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Workshop no. 105, FATS IIWorkshop no. 105, FATS II,,
Binzen, 26Binzen, 26--28 September, 200628 September, 2006
M. Peglow1, S. Heinrich2 (speaker), E. Tsotsas1
1: Chair for Thermal Process Engineering, Otto-von-Guericke University
2: Chair for Chemical Apparatus Design, Otto-von-Guericke University
Magdeburg, Germany
Population balance modeling
as a tool for understanding
fluidized bed coating
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Some granulated fluidized bed products
of the University of Magdeburg
protein
granulation from
suspensiongranulation from
melt
urea
granulation from
solution
potash
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Population BalancesDistributed Properties
Disperse phase – state variables:
Density distributions
k jf f(t,r , x )=
x1 x
Description of disperse systemsq 60% of all products in chemical industry are disperse systems
q Traditional approaches for modeling: averages
q Detailed modeling by means of population balances
Spatial distributed
disperse system
properties xj: x1, ..., xN
(internal coordinates)
spatial coordinates r k: r 1, r 2, r 3(external coordinates)
, z.B. Partikelgröße
r 2*
r 1*
f , z.B. Anzahldichte
r 1
r 2
r 3
r 3*
e.g. particle size
r 2*
r 1*
f e.g. number density
r 1
r 2
r 3
r 3*Continuous phase – state variables:
density
pressure
concentrations of species; enthalpy
velocity in direction r m
( , )
( , )
( , )
v v ( , )
====
k
k
l l k
m m k
t r
p p t r
t r
t r
ρ ρ
φ φ
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Population BalancesDisperse phase, f = (f
i )
Challenges:q multidimensional
(regarding f i resp. x j)
q efficient calculation of the
integral sink and source
terms
Coupling with continuous phase
growth aggregationbreakage
( ) (
) xagg i xbr i
f
k iik
k
iij
j
i dV f hdV f h j f r
f G xt
f
x x
~,~,, )v,,( )v,,(v )v,( ∫ ∫ ΩΩ
+=+∂∂
+∂∂
+∂∂
φφφ
con-
vection
Population balance:
accu-
mulation
diffu-
sion
property x j
f f
property x j
f
property x j
nucleation
Boundary conditions:
)()()v,( ,0, φφ jnuc jiij B x f G =⋅
property x j
f
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Droplets
(H2O+Binder)
gas inlet (dry, warm)
gas outlet (humid, cold)
Liquid
(Binder)
Primary
particles
Objective:Example 1: Description of simultaneous agglomeration and drying
using population balances
Agglomerate
Agglomeration Drying
Population BalancesFluidized bed spray agglomeration
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Drying of agglomerate
Determination of moisture content X, temperature ϑ andsize v of agglomerates
( ) ( )ps p g g eq P sM A Y X, Y = ⋅β ⋅ ρ ϑ − ⋅ ν η & &
Heating/Cooling of agglomerate
( )ps p ps p sQ A= ⋅ α ⋅ ϑ − ϑ&
P
New: moisture content and temperature of agglomerates are
- similar to size of agglomerate – distributed properties.Ñ
+Formation of agglomerate
Population BalancesDefinition of particle properties
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Population phenomena:+ Agglomeration
+ Breakage
+ Growth
Attrition/Layering,
Drying/Wetting
Heating/Cooling+ Nucleation
M a s s o f l i q u i d
Solid particle volume
l1+ l2
l1
l2
v1 v2 v1+ v2
AgglomerationDrying
Wetting
(Not shown: heat and mass transfer)
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
v l
0 0
0 0
f t,v,l G v,l f t,v,l 1t,v u,u,l , f t,v u,l f t,u, d dut l 2
t,v,u,l, f t,v,l f t,u, d du∞ ∞
∂ ∂ ⋅+ = β − − γ γ ⋅ − − γ ⋅ γ γ ∂ ∂
− β γ ⋅ ⋅ γ γ
∫ ∫
∫ ∫
partial integro differential equation
Population BalancesMultidimensional population balance
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• High numerical effort (exponential in number of properties N)
• Model reduction using marginal distributions (linear in N)• Assumption: influence of moisture content of particles on agglomeration is
only considered by the pre-factor of agglomeration kernel β
Number distribution:
Liquid distribution:
( ) ( )0
n t,v f t,v,l dl∞
= ∫
( ) ( )l
0
m t,v l f t,v,l dl∞
= ⋅∫
( ) ( ) ( ) ( )' *0t,v,u,l, t,v,u t v,uβ γ = β = β ⋅β
Model reduction Pre-factor
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
v l
0 0
0 0
f t,v,l G v,l f t,v,l 1 t,v u,u,l , f t,v u,l f t,u, d dut l 2
t,v,u,l, f t,v,l f t,u, d du∞ ∞
∂ ∂ ⋅+ = β − − γ γ ⋅ − − γ ⋅ γ γ ∂ ∂
− β γ ⋅ ⋅ γ γ
∫ ∫
∫ ∫
Population BalancesReduction of multidimensional population balance
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Heterogeneous fluidized bed model with
active bypass for gas phase.
Number distribution
Liquid distribution
Agglomeration Wetting/Drying
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
v* *
0
0 0
n v 1t u,v u n u n v u du n v u,v n u du
t 2
∞ ∂= β β − ⋅ ⋅ − − β ⋅ ∂
∫ ∫
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )v
l * *
0 l l np ps
0 0
m v t u,v u n u m v u du n v u,v m u du M Mt v
∞
∂ ∂= β β − ⋅ ⋅ − − β ⋅ + − ∂ ∂ ∫ ∫ & &
(Not shown: Enthalpy-distribution)
Coupling
Influence of moisture only onpre-factor!
Coupling
Population BalancesReduction of multidimensional population balance
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Ñ
P Applicable for any grid type, moments to be conserved can be chosen
Difficult to program
Cell Average, Fixed Pivot, Moving Pivot
P
Only applicable for geometric grids
Easy to program, very common
Only existing formulation for reduced multidimensional PBE!
(Hounslow‘s DTMD)
Ñ
P
Hounslow’s DPBE (Extension by Litster and Wynn)
Population BalancesCommon discretization schemes
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Linear grid Geometric grid
( ) ( ) ( ) ( ) ( ) ( ) ( )v
0 0
n v 1 u,v u n u n v u du n v u,v n u dut 2
∞
∂ = β − ⋅ ⋅ − − β ⋅∂ ∫ ∫ Method of Lines
Primary particle:
Agglomerate:
d0
dI= 40 d
0
Example:
( )3
0
3
0
40dI 64000
d= =
( )3
0
3
0
40dln
dI 17
ln2= ≈ PÑ
Number of grip points:
Population BalancesDiscretization schemes
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Number distribution – Application of Hounslow‘s DPBE
Liquid distribution – Application of Hounslow‘s DTMD
Agglomeration Wetting/Drying
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
v* *
0
0 0
n v 1t u,v u n u n v u du n v u,v n u du
t 2
∞ ∂= β β − ⋅ ⋅ − − β ⋅
∂ ∫ ∫
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )v
l * *
0 l l np ps
0 0
m v t u,v u n u m v u du n v u,v m u du M Mt v
∞
∂ ∂= β β − ⋅ ⋅ − − β ⋅ + − ∂ ∂ ∫ ∫ & &
Coupling
(Not shown: Enthalpy density distribution)
Problem:
How will the moisture distribution X(v) change during agglomeration
when mass transfer is neglected?…or…
Can intensive material properties (moisture content/temperature) be calculated?
Ñ
Population BalancesLack of Hounslow’s discretization
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M o
i s t u r e
a n
d m a s s o
f p a r t
i c l e s
Solid volume of a particle
Initial condition
Final distributions are not identical!
Problem:
How will the moisture distribution X(v) change during agglomeration
when mass transfer is neglected?
( ) ( )l pm v m v=
( ) ( )l pm v m v≠
Population BalancesLack of Hounslow’s discretization
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M o
i s t u r e c o n
t e n t
Moisture content has
been changed
Hounslow’s approach is not
consistent with intensive
properties!
Coupling with gas phase not
possible!
Reason: particle mass (DPBE) is
assigned in a different way than
moisture (DTMD).
( )X v 1=
( )X v 1≠
( )( )
( )
( )
( )l l
p
m v m vX v
k n v m v= =
⋅
Problem:
How will the moisture distribution X(v) change during agglomeration
when mass transfer is neglected?
Population BalancesLack of Hounslow’s discretization
Initial condition
Solid volume of a particle
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Inconsistency can be removed by introduction of correction factors(Peglow et. al, AIChE J. 2006)
Modification is identical with Hounslow’s formulation of DTMDfor K = 1
Population BalancesModification of Hounslow’s discretization
Peglow, M., Kumar, J., Warnecke, G., Heinrich, S., Tsotsas, E., Mörl, L., Hounslow, M.: An improved discretized tracer massdistribution of Hounslow et al., AIChE J. 52 (2006) 4, 1326-1332.
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Modification is consistent with
intensive properties!
Coupling with gas phase ispossible!
Basic concept can be adapted to
other numerical methods such as
Cell Average or Fixed Pivot.
Final distribution
( ) ( )l pm v m v=
( ) ( )l pm v m v=
Problem:
How will the moisture distribution X(v) change during agglomeration
when mass transfer is neglected?
Population BalancesModification of Hounslow’s discretization
Initial condition
Solid volume of a particle
M o
i s t u r e
a n
d m a s s o
f p a r
t i c l e s
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Intensive properties of solid phasein agglomeration processes
can be described
Formulation of a fluidized bed model for simultaneous agglomeration and
drying
Population BalancesTwo-phase fluidized bed model
Model assumptions
• Plug flow of gas phase
• Total back-mixing of solid phase
• Heat and mass transfer between
+ solid and suspension gas
+ suspension and bypass gas
• Agglomeration, no breakage
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Population BalancesSolid phase balance equations
• Number distribution
• Water distribution
• Enthalpy distribution
( )( ) ( ) ( ) ( ) ( ) ( )
v
0 0
n v,t 1t,u,v u n u,t n v u,t du n v,t t,u,v n u,t du
t 2
∞∂= β − − − β
∂ ∫ ∫
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
vw,l
w,l w,l
0
p p
0
s n
m v,tt,u,v u n u,t m v u,t du m v,t t,u,v n u,t du
tM M
v
∞∂ ∂= β − − − − +β +
∂ ∂∫ ∫ & &
( )( ) ( ) ( ) ( ) ( ) ( ) ( )ps np pw s
vp
p p
0 0
p
h v,tt,u,v u n u,t h v u,t du h v,t t,u,v n u,t du
t vH H Q Q
∞∂ ∂= β − − − β +
∂ ∂− + − +∫ ∫ & && &
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• Number distribution
• Water distribution
• Enthalpy distribution
• Mass transfer particle – gas
• Heat transfer particle - gas
( )( ) ( ) ( ) ( ) ( ) ( )
v
0 0
n v,t 1t,u,v u n u,t n v u,t du n v,t t,u,v n u,t du
t 2
∞∂= β − − − β
∂ ∫ ∫
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
vw,l
w,l w,l
0
p p
0
s n
m v,tt,u,v u n u,t m v u,t du m v,t t,u,v n u,t du
tM M
v
∞∂ ∂= β − − − − +β +
∂ ∂∫ ∫ & &
( )( ) ( ) ( ) ( ) ( ) ( ) ( )ps np pw s
vp
p p
0 0
p
h v,tt,u,v u n u,t h v u,t du h v,t t,u,v n u,t du
t vH H Q Q
∞∂ ∂= β − − − β +
∂ ∂− + − +∫ ∫ & && &
[ ]g pg p pes qp sX,M A Y Y )X( ) ( = ρ β − ν η ϑ &&
( )pgps spp AQ = ϑα − ϑ&
Particle properties:
• adsorption isotherm
• normalized drying curve)
Population BalancesSolid phase balance equations
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• Moisture distribution
• Enthalpy distribution
Population BalancesGas phase balance equations
( ) ( ) ( )p
g s sg sbs
dM Y Y1 1 M M
d tM
∂ ∂ ∂− ν = − − ν + −
ξ ∂ ∂ξ ∂ξ&& &
( ) ( ) ( )ps pg s s
g sb bs swsdM h h1 1 M H Q Qd t
H Q∂ ∂ ∂− ν = − − ν + − + −ξ ∂ ξ ∂ξ
−∂
& &&&& &
Suspension gas phase
• Moisture distribution
• Enthalpy distribution
Bypass gas phase
g b b sbg
dM Y Y dMM
d t d
∂ ∂ ν = −ν +
ξ ∂ ∂ξ ξ
&&
( )g b bg sb bs bw
dM h hM d H Q Q
d t
∂ ∂ ∂ ν = −ν ξ+ − −
ξ ∂ ∂ξ ∂ξ& && &
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Number distribution Moisture content
Particle temperature
X [ g
/ k g ]
ϑ
[ ° C ]
N
[ - ]
d [mm]t [s] d [mm]t [s]
d [mm]t [s]
Simulation of simultaneous
agglomeration, wetting and drying
3 stages
Population BalancesFluidized bed model – Simulation results
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X [ g
/ k g ]
ϑ
[ ° C ]
N
[ - ]
d [mm]t [s] d [mm]t [s]
d [mm]t [s]
Stage 1 – Pre-drying• Drying of solid material, decrease of
particle moisture content
• Heating of particles
• Particles size is constant
Number distribution Moisture content
Particle temperature
Population BalancesFluidized bed model – Simulation results
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X [ g
/ k g ]
ϑ
[ ° C ]
N
[ - ]
d [mm]t [s] d [mm]t [s]
d [mm]t [s]
Stage 2 – Spraying:• Wetting of solid material, increase of
particle moisture content
• Cooling of particles
• Change of particle size distribution by
agglomeration
Population BalancesFluidized bed model – Simulation results
Number distribution Moisture content
Particle temperature
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X [ g
/ k g ]
X [ g
/ k g ]
ϑ ϑ
[ [ ° ° C ] C ]
N
[
N
[ - - ] ]
d [mm]d [mm]t [s]t [s] d [mm]d [mm]t [s]t [s]
d [mm]d [mm]t [s]t [s]
Stage 3 – Drying:• Drying of particles
• Heating of particles
• Particles size is constant
Population BalancesFluidized bed model – Simulation results
Number distribution Moisture content
Particle temperature
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Lab scale fluidized bed: GLATT GPCG 1.1
Particle system
• Primary particles: Microcrystalline cellulose
• Binder: Pharamcoat 606 (HPMC-Binder)
Characterization
• Adsorption isotherm
• Drying curve
Measurement
• Particle size distribution
• Mean particle moisture content (no resolution!)• Temperature and moisture of gas
• Spraying ratePrimary
particles
Population BalancesExperimental results
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Adsorption isotherm Desorption isotherm
System: DVS 1 Firma PorotechPrinciple: Balance, measurement of
change of sample mass
Model: Fitting using the BET-Isotherm
(3 fitting parameters)
Population BalancesMaterial properties
eqweq
eq
p
pg
X,p ( )MY
P p (X, )M=
−
ϑ
ϑ
%
%
eq satp p eq pp ( ) p ( ) (X, X ),ϑ φϑ ϑ=
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Normalized drying curve
Population BalancesMaterial properties
System: DVS 1 Firma Porotech
Principle: Balance, measurement of change of sample mass
ps
ps,I
M
M
( )( )
( )
ηη
η
ν =&
&
&
hyg
cr hyg
X XX X−
η −=
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Gas outlet temperature Mean particle moistureGas outlet humidity
Population BalancesExperimental results – Influence of gas flow rate
High gas flow rate
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Gas outlet temperature Mean particle moistureGas outlet humidity
Population BalancesExperimental results – Influence of gas flow rate
Low gas flow rate
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Population BalancesExperimental results – Influence of gas flow rate
High gas flow rate Low gas flow rate
4
0 5.4 10−β = ⋅ 4
0 7.4 10−β = ⋅
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Restrictions:• Empirical agglomeration kernel
• Fitting of kinetic constants
to measurements,
thus solely descriptive model
• Only one way coupling between
drying and agglomeration model!
Population BalancesExperimental results
( )
( )
( )
( )
a 0.7105
0 0b 0.0621
u v u v
u v u v
+ +β = β = β
⋅ ⋅
Kernel ( )v,uβ Lit.
1 [1]
u v+ [2]
u v⋅
( ) ( )a b
u v u v+ ⋅ [3]
( ) ( )2/3 2/3u v 1/ u 1/ v+ + [4]
1 für 1t t≤
u v+ für 1t t>
[5]
1 für *W W≤
0 für *W W>
mit ( ) ( )a b
W u v u v= + ⋅
[6]
( )3
1/3 1/ 3u v+ [7]
( )2
1/ 3 1/3u v 1 u 1 v+ + [8]
Sim Exp
0,i 0,i
Simi 0,ii
q (t) q (t)RE(t)q (t)−= ∑ ∑
[1] Kapur, P.C., Fürstenau, D.W.: A coalescence model for granulation,Ind. Eng. Chem. Process Des. Dev. 8 (1969), 56-62.
[2] Golovin, A.M.: The solution of the coagulation equation for raindrops,Sov. Phys. Dokl. 8 (1963), 191-193.
[3] Kapur, P.C.: Kinetics of granulation by non-random coalescencemechanism, Chem. Eng. Sci. 27 (1972), 1863-1869.
[4] Sastry, K.V.S.: Similarity size distribution of agglomerates during their growth by coalescence in granulation or green pelletization, Int. J. Min.Proc. 2 (1975), 187-203.
[5] Adetayo, A.A., Litster, J.D., Pratsinis, S.E., Ennis, B.J.: Populationbalance modelling of drum granulation of materials with wide sizedistribution, Powder Technol. 82 (1995), 37-50.
[6] Adetayo, A.A., Ennis, B.J.: Unifying approach to modelling granulecoalescence mechanisms, AIChE J. 43 (1997), 927-934.
[7] Smoluchowski, M.V.: Mathematical theory of the kinetics of thecoagulation of colloidal solutions, Z. Phys. Chem. 92 (1917), 129.
[8] Hounslow, M.J.: The population balance as a tool for understanding particle rate processes, in Kona. 1998.
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P l ti B l
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N u m
b e r o
f P P
o f B
N u m b e r o f P P o f A
Agglomeration of two types of primary particles (component A and B)
Parameters for calculation
• 22 x 22 geometric grid
• Size independent kernel
• Degree of aggregation
( )β γ = β0t,v,u,l,
agg 0I 1 N N 0.98= − =
Initial distribution
N u
m b e r
d e n s
i t y
n
component A
component B
Populations BalancesSimulation results – Test case 1
P l ti B l
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( )β γ = β0t,v,u,l,
agg 0I 1 N N 0.98= − =
Component A
Component B
M o m e n t
Dimensionless time
0th moment (number)
1st moment (mass)
Agglomeration of two types of primary particles (component A and B)
Parameters for calculation
• 22 x 22 geometric grid
• Size independent kernel
• Degree of aggregation
Populations BalancesSimulation results – Test case 1
P l ti B l
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PP BPP A
Final distribution (Numerical)
PP A PP B
Agglomeration of two types of primary particles (component A and B)
N u
m b e r
d e n s
i t y
n
N u m
b e r
d e n s
i t y
n
Final distribution (Analytical)
Populations BalancesSimulation results – Test case 1
P l ti B l
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Populations BalancesSimulation results – Test case 2
Parameters for computation
• 13 x 13 geometric grid• Constant kernel
• Degree of aggregation
( )β γ = β0t,v,u,l,
agg 0I 1 N N 0.98= − =
Agglomeration of mono-disperse particlessame amount of both properties in each particle
Property 1
P r o p e r t y 2
Populations Balances
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Populations BalancesSimulation results – Test case 2
Agglomeration of mono-disperse particlessame amount of both properties in each particle
Analytical Solution
Property 1
P r o p e r t y 2
P r o p e r t y 2
Property 1
number density
Fixed Pivot Technique Cell Average Technique
number density
Population Balances
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n u m b e r d
e n s i t y
c o m p o n e n t A c o m p o n
e n t B
n u m b e r d e n s i t y
c o m p o n e n t A c o m p o n
e n t B
Coupling of agglomeration and drying
P drying kinetics depend on particle sizeP Influence of particle moisture on agglomeration kernel (but only in pre-factor)
Ñ Bilateral coupling of agglomeration and drying
Experimental results
P Influence of mean moisture content
Ñ Measurement of particle size depended moisture or single particle moisture
Multidimensional population balanceP Model reduction using marginal distributions
P Development of discretization methods for intensive properties of solid phase
Ñ Extension of Cell-Average Methode to 2D problems (first simulation results)
Test simulation2D-PBE for agglomeration
Population BalancesSummary and Outlook
Population Balances
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Population BalancesOverview of new published literature
§ Heinrich, S., Peglow, M., Ihlow, M., Henneberg, M., Mörl, L.: Analysis of the start-up process in continuous fluidized bed spray
granulation by population balance modelling, Chem. Eng. Sci. 57 (2002) 20, 4369-4390
§ Peglow, M., Kumar, J., Warnecke, G., Heinrich, S., Mörl, L.: A new technique to determine rate constants for growth and agglomeration with size and time dependent nuclei formation, Chem. Eng. Sci. 61 (2006) 1,Special issue: Advances in population balance modelling, 282-292
§ Peglow, M., Kumar, J., Warnecke, G., Heinrich, S., Tsotsas, E., Mörl, L., Hounslow, M.: An improved discretized tracer massdistribution of Hounslow et al., AIChE J. 52 (2006) 4, 1326-1332
§ Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., Mörl, L.: Improved accuracy and convergence of discretized populationbalance for aggregation: The cell average technique, Chem. Eng. Sci. 61 (2006), 3327-3342
§ Radichkov, R., Müller, T., Kienle, A., Heinrich, S., Peglow, M., Mörl, L.: A numerical bifurcation analysis of continuous fluidized bed spray granulation with external product classification, Chem. Eng. Proc. 45 (2006) 10, 826-837
§ Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., Mörl, L.: A discretized model for tracer population balance equation:Improved accuracy and convergence, Comp. Chem. Eng. 30 (2006), 1278-1292
§ Mörl, L., Heinrich, S., Peglow, M. (Eds.: Salman, A., Hounslow, M., Seville, J.P.K.): Fluidized bed spray granulation,Granulation (Handbook of Powder Technology, Volume 11), Chapter: The Macro Scale I: Processing for Granulation,Elsevier Science, 169 Seiten, in press, ISBN 0-444-51871-1
§Peglow, M., Heinrich, S., Tsotsas, E.: Towards a complete population balance model for fluidized bed spray granulation:Simultaneous drying and particle formation, Glatt International Times, 22 (2006) June, 7-13
§ Peglow, M., Kumar, J., Heinrich, S., Warnecke, G., Mörl, L., Wolf, B.: A generic population balance model for simultaneous agglomeration and drying in fluidized beds, Chem. Eng. Sci., Special issue: Applications of fluidization, 51 pages (in press)
§ Kumar, J., Peglow, M., Heinrich, S., Tsotsas, E., Warnecke, G., Hounslow, M.J. (Eds.: Tsotsas, E., Mujumdar, A.S.):Chapter 4: Numerical methods for solving population balances, Modern Drying Technology, Volume 1: Computational tools atdifferent scales, WILEY-VCH, 57 pages (submitted)
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