Question

Consider the flow of oil 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

Answer: In laminar flow, the hydrodynamic and thermal entry lengths show a direct relationship with Reynolds number and pipe diameter, and with Reynolds number, Prandtl number, and pipe diameter, respectively. In contrast, in turbulent flow, the hydrodynamic and thermal entry lengths show a relationship with the square root of Reynolds number and the power of Prandtl number. The magnitudes of entry lengths may differ significantly in turbulent flow compared to laminar flow due to the different relationships with the involved parameters.

Step-by-step solution

Understand the entry lengths

Entry length is defined as the distance along the flow direction at which a developing flow is affected by the boundary conditions at the pipe entrance and reaches its fully developed state. There are two types of entry lengths: hydrodynamic entry length and thermal entry length. The hydrodynamic entry length is the distance required for velocity profiles to be fully developed, while the thermal entry length is the distance required for temperature profiles to be fully developed.

Laminar flow's entry lengths

For laminar flow, the hydrodynamic entry length (Lh) and thermal entry length (Lt) can be estimated using the following formulas: Lh = 0.05 * Re * D Lt = 0.05 * Re * Pr * D Where Re is the Reynolds number, Pr is the Prandtl number, and D is the diameter of the pipe.

Turbulent flow's entry lengths

For turbulent flow, the hydrodynamic entry length (Lh) and thermal entry length (Lt) can be estimated using the following formulas: Lh = 4.4 * (Re^0.5) * D Lt = 0.8 * (Re^0.5) * Pr^0.4 * D

Compare the entry lengths in laminar and turbulent flows

Comparing the entry lengths in laminar and turbulent flows, we can see that for laminar flow, both the hydrodynamic and thermal entry lengths are directly proportional to the product of Reynolds number (Re) and pipe diameter (D). However, in turbulent flow, the entry lengths depend on the square root of Reynolds number and the power of Prandtl number (0.4 for thermal entry length). As a result, the relationship between hydrodynamic and thermal entry lengths would be different for laminar and turbulent flow regimes.

Conclusion

In laminar flow, the hydrodynamic and thermal entry lengths show a direct relationship with Reynolds number and pipe diameter, and with Reynolds number, Prandtl number, and pipe diameter, respectively. Whereas in turbulent flow, the hydrodynamic and thermal entry lengths show a relationship with the square root of Reynolds number and the power of Prandtl number. They do not exhibit the same relationship as in laminar flow, and their magnitudes may differ significantly in turbulent flow compared to laminar flow.