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The Knudsen compressor, based on the rarefied flow phenomenon of thermal creep (or thermal transpiration), is an unconventional nano/micro/mesoscale gas pump or compressor with no moving parts. Recent experimental studies of single micro/mesoscale stages have indicated that, at low pressures (les 100 Pa), stage performance was decreased in qualitatively somewhat understood and quantitatively not-well-understood ways. The investigation reported in this paper was undertaken to establish a detailed picture of the thermally driven flow's behavior for a range of representative Knudsen compressor stage configurations and flow conditions. The objectives of the study were to identify, quantify, and ultimately understand the sources of negative impacts on the performance of single stages using the direct simulation Monte Carlo technique. The findings suggest that once the Knudsen number in the cooling region is less than 0.1, the maximum pressure gain from a single stage of a Knudsen compressor can be carried forward to the next stage with relatively minor performance losses. Simultaneously, for Knudsen numbers below 0.1, the working gas' temperature can be reduced to within a few percent of its initial temperature entering the stage. What was found relates to a necessary characteristic of successful Knudsen compressors, with the same occurrences being anticipated for thermal-creep-driven gas transport channels in other nano/microscale devices.