PRESSURE AND PIEZOMETRY (PRESSURE MEASUREMENT) What is pressure? .......................................................................................................................................... 1 Pressure unit: the pascal ............................................................................................................................ 3 Pressure measurement: piezometry ........................................................................................................... 4 Vacuum ......................................................................................................................................................... 5 Vacuum generation ................................................................................................................................... 6 Hydrostatic pressure ...................................................................................................................................... 8 Atmospheric pressure in meteorology ...................................................................................................... 8 Liquid level measurement ......................................................................................................................... 9 Archimedes' principle. Buoyancy ............................................................................................................. 9 Weighting objects in air and water ..................................................................................................... 10 Siphons .................................................................................................................................................... 10 Pressure in the kinetic theory of gases ........................................................................................................ 11 Pressure in fluid flow in ducts..................................................................................................................... 12 Bernoulli equation ................................................................................................................................... 13 Dynamic pressure and total pressure ...................................................................................................... 14 Pitot tube ................................................................................................................................................. 14 Venturi effect .......................................................................................................................................... 14 Cavitation ................................................................................................................................................ 15 Exit pressure. Jets .................................................................................................................................... 15 Water hammer and ram pump ................................................................................................................. 16 Pressure sensors .......................................................................................................................................... 18 Liquid head transducers: the U-tube ....................................................................................................... 19 Mechanical transducers: the Bourdon tube ............................................................................................. 20 Electrical transducers: membrane sensors .............................................................................................. 21 Electrical transducers: piezoeresistive sensors ....................................................................................... 21 Electrical transducers: piezoelectric sensors ........................................................................................... 22 Non-contact pressure transducers and pressure mapping ....................................................................... 22 Laboratory practice ................................................................................................................................. 23 Pressure cookers and boilers ....................................................................................................................... 24 Pressure cooker safety valves ............................................................................................................. 24 Acoustic pressure: microphones ................................................................................................................. 26 Blood pressure............................................................................................................................................. 28 References ................................................................................................................................................... 29 WHAT IS PRESSURE? Contrary to temperature (see Thermometry), pressure is not chosen as a basic magnitude in the International System, but as a derived magnitude; pressure is surface force divided by surface area, both force and area being derived from the basic magnitudes: time, length, and mass. The metrological implications are obvious: pressure measurement reduces to force measurement on a well-defined area. In its turn, force can be measured by the acceleration of a mass or by the deformation of a body. In summary, fundamental pressure metrology is a subfield of mass metrology (usually split in two subfields: mass and density, and force and pressure). Pressure and piezometry (pressure measurement) 1 Furthermore, as forces can ultimately be measured only in isolated bodies (that is why free-body diagrams are fundamental in Mechanics), most pressure transducers measure the difference between the pressure acting on the probe active-surface and the pressure acting on the probe rear-surface, usually the surrounding ambient pressure, yielding what is termed gauge pressure (gauge means standard); if the rear side is evacuated, then the measure is termed absolute pressure, and, if the rear side is filled and sealed to a calibrated pressure, it is termed differential pressure (the term differential pressure also applies when measuring differences between two locations within the system, as in Pitot and Venturi probes). Notice that the gauge pressure in a system varies with atmospheric conditions even when the internal pressure is constant. Sometimes the term 'relative pressure' is applied to both, gauge pressure and differential pressure; see Fig. 1 for a summary. Fig. 1. Types of pressure references. The common practice of measuring pressure differences do not imply that pressure is a relative magnitude like position or time, which depends on the reference frame or start-point chosen by the observer; pressure, like temperature (or period, length, or mass), is an absolute magnitude, as will be seen below (e.g., the length occupied by a known amount of an ideal gas inside an ideal cylinder-piston device, can be used either as a thermometer or as a piezometer). Moreover, there are no 'deltas' in the ideal gas equation pV=mRT; all variables there are absolute. By the way, how can one change the pressure of a system? In a gaseous system, equation pV=mRT already shows several ways: by increasing the mass of gas inside, by increasing its temperature, or by decreasing its volume. Those are static ways, as piling up more fluid or fitting a tight weight may do in a liquid. However, the common way to increase pressure in a fluid is with a rotodynamic pump, by first giving it momentum with an impeller, and then decelerating it in a diffuser. Large pressure outputs, however, usually demand volumetric-type pumps, where the momentum transfer can be performed quasi-statically. In spite of pressure being defined in terms of force (a vector magnitude), pressure is a scalar magnitude by definition: • For contacting solids, pressure it is just the normal force (not the total force, which may include shear) divided by the contacting area. Notice the averaging process implied, since the contact between two solids is at a wide variety of microscopic crests (varying with pressure) with air Pressure and piezometry (pressure measurement) 2 trapped in between (it the contact were perfect, the two solids would weld together). The concept of pressure, however, does only applies to fluids and their interfaces with solids, since deformed solids are not in proper thermodynamic equilibrium and their fundamental equation, in the elastic realm, is dU=++ Tds∑∑τµi d( cV i0 ) ii dn , instead of dU=−+ Tds pdV∑ µii dn , where τι are the components of the stress tensor (six, because it is symmetric), ci are the components of the displacement tensor (six, because it is symmetric), and V0 the un-deformed volume. • In Thermodynamics, pressure is defined for an isotropic media at equilibrium, as p≡T∂S/∂V|U,ni, i.e. pressure is the sensitivity of entropy to changes in volume (or the escaping force of compression-work energy). Mechanical stability means that pressure can only be positive because, otherwise, the tendency of entropy in isolated systems to increase would mean that its volume would disappear; dV<0 if p<0 in p≡T∂S/∂V|U,ni. On the opposite, the natural tendency for systems to expand is compensated by the restrictions imposed by the surroundings. In absence of external fields, the same pressure level would be shared by any part of a system at equilibrium, but in the presence