Adaptive Multiple Size-Group (AMUSIG)

The adaptive multiple size-group (AMUSIG) model predicts the size distribution of droplets or bubbles in a dispersed flow regime of a multiphase flow.

The collection of droplets or bubbles is modeled using a number of groups of representative computational particles. Each group consists of identical particles, has its own size, and optionally can have its own velocity. A population balance is solved for each size-group in each cell. Breakup and coalescence models provide the principal contributions to the birth and death source terms for each balance. Other relevant effects include particle diffusion, acceleration, separation, and phase density changes. The solution to the balance equation gives the local particle number density for the group. The particle diameter is computed from the number density and the volume fraction for the group.

The method is locally adaptive so that each size-group carries approximately the same share of the phase mass or volume. This method is efficient and benefits fully from the computational expense of multi-speed modeling. Such modeling is necessary when analysing strong interactions between the size distribution and the flow. However, adaptation also means that the particle size that is associated with a group can vary gradually from cell to cell. Take care when you are comparing or integrating group results from different parts of the domain.

Multi-Speed AMUSIG

You use multi-speed AMUSIG for modeling flow-size interactions in full. Mass and momentum equations are solved individually for each particle size-group. This option is more expensive, but is justified where there is strong interaction between the flow and the local size distribution. For example, this option is relevant for flows that separate particles by size. It is also relevant for flows where the interaction of buoyancy, turbulence, coalescence, and breakup cause different secondary flow patterns for different particle sizes, and therefore further promote overall turbulence and mixing.

In order to account for the multidisperse nature of the flow, the dispersed phase is split into $M$ size-groups. From the modeling point of view, each group can be considered a separate phase in every aspect but the name. Each group moves with its own velocity and exchanges mass, momentum, and energy with the other groups and with the continuous phase [504].

The Reynolds-averaged mass conservation equation for the $i^{t h}$ group reads:

Figure 1. EQUATION_DISPLAY

\frac{\partial ρ_{p} {\bar{α}}_{i}}{\partial t} + \nabla\cdot (ρ_{p} {\bar{α}}_{i} 〈 v_{i} 〉) = m_{i j} - m_{j i}

(2238)

where:

$ρ_{p}$ is the density of the dispersed phase.
${\bar{α}}_{i}$ is the Reynolds-averaged volume fraction of the $i^{t h}$ group.
$m_{i j}$ and $m_{j i}$ are the (averaged) mass fluxes from the $j^{t h}$ group to the $i^{t h}$ group, and from the $i^{t h}$ group to the $j^{t h}$ group, respectively.
$〈 v_{i} 〉$ is the phase-averaged velocity of the group [459]:

Figure 2. EQUATION_DISPLAY

〈 v_{i} 〉 = \frac{〈 α_{i} v_{i} 〉}{{\bar{α}}_{i}}

(2239)

Where:

$α_{i}$ is the instantaneous value of volume fraction.
$v_{i}$ is the instantaneous value of velocity.
Angular brackets mean Reynolds averaging.

The Reynolds-averaged momentum conservation equation for the $i^{t h}$ group reads:

Figure 3. EQUATION_DISPLAY

\frac{\partial ρ_{p} {\bar{α}}_{i} 〈 v_{i} 〉}{\partial t} + \nabla\cdot (ρ_{p} {\bar{α}}_{i} 〈 v_{i} 〉 〈 v_{i} 〉) = - {\bar{α}}_{i} \nabla P - \nabla\cdot τ_{i} + 〈 F_{i} 〉 + m_{i j} 〈 v_{j} 〉 - m_{j i} 〈 v_{i} 〉

(2240)

Where:

$τ_{i}$ is the Reynolds stress.
$F_{i}$ is the interaction force between the continuous phase and the $i^{t h}$ group.

In order to calculate the phase-interaction forces, the size of the particles in the $i^{t h}$ group has to be specified. When a constant size for the group is prescribed, the MUSIG method is obtained [504]. If the particle size distribution varies significantly across the system, the fixed discretization in the size space is not efficient from the numerical point of view. To track the size distribution adaptively, Eqn. (2239) and Eqn. (2240) are augmented by an equation for the number density of the $i^{t h}$ group.

Having the mean volume fraction and number density, the volume of the particles in the $i^{t h}$ group can be postulated as:

Figure 4. EQUATION_DISPLAY

{\bar{v}}_{i} = \frac{{\bar{α}}_{i}}{{\bar{n}}_{i}}

(2241)

The Reynolds averaged equation for the number density reads:

Figure 5. EQUATION_DISPLAY

\frac{\partial {\bar{n}}_{i}}{\partial t} + \nabla\cdot {\bar{n}}_{i} {〈 v_{i} 〉 + D_{T} \nabla (\ln {\bar{α}}_{i} - \ln {\bar{n}}_{i})} = 〈 S_{i} 〉

(2242)

Where:

$〈 S_{i} 〉$ is a source term that is provided by the AMUSIG model.
$D_{T}$ is the number density diffusion coefficient.
This value is calculated from the kinematic turbulent viscosity and the Number Density Turbulent Prandtl number, as:

$D_{T} = ν_{t} / σ_{n}$ .

If the volume of the group ( ${\bar{α}}_{i} / {\bar{n}}_{i}$ ) is constant, the term $\nabla (\ln {\bar{α}}_{i} - \ln {\bar{n}}_{i}) = 0$ , and so the diffusive velocity in Eqn. (2242) vanishes.

Single-Speed AMUSIG

You use single-speed AMUSIG for modeling flow-size interactions in the mean. Mass and momentum equations are solved at the phase level, and all of the particle size groups are assumed to be convected at phase velocity. This option allows Simcenter STAR-CCM+ to model the effect of the flow and turbulence on the size distribution. The predicted mean particle size is passed back to the flow calculation.

The single-speed AMUSIG model is a simplified version of the multi-speed model. This model assumes that all size-groups are moving with the same velocity. The mass and momentum equations are solved at the phase level and then translated to the size-group level as:

Figure 6. EQUATION_DISPLAY

\begin{matrix} {\bar{α}}_{i} = \frac{{\bar{α}}_{p h a s e}}{M} \\ v_{i} = v_{p h a s e} \end{matrix}

(2243)

The only equation that is solved at the size-group level is Eqn. (2242).

Population Balance Algorithm

In order to close the model that is described by Eqn. (2239), Eqn. (2240), Eqn. (2241), and Eqn. (2242) the AMUSIG population balance algorithm specifies $m_{i j}$ , $m_{j i}$ and $〈 S_{i} 〉$ .

There are five physical and numerical factors that contribute to $m_{i j}$ , $m_{j i}$ and $〈 S_{i} 〉$ :

Breakup
The probability of a particle breaking up is given by the corresponding Breakage Rate. When particle breakup occurs, the fragments are distributed according to the Daughter Particles Size distribution. The mass and number density of the fragments are distributed among the other size groups. However, unlike the method of classes, breakup of the smallest size-group is allowed.
Coalescence
The probability of two particles colliding is given by the corresponding Collision Rate. When two particles collide, the probability that the particles coalesce is given by the Coalescence Efficiency. The mass and number density of the resulting particle are distributed among other size-groups. Once again, due to the adaptive nature of the method, the biggest size-group can coalesce. Coalescence between particles within the same size-group is also allowed.
Inlet
The AMUSIG model requires the inlet boundary conditions to be specified at the phase level. This specification is required because the sizes of the particles at an inlet are not necessarily equal to the sizes of the particles in the nearby cells. The fresh particles that are coming through the inlet are mixing with the particles that are already present in the stream. The mixing is performed using the same procedure as coalescence and breakup: the mass and number density of the incoming particles are distributed among the existing size-groups.
Redistribution
Redistribution is a counter-intuitive term that has no physical counterpart, but plays a central role in the AMUSIG method. When there is a strong breakup, you would intuitively assume that the volume fraction of the large size-groups would decrease, while the smallest size-groups would gain more mass. However, if you plot the volume fraction of the different size-groups, you can see that they are nearly equal in each cell. Instead, the diameters of all the size-groups decrease: this reduction is due to the redistribution. The redistribution algorithm does not require any user input, but its effect is apparent and has a strong impact on the performance of the AMUSIG method.
Number density diffusion
It is known that the diffusive flux in Eqn. (2242) produces so-called spurious dissipation proportional to $D_{T} {| \nabla {\bar{v}}_{i} |}^{2}$ ([458]). The spurious dissipation implies that the predicted size distribution is less broad than it should be — the standard deviation of the size distribution is "dissipated" by the turbulent diffusion. In order to avoid the spurious dissipation, the AMUSIG model has a special treatment to the number density diffusion term.

However, this term sometimes leads to instability, so the Size Distribution Diffusion Limiter ( $λ_{D}$ ) property is available to experienced users. For more information, see Adaptive Multiple Size-Group Model Properties.