Heat Toolbox

1. Models

The model used by default with the heat toolbox is represented by this equation

$\rho C_p \frac{\partial T}{\partial t} + \nabla \cdot \boldsymbol{q}= Q, \quad \text{ in } \Omega$

with the following properties :

• $q=-k \nabla T$ is the conductive heat flux,

• $Q=0$ is heat source by default equal to 0,

• $\Omega$ the domain where the model is applied (.i.e. the materials).

For a steady-state problem, the temperature does not change with time, and the first term disappears.

1.1. Heat sources

The source term $Q$ in the heat equation is used to model internal heat generation and absorption with positive and negative values respectively.

For adding a heat source term in the model, a JSON key called `heat-sources` must be added and the JSON value associated can be

• a string: explicit expression of $Q$,

• a JSON object: setup one heat source,

• an array of JSON object : $Q$ is composed of several heat sources.

We also consider two ways for defined a heat source :

• from an explicit expression of the heat source

• from an expression of the heat rate $P$. In this case, $Q = P/V$, where $V$ is the volume of the material belongs to this model.

The setup of a heat source is defined from the properties listed in the next table.

Keyword Description Value Type Default Value

`type`

possible values : `heat-source`, `heat-rate`

string

`heat-source`

`expr`

expression with respect to the type used

string

The term $Q$ can be also split into $N$ terms (for example due to the multiple physics). The term $Q$ can be written as $Q=\sum_{i=1}^N Q_i$. This setup can be done by using an array of JSON objects for the `heat-sources` entry. And each JSON object corresponds to the previous description of one heat source.

``````{
"Models": {
"heat": [
{ (1)
"name": "heat1",
"materials": [
"mymatA",
"mymatB"
],
"setup": {
"heat-sources": [ (2)
{ (3)
"type": "heat-source",
"expr": "3"
},
{ (4)
"type": "heat-rate",
"expr": "2*exp(t):t"
}
]
}
},
{ (5)
"name": "heat2",
"materials": [
"mymatC"
],
"setup": {
"heat-sources": { (6)
"expr": "30"
}
}
}
]
}
}``````
 1 setup of first model called `heat1` on materials `mymatA` and `mymatB` 2 heat sources will be composed of 2 terms that we will call $Q_1$ and $Q_2$ (with $Q=Q_1+Q_2$) 3 we define $Q_1=3$ 4 we define $Q_2=2*exp(t)/V$ with $V$ the measure of materials `mymatA` and `mymatB` 5 setup of second model called `heat2` on materials `mymatC` 6 we define one heat source with $Q=30$

1.2. Convective heat transfert

Modeling heat transfer in fluid requires taking into account the convective effect in the heat equation. The heat equation is written as :

$\rho C_p \frac{\partial T}{\partial t} + \underbrace{\rho C_p \boldsymbol{u} \cdot \nabla T}_{\text{new term}} + \nabla \cdot \mathbf{q}= Q, \quad \text{ in } \Omega$

with $\boldsymbol{u}$ the fluid velocity.

For adding this feature to the model, a `convection` section should be added. The value can be a string corresponding to velocity expression.

``````{
"Models": {
"heat": {
"setup": {
"convection": "{-30*(y+-0.7)*(y-0.5)*(y>0.5)*(y<0.7),0}:y"
}
}
}
}``````

2. Materials

Table 1. Materials properties defined in the heat toolbox (symbols are given without components suffix)
Name Symbol Shape Description

thermal-conductivity

k

scalar or matrix

thermal conductivity

density

rho

scalar

density

thermal-expansion

beta

scalar

thermal expansion

specific-heat-capacity

Cp

scalar

specific heat capacity

3. Fields and symbols expressions

Table 2. Fields available
Name Description Shape

temperature

the temperature field

scalar

Table 3. Symbols expressions by considering `heat` as the keyword used with the toolbox
Symbol Expression Description

heat_T

$T$

evaluate the temperature

$\frac{\partial T}{\partial x}$

evaluate the first component of gradient of temperature

$\frac{\partial T}{\partial y}$

evaluate the second component of gradient of temperature

$\frac{\partial T}{\partial z}$

evaluate the third component of gradient of temperature

heat_dn_T

$\nabla T \cdot \boldsymbol{n}$

evaluate the normal derivative of temperature

heat_nflux

$-k \nabla T \cdot \boldsymbol{n}$

evaluate the normal flux

4. Boundary Conditions

4.1. Temperature imposed

$T = g \quad \text{ on } \Gamma$

JSON keywords : `temperature`, `temperature_imposed`

Table 4. JSON properties of boundary condition temperature imposed
Property Description Value Type Is Optional Default Value

`markers`

mesh marker where the BC is applied

`string`, `array of string`, `JSON`

yes

name given to the BC

`expr`

expression of `g`

`string`, `number`

false

``````{
"BoundaryConditions": {
"heat": { (1)
"temperature": { (2)
"bc1": { (3)
"markers": "gamma1", (4)
"expr": "293-10*(y-1)*(y+1):y" (5)
},
"bc2": { (6)
"expr": "293+0.38*t:t"
}
}
}
}
}``````
 1 the toolbox identity 2 JSON keyword for defining this kind of boundary condition 3 name given to a boundary condition of type temperature imposed 4 marker names where the boundary condition is applied 5 the expression of the temperature imposed 6 another boundary condition of type temperature imposed where the marker used will be the name (i.e. `bc2`)
 The temperature imposed can be applied on all kind of marked entities (elements,faces,edges,points)

4.2. Heat flux

$k \nabla T \cdot \boldsymbol{n} = g \quad \text{ on } \Gamma$

JSON keywords : `heat_flux`, `flux`

Table 5. JSON properties of boundary condition heat flux
Property Description Value Type Is Optional Default Value

`markers`

mesh marker where the BC is applied

`string`, `array of string`, `JSON`

yes

name given to the BC

`expr`

expression of `g`

`string`, `number`

false

``````{
"BoundaryConditions": {
"heat": {
"flux": {
"bc1": {
"markers": [
"MarkerNeumann_1",
"MarkerNeumann_2"
],
"expr": "34*t:t"
}
}
}
}
}``````
 The expression given to the heat flux boundary condition can depend on the unknown (i.e. any symbols depending on the temperature).

4.3. Convective heat flux

$-k \nabla T \cdot \boldsymbol{n} = h \left( T - g \right) \quad \text{ on } \Gamma$

JSON keywords : `convective_heat_flux`

Table 6. JSON properties of boundary condition convective heat flux
Property Description Value Type Is Optional Default Value

`markers`

mesh marker where the BC is applied

`string`, `array of string`, `JSON`

yes

name given to the BC

`h`

expression of `h`

`string`, `number`

false

`Text`

expression of `g`

`string`, `number`

false

``````{
"BoundaryConditions": {
"heat": {
"convective_heat_flux": {
"bc1": {
"h": "16",
"Text": "287"
}
}
}
}
}``````
 The expressions given to the convective heat flux boundary condition can depend on the unknown but the automatic differentiation is not implemented here, so the jacobian will be inexact if a Newton solver is used.

 Radiative heat transfer is not yet available in the toolboxes. An application implementing radiative heat is currently available in feelpp/doc/manual/heat.

New sections in the JSON configuration file are necessary. Firstly, the "Coating" section, which contains information about the radiative properties of the emitting surfaces, especially the value of the emissivity constant $\epsilon$.

Table 7. JSON properties of Coating section
Property Description Value Type Is Optional

`markers`

list of mesh markers associated to a radiative surface

`string`, `array of string`

no

`epsilon`

value of the emissivity of the radiative surface

`string`

no

``````{
"Coating":{
"materialName": {
"markers": [
"markerName"
],
"epsilon": "0.5"
}
}
}``````

Secondly, the "radiative_enclosure_heat_flux" and "radiative_blackbody_heat_flux" sections, inside the "BoundaryConditions" section, which specify the decomposition of the radiating boundaries into cavities, independent of each other, open or closed, where the radiative boundary conditions are imposed.

Table 8. JSON properties of the "radiative_enclosure_heat_flux/CavityName" section
Property Description Value Type Values Is Optional

`enclosure`

Specifies if the cavity is isolated (closed) or if it communicates with a black body (open)

`string`

`open`, `closed`

no

`markers`

list of mesh markers associated to a radiative surface

`string`, `array of string`

no

`sigma`

Stefan-Boltzmann constant

`string`, `array of string`

no

`Tref`

Reference temperature of the black body (only for open enclosures)

`string`

yes

`viewfactors`

JSON

no

Table 9. JSON properties of the "radiative_enclosure_heat_flux/CavityName/viewfactors" section
Property Description Value Type Values Is Optional

`status`

Specifies if view factors have to be computed or loaded from file

`string`

`compute`, `load`

no

`filename`

File name for the computation of view factors (JSON format) or loading (CSV)

`string`

no

Table 10. JSON properties of the "radiative_blackbody_heat_flux/CavityName" section
Property Description Value Type Values Is Optional

`markers`

list of mesh markers associated to a radiative surface

`string`, `array of string`

no

`sigma`

Stefan-Boltzmann constant

`string`, `array of string`

no

`Tref`

Reference temperature of the black body (only for open enclosures)

`string`

yes

``````{
"BoundaryConditions": {
"heat": {
"Cavity": {
"enclosure": "closed",
"markers": ["InnerTop", "LowerTop","LeftTop","RightTop"],
"sigma": "sigma:sigma",
"Tref": "Tref_C+273.15:Tref_C",
"viewfactors": {
"filename":"$cfgdir/VF_Matrix_Cavity_1.csv" // "status":"compute", // "filename":"$cfgdir/rectangular_cavity_vf.json"
}
}
},
"Gamma_BBC_1": {
"markers": ["Gamma_B_11","Gamma_B_12"],
"Tref": "Tref_C+273.15:Tref_C",
"sigma": "sigma:sigma"
}
}
}
}
}``````

Finally, an additional JSON file per cavity is necessary if the view factor matrix needs to be computed. This file will specify the algorithm to be use in order to compute the view factors, and the markers forming the cavity.

Table 11. JSON properties for the computation of view factors
Property Description Value Type Values Is Optional

`type`

Choose between raytracing (obstructed and non-obstructed cavities) and quadrature (only non-obstructed cavities)

`string`

`Raytracing`, `UnobstructedPlanar`

no

`algorithm`

Choose the algorithm (only for UnobstructedPlanar) between DoubleAreaIntegration (2D) and SingleAreaIntegration (3D)

`string`

`DoubleAreaIntegration`, `SingleAreaIntegration`

yes

`quadrature_order`

`int`

yes

`markers`

list of mesh markers associated to the cavity

`string`, `array of string`

no

``````{
"viewfactor": {
"type": "UnobstructedPlanar",
"algorithm":"DoubleAreaIntegration",
"description": "The viewfactor of a closed triangular cavity."
}
}``````

6. Post Processing

6.1. Exports

Table 12. Fields allowed to be exported in the `fields` section are:
Name Description

temperature

the temperature field

velocity-convection

pid

the mesh partitioning

all

all fields available

 All materials properties given in the section Materials can be also exported by specifying the name in the `fields` entry.

6.2. Measures

6.2.1. Normal-Heat-Flux

Example of a `Parameters` section
``````{
"Normal-Heat-Flux": { (1)
"bottom": { (2)
"markers": [ (3)
"bottom1",
"bottom2"
],
"direction": "inward" (4)
},
"top": { (5)
"direction": "inward"
}
}
}``````
 1 kind of measure 2 name of measure 3 surface markers where the measure is computed 4 direction : `inward` or `outward` 5 another measure applied on surface marker called top (because when markers is no present, the name is used as marker name)