Fullydeveloped flow and heat transfer
From ThermalFluidsPedia
In this section, we consider the case of fully developed laminar flow and constant properties in a circular tube with a fully developed temperature and concentration profiles. We first consider the case of constant heat rate per unit surface area for steady, laminar, fully developed flow. The energy equation in a circular tube, by neglecting axial heat conduction and viscous dissipation terms, is:

For a fully developed flow with constant wall heat flux, eq. (5.20) can be substituted into eq. (5.36) to obtain

The boundary conditions are

Integrating eq. (5.37) twice and applying the boundary conditions in eq. (5.38) to get the temperature distribution gives us

Using the definition of mean temperature presented in the last section with the above profile for temperature, and assuming constant properties:
Substituting eq. (5.39) into the above expression yields:

The heat flux at the wall can be obtained using the above relation for Tm

The heat flux at the wall can also be calculated using eq. (5.39) for the temperature profile and Fourier’s law of heat conduction

Combining eqs. (5.41) and (5.42) and solving for the heat transfer coefficient, h, yields h = 4.364 k / D or in terms of the Nusselt number,
Nu = 4.364 