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# Lecture flow

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### Lecture flow

1. 1. Measurements Course 4600:483-001Lecture 7 – Pressure and Velocity MeasurementsInformation about this course can be found at the home page of the instructor:http://gozips.uakron.edu/~dorfi/IntroductionThis lecture focuses on standard pressure and velocity measurement techniques in fluid flow applicationsand the instrumentation necessary to perform the measurements. Fluid flow is governed by theeconservation equations: • Conservation of Mass • Conservation of Momentum • Conservation of EnergyThese equations govern the fluid flow and its field variables: fluid pressure, velocity and temperature.Pressure ConceptsPressure is defined as contact force per unitarea. It always acts inward and is thereforealways positive (for absolute pressure). Since Gagepressure is the macroscopic manifestation of Pressurethe kinetic energy of fluid molecules, zero Referencepressure is found in absolute vacuum or at Pressure Absoluteabsolute zero temperature. In fluid Standard System Atmospheric Pressureapplications, pressure is a local scalar quantity. Pressure 101.3 KPa absThe pressure relative to a reference pressure 14.696 psiasuch as the current atmospheric pressure is Perfectcalled gage pressure. Gage pressure can be Vacuumboth positive and negative, since it is simply a Pressure Scalespressure difference.Bernoulli EquationThe Bernoulli Equation is the conservation of momentum equation for steady-state, inviscid (no wallfriction) flow for an incompressible fluid p1 v12 p v2 + + gh1 = 2 + 2 + gh2 = const ρ 2 ρ 2where p is the absolute pressure, v is the fluid velocity, g gravity and h the elevation at arbitrary locations 1and 2 on the stream line. The Bernoulli equation states that the kinetic and potential energy of the fluidremains constant along a stream line.It is important to remember the assumptions made in deriving this equation: • steady flow • incompressible flow • frictionless flow • flow along a streamlineDr. Hans R. Dorfi 11/03/03 Page 1of 4
2. 2. Measurements Course 4600:483-001ManometerThe manometer is an instrument, which converts p1 p1 p2pressure differences into elevation differences (flowvelocity is zero). If we define the specific gravity of a Specific xfluid as γ=ρ/g and set the velocity to zero, the Gravity ofBernoulli equation simplifies to Gas: γ p1 + γ h1 = p2 + γ h2 θ L=H /sinθ HApplying the Bernoulli equation to the U-tube Specific Gravitymanometer as shown on the right gives the equation of Measurement p1 + γ ( x + H ) = p2 + γ x + γ m H Fluid: γmSolving for the pressure difference yields p1 − p2 = (γ m − γ ) HThe pressure difference is directly proportional to theheight difference H. The difference in specific weight θof the measurement fluid and the gas is theproportionality factor. Usually the specific weight of U-tube Manometer (optionally with inclined tube)the measurement fluid is several orders of magnitudelarger than the specific weight of the gas.In order to increase the sensitivity of the instrument, one of the arms of the manometers can be inclined byan angle θ. The pressure difference for an inclined manometer is then found from p1 − p2 = p = (γ m − γ ) L sin θUncertainty in inclined manometer measurementsThe above equation shows that the measurement of pressure differentials p with inclined manometers isaffected by the uncertainty in the fluid specific weight (density/gravity), the ability to read the manometer(uncertainty in L) and the uncertainty for the inclination angle θ. The normalized uncertainty equation forthe manometer is thus given by e p = eγ2m + eγ2 + eL + eθ2 2Example: An inclined manometer with indicating leg at 30° is used at 20°C to measure a gas pressure ofnominal magnitude of 100N/m2 relative to ambient. The specific weight of the measurement fluid is 9770N/m3. The manometer resolution is 1mm and the specific gravity of the gas is 11.5 N/m3. The relativeuncertainty in the specific gravities is assumed to be 1%. We can measure the angle of the inclined tube towithin 1° resolution.What is the design state uncertainty for the pressure measurement with this manometer?From the uncertainty analysis we know that the relative uncertainty ei for parameter i is 1 dp ei = Ui p dxiSubstitution of the relative uncertainty equation for the specific gravities yields 1 dp γ m Uγ m 9770 eγ m = Uγ m = = * .01 ≈ .01 p dγ m γ m − γ γ m 9770 − 11.5 1 dp Uγ γ Uγ 11.5 eγ = Uγ = = = * .01 ≈ 10 −5 p dγ γm −γ γm −γ γ 9770 − 11.5Notice how the uncertainty due to the specific gravity of the gas is negligibly small. This is because thespecific gravity of the measurement fluid is about 1000 times larger than the gas.Dr. Hans R. Dorfi 11/03/03 Page 2of 4
3. 3. Measurements Course 4600:483-001The relative uncertainty in the length is 1 dp U eL = UL = L p dL LIn order to calculate the relative uncertainty, we need to calculate the Length L of the fluid column themanometer at the nominal pressure of 100N/m2. p 100 N / m2 L= = = 0.0205m = 20.5mm (γ m − γ ) sin θ (9770 − 11.5) sin 30o N / m3The manometer resolution is 1mm, therefore the zero order uncertainty is 0.5mm. The relative uncertaintyin length L is therefore UL 0.5mm eL = = = 0.024 L 20.5mmThe resolution in the angle measurement is 1°. Therefore the zero order uncertainty is 0.5° or 0.0087radians. The relative uncertainty in the angle is found from 1 dp Uθ 0.5o 0.0087 rad eθ = Uθ = = = = 0.015 p dθ tan θ tan 30 o 0.5774Putting all uncertainties together yields e p = 10 −2 * 12 + 10 −3*2 + 2.4 2 + 1.52 = 10 −2 * 1 + 10 −6 + 5.76 + 2.25 = 0.03 = 3%The uncertainty in the length measurement L is the primary contributor to the uncertainty in the pressuremeasurement. For the nominal pressure of 100N/m2 the design stage uncertainty is 3% or 3N/m2.Pitot-Static TubesThe Bernoulli equation can also be used tomeasure the fluid velocity. Recall the equation :for zero elevation change p1 is sensed normal to p1v12 p2 v2 2 Point 1: flow direction + = + ρ 2 ρ 2 p1,v1If we could slow down the fluid flow to zero atpoint 2, we can solve for the fluid velocity atpoint 1. = Point 2 2( p2 − p1 ) 2∆p p2,v2=0 v1 = = ρ ρThis is the basic idea of the Pitot-Static tube. ThePitot-Static (P-S) tube consists basically of twoconcentric tubes, with the end turned through aright angle so that the tip can be faced into theairstream after insertion through the duct wall.The modified ellipsoidal nose form has a singleforward facing hole for sensing Total Pressureand a ring of side holes for sensing the StaticPressure. Both these inlets are individuallyconnected to tapping outlets at the tail of the unit.p2 is called the stagnation or total pressure atthe forward facing inlet to the Pitot-static probewhere the velocity becomes zero, p1 or pstatic is Schematic of Pitot-Static Tubethe static pressure along the sides of the Pitot-Dr. Hans R. Dorfi 11/03/03 Page 3of 4
4. 4. Measurements Course 4600:483-001static probe where the velocity is unchanged from the upstream duct velocity v. The pressure difference,∆P, is called the dynamic pressure because it is related to the change in fluid velocity. We can calculatethe duct velocity from the dynamic pressure as 2∆P v= ρ fluidNote that this expression is only accurate if the P-S tube points directly into v1 such that all of v1 isstagnated. If the P-S tube is misaligned, the measured velocity will be too low. Remember that thisequation was derived based on the Bernoulli equation, which assumes laminar and incompressible fluidflow. For flow velocities greater than 30% of the sonic velocity, the fluid must be treated as compressible.Then the simplified equation for the P-S tube no longer holds.Textbook descriptions of Pitot-static probes usually describe their use in a laminar flow. What happenswhen Pitot-static probes are used in time-varying turbulent flows? The pressure difference associated withthe fluctuation velocity must move a mass in the pressure sensor to measure the pressure change associatedwith a given velocity change. The measurement devices are thus second-order mechanical systems withtheir own natural frequency and damping ratio. If the frequency of the velocity fluctuation is much fasterthan the natural frequency of the measuring system, then it will display the average value of the fluctuatingsignal. This will only hold true for moderately turbulent flows (less than 10% turbulence intensity) becausethe velocity vector must remain approximately parallel to the Pitot-static probe. Duct flows typically havelow enough turbulence intensities that the effect of turbulence can be neglected, but disturbed regions offlow near sharp edges or area changes can prevent good readings.Uncertainty in Pitot-Static TubesSee the example in the lecture on uncertainty.Dr. Hans R. Dorfi 11/03/03 Page 4of 4