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 understand why the Nusselt number for constant wall heat flux (Nu_q) is greater than the Nusselt number for constant wall temperature (Nu_T) under fully developed laminar flow conditions in a circular pipe, we can analyze the temperature profiles and the definition of the Nusselt number.

The Nusselt number (Nu) represents the ratio of convective to conductive heat transfer at a boundary and is defined as:
$$\mathrm{Nu}=\frac{hD}{k}$$
where h is the convective heat transfer coefficient, D is the pipe diameter, and k is the thermal conductivity of the fluid. Since D and k are constants for a given pipe and fluid, the Nusselt number is directly proportional to the heat transfer coefficient h.

The local heat transfer coefficient h is defined using Newton's law of cooling:
$$q=h\left(T_w-T_m\right)$$
where q is the wall heat flux, T_w is the wall temperature, and T_m is the mean (bulk) fluid temperature. Rearranging for h gives:
$$h=\frac{q}{T_w-T_m}$$

For a hydrodynamically and thermally fully developed laminar flow in a circular tube, exact analytical solutions for the temperature profile can be obtained:
1. Under the constant wall heat flux (q = constant) boundary condition, the temperature profile develops in a way that yields a constant temperature difference between the wall and the bulk fluid along the pipe length. Solving the energy equation for this case gives a constant Nusselt number value of:
$${\mathrm{Nu}}_q=\frac{48}{11}\approx 4.364$$
2. Under the constant wall temperature (T_w = constant) boundary condition, the temperature profile adjusts differently. The difference between the wall temperature and the bulk fluid temperature decreases exponentially along the flow direction. Solving the energy equation (which leads to an eigenvalue problem resolved via the Graetz solution) gives a constant Nusselt number value of:
$${\mathrm{Nu}}_T\approx 3.66$$

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

Physically, this occurs because a constant heat flux forces a more active thermal development and a steeper temperature gradient near the wall compared to the constant wall temperature case, where the temperature gradient near the wall relaxes more rapidly as the fluid temperature approaches the wall temperature. A steeper gradient at the wall results in a higher convective heat transfer coefficient, and thus a higher Nusselt number.