Bernoulli's Equation Explained

First we discussed Euler’s equation for fluid flow, and then we integrated it for ideal fluid flow along streamlines to obtain the energy equation for fluid flow. We called it Bernoulli’s Equation. It is a sort of energy conservation equation, the value of which remains constant along a streamline in an ideal fluid flow. Each term of the equation has the dimensions of energy per unit weight, that is, length. These terms represents different types of energy associated with the fluid particle.

Bernoulli’s equation for an ideal fluid flow is written as:

z + p/ρg + v²/2g = constant

Let us first recall and make it clear under what conditions the Bernoulli’s Equation is applicable. It is applicable for a flow

• Along a streamline
• Continuous
• Steady
• Incompressible
• Non-viscous, Invinsid, frictionless

The ‘z’ term stands for the elevation of the point under consideration from a reference level, called a datum, where we fix the potential energy to be zero. ‘p’ is the pressure of the fluid at that point. ‘ρ’ is density of the fluid at the point and for the conditions of applicability of the Bernoulli’s equation the density of the fluid is constant. ‘g’ is the acceleration due to gravity and this is also taken as constant assuming that the variation in the elevation is such that its value can be considered as constant. And ‘V’ is the average velocity of the fluid at the point of consideration.

‘z’ the elevation of the fluid particle gives the potential energy of the particle at that point with reference to the datum. ‘z’ multiplied by the weight of the fluid particle gives the potential energy. Thus, ‘z’ is the potential energy per unit weight of the fluid particle. Looked at dimensionally, the dimension of ‘z’ is m (meters), the dimension of weight is N (newtons), thus, the dimensions of potential energy is Nm and the dimensions of potential energy per unit weight is Nm/N, that is, m. This is also called as the potential head of the fluid.

The second term of Bernoulli’s Equation, ‘p/ρg’, is related to the pressure energy. Pressure energy is the work done by fluid pressure like the pV work done by the pressure in the cylinder to displace the piston. ‘pV’ is the work done and ‘p/ρg’ is the work done per unit weight of the fluid. ‘V’ here is the volume of the fluid displaced, which can be written as W/ρg, where W is the weight of fluid particle. So, pV can be written as pW/ρg and work done per unit weight is written as p/ρg. This is pressure energy of the fluid per unit weight or pressure head.

V²/2g term has square of velocity in numerator which hints of the kinetic energy. But as we know kinetic energy is ½ mV² so the third term in the Bernoulli’s Equation is the kinetic energy per unit weight. Mass ‘m’ can be written as W/g so ½ mV² is equal to ½ WV²/g or V²/2g. It has the dimensions of Nm/N or m, the kinetic energy per unit weight of fluid particle, which is also called velocity head.

This was what each term in Bernoulli’s Equation stands for and their interpretations. In the next article we will discuss some applications of Bernoulli’s equation for hydraulics in Civil Engineering.