The energy equation for ideal flow has a kinetic energy term with a velocity square factor. We consider ideal flows to be inviscid. For inviscid flow shear effects are not present, hence the flow is uniform across the area of cross section of the flow. It means that the velocity is uniform or the same across the cross section of flow. Therefore the average velocity is equal to the velocity at any point on the section. The total kinetic energy at the section can be written for the ideal flow using the average velocity.
We prefer to use average velocity to calculate kinetic energy because it is easy to find average velocity. Average velocity at any cross section of flow is equal to the rate of flow divided by the area of flow.
Effect of Viscosity on Velocity
In real flow conditions we cannot neglect the effect of viscosity. Viscosity generates shear stresses acting between the layers or particles of fluid flowing. The interaction between the fluid layers and particles and the interaction of fluid with the boundary of conduit makes the velocity of flow nonuniform across the section of flow. For real cases average velocity does not give the correct total kinetic energy at the section.
But still we want to use the average velocity to calculate the total kinetic energy because it is simple to find the average velocity of flow at the section of flow by simply dividing the rate of flow by the area of the section. To make it possible to use average velocity in the energy equation for real flow a correction factor known as the kinetic energy correction factor or kinetic energy coefficient is introduced. It is multiplied to the kinetic energy term to obtain actual total kinetic energy across the section using average velocity.
How to Find the Correction Factor
To find the correction factor we should first find the total kinetic energy across the cross section of flow for real flow. Then we compare the actual total kinetic energy with the total kinetic energy found using the average velocity. Both terms can be equated by multiplying the correction factor on the average velocity side. Hence, the correction factor is equal to the actual total kinetic energy divided by the kinetic energy found using average velocity.
Finding Real Total Kinetic Energy
For real flow, velocities of fluid particles are different across the section of the flow. Therefore we cannot simply use the average velocity to find the total kinetic energy at the cross section of flow. To find the total kinetic energy for real flow consider an elemental area dA on the section of flow, the velocity of flow for this area be v. Mass of fluid passing through the elemental area per second would be ρ_vdA_. Kinetic energy of mass passing through the elemental area would be ½ ρ_vdAv2_. Total kinetic energy at the section is the integration of the term for elemental area over the area of the section of flow.
Modified Energy Equation
The total kinetic energy at the section using average velocity is given by ½ρV3A. Therefore, the correction factor α is (A∫½ ρ_v3dA)/(½ρV3A) or 1/AA∫(v/V)3_dA. If we have a defined velocity distribution function we can find the correction factor analytically. For inviscid flows it is 1, for laminar flow it is 2 and for turbulent flows it varies between 1.03 and 1.3 with Reynolds Number.
After this and the last article for real flows Bernoulli Equation will be written as
z1 + p1/ρg + α1v12/2g = z2 + p2/ρg + α2v22/2g + Losses
In the coming articles we will see how the analysis for real flow can be utilized in some measurements and calculations for real flows.
This post is part of the series: Energy Equation for Real Fluid Flow
For analysis of fluid flow in real conditions we have take into account the effects of viscosity, frictional losses and sudden changes in geometry of flow path. In this article series we will derive the energy equation for real fluid flows while discussing different aspects of real fluid flows.