For energy sources due to convective heat transfer, you can limit the source terms by setting Option to Correction Factor Bounding.

You can use the Stop Bounding after Iteration option to limit the number of iterations that have source term bounding. The default value for this option is 40 iterations.

The particle convective energy source in the gas phase can be written as:

(8–5)

where

is the particle temperature,

is the local fluid temperature, and

is a transfer coefficient.

Equation 8–5 is solved in the particle tracker at each particle iteration step and the accumulated sources are stored at a place where they are later on picked up in the energy assembly of the gas phase. By default, particles are not tracked after every fluid flow simulation, but in user defined intervals (default: every 5^th fluid step in a steady-state run). Whenever the particle tracker is run, new values for the particle sources are generated and these sources persist up to the next time particles are tracked. If the particle sources are just large enough, it might not be sufficient to just limit the sources once, when they are generated, but also every time they are picked up and used by the gas phase energy equation. This somehow mimics a ‘low cost’ particle tracking step, where only the energy sources are updated to account for changes in the fluid temperature (enthalpy) due to energy sources applied in previous flow solver steps.

The basis of the suggested modification is the equation for the particle convective energy source, Equation 8–5. This equation is reformulated as follows:

(8–6)

where

is the (average) particle temperature,

the fluid temperature, and

is a transfer coefficient.

Building the ratio of allows approximating the new value of the particle source term for the current iteration, taking the fluid temperature change from the ‘old’ () to the current time step () into account:

(8–7)

Note that in the above formulation the average particle temperature, from the last tracking step is used. This quantity is not changing in between successive tracking steps. and are updated every time step.

The correction factor, , is limited to be in the range of [0,1]. This correction is applied before the particle source term is used by the gas phase energy equation.

For momentum sources due to fluid drag, you can limit the source terms by setting Option to Correction Factor Bounding.

You can use the Stop Bounding after Iteration option to limit the number of iterations that have source term bounding. The default value for this option is 40 iterations.

Similar considerations as outlined in Particle Heat Source Bounding, can be made to limit the particle momentum sources to the gas phase. Under the assumption that the particle drag is the major contributor to the particle momentum source term, it is possible to write the particle momentum source as:

(8–8)

where

is the particle drag coefficient,

is the particle slip velocity,

is the fluid velocity vector,

is the particle velocity vector, and

is a coefficient, dependent on the fluid density, .

Forming the ratio of and rearranging the resulting equation for gives:

(8–9)

Assuming that the drag coefficient, , is determined by the Schiller-Naumann drag law

(8–10)

with

being the particle Reynolds number computed from the slip velocity between particles and fluid, the particle diameter and the fluid kinematic viscosity. Inserting the particle Reynolds number into Equation 8–10 it is possible to simplify Equation 8–9 to give:

(8–11)

Equation 8–11 is used to determine a factor that limits the particle momentum source by taking fluid velocity and density changes into account. The correction factor is limited to be in the range of [-1,1].

In Ansys CFX, particle mass sources are, by default, not linearized with respect to the fluid mass fraction (mass fraction equations) or the fluid pressure (continuity equation). For cases with relatively small mass transfer rates, linearization of the particle mass sources may not play an important role. This may be different, though, for cases where the particle mass transfer is large enough to significantly affect the fluid energy and/or the fluid momentum. In these cases, the use of mass source linearization may be crucial for a successful simulation run.

Linearization of particle mass sources can be included by setting Option to Source Coefficient with respect to Mass Fraction. With this setting, the derivatives in the following subsections are included in the linearization of sources in the equation for mass of the particle component (see Interphase Transfer Through Source Terms in the CFX-Solver Theory Guide).

 

8.10.8.3.1. Simple Mass Transfer Model

Each component of mass being transferred between the continuous and particle phases satisfies the equation:

(8–12)

In Equation 8–12, is the mass of the constituent in the particle, is the mass fraction of component in the particle, is the mass fraction of component in the surrounding fluid, is the equilibrium mass fraction ratio, is the dynamic diffusivity of the mass fraction in the continuum, and is the Sherwood number.

For the mass fraction equation of species, , the derivative of Equation 8–12 with respect to is required. This derivative can be written as:

(8–13)

Equation 8–13 is used to compute the source term coefficient in the gas phase mass fraction equation.

The derivative of the mass transfer rate with respect to the fluid pressure is zero, because the mass transfer rate does not explicitly depend on the fluid pressure.

 

8.10.8.3.2. Liquid Evaporation Model

The liquid evaporation model is a model for particles with heat transfer and mass transfer, and in which the continuous gas phase is at a higher temperature than the particles. The model uses two mass transfer correlations depending on whether the droplet is above or below the boiling point. This is determined through an Antoine equation.

 

8.10.8.3.2.1. Droplet Temperature Below Boiling Point

The mass transfer rate from the particle to the gas phase is determined by:

(8–14)

where is the mass of the constituent in the particle, is the dynamic diffusivity of the mass fraction in the continuum, and is the Sherwood number. and are the molecular weights of the vapor and the mixture in the continuous phase, is the molar fraction in the gas phase, and is the equilibrium mole fraction at the droplet surface defined as the component vapor pressure divided by the pressure in the continuous phase.

The derivative of Equation 8–14 with respect to the fluid component mass fraction is:

(8–15)

As with the simple mass transfer model, there is no linearization applied with respect to the fluid continuity and volume fraction equations.

 

8.10.8.3.2.2. Droplet Temperature Above Boiling Point

If the particle is boiling, the mass transfer into the gas phase is driven by convective heat transfer and radiative heat transfer to the particle:

(8–16)

and are the rates of convective heat transfer and radiative heat transfer to the particle, respectively. is the latent heat of vaporization. The mass transfer rate in the boiling regime does not depend on the fluid component mass fraction, , nor on the fluid pressure, . Therefore both derivatives are zero:

(8–17)