We know that for constant heat flux $\dot{q}$ Nusselt number can be written as
$Nu = \frac{hL}{k}$ with $\dot{q}=h(T_f-T_s)$, therefore $Nu =\frac{\dot{q}L}{k(T_f-T_s)}$
However, I'm a little bit stuck on writing a definition for a constant wall temperature, lets say in a duct with $T_s=const.=500K$ where $\dot{q}$ is not a known constant from boundary conditions.
How can one derive $Nu$ in this case?
The Nusselt number is the ratio between convective and conductive heat transfer, which cannot refer to the duct wall (as it is solid and no convection). So the thermal conductivity $k$, should be for the fluid in the duct (air?) not the duct wall.
The heat transfer coefficient $h$, is across the boundary layer. This can be calculated if you know the heat transfer rate, but otherwise there are correlations to estimate it. Frequently these correlations actually calculate the Nusselt number itself.
The correlations are usually specific to a particular geometry and flow type, for example for circular ducts and turbulent flow, the Dittus-Boelter is often used.
If heat transport is mainly by diffusion (conduction), a general definition of heat transport is: $$\left.q\right|_w=-k \left. {{dT}\over{dx}}\right|_w = h\left(\left.T\right|_w-T_f\right)$$ The subscript $w$ implies that the derivative and the temperature is defined at the wall. Furthermore, by convention, it is assumed the temperature difference is positive, i.e. the wall temperature is higher than the fluid temperature.
If the heat flux is unknown but you are able to measure the temperature field then you may calculate the Nusselt number through:
$$\mathrm{Nu}=\frac{hL}{k}=-\frac{L}{\left.T\right|_w-T_f}\left.{{dT}\over{dx}}\right|_w$$
Another way is to use you original equation and measure the heat input into the system by a heat balance. This is done by integrating the total energy at the entrance (usually known from boundary conditions, i.e. imposed enthalpy, temperature, etc.), the total energy at the outlet and finding the difference which must be the energy input into the system by the walls. This, divided by the heat transport surface area, gives you the mean heat flux and from it can be determined a mean Nusselt number.
