6. Flow Of Fluid And Bernoulli’s Equation

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6. Flow of Fluid and Bernoulli’s EquationChapter Objectives Define various types of flow, volume flow rate,weight flow rate, and mass flow rate and their units.Define steady flow and the principle of continuity.Write the continuity equation, and use it to relate thevolume flow rate, area, and velocity of flow betweentwo points in a fluid flow system.Describe five types of commercially available pipeand tubing: steel pipe, ductile iron pipe, steel tubing,copper tubing, and plastic pipe and tubing.Specify the desired size of pipe or tubing for carryinga given flow rate of fluid at a specified velocity.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationLaminar and Turbulent flow As the water flows from a faucet at a very lowvelocity, the flow appears to be smooth andsteady. The stream has a fairly uniformdiameter and there is little or no evidence ofmixing of the various parts of the stream. This iscalled laminar flow. When the faucet is nearly fully open, the waterhas a rather high velocity. The elements of fluidappear to be mixing chaotically within thestream. This is a general description ofturbulent flow.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationChapter Outline1. Fluid Flow Rate and the Continuity Equation2. Commercially Available Pipe and Tubing3. Recommended Velocity of Flow in Pipe andTubing4. Conservation of Energy – Bernoulli’s Equation5. Interpretation of Bernoulli’s Equation6. Restrictions on Bernoulli’s Equation7. Applications of Bernoulli’s Equation8. Torricelli’s Theorem9. Flow Due to a Falling Headã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation The quantity of fluid flowing in a system per unit timecan be expressed by the following three differentterms: Q The volume flow rate is the volume of fluid flowingpast a section per unit time. W The weight flow rate is the weight of fluid flowingpast a section per unit time. M The mass flow rate is the mass of fluid flowing pasta section per unit time.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation The most fundamental of these three terms is thevolume flow rate Q, which is calculated from where A is the area of the section and ν is theaverage velocity of flow. The units of Q can bederived as follows, using SI units for illustration:ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation The weight flow rate W is related to Q by where γ is the specific weight of the fluid. The units ofW are then The mass flow rate M is related to Q byã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation The units of M are then Table 6.1 shows the flow rates. Useful conversions areã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equationã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation Table 6.2 shows the typical volume flow rates.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.1Convert a flow rate of 30 gal/min to ft3/s.The flow rate isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.2Convert a flow rate of 600 L/min to m3/s.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.3Convert a flow rate of 600 L/min to m3/s.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation The method of calculating the velocity of flow of a fluidin a closed pipe system depends on the principle ofcontinuity. Fig 6.1 shows the portion of a fluid distribution systemshowing variations in velocity, pressure, andelevation. This can be expressed in terms of the mass flow rateasã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equationã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation As M ρAv, we have Equation (6–4) is a mathematical statement of theprinciple of continuity and is called the continuityequation. It is used to relate the fluid density, flow area, andvelocity of flow at two sections of the system in whichthere is steady flow. It is valid for all fluids, whether gas or liquid.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.1 Fluid Flow Rate and the Continuity Equation If the fluid in the pipe in Fig. 6.1 is a liquid that can beconsidered incompressible, then the terms ρ1 and ρ2is the same. Since Q Av, Equation (6–5) is the continuity equation as applied toliquids; it states that for steady flow the volume flowrate is the same at any section.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.4In Fig. 6.1 the inside diameters of the pipe at sections 1and 2 are 50 mm and 100 mm, respectively. Water at isflowing with an average velocity of 8 m/s at section 1.Calculate the following:(a) Velocity at section 2(b) Volume flow rate(c) Weight flow rate(d) Mass flow rateã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.4(a) Velocity at section 2.From Eq. (6–5) we haveã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.4Then the velocity at section 2 isNotice that for steady flow of a liquid, as the flow areaincreases, the velocity decreases.This is independent of pressure and elevation.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.4(b) Volume flow rate Q.From Table 6.1, Q vA. Because of the principle ofcontinuity we could use the conditions either at section 1or at section 2 to calculate Q. At section 1 we haveã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.4(c) Weight flow rate W.From Table 6.1, W γQ. At 70 C, the specific weight ofwater is 9.59kN/m3. Then the weight flow rate isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.4(d) Mass flow rate M.From Table 6.1, M ρQ. At the density of water is978 kg/m3. Then the mass flow rate isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.5According to the continuity equation for gases, Eq. (6–4), we haveThen, we can calculate the area of the two sections andsolve for ρ2ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.5(a)Then, the density of the air in the round section is(b) The weight flow rate can be found at section 1 from .Then, the weight flow rate isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2 Commercially Available Pipe and Tubing The nominal sizes for commercially available pipe stillrefer to an “inch” size even though the transition to theSI system is an international trend. For many applications, codes and standards must befollowed as established by governmental agencies ororganizations such as the following:ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.1 Steel Pipe General-purpose pipe lines are often constructed ofsteel pipe. Standard pipe sizes are designated by the nominalsize and schedule number. Schedule numbers are related to the permissibleoperating pressure of the pipe and to the allowablestress of the steel in the pipe.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.1 Steel PipeNominal Pipe Sizes in Metric Units Because of the long experience with manufacturingstandard pipe according to the standard schedulenumbers, they continue to be used often even whenthe piping system is specified in metric units. The following set of equivalents has been establishedby the International Standards Organization (ISO).ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.1 Steel Pipeã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.2 Steel Tubing Standard steel tubing is used in fluid power systems,condensers, heat exchangers, engine fuel systems,and industrial fluid processing systems. Sizes are designated by outside diameter and wallthickness.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.3 Copper Tubing There are six types of copper tubing offered, and thechoice of which to use depends on the application,considering the environment, fluid pressure, and fluidproperties. Copper tubing is available in either a soft, annealedcondition or hard drawn. Drawn tubing is stiffer and stronger, maintains astraight form, and can carry higher pressures. Annealed tubing is easier to form into coils and otherspecial shapes.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.3 Ductile Iron Pipe Water, gas, and sewage lines are often made ofductile iron pipe because of its strength, ductility, andrelative ease of handling. It has replaced cast iron in many applications.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.3 Plastic Pipe and Tubing Plastic pipe and tubing are being used in a widevariety of applications where their light weight, ease ofinstallation, corrosion and chemical resistance, andvery good flow characteristics present advantages.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.2.4 Hydraulic Hose Hose materials include butyl rubber, synthetic rubber,silicone rubber, thermoplastic elastomers, and nylon. Braided reinforcement may be made from steel wire,Kevlar, polyester, and fabric. Industrial applications include steam, compressed air,chemical transfer, coolants, heaters, fuel transfer,lubricants, refrigerants, paper stock, power steeringfluids, propane, water, foods, and beverages.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.3 Recommended Velocity of Flow in Pipe and Tubing Many factors affect the selection of a satisfactoryvelocity of flow in fluid systems. Some of the important ones are the type of fluid, thelength of the flow system, the type of pipe or tube, thepressure drop that can be tolerated, the devices (suchas pumps, valves, etc.) that may be connected to thepipe or tube, the temperature, the pressure, and thenoise.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.3 Recommended Velocity of Flow in Pipe and Tubing The resulting flow velocities from the recommendedpipe sizes in Fig. 6.2 are generally lower for thesmaller pipes and higher for the larger pipes, asshown for the following data.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.3 Recommended Velocity of Flow in Pipe and TubingRecommended Flow Velocities for SpecializedSystemsã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.3 Recommended Velocity of Flow in Pipe and Tubing For example, recommended flow velocities for fluidpower systems are as follows:ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.3 Recommended Velocity of Flow in Pipe and Tubing The suction line delivers the hydraulic fluid from thereservoir to the intake port of the pump. A discharge line carries the high-pressure fluid fromthe pump outlet to working components such asactuators or fluid motors. A return line carries fluid from actuators, pressurerelief valves, or fluid motors back to the reservoir.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.6Determine the maximum allowable volume flow rate inL/min that can be carried through a standard steel tubewith an outside diameter of 1.25in and a 0.065 in wallthickness if the maximum velocity is to be 3.0 m/s.Using the definition of volume flow rate, we haveã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.7Because Q and v are known, the required area can befound fromFirst, we must convert the volume flow rate to the unitsof m3/s:ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.7This must be interpreted as the minimum allowable areabecause any smaller area would produce a velocityhigher than 6.0 m/s. Therefore, we must look inAppendix F for a standard pipe with a flow area justlarger than 8.88 x 10-3 m2. A standard 5-in Schedule 40steel pipe, with a flow area of 1.291 x 10-2m2 is required.The actual velocity of flow when this pipe carries 0.0533m3/s of water isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.7If the next-smaller pipe (a 4-in Schedule 40 pipe) isused, the velocity isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.8Entering Fig. 6.2 at Q 400 gal/min, we select thefollowing:The actual average velocity of flow in each pipe isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.8 - CommentsAlthough these pipe sizes and velocities should beacceptable in normal service, there are situationswhere lower velocities are desirable to limit energylosses in the system. Compute the velocities resultingfrom selecting the next-larger standard Schedule 40 pipesize for both the suction and discharge lines:ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s EquationExample 6.8 - CommentsThe actual average velocity of flow in each pipe isIf the pump connections were the 4-in and 3-in sizesfrom the initial selection, a gradual reducer and gradualenlargement could be designed to connect these pipesto the pump.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation In physics you learned that energy can be neithercreated nor destroyed, but it can be transformed fromone form into another. This is a statement of the law of conservation ofenergy. Fig 6.3 shows the element of a fluid in a pipe.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation There are three forms of energy that are alwaysconsidered when analyzing a pipe flow problem.1. Potential Energy. Due to its elevation, the potentialenergy of the element relative to some referencelevel iswhere w is the weight of the element.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation2. Kinetic Energy. Due to its velocity, the kinetic energyof the element is3. Flow Energy. Sometimes called pressure energy orflow work, this represents the amount of worknecessary to move the element of fluid across acertain section against the pressure p. Flow energyis abbreviated FE and is calculated fromã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation Equation (6–8) can be derived as follows.The work done iswhere V is the volume of the element. The weight ofthe element w iswhere γ is the specific weight of the fluid. Then, thevolume of the element isã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation And we have Eq. (6-8) Fig 6.4 shows the flow energy.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation The total amount of energy of these three formspossessed by the element of fluid is the sum E, Each of these terms is expressed in units of energy,which are Newton-meters (Nm) in the SI unit systemand foot-pounds (ft-lb) in the U.S. CustomarySystem.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation Fig 6.5 shows the fluid elements used in Bernoulli’sequation.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation At section 1 and 2, the total energy is If no energy is added to the fluid or lost betweensections 1 and 2, then the principle of conservationof energy requires thatã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.4 Conservation of Energy – Bernoulli’s Equation The weight of the element w is common to all termsand can be divided out.The equation then becomes This is referred to as Bernoulli’s equation.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.5 Interpretation of Bernoulli’s Equation Each term in Bernoulli’s equation, Eq. (6–9), resultedfrom dividing an expression for energy by the weightof an element of the fluid.Each term in Bernoulli’s equation is one form ofthe energy possessed by the fluid per unitweight of fluid flowing in the system.The units for each term are “energy per unit weight.”In the SI system the units are Nm/N and in the U.S.Customary System the units are lb.ft/lb.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.5 Interpretation of Bernoulli’s Equation Specifically, Fig 6.6 shows the pressure head, elevation head,velocity head, and total head.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.5 Interpretation of Bernoulli’s Equationã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.5 Interpretation of Bernoulli’s Equation In Fig. 6.6 you can see that the velocity head atsection 2 will be less than that at section 1. This canbe shown by the continuity equation, In summary,Bernoulli’s equation accounts for the changes inelevation head, pressure head, and velocity headbetween two points in a fluid flow system. It isassumed that there are no energy losses oradditions between the two points, so the totalhead remains constant.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.6 Restriction on Bernoulli’s Equation Although Bernoulli’s equation is applicable to a largenumber of practical problems, there are severallimitations that must be understood to apply itproperly.1. It is valid only for incompressible fluids because thespecific weight of the fluid is assumed to be thesame at the two sections of interest.2. There can be no mechanical devices between thetwo sections of interest that would add energy to orremove energy from the system, because theequation states that the total energy in the fluid isconstant.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.6 Restriction on Bernoulli’s Equation3. There can be no heat transferred into or out of thefluid.4. There can be no energy lost due to friction. In reality no system satisfies all these restrictions. However, there are many systems for which only anegligible error will result when Bernoulli’s equationis used. Also, the use of this equation may allow a fastestimate of a result when that is all that is required.ã2005 Pearson Education South Asia Pte Ltd

6. Flow of Fluid and Bernoulli’s Equation6.7 Applications of Bernou

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 .

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