These tests check 2D and 3D heat equation elements with linear conduction and convection. In addition, expressions for the convection law are tested, too.

Given a concrete wall that extends infinitely in the y- and
z-directions, compute the temperature through the wall. The wall at x=0 is
exposed to air at 270K and the wall at x=L= to air at 330K. Both conditions
are formulated with convection overlay elements. The convection coefficient
for x=0 is h_{c1}=40W/m^{2}/K ,
modelling an air current and for x=L
h_{c2}=10[W/m^{2}/K, modelling
still air. The conduction coefficient k of concrete is set to
1.8W/m/K.

The theoretical solution for the heat flux through the wall is (see Kreith & Bohn, Principles of Heat Transfer)

*q=
(T _{2}-T_{1}) /
(R_{1}+R_{2}+R_{3})*

with

*R _{1}=
1/h_{c1}*,

*R*,

_{2}= L/k*R*

_{3}= 1/h_{c2}With the constants from above the heat flow through the wall becomes

*q= (270-330) /(1/40 + 0.2/1.8
1/10)* = -254.1

i.e. the heat flows from the right to the left in the figure below. Both Q and HE elements are tested, the problem being the same in case of 2D or 3D elements. The mesh for the 2D element model with Q4 conduction elements and L2 convection elements is displayed below:

For the 2D cases the convection conditions are modelled with L2 or L3 convection overlay line elements along the vertical edges of the above model. Please note that, for a 2D model the thickness, i.e. the extension of the model in the z-direction, must be specified both for the conduction and the convection overlay elements.

For the 3D cases the conduction in the solid is modelled with HE8 or TE4 elements and the convection conditions with Q4 or T3 convection overlay surface elements along the vertical faces of the model. Please note that the thickness of conduction and convection overlay elements is ignored!

All tests give the same result for the heat flow, i.e. the element gradient in the x-direction, with a value of -254.1. The temperature distribution is, as expected, linear: