Velocity Profile in a Pipe of Constant Area

When a fluid flows through a pipe of constant cross-sectional area, the velocity of the fluid varies across the pipe's diameter. This variation in velocity is known as the velocity profile. The profile is influenced by the type of flow—laminar or turbulent—and the physical properties of the fluid.

Laminar vs. Turbulent Flow

The flow of fluid in a pipe can be characterized as either laminar or turbulent. The type of flow is determined by the Reynolds number, which is a dimensionless quantity used to predict flow patterns in different fluid flow situations.

Laminar Flow

In laminar flow, fluid particles move in smooth, orderly layers or laminae, with minimal mixing between the layers. This type of flow typically occurs at lower velocities and is characterized by a Reynolds number less than 2000.

Turbulent Flow

In turbulent flow, fluid particles move in a chaotic manner, with significant mixing and eddies. Turbulent flow usually occurs at higher velocities and is characterized by a Reynolds number greater than 4000.

Transitional Flow

Between laminar and turbulent flow, there is a transitional regime where the flow can switch between the two states. This occurs in the Reynolds number range of 2000 to 4000.

Velocity Profile Formulas

The velocity profile in a pipe can be described mathematically. The most common profiles are for laminar and turbulent flows.

Laminar Flow Velocity Profile

For laminar flow, the velocity profile is parabolic. The maximum velocity occurs at the center of the pipe and decreases towards the walls. The velocity profile can be described by the Hagen-Poiseuille equation:

$$ v(r) = \frac{1}{4\mu} \left( \frac{\Delta P}{L} \right) (R^2 - r^2) $$

where:

• ( v(r) ) is the fluid velocity at a distance ( r ) from the center of the pipe,
• ( \mu ) is the dynamic viscosity of the fluid,
• ( \Delta P ) is the pressure drop over a length ( L ) of the pipe,
• ( R ) is the radius of the pipe.

Turbulent Flow Velocity Profile

For turbulent flow, the velocity profile is flatter in the center and drops sharply near the walls. The profile can be approximated by the power-law equation:

$$ v(r) = V_{max} \left(1 - \frac{r}{R}\right)^{1/n} $$

where:

• ( V_{max} ) is the maximum velocity at the center of the pipe,
• ( n ) is an empirical constant that depends on the Reynolds number.

Differences Between Laminar and Turbulent Flow Velocity Profiles

Here is a table summarizing the differences between laminar and turbulent flow velocity profiles:

Feature	Laminar Flow	Turbulent Flow
Reynolds Number	< 2000	> 4000
Flow Pattern	Orderly layers	Chaotic eddies
Velocity Profile Shape	Parabolic	Flatter with sharp drop near walls
Maximum Velocity Location	Center of the pipe	Center of the pipe
Mathematical Model	Hagen-Poiseuille equation	Power-law equation

Example: Calculating Velocity Profile for Laminar Flow

Let's calculate the velocity profile for water flowing in a laminar regime through a pipe with a radius of 0.05 m and a length of 10 m. Assume the pressure drop is 100 Pa and the dynamic viscosity of water is ( 1 \times 10^{-3} ) Pa·s.

Using the Hagen-Poiseuille equation:

$$ v(r) = \frac{1}{4 \times 1 \times 10^{-3}} \left( \frac{100}{10} \right) (0.05^2 - r^2) $$

At the center of the pipe (( r = 0 )):

$$ v(0) = \frac{1}{4 \times 1 \times 10^{-3}} \left( \frac{100}{10} \right) (0.05^2) $$ $$ v(0) = 0.625 \text{ m/s} $$

At ( r = 0.025 ) m (halfway to the wall):

$$ v(0.025) = \frac{1}{4 \times 1 \times 10^{-3}} \left( \frac{100}{10} \right) (0.05^2 - 0.025^2) $$ $$ v(0.025) = 0.46875 \text{ m/s} $$

This example illustrates how the velocity decreases from the center of the pipe towards the walls in laminar flow.

Understanding the velocity profile in a pipe of constant area is crucial for various engineering applications, including fluid transport and heat exchange systems. It helps in designing efficient systems and predicting the behavior of the fluid flow under different conditions.