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Discrete Particle Model

Within the discrete particle model, the flow of the dispersed phase is modelled by tracking a specified number of particles through the continuous fluid phase. In Simerics-MP, the model has the following assumptions/restrictions:

• The discrete phase is composed of definite number of spherical particles through the continuous fluid flow phase. The particles are defined to be either having a mass, Has Mass or Massless.

• The size of a particle is determined by a specified radius at the release position and time and remains unchanged. The particle-particle interaction is assumed to be negligible.

• The particles interact with both the fluid flow and wall boundaries, but the volume of a particle neither displaces any fluid (low volume fractions in the particle phase) nor interferes with the geometry (i.e. an oversize particle could fit through a smaller gap).

• Between the continuous fluid phase and the particles there are no heat and mass transfer. The particle temperature is assumed to be the same as the local temperature of the fluid flow.

Under these assumptions, the motion of each individual particle is tracked using a Lagrangian approach. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated over time to calculate the behavior of the particles as they traverse the flow domain. The particle modelling approach in Simerics-MP has the following salient characteristics:

• The discrete particle model follows the Euler-Lagrange approach. The fluid phase is treated as a continuum by solving the continuity and Navier-Stokes equations, while the dispersed phase is solved by tracking the motion of each individual particle using a Lagrangian approach. The fraction of volume taken by the particles is not included in the continuous phase calculation.

• For the option of Massless, the particles are moving along with the fluid flow, or following the streamlines of the flow field. In this case, the size (radius) of the particles does not have any influence on both the flow and particles and is used for the purpose of display only.

• For the particles with mass (Has Mass), the mass of a particle is dictated by the specified particle radius/ diameter and particle density. The forces acting on a particle, which determine the motion of particles, include particle-fluid drag (inertial force) and gravity. The turbulence dispersion forces on particles are not considered. In this option, the size of the particles influences both the particle-fluid drag forces as well as post processing.

• The momentum exchange between the fluid phase and the discrete particle phases are modelled by one-way coupling in which only the fluid phase affects the motions of particles or two-way coupling where the particles also affect the fluid flow through particle-fluid drag forces.

• The wall-particle interactions are modelled using a variety of particle wall models including stick, perfect bounce and partial bounce.

• While the fluid phase can be both steady and unsteady, the particle tracking is a transient process that involves the integration of particle paths through the discretized domain. In this approach, individual particles are released/injected from the specified locations at different times. Each particle is then tracked from its release position to final destination (either escapes the domain or meet certain integration limits). Finally, an average of all particle tracks can be obtained, and the particle-fluid interactions are calculated as source terms to the fluid phase momentum equations.

• The path travelled by particles can be displayed using the related Streakline Tracking Method available in the Particle module.

 

Particle Motion Theory

In the Lagrange approach, the particle motion is determined by the force balance on the particle and the conditions under which the particle is released (initial conditions). To model the discrete particle phase, therefore, the equations of motion for particles are first formed based on force balance. Then the boundary and initial conditions for particles are specified. Finally, the integration of particle equation of motion is carried out for particle tracking.

 

Equations of Motion for Particles

Particle Force Balance

For a discrete particle traveling in a continuous fluid medium, the motion of the particle is determined by the net force acting on it. According to Newton’s second law, the force balance on the particle can be written in the following Lagrange form:

5.513

where

	Particle mass (kg)
	Particle velocity (m/s)
	Net force exerted on the particle (N), which affects the particle acceleration.

In a Cartesian coordinate system, let point represent the location of the particle, and be the particle velocity components. With the Lagrangian approach, the particle velocity is defined as:

 

5.514

 

For a spherical particle occupying a volume with density  and diameter (Simerics-MP accepts radius as input), the particle mass is computed as:

5.515

As for the net force there are a number of contributing factors such as, the fluid-to-particle drag force, the gravity effect, the forces due to domain rotation (centripetal and Coriolis forces), and other forces due to the difference in velocity between the particle and fluid as well as to the displacement of the fluid by the particle. In Simerics-MP may be expressed as follows:

5.516

where

	Drag force (N)
	Gravity force (N)
	Centripetal and Coriolis forces (N)
	Other forces such as virtual mass force, pressure gradient force, lift force, etc. specified by the user (N)

By default, only the drag force on the particle is considered.

 

Substituting equation 5.516 into equation 5.513 and dividing it by , the solved force balance equation for a particle has the following form:

5.517

To close equation 5.517, the contribution of each individual force needs to be calculated. The sub-models or formulations adopted in Simerics-MP are described in the following subsections.

Drag Force on Particles

The aerodynamic drag force on a particle is proportional to the phase slip velocity, the difference between the fluid and particle velocities. Assuming that at the same space where the particle is located at a given time, the fluid flow velocity equals , then we have the drag force in the form:

5.518

where is the fluid phase density; and is the area of the particle projected in the flow direction. For a spherical particle with diameter , is the maximum area of cross section:

5.519

The term is the drag coefficient, which depends on the relative Reynolds number, :

5.520

where is the fluid dynamic viscosity (Pa-s).

The drag coefficient is introduced to account for experimental results on the viscous drag of a solid sphere. Various models or empirical correlations have been developed to determine the drag function () to estimate the fluid-particle exchange. For smooth spherical particles, among many models, the most complete function is the corrections by Morsi and Alexander1S. A. Morsi and A. J. Alexander, "An Investigation of Particle Trajectories in Two-Phase Flow Systems", J. Fluid Mech., 55(2) 193–208, September 26 1972., which has the general expression:

5.521

where , and are model constants, whose values depend on the relative Reynolds number, as shown in Table 5.42.

From Table 5.42, one can see that at very low particle Reynolds numbers (the viscous regime),  , the drag coefficient for flow past spherical particles returns to the Stokes’ law:

5.522

On the other hand, when is sufficiently large so that the inertial effects dominate the viscous effects, the fluid-particle flow is in the inertial or Newton regime. From Table 5.42, it is clear that the drag coefficient becomes less and less dependent on the relative Reynolds number. In fact, a constant of value is often used instead of the full Morsi and Alexander model:

5.523

In the transitional region between the viscous and inertial regimes, , for spherical particles, both viscous and inertial effects are important. Therefore, the drag coefficient is a complex function of the relative Reynolds number, which can be estimated by Morsi and Alexander model or other correlations. For example, according to the Schiller and Naumann model2L. Schiller and Z. Naumann, "Z. Ver. Deutsch. Ing. 77. 318. 1935.:

5.524

Finally, to simplify the expression of the drag force term in equation 5.517, the “particulate relaxation time”, , is introduced:

5.525

By combing equations equation 5.515, equation 5.518 - equation 5.520 and equation 5.525, the drag force per unit particle mass has the formulation:

	5.526

Therefore, the default (only the drag force is considered) particle force balance equation has the form:

5.527

 

Inclusion of the Gravity Term

The gravity term is by default not included in the particle force balance equation. However, it can be turned on as an option in Simerics-MP. For a particle immersed in fluid flow, the effect of gravity results in a buoyancy force, which is equal to the weight of the displaced fluid by the particle. Assuming is the fluid mass displaced by the particle and is the gravity vector, the resulting force is given as:

5.528

Or the force per unit particle mass is given as:

5.529

And the force balance equation has the form:

5.530

 

Rotation Force on Particles

When modelling fluid flows in a rotating reference frame, the rotation-induced additional force term, , is an intrinsic part of the particle acceleration and consists of the effect of Coriolis and centripetal forces:

	5.531

Or the rotation force per unit particle mass is given as:

5.532

where is the angular velocity of the rotating reference frame and is the vector connecting the axis center and particle location. With the addition of this force term, the particle balance equation is:

5.533

equation 5.533 governs the motion of a particle in a Lagrange system when the flow is solved in a rotating reference frame.

Boundary and Initial Conditions for Particles

In the Lagrange approach, particle tracking is a transient procedure. Therefore, both boundary and initial conditions are required to calculate the trajectories of the particles. The boundary conditions define the particle behavior at the boundaries of the computational domain, in particular, the particle-wall interactions. The initial conditions concern the particle release from boundaries including the release position, frequency, velocity, particle type and size (radius), and the number of particles. In this section, both the initial and boundary conditions are described.

 

Boundary Conditions

Simerics-MP applies a discrete phase boundary condition to determine the behavior of particles at a boundary. When a particle reaches a boundary of the flow domain (including physical boundary and solid-fluid interface), for example a wall or an inlet boundary, one of several scenarios may occur at the boundary:

• The particle may be reflected via an elastic or inelastic collision.
• The particle may escape through the boundary. In such a case, the particle is lost from the calculation at the point of impact with the boundary.
• The particle may be trapped at the wall and is lost from the calculation at the point of impact with the boundary.
• The particle may pass through an internal boundary zone, such as fan or porous jump.
• The particle-boundary interaction may also be determined by user-defined methods to model the particle behavior when hitting the boundary.

Based on the particle behavior at boundaries, the flow boundary conditions and interfaces are regrouped into three types of discrete particle boundary conditions: Open, Symmetry and Wall.

 

Open Particle Boundary

Open discrete boundary condition allows particles or streamlines to exit the computational domain. For the discrete particle phase, an open boundary is usually an inlet or outlet boundary of the fluid flow phase in the Euler system, but it can also apply to the other types of flow boundaries such as wall and symmetry. At an open particle boundary, the particle can either leave from or enter into the domain depending on the particle velocity direction.

Let be the unit normal vector to the open boundary which points in the direction away from the computational domain, with the particle boundary velocity . If , the velocity vector points away from the computational domain, indicating that the particle escapes through the boundary. The particle is then lost from the calculation at the point of impact with the boundary.

 

Symmetry Particle Boundary

When a particle or streakline in the computational domain hits a discrete symmetry boundary, the boundary condition only allows it to reflect back into the domain. For the discrete particle phase, a symmetry particle boundary typically corresponds to a flow symmetry in the Euler system. It can also be served as a location for particle release.

Let be the normal-to-symmetry unit vector at point “” of the symmetry, with its direction pointing away from the symmetry to the computational domain. Further, and are introduced to indicate the particle impact velocity angle at the particle symmetry boundary, as shown in Figure 5.215. With the particle reflecting from the symmetry, its total kinetic energy is conserved: the tangential velocity remains the same while the normal velocity component only changes the sign. Mathematically, the particle symmetry boundary condition can be expressed as:

5.534

where

	Angle at the point “": of the symmetry (deg)
	Magnitude of particle incident velocity (m/s)
	Magnitude of particle reflected velocity (m/s)

 

Wall Particle Boundary

Particle-wall interaction involves very complex physical phenomena and not all aspects are well understood. For example, for liquid droplets, dimensional analysis shows that the droplet-wall interaction depends on the wall temperature, wall material and roughness, impact angle and impact velocity, the existence of a wall film, and various other parameters. As a result, there are a range of sub-models developed to reproduce the different regimes of wall-particle interactions as well as to account for the impacts of flow parameters and wall boundary conditions.

In the present discrete particle model, the shape, size and mass of the particles are assumed to remain unchanged. Also, the fluid and particles are considered to be in thermal equilibrium. Therefore, a very simplistic approach is used to describe the process of particles (has mass) colliding with walls: During the collision process, particles exchange momentum only with the wall, and the particles can only have one of three ways to interact with the wall: perfect bounce, stick and partial bounce, as shown in Figure 5.216.