A variational upper bound principle is used to derive equations for the Knudsen void gas thermal conductivity in three model fiber beds made up of randomly placed. freely overlapping, long right circular cylinders with their central axes mutually parallel, with each cylinder axis independently oriented in two dimensions or with each axis independently oriented in three dimensions. In order to establish the importance of dispersed solid shape and spatial distribution in predicting the low pressure gas conductivities in cryogenic insulation, the five fiber bed conductivities are compared with sphere bed and parallel plate Knudsen thermal conductivity equations for various values of the thermal accommodation coefficient.