A piezometer consists of a vertical tube open to atmospheric pressure at one end. A force balance determines the height of the fluid:
$$ P_{f} = P_{atm} + \gamma_{f} h $$
where \( P_{f} \) is the absolute pressure of the fluid, \( P_{atm} \) is atmospheric pressure (101.3 kPa), \( \gamma_{f} \) is specific weight of the fluid, and \( h \) the height of fluid in the piezometer. Specific weight is:
$$ \gamma_{f} = \rho_{f} g $$
where \( \rho_{f} \) is fluid density and \( g \) is the gravitational constant, 9.81 m/s2.
Unlike the piezometer, a U-tube manometer has two components: the fluid being measured and the manometer fluid, which is typically a dense, non-volatile liquid like mercury. A U-tube manometer can measure the pressure of a gas or a liquid, whereas a piezometer can only measure liquid pressure, because gas would escape the manometer. For a U-tube manometer:
$$ P_{f} + \gamma_{f} h = P_{atm} + \gamma_{m} h $$
where \( \gamma_{m} \) is the specific gravity of the manometer fluid. Gauge pressure is:
$$ P_{g} = h \left( \gamma_{m} - \gamma_{f} \right) $$
Where \( P_{g} \) is gauge pressure. For measurements of gas pressure, \( \gamma_{f} ≪ \gamma_{m} \), so gauge pressure is
$$ P_{g} = \gamma_{m} h $$
For an inclined manometer, the fluid height \( h \) is:
$$ h = L \; \mathrm{sin} \theta $$
where \( L \) is the length of manometer fluid and \( \theta \) is the angle of the manometer tube relative to the horizontal. An inclined-tube manometer is used because it provides more precise measurements than an otherwise identical U-tube manometer.
References:
- B. R. Munson, T. H. Okiishi and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2009.
- Cal Poly Pomona. Force Balance on Inclined Manometer [Video]. (Mar 28, 2017) www.youtube.com/watch?v=wHGUJTSMjOo&t=468s.
- University of Colorado Boulder. Manometer Example [Video]. (Mar 28, 2017) www.youtube.com/watch?v=Q1opScBlMkA.
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