380Chapter9PressureandVelocityMeasurementsClosed endMercury vapor atvapor pressureMeniscusScaleLiquidmercuryTTTTTTTTTTGlasstubeGlass cylinderZeroingreservoirpointerAdjustingscrewFigure9.4Fortin barometer.change the vapor pressure,for temperature and altitude effects on the weight ofmercury,and fordeviations from standard gravity (9.80665 m/s2 or 32.17405 f/s).Correction curves are providedbyinstrumentmanufacturersBarometers areusedaslocal standardsforthemeasurementof absoluteatmosphericpressure.Under standard conditions for pressure temperature and gravity, the mercury rises 760 mm (29.92in.)above the reservoir surface.The U.S.National Weather Service alwaysreports a barometricpressure that has been corrected to sea-level elevation.ManometerA manometeris an instrumentusedto measure differential pressure based onthe relationshipbetween pressure and the hydrostatic equivalent head of fluid. Several design variations are
E1C09 09/14/2010 15:4:52 Page 380 change the vapor pressure, for temperature and altitude effects on the weight of mercury, and for deviations from standard gravity (9.80665 m/s2 or 32.17405 ft/s2 ). Correction curves are provided by instrument manufacturers. Barometers are used as local standards for the measurement of absolute atmospheric pressure. Under standard conditions for pressure temperature and gravity, the mercury rises 760 mm (29.92 in.) above the reservoir surface. The U.S. National Weather Service always reports a barometric pressure that has been corrected to sea-level elevation. Manometer A manometer is an instrument used to measure differential pressure based on the relationship between pressure and the hydrostatic equivalent head of fluid. Several design variations are Glass cylinder reservoir Zeroing pointer Adjusting screw Mercury vapor at vapor pressure Closed end Meniscus Liquid mercury Glass tube Scale Figure 9.4 Fortin barometer. 380 Chapter 9 Pressure and Velocity Measurements
9.3381PressureReferenceInstruments+日上Figure 9.5 U-tube manometer.available,allowingmeasurementsrangingfrom theorder of0.001mmofmanometerfluid to severalmeters.The U-tube manometer in Figure 9.5 consists of a transparent tube filled with an indicatingliquid of specific weight Ym.This forms two free surfaces of the manometer liquid. The difference inpressures P, and p2 applied across the two free surfaces brings about a deflection, H, in the level ofthe manometer liquid.For a measured fluid of specific weight ,the hydrostatic equation can beapplied tothemanometerof Figure9.5asPi =P2+x+mH-(H +x)which yields the relation between the manometer deflection and applied differential pressure,(9.5)Pi-P2=(Ym-)HFrom Equation 9.5,the static sensitivity of the U-tube manometer is given byK =1/(m-).To maximize manometer sensitivity,we want to choose manometer liquidsthat minimize the value of (Ym-).From a practical standpoint the manometer fluid must notbe soluble with the working fluid. The manometer fluid should be selected to provide adeflection that is measurable yet not so great that it becomes awkward to observe.A variation in the U-tubemanometer is the micromanometer shown in Figure 9.6.Thesespecial-purpose instruments are used to measure very small differential pressures,down to0.005mmH,O (0.0002 in.H,O).In the micromanometer,themanometer reservoirismovedup or down until the level of the manometer fluid within the reservoir is at the same level as a setmark within a magnifying sight glass. At that point the manometer meniscus is at the set mark, andthis serves as a reference position.Changes in pressure bring about fluid displacement so that thereservoir mustbe moved upor down tobring the meniscus back to the setmark.The amount of thisrepositioning is equal to the change in the equivalent pressure head. The position of the reservoir iscontrolled by a micrometer or other calibrated displacement measuring device so that relativechanges in pressurecan bemeasured withhigh resolution
E1C09 09/14/2010 15:4:53 Page 381 available, allowing measurements ranging from the order of 0.001 mm of manometer fluid to several meters. The U-tube manometer in Figure 9.5 consists of a transparent tube filled with an indicating liquid of specific weight gm. This forms two free surfaces of the manometer liquid. The difference in pressures p1 and p2 applied across the two free surfaces brings about a deflection, H, in the level of the manometer liquid. For a measured fluid of specific weight g, the hydrostatic equation can be applied to the manometer of Figure 9.5 as p1 ¼ p2 þ gx þ gmH gðH þ xÞ which yields the relation between the manometer deflection and applied differential pressure, p1 p2 ¼ ð Þ gm g H ð9:5Þ From Equation 9.5, the static sensitivity of the U-tube manometer is given by K ¼ 1=ðgm gÞ. To maximize manometer sensitivity, we want to choose manometer liquids that minimize the value of (gm g). From a practical standpoint the manometer fluid must not be soluble with the working fluid. The manometer fluid should be selected to provide a deflection that is measurable yet not so great that it becomes awkward to observe. A variation in the U-tube manometer is the micromanometer shown in Figure 9.6. These special-purpose instruments are used to measure very small differential pressures, down to 0.005 mm H2O (0.0002 in. H2O). In the micromanometer, the manometer reservoir is moved up or down until the level of the manometer fluid within the reservoir is at the same level as a set mark within a magnifying sight glass. At that point the manometer meniscus is at the set mark, and this serves as a reference position. Changes in pressure bring about fluid displacement so that the reservoir must be moved up or down to bring the meniscus back to the set mark. The amount of this repositioning is equal to the change in the equivalent pressure head. The position of the reservoir is controlled by a micrometer or other calibrated displacement measuring device so that relative changes in pressure can be measured with high resolution. H x p1 p2 m Figure 9.5 U-tube manometer. 9.3 Pressure Reference Instruments 381
382Chapter9PressureandVelocityMeasurementsPiInclined tubeMeniscMicrometeradjustingscrewSetmark王ReservoirFlexibletubeFigure 9.6 Micromanometer.The inclined tube manometer is also used to measure small changes in pressure.It is essentiallya U-tube manometer with one leg inclined at an angle , typically from 10 to 30 degrees relative tothe horizontal.As indicated in Figure 9.7,a change in pressure equivalent to a deflection of height Hin a U-tube manometer would bring about a change in position ofthemeniscus in the inclined leg ofL =H/sin e. This provides an increased sensitivity over the conventional U-tube by the factor of1/sin 0.A number of elemental errors affect the instrument uncertainty of all types of manometers.These include scale and alignment errors, zero error, temperature error,gravity error, and capillaryand meniscus errors.The specific weight of the manometer fluid varies with temperaturebutcan becorrected. For example, the common manometer fluid of mercury has a temperature dependenceapproximatedby133.084848.707[1b/ft3]Yig= 1 +0.000 [N/m] 1+ 0.000101(T 32)with T in C or °F, respectively.A gravity correction for elevation z and latitude corrects forgravity erroreffects using the dimensionless correction,(9.6a)ej = -(2.637 ×10-3 cos2Φ + 9.6 × 108 z + 5 × 10-5)us=-(2.637×10-3cos2Φ+2.9×10-8z+5×10-5(9.6b)where Φ is in degrees and z is in feet for Equation 9.6a and meters in Equation 9.6b. Tube-to-liquidcapillary forces lead to the development of a meniscus.Although the actual effect varies with purityof themanometer liquid, these effects can be minimized by using manometer tubebores of greaterthan about 6 mm (0.25 in.).In general, the instrument uncertainty in measuring pressure can be aslowas0.02%to0.2%ofthereading
E1C09 09/14/2010 15:4:53 Page 382 The inclined tube manometer is also used to measure small changes in pressure. It is essentially a U-tube manometer with one leg inclined at an angle u, typically from 10 to 30 degrees relative to the horizontal. As indicated in Figure 9.7, a change in pressure equivalent to a deflection of height H in a U-tube manometer would bring about a change in position of the meniscus in the inclined leg of L ¼ H/sin u. This provides an increased sensitivity over the conventional U-tube by the factor of 1/sin u. A number of elemental errors affect the instrument uncertainty of all types of manometers. These include scale and alignment errors, zero error, temperature error, gravity error, and capillary and meniscus errors. The specific weight of the manometer fluid varies with temperature but can be corrected. For example, the common manometer fluid of mercury has a temperature dependence approximated by gHg ¼ 133:084 1 þ 0:00006T N=m3 ¼ 848:707 1 þ 0:000101ð Þ T 32 lb=ft3 with T in C or F, respectively. A gravity correction for elevation z and latitude f corrects for gravity error effects using the dimensionless correction, e1 ¼ ð2:637 103 cos2f þ 9:6 108 z þ 5 105 ÞUS ð9:6aÞ ¼ ð2:637 103 cos2f þ 2:9 108 z þ 5 105 Þmetric ð9:6bÞ where f is in degrees and z is in feet for Equation 9.6a and meters in Equation 9.6b. Tube-to-liquid capillary forces lead to the development of a meniscus. Although the actual effect varies with purity of the manometer liquid, these effects can be minimized by using manometer tube bores of greater than about 6 mm (0.25 in.). In general, the instrument uncertainty in measuring pressure can be as low as 0.02% to 0.2% of the reading. Inclined tube Meniscus Set mark Micrometer adjusting Reservoir screw Flexible tube H p1 p2 Figure 9.6 Micromanometer. 382 Chapter 9 Pressure and Velocity Measurements
9.3383PressureReferenceInstrumentsP1InclinedtubeH=LsineTFigure 9.7 Inclined tubeLOmanometer.Example 9.2A high-quality U-tube manometer is a remarkably simple instrument to make. It requires only atransparent U-shaped tube, manometer fluid, and a scale to measure deflection. While a U-shapedglass tube of 6 mm or greater internal bore is preferred, a length of 6-mm i.d. (inside diameter) thick-walled clear tubing from the hardware store and bent to a U-shape works finefor many purposes.Water, alcohol, or mineral oil are all readily available nontoxic manometer fluids with tabulatedproperties. A sheet of graph paper or a ruler serves to measure meniscus deflection. There is a limitto the magnitude of pressure differential that can be measured, although use of a step stool extendsthis range.Tack the components to a board and the resulting instrument is accurate for measuringmanometer deflections down to one-half the resolution of the scale in terms of fluid used!The U-tube manometer is a practical tool useful for calibrating other forms of pressuretransducers in the pressure range spanning atmospheric pressure levels. As one example, the ad-hocU-tube described is convenient for calibrating surgical pressure transducers over the physiologicalpressure ranges.Example9.3An inclined manometer with the inclined tube set at 30 degrees is to be used at 20°C to measure anair pressureof nominal magnitudeof 100N/mrelative to ambient.Manometer"unity"oil (S=1)is to be used.The specific weight of the oil is 9770 ± 0.5% N/m(95%)at 20°C, the angle ofinclination can be set to within 1 degree using a bubble level, and the manometer resolution is 1 mmwith a manometer zero error equal to its interpolation error.Estimate the uncertainty in indicateddifferential pressure at thedesign stage.KNOWNp=100N/m~(nominal)ManometerResolution:1mmZeroerror:0.5mm = 30 ± 1° (95% assumed)Ym=9770±0.5%N/m2(95%)
E1C09 09/14/2010 15:4:53 Page 383 Example 9.2 A high-quality U-tube manometer is a remarkably simple instrument to make. It requires only a transparent U-shaped tube, manometer fluid, and a scale to measure deflection. While a U-shaped glass tube of 6 mm or greater internal bore is preferred, a length of 6-mm i.d. (inside diameter) thickwalled clear tubing from the hardware store and bent to a U-shape works fine for many purposes. Water, alcohol, or mineral oil are all readily available nontoxic manometer fluids with tabulated properties. A sheet of graph paper or a ruler serves to measure meniscus deflection. There is a limit to the magnitude of pressure differential that can be measured, although use of a step stool extends this range. Tack the components to a board and the resulting instrument is accurate for measuring manometer deflections down to one-half the resolution of the scale in terms of fluid used! The U-tube manometer is a practical tool useful for calibrating other forms of pressure transducers in the pressure range spanning atmospheric pressure levels. As one example, the ad-hoc U-tube described is convenient for calibrating surgical pressure transducers over the physiological pressure ranges. Example 9.3 An inclined manometer with the inclined tube set at 30 degrees is to be used at 20C to measure an air pressure of nominal magnitude of 100 N/m2 relative to ambient. Manometer ‘‘unity’’ oil (S ¼ 1) is to be used. The specific weight of the oil is 9770 0.5% N/m2 (95%) at 20C, the angle of inclination can be set to within 1 degree using a bubble level, and the manometer resolution is 1 mm with a manometer zero error equal to its interpolation error. Estimate the uncertainty in indicated differential pressure at the design stage. KNOWN p ¼ 100 N/m2 (nominal) Manometer Resolution: 1 mm Zero error: 0.5 mm u ¼ 30 1 (95% assumed) gm ¼ 9770 0.5% N/m3 (95%) p2 p1 Inclined tube 0 L H = L sin Figure 9.7 Inclined tube manometer. 9.3 Pressure Reference Instruments 383
384Chapter9PressureandVelocityMeasurementsASSUMPTIONSTemperature and capillary effects in themanometer and gravity error in thespecific weights of the fluids are negligible.The degrees of freedom in the stated uncertainties arelarge.FINDudSOLUTION The relation between pressure andmanometerdeflection is given byEquation9.5with H=L sin e:Ap=Pi-P2= L(m=)sin ewhere P2 is the ambient pressure so that Ap is the nominal pressure relative to the ambient.For anominalAp=10oN/m2,thenominal manometerriseLwouldbeApApL=21mmmsin(m-)sinwhere Ym 》 and the value for and its uncertainty are neglected. For the design stage analysis,p =f(m,L, ), so that the uncertainty in pressure, Ap, is estimated byToApTo4p++(ud),(ud)(ud),alaeAt assumed 95% confidence levels,the manometer specific weight uncertainty and angleuncertaintyareestimatedfromtheproblemas(ua)m= (9770N/m2)(0.005)~49N/m3(ua)。=1degree = 0.0175 radThe uncertainty in estimating the pressure from the indicated deflection is due both to themanometer resolution, uo, and the zero point offset error, which we take as its instrument error, ue.Using the uncertainties associated with these errors,(ud)z= /ug +u = V(0.5 mm)2 + (0.5mm)2= 0.7mmEvaluating the derivatives and substituting values gives a design-stage uncertainty interval of ameasuredAp of(ua)ap= (0.26)2 + (3.42) + (3.10) = ±4.6N/m2 (95%)COMMENTAta 30-degree inclination and for this pressure, theuncertainty in pressure isaffected almostequallybytheinstrument inclination andthedeflection uncertainties.Asthemanometer inclination is increased to a more vertical orientation, that is, toward the U-tubemanometer,inclination uncertainty becomesless important andis negligible near 90 degrees.However,foraU-tubemanometer,thedeflectionisreducedtolessthan11mm,a50%reductioninmanometer sensitivity,with an associated design-stage uncertainty of 6.8N/m(95%).DeadweightTestersThe deadweight tester makes direct use of thefundamental definition of pressure as a force perunit area to create and to determine the pressure within a sealed chamber.These devices are a
E1C09 09/14/2010 15:4:53 Page 384 ASSUMPTIONS Temperature and capillary effects in the manometer and gravity error in the specific weights of the fluids are negligible. The degrees of freedom in the stated uncertainties are large. FIND ud SOLUTION The relation between pressure and manometer deflection is given by Equation 9.5 with H ¼ L sin u: Dp ¼ p1 p2 ¼ Lðgm gÞ sin u where p2 is the ambient pressure so that Dp is the nominal pressure relative to the ambient. For a nominal Dp ¼ 100 N/m2 , the nominal manometer rise L would be L ¼ Dp ð Þ gm g sin u Dp gmsin u ¼ 21 mm where gm g and the value for g and its uncertainty are neglected. For the design stage analysis, p ¼ fðgm; L; uÞ, so that the uncertainty in pressure, Dp, is estimated by ð Þ ud p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qDp qgm ð Þ ud gm 2 þ qDp qL ð Þ ud L 2 þ qDp qu ð Þ ud u 2 s At assumed 95% confidence levels, the manometer specific weight uncertainty and angle uncertainty are estimated from the problem as ð Þ ud gm ¼ 9770 N=m3 ð Þð Þ 0:005 49 N=m3 ð Þ ud u ¼ 1 degree ¼ 0:0175 rad The uncertainty in estimating the pressure from the indicated deflection is due both to the manometer resolution, uo, and the zero point offset error, which we take as its instrument error, uc. Using the uncertainties associated with these errors, ð Þ ud L ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 o þ u2 c q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:5 mmÞ 2 þ ð0:5 mmÞ 2 q ¼ 0:7 mm Evaluating the derivatives and substituting values gives a design-stage uncertainty interval of a measured Dp of ð Þ ud Dp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:26Þ 2 þ ð3:42Þ 2 þ ð3:10Þ 2 q ¼ 4:6N=m2 ð95%Þ COMMENT At a 30-degree inclination and for this pressure, the uncertainty in pressure is affected almost equally by the instrument inclination and the deflection uncertainties. As the manometer inclination is increased to a more vertical orientation, that is, toward the U-tube manometer, inclination uncertainty becomes less important and is negligible near 90 degrees. However, for a U-tube manometer, the deflection is reduced to less than 11 mm, a 50% reduction in manometer sensitivity, with an associated design-stage uncertainty of 6.8 N/m2 (95%). Deadweight Testers The deadweight tester makes direct use of the fundamental definition of pressure as a force per unit area to create and to determine the pressure within a sealed chamber. These devices are a 384 Chapter 9 Pressure and Velocity Measurements