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

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 reaction solid products material � binder on solid � pores � binder in pores binder in pores binder on solid � reaction products pores • particle shape: sphere 5 independent variables for particle description 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 source: http://www.loedige.de/ • So what? – Run simulations! Not quite! e.g. K 0 , k reac x 1 ... x K reaction reaction solid solid products products material material Simulation binder in pores binder in pores binder on solid binder on solid results pores pores Establish x ’s through experiments Andreas Braumann ab589@cam.ac.uk

Unknown parameters • unknown parameters: – coalescence rate constant K 0 = ? – compaction rate constant k comp = ? – breakage rate constant k att = ? – reaction rate constant k reac = ? Andreas Braumann ab589@cam.ac.uk

The idea Granulation model Granulation model with uncertainties with uncertainties Mathematical model Mathematical model Granulator Granulator • particle properties • particle properties • state variables • state variables • bulk properties • bulk properties • governing equations • governing equations • etc. • etc. • model parameters • model parameters Experimental Experimental Model predictions Model predictions observations observations + a priori uncertainties + a priori uncertainties + experimental uncertainties + experimental uncertainties Determine Determine Determine • unknown model parameters • unknown model parameters • unknown model parameters • uncertainties of model parameters • uncertainties of model parameters • uncertainties of model parameters Andreas Braumann ab589@cam.ac.uk

Theory • Experimental data η = η ± σ exp exp exp 0 • Model response with paramaters x ( ) ( ) η = η = x ,..., x with x x K 1 = + ξ x x c • and with uncertainty factor c and 0 standard normally distributed ξ For a simple linear model, K =1 ( ) ( ) ( ) η ξ = + + ξ η = + x , c , A B x c x A B x 0 [ ] ( ) ( ( ) ) ( ) ( ) µ = η ξ = + σ = η ξ = 2 2 x E x , c , A B x c Var x , c , B c 0 0 0 0 Andreas Braumann ab589@cam.ac.uk

Theory II • Optimal values for parameters and associated uncertainties { } ∗ ∗ = Φ ( , ) arg min ( , ) x c x c 0 0 , x c 0 • with objective function ( ) N ∑ Φ = η − µ + σ − σ exp 2 exp 2 ( x , c ) [ ( x , c )] [ ( x , c )] 0 0 0 0 , i i i i = 1 i • and constraints ≤ ≤ = K x x x ( k , , K ) 1 , k , low , k , k , up 0 0 0 ≤ c ≤ ( 0 ) c 0 Andreas Braumann ab589@cam.ac.uk

Get on with it • Problem: Running the optimisation on the complex model is computationally expensive find substitute: Response surfaces = approximation of process behaviour η = + ε ( ) ( ) x y x sim • How to use? Andreas Braumann ab589@cam.ac.uk

Response surfaces • approximate process (model) behaviour locally • e.g., 1 st order response surface K ∑ η = β + β ( x ) x 0 k k = k 1 η = with ( x ) response surface = K number of variables β β = , parameters of surface k 0 β x • : process sensitivities with respect to k k Andreas Braumann ab589@cam.ac.uk

Methodology in short response surface y y y y y y scenario 1 y y y y y y scenario 2 y y x 2 x 2 x 2 x 2 -1 -1 -1 -1 x 1 x 1 x 1 x 1 1 1 1 1 -1 -1 -1 -1 x 2 x 2 x 2 x 2 1 1 1 1 experiment 1 y y y y x 2 x 2 x 2 x 2 x 2 x 2 scenario 3 scenario 4 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 x 2 x 2 x 2 x 2 -1 -1 -1 -1 x 1 x 1 x 1 x 1 -1 -1 1 1 -1 -1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 K ∑ η = β + β ( x ) k x parameters: K 0 , k reac k 0 = k 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 K 0 = (1.18 ± 0.00) ·10 -10 m 3 k comp = 0.20 ± 0.04 s/m k att = (4.8 ± 1.0) ·10 7 s/m 5 k reac =(3.6 ± 0.0) ·10 -9 m/s Andreas Braumann ab589@cam.ac.uk

Example – all data 900 rpm 1200 rpm Parameters and uncertainties using all 8 sets K 0 = (1.23 ± 0.00) ·10 -10 m 3 , k comp = 0.20 ± 0.05 s/m k att = (4.9 ± 0.0) ·10 7 s/m 5 , k reac =(4.000 ± 0.003) ·10 -9 m/s Andreas Braumann ab589@cam.ac.uk

All data – a closer look outlier ? revised set of parameters exclude from procedure 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 Andreas Braumann ab589@cam.ac.uk