Matrix or Vector | Geometry | Shape Functions | Integration Points |
---|---|---|---|
Conductivity Matrix and Heat Generation Load Vector | Quad | Equation 11–128 | 2 x 2 |
Triangle | Equation 11–108 |
1 if planar | |
Specific Heat Matrix | Same as conductivity matrix. Matrix is diagonalized as described in Lumped Matrices. | Same as conductivity matrix | |
Convection Surface Matrix and Load Vector | Same as conductivity matrix evaluated at the face | 2 |
Heat Flow describes the derivation of the element matrices and load vectors as well as heat flux evaluations. SOLID70 - 3-D Thermal Solid describes fluid flow in a porous medium, accessed in PLANE55 with KEYOPT(9) = 1.
If KEYOPT(8) > 0, the mass transport option is included as described in Heat Flow Fundamentals with Equation 6–1 and by of Equation 6–22. The solution accuracy is dependent on the element size. The accuracy is measured in terms of the non-dimensional criteria called the element Peclet number (Gresho([58])):
(13–98) |
where:
V = magnitude of the velocity vector |
L = element length dimension along the velocity vector direction |
ρ = density of the fluid (input as DENS on MP command) |
C_{p} = specific heat of the fluid (input as C on MP command) |
K = equivalent thermal conductivity along the velocity vector direction |
The terms V, L, and K are explained more thoroughly below:
(13–99) |
where:
V_{x} = fluid velocity (mass transport) in x direction (input as VX on R command) |
V_{y} = fluid velocity (mass transport) in y direction (input as VY on R command) |
Length L is calculated by finding the intersection points of the velocity vector which passes through the element origin and intersects at the element boundaries.
For orthotropic materials, the equivalent thermal conductivity K is given by:
(13–100) |
where:
K_{x}, K_{y} = thermal conductivities in the x and y directions (input as KXX and KYY on MP command) |
For the solution to be physically valid, the following condition must be satisfied (Gresho([58])):
(13–101) |
This check is carried out during the element formulation and an error message is printed out if Equation 13–101 is not satisfied. When this error occurs, the problem should be rerun after reducing the element size in the direction of the velocity vector.