The entry region should be separately calculated using the appropriate entry region heat transfer augmentation equation for the flow system. The developed flow region is generally simple. In this case the entry region is clearly significant part of the total.

Your problem may be complicated by the development of both velocity and temperature profiles with relative downstream rate of development depending on the Prandtl number Pr. Also, you have not specified whether the flow is low Re or high Re or laminar or turbulent, which will make a huge difference. In the case of turbulent flow you can probably neglect development of the velocity profile but you still have to do a calculation of the developing temperature profile. There is a lot of literature on this type of problem e.g. in the International Journal of Heat and Mass Transfer.

Consider Reynold's Analogy: the Stanton number is equal to the friction factor over two. There are experimental correlations for entrance effects. Combining these two will allow you to correct the Nusselt number for entrance effects, roughness, etc.