Heat Transfer

When particles make contact with other particles or with the wall, heat is transferred through conduction. The rate of heat transfer is determined by the temperature difference, the effective contact area and the thermal conductivity of the materials.

Conduction

Two bodies are assumed to exchange heat through conduction when they make physical contact. The two bodies can be two particles or a particle and a wall.

There are four cases:

Particle-particle (see [719]):
The heat exchange $q_{i j}$ from particle i to particle j is:
Figure 1. EQUATION_DISPLAY

$q_{i j} = 4 r_{c} k (T_{j} - T_{i})$

(3299)

where:
- $r_{c}$ is the contact area radius.
- k is the equivalent thermal conductivity of the two particles, where
  
  $\frac{1}{k} = \frac{1}{k_{i}} + \frac{1}{k_{j}}$  
  
  where $k_{i}$ and $k_{j}$ are the thermal conductivities for particles i and j.
- $T_{i}$ and $T_{j}$ are the temperatures of particle i and particle j.
Particle-wall, adiabatic wall: heat exchange is zero.
Particle-wall, fixed wall temperature:
The heat exchange $q_{i w}$ from the wall to particle i is:
Figure 2. EQUATION_DISPLAY

$q_{i w} = 4 r_{c} k (T_{w} - T_{i})$

(3300)

where:
- k is the equivalent thermal conductivity, equal to $k_{i}$ .
- $T_{i}$ and $T_{w}$ are the temperatures of particle i and the wall.
Particle-wall, fixed wall heat flux:
The heat exchange $q_{i w}$ from the wall to particle i is:
Figure 3. EQUATION_DISPLAY

$q_{i w} = A_{c} q_{w}$

(3301)

where:
- $A_{c}$ is the contact area between the particle and the wall.
- $q_{w}$ is the heat flux density of the wall.

If the simulation includes a continuous phase, its energy state is unaffected by particle-particle and particle-wall heat conduction unless you activate the Two-Way Coupling model. With this model active, the Lagrangian energy source term is added to the energy equation of the continuous phase.

The linearized wall heat flux, $q_{w}$ , is expressed as:

Figure 4. EQUATION_DISPLAY

q_{w} = a + b T_{c} + c T_{w} + {d T_{w}}^{4}

(3302)

where:

$T_{c}$ and $T_{w}$ are the cell and wall temperatures, respectively.
$a$ , $b$ , $c$ , and $d$ are wall heat flux coefficients.

The contribution of DEM particle-wall conductive heat flux to the wall heat flux can be made by modifying wall heat flux coefficients $a$ and $c$ as follows:

For fixed wall heat flux conditions, only $a$ needs to be modified:

Figure 5. EQUATION_DISPLAY

a = \sum \frac{A_{c}}{A} q_{w}

(3303)

where:

$A_{c}$ is the particle-wall contact area.
$A$ the boundary face area.

For fixed wall temperature conditions, $a$ and $c$ are modified as:

Figure 6. EQUATION_DISPLAY

\begin{matrix} a = \sum 4 r_{c} {k T}_{p} \\ c = - \sum 4 r_{c} k \end{matrix}

(3304)

where:

$k$ is the particle thermal conductivity.
$r_{c}$ the contact radius.
$T_{p}$ the particle temperature.

Impact Heat Model

The impact heat model is a widely used approximate model. It calculates the rate of heat production that results from friction and damping in DEM particles using a linear formulation.

The rate of heat generation is:

Figure 7. EQUATION_DISPLAY

q_{r} = c_{t} F_{t} v_{t} + c_{n} F_{n} v_{n}

(3305)

where:

$c_{t}$ and $c_{n}$ are, respectively, the fractions of frictional work and damping work that are converted to heat.
$F_{t}$ and $F_{n}$ are, respectively, the frictional (tangential) forces and damping (normal) forces on the particle.
$v_{t}$ and $v_{n}$ are, respectively, the relative tangential and normal impact velocities.

The values for $c_{t}$ and $c_{n}$ must be estimated for the individual setup being simulated. The other factors are derived from other inputs in the DEM models. For examples, see Iwasaki [725] and Rojek [732].