Elementary Fluid Dynamics:The Bernoulli EquationBernoulliAlong a StreamlineBernoulliAlong a StreamlineIntegrate F=ma Along a StreamlineBernoulli EquationBernoulli EquationBernoulli Equation: Simple Case (V = 0)Hydraulic and Energy Grade Lines (neglecting losses for now)Bernoulli Equation: Simple Case (p = 0 or constant)Bernoulli Equation Application:Stagnation TubeBernoulli Equation Application:Stagnation TubeStagnation TubePitot TubesPitot TubeRelaxed Assumptions for Bernoulli EquationBernoulli Normal to the StreamlinesBernoulli Normal to the StreamlinesIntegrate F=ma Normal to the StreamlinesPressure Change Across StreamlinesEnd of pipeline?Nozzle Flow Rate: Find QSolution to Nozzle FlowSolution to Nozzle Flow (continued)Incorrect technique…Bernoulli Equation ApplicationsPing Pong BallSummaryJet ProblemJet SolutionWhat is the radius of curvature at the end of the pipe?Example: VenturiExample: VenturiExample VenturiMonroe L. Weber-Shirk School of Civil andEnvironmental EngineeringElementary Fluid Dynamics:The Bernoulli EquationCEE 331June 8, 2006BernoulliAlong a Streamline j izyx kˆsˆnˆpgakρρ−∇ = +Separate acceleration due to gravity. Coordinate system may be in any orientation!k is vertical, s is in direction of flow, n is normal.szpssdda gρρ∂−= +∂Component of g in s directionNote: No shear forces! Therefore flow must be frictionless. Steady state (no change in p wrt time)BernoulliAlong a StreamlinesdV Vadt s∂==∂spdzasdsργ∂−= +∂Can we eliminate the partial derivative?ppdp ds dnsn∂∂=+∂∂0 (n is constant along streamline, dn=0)dp dV dzVds ds dsργ−= +Write acceleration as derivative wrt sChain ruledsdt=dp ds p s∴=∂ ∂ dV ds V s=∂∂andVVs∂∂Integrate F=ma Along a Streamlinedp dV dzVds ds dsργ−= +Eliminate ds0dp VdV dzργ++=But density is a function of ________.pressureNow let’s integrate…0dpVdV g dzρ++=∫∫ ∫212pdpVgzCρ++=∫If density is constant…2'12ppVzCργ++=Bernoulli Equation¾ Assumptions needed for Bernoulli Equation¾ Eliminate the constant in the Bernoulli equation? _______________________________________¾ Bernoulli equation does not include¾ ___________________________¾ ___________________________Apply at two points along a streamline.Mechanical energy to thermal energyHeat transfer, Shaft Work³Frictionless³Steady³Constant density (incompressible)³Along a streamlineBernoulli EquationThe Bernoulli Equation is a statement of the conservation of ____________________Mechanical Energyp.e. k.e.212ppgzVCρ++=2"2ppVzCgγ++ =Pressure headpγ=z=pzγ+=22Vg=Elevation headVelocity headPiezometric head22pVzgγ++=Total headEnergy Grade LineHydraulic Grade LineBernoulli Equation: Simple Case (V = 0)z¾Reservoir (V = 0)¾Put one point on the surface, one point anywhere else12Pressure datumElevation datum2"2ppVzCgγ++ =1212ppzzγγ+= +We didn’t cross any streamlines so this analysis is okay!212pzzγ−=Same as we found using staticsHydraulic and Energy Grade Lines (neglecting losses for now)Mechanical energyMechanical Energy ConservedThe 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)?pγz22VgElevation datumzPressure datum? __________________Atmospheric pressurez22Vg2"2ppVzCgγ++ =TeamsBernoulli Equation: Simple Case (p = 0 or constant)¾What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________22112 21222pVp Vzzggγγ++ = ++()221212VgzzV=−+22121222VVzzgg+=+Free jetBernoulli Equation Application:Stagnation Tube¾ What happens when the water starts flowing in the channel?¾ Does the orientation of the tube matter? _______¾ How high does the water rise in the stagnation tube?¾ How do we choose the points on the streamline?2p"C2pVzgγ++ =Stagnation pointYes!Bernoulli Equation Application:Stagnation Tube2p"C2pVzgγ++ =2a1a2b1b¾1a-2a¾_______________¾1b-2a¾_______________ ¾1a-2b¾____________________________Same streamlineCrosses || streamlinesDoesn’t cross streamlines22112 21222pVp Vzzggγγ++ = ++z2122Vzg=212Vgz=V = f(∆p)V = f(z2)V = f(∆p)In all cases we don’t know p1Stagnation Tube¾Great for measuring __________________¾How could you measure Q?¾Could you use a stagnation tube in a pipeline?¾What problem might you encounter?¾How could you modify the stagnation tube to solve the problem?EGL (defined for a point)QVdA=⋅∫Pitot Tubes¾Used to measure air speed on airplanes¾Can connect a differential pressure transducer to directly measure V2/2g¾Can be used to measure the flow of water in pipelinesPoint measurement!Pitot TubeVV1=12z1= z2()122Vppρ=−Static pressure tapStagnation pressure tap022112 21222pVp Vzzggγγ++ = ++Connect two ports to differential pressure transducer. Make sure Pitot tube is completely filled with the fluid that is being measured.Solve for velocity as function of pressure differenceRelaxed Assumptions for Bernoulli Equation¾Frictionless (velocity not influenced by viscosity)¾Steady¾Constant density (incompressible)¾Along a streamlineSmall energy loss (accelerating flow, short distances)Or gradually varyingSmall changes in densityDon’t cross streamlinesBernoulliNormal to the StreamlinesSeparate acceleration due to gravity. Coordinate system may be in any orientation!ˆpgakρρ−∇ = +nzpnndda gρρ∂−= +∂Component of g in n direction kˆsˆnBernoulliNormal to the Streamlinesnpdzagndnρρ∂−= +∂Ris local radius of curvature2nVaR=ppdp ds dnsn∂∂=+∂∂0 (s is constant normal to streamline)dp dn p n∴=∂ ∂n is toward the center of the radius of curvature2dp V dzgdn R dnρρ−= +Integrate F=ma Normal to the Streamlines2dp V dzgdn R dnρρ−= +Multiply by dn2ndp Vdngdz CRρ++=⌠⌠⎮⎮⌡⌡∫Integrate2npVdngzCRρ++=⌠⎮⌡(If density is constant)2"nVpdn gz CRρρ++=⌠⎮⌡Pressure Change Across Streamlines2'1npVdn z CgRγ++=⌠⎮⌡2"nVpdn gz CRρρ++=⌠⎮⌡If you cross streamlines that are straight and parallel, then ___________ and the pressure is ____________.pgz Cρ+=hydrostaticnr1()Vr Cr=21"npCrdrgzCρρ−+=∫dn dr=−221"2nCprgzCρρ−+=As r decreases p ______________decreasesEnd of pipeline?End of pipeline?¾What must be happening when a horizontal pipe discharges to the atmosphere?2"nVpdn gz CRρρ++=⌠⎮⌡(assume straight streamlines)Try applying statics…Streamlines must be curved!Nozzle Flow Rate: Find QD1=30 cmD2=10 cmQ90 cm12313,0pp=23,0zz=1zh=zPressure datum____________Crossing streamlinesCoordinate systemhh=105