Tackling the inverse problem in multidimensional granulation - - PowerPoint PPT Presentation

Tackling the inverse problem in multidimensional granulation modelling

Andreas Braumann, Markus Kraft

Mannheim, 8 September 2009

Andreas Braumann ab589@cam.ac.uk

+

Wet granulation – A start

• Products
• “Lump” small particles into bigger entities

– Improve handling (storage, transport, safety,...) – Creation of micro mixtures (segregation, distribution of active components)

• Here: Model the process with

multidimensional population balance model

Andreas Braumann ab589@cam.ac.uk

Model – Particle description

• composition of a granule

solid material binder on solid pores binder in pores reaction products

• particle shape: sphere

5 independent variables for particle description

solid material binder on solid pores binder in pores reaction products

Andreas Braumann ab589@cam.ac.uk

Model - Transformations

• Addition of binder
• Coalescence
• Compaction
• Breakage
• Penetration
• Reaction

Andreas Braumann ab589@cam.ac.uk

Using the model

• So what? – Run simulations!

Establish x’s through experiments Not quite!

solid material binder on solid pores binder in pores reaction products

Simulation results x1...xK

source: http://www.loedige.de/

solid material binder on solid pores binder in pores reaction products

e.g. K0, kreac

Andreas Braumann ab589@cam.ac.uk

Unknown parameters

• unknown parameters:

– coalescence rate constant K0 = ? – compaction rate constant kcomp = ? – breakage rate constant katt = ? – reaction rate constant kreac = ?

Andreas Braumann ab589@cam.ac.uk

The idea

Mathematical model Experimental

• bservations
• state variables
• governing equations
• model parameters

+ experimental uncertainties + a priori uncertainties

• particle properties
• bulk properties
• etc.
• unknown model parameters
• uncertainties of model parameters

Determine Model predictions Granulator

Granulation model

with uncertainties Mathematical model Experimental

• bservations
• state variables
• governing equations
• model parameters

+ experimental uncertainties + a priori uncertainties

• particle properties
• bulk properties
• etc.
• unknown model parameters
• uncertainties of model parameters

Determine

• unknown model parameters
• uncertainties of model parameters

Determine Model predictions Granulator

Granulation model

with uncertainties

Andreas Braumann ab589@cam.ac.uk

Theory

• Experimental data
• Model response with paramaters x
• and

( )

x B A x + = η

For a simple linear model, K=1

( ) ( )

ξ ξ η c x B A c x + + = , ,

( ) ( ) [ ]

x B A c x E x + = = ξ η µ , ,

( ) ( ) ( )

2 2

c B c x c = = ξ η σ , , Var

exp exp exp

σ η η ± = ξ c x x + =

( )

x η η =

( )

K

x x ,...,

1

= x with

with uncertainty factor c and standard normally distributed ξ

Andreas Braumann ab589@cam.ac.uk

Theory II

• Optimal values for parameters and associated uncertainties
• with objective function
• and constraints

{ }

) , ( min arg ) , (

,

c x c x

c x

Φ =

∗ ∗

( )

∑

=

− + − = Φ

N i i i i i 1 2 exp 2 exp ,

)] ( [ )] ( [ ) , ( c , x c , x c x σ σ µ η ) , , (

up , , , low , ,

K k x x x

k k k

K 1 = ≤ ≤

) (0

c c ≤ ≤

Andreas Braumann ab589@cam.ac.uk

Get on with it

• Problem: Running the optimisation on the

complex model is computationally expensive

• How to use?

find substitute: Response surfaces = approximation of process behaviour

ε η + = ) ( ) ( x y x

sim

Andreas Braumann ab589@cam.ac.uk

Response surfaces

• approximate process (model) behaviour locally
• e.g., 1st order response surface
• : process sensitivities with respect to

k

β

surface

• f

parameters , variables

• f

number surface response ) ( with ) ( = = = + =

∑

= k K k k k

K x x x β β η β β η

1

k

x

Andreas Braumann ab589@cam.ac.uk

Methodology in short

x1 x2 y

• 1
• 1

1 1 x1 x2 y

• 1
• 1

1 1

∑

=

+ =

K k k k x

x

1

β β η ) (

x1 x2 y

• 1
• 1

1 1 x1 x2 y

• 1
• 1

1 1

parameters: K0, kreac

scenario 4 scenario 1 scenario 2 scenario 3

x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1

x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1 x1 x2 y

• 1
• 1

1

response surface experiment 1

Andreas Braumann ab589@cam.ac.uk

Example

• experimental data from Simmons et al. (2006)

– bench scale granulation of non-pareils (sugar) with PEG4000/water binder – run at 900 and 1200 rpm impeller speed – examine amount of agglomerates – 4 sampling times

8 experimental data sets

Andreas Braumann ab589@cam.ac.uk

Example – 900 rpm data

Parameters and uncertainties using all 4 sets K0 = (1.18 ± 0.00) ·10-10 m3 kcomp = 0.20 ± 0.04 s/m katt = (4.8 ± 1.0) ·107 s/m5 kreac =(3.6 ± 0.0) ·10-9 m/s

Andreas Braumann ab589@cam.ac.uk

Example – all data

Parameters and uncertainties using all 8 sets K0 = (1.23 ± 0.00) ·10-10 m3 , kcomp = 0.20 ± 0.05 s/m katt = (4.9 ± 0.0) ·107 s/m5 , kreac =(4.000 ± 0.003) ·10-9 m/s

900 rpm 1200 rpm

Andreas Braumann ab589@cam.ac.uk

All data – a closer look

• utlier ?

exclude from procedure revised set of parameters

Andreas Braumann ab589@cam.ac.uk

Summary

• Population balance model for granulation

– 5 dimensional particle space – several subprocesses

• Approach for parameter and uncertainty

estimation using experimental data

• Falsification of models

Andreas Braumann ab589@cam.ac.uk

Further information

CoMo Group preprints

http://como.cheng.cam.ac.uk/index.php?Page=Preprints