Entrance Region in Pipe Flow

The entrance region refers to that portion of pipe until the velocity profile is fully developed. When a fluid is entering a pipe at a uniform velocity, the 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 viscosity of the fluid, this layer in contact with the pipe surface, resists the motion of adjacent layers and slows them down gradually. For the conservation of mass to hold true the velocity of middle layers of the fluid in the pipe increases (since the layers of fluid near the pipe surface have reduced velocities). This develops a velocity gradient across the cross section of the pipe.

Boundary layer

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

1. Boundary layer region: The region in which viscous effects and the velocity changes are significant.
2. The irrotational (core) flow region: The region in which viscous effects and velocity changes are negligible.

When the fluid just enters the pipe, the thickness of the boundary layer gradually increases from zero as we move in direction of fluid flow and eventually it reaches the pipe centre and fills the entire pipe. This region from the entrance of pipe to the point where the boundary layer covers the entire pipe is termed as the hydrodynamic entrance region and 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 hydrodyanamically developing flow. After this region, the velocity profile is fully developed and continues unchanged. This region is termed as Hydrodyamically fully developed region. But this is not the fully developed fluid flow until the normalised temperature profile also becomes constant.

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 (or fuller) 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  : ${\frac {\partial u(r,x)}{\partial x}}=0\quad \Rightarrow u=u(r)$

We can see in this image the developing velocity profile of a fluid entering a pipe.

Shear stress $(\tau _{w})$

In the hydrodynamic entrance region, the wall shear stress (τw ) is highest at the pipe inlet where the boundary layer thickness is the smallest and it decreases along the flow direction. That is why the pressure drop is the highest in entrance region of a pipe and hence it always increases the average friction factor for the whole pipe. This increase in the friction factor is negligible for long pipes.
In fully developed region the pressure gradient and the shear stress in flow are in balance.

In this image we can see that the Shear Stress is maximum right at the entry point and becomes uniform as the fluid flow develops.

Entry Length

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

$L_{h,laminar}=0.05R_{e}D$

Where, $R_{e}$ is The Reynold's number and $D$ is the diameter of the pipe.
But in the case of turbulent flow,

$L_{h,turbulant}=1.359D(R_{e})^{1/4}$

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

$L_{h,turbulant}\approx 10D$

Entry Length For Pipes with Non-circular Cross-Section

In 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 until the cross sectional area shape is not too exaggerated. Hydraulic Diameter is defined as:

$D_{h}={\frac {4A}{P}}$

Where, $A$ is the area of cross section and $P$ is the Perimeter of the wet part of the pipe

Average velocity

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. The velocity as the function of radius comes out to be:

$u(r)=-{\frac {R^{3}}{4\mu }}{\frac {dP}{dx}}(1-{\frac {r^{2}}{R^{2}}})$
Where

${\frac {dP}{dx}}=constant$

By definition of Average velocity,

$V_{avg}={\frac {\int u\mathrm {d} A}{A_{c}}}$ where $A_{c}$ is Area of cross section

Thus,

$V_{avg}={\frac {2}{R^{2}}}\int _{0}^{R}u(r)r\mathrm {d} r$

$=-{\frac {2}{R^{2}}}\int _{0}^{R}{\frac {R^{2}}{4\mu }}{\frac {dP}{dx}}(1-{\frac {r^{2}}{R^{2}}})r\mathrm {d} r$

$=-{\frac {R^{2}}{8\mu }}{\frac {dP}{dx}}$

In a fully developed flow, the maximum velocity will be at r=0.
Thus,

$U_{max}=2V_{avg}$

References

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