Initial and Boundary Particle Size Distributions

Simcenter STAR-CCM+ samples the initial conditions once, and then attempts to approximate the cumulative distribution for each method with a few groups. It resamples the inflow conditions at every iteration, and can contribute to different size-groups depending on the conditions inside the converging flow next to the boundary.

The AMUSIG method is volume-based, that is, at each point all size-groups have approximately the same volume fractions. The initial and boundary conditions are also formulated in terms of volume distributions. The volume fraction of all the particles (droplets, bubbles) that have a diameter less than or equal to $d$ is:

where ${\alpha }_d$ is the volume fraction of the dispersed phase, and $F\left(d\right)$ is the cumulative size distribution (CSD).

The AMUSIG method represents the dispersed phase as an ensemble of the size-groups, so the CSD of the method has a staircase shape, as shown below.

Table 1. Rosin-Rammler distribution and its approximation by the AMUSIG method: 5 size-groups (left), and 20 size-groups (right).

The position of an increment of $F$ corresponds to the group diameter of the corresponding size-group; all increments of $F$ have the same height.

In order to approximate a continuous distribution by the AMUSIG method, it is assumed that the middle of each increment (the dots in the 5 size-groups figure, above left) belongs to the curve. For $M$ size-groups, the diameter of the $i^{th}$ size-group is calculated as the solution of the following equation:

The quality of the approximation increases with the number of size-groups. However, you can verify that even a small number of groups (for example, $M=5$) captures the main characteristics of the distribution, for example, $d_{43}$, $d_{32}$.

The distributions that are used in the AMUSIG method are described below.

Uniform distribution

This distribution assumes that the particles are uniformly distributed between a minimum size and a maximum size:

$$F\left(d\right)=\{\begin{array}{ccc}0& ,& d<d_{\min }\\ \frac{(d-d_{\min })}{(d_{\max }-d_{\min })}& ,& d_{\min }\le d\le d_{\max }\\ 1& ,& d>d_{\max }\end{array}$$

(2296)

where $d_{\min }$ is the minimum diameter and $d_{\max }$ is the maximum diameter.

The uniform distribution is recommended in most cases, because it allows for easy control of the size distribution. In many cases, the initial (inlet) distribution is known only approximately and has a minor effect on the solution, so specifying the maximum size and the minimum size is sufficient.

The mean diameters of the uniform distribution are:

$$d_{pq}=\sqrt[p-q]{\frac{(q-2)(d_{\max }^{p-2}-d_{\min }^{p-2})}{(p-2)(d_{\max }^{q-2}-d_{\min }^{q-2})}}$$

(2297)

In particular:

$$\begin{array}{cc}{d}_{43}=\frac{d_{\max }+d_{\min }}{2},& d_{32}=\frac{d_{\max }+d_{\min }}{\ln \left(d_{\max }\right)-\ln \left(d_{\min }\right)}\end{array}$$

(2298)

Rosin-Rammler distribution

The Rosin-Rammler distribution, also known as the Weibull distribution, is:

$$F\left(d\right)=1-\exp (-{\left[\frac{d}{D_{\text{ref}}}\right]}^k)$$

(2299)

where $k$ is the shape parameter and $D_{\text{ref}}$ is the size parameter.

This distribution is widely used in particle size analysis of comminution processes. In the particular case $k=1$ , it becomes an exponential distribution. The parameter $D_{\text{ref}}$ has the dimension of length and specifies the characteristic size, while the parameter $k$ is dimensionless and controls the spread of the distribution (a higher $k$ implies a narrower distribution).

The mean diameters of the Rosin-Rammler distribution are:

$$d_{pq}=D_{\text{ref}}\sqrt[p-q]{\frac{\mathrm{\Gamma }\left(\frac{p-1}{k}\right)}{\mathrm{\Gamma }\left(\frac{q-1}{k}\right)}}$$

(2300)

where $\mathrm{\Gamma }$ is the Euler gamma function. It can be seen that $d_{pq}$ is determined only for $\begin{array}{cc}p>1,& q>1\end{array}$ . In particular, $d_{10}$ and $d_{30}$ are not determined for the Rosin-Rammler distribution.

Log-normal distribution

The log-normal distribution implies that the logarithm of diameters has a normal distribution with mean $\mu$ and standard deviation $\sigma$. The formula for the CSD reads:

$$F\left(d\right)=\frac{1}{2}+\frac{1}{2}\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{\ln d-\mu }{\sqrt{2}\sigma }\right)$$

(2301)

The mean diameters of the log-normal distribution are:

$$d_{pq}=\exp (\mu +\frac{1}{2}(p+q-6){\sigma }^2)$$

(2302)

The equation above provides an easy way to fit the log-normal distribution to the experimental data. Knowing two different diameters, for example $d_{pq}$ and $d_{st}$, the following equations are obtained:

$$\begin{array}{cc}\frac{d_{pq}}{d_{st}}=\exp \left(\frac{1}{2}(p+q-s-t){\sigma }^2\right),& {\sigma }^2=2\frac{\ln \left(\frac{d_{pq}}{d_{st}}\right)}{p+q-s-t}\end{array}$$

(2303)

Substitution of ${\sigma }^2$ into the previous equation yields (after some algebra):

$$\mu =\frac{6-s-t}{p+q-s-t}\ln \left(d_{pq}\right)+\frac{p+q-6}{p+q-s-t}\ln \left(d_{st}\right)$$

(2304)

User Specified Diameter Distribution

The cumulative size distribution (CSD) can be specified as a table. The first column of the table contains diameters [ $d_0$ ; $d_1$ ;…; $d_i$ ;…; $d_n$ ]. The second column contains the values of the CSD [ $F_0$ ; $F_1$ ;…; $F_i$ ;…; $F_n$ ]. Both columns must increase monotonically.

It follows from the definition of the cumulative size distribution that $F_0=0$ and $F_n=1$ . The resulting CSD is piece-wise linear:

$$F\left(d\right)=\{\begin{array}{cc}0& d<d_0\\ \frac{d-d_0}{d_1-d_0}(F_1-F_0)+F_0& d_0\le d<d_1\\ \frac{d-d_1}{d_2-d_1}(F_2-F_1)+F_1& d_1\le d<d_2\\ \mathrm{...}& \mathrm{...}\\ \frac{d-d_i}{d_{i+1}-d_i}(F_{i+1}-F_i)+F_0& d_i\le d<d_{i+1}\\ \mathrm{...}& \mathrm{...}\\ 1& d>d_n\end{array}$$

(2305)