We now consider how the size distribution is discretized into a predefined set of size groups. Note that the group sizes are represented by mass rather than diameter or volume, because this leads to simpler equations for variable-density fluids. However, for setup convenience, the size group discretization is controlled by user-defined diameters; these diameters are converted to masses using the fluid density (for constant density fluids) or a user-specified reference density (for variable density fluids). The diameter and mass represented by a particular group are related by:
 | (5–155) |
There are three built-in recipes for creating size groups from a user-defined minimum and maximum diameter: equal mass, equal diameter, and geometric.
In this case, the minimum and maximum mass are calculated from
the minimum and maximum diameter using Equation 5–155. The mass represented by group 
is calculated
from:
 | (5–156) |
where 
is the minimum particle mass,
and
 | (5–157) |
where 
is the maximum particle mass,
and 
is the number of groups.
In this case, the diameter represented by group 
is calculated
from:
 | (5–158) |
where 
is the minimum particle diameter,
and
 | (5–159) |
where 
is the maximum particle diameter,
and 
is the number of groups.
In this case, the mass represented by group 
is calculated
from:
 | (5–160) |
where 
is the mass represented by group 
, 
is the minimum
particle mass, 
is the maximum particle mass,
and 
is the number of groups.
The following table compares the discretization options for
a polydispersed fluid having 
and 
.
Table 5.1: Representative Particle Diameters in each Size Group
|
Group |
Equal Mass Discretization (mm) |
Equal Diameter Discretization (mm) |
Geometric Mass Discretization (mm) |
|---|---|---|---|
|
1 |
0.5850 |
0.050 |
0.0225 |
|
2 |
0.8430 |
0.150 |
0.0284 |
|
3 |
1.000 |
0.250 |
0.0358 |
|
4 |
1.118 |
0.350 |
0.0451 |
|
5 |
1.216 |
0.450 |
0.0568 |
|
6 |
1.300 |
0.550 |
0.0715 |
|
7 |
1.375 |
0.650 |
0.0901 |
|
8 |
1.442 |
0.750 |
0.1140 |
|
9 |
1.503 |
0.850 |
0.1430 |
|
10 |
1.560 |
0.950 |
0.1800 |
|
11 |
1.613 |
1.050 |
0.2270 |
|
12 |
1.663 |
1.150 |
0.2860 |
|
13 |
1.710 |
1.250 |
0.3600 |
|
14 |
1.754 |
1.350 |
0.4540 |
|
15 |
1.796 |
1.450 |
0.5720 |
|
16 |
1.837 |
1.550 |
0.7210 |
|
17 |
1.875 |
1.650 |
0.9080 |
|
18 |
1.912 |
1.750 |
1.1440 |
|
19 |
1.948 |
1.850 |
1.4420 |
|
20 |
1.983 |
1.950 |
1.8170 |
Each size group represents a range of sizes, and therefore masses. The solver requires not only the size and mass represented by each size group, but also each size group boundary. The manner in which the masses represented by the group boundaries are calculated depends on the size group discretization method:
Equal diameter discretization: the diameter represented by the group boundary is assumed to be midway between the diameters represented by the adjacent groups
Other discretizations (including user-defined): the mass represented by the group boundary is assumed to be midway between the masses represented by the adjacent groups
The mass represented by the upper limit of the largest-size group is based on the diameter represented by that limit; for user-defined groups, that diameter is not available and so is derived based on an extrapolation of the two largest groups. The mass represented by the lower limit of the smallest-size group is assumed to be zero.