Accounting for Gravitational Potential, Kinetic and Pressure Energies

You might have already sensed from the previous descriptions, that these three forms of energy are often converted from one form to the other. (Does this sound familiar? If it doesn’t, you can look back at the definition of the conservation of energy).

Let us consider the following situations in a little more detail:

  • Pressure energy in a balloon can be converted into kinetic energy by releasing the pressurised air.

  • Gravitation potential energy of a bucket of water at an elevated height can be converted into kinetic energy if the water loses height by being poured down a pipe.

To ensure that we can account for the conservation of these three energies, we can quantify them using Bernoulli’s equation (named after the originator of the equation, Daniel Bernoulli.)

Daniel Bernoulli (1700-1782) was a Swiss mathematician and physicist who was persuaded by his father, himself an eminent mathematician, to study business and medicine (he gained a PhD in medicine along the way). In return, his father taught him mathematics. He put the latter to use in developing a conservation of energy equation, for fluids, which was ultimately named after him.

Bernoulli proposed that the three forms of energy, gravitational potential, kinetic and pressure energies, would always be conserved in a way that would sum to a constant value:

G r a v. space p o t e n t i a l space e n e r g y space plus space k i n e t i c space e n e r g y space plus space p r e s s u r e space e n e r g y space equals space c o n s tan t

The equations for quantifying these three forms of energy are given in the following expression:

rho g h space plus space 1 half space rho u squared space plus space P space equals space c o n s tan t

Where ρ is the fluid density, g is gravity, h is the vertical height of the fluid, u is the fluid velocity and P is the fluid pressure.

(Strictly speaking, this equation assumes no gain or loss of heat or work and also a fixed temperature).

Bernoulli’s equation has many important applications in the physical world. For example, it explains the mechanism for how aerofoil shapes generate lift for an aircraft and also the swing of a cricket ball, or the swerve of a football. Take some time to do further reading in these fields.

On the next page we’ll look at a worked example of Bernoulli’s equation applied to a brewery situation.

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