16 Modeling, Simulation and Optimization of Process Chains
575
Symbols Used
Abbreviations
ADJ Adjoint equation
FBSG Fluidized bed spray granulation
PBE Population balance equation PRI
Primal equation
PSD Particle size distribution
RSD
Relative standard deviation
Greek Letters
γ [a.u.] Weight function in nonlocal term
μ [m] Mean particle size
ξ [a.u.] Solution of characteristic equation π [–] 3.1415 . . .
σ [m]
Standard deviation
[–] Grade efficiency function
χ [–]
Characteristic function
[–] Domain of interest
Symbols
∂
[–]
Partial derivative
d
dz
[–]
Total derivative (to z)
∇
[–]
Gradient
t
[s]
Time
t f
[s]
Fnal time
n
[–]
Dimension of disperse properties
x
[m]
Particle radius
x
[a.u.]
Vector of disperse properties
G
[a.u.]
One-dimensional growth function
G
[a.u.]
Multi-dimensional growth function
G Ost
[–]
Ostwald ripening function
u G
[a.u.]
Time-dependent parameter in G
u G
[a.u.]
Time-dependent parameter in G
u in
[–]
Time-dependent influx rate
u out
[–]
Time-dependent extraction rate
u 0
[–]
Scaling factor of initial PSD
u
[a.u.]
Lower bound on process condition u
u
[a.u.]
Upper bound on process condition u
q
[a.u.]
PSD
q 3
[a.u.]
Mass-related PSD
q in
[a.u.]
PSD of external nuclei
q in,3
[a.u.]
Mass-related PSD of external nuclei
q 0
[a.u.]
Initial PSD
q d
[a.u.]
Desired PSD
W q
[a.u.]
Nonlocal term in (1)
x min
[a.u.]
Vector of minimal disperse properties
x min
[m]
Minimal radius
p
[a.u.]
Solution of (3)
Y p
[a.u.]
Nonlocal term in (3)
J
[a.u.]
Cost functional
U ad
[–]
Set of admissible process conditions
ˆ
ξ
[m]
Semi-discr. of ξ
q
[a.u.]
Semi-discr. of q
p
[a.u.]
Semi-discr. of p
ˆ
W q
[a.u.]
Approx. of W q
ˆ
q
[a.u.]
Approximation of q
ˆ
Y p
[a.u.]
Approx. of Y p
ˆ
p
[a.u.]
Approximation of p
d(u)
[–]
Number of discretization points of u
d
[–]
Number of discretization points
K
[–]
number of grid points
x k
[a.u.]
Grid point
x L
[a.u.]
Largest grid point
E n
[a.u.]
n-th moment
E
[a.u.]
First moment
μ d
[m]
Desired mean particle size
μ in
[m]
Mean particle size of entering PSD
μ out
[m]
Mean particle size of outgoing PSD
P
[W]
Milling power
q tot
[m −1 ]
PSD of total product
q crush
[m −1 ] PSD of crushed particles
q crush,3
[m −1 ]
Mass-related PSD of crushed particles
x cut
[m]
Cut size
˙
m
[kg·s −1 ] Mass flow
Norms
· · L p ((0,tf)) Norm on Lebesgue space of p-integrable functions on (0, t f )
· · H 1 ((0,tf)) Norm on Sobolev space of functions f ∈ L 2 ((0, t f )) with ˙
f ∈ L 2 ((0, t f ))
575
Symbols Used
Abbreviations
ADJ Adjoint equation
FBSG Fluidized bed spray granulation
PBE Population balance equation PRI
Primal equation
PSD Particle size distribution
RSD
Relative standard deviation
Greek Letters
γ [a.u.] Weight function in nonlocal term
μ [m] Mean particle size
ξ [a.u.] Solution of characteristic equation π [–] 3.1415 . . .
σ [m]
Standard deviation
[–] Grade efficiency function
χ [–]
Characteristic function
[–] Domain of interest
Symbols
∂
[–]
Partial derivative
d
dz
[–]
Total derivative (to z)
∇
[–]
Gradient
t
[s]
Time
t f
[s]
Fnal time
n
[–]
Dimension of disperse properties
x
[m]
Particle radius
x
[a.u.]
Vector of disperse properties
G
[a.u.]
One-dimensional growth function
G
[a.u.]
Multi-dimensional growth function
G Ost
[–]
Ostwald ripening function
u G
[a.u.]
Time-dependent parameter in G
u G
[a.u.]
Time-dependent parameter in G
u in
[–]
Time-dependent influx rate
u out
[–]
Time-dependent extraction rate
u 0
[–]
Scaling factor of initial PSD
u
[a.u.]
Lower bound on process condition u
u
[a.u.]
Upper bound on process condition u
q
[a.u.]
PSD
q 3
[a.u.]
Mass-related PSD
q in
[a.u.]
PSD of external nuclei
q in,3
[a.u.]
Mass-related PSD of external nuclei
q 0
[a.u.]
Initial PSD
q d
[a.u.]
Desired PSD
W q
[a.u.]
Nonlocal term in (1)
x min
[a.u.]
Vector of minimal disperse properties
x min
[m]
Minimal radius
p
[a.u.]
Solution of (3)
Y p
[a.u.]
Nonlocal term in (3)
J
[a.u.]
Cost functional
U ad
[–]
Set of admissible process conditions
ˆ
ξ
[m]
Semi-discr. of ξ
q
[a.u.]
Semi-discr. of q
p
[a.u.]
Semi-discr. of p
ˆ
W q
[a.u.]
Approx. of W q
ˆ
q
[a.u.]
Approximation of q
ˆ
Y p
[a.u.]
Approx. of Y p
ˆ
p
[a.u.]
Approximation of p
d(u)
[–]
Number of discretization points of u
d
[–]
Number of discretization points
K
[–]
number of grid points
x k
[a.u.]
Grid point
x L
[a.u.]
Largest grid point
E n
[a.u.]
n-th moment
E
[a.u.]
First moment
μ d
[m]
Desired mean particle size
μ in
[m]
Mean particle size of entering PSD
μ out
[m]
Mean particle size of outgoing PSD
P
[W]
Milling power
q tot
[m −1 ]
PSD of total product
q crush
[m −1 ] PSD of crushed particles
q crush,3
[m −1 ]
Mass-related PSD of crushed particles
x cut
[m]
Cut size
˙
m
[kg·s −1 ] Mass flow
Norms
· · L p ((0,tf)) Norm on Lebesgue space of p-integrable functions on (0, t f )
· · H 1 ((0,tf)) Norm on Sobolev space of functions f ∈ L 2 ((0, t f )) with ˙
f ∈ L 2 ((0, t f ))