Heat Toolbox :: Feel++ Docs

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

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

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
heat_grad_T_0	\(\frac{\partial T}{\partial x}\)	evaluate the first component of gradient of temperature
heat_grad_T_1	\(\frac{\partial T}{\partial y}\)	evaluate the second component of gradient of temperature
heat_grad_T_2	\(\frac{\partial T}{\partial z}\)	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.

5. Initial Conditions

6. Post Processing

6.1. Exports

Table 7. Fields allowed to be exported in the `fields` section are:
Name	Description
temperature	the temperature field
velocity-convection	evaluate the gradient of temperature
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)

Heat Toolbox