The steady-state turbulent heat transfer coefficients in a vertical circular Platinum (Pt) test tube for the flow velocities (u=4.22 to 21.45 m/s), the inlet liquid temperatures (T_in=308.20 to 311.77 K), the inlet pressures (P_in=834.04 to 910.23 kPa) and the increasing heat inputs (Q₀exp(t/τ), exponential periods, τ, of 6.04 to 32.13 s) were systematically measured by an experimental water loop comprised of a multistage canned-type circulation pump with high pump head. Measurements were made on Pt test tubes of 3, 6 and 9 mm inner diameters, 32.7, 69.6 and 49.6 mm heated lengths and 0.5, 0.4 and 0.3 mm thicknesses, respectively. Theoretical equations for turbulent heat transfer in circular tubes of 3, 6 and 9 mm in diameter and 492, 636 and 616 mm long were numerically solved for heating of water with heated sections of 3, 6 and 9 mm in diameter and 33, 70 and 50 mm long by using PHOENICS code under the same condition as the experimental one considering the temperature dependence of thermo-physical properties concerned. The surface heat flux, q, and the average surface temperature, T_s,av, on the circular tubes solved theoretically under the flow velocities, u, of 4.22 to 21.45 m/s were compared with the corresponding experimental values on heat flux, q, versus the temperature difference between average inner surface temperature and liquid bulk mean temperature, ΔT_L [=T_s,av-T_L, T_L=(T_in+T_out)/2], graph. The numerical solutions of q and ΔT_L are almost in good agreement with the corresponding experimental values of q and ΔT_L with the deviations less than ±10 % for the range of ΔT_L tested here. The numerical solutions of local surface temperature, (T_s)_z, and local average liquid temperature, (T_f,av)_z, are within ±10 % of the corresponding experimental data on (T_s)_z and (T_f,av)_z. The thickness of the conductive sub-layer, δ_CSL [=(Δr)_out/2], and the non-dimensional thickness of conductive sub-layer, y⁺_CSL [=(f_F/2)^0.5ρ_luδ_CSL/µ_l], for the turbulent heat transfer in various vertical tubes are clarified based on the numerical solutions. It was confirmed in this study that authors' steady-state turbulent heat transfer correlation based on the experimental data [Hata and Noda, 2008] can not only describe the experimental data of steady-state turbulent heat transfer but also the numerical solutions within ±10 % difference for the wide ranges of tube inner diameters (3, 6 and 9 mm), temperature differences between average inner surface temperature and liquid bulk mean temperature (ΔT_L=8 to 145 K) and flow velocities (u=4.22 to 21.45 m/s).