BERNOULLI Principles- Exercises

The scientist, Daniel Bernoulli, raised in 1738, a principle that bears his name, which establishes the relationship of the speed of a fluid and the pressure it exerts, when the fluid is in motion. Fluids tend to increase their velocity in narrow pipes.

It also proposes that, for a fluid in motion, the energy is transformed each time the cross-sectional area of ​​the pipe changes, presenting in the Bernoulli Equation, the mathematical relationship between the forms of energy that the fluid in motion presents.

The use of the Bernoulli principle has a wide variety of household, commercial and industrial applications, such as in chimneys, insecticide sprays, flow meters, Venturi tubes, engine carburettors, suction cups, aircraft lift, water ozonators , dental equipment, among others. It is the basis for the study of hydrodynamics and fluid mechanics.

BASIC CONCEPTS to understand Bernoulli's Principles

I invited themLet's see the article of The heat of Joule's Law "Applications - Exercises"

Fluid:

Set of randomly distributed molecules that are held together by weak cohesive forces and by forces exerted by the walls of a container, without a defined volume. Both liquid and gases are considered fluids. In the study of the behavior of fluids, the study of fluids in a state of rest (hydrostatic) and fluids in motion (hydrodynamics) is usually carried out. See figure 1.

We invite you to see the article Thermodynamic Principles

Mass:

Measure of the inertia or resistance to change the movement of a fluid body. Measurement of the amount of fluid, it is measured in kg.

Weight:

Force with which the fluid is attracted to the earth by the action of gravity. It is measured in N, lbm.ft / s².

Density:

Amount of mass per unit volume of a substance. It is measured in kg / m³.

Flow:

Volume per unit of time, in m3 / s.

Pressure:

Amount of force exerted on a unit area of ​​a substance, or on a surface. It is measured in Pascal or psi, among other units.

Viscosity:

Resistance of fluids to flow, due to internal friction. The higher the viscosity, the lower the flow. It varies with pressure and temperature.

Energy Conservation Law:

Energy is neither created nor destroyed, it is transformed into another type of energy.

Continuity equation:

In a pipe with different diameters, with constant flow, there is a relationship between the areas and the speed of the fluid. The velocities are inversely proportional to the cross-sectional areas of the pipe. [1]. See figure 2.

Bernoulli's principle

Statement of Bernoulli's Principle

Bernoulli's principle establishes the relationship between the velocity and the pressure of a moving fluid. Bernoulli's principle states that, in a fluid in motion, as the speed of a fluid increases, the pressure decreases. Higher speed points will have less pressure. [two]. See figure 2.

When a fluid moves through a pipe, if the pipe has a reduction (smaller diameter), the fluid has to increase its speed to maintain the flow, and its pressure decreases. See figure 4.

Uses of Bernoulli's Principle

Carburetor:

Device, in gasoline-powered engines, where air and fuel are mixed. As the air passes through the throttle valve, its pressure decreases. With this decrease in pressure the gasoline begins to flow, at such a low pressure it vaporizes and mixes with the air. [3]. See figure 5.

Planes:

For the flight of airplanes, the wings are designed so that a force called "lift" is produced, creating a pressure difference between the upper and lower part of the wings. In figure 6 you can see one of the designs of the airplane wings. The air that passes under the wing of the aircraft tends to separate creating greater pressure, while the air that passes over the wing travels greater distance and greater speed. Since the high pressure is under the wing, a lift force results that propels the wing upward.

Boat propeller:

It is a device used as a propellant on ships. The propellers consist of a series of blades designed so that when the propeller rotates, a speed difference is generated between the faces of the blades, and therefore a pressure difference (Bernoulli effect). Al. The pressure difference produces a thrust force, perpendicular to the plane of the propeller, which propels the boat. See figure 7.

Swimming:

When you move your hands when swimming, there is a pressure difference between the palm and the back of the hand. In the palm of the hand, the water passes at low speed and high pressure (Bernoulli's principle), originating a “lift force” that depends on the pressure difference between the palm and the back of the hand. See figure 8.

Equation for Bernoulli's principle

Bernoulli's equation allows to mathematically analyze fluids in motion. Bernoulli's principle arises, mathematically, based on the conservation of energy, which states that energy is not created or destroyed, it is transformed into another type of energy. Kinetic, potential and flow energy are considered:

• Kinetics: which depends on the speed and mass of the fluid
• Potential: due to height, relative to a reference level
• Flow or pressure: energy carried by the molecules of the fluid as they move along the pipe. See figure 9.

The total energy that a fluid has in motion is the sum of the energy of the flow pressure, the kinetic energy and the potential energy. By the Law of Conservation of Energy, the energy of a fluid through a pipe is equal to the inlet and outlet. The sum of the energies at the initial point, at the entrance of the pipe, is equal to the sum of the energies at the outlet. [1]. See figure 10.

Constraints of the Bernoulli Equation

• It is only valid for incompressible fluids.
• It does not take into account devices that add power to the system.
• Heat transfer is not taken into account (in the basic equation).
• The surface material is not taken into account (There is no friction loss).

Exercise

To bring water to a second floor of a house, a pipe like the one shown in figure 11 is used. It is desired that, at the outlet of the pipe, located 3 meters above the ground, the water has a velocity of 5 m / s, with a pressure equal to 50.000 Pa. What must be the speed and pressure at which the water must be pumped? In figure 10 the water inlet is marked as point 1 and the water outlet in the narrower pipe as point 2.

Solution

To determine the velocity v1, the continuity equation is used at the pipe inlet. See figure 12.

Bernoulli's equation will be used to calculate the pressure at the inlet P1, as shown in figure 13.

Conclusions of Bernoulli's Principle

Bernoulli's principle states that, in a fluid in motion, when its speed increases, the lower the pressure it exerts. The energy is transformed each time the cross-sectional area of ​​the pipe changes.

Bernoulli's equation is a consequence of the conservation of energy for fluids in motion. It states that the sum of the fluid pressure, the kinetic energy and the potential energy, remains constant throughout the entire path of the fluid.

This principle has multiple applications such as in the lift of airplanes, or of a person when swimming, as well as in the design of equipment for the transport of fluids, among many others, its study and understanding being of great importance.

REFERENCES

[1] Mott, Robert. (2006). Fluid mechanics. 6th edition. Pearson Education
[2]
[3]