13.14 - Bernoulli’s equation
Bernoulli’s equation applies to ideal fluids. It was developed by the Swiss
mathematician and physicist Daniel Bernoulli (1700-1782). The equation is used to
analyze fluid flow at different points in a closed system. It states that the sum of the
pressure, the KE per unit volume, and the PE per unit volume has a constant value.
Concept 1 shows an idealized apparatus for determining these three values at various
points in such a system.
A simplified form of Bernoulli’s equation is shown in Equation 1. It applies to horizontal
flow, in which the PE of the fluid is everywhere the same. In such a system, the sum of
just the pressure and the kinetic energy per unit volume is constant. Since the
expression for the “KE” uses the density ȡ of the fluid in place of mass, it describes
energy per unit volume: the kinetic energy density.
To illustrate the simplified equation, we use the horizontal-flow configuration shown in
the diagram of Equation 1. The pressure of the fluid is measured where it passes the
gauges. If the speed of the fluid is known at the first gauge, its speed at the second
gauge can be calculated using Bernoulli’s equation.
When the sum of the pressure and kinetic energy density equals a constant in a
system, as in the case of horizontal flow, it is often useful to set the sum of these values
at one point equal to the sum of the values at another point. This is stated for points P 1
and P 2 in Equation 2. An implication of the simplified form of Bernoulli’s equation is the
Bernoulli effect: When a fluid flows faster, its pressure decreases.
Airplane wings, like the one shown in Equation 2, take advantage of the Bernoulli effect.
Air travels faster over the upper surface of the wing than the lower, because it must
traverse a longer path in the same amount of time. A faster fluid is a lower pressure
fluid; the result is that there is more pressure below the wing than above. This causes a
net force up, which is called lift. (The Bernoulli effect is only one way to explain how
wings work. Lift can also be explained using Newton’s third law in conjunction with a
fluid phenomenon called the Coanda effect. The topic of wing lift engenders much
discussion.)
In Equation 3 you see a general form of Bernoulli’s equation, which also accounts for
differences in height and the resulting differences in potential energy density. It states
that the sum of the pressure and the kinetic energy density, plus the potential energy
density, is constant in a closed system. At a higher point in such a system, the potential
energy density of the fluid is greater. At that point either the pressure or the kinetic
energy density, or both, must be less than they are at lower points.
Bernoulli’s equation
The sum of:
equals a constant in a closed system
·pressure
·KE / unit volume
·PE / unit volume