118
4. SIMULATING GROWTH AND FORM
(a)
~~
~
(e)
(b)
(f)
(c)
(g)
...
..
~.~
7~
~
.. '.
(d)
. ~
(h)
Fig.4.24a-h. Slices through the middle
of the aggregates in the xy-plane at
zmax f z, mono directional flow experiment : (a-h) , Peeler number increases
from approximately 0.0 to 3.0.
nutrient distribution and aggregate; in this experiment Peis set to the value
3.000. In this picture the aggregate is seen from above.
The results of the simulation experiments are summarized in Table 4.2.
We have estimated error bounds on the various measurements by repeating
the simulations at Pe = 0.0150 and Pe = 3.000 four times. The ratio R of the
total sink nodes to the total cluster size indicates the compactness of the aggregates. The typical cluster size is about 6-4 x 10
4 • The fractal dimension Dt,ox
ofthe surface ofthe aggregate was determined using a three-dimensional version of the (cube) box-counting method described by Feder (1988) (compare
the two-dimensional version in (j.ioj), Dt,ox measures the space-filling properties of the surface of the aggregate. In three dimensions its value varies
from a minimum of 2, for a solid object with a perfectly smooth surface, to
a maximum of 3 for a solid with a space-filling surface. In the simulations we
have determined the number of sink nodes that have a nutrient absorption
between a and a + L\a for the aggregates obtained at various Peeler numbers.
The number of sink nodes with absorption rate a, N(a), can be related to a
by a power law:
N(a) - a-Dabs
(4.19)
Pe
0.0150
0.1322 0.2521 0.4918 1.0000 2.0000 2·5000
3·0000
Table 4.2. Ratio R of the total sink nodes
R
2.35 ± 0.10
2.21
1.88
1.64
1.29
1.20
1.31
1.27 ± 0.06
to the total cluster size, the 3D fractal diDj,ox 2.27 ± 0.02
2.20
2.27
2.20
2.16
2.10
2.09
2.05 ± 0.04
mension Dj,ox' the average absorption a, ii
0.04 ± 0.01
0.85
1.75
3·2
5·5
11.0
13
15
± 5
the measure Dab s of the uniformity of
Dabs
1.7 ± 0.2
1.0
1.0
0·9
0.8
0·7
0.8
0·7 ± 0.1
the nutrient distribution, and the averx
0.47 ± 0.03
0·38
0·33
0·32
0·32
0.24
0.24
0.24 ± 0.01
age x, y, z coordinate of the aggregate y
0.51 ± 0.04
0·39
0·34
0·33
0·30
0.19
0.18
0.16 ± 0.01
in the mono directional flow experiment
Ii
0.48 ± 0.02
0·53
0·50
0·49
0·50
0·49
0·5
0.49 ± 0.02
for a series of increasing Pe numbers
4. SIMULATING GROWTH AND FORM
(a)
~~
~
(e)
(b)
(f)
(c)
(g)
...
..
~.~
7~
~
.. '.
(d)
. ~
(h)
Fig.4.24a-h. Slices through the middle
of the aggregates in the xy-plane at
zmax f z, mono directional flow experiment : (a-h) , Peeler number increases
from approximately 0.0 to 3.0.
nutrient distribution and aggregate; in this experiment Peis set to the value
3.000. In this picture the aggregate is seen from above.
The results of the simulation experiments are summarized in Table 4.2.
We have estimated error bounds on the various measurements by repeating
the simulations at Pe = 0.0150 and Pe = 3.000 four times. The ratio R of the
total sink nodes to the total cluster size indicates the compactness of the aggregates. The typical cluster size is about 6-4 x 10
4 • The fractal dimension Dt,ox
ofthe surface ofthe aggregate was determined using a three-dimensional version of the (cube) box-counting method described by Feder (1988) (compare
the two-dimensional version in (j.ioj), Dt,ox measures the space-filling properties of the surface of the aggregate. In three dimensions its value varies
from a minimum of 2, for a solid object with a perfectly smooth surface, to
a maximum of 3 for a solid with a space-filling surface. In the simulations we
have determined the number of sink nodes that have a nutrient absorption
between a and a + L\a for the aggregates obtained at various Peeler numbers.
The number of sink nodes with absorption rate a, N(a), can be related to a
by a power law:
N(a) - a-Dabs
(4.19)
Pe
0.0150
0.1322 0.2521 0.4918 1.0000 2.0000 2·5000
3·0000
Table 4.2. Ratio R of the total sink nodes
R
2.35 ± 0.10
2.21
1.88
1.64
1.29
1.20
1.31
1.27 ± 0.06
to the total cluster size, the 3D fractal diDj,ox 2.27 ± 0.02
2.20
2.27
2.20
2.16
2.10
2.09
2.05 ± 0.04
mension Dj,ox' the average absorption a, ii
0.04 ± 0.01
0.85
1.75
3·2
5·5
11.0
13
15
± 5
the measure Dab s of the uniformity of
Dabs
1.7 ± 0.2
1.0
1.0
0·9
0.8
0·7
0.8
0·7 ± 0.1
the nutrient distribution, and the averx
0.47 ± 0.03
0·38
0·33
0·32
0·32
0.24
0.24
0.24 ± 0.01
age x, y, z coordinate of the aggregate y
0.51 ± 0.04
0·39
0·34
0·33
0·30
0.19
0.18
0.16 ± 0.01
in the mono directional flow experiment
Ii
0.48 ± 0.02
0·53
0·50
0·49
0·50
0·49
0·5
0.49 ± 0.02
for a series of increasing Pe numbers