Theoretical Background of the Different Formulations
Three formulations are available for the modeling of heat transfer in thin structures defined as boundaries:
The general formulation, using the Extra Dimension tool to solve the equations into the boundaries and through the thin structure’s thickness
The thermally thin approximation, a lumped formulation assuming that heat transfer mainly follows the tangential direction of the thin structure
The thermally thick approximation, a lumped formulation assuming that heat transfer is dominant in the direction normal to the thin structure
They all derive from the energy equation established in Equation 4-13, and recalled here below:
where E is the variable for the internal energy.
General Formulation
The general formulation uses the Extra Dimension tool to solve the equations through the thin structure’s thickness. The thin structure has its domain represented by the product space between the lumped boundary and the additional dimension for the thickness. Applying the split of the gradient operator given earlier at Equation 4-36, the energy equation becomes
(4-37)
The t operator is the tangential derivative in the thin structure boundary, and the n operator is the derivation operator along the extra dimension which is normal to the thin structure (see Tangential and Normal Gradients). The subscript s appended on E (and T in the following) is here to recall that this variable lives in the product space of the thin structure.
Equation 4-37 comes along with Fourier’s law of conduction:
(4-38)
and constraints on the temperature at the extremities of the extra dimension:
(4-39)
Here, ds is the length of the extra dimension, or equivalently the thickness of the thin structure, and Tu and Td are the temperature at the upside and the downside of the thin structure.
Thermally Thin Approximation
This formulation applies to a thin structure where heat transfer mainly follows the tangential direction. The gradient operator is then simplified to
This assumption is often valid for thin structures that are good thermal conductors compared to the adjacent domains, and/or with fast convection along the tangential direction.
With these assumptions, Equation 4-13 becomes:
(4-40)
(4-41)
where ds is the layer thickness (SI unit: m). The heat source Q is a density distributed in the layer while q0 is the received out-of-plane heat flux.
In 2D, Equation 4-40 and Equation 4-41 have an additional factor, dz, to account for the out-of-plane thickness.
When Equation 4-40 is solved in a boundary adjacent to a domain modeling heat transfer, the two entities exchange a certain amount of heat flux according to:
In this coupling relation, the outgoing heat flux n ⋅ q leaves the domain and is received in the source term q0 by the adjacent thin layer modeled as a boundary. From the point of view of the domain, and neglecting thermoelastic effects, the following heat source is received from the thin structure:
(4-42)
Equations for all supported types of medium are presented in the next sections, Thin Layer, Thin Film, Fracture, and Thin Rod.
Thermally Thick Approximation
This formulation applies to a thin structure where heat transfer mainly follows the normal direction. The gradient operator is then simplified to
This assumption is often valid for thin structures that are thermally resistive compared to the adjacent domains.
With these assumptions, Equation 4-13 becomes:
(4-43)
(4-44)
where ds is the layer thickness (SI unit: m). The heat source Q is a density distributed in the layer while q0 is the received out-of-plane heat flux.
In 2D, Equation 4-43 and Equation 4-44 have an additional factor, dz, to account for the out-of-plane thickness.
When Equation 4-43 is solved in a boundary adjacent to a domain modeling heat transfer, the two entities exchange a certain amount of heat flux according to:
In this coupling relation, the outgoing heat flux n ⋅ q leaves the domain and is received in the source term q0 by the adjacent thin layer modeled as a boundary. From the point of view of the domain, and neglecting thermoelastic effects, the following heat source is received from the thin structure:
(4-45)
To evaluate the normal gradient operation, n, temperatures Tu and Td are introduced for the upside and downside of the thin structure boundary. They are defined from the heat flux across the thin resistive structure. At the middle of the thickness, the temperature, T1 ⁄ 2, is approximated by (1 ⁄ 2)(Tu + Td). The term n ⋅ (−dsknT) is then given by:
which can be seen as the sum of two contributive sources on the upside and on the downside of the boundary that compensate:
Similarly, the time-dependent term can be expressed using Tu and Td by:
leading to:
(4-46)
(4-47)
Equations for all supported types of medium are presented in the next sections, Thin Layer, Thin Film, Fracture, and Thin Rod.
Upside, Downside, and Exterior Temperatures
This formulation is provided by the Thermally thick approximation option of the Thin Layer (Heat Transfer interface) and Solid (Heat Transfer in Shells interface) feature. Figure 4-4 shows how Thin Layer (Heat Transfer interface) and Solid (Heat Transfer in Shells interface) splits the temperature into Tu and Td on interior boundaries:
Figure 4-4: Upside and downside temperatures at a thin layer applied on an interior boundary. The thin layer is represented by the gray domain.
On exterior boundaries, it introduces a new degree of freedom represented by the variable TextFace. Depending on whether the heat domain is on the upside or the downside of the boundary, TextFace, is equal to Tu or Td and the same thing goes for the dependent variable T. An example is illustrated in the figure below:
Figure 4-5: Upside and downside temperatures at a thin layer applied on an exterior boundary.
Formulations Available Within the Features
Table 4-3 summarizes the formulations available within the thin structure features of the Heat Transfer (ht) and Heat Transfer in Shells (htlsh) interfaces.