.TAeZoSysPro.HeatTransfer.Functions.MeshGrid.biotAndGeometricalGrowthGrid

Information

This functions gives the position of the vertices that form segments. The position is computed to insure a geometrical increase of the distance between a vertex and its next expected for one of the boundary segment (x[2] - x[1]) if symmetricalMesh is false and both otherwise, where the size of the segment is computed for having a Biot number equal to 0.1.

For the special case where the number of segments N equal 2, the length of the segment is half of the length of the domain and biot layers are ignored.

In order to compute the size of the 'Biot' layer, the user has to supply as input of the function a decouple value for the heat transfer coefficient at the boundary(ies) and the thermal conductivity of the material.

if symmetricalMesh is true, the mesh is symmetrical on both sides of the middle of the domain. In that case, one of the two different processes are run depending on whether the number of nodes (segments) N is even or not.

The common equations derives:

The distance between vertices derives for odd number of segments:

if symmetricalMesh is false, the mesh grows from the start to the end of the domain.
The distance between vertices derives:

Where :

• symmetricalMesh is a boolean to select if the mesh is symetric to the middle of the domain
• x is the vector of size N+1 of position of the vertex that form the segment of the mesh of the domain
• dx_biot is length of the biot layer segment
• Biot is the Biot number
• λ is the thermal conductivity os the material
• h is the heat exchange coefficient at the boundary(ies)
• dx is length of the first segment after the Biot layer
• q is the geometric progression of the length of the segment. q ≠ 1
• N is the number of segments
• n is the integer part of the division of segments by 2
• L is the length of the domain to mesh

A mesh divided into N segments contains N+1 vertices. That is why the size of x is N+1.

Interface

function biotAndGeometricalGrowthGrid
  input Modelica.SIunits.Length L "Length of the domain to mesh";
  input Integer N "number of segments";
  input Real q "Growth rate";
  input Modelica.SIunits.CoefficientOfHeatTransfer h "Decoupled value of the heat transfer coefficient";
  input Modelica.SIunits.ThermalConductivity k "Decoupled value of thermal conductivity";
  input Boolean symmetricalMesh = true "Axial symmetry mesh where the axis is the middle of the domain";
  output Modelica.SIunits.Position x[N + 1] "Vector of vertice position";
end biotAndGeometricalGrowthGrid;