Chapter 5 MASS, BERNOULLI AND ENERGY EQUATIONS

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Fluid Mechanics: Fundamentals and Applications, 2nd EditionYunus A. Cengel, John M. CimbalaMcGraw-Hill, 2010Chapter 5MASS, BERNOULLI ANDENERGY EQUATIONSLecture slides byHasan HacışevkiCopyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Wind turbine “farms” are being constructed all over the world toextract kinetic energy from the wind and convert it to electricalenergy. The mass, energy, momentum, and angular momentumbalances are utilized in the design of a wind turbine. TheBernoulli equation is also useful in the preliminary design stage.2

Objectives Apply the conservation of mass equation tobalance the incoming and outgoing flow rates ina flow system. Recognize various forms of mechanical energy,and work with energy conversion efficiencies. Understand the use and limitations of theBernoulli equation, and apply it to solve avariety of fluid flow problems. Work with the energy equation expressed interms of heads, and use it to determine turbinepower output and pumping power requirements.3

5–1 INTRODUCTIONYou are already familiar withnumerous conservation lawssuch as the laws ofconservation of mass,conservation of energy, andconservation of momentum.Historically, the conservationlaws are first applied to a fixedquantity of matter called aclosed system or just a system,and then extended to regionsin space called controlvolumes.The conservation relations arealso called balance equationssince any conserved quantitymust balance during a process.4

Conservation of MassThe conservation of mass relation for a closed system undergoing achange is expressed as msys constant or dmsys/dt 0, which is thestatement that the mass of the system remains constant during aprocess.Mass balance for a control volume (CV) in rate form:the total rates of mass flow intoand out of the control volumethe rate of change of mass within thecontrol volume boundaries.Continuity equation: In fluid mechanics, the conservation ofmass relation written for a differential control volume is usuallycalled the continuity equation.5

The Linear Momentum EquationLinear momentum: The product of the mass and the velocity of abody is called the linear momentum or just the momentum of thebody.The momentum of a rigid body of mass m moving with a velocityV is mV.Newton’s second law: The acceleration of a body is proportionalto the net force acting on it and is inversely proportional to itsmass, and that the rate of change of the momentum of a body isequal to the net force acting on the body.Conservation of momentum principle: The momentum of asystem remains constant only when the net force acting on it iszero, and thus the momentum of such systems is conserved.Linear momentum equation: In fluid mechanics, Newton’ssecond law is usually referred to as the linear momentumequation.6

Conservation of EnergyThe conservation of energy principle (the energy balance): Thenet energy transfer to or from a system during a process be equal tothe change in the energy content of the system.Energy can be transferred to or from a closed system by heat or work.Control volumes also involve energy transfer via mass flow.the total rates of energy transfer intoand out of the control volumethe rate of change of energywithin the control volume boundariesIn fluid mechanics, we usually limit our consideration tomechanical forms of energy only.7

5–2 CONSERVATION OF MASSConservation of mass: Mass, like energy, is a conserved property,and it cannot be created or destroyed during a process.Closed systems: The mass of the system remain constant duringa process.Control volumes: Mass can cross the boundaries, and so we mustkeep track of the amount of mass entering and leaving the controlvolume.Mass is conserved even during chemical reactions.Mass m and energy E can be converted to each other:c is the speed of light in a vacuum, c 2.9979 108 m/sThe mass change due to energy change is negligible.8

Mass and Volume Flow RatesMass flow rate: The amount of mass flowingthrough a cross section per unit time.The differential mass flow ratePoint functions have exact differentialsPath functions have inexact differentialsThe normal velocity Vn for asurface is the component ofvelocity perpendicular to the9surface.

Average velocityMass flow rateThe average velocity Vavg isdefined as the average speedthrough a cross section.Volume flow rateThe volume flow rate is thevolume of fluid flowing througha cross section per unit time.10

Conservation of Mass PrincipleThe conservation of mass principle for a control volume: The net mass transferto or from a control volume during a time interval t is equal to the net change(increase or decrease) in the total mass within the control volume during t.the total rates of massflow into and out of thecontrol volumethe rate of change of masswithin the control volumeboundaries.Mass balance is applicable toany control volume undergoingany kind of process.Conservation of mass principlefor an ordinary bathtub.11

The differential control volumedV and the differential controlsurface dA used in thederivation of the conservation ofmass relation.12

The time rate of change of mass within the controlvolume plus the net mass flow rate through the controlsurface is equal to zero.The conservationof mass equationis obtained byreplacing B in theReynoldstransport theoremby mass m, and bby 1 (m per unitmass m/m 1).A control surface shouldalways be selected normal tothe flow at all locations whereit crosses the fluid flow toavoid complications, eventhough the result is the same.13

Moving or Deforming Control Volumes14

Mass Balance for Steady-Flow ProcessesDuring a steady-flow process, the total amount of mass contained within acontrol volume does not change with time (mCV constant).Then the conservation of mass principle requires that the total amount of massentering a control volume equal the total amount of mass leaving it.For steady-flow processes, we areinterested in the amount of mass flowing perunit time, that is, the mass flow rate.Multiple inletsand exitsSinglestreamMany engineering devices such as nozzles,diffusers, turbines, compressors, andpumps involve a single stream (only oneinlet and one outlet).Conservation of mass principle for a twoinlet–one-outlet steady-flow system.15

Special Case: Incompressible FlowThe conservation of mass relations can be simplified even further whenthe fluid is incompressible, which is usually the case for flow (single stream)There is no such thing as a “conservation ofvolume” principle.However, for steady flow of liquids, the volume flowrates, as well as the mass flow rates, remainconstant since liquids are essentially incompressiblesubstances.During a steady-flow process, volumeflow rates are not necessarily conservedalthough mass flow rates are.16

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5–3 MECHANICAL ENERGY AND EFFICIENCYMechanical energy: The form of energy that can be converted tomechanical work completely and directly by an ideal mechanicaldevice such as an ideal turbine.Mechanical energy of a flowing fluid per unit mass:Flow energy kinetic energy potential energyMechanical energy change: The mechanical energy of a fluid does not change during flow ifits pressure, density, velocity, and elevation remain constant. In the absence of any irreversible losses, the mechanical energychange represents the mechanical work supplied to the fluid (if emech 0) or extracted from the fluid (if emech 0).20

Mechanical energy is a useful conceptfor flows that do not involve significantheat transfer or energy conversion, suchas the flow of gasoline from anunderground tank into a car.21

Mechanical energy is illustrated by an ideal hydraulic turbine coupled with anideal generator. In the absence of irreversible losses, the maximum producedpower is proportional to (a) the change in water surface elevation from theupstream to the downstream reservoir or (b) (close-up view) the drop in waterpressure from just upstream to just downstream of the turbine.22

The available mechanical energy of water at thebottom of a container is equal to the avaiablemechanical energy at any depth including the freesurface of the container.23

Shaft work: The transfer of mechanical energy is usually accomplished by arotating shaft, and thus mechanical work is often referred to as shaft work.A pump or a fan receives shaft work (usually from an electric motor) and transfersit to the fluid as mechanical energy (less frictional losses).A turbine converts the mechanical energy of a fluid to shaft work.Mechanical efficiencyof a device or processThe effectiveness of the conversion process between the mechanical worksupplied or extracted and the mechanical energy of the fluid is expressed by thepump efficiency and turbine efficiency,24

The mechanical efficiencyof a fan is the ratio of thekinetic energy of air at thefan exit to the mechanicalpower input.25

l efficiencyTurbine-Generator overall efficiency:The overall efficiency of a turbine–generator is the product of theefficiency of the turbine and theefficiency of the generator, andrepresents the fraction of themechanical energy of the fluidconverted to electric energy.26

The efficiencies just defined range between 0 and 100%.0% corresponds to the conversion of the entiremechanical or electric energy input to thermal energy, andthe device in this case functions like a resistance heater.100% corresponds to the case of perfect conversion withno friction or other irreversibilities, and thus no conversionof mechanical or electric energy to thermal energy (nolosses).For systems that involve only mechanicalforms of energy and its transfer as shaftwork, the conservation of energy isEmech, loss : The conversion of mechanicalenergy to thermal energy due toirreversibilities such as friction.Many fluid flow problems involvemechanical forms of energy only, andsuch problems are conveniently solvedby using a mechanical energy balance.27

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5–4 THE BERNOULLI EQUATIONBernoulli equation: An approximate relation between pressure,velocity, and elevation, and is valid in regions of steady,incompressible flow where net frictional forces are negligible.Despite its simplicity, it has proven to be a very powerful tool in fluidmechanics.The Bernoulli approximation is typically useful in flow regions outsideof boundary layers and wakes, where the fluid motion is governed bythe combined effects of pressure and gravity forces.The Bernoulli equation is anapproximate equation that is validonly in inviscid regions of flowwhere net viscous forces arenegligibly small compared toinertial, gravitational, or pressureforces. Such regions occuroutside of boundary layers andwakes.32

Acceleration of a Fluid ParticleIn two-dimensional flow, the acceleration can be decomposed into twocomponents:streamwise acceleration as along the streamline andnormal acceleration an in the direction normal to the streamline, which isgiven as an V2/R.Streamwise acceleration is due to a change in speed along a streamline,and normal acceleration is due to a change in direction.For particles that move along a straight path, an 0 since the radius ofcurvature is infinity and thus there is no change in direction. The Bernoulliequation results from a force balance along a streamline.Acceleration in steadyflow is due to the changeof velocity with position.During steady flow, a fluid may notaccelerate in time at a fixed point,but it may accelerate in space.33

Derivation of the Bernoulli EquationSteady flow:The forces acting on a fluidparticle along a streamline.The sum of the kinetic, potential, andflow energies of a fluid particle isconstant along a streamline duringsteady flow when compressibility andfrictional effects are negligible.Steady, incompressible flow:BernoulliequationThe Bernoulli equation between anytwo points on the same streamline:34

The incompressible Bernoulli equation isderived assuming incompressible flow,and thus it should not be used for flowswith significant compressibility effects.35

The Bernoulli equationstates that the sum of thekinetic, potential, and flowenergies of a fluid particle isconstant along a streamlineduring steady flow. The Bernoulli equation can be viewed as the“conservation of mechanical energy principle.” This is equivalent to the general conservationof energy principle for systems that do notinvolve any conversion of mechanical energyand thermal energy to each other, and thusthe mechanical energy and thermal energy areconserved separately. The Bernoulli equation states that duringsteady, incompressible flow with negligiblefriction, the various forms of mechanicalenergy are converted to each other, but theirsum remains constant. There is no dissipation of mechanical energyduring such flows since there is no friction thatconverts mechanical energy to sensiblethermal (internal) energy. The Bernoulli equation is commonly used inpractice since a variety of practical fluid flowproblems can be analyzed to reasonableaccuracy with it.36

Force Balance across StreamlinesForce balance in the direction n normal to the streamline yields the followingrelation applicable across the streamlines for steady, incompressible flow:For flow along a straight line, R andthis equation reduces to P/ρ gz constantor P ρgz constant, which is anexpression for the variation of hydrostaticpressure with vertical distance for astationary fluid body.Pressure decreases towards thecenter of curvature whenstreamlines are curved (a), butthe variation of pressure withelevation in steady,incompressible flow along astraight line (b) is the same asthat in stationary fluid.37

Unsteady, Compressible FlowThe Bernoulli equation for unsteady, compressible flow:38

Static, Dynamic, and Stagnation PressuresThe kinetic and potential energies of the fluid can be converted to flowenergy (and vice versa) during flow, causing the pressure to change.Multiplying the Bernoulli equation by the density givesP is the static pressure: It does not incorporate any dynamic effects; itrepresents the actual thermodynamic pressure of the fluid. This is the sameas the pressure used in thermodynamics and property tables.ρV2/2 is the dynamic pressure: It represents the pressure rise when thefluid in motion is brought to a stop isentropically.ρgz is the hydrostatic pressure: It is not pressure in a real sense since itsvalue depends on the reference level selected; it accounts for the elevationeffects, i.e., fluid weight on pressure. (Be careful of the sign—unlikehydrostatic pressure ρgh which increases with fluid depth h, the hydrostaticpressure term ρgz decreases with fluid depth.)Total pressure: The sum of the static, dynamic, andhydrostatic pressures. Therefore, the Bernoulli equationstates that the total pressure along a streamline is constant.39

Stagnation pressure: The sum of the static and dynamic pressures. It representsthe pressure at a point where the fluid is brought to a complete stop isentropically.Close-up of a Pitot-static probe,showing the stagnation pressure holeand two of the five static circumferentialpressure holes.The static, dynamic, andstagnation pressures measuredusing piezometer tubes.40

Careless drilling ofthe static pressuretap may result in anerroneous readingof the staticpressure head.Streaklines produced bycolored fluid introducedupstream of an airfoil; sincethe flow is steady, thestreaklines are the same asstreamlines and pathlines.The stagnation streamlineis marked.41

Limitations on the Use of the Bernoulli Equation1. Steady flow The Bernoulli equation is applicable to steady flow.2. Frictionless flow Every flow involves some friction, no matter how small,and frictional effects may or may not be negligible.3. No shaft work The Bernoulli equation is not applicable in a flow section thatinvolves a pump, turbine, fan, or any other machine or impeller since suchdevices destroy the streamlines and carry out energy interactions with thefluid particles. When these devices exist, the energy equation should beused instead.4. Incompressible flow Density is taken constant in the derivation of theBernoulli equation. The flow is incompressible for liquids and also by gasesat Mach numbers less than about 0.3.5. No heat transfer The density of a gas is inversely proportional totemperature, and thus the Bernoulli equation should not be used for flowsections that involve significant temperature change such as heating orcooling sections.6. Flow along a streamline Strictly speaking, the Bernoulli equation isapplicable along a streamline. However, when a region of the flow isirrotational and there is negligibly small vorticity in the flow field, theBernoulli equation becomes applicable across streamlines as well.42

Frictional effects, heat transfer, and componentsthat disturb the streamlined structure of flow makethe Bernoulli equation invalid. It should not be usedin any of the flows shown here.When the flow is irrotational, the Bernoulli equation becomes applicablebetween any two points along the flow (not just on the same streamline).43

Hydraulic Grade Line (HGL) and Energy Grade Line (EGL)It is often convenient to represent the level of mechanical energy graphically usingheights to facilitate visualization of the various terms of the Bernoulli equation.Dividing each term of the Bernoulli equation by g givesP/ρg is the pressure head; it represents the height of a fluid columnthat produces the static pressure P.V2/2g is the velocity head; it represents the elevation needed for afluid to reach the velocity V during frictionless free fall.z is the elevation head; it represents the potential energy of the fluid.An alternative form of theBernoulli equation is expressedin terms of heads as: The sumof the pressure, velocity, andelevation heads is constantalong a streamline.44

Hydraulic grade line (HGL), P/ρg z The line that represents the sum ofthe static pressure and the elevation heads.Energy grade line (EGL), P/ρg V2/2g z The line that represents thetotal head of the fluid.Dynamic head, V2/2g The difference between the heights of EGL and HGL.The hydraulicgrade line (HGL)and the energygrade line (EGL)for free dischargefrom a reservoirthrough ahorizontal pipewith a diffuser.45

Notes on HGL and EGL For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide withthe free surface of the liquid.The EGL is always a distance V2/2g above the HGL. These two curves approacheach other as the velocity decreases, and they diverge as the velocity increases.In an idealized Bernoulli-type flow, EGL is horizontal and its height remainsconstant.For open-channel flow, the HGL coincides with the free surface of the liquid, andthe EGL is a distance V2/2g above the free surface.At a pipe exit, the pressure head is zero (atmospheric pressure) and thus theHGL coincides with the pipe outlet.The mechanical energy loss due to frictional effects (conversion to thermalenergy) causes the EGL and HGL to slope downward in the direction of flow. Theslope is a measure of the head loss in the pipe. A component, such as a valve,that generates significant frictional effects causes a sudden drop in both EGL andHGL at that location.A steep jump/drop occurs in EGL and HGL whenever mechanical energy isadded or removed to or from the fluid (pump, turbine).The (gage) pressure of a fluid is zero at locations where the HGL intersects thefluid. The pressure in a flow section that lies above the HGL is negative, and the46pressure in a section that lies below the HGL is positive.

In an idealized Bernoulli-type flow,EGL is horizontal and its heightremains constant. But this is notthe case for HGL when the flowvelocity varies along the flow.A steep jump occurs in EGL a

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.

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