Mean Particle Diameters

For many practical applications, the size distribution is too detailed a description of the dispersed phase, and more integral characteristics are needed. The main output from the population balance calculation is the mean particle diameter. There are different definitions of the mean diameter depending on the weight function of the averaging.

Differences in values between these diameters are indicators of non-uniformity in the distribution. Variance can be quantified from these differences.

The general form of the formula for the mean diameter, based on moments of the number density, is:

Figure 1. EQUATION_DISPLAY

d_{p q} = \sqrt[(p - q)]{\frac{\sum n_{i} d_{i}^{p}}{\sum n_{i} d_{i}^{q}}}

(2284)

Choosing appropriate values for $p$ and $q$ allows you to obtain the various alternative definitions for diameter. The particular and most important types of mean diameter that are implemented in the AMUSIG model are:

Surface Mean Diameter (commonly known as the Sauter mean diameter): Figure 2. EQUATION_DISPLAY

$d_{32} = \frac{\sum n_{i} d_{i}^{3}}{\sum n_{i} d_{i}^{2}}$

(2285)

For the AMUSIG model, the mean diameter $d_{32}$ is labeled as Surface Mean Diameter to avoid a name clash with the existing usage in the S-Gamma model.
Volume Mean Diameter: Figure 3. EQUATION_DISPLAY

$d_{43} = \frac{\sum n_{i} d_{i}^{4}}{\sum n_{i} d_{i}^{3}}$

(2286)
Volume-Based Diameter: Figure 4. EQUATION_DISPLAY

$d_{30} = \sqrt[3]{\frac{\sum n_{i} d_{i}^{3}}{\sum n_{i}}}$

(2287)

Mean Moments

The moments $m_{n}$ of a particle distribution are given by:

Figure 5. EQUATION_DISPLAY

m_{n}^{k} = \sum_{i = 0}^{M} n_{i} d_{i}^{k}

(2288)

where $k$ is the order of the moment and $M$ is the number of size-groups.

The zero'th moment, $m_{0}$ , is the total number or concentration of particles.
The first moment, $m_{1}$ , is the average particle diameter times $m_{0}$ .
The second moment, $m_{2}$ , is the total surface area of the particles divided by $π$ (for spherical particles).
The third moment, $m_{3}$ , is proportional to the volume fraction of particles.

Surface Moment: Figure 6. EQUATION_DISPLAY

$m_{n}^{k} = \sum_{c} W_{c} (\sum_{i = 0}^{M} n_{i c} {(d_{i c} - d_{10})}^{k})$

(2289)
Volume Moment: Figure 7. EQUATION_DISPLAY

$m_{α}^{k} = \sum_{c} W_{c} (\sum_{i = 0}^{M} α_{i c} {(d_{i c} - d_{43})}^{k})$

(2290)

The weights $W_{c}$ are:

For a 3D region, the volume of the cell $V_{c}$
For a surface, the area of the face $A_{f}$
For a flow boundary (interface), the volumetric flow rate $u \cdot A_{f}$ .

where $d_{10}$ and $d_{43}$ are the corresponding mean diameters of the same region.

In general, the expression for $d_{p q}$ is:

Figure 8. EQUATION_DISPLAY

d_{p q} = \sqrt[p - q]{\frac{\sum_{c} W_{c} (\sum_{i = 0}^{M} n_{i c} d_{i c}^{p})}{\sum_{c} W_{c} (\sum_{i = 0}^{M} n_{i c} d_{i c}^{q})}}

(2291)