Particle-Particle Model: Heat Conduction

Heat flow between colliding particles due to conduction. From https://doi.org/10.1016/j.oceram.2021.100182, Equation 6 - Equation 8.

\dot{Q}_{cond} = 2 R_{c} k f_{res,eff} \kappa_{eq} \left( T_{1} - T_{2} \right)

R_{c} = 2 \sqrt{R_{eq} \delta_{n}}

f_{res,eff} = \begin{cases}
f_{res} &\delta_{n} \le 2 \lambda_{min} R_{eq} \\
f_{res} + \frac{1 - f_{res}}{\lambda_{max} - \lambda_{min}} \left( \frac{\delta_{n}}{2R_{eq}} - \lambda_{min} \right) &2\lambda_{min}R_{eq} < \delta_{n} < 2\lambda_{max}R_{eq} \\
1 &2\lambda_{max}R_{eq} < \delta_{n}
\end{cases}

R_{eq} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}

\kappa_{eq} = \frac{\kappa_{1}\kappa_{2}}{\kappa_{1}+\kappa_{2}}

Integration of heat is done as:

\dot{Q}_{1} = -\dot{Q}_{cond}

\dot{Q}_{2} = +\dot{Q}_{cond}

T_{1,2} = T_{1,2} + \frac{\dot{Q}_{1,2}}{C_{1,2} m_{1,2}} \Delta t_{sim}

Symbol	Parameter name	Description
$\delta_{n}$		Normal overlap [m]
$\kappa_{1}$, $\kappa_{2}$		Particle thermal conductivities [W/m/K]
$\kappa_{eq}$		Equivalent thermal conductivity in contact [W/m/K]
$C_{1}$, $C_{2}$		Particle Heat capacities from materials editor [J/kg/K]
$\lambda_{max}$	MAX_OVERLAP	Maximum overlap factor [-]
$\lambda_{min}$	MIN_OVERLAP	Minimum overlap factor [-]
$f_{res}$	RESISTIVITY_FACTOR	Thermal conduction resistivity factor [-]
$f_{res,eff}$		Effective thermal conduction resistivity factor [-]
$k$	CONDUCTION_SCALING_FACTOR	Scaling factor for conductivity [-]
$m_{1}$, $m_{2}$		Particle masses [kg]
$\dot{Q}_{1}$, $\dot{Q}_{2}$		Particle heat flows [W]
$\dot{Q}_{cond}$		Heat flow due to conduction [W]
$R_{1}$, $R_{2}$		particle radii [m]
$R_{c}$		Contact radius [m]
$R_{eq}$		Equivalent particle radius [m]
$\Delta t_{sim}$		Simulation time step [s]
$T_{1}$, $T_{2}$		Particle temperatures [K]