Elementary Fluid Dynamics: The Bernoulli EquationBernoulli Along a StreamlineSlide 3Integrate F=ma Along a StreamlineBernoulli EquationSlide 6Bernoulli 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 TubeSlide 11Stagnation TubePitot TubesPitot TubeRelaxed Assumptions for Bernoulli EquationBernoulli Normal to the StreamlinesSlide 17Integrate 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: VenturiSlide 32Example VenturiMonroe L. Weber-Shirk School of Civil and Environmental EngineeringElementary Fluid Dynamics:The Bernoulli EquationElementary Fluid Dynamics:The Bernoulli EquationCEE 331January 14, 2019BernoulliAlong a Streamlinejizyxkˆsˆnˆp gr r- � = +a kSeparate acceleration due to gravity. Coordinate system may be in any orientation!k is vertical, s is in direction of flow, n is normal.szpssdda gr r�- = +�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�= =�sp dzas dsr g�- = +�p pdp ds dns n� �= +� �0 (n is constant along streamline, dn=0)dp dV dzVds ds dsr g- = +Write acceleration as derivative wrt sChain ruledsdt=dp ds p s\ =� �dV ds V s=� �andCan we eliminate the partial derivative?VVs��Integrate F=ma Along a Streamline0dp VdV dzr g+ + =0dpVdV g dzr+ + =� � �212pdpV gz Cr+ + =�2'12pp V z Cr g+ + =If density is constant…But density is a function of ________.pressureEliminate dsNow let’s integrate…dp dV dzVds ds dsr g- = +Assumptions needed for Bernoulli EquationEliminate the constant in the Bernoulli equation? _______________________________________Bernoulli equation does not include ___________________________ ___________________________Bernoulli EquationApply at two points along a streamline.Mechanical energy to thermal energyHeat transfer, Shaft WorkFrictionlessSteadyConstant density (incompressible)Along a streamlineBernoulli EquationThe Bernoulli Equation is a statement of the conservation of ____________________Mechanical Energyp.e. k.e.212ppgz V Cr+ + =2"2pp Vz Cgg+ + =Pressure headpg=z =pzg+ =22Vg=Elevation headVelocity headPiezometric head22p Vzgg+ + =Total headEnergy Grade LineHydraulic Grade LineBernoulli Equation: Simple Case (V = 0)Reservoir (V = 0)Put one point on the surface, one point anywhere elsezElevation datum21 2pz zg- =Pressure datum12Same as we found using statics 2"2pp Vz Cgg+ + =1 21 2p pz zg g+ = +We didn’t cross any streamlines so this analysis is okay!Mechanical Energy ConservedHydraulic and Energy Grade Lines (neglecting losses for now)The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)?pgz22VgElevation datumzPressure datum? __________________Atmospheric pressureTeamsz22Vg2"2pp Vz Cgg+ + =Mechanical energyBernoulli Equation: Simple Case (p = 0 or constant)What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________2 21 1 2 21 22 2p V p Vz zg gg g+ + = + +( )22 1 2 12V g z z V= - +2 21 21 22 2V Vz zg g+ = +Free jetBernoulli Equation Application:Stagnation TubeWhat happens when the water starts flowing in the channel?Does the orientation of the tube matter? _______How high does the water rise in the stagnation tube?How do we choose the points on the streamline?2p"C2p Vzgg+ + =Stagnation pointYes!2p"C2p Vzgg+ + =2a1a2b1bBernoulli Equation Application:Stagnation Tube1a-2a_______________1b-2a_______________ 1a-2b____________________________Same streamlineCrosses || streamlinesDoesn’t cross streamlines2 21 1 2 21 22 2p V p Vz zg gg g+ + = + +z2122Vzg=212V gz=V = f(p)V = f(z2)V = f(p)In all cases we don’t know p1Stagnation TubeGreat for measuring __________________How could you measure Q?Could you use a stagnation tube in a pipeline?What problem might you encounter?How could you modify the stagnation tube to solve the problem?EGL (defined for a point)Q V dA= ��Pitot TubesUsed to measure air speed on airplanesCan connect a differential pressure transducer to directly measure V2/2gCan be used to measure the flow of water in pipelinesPoint measurement!Pitot TubeVV1 =12Connect 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 differencez1 = z2( )1 22V p pr= -Static pressure tapStagnation pressure tap02 21 1 2 21 22 2p V p Vz zg gg g+ + = + +Relaxed Assumptions for Bernoulli EquationFrictionless (velocity not influenced by viscosity)SteadyConstant density (incompressible)Along a streamlineSmall energy loss (accelerating flow, short distances)Or gradually varyingSmall changes in densityDon’t cross streamlinesBernoulli Normal to the Streamlineskˆsˆnˆp gr r- � = +a kSeparate acceleration due to gravity. Coordinate system may be in any orientation!nzpnndda gr r�- = +�Component of g in n directionBernoulli Normal to the Streamlines2nVaR=np dza gn dnr r�- = +�p pdp ds dns n� �= +� �0 (s is constant normal to streamline)2dp V dzgdn R dnr r- = +R is local radius of curvaturedp dn p n\ =� �n is toward the center of the radius of curvatureIntegrate F=ma Normal to the Streamlines2ndp Vdn gdz CRr+ + =�������(If density is constant)2dp V dzgdn R dnr r- = +2np Vdn gz CRr+ + =���2"nVp dn gz CRr r+ + =���Multiply by dnIntegratenPressure Change Across Streamlines1( )V r C r=If you cross streamlines that are straight and parallel, then ___________ and the pressure is ____________.p gz Cr+ =hydrostatic2"nVp dn gz CRr r+ + =���21 "np C rdr gz Cr r- + =�221"2nCp r gz Crr- + =dn dr=-rAs r decreases p ______________decreases2'1np Vdn z Cg Rg+ + =���End of pipeline?End of pipeline?What must be happening when a horizontal pipe discharges to the atmosphere?2"nVp dn gz CRr r+ + =���Try applying statics…Streamlines must be curved!(assume straight streamlines)Nozzle Flow Rate: Find QD1=30 cmD2=10 cmQ90
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