heatTransferRadiation

Template Info

Target Parameters: epsilon_p, Qabs_p, Qsca_p
Contact Models: normal hertz, tangential history
Particle Shapes: sphere

This case simulates the heat transfer in a compacted particle bed to calibrate three parameters of the P1CFDEM radiation heat transfer model: (i) particle emissivity epsilon_p, (ii) particle absorption efficiency Qabs_p, and (iii) particle scattering efficiency Qsca_p. The P1CFDEM model is described in the CFDEMcoupling documentation.

_images/radiation.png — Figure 1: Simulation result obtained from `heatTransferRadiation` calibration template.

Simulation phases

Initialization

During the initialization, particles are inserted into the domain and pressed by a flat plate to form a compacted particle bed: see the illustration in the figure above. The simulation automatically stops when the pressure in the bed reaches a certain value. This is achieved by monitoring the time evolution of the height of the flat plate used to compress the bed, i.e. the height of the particle bed. Once the relative change of the height is smaller than a certain value, set by the pressureConvTol parameter, the initialization phase is completed.

Optimization

The optimization run is a CFDEMcoupling simulation that is performed until the effective heat transfer coefficient (see below) reaches a steady state. A constant heat flux is imposed at the top plate, whereas a constant temperature boundary condition is used at the bottom. The values of the top-wall heat flux and bottom-wall temperature are set by the parameters heatFlux and T_w0, respectively.

The effective heat transfer coefficient is defined by

$\lambda = \dot{q}_{top} \Delta x / \Delta \bar{T},$

where $\dot{q}_{top}$ [W / m²] is the heat flux that is imposed at the top wall , $\Delta x$ [m] is the compacted bed height and $\Delta \bar{T}$ [K] is the difference between the (calculated) averaged particle temperature and the (imposed) bottom-wall temperature.

In fact, the region where the particle-based averaging is performed, the probe region, can be modified by the non-dimensional parameter heatSourceFraction, which also modifies the value of $\Delta x$ [m] used in the expression above. This parameter is a fraction of the compacted bed height defining a distance, from the top wall downwards, where heat transfer is imposed. Heat transfer is applied in a volume region, instead of the wall surface, to avoid numerical issues related to particle-wall heat conduction. For example, if heatSourceFraction is set to 0.05 (the default value), $\Delta x$ will be 0.95 times the bed height, defining a probe region tha goes from the bottom plate up to $\Delta x$ .

The parameter heatSourceBot is used in the calculation of the probe region where the temperature averaging (measurement) is performed. It is a fraction of the compacted bed height and it is used to define the distance from the bottom where the probe region starts. For example, a zero heatSourceBot results in a probe region starting at the bottom wall. Non-zero values lead to probe regions starting above the bottom wall.

The load balancing is disabled by default due to the non-deterministic nature of the algorithm which can effect the results of calibration. If the number of particles is sufficiently high ( $10^5$ and above), the non-deterministic effect induced by the load balancing algorithm is negligible and load balancing should be enabled by setting the parameter useLoadBalancing to 1.

Quality Function

The quality function is the percentual difference between the calculated and reference effective heat transfer coefficient:

$\phi = (\lambda - \lambda_{ref}) / \lambda_{ref}.$

The reference value $\lambda_{ref}$ is set by the variable lambdaRef.

Measurement file

The relevant information for the calibration is the parameter lambdaRef. However, due to current restrictions of the framework, Aspherix® Calibration needs a measurement file to be present. The content of this file is irrelevant and will not affect calibration.

Template parameters

General settings

Parameter	Default
NPROC	4
dumpT	0.01 s
dump	0
useLoadBalancing	0

Specific settings

Variable	Default	Meaning
CG	1.0	coarse graining factor
DEMts	1e-6 [s]	time step of DEM simulation in the Initialization phase
DEM_writestep	0.001 [s]	time step for data output of DEM simulation
CFD_timestep	0.002 [s]	time step of CFD simulation
CFD_endtime	100.0 [s]	maximum allowed time interval of the CFD simulation
nCellsLength	16	number of vertical CFD cells
nCellsWidth	4	number of horizontal CFD cells
parcelsPerCell	3	number of parcels per CFD cell: is used to calculate the size of the DEM simulation domain
targetVolumeFraction	0.65	insertion target volume fraction (pre-compaction)
pressureConvTol	1e-4	percentage-variation of the bed height over time during compaction: defines a steady state criteria to stop the DEM simulation in the Initialization phase
servo_ctrl_Kp	0.1	proportional constant `kp` of the `mesh_module servo` for the bed compaction at the Initialzation phase (see the Aspherix Solver documentation)
servo_ctrl_Ki	0.0	integral constant `ki` of the `mesh_module servo` for the bed compaction at the Initialzation phase (see the Aspherix Solver documentation)
servo_ctrl_Kd	0.0	differential constant `kd` of the `mesh_module servo` for the bed compaction at the Initialzation phase (see the Aspherix Solver documentation)
servo_ctrl_max_vel	5.0 [m / s]	servo control PID velocity (compaction)
heatSourceFraction	0.05	fraction of the compacted bed height defining the distance, from the top wall downwards, where heat transfer is imposed
probeFraction	0.05	fraction of the compacted bed height defining the distance, from the bottom wall upwards, where the particle-based temperature averaging is performed: defines the probe region
convergenceIters	100	used to calculate the number of DEM timesteps per convergence loop in the Optimization phase (= convergenceIters * couplingInterval)
convergenceRelTol	1e-4	relative tolerance for lambda convergence
couplingInterval	10	used in the calculation of the DEM time step (= CFD_timestep/couplingInterval) in the optmization phase
radiationCouplingInterval	1	number of CFD timesteps per radiation solution
radiation	true	switch radiation modeling on/off

Some additional parameters have to be provided in the .casx calibration file, either as variable, param_calibration or param_fixed. They are:

r_primary	0.0005 [m]	particle radius before any coarse graining is applied
heatFlux	10e3 [W / m²]	heat flux applied at the top plate
lambdaRef	0.15 [W / (m K)]	reference effective bed conductivity
sigma_n	200.0 [W / (m K)]	compaction normal pressure [Pa]
T_w0	300.0 [K]	bottom wall temperature
T_f0	300.0 [K]	initial fluid temperature
T_p0	300.0 [K]	initial particle temperature
absorptivity_f	0.0 [1/m]	fluid absorption coefficient
emissivity_f	0.0 [1/m]	fluid emissivity
emission_f	0.0 [kg / (m s³)]	radiation intensity emitted by the fluid
emissivity_w	0.8 [1/m]	wall emissivity
Cp_p	1.0 [J/K]	particle thermal capacity
lambda_p	0.2 [W / (m K)]	particle thermal conductivity
Y_p	5e7 [Pa]	particle Young’s Modulus
poissons_p	0.3	particle Poisson’s ratio
rho_f	1.2 [kg / m³]	fluid density
Pr_f	0.7	fluid Prandtl number
nu_f	1.5e-5 [m / s²]	fluid kinematic viscosity

Some enviroment variables need to be set in order to run the CFDEMcoupling simulation: please see CFDEMcoupling intallation instructions for more details.