FAQ:General:AveragingofConductivitiesatElementBoundaries
(22): Averaging of Conductivities at Element Boundaries
For the numerical solution of the transport equations the component must be divided
into a series of grid elements for whose midpoints the resulting temperatures and water
contents are computed at each time step, and across whose element boundaries the heat
and moisture fluxes required by the equations are flowing. In order to arrive at correct
fluxes across the boundaries, effective conductivities have to be assigned to the
boundaries which represent the integral effect of the conductivities between the
midpoints of the two elements. The investigations reported in [1] show that the results
do in fact depend on the way these effective heat and moisture conductivities are
determined from the real conductivities of the two elements: obviously linear
interpolation between the neighboring conductivities is preferable within a material
layer, and a resistance formulation is more appropriate for boundaries between layers
with different materials. How does WUFI treat these element boundary
conductivities?
It is obvious that at element boundaries where materials with possibly very different
conductivities are in contact with each other a simple average of the conductivities
(or resistances) cannot result in a realistic effective conductivity to describe the
fluxes between the elements. Take as an example a material with very low resistance
which borders on a material with very high resistance. The flux flowing between the
midpoints of the two elements is determined by the sum of the two successively
encountered resistances, not by the arithmetical average of the conductivities.
One might suppose now that this physically motivated reasoning also applies to
smaller differences between the neighboring elements and that therefore the resistance
formulation (i.e. the harmonic mean of the conductivities) should always be used
within the entire component. However, test calculations during the development
of WUFI's numerics showed that this is not the case. Within a material the
arithmetical mean of the conductivities yielded better results (compared with
experimental data), so that WUFI uses harmonic averages at material boundaries and
arithmetical averages within a material, in agreement with the cited investigation.
The derivation of the resistance formulation assumes equal fluxes in the two element
halves, but this need not be the case if transient processes in materials with heat
or moisture storage capacities are considered.
[1] Galbraith, G.H. et al.: Evaluation of Discretized Transport Properties for
Numerical Modelling of Heat and Moisture Transfer in Building Structures,
Journal of Thermal Env. & Bldg. Sci., Vol. 24, Jan. 2001
