Chapter 4 : Conservation Equations and Analysis of Finite Control Volume Lecture 13 : Bernoulli's Equation

Energy Equation of an ideal Flow along a Streamline

Euler’s equation (the equation of motion of an inviscid fluid) along a stream line for a steady flow with gravity as the only body force can be written as (13.6)

Application of a force through a distance ds along the streamline would physically imply work interaction. Therefore an equation for conservation of energy along a streamline can be obtained by integrating the Eq. (13.6) with respect to ds as  (13.7)

Where C is a constant along a streamline. In case of an incompressible flow, Eq. (13.7) can be written as (13.8)

The Eqs (13.7) and (13.8) are based on the assumption that no work or heat interaction between a fluid element and the surrounding takes place. The first term of the Eq. (13.8) represents the flow work per unit mass, the second term represents the kinetic energy per unit mass and the third term represents the potential energy per unit mass. Therefore the sum of three terms in the left hand side of Eq. (13.8) can be considered as the total mechanical energy per unit mass which remains constant along a streamline for a steady inviscid and incompressible flow of fluid. Hence the Eq. (13.8) is also known as Mechanical energy equation.

This equation was developed first by Daniel Bernoulli in 1738 and is therefore referred to as Bernoulli’s equation. Each term in the Eq. (13.8) has the dimension of energy per unit mass. The equation can also be expressed in terms of energy per unit weight as (13.9)

In a fluid flow, the energy per unit weight is termed as head. Accordingly, equation 13.9 can be interpreted as (13.10) 