The ablation and acceleration of diamond-like high-density carbon foils irradiated by thermal X-ray radiations are investigated with radiation hydrodynamics simulations. The time-dependent front of the ablation wave is given numerically for radiation temperatures in the range of 100-300 eV. The mass ablation rates and ablation pressures can be derived or implied from the coordinates of ablation fronts, which agree well with reported experiment results of high-density carbon with radiation temperatures Trad in the range of 160-260 eV. It is also found that the T3 rad scaling law for ablation rates does not apply to Trad above 260 eV. The trajectories of targets and hydrodynamic efficiencies for different target thicknesses can be derived from the coordinates of ablation fronts using a rocket model and the results agree well with simulations. The peak hydrodynamic efficiencies of the acceleration process are investigated for different foil thicknesses and radiation temperatures. Higher radiation temperatures and target thicknesses result in higher hydrodynamic efficiencies. The simulation results are useful for the design of fusion capsules. © The Author(s) 2020.