A water-$Al_2O_3$ nanofluid was manufactured, and its thermal conductivity was measured in this study. The measurement was performed at volumetric concentrations of 0.5%, 1%, 2%, and 3%, and the nanoparticle sizes were 20 nm and 70 nm. Experimental test equipment, using the transient hot wire method, was installed to measure the thermal conductivity of the nanofluid, and the measured results were confirmed by measuring pure water with a measurement error of 0.92% at $20^{circ}C$. The thermal conductivity enhancement ranged from 4.8% to 13.6% for the 20 nm particle size, and from 3.1% to 8.8% for the 70 nm particle size at a concentration range of 0.5% to 3%. The enhancement increased with a decrease in particle size and an increase in concentration. With the elapse of time after manufacturing the nanofluid, the thermal conductivity enhancement decreased significantly from 5 to 9 h, and this trend was measured under all of the measurement conditions. After 24 h, the enhancement ranged from 1.2% to 3.5% for the 20 nm particles, and from 0.6% to 2.3% for the 70 nm particles. The enhancement trends with the elapse of time were almost identical with and without stirring the nanofluid. SDBS (Sodium Dodecyl Benzene Sulfonate) was added as a dispersing agent, and the decrease in the thermal conductivity enhancement was delayed.