How to prepare for JEE advanced for Bernoulli’s theorem

How to prepare for JEE advanced

In this post we will learn How to prepare for JEE advanced for Bernoulli’s theorem? “Bernoulli’s Principle”, also known as “Bernoulli’s Equation” or “Bernoulli’s Trinomial”, describes the behavior of a fluid at rest moving along a water current. It was first exposed by Daniel Bernoulli in his paper “Hydrodynamics” (1738) and it shows that while an ideal fluid (meaning that it has no viscosity and no friction) moves in a circulation regime in a closed conduit, the energy that the fluid posses remains constant during all it’s path. The energy of a fluid in any given time has three components, which are:

1. Kinetics: is the energy that comes from the velocity of the fluid.

2. Gravitational Potential: is the energy that comes from how high the fluid is from the reference.

3. Flow Energy: is the energy that a fluid has due to the pressure it possesses.

The next equation, known as “Bernoulli’s Equation” (Bernoulli’s Trinomial) has this exact terms:

(V2 . ρ / 2) + P + ρ . g . z = constant

where:

V = fluid’s velocity at a given section.

ρ = fluid’s density.

P = pressure along the current line.

g = gravitational acceleration.

z = height (along the gravitational direction) from a reference.

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To be able to use this equation, we must first make the following assumptions:

  1. Viscosity (internal friction) = 0. Meaning that, we consider that the current line in which we are working is a “non viscous” fluid zone.

                2. Flow = constant.

                3. The fluid is compressible, and ρ is constant.

                4. The equation is applied along a current line or in a rotational flow.

Even though this particular equation has Bernoulli’s name, the way it is now presented was first written by Leonhard Euler.

The fact that the sum of the three energies remains constant means that, if there is a variation in any of them, one (or both) of the remaining two must change to compensate, so that the total sum remains constant.

An example of the application of this principle can be found in the water flow of a pipe.

For example, if we modify the velocity of a fluid without modifying it’s height, then the pressure must change to compensate. If we increase the velocity, by decreasing the area of the pipe for example, the pressure exerted by the fluid will decrease. If we decrease the velocity, by increasing the area of the pipe for instance, the pressure will increase.

Every one of the terms of this equation has units of length, and at the same time represent different types of energy, in hydraulics it’s common to express energy in terms of length, and it’s commonly referred as “head” or “height”. So, in Bernoulli’s equation, the terms are usually called velocity head, pressure head and hydraulic head; the z term is generally grouped with P/γ (where γ = ρg) to give place to the so called piezometric height, also known as, piezometric charge.

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Characteristics and consequences

Velocity head + piezometric height or charge = Hydraulic head

V2 / 2 . g + P / γ + z = H

We can also rewrite this principle in the shape of the sum of pressures, multiplying all the equation by γ, in this way the term relative to velocity will be called dynamic pressure, and the terms regarding pressure and height will be joined in the static pressure.

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Shown above, an schematic on Venturi’s Effect.

Dynamic pressure + Static pressure = constant

V2 ρ / 2 + P + γ . z = constant

or, written in a simpler way:

q + p = p0

where:

  • q = V2 ρ / 2
  • p = P + γ. z
  • p0= constant

Similarly, we can write the same equation as the sum of the kinetic energy, the flow energy and the gravitational potential energy per mass unit:

kinetic energy + flow energy + grav. Pot. en. = constant

V2 / 2 + P /ρ + g . z = constant

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Bernoulli’s Theorem Applications

Chimney

Chimneys are high to take advantage of the fact that wind velocity is greater and constant at higher altitudes. The faster the wind flows over the top of a chimney, the lower the pressure, and the difference of pressure between the base and the top of the chimney is greater. In consequence,
the combustion gases are easily extracted.

Pipes

Bernoulli’s equation and the Continuity equation also tells us that if we reduce the traversal area of a pipe, so as to increase the velocity of the fluid that goes through it, the pressure will reduce.

Swimming

The application in this sport is implied in the way a swimmer’s hands cut the water. He or she does it in a way that generates the least amount of pressure, and the greatest amount of propulsion.

Engine carburetor

In a car’s engine carburetor, the pressure from the air that goes through the carburetor’s body decreases every time it goes through a constriction of the pipe. Each time the pressure diminishes, the gasoline flows in, vaporizes and mixes with the air flow.

Fluid flow from a tank

The rate of the flow is given by Bernoulli’s equation.

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Venturi’s devices

In oxygen therapy, most of the systems that regulate high debit supply use devices that are similar to Venturi’s device, which is based in Bernoulli’s principle.

Aviation

In planes, the superior part of the wing has a curve that is more pronounced that the inferior part of the wing. This causes the air that flows over the superior part of the wing to increase it’s velocity, which in turn diminishes it’s pressure, creating a suction that helps to keep the plane in flight.

The velocity in the superior part of the wing increases as the law of the continuity of mass predicts, where the entry flow must equal the exit flow. By raising the velocity at the top, the kinetic energy of the fluid increases and, according to Bernoulli’s principle, so as to make the sum remain constant, either the height or the pressure must vary with it. The height can not change, so the fluid pressure decreases.

If you want to learn more about Bernauli’s Theorem you can read from this book.

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