# Bernoulli’s Theorem in Chemical Engineering

**Bernoulli’s Theorem** in **fluid dynamics** is one of the important discoveries. The relation among pressure, velocity, and elevation in a moving fluid, the compressibility and viscosity of which are negligible and the flow of which is steady or laminar. The Theorem was first derived in 1738 by Swiss mathematician **Daniel Bernoulli**. The theorem states that, in effect, that the total mechanical energy of the flowing fluid, comprising the energy associated with fluid pressure, the gravitational potential energy of elevation and the kinetic energy of fluid motion, remains constant. Bernoulli’s theorem is the principle of energy conservation for ideal fluids in steady or streamlines flow.

**Bernoulli’s principle** can be applied to various types of fluid flow. There are different applications of **Bernoulli’s Theorem**. The theorem can be derived through the law of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir the sum of all forms of energy is the same on all the streamlines because in a reservoir the energy per unit mass is the same everywhere.

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can occur only because the fluid on that section has moved from a region of a higher pressure to a region of lower pressure and if its speed decreases, it can only occur because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

In most liquid flows and gas flows at high speeds, the mass density of a fluid parcel can be considered to be constant, regardless, of pressure variations in the flow. For this reason, the fluid in such flows can be considered to be incompressible and these flows can be described as an incompressible flow. Bernoulli performed experiments on liquids and his equation in its original form is valid only in incompressible flow. A common form of Bernoulli’s equation, valid at any arbitrary point along a streamline where gravity is constant, is:

V^{2}/2 + gz + p/ρ = constant

Where,

v is the fluid flow speed at a point on a streamline

g is the acceleration due to gravity

z is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration

p is the pressure of the fluid at all the points in the fluid

ρ is the density of the fluid at all points in the fluid

For conservative force fields, Bernoulli’s equation can be generalized as

V^{2}/2 + Ψ + p/ρ = constant

Where, Ψ is the force potential at the point considered on the streamline.

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