Question

Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and velocity profiles. The Nusselt number for this flow depends on

• A. neither the Reynolds number nor the Prandtl number
• B. both the Reynolds and Prandtl numbers
• C. the Reynolds number but not the Prandtl number
• D. the Prandtl number but not the Reynolds number

Hint:

This is a classic result in convection heat transfer that is important to memorize. For fully-developed laminar flow in a circular tube:
- \(Nu = 3.66\) for constant wall temperature.
- \(Nu = 4.36\) for constant wall heat flux.
Remember that this independence from Re and Pr only applies to the fully-developed region. In the developing region, Nu depends on both.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks about the dependencies of the Nusselt number (\(Nu\)) for a very specific and important case in heat transfer: fully-developed laminar flow in a tube with constant wall temperature.
- Fully-developed velocity profile: The velocity profile is parabolic and no longer changes in the direction of flow. This happens after the hydrodynamic entrance length.
- Fully-developed temperature profile: The shape of the dimensionless temperature profile \(\frac{T_w - T(r)}{T_w - T_m}\) no longer changes in the direction of flow. This happens after the thermal entrance length.
- Nusselt Number (\(Nu\)): A dimensionless number representing the ratio of convective to conductive heat transfer across a boundary. For internal flow, \(Nu = \frac{hD}{k}\).
Step 2: Detailed Explanation:
1. In the thermal entrance region of a tube, where the temperature profile is still developing, the local heat transfer coefficient \(h_x\) and thus the local Nusselt number \(Nu_x\) are functions of both the Reynolds number (\(Re\)) and the Prandtl number (\(Pr\)). They are high at the entrance and decrease as the flow develops.
2. However, the question specifies that the flow is fully-developed in terms of both velocity and temperature.
3. For this specific condition, the analysis of the convection-diffusion equation shows that the heat transfer coefficient \(h\) becomes a constant value that depends only on the fluid's thermal conductivity (\(k\)), the tube's geometry (diameter \(D\)), and the thermal boundary condition.
4. As a result, the Nusselt number, \(Nu = hD/k\), also becomes a constant value.
5. For fully-developed laminar flow in a circular tube:
- With a constant surface temperature (isothermal) boundary condition, the theoretical value is \(\mathbf{Nu_D = 3.66}\).
- With a constant surface heat flux boundary condition, the theoretical value is \(\mathbf{Nu_D = 4.36}\).
6. Since the Nusselt number is a constant (3.66 in this case), its value does not depend on the flow velocity (and thus not on the Reynolds number) or the fluid properties encapsulated by the Prandtl number. The conditions of the problem have already fixed the state for which \(Nu\) becomes independent of \(Re\) and \(Pr\).
Step 3: Final Answer:
The Nusselt number for this flow depends on neither the Reynolds number nor the Prandtl number.
Step 4: Why This is Correct:
The state of "fully-developed laminar flow and temperature profiles" is a special case where the Nusselt number converges to a constant value determined solely by the geometry and boundary condition type. Therefore, it is independent of \(Re\) and \(Pr\).