Two-Way Coupling with the Continuous Phase

With one-way coupling, only the continuous phase influences the dispersed phase, but not in the reverse direction. With two-way coupling, the effects of the dispersed phase on the continuous phase such as displacement, interphase momentum, mass, and heat transfer are taken into account.

The displacement of the continuous phase by the dispersed phase is accounted for through the volume fraction. The volume fraction of a Lagrangian phase is the fraction of the local cell volume which that phase occupies. It is calculated for both discrete material particles and DEM particles.

The volume fraction

ϕ_{c}

of all Lagrangian phases for which the two-way coupling is active is accumulated. This volume is seen as a void by the continuous phase: the volume available for the continuous phase decreases by this volume fraction. The available volume fraction for the continuous phase (corresponding to the void fraction)

η

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). The volume fraction of the Lagrangian phase is related to the available volume fraction for the continuous phase as:

Figure 1. EQUATION_DISPLAY

ϕ_{c} = 1 - η

(3040)

For more information on how the dispersed phase influences the continuous phase flow equations, see Volume Partitioning.

In practice, $ϕ_{c}$ is under-relaxed to promote stability:

Figure 2. EQUATION_DISPLAY

ϕ_{c} = α (\sum ϕ) + {(1 - α) ϕ}_{c, o l d}

(3041)

where $α$ is an under-relaxation factor of the two-way coupling solver and $\sum ϕ$ is the sum of the volume fractions of the relevant Lagrangian phases. The value of $ϕ_{c}$ which is contributed to the continuous phase void fraction is limited to $m i n (ϕ_{c}, ϕ_{c, m a x})$ , again to promote stability.

The Lagrangian dispersed phase equations, when integrated over a cell, yield the changes in the momentum, mass, and energy of each particle between its entry and exit. The sum of these changes for all particles crossing the volume provides the net momentum, mass, and energy that is exchanged with the continuous phase. These net exchanges enter the continuous phase equations as source terms. For massless particles, the source terms are not computed as these particles do not influence the continuous phase.

Momentum Transfer

The rate of momentum transfer to a single particle from the continuous phase is $F_{s} + {\dot{m}}_{p} v_{p}$ , where $F_{s}$ is the force acting on surface of the particle, defined in Eqn. (2957), and ${\dot{m}}_{p}$ the rate of mass transfer to the particle.

In unsteady simulations, the rate of momentum transfer from all particles in a cell c to the continuous phase is:

Figure 3. EQUATION_DISPLAY

S_{v} = - \frac{1}{Δ t} \sum_{π}^{} (\int_{t}^{t + ▵ t} \int_{V_{c}}^{} δ (r - r_{π}) n_{π} (F_{s} + {\dot{m}}_{p} v_{p}) dV dt)

(3042)

where the volume integral is over the cell; the Dirac delta function filters out parcels which are not in the cell. The summation is over all parcels for which two-way coupling is active. The discrete form of Eqn. (3042) is:

Figure 4. EQUATION_DISPLAY

S_{v} = - \frac{1}{Δ t} \sum_{π}^{} \sum_{δ t_{p}}^{} n_{π} (F_{s} + {\dot{m}}_{p} v_{p}) δ t_{p}

(3043)

where the second summation is over all $δ t_{p}$ for which parcel $π$ is in cell c.

In steady simulations, the rate of momentum transfer from all particles in a cell c to the continuous phase is

Figure 5. EQUATION_DISPLAY

S_{v} = - \sum_{π}^{} (\int_{0}^{\infty} \int_{V_{c}}^{} δ (r - r_{π}) {\dot{n}}_{π} (F_{s} + {\dot{m}}_{p} v_{p}) dV dt)

(3044)

or, in discrete form,

Figure 6. EQUATION_DISPLAY

S_{v} = - \sum_{π}^{} \sum_{δ t_{p}}^{} {\dot{n}}_{π} (F_{s} + {\dot{m}}_{p} v_{p}) δ t_{p}

(3045)

The field $S_{v}$ is applied in the continuous phase momentum equation.

Mass Transfer

The rate of mass transfer to a single particle from the continuous phase is

{\dot{m}}_{p}

.

In unsteady simulations, the rate of mass transfer from all particles in a cell c to the continuous phase is

Figure 7. EQUATION_DISPLAY

S_{m} = - \frac{1}{Δ t} \sum_{π}^{} (\int_{t}^{t + ▵ t} \int_{V_{c}}^{} δ (r - r_{π}) n_{π} {\dot{m}}_{p} dV dt)

(3046)

where the volume integral is over the cell; the Dirac delta function filters out parcels which are not in the cell. The summation is over all parcels for which two-way coupling is active. The discrete form of Eqn. (3046) is

Figure 8. EQUATION_DISPLAY

S_{m} = - \frac{1}{Δ t} \sum_{π}^{} \sum_{δ t_{p}}^{} n_{π} {\dot{m}}_{p} δ t_{p}

(3047)

where the second summation is over all $δ t_{p}$ for which parcel $π$ is in cell c.

In steady simulations, the rate of mass transfer from all particles in a cell c to the continuous phase is

Figure 9. EQUATION_DISPLAY

S_{m} = - \sum_{π}^{} (\int_{0}^{\infty} \int_{V_{c}}^{} δ (r - r_{π}) {\dot{n}}_{π} {\dot{m}}_{p} dV dt)

(3048)

or, in discrete form,

Figure 10. EQUATION_DISPLAY

S_{m} = - \sum_{π}^{} \sum_{δ t_{p}}^{} {\dot{n}}_{π} {\dot{m}}_{p} δ t_{p}

(3049)

The field $S_{m}$ is applied in the continuous phase continuity equation.

Energy Transfer

The rate of total energy transfer to a single particle from the continuous phase is

Figure 11. EQUATION_DISPLAY

Q_{t} + F_{s} \cdot v_{p} + \frac{1}{2} {\dot{m}}_{p} v_{p}^{2} + {\dot{m}}_{p} h

(3050)

where $Q_{t}$ is the surface heat transfer defined in Eqn. (3020), $F_{s}$ is the surface force that is defined in Eqn. (2957), ${\dot{m}}_{p}$ the rate of mass transfer to the particle and $h$ the enthalpy of transferred material.

In unsteady simulations, the rate of energy transfer from all particles in a cell c to the continuous phase is

Figure 12. EQUATION_DISPLAY

S_{E} = - \frac{1}{Δ t} \sum_{π}^{} (\int_{t}^{t + ▵ t} \int_{V_{c}}^{} δ (r - r_{π}) n_{π} (Q_{t} + F_{s} \cdot v_{p} + \frac{1}{2} {\dot{m}}_{p} v_{p}^{2} + {\dot{m}}_{p} h) dV dt)

(3051)

where the volume integral is over the cell; the Dirac delta function filters out parcels which are not in the cell. The summation is over all parcels for which two-way coupling is active. The discrete form of Eqn. (3051) is

Figure 13. EQUATION_DISPLAY

S_{E} = - \frac{1}{Δ t} \sum_{π}^{} \sum_{δ t_{p}}^{} n_{π} (Q_{t} + F_{s} \cdot v_{p} + \frac{1}{2} {\dot{m}}_{p} v_{p}^{2} + {\dot{m}}_{p} h) δ t_{p}

(3052)

where the second summation is over all $δ t_{p}$ for which parcel $π$ is in cell c.

In steady simulations, the rate of energy transfer from all particles in a cell c to the continuous phase is

Figure 14. EQUATION_DISPLAY

S_{E} = - \sum_{π}^{} (\int_{0}^{\infty} \int_{V_{c}}^{} δ (r - r_{π}) {\dot{n}}_{π} (Q_{t} + F_{s} \cdot v_{p} + \frac{1}{2} {\dot{m}}_{p} v_{p}^{2} + {\dot{m}}_{p} h) dV dt)

(3053)

or, in discrete form,

Figure 15. EQUATION_DISPLAY

S_{E} = - \sum_{π}^{} \sum_{δ t_{p}}^{} {\dot{n}}_{π} (Q_{t} + F_{s} \cdot v_{p} + \frac{1}{2} {\dot{m}}_{p} v_{p}^{2} + {\dot{m}}_{p} h) δ t_{p}

(3054)

The field $S_{E}$ is applied in the continuous phase energy equation.