Lagrangian Approach for the Post-Processor Simulation

Generally, in this case the particles mass flow rate should be lower than about 30% of the fluid mass flow rate. To simulate dilute two-phase flows, where flows of gases or liquids contaminated with particles, the Lagrangian approach is used.

The particles of a specified (liquid or solid) material and constant mass are assumed to be spherical. The particles trajectories are determined by numerical integration of the equation:

where mp is the particle mass, is the particle velocity vector, d is the particle diameter, CD is the particle resistance coefficient, is the gravitational acceleration vector, and Re is the Reynolds number based on the particle diameter, the relative velocity, and freestream density and viscosity. The particle resistance coefficient is calculated with Henderson’s formula (Ref. 7), derived for continuum and rarefied, subsonic and supersonic, laminar, transient, and turbulent flows over the particles, and taking into account the temperature difference between the fluid and the particle. The particles’ rotation, their interaction with each other, Brown motion and additional forces are not taken into account.

For subsonic flow (M ≤ 1):

For supersonic flow (M ≥ 1.75):

For the flow regimes with Mach between 1 and 1.75, a linear interpolation is to be used:

where is the drag coefficient calculated using the correlation for subsonic flow with M = 1, is the drag coefficient calculated using the correlation for supersonic flow with M = 1.75, M is the Mach number based on the relative velocity between the particle and the fluid flow, is the molecular speed ratio, γ is the specific heat ratio, T is the fluid temperature in freestream, Tp is the particle temperature, and subscript “∞” refers to freestream conditions.

Thermal energy equation for a particle is given by:

where Cp is the specific heat of particle, Tp is the particle temperature, k is the thermal conductivity of fluid, and Nu is the Nusselt number. The particle/fluid heat transfer coefficient is calculated with the formula proposed in Ref. 1:

If necessary, the gravity is taken into account. Since the particle mass is assumed constant, the particles cooled or heated by the surrounding fluid change their size.

The interaction of particles with the model surfaces is taken into account by specifying either full absorption of the particles (that is typical for liquid droplets impinging on surfaces at low or moderate velocities) or ideal or non-ideal reflection (that is typical for solid particles). The ideal reflection denotes that in the impinging plane defined by the particle velocity vector and the surface normal at the impingement point, the particle velocity component tangent to surface is conserved, whereas the particle velocity component normal to surface changes its sign. A non-ideal reflection is specified by the two particle velocity restitution (reflection) coefficients, en and eτ, determining values of these particle velocity components after reflection, V2,n and V2,τ, as their ratio to the ones before the impingement, V1,n and V1,τ:

As a result of particles impingement on a solid surface, the total erosion mass rate (RΣe) and the total accretion mass rate (RΣa) are determined as follows:

where:

N is the number of fractions of particles specified by user as injections in Flow Simulation (the user may specify several fractions of particles, also called injections, so that the particle properties at inlet, i.e. temperature, velocity, diameter, mass flow rate, and material, are constant within one fraction),

i is the fraction number,

Mpi is the mass impinging on the model walls in unit time for the i-th particle fraction,

Ki is the impingement erosion coefficient specified by user for the i-th particle fraction,

Vpi is the impingement velocity for the i-th particle fraction,

b is the user-specified velocity exponent (b = 2 is recommended),

f1i (αpi) is the user-specified dimensionless function of particle impingement angle αpi,

f2i (dpi) is the user-specified dimensionless function of particle diameter dpi.