Bernoulli's Equation

Definition :

The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids.

The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the lowering of fluid pressure in regions where the flow velocity is increased.

This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density.

In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy.

where,

· points 1 and 2 lie on a streamline,

· the fluid has constant density,

· the flow is steady, and

· there is no friction.

Although these restrictions sound severe, the Bernoulli

equation is very useful, partly because it is very simple to use

and partly because it can give great insight into the balance between pressure, velocity and elevation.

Examples

Water circulates throughout a house in a hot-water heating system. If water is pumped out at a speed of 0.50 m/s through a 4.0 cm diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6 cm diameter pipe on the second floor 5.0 m above? Assume pipes do not divide into branches.

Solution:

First calculate the flow speed on the second floor, calling it v₂, using the equation of continuity. We can call the basement point 1.

v₂ = (v₁A₁) / (A₂) = (v₁π(r₁)²) / (π(r₂)²) = (0.50 m/s)(0.020 m)² / (0.013 m)² = 1.2 m/s

To find pressure, we use Bernoulli’s equation:

P₂ = P₁ + ρg(h₁– h₂) + ½ρ((v₁)² – (v₂)²)

P₂ = (3.0 x 10⁵ N/m²) + (1.0 x 10³ kg/m³) [ (0.50 m/s)² – (1.2 m/s)²]

P₂ = (3.0 x 10⁵ N/m²) – (4.9 x 10⁴ N/m²) – (6.0 x 10² N/m²)

P2 = 2.5 x 10⁵N/m²

What is the lift (in newtons) due to Bernoulli’s principle on a wing of area 86 m2 if the air passes over the top and bottom surfaces at speeds of 340 m/s and 290 m/s, respectively?

Solution: