Physics
Application of Bernoulli's Equation
The application of Bernoulli's equation involves using the principle of conservation of energy to analyze the behavior of fluids in various situations. It is commonly used to study the flow of fluids through pipes, in airfoil design for aircraft, and in understanding the dynamics of blood flow in the human body. By applying Bernoulli's equation, engineers and scientists can make predictions about fluid behavior and design efficient systems.
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11 Key excerpts on "Application of Bernoulli's Equation"
- Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
Several applications of the Bernoulli equation are discussed. In some flow situations, such as the use of a Pitot-static tube to measure fluid velocity or the flow of a liquid as a free jet from a tank, a Bernoulli equation alone is sufficient for the analysis. In other instances, such as confined flows in tubes and flowmeters, it is necessary to use both the Bernoulli equation and the continuity equation, which is a statement of the fact that mass is conserved as fluid flows. The following checklist provides a study guide for this chapter. When your study of the entire chapter and end-of-chapter exercises has been completed, you should be able to ■ write out meanings of the terms listed here in the margin and understand each of the related concepts. These terms are particularly important and are set in italic, bold, and color type in the text. ■ explain the origin of the pressure, elevation, and velocity terms in the Bernoulli equation and how they are related to Newton’s second law of motion. ■ apply the Bernoulli equation to simple flow situations, including Pitot-static tubes, free jet flows, confined flows, and flowmeters. ■ use the concept of conservation of mass (the continuity equation) in conjunction with the Bernoulli equation to solve simple flow problems. ■ apply Newton’s second law across streamlines for appropriate steady, inviscid, incompress-ible flows. ■ use the concepts of pressure, elevation, velocity, and total heads to solve various flow problems. ■ explain and use the concepts of static, stagnation, dynamic, and total pressures. ■ use the energy line and the hydraulic grade line concepts to solve various flow problems. ■ explain the various restrictions on use of the Bernoulli equation. The Bernoulli equation is not valid for flows that involve pumps or turbines.
- John I. Hochstein, Andrew L. Gerhart(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
72 LEARNING OBJECTIVES After completing this chapter, you should be able to: CHAPTER 3 Elementary Fluid Dynamics— The Bernoulli Equation • discuss the application of Newton’s second law to fluid flows. • explain the development, uses, and limitations of the Bernoulli equation. • use the Bernoulli equation (stand-alone or in combination with the simple continuity equation) to solve flow problems. In this chapter we investigate some typical fluid motions (fluid dynamics) in an elementary way. We will discuss in some detail the use of Newton’s second law (F = ma) as it is applied to fluid particle motion that is “ideal” in some sense. We will obtain the celebrated Bernoulli equation and apply it to various flows. Although this equation is one of the oldest in fluid mechanics and the assumptions involved in its derivation are numerous, it can be used effectively to predict, understand, and analyze a variety of flow situations. However, if the equation is applied with- out proper respect for its restrictions, serious errors can arise. Indeed, the Bernoulli equation is appropriately called the most used and the most abused equation in fluid mechanics. A thorough understanding of the elementary approach to fluid dynamics involved in this chapter will be useful on its own. It also provides a good foundation for the material in the fol- lowing chapters where some of the present restrictions are removed and “more nearly exact” results are presented. 3.1 Newton’s Second Law Consider a really tiny volume of fluid, which is still large enough to contain a significant num- ber of molecules. This volume is called a fluid particle (with further description in Section 4.1). As the fluid particle moves from one location to another, it usually experiences an acceleration or deceleration.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Making Connections: Take-Home Investigation with a Sheet of Paper Hold the short edge of a sheet of paper parallel to your mouth with one hand on each side of your mouth. The page should slant downward over your hands. Blow over the top of the page. Describe what happens and explain the reason for this behavior. Chapter 12 | Fluid Dynamics and Its Biological and Medical Applications 439 Bernoulli’s Equation The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant: (12.17) P + 1 2 ρv 2 + ρgh = constant, where P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes (12.18) P 1 + 1 2 ρv 1 2 + ρgh 1 = P 2 + 1 2 ρv 2 2 + ρgh 2 . Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with m replaced by ρ . In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting ρ = m / V into it and gathering terms: (12.19) 1 2 ρv 2 = 1 2 mv 2 V = KE V . So 1 2 ρv 2 is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find (12.20) ρgh = mgh V = PE g V , so ρgh is the gravitational potential energy per unit volume. Note that pressure P has units of energy per unit volume, too.
- Ahlam I. Shalaby(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
p between points 1 and 2 in the pipe. Therefore, if the actual difference in pressure as measured by the difference in piezometric heights at points 1 and 2 is small in comparison to the magnitude of pressure at points 1 and 2, then one can assume that the viscous force is negligible in comparison to the pressure, gravitational, and inertial forces. Finally, in its simplest application, the Bernoulli equation illustrates no conversion of energy forms.4.5.3.1Applications of the Bernoulli EquationApplication of the Bernoulli equation represents the assumption of ideal flow, which may be generally classified as internal flow (pipe or open channel), external flow around an object, or flow from a tank (or a water source). As such, there are numerous applications of the Bernoulli equation, which include: (1) flowrate measurement for pipe flow using ideal flow meters (ideal pipe flow); (2) flowrate measurement for open channel flow using ideal flow meters (ideal open channel flow); (3) gradual (vertical or horizontal) channel contractions/transitions (ideal open channel flow); 4) velocity (and pressure) measurement for both internal and external flow using a pitot-static tube (ideal pipe flow, ideal open channel flow, and ideal external flow); and (5) flow from a tank (or a water source) through a jet, siphon, or some other opening/connection to the tank (ideal flow from a tank). Application of the Bernoulli equation for velocity (and pressure) measurement for both internal and external flow using a pitot-static tube demonstrates how the dynamic pressure is modeled by a pressure rise, as presented in Section 4.5.3.2 . The use of several pitot-static tubes along a pipe or a channel section allows schematic illustrations of the energy equation, which are known as the energy grade line, EGL , and the hydraulic grade line, HGL . Furthermore, application of the Bernoulli equation for ideal flow meters for internal flow, for ideal gradual channel contraction, and for ideal flow from a tank demonstrates how the dynamic pressure is modeled by a pressure drop, as presented in Section 4.5.3.3 . Illustrations of the EGL and HGL are presented in Sections 4.5.3.4 and 4.5.3.5 , while example problems illustrating the applications of the Bernoulli equation are presented in Sections 4.5.6 and 4.5.7
eBook - ePub- Joseph Katz(Author)
- 2016(Publication Date)
- Wiley(Publisher)
Of course, in reality the flow could have a much more complex three-dimensional velocity distribution. Figure 3.1 One- and two-dimensional velocity distributions The basic equations used for such one-dimensional flows are the integral form of the continuity and momentum equations (as developed in Section 2.6). However, an additional equation, called the Bernoulli equation, is frequently used and therefore we must derive it first. 3.2 The Bernoulli Equation In the earlier discussion about the kinetic theory of gases (see Fig. 1.17), we speculated how Daniel Bernoulli connected (in the mid-1700s) velocity with pressure variations in a moving fluid. This equation is widely used in numerous engineering applications, but it is initially limited to flows without friction (this can be modified later by including a head loss term). The objective of this section is to introduce the Bernoulli equation for inviscid flows, and a more detailed discussion of its limitation will follow in Section 6.3. To clarify the applicability of this equation, consider the flow over a moving vehicle, as shown in Fig. 3.2. We may follow a particle moving along a streamline as shown, but we assume that there are no losses such as friction along this path. This assumption suggests that we must use the inviscid form of the momentum equation (Eq. 2.41), called the Euler equation. (2.41) Figure 3.2 Frictionless flow along a streamline The x component of the Euler equation may be rewritten (recall that the discussion is on one-dimensional flow only) as: (3.1) The only body force we consider for this case is gravity, which acts in the vertical direction: The negative sign reflects that gravity acts opposite to the z (or h) direction. This force is conservative and the work along the streamline depends only on h. Consequently we can write that the force is a gradient of a conservative potential (− gh) Again, note that here (see Fig
- Andrew L. Gerhart, John I. Hochstein, Philip M. Gerhart(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
At high speeds, compressibility becomes important (the density is not constant) and other phenomena occur. Some of these ideas are discussed in Section 3.8, while others (such as shockwaves for supersonic Pitot-tube applications) are discussed in Chapter 11. The concepts of static, dynamic, stagnation, and total pressure are useful in a variety of flow problems. These ideas are used more fully in the remainder of the book. 3.6 Examples of Use of the Bernoulli Equation In this section we illustrate various additional applications of the Bernoulli equation. Between any two points, (1) and (2), on a streamline with steady, inviscid, incompressible flow, the Bernoulli equation can be applied in the form p 1 + 1 _ 2 ρV 1 2 + γ z 1 = p 2 + 1 _ 2 ρV 2 2 + γ z 2 (3.17) Obviously, if five of the six variables are known, the remaining one can be determined. In many instances it is necessary to introduce other equations, such as the conservation of mass. Such considerations will be discussed briefly in this section and in more detail in Chapter 5. 3.6.1 Free Jets One of the oldest equations in fluid mechanics deals with the flow of a liquid from a large reservoir. A modern version of this type of flow involves the flow of coffee from a coffee urn as indicated by the adjacent figure. The basic principles of this type of flow are shown in Fig. 3.11, where a jet of liquid of diameter d flows from the nozzle with velocity V. (A nozzle is a device shaped to accelerate a fluid.) Application of Eq. 3.17 between points (1) and (2) on the stream- line shown gives γh = 1 _ 2 ρV 2 We have used the facts that z 1 = h, z 2 = 0, the reservoir is large ( V 1 ≅ 0) and open to the atmo- sphere ( p 1 = 0 gage), and the fluid leaves as a free jet ( p 2 = 0). Thus, we obtain V = √ _ 2 γh _ ρ = √ _ 2 g h (3.18) which is the modern version of a result obtained in 1643 by Torricelli (1608–1647), an Italian physicist.
- Andrew L. Gerhart, John I. Hochstein, Philip M. Gerhart(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
In general, this equation is valid for both planar and nonpla- nar (three-dimensional) flows, provided it is applied along the streamline. We will provide many examples to illustrate the correct use of the Bernoulli equation and will show how a violation of the basic assumptions used in the derivation of this equation can lead to erroneous conclusions. The constant of integration in the Bernoulli equation can be evaluated if sufficient information about the flow is known at one location along the streamline. V4.2 Balancing ball Video V4.3 Flow past a biker Video EXAMPLE 4.2 The Bernoulli Equation Given Consider the flow of air around a bicyclist moving through still air with velocity V 0 , as is shown in Fig. E4.2. Find Determine the difference in the pressure between points (1) and (2). Solution In a coordinate fixed to the ground, the flow is unsteady as the bicyclist rides by. However, in a coordinate system fixed to the bike, it appears as though the air is flowing steadily toward the bicy- clist with speed V 0 . Since use of the Bernoulli equation is restricted to steady flows, we select the coordinate system fixed to the bike. If the assumptions of Bernoulli’s equation are valid (steady, incompressible, inviscid flow), Eq. 4.7 can be applied as follows along the streamline that passes through (1) and (2) p 1 + 1 _ 2 ρV 1 2 + γ z 1 = p 2 + 1 _ 2 ρV 2 2 + γ z 2 We consider (1) to be in the free stream so that V 1 = V 0 and (2) to be at the tip of the bicyclist’s nose and assume that z 1 = z 2 and V 2 = 0 (both of which, as is discussed in Section 4.5, are reasonable assump- tions). It follows that the pressure at (2) is greater than that at (1) by an amount p 2 − p 1 = 1 _ 2 ρV 1 2 = 1 _ 2 ρV 0 2 (Ans) Comments A similar result was obtained in Example 4.1 by integrating the pressure gradient, which was known because the velocity distribution along the streamline, V (s), was known. The Bernoulli equation is a general integration of F = ma.
eBook - PDF- Donald F. Elger, Barbara A. LeBret, Clayton T. Crowe, John A. Roberson(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
3. Applying Eq. (R1) and doing term-by-term analysis gives 108 CHAPTER 4 • THE BERNOULLI EQUATION AND PRESSURE VARIATION 4. The calculations are 4.6 The Bernoulli Equation along a Streamline This section presents the Bernoulli equation, which is one of the five most commonly used equations in fluid mechanics. The Bernoulli equation can be confusing to learn because you can find several different forms of this equation in the literature because the form of the Bernoulli equation depends on the assumptions that are used to derive it. This section introduces the form of the Bernoulli equation that applies to steady flow between two points on a streamline. Derivation of the Bernoulli Equation Select a particle on a streamline (Fig. 4.26). The position coordinate s gives the particle’s posi- tion. The unit vector u t is tangent to the streamline, and the unit vector u n is normal to the streamline. Assume steady flow so that the velocity of the particle depends on position only. That is, V = V(s). Assume that viscous forces on the particle can be neglected. Then, apply Euler’s equation (Eq. 4.15) to the particle in the u t direction: − s ( p + γz) = ρa t (4.17) Acceleration is given by Eq. (4.11). Because the flow is steady, V/ t = 0, and Eq. (4.11) gives a t = V V s + V t = V V s (4.18) Because p, z, and V in Eqs. (4.17) and (4.18) depend only on position s, the partial derivatives become ordinary derivatives (i.e., functions only of a single variable). Thus, write these deriva- tives as ordinary derivatives, and combine Eqs. (4.17) and (4.18) to give − d ds ( p + γz) = ρV dV ds = ρ d ds ( V 2 2 ) (4.19) Move all the terms to one side: d ds ( p + γz + ρ V 2 2 ) = 0 (4.20) FIGURE 4.26 Sketch used for the derivation of the Bernoulli equation. s u t u n V(s) Streamline 1 2 The Bernoulli Equation along a Streamline 109 When the derivative of an expression is zero, the expression is equal to a constant.
eBook - ePubFluid Mechanics
Analytical Methods
- Michel Ledoux, Abdelkhalak El Hami(Authors)
- 2017(Publication Date)
- Wiley-ISTE(Publisher)
B at the top of the jet, we obtain:[4.189]By applying the values given with the problem:[4.190]The following is immediately apparent:[4.191]This can be interpreted easily: the fluid in the reservoir has a potential volume energy of ρgH. This energy is transformed into kinetic energy in the jet. At the top of the jet, any kinetic energy is returned to its potential form. The “kinetic” and “gravitational” energies are exchanged, as the pressure energy is constant along the whole jet submerged in the air, which is at atmospheric pressure.2) We apply the Bernoulli on a current line between the point A of the free surface in the cave and the point B at the top of the jet.At the altitude hLav, the velocityV is such that:[4.192]Therefore:[4.193]The flow is therefore:[4.194]where is the cross-sectional area of the tube.3) Numerical application. h = H = 30 m:[4.195][4.196][4.197]NOTE.– This flow comes out as qV = 136 liters.mn−1 , which can quite understandably be considered a bit excessive (a household faucet has a flow of 10 liters.mn−1 ). This is an inherent problem involving the blind application of the Bernoulli theorem. Under real conditions, head losses must also be considered.EXAMPLE 4.16 (Injection problems).– Here we look at the different phases involved as a nurse performs the injection of a medication into a patient. Part A. – During the injection:A nurse must inject a patient with a quantity of medication equal to 8 cm3 . Despite its high price, this medication is made up mainly of distilled water, and as such has a density ρ = 1000 kg.m–3 . To do this, she inserts the needle of the syringe into the patient at the relevant location. We assume that inside the patient the pressure is equal to atmospheric pressure Papresent outside the patient. The internal component of the syringe is a cylinder of diameter D = 2 cm. The internal diameter of the needle where the liquid exits is equal to d = 0,5 mm
eBook - PDF- Richard W. Johnson(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
∂ + + + + ∂ ∂ φ θ λ θ μ θ θ θ θ φ u r u r r r u r cot cot sin sin div u + ∂ ∂ 1 r u sin θ φ θ (4.89) where div u = ∂ ∂ ( ) + ∂ ∂ + ∂ ∂ 1 1 1 2 2 r r r u r u r u r sin ( sin ) sin θ θ θ θ φ θ φ (4.90) 4.1.6 Bernoulli Equations The Bernoulli equations are mechanical energy equations that can be obtained from the energy equation (Panton, 1984) for an inviscid flow with a conservative body force, or by integrating the inviscid momentum equation (Euler equation) in the stream direction. They are first integrals of the equations of motion. The conservative body force is assumed to be given by Equation 4.5. Various forms of the Bernoulli equation are listed below, along with the assumptions under which each form is valid. Other forms are given in standard texts (Yih, 1969; Panton, 1984). In the following equations, q 2 = u · u is the square of the magni-tude of the total velocity. 1. Steady flow, valid along a streamline dp q gz C ρ ψ + + = ∫ 2 2 ( ) (4.91) where C ( ψ ) is a constant that can vary from one stream-line to another; if ω × u = 0 and the flow is homentropic, then, this equation is valid everywhere. 2. Unsteady, irrotational flow, ω = curl u = 0, u = grad Φ , ∂ ∂ + + + = ∫ Φ t dp q gz K t ρ 2 2 ( ) (4.92) where K ( t ) is a constant that can vary with time and Φ is the velocity potential; for steady flow, the constant is the same everywhere. 3. Steady flow with respect to a frame rotating with a con-stant angular velocity Ω about the z -direction, using polar cylindrical coordinates ( r , θ , z ) dp q gz r C ρ ψ + + -= ∫ 2 2 2 2 2 Ω ( ) (4.93) where C ( ψ ) is a constant that can vary from one stream-line to another. For a perfect gas dp p ρ γ γ ρ = -∫ 1 (4.94) while for an incompressible fluid, the integral in Equation 4.94 is p / ρ .
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
3 Basic Equations of Fluid MechanicsBefore we begin a discussion of the equations of fluid mechanics, it is important to examine types of flows and how they can be characterized. Closed-conduit flows are completely enclosed by restraining solid surfaces; examples are flow through a pipe and flow between parallel plates where there is no free surface. Open-channel flows have one surface exposed to atmospheric pressure; examples are flow in a river and flow in a spillway. In unbounded flows , the fluid is not in contact with any solid surface; examples are the jet that issues from a household faucet and the jet from a can of spray paint.After completing this chapter, you should be able to:- Discuss the behavior of flowing fluids in various situations;
- Recognize the unified mathematical approach to solving fluid flow problems;
- Derive the equations of fluid mechanics;
- Apply the continuity, momentum, energy and Bernoulli equations to problems.
3.1 KINEMATICS OF FLOW
In flow situations, solution of a problem often requires determination of a velocity. As we will see, however, velocity varies in the flow field. Moreover, the flow can be classified according to how the velocity varies. If the parameters of both the fluid and the flow are constant at any cross section normal to the flow, or if they are represented by average values over the cross section, the flow is said to be one-dimensional . Velocity distributions for one-dimensional flow are illustrated in Figure 3.1 . Although flow velocity may change from point to point, it is constant at each location.FIGURE 3.1 One-dimensional flow where velocity and pressure are uniform at any cross section.A flow is said to be two-dimensional if the fluid or the flow parameters have gradients in two directions. In Figure 3.2 a—flow in a pipe—the velocity at any cross section is parabolic. The velocity is thus a function of the radial coordinate; a gradient exists. In addition, a pressure gradient exists in the axial direction that maintains the flow. That is, a pressure difference from inlet to outlet is imposed on the fluid that causes flow to occur. The flow is one-directional, but because we have both a velocity and a pressure gradient, it is two-dimensional. Another example of two-dimensional flow is given in Figure 3.2 b. At the constant-area sections, the velocity is a function of one variable. At the convergent section, velocity is a function of two space variables. In addition, a pressure gradient exists that maintains the flow. Figure 3.2
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