Thin Layer

Thin layers of solid materials can be considered as boundaries when their thickness is significantly smaller than the typical lengths of the adjacent domains.

General Formulation

With this formulation, multiple sandwiched layers with different material properties and thicknesses can be modeled. An additional 1D segmented line represents the multiple layers in the thin structure. In this extra dimension, the governing equation is derived from Equation 4-59 to give:

(4-70)

(4-71)

where Ts is an auxiliary dependent variable defined on the product space. The remaining quantities are recalled below:

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ρsi is the density of layer i (SI unit: kg/m3)

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Cp, si is the heat capacity of layer i (SI unit: J/(kg·K))

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ksi is the thermal conductivity of layer i (SI unit: W/(m·K))

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Qsi is the heat source applied to layer i (SI unit: W/m3)

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ds is the shell thickness (SI unit: m)

The constraint T = Ts is specified on each side of the extra dimension to connect T to Ts.


	See Thin Layer (Heat Transfer Interface) and Solid (Heat Transfer in Shells Interface) with Layer type set as General or more information about the boundary feature solving Equation 4-70 and Equation 4-71.

Thermally Thin Approximation

The Heat Transfer Module supports heat transfer in thermally thin structures in 3D, 2D, and 2D axisymmetry. The material in the thin structure might be a good thermal conductor for this approximation to be valid. For example, in a printed circuit with copper traces, where the traces are often good thermal conductors compared to the board’s substrate material.

The thermally thin approximation is derived from Equation 4-62 to Equation 4-64. Inside the thin layer, the heat equation becomes:

(4-72)

(4-73)

where ds is the layer thickness (SI unit: m).

The heat source Qs is a density distributed in the layer while q0 is the received out-of-plane heat flux.


	In 2D, Equation 4-72 and Equation 4-73 have an additional factor, dz, to account for the out-of-plane thickness.

From the point of view of the domain, the following heat source, derived from Equation 4-64, is received from the layer:

(4-74)

This formulation may be used to model efficiently a shell with multiple sandwiched layers with different material properties and thicknesses, by approximating the homogenized material properties as weighted averages of constant properties within each layer:

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The homogenized density is defined as

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The homogenized heat capacity is defined as

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The homogenized thermal conductivity is defined as

with roti the rotation tensor of layer i:

and A the transformation matrix between the local and global coordinates systems:


	See Thin Layer (Heat Transfer Interface) and Solid (Heat Transfer in Shells Interface) with Layer type set as Thermally thin approximation for more information about the boundary feature solving Equation 4-74. See The Heat Transfer in Shells Interface for more information about the physics interface solving Equation 4-72.

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Heat Transfer in a Surface-Mount Package for a Silicon Chip: Application Library path

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Silica Glass Block Coated with a Copper Layer: Application Library path

Thermally Thick approximation

When a thin layer is formed of one or more thermally resistive materials, it can be defined through its thermal resistance:

The heat flux across the thermally thick structure is derived from Equation 4-67 and gives

(4-75)

(4-76)

where the u and d subscripts refer to the upside and downside of the layer, respectively.

When the material has a multilayer structure, ks and ds in the expressions above are replaced by dtot and ktot, which are defined according to Equation 4-77 and Equation 4-78:

(4-77)

(4-78)

where n is the number of layers.


	See Thin Layer (Heat Transfer Interface) and Solid (Heat Transfer in Shells Interface) with Layer type set as Thermally thick approximation for more information about the boundary feature solving Equation 4-75 and Equation 4-76.