5 Development of a Dynamic-Physical Process Model for Sieving
181
models, which consider the whole fine material as one lumped undersized fraction, by
j
k=1 |m sim (k) − m mod (k)|
/j, where j is the total number of considered time steps k. For models, which consider the different undersized particle
classes i as fractions, the average of the obtained fractional deviations is considered by
l
i=1
j
k=1 |m sim (i, k) − m mod (i, k)|
/( j · l), where l is the number of
undersized fractions.
In Fig. 17 the summed up deviations of all performed simulations using spherical
particles (Fig. 17a, b), double cones (Fig. 17c, d) and volume equivalent cylinders
(Fig. 17e, f) for varying amplitudes (Fig. 17a, c, e) and frequencies (Fig. 17b, d, f)
for all considered process models are presented. Almost all models have problems
for representing flat residual mass curves, which are caused by small amplitudes or
low and high vibration frequencies. The kinetic models by Andreev et al. (model
β), Standish (model δ) and Subasinghe et al. (model ζ) as well as the probabilistic
models by Subasinghe et al. (model θ) and Ferrara et al. (model ν) as well as the fractional complex model by Soldinger (model π) demonstrate the best overall results.
However, they rely on empirical model parameters (model β) or, because of their
non-explicit functional form, require long adjustment times during fitting (model ν).
At different amplitudes, the models β and γ show particularly suitable results. The
functional forms with an additional adjustable parameter as exponent of the time t
can compensate well for the variations in residual mass on the screen caused by different amplitudes. Due to optimization for continuous screening and due to a simple
parameter structure, deviations from the model by Trumic and Magdalinovic (model
γ) are largest, although acceptable results for some small amplitudes and frequencies of e.g. 6.9, 20.7 and 55.2 Hz are obtained. However, model γ has difficulties to
represent simulations with strong residual mass decrease and complete depletion of
material caused by large amplitudes and passage optimized frequencies, respectively.
In all investigated cases (Fig. 17) the models α, ε, η, θ, κ, λ and μ indicate deviations
of the same order of magnitude. Models κ, λ and μ rely directly on the screening
efficiency of the rate law (model α), where only the ranges of the model parameters
are changed. These altered ranges may affect the setting of the model parameters
(easier guess of initial values and quicker convergence) or improve their physical
meaning as e.g. in model κ, where passage probabilities are introduced, but overall
model accuracy is unaffected.
In addition to quantitatively better results, nearly all models represent the simulations with complex shaped particles qualitatively similar to those with spheres.
Total deviations are smaller for double cones in comparison with volume equivalent
cylinders. Model ρ by Yoshida et al. shows despite its complexity comparatively
large deviations, because it is derived for a batch simulation setup differing from the
setup used here. From the comparison of the two models by Soldinger (models ξ
and π), it can be concluded that a representation of different fractions in a lumped
way can reduce model accuracy, especially in complex models where stratification
is considered in detail.
181
models, which consider the whole fine material as one lumped undersized fraction, by
j
k=1 |m sim (k) − m mod (k)|
/j, where j is the total number of considered time steps k. For models, which consider the different undersized particle
classes i as fractions, the average of the obtained fractional deviations is considered by
l
i=1
j
k=1 |m sim (i, k) − m mod (i, k)|
/( j · l), where l is the number of
undersized fractions.
In Fig. 17 the summed up deviations of all performed simulations using spherical
particles (Fig. 17a, b), double cones (Fig. 17c, d) and volume equivalent cylinders
(Fig. 17e, f) for varying amplitudes (Fig. 17a, c, e) and frequencies (Fig. 17b, d, f)
for all considered process models are presented. Almost all models have problems
for representing flat residual mass curves, which are caused by small amplitudes or
low and high vibration frequencies. The kinetic models by Andreev et al. (model
β), Standish (model δ) and Subasinghe et al. (model ζ) as well as the probabilistic
models by Subasinghe et al. (model θ) and Ferrara et al. (model ν) as well as the fractional complex model by Soldinger (model π) demonstrate the best overall results.
However, they rely on empirical model parameters (model β) or, because of their
non-explicit functional form, require long adjustment times during fitting (model ν).
At different amplitudes, the models β and γ show particularly suitable results. The
functional forms with an additional adjustable parameter as exponent of the time t
can compensate well for the variations in residual mass on the screen caused by different amplitudes. Due to optimization for continuous screening and due to a simple
parameter structure, deviations from the model by Trumic and Magdalinovic (model
γ) are largest, although acceptable results for some small amplitudes and frequencies of e.g. 6.9, 20.7 and 55.2 Hz are obtained. However, model γ has difficulties to
represent simulations with strong residual mass decrease and complete depletion of
material caused by large amplitudes and passage optimized frequencies, respectively.
In all investigated cases (Fig. 17) the models α, ε, η, θ, κ, λ and μ indicate deviations
of the same order of magnitude. Models κ, λ and μ rely directly on the screening
efficiency of the rate law (model α), where only the ranges of the model parameters
are changed. These altered ranges may affect the setting of the model parameters
(easier guess of initial values and quicker convergence) or improve their physical
meaning as e.g. in model κ, where passage probabilities are introduced, but overall
model accuracy is unaffected.
In addition to quantitatively better results, nearly all models represent the simulations with complex shaped particles qualitatively similar to those with spheres.
Total deviations are smaller for double cones in comparison with volume equivalent
cylinders. Model ρ by Yoshida et al. shows despite its complexity comparatively
large deviations, because it is derived for a batch simulation setup differing from the
setup used here. From the comparison of the two models by Soldinger (models ξ
and π), it can be concluded that a representation of different fractions in a lumped
way can reduce model accuracy, especially in complex models where stratification
is considered in detail.