464
S. Schiele et al.
where x is a uniformly distributed random number between 0 and 1. The calculation
of the time steps for the shear rate dependent agglomeration kernel was more complex
and explained in detail by Briesen [48] and Kovaˇ cevi´ c and Briesen [6].
With the time step size at hand, growth of the particle population was calculated.
For the simple and complex approach this was done in a straightforward way with
h C,new = h C + tG h for each particle in the simple approach or with h C,part new,i =
h C,part,i + tG h for each face distance i of each primary particle for the complex
approach.
The population balance of the 2D approach did not directly contain the face
distances. The growth was modeled based on the growth of the volume and surface
area directly: V new = V + tG V and A new = A + tG A . The growth rate of the
volume is given as G V = AG h . Until this point G A remained unknown. An empirical
relation with the parameters p 1 and p 2 was proposed:
G A = p 1 A
p 2 V
1−2p 2
3
G h
(36)
Kovaˇ cevi´ c and Briesen [6] derived that for single crystals p 1 is in fact a function
of p 2 . For octahedral particles the following function was found:
p 1 =
24
√
3
12
√
3
p 2
4
√
3
1−2p 2
3
(37)
p 1 and p 2 can be interpreted as geometric factors that are specific for certain geometries and do not only differ depending on whether a particle is an agglomerate or not,
but also on the geometry of each individual agglomerate. Kovaˇ cevi´ c and Briesen
[6] studied various values for p 1 and p 2 and find that—even though the values
slightly vary for different simulation cases—reasonable average values can be found
to describe particles in a process. To study different parameters, they used the complex method to generate 1000 isolated agglomerates. The term isolated stems from
the fact that they are not simulated within a reaction environment. p 1 and p 2 are only
dependent on the geometry. How a certain geometry is reached is not important for
their calculation. To produce a set of agglomerates, two particles were brought to
contact. Subsequently their face distances were displaced in several steps without
consideration of a supersaturation. Therefore, the time-consuming calculation of the
mass balances was not necessary. After each face displacement step, both volume and
area were calculated and then allowed a correlation between growth rate by means
of face displacement, and the change of the surface area. The procedure is illustrated
in Fig. 25.
With p 2 = 1.135 a hypothetical crystallization process was simulated with the
scheme displayed in Fig. 24. The results are shown in Fig. 26. The agglomeration
kernel was chosen relatively high compared to a real crystallization problem to make
the effect of agglomeration clearer.
S. Schiele et al.
where x is a uniformly distributed random number between 0 and 1. The calculation
of the time steps for the shear rate dependent agglomeration kernel was more complex
and explained in detail by Briesen [48] and Kovaˇ cevi´ c and Briesen [6].
With the time step size at hand, growth of the particle population was calculated.
For the simple and complex approach this was done in a straightforward way with
h C,new = h C + tG h for each particle in the simple approach or with h C,part new,i =
h C,part,i + tG h for each face distance i of each primary particle for the complex
approach.
The population balance of the 2D approach did not directly contain the face
distances. The growth was modeled based on the growth of the volume and surface
area directly: V new = V + tG V and A new = A + tG A . The growth rate of the
volume is given as G V = AG h . Until this point G A remained unknown. An empirical
relation with the parameters p 1 and p 2 was proposed:
G A = p 1 A
p 2 V
1−2p 2
3
G h
(36)
Kovaˇ cevi´ c and Briesen [6] derived that for single crystals p 1 is in fact a function
of p 2 . For octahedral particles the following function was found:
p 1 =
24
√
3
12
√
3
p 2
4
√
3
1−2p 2
3
(37)
p 1 and p 2 can be interpreted as geometric factors that are specific for certain geometries and do not only differ depending on whether a particle is an agglomerate or not,
but also on the geometry of each individual agglomerate. Kovaˇ cevi´ c and Briesen
[6] studied various values for p 1 and p 2 and find that—even though the values
slightly vary for different simulation cases—reasonable average values can be found
to describe particles in a process. To study different parameters, they used the complex method to generate 1000 isolated agglomerates. The term isolated stems from
the fact that they are not simulated within a reaction environment. p 1 and p 2 are only
dependent on the geometry. How a certain geometry is reached is not important for
their calculation. To produce a set of agglomerates, two particles were brought to
contact. Subsequently their face distances were displaced in several steps without
consideration of a supersaturation. Therefore, the time-consuming calculation of the
mass balances was not necessary. After each face displacement step, both volume and
area were calculated and then allowed a correlation between growth rate by means
of face displacement, and the change of the surface area. The procedure is illustrated
in Fig. 25.
With p 2 = 1.135 a hypothetical crystallization process was simulated with the
scheme displayed in Fig. 24. The results are shown in Fig. 26. The agglomeration
kernel was chosen relatively high compared to a real crystallization problem to make
the effect of agglomeration clearer.