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1

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



2

Some granulated fluidized bed products

of the University of Magdeburg

protein

granulation from

suspensiongranulation from

melt

urea

granulation from

solution

potash



3

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



4

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



5

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



6

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



7

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



8

• 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



9

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



10

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



11

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



12

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



13

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:


when mass transfer is neglected?

( ) ( )l pm v m v=

( ) ( )l pm v m v≠




14

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:




Initial condition




15

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.



16

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:



Population BalancesModification of Hounslow’s discretization

Initial condition


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



17

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



18

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 +

∂ ∂− + − +∫ ∫ & && &



19

• Number distribution

• Water 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



20

• Moisture 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


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& && &



21

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



22

X [ g

/ k g ]

ϑ

[ ° C ]

N

[ - ]


d [mm]t [s]

Stage 1 – Pre-drying• Drying of solid material, decrease of

particle moisture content

• Heating of particles

• Particles size is constant






23

X [ g

/ k g ]

ϑ

[ ° C ]

N

[ - ]


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






24

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






25

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



26

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=



27

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 −=



28

Gas outlet temperature Mean particle moistureGas outlet humidity

Population BalancesExperimental results – Influence of gas flow rate

High gas flow rate



29

Gas outlet temperature Mean particle moistureGas outlet humidity


Low gas flow rate



30


High gas flow rate Low gas flow rate

4

0 5.4 10−β = ⋅ 4

0 7.4 10−β = ⋅



31

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.



P l ti B l



33

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



34

( )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)


Parameters for calculation

• 22 x 22 geometric grid

• Size independent kernel



P l ti B l



35

PP BPP A

Final distribution (Numerical)

PP A PP 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)


P l ti B l



36


Parameters for computation

• 13 x 13 geometric grid• Constant kernel


( )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



37


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



38

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



39

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)



40

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heinrich stefan coating

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