Introduction

In fluid mechanics, the kinetic energy correction factor is a dimensionless parameter that accounts for the deviation of actual flow velocity from the average velocity in a pipe. It is commonly used in the calculation of pressure drop in pipe flow.

Explanation

The kinetic energy correction factor, denoted as "C", is given by the equation:

C = (2ln(D/2r) - 1)/(ln(D/r) - 1)

where D is the pipe diameter and r is the hydraulic radius (half of the pipe diameter in the case of a circular pipe).

Derivation

To understand why the kinetic energy correction factor is 2 for laminar flow through a circular pipe, let's consider the flow profile within the pipe.

In laminar flow, the velocity distribution across the pipe is parabolic, with the maximum velocity occurring at the centerline and decreasing towards the pipe walls. The average velocity is obtained by integrating the velocity profile across the pipe cross-section.

Key Points:
- Laminar flow has a parabolic velocity profile.
- Maximum velocity occurs at the centerline.
- Average velocity is obtained by integrating the velocity profile.

Integration of the Velocity Profile

To calculate the average velocity, we need to integrate the velocity profile across the pipe cross-section. The velocity at any radial position, "r", can be given by the equation:

v(r) = (2 * Vmax) * (1 - (r/R)^2)

where Vmax is the maximum velocity at the centerline and R is the pipe radius.

Key Points:
- Velocity at any radial position, "r", is given by v(r) = (2 * Vmax) * (1 - (r/R)^2).
- Vmax is the maximum velocity at the centerline.
- R is the pipe radius.

Calculation of the Average Velocity

To calculate the average velocity, we need to integrate the velocity profile across the pipe cross-section. The integration can be performed from r = 0 to r = R, which gives:

Vavg = (2/3) * Vmax

Key Points:
- Average velocity is given by Vavg = (2/3) * Vmax.

Calculating the Kinetic Energy Correction Factor

The kinetic energy correction factor, C, is defined as the ratio of the actual kinetic energy to the kinetic energy calculated using average velocity. It can be calculated using the equation:

C = (actual kinetic energy)/(kinetic energy using average velocity)

The actual kinetic energy is given by:

KE_actual = (1/2) * ρ * Vmax^2 * (π * R^2)

The kinetic energy using average velocity is given by:

KE_avg = (1/2) * ρ * Vavg^2 * (π * R^2)

Substituting the values of Vmax and Vavg, we get:

C = (2/3)^2 = 4/9 ≈ 0.44

Key Points:
- Kinetic energy correction factor, C, is the ratio of actual kinetic energy to kinetic energy using average velocity.
- Actual kinetic energy is (1/2) * ρ * Vmax^2 * (π * R^2).
- Kinetic energy using average velocity