You're correct that the standard Dittus-Boelter equation, $Nu = 0.023Re^{0.8}Pr^{0.3}$, applies only to fully developed turbulent flow, which is not the case in your entrance region. To address this, we need a formulation for the local Nusselt number, $Nu_x$, that accounts for the developing thermal boundary layer. A suitable form is $Nu_x = Nu_{\infty} [1 + f(x/D, Re, Pr)]$, where $Nu_{\infty}$ is the fully developed Nusselt number (obtainable from Dittus-Boelter), $x$ is the distance along the plate, $D$ is a characteristic length, and $f$ is a function capturing the entrance length effects.
This function $f$ is typically expressed as $f(x/D, Re, Pr) = C(Re, Pr) \cdot (x/D)^{-m(Re, Pr)}$. Here, $C$ and $m$ are functionals of the Reynolds and Prandtl numbers. They can be determined experimentally or through a similarity analysis of the boundary layer equations.
The literature provides various parameterizations for $C$ and $m$, often piecewise functions over different $Re$ and $Pr$ ranges. For example, you might find $C(Re, Pr) = a_1 Re^{b_1} Pr^{c_1}$ and $m(Re, Pr) = a_2 Re^{b_2} Pr^{c_2}$ for specific $Re$ and $Pr$ intervals, where $a_i$, $b_i$, and $c_i$ are empirically determined constants.
Finally, with $Nu_x$ determined, the local heat transfer coefficient is simply $h_x = Nu_x k/D$, where $k$ is the thermal conductivity of your helium. This approach provides a more accurate estimate of $h$ in the entrance region by incorporating the physics of the developing thermal boundary layer.
