23.3.13 Using the Rosin-Rammler Diameter Distribution Method

For liquid sprays, a convenient representation of the droplet size distribution is the Rosin-Rammler expression. The complete range of sizes is divided into an adequate number of discrete intervals; each represented by a mean diameter for which trajectory calculations are performed. If the size distribution is of the Rosin-Rammler type, the mass fraction of droplets of diameter greater than $d$ is given by

where $\bar{d}$ is the size constant and $n$ is the size distribution parameter.

By default, you will define the size distribution of particles by inputting a diameter for the first and last points and using the linear equation ( 23.3-1) to vary the diameter of each particle stream in the group. When you want a different mass flow rate for each particle/droplet size, however, the linear variation may not yield the distribution you need. Your particle size distribution may be defined most easily by fitting the size distribution data to the Rosin-Rammler equation. In this approach, the complete range of particle sizes is divided into a set of discrete size ranges, each to be defined by a single stream that is part of the group. Assume, for example, that the particle size data obeys the following distribution:

Diameter Range ( $\mu$m)	Mass Fraction in Range
0-70 70-100 100-120 120-150 150-180 180-200	0.05 0.10 0.35 0.30 0.15 0.05

The Rosin-Rammler distribution function is based on the assumption that an exponential relationship exists between the droplet diameter, $d$, and the mass fraction of droplets with diameter greater than $d$, $Y_d$:

ANSYS FLUENT refers to the quantity $\overline{d}$ in Equation  23.3-3 as the Mean Diameter and to $n$ as the Spread Parameter. These parameters are input by you (in the Set Injection Properties dialog box under the First Point heading) to define the Rosin-Rammler size distribution. To solve for these parameters, you must fit your particle size data to the Rosin-Rammler exponential equation. To determine these inputs, first recast the given droplet size data in terms of the Rosin-Rammler format. For the example data provided above, this yields the following pairs of $d$ and $Y_d$:

Diameter, $d$ ( $\mu$m)	Mass Fraction with Diameter Greater than $d$, $Y_d$
70 100 120 150 180 200	0.95 0.85 0.50 0.20 0.05 (0.00)

A plot of $Y_d$ vs. $d$ is shown in Figure  23.3.6.

Figure 23.3.6: Example of Cumulative Size Distribution of Particles

Next, derive values of $\overline{d}$ and $n$ such that the data in Figure  23.3.6 fit Equation  23.3-3. The value for $\overline{d}$ is obtained by noting that this is the value of $d$ at which $Y_d = e^{-1} \approx 0.368$. From Figure  23.3.6, you can estimate that this occurs for $d \approx 131$ $\mu$m. The numerical value for $n$ is given by

By substituting the given data pairs for $Y_d$ and $d/\overline{d}$ into this equation, you can obtain values for $n$ and find an average. Doing so yields an average value of $n$ = 4.52 for the example data above. The resulting Rosin-Rammler curve fit is compared to the example data in Figure  23.3.7. You can input values for $\overline{d}$ and $n$, as well as the diameter range of the data and the total mass flow rate for the combined individual size ranges, using the Set Injection Properties dialog box.

This technique of fitting the Rosin-Rammler curve to spray data is used when reporting the Rosin-Rammler diameter and spread parameter in the Discrete Phase Summary dialog box in Section  23.7.8.

Figure 23.3.7: Rosin-Rammler Curve Fit for the Example Particle Size Data

A second Rosin-Rammler distribution is also available based on the natural logarithm of the particle diameter. If in your case, the smaller-diameter particles in a Rosin-Rammler distribution have higher mass flows in comparison with the larger-diameter particles, you may want better resolution of the smaller-diameter particle streams, or "bins''. You can therefore choose to have the diameter increments in the Rosin-Rammler distribution done uniformly by $\ln d$.

In the standard Rosin-Rammler distribution, a particle injection may have a diameter range of 1 to 200 $\mu$m. In the logarithmic Rosin-Rammler distribution, the same diameter range would be converted to a range of $\ln 1$ to $\ln 200$, or about 0 to 5.3. In this way, the mass flow in one bin would be less-heavily skewed as compared to the other bins.

When a Rosin-Rammler size distribution is being defined for the group of streams, you should define (in addition to the initial velocity, position, and temperature) the following parameters, which appear under the heading for the First Point:

• Total Flow Rate

  This is the total mass flow rate of the $N$ streams in the group. Note that in axisymmetric problems this mass flow rate is defined per $2\pi$ radians and in 2D problems per unit meter depth.

• Min. Diameter

  This is the smallest diameter to be considered in the size distribution.

• Max. Diameter

  This is the largest diameter to be considered in the size distribution.

• Mean Diameter

  This is the size parameter, $\overline{d}$, in the Rosin-Rammler equation ( 23.3-3).

• Spread Parameter

  This is the exponential parameter, $n$, in Equation  23.3-3.

The Stochastic Rosin-Rammler Diameter Distribution Method

For atomizer injections, a Rosin-Rammler distribution is assumed for the particles exiting the injector. In order to decrease the number of particles necessary to accurately describe the distribution, the diameter distribution function is randomly sampled for each instance where new particles are introduced into the domain.

The Rosin-Rammler distribution can be written as

where $Y$ is the mass fraction smaller than a given diameter $D$, $\bar{d}$ is the Rosin-Rammler diameter and $n$ is the Rosin-Rammler exponent. This expression can be inverted by taking logs of both sides and rearranging,

Given a mass fraction $Y$ along with parameters $\bar{d}$ and $n$, this function will explicitly provide a diameter, $D$. Diameters for the atomizer injectors described in Section  23.3.7 are obtained by uniformly sampling $Y$ in equation 23.3-5.