STATIC, STAGNATION, AND DYNAMIC PRESSURE

1 STATIC, STAGNATION, AND DYNAMIC PRESSURESB ernoulli equation is Inthisequationpiscalledstaticpressure, ,iftheinstrumentwerestaticwithrespecttof lowingfluid, , , tap ,drilledcarefullyinthewall, gzVp ME304 51 Inafluidstreamfarfromawall,orwherestream linesarecurved,accuratestaticpressuremea surementscanbemadebycarefuluseofastaticp ressureprobe, , (Pitottube).Inincompressibleflow,applyin gBernoulliequationbetweenpointsinthefree streamandatthenoseoftubeandtakingz=0atth etubecenterline,wegetwhereP0isthestagnat ionpressure, ,weget,and for the speed 2220200 VpVp 2021 Vpp 221V ppV 0221 ppV 02ME304 52 The static PRESSURE corresponds to a point Ais read from the wall static PRESSURE tap.

2 The stagnation pressureis measured directly at Aby the total head probes are combined as in pitot-static tube. The inner tube is used to measure the stagnation pressureat point Bwhile the static PRESSURE at Cis measured by the small holes in the 53 , , 54To be completed in classME304 55 RELATION BETWEEN THE FIRST LAW OF THERMODYNAMICS AND THE BERNOULLI equation (Energy equation)Restrictions:1) 2) 3) 4) Steady flow5) Uniform flow and properties at each sectionUnder these restrictionsBut from continuity under these restrictionsor CSCothershearsAdVpedetWWWQ 00000 sW 0 shearW 0 otherW QAVgzVpuAVgzVpu 222222222111121111220 CSCAdVdt 00 2221110 AVAV ME304 56 Thatis,Also,Thus, ,undertheadditionalrestrictions,6)incomp ressibleflow7)Theenergyequationreducesto Before,theBernoulliequationwasderivedfro mmomentumconsiderations(Newton ssecondlaw)

3 ,andisvalidforsteady,incompressible, ,theBernoulliequationwasobtainedbyapplyi ngthefirstlawofthermodynamicstoastreamtu becontrolvolume, mdmQdtdmdmQtQQ mdmQuumgzVpgzVp 121211122222220 dmQuugzVpgzVp 12222221211122 121 dmQuugzVpgzVp 122222121122constant121 012 dmQuu constant2222221211 gzVpgzVp ME304 57 Example:Consider the frictionless, incompressible flow with heat transfer. Show thatdmQuu 12ME304 58To be completed in classENERGY GRADE LINE AND HYDRAULIC GRADE LINEO ften it is convenient to represent the mechanical energy level of a flow graphically. The energy equation, that is Bernoulli equation, suggests such a representation.

4 Dividing Bernoulli equation by g, we obtainEach term has dimensions of length, or head of flowing fluid. The individual terms are is the head due to local static pressureisthe head due to local DYNAMIC pressurezis elevation headH is the total head of the flowThe energy grade line (EGL):The locus of points at a vertical distance,, measured above a horizontal datum, which is the total head of the hydraulic grade line (HGL):The locus of points at a vertical distance, , measured above a horizontal difference is heights between the EGL and HGL represents, the DYNAMIC (velocity) head, .constant22 HzgVgp gp gV22zgVgpH 22 zgp gV22ME304 59ME304 510 UNSTEADY BERNOULLI EQUATION INTEGRATION OF EULER S EQUATION ALONG A STREAMLINEC onsider the streamwiseEuler equation in streamline coordinatesThe above equation may now be integrated along an instantaneous streamline from point 1 to point 2 to yieldFor an incompressible flow, it becomesRestrictions.

5 1) Incompressible flow2) Frictionless flow3) Flow along a streamline01 tVszgspsVV 0121212121 dstVdsszgdsspdssVV 212222121122dstVgzVpgzVps ME304 511 , 512To be completed in classME304 513ME304 514 FLOW MEASUREMENTFlow measurement refers to the ability to measure the velocity, volume flow rate, or mass flow rate of any liquid are many types of devices used for flow measurement. Many of these devices use the rinciple of Bernoulli choice of a flow meter is influenced by accuracy required, range, cost, complication, ease of reading or data reduction, and service Measurement TechniquesIn general devices used for flow measurementcanbegrouped depending on the nature of the data obtained by the device.

6 Based on this, flow measurement devices can be grouped as follows:A)Measurement of Integral Properties of Flows (Mass and volume flow measurement)1) Restriction flow meters for Internal Flowsa) Orifice meterb) Flow nozzle c) Venturimeter2) Rotameter3) Turbine flow meter4) Coriolis techniqueB) Measurement of Local Flow Parameters (Local VelocityMeasurement)1) Pitot-static tube2) Hot wire anemometer3) Lase doppler anemometry (LDA)4) Particle image velocimery (PIV)5) Ultrosonic technique6) Magnetic techniqueME304 515 MASS AND VOLUME FLOW MEASUREMENTR estriction Flow MetersAn easy and cheap way to measure flow rate through a pipe is to place some type of restriction within the pipe as shown in the figure below: Orifice meter-Head loss high-Initial cost lowFlow nozzle meter-Head loss intermediate-Initial cost intermediateVenturi meter-Head loss low-Initial cost highThe operation of each of these devices is based on the same principles, due to restriction velocityincreases and PRESSURE decreases.

7 We assume the flow is horizontal (z1=z2), steady, frictionless and incompressible between points 1and 2:ME304 516 Duetosharpedgeofflownozzleandorifice, venacontracta , ,theflowareaisminimum,streamlinesarestra ight, :To be completed in classME304 517 APPAAA mltheoretica21212221 Thisequationshowsthatunderoursetofassump tions,foragivenfluid( )andflowmetergeometry(A1andA2),theflowra teisdirectlyproportionaltothepressuredro pacrossthemetertabs, (especiallydownstreamfromthemeter.) , ,theactualflowrateisdifferentfromthetheo reticalflowrategivenbyEq.(A).Hence,Eq.(A )isadjustedforReynoldsnumberanddiameterr atio(Dt/D1)bydefiningandempiricaldischar gecoefficientC,asfollows:ME304 518 212121 PPAACAmCmttltheoreticaactual ,,441211 ThereforeDDAA thenDDLettingttt 21421 PPCA mtactual ""114factorapproachofvelocityasknownis Discharge coefficient Cand velocity of approach factor are combined into a single flow coefficient Kas41 CKIn terms of flow coefficient, actual mass flow arte is expressed as, 212 PPKA mtactual Forstandardizedmeters, (Re>4000),dischargecoeffientandflowcoeff icientmaymeexpressedasfollows.

8 NDbKK1Re114 nDbCC1Re ME304 519In the above equations, subscript denotes the coefficient at infinite Reynolds number; constants band nallow for scaling to finite Reynolds equations and curves of coefficients versus reynolds number are given for orifice plate, flow nozzle and venturi (Plate)The correlating equation recommended for a concentric orifice with corner tabs is Thisequationpredictsthedischargecoeffici entCwithin Re : PRESSURE tabs for orifices may be placed in several locations as shown in above coefficient for corner concentric orifices with corner 520 Flow NozzleFlow nozzle may be used as metering elements in either plenums or ducts as shown in equation recommended for an ASME long-radius flow nozzle is Thisequationpredictsthedischargecoeffici entCfortheflownozzlewithin Re.

9 Flow coefficients for ASME long-radius flow plenum nozle = coefficient K is in the range of K ME304 521 VenturiMeterExperimental data show that discharge coefficients for venturimeters range from to high Reynolds numbers (ReD1>2x105) Thus, C= be used to measure the mass flow rate within about 1 percent at high Reynolds p, :ME304 522 be completed in classME304 523 RotameterTurbine Flow of a multi-bladed rotor mounted at right angles to the flow & suspended in the fluid stream on a free-running diameter of the rotor is slightly less than the inside diameter of the flow metering of rotation of rotor proportional to the volumetric flow free moving float is balanced inside a vertical tapered the fluid flows upward the float remains steady when the DYNAMIC forces acting on it are flow rate indicated by the position of the float relative to a calibrated 524 LOCAL VELOCITY MEASUREMENT1)Pitot-static tube2)Hot wire anemometer3)

10 Lase doppler anemometry (LDA)4)Particle image velocimery (PIV)5)Ultrosonic technique6)Magnetic techniquePitot-Static TubeThe static PRESSURE corresponds to a point Ais read from the wall static PRESSURE tap. The stagnation pressureis measured directly at Aby the total head probes are combined as in pitot-static tube. The inner tube is used to measure the stagnation pressureat point Bwhile the static PRESSURE at Cis measured by the small holes in the PRESSURE measurement technique aboveME304 525 Example:A pitot-static tube is used to measure the speed of air at standard conditions at a point in a flow. The manometer deflection in millimeters of water is measured as 63mm.