Defaults (Re=50000, Pr=7.0, k=0.60 W/(m·K), D=25 mm, heating) give Nu ≈ 287.7, h ≈ 6905 W/(m²·K), heat flux at ΔT=10 K ≈ 69.0 kW/m², thermal boundary-layer ≈ 0.087 mm. Valid range is Re > 4000 and 0.6 < Pr < 160; below Re = 4000 the flow is laminar and a different correlation is required.
Live view of heat conducted from the hot wall (red) into the fluid. Particles pick up heat near the wall (turning red) and carry it downstream. In laminar flow the motion is ordered with a thick thermal boundary layer and low Nu; in turbulent flow it is mixed, the boundary layer is thin and Nu is high. Raise Re (flow rate) to see the regime transition, the boundary layer thin out, and Nu and h rise.
Horizontal: Reynolds number Re (4000 to 200000, log10) / Vertical: Nusselt number Nu (log10) / Blue line: Dittus-Boelter prediction with slope 0.8 / Yellow marker: current (Re, Nu) operating point / Changing Pr shifts the whole line vertically; larger Pr gives larger Nu and stronger convective heat transfer.
The Dittus-Boelter correlation is the most basic empirical equation for forced-convection heat transfer in turbulent pipe flow (Re > 4000, 0.6 < Pr < 160):
$$\mathrm{Nu} = 0.023\,\mathrm{Re}^{0.8}\,\mathrm{Pr}^{n}$$$n = 0.4$ for heating (fluid receives heat from the wall, $T_w > T_b$) and $n = 0.3$ for cooling. The Reynolds and Prandtl numbers are defined as:
$$\mathrm{Re} = \frac{\rho U D}{\mu},\quad \mathrm{Pr} = \frac{\mu c_p}{k}$$The heat-transfer coefficient $h$ follows from the definition $\mathrm{Nu} = hD/k$:
$$h = \frac{\mathrm{Nu}\,k}{D}$$The wall heat flux follows from Newton's law of cooling, and the thermal boundary-layer thickness is $\delta_T \approx D/\mathrm{Nu}$:
$$q = h\,(T_w - T_b),\quad \delta_T \approx \frac{D}{\mathrm{Nu}}$$$\rho$ is density, $U$ is the mean velocity, $D$ is the pipe diameter, $\mu$ is viscosity, $c_p$ is the specific heat at constant pressure, $k$ is the fluid thermal conductivity, $T_w$ is the wall temperature and $T_b$ is the bulk (mixing-cup) temperature.