14 Numerical Methods for Coupled Population Balance Systems …
497
As final simulation time, t end = 1800 s = 30 min was set, such that 36,000 time
steps had to be performed. For the flow, the mass flow rate at the inlet was 56 kg /h in
the whole time interval. The flow field was allowed to develop in the first 30 s. Then,
the crystals were inserted in the flow during the time interval [30 s, 40 s]. In contrast
to the experiment, where all crystals are inserted into the crystallizer basically at the
same time, the crystals enter in the simulations during a short time interval. There is
an algorithmic reason, since the SPS method works better if there is a rather uniform
distribution of crystals. The seed mass of the crystals was 10
−4 kg. It was divided
equally into two parts, one with crystals of diameter 75 μm and one of crystals with
diameter 125 μm. Both parts were represented via a log-normal distribution with
25 μm standard deviation.
Storage for 256 computational crystals was assigned to each mesh cell of the flow
domain Ω. As already mentioned in Sect. 2.4.1, each computational crystal represents
a number of physical crystals. In preliminary simulations, 5.0 × 10
8 # physical crystals /m
3
was found to be an upper bound for the concentration of physical crystals away from
the bottom of the device. In this region, a linear conversion to computational crystals
was used such that this upper bound corresponds to 256 computational crystals.
Close to the bottom, the concentration of physical crystals was often higher, due to
sedimentation. In this region, still a linear conversion was applied, but the conversion
factors were increased by 10 below 0.1 m and by 100 below 0.05 m. The choice of
the conversion factor is a purely numerical issue. It influences the computational
cost and the numerical precision, but otherwise it has no effect on the results for the
physical quantities. This setup led to roughly 150,000 computational crystals in Ω
after having completed the insertion at 40 s. This number is typically reduced by
around 50% at the end of a simulation because of agglomeration and in addition
since, as explained in Sect. 2.4.1, crystals that would leave through the inlet due to
sedimentation were removed from the simulations.
The coupling term for the temperature equation (1.2) is given by
F ener,growth (c, T , f ) = g T
Ω m
Gf (t, x, m) dm,
where the model for the growth term will be discussed below. Table 1 presents all other
coefficients for this equation. The Dirichlet boundary data for the temperature were
linearly interpolated in the time interval [0, t end ], where the initial temperature was
T (0 s) = 288.95 K, i.e. 15.8
◦ C, and the final temperature was T (3600 s) = 288.35 K,
which is 15.2
◦ C. The right-hand side of the concentration equation (1.3) is given by
F conc,growth (c, T , f ) = g c
Ω m
Gf (t, x, m) dm.
Also the coefficients for this equation are provided in Table 1. As initial condition
c(0 s) = 207 mol /m
3 was chosen, which corresponds to the saturation concentration
at 17
◦ C.
497
As final simulation time, t end = 1800 s = 30 min was set, such that 36,000 time
steps had to be performed. For the flow, the mass flow rate at the inlet was 56 kg /h in
the whole time interval. The flow field was allowed to develop in the first 30 s. Then,
the crystals were inserted in the flow during the time interval [30 s, 40 s]. In contrast
to the experiment, where all crystals are inserted into the crystallizer basically at the
same time, the crystals enter in the simulations during a short time interval. There is
an algorithmic reason, since the SPS method works better if there is a rather uniform
distribution of crystals. The seed mass of the crystals was 10
−4 kg. It was divided
equally into two parts, one with crystals of diameter 75 μm and one of crystals with
diameter 125 μm. Both parts were represented via a log-normal distribution with
25 μm standard deviation.
Storage for 256 computational crystals was assigned to each mesh cell of the flow
domain Ω. As already mentioned in Sect. 2.4.1, each computational crystal represents
a number of physical crystals. In preliminary simulations, 5.0 × 10
8 # physical crystals /m
3
was found to be an upper bound for the concentration of physical crystals away from
the bottom of the device. In this region, a linear conversion to computational crystals
was used such that this upper bound corresponds to 256 computational crystals.
Close to the bottom, the concentration of physical crystals was often higher, due to
sedimentation. In this region, still a linear conversion was applied, but the conversion
factors were increased by 10 below 0.1 m and by 100 below 0.05 m. The choice of
the conversion factor is a purely numerical issue. It influences the computational
cost and the numerical precision, but otherwise it has no effect on the results for the
physical quantities. This setup led to roughly 150,000 computational crystals in Ω
after having completed the insertion at 40 s. This number is typically reduced by
around 50% at the end of a simulation because of agglomeration and in addition
since, as explained in Sect. 2.4.1, crystals that would leave through the inlet due to
sedimentation were removed from the simulations.
The coupling term for the temperature equation (1.2) is given by
F ener,growth (c, T , f ) = g T
Ω m
Gf (t, x, m) dm,
where the model for the growth term will be discussed below. Table 1 presents all other
coefficients for this equation. The Dirichlet boundary data for the temperature were
linearly interpolated in the time interval [0, t end ], where the initial temperature was
T (0 s) = 288.95 K, i.e. 15.8
◦ C, and the final temperature was T (3600 s) = 288.35 K,
which is 15.2
◦ C. The right-hand side of the concentration equation (1.3) is given by
F conc,growth (c, T , f ) = g c
Ω m
Gf (t, x, m) dm.
Also the coefficients for this equation are provided in Table 1. As initial condition
c(0 s) = 207 mol /m
3 was chosen, which corresponds to the saturation concentration
at 17
◦ C.