mesh_module heattransfer

Purpose

Command for enabling heat transfer between mesh elements, particles and fluid.

Syntax

mesh_module heattransfer keyword value

Keywords:

Keyword	Description
id	obligatory, user-defined name for the mesh module
heat_transfer_mode	available options: `legacy`, `coefficient` (recommended), `constant`, `flux` or `power` mode dependent keywords can be found below default: `legacy`
initial_temperature	obligatory except for restart; initial temperature of all wall elements in temperature units units: [temperature]
include_tangential_heatflux	`yes` or `no`, include / do not include the mesh internal heat transfer `yes` requires non-zero `wall_thickness` / `thickness_layers_wall` (see keywords below) default: `yes`
wall_density	density of material that the wall is made of (optional, derived from mesh material if not provided) units: [mass/length^3]
output_detailed_heatflux	available options: `yes` or `no` default: `no`
enable_radiation	available options: `yes` or `no` default: `no`
add_external_heatflux	available options: `yes` or `no` default: `no`

Keyword	Description
htc_exterior_fluid	mandatory; wall-fluid heat transfer coefficient units: [power/(length^2*temperature)]
temperature_exterior_fluid	mandatory; temperature of exterior (far field) fluid units: [temperature]
wall_thickness *	thickness of each mesh element units: [length]
thickness_layers_wall *	list of thickness of layers {thickness_layer_1, thickness_layer_2, …} units: [length]
kappa_layers_wall	list of thermal conductivities kappa of layers {kappa_layer_1, kappa_layer_2, …} only usable in combination with `thickness_layers_wall` units: [power/(length*temperature)]
active_region	ID of region to select the mesh elements where the exterior temperature is applied

* The keywords wall_thickness and thickness_layers_wall are mutually exclusive, one of them is mandatory.

No keywords, the mesh will remain at the initial temperature.

Keyword	Description
flux_value	heat flux density units: [power/length^2]
wall_thickness	thickness of each mesh element; mandatory if `include_tangential_heatflux` is `yes` (default) units: [length]

Keyword	Description
power_value	heat flux units: [power]
wall_thickness, mandatory if `include_tangential_heatflux` is `yes` (default)	thickness of each mesh element units: [length]
active_region	ID of region to select the mesh elements where the exterior temperature is applied

Keyword	Description
wall_thickness	mandatory; thickness of each mesh element units: [length]
heat_transfer_coefficient_fluid	wall-fluid heat transfer coefficient units: [power/(length^2*temperature)]; default: 0
temperature_fluid	temperature of fluid surrounding each mesh element units: [temperature]; default: 0
heat_flux_density	heat flux density units: [power/length^2]

Keyword	Description
radiation_update_timestep	filename of the view factors file units: [time]; default: 1e-2
view_factors_file	filename of the view factors file default: same file as in enable_heat_transfer

Associated material properties

Material properties

thermalConductivity ( $k$ ): thermal conductivity of a material [ $W/m\,K$ ]
thermalCapacity ( $c$ ): thermal (specific) capacity of a material [ $J/kg K$ ]
youngsModulusOriginal ( $Y^*_{orig}$ ): original Young’s modulus of each material [ $Pa$ ] (required if area_correction yes is used in the wall_contact_model command)

Material interaction properties

ht_modification ( $\alpha$ ): modifier for contact area calculation [ $\cdot$ ] (required if modified_area_correction yes is used in the wall_contact_model command)

Examples

mesh_module heattransfer id my_heattransfer wall_density 500. wall_thickness 0.01 initial_temperature 300. &
   temperature_fluid 200. heat_transfer_coefficient_fluid 0.01
mesh_module heattransfer id my_heattransfer heat_transfer_mode coefficient initial_temperature 300 &
  htc_exterior_fluid 0.01 temperature_exterior_fluid 400 thickness_layers_wall {0.01,0.02} &
  kappa_wall_layers {0.5,1.0} include_tangential_heatflux yes
mesh_module heattransfer id my_heattransfer heat_transfer_mode constant initial_temperature 300
mesh_module heattransfer id my_heattransfer heat_transfer_mode flux flux_value 10 initial_temperature 300 &
  wall_thickness 0 include_tangential_heatflux no
mesh_module heattransfer id my_heattransfer heat_transfer_mode power power_value 1 initial_temperature 300 &
  wall_thickness 0.01
mesh_module heattransfer id my_heattransfer heat_transfer_mode coefficient &
  initial_temperature 300 htc_exterior_fluid 0.01 temperature_exterior_fluid 400 thickness_layers_wall {0.01,0.02} &
  kappa_wall_layers {0.5,1.0} include_tangential_heatflux yes
mesh_module heattransfer id my_heattransfer heat_transfer_mode coefficient wall_thickness 0.18 &
  temperature_exterior_fluid function(temp.csv,x) htc_exterior_fluid 0.01 initial_temperature 500 id mm1 &
  active_region my_active_region
mesh_module heattransfer id my_heattransfer heat_transfer_mode constant initial_temperature 300 enable_radiation yes

Description

This mesh module can solve for three different mechanisms of heat transfer associated to the mesh:

heat transfer between particles and mesh elements
heat transfer between mesh elements and the exterior fluid
heat transfer between mesh elements

The heat transfer between particles and mesh is identical to the implementation in the enable_heat_transfer command. The thermal capacity and conductivity values for particle and wall material are defined via the material_properties commands.

Warning

Although heat_transfer_mode legacy is the default setting (for backward compatibility), any other heat transfer modes is preferred.

With heat_transfer_mode coefficient the heat transfer between the mesh elements and an exterior fluid is included. It is assumed that the exterior fluid is located at one side of the mesh, is at rest and has constant (in time) temperature; hence, only conductive heat transfer is considered. Note that the temperature_exterior_fluid can be set to a scalar value, or provided via a csv data file (to make it location dependent). In the latter case, the following formatting has to be used in the input file:

temperature_exterior_fluid function(my_temp_data.csv,spatial_direction)

Here ‘my_temp_data.csv’ is a file with two columns: the first column contains a spatial position (along the ‘spatial_direction’ axis, which must be set to ‘x’, ‘y’, or ‘z’) and the second column the exterior temperature at the position. The exterior temperature of mesh elements is derived (interpolated based on the center point of the mesh element) from the provided values at initialization via a precomputed lookup table. In order to apply the exterior temperature condition only to certain mesh elements, the active_region can be set. This will ensure only mesh elements with their center point within the active region will apply the temperature boundary condition. Note that the active region does not affect the (tangential) heat conduction within a mesh, which is enabled by default via the include_tangential_heatflux option. If no active region is set, the temperature boundary condition will be applied to all mesh elements.

From the heat transfer coefficient of the exterior fluid and the thickness and thermal conductivity (kappa) values of the wall layers (defined by thickness_layers_wall and kappa_layers_wall), a total heat transfer coefficient is determined according to:

$h_t = \frac{1}{R_t} = \frac{1}{\Sigma_i^n \frac{d_i}{\kappa_i} + htc},$

where $R_t$ is the total thermal resistance, $d_i$ and $\kappa_i$ are the thickness and thermal conductivity of wall layer $i$ (there are in total $n$ layers), and $htc$ is the heat transfer coefficient of the exterior fluid.

For heat_transfer_mode legacy the total heat transfer coefficient is determined as:

$h_t = \frac{2 h_f}{d_{wall}},$

where $h_f$ is the wall-fluid heat transfer coefficient (note that its dimensions are different from $htc$ , so a more proper naming would be thermal conductivity) and $d_{wall}$ is the wall thickness.

$Q = (T_f - T_m) h_t A_m,$

where $Q$ is the heat flux, $T_f$ is the (exterior) fluid temperature, $T_m$ is the temperature of the mesh element, $h_t$ is the total heat transfer coefficient, and $A_m$ is the area of the mesh element.

For heat transfer between mesh elements, a similar formula is used:

$Q = (T_i - T_j) h A_{ij},\\ A_{ij} = d_{wall} l_{ij}$

where $l_{ij}$ is the length of the edge that is shared between mesh elements i and j, and $d_{wall}$ is the wall thickness, that has to be specified by the user. The thermal conductivity is taken from the material_properties commands defining the thermal conductivity for the wall material (also for a layered wall), see enable_heat_transfer.

The mesh can be used as a heat source/sink with a defined heat flux density $\bar{Q}$ (which is the flux_value for heat_transfer_mode flux). A heat flux of

$Q_{source} = \bar{Q} A_m$

with $A_m$ the mesh element area is then added to each triangle. For heat_transfer_mode power the power_value is directly used as $Q_{source}$ , without being effected by the mesh element area.

For the temperature update of each mesh element initial_temperature is used as initial condition, which can be set to a scalar value or provided via a csv data file (by a function to make it location dependent, see the description for the temperature_exterior_fluid). The mass of each mesh element is calculated from its area, the value for wall_thickness and the density, which is taken from the material_properties commands (also for a layered wall), but can be overridden via the wall_density keyword for backwards compatibility.

Please note that the initial_temperature keyword is obligatory only if the simulation is started from scratch (in that case it replaces the temperature keyword of the mesh command). For restarted simulations it is an optional keyword, which is used to (re)set the temperature in each mesh element. If the heat transfer mesh module was not enabled when generating the restart file and the initial_temperature keyword is omitted, the temperature in each mesh element will be set to zero. If the initial_temperature is used in combination with a simulation restart, a warning will be written to the screen/output file.

With output_detailed_heatflux it is possible to include detailed heat flux information (for each component contributing to the heat flux) to the mesh output as done by output_settings.

Additionally, this keyword also creates two additional per-particle properties heatFluxWallConduction and heatFluxWallRadiation which output the instanteaneous heat flux a particle experiences due to the heat flux coming from a mesh for both conduction and radiation, respectively.

Warning

Mesh module heattransfer implements a very simplistic way of computing heat transfer on meshes, and it might lead to mesh-dependent results, especially in the case of (a) skewed triangles and (b) very fine meshes. When in doubt about the results, try using a mesh with larger, (preferrably) equilateral triangles.

Warning

When using this module and periodic boundary conditions the mesh needs to fulfill the following properties: (i) a triangle is not allowed to be in contact with itself through periodic boundaries, (ii) if two triangles are in contact inside the domain, they are not allowed to be in contact through periodic boundaries. Note that these restrictions are not validated by Aspherix(R) and must be ensured by the user.

Warning

It is not possible to have heat transfer between separate meshes. Only particle-particle, particle-wall and wall internal heat transfer is possible. Note also that mesh elements only exchange heat with each other if the elements share an edge. Therefore, a cylinder with baffles will only exchange heat between the cylinder and baffles if they are part of the same mesh (stl) and if the baffles share an edge with one of the elements of the cylinder.

In case enable_radiation is set to yes a radiation model based on local void fractions and particle-wall distance is employed using view factors read from a file.

The model that was implemented is based on the work of Johnson et al. [2]. It defines the heat transferred from a wall element $j$ to a particle $i$ as

$q_{r,ij} = \epsilon_j A_{ij} \sigma D_{ij} (T_j^4 - T_i^4).$

For a description of the involved quantities see the particle-particle heat transfer including the description of the view_factors_file. As the view factor depends on the volume which can be reduced by the presence of a wall we use the integral theorem to calculate it, i.e.

$V_i = \iiint 1 dV = \frac{1}{3} \iint x \cdot dS$

This surface integral can then be approximated via a ray summation, i.e. 128 equally spaced rays are cast from the center to the surface of the influence sphere and the volume is then approximated via

$V_i = \sum_j x_j \cdot n_j A_j,$

which, if the ray hits no triangle, reduces to

$r_{max} \frac{4 \pi r_{max}^2}{N},$

where $N$ is the number of rays and to

$x_j \cdot n_j \frac{4 \pi \|x_j\|^2}{N},$

where $x_j$ is the intersection point of ray with triangle and $n_j$ is the normal of the triangle.

The update of this local influence volume of each particle is done every radiation_update_timestep time units. The void fraction of a particle close to a wall is then computed using the sum of the volume of all particles in its sphere of influence divided by $V_i$ .

The lookup table present in the view factors file is for a plane which has a certain distance to the particle. Thus, if the wall is discretized using multiple triangles not every triangle should contribute the full view factor. Instead there must be a weight which is based on the percentage of the triangle that covers this plane. We decided that this weight function should be

$\omega_j = \frac{1}{I} \iint_j \left( 1 - \frac{2 \phi}{\pi} \right),$

where the integral over the full plane is

$I = \iint_j \left( 1 - \frac{2 \phi}{\pi} \right) = (\pi r + 2 d_n)r - 2 r_{max}^2 \tan^{-1}\left({\frac{r}{d_n}}\right),$

with $r = \sqrt{r_{max}^2 - d_n^2}$ , $r_{max}$ the radius of the sphere of influence and $d_n$ the normal distance to the triangle. The weight can then be approximated as the sum over all rays using

$\omega_j \approx \frac{1}{I \xi} \sum_j \frac{\pi r^2}{N_{guess}}\left( 1 - \frac{2 \phi}{\pi} \right),$

where $\xi$ is an empirical correction factor that corrects a systematic approximation error via

$\xi = \frac{19}{30 \sqrt{x}} - \frac{4}{3}x - 0.6 + 0.2 \sin\left(\pi x\right),$

with $x = d_n/r_{max}$ . The number of estimated ray hits of a plane with a certain distance $d_n$ is given as

$N_{guess} = N \frac{2 \pi r_{max}(r_{max} - d_n)}{4 \pi r_{max}^2}.$

Thus, the final equation reads

$q_{r,ij} = \epsilon_j \pi r_i^2 \sigma \omega_j D_{ij} (T_j^4 - T_i^4),$

where $D_{ij}$ depends on the volume fraction, which is corrected by the approximated $V_j$ .

External heat flux

In case add_external_heatflux is set to yes an external heat flux is imposed on the mesh

This keyword allows using an external application to impose an additional heat flux on a mesh.

To see how the external application can interact with Aspherix see the enable heat_transfer command. To allow using this keyword the mesh_properties keyword in the output_settings command must contain the heatFluxExternal value. The per-triangle heat flux written by the external application is then read in by Aspherix and applied to the mesh.

Additional information

This mesh module writes information to binary restart files to be able to continue correctly after restart. None of the modify_command options are relevant to this module.

This module stores per-mesh-element properties (“Temp”: temperature and “heatFlux”: aggregated heat flux) which can be accessed via output_settings. Moreover, it stores the average mesh temperature accessible via the entry avg_temp for the mesh. If the exterior temperature is set via a function, the TempAmbient per-mesh-element property is available for output as well. If an active_region is used, also the active per-mesh-element property is available. If add_external_heatflux yes is used, the heatFluxExternal per-mesh-element property will be available. If output_detailed_heatflux is used, one or more of the following per-mesh-element properties (depending on the settings of this command) are available for output as well: heatFluxExteriorConduction, heatFluxParticleConduction, heatFluxParticleRadiation, heatFluxSource, heatFluxWallConduction.

Restrictions

This module specification must be followed by the enable_heat_transfer.

mesh_module heattransfer

Syntax

Associated material properties

Material properties

Material interaction properties

Examples

Description

Additional information

Restrictions

Related commands