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