In fluid dynamics, the entrance length is the distance a flow travels after entering a pipe before the flow becomes fully developed.^[1] Entrance length refers to the length of the entry region, the area following the pipe entrance where effects originating from the interior wall of the pipe propagate into the flow as an expanding boundary layer. When the boundary layer expands to fill the entire pipe, the developing flow becomes a fully developed flow, where flow characteristics no longer change with increased distance along the pipe. Many different entrance lengths exist to describe a variety of flow conditions. Hydrodynamic entrance length describes the formation of a velocity profile caused by viscous forces propagating from the pipe wall. Thermal entrance length describes the formation of a temperature profile.^[2] Awareness of entrance length may be necessary for the effective placement of instrumentation, such as fluid flow meters.^[3]

Hydrodynamic entrance length

The hydrodynamic entrance region refers to the area of a pipe where fluid entering a pipe develops a velocity profile due to viscous forces propagating from the interior wall of a pipe.^[1] This region is characterized by a non-uniform flow.^[1] The fluid enters a pipe at a uniform velocity, then fluid particles in the layer in contact with the surface of the pipe come to a complete stop due to the no-slip condition. Due to viscous forces within the fluid, the layer in contact with the pipe surface resists the motion of adjacent layers and slows adjacent layers of fluid down gradually, forming a velocity profile.^[4] For the conservation of mass to hold true, the velocity of layers of the fluid in the center of the pipe increases to compensate for the reduced velocities of the layers of fluid near the pipe surface. This develops a velocity gradient across the cross-section of the pipe.^[5]

Boundary layer

The layer in which the shearing viscous forces are significant, is called the boundary layer.^[6] This boundary layer is a hypothetical concept. It divides the flow in pipe into two regions:^[6]

1. Boundary layer region: The region in which viscous effects and the velocity changes are significant.^[6]
2. The irrotational (core) flow region: The region in which viscous effects and velocity changes are negligible, also known as the inviscid core.^[2]

When the fluid just enters the pipe, the thickness of the boundary layer gradually increases from zero moving in the direction of fluid flow and eventually reaches the pipe center and fills the entire pipe. This region from the entrance of the pipe to the point where the boundary layer covers the entire pipe is termed as the hydrodynamic entrance region and the length of the pipe in this region is termed as the hydrodynamic entry length. In this region, the velocity profile develops and thus the flow is called the hydrodynamically developing flow. After this region, the velocity profile is fully developed and continues unchanged. This region is called the hydrodynamically fully developed region. But this is not the fully developed fluid flow until the normalized temperature profile also becomes constant.^[6]

In case of laminar flow, the velocity profile in the fully developed region is parabolic but in the case of turbulent flow it gets a little flatter due to vigorous mixing in radial direction and eddy motion.

The velocity profile remains unchanged in the fully developed region.

Hydrodynamic Fully Developed velocity profile Laminar Flow :

^[6]

where ${\displaystyle x}$ is in the flow direction.

Shear stress

In the hydrodynamic entrance region, the wall shear stress, ${\displaystyle \tau _{w}}$, is highest at the pipe inlet, where the boundary layer thickness is the smallest. Shear stress decreases along the flow direction.^[6] That is why the pressure drop is highest in the entrance region of a pipe, which increases the average friction factor for the whole pipe. This increase in the friction factor is negligible for long pipes.^[6] In a fully developed region, the pressure gradient and the shear stress in flow are in balance.^[6]

Calculating hydrodynamic entrance length

The length of the hydrodynamic entry region along the pipe is called the hydrodynamic entry length. It is a function of Reynolds number of the flow. In case of laminar flow, this length is given by:

^[2]

where ${\displaystyle R_{e}}$ is the Reynolds number and ${\displaystyle D}$ is the diameter of the pipe.
But in the case of turbulent flow,

^[8]

Thus, the entry length in turbulent flow is much shorter as compared to laminar one. In most practical engineering applications, this entrance effect becomes insignificant beyond a pipe length of 10 times the diameter and hence it is approximated to be:

^[6]
Other authors give much longer entrance length, e.g.

• Nikuradse recommends ${\displaystyle 40D}$^[9] and
• Lien et al. recommend ${\displaystyle 150D}$ for high Reynolds flows.^[10]

Entry length for pipes with non-circular cross-sections

In the case of a non-circular cross-section of a pipe, the same formula can be used to find the entry length with a little modification. A new parameter “hydraulic diameter” relates the flow in non-circular pipe to that of circular pipe flow. This is valid as long as the cross-sectional area shape is not too exaggerated. Hydraulic Diameter is defined as:

^[6]

where ${\displaystyle A}$ is the area of cross-section and ${\displaystyle P}$ is the Perimeter of the wet part of the pipe

Average velocity of fully developed flow

By doing a force balance on a small volume element in the fully developed flow region in the pipe (Laminar Flow), we get velocity as function of radius only i.e. it does not depend upon the axial distance from the entry point.^[6] The velocity as the function of radius comes out to be:

^[6]
where ${\textstyle {\frac {\mathrm {d} P}{\mathrm {d} x}}}$ is constant.

By definition of average velocity is given by

where ${\displaystyle A_{c}}$ is cross-sectional area.

Thus,

^[6]

For fully developed flow, the maximum velocity will be at ${\displaystyle r=0}$.
Thus,

^[6]

Zdroj:https://en.wikipedia.org?pojem=Entrance_length_(fluid_dynamics)
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