Pressure Measurement Sensors
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Pressure measurement in liquids or gases is common, particularly in process control. Pressure ...
Pressure measurement in liquids or gases is common, particularly in process control. Pressure is defined as the force per unit area. To measure a pressure, it is either compared with a known force or its effect on an elastic element is measured (deflection measurement). Table 1.5 shows some of the methods available.
In liquid column manometers such as the U-tube in Figure 1.12, second- order effects are disregarded in comparing the pressure to be measured with a reference pressure; the result is a difference h of liquid level
P – P ref
h = –-------------
where p is the density of the liquid and g is the acceleration of gravity. A level sensor (photoelectric, float, etc.) yields an electric output signal.
A pressure applied to an elastic element deforms it until the internal stress balances the applied pressure. Depending on the material and its geometry, the resulting displacement or deformation may be small or large, which determines which sensors can be applied (Table 1 .5). The usual devices utilize the I3ourdon tube or the clamped or bonded diaphragm.
The Bourdon tube was developed by Eugene Bourdon in 1849. It consists of a metallic tube with a non-circular cross section obtained by deforming a tube having a circular cross section. A pressure difference applied across the wall causes the tube to tend to recover its original circular section. Figure 1.13 shows that if one of the tube ends is closed and the other one is firmly held, that tendency to recover the original shape displaces the free end. This displacement is not linear along its entire range but is linear enough in short ranges. The configurations that offer the greater displacements have the drawbacks of their greater compliance and length that results in a small frequency passband. Displacement sensors are used to obtain an electric output signal.
A diaphragm is a flexible circular plate consisting of a taut membrane or a clamped sheet that strains under the action of the pressure difference to be measured. The transduction is then made by detecting the displacement of the central part of the diaphragm, its global deformation, or the local strain (in this case using strain gages; Section 2.2).
For a thin plate with thickness t and radius R experiencing a pressure difference P between both sides, if the maximal deformation z is less than one-third of the thickness, we have 
16E14 z P –-----------------------------
= 3R(l — v2) + 0.488 (-)]
where E is Young’s modulus and v the Poisson’s ratio for the plate material. If piezoresistive sensors are to be applied, it is necessary to know the
mechanical stress at the different points across the plate. At all points at a distance r from the center, the radial stress is
The tangential stress is
3PR2v 1/I \ /1 \fr\21
St = 8t [v— + I) — ± 3)) j (1.30)
Across the diaphragm there are tensions and compressions. This requires the placement of several strain gages as well as combinations of these gages in a measurement bridge to benefit from additive effects and provide temperature compensation (Section 3.4.4).
Some of the elastic materials used are beryllium-copper, stainless steel. nickel-copper alloys, and even silicon if the diaphragm is going to incorporate silicon strain gages.
If the resulting displacement in a single diaphragm is not large enough. capsules and bellows provide larger displacements. Figure 1.14 shows a capsule that consists of twin diaphragms joined by their external border and placed on opposite sides of the pressurized chamber. Figure 1.15 shows bellows, which are flexible chambers with axial elongation that undergo even larger deflections than capsules, even reaching in some cases 10% of their length. But both devices are vibration- and acceleration-sensitive and do not withstand high over pressures.
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