Use of general finite element software
The use of
finite element analysis/finite element method (FEA/FEM) can be
used as the basis for heat flow calculations.
FEA was first developed in 1943 by Richard Courant to
obtain approximate solutions to vibration systems. Due to
the cost of computers its early uses were limited to aeronautics,
automotive, defence and nuclear industries, but since the rapid decline
in the cost of computers FEA has become ever more widespread and important.
FEA is particularly well suited to dealing with heat
transfer problems as it is able to deal with the complex geometries
and combined modes of transport (conduction, convection etc) that are
commonplace.
With this technique systems are described by mathematical
equations. The number of equations will depend on the complexity
of the system. For a simple object the equations can be derived,
however this is not practical when trying to find a solution that
describes a complex structure. FEA deals with this problem
by splitting a complex system into smaller sections. Now the
solution for each section can be represented by an equation that is
much simpler than that which describes the entire system. Also In
this way a nonlinear problem becomes is split down into a series of linear
ones that closely approximate the real solution.
The type of programme used will determine how the system
is split up (into finite elements). Bisco uses a triangular
grid whilst Trisco uses a rectangular grid. A triangular grid
has the advantage that it can better approximate irregular geometries.
The fineness of the mesh will determine the accuracy of the calculation
– a fine mesh will be more accurate but will take a longer time to
compute, whilst a coarse mesh will be quicker but less accurate.
The junction
between the smaller elements is called a node. Solutions to
the equations at the nodes are found (based on minimising an energy
functional, usually the potential energy) and as such a approximation
for the whole system can be given.
The equations must satisfy several conditions at the
nodes. Firstly the heat flow into each node from surrounding
elements is equal to the heat flow out of the node. The second
factor relates to the boundary conditions. The boundary conditions
will dictate the temperature or heat flow at certain nodes. For
example consider the system in the following figure.
In this example
there are several boundary conditions. Firstly, the adiabatic
line on the left and right hand side means there is zero heat flow
across the nodes at those points. The second stipulation relates
to the environment conditions. The temperature and surface resistance
of the nodes next to the internal environment will be 20˚C and 0.13
W/m 2K respectively, while those next to the external environment
will be 0˚C and 0.04
W/m2K.
These boundary conditions are limited to this example -
they will change depending on the system under consideration.
Once the solutions at the nodes have been calculated,
the intermediate answers (be it displacement, heat flow etc) can
be found at any point within the surrounding elements and so an approximate
solution to the complete system is found
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