Question

Consider the flow of mercury (a liquid metal) in a tube. How will the hydrodynamic and thermal entry lengths compare if the flow is laminar? How would they compare if the flow were turbulent?

Short answer

Question: Compare the hydrodynamic and thermal entry lengths for the flow of mercury in a tube under laminar and turbulent flow conditions. Answer: In both laminar and turbulent flows of mercury, the hydrodynamic entry length is longer than the thermal entry length. This is due to the small Prandtl number of mercury, which results in shorter thermal entry lengths.

Step-by-step solution

Understand Hydrodynamic and Thermal Entry Lengths

The hydrodynamic entry length is the distance from the tube inlet where the fluid velocity profile becomes fully developed. In other words, it's the distance it takes for the fluid velocity profile to stabilize after entering the tube. The thermal entry length is the distance from the inlet where the temperature profile of the fluid becomes fully developed. Similar to the hydrodynamic entry length, it's the distance it takes for the fluid temperature profile to stabilize after entering the tube.

Hydrodynamic Entry Length in Laminar and Turbulent Flows

The hydrodynamic entry length can be estimated using the following relation: L_h = 0.05 * Re * D, where L_h is the hydrodynamic entry length, Re is the Reynolds number, and D is the tube diameter. In laminar flow (Re < 2100), the entry length is relatively short, while for turbulent flow (Re > 4000), the entry length becomes much larger.

Thermal Entry Length in Laminar and Turbulent Flows

The thermal entry length can be estimated using the following relation: L_t = 0.05 * Re * Pr * D, where L_t is the thermal entry length, Re is the Reynolds number, Pr is the Prandtl number (dimensionless number relating the relative importance of momentum and thermal diffusion in a fluid), and D is the tube diameter. In laminar flow, the Prandtl number for mercury is around 0.02, indicating that the thermal entry length is considerably shorter than the hydrodynamic entry length. In turbulent flow, the Prandtl number for mercury remains small, meaning that the thermal entry length is still shorter than the hydrodynamic entry length.

Comparing Entry Lengths in Laminar and Turbulent Flows

Based on the equations for hydrodynamic and thermal entry lengths and properties of mercury, we can conclude the following: - In laminar flow, the hydrodynamic entry length will be longer than the thermal entry length, because the Prandtl number of mercury is small (0.02), making the thermal entry length shorter. - In turbulent flow, the hydrodynamic entry length will still be longer than the thermal entry length, as the Prandtl number for mercury remains small. So, in both laminar and turbulent flows of mercury, the hydrodynamic entry length is longer than the thermal entry length.