enable_particle_melting command

Purpose

Addition to the enable_heat_transfer command to allow for melting particles.

Syntax

enable_particle_melting keyword value

Keywords:

Keyword	Description
minimum_volume *	minimum particle volume before particles get deleted from the simulation units: [length^3]
minimum_radius *	minimum particle radius before particles get deleted from the simulation units: [length]
disable_conduction	`yes` or `no`; calculation of thermal conduction between particles can be switched off default: `no`
enable_liquid_solidification	`yes` or `no`; allow solidification of liquid on particles that are colder than the melting temperature. Requires the enable_cfd_coupling command and the melting CFD forceModel. default: `no`
attenuate_solidification	scalar to reduce (or increase) the amount of liquid solidifying. Must be larger than 1. default: `1`

* minimum_volume and minimum_radius are mutually exclusive, the usage of one of them is mandatory.

Examples

enable_heat_transfer initial_particle_temperature 273.15
enable_particle_melting minimum_volume 1e-11

Associated material properties

Material properties

latentHeat ( $L$ ): specific latent heat of the material [ $J/Kg$ ]
Tmelt ( $T_{melt}$ ): melting temperature of the material [ $K$ ]
thermalCapacity ( $c$ ): specific thermal capacity of the material [ $J/(Kg K)$ ]

Description

This model implements the melting of particles when the melting temperature $T_{melt}$ [K] is exceeded. Depending on the choice of the internal temperature model (see enable_heat_transfer) two different melting rate models are available, as described below.

The phase change is modelled in two parts

$\frac{m^{n+1} - m^n}{\Delta t} = -\frac{q}{L},\; \mbox{ if } T > T_{melt} \mbox{ and } q > 0 \quad (1) \frac{m^{n+1} - m^n}{\Delta t} = - m \frac{c}{L} \frac{T-T_{melt}}{\Delta t},\; \mbox{ if } T > T_{melt} \quad (2)$

where $m$ [Kg] is the particle mass, $n$ the time step, $q$ [J/s] is the total heat transfer rate into the particle, $c$ [J/(KgK)] is the specific thermal capacity and $\Delta t$ [s] the time step size. (1) is a term due to heat flux from outside, (2) is a term that assures that the particle temperature $T$ [K] satisfies $T \leq T_{melt}$ in case the particle temperature is changed e.g. by the set command.

The phase change is modelled by the following equation:

$\dot{m} = \frac{4 \pi r^2}{L}\left( -\lambda \frac{\partial T}{\partial r} + \dot{q}\right)$

where $\dot{m}$ [Kg/s] is the melting rate, $r$ [m] is the particles radius (volume equivalent radius for non-spherical particles) and $\dot{q}$ [J/(s m²)] is the external heat flux. The temperature gradient is taken at the particle boundary and it is calculated explicitly using the temperature from the last two grid points inside the particle:

$\frac{\partial T}{\partial r} = \frac{T_{N}^{n} - T_{N-1}^{n}}{\Delta r}$

where $n$ is current time step and $N$ is the number of the grid points inside the particle. For all this grid points inside, the shell model is applied without changes as in enable_heat_transfer. The temperature field at the rest of the grid points outside of the shrunken particle is overwritten by the melting temperature.

The cooling effect of the phase change is depicted by adjusting the heat transfer rate via

$q^{n+1} = q^n - L \frac{m^{n+1} - m^n}{\Delta t},$

where $n$ is the time step. As soon as the particle volume is lower than the limit specified by minimum_volume or the minimum_radius they are deleted from the simulation. When the particle is deleted, more accurate values of the corresponding melted mass are obtained as the value of minimum_volume is decreased.

Additional solification of liquid around a particle may occur if the enable_liquid_solidification is set:

The amount of solidified liquid is determined by the incoming heat transfer rate:

$\frac{m^{n+1} - m^n}{\Delta t} = -\frac{q}{L} \alpha$

Solidification only occurs if the particle temperature is lower than the melting temperature and the surrounding fluid is hotter than the particle, i.e. the heat transfer rate $q$ [J/s] in above equation is positive. Furthermore, the amount of solidified mass is scaled with the non-dimensional attenuation parameter $\alpha$ , set by the attenuate_solidification keyword. The amount of solidified liquid mass is further limited by the available liquid in vicinity of the particle as communicated from the CFD side, see enable_cfd_coupling command.

At the end of the phase change process, the heat transfer rate is updated considering the heat released by the solidified liquid.

Note

This approach to solification assumes that particle and liquid consist of the same material and that the material properties are identical in solid and liquid state.

Additional information

The initial particle radius is written to binary restart files so simulations can continue properly.

This fix computes a value which can be accessed by various output commands. This value is the total (accumulated) molten mass which can be accessed via id_ID.molten_mass where ID is the id of the corresponding enable_heat_transfer command.

Restrictions

none