Continuing with the fundamental principles of chemical engineering today I will try to explain the basic principle of Bernoulli's Law, which is the main mathematical formula to which the different calculations relate to the transient phenomena where matter moves, which they are transfer of quantity of movement, transfer of quantity of energy and transfer of quantity of mass.

Scheme of Bernoulli's Principle

Bernoulli's principle, also called the Bernoulli equation or Bernoulli's trinomial, describes the behavior of a fluid moving along a line of current. It was exposed by Daniel Bernoulli in his work Hidrodinámica (1738) and states that in an ideal fluid (without viscosity or friction) in a circulation system through a closed conduit, the energy that the fluid has remains constant along its path.

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The energy of a fluid at any time consists of three components:

Kinetic: is the energy due to the speed of the fluid.
Gravitational potential: is the energy due to the altitude that a fluid possesses.
Flow energy: is the energy that a fluid contains due to the pressure it has.

The following equation known as the "Bernoulli equation" (Bernoulli's trinomial) consists of these same terms.

Where:
: Velocity of the fluid in the section considered.
: Gravitational acceleration
: Height in the direction of gravity from a reference height.
: Pressure along the current line.
: Density

To apply the equation, the following assumptions must be made:

Viscosity (internal friction) = 0 That is, it is considered that the current line on which it is applied is in a 'non-viscous' area of the fluid.
Constant flow
Incompressible fluid, where ρ is constant.
The equation is applied along a current line.

Although the name of the equation is due to Bernoulli, the form presented above was presented in the first place by Leonhard Euler.
An example of application of the principle is found in the flow of water in pipes.

Characteristics and consequences

Each of the terms in this equation have units of length, and at the same time they represent different forms of energy; in hydraulics it is common to express energy in terms of length, and we speak of height or head, Thus in the Bernoulli equation terms are often called heights or heads of speed, pressure and hydraulic head, the term z is usually grouped with P / γ to give rise to the so-called piezometric height or also piezometric load.

We can also rewrite this principle as a sum of pressures multiplying the whole equation by γ, so the term relative to speed will be called dynamic pressure, the terms of pressure and height are grouped in the static pressure.

Thus the Bernoulli principle can be seen as another form of the law of conservation of energy, that is, in a line of current each type of energy can rise or fall by virtue of the decrease or increase of the other two.
This equation allows explaining phenomena such as the Venturi effect, since the acceleration of any fluid in an equipotential path (with equal potential energy) would imply a decrease in pressure. This effect explains why light things often tend to get out of a moving car when windows are opened. The air pressure is less outside because it is moving with respect to that which is inside, where the pressure is necessarily greater. In a way, apparently, contradictory air enters the vehicle but this occurs due to turbulence and boundary layer phenomena.

Bernoulli equation and the First Law of Thermodynamics

From the first law of thermodynamics we can conclude an equation aesthetically similar to the Bernouilli equation previously mentioned, but conceptually different. The fundamental difference lies in the limits of operation and in the formulation of each formula. The Bernoulli equation is a balance of forces on a particle of fluid that moves through a current line, while the first law of thermodynamics consists of an energy balance between the limits of a given volume of control, by which is more general since it allows to express the energy exchanges along a fluid current, as are the friction losses that subtract energy, and the pumps or fans that add energy to the fluid. The general form of this, let's call it, "energetic form of the Bernoulli equation" is:

Where:
: Is the specific weight
: it is a measure of the energy that is supplied to the fluid
: Is a measure of the energy used to overcome friction forces through the fluid path.
: Gravitational acceleration.

Assumptions

The equation above is a derivative of the first law of thermodynamics for fluid flows with the following characteristics.

The working fluid, that is, the one that flows and that we are considering, has a constant density.
There is no internal energy change.

Applications of the Bernoulli Principle

Airsoft

The replicas used in this game usually include a system called HopUp that causes the ball to be projected with a circular effect, which increases the effective range of the replica. This effect is known as the Magnus effect, the rotation of the ball causes the speed of the flow above it to be greater than below, and with it the appearance of a difference of pressures that creates the sustaining force, which makes the ball take more time to fall.

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Chimney

The chimneys are high to take advantage of the fact that the wind speed is more constant and higher at higher altitudes. The faster the wind blows over the mouth of a chimney, the lower the pressure and the greater the pressure difference between the base and the mouth of the chimney, consequently, the combustion gases are extracted better.

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Pipeline

The Bernoulli equation and the continuity equation also tell us that if we reduce the cross-sectional area of a pipe to increase the velocity of the fluid passing through it, the pressure will be reduced.

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Swimming

The application within this sport is reflected directly when the swimmer's hands cut the water generating less pressure and more propulsion.

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Aircraft support

The Bernoulli effect is also partly the origin of aircraft support. Thanks to the shape and orientation of the aerodynamic profiles, the wing is curved on its upper face and angled with respect to the incident current lines. Therefore, the upstream lines of the wing are closer together than below, so the air velocity is higher and the pressure is lower above the wing; as the pressure below the wing is greater, a net upward force called sustentation is generated.
Movement of a ball or ball with effect If we throw a ball or a ball with effect, that is, by rotating on itself, it deflects to the side. Also for the known Magnus effect, typical is the chopped ball, when the player puts the instep under the ball causing a rotating effect so that it traces a parabolic trajectory. It is what we know as Vaseline.

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Car carburetor

In a car carburetor, the air pressure that passes through the body of the carburetor decreases when it passes through a throttle. When the pressure decreases, gasoline flows, vaporizes and mixes with the air stream.