### 5.8.2. Size Group Discretization

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.

#### 5.8.2.1. Equal Mass Discretization

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.

#### 5.8.2.2. Equal Diameter Discretization

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.

#### 5.8.2.3. Geometric Mass Discretization

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.

#### 5.8.2.4. Comparison

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

#### 5.8.2.5. Size Group Boundaries

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.

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