DPM Theory -- Governing Equations, Parcels, and Particle Force Balance

#+CATEGORY: cfd

#+TAGS: DPM Lagrangian particle-tracking multiphase

#+MATURITY: growing

DPM Theory :: Governing Equations, Parcel Concept, and Particle Force Balance

The Discrete Phase Model (DPM) in ANSYS Fluent solves the continuous phase on an Eulerian mesh and tracks each particle on a Lagrangian path. The two phases exchange momentum, mass, and energy. The particle equation is a force balance integrated along the trajectory. This page collects the equations, the parcel concept, and the assumptions that determine when DPM is valid.

The Parcel Concept

A real smoke stream contains billions of particles. Tracking each one is impossible. The parcel concept groups identical particles into a single computational particle. One parcel represents $N_p$ real particles that share the same diameter, density, velocity, and temperature. The solver integrates one trajectory per parcel, then multiplies the result by $N_p$ to recover statistics.

The parcel count is the primary accuracy knob. Too few parcels --> noisy concentration fields. Too many parcels --> long run times with no extra accuracy. For PM2.5 smoke in a 3D geometry, expect 5 000--20 000 parcels per injection for convergence.

Governing Equation -- Particle Momentum

The particle momentum equation (Basset-Bouenette-Oseen form, simplified in Fluent) is:

where:

• $\mathbf{u}_p$ = particle velocity [m/s]
• $\mathbf{u}$ = continuous-phase (fluid) velocity at the particle location [m/s]
• $\rho_p$ = particle density [kg/m^3]
• $\rho$ = fluid density [kg/m^3]
• $\mathbf{g}$ = gravitational acceleration [m/s^2]
• $F_D$ = drag force per unit particle mass [1/s]
• $\mathbf{F}_{\text{other}}$ = all additional forces (see Additional Forces)

Drag Force

The drag term is the dominant force for most DPM applications. Fluent writes it as:

where:

• $\mu$ = fluid dynamic viscosity [Pa.s]
• $d_p$ = particle diameter [m]
• $C_D$ = drag coefficient (function of particle Reynolds number)
• $Re_p$ = particle Reynolds number

The particle Reynolds number is:

Fluent ships several drag-law options. The selection depends on particle shape and Reynolds regime:

For PM2.5 in a shisha simulation, the particles are sub-micron to 2.5 \mu m. The particle Reynolds number is well below unity. Stokes or Schiller-Naumann is correct.

See DPM Drag Models -- Full Reference for details on every law.

Additional Forces

Fluent computes $\mathbf{F}_{\text{other}}$ from the Physical Models tab. The forces available are:

Particle force ranking for shisha PM2.5: 1. Drag (dominant) 2. Gravity + buoyancy (significant -- particle settling) 3. Brownian motion (significant for $d_p < 1$ \mu m) 4. Thermophoretic force (moderate near bowl and hose) 5. Saffman lift (minor)

See DPM Physical Models -- Fluent Panel Walkthrough for panel-by-panel setup.

Heat Transfer -- Particle Energy

The particle temperature equation is:

where:

• $m_p$ = particle mass [kg]
• $c_p$ = particle specific heat [J/(kg.K)]
• $h$ = convective heat transfer coefficient [W/(m^2.K)]
• $A_p$ = particle surface area [m^2]
• $T$, $T_p$ = fluid and particle temperature [K]
• $\epsilon_p$ = particle emissivity (0--1)
• $\sigma$ = Stefan-Boltzmann constant
• $Q_{\text{reaction}}$ = heat from chemical reaction (combusting particles only)

Fluent uses the Ranz-Marshall correlation for convective heat transfer:

For a stationary or very fine particle ($Re_p \approx 0$), this reduces to Nu = 2, the pure-conduction limit.

Shisha smoke particles are cool. Unless the bowl chamber exceeds 200 degrees C, heat transfer is a secondary concern. Enable it for completeness but do not expect large temperature changes.

See DPM Heat and Mass Transfer for evaporation and boiling laws.

Mass Transfer (Inert Particles: None)

Inert particles do not evaporate, boil, or react. Their mass is constant during the trajectory. For shisha smoke modeled as inert PM2.5, this is correct. The smoke aerosol is solid-phase carbon and tar particles. They do not evaporate at ambient temperatures.

If you later model liquid droplets (water mist, fuel spray), activate the Droplet particle type. Fluent then applies Law 2 (evaporation) or Law 3 (boiling) based on the local saturation pressure.

Turbulence Interaction -- Discrete Random Walk

Real smoke does not follow smooth streamlines. Turbulent eddies scatter particles randomly. Fluent models this with the Discrete Random Walk (DRW) model.

The DRW model works as follows:

1. The solver reads the local turbulent kinetic energy ($k$) and dissipation rate ($\epsilon$) from the turbulence model. 2. It generates a random Gaussian velocity fluctuation $u'$ for the fluid seen by the particle. 3. The interaction time is the shorter of:

• Eddy lifetime: $t_e = L_e / |u'|$
• Eddy crossing time: $t_c = -\tau_p \ln(1 - L_e / (\tau_p\,|\mathbf{u} - \mathbf{u}_p|))$

4. After the interaction time, the particle meets a new eddy and gets a new random fluctuation.

where:

• $L_e$ = eddy length scale
• $\tau_p$ = particle relaxation time

Stochastic tracking: Fluent runs Number of Tries random trajectories for each parcel and averages the results. Set this to 10--20 for PM2.5 to get statistically converged concentration fields.

One-way turbulence coupling: Particles modify turbulence by default. Toggle Two-way turbulence coupling in the Physical Models tab. For dilute smoke, one-way is sufficient.

See DPM Turbulence Interaction -- DRW Deep Dive for full mathematics and code validation.

Coupling -- One-Way vs Two-Way

Volume fraction check for shisha PM2.5:

Smoke concentration in a hookah chamber is on the order of 0.1--10 mg/m^3. For PM2.5 particles at 1200 kg/m^3 density, the volume fraction is:

This is well below the $10^{-6}$ threshold for two-way coupling. Start with one-way coupling. If you later increase the smoke concentration (e.g., dense chamber filling), switch to two-way.

Validity of the Dilute Assumption

DPM is valid when: 1. Particle volume fraction $\alpha_p \ll 1$ (dilute limit) 2. Particle-particle collisions are negligible (or handled by DDPM/DEM) 3. Inter-particle distance is much larger than particle diameter 4. Stokes number varies -- particles may be tightly coupled to flow or independent

If $\alpha_p$ exceeds 10--15%, switch to the Dense Discrete Phase Model (DDPM) or the Eulerian model. See Dense Discrete Phase Model.

Summary

1. The parcel concept groups identical particles into one computational particle. 2. The momentum equation is a drag-dominant force balance. 3. For shisha PM2.5, drag >> gravity > Brownian > thermophoresis >> lift. 4. The DRW model adds stochastic turbulent dispersion. 5. Start with one-way coupling; switch to two-way if mass loading exceeds 1. 6. Use stochastic tracking with 10--20 tries for statistical convergence. 7. DPM is valid for volume fraction well below 10%.

See Also

• ANSYS Fluent Multiphase Modeling (parent overview)
• DPM Drag Models
• DPM Turbulence Interaction -- DRW
• DPM Heat and Mass Transfer
• DPM Coupling -- One-Way vs Two-Way
• DPM Setup and Activation in Fluent
• Shisha PM2.5 DPM Case Study