1. Bernoulli’s Principle
Engineering Problem
ENGR 1201

2. Bernoulli’s Principle
Relates the pressure of a fluid to its elevation and speed
Bernoulli’s equation is used to approximate parameters in water, air, or any fluid with low viscosity
Examples of applications:
Aerospace wing design
Hydroelectric plant pipe design
Design for high wind force on a tall building
Determine pressure through a chemical reactor

3. Fluid Mechanics v1 v2
Ground (h = 0) h1 h2 P1 P2
As the water loses elevation from the high end of the pipe to the low end, it gains velocity.
To find the exact value of any parameter, we apply the Bernoulli equation to two points anywhere along the same streamline.
streamline

4. Bernoulli Equation Derivation
The Bernoulli equation is an expression of conservation of energy KE1 + PE1 + W1 = KE2 + PE2 + W2
where KE is kinetic energy, PE is potential energy, and W is the work done on the system. Imagine a block of ice sliding down the water slide at some velocity. The block has a kinetic energy equal to one-half its mass times its velocity squared, or
KE = ½ mv2
where KE is kinetic energy and m is mass. The block also has a certain potential energy described by
PE = mgh
where PE is the potential energy and h is the height.

5. Derivation (continued)
The work is being done on the block by the force of the water pressure behind it (with a force F=PA, where P is water pressure, and A is the area of the face of the ice that is getting pushed along). When the block is pushed a distance equivalent to its own width, Δx, then the work done on the block is
Wext = F Dx = PA Dx = PV1
Dividing this equation by the volume, and recalling that density, ρ equals mass divided by volume, it reduces to
½ mv12 + mgh1 + P1V = ½ mv22 + mgh2 + P2V
where V is volume. The equation for conservation of energy becomes
½ rv12 + rgh1 + P1 = ½ rv22 + rgh2 + P2
which is the Bernoulli equation.

6. Bernoulli Equation For an “ideal fluid,” Zero viscosity
Constant density
Steady/constant flow
The sum of kinetic energy and potential energy is constant (see derivation)
Where v is fluid velocity
ρ is fluid density
h is relative height
P is pressure
g is gravitational acceleration, or 9.8 m/s2

7. Bernoulli Equation Example
The water at the top of the reservoir starts at rest, so v1 is zero, and the first term drops out.
Since the final height (h2) is also zero, this term drops out, too.
Lastly, P1 = P2, which is atmospheric pressure, so these terms drop out as well.
Plugging in the remaining the known parameters:
rwater g (250 m) = ½ rwater v22
Now the rwater terms can be cancelled out.
Using g = 9.8 m/s2 and solving for v2, we have
v2 = sqrt (2*9.8 m/s2 * 250 m)
v2 = 70 m/s
Now we know how fast the water will flow through the energy-generating turbines!
h1 = 250 m
P1 = atmospheric
h2 = 0 m
P2 = atmospheric