In this paper, thermal characteristics of a nitrous oxide ($N_2O$) catalytic reactor with packed-bed geometry are theoretically and numerically investigated. Several researchers experimentally presented that catalytic decomposition of $N_2O$ in a packed bed generates about 82kJ/mole in the exothermic reaction. Based on the results they have studied the catalytic decomposition of $N_2O$ in a packed bed to use it not only as a mono-propellant thrust for small satellites but also as an igniter system for hybrid rockets. So we aim to identify important parameters existing in an $N_2O$ packed-bed geometry, and to clarify its critical effect on thermal characteristics of the catalytic igniter using a porous medium approach.