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The application of Appropriate Technology

# Part 5: Energy in a Perfect System – The Bernoulli Equation

Anywhere in a perfect system (i.e. there are no frictional effects), for an incompressible fluid there are three types of energy existing:

1. Pressure Energy. Example: If you blow up the tyre of a car with a pump you are turning your physical energy of working the pump into pressure energy in the tyre.
2. Kinetic Energy. Example: This is the energy contained in a moving fluid. If a wave hits you at the beach, you feel the kinetic energy contained within it.
3. Potential Energy. Example: Gravity is trying to pull water to the lowest point on the earth’s surface. So when water is at a high point it contains energy, which can “potentially” allow it to flow down.

At any point in a perfect system the sum of these “bits” of energy in different forms (Pressure, Kinetic and Potential) must equal the sum of these “bits” at any other point in the system.

This is because energy cannot be “created” or “destroyed”, it can only change its form. This is what the BERNOULLI EQUATION expresses.

Appendix 1. gives the derivation of the Kinetic Energy in terms of a pressure and Appendix 2. gives the derivation of the Potential Energy in terms of a pressure.

These different forms of energy are expressed mathematically (as pressures) in the Bernoulli Equation (for a perfect system) shown below:

$P_1 + \frac{1}{2}.\rho .v_1^2 + \rho .g.h_1 = P_2 + \frac{1}{2}.\rho .v_2^2 + \rho .g.h_2$

(12)

The terms on the left hand side of the Equation are as follows:

P1 is the pressure energy at point 1 (expressed as a pressure). [Units are N/m2 or Pa]
ρ is the density of the fluid.[Units are Kg/m3]
v1 is the velocity of the fluid at point 1. [Units are m/s]
g is the acceleration due to earth’s gravity (9.81 m/s/s).[Units are m/s/s]
h1 is the height (from a given datum) of the fluid at point 1.[Units are m]

The terms are similar on the right hand side of the Equation, but for point 2.

The left hand side of the equation represents all the “bits” of energy (expressed as pressures) at a point 1 in a perfect system and the right hand side all the “bits” of energy (expressed as pressures) at another point 2.