Chapter 3 Bernoulli Equation - University Of Iowa
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Chapter 31Chapter 3 Bernoulli Equation3.1 Flow Patterns: Streamlines, Pathlines, Streaklines1) A streamline, is a line that is everywhere tangent to the velocityvector at a given instant.Examples of streamlines around an airfoil (left) and a car (right)2) A pathline is the actual path traveled by a given fluid particle.An illustration of pathline (left) and an example of pathlines, motion of water induced by surface waves (right)3) A streakline is the locus of particles which have earlier passed through aparticular point.An illustration of streakline (left) and an example of streaklines, flow past a full-sized streamlined vehicle in the GM aerodynamics laboratory wind tunnel, and 18-ft by 34-ft test section facilility by a 4000-hp, 43-ft-diameter fan (right)
Chapter 3257:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Note:1. For steady flow, all 3 coincide.2. For unsteady flow,pattern changes with time, whereas pathlines andstreaklines are generated as the passage of timeStreamline:0 which upon expansion yields theBy definition we must haveequation of the streamlines for a given timewhere integration parameter. So if ( , , ) know, integrate with respect towith I.C. ( , , , ) at0 and then eliminate .forPathline:The path line is defined by integration of the relationship between velocityand displacement.Integrate , ,with respect to using I.C. ( ,,,) then eliminate .Streakline:To find the streakline, use the integrated result for the pathline retainingtime as a parameter. Now, find the integration constant which causes the path. Then eliminate .line to pass through ( , , ) for a sequence of time
Chapter 3357:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20083.2 Streamline CoordinatesEquations of fluid mechanics can be expressed in different coordinate systems, which are chosen for convenience, e.g., application of boundary conditions:Cartesian ( , , ) or orthogonal curvilinear (e.g., , , ) or non-orthogonal curvilinear. A natural coordinate system is streamline coordinates ( , , ℓ); however,difficult to use since solution to flow problem (V) must be known to solve forsteamlines.For streamline coordinates, since V is tangent to there is only one velocitycomponent.Vwhere,,,0 by definition.Figure 4.8 Streamline coordinate system for two-dimensional flow.The acceleration isVVVwhere,;VV
Chapter 3457:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008VVFigure 4.9 Relationship between the unit vector along the streamline, , and the radius ofcurvature of the streamline,Space incrementNormal to ̂Time increment
Chapter 3557:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008θ or,where, localin ̂ direction localindirection convectivedue to spatial gradient of Vi.e. convergence /divergence convectivedue to curvature of: centrifugal accerleration
Chapter 3657:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20083.3 Bernoulli EquationConsider the small fluid particle of size byin the plane of the figure andnormal to the figure as shown in the free-body diagram below. Forsteady flow, the components of Newton’s second law along the streamline andnormal directions can be written as following:1) Along a streamline where,VV sin2V1st order Taylor SeriesThus,VVV sin
Chapter 3757:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008sinsin change in speed due toand(i.e.along )2) Normal to a streamline where,V2V cos2V1st order Taylor SeriesThus,V22VV coscoscos streamline curvature is due toand(i.e.along )
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Chapter 38In a vector form:(Euler equation)orSteady flow, constant, equation02Steady flow, constant,equationFor curved streamlines( constant for static fluid) decreases in therection, i.e. towards the local center of curvature.di-It should be emphasized that the Bernoulli equation is restricted to the following: inviscid flowsteady flowincompressible flowflow along a streamlineNote that if in addition to the flow being inviscid it is also irrotational, i.e.V 0, the Bernoulli constant is same for all ,rotation of fluid vorticity as will be shown later.
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Chapter 393.4 Physical interpretation of Bernoulli equationIntegration of the equation of motion to give the Bernoulli equation actually corresponds to the work-energy principle often used in the study of dynamics.This principle results from a general integration of the equations of motion for anobject in a very similar to that done for the fluid particle. With certain assumptions, a statement of the work-energy principle may be written as follows:The work done on a particle by all forces acting on the particle is equal tothe change of the kinetic energy of the particle.The Bernoulli equation is a mathematical statement of this principle.In fact, an alternate method of deriving the Bernoulli equation is to use thefirst and second laws of thermodynamics (the energy and entropy equations), rather than Newton’s second law. With the approach restrictions, the generalenergy equation reduces to the Bernoulli equation.An alternate but equivalent form of the Bernoulli equation is2along a streamline.Pressure head:Velocity head:Elevation head:The Bernoulli equation states that the sum of the pressure head, the velocityhead, and the elevation head is constant along a streamline.
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Chapter 3103.5 Static, Stagnation, Dynamic, and Total Pressure12along a streamline.Static pressure:Dynamic pressure:Hydrostatic pressure:Stagnation points on bodies in flowing fluids.Stagnation pressure:(assuming elevation effects are negligible) whereand are the pressure and velocity of the fluid upstream of stagnationpoint. At stagnation point, fluid velocity becomes zero and all of the kinetic energy converts into a pressure rize.Total pressure:(along a streamline)
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008The Pitot-static tube (left) and typical Pitot-static tube designs (right).Typical pressure distribution along a Pitot-static tube.Chapter 311
Chapter 31257:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20083.6 Applications of Bernoulli Equation1) Stagnation Tube222,022Limited by length of tube and needfor free surface reference
Chapter 31357:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20082) Pitot Tube222where,0 and piezometric head2from manometer or pressure gageFor gas flow Δ Δ2Δ
Chapter 31457:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20083) Free JetsVertical flow from a tankApplication of Bernoulli equation between points (1) and (2) on the streamlineshown gives12Since,0,0,120,0, we have1222Bernoulli equation between points (1) and (5) gives2
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20084) Simplified form of the continuity equationSteady flow into and out of a tankObtained from the following intuitive arguments:Volume flow rate:Mass flow rate:Conservation of mass requiresFor incompressible flowor, we haveChapter 315
Chapteer 31657:020 Mechanics of FluidsFand Transport ProceessesProfessor Fred Stern FallF 20085) Voluume Rate of Flow (flowrate,(, dischargge)1. Crosss-sectionaal area orriented noormal to velocityvveector(simpple case whhere constannt: volume flux [m/s m2 m3/s]/constaant:Similarlyy the masss flux 2. Geneeral caseVV cosVAveerage veloccity:
Chapter 31757:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Example:At low velocities the flow through a long circular tube, i.e. pipe, has a parabolic velocity distribution (actually paraboloid of revolution).1 centerline velocitywhere,a) findandV2where,2,2and not ,21212
Chapter 31857:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20086) Flowrate measurementVarious flow meters are governed by the Bernoulli and continuity equations.Typical devices for measuring flowrate in pipes.Three commonly used types of flow meters are illustrated: the orifice meter, the nozzle meter, and the Venturi meter. The operation of each is based onthe same physical principles—an increase in velocity causes a decrease in pressure. The difference between them is a matter of cost, accuracy, and how closelytheir actual operation obeys the idealized flow assumptions.), steady, inviscid, and incomWe assume the flow is horizontal (pressible between points (1) and (2). The Bernoulli equation becomes:1212
Chapter 31957:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008If we assume the velocity profiles are uniform at sections (1) and (2), the continuity equation can be written as:is the small () flow area at section (2). Combination of thesewheretwo equations results in the following theoretical flowrate21 assumed vena contracta 0, i.e., no viscous effects. Otherwise,21where contraction coefficientA smooth, well-contoured nozzle (left) and a sharp corner (right)The velocity profile of the left nozzle is not uniform due to differences inelevation, but in generaland we can safely use the centerline velocity, ,as a reasonable “average velocity.”For the right nozzle with a sharp corner, will be less than . This phenomenon, called a vena contracta effect, is a result of the inability of the fluid toturn the sharp 90 corner.
57:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Chapter 320Figure 3.14 Typical flow patterns and contraction coefficientsThe vena contracta effect is a function of the geometry of the outlet. Sometypical configurations are shown in Fig. 3.14 along with typical values of the expe , where andare therimentally obtained contraction coefficient,areas of the jet a the vena contracta and the area of the hole, respectively.
Chapter 32157:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Other flow meters based on the Bernoulli equation are used to measureflowrates in open channels such as flumes and irrigation ditches. Two of thesedevices, the sluice gate and the sharp-crested weir, are discussed below underthe assumption of steady, inviscid, incompressible flow.Sluice gate geometryWe apply the Bernoulli and continuity equations between points on the free surfaces at (1) and (2) to give:1212and0:With the fact that21 In the limit of, then2:2
Chapter 32257:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2008Rectangular, sharp-crested weir geometryFor such devices the flowrate of liquid over the top of the weir plate is dependent on the weir height, , the width of the channel, , and the head, , ofthe water above the top of the weir. Between points (1) and (2) the pressure andgravitational fields cause the fluid to accelerate from velocity to velocity . At, while at (2) the pressure is essentially atmospheric,(1) the pressure is0. Across the curved streamlines directly above the top of the weir plate(section a–a), the pressure changes from atmospheric on the top surface to somemaximum value within the fluid stream and then to atmospheric again at the bottom surface.For now, we will take a very simple approach and assume that the weir flowis similar in many respects to an orifice-type flow with a free streamline. In thisinstance we would expect the average velocity across the top of the weir to beand the flow area for this rectangular weir to be proporproportional to 2tional to. Hence, it follows that22
Chapter 32357:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20083.7 Energy grade line (EGL) and hydraulic grade line (HGL)This part will be covered later at Chapter 5.3.8 Limitations of Bernoulli EquationAssumptions used in the derivation Bernoulli Equation:(1) Inviscid(2) Incompressible(3) Steady(4) Conservative body force1) Compressibility Effects:The Bernoulli equation can be modified for compressible flows. A simple,although specialized, case of compressible flow occurs when the temperature of aperfect gas remains constant along the streamline—isothermal flow. Thus, weconsider, where is constant (In general, , , and will vary). An equation similar to the Bernoulli equation can be obtained for isentropic flow of aperfect gas. For steady, inviscid, isothermal flow, Bernoulli equation becomes12The constant of integration is easily evaluated iflocation on the streamline. The result is2ln,2, andare known at some
Chapter 32457:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 20082) Unsteady Effects:The Bernoulli equation can be modified for unsteady flows. With the inclusion of the unsteady effect ( 0) the following is obtained:0 (along a streamline)For incompressible flow this can be easily integrated between points (1) and (2) togive(along a streamline)3) Rotational EffectsCare must be used in applying the Bernoulli equation across streamlines. Ifthe flow is “irrotational” (i.e., the fluid particles do not “spin” as they move), it isappropriate to use the Bernoulli equation across streamlines. However, if theflow is “rotational” (fluid particles “spin”), use of the Bernoulli equation is restricted to flow along a streamline.4) Other RestrictionsAnother restriction on the Bernoulli equation is that the flow is inviscid. TheBernoulli equation is actually a first integral of Newton's second law along astreamline. This general integration was possible because, in the absence of viscous effects, the fluid system considered was a conservative system. The totalenergy of the system remains constant. If viscous effects are important the system is nonconservative and energy losses occur. A more detailed analysis isneeded for these cases.The Bernoulli equation is not valid for flows that involve pumps or turbines.The final basic restriction on use of the Bernoulli equation is that there are nomechanical devices (pumps or turbines) in the system between the two pointsalong the streamline for which the equation is applied. These devices representsources or sinks of energy. Since the Bernoulli equation is actually one form ofthe energy equation, it must be altered to include pumps or turbines, if these arepresent.
Chapter 3 1 Chapter 3 Bernoulli Equation 3.1 Flow Patterns: Streamlines, Pathlines, Streaklines 1) A streamline ð k T, o is a line that is everywhere tangent to the velocity vector at a given instant. Examples of streamline
Chapter Outline 1. Fluid Flow Rate and the Continuity Equation 2. Commercially Available Pipe and Tubing 3. Recommended Velocity of Flow in Pipe and Tubing 4. Conservation of Energy –Bernoulli’s Equation 5. Interpretation of Bernoulli’s Equation 6. Restrictions on Bernoulli’s Equation 7. Applications of Bernoulli’s Equation 8 .
Chapter 5 Flow of an Incompressible Ideal Fluid Contents 5.1 Euler’s Equation. 5.2 Bernoulli’s Equation. 5.3 Bernoulli Equation for the One- Dimensional flow. 5.4 Application of Bernoulli’s Equation. 5.5 The Work-Energy Equation. 5.6 Euler’s Equation for Two- Dimensional Flow. 5.7 Bernoulli’s Equation for Two- Dimensional Flow Stream .
Derive the Bernoulli (energy) equation. Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow. Understand the use of hydraulic and energy grade lines. Apply Bernoulli Equation to solve fluid mechanics problems (e.g. flow measurement). K. ALASTAL 2 CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONS FLUID MECHANICS, IUG
Chapter 7 The Energy Equation 7.1 Energy, Work, and Power When matter has energy, the matter can be used to do work. A fluid can have several forms of . 7.5 Contrasting the Bernoulli Equation and the Energy Equation The Bernoulli equation and the energy equation are derived in different ways.
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
Chapter 5 Venturimeter & Orificemeter Applications of the Bernoulli Equation The Bernoulli equation can be applied to a great many situations not just the pipe flow we have been considering up to now. In the following sections we will see some examples of its application to flow measurement from tanks, within pipes as well as in open channels. 1.
MASS, BERNOULLI, AND ENERGY EQUATIONS This chapter deals with three equations commonly used in fluid mechanics: the mass, Bernoulli, and energy equations. The mass equa- tion is an expression of the conservation of mass principle. The Bernoulli equationis concerned with the conservation of kinetic, potential, and flow energies of a fluid stream and their conversion to each other in
Chapter 5 MASS, BERNOULLI AND ENERGY EQUATIONS Lecture slides by Hasan Hacışevki. . Bernoulli equation is also useful in the preliminary design stage. 3. Objectives Apply the conservation of mass equation to balance the incoming and outgoing flow rates in a flow system.
