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22.9.1 Coupling Between the Discrete and Continuous Phases

As the trajectory of a particle is computed, FLUENT keeps track of the heat, mass, and momentum gained or lost by the particle stream that follows that trajectory and these quantities can be incorporated in the subsequent continuous phase calculations. Thus, while the continuous phase always impacts the discrete phase, you can also incorporate the effect of the discrete phase trajectories on the continuum. This two-way coupling is accomplished by alternately solving the discrete and continuous phase equations until the solutions in both phases have stopped changing. This interphase exchange of heat, mass, and momentum from the particle to the continuous phase is depicted qualitatively in Figure  22.9.1.

Figure 22.9.1: Heat, Mass, and Momentum Transfer Between the Discrete and Continuous Phases
figure



Momentum Exchange


The momentum transfer from the continuous phase to the discrete phase is computed in FLUENT by examining the change in momentum of a particle as it passes through each control volume in the FLUENT model. This momentum change is computed as


 F = \sum \left( \frac{18 \mu C_D {\rm Re}}{\rho_p d^{2}_{p}24} ({u}_p - u) + F_{\rm other} \right) \dot{m}_p \Delta t (22.9-1)


where      
  $\mu$ = viscosity of the fluid
  $\rho_p$ = density of the particle
  $d_p$ = diameter of the particle
  Re = relative Reynolds number
  $u_{p}$ = velocity of the particle
  $u$ = velocity of the fluid
  $C_D$ = drag coefficient
  $\dot{m}_p$ = mass flow rate of the particles
  $\Delta t$ = time step
  $F_{\rm other}$ = other interaction forces

This momentum exchange appears as a momentum sink in the continuous phase momentum balance in any subsequent calculations of the continuous phase flow field and can be reported by FLUENT as described in Section  22.16.



Heat Exchange


The heat transfer from the continuous phase to the discrete phase is computed in FLUENT by examining the change in thermal energy of a particle as it passes through each control volume in the FLUENT model. In the absence of a chemical reaction (i.e., for all particle laws except Law 5) the heat exchange is computed as


 Q = ({m_p}_{\rm in} - {m_p}_{\rm out})[-{H_{lat}}_{\rm ref} ... ...{m_p}_{\rm in}\int^{{T_p}_{\rm in}}_{T_{\rm ref}} c{_{p}}_p dT (22.9-2)


where      
  ${m_p}_{\rm in}$ = mass of the particle on cell entry (kg)
  ${m_p}_{\rm out}$ = mass of the particle on cell exit (kg)
  ${c_p}_p$ = heat capacity of the particle (J/kg-K)
  $H_{\rm pyrol}$ = heat of pyrolysis as volatiles are evolved (J/kg)
  ${T_p}_{\rm in}$ = temperature of the particle on cell entry (K)
  ${T_p}_{\rm out}$ = temperature of the particle on cell exit (K)
  $T_{\rm ref}$ = reference temperature for enthalpy (K)
  ${H_{lat}}_{\rm ref}$ = latent heat at reference conditions (J/kg)

The latent heat at the reference conditions ${H_{lat}}_{\rm ref}$ for droplet particles is computed as the difference of the liquid and gas standard formation enthalpies, and can be related to the latent heat at the boiling point as follows:


 {H_{lat}}_{\rm ref} = H_{lat} - \int^{{T_{bp}}}_{T_{\rm ref}} c{_{p}}_g dT + \int^{{T_{bp}}}_{T_{\rm ref}} c{_{p}}_p dT (22.9-3)


where      
  ${c_p}_g$ = heat capacity of gas product species (J/kg-K)
  $T_{bp}$ = boiling point temperature (K)
  $H_{lat}$ = latent heat at the boiling point temperature (J/kg)

For the volatile part of the combusting particles, some constraints are applied to ensure that the enthalpy source terms do not depend on the particle history. The formulation should be consistent with the mixing of two gas streams, one consisting of the fluid and the other consisting of the volatiles. Hence ${H_{lat}}_{\rm ref}$ is derived by applying a correction to $H_{lat}$, which accounts for different heat capacities in the particle and gaseous phase:


 {H_{lat}}_{\rm ref} = H_{lat} - \int^{{T_{p,init}}}_{T_{\rm ... ... c{_{p}}_g dT + \int^{{T_{p,init}}}_{T_{\rm ref}} c{_{p}}_p dT (22.9-4)


where      
  $T_{p,init}$ = particle initial temperature (K)



Mass Exchange


The mass transfer from the discrete phase to the continuous phase is computed in FLUENT by examining the change in mass of a particle as it passes through each control volume in the FLUENT model. The mass change is computed simply as


 M = \frac{\Delta m_p}{m_{p,0}} \;\; \dot{m}_{p,0} (22.9-5)

This mass exchange appears as a source of mass in the continuous phase continuity equation and as a source of a chemical species defined by you. The mass sources are included in any subsequent calculations of the continuous phase flow field and are reported by FLUENT as described in Section  22.16.



Under-Relaxation of the Interphase Exchange Terms


Note that the interphase exchange of momentum, heat, and mass is under-relaxed during the calculation, so that


 F_{\rm new} = F_{\rm old} + \alpha (F_{\rm calculated} - F_{\rm old}) (22.9-6)


 Q_{\rm new} = Q_{\rm old} + \alpha (Q_{\rm calculated} - Q_{\rm old}) (22.9-7)


 M_{\rm new} = M_{\rm old} + \alpha (M_{\rm calculated} - M_{\rm old}) (22.9-8)

where $\alpha$ is the under-relaxation factor for particles/droplets that you can set in the Solution Controls panel. The default value for $\alpha$ is 0.5. This value may be reduced to improve the stability of coupled calculations. Note that the value of $\alpha$ does not influence the predictions obtained in the final converged solution.

Two options exist when updating the new particle source terms $F_{\rm new}$, $Q_{\rm new}$ and $M_{\rm new}$. The first option is to compute the new source terms and the particle source terms, $F_{\rm calculated}$, $Q_{\rm calculated}$ and $M_{\rm calculated}$, at the same time. The second option is to update the new source terms, $F_{\rm new}$, $Q_{\rm new}$ and $M_{\rm new}$, every flow iteration, while the particle source terms, $F_{\rm calculated}$, $Q_{\rm calculated}$ and $M_{\rm calculated}$, are calculated every Discrete Phase Model iteration. The latter option is recommended for transient flows, where the particles are updated once per flow time step.



Interphase Exchange During Stochastic Tracking


When stochastic tracking is performed, the interphase exchange terms, computed via Equations  22.9-1 to  22.9-8, are computed for each stochastic trajectory with the particle mass flow rate, $\dot{m}_{p0}$, divided by the number of stochastic tracks computed. This implies that an equal mass flow of particles follows each stochastic trajectory.



Interphase Exchange During Cloud Tracking


When the particle cloud model is used, the interphase exchange terms are computed via Equations  22.9-1 to  22.9-8 based on ensemble-averaged flow properties in the particle cloud. The exchange terms are then distributed to all the cells in the cloud based on the weighting factor defined in Equation  22.2-55.


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