Why does the Nusselt Number for a Constant Surface Temperature, Thermally Fully Developed Circular Pipe Converge To a Constant of 3.66? I have flow in a pipe with a constant surface temperature boundary condition. The velocity profile is fully developed and the temperature profile is also fully developed. In my textbook, it says that the Nusselt number stays at a constant 3.66 value when these conditions exist.
I know that the Nusselt number is dependent on the derivative of the temperature in the radial direction at the pipe wall and the mean temperature of the fluid.
The further the fluid travels up the pipe,  the derivative decreases and the mean temperature increases. 
Shouldn't the derivative converge to zero because  the temperature of the fluid eventually reaches the wall temperature? Therefore, shouldn't the Nusselt number keep decreasing past 3.66 and eventually to 0?
If so, then what does thermally developed mean if the temperature is still changing?
 A: The Nusselt number in generally defined as 
$$Nu = \frac{hL}{k}$$
In this case $L=D$ and k is constant.  What you're looking for is the heat transfer coefficient $h$. This guy basically captures the effectiveness of the convective process at play.  From it you get a heat flux proportional to the temperature difference between the solid and the bulk flow.
$$q'' = h(T_f -T_s)$$
So your sense of things is right for the pipe flow situation.  The fluid will continually approach the temperature of the pipe and the heat flux will drop as a result, but h will not change.
If you're following certain texts on the subject (e.g. Incropera and DeWitt) you'll see them using temperature to determine $h$.  This special case for laminar pipe flow happens to admit a relatively easy solution.  This general solution, along with the definition of $T_m$, can be expressed as a value of $h$ or $Nu$. Although temperature is used in the calculation, the numerical values of $T$ don't actually change the resulting $Nu$.
