Dispersed Multiphase

The Dispersed Multiphase (DMP) model simulates the flow of dispersed particles in a continuous phase using a Eulerian approach. As such, for simulating these types of flow, the DMP model is an alternative to using the Lagrangian Multiphase (LMP) model, which uses a Lagrangian approach.

By default, DMP uses one-way coupling: the continuous phase influences the dispersed phase particles through terms such as drag in the momentum equation and heat transfer in the energy equation, but there is no reverse effect. When the optional Two-Way Coupling model is activated in DMP, the reverse effect is accounted for, and dispersed phase source terms appear in continuous phase equations.

In two-way coupled simulations, drag force and heat transfer contribute to both the dispersed phase and the continuous phase, but with different signs. For example, the drag force between the two phases is an action-reaction force that satisfies Newton's third law. Activating two-way coupling ensures that the net drag force and heat transfer flux is zero in the continuous phase–dispersed phase system. Two-way coupling also affects the available volume from the point of view of the continuous phase: the volume that the dispersed phase occupies is accounted for, hence the available volume for the continuous phase is decreased. The available volume fraction for the continuous phase (corresponding to the void fraction) $\eta$ is defined as the ratio of volume that is occupied by all continuous fluid phases to the total cell volume as given by Eqn. (872). Typically, this effect becomes more important at higher particle loadings. However, with high particle loadings, you approach the limits of model applicability. The DMP model generally assumes a dilute multiphase mixture. For more information on how the dispersed phase influences the continuous phase flow equations, see Volume Partitioning.

For each dispersed phase, Simcenter STAR-CCM+ solves transport equations for the conservation of mass and momentum.

Volume Fraction

The share of the flow domain that is occupied by each phase is given by its volume fraction.

The volume of a phase $i$ is given by:

$$V_i=\underset{V}{\int }{\alpha }_{i}dV$$

(2843)

where ${\alpha }_i$ is the volume fraction of phase $i$. The available volume fraction of the continuous phase is related to the dispersed phase volume fraction as:

$$\eta ={\eta }^{*}+(1-\frac{V_i}{V})$$

(2844)

where ${\eta }^{*}$ is the available volume fraction from the previous iteration or time-step and $V$ is the volume of the cell.

Continuity Equation

The continuity equation for a phase $i$ is given as:

$$\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i{\rho }_i\mathrm{dV}+\underset{A}{\oint }{\alpha }_i{\rho }_i{\mathbf{\text{v}}}_i\cdot da=S_{u,i}$$

(2845)

where:

• ${\alpha }_i$ is the volume fraction of dispersed phase $i$
• ${\rho }_i$ is the density of dispersed phase $i$
• $v_i$ is the velocity of dispersed phase $i$
• $S_{u,i}$ is a user-defined mass source

Momentum Equation

The momentum equation is given as:

$$\frac{\partial }{\partial t}\int {\alpha }_i{\rho }_i{\mathbf{\text{v}}}_{i}dV+\underset{A}{\oint }{\alpha }_i{\rho }_i{\mathbf{\text{v}}}_i{\otimes \mathbf{\text{v}}}_i\cdot da=\underset{V}{-\int }{\alpha }_i\nabla pdV+F_{\mathrm{ij}}^D+S_{u,i}$$

(2846)

where:

• $p$ is the static pressure. This term contains the buoyancy force $-{\rho }_{c}g$ through the definition of the hydrostatic pressure as $p_{hs}=-{\rho }_{c}gh$.
• $F_{\mathrm{ij}}^D$ is the drag force that acts on the dispersed phase $i$ due to the drag of phase $j$
• $S_{u,i}$ is a user-defined momentum source

Energy Equation

$$\begin{array}{c}\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i{\rho }_i{E}_{i}dV+\underset{A}{\oint }{\alpha }_i{\rho }_i{H}_i{v}_i\cdot da+\underset{A}{\oint }{\alpha }_{i}pda=\underset{A}{\oint }{\alpha }_i{k}_{\mathrm{eff},i}\nabla T_{i}da+\underset{A}{\oint }{\mathbf{\text{T}}}_i\cdot {\mathbf{\text{v}}}_{i}da\\ +\underset{V}{\int }{\mathbf{\text{f}}}_i\cdot {\mathbf{\text{v}}}_{i}dV+\underset{V}{\int }\sum _{j\ne i}{Q}_{\mathrm{ij}}dV+\underset{V}{\int }{S}_{u,i}dV\end{array}$$

(2847)

where:

• $E_i$ is the total energy
• $H_i$ is the total enthalpy
• ${\mathbf{\text{T}}}_i$ is the viscous stress tensor
• $T_i$ is the temperature.
• $k_{\mathrm{eff},i}$ is the effective thermal conductivity, from Eqn. (1908).
• ${\mathbf{\text{f}}}_i$ is the body force vector.
• $Q_{ij}$ is the interphase heat transfer rate to phase $i$ from phase $j$. It refers to the heat transfer between phases due to a temperature difference between the phases.