Particle Injection

Particles enter the computational domain through injectors at one or more discrete locations. An injector defines the size and the velocity vector distribution of the particles, and for unsteady simulations, the frequency. For heat and mass transfer effects, particle temperature and composition are also specified.

Simcenter STAR-CCM+ provides several types of injectors. For some of them, a statistical particle size distribution is specified.

Particle Size Distributions

A range of particle sizes that are generated by an injector are represented statistically by a particle size distribution.

In Simcenter STAR-CCM+, a particle size distribution is quantified using a cumulative distribution function (CDF) $F (D)$ . Here, size means diameter or mass, depending on the particle size specification. The symbol $D$ is used for either. The flow rate specification of the injector determines the precise definition of the cumulative distribution function:

When the mass flow rate is specified, the cumulative distribution function $F_{m} (D)$ gives the fraction of the mass flow rate of the injector with a size smaller than $D$ . Hence $F (D_{M}) = 0.5$ defines $D_{M}$ as the mass-median size.
When the volume flow rate is specified, the cumulative distribution function $F_{V} (D)$ gives the fraction of the volume flow rate of the injector with a size smaller than $D$ . Hence $F (D_{M}) = 0.5$ defines $D_{M}$ as the volume-median size.
When the particle flow rate is specified, the cumulative distribution function $F_{N} (D)$ gives the fraction of the particle flow rate of the injector with a size smaller than $D$ . Hence $F (D_{M}) = 0.5$ defines $D_{M}$ as the number-median size.

This definition provides three essential properties of a cumulative distribution function:

Figure 1. EQUATION_DISPLAY

\begin{matrix} lim_{D \to 0} F (D) = 0 \\ lim_{D \to \infty} F (D) = 1 \\ \frac{d F (D)}{d D} \geq 0 \end{matrix}

(3055)

Simcenter STAR-CCM+ provides four internal cumulative distribution functions:

Log-Normal
Normal
Nukiyama-Tanazawa
Rosin-Rammler

In addition, the function is tabulated as pairs of ( $D$ , $F (D)$ ) values. Make sure that these values respect the properties in Eqn. (3055), with the additional constraint that CDF values must be strictly increasing—duplicate values are not permitted.

Given a cumulative distribution function (CDF), Simcenter STAR-CCM+ generates parcel sizes by dividing the distribution into ranges of equal mass, volume, or number (depending on how the flow rate is specified). The number of ranges is the number of parcel streams for the injector.

Consider an injector with a given mass flow rate and number of injected parcels, but with a diameter specified by a distribution. The mass injected in each step is divided by the number of injected parcels so that each injected parcel has an equal mass. Each parcel has a particle diameter that is stochastically sampled from the distribution, then the particle count (particles per parcel) is adjusted to ensure each parcel has the same mass.

If Volume Flow Rate is specified, the particle volume is divided equally among the parcel streams. Likewise if a Particle Flow Rate is specified, the number of particles is divided equally among the parcel streams. Note that if the Particle Flow Rate is specified and the diameter is stochastically sampled, the particle count is then constrained and the parcel mass varies instead. The number of parcel streams is specified for the Lagrangian injector under Physics Values. Each parcel represents a statistical sample, one sample per parcel.

Log-Normal

The log-normal size distribution [672] is a normal (Gaussian) distribution which uses the logarithm of the particle size as the independent variable. Its cumulative distribution function is

Figure 2. EQUATION_DISPLAY

F (D) = \frac{1}{2} [1 + e r f (\frac{\ln D - \ln \bar{D}}{σ \sqrt{2}})]

(3056)

in which the parameters are the standard deviation $σ$ and the mean size $\bar{D}$ .

Normal

The normal size distribution is a normal (Gaussian) distribution which uses the particle size as the independent variable. Its cumulative distribution function is

Figure 3. EQUATION_DISPLAY

F (D) = \frac{1}{2} [1 + e r f (\frac{D - \bar{D}}{σ \sqrt{2}})]

(3057)

in which the parameters are the standard deviation $σ$ and the mean size $\bar{D}$ .

Nukiyama-Tanazawa

The Nukiyama-Tanazawa size distribution is a probability density function that is based on empirical studies of liquid droplet size in spray injectors. The size probability distribution function is:

Figure 4. EQUATION_DISPLAY

F (D) = P (a, b)

(3058)

where P is defined using the complete ( $Γ (a)$ ) and lower incomplete ( $γ (a, b)$ ) gamma functions:

$P (a, b) = \frac{γ (a, b)}{Γ (a)} = \frac{\int_{0}^{b} e^{- t} t^{a - 1} d t}{\int_{0}^{\infty} e^{- t} t^{a - 1} d t}$
$a = \frac{α + 1}{β}$
$b = B D^{β}$
where:
$B = {(\frac{1}{\bar{D}} \frac{Γ (\frac{α + 4}{β})}{Γ (\frac{α + 3}{β})})}^{β}$

$\bar{D}$ is the mean size. The values of $α$ and $β$ can be set in the model. By default, $α = 2$ and $β = 1$ .

Rosin-Rammler

The Rosin-Rammler distribution [672] was developed to describe the volume distribution of particles as a function of their diameter, $F_{V} (D_{p})$ . In Simcenter STAR-CCM+, it is extended to be a generic size distribution, with cumulative distribution function

Figure 5. EQUATION_DISPLAY

F (D) = 1 - \exp [- {(\frac{D}{D_{r e f}})}^{q}]

(3059)

in which the parameters are the exponent $q$ and the reference size $D_{r e f}$ . This form identifies the Rosin-Rammler distribution as a Weibull distribution.

As noted above, the Rosin-Rammler distribution is a cumulative mass, volume, or number distribution, depending on the flow rate specification of the injector.