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This is called Bernoulli’s equation. Daunting but familiar.
Bernoulli’s equation is a conservation of energy statement for fluids. It sets equal the initial energies of the fluid and the final energies of the fluid, where:
P is the pressure
ρgy is the gravitational potential energy
0.5ρv2 is the kinetic energy
This equation, essentially, treats pressure as a kind of energy.
Use the simulation to explore the following ideas.
Idea #1: ∆h → ∆ U → ∆P
Idea #1: ∆h → ∆ U → ∆P
The pipe system shown is full of fast moving water traveling to the right.
These two sections have the same radius. Due to the continuity equation (Av = Av), they must have the same velocity & kinetic energy.
The second section has less gravitational potential energy. So, it must have more pressure.
Kinetic energy is determined by radius
Kinetic energy is determined by radius
When you solving a problem involving the energies within a pipe, it is important to remember that the kinetic energy will be determined by the continuity equation. Then, you can use conservation of energy to see if the pressure increased, decreased, or didn’t change.
Idea #2: ∆r → ∆v → ∆P
Idea #2: ∆r → ∆v → ∆P
A round pipe full of water flowing to the right has two sections, A & B. Section A has a radius of 0.25 m and the water’s velocity in it is 9 m/s. Section B has a radius of 0.75 m. The centers of the two pipes are at the same height.
The density of water is 1,000 kg/m3. If there is 12,000 N/m2 of pressure in section B, calculate the pressure in section A.
Determine the velocity in B.
A1v1 = A2v2
πr12v1 = πr22v2
(0.25)2(9) = (0.75)2v2
v2 = 1 m/s
Determine pressure in B.
P1 + K1 + Ug1 = P2 + K2 + Ug2
12,000 + 40,500 + 0 = P2 + 500 + 0
P2 = 52,000 N/m2
Idea #3: ∆r + ∆h → ∆P
Idea #3: ∆r + ∆h → ∆P
A round pipe full of water flowing to the right has two sections, A & B. Section A has a radius of 2 m and the water’s velocity in it is 0.5 m/s. Section B has a radius of 1 m and it’s center is located 5 cm vertically above section A’s center.
The density of water is 1,000 kg/m3. If there is 5,000 N/m2 of pressure in section A, calculate the pressure in section B.
Determine the velocity in B.
A1v1 = A2v2
πr12v1 = πr22v2
(2)2(0.5) = (1)2v2
v2 = 2 m/s
Determine pressure in B.
P1 + K1 + Ug1 = P2 + K2 + Ug2
5,000 + 125 + 0 = P2 + 2,000 + 500
P2 = 2,625 N/m2
The Venturi tube or venturimeter is an instrument for measuring with accuracy the flow rate of fluids in pipes. A typical arrangement of a section through such a device is shown in Figure 49.3, and consists of a short converging conical tube called the inlet or upstream cone leading to a cylindrical portion called the throat. This is followed by a diverging section called the outlet or recovery cone. The entrance and exit diameter is the same as that of the pipeline into which it is installed. Angle β is usually a maximum of 21°, giving a taper of β/2 of 10. The length of the throat is made equal to the diameter of the throat. Angle α is about 5° to 7° to ensure a minimum loss of energy but where this is unimportant α can be as large as 14° or 15°.
Venturi tubes have three sections with different radii. Which of the following is true about manometer readings?
Bernoulli Problems
Bernoulli Problems
Water enters a horizontal pipe of non-uniform cross-section with a velocity of 0.6m/s and leaves the other end with a velocity of 0.4m/s. At the first end, pressure of water is 1600N/m2. Calculate the pressure of water at the other end. Density of water = 1000 kg/m3? P = 1700 N/m2
2. Calculate the minimum pressure required to force the blood from the heart to the top of the head (vertical distance 0.5m). Assume the density of blood to be 1040kg/m3. Friction is to be neglected ? ∆P=5200 Pa
3. An airplane wing is designed so that the speed of the air across the top of the wing is 251m/s when the speed of the air below the wing is 225m/s. The density of air is 1.29kg/m3. What is the lifting force on a wing of area 24m 2 ? F = 1.92 × 105 N
4. Water is circulating through a closed system of pipes in a two floor apartment. On the first floor, the water has a gauge pressure of 3.4 × 10 5 pa and a speed of 2.1m/s. However, on the second floor, which is 4m higher, the speed of the water is 3.7m/s. The speeds are different because the pipe diameters are different. What is the gauge pressure of the water on the second floor? P = 295360 Pa = 3x10^5 Pa
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