Bernoulli-Equation

Bernoulli’s Equation

Bernoulli’s Equation

Thermodynamics is the branch of physics that describes the macro range properties of a fluid. One of the principal results of the study of thermodynamics is the conservation of energy.

Within a system, energy is neither created nor destroyed but might be converted from one form to another. The Bernoulli Equation can be thought about to be a statement of the conservation of energy principle suitable for flowing liquids.

Statement of Bernoulli’s Equation

In the 1700s, Daniel Bernoulli investigated the forces existing in a flowing liquid. The equation states:

“The static pressure ps and the dynamic pressure in the flow, one-half of the density r times the velocity V squared, is equal to a constant throughout the flow. We call this constant the total pressure p of the flow”.

While the Bernoulli equation is mentioned in terms of generally valid ideas like conservation of energy and the key concepts of pressure, kinetic energy as well as potential energy, its application in the above type is limited to situations of stable flow.

For flow through a tube or pipe, such flow can be pictured as laminar flow, which is still an idealization, yet if the flow is to a great estimate laminar, after that the kinetic energy of flow at any point of the liquid can be designed and computed.

The kinetic energy per volume term in the equation is the one that needs rigorous restraints for the Bernoulli equation to apply – it basically is the assumption that all the kinetic energy of the fluid is adding straight to the forward flow process of the liquid.

Derivation of Bernoulli’s Equation

Bernoulli’s Equation is the essential formula in fluid dynamics that relates pressure to liquid speed as well as height. In deriving this formula, we suppose that the fluid is incompressible, non-viscous, and streams in a constant state. Now lets suppose the flow of the fluid with the pipe in time t,

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Bernoulli-Equation-Der

The force on the upper end of the fluid is P1A1 where P1 is the pressure and A1 is the area of the cross-section at the upper end. The work done on the fluid, by the fluid behind it, in moving through a distanceBernoulli-Equation-6, will be

Bernoulli-Equation-2

In this equation, P2 is the pressure, A2 is the area of cross-section, and is the distance traveled by the fluid in interval time t, at the lower end. The work W2 is taken to be negative as this is against the fluid force.

The net work done = W = W1 + W2

Bernoulli-Equation-3

If v1 and v2 are velocities at the upper and lower ends respectively; then

W = P1A1 v1t – P2A2 v2t

From the equation of continuity,

A1v1 = A2v2

A1 v1 x t = A2 v2 x t

W= (P1 – P2) V

Now if mass m, and  is density then

Bernoulli-Equation-4

Some part of this work is utilized by the fluid in changing its K.E and some part is used in changing its gravitational P.E. This is given as,

Change in K.E = ½ mv22 – ½ mv21

Change in P.E = mgh2 – mgh1

Applying the law of conservation of energy to this volume of the fluid we will obtain

Bernoulli-Equation-5

 

MCQs

  • What is Bernoulli’s Equation primarily related to?
    • A) Conservation of mass
    • B) Conservation of energy
    • C) Conservation of momentum
    • D) Conservation of volume
    • Answer: B) Conservation of energy
  • In Bernoulli’s Equation, what is represented by “ps”?
    • A) Static pressure
    • B) Dynamic pressure
    • C) Total pressure
    • D) Flow pressure
    • Answer: A) Static pressure
  • What does the term “one-half of the density r times the velocity V squared” represent in Bernoulli’s Equation?
    • A) Dynamic pressure
    • B) Total pressure
    • C) Potential energy
    • D) Kinetic energy
    • Answer: D) Kinetic energy
  • According to Bernoulli’s Equation, what is the condition for the kinetic energy term to be applied?
    • A) The flow must be turbulent
    • B) The flow must be unstable
    • C) The flow must be laminar
    • D) The flow must be compressible
    • Answer: C) The flow must be laminar
  • Which physicist investigated the forces existing in a flowing liquid leading to Bernoulli’s Equation?
    • A) Isaac Newton
    • B) Daniel Bernoulli
    • C) Albert Einstein
    • D) Galileo Galilei
    • Answer: B) Daniel Bernoulli
  • What does Bernoulli’s Equation relate to in fluid dynamics?
    • A) Pressure to volume
    • B) Pressure to temperature
    • C) Pressure to velocity and height
    • D) Pressure to density
    • Answer: C) Pressure to velocity and height
  • In the derivation of Bernoulli’s Equation, what assumption is made about the fluid?
    • A) Incompressible and viscous
    • B) Compressible and viscous
    • C) Incompressible and non-viscous
    • D) Compressible and non-viscous
    • Answer: C) Incompressible and non-viscous
  • What is the condition of the fluid’s flow assumed during the derivation of Bernoulli’s Equation?
    • A) Turbulent
    • B) Irregular
    • C) Laminar
    • D) Compressible
    • Answer: C) Laminar
  • In Bernoulli’s Equation, what does “P1A1” represent?
    • A) Pressure at upper end
    • B) Pressure at lower end
    • C) Area at upper end
    • D) Area at lower end
    • Answer: A) Pressure at upper end
  • Which term represents the area of cross-section at the lower end in Bernoulli’s Equation?
    • A) P1
    • B) P2
    • C) A1
    • D) A2
    • Answer: D) A2
  • What does “W1 + W2” represent in Bernoulli’s Equation?
    • A) Total work done
    • B) Work done on upper end
    • C) Work done on lower end
    • D) Net work done
    • Answer: D) Net work done
  • According to Bernoulli’s Equation, what happens to the net work done?
    • A) It is always positive
    • B) It is always negative
    • C) It depends on the fluid viscosity
    • D) It depends on the flow rate
    • Answer: D) It depends on the flow rate
  • What principle is applied to derive Bernoulli’s Equation?
    • A) Law of Thermodynamics
    • B) Law of Conservation of Energy
    • C) Newton’s Law of Motion
    • D) Archimedes’ Principle
    • Answer: B) Law of Conservation of Energy
  • What aspect of the fluid does Bernoulli’s Equation relate pressure to?
    • A) Temperature
    • B) Velocity
    • C) Density
    • D) Viscosity
    • Answer: B) Velocity
  • In Bernoulli’s Equation, what is assumed about the flow of the fluid?
    • A) It is reversible
    • B) It is compressible
    • C) It is continuous
    • D) It is turbulent
    • Answer: C) It is continuous
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Summary

At the heart of fluid dynamics lies Bernoulli’s Equation, a fundamental principle in physics that elucidates the relationship between pressure, velocity, and height within a flowing liquid. Stemming from the conservation of energy principle in thermodynamics, Bernoulli’s Equation posits that the sum of static and dynamic pressures remains constant throughout a fluid’s flow.

Daniel Bernoulli’s 18th-century investigations into flowing liquids culminated in the formulation of this equation, which states that the sum of static pressure and dynamic pressure, proportional to the square of velocity, equals the total pressure along the flow. While applicable to stable flow situations, its accurate use necessitates laminar flow conditions and rigorous constraints on kinetic energy assumptions.

In the derivation of Bernoulli’s Equation, assumptions of incompressibility, non-viscosity, and constant state underpin its development. Through a series of steps involving the balance of forces and energies within the fluid, the equation emerges as a concise expression relating pressure, velocity, and height.

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Ultimately, Bernoulli’s Equation serves as a cornerstone in understanding fluid behavior, facilitating analyses ranging from fluid flow through pipes to the dynamics of air over an aircraft wing. Its elegance lies in its ability to distill complex fluid phenomena into a simple mathematical relationship, making it an indispensable tool in various engineering and scientific applications.