particles occurs These particles grow fur-ther by coagulations (condensation and surface reaction ofsome other growth mechanisms Kalani and Panagiotis3)In aerosol processes the growth or coagulation rate whichis affected additively by Brownian and turbulent shearforces influences strongly on particle size size distributionand morphology And therefore it is of fundamental inter-est in the synthesis of nanoparticles by aerosol processesNumerical simulation provides a means of understandingand ultimately controlling the various dynamics trans-port and mass and energy coupling processes in aerosolsystemsA high temperature reactor for producing spherical
nanoparticles from vapor phase is a multi-compositionsystem varying appreciably in the flow and temperaturefields Nanoparticle synthesized in high temperature reac-tor often carries certain charges which can be further uti-lized in the electrostatic manipulation measurement andcontrol of such particles The nanoparticles produced athigh temperature often undergo rapid coalescence withcomplex rate laws Michael and Markus4 There is a needfor detailed understanding of nanoparticle formation espe-cially the material synthesis and combustion in high tem-perature processesThe dynamic model of aerosol processes are typically
obtained from the application of population material andenergy balances and consist of partial integro-differentialequation systems (where the independent variables aretime space and one or more internal particle coordinatessuch as particle volume and shape) Nonlinearity usu-ally arises from complex reaction nucleation condensa-tion and coagulation rates and their nonlinear dependenceon temperature The complex nature of aerosol processmodels has motivated an extensive research activity on thedevelopment of numerical methods for the accurate com-putation of their solution Numerical models can be viewedas mathematical frameworks that permit the interaction ofcomplex physical processes to be simulated The first stepin developing a numerical aerosol model is to assembleexpressions for the relevant physical processes such aschemical reactions nucleation condensation coagulation
etc The next step is to approximate the particle size distri-bution with a mathematical size distribution function Thefollowing are examples of solution methodsSectional methods (SM) are widely used to solve pop-
ulation equations In these methods the size spectrumis divided into a set of size classes In so doing dis-tinguishes between zero-order and higher-order methodsHigher-order methods use low order polynomials to repre-sent the particles within each section and can be regardedas a simple form of finite elements methods and will bediscussed in the next section They can suffer from stabil-ity problems and artificial dispersion whereas zero-ordermethods are more robust Using sectional methods com-putational domain is divided into rather small intervalsin which the solution is approximated by step functionsFor each interval one obtained an ordinary differentialequation which is coupled to neighbours depending on thediscretization scheme used Debra and Sonia5 described asingle moment sectional model to simulate the evolution ofan aerosol distribution that contains more than one chem-ical component The proposed method is based on divid-ing the particle domain into X sections with time variantsections boundaries To alleviate the problem with numeri-cal diffusion in the presence of the surface growth Kumarand Ramkrishna6ndash8 introduced a pivot technique combinedwith a moving grid and also the method of characteristicsAn additional method set of equation is solved to ensurethat a chosen set of properties is conserved Vanni9 cou-pled a sectional model to a detailed gas phase mechanismto calculate the soot particle size distribution in a contin-uous stirred tank reactor (CSTR)An alternative to sectional methods are the more
sophisticated finite element methods In the finite ele-ment approach the solution of the population balance isexpanded in series of polynomials For the coefficients ofthis expansion a set of equations has to be solved and thisis obtained by inserting the expansion into the populationbalance equation Various methods can be derived by dif-ferent nodes functions and time stepping schemes Themathematical discipline of functional analysis provides thetheoretical framework with which errors can be estimatedThis is of course a very attractive feature of finite elementmethodsAnother alternative to sectional methods for solving
the population balance equation are Monte Carlo (MC)methods They are easy to implement can account for fluc-tuations and can easily incorporate several internal coor-dinates In the case of nanoparticle modelling the numberof particles is so large that the fluctuations in particle num-bers can be neglectedOne of the applications of Monte Carlo methods is the
stochastic Monte Carlo (MC) method which is based onthe principle that the dynamic evolution of an extremelylarge population of particles Np(t) can be followed bytracking down the relevant particle events (ie growth
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aggregation nucleation) Meimaroglou et al10 used thismethodJames and Panagiotis11 proposed a finite-dimensional
approximation and control of nonlinear parabolic partialdifferential equation (PDE) systems by combiningGalerkinrsquos method with the concept of approximate initialmanifolds known as non-linear Galerkinrsquos method Aleckand Costas12 used orthogonal collection of finite elementmethods to solve a continuous form of general popula-tion balance equation (PBE) Hailian et al13 developeda predictive controller for parabolic convectionndashdiffusionndashreaction systems operating in convection-dominatedregimes using the combination of finite differencesapproximation of the diffusion term and the method ofcharacteristics Dan et al14 designed a control algorithmon the basis of finite dimensional models to capture thedominant dynamics of particulate processes The leastsquares method (LSM) is a well-established technique forsolving a wide range of mathematical problems The basicidea in the LSM is to minimize the integral of the squareof the residual over the computational domain in thecase when the exact solutions are sufficiently smooth theconvergence rate is exponential Daora and Jakobosen15
applied the least square method to solve the populationbalance equationThe method of moments (MOM) is computationally the
most efficient approach to obtain a numerical approxi-mation to the moments of population balance For thisreason this method is often used when simulating prob-lems where transport of particles in a flow with com-plex geometry is essential In the area of nanoparticlemodelling two techniques have been used so far Oneof the techniques is the quadrature method of moments(QMOM) which is a more recent technique Dorao andJakobson16 derived (QMOM) in two ways ie the stan-dard quadrature method of moments which is a numer-ical closure for method of moments (MOM) and themethod of moments in the method of weighted residuals(MOM-MWR)The second technique is the method of moment
with internal closure (MOMIC) Diemer and Ehrman17
developed a design for the comparison of reconstructeddistributions from moment with direct calculation via sec-tional method The design was used to probe sensitiv-ity of distribution reconstruction and problem solution tomodel size for both the MOMIC and QMOM approachesBarret and Webb18 compared some approximation meth-ods for solving the aerosol dynamic equation The methodscompared are quadrature method of moments the finiteelement method (FEM) Luguerre quadrature and Associ-ated Laguerre quadrature Suddah and Mark19 also usedquadrature method of moments to solve the problem ofobtaining closure of the moment equations for the coag-ulation growth diffusion and thermophoretic terms to beexpressed in their original forms
In the first instance MOMIC has been developed todescribe the formation and oxidation of soot particlesIn its early form the method is based on univariatedescription of spherical soot particles in the free molec-ular regime An alternative approach for obtaining themoments of the PSD is the quadrature method of moments(QMOM) In this method the moments are calculatedassuming the PSD can be represented as weighted multi-dimensional Dirac delta function The weights and thenodes are then chosen to satisfy the transport equationsfor the moments of the PSD The advantage of thisapproach is that due to the choice of delta functions thereexists no closure problem Gerber and Mousavi20 appliedthe quadrature method of moments to the polydisperseddroplet spectrum in transonic steam flows with primaryand secondary nucleationThe method of moments (MOM) is computationally the
most efficient approach to obtain a numerical approxima-tion to the moments of population balance For this rea-son this method is often used when simulating problemswhere transport of particles in a flow with complex geome-try is essential The dominant dynamic behaviour of manyaerosol processes can be accurately captured by a modelthat describes the evolution of the three leading momentsof aerosol size distribution21 The aim of this work isto investigate the sensitivity of the process model to theinfluence of disturbances and to use the reduced popu-lation balance model for the control of nanoparticle sizedistribution The measurement of particle size distributionis a distinguishing feature in production of nanoparticlesbecause the particle size provides the critical link betweenthe product quality indices and the operating variables2
Thus the ability to effectively control the shape of thePSD is essential for regulating the end product quality ofthe process Many applications require a close control ofthis distributed particle length scale in order to achieve thehighest performance
2 PROCESS DESCRIPTION
The premixed preheated reactants (titanium tetrachlorideand oxygen gas) are injected into the reactor where fol-lowing exothermic reaction takes place The products ofthese reactions are titania monomers and chlorine gas
TiCl4g+O2grarr TiO2s+2Cl2g (1)
The size of a single TiO2 molecule (monomer) is largerthan the thermodynamic critical cluster size As a resultthe difference between chemical reaction and nucleationcannot be noticed thereby implying that the rapid chemi-cal reaction leads to nucleation burst The coagulation ofTiO2 monomers leads to an increase in the average particlesize and a decrease in particle concentration
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
21 Population Balance Model
The population balance equation consists of the followingnonlinear partial integro-differential equation22
n
t+ Gv z xn
v+ cz
n
zminus Ivlowastvminusvlowast
= 12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (2)
The first term on the left hand side of Eq (2) describethe change in the number concentration of particle vol-ume interval v v+dv and in the spatial interval z z+dznv z t denotes the particle size distribution function vis particle volume t is time z isin 0L is the spatial coor-dinate L is the length of the process The second termon the left hand side gives the loss or gain of particles bycondensational growth the third term on the left hand sidewhich is cznz corresponds to the convective trans-port of aerosol particles at fluid velocity cz and the fourthterm on the left hand side accounts for the formation ofnew particles of critical volume vlowast by nucleation rate I Ivlowastvminusvlowast also accounts for gain and loss of particlesby condensation Gv z x Ivlowast and v vminus v x arethe nonlinear scalar functions and is the standard Diracfunction The mass and energy balance model which pre-dicts the spatio-temporal evolution of the concentrations ofspecies and temperature of the gas phase given by Ashishand Panagiotis23 has the following form
dx
dt= A
dx
dz+ f x+ gxbzut
+ Aint
0a v xdv (3)
where xz t is an n-dimensional vector of state variablesthat depends on space and time A A are constant matri-ces f x gx a v x are nonlinear vector functionsut is the axially distributed manipulated input (eg walltemperatures Tw1 and Tw2 and bz is a known functionwhich determines how control action ut is distributedin space The last term on the right hand side of Eq (3)accounts for mass and heat transfer from the continuousphase to all the particles in population The gain and lossof particles by Brownian coagulation is described by thefirst and second term on the right hand side of Eq (2)respectively
12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (4)
Gv z x and are the condensational growth andcollision frequency function respectively for which two
different expressions are used for free molecule size andcontinuum size regimes14 The free molecule size regimetakes the following form
GFMx v z= B1v13Sminus1 where
B1 = 3613v1nskBT 2m112
FMx v z= B2
(1v+ 1
v
)12
v13+ v132
B2 = 34166kBTv1m112 (5)
And the continuum size regime takes the followingform
GCx v z= B3v13Sminus1 where
B3 = 48213Dv1ns D = 8kBT m1123
C = B4
(Cv
v13+ Cv
v13
)v13+ v13 B4 =
2kBT3
(6)
In Eqs (5) and (6) T is the temperature S is the sat-uration ratioD is the condensable vapour diffusivity isthe mean free path of the gas is the fluid viscosity ns
is the monomer concentration at saturation (ns = PskBT where Ps is the saturation pressure) m1 is the monomermass v1 is the monomer volume r is the particle radiusCv= 1+B5r is the Cunningham correction factor andB5 = 113257 Lastly the nucleation rate Ivlowast is assumedto follow the classical Becker-Doring theory given by theexpression below (Pratisinis)24
I = n2s s1kBT 2m1
12S22913
times12sum
expminusklowastInS2 (7)
where s1 is the monomer surface area and klowast is the numberof monomer in critical nucleus and is given by
klowast =
6
(4sum
InS
)3
where
sum = v231 kBT and is the surface tension (8)
3 LOGNORMAL AEROSOL SIZEDISTRIBUTION
The population balance model in Eq (2) is highly complexand does not allow the direct use for numerical com-putation of the size distribution in real-time To over-come this problem and to accelerate the computationsmethod of moments was employed to reduce the popu-lation balance model to a set of ODEs for the momentsof the size distribution In order to describe the spatio-temporal evolution of the three leading moments of thevolume distribution (which describes the exact evolutionof the lognormal aerosol size) a lognormal function wasemployed in moment model which was applied to popu-lation balance model
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
31 Moment Model
We assumed that the aerosol size distribution can beadequately represented by lognormal function which isdescribed as
nv z t= 13v
1radic2In
exp[minus In2vvg
18In2
](9)
where vg is the geometric average particle volume and is the standard deviation The kth moment of the distribu-tion is defined as
Mkt=int
0vknv z tdv (10)
According to the properties of a lognormal functionany moment can be written in terms of M0 vg and asfollows
Mk =M0vkg exp
(92k2In2
)(11)
If Eq (11) is written for k= 01 and 2 then vg and canexactly be expressed in terms of the first three momentsof the distribution according to the following relations
In2 = 19In
(M0M2
M21
)and vg =
M21
M320 M
122
(12)
In this subsection the ODEs describing the temporal evo-lution of the three leading moments of the size distributionfor the free molecule size and continuum size regime arepresented
311 Free Molecule Size Regime
The ODE system that describes the spatio-temporal sizedistribution of the kth moment of the aerosol size distri-bution is computed by substituting Eq (5) into Eq (2)multiplying by vk and integrating over all particle sizesThat gives the temporal evolution of the zeroth momentwhich is affected by nucleation and coagulation
dM0
dt= minuscz
dMo
dz+I
minusB2b0M0M16+2M13Mminus16+M23Mminus12 (13)
where the coefficient b0 is used for the relationship
(1v+ 1
v
)12
= b0
(1
v12+ 1
v12
)(14)
and it was computed by the expression b0 = 013633+0130922minus0130223 in publication of Pratisinis24 The evo-lution of M1 (aerosol volume) which is affected by con-densation is given by
dM1
dt=minuscz
dM1
dz+ Ivlowast +B1Sminus1M23 (15)
And the second moment M2 depends on nucleation andcoagulation according to the formula
dM2
dt= minuscz
dM2
dz+ Ivlowast2+2B1Sminus1M53
+2b2B2M76M1+2M43M56+M12M53 (16)
where b2 is used as b0 but for coagulation kernel of thesecond moment b2 is computed by the expression b2 =01339+0135 minus0132142+0130293 (Pratisinis)24
312 Continuum Size Regime
The spatio-temporal evolution of the kth moment of theaerosol size distribution in the continuum regime is com-puted by substituting Eq (6) into Eq (2) multiplying byvk and integrating over all particle sizes gives the temporalevolution of the zeroth moment M0 M1 and M2
dM0
dt
=minusczdM0
dz+ I minusB4
[M2
0 +M13Mminus13+B5
(43
)13
middot M0Mminus13+M13Mminus23
](17)
dM1
dt=minuscz
dM1
dz+ Ivlowast +B3Sminus1M13 (18)
dM2
dt=minuscz
dM2
dz+ Ivlowast2+2B3Sminus1M43
+2B4
[M2
1M43M23+B5
(43
)13
middot M1M23+M13M43
](19)
4 ANALYSIS OF RESULTS ANDDISCUSSION
This section describes the application of moment modelof the aerosol flow reactor for the purpose of nonlinearcontrol of the reactor Under the assumption of lognor-mal aerosol size distribution the mathematical model thatdescribes the evolution of the first three moments of dis-tribution together with the monomer and reactant concen-tration and temperature takes the following form
dN
d=minusczl
dN
dz+ I prime minusN 2
dV
d=minusczl
dV
dz+ I primeklowast +Sminus1N
dV2
d=minusczl
dV2
dz+ I primeklowast2+2Sminus1V +2V 2
dS
d=minusczl
dS
dz+CC1C2minus I primeklowast minusSminus1N
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
dC1
d=minusczl
dC1
dminusA1C1C2
dC2
d=minusczl
dC2
dminusA2C1C2
dT
d=minusvzl
dT
dz+BC1C2T +ET Tw minus T (20)
= FMCFM +C
C = K
[1+ expIn2+B5Kn1
r primeg
times exp(12In2
)1+ exp2In2
]
FM = rprime12g b0
[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)](21)
= FMC
FM +C
FM = vprime23g exp2In2 and
C = 4Kn1
3vprime13g exp
(12In2
)(22)
= FMC
FM +C
C = 4Kn1
3vprime13g exp
(72In2
)and
FM = vprime23g exp8In2 (23)
= FMCFM + C
FM = r prime12g b2 exp(32In2
)
times[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)]
C = K
[1+ expIn2+B5Kn1
r primegtimes exp(minus12In2
)
times(1+ expminus2In2
)](24)
where C1 and C2 are the dimensionless concentrations ofthe reactants Tw T are the dimensionless reactor and walltemperature respectively A1 A2 B C E are the dimen-sionless quantities Table II gives the process parametersused in the simulationDimensionless quantities for the model of Eq (20)
according to Ashish and Panagiotis23
A1 = kP0y20RT0 A2 = kP0y10RT0
B = P0kHRy10y20RT20 CP Ci = yiy10T
C = Navky10y20P0RT02ns0 E = 4URT0DCpP0
Table I Dimensionless variables by Ashish and Panagioti
N =M0ns V =M1nsv1 Aerosol concentration and volumeV2 =M2nsv
21 Second aerosol moment
= 2m1kBT 12nss1 Characteristic time for particle growth
K = 2kBT 3ns Coagulation coefficient and Nucleation rateI prime = Ins
Kn1= r1 Knudsen number
vprimeg = vgv1 Dimensionless geometric volumer primeg = rgr1 Dimensionless geometric radiusz= zL Dimensionless distanceczl = czL = t Dimensionless velocity and time = t Dimensionless time
Source Reprinted with permission from [23] K Ashish and D C PanagiotisChem Engng Sci 54 2669 (1999) copy 1999 Elsevier
T = T T0 and Tw = TwT0
In this chapter the process model of the equations inEq (20) was numerically solved in by using simulinka part of (Matlab) soft ware environment it is an inter-active computing package for simulating and analysingdifferential equations mathematical models and dynamicsystems The computation was done using a multi stepsolver known as ode 15s (stiffNDF) solver ode 15s isa variable-order solver based on the numerical differenti-ation formulas (NDFs) Optionally it uses the backwarddifferential formulas BDFs Sergey25 Table II shows theprocess variable parameters used for solving the processmodel of Eq (20)Since the objective of this study is to control nanopar-
ticle growth with desired particle distribution in high tem-perature reactor we studied the effect of wall temperature
Table II Process model parameters for the simulation study
L= 1135 m Reactor lengthD = 01301 m Reactor diameterP0 = 101000 pa Process pressureT0 = 2000 K Inlet temperaturey10 = 0134 Inlet mole fractions of O2
y20 = 0136 Inlet mole fractions of TiCl4U = 160 J mminus2 sminus1 Kminus1 Overall coefficient of heat transferHR = 88000 J molminus1 Heat of reactionCP = 16151325 Jmolminus1Kminus1 Heat capacity of process fluidMWg = 14130times10minus3 kg molminus1 Mol wt of process fluidK = 11134 m3 molminus1sminus1 Reaction rate constant= 6137times10minus5 kg mminus1 sminus1 Viscosity of process fluidlogPsmmHg PVT relation
=minus4644T +013906 logTminus01300162T +913004
= 01308 N mminus1 Surface tensionv1 = 51333times10minus29 m3 Monomer volumeNav = 613023times1023molminus1 Avogadrorsquos constantR= 813314 J molminus1 Kminus1 Universal gas constantkB = 11338times10minus23 JKminus1 Boltzmannrsquos constant = 0135 Sigmarg = 1135e06 m Geometric radiusr1 = 0135e06 m Monomer radiusvg = dp36 Geometric volumes0 = 1131587Eminus19 m Monomer surface areaTw = 800 K 500 K Wall temperatureRe = 2000 Reynolds number
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
aggregation nucleation) Meimaroglou et al10 used thismethodJames and Panagiotis11 proposed a finite-dimensional
approximation and control of nonlinear parabolic partialdifferential equation (PDE) systems by combiningGalerkinrsquos method with the concept of approximate initialmanifolds known as non-linear Galerkinrsquos method Aleckand Costas12 used orthogonal collection of finite elementmethods to solve a continuous form of general popula-tion balance equation (PBE) Hailian et al13 developeda predictive controller for parabolic convectionndashdiffusionndashreaction systems operating in convection-dominatedregimes using the combination of finite differencesapproximation of the diffusion term and the method ofcharacteristics Dan et al14 designed a control algorithmon the basis of finite dimensional models to capture thedominant dynamics of particulate processes The leastsquares method (LSM) is a well-established technique forsolving a wide range of mathematical problems The basicidea in the LSM is to minimize the integral of the squareof the residual over the computational domain in thecase when the exact solutions are sufficiently smooth theconvergence rate is exponential Daora and Jakobosen15
applied the least square method to solve the populationbalance equationThe method of moments (MOM) is computationally the
most efficient approach to obtain a numerical approxi-mation to the moments of population balance For thisreason this method is often used when simulating prob-lems where transport of particles in a flow with com-plex geometry is essential In the area of nanoparticlemodelling two techniques have been used so far Oneof the techniques is the quadrature method of moments(QMOM) which is a more recent technique Dorao andJakobson16 derived (QMOM) in two ways ie the stan-dard quadrature method of moments which is a numer-ical closure for method of moments (MOM) and themethod of moments in the method of weighted residuals(MOM-MWR)The second technique is the method of moment
with internal closure (MOMIC) Diemer and Ehrman17
developed a design for the comparison of reconstructeddistributions from moment with direct calculation via sec-tional method The design was used to probe sensitiv-ity of distribution reconstruction and problem solution tomodel size for both the MOMIC and QMOM approachesBarret and Webb18 compared some approximation meth-ods for solving the aerosol dynamic equation The methodscompared are quadrature method of moments the finiteelement method (FEM) Luguerre quadrature and Associ-ated Laguerre quadrature Suddah and Mark19 also usedquadrature method of moments to solve the problem ofobtaining closure of the moment equations for the coag-ulation growth diffusion and thermophoretic terms to beexpressed in their original forms
In the first instance MOMIC has been developed todescribe the formation and oxidation of soot particlesIn its early form the method is based on univariatedescription of spherical soot particles in the free molec-ular regime An alternative approach for obtaining themoments of the PSD is the quadrature method of moments(QMOM) In this method the moments are calculatedassuming the PSD can be represented as weighted multi-dimensional Dirac delta function The weights and thenodes are then chosen to satisfy the transport equationsfor the moments of the PSD The advantage of thisapproach is that due to the choice of delta functions thereexists no closure problem Gerber and Mousavi20 appliedthe quadrature method of moments to the polydisperseddroplet spectrum in transonic steam flows with primaryand secondary nucleationThe method of moments (MOM) is computationally the
most efficient approach to obtain a numerical approxima-tion to the moments of population balance For this rea-son this method is often used when simulating problemswhere transport of particles in a flow with complex geome-try is essential The dominant dynamic behaviour of manyaerosol processes can be accurately captured by a modelthat describes the evolution of the three leading momentsof aerosol size distribution21 The aim of this work isto investigate the sensitivity of the process model to theinfluence of disturbances and to use the reduced popu-lation balance model for the control of nanoparticle sizedistribution The measurement of particle size distributionis a distinguishing feature in production of nanoparticlesbecause the particle size provides the critical link betweenthe product quality indices and the operating variables2
Thus the ability to effectively control the shape of thePSD is essential for regulating the end product quality ofthe process Many applications require a close control ofthis distributed particle length scale in order to achieve thehighest performance
2 PROCESS DESCRIPTION
The premixed preheated reactants (titanium tetrachlorideand oxygen gas) are injected into the reactor where fol-lowing exothermic reaction takes place The products ofthese reactions are titania monomers and chlorine gas
TiCl4g+O2grarr TiO2s+2Cl2g (1)
The size of a single TiO2 molecule (monomer) is largerthan the thermodynamic critical cluster size As a resultthe difference between chemical reaction and nucleationcannot be noticed thereby implying that the rapid chemi-cal reaction leads to nucleation burst The coagulation ofTiO2 monomers leads to an increase in the average particlesize and a decrease in particle concentration
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
21 Population Balance Model
The population balance equation consists of the followingnonlinear partial integro-differential equation22
n
t+ Gv z xn
v+ cz
n
zminus Ivlowastvminusvlowast
= 12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (2)
The first term on the left hand side of Eq (2) describethe change in the number concentration of particle vol-ume interval v v+dv and in the spatial interval z z+dznv z t denotes the particle size distribution function vis particle volume t is time z isin 0L is the spatial coor-dinate L is the length of the process The second termon the left hand side gives the loss or gain of particles bycondensational growth the third term on the left hand sidewhich is cznz corresponds to the convective trans-port of aerosol particles at fluid velocity cz and the fourthterm on the left hand side accounts for the formation ofnew particles of critical volume vlowast by nucleation rate I Ivlowastvminusvlowast also accounts for gain and loss of particlesby condensation Gv z x Ivlowast and v vminus v x arethe nonlinear scalar functions and is the standard Diracfunction The mass and energy balance model which pre-dicts the spatio-temporal evolution of the concentrations ofspecies and temperature of the gas phase given by Ashishand Panagiotis23 has the following form
dx
dt= A
dx
dz+ f x+ gxbzut
+ Aint
0a v xdv (3)
where xz t is an n-dimensional vector of state variablesthat depends on space and time A A are constant matri-ces f x gx a v x are nonlinear vector functionsut is the axially distributed manipulated input (eg walltemperatures Tw1 and Tw2 and bz is a known functionwhich determines how control action ut is distributedin space The last term on the right hand side of Eq (3)accounts for mass and heat transfer from the continuousphase to all the particles in population The gain and lossof particles by Brownian coagulation is described by thefirst and second term on the right hand side of Eq (2)respectively
12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (4)
Gv z x and are the condensational growth andcollision frequency function respectively for which two
different expressions are used for free molecule size andcontinuum size regimes14 The free molecule size regimetakes the following form
GFMx v z= B1v13Sminus1 where
B1 = 3613v1nskBT 2m112
FMx v z= B2
(1v+ 1
v
)12
v13+ v132
B2 = 34166kBTv1m112 (5)
And the continuum size regime takes the followingform
GCx v z= B3v13Sminus1 where
B3 = 48213Dv1ns D = 8kBT m1123
C = B4
(Cv
v13+ Cv
v13
)v13+ v13 B4 =
2kBT3
(6)
In Eqs (5) and (6) T is the temperature S is the sat-uration ratioD is the condensable vapour diffusivity isthe mean free path of the gas is the fluid viscosity ns
is the monomer concentration at saturation (ns = PskBT where Ps is the saturation pressure) m1 is the monomermass v1 is the monomer volume r is the particle radiusCv= 1+B5r is the Cunningham correction factor andB5 = 113257 Lastly the nucleation rate Ivlowast is assumedto follow the classical Becker-Doring theory given by theexpression below (Pratisinis)24
I = n2s s1kBT 2m1
12S22913
times12sum
expminusklowastInS2 (7)
where s1 is the monomer surface area and klowast is the numberof monomer in critical nucleus and is given by
klowast =
6
(4sum
InS
)3
where
sum = v231 kBT and is the surface tension (8)
3 LOGNORMAL AEROSOL SIZEDISTRIBUTION
The population balance model in Eq (2) is highly complexand does not allow the direct use for numerical com-putation of the size distribution in real-time To over-come this problem and to accelerate the computationsmethod of moments was employed to reduce the popu-lation balance model to a set of ODEs for the momentsof the size distribution In order to describe the spatio-temporal evolution of the three leading moments of thevolume distribution (which describes the exact evolutionof the lognormal aerosol size) a lognormal function wasemployed in moment model which was applied to popu-lation balance model
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
31 Moment Model
We assumed that the aerosol size distribution can beadequately represented by lognormal function which isdescribed as
nv z t= 13v
1radic2In
exp[minus In2vvg
18In2
](9)
where vg is the geometric average particle volume and is the standard deviation The kth moment of the distribu-tion is defined as
Mkt=int
0vknv z tdv (10)
According to the properties of a lognormal functionany moment can be written in terms of M0 vg and asfollows
Mk =M0vkg exp
(92k2In2
)(11)
If Eq (11) is written for k= 01 and 2 then vg and canexactly be expressed in terms of the first three momentsof the distribution according to the following relations
In2 = 19In
(M0M2
M21
)and vg =
M21
M320 M
122
(12)
In this subsection the ODEs describing the temporal evo-lution of the three leading moments of the size distributionfor the free molecule size and continuum size regime arepresented
311 Free Molecule Size Regime
The ODE system that describes the spatio-temporal sizedistribution of the kth moment of the aerosol size distri-bution is computed by substituting Eq (5) into Eq (2)multiplying by vk and integrating over all particle sizesThat gives the temporal evolution of the zeroth momentwhich is affected by nucleation and coagulation
dM0
dt= minuscz
dMo
dz+I
minusB2b0M0M16+2M13Mminus16+M23Mminus12 (13)
where the coefficient b0 is used for the relationship
(1v+ 1
v
)12
= b0
(1
v12+ 1
v12
)(14)
and it was computed by the expression b0 = 013633+0130922minus0130223 in publication of Pratisinis24 The evo-lution of M1 (aerosol volume) which is affected by con-densation is given by
dM1
dt=minuscz
dM1
dz+ Ivlowast +B1Sminus1M23 (15)
And the second moment M2 depends on nucleation andcoagulation according to the formula
dM2
dt= minuscz
dM2
dz+ Ivlowast2+2B1Sminus1M53
+2b2B2M76M1+2M43M56+M12M53 (16)
where b2 is used as b0 but for coagulation kernel of thesecond moment b2 is computed by the expression b2 =01339+0135 minus0132142+0130293 (Pratisinis)24
312 Continuum Size Regime
The spatio-temporal evolution of the kth moment of theaerosol size distribution in the continuum regime is com-puted by substituting Eq (6) into Eq (2) multiplying byvk and integrating over all particle sizes gives the temporalevolution of the zeroth moment M0 M1 and M2
dM0
dt
=minusczdM0
dz+ I minusB4
[M2
0 +M13Mminus13+B5
(43
)13
middot M0Mminus13+M13Mminus23
](17)
dM1
dt=minuscz
dM1
dz+ Ivlowast +B3Sminus1M13 (18)
dM2
dt=minuscz
dM2
dz+ Ivlowast2+2B3Sminus1M43
+2B4
[M2
1M43M23+B5
(43
)13
middot M1M23+M13M43
](19)
4 ANALYSIS OF RESULTS ANDDISCUSSION
This section describes the application of moment modelof the aerosol flow reactor for the purpose of nonlinearcontrol of the reactor Under the assumption of lognor-mal aerosol size distribution the mathematical model thatdescribes the evolution of the first three moments of dis-tribution together with the monomer and reactant concen-tration and temperature takes the following form
dN
d=minusczl
dN
dz+ I prime minusN 2
dV
d=minusczl
dV
dz+ I primeklowast +Sminus1N
dV2
d=minusczl
dV2
dz+ I primeklowast2+2Sminus1V +2V 2
dS
d=minusczl
dS
dz+CC1C2minus I primeklowast minusSminus1N
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
dC1
d=minusczl
dC1
dminusA1C1C2
dC2
d=minusczl
dC2
dminusA2C1C2
dT
d=minusvzl
dT
dz+BC1C2T +ET Tw minus T (20)
= FMCFM +C
C = K
[1+ expIn2+B5Kn1
r primeg
times exp(12In2
)1+ exp2In2
]
FM = rprime12g b0
[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)](21)
= FMC
FM +C
FM = vprime23g exp2In2 and
C = 4Kn1
3vprime13g exp
(12In2
)(22)
= FMC
FM +C
C = 4Kn1
3vprime13g exp
(72In2
)and
FM = vprime23g exp8In2 (23)
= FMCFM + C
FM = r prime12g b2 exp(32In2
)
times[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)]
C = K
[1+ expIn2+B5Kn1
r primegtimes exp(minus12In2
)
times(1+ expminus2In2
)](24)
where C1 and C2 are the dimensionless concentrations ofthe reactants Tw T are the dimensionless reactor and walltemperature respectively A1 A2 B C E are the dimen-sionless quantities Table II gives the process parametersused in the simulationDimensionless quantities for the model of Eq (20)
according to Ashish and Panagiotis23
A1 = kP0y20RT0 A2 = kP0y10RT0
B = P0kHRy10y20RT20 CP Ci = yiy10T
C = Navky10y20P0RT02ns0 E = 4URT0DCpP0
Table I Dimensionless variables by Ashish and Panagioti
N =M0ns V =M1nsv1 Aerosol concentration and volumeV2 =M2nsv
21 Second aerosol moment
= 2m1kBT 12nss1 Characteristic time for particle growth
K = 2kBT 3ns Coagulation coefficient and Nucleation rateI prime = Ins
Kn1= r1 Knudsen number
vprimeg = vgv1 Dimensionless geometric volumer primeg = rgr1 Dimensionless geometric radiusz= zL Dimensionless distanceczl = czL = t Dimensionless velocity and time = t Dimensionless time
Source Reprinted with permission from [23] K Ashish and D C PanagiotisChem Engng Sci 54 2669 (1999) copy 1999 Elsevier
T = T T0 and Tw = TwT0
In this chapter the process model of the equations inEq (20) was numerically solved in by using simulinka part of (Matlab) soft ware environment it is an inter-active computing package for simulating and analysingdifferential equations mathematical models and dynamicsystems The computation was done using a multi stepsolver known as ode 15s (stiffNDF) solver ode 15s isa variable-order solver based on the numerical differenti-ation formulas (NDFs) Optionally it uses the backwarddifferential formulas BDFs Sergey25 Table II shows theprocess variable parameters used for solving the processmodel of Eq (20)Since the objective of this study is to control nanopar-
ticle growth with desired particle distribution in high tem-perature reactor we studied the effect of wall temperature
Table II Process model parameters for the simulation study
L= 1135 m Reactor lengthD = 01301 m Reactor diameterP0 = 101000 pa Process pressureT0 = 2000 K Inlet temperaturey10 = 0134 Inlet mole fractions of O2
y20 = 0136 Inlet mole fractions of TiCl4U = 160 J mminus2 sminus1 Kminus1 Overall coefficient of heat transferHR = 88000 J molminus1 Heat of reactionCP = 16151325 Jmolminus1Kminus1 Heat capacity of process fluidMWg = 14130times10minus3 kg molminus1 Mol wt of process fluidK = 11134 m3 molminus1sminus1 Reaction rate constant= 6137times10minus5 kg mminus1 sminus1 Viscosity of process fluidlogPsmmHg PVT relation
=minus4644T +013906 logTminus01300162T +913004
= 01308 N mminus1 Surface tensionv1 = 51333times10minus29 m3 Monomer volumeNav = 613023times1023molminus1 Avogadrorsquos constantR= 813314 J molminus1 Kminus1 Universal gas constantkB = 11338times10minus23 JKminus1 Boltzmannrsquos constant = 0135 Sigmarg = 1135e06 m Geometric radiusr1 = 0135e06 m Monomer radiusvg = dp36 Geometric volumes0 = 1131587Eminus19 m Monomer surface areaTw = 800 K 500 K Wall temperatureRe = 2000 Reynolds number
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
14 J Comput Theor Nanosci 8 8ndash16 2011
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
J Comput Theor Nanosci 8 8ndash16 2011 15
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
21 Population Balance Model
The population balance equation consists of the followingnonlinear partial integro-differential equation22
n
t+ Gv z xn
v+ cz
n
zminus Ivlowastvminusvlowast
= 12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (2)
The first term on the left hand side of Eq (2) describethe change in the number concentration of particle vol-ume interval v v+dv and in the spatial interval z z+dznv z t denotes the particle size distribution function vis particle volume t is time z isin 0L is the spatial coor-dinate L is the length of the process The second termon the left hand side gives the loss or gain of particles bycondensational growth the third term on the left hand sidewhich is cznz corresponds to the convective trans-port of aerosol particles at fluid velocity cz and the fourthterm on the left hand side accounts for the formation ofnew particles of critical volume vlowast by nucleation rate I Ivlowastvminusvlowast also accounts for gain and loss of particlesby condensation Gv z x Ivlowast and v vminus v x arethe nonlinear scalar functions and is the standard Diracfunction The mass and energy balance model which pre-dicts the spatio-temporal evolution of the concentrations ofspecies and temperature of the gas phase given by Ashishand Panagiotis23 has the following form
dx
dt= A
dx
dz+ f x+ gxbzut
+ Aint
0a v xdv (3)
where xz t is an n-dimensional vector of state variablesthat depends on space and time A A are constant matri-ces f x gx a v x are nonlinear vector functionsut is the axially distributed manipulated input (eg walltemperatures Tw1 and Tw2 and bz is a known functionwhich determines how control action ut is distributedin space The last term on the right hand side of Eq (3)accounts for mass and heat transfer from the continuousphase to all the particles in population The gain and lossof particles by Brownian coagulation is described by thefirst and second term on the right hand side of Eq (2)respectively
12
int v
ov vminus v xnv z tnvminus v z tdv
minusnv z tint
ov vnv z tdv (4)
Gv z x and are the condensational growth andcollision frequency function respectively for which two
different expressions are used for free molecule size andcontinuum size regimes14 The free molecule size regimetakes the following form
GFMx v z= B1v13Sminus1 where
B1 = 3613v1nskBT 2m112
FMx v z= B2
(1v+ 1
v
)12
v13+ v132
B2 = 34166kBTv1m112 (5)
And the continuum size regime takes the followingform
GCx v z= B3v13Sminus1 where
B3 = 48213Dv1ns D = 8kBT m1123
C = B4
(Cv
v13+ Cv
v13
)v13+ v13 B4 =
2kBT3
(6)
In Eqs (5) and (6) T is the temperature S is the sat-uration ratioD is the condensable vapour diffusivity isthe mean free path of the gas is the fluid viscosity ns
is the monomer concentration at saturation (ns = PskBT where Ps is the saturation pressure) m1 is the monomermass v1 is the monomer volume r is the particle radiusCv= 1+B5r is the Cunningham correction factor andB5 = 113257 Lastly the nucleation rate Ivlowast is assumedto follow the classical Becker-Doring theory given by theexpression below (Pratisinis)24
I = n2s s1kBT 2m1
12S22913
times12sum
expminusklowastInS2 (7)
where s1 is the monomer surface area and klowast is the numberof monomer in critical nucleus and is given by
klowast =
6
(4sum
InS
)3
where
sum = v231 kBT and is the surface tension (8)
3 LOGNORMAL AEROSOL SIZEDISTRIBUTION
The population balance model in Eq (2) is highly complexand does not allow the direct use for numerical com-putation of the size distribution in real-time To over-come this problem and to accelerate the computationsmethod of moments was employed to reduce the popu-lation balance model to a set of ODEs for the momentsof the size distribution In order to describe the spatio-temporal evolution of the three leading moments of thevolume distribution (which describes the exact evolutionof the lognormal aerosol size) a lognormal function wasemployed in moment model which was applied to popu-lation balance model
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
31 Moment Model
We assumed that the aerosol size distribution can beadequately represented by lognormal function which isdescribed as
nv z t= 13v
1radic2In
exp[minus In2vvg
18In2
](9)
where vg is the geometric average particle volume and is the standard deviation The kth moment of the distribu-tion is defined as
Mkt=int
0vknv z tdv (10)
According to the properties of a lognormal functionany moment can be written in terms of M0 vg and asfollows
Mk =M0vkg exp
(92k2In2
)(11)
If Eq (11) is written for k= 01 and 2 then vg and canexactly be expressed in terms of the first three momentsof the distribution according to the following relations
In2 = 19In
(M0M2
M21
)and vg =
M21
M320 M
122
(12)
In this subsection the ODEs describing the temporal evo-lution of the three leading moments of the size distributionfor the free molecule size and continuum size regime arepresented
311 Free Molecule Size Regime
The ODE system that describes the spatio-temporal sizedistribution of the kth moment of the aerosol size distri-bution is computed by substituting Eq (5) into Eq (2)multiplying by vk and integrating over all particle sizesThat gives the temporal evolution of the zeroth momentwhich is affected by nucleation and coagulation
dM0
dt= minuscz
dMo
dz+I
minusB2b0M0M16+2M13Mminus16+M23Mminus12 (13)
where the coefficient b0 is used for the relationship
(1v+ 1
v
)12
= b0
(1
v12+ 1
v12
)(14)
and it was computed by the expression b0 = 013633+0130922minus0130223 in publication of Pratisinis24 The evo-lution of M1 (aerosol volume) which is affected by con-densation is given by
dM1
dt=minuscz
dM1
dz+ Ivlowast +B1Sminus1M23 (15)
And the second moment M2 depends on nucleation andcoagulation according to the formula
dM2
dt= minuscz
dM2
dz+ Ivlowast2+2B1Sminus1M53
+2b2B2M76M1+2M43M56+M12M53 (16)
where b2 is used as b0 but for coagulation kernel of thesecond moment b2 is computed by the expression b2 =01339+0135 minus0132142+0130293 (Pratisinis)24
312 Continuum Size Regime
The spatio-temporal evolution of the kth moment of theaerosol size distribution in the continuum regime is com-puted by substituting Eq (6) into Eq (2) multiplying byvk and integrating over all particle sizes gives the temporalevolution of the zeroth moment M0 M1 and M2
dM0
dt
=minusczdM0
dz+ I minusB4
[M2
0 +M13Mminus13+B5
(43
)13
middot M0Mminus13+M13Mminus23
](17)
dM1
dt=minuscz
dM1
dz+ Ivlowast +B3Sminus1M13 (18)
dM2
dt=minuscz
dM2
dz+ Ivlowast2+2B3Sminus1M43
+2B4
[M2
1M43M23+B5
(43
)13
middot M1M23+M13M43
](19)
4 ANALYSIS OF RESULTS ANDDISCUSSION
This section describes the application of moment modelof the aerosol flow reactor for the purpose of nonlinearcontrol of the reactor Under the assumption of lognor-mal aerosol size distribution the mathematical model thatdescribes the evolution of the first three moments of dis-tribution together with the monomer and reactant concen-tration and temperature takes the following form
dN
d=minusczl
dN
dz+ I prime minusN 2
dV
d=minusczl
dV
dz+ I primeklowast +Sminus1N
dV2
d=minusczl
dV2
dz+ I primeklowast2+2Sminus1V +2V 2
dS
d=minusczl
dS
dz+CC1C2minus I primeklowast minusSminus1N
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
dC1
d=minusczl
dC1
dminusA1C1C2
dC2
d=minusczl
dC2
dminusA2C1C2
dT
d=minusvzl
dT
dz+BC1C2T +ET Tw minus T (20)
= FMCFM +C
C = K
[1+ expIn2+B5Kn1
r primeg
times exp(12In2
)1+ exp2In2
]
FM = rprime12g b0
[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)](21)
= FMC
FM +C
FM = vprime23g exp2In2 and
C = 4Kn1
3vprime13g exp
(12In2
)(22)
= FMC
FM +C
C = 4Kn1
3vprime13g exp
(72In2
)and
FM = vprime23g exp8In2 (23)
= FMCFM + C
FM = r prime12g b2 exp(32In2
)
times[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)]
C = K
[1+ expIn2+B5Kn1
r primegtimes exp(minus12In2
)
times(1+ expminus2In2
)](24)
where C1 and C2 are the dimensionless concentrations ofthe reactants Tw T are the dimensionless reactor and walltemperature respectively A1 A2 B C E are the dimen-sionless quantities Table II gives the process parametersused in the simulationDimensionless quantities for the model of Eq (20)
according to Ashish and Panagiotis23
A1 = kP0y20RT0 A2 = kP0y10RT0
B = P0kHRy10y20RT20 CP Ci = yiy10T
C = Navky10y20P0RT02ns0 E = 4URT0DCpP0
Table I Dimensionless variables by Ashish and Panagioti
N =M0ns V =M1nsv1 Aerosol concentration and volumeV2 =M2nsv
21 Second aerosol moment
= 2m1kBT 12nss1 Characteristic time for particle growth
K = 2kBT 3ns Coagulation coefficient and Nucleation rateI prime = Ins
Kn1= r1 Knudsen number
vprimeg = vgv1 Dimensionless geometric volumer primeg = rgr1 Dimensionless geometric radiusz= zL Dimensionless distanceczl = czL = t Dimensionless velocity and time = t Dimensionless time
Source Reprinted with permission from [23] K Ashish and D C PanagiotisChem Engng Sci 54 2669 (1999) copy 1999 Elsevier
T = T T0 and Tw = TwT0
In this chapter the process model of the equations inEq (20) was numerically solved in by using simulinka part of (Matlab) soft ware environment it is an inter-active computing package for simulating and analysingdifferential equations mathematical models and dynamicsystems The computation was done using a multi stepsolver known as ode 15s (stiffNDF) solver ode 15s isa variable-order solver based on the numerical differenti-ation formulas (NDFs) Optionally it uses the backwarddifferential formulas BDFs Sergey25 Table II shows theprocess variable parameters used for solving the processmodel of Eq (20)Since the objective of this study is to control nanopar-
ticle growth with desired particle distribution in high tem-perature reactor we studied the effect of wall temperature
Table II Process model parameters for the simulation study
L= 1135 m Reactor lengthD = 01301 m Reactor diameterP0 = 101000 pa Process pressureT0 = 2000 K Inlet temperaturey10 = 0134 Inlet mole fractions of O2
y20 = 0136 Inlet mole fractions of TiCl4U = 160 J mminus2 sminus1 Kminus1 Overall coefficient of heat transferHR = 88000 J molminus1 Heat of reactionCP = 16151325 Jmolminus1Kminus1 Heat capacity of process fluidMWg = 14130times10minus3 kg molminus1 Mol wt of process fluidK = 11134 m3 molminus1sminus1 Reaction rate constant= 6137times10minus5 kg mminus1 sminus1 Viscosity of process fluidlogPsmmHg PVT relation
=minus4644T +013906 logTminus01300162T +913004
= 01308 N mminus1 Surface tensionv1 = 51333times10minus29 m3 Monomer volumeNav = 613023times1023molminus1 Avogadrorsquos constantR= 813314 J molminus1 Kminus1 Universal gas constantkB = 11338times10minus23 JKminus1 Boltzmannrsquos constant = 0135 Sigmarg = 1135e06 m Geometric radiusr1 = 0135e06 m Monomer radiusvg = dp36 Geometric volumes0 = 1131587Eminus19 m Monomer surface areaTw = 800 K 500 K Wall temperatureRe = 2000 Reynolds number
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Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
16 J Comput Theor Nanosci 8 8ndash16 2011
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Application of Reduced Population Balance Model II Sadiku and Kolesnikov
31 Moment Model
We assumed that the aerosol size distribution can beadequately represented by lognormal function which isdescribed as
nv z t= 13v
1radic2In
exp[minus In2vvg
18In2
](9)
where vg is the geometric average particle volume and is the standard deviation The kth moment of the distribu-tion is defined as
Mkt=int
0vknv z tdv (10)
According to the properties of a lognormal functionany moment can be written in terms of M0 vg and asfollows
Mk =M0vkg exp
(92k2In2
)(11)
If Eq (11) is written for k= 01 and 2 then vg and canexactly be expressed in terms of the first three momentsof the distribution according to the following relations
In2 = 19In
(M0M2
M21
)and vg =
M21
M320 M
122
(12)
In this subsection the ODEs describing the temporal evo-lution of the three leading moments of the size distributionfor the free molecule size and continuum size regime arepresented
311 Free Molecule Size Regime
The ODE system that describes the spatio-temporal sizedistribution of the kth moment of the aerosol size distri-bution is computed by substituting Eq (5) into Eq (2)multiplying by vk and integrating over all particle sizesThat gives the temporal evolution of the zeroth momentwhich is affected by nucleation and coagulation
dM0
dt= minuscz
dMo
dz+I
minusB2b0M0M16+2M13Mminus16+M23Mminus12 (13)
where the coefficient b0 is used for the relationship
(1v+ 1
v
)12
= b0
(1
v12+ 1
v12
)(14)
and it was computed by the expression b0 = 013633+0130922minus0130223 in publication of Pratisinis24 The evo-lution of M1 (aerosol volume) which is affected by con-densation is given by
dM1
dt=minuscz
dM1
dz+ Ivlowast +B1Sminus1M23 (15)
And the second moment M2 depends on nucleation andcoagulation according to the formula
dM2
dt= minuscz
dM2
dz+ Ivlowast2+2B1Sminus1M53
+2b2B2M76M1+2M43M56+M12M53 (16)
where b2 is used as b0 but for coagulation kernel of thesecond moment b2 is computed by the expression b2 =01339+0135 minus0132142+0130293 (Pratisinis)24
312 Continuum Size Regime
The spatio-temporal evolution of the kth moment of theaerosol size distribution in the continuum regime is com-puted by substituting Eq (6) into Eq (2) multiplying byvk and integrating over all particle sizes gives the temporalevolution of the zeroth moment M0 M1 and M2
dM0
dt
=minusczdM0
dz+ I minusB4
[M2
0 +M13Mminus13+B5
(43
)13
middot M0Mminus13+M13Mminus23
](17)
dM1
dt=minuscz
dM1
dz+ Ivlowast +B3Sminus1M13 (18)
dM2
dt=minuscz
dM2
dz+ Ivlowast2+2B3Sminus1M43
+2B4
[M2
1M43M23+B5
(43
)13
middot M1M23+M13M43
](19)
4 ANALYSIS OF RESULTS ANDDISCUSSION
This section describes the application of moment modelof the aerosol flow reactor for the purpose of nonlinearcontrol of the reactor Under the assumption of lognor-mal aerosol size distribution the mathematical model thatdescribes the evolution of the first three moments of dis-tribution together with the monomer and reactant concen-tration and temperature takes the following form
dN
d=minusczl
dN
dz+ I prime minusN 2
dV
d=minusczl
dV
dz+ I primeklowast +Sminus1N
dV2
d=minusczl
dV2
dz+ I primeklowast2+2Sminus1V +2V 2
dS
d=minusczl
dS
dz+CC1C2minus I primeklowast minusSminus1N
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Sadiku and Kolesnikov Application of Reduced Population Balance Model II
dC1
d=minusczl
dC1
dminusA1C1C2
dC2
d=minusczl
dC2
dminusA2C1C2
dT
d=minusvzl
dT
dz+BC1C2T +ET Tw minus T (20)
= FMCFM +C
C = K
[1+ expIn2+B5Kn1
r primeg
times exp(12In2
)1+ exp2In2
]
FM = rprime12g b0
[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)](21)
= FMC
FM +C
FM = vprime23g exp2In2 and
C = 4Kn1
3vprime13g exp
(12In2
)(22)
= FMC
FM +C
C = 4Kn1
3vprime13g exp
(72In2
)and
FM = vprime23g exp8In2 (23)
= FMCFM + C
FM = r prime12g b2 exp(32In2
)
times[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)]
C = K
[1+ expIn2+B5Kn1
r primegtimes exp(minus12In2
)
times(1+ expminus2In2
)](24)
where C1 and C2 are the dimensionless concentrations ofthe reactants Tw T are the dimensionless reactor and walltemperature respectively A1 A2 B C E are the dimen-sionless quantities Table II gives the process parametersused in the simulationDimensionless quantities for the model of Eq (20)
according to Ashish and Panagiotis23
A1 = kP0y20RT0 A2 = kP0y10RT0
B = P0kHRy10y20RT20 CP Ci = yiy10T
C = Navky10y20P0RT02ns0 E = 4URT0DCpP0
Table I Dimensionless variables by Ashish and Panagioti
N =M0ns V =M1nsv1 Aerosol concentration and volumeV2 =M2nsv
21 Second aerosol moment
= 2m1kBT 12nss1 Characteristic time for particle growth
K = 2kBT 3ns Coagulation coefficient and Nucleation rateI prime = Ins
Kn1= r1 Knudsen number
vprimeg = vgv1 Dimensionless geometric volumer primeg = rgr1 Dimensionless geometric radiusz= zL Dimensionless distanceczl = czL = t Dimensionless velocity and time = t Dimensionless time
Source Reprinted with permission from [23] K Ashish and D C PanagiotisChem Engng Sci 54 2669 (1999) copy 1999 Elsevier
T = T T0 and Tw = TwT0
In this chapter the process model of the equations inEq (20) was numerically solved in by using simulinka part of (Matlab) soft ware environment it is an inter-active computing package for simulating and analysingdifferential equations mathematical models and dynamicsystems The computation was done using a multi stepsolver known as ode 15s (stiffNDF) solver ode 15s isa variable-order solver based on the numerical differenti-ation formulas (NDFs) Optionally it uses the backwarddifferential formulas BDFs Sergey25 Table II shows theprocess variable parameters used for solving the processmodel of Eq (20)Since the objective of this study is to control nanopar-
ticle growth with desired particle distribution in high tem-perature reactor we studied the effect of wall temperature
Table II Process model parameters for the simulation study
L= 1135 m Reactor lengthD = 01301 m Reactor diameterP0 = 101000 pa Process pressureT0 = 2000 K Inlet temperaturey10 = 0134 Inlet mole fractions of O2
y20 = 0136 Inlet mole fractions of TiCl4U = 160 J mminus2 sminus1 Kminus1 Overall coefficient of heat transferHR = 88000 J molminus1 Heat of reactionCP = 16151325 Jmolminus1Kminus1 Heat capacity of process fluidMWg = 14130times10minus3 kg molminus1 Mol wt of process fluidK = 11134 m3 molminus1sminus1 Reaction rate constant= 6137times10minus5 kg mminus1 sminus1 Viscosity of process fluidlogPsmmHg PVT relation
=minus4644T +013906 logTminus01300162T +913004
= 01308 N mminus1 Surface tensionv1 = 51333times10minus29 m3 Monomer volumeNav = 613023times1023molminus1 Avogadrorsquos constantR= 813314 J molminus1 Kminus1 Universal gas constantkB = 11338times10minus23 JKminus1 Boltzmannrsquos constant = 0135 Sigmarg = 1135e06 m Geometric radiusr1 = 0135e06 m Monomer radiusvg = dp36 Geometric volumes0 = 1131587Eminus19 m Monomer surface areaTw = 800 K 500 K Wall temperatureRe = 2000 Reynolds number
J Comput Theor Nanosci 8 8ndash16 2011 13
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IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
14 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
J Comput Theor Nanosci 8 8ndash16 2011 15
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
16 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Sadiku and Kolesnikov Application of Reduced Population Balance Model II
dC1
d=minusczl
dC1
dminusA1C1C2
dC2
d=minusczl
dC2
dminusA2C1C2
dT
d=minusvzl
dT
dz+BC1C2T +ET Tw minus T (20)
= FMCFM +C
C = K
[1+ expIn2+B5Kn1
r primeg
times exp(12In2
)1+ exp2In2
]
FM = rprime12g b0
[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)](21)
= FMC
FM +C
FM = vprime23g exp2In2 and
C = 4Kn1
3vprime13g exp
(12In2
)(22)
= FMC
FM +C
C = 4Kn1
3vprime13g exp
(72In2
)and
FM = vprime23g exp8In2 (23)
= FMCFM + C
FM = r prime12g b2 exp(32In2
)
times[exp
(258In2
)+2 exp
(58In2
)
+ exp(18In2
)]
C = K
[1+ expIn2+B5Kn1
r primegtimes exp(minus12In2
)
times(1+ expminus2In2
)](24)
where C1 and C2 are the dimensionless concentrations ofthe reactants Tw T are the dimensionless reactor and walltemperature respectively A1 A2 B C E are the dimen-sionless quantities Table II gives the process parametersused in the simulationDimensionless quantities for the model of Eq (20)
according to Ashish and Panagiotis23
A1 = kP0y20RT0 A2 = kP0y10RT0
B = P0kHRy10y20RT20 CP Ci = yiy10T
C = Navky10y20P0RT02ns0 E = 4URT0DCpP0
Table I Dimensionless variables by Ashish and Panagioti
N =M0ns V =M1nsv1 Aerosol concentration and volumeV2 =M2nsv
21 Second aerosol moment
= 2m1kBT 12nss1 Characteristic time for particle growth
K = 2kBT 3ns Coagulation coefficient and Nucleation rateI prime = Ins
Kn1= r1 Knudsen number
vprimeg = vgv1 Dimensionless geometric volumer primeg = rgr1 Dimensionless geometric radiusz= zL Dimensionless distanceczl = czL = t Dimensionless velocity and time = t Dimensionless time
Source Reprinted with permission from [23] K Ashish and D C PanagiotisChem Engng Sci 54 2669 (1999) copy 1999 Elsevier
T = T T0 and Tw = TwT0
In this chapter the process model of the equations inEq (20) was numerically solved in by using simulinka part of (Matlab) soft ware environment it is an inter-active computing package for simulating and analysingdifferential equations mathematical models and dynamicsystems The computation was done using a multi stepsolver known as ode 15s (stiffNDF) solver ode 15s isa variable-order solver based on the numerical differenti-ation formulas (NDFs) Optionally it uses the backwarddifferential formulas BDFs Sergey25 Table II shows theprocess variable parameters used for solving the processmodel of Eq (20)Since the objective of this study is to control nanopar-
ticle growth with desired particle distribution in high tem-perature reactor we studied the effect of wall temperature
Table II Process model parameters for the simulation study
L= 1135 m Reactor lengthD = 01301 m Reactor diameterP0 = 101000 pa Process pressureT0 = 2000 K Inlet temperaturey10 = 0134 Inlet mole fractions of O2
y20 = 0136 Inlet mole fractions of TiCl4U = 160 J mminus2 sminus1 Kminus1 Overall coefficient of heat transferHR = 88000 J molminus1 Heat of reactionCP = 16151325 Jmolminus1Kminus1 Heat capacity of process fluidMWg = 14130times10minus3 kg molminus1 Mol wt of process fluidK = 11134 m3 molminus1sminus1 Reaction rate constant= 6137times10minus5 kg mminus1 sminus1 Viscosity of process fluidlogPsmmHg PVT relation
=minus4644T +013906 logTminus01300162T +913004
= 01308 N mminus1 Surface tensionv1 = 51333times10minus29 m3 Monomer volumeNav = 613023times1023molminus1 Avogadrorsquos constantR= 813314 J molminus1 Kminus1 Universal gas constantkB = 11338times10minus23 JKminus1 Boltzmannrsquos constant = 0135 Sigmarg = 1135e06 m Geometric radiusr1 = 0135e06 m Monomer radiusvg = dp36 Geometric volumes0 = 1131587Eminus19 m Monomer surface areaTw = 800 K 500 K Wall temperatureRe = 2000 Reynolds number
J Comput Theor Nanosci 8 8ndash16 2011 13
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
14 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
J Comput Theor Nanosci 8 8ndash16 2011 15
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
16 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 1 Average particle diameter as function of the wall temperature
on average particle diameter It is a variable that could beused in industry to control the aerosol size distributionThe effect of wall temperature on average particle diame-ter was investigated in the open-loop simulation in orderto use the model as a basis to synthesize feedback con-trollers which manipulated the input (wall temperatures)in order to achieve control of nanoparticle size and itsdistribution From the Figure 1 it was found that the parti-cle size increases with increasing wall temperature whichclearly shows that the wall temperature is a variable thathas significant effect on the average particle diameterIt is important to point out that in our studies we have
calculated the dimensionless time for the process in orderto observe the sensitivity of the disturbance on average par-ticle diameter (Table I shows the dimensionless variable)The residence time of particle distribution is intimatelyrelated to the average flow velocity and characteristics timeof particle growth and they are useful because they areused in calculating the dimensionless time ( The dimen-sionless time ( t is equal to 7e06 Figure 2 showsthe average particle diameter as function of dimensionlesstimeThe dimensionless time affect the particle aerosol char-
acteristics [average size] The average particle diame-ter here increases with increase of the dimensionlesstime One can see from this graph that when particlesare nucleated a primary particle with diameter of about400 nm is produced at constant time After a certainnumber of particles have been produced the frequency of
Fig 2 Average particle diameter as function of dimensionless time
bi-particle collision increases resulting in a sharp increasein particle diameter Several simulation runs were per-formed to investigate the effect of disturbances of TiCl4O2 flow rate on average particle diameter with respect tothe model parameters The disturbances actually affect themodel because it made the system to be unstable Andbecause the process model is highly nonlinear the influ-ence of disturbance on average particle diameter is non-stationary (random walk) Figure 3 shows the effect of
Fig 3 Top plot effect of TiCl4 and O2 disturbances on average par-ticle diameter Middle plot effect of O2 disturbance on average particlediameter and bottom plot effect of TiCl4 on average particle diameter
14 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
J Comput Theor Nanosci 8 8ndash16 2011 15
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
16 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Sadiku and Kolesnikov Application of Reduced Population Balance Model II
Fig 4 Comparison of average particle diameter with and withoutdisturbance
disturbances of TiCl4 O2 flow rate on average particlediameter For this dimensionless time one could observedthat disturbances to have great effect on the outputs thusthe need to control particle size distributionFigure 4 shows the comparison of the average par-
ticle diameter with and without disturbances It can beobserved that the influence of disturbance resulted tounstable process The result of average particle diameterwithout distubance shows the process is not distort whilethat of disturbance shows instability this attributed to thefact that the disturbance distort the process
41 Closed Loop Simulation
The control problem was formulated as one of trackingthe average particle diameter of the aerosol system alonga time-varying profile by manipulating the wall tempera-ture ie yt = dpt and ut = Tw The effect of walltemperature on average particle diameter was investigatedthrough the open loop simulation and was found that theprocess model could be used to control particle size dis-tribution Therefore the model of Eq (20) was used as
Fig 5 Closed-loop profiles of average particle diameter in the outletof the reactor with minus023 error
Fig 6 Closed-loop profiles of average particle diameter in the outletof the reactor with minus003 error
the basis for the synthesis of proportionalndashintegral con-troller utilizing the control method with k = 0131 and =4times10minus10 (k and were computed after extensive trial anderrors) The proposed controller regulate the particle diam-eter with minimum percentage error to its new set pointvalue Figures 5 and 6 shows the controlled output of aver-age particle diameter (from t= 0 to 775e06) with minus023and 003 error respectively The wall temperature is notmanipulated directly but indirectly through manipulationof inlet flow rate and dimensionless temperature To thisend controller should be designed base on an ODE modelthat describes the dynamics that operates in an internalloop to manipulate the inlet flowrate and dimensionlesstemperature to ensure that the average particle diameterobtains the values computed by the controllerThe objective of closed loop simulation is to show that
the use of feedback control allows producing an aerosolproduct with a desired average particle diameter (dp =1800 nm) It is clear that the use of feedback control allowsproducing an aerosol product with particle diameter thatexactly matches the desired one and sometimes almostmatches the desired one with minimal error
Fig 7 Manipulated input profile
J Comput Theor Nanosci 8 8ndash16 2011 15
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
16 J Comput Theor Nanosci 8 8ndash16 2011
Delivered by Ingenta toSophie Itumeleng Segwale
IP 1681720254Sat 12 Feb 2011 173331
RESEARCH
ARTIC
LE
Application of Reduced Population Balance Model II Sadiku and Kolesnikov
Fig 8 Manipulated input profile
The manipulated input profile of Figures 7 and 8 aresmooth functions of dimensionless time It is importantto note that the wall temperature was manipulated inthe last equation of Eq (20) which is the dimensionlesstemperature thus the effect of the manipulated input pro-file is shown in the dimensionless temperature
5 CONCLUSION
In this work we used the application of the reduced pop-ulation balance model for the control of nanoparticle sizedistribution The model accounts for simultaneous nucle-ation condensation and coagulation for the control of sizedistribution Under the assumption of lognormal size dis-tribution the method of moment was employed to reducethe population balance model into the simpler modeldescribing the evolution of the first three leading momentThis simplified model is a set of ODEs The effect ofwall temperature on average particle diameter was inves-tigated in order to synthesize nonlinear output feedbackcontrollers for titanium aerosol reactor that attain size dis-tributions with desired characteristics The average parti-cle diameter can be increased by increasing the reactorwall temperature The performance of the open loop sim-ulation was verified through computer simulation usinganother soft ware Based on the sensitivity of wall tem-perature upon the particle diameter dp we investigated the
process model sensitivity to the influence of disturbanceThe process model was subsequently used as a basis tosynthesize a feedback controller which manipulates thewall temperature of the reactor to control the aerosol sizedistribution in the outlet of the reactor with desired averageparticle diameter
References
1 M Zabarjadi K Esfarjani A Shakori Z Bian J Bahk Z GehongJ Bowers H Lu J Zide and A Gossard J Electron Mater 38 954(2009)
2 D C Panagiotis Model Based Control of Particulate ProcessesSpringer New York (2002)
3 A Kalani and D C Panagiotis Aerosol Sci Technol 32 369 (2000)4 G Michael and K Markus Journal of Computational Physics
183 210 (2002)5 Y H Debra and M K Sonia Journal of Atmospheric Environment
32 1701 (1998)6 S Kumar and D Ramkrishna Chem Eng Sci 51 1311 (1996)7 S Kumar and D Ramkrishna Chem Eng Sci 51 1333 (1996)8 S Kumar and D Ramkrishna Chem Eng Sci 52 4659 (1997)9 M Vanni J Colloid Interface Sci 221 143 (2000)
10 D Meimaroglou A I Roussos and C Kiparissides Chem EngSci 61 5620 (2006)
11 B James and D C Panagiotis International Journal of Control73 439 (2000)
12 A H Aleck and A K Costas Journal of Chemical EngineeringScience 60 4157 (2005)
13 S Hailian J F Fraser and G Martin Journal of Process Control17 379 (2007)
14 S Dan H E F Nael L Mingheng M Prashant and D CPanagiotis Chem Engng J 6 268 (2006)
15 C A Dorao and H A Jakobsen Comput Chem Eng 30 535 (2006)16 C A Dorao and H A Jakobsen Chem Eng Sci 61 7795 (2006)17 R B Diemer and S H Ehrman Journal of Powder Technology
61 7795 (2006)18 J C Barret and N A Webb J Aerosol Sci 29 31 (1998)19 A T Suddha and T M Mark Aerosol Science 35 889 (2004)20 A G Gerber and A Mousavi Applied Mathematical Modelling
31 1518 (2007)21 J A Schwarz Dekker Encyclopedia of Nanoscience and Technol-
ogy 1st edn Mercel Dekker Inc New York (2004)22 K F Sheldon Smoke Dust and Haze Fundamentals of Aerosol
Dynamics 2nd edn Oxford New York (2000)23 K Ashish and D C Panagiotis Chem Engng Sci 54 2669 (1999)24 S E Pratsinis J Collide Interface Sci 124 416 (1988)25 E L Sergey Engineering and Scientific Computation Using Matlab
Rochester Institute of Technology Wiley Interscience New Jersey(2003)
Received 9 February 2010 Accepted 30 March 2010
16 J Comput Theor Nanosci 8 8ndash16 2011
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