Question

How is the hydrodynamic entry length defined for flow in a tube? Is the entry length longer in laminar or turbulent flow?

Short answer

Answer: Turbulent flow has a longer hydrodynamic entry length compared to laminar flow. This is because the chaotic and disordered motion of fluid particles in turbulent flow requires more distance to become fully developed than the smooth, parallel layers in laminar flow.

Step-by-step solution

Definition of Hydrodynamic Entry Length

Hydrodynamic entry length is the distance along the tube from the inlet where the flow becomes fully developed. In fully developed flow, the flow velocity profile doesn't change along the pipe, and the gradients in the axial direction cease to exist.

Laminar and Turbulent Flows

There are two types of flows in a pipe - laminar and turbulent. Laminar flow occurs when the fluid particles move in smooth, parallel layers due to low velocity, and there is no mixing between these layers. On the other hand, turbulent flow occurs when the fluid is moving at a high velocity, resulting in chaotic and disordered motion of fluid particles and significant mixing between the layers.

Laminar Entry Length

In laminar flow, the hydrodynamic entry length can be calculated using the following formula: \[L_{e_{laminar}}=0.05\cdot Re\cdot D\] where \(L_{e_{laminar}}\) is the entry length for laminar flow, \(Re\) is the Reynolds number, and \(D\) is the diameter of the tube.

Turbulent Entry Length

In turbulent flow, the hydrodynamic entry length can be calculated using the following formula: \[L_{e_{turbulent}}=4.4 \cdot (Re)^{1/6}\cdot D\] where \(L_{e_{turbulent}}\) is the entry length for turbulent flow, \(Re\) is the Reynolds number, and \(D\) is the diameter of the tube.

Comparing Entry Lengths in Laminar and Turbulent Flows

Typically, the entry length is longer for turbulent flow than for laminar flow because the chaotic and disordered motion of fluid particles in turbulent flow requires more distance to become fully developed, as compared to the smooth, parallel layers in laminar flow. By comparing the formulas for laminar and turbulent entry lengths, you can observe that the entry length in turbulent flow increases with the Reynolds number raised to the power of 1/6, whereas the entry length in laminar flow increases linearly with the Reynolds number.