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A kinetic-empirical model for particle size distribution evolution during pulverised fuel combustion

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/223427199
A kinetic-empirical model for particle sizedistribution evolution during pulverised fuelcombustion
Article
in
Fuel · September 2010
DOI: 10.1016/j.fuel.2009.12.013
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Mariusz K. CieplikEnergy Research Centre of the Netherlands
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A kinetic-empirical model for particle size distribution evolution duringpulverised fuel combustion
Kalpit V. Shah
a,
*
, Mariusz K. Cieplik
b
, Christine I. Betrand
b
, Willem L. van de Kamp
b
, Hari B. Vuthaluru
a
a
Department of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth 6001, Australia
b
Energy Research Centre of Netherlands, P.O. Box 1, 1755 ZG Petten, The Netherlands
a r t i c l e i n f o
Article history:
Received 14 August 2009Receivedinrevisedform20November 2009Accepted 15 December 2009Available online 29 December 2009
Keywords:
PF combustionAsh formationParticle-population balance modelParticle sizeMathematical modelling
a b s t r a c t
Particle size is an essential parameter in pulverised fuel (PF) combustion as many of the problems or fur-ther areas of development in these systems are strongly inﬂuenced by the fuel and ash size distribution.This is particularly true for dynamic processes like pollutant formation, corrosion, erosion, slagging andfouling and the related decrease of the combustion and boiler efﬁciency. The evolution of particle sizedistribution (PSD) is a complex interaction of various competing chemical and physical transformations.Char oxidation, devolatilization and fragmentation, etc. represent ﬁrst line physical and chemical trans-formations which can amend the particle size in the radiation zone. The evolution of the PSD representsthe convolution of all of these physical and chemical transformations, operating over the entire size dis-tribution. As a consequence, it is difﬁcult to extract the relative importance of all competing size alteringprocesses from the experiments. Various models such as break-up, thermal stress, shrinking core, perco-lation and particle-population model have been developed by incorporating numerous ash transforma-tion mechanisms to predict the particle size evolution during the pulverised fuel combustion. Thepresent workdescribes anadaptationof thenumerical kinetic-based particle-populationbalancefor pre-dicting particle size evolution during PF combustion developed by Dunn-Rankin andMitchell. The modelisfurthersimpliﬁedanalyticallyandvalidatedagainstexperimentalresults.Severalempiricalparametersderived from the experiments are incorporated into the model. The resulting simpliﬁed PSD evolutionmodel shows good agreement with literature and experimental results, with maximum 10% absolutestandard deviation.Crown Copyright
2009 Published by Elsevier Ltd. All rights reserved.
1. Introduction
Over the past decades signiﬁcant progress has been made inunderstanding and quantifying the processes governing the ashformation during pulverised fuel combustion [1]. Processes likethe char oxidation, devolatilization and fragmentation are consid-ered as the ﬁrst line physical transformations responsible forcoarse ash formation in the radiation zone of the boiler. Otherphysical transformations such as nucleation, coagulation and con-densation of devolatilized inorganic gaseous species are responsi-ble mainly for submicron aerosol formation. Experimental andtheoretical investigations indicate that particle shape, size anddensity inﬂuence particle dynamics, including drying, heatingand conversion rates. Therefore, their effect on ﬁrst line physicaltransformations in the radiation zone, will be quite signiﬁcant[1,2]. These transformations compete with each other in theradiant zone of the PF furnace [3]. From the experiments it isobserved that devolatilization (of both organics and inorganics) issigniﬁcant, even at the char combustion phase. Fragmentation of thecharparticledependsoncharburnoutandthermalstress.Frag-mentation starts from 10% burnout and occurs throughout in bothdiffusion (char burnout) and chemically kinetic controlled regimes[4]. Overall, the evolution of particle sizes in a combustion systemis a convolution of all such various competing physicaltransformations.Particle size after combustion is a very important parameter inpulverised coal combustion systems as processes like pollutantformation, corrosion, erosion, slagging and fouling are stronglyinﬂuenced by the ﬂy ash size distribution after combustion [4].Furthermore, ash particle size after combustion has been foundto affect the ash transport behaviour to a great extent. Largeash particles tend to impact onto boiler heat transfer surfacesby inertia, whereas ﬁne ash particles tend to reach wall surfacesby thermophoresis or Brownian motion. For instance, a 60
l
m ashparticle was estimated to reach the deposit surface with higherprobability compared to 30
l
m particle primarily due to inertialeffect [1].
0016-2361/$ - see front matter Crown Copyright
2009 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.fuel.2009.12.013
*
Corresponding author. Tel.: +61 892667911; fax: +61 892662681.
E-mail address:
k.shah6@postgrad.curtin.edu.au (K.V. Shah).Fuel 89 (2010) 2438–2447
Contents lists available at ScienceDirect
Fuel
journal homepage: www.elsevier.com/locate/fuel
Evolution of particle sizes after combustion has been describedby numerousresearchers and details of their ﬁndings canbe foundelsewhere [5]. Various mechanisms have been studied in depth tounderstand the overall ash formation process [6,7]. Explanation of why and how char oxidation, devolatilization and fragmentationare likely to occur, have been illustrated [1]. Separate mathemati-cal models for prediction of char oxidation, devolatilization andfragmentation have been developed. The integration of the mech-anisms alongside with the mineral matter distribution, particlesize, shape and density has been incorporated in the models withlinear, nonlinear, deterministic or probabilistic relationships. Themodels developed [5] are break-up [8], thermal stress [9], shrink-
ing core [10–12], percolation [13–16] and particle-population
[20–23] models, etc.
This paper describes an adaptation of a kinetic population bal-ancemodel,whichpredictsPSDevolutionofparticlesafterPFcom-bustion in the radiation zone mainly working with three ashtransformations i.e. char oxidation, devolatilization and fragmen-tation. The model is a set of ﬁrst order linear ordinary differentialequations and therefore particle–particle interaction in the spaceis neglected. Also, other physical transformations such as nucle-ation, coagulation, homogeneous and heterogeneous condensationwithgaseousphasechemicalreactionsarealsonotincludedinthismodel. The fragmentation and burning rate constants are derivedfrom lab-scale experiments and incorporated into the presentmodel. These experiments have been performed in a Lab-scaleCombustion Simulator (LCS), under very well-deﬁned and con-trolled conditions. Furthermore, the model is simpliﬁed analyti-cally. In order to do so, instead of predicting the full particle sizedistributions at every time step, fragmentation into two distinctparticle diameters within each particle size class/bins are solvedanalytically. Particle shape and density changes are also neglectedin the present model.
1.1. Background
Kinetic models (or populationbalances) are applied in the anal-ysis of many size degradation and size enhancements processes.Ballauff and Wolf [17] gives exact solutions of ﬁrst order kineticequations describing the degradation of chain molecules by ran-domscission. Intheirformulation,afragmentationeventgivesriseto a pair of daughter fragments while all fragment pairs are as-sumed equally probable. The solutions give the molecular weightdistribution as a function of time. The kinetic simulations of Ball-auff and Wolf [17] assume that single fragmentation event pro-duces only two daughter fragments. However, other kineticsimulations presume that a single fragmentation event producesa family of fragments. Austin et al. [18] reported similar solutionsofsize-continuousformofkineticequationswithspeciﬁcbreakagefunctions and fragmentation families for grinding process. Waldieand Wilkimson [19] employs the population balance method withfamilial fragmentation to simulate competing processes of particlegrowth, attrition and fragmentationduring palletisation in a rotat-ingdrum.Dunn-Rankin[3,4]introducedakineticmodelusingpar-
ticle-population balance approach to simulate PSD evolutionduring the oxidation and fragmentation of char. This model how-ever, does not include the densitychangesoccurreddueto particleswelling. Later on Mitchell [20] modiﬁed the particle-populationbalance model by incorporating these density changes. Their ex-tended model was then used to evaluate PSD as a result of frag-mentation occurring during both coal devolatilization as well asthe char oxidation. Recently, Syred et al. [5] simpliﬁed the presentmodel analytically for two size classes for fragmentation only. Thepresent model is an extended version of Syred’s work with theinclusion of burning alongside with fragmentation.
2. Mathematical modelling
2.1. Overall mass balance equation
The present simpliﬁed model predicts the cumulative massfraction alongside with a PSD derived from the particle numbercalculations at different distinct time steps. The initial particle sizedistribution represents different size bins as shown in Fig. 1. The
c
(also shown in Fig. 1) is a very important parameter, deﬁning theupper and the lower cut-offs of each interval and can be expressedas a ratio of larger and smaller size particle for each size bin. Theoverall mass balance for each size bin has been considered asfollows:
M
0
ð
1
G
Þ¼
m
1
ð
t
Þ
N
1
ð
t
Þ
þ
m
2
ð
t
Þ
N
2
ð
t
Þ
:
ð
1
Þ
The
M
0
is the initial mass number of the each size bin having
m
1(0)
and
m
2(0)
weighted particles with
N
1(0)
and
N
2(0)
particle numbers,respectively. Eq. (1) implies that the residual mass after conversion
G
isdividedintotwosizeclasses.The‘
G
’isthetotalcharconversionof the particular size bin into the gas-phase due to devolatilizationand char oxidation. Burning (chemical conversion) and fragmenta-tion are the main cause for the two resultant size classes.The particle numbers
N
1(
t
)
and
N
2(
t
)
after burning and fragmen-tationhavebeencalculatedbysolvingpopulationbalanceequationof Mitchell [20] analytically for two size classes.
2.2. Population balance equation
The structure of the particle-population balance model byMitchell [21] is presented as a set of differential equations havingthe following form (Eq. (2)):
dN
i
;
k
dt
¼
S
i
;
k
N
i
;
k
þ
X
i j
¼
1
X
K
k
¼
1
ð
b
ij
;
k
k
S
j
;
k
N
j
;
k
Þ
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Fragmentation
C
i
;
k
N
i
;
k
þ
C
i
1
;
k
N
i
1
;
k
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Size
D
i
;
k
N
i
;
k
þ
D
i
;
k
1
N
i
;
k
1
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Density
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Burning
:
ð
2
Þ
Theindices
i
and
k
refertosize-classanddensityclass. Theﬁrsttwoterms on the right-hand side of the Eq. (1) represent the rates atwhich particles leave and enter a particular class (
i
,
k
), as a resultoffragmentation.Thethirdandthefourthtermsrepresenttheratesatwhichparticlesleaveandentertheclass, as aresultofchangesinsizeduetoburning.Thelasttwotermsrepresenttherates,atwhichparticles leave and enter the class, as a result of changes in densitydue to burning. Thus,
N
i
,
k
is the number of particles in size-class
i
and density-class
k. S
i
,
k
is the fragmentation rate constant and
C
i
,
k
and
D
i
,
k
are the burning rate constants. The
b
i,j
are elements of the fragmentation progeny matrix, which specify the number of fragments that enter higher size bin
i
per particle that fragmentsin lower size bin
j
. Particles fragmenting in bin
j
can produce frag-mentsonlyinbin
i
where
i > j
, therefore,
b
ij
=
0for
i < j.
Theprogenyelements were determined for each type of fragmentation consid-ered. Three kinds of fragmentation described are considered i.e.,attrition, breakage and fragmentation. The fragmentation modesused in the present model are described below [5].Attrition is incorporated in the model by assuming 0.01% vol-ume of largest particles to fragment attritively to lowest size bins.Progenymatrixfor break-up fragmentation canbe expressed as(Eq. (3)):
b
ij
¼
0
i
6
j
c
3
i
¼
j
þ
10
i
>
j
þ
1
:
8><>:
ð
3
Þ
K.V. Shah et al./Fuel 89 (2010) 2438–2447
2439
Progenymatrix(
b
i,j
)forpercolativefragmentationcanbedeﬁnedas(Eq. (4)):
b
ij
¼
c
3
ð
1
j
Þ
=
ð
n
j
þ
1
Þ
i
P
j
0 Otherwise
:
(
ð
4
Þ
The present work uses the population balance approach developedby Mitchell [21] to simulate the evolution of the particle size dis-tribution during char combustion. The simulation includes boththe burning and the fragmentation. The burning includes bothchar oxidation and devolatilization. Therefore, this model predictsthe PSD evolution during the combustion by taking into accountall three important ﬁrst line physical transformations: i.e. charoxidation, devolatilization and fragmentation and their revalua-tions with size changes. The model is a set of isolated ﬁrst orderlinear ordinary differential equations therefore particle–particleinteraction in the space is neglected. Other physical transforma-tions for gaseous phase such as nucleation, coagulation, homoge-neous and heterogeneous condensation with chemical reactionsare also not included in this model. Although the present modelis considering only ash transformation mechanisms occurring inthe radiation zone and lot of simpliﬁcations have been made inthe numerical approach by selecting ODE structure instead of PDE, it is still analytically to bulky to incorporate Mitchell’s modelinto simple visual basic or even Excel-based engineering modelsand CFD routines. For this reason, this model is further simpliﬁedand solved analytically as below. Several parameters obtainedfrom the ash formation experiments conducted at ECN are usedin the simpliﬁed model.
2.3. Simpliﬁcations
In the present kinetic model, it is assumed that the particlecombustion rate depends on the instantaneous particle diameter.Changes in the particle density, due to the steady diameter charoxidation or to cenospheres formation are neglected. In result,the Mitchell’s [21] model is simpliﬁed as shown in Eq. (5), overall
resembling much the srcinal model of Dunn-Rankin [3,4], but
with the progeny matrix from Mitchell’s model [5,21] described
in Eqs. (3) and (4).
dN
i
dt
¼
S
i
N
i
þ
X
i j
¼
1
b
ij
S
j
N
j
C
i
N
i
þ
C
i
1
N
i
1
:
ð
5
Þ
Thedescribedsimpliﬁcationsareconsideredwhilesolvingthemod-el equation analytically for two size class for each size bin. Insteadof using a PSD classiﬁcation for the combustion, the particle sizebins before and after combustion are classiﬁed only in two sizes,having a higher (
m
1
) and a lower (
m
2
) particles masses. Therefore,everycombustiontimestepineachsizebinwouldcreatenewchildparticles classiﬁed in higher and lower mass sizes. The values of (
m
1
) and(
m
2
) are time-dependent, however, their change is limitedwithintheratiobetweenthehigherandthelowerparticlemass(
c
3
)in each size bin, assumed to be constant during the process. Thissimpliﬁcation is the same as proposed by Syred et al. [5], who sim-
pliﬁed and solved the model equation analytically for pure frag-mentation. However, pure fragmentation is an incompleterepresentationof charoxidation,sincetheseparticlesmustburnoutin a ﬁnite time. Therefore, instead of only the fragmentation, thepresent adaptationof themodel is extendedontoburning. Theana-lytical solutions of the above Eq. (5) for burning and fragmentation
302013PARTICLE SIZE (µm)1006744Size bin123456M
0
* GChar conversion to Gaseous phase20Over all mass balance for each size bin443010067PARTICLE SIZE (µm)150PSD of Fuel at t=0N
1(0)
of m
1(0)
N
2(0)
of m
2(0)
PSD of fuel after Fragmentation+Burning at t = tN
1(t)
of m
1(t)
N
2(t)
of m
2(t)
++++++++++++
PPMMODEL
Burnning + Fragmentation
M
0
m
1
N
1
m
2
N
2
M
0
G
++
γ γ
Fig. 1.
Modelling chart of PSD evolution during pf combustion.2440
K.V. Shah et al./Fuel 89 (2010) 2438–2447
are derived in Section 3. In contrast to Dunn-Rankin’s, Syred’s andMitchell’s model, fragmentation rate and burning rate constantsare derived empirically from dedicated experiments. Throughout,the particle shape is considered to be spherical, in order to avoidcomplexityandnoshapefactor is includedinthedevelopedmodel.
2.4. Empirical parameters
Apart from the progeny matrix, the burning and fragmentationrate constants are the two unknown values in the model equation.Fragmentation and burning are the two parallel ash transforma-tions which are responsible for PSD evolution after combustion.Both rate constants are derived from experiments and incorpo-rated into the model. The detail of the derivation of the constantsfrom the experiments is explained in the Section 4.
2.4.1. Burning rate constant
The overall spherical particle burning rate [23] is deﬁned as themass loss rate per unit of external surface area and can be ex-pressed as described below:
Q
¼
1
P
D
2
p
dm
c
dt
¼
q
pc
2
dD
p
dt
þ
D
p
6
d
q
pc
dt
;
ð
6
Þ
where
m
c
and
q
pc
arethemass andapparent densityrespectivelyof theparticlediameter
D
p
. Theﬁrst termontheRHSof theEq. (6)canbedeﬁnedastheapparentexternalburningrateduetosizechangeswith time and second term as the apparent internal burning ratedue to density changes with time. As mentioned in the simpliﬁca-tion section above, this model assumes shrinking core burning,hence density changes are neglected in the present model, thuszeroing the second term. The Eq. (6) is then reduced to:
Q
¼
1
P
D
2
p
dm
c
dt
¼
q
pc
2
dD
p
dt
:
ð
7
Þ
So, from the above equation particle size changes due to burningcan be derived. For this, burning rate constant
C
i
is calculated [23]as below:
A
¼
C
i
¼
dDpdt
ð
x
i
x
i
þ
1
Þ
x
i
þ
1
:
ð
8
Þ
Theburningrateisatimedependentfunction,whichdescribeshowrapidlyparticlesleavegivensizeclassduetotheoverallburning.Totreat this rate as a constant for particular time step, its value is cal-culated using experiments for that selected time step.The burning rate constant will be a single value derived fromexperiment for each size class for the deﬁned time step (as
C
1
=
C
2
) and can be termed as A for further calculations.
2.4.2. Fragmentation rate constant
The physical signiﬁcance of a fragmentation event is made evi-dent by solving, without burning, the basic kinetic equation onlyfor larger particles.
dN dt
¼
SN
:
B
¼
Z
t
0
Sdt
¼
Ln N
ð
0
Þ
=
N
ð
t
Þ
:
ð
9
Þ
where
N
(0)
is the initial number of largest particles (>30
l
m), and
N
(
t
)
is the particle number at time step
t
. Thus,
S
is directly relatedto the fractionof the largest particles that ultimately fragment dur-ing the simulation. In the described study, though
S
is a function of time, it is obtained from the experiments for different residencetimes and incorporated into the model. Therefore, it can be takenas a distinct value/constant for that selected time step.Fragmentation rate constant will have a single value for all sizeclasses (>30
l
m) for particular time step for each size bin and canbe termed as
B
for further calculations.As, S
i
and C
i
will be a value for particular time step, analyticalsolution of Eq. (2) for that corresponding time step is possibleand derived in Section 3.
2.4.3. Mode of fragmentation
In several studies [22,23], it was observed that the char com-
bustion will be in a kinetic-diffusion controlled regime (wheremass loss rates due to pore diffusion and chemical reactions arecomparable), even with extended residence time under typical PFﬁring conditions. Three kinds of fragmentations are considered inthis model: attrition, breakage and percolation. Initially, a particlewill be forming small particles from its outer surface, which isessentiallysimplythephenomenonofattrition.Assoonasthepar-ticle starts devolatilizing and oxidizing, thermal stress within theparticleincreases,duetorapidvaporizationandtheincreasedtem-perature. This in turn causes the particle to break into relativelylarge particles, which process is called breakage. After a certainconversion, due to very high thermal stress, particle fragment per-colatively into smaller and larger particles excessively. It is ob-served that signiﬁcant percolative fragmentation does not occuruntil substantial chemical conversion (60–70%) of the fuel[24,25]. To quantitatively incorporate the possibility of attrition,breakage and delayed excessive percolative fragmentation, prog-enymatrixdiscussedpreviously(Section2)areusedandanalytical
Fig. 2.
Particle surface regression computed with the one dimensional coalcombustor program 1-DICOG [4].
K.V. Shah et al./Fuel 89 (2010) 2438–2447
2441

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