This study investigates the effects of nanoparticle size and temperature on the thermal conductivity enhancement of water-based alumina ($Al_2O_3$) nanofluids, using the centrifuging method and relative centrifugal forces of differing magnitude to produce nanofluids of three different particles without involving any dispersants or surfactants. We determined the coupling dependency in thermal conductivity enhancement relative to nanoparticle size and temperature of the alumina nanofluids and also experimentally showed that the effect of temperature on thermal conductivity is strongly dependent on nanoparticle size. Also, our experimental data presented that the effective medium theory models such as the Maxwell model and Hasselman and Johnson model are not sufficient to explain the thermal conductivity of nanofluids since they cannot account for the temperature- and size-dependent nature of water-based alumina nanofluids.