Bernoulli Equation
The Bernoulli equation, named after Swiss mathematician Daniel Bernoulli, describes the behavior of fluid flow along a streamline. It relates the pressure, velocity, and elevation of a fluid at any two points along its path. The equation is fundamental in fluid mechanics and is used to analyze and predict the behavior of fluids in various engineering applications.
6 Key excerpts on "Bernoulli Equation"
eBook - ePubIntroduction to Engineering Mechanics
A Continuum Approach, Second Edition
- Jenn Stroud Rossmann, Clive L. Dym, Lori Bassman(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
...If we remember that streamlines are everywhere tangent to the velocity vector, we can also think of a set of coordinates defined relative to the streamlines—for two-dimensional flow, one coordinate s directed along the streamline, and n defined normal to the streamline. We could then write the component equations of motion in the s and n directions. The resulting equation in the s direction, which states ∑ F = m a along a streamline, may be integrated to yield the following equation: p ρ + g z + 1 2 V 2 = constant along a streamline, (18.34) where we have assumed that gravity acts in the negative z -direction, and where V is the velocity in the s direction, simply the magnitude of the velocity vector since V is in the s direction. This equation is known as the Bernoulli Equation, and it is true for steady flow of an incompressible fluid under inviscid conditions. For convenience, we write the equation together with its restrictions: p ρ + g z + 1 2 V 2 = constant • On a streamline • For steady flow • For incompressible fluid • If viscous effects neglected Many problems can be solved using the Bernoulli Equation, allowing us to dodge having to solve the full Euler or Navier–Stokes equations. It should not escape our notice that the Bernoulli Equation, derived from ∑ F = m a, looks like an energy conservation equation. This is even easier to see if we multiply through by the (assumed constant) density: Equation 18.34 becomes p + ρ g z + 1 2 ρ V 2 = constant, (18.35) and we can think of pressure p as a measure of flow work, ρ gz as a gravitational potential energy, and 1 2 ρ V 2 as a kinetic energy, all per unit volume of fluid. Daniel Bernoulli actually first arrived at Equation 18.34 by performing an energy balance, even though the concept of energy was still a bit fuzzy in 1738. One of the most useful applications of the Bernoulli Equation is a device known as a Pitot * tube, and its cousin the Pitot-static tube, used to measure flow velocities...
eBook - ePub- Rose G Davies(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...Equation (2.10) is called Bernoulli’s equation. The conditions required to obtain this equation can be summarized as follows: incompressible fluid particle flows along a streamline; the flow is in steady state; and consider no friction. Each term of the equations represents a type of energy per unit mass (2.10a), or per unit volume (2.10b) of the fluid along the streamline. The statement of the Bernoulli’s equation is that for a flow of an incompressible ideal fluid along its streamline the sum of the potential energy (ghρ), kinetic energy (ρ (v 2 / 2)) and the capability (p) to produce work due to pressure at any point on the streamline remains constant – Bernoulli’s Theorem. Take the fluid flow shown in Figure 2.1 as an example, there are two labeled sections: upstream section 1 and downstream section 2. p 1, and p 2 are the pressure at these two sections, v 1, and v 2 are the speed of the fluid, and h 1, and h 2 are the height at these two sections respectively. According to Bernoulli’s Theorem, the sum of the energies mentioned above remains constant; therefore, we can obtain: g h 1 ρ + p 1 + ρ 2 v 1 2 = g h 2 ρ + p 2 + ρ 2 v 2 2 (2.11) From Equation (2.11) we learn that the speed of the fluid flow will decrease if the fluid flows up if the fluid pressure is the same at both sections; or when flow speed increases, the pressure will decrease if these two sections are at the same level. Each term of Bernoulli’s Equation (2.11) has the unit of pressure, [Pa]. We can use P as the constant in Equation (2.10): g h ρ + p + ρ 2 v 2 = P (2.12) in which, ghρ – “Potential” pressure ; p – Static pressure ; ρ 2 v 2 – Dynamic pressure ; P – Total pressure. (Sometimes p t is used) Divide each term of Equation (2.12) by gρ, assuming the fluid is incompressible, and it becomes: h + p g ρ + v 2 2 g = H (2.13) The terms of Equation (2.13) have the unit of length/height, [m]. In Equation (2.13), h – (“Natural” height) Head ; p g ρ – Static...
- Amithirigala Widhanelage Jayawardena(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
...Chapter 13 Applications of basic fluid flow equations 13.1 Introduction There are many problems in nature that involve principles of fluid mechanics. Solutions to such problems are obtained by applying the governing equations of fluid flow or their approximations. The three basic equations involved are the continuity equation, the momentum equation and the energy equation. Assumptions and/or simplifications necessary to apply the basic equations to certain types of practical problems include incompressibility, steady state condition, ideal fluid, and sometimes reducing the dimensionality of the problem. In this chapter, how the governing equations of fluid flow are applied to some typical practical problems in nature are highlighted. 13.2 Kinetic energy correction factor When flows in open channels or pipes are considered, it is generally assumed to be one-dimensional with an average velocity at each section. The kinetic energy is V 2 2 g per unit weight, but is different from ∫ cross section v 2 2 g. Therefore, a correction factor, α, is used for V 2 / 2 g so that α V 2 2 g is the average kinetic energy for unit weight, passing the section. Referring to Figure 13.1, the kinetic energy passing δ A per unit time is Figure 13.1 Velocity profile in pipe flow. ∫ v 2 2 (ρ v d A) where ρvdA is the mass. Therefore, α V 2 2 ρ V A = ∫ A v 2 2 (ρ v d A) which leads to α = 1 A ∫ A (v V) 3 d A (13.1) The. Bernoulli Equation then becomes z + p ρ g + α V 2 2 g = C o n s t a n t For laminar flow in pipes α = 2 ; for turbulent flows in pipes, α varies from 1.01 to 1.10 and is usually ignored. All terms in the Bernoulli Equation are available energy, and, for real fluids flowing through a system, the available energy decreases in the downstream direction...
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...We conclude that for both equations to be identical, any change in internal energy of the fluid must equal the amount of heat transferred. It can be seen that for an incompressible flow with no work, no heat transfer, and no changes in internal energy, the energy equation and the Bernoulli Equation derived from the momentum equation become identical. Thus, under certain flow conditions, the energy and momentum equations reduce to the same expression. Hence, the Bernoulli Equation is referred to as the mechanical energy equation. For many flow problems, only the continuity and Bernoulli Equations are required for a description of the flow. Example 3.13 A water jet issues from a sink and falls vertically downward. The water flow rate is such that it will fill a 250 ml cup in 8 seconds. The faucet is 30 cm above the sink, and at the point of impact, the jet diameter is 0.3 cm. What is the jet diameter at the faucet exit? Solution: A jet exiting a faucet and impacting a flat surface; we locate section 1 at the faucet exit and section 2 at the sink. Continuity Q = A 1 V 1 = A 2 V 2 assuming one-dimensional, steady flow. The volume flow rate is Q = 0.251 Liters/s 8 s = 0.031 25 × 10 − 3 m 3 /s Substituting for flow rate and area gives the velocity at each section. as V 1 = Q A 1 = 4 Q π D 1 2 = 4 0.031 25 × 10 − 3 π D 1 2 or V 1 = 3.98 × 10 − 5 D 1 2 or V 2 = 4 0.031 25 × 10 − 3 π 0.003 2 = 4.42 m/s The Bernoulli Equation applied to these two sections. is p 1 ρ + V 1 2 2 g c + g z 1 g c = p 2 ρ + V 2 2 2 g c + g z 2 g c Evaluating properties: p 1 = p 2 = p atm z 1 = 0.3 m z 2 = 0 The Bernoulli Equation becomes, after. simplification, V 1 2 2 g + z 1 = V 2 2 2 g Substituting, 3.98 × 10 − 5 D 1 2 2 1 2 9.81 + 0.3 = 4.42 2 2 9.81 Solving, D 1 4 = 1.16 × 10 − 10 and D 1 = 3.28 × 10 − 3 m = 0.328 cm ¯ Example 3.14 Consider the flow of water through a venturi meter, as shown in Figure 3.24...
eBook - ePub- Colin Salter(Author)
- 2021(Publication Date)
- Pavilion(Publisher)
...Daniel’s particular interest was the theory of conservation of energy – he noted the principle of a change from kinetic to potential energy in a moving body which accompanied a gain in height, and applied it to fluids, in which kinetic energy was exchanged for pressure. Daniel Bernoulli published his magnum opus, Hydrodynamica, in 1738. The name was a word of his own invention and was adopted for the new field of engineering, hydrodynamics, which the book opened up. In it he took the conservation of energy as his starting point and considered the efficiency of hydraulic machines. It contains the first exposition of the kinetic theory of gases. Daniel Bernoulli was actually born in the Netherlands, then under Spanish rule. The talented family of mathematicians moved to Basel in Switzerland to escape Spanish persecution of Protestants. Above all it includes Bernoulli’s Principle, the rule that an increase in the speed of a fluid goes hand in hand with a decrease in pressure or a decrease in the fluid’s potential energy. Leonhard Euler devised the equation that goes with it. Besides its application to hydrodynamic engineering, Bernoulli’s Principle is central to aerodynamics. It is the reason that aeroplane wings have their distinctive cross-section, which encourages lift and flight. Daniel’s father was so jealous of Hydrodynamica that he plagiarized it in his own 1739 book, Hydraulica, which he backdated to 1732 so that it appeared that he, Johann Bernoulli, had thought of Bernoulli’s Principle first. Johann harboured this resentment of his son’s success until the day he died. A pattern of air flow demonstrating the Bernoulli Principle, which states that the internal pressure of a gas is lowered the faster it travels. This principle gives an aeroplane wing lift. Bernoulli’s great work was shamelessly plagiarized by his father....
eBook - ePub- Roger Legg(Author)
- 2017(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 13 Fluid Flow: General Principles Abstract In this chapter, the general principles of airflow in ducts are explained, the fluid being treated as incompressible. The relevant equations are given and the general characteristics of flow described. The methods of calculating pressure losses in straight ducts and in fittings are illustrated, together with the pressure distributions. The concept of resistance is explained, and the chapter concludes by giving some relevant methods for measuring flow rates. An understanding of these topics is important before going on in subsequent chapters to consider the design and sizing of ductwork systems, fan selection, and on-site balancing procedures. Keywords Bernoulli Equation; Conservation of energy; Measurement of flow rate; Moody chart; Orifice plate; Pitot-static tube; Pressure losses in fittings; Pressure losses in straight ducts; Resistance; Reynolds number Symbols A duct or pipe cross section A o orifice plate area b duct breadth C calibration coefficient for Pitot-static tube D diameter D e equivalent diameter of rectangular duct d diameter of orifice or conical inlet f friction factor K pressure loss coefficient K b bend pressure loss coefficient K e expansion pressure loss coefficient K f straight duct pressure loss coefficient k roughness coefficient of duct wall L duct length m. mass flow rate p at atmospheric pressure p s static...
