[PPT] - Numerical Solutions of Population-Balance Models in Particulate PowerPoint Presentation

SLIDE 1

0.1cm

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 2

0.1cm

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 3

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Aim Applications

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 4

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Aim Applications

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 5

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Aim Applications

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 6

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Aim Applications

Industrial Applications

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 7

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 8

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

General Population Balance Equation (PBE)

∂f(t, x) ∂t + ∂[G(t, x)f(t, x)] ∂x = Q±

agg(t, x) + Q± break(t, x) + Q+ nuc(t, x)

f(0, x) = f0 , x ∈ R+ :=]0, +∞[, t ≥ 0

1

f(t, x) is the number density function,

2

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.

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 9

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

phase boundary

˙ Qout ˙ Qin Q±

agg

Q±

break ∂[Gf] ∂x

Q−

dis

Q+

nuc

Figure: A schematic representation of different particulate processes

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 10

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Q±

agg(t, x) =1

2

x

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′) ∈ R2

+ .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 11

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Q±

break(t, x) = ∞

x

b(t, x, x′) S(x′) f(t, x′)dx′ − S(x) f(t, 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: µi(t) =

∞

xif(t, x)dx ,

i = 0, 1, 2, · · · , µ0(T) =total number of particles, µ1(t) =total volume of particles

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 12

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 13

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Reformulation of PBE

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) x = −∂Fagg(t, x) ∂x + ∂Fbreak(t, x) ∂x + ˜ Qnuc , ˜ f(0, x) = ˜ f0 , x ∈ R+ , t ≥ 0 , where ˜ f(t, x) := xf(t, x), ˜ Qnuc = xQ+

nuc and

Fagg(t, x) = −

x

∞
x−u

u β(t, u, v) f(t, u) f(t, v) dvdu (Filbet & Laurencot, 2004) Fbreak(t, x) =

x

∞
x

u b(t, u, v) S(v) f(t, v) dvdu .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 14

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 15

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Amino acid enantiomers

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 16

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Ternary Phase Diagram

Metastable zone Solubility curves seeds Equilibrium point Seeding with Real trajectories after seeding with Solvent

E E E1 E1 E1 E2 A M Tcryst Tcryst + ∆T

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 17

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Preferential Batch Crystallizer With Fines Dissolution

Loop

Fines Dissolution Settling Zone Annular Tank A (unseeded) counter crystals preferred crystals (seeded) F(k)(x, t) τ1, V1 τ2, V2 m(k)

liq,2, mw,2

m(k)

liq,1, mw,1

˙ V

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 18

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

Model for Preferential Crystallization

Balance for solid phase ∂f (k)(t, x) ∂t = −G(k)(t)∂f (k)(t, x) ∂x − 1 τ1 h(x)f (k)(t, x) , k ∈ [p, c]. Mass balance for liquid phase in cyrstallizer dm(k)(t) dt = ˙ m(k)

in (t) − ˙

m(k)

ut(t) − 3ρkvG(k)(t)

∞ x2f (k)(t, x)dx. f (k)(t, 0) = B(k)(t) G(k)(t) , w(k)(t) = m(k)(t) m(p)(t) + m(c)(t) + mW(t) S(k)(t) = w(k)(t) w(k)

eq

− 1, G(k)(t) = kg [S(k)(t)]α, kg ≥ 0, α ≥ 1 .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 19

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results General Population Balance Equation (PBE) Reformulation of PBE Preferential Crystallization Model

B(p)

0 (t) = k(p) b

S(p)(t)

b(p) µ(p)

3 (t)

B(c)

0 (t) = k(c) b e −

b(c) ln(S(c)(t)) 2

˙ m(k)

ut(t) = w(k)(t) ρliq(T)

˙ m(k)

in (t) = ˙

m(k)

ut(t − τ2) + kvρ

τ1 ∞ x3h(x)f (k)(t − τ2, x) dx .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 20

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 21

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

Domain Discretization

Regular/Irregular grid: Let N be a large integer and denote by (xi− 1

2 )i∈{1,··· ,N+1} a mesh of [xmin, xmax]. We set

x1/2 = xmin , xN+1/2 = xmax , xi+1/2 = xmin + i · ∆xi , ∀ i = 1, 2, · · · N − 1 . xi−1 xi xi+1 xi− 1

2

xi+ 1

2

x Here xi = (xi−1/2 + xi+1/2)/2 , ∆xi = xi+1/2 − xi−1/2 .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 22

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

Geometric grid: x1/2 = xmin , xi+1/2 = xmin + 2(i−N)/q(xmax − xmin) , ∀ i = 1, 2, · · · , N where the parameter q is any positive integer. Let Ωi =

xi−1/2, xi+1/2
for i ≥ 0. We approximate the initial data

f0(x) in each grid cell by fi = 1 ∆xi

Ωi

f0(x)dx .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 23

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 24

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

Method 1: Combination of MOC and FVS

Let us substitute the growth rate G(t, x) by dx dt := ˙ x(t) = G(t, x) . Then we have to solve: d˜ fi dt = − 1 ∆xi(t)

(Fagg)i+ 1

2 − (Fagg)i− 1 2

+

1 ∆xi(t)

(Fbreak)i+ 1

2 − (Fbreak)i− 1 2

+

Gi+ 1

2

˜ fi xi(t) −

Gi+ 1

2 − Gi− 1 2

˜

fi ∆xi(t) + ˜ Qi dxi+ 1

2

dt = Gi+ 1

2 ,

∀ i = 1, 2, · · · , N with i.c. ˜ f(0, xi) = ˜ f0(xi)

where

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 25

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

(Fagg)i+1/2 =

i

k=0

∆xk(t)˜ fk       

N

j=αi,k
Λh

j

β(x′, xk) x′ dx′˜ fj +

αi,k−1/2

xi+1/2−xk

β(x′, xk) x′ dx′˜ fαi,k−1        , (Fbreak)i+1/2 =

i

k=0
Ωk

x∗   

N

j=i+1

˜ fj

Ωj

b(x∗, x′) S(x′) x′ dx′    dx∗ + O(∆x3) . Here, the integer αi,k corresponds to the index of the cell such that xi+1/2(t) − xk(t) ∈ Ωαi,k−1(t).

A standard ODE-solver can be used to solve the above ODEs.

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 26

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

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

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 27

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

Method 2: Semidiscrete HR-schemes

Integration of PBE over the control volume Ωi =

xi− 1

2 , xi+ 1 2

implies
Ωi

∂˜ f(t, x) ∂t dx+

Ωi

∂[G(t, x)˜ f(t, x)] ∂x dx −

Ωi

G(t, x)˜ f(t, x) x dx = −

Ωi

∂Fagg(t, x) ∂x dx +

Ωi

∂Fbreak(t, x) ∂x dx +

Ωi

˜ Q(t, x) dx . Let ˜ fi = ˜ fi(t) and ˜ Qi = ˜ Qi(t) be the averaged values, then we have ∂˜ fi ∂t = − 1 ∆x

Fi+ 1

2 − Fi− 1 2

− 1

∆x

(Fagg)i+ 1

2 − (Fagg)i− 1 2

+
(Fbreak)i+ 1

2 − (Fbreak)i− 1 2

+

Gi+ 1

2

˜ fi xi + ˜ Qi , where Fi+ 1

2 = (G˜

f)i+ 1

2 and (Fagg) & (Fbreak) are as given in Method 1.

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 28

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Domain Discretization Numerical Method 1: Combination of MOC and FVS Numerical method 2: Semi-Discrete HR-Schemes

The flux Fi+ 1

2 at the right cell interface is given as (Koren, 1993):

Fi+ 1

2 =

Fi + 1

2Φ

ri+ 1

2

(Fi − Fi−1)
and Φ is defined as:

Φ(ri+ 1

2 ) = max

0, min
2ri+ 1

2 , min

1 3 + 2 3ri+ 1

2 , 2

.

The argument ri+ 1

2 of the function Φ is given as

ri+ 1

2 = Fi+1 − Fi + ε

Fi − Fi−1 + ε . Analogously, one can formulate the flux Fi− 1

2 . 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 2002, Koren 1993).

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 29

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Example 1: All Processes

The initial data: f(0, x) =

100

for 0.4 ≤ x ≤ 0.6 , 0.01 elsewhere . B.C.: f(t, 0) = 100 + 106 exp(−104(t − 0.215)2) . G = 1.0, β = 1.5 · 10−5, b(t, x, x′) = 2

x′ and S(x) = x2. The exact

solution in growth and nucleation case is: f(t, x) =    102 + 106 exp(−104((G′t − x) − 0.215)2) for 0 ≤ x ≤ Gt , 102 for 0.4 ≤ x − Gt ≤ 0.6 , 0.01 elsewhere . tmax = 0.5 and N = 200.

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 30

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Results of Method 1: MOC+FVM

0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Pure Growth and Nucleation

MOC Analytical Pure Growth and Nucleation 0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Nucleation + Growth + Aggregation

MOC+FVM Analytical Pure Growth and Nucleation 0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Nucleation+Growth + Breakage

MOC+FVM Analytical Pure Growth and Nucleation 0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Nucleation + Growth + Aggregation + Breakage

MOC+FVM Anaytical Pure Growth and Nucleation

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 31

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Results of Method 2: FVM

0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Pure Growth and Nucleation

FVM Analytical Pure Growth and Nucleation 0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Nucleation + Growth + Aggregation

FVM Analytical Pure Growth and Nucleation 0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Nucleation + Growth + Breakage

FVM Analytical Pure Growth and Nucleation 0.5 1 1.5 2 10

−2

10 10

2

10

4

10

6

volume number density

Nucleation +Growth + Aggregation + Breakage

FVM Analytical Pure growth and Nucleation

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 32

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Example 2: Pure Growth

The initial data are: f(0, x) = 1 × 1010 if 10 < x < 20 , elsewhere . with G = 1.0, N = 100, t = 60.

20 40 60 80 100 2 4 6 8 10 x 10

9

Uniform Mesh

crystal size particle density

Exact First Order LeVeque Koren 20 40 60 80 100 2 4 6 8 10 x 10

9

Adaptive Mesh

crystal size particle density

Exact First Order LeVeque Koren

For mesh adaptation we have used a moving mesh technique

f T. Tang et al. (2003)
S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 33

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Preferential Crystallization

Isothermal Case Temperature = 33Co. Non-isothermal Case T(t)[Co] = −1.24074e−7t3+4.50926e−5t2−0.00405556t+33 .

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 34

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Example 3: Preferential Crystallization

The initial data: f (p)(0, x) = 1 √ 2πσIa · 1 x · exp

−1

2 · ln(x) − µ σ 2 , f (c)(0, x) = 0 , with Ia = kV · ρs Ms µ(p)

3 (0) .

Here Ms = 2.5 · 10−3kg is the mass of initial seeds. The maximum crystal size is xmax = 0.005 m with N = 500 for t = 600 min.

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 35

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Example 3: Preferential Crystallization

1 2 3 4 5 x 10

−3

2 4 6 8 10 x 10

8 Without Fines Dissolution

crystal size [m] particle density [#/m]

Initial Data First Order HR−κ=−1 HR−κ=1/3 1 2 3 4 5 x 10

−3

2 4 6 8 10 x 10

8

With Fines Dissolution

crystal size [m] particle density [#/m]

Initial Data First Order HR−κ=−1 HR−κ=1/3 1 2 3 4 5 x 10

−3

1 2 3 4 5 x 10

9 Without Fines Dissolution

crystal size [m] particle density [#/m]

Initial Data First Order HR−κ=−1 HR−κ=1/3 1 2 3 4 5 x 10

−3

1 2 3 4 5 6 x 10

9

With Fines Dissolution

crystal size [m] particle density [#/m]

Initial Data First Order HR−κ=−1 HR−κ=1/3

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 36

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Example 3: Preferential Crystallization

100 200 300 400 500 600 1 2 3 4 5 6 x 10

−6

Isothermal Case

time [min] growth rate [m/min]

p−without fines c−without fines p−with fines c−with fines 100 200 300 400 500 600 1 2 3 4 5 6 x 10

−6

Non−isothermal Case

time [min] growth rate [m/min]

p−without fines c−without fines p−with fines c−with fines 100 200 300 400 500 600 1000 2000 3000 4000 5000

Isothermal Case

time [min] nucleation rate [m/min]

p−without fines c−without fines p−with fines c−with fines 100 200 300 400 500 600 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 x 10

4

Non−isothermal Case

time [min] nucleation rate [m/min]

p−without fines c−without fines p−with fines c−with fines

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 37

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results 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 2.2 4.4 HR-κ = 1/3 3.813 3.814 4.736 4.737 2.3 4.6 MOC 2.604 1.844 3.792 2.917 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 2.801 2.838 2.841 2.904 2.3 5.5 HR-κ = −1 2.873 2.875 2.962 2.965 3.1 7.5 HR-κ = 1/3 2.875 2.876 2.965 2.967 3.5 7.7 MOC 1.823 1.30 2.055 1.086 0.39 0.71

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 38

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Current Project

Results of isothermal (30 ◦C) seeded growth experiments with mandelic acid in water. Left:without counter enantiomer; Right: with counter-enantiomer (Lorenz et al., 2006).

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 39

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

For 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, 2006.
S. Qamar, and G. Warnecke

Solving population balance equations for two-component aggregation by a finite volume scheme.

Chem. Sci. Eng., 62, 679-693, 2006.
S. Qamar, and G. Warnecke

Numerical solution of population balance equations for nucleation growth and aggregation processes.

Compt. & Chem. Eng. (in press), 2007.
S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg

SLIDE 40

0.1cm

Motivation Mathematical Model Numerical Procedure Numerical Results Further Reading

Thanks for your Attention

S. Qamar, G. Warnecke

Otto-von-Guericke University Magdeburg