Transient heat conduction with uniform rate of internal heat generation has been numerically investigated in infinitely long eccentrically hollow cylinders. Two combinations of boundary conditions of the first and second kinds have been considered. The first of these combinations corresponds to the case of one surface being heated isothermally while the opposite surface is maintained at ambient temperature. The second combination of boundary conditions is with one surface maintained isothermal while the opposite surface is adiabatic. Results are presented for a radius ratio 0.5 with dimensionless eccentricities ranging from 0.1 to 0.8 and various values of internal heat generation.