## Kinetic energy and Momentum correction factors

Introduction: we have assumed in the derivation of Bernoulli equation that the velocity at the end sections (1) and (2) is uniform. But in a practical situation this may not be the case and the velocity can very across the cross section. A remedy is to use a correction factor for the kinetic energy term in the equation. If is the average velocity at an end section then we can write for energy,

After simplification we find that

Consequently,

Where is the Kinetic Energy Factor. Its value for a fully developed laminar pipe flow is around 2, whereas for a turbulent pipe flow it is between 1.04 to 1.11. It is usual to take it is 1 for a turbulent flow. It should not be neglected for a laminar flow.

**Momentum Correction:** The concept of momentum correction is that

(New momentum component) = (old momentum component) X (correction factor)

The most important job for momentum correction is to find the correction factor. I will find the correction-factor using the method of g8b (Charles Hanretty). That is, the correction-factors are decided as the pull distributions of the kinematic fitting are optimized. As the first step, I have fixed the bins for momentum correction. I made the momentum pull distributions of protons, π , and π- in each bins. I have fitted these pulls using the gaussian. I had the means, sigmas, entries from these gaussian fitting.

The correction factor for V1.0 is made from the following equation:

Cor fac = 1 mean / 100

And then the correction factors for the version more than V1.1 are taken from the Charles's code

**Energy and Momentum Coefficients:** Generally, in the energy and momentum equations the velocity is assumed to be steady .

**Uniform and non-varying vertically. **This assumption does not introduce any appreciable error in case of steady (or nearly uniform) flows. However, the boundary resistance modifies the velocity distribution. The velocity at the boundaries is less than the velocity at a distance from the boundaries. Further, in cases where the velocity distribution is distorted such as in flow through sudden expansions/contractions or through natural channels or varying cross sections, error is introduced.

When the velocity varies across the section, the true mean velocity head across the section, (v^{2}/2g) , (the subscript m indicating the mean value) need not necessarily be equal to V^{-2}/2g . Hence, a correction factor is required to be used for both in energy and momentum equations (See Box). The mean velocity is usually calculated using continuity equation.

Keulegan presented a complete theoretical derivation of energy coefficient α and proved that the selection of α and β (Momentum coefficient) depends solely on the concept of the coefficient of friction which is adopted. If the equation of motion is derived by the energy method, the concept underlying the friction coefficient in that equation is that of energy dissipation in the fluid per unit length of channel and α is the proper factor to use. To understand proper use of factors α and β and the energy principle or momentum principle is used appropriately.

**Box: **