For a hydrodynamically and thermally fully developed laminar flow through a circular pipe of constant cross section, the Nusselt number at constant wall heat flux (Nu_q) and that at constant wall temperature (Nu_T) are related as

The correct option is Nu_q > Nu_T.

To establish the relationship between the Nusselt numbers under the given thermal boundary conditions, we examine a hydrodynamically and thermally fully developed laminar flow of a fluid through a circular pipe of constant cross-section.

For a fully developed laminar flow, the velocity profile is parabolic and steady, while the temperature profile, when normalized, becomes independent of the axial position. Under these conditions, the energy equation can be solved analytically for two standard boundary conditions:
1. Constant Wall Heat Flux (Nu_q): When the heat flux at the wall is constant along the axial direction, the analytical solution yields a constant Nusselt number value:
${\mathrm{Nu}}_q=\frac{48}{11}\approx 4.36$
2. Constant Wall Temperature (Nu_T): When the wall temperature is maintained constant along the axial direction, the analytical solution yields:
${\mathrm{Nu}}_T\approx 3.66$

Comparing these two constant values:
$4.36>3.66$
Therefore, we have:
${\mathrm{Nu}}_q>{\mathrm{Nu}}_T$

Physical Explanation:
For the constant wall heat flux boundary condition, the temperature difference between the wall and the bulk fluid remains constant along the flow direction. For the constant wall temperature boundary condition, this temperature difference decreases as the fluid heated downstream approaches the wall temperature. This leads to a steeper temperature gradient at the wall under constant heat flux, resulting in a higher heat transfer coefficient and consequently a larger Nusselt number.