Can one model fluid flow only by using the convection–diffusion equation?

Modeling fluid flow in a coil seems very complicated, especially for turbulent flow. Having Navier-Stokes equations seems to be the right way to go, but I was wondering… can one use the convection-diffusion equation?

$$
rho C_p frac{partial T}{partial t}+rho C_p vec{u} nabla T – nabla cdot (-k nabla T) = 0
$$

If one would assume that the velocity in a coil is 5 m/s in all directions and doesn’t change (I know this is a stretch), what would be the limitations of such a model? I’m trying to model cooling of a metal by a coil and having flow equations is too complicated at this point 🙂 In theory, it should be fine, since such an equation considers the transport of a substance or quantity by bulk motion.

EDIT: The system would look like this:

The cold side has a constant hot temperature BC which is 25K lower than the initial temperature of the entire system, so the hot side should take the heat from the metal matrix and release it in the hot side. All the other sides are insulated.