Chapter 9. Pressure. Pressure. Pressure and Velocity Measurements

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1 Chapter 9 Pressure and Velocity Measurements Material from Theory and Design for Mechanical Measurements; Figliola, Third Edition Pressure Pressure is force per unit area. It acts inward or outward, normal to the surface of any physical boundary. Pressure P g = P abs P o Common P o is P abs at local atmospheric conditions. Absolute pressure will be a positive number. Gauge pressure can be positive or negative depending on the value of measured pressure relative to the reference pressure. A differential pressure, such as p1- p2, is a relative measure and can t be written as an absolute pressure. 1

2 Pressure Pressure can also be described in terms of the pressure exerted on a surface submerged in a column of fluid at a depth, h, as depicted. The pressure at any depth within a fluid of specific weight γ can be written as: p abs (h) = p 0 (h 0 ) + γh p 0 is pre-determined and h is measured from an arbitrary datum line, h 0. The fluid specific weight is given by γ = ρg / g c. The equivalent head of fluid of depth, h = (p abs p) / γ Pressure Pressure transducers are usually calibrated using a reference pressure instrument such as: McLeod Gauge Barometer Manometer McLeod Gauge The McLeod gauge is a pressure-measuring instrument and laboratory reference standard used to establish gas pressures in the subatmospheric range of 1 mm Hg abs down to 0.1 µm Hg abs. The glass tubing is arranged so that a sample of the gas at the low pressure to be determined can be trapped by inverting the gauge from the sensing position to that of the measuring position. 2

3 McLeod Gauge The trapped gas within the capillary is isothermically compressed by a rising column of mercury. Boyle s law is then used to relate the two pressures on either side of the mercury to the distance of travel of the mercury within the capillary. Barometer A barometer consists of an initially evacuated tube that is closed on one end. The open end is inverted and immersed within a liquid-filled reservoir as shown on the next slide. The reservoir is open to atmospheric pressure, which forces the liquid to rise up the tube. The resulting height of the liquid column above the reservoir free surface is a measure of the absolute atmospheric pressure in equivalent head. 3

4 U-tube Manometer 1. Transparent tubes 2. Indicating liquid of γ m specific weight 3. Working fluid of γ specific weight 4. Hydrostatic Equation: P 1 = P 2 + γ x + γ m H - γ (H+x) Then, P 1 P 2 = (γ m - γ) H Static sensitivity K = 1 / (γ m - γ) Indicating fluid should be denser than working fluid and not soluble with working fluid. Pressure Transducers Converts measured pressure into a mechanical or electrical signal. Pressure transducers are subject to: Elemental error - hysterism Resolution drift Zero-shift error Sensitivity error Bellows and Potentiometer Diaphragm magnitude of deformation is proportional to pressure difference. Membrane made of metal, plastic, neoprene Well-suited for both static and dynamic pressure Good linearity and resolution over useful range Have very wide frequency responses and very short rise time Most common sensing technology is the strain gauge diaphragm with bridge 4

5 Strain Gauge Elements The most common method for converting diaphragm displacement into a measurable signal is to sense the strain induced on the diaphragm surface as it is displaced. Strain gauges, devices whose measurable resistance is proportional to their sensed strain, can be bonded directly onto the diaphragm or onto a deforming element. Strain Gauge Elements The use of semiconductor technology in pressure transducer construction has led to the development of a variety of very fast, very small, highly sensitive strain gauge diaphragm transducers. Silicone piezoresistive strain gauges can be diffused into a single crystal of silicone wafer, which forms the diaphragm. Semiconductor strain gauges have a sensitivity that is 50 times greater than conventional metallic strain gauges. 5

6 Capacitance Elements When one or more fixed metal plates are placed directly above or below a metallic diaphragm, a capacitor is created that forms an effective secondary element. Such a transducer using this method is shown next. Pressure Pressure causes plates to move C = cεa/τ C = proportionality constant ε = dielectric constant of medium (air ε = 1) A = area of plate overlap τ = gap Piezoelectric Pressure Transducers Piezoelectric pressure transducers: Very fast Very small High sensitivity Immune to thermoelastic strain 6

7 Piezoelectric Crystal Elements Pressure acts normal to the crystal axis and changes the crystal thickness, t, by a small amount, t. This sets up a charge: q = K q pa Where p is the pressure acting over the electrode area A and K q is the crystal charge sensitivity, a material property. Piezoelectric Crystal Elements The voltage developed across the electrodes is given by: E o = q / C Where C is capacitance of the crystal-electrode combination. The capacitance can be obtained to yield: E o = K q pt / cε = K E pt Where K E is the voltage sensitivity of the transducer. The crystal sensitivities for quartz, the most common material used, are K q = 2.2 * 10-9 C/N and K E = V m/n. A charge amplifier is used to convert charge to voltage. Static and Dynamic Calibration Static Calibration Direct comparison to any standard pressure reference instrument Dynamic Calibration Rise time is found by step change in input Switching value Shock tube Frequency response found by application of periodic input signal Reciprocating piston generates variation in pressure 7

8 Pressure Flow in Moving Liquids u 1 = u 3 u 4 > u 3 (conservation of mass) From conservation of energy, path 1 and 2: P 1 + ρ(u 12 /2g c ) = P 2 + ρ(u 22 /2g c ) Path 3 and 4: P 3 + ρ(u 32 /2g c ) = P 4 + ρ(u 42 /2g c ) At stagnation point: u 2 = 0 ; P 2 = P 1 + ρ(u 12 /2g c ) = P T Pressure Flow in Moving Liquids Thus, P 2 > P 1 by u 12 /2gc, which is the kinetic energy of the flow that must be brought to rest along path A. The pressure at point 2 is called the stagnation pressure or total pressure. Pressure at points 1, 3, and 4 are known as the stream pressure, or static pressure. Due to conservation of mass, the flow accelerates around the mass such that P 3 P 4. Static pressure is the pressure sensed by the fluid particle as it moves with the same velocity as the local flow. Total pressure is measured with an impact probe. A small hole in the impact probe is aligned with the flow to cause the flow to come to a stop at the hole. Alignment is important--- ± 7 o on type a and b. The Kiel probe uses a shroud and is insensitive to misalignments up to ± 40 8

9 Static Pressure Static Pressure Theory: Due to the boundary layer theory, we can measure static pressure at a point normal to the flow direction as long as there is no disturbance in the flow in the region of the tap. Within ducted flows, static pressure is sensed by wall taps, small holes drilled into the duct wall perpendicular to the flow direction at the measurement point. The tap is fitted with a hose or tube, which is connected to a pressure gauge or transducer. Static Pressure Tap Tap hole 1% to 10% of pipe diameter Normal to flow No obtrusion into flow, or burrs Stay away from bends in pipe Design and Installation: Transmission Effects The size of the pressure tap diameter and the length of tubing between a pressure tap and the connecting transducer form a pressure-measuring system that can have dynamic response characteristics very different from the pressure transducer itself. 9

10 Transducer Tap 1. Tube length L of diameter d connects pressure tap to the transducer of dead volume V. Design and Installation: Transmission Effects It is assumed that the transducer has frequency response and rise time characteristics that exceed those of the entire pressure measuring system (tubing plus transducer). Consider the configuration depicted previously, in which a rigid tube of length L and diameter d is used to connect a pressure tap to a pressure transducer of internal dead volume V. Design and Installation: Transmission Effects Initially, we can assume that under static conditions the input pressure at the tap will be indicated by the pressure transducer, but if the pressure tap is exposed to a timedependent pressure, p a (t), the response behavior of the tubing will cause the system output from the transducer, p(t), to lag the input. 10

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