Numerical Solutions of Population-Balance Models in Particulate Systems

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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 5, 007

Outline Outline 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Outline Aim Applications 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Aim Applications 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

Outline Aim Applications 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Industrial Applications Aim Applications Applications Pharmaceutical Chemical industries Biomedical science Aerosol formation Atmospheric physics Food industries

Outline 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

f(t, x) t [G(t, x)f(t, x)] + = Q ± agg x (t, x) + Q± break (t, x) + Q+ nuc (t, x) f(0, x) = f 0, x R + :=]0, + [, t 0 1 f(t, x) is the number density function, t denotes the time and x is an internal coordinate 3 G(t, x) is the growth/dissolution rate along x, 4 Q ± α (t, x) are the aggregation, breakage and nucleation terms for α = {agg, break, nuc}. 5 The entities in the population density can be crystals, droplets, molecules, cells, and so on.

Q out phase boundary Q + nuc Q ± agg [Gf] x Q dis Q ± break Q in Figure: A schematic representation of different particulate processes

Q ± agg(t, x) = 1 x 0 0 β(t, x, x x ) f(t, x ) f(t, x x)dx β(t, x, x ) f(t, x) f(t, x )dx. 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, 0 β(t, x, x ) = β(t, x, x), x ]0, x[, (x, x ) R +.

Q ± break (t, x) = b(t, x, x ) S(x ) f(t, x )dx S(x) f(t, x). 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. Moments: µ i (t) = 0 x i f(t, x)dx, i = 0, 1,,, µ 0 (T) =total number of particles, µ 1 (t) =total volume of particles

Outline 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Multiply the original PBE with x and re-arrange the terms, we get f(t, x) t + [(G f)(t, x)] x (G f)(t, x) = F agg(t, x) + F break(t, x) + Q nuc, x x x f(0, x) = f0, x R +, t 0, where f(t, x) := xf(t, x), Q nuc = xq + nuc and x F agg (t, x) = 0 F break (t, x) = x u x 0 u β(t, u, v) f(t, u) f(t, v) dvdu (Filbet & Laurencot, 004) x u b(t, u, v) S(v) f(t, v) dvdu.

Outline 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Amino acid enantiomers

Ternary Phase Diagram Metastable zone Equilibrium point Solvent Seeding with E 1 seeds T cryst E Solubility curves M A T cryst + T E 1 E Real trajectories after seeding with E 1

Preferential Batch Crystallizer With Fines Dissolution preferred crystals (seeded) counter crystals (unseeded) F (k) (x, t) m (k) liq,1, mw,1 τ 1, V 1 V Fines Dissolution Loop m (k) liq,, mw, 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 11111111110000000000 Annular Settling Zone τ, V Tank A

Model for Preferential Crystallization Balance for solid phase f (k) (t, x) t = G (k) (t) f (k) (t, x) x 1 τ 1 h(x)f (k) (t, x), k [p, c]. Mass balance for liquid phase in cyrstallizer dm (k) (t) dt = ṁ (k) in (t) ṁ(k) out (t) 3ρk vg (k) (t) 0 x f (k) (t, x)dx. f (k) (t, 0) = B(k) (t) G (k) (t), w(k) (t) = S (k) (t) = w(k) (t) w (k) eq m (k) (t) m (p) (t) + m (c) (t) + m W (t) 1, G (k) (t) = k g [S (k) (t)] α, k g 0, α 1.

B (p) 0 B (c) 0 ( ) b (t) = k(p) b S (p) (p) (t) µ (p) 3 (t) (t) = k(c) b b (c) e ln(s (c) (t)) ṁ (k) out (t) = w(k) (t)ρ liq (T) ṁ (k) in (t) = ṁ(k) out (t τ ) + k vρ τ 1 0 x 3 h(x)f (k) (t τ, x) dx.

Outline Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Domain Discretization Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes Regular/Irregular grid: Let N be a large integer and denote by ) i {1,,N+1} a mesh of [x min, x max ]. We set (x i 1 x 1/ = x min, x N+1/ = x max, x i+1/ = x min + i x i, i = 1,, N 1. x i 1 x i+ 1 x i 1 x i x i+1 x Here x i = (x i 1/ + x i+1/ )/, x i = x i+1/ x i 1/.

Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes Geometric grid: x 1/ = x min, x i+1/ = x min + (i N)/q (x max x min ), i = 1,,, N where the parameter q is any positive integer. Let Ω i = [ ] x i 1/, x i+1/ for i 0. We approximate the initial data f 0 (x) in each grid cell by f i = 1 f 0 (x)dx. x i Ω i

Outline Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes Method 1: Combination of MOC and FVS Let us substitute the growth rate G(t, x) by Then we have to solve: d f i dt = 1 [ (F agg ) x i (t) i+ 1 + G i+ fi ( 1 x i (t) G i+ 1 dx dt := ẋ(t) = G(t, x). (F agg ) i 1 G i 1 ] + 1 [ (F break ) x i (t) i+ 1 ) fi x i (t) + Q i (F break ) i 1 ] dx i+ 1 dt where = G i+ 1, i = 1,,, N with i.c. f(0, xi ) = f 0 (x i )

Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes i N β(x α i,k 1/, x k ) (F agg ) i+1/ = x k (t) f k x k=0 dx fj β(x, x k ) + x dx fαi,k 1, j=α i,k Λ h x i+1/ x k j i (F break ) i+1/ = x N S(x ) fj x dx dx + O( x 3 ). k=0 Ω k j=i+1 Ω j b(x, x ) Here, the integer α i,k corresponds to the index of the cell such that x i+1/ (t) x k (t) Ω αi,k 1(t). A standard ODE-solver can be used to solve the above ODEs.

Outline Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 1 Aim Application Areas 3 Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes 4

Method : Semidiscrete HR-schemes [ Integration of PBE over the control volume Ω i = Ω i f(t, x) t dx+ Ω i [G(t, x) f(t, x)] x = Ω i F agg (t, x) x Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes dx Ω i dx + Ω i x i 1 G(t, x) f(t, x) x F break (t, x) x, x i+ 1 dx ] implies dx + Let f i = f i (t) and Q i = Q i (t) be the averaged values, then we have f i t = 1 x + [ F i+ 1 [ (F break ) i+ 1 where F i+ 1 = (G f) i+ 1 F i 1 ] 1 x (F break ) i 1 [ (F agg ) i+ 1 (F agg ) i 1 ] + G i+ fi 1 + Q i, x i Ω i Q(t, x) dx. and (F agg ) & (F break ) are as given in Method 1. ]

Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method : Semi-Discrete HR-Schemes The flux F i+ 1 at the right cell interface is given as (Koren, 1993): (F i + 1 ( Φ F i+ 1 and Φ is defined as: = Φ(r i+ 1 ) = max ( ( 0, min r i+ 1 r i+ 1, min ) ) (F i F i 1 ) The argument r i+ 1 of the function Φ is given as r i+ 1 = F i+1 F i + ε F i F i 1 + ε. ( 1 3 + ))) 3 r i+,. 1 Analogously, one can formulate the flux F i 1. Here, ε = 10 10. There are several other limiting functions, namely, minmod, superbee and MC limiters, etc. Each of them leeds to a different HR-scheme (LeVeque 00, Koren 1993).

Example 1: All Processes Further Reading The initial data: f(0, x) = { 100 for 0.4 x 0.6, 0.01 elsewhere. B.C.: f(t, 0) = 100 + 10 6 exp( 10 4 (t 0.15) ). G = 1.0, β = 1.5 10 5, b(t, x, x ) = x and S(x) = x. The exact solution in growth and nucleation case is: 10 + 10 6 exp( 10 4 ((G t x) 0.15) ) for 0 x Gt, f(t, x) = 10 for 0.4 x Gt 0.6, 0.01 elsewhere. t max = 0.5 and N = 00.

Further Reading Results of Method 1: MOC+FVM 10 6 Pure Growth and Nucleation MOC 10 6 Nucleation + Growth + Aggregation MOC+FVM Analytical Pure Growth and Nucleation Analytical Pure Growth and Nucleation 10 4 10 4 number density 10 number density 10 10 0 10 0 10 0 0.5 1 1.5 volume 10 0 0.5 1 1.5 volume 10 6 10 4 Nucleation+Growth + Breakage MOC+FVM Analytical Pure Growth and Nucleation Nucleation + Growth + Aggregation + Breakage 10 6 MOC+FVM Anaytical Pure Growth and Nucleation 10 4 number density 10 10 0 number density 10 10 0 10 10 0 0.5 1 1.5 volume 0 0.5 1 1.5 volume

Results of Method : FVM Further Reading 10 6 Pure Growth and Nucleation FVM 10 6 Nucleation + Growth + Aggregation FVM Analytical Pure Growth and Nucleation Analytical Pure Growth and Nucleation 10 4 10 4 number density 10 number density 10 10 0 10 0 10 0 0.5 1 1.5 volume 10 0 0.5 1 1.5 volume 10 6 10 4 Nucleation + Growth + Breakage FVM Analytical Pure Growth and Nucleation Nucleation +Growth + Aggregation + Breakage 10 6 FVM Analytical Pure growth and Nucleation 10 4 number density 10 10 0 number density 10 10 0 10 10 0 0.5 1 1.5 volume 0 0.5 1 1.5 volume

Example : Pure Growth Further Reading The initial data are: f(0, x) = with G = 1.0, N = 100, t = 60. particle density 10 8 6 4 x 10 9 Exact First Order LeVeque Koren Uniform Mesh { 1 10 10 if 10 < x < 0, 0 elsewhere. particle density 10 8 6 4 x 10 9 Exact First Order LeVeque Koren Adaptive Mesh 0 0 0 40 60 80 100 crystal size 0 0 40 60 80 100 crystal size For mesh adaptation we have used a moving mesh technique of T. Tang et al. (003)

Preferential Crystallization Further Reading Isothermal Case Temperature = 33C o. Non-isothermal Case T(t)[C o ] = 1.4074e 7 t 3 +4.5096e 5 t 0.00405556t+33.

Further Reading Example 3: Preferential Crystallization The initial data: f (p) (0, x) = [ 1 1 πσia x exp 1 ( ) ] ln(x) µ, σ f (c) (0, x) = 0, with I a = k V ρ s µ (p) M 3 (0). s Here M s =.5 10 3 kg is the mass of initial seeds. The maximum crystal size is x max = 0.005 m with N = 500 for t = 600 min.

Further Reading Example 3: Preferential Crystallization particle density [#/m] x 10 8 10 8 6 4 Without Fines Dissolution Initial Data First Order HR κ= 1 HR κ=1/3 particle density [#/m] 10 x 108 With Fines Dissolution Initial Data First Order 8 HR κ= 1 HR κ=1/3 6 4 particle density [#/m] 0 0 1 3 4 5 crystal size [m] x 10 3 x 10 9 5 4 3 1 Without Fines Dissolution Initial Data First Order HR κ= 1 HR κ=1/3 particle density [#/m] 0 0 1 3 4 5 crystal size [m] x 10 3 6 x 109 With Fines Dissolution Initial Data First Order 5 HR κ= 1 HR κ=1/3 4 3 1 0 0 1 3 4 5 crystal size [m] x 10 3 0 0 1 3 4 5 crystal size [m] x 10 3

Further Reading Example 3: Preferential Crystallization growth rate [m/min] 6 x 10 6 Isothermal Case p without fines c without fines 5 p with fines c with fines 4 3 1 growth rate [m/min] 6 x 10 6 Non isothermal Case p without fines c without fines 5 p with fines c with fines 4 3 1 nucleation rate [m/min] 5000 4000 3000 000 1000 0 0 100 00 300 400 500 600 time [min] Isothermal Case p without fines c without fines p with fines c with fines 0 0 100 00 300 400 500 600 time [min] nucleation rate [m/min] 3.5.5 1.5 0.5 0 0 100 00 300 400 500 600 time [min] 4 x 104 Non isothermal Case 3 1 0 p without fines c without fines p with fines c with fines 0.5 0 100 00 300 400 500 600 time [min]

Further Reading Example 3: Mass Preservation in the Schemes Table: Percentage errors in mass preservation without fines dissolution. Method Isothermal Non-isothermal CPU time (s) (isothermal) N=500 N=1000 N=500 N=1000 N=500 N=1000 First order 3.737 3.775 4.460 4.669 1.5 3.1 HR-κ = 1 3.811 3.813 4.733 4.736. 4.4 HR-κ = 1/3 3.813 3.814 4.736 4.737.3 4.6 MOC.604 1.844 3.79.917 0.34 0.41 Table: Percentage errors in mass preservation with fines dissolution. Method Isothermal Non-isothermal CPU time (s) (isothermal) N=500 N=1000 N=500 N=1000 N=500 N=1000 First order.801.838.841.904.3 5.5 HR-κ = 1.873.875.96.965 3.1 7.5 HR-κ = 1/3.875.876.965.967 3.5 7.7 MOC 1.83 1.30.055 1.086 0.39 0.71

Current Project Further Reading Results of isothermal (30 C) seeded growth experiments with mandelic acid in water. Left:without counter enantiomer; Right: with counter-enantiomer (Lorenz et al., 006).

For Further Reading Further Reading S. Qamar, M. Elsner, I. Angelov, G. Warnecke and A. Seidel-Morgenstern A comparative study of high resolution schemes for solving population balances in crystallization. Compt. & Chem. Eng., Vol. 30, 1119-1131, 006. S. Qamar, and G. Warnecke Solving population balance equations for two-component aggregation by a finite volume scheme. Chem. Sci. Eng., 6, 679-693, 006. S. Qamar, and G. Warnecke Numerical solution of population balance equations for nucleation growth and aggregation processes. Compt. & Chem. Eng. (in press), 007.

Further Reading Thanks for your Attention

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