Outline Numerical Solutions of Population-Balance Models in Particulate Systems Shamsul Qamar Gerald Warnecke Institute for Analysis and Numerics Otto-von-Guericke University Magdeburg, Germany In collaboration with Max-Planck Institute for Dynamical Systems, Magdeburg Harrachov, August 19–25, 2007 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Outline Outline 1 Motivation Aim Application Areas 2 Mathematical Model General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model 3 Numerical Procedure Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes 4 Numerical Results 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation Mathematical Model Aim Numerical Procedure Applications Numerical Results Outline 1 Motivation Aim Application Areas 2 Mathematical Model General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model 3 Numerical Procedure Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes 4 Numerical Results 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation Mathematical Model Aim Numerical Procedure Applications Numerical Results Motivation Aim To model and simulate nucleation, growth, aggregation and Breakage phenomena in processes engineering by solving population balance equations (PBEs). Numerical Methods To solve population balance models we use the high resolution finite volume schemes as well as their combination with the method of characteristics 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation Mathematical Model Aim Numerical Procedure Applications Numerical Results Outline 1 Motivation Aim Application Areas 2 Mathematical Model General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model 3 Numerical Procedure Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes 4 Numerical Results 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation Mathematical Model Aim Numerical Procedure Applications Numerical Results Industrial Applications Applications Pharmaceutical Chemical industries Biomedical science Aerosol formation Atmospheric physics Food industries 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results Outline 1 Motivation Aim Application Areas 2 Mathematical Model General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model 3 Numerical Procedure Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes 4 Numerical Results 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results General Population Balance Equation (PBE) ∂ f ( t , x ) + ∂ [ G ( t , x ) f ( t , x )] agg ( t , x ) + Q ± break ( t , x ) + Q + nuc ( t , x ) = Q ± ∂ t ∂ x f ( 0 , x ) = f 0 , x ∈ R + :=] 0 , + ∞ [ , t ≥ 0 f ( t , x ) is the number density function, 1 t denotes the time and x is an internal coordinate 2 G ( t , x ) is the growth/dissolution rate along x , 3 α ( t , x ) are the aggregation, breakage and nucleation Q ± 4 terms for α = { agg , break , nuc } . 5 The entities in the population density can be crystals, droplets, molecules, cells, and so on. 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results ˙ Q out phase boundary Q + nuc Q ± agg ∂ [ Gf ] ∂ x Q − Q ± dis break ˙ Q in Figure: A schematic representation of different particulate processes 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results x agg ( t , x ) = 1 � β ( t , x ′ , x − x ′ ) f ( t , x ′ ) f ( t , x ′ − x ) dx ′ Q ± 2 0 ∞ � β ( t , x , x ′ ) f ( t , x ) f ( t , x ′ ) dx ′ . − 0 Where: β = β ( t , x , x ′ ) is the rate at which the aggregation of two particles with respective volumes x and x ′ produces a particle of volume x + x ′ and is a nonnegative symmetric function, x ′ ∈ ] 0 , x [ , 0 ≤ β ( t , x , x ′ ) = β ( t , x ′ , x ) , ( x , x ′ ) ∈ R 2 + . 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results ∞ � b ( t , x , x ′ ) S ( x ′ ) f ( t , x ′ ) dx ′ − S ( x ) f ( t , x ) . break ( t , x ) = Q ± x b := b ( t , x , x ′ ) is the probability density function for the formation of particles of size x from particle of size x ′ . The selection function S ( x ′ ) describes the rate at which particles are selected to break. ∞ � x i f ( t , x ) dx , Moments: µ i ( t ) = i = 0 , 1 , 2 , · · · , 0 µ 0 ( T ) = total number of particles, µ 1 ( t ) = total volume of particles 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results Outline 1 Motivation Aim Application Areas 2 Mathematical Model General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model 3 Numerical Procedure Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes 4 Numerical Results 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results Reformulation of PBE Multiply the original PBE with x and re-arrange the terms, we get ∂ ˜ f ( t , x ) + ∂ [( G ˜ f )( t , x )] − ( G ˜ f )( t , x ) = − ∂ F agg ( t , x ) + ∂ F break ( t , x ) + ˜ Q nuc , ∂ t ∂ x x ∂ x ∂ x f ( 0 , x ) = ˜ ˜ f 0 , x ∈ R + , t ≥ 0 , where ˜ f ( t , x ) := xf ( t , x ) , Q nuc = x Q + ˜ nuc and x ∞ � � F agg ( t , x ) = − u β ( t , u , v ) f ( t , u ) f ( t , v ) dvdu ( Filbet & Laurencot, 2004 ) x − u 0 x ∞ � � F break ( t , x ) = u b ( t , u , v ) S ( v ) f ( t , v ) dvdu . x 0 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results Outline 1 Motivation Aim Application Areas 2 Mathematical Model General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model 3 Numerical Procedure Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes 4 Numerical Results 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results Amino acid enantiomers 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
Motivation General Population Balance Equation (PBE) Mathematical Model Reformulation of PBE Numerical Procedure Preferential Crystallization Model Numerical Results Ternary Phase Diagram Solvent Metastable Equilibrium point zone Seeding with T cryst E 1 seeds Solubility E E curves M A T cryst + ∆ T E 1 E 2 E 1 Real trajectories after seeding with 0.1cm S. Qamar, G. Warnecke Otto-von-Guericke University Magdeburg
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