It is well-known that the analysis of temperature distribution is substantilly important in optimal design of quenching process. The unsteady state temperature gradients generated during the quenching process were numerically calculated by the Finite Element Method(F. E. M.). Formulations of F. E. M. based weighted residural method were presented for the analysis of the two dimensional heat conduction problem. In the process of calculation, the temperature dependency of physical properties of the material was in consideration. At early stage of the quenching process, the abrupt temperature gradient has been shown in the surface of the carbon steel(SM45C).