Eulerian Multiphase (EMP) Flow

The two-fluid model that is referred to as the Eulerian Multiphase (EMP) model in Simcenter STAR-CCM+, and whose initial development was driven largely by nuclear industry, is generally used for modeling dispersed flows.

The Eulerian Multiphase (EMP) model treats each phase as inter-penetrating continua. It solves transport equations for mass, momentum and energy of each phase while all the phases share a common pressure field. Eulerian averaging [480] [566] [454] of the transport equations results in additional interaction between the phases. These interactions require models for closure like drag force, lift force, interphase heat transfer, and so on.

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:

Figure 1. EQUATION_DISPLAY

V_{i} = \int_{V} α_{i} d V

(1902)

where $α_{i}$ is the volume fraction of phase $i$ . The volume fractions of each Eulerian phase must fulfill the requirement:

Figure 2. EQUATION_DISPLAY

\sum_{i = 1}^{n} α_{i} = 1

(1903)

where $n$ is the number of all phases.

Continuity Equation

The conservation of mass for a generic phase $i$ is given by:

Figure 3. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} α_{i} ρ_{i} d V + \oint_{A} α_{i} ρ_{i} v_{i} \cdot d a = \int_{V} \sum_{j \neq i} (m_{i j} - m_{j i}) d V + \int_{V} S_{i}^{α} d V

(1904)

where:

$α_{i}$ is the volume fraction
$ρ_{i}$ is the density
$v_{i}$ is the velocity
$m_{i j}$ is the mass transfer rate to phase i, from phase j ( $m_{i j} \geq 0$ )
$m_{j i}$ is the mass transfer rate to phase j, from phase i ( $m_{j i} \geq 0$ )
$S_{i}^{α}$ is a user-defined phase mass source term

Momentum Equation

The momentum balance for the generic phase $i$ reads:

Figure 4. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} \int_{V} α_{i} ρ_{i} v_{i} d V + \oint_{A} α_{i} ρ_{i} v_{i} \otimes v_{i} \cdot d a = - \int_{V} α_{i} \nabla p d V \\ + \int_{V} α_{i} ρ_{i} g d V + \oint_{A} [α_{i} (T_{i} + T_{i}^{t})] \cdot d a + \int_{V} M_{i} d V \\ + \int_{V} {(F_{i n t})}_{i} d V + \int_{V} S_{i}^{α} d V + \int_{V} \sum_{i = 1}^{n} (m_{i j} v_{j} - m_{j i} v_{i}) d V \end{matrix}

(1905)

where:

$p$ is the pressure, assumed to be equal in all phases
$g$ is the gravity vector
$T_{i}$ and ${T_{i}}^{t}$ are the molecular and turbulent stresses, respectively
$M_{i}$ is the interphase momentum transfer per unit volume
${(F_{i n t})}_{i}$ represents internal forces (such as the solid pressure force between particles or a user-defined potential force)
$S_{i}^{α}$ is the phase momentum source term
$m_{i j}$ is the mass transfer rate from phase j to phase i.
$m_{j i}$ is the mass transfer rate from phase i to phase j.

The interphase momentum transfer represents the sum of all the forces the phases exert on one another and satisfies the equation:

Figure 5. EQUATION_DISPLAY

\sum_{i}^{} M_{i} = 0

(1906)

Energy Equation

Energy is conserved for a Eulerian phase according to the following equation:

Figure 6. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} \int_{V} α_{i} ρ_{i} E_{i} d V + \oint_{A} α_{i} ρ_{i} H_{i} v_{i} \cdot d a = \oint_{A} α_{i} k_{e f f, i} \nabla T_{i} d a + \oint_{A} T_{i} \cdot v_{i} \cdot d a \\ + \int_{V} f_{i} \cdot v_{i} d V + \int_{V} \sum_{j \neq i} Q_{ij} d V + \int_{V} \sum_{(i j)}^{} {Q_{i}}^{(i j)} d V + \int_{V} S_{u, i} d V + \int_{V} \sum_{j \neq i}^{} (m_{i j} - m_{j i}) h_{i} (T_{i j}) d V \end{matrix}

(1907)

where:

$E_{i}$ is the total energy
$H_{i}$ is the total enthalpy
$T_{i}$ is the viscous stress tensor
$α_{i}$ , $ρ_{i}$ and $v_{i}$ are the volume fraction, density, and velocity of phase $i$ , as in Eqn. (1904).
$T_{i}$ is the temperature.
$k_{e f f, i}$ is the effective thermal conductivity, from Eqn. (1908).
$f_{i}$ is the body force vector.
$Q_{i j}$ 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.
$Q_{i}^{(i j)}$ is the heat transfer rate to phase i from phase pair interface (ij). It refers to the heat transfer phenomena such as evaporation, condensation, and boiling where heat transfer occurs between the phase and the interface.
$S_{u, i}$ is the energy source.
$h_{i} (T_{i j})$ is the phase i enthalpy that is evaluated at the interface temperature $T_{i j}$ .

The effective thermal conductivity $k_{e f f}$ is:

Figure 7. EQUATION_DISPLAY

k_{e f f, i} = k_{i} + \frac{μ_{t, i} C_{p, i}}{σ_{t, i}}

(1908)

where:

$k_{i}$ is the thermal conductivity.
$μ_{t, i}$ is the turbulent viscosity.
$C_{p, i}$ is the specific heat.
$σ_{t, i}$ is the turbulent thermal diffusion Prandtl number.

Now, total energy, $E$ , is related to the total enthalpy, $H$ , by

Figure 8. EQUATION_DISPLAY

E_{i} = H_{i} - \frac{p}{ρ_{i}}

(1909)

where:

Figure 9. EQUATION_DISPLAY

H_{i} = h_{i} + \frac{{| v_{i} |}^{2}}{2}

(1910)

and

Figure 10. EQUATION_DISPLAY

h_{i} (T_{i}) = h_{i}^{R E F} + \int_{T_{i}^{R E F}}^{T_{i}} C_{p, i} (T′) d T′

(1911)

where $h_{i}^{R E F}$ is the heat of formation for phase $i$ and $T_{i}^{R E F}$ is the standard state temperature for phase $i$ .

When heat is transferred only by thermal diffusion between phases, the interphase heat transfer rates satisfy $Q_{i j} = - Q_{j i}$ . Where heat and mass are only transferred by phase change (bulk boiling or condensation), the heat transfer rates between each phase i,j, and a phase pair interface (ij) satisfy a heat balance:

Figure 11. EQUATION_DISPLAY

Q_{i}^{(i j)} + Q_{j}^{(i j)} + (m_{i j} - m_{j i}) Δ h_{i j} = 0

(1912)

where $Δ h_{i j}$ is the heat required to produce phase i from phase j. This is a signed quantity so that $Δ h_{i j} = - {Δ h}_{j i}$ . The heat of phase change is defined from the phase enthalpy definitions that are evaluated at the interface temperature $T_{i j}$ :

Figure 12. EQUATION_DISPLAY

Δ h_{i j} = h_{j} (T_{i j}) - h_{i} (T_{i j})

(1913)